Condensed Matter Physics

506
Condensed Matter Physics I Peter S. Riseborough November 21, 2002 Contents 1 Introduction 9 1.1 The Born-Oppenheimer Approximation .............. 9 2 Crystallography 13 3 Structures 13 3.1 Fluids ................................. 14 3.2 Crystalline Solids ........................... 18 3.3 The Direct Lattice .......................... 18 3.3.1 Primitive Unit Cells ..................... 19 3.3.2 The Wigner-Seitz Unit Cell ................. 19 3.4 Symmetry of Crystals ........................ 21 3.4.1 Symmetry Groups ...................... 21 3.4.2 Group Multiplication Tables ................ 22 3.4.3 Point Group Operations ................... 23 3.4.4 Limitations Imposed by Translational Symmetry ..... 24 3.4.5 Point Group Nomenclature ................. 24 3.5 Bravais Lattices ............................ 28 3.5.1 Exercise 1 ........................... 28 3.5.2 Cubic Bravais Lattices. ................... 28 3.5.3 Tetragonal Bravais Lattices.................. 30 3.5.4 Orthorhombic Bravais Lattices................ 31 3.5.5 Monoclinic Bravais Lattice. ................. 32 3.5.6 Triclinic Bravais Lattice.................... 32 3.5.7 Trigonal Bravais Lattice.................... 33 3.5.8 Hexagonal Bravais Lattice. ................. 33 3.5.9 Exercise 2 ........................... 35 3.6 Point Groups ............................. 36 3.7 Space Groups ............................. 36 3.8 Crystal Structures with Bases. ................... 39 3.8.1 Diamond Structure ...................... 39 3.8.2 Exercise 3 ........................... 40 3.8.3 Graphite Structure ...................... 40 1

Transcript of Condensed Matter Physics

Page 1: Condensed Matter Physics

Condensed Matter Physics I

Peter S. Riseborough

November 21, 2002

Contents

1 Introduction 91.1 The Born-Oppenheimer Approximation . . . . . . . . . . . . . . 9

2 Crystallography 13

3 Structures 133.1 Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Crystalline Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 The Direct Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.1 Primitive Unit Cells . . . . . . . . . . . . . . . . . . . . . 193.3.2 The Wigner-Seitz Unit Cell . . . . . . . . . . . . . . . . . 19

3.4 Symmetry of Crystals . . . . . . . . . . . . . . . . . . . . . . . . 213.4.1 Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . 213.4.2 Group Multiplication Tables . . . . . . . . . . . . . . . . 223.4.3 Point Group Operations . . . . . . . . . . . . . . . . . . . 233.4.4 Limitations Imposed by Translational Symmetry . . . . . 243.4.5 Point Group Nomenclature . . . . . . . . . . . . . . . . . 24

3.5 Bravais Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5.1 Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5.2 Cubic Bravais Lattices. . . . . . . . . . . . . . . . . . . . 283.5.3 Tetragonal Bravais Lattices. . . . . . . . . . . . . . . . . . 303.5.4 Orthorhombic Bravais Lattices. . . . . . . . . . . . . . . . 313.5.5 Monoclinic Bravais Lattice. . . . . . . . . . . . . . . . . . 323.5.6 Triclinic Bravais Lattice. . . . . . . . . . . . . . . . . . . . 323.5.7 Trigonal Bravais Lattice. . . . . . . . . . . . . . . . . . . . 333.5.8 Hexagonal Bravais Lattice. . . . . . . . . . . . . . . . . . 333.5.9 Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6 Point Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.7 Space Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.8 Crystal Structures with Bases. . . . . . . . . . . . . . . . . . . . 39

3.8.1 Diamond Structure . . . . . . . . . . . . . . . . . . . . . . 393.8.2 Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.8.3 Graphite Structure . . . . . . . . . . . . . . . . . . . . . . 40

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3.8.4 Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.8.5 Hexagonal Close-Packed Structure . . . . . . . . . . . . . 413.8.6 Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.8.7 Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.8.8 Other Close-Packed Structures . . . . . . . . . . . . . . . 433.8.9 Sodium Chloride Structure . . . . . . . . . . . . . . . . . 453.8.10 Cesium Chloride Structure . . . . . . . . . . . . . . . . . 453.8.11 Fluorite Structure . . . . . . . . . . . . . . . . . . . . . . 473.8.12 The Copper Three Gold Structure . . . . . . . . . . . . . 473.8.13 Rutile Structure . . . . . . . . . . . . . . . . . . . . . . . 483.8.14 Zinc Blende Structure . . . . . . . . . . . . . . . . . . . . 483.8.15 Zincite Structure . . . . . . . . . . . . . . . . . . . . . . . 493.8.16 The Perovskite Structure . . . . . . . . . . . . . . . . . . 503.8.17 Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.9 Lattice Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.9.1 Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.9.2 Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.10 Quasi-Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Structure Determination 564.1 X Ray Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.1.1 The Bragg conditions . . . . . . . . . . . . . . . . . . . . 564.1.2 The Laue conditions . . . . . . . . . . . . . . . . . . . . . 574.1.3 Equivalence of the Bragg and Laue conditions . . . . . . . 594.1.4 The Ewald Construction . . . . . . . . . . . . . . . . . . . 604.1.5 X-ray Techniques . . . . . . . . . . . . . . . . . . . . . . . 614.1.6 Exercise 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 624.1.7 The Structure and Form Factors . . . . . . . . . . . . . . 624.1.8 Exercise 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1.9 Exercise 12 . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1.10 Exercise 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 704.1.11 Exercise 14 . . . . . . . . . . . . . . . . . . . . . . . . . . 704.1.12 Exercise 15 . . . . . . . . . . . . . . . . . . . . . . . . . . 714.1.13 Exercise 16 . . . . . . . . . . . . . . . . . . . . . . . . . . 724.1.14 Exercise 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2 Neutron Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3 Theory of the Differential Scattering Cross-section . . . . . . . . 76

4.3.1 Time Dependent Perturbation Theory . . . . . . . . . . . 774.3.2 The Fermi-Golden Rule . . . . . . . . . . . . . . . . . . . 784.3.3 The Elastic Scattering Cross-Section . . . . . . . . . . . . 804.3.4 The Condition for Coherent Scattering . . . . . . . . . . . 834.3.5 Exercise 18 . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3.6 Exercise 19 . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3.7 Exercise 20 . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3.8 Anti-Domain Phase Boundaries . . . . . . . . . . . . . . . 864.3.9 Exercise 21 . . . . . . . . . . . . . . . . . . . . . . . . . . 87

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4.4 Elastic Scattering from Quasi-Crystals . . . . . . . . . . . . . . . 884.5 Elastic Scattering from a Fluid . . . . . . . . . . . . . . . . . . . 90

5 The Reciprocal Lattice 935.0.1 Exercise 22 . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.1 The Reciprocal Lattice as a Dual Lattice . . . . . . . . . . . . . . 945.1.1 Exercise 23 . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2 Examples of Reciprocal Lattices . . . . . . . . . . . . . . . . . . . 975.2.1 The Simple Cubic Reciprocal Lattice . . . . . . . . . . . . 975.2.2 The Body Centered Cubic Reciprocal Lattice . . . . . . . 985.2.3 The Face Centered Cubic Reciprocal Lattice . . . . . . . 985.2.4 The Hexagonal Reciprocal Lattice . . . . . . . . . . . . . 995.2.5 Exercise 24 . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3 The Brillouin Zones . . . . . . . . . . . . . . . . . . . . . . . . . 1005.3.1 The Simple Cubic Brillouin Zone . . . . . . . . . . . . . . 1005.3.2 The Body Centered Cubic Brillouin Zone . . . . . . . . . 1015.3.3 The Face Centered Cubic Brillouin Zone . . . . . . . . . . 1015.3.4 The Hexagonal Brillouin Zone . . . . . . . . . . . . . . . . 102

6 Electrons 103

7 Electronic States 1037.1 Many Electron Wave Functions . . . . . . . . . . . . . . . . . . . 104

7.1.1 Exercise 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.2 Bloch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 1147.4 Plane Wave Expansion of Bloch Functions . . . . . . . . . . . . . 1167.5 The Bloch Wave Vector . . . . . . . . . . . . . . . . . . . . . . . 1187.6 The Density of States . . . . . . . . . . . . . . . . . . . . . . . . 120

7.6.1 Exercise 26 . . . . . . . . . . . . . . . . . . . . . . . . . . 1227.7 The Fermi-Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8 Approximate Models 1258.1 The Nearly Free Electron Model . . . . . . . . . . . . . . . . . . 125

8.1.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . 1258.1.2 Non-Degenerate Perturbation Theory . . . . . . . . . . . 1268.1.3 Degenerate Perturbation Theory . . . . . . . . . . . . . . 1298.1.4 Empty Lattice Approximation Band Structure . . . . . . 1328.1.5 Exercise 27 . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.1.6 Degeneracies of the Bloch States . . . . . . . . . . . . . . 1378.1.7 Exercise 28 . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.1.8 Brillouin Zone Boundaries . . . . . . . . . . . . . . . . . . 1468.1.9 The Geometric Structure Factor . . . . . . . . . . . . . . 1488.1.10 Exercise 29 . . . . . . . . . . . . . . . . . . . . . . . . . . 1508.1.11 Exercise 30 . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.1.12 Exercise 31 . . . . . . . . . . . . . . . . . . . . . . . . . . 153

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8.1.13 Exercise 32 . . . . . . . . . . . . . . . . . . . . . . . . . . 1548.1.14 Exercise 33 . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.2 The Pseudo-Potential Method . . . . . . . . . . . . . . . . . . . . 1558.2.1 The Scattering Approach . . . . . . . . . . . . . . . . . . 1598.2.2 The Ziman-Lloyd Pseudo-potential . . . . . . . . . . . . . 1608.2.3 Exercise 34 . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8.3 The Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . . 1628.3.1 Tight-Binding s Band Metal . . . . . . . . . . . . . . . . . 1678.3.2 Exercise 35 . . . . . . . . . . . . . . . . . . . . . . . . . . 1708.3.3 Exercise 36 . . . . . . . . . . . . . . . . . . . . . . . . . . 1708.3.4 Exercise 37 . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.3.5 Exercise 38 . . . . . . . . . . . . . . . . . . . . . . . . . . 1728.3.6 Exercise 39 . . . . . . . . . . . . . . . . . . . . . . . . . . 1728.3.7 Exercise 40 . . . . . . . . . . . . . . . . . . . . . . . . . . 1738.3.8 Wannier Functions . . . . . . . . . . . . . . . . . . . . . . 1738.3.9 Exercise 41 . . . . . . . . . . . . . . . . . . . . . . . . . . 175

9 Electron-Electron Interactions 1769.1 The Landau Fermi Liquid . . . . . . . . . . . . . . . . . . . . . . 176

9.1.1 The Scattering Rate . . . . . . . . . . . . . . . . . . . . . 1779.1.2 The Quasi-Particle Energy . . . . . . . . . . . . . . . . . 1779.1.3 Exercise 42 . . . . . . . . . . . . . . . . . . . . . . . . . . 180

9.2 The Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . 1809.2.1 The Free Electron Gas. . . . . . . . . . . . . . . . . . . . 1849.2.2 Exercise 43 . . . . . . . . . . . . . . . . . . . . . . . . . . 192

9.3 The Density Functional Method . . . . . . . . . . . . . . . . . . . 1929.3.1 Hohenberg-Kohn Theorem . . . . . . . . . . . . . . . . . . 1939.3.2 Functionals and Functional Derivatives . . . . . . . . . . 1959.3.3 The Variational Principle . . . . . . . . . . . . . . . . . . 1989.3.4 The Electrostatic Terms . . . . . . . . . . . . . . . . . . . 2009.3.5 The Kohn-Sham Equations . . . . . . . . . . . . . . . . . 2029.3.6 The Local Density Approximation . . . . . . . . . . . . . 204

9.4 Static Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2069.4.1 The Thomas-Fermi Approximation . . . . . . . . . . . . . 2089.4.2 Linear Response Theory . . . . . . . . . . . . . . . . . . . 2119.4.3 Density Functional Response Function . . . . . . . . . . . 2139.4.4 Exercise 44 . . . . . . . . . . . . . . . . . . . . . . . . . . 2159.4.5 Exercise 45 . . . . . . . . . . . . . . . . . . . . . . . . . . 215

10 Stability of Structures 21710.1 Momentum Space Representation . . . . . . . . . . . . . . . . . . 21710.2 Real Space Representation . . . . . . . . . . . . . . . . . . . . . . 222

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11 Metals 22811.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

11.1.1 The Sommerfeld Expansion . . . . . . . . . . . . . . . . . 22911.1.2 The Specific Heat Capacity . . . . . . . . . . . . . . . . . 23111.1.3 Exercise 46 . . . . . . . . . . . . . . . . . . . . . . . . . . 23411.1.4 Exercise 47 . . . . . . . . . . . . . . . . . . . . . . . . . . 23411.1.5 Pauli Paramagnetism . . . . . . . . . . . . . . . . . . . . 23411.1.6 Exercise 48 . . . . . . . . . . . . . . . . . . . . . . . . . . 23711.1.7 Exercise 49 . . . . . . . . . . . . . . . . . . . . . . . . . . 23711.1.8 Landau Diamagnetism . . . . . . . . . . . . . . . . . . . . 23811.1.9 Landau Level Quantization . . . . . . . . . . . . . . . . . 23911.1.10The Diamagnetic Susceptibility . . . . . . . . . . . . . . . 241

11.2 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . 24411.2.1 Electrical Conductivity . . . . . . . . . . . . . . . . . . . 24411.2.2 Scattering by Static Defects . . . . . . . . . . . . . . . . . 24411.2.3 Exercise 50 . . . . . . . . . . . . . . . . . . . . . . . . . . 25111.2.4 The Hall Effect and Magneto-resistance. . . . . . . . . . . 25211.2.5 Multi-band Models . . . . . . . . . . . . . . . . . . . . . . 260

11.3 Electromagnetic Properties of Metals . . . . . . . . . . . . . . . . 26311.3.1 The Longitudinal Response . . . . . . . . . . . . . . . . . 26611.3.2 Electron Scattering Experiments . . . . . . . . . . . . . . 27411.3.3 Exercise 51 . . . . . . . . . . . . . . . . . . . . . . . . . . 27811.3.4 Exercise 52 . . . . . . . . . . . . . . . . . . . . . . . . . . 28011.3.5 The Transverse Response . . . . . . . . . . . . . . . . . . 28511.3.6 Optical Experiments . . . . . . . . . . . . . . . . . . . . . 28811.3.7 Kramers-Kronig Relation . . . . . . . . . . . . . . . . . . 28911.3.8 Exercise 53 . . . . . . . . . . . . . . . . . . . . . . . . . . 29011.3.9 Exercise 54 . . . . . . . . . . . . . . . . . . . . . . . . . . 29111.3.10The Drude Conductivity . . . . . . . . . . . . . . . . . . . 29111.3.11Exercise 55 . . . . . . . . . . . . . . . . . . . . . . . . . . 29611.3.12Exercise 56 . . . . . . . . . . . . . . . . . . . . . . . . . . 29611.3.13The Anomalous Skin Effect . . . . . . . . . . . . . . . . . 29711.3.14 Inter-Band Transitions . . . . . . . . . . . . . . . . . . . . 299

11.4 Measuring the Fermi-Surface . . . . . . . . . . . . . . . . . . . . 30011.4.1 Semi-Classical Orbits . . . . . . . . . . . . . . . . . . . . 30111.4.2 de Haas - van Alphen Oscillations . . . . . . . . . . . . . 30511.4.3 Exercise 57 . . . . . . . . . . . . . . . . . . . . . . . . . . 30711.4.4 The Lifshitz-Kosevich Formulae . . . . . . . . . . . . . . . 30811.4.5 Other Fermi-Surface Probes . . . . . . . . . . . . . . . . . 31311.4.6 Cyclotron Resonances . . . . . . . . . . . . . . . . . . . . 315

11.5 The Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . 31911.5.1 The Integer Quantum Hall Effect . . . . . . . . . . . . . . 31911.5.2 Exercise 58 . . . . . . . . . . . . . . . . . . . . . . . . . . 32511.5.3 The Fractional Quantum Hall Effect . . . . . . . . . . . . 32611.5.4 Quasi-Particle Excitations . . . . . . . . . . . . . . . . . . 32811.5.5 Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

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11.5.6 Composite Fermions . . . . . . . . . . . . . . . . . . . . . 338

12 Insulators 34112.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

12.1.1 Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34512.1.2 Intrinsic Semiconductors . . . . . . . . . . . . . . . . . . . 34712.1.3 Extrinsic Semiconductors . . . . . . . . . . . . . . . . . . 34912.1.4 Exercise 59 . . . . . . . . . . . . . . . . . . . . . . . . . . 352

12.2 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . 35312.3 Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 353

13 Phonons 354

14 Harmonic Phonons 35414.1 Lattice with a Basis . . . . . . . . . . . . . . . . . . . . . . . . . 36014.2 A Sum Rule for the Dispersion Relations . . . . . . . . . . . . . . 360

14.2.1 Exercise 60 . . . . . . . . . . . . . . . . . . . . . . . . . . 36314.3 The Nature of the Phonon Modes . . . . . . . . . . . . . . . . . . 363

14.3.1 Exercise 61 . . . . . . . . . . . . . . . . . . . . . . . . . . 36414.3.2 Exercise 62 . . . . . . . . . . . . . . . . . . . . . . . . . . 36414.3.3 Exercise 63 . . . . . . . . . . . . . . . . . . . . . . . . . . 36514.3.4 Exercise 64 . . . . . . . . . . . . . . . . . . . . . . . . . . 36514.3.5 Exercise 65 . . . . . . . . . . . . . . . . . . . . . . . . . . 36614.3.6 Exercise 66 . . . . . . . . . . . . . . . . . . . . . . . . . . 36614.3.7 Exercise 67 . . . . . . . . . . . . . . . . . . . . . . . . . . 367

14.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 36814.4.1 The Specific Heat . . . . . . . . . . . . . . . . . . . . . . 37014.4.2 The Einstein Model of a Solid . . . . . . . . . . . . . . . . 37114.4.3 The Debye Model of a Solid . . . . . . . . . . . . . . . . . 37214.4.4 Exercise 68 . . . . . . . . . . . . . . . . . . . . . . . . . . 37414.4.5 Exercise 69 . . . . . . . . . . . . . . . . . . . . . . . . . . 37414.4.6 Exercise 70 . . . . . . . . . . . . . . . . . . . . . . . . . . 37414.4.7 Exercise 71 . . . . . . . . . . . . . . . . . . . . . . . . . . 37514.4.8 Lindemann Theory of Melting . . . . . . . . . . . . . . . . 37514.4.9 Thermal Expansion . . . . . . . . . . . . . . . . . . . . . 37714.4.10Thermal Expansion of Metals . . . . . . . . . . . . . . . . 379

14.5 Anharmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37914.5.1 Exercise 72 . . . . . . . . . . . . . . . . . . . . . . . . . . 380

15 Phonon Measurements 38115.1 Inelastic Neutron Scattering . . . . . . . . . . . . . . . . . . . . . 381

15.1.1 The Scattering Cross-Section . . . . . . . . . . . . . . . . 38215.2 The Debye-Waller Factor . . . . . . . . . . . . . . . . . . . . . . 38715.3 Single Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . 38915.4 Multi-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . 390

15.4.1 Exercise 73 . . . . . . . . . . . . . . . . . . . . . . . . . . 391

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15.4.2 Exercise 74 . . . . . . . . . . . . . . . . . . . . . . . . . . 39115.4.3 Exercise 75 . . . . . . . . . . . . . . . . . . . . . . . . . . 391

15.5 Raman and Brillouin Scattering of Light . . . . . . . . . . . . . . 391

16 Phonons in Metals 39416.1 Screened Ionic Plasmons . . . . . . . . . . . . . . . . . . . . . . . 395

16.1.1 Kohn Anomalies . . . . . . . . . . . . . . . . . . . . . . . 39616.2 Dielectric Constant of a Metal . . . . . . . . . . . . . . . . . . . . 39616.3 The Retarded Electron-Electron Interaction . . . . . . . . . . . . 39916.4 Phonon Renormalization of Quasi-Particles . . . . . . . . . . . . 40016.5 Electron-Phonon Interactions . . . . . . . . . . . . . . . . . . . . 40216.6 Electrical Resistivity due to Phonon Scattering . . . . . . . . . . 403

16.6.1 Umklapp Scattering . . . . . . . . . . . . . . . . . . . . . 40816.6.2 Phonon Drag . . . . . . . . . . . . . . . . . . . . . . . . . 409

17 Phonons in Semiconductors 41017.1 Resistivity due to Phonon Scattering . . . . . . . . . . . . . . . . 41017.2 Polarons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41117.3 Indirect Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 411

18 Impurities and Disorder 41318.1 Scattering By Impurities . . . . . . . . . . . . . . . . . . . . . . . 41618.2 Virtual Bound States . . . . . . . . . . . . . . . . . . . . . . . . . 41818.3 Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42118.4 Coherent Potential Approximation . . . . . . . . . . . . . . . . . 42218.5 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

18.5.1 Anderson Model of Localization . . . . . . . . . . . . . . . 42418.5.2 Scaling Theories of Localization . . . . . . . . . . . . . . . 425

19 Magnetic Impurities 42819.1 Localized Magnetic Impurities in Metals . . . . . . . . . . . . . . 42819.2 Mean Field Approximation . . . . . . . . . . . . . . . . . . . . . 428

19.2.1 The Atomic Limit . . . . . . . . . . . . . . . . . . . . . . 43119.3 The Schrieffer-Wolf Transformation . . . . . . . . . . . . . . . . . 431

19.3.1 The Kondo Hamiltonian . . . . . . . . . . . . . . . . . . . 43419.4 The Resistance Minimum . . . . . . . . . . . . . . . . . . . . . . 435

20 Collective Phenomenon 440

21 Itinerant Magnetism 44021.1 Stoner Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

21.1.1 Exercise 76 . . . . . . . . . . . . . . . . . . . . . . . . . . 44221.1.2 Exercise 77 . . . . . . . . . . . . . . . . . . . . . . . . . . 442

21.2 Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . 44221.3 Magnetic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 44421.4 Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

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21.5 The Heisenberg Model . . . . . . . . . . . . . . . . . . . . . . . . 449

22 Localized Magnetism 45022.1 Holstein - Primakoff Transformation . . . . . . . . . . . . . . . . 45122.2 Spin Rotational Invariance . . . . . . . . . . . . . . . . . . . . . . 454

22.2.1 Exercise 78 . . . . . . . . . . . . . . . . . . . . . . . . . . 45722.3 Anti-ferromagnetic Spinwaves . . . . . . . . . . . . . . . . . . . . 458

22.3.1 Exercise 79 . . . . . . . . . . . . . . . . . . . . . . . . . . 460

23 Spin Glasses 46023.1 Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 46223.2 The Sherrington-Kirkpatrick Solution. . . . . . . . . . . . . . . . 463

24 Magnetic Neutron Scattering 46724.1 The Inelastic Scattering Cross-Section . . . . . . . . . . . . . . . 467

24.1.1 The Dipole-Dipole Interaction . . . . . . . . . . . . . . . . 46724.1.2 The Inelastic Scattering Cross-Section . . . . . . . . . . . 467

24.2 Time Dependent Spin Correlation Functions . . . . . . . . . . . . 47124.3 The Fluctuation Dissipation Theorem . . . . . . . . . . . . . . . 47324.4 Magnetic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 475

24.4.1 Neutron Diffraction . . . . . . . . . . . . . . . . . . . . . 47524.4.2 Exercise 80 . . . . . . . . . . . . . . . . . . . . . . . . . . 47624.4.3 Exercise 81 . . . . . . . . . . . . . . . . . . . . . . . . . . 47724.4.4 Spin Wave Scattering . . . . . . . . . . . . . . . . . . . . 47724.4.5 Exercise 82 . . . . . . . . . . . . . . . . . . . . . . . . . . 47824.4.6 Critical Scattering . . . . . . . . . . . . . . . . . . . . . . 478

25 Superconductivity 48025.1 Experimental Manifestation . . . . . . . . . . . . . . . . . . . . . 480

25.1.1 The London Equations . . . . . . . . . . . . . . . . . . . . 48125.1.2 Thermodynamics of the Superconducting State . . . . . . 483

25.2 The Cooper Problem . . . . . . . . . . . . . . . . . . . . . . . . . 48525.3 Pairing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

25.3.1 The Pairing Interaction . . . . . . . . . . . . . . . . . . . 48925.3.2 The B.C.S. Variational State . . . . . . . . . . . . . . . . 49125.3.3 The Gap Equation . . . . . . . . . . . . . . . . . . . . . . 49325.3.4 The Ground State Energy . . . . . . . . . . . . . . . . . . 494

25.4 Quasi-Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49625.4.1 Exercise 83 . . . . . . . . . . . . . . . . . . . . . . . . . . 499

25.5 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 49925.6 Perfect Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 50125.7 The Meissner Effect . . . . . . . . . . . . . . . . . . . . . . . . . 503

26 Landau-Ginsberg Theory 504

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1 Introduction

Condensed Matter Physics is the study of materials in Solid and Liquid Phases.It encompasses the study of ordered crystalline phases of solids, as well as disor-dered phases such as the amorphous and glassy phases of solids. Furthermore,it also includes materials with short-ranged order such as conventional liquids,and liquid crystals which show unconventional order intermediate between thoseof a crystalline solid and a liquid. Condensed matter has the quite remarkableproperty that, due to the large number of particles involved, the behavior ofthe materials may be qualitatively distinct from those of the individual con-stituents. The behavior of the incredibly large number of particles is governedby (quantum) statistics which, through the chaotically complicated motion ofthe particles, produces new types of order. These emergent phenomena are bestexemplified in phenomenon such as magnetism or superconductivity where thecollective behavior results in transitions to new phases.

In surveying the properties of materials it is convenient to separate theproperties according to two (usually) disparate time scales. One time scaleis a slow time scale which governs the structural dynamics, and a faster timescale that governs the electronic motion. The large difference between the timescales is due to the large ratio of the nuclear masses to the electronic mass,Mn/me ∼ 103. The long-ranged electromagnetic force binds these two con-stituents of different mass into electrically neutral material. The slow movingnuclear masses can be considered to be quasi-static, and are responsible fordefining the structure of matter. In this approximation, the fast moving elec-trons equilibrate in the quasi-static potential produced by the nuclei.

1.1 The Born-Oppenheimer Approximation

The difference in the relevant time scales for electronic and nuclear motion allowsone to make the Born-Oppenheimer Approximation. In this approximation, theelectronic states are treated as if the nuclei were at rest at fixed positions. How-ever, when treating the slow motions of the nuclei, the electrons are consideredas adapting instantaneously to the potential of the charged nuclei, thereby min-imizing the electronic energies. Thus, the nuclei charges are dressed by a cloudof electrons forming ionic or atomic-like aggregates.

A qualitative estimate of the relative energies of nuclear versus electronicmotion can be obtained by considering metallic hydrogen. The electronic en-ergies are calculated using only the Bohr model of the hydrogen atom. Theequation of motion for an electron of mass me has the form

− Z e2

a2= − me v

2

a(1)

where Z is the nuclear charge and a is the radius of the atomic orbital. The stan-

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dard semi-classical quantization condition due to Bohr and Sommerfeld restrictsthe angular momentum to integral values of h

me v a = n h (2)

These equations can be combined to find the quantized total electronic energyof the hydrogen atom

Ee = −Z e2

a+

me v2

2

= −Z e2

2 a

= −me Z2 e4

2 n2 h2 (3)

which is a standard result from atomic physics. Note that the kinetic energyterm and the electrostatic potential term have similar magnitudes.

Now consider the motion of the nuclei. The forces consist of Coulomb forcesbetween the nuclei and electrons, and the quantum mechanical Pauli forces. Theelectrostatic repulsions and attractions have similar magnitudes, since the inter-nuclear separations are of the same order as the Bohr radius. In equilibrium,the sum of the forces vanish identically. Furthermore, if an atom is displacedfrom the equilibrium position by a small distance equal to r, the restoring forceis approximately given by the dipole force

− αZ e2

a3r (4)

where α is a dimensionless constant. Hence, the equation of motion for thedisplacement of a nuclei of mass Mn is

− αZ e2

a3r = Mn

d2 r

dt2(5)

which shows that the nuclei undergo harmonic oscillations with frequency

ω2 = αZ e2

Mn a3(6)

The semi-classical quantization condition∮dr Mn v = n h (7)

yields the energy for nuclear motion as

EN = n h ω

= nme Z

2 e4

h2 α12

(me

Mn

) 12

(8)

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Thus, the ratio of the energies of nuclear motion to electronic motion are givenby the factor

EN

Ee∼(me

Mn

) 12

(9)

Since the ratio of the mass of electron to the proton mass is 12000 , the nuclear

kinetic energy is negligible when compared to the electronic kinetic energy. Amore rigorous proof of the validity of the Born-Oppenheimer approximation wasgiven by Migdal.

In the first part of the course it is assumed that the Born-Oppenheimer ap-proximation is valid.

First, the subject of Crystallography shall be discussed, and the charac-ters of the equilibrium structures of the dressed nuclei in matter are described.An important class of such materials are those which posses long-ranged peri-odic translational order and other symmetries. It shall be shown how these longrange ordered and amorphous structures can be effectively probed by variouselastic scattering experiments, in which the wave length of the scattered parti-cles is comparable to the distance between the nuclei.

In the second part, the properties of the Electrons shall be discussed. Onassuming the validity of the Born-Oppenheimer approximation, the nature ofthe electronic states that occur in the presence of the potential produced bythe nuclei shall be discussed. One surprising result of this approach is that,even though the strength of the ionic potential is quite large (of the order ofRydbergs), in some metals the highest occupied electronic states bear a closeresemblance to the states expected if the ionic potential was very weak or neg-ligible. In other materials, the potential due to the ionic charges can producegaps in the electronic energy spectrum. Using Bloch’s theorem, it shall beshown how periodic long-ranged order can produce gaps in the electronic spec-trum. Another surprising result is that, in most metals, it appears as thoughthe electron-electron interactions can be neglected, or more precisely can bethought of sharing the properties of a non-interacting electron gas, albeit withrenormalized masses or magnetic moments.

The thermodynamic properties of electrons in these Bloch states shall betreated using Fermi-Dirac statistics. Furthermore, the concepts of the Fermi-energy and Fermi-surface of metals are introduced. It shall be shown how theelectronic transport properties of metals are dominated by states with energiesclose to the Fermi-surface, and how the Fermi-surface can be probed.

The third part concerns the motion of the ions or nuclei. In particular, itwill be considered how the fast motion of the electrons dress or screen the inter-nuclear potentials. The low energy excitations of the dressed nuclear or ionicstructure of matter give rise to harmonic-like vibrations. The elementary exci-

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tation of the quantized vibrations are known as Phonons. It shall be shownhow these phonon excitations manifest themselves in experiments, in thermo-dynamic properties and, how they participate in limiting electrical transport.

The final part of the course concerns some of the more striking examplesof the Collective Phenomenon such as Magnetism and Superconductivity.These phenomena involve the interactions between the elementary excitationsof the solid, and through collective action, they spontaneously break the sym-metry of the Hamiltonian. In many cases, the spontaneously broken symmetryis accompanied by the formation of a new branch of low energy excitations.

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2 Crystallography

Crystallography is the study of structure of ordered solids, disordered solids andalso liquids. In this section, it shall be assumed that the nuclei are static, frozeninto their average positions. Due to the large nuclear masses and strong inter-actions between the nuclei (dressed by their accompanying clouds of electrons),one may assume that the nuclear or ionic motion can be treated classically. Themost notable failure of this assumption occurs with the very lightest of nuclei,such as He. In the anomalous case of He, where the separation between ions,d, is of the order of angstroms, the uncertainty of the momentum is given by h

dand the kinetic energy EK for this quantum zero point motion is given by

EK =h2

2 M d2(10)

The kinetic energy is large since the mass M of the He atom is small. Themagnitude of the kinetic energy of the zero point fluctuations is larger than theweak van der Waals or London force between the He ions. Thus, the inter-ionicforces are insufficient to bind the He ion into a solid and the material remains ina liquid-like state, until the lowest temperatures. For these reasons, He behaveslike a quantum fluid. However, for the heavier nuclei, the quantum nature ofthe particles manifest themselves in more subtle ways.

First, the various types of structures and the symmetries that can be foundin Condensed Matter are described and then the various experimental methodsused to observe these structures are discussed.

3 Structures

The structure of condensed materials is usually thought about in terms of den-sity of either electrons or nuclear matter. To the extent that the regions ofnon-zero density of the nuclear matter are highly localized in space, with lin-ear dimensions of 10−15 meters, the nuclei can be discussed in terms of pointobjects. The electron density is more extended and varies over length scales of10−10 meters. The length scale for the electronic density in solids and fluids isvery similar to the length scale over which the electron density varies in isolatedatoms. The similarity of scales occurs as electrons are partially responsible forthe bonding of atoms into a solid. That is, the characteristic atomic length scaleis almost equal to the characteristic separation of the nuclei in condensed mat-ter. Due to the near equality of these two length scales, the electron density insolids definitely cannot be represented in terms of a superposition of the densityof well defined atoms. However, the electron density does show a significantvariation that can be interpreted in terms of the electron density of isolatedatoms, subject to significant modifications when brought together. As the elec-tron density for isolated atoms is usually spherically symmetric, the structure

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in the electronic density may, for convenience of discussion, be approximatelyrepresented in terms of a set of spheres of finite radius.

3.1 Fluids

Both liquids and gases are fluids. The macroscopic characteristics of fluids arethat they are spatially uniform and isotropic, which means that the average en-vironment of any atom is identical to the average environment of any other atom.

The density is defined by the function

ρ(r) =∑

i

δ3( r − ri ) (11)

in which ri is the instantaneous position of the i-th atom. A measurement ofthe density usually results in the time average of the density which correspondsto the time averaged positions of the atoms.

In particular, for a fluid, spatial homogeneity ensures that the average den-sity ρ(r) at position r is equal to the average density at a displaced positionr +R,

ρ(r) = ρ(r +R) (12)

The value of the displacement R is arbitrary, so the average density is inde-pendent of r and can be expressed as ρ(0). This just means that the averageposition of an individual atom is undetermined.

The operations which leave the system unchanged are the symmetry oper-ations. For a fluid, the symmetry operations consist of the continuous transla-tions through an arbitrary displacement R, rotations through an arbitrary angleabout an arbitrary axis, and also reflections in arbitrary mirror planes.

The set of symmetry operations form a group called the symmetry group.For a fluid, the symmetry group is the Euclidean group. Fluids have the largestpossible number of symmetry operators and have the highest possible symmetry.All other materials are invariant under a smaller number of symmetry opera-tions.

Nevertheless, fluids do have short-ranged structure which is exemplified bylocating one atom and then examining the positions of the neighboring atoms.The local spatial correlations are expressed by the density - density correlationfunction which is expressed as an average

C(r, r′) = ρ(r) ρ(r′)

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=∑i,j

δ3( r − ri ) δ3( r′ − rj )

(13)

Since fluids are homogeneous, the correlation function is only a function of thedifference of the positions r − r′. Furthermore, since the fluids are isotropicand invariant under rotations, the correlation function is only a function of thedistance separating the two regions of space | r − r′ |. At sufficiently largeseparation distances, the positions of the atoms become uncorrelated, thus,

limr−r′ → ∞

C(r, r′) → ρ(r) ρ(r′)

→ ρ(0) ρ(r − r′) (14)

That is, at large spatial separations, the density - density correlation functionreduces to the product of the independent average of the density at the originand the average density at a position r. From the homogeneity of the fluid, ρ(r)is identical to the average of ρ(0).

The density - density correlation function contains the correlation betweenthe same atom, that is, there are terms with i = j. This leads to a contributionwhich shows up at short distances,∑

i=j

δ3( r − ri ) δ3( r′ − ri ) = δ3( r − r′ )∑

i

δ3( r − ri )

= δ3( r − r′ ) ρ(r) (15)

which is proportional to the density.

The pair distribution function g(r− r′) is defined as the contribution to thedensity - density correlation function which excludes the correlation between anatom and itself,

g( r − r′ ) = C( r − r′ ) − δ3( r − r′ ) ρ(r) (16)

For a system which possesses continuous translational invariance, the pair dis-tribution function can be evaluated as

g( r − r′ ) =∑i 6=j

δ3( r − ri ) δ3( r′ − rj )

=1V

∫d3R

∑i 6=j

δ3( r − ri − R ) δ3( r′ − rj − R )

=1V

∑i 6=j

δ3( r − r′ − ri + rj ) (17)

Since the sum over i runs over each of the inter-atomic separations rj − ri for

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each value of j, spatial homogeneity demands that the contribution from eachdifferent j value is identical. There are N such terms, and this leads to theexpression for the pair distribution function involving an atom at the centralsite r0, and the others at sites i

g(r) = ρ(0)∑

i

δ3( r − ri + r0 ) (18)

whereρ(0) =

N

V(19)

As this only depends on the radial distance | r | this is called the radial distri-bution function g(r). For large r, the pair distribution function, like C(r, 0),approaches ρ(0)

2, or

limr → ∞

g(r) → ρ(0)2

(20)

Liquids are defined as the fluids that have high densities. The liquid phaseis not distinguished from the higher temperature gaseous phase by a changein symmetry, unlike most other materials. In the liquid phase the density ishigher, the inter-atomic forces play a more important role than in the low densitygaseous phase. The interaction forces are responsible for producing the shortranged correlation in the density - density function. A model potential thatis representative of typical inter-atomic force between two neutral atoms is theLennard-Jones potential.

V (r) = 4 V0

[ (a

r

)12

−(a

r

)6 ](21)

The potential has a short ranged repulsion between the atoms caused by theoverlap of the electronic states, and the long-ranged van der Waals attractioncaused by fluctuation induced electric polarizations of the atoms. The resultingpotential falls to zero at r = a and has a minimum at r = 2

16 a. The potential

at the minimum of the well is given by − V0. Another model potential that isof use is the hard sphere potential which excludes the center of another atomfrom the region of radius 2 a centered on the central atom.

As the repulsion between atoms dominates the structure of liquids, theBernal model of random close packing of hard spheres is responsible for mostof the structure of a liquid. On randomly packing spheres, one finds a packingfraction of atoms given by 0.638. The packing fraction is defined as the totalvolume of the hard spheres divided by the (minimum) volume that contains allthe spheres. Random packings of hard spheres can be used to calculate theradial distribution function g(r). The model shows that there are strong short-ranged correlations between the closest atoms and that there are 12 in threedimensions at radial distances 2 a. These correlations show up as a strong peak

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in the radial distribution function at 2 a, and there are other peaks correspond-ing to the next few shells of neighboring atoms.

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3.2 Crystalline Solids

A perfect crystal consists of a space filling periodic array of atoms. It canbe partitioned into identical individual structural units that can be repeatedlystacked together to form the crystal. The structural unit is called the unit cell.There are many alternate ways of performing the partitioning and, therefore,there are many alternate forms for the unit cells. The unit cells which have thesmallest possible volume are called primitive unit cells. A unit cell may containone or more atoms. The positions of the atoms, when referenced to a specificpoint in the unit cell, composes the basis of the lattice.

3.3 The Direct Lattice

Equivalent points taken from each unit cell in a perfect crystal form a periodiclattice. The points are called lattice points. Any lattice point can be reachedfrom any other by a translation R that is a combination of an integer multipleof three primitive lattice vectors a1, a2, a3,

R = n1 a1 + n2 a2 + n3 a3 (22)

Here, n1, n2 and n3 are integers that determine the magnitudes of three com-ponents of a three-dimensional vector. The set of integers (n1, n2, n3) can beused to represent a lattice point in terms of the primitive lattice vectors. Theset (n1, n2, n3) runs through all the positive and negative integers. The set oftranslations R is closed under addition and, therefore, the translation operationsform a group.

Given any lattice, there are many choices for the primitive lattice vectorsa1, a2, a3.

The array of lattice points have arrangements and orientations which areidentical in every respect when viewed from origins centered on different latticepoints. For example, on translating the origin through a lattice vector Rm, thedisplacements in the primed reference frame are related to displacements in theunprimed reference frame via

r′ = r + m1 a1 + m2 a2 + m3 a3 (23)

and the lattice points in the two frames are related via

n′i = ni + mi (24)

and as the numbers ni and n′i take on all possible integer values, the set of alllattices are identical in the two reference frames.

A crystal structure is composed of a lattice in which a basis of atoms isattached to each lattice point. That is, a complete specification of a crystal

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requires specifying the lattice and the distribution of the various atoms aroundeach lattice point. The basis is specified by giving the number of atoms andtypes of the atoms in the basis (j) together with their positions relative to thelattice points. The position of the j-th atom relative to the lattice point, rj , isdenoted by

rj =(

xj a1 + yj a2 + zj a3

)(25)

where the set (xi, yi, zi) may be non-integer numbers.

The choice of lattice and, therefore, the basis, is non-unique for a crystalstructure. An example of this is given by a two dimensional crystal structurewhich can be represented many different ways including the possibilities of arepresentation either as a lattice with a one atom basis or as a lattice with atwo atom basis.

3.3.1 Primitive Unit Cells

The parallelepiped defined by the primitive lattice vectors forms a primitiveunit cell. When repeated a primitive unit cell will fill all space. The primitiveunit cell is also a unit cell with the minimum volume. Although there are anumber of different ways of choosing the primitive lattice vectors and unit cells,the number of basis atoms in a primitive cell is unique for each crystal structure.No basis contains fewer atoms than the basis associated with a primitive unitcell.

There is always just one lattice point per primitive unit cell.

If the primitive unit cell is a parallelepiped with lattice points at each of theeight corners, then each corner is shared by eight cells, so that the total numberof lattice points per cell is unity as 8 × 1

8 = 1.

The volume of the parallelepiped is given in terms of the primitive latticevectors via

Vc = | a1 . ( a2 ∧ a3 ) | (26)

The primitive unit cell is a unit cell of minimum volume.

3.3.2 The Wigner-Seitz Unit Cell

An alternate method of constructing a unit cell is due to Wigner and Seitz.The Wigner-Seitz cell has the important property that there are no arbitrarychoices made in defining the unit cell. The absence of any arbitrary choice hasthe consequence that the Wigner-Seitz unit cell always has the same symmetryas the lattice. The Wigner-Seitz unit cell is constructed by forming a set of

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planes which bisect the lines joining a central lattice point to all other latticepoints. The region of space surrounding the central lattice point, of minimumvolume, which is completely enclosed by a set of the bisecting planes consti-tutes the Wigner-Seitz cell. Thus, the Wigner-Seitz cell consists of the volumecomposed of all the points that are closer to the central lattice site than to anyother lattice site.

The equations of the planes bisecting the vector from the central point tothe i-th lattice point is given by(

r − 12Ri

). Ri = 0 (27)

where Ri is the lattice vector. The sections of planes closest to the origin formthe surface of the Wigner Seitz-cell.

As the definition does not involve any arbitrary choice of primitive latticevectors, the Wigner-Seitz cell possesses the full symmetry of the lattice. Fur-thermore, the Wigner-Seitz cell is space filling, since every point in space mustlie closer to one lattice point than any other.

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3.4 Symmetry of Crystals

A symmetry operation acts on a crystal producing a new crystal, shifting theatoms to new positions such that the new crystal is identical in appearance tothe original crystal. That is, the positions of the atoms in the new crystal coin-cide with the positions of similar atoms in the original crystal. The symmetryoperations may consist of :-

(i) Translation operations which leave no point unchanged.

(ii) Symmetry operations which leave one point unchanged.

(iii) Combinations of the above two types of operations.

3.4.1 Symmetry Groups

A set of symmetry operations form a group if, when the symmetry operationsare combined, the following properties are satisfied :

(I) The product of any two symmetry operators from the set, say A and B,defined as A B = C then C is also in the set. That is, the set of symmetryoperations is closed under composition.

(II) The composition of any three elements is associative, which means thatthe symmetry operation is independent of whether the first and second operatorsare combined before they are combined with the third, or whether the secondand third operators are combined before they are combined with the first.

A ( B C ) = ( A B ) C (28)

(III) There exists a symmetry operator which leaves all the atoms in theiroriginal places, called the identity operator E. The product of an arbitrarilychosen symmetry operator of the group with the identity gives back the arbi-trarily chosen operator.

A E = E A = A (29)

(IV) For each operator in the group, there exists a unique inverse operatorsuch that when the operator is combined with the inverse operator, they producethe identity.

A A−1 = A−1 A = E (30)

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A group of symmetry operators may contain a sub-set of symmetry operatorswhich also form a group. That is, the group laws are obeyed for all the elementsof the sub-set. This sub-set of elements forms a sub-group of the group, but isonly a sub-group if the elements are combined with the same law of compositionas the group.

The symmetry group of the direct lattice contains at least two sub-groups.These are the sub-group of translations and the point group of the lattice. Un-der a translation which is not the identity, no point remains invariant. Thepoint group of the lattice consists of the set of symmetry operations in whichat least one point of the lattice is invariant.

3.4.2 Group Multiplication Tables

The properties of a group are concisely represented by the group multiplicationtable. The number of elements in the group is called the order of the group, sothe general group with n operators is of order n. The group multiplication tableconsists of an n by n array. The group multiplication table has the conventionthat if A × B = C then the operator A which is the first element of theproduct is located on the left most column of the table, and the operator Bwhich is the second element is located in the uppermost row. The product C isentered in the same row as the element A and the same column as element B.

E . . B .. . . . .A . . C .. . . . .. . . . .

(31)

In general the symmetry operations do not commute, that is, A × B 6= B × A.The identity operator is placed as the first element of the series of symmetryoperators, so the first row and first column play the dual role as the list of groupelements and also are the elements found by compounding the elements withthe identity. Every operator appears once, and only once, in each row or columnof the group table. The fact that each operator occurs only once in any row, orany column, is a consequence of the uniqueness of the inverse.

As an example, consider the point group for a single H2O molecule. Thegroup contains a symmetry operation which is a rotation by π about an axis inthe plane of the molecule that passes through the O atom and bisects the linebetween the two H atoms. This is a two-fold axis since a second rotation by π isequivalent to the identity. The two-fold rotation is labelled as C2. In this case,the two-fold axis is the rotation axis of highest order and, thus, is consideredto define the vertical direction. In addition to the two-fold axis, there are twomirror planes. It is conventional to denote a mirror plane that contains the n-fold axis of rotation (Cn) with highest n as a vertical plane. The H2O molecule

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is symmetric under reflection in a mirror plane passing through the two-fold axisin the plane which contains the molecule. That is, the mirror plane is the planepassing through all three atoms. This is a vertical mirror symmetry operationand is denoted by σv. The second mirror symmetry operation is a reflection inanother vertical plane passing through the C2 axis, but this time, the mirrorplane is perpendicular to the plane of the molecule, and is denoted by σ′v. Thesymmetry group contains the elements E, C2, σv, σ′v. The group is of order 4.The group table is given by

E C2 σv σ′v

C2 E σ′v σv

σv σ′v E C2

σ′v σv C2 E

Since all the operations in this group commute, the group is known as an Abeliangroup. Inspection of the table immediately shows that σv × C2 = σ′v.

The symmetry group of a crystal has at least two sub-groups. One sub-groupis the group of translations through the set lattice vectors R. A general trans-lation which is not the identity, leaves no point unchanged by the translations.A second sub-group is formed by the set of all transformations which leave thesame point of the crystal untransformed. This sub-group is the point group.

3.4.3 Point Group Operations

The crystallographic point group consists of the symmetry operations that leaveat least one point untransformed.

The possible symmetry elements of the point group are:-

Rotations through integer multiples of 2 πn around an axis. The n-fold rota-

tions are denoted as Cn.

Reflections that take every point into its mirror image with respect to a planeknown as the mirror plane. Reflections are denoted by σ.

Inversions that take every point r, as measured from an origin, into the point− r. The inversion operator is denoted by I.

Rotation Reflections which are rotations about an axis through integer mul-tiples of 2 π

n followed by reflection in a plane perpendicular to the axis. Then-fold rotation reflections are denoted by Sn. For even n, (Sn)n = E, while

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for odd n, (Sn)n = σ.

Rotation Inversions which are rotations about an axis through integer mul-tiples of 2 π

n followed by an inversion through an origin. The Internationalnotation for a rotation reflection is n. The rotation inversion and rotation re-flection operations are related for example, 3 = S−1

6 , 4 = S−14 and 6 = S−1

3 .

Since at least one point is invariant under all the transformations of the pointgroup, the rotation axes and mirror planes must all intersect at these points.

3.4.4 Limitations Imposed by Translational Symmetry

Although all point group operations are allowable for isolated molecules, certainpoint group symmetries are not allowed for periodic lattices. The limitations onthe possible types of point group symmetry operations can be seen by examiningthe effect of an n-fold axis in a plane perpendicular to a line through latticepoints A − B . . . C − D, with 1 + m1 lattice points on it. The directionof the line will be chosen as the direction of the primitive lattice vector a1, andthe line is assumed to have a length m1 a1. A clockwise rotation of 2 π

n aboutthe n-fold axis of rotation through point A will generate a new line A − B′.Likewise, a counter clockwise rotation of 2 π

n about the n-fold axis of rotationthrough point D will generate a new line D − C”. The line constructed throughB′ − C” is parallel to the initial line A − D. The length of the line B′ − C”is equal to m1 a1 − 2 cos 2 π

n a1 and must be equal to an integer multiple ofa1, say m′

1. Then

cos2 πn

=m1 − m′

1

2(32)

Thus, cos 2 πn must be integer or half odd integer which is in the set ±1, 0,± 1

2.This restriction limits the possible n-fold rotation axis to be of order n =1 , 2 , 3 , 4 , 6 and allows no others. Thus, a crystalline lattice can onlycontain a two, three-fold, four-fold or six-fold axis of rotation. However, theredo exist solids that possess five-fold symmetry, such as quasi-crystals. Quasi-crystals are not crystals as they do not possess periodic translational invariance.

3.4.5 Point Group Nomenclature

The point groups are referred to by using two different notation schemes, theSchoenflies and the International notation. In the following examples, first thegroups are labelled by their Schoenflies designation and then their Internationaldesignation is given.

The point groups are:

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Cn The groups Cn only contain an n-fold rotation axis. The group containsas many elements as the order of the axis. The international symbol is n.

Cn,v The groups Cn,v contain the n-fold rotation axis and have vertical mir-ror planes which contains the axis of rotation. The effect of the n-fold axis, if nis odd, is such that it produces a set of n equivalent mirror planes. This yields2 n symmetry operations, which are the n rotations and the reflections in the nmirror planes. If n is even, the effect of repeating Cn only produces n

2 equivalentmirror planes. The other n

2 rotations merely bring the mirror plane into coinci-dence with itself, but with the two surfaces of the mirror interchanged. A mirrorplane is equivalent to its partner mirror plane found by rotating it through πsince, by definition, a mirror plane is two-sided. However, for even n, the effectof the compounded operation Cn σv acting on an arbitrary point P producesa point P ′ which is identical to the point P ′ produced by reflection of P inthe mirror plane that bisects the angle between two equivalent mirror planesσv. Thus, the symmetry element given by the product Cn σv is equivalent to amirror reflection in the bisecting (vertical) mirror plane. The effect of Cn is totransform this bisecting mirror planes into a set of n

2 equivalent bisecting mirrorplanes. Thus, if n is even, there are also 2 n symmetry operations. These 2 nsymmetry operations are the set of n rotations and the two sets of n

2 reflections.Mirror planes which are not perpendicular to the rotation axis are recorded asm without any special marking. For even n, the International symbol for Cn,v

is nmm. The two m’s refer to two distinct sets of mirror planes: one from theoriginal vertical mirror plane and the second m refers to vertical the mirrorplanes which bisect the first set. For odd n, the international symbol is justnm, as the group only contain one set of mirror planes and does not contain aset of bisecting mirror planes.

Cn,h The groups Cn,h contain the n-fold rotation, and have a horizontal mir-ror plane which is perpendicular to the axis of rotation. These groups contain2 n elements and, if n is even, the group contains C

n2

n . σh = C2 . σh = I whichis the inversion operator. The International notation usually refers to these asn/m. The diagonal line indicates that the symmetry plane is perpendicular tothe axis of rotation. The only exception is C3h or 6. The international symbolsignifies that C3h is relegated to the group of rotation reflections which are, ingeneral, designated by n.

Sn The groups Sn only contain the n-fold rotation - reflection axis. For evenn, the group contains only n elements as (Sn)n = E. For odd n, (Sn)n = σ,so the group must contain 2 n elements. The International notation is givenby the equivalent rotation inversion group n. For example, S6 ≡ 3, S4 ≡ 4,S3 ≡ 6.

Dn The groups Dn contain an n-fold axis of rotation and a two-fold axiswhich is perpendicular to the n-fold axis. The effect of the n-fold rotation is

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to produce a set of equivalent two-fold axes. If n is odd there are n equivalenttwo-fold axes. If n is even, the n-fold rotation produces n

2 equivalent two-foldaxes which are two sided. When n is even, the action of a two-fold rotationfollowed by an n-fold rotation is equivalent to a new two-fold rotation about anaxis that bisects the original sets of two-fold axes. This can be seen by followingthe action of an arbitrary point P with coordinates (x, y, z) under the two-foldrotation about a horizontal axis, say the x axis. The rotation by π about thex axis sends z → − z and y → − y. A further rotation of 2 π

n about thez axis, sends the point (x,−y,−z) to the final image point P ′. Note that thez coordinate of point P ′ is − z. Construct the line joining P and P ′. Themid-point lies on the plane z = 0, and subtends an angle of π

n with the x axisand, therefore, lies on the bisecting rotation axis. As the bisecting axis passesthrough the mid-point of line P − P ′, this shows that the arbitrary point P canbe sent to P ′ via a π rotation about the bisecting axis. Thus, for even n, thereare n

2 bisecting two-fold axes, and the n2 two-fold axes. In case of either even

or odd n, the group contains 2 n elements consisting of the n-fold rotations anda total of n two-fold rotations. The International designation for Dn is eithern22 or just n2, depending on whether n is even or odd. These two designationsoccur for similar reasons as to why there are two International designations forCn,v. For odd n the designation n2 indicates that there is one n-fold axis andone set of equivalent two-fold axes. For even n, the symbol n22 indicates theexistence of an n-fold axes and two inequivalent sets of two-fold axes.

Dnh The groups Dnh contains all the elements of Dn and also contain a hor-izontal mirror plane perpendicular to the n-fold axis. The effect of a rotationabout a two-fold axis followed by the reflection σh is equivalent to a reflectionabout a vertical plane σv passing through the two-fold axis. Since rotating σv

about the Cn axis produces a set of n vertical mirror planes, adding a horizontalmirror plane to Dn produces n vertical mirror planes σv. The group has 4 nelements which are formed from the 2 n rotations of Dn, the n reflections in thevertical mirror planes, and n rotation reflections Ck

n σh. For even n, the Inter-national symbol is n

m2m

2m which is often abbreviated to n/mmm. The symbol

indicates that the n-fold axis has a perpendicular mirror plane, and also the twosets of two-fold axes also have their perpendicular mirror planes. For odd n, theInternational symbol for the group acknowledges the 2n-fold rotation inversionsymmetry and is labelled as 2n2m.

Dnd The groups Dnd contains all the elements of Dn and mirror planeswhich contain the n-fold axis and bisect the two-fold axes. The effect of thetwo-fold rotations generate a total of n vertical reflection planes. There are4 n elements, the 2 n rotations of Dn, n mirror reflections σd in the n verticalplanes. The remaining n elements are rotation reflections about the principleaxis of the form S2k+1

2n where k = 0 , 1 , 2 , . . . , ( n − 1 ). The principle axisis, therefore, a 2n-fold rotation reflection axis. The International symbol is n2mindicating a n-fold axis, a perpendicular two fold axis and a vertical mirror plane.

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T The tetrahedral group corresponds to the group of rotations of the reg-ular tetrahedron. The elements are comprised of four three-fold rotation axespassing through one vertex and the center of the opposite faces. The compoundaction of two of the three-fold rotations yields a rotation about a two-fold axis.There are three such two-fold axes passing through the midpoints of oppositeedges of the tetrahedron. The tetrahedral group has 12 elements. The symme-try operations can also be found in a cube, if the three four-fold rotation axespresent in the the cube are discarded. The group has the International symbolof 23.

Td The group Td corresponds to the tetrahedral group adjoined by a re-flection plane passing through one edge and the mid point of the opposite edgeof the tetrahedron. The reflection planes bisects a pair of two-fold axes of T .There are six mirror planes σd. For the cube, these mirror planes are the diago-nal planes which motivates the use of the subscript d. The mirror planes convertthe two-fold axes to produce four-fold rotation reflection axes S4. The groupTd contains 24 elements. The group Td has the International designation as 43m

Th The group consists of the tetrahedral group adjoined by a mirror planewhich bisects the angle between the three-fold axes. For the tetrahedron, thisgroup is equivalent to Td, but for the cube, the mirror plane is parallel to op-posite faces. There are three such horizontal planes. The planes bisect theangles between the three-fold axes, and, therefore, converts them into six-foldrotation reflection axes. Since the group contains S6, it also contains I. Hence,Th = Ti × CI . The group has 24 elements. The International designation forthe group Th is either 2

m3 or m3.

O The octahedral group has three mutually perpendicular four-fold axes.There are four three-fold axes, and six two-fold axes. It has 24 elements. It hasan International designation of 432.

Oh On adjoining a mirror plane to the octahedral group one obtains Oh.Adding a vertical mirror plane generates three other mirror planes. The effectof a reflection in the vertical mirror plane followed by a rotation C4 is equiva-lent to a reflection in a diagonal mirror plane. The C3 axes becomes S6 axes,just as in the case of Th. The group contains 48 elements. The Internationaldesignation is either 4

m3 2m or m3m.

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3.5 Bravais Lattices

There are an infinite number of choices for the primitive lattice vectors, however,only a few special lattices are invariant under point group operations. Theseare the Bravais lattices. In three dimensions there are 14 Bravais lattices types.The 14 Bravais lattices are organized according to seven crystal systems. TheBravais lattices can be categorized in terms of the number of symmetry opera-tions.

The unit cells have lattice vectors a, b, and c, of length a, b and c, as shownin the figure. The angles between the vectors are denoted by α, β and γ, suchthat α is the angle between b and c, etc. That is, α ( 6 b , c ), β ( 6 a , c ),and γ ( 6 a , b ).

——————————————————————————————————

3.5.1 Exercise 1

Show that the volume of a primitive unit cell, Vc is given by

Vc = a b c

[1 + 2 cos α cos β cos γ − cos2 α − cos2 β − cos2 γ

] 12

(33)

——————————————————————————————————

If the point group contains four three-fold axes C3 or (3), the system is cu-bic. It is possible to choose three coordinate axes which are orthogonal to eachother and are perpendicular to the faces of a cube that has the four three-foldaxes as the body diagonals.

3.5.2 Cubic Bravais Lattices.

The cubic Bravais lattices have the highest symmetry. The simple cube (P) hasthree four-fold rotation axes and four three-fold axes, along with six two-foldaxes. There are three mirror planes can be adjoined, to the set of rotationalsymmetry operations. The three four-fold rotation axes (C4) are mutually per-pendicular and pass through the centers of opposite faces of the cube. Anyrotation which is an integer multiple of 2 π

4 will bring the cube into coincidencewith itself. The four three-fold axes (C3) pass through pairs of opposite cornersof the cube. A rotation of any multiple of 2 π

3 will bring the cube into coinci-dence with itself, as can be seen by inspection of the three edges at the vertexwhich the rotation axis passes through. The six two-fold axes (C2) join thecenters of opposite edges. The highest symmetry group when mirror symmetry

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is not included is the octahedral group O. The octahedral group O contains 24symmetry operations. On adjoining a mirror plane to the set of rotations of theoctahedral group, one has the highest symmetry point group which is labelledas Oh or m3m and has 48 symmetry elements.

The reason that the cubic group is called the octahedral group is explainedby the following observation. The group of symmetry operations of the cubeis equivalent to the group of symmetry operations on the regular octahedron.This can be seen by inscribing an octahedron inside a cube, where each vertexof the octahedron lies on the center of the faces of the cube. Thus, the cubicpoint group is called the octahedral group O.

There are three types of cubic Bravais lattices: the simple cubic (P), thebody centered cubic (I) and face centered cubic (F) Bravais lattices.

The primitive lattice vectors for the simple cubic lattice (P) can be takenas the three orthogonal vectors which form the smallest cube with the latticepoints as vertices. The three primitive lattice vectors have equal length, a, andare orthogonal. The vertices of the cube can be labelled as (0, 0, 0), (0, 0, 1),(0, 1, 0), (1, 0, 0), (1, 1, 0), (1, 0, 1), (0, 1, 1) and (1, 1, 1). The primitive cell is thecube which contains one lattice point and has a volume a3.

The body centered cubic Bravais lattice (I) has a lattice point at the verticesof the cube and also one at the central point which is a

2 (1, 1, 1) when specified interms of the Cartesian coordinates formed by the edges of the conventional non-primitive unit cell (which is the cube). The primitive lattice vectors are givenin terms of the Cartesian coordinates by a1 = a

2 (1, 1,−1), a2 = a2 (−1, 1, 1),

a3 = a2 (1,−1, 1). These are the three vectors from any lattice point joining

three neighboring body centers. The conventional unit cell contains two latticesites and has a volume a3, where a is the length of the side of the cube. Theprimitive unit cell is a rhombohedron of edge a

√3

2 which contains one latticesite and has a volume 1

8 4 a3. The angles between the primitive lattice vectorsis given by cos γ = − 1

3 . In the primitive cell, each body center of the conven-tional unit cell is connected by three primitive lattice vectors to three verticesof the conventional cell.

The Wigner-Seitz cell for the body centered cubic lattice is a truncated oc-tahedron. It is made of eight hexagonal planes which are bisectors of the linesjoining the body center to the vertices. These eight planes are truncated by theplanes of the cube which coincide with the bisectors of the lines between theneighboring body centers. The truncation produces the six square faces of thebody centered cubic Wigner-Seitz cell.

The face centered cubic Bravais lattice (F) consists of the lattice points atthe vertices of the cube, and lattice points at the centers of the six faces. The

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lattice points at the face centers are located at a2 (1, 1, 0), a

2 (1, 0, 1), a2 (0, 1, 1),

a2 (1, 1, 2), a

2 (1, 2, 1) and a2 (2, 1, 1). The primitive lattice vectors point from the

vertex centered at (0, 0, 0) to the three closest face centers, a1 = a2 (1, 1, 0),

a2 = a2 (1, 0, 1), a3 = a

2 (0, 1, 1). Since each face is shared by two consecutivenon-primitive unit cells there are 4 lattice sites in the conventional non-primitivecubic unit cell. The primitive unit cell is a rhombohedron with side a√

2. The

edges of the primitive unit cell connect two opposite vertices of the cube viathe six face centers. The edges of the primitive cell are found by connecting thevertex to the three neighboring face centers. The volume of the primitive unitcell is found to be 1

4 a3. The angles between the primitive lattice vectors are π

3 .

The Wigner-Seitz cell for the face centered cubic lattice is best seen by trans-lating the conventional unit cell by a

2 along one axis. After the translation hasbeen performed, the unit cell has the appearance of being a cube which haslattice sites at the body center and at the mid-points of the twelve edges ofthe cube. The Wigner-Seitz cell can then be constructed by finding the twelveplanes bisecting the lines from the body center to the mid-points of the edges.The resulting figure is a rhombic dodecahedron.

——————————————————————————————————

The presence of either one four-fold C4, (4) or four-fold inversion rotationaxes S4, (4), makes it possible to choose three vectors such that a = b 6= c,α = β = γ = π

2 and c is parallel to the C4 or S4 axes. This is the tetragonalsystem.

3.5.3 Tetragonal Bravais Lattices.

The tetragonal Bravais lattice can be considered to be formed from the cubicBravais lattices by deforming the cube, by stretching it, or contracting it alongone axis. This special axis is denoted as the c axis. Thus, the conventional unitcell can be constructed, starting with a square base of side a, by constructingedges of length c 6= a parallel to the normals of the base, from each corner.

The simple tetragonal Bravais Lattice has a four-fold rotational axis andtwo orthogonal two-fold axes. These symmetry elements generate the groupD4. On adding a horizontal mirror plane to D4, one obtains the highest symme-try tetragonal point group which is D4h or 4/mmm with 16 symmetry elements.

There are two tetragonal Bravais lattices: the simple tetragonal Bravais lat-tice (P) and the body centered tetragonal Bravais lattice (I).

The face centered tetragonal lattice is equivalent to the body centered tetrag-onal lattice. This can be seen by considering a body centered tetragonal latticein which the conventional unit cell can be described in terms of a side of length c

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perpendicular to the square base of side a and area a2. Consider the view alongthe c axis which is perpendicular to the square base. By taking a new base ofarea 2 a2 and sides

√2 a which are the diagonals of the original base, one finds

that the body centers can now be positioned as the face centers. That is, thebody centered tetragonal is equivalent to the face centered tetragonal unit cell.

The equivalence between the body centered and face centered structures doesnot apply to the cubic system. However, the conventional body centered cubicunit cell is equivalent to a face centered tetragonal unit cell in which the heightalong the c-axis has a special relation to the side of the base. Namely, the c-axisheight is a and the side of the base is

√2 a. Using the converse construction, the

face centered cubic unit cell can be shown to be equivalent to the body centeredtetragonal lattice with a particular length of the c-axis.

——————————————————————————————————

The orthorhombic system has three mutually perpendicular two-fold rota-tion axes. The existence of the three mutually perpendicular two-fold rotationaxes is compatible with the point groups D2, C2v and D2h. It is possible toconstruct a unit cell α = β = γ = π

2 .

3.5.4 Orthorhombic Bravais Lattices.

The conventional orthorhombic unit cell can be considered to be formed by de-forming the tetragonal unit cell by stretching the base along an axis in the basalplane. Thus, the base can be viewed as consisting of a rectangle of side a 6= b.The unit cell has another set of edges which are parallel to the normal to thebase and have lengths c. Thus, the conventional unit cell has edges which areparallel to three orthogonal unit vectors.

The simple orthorhombic lattice (P) only has two-fold rotation axes. Thetwo-fold axes are perpendicular, so the rotational group is D2. The effect of ad-joining a horizontal mirror plane converts D2 into the orthorhombic point groupwith highest symmetry which is D2h or 2/mmm with 4 symmetry operations.

There are four inequivalent orthorhombic Bravais lattices. These are the sim-ple orthorhombic lattice (P), the body centered orthorhombic (I), face centeredorthorhombic (F) and a new type of lattice, the base centered orthorhombiclattice (C).

The base centered orthorhombic lattice (C) can be constructed from thetetragonal lattice in the following manner. View the square net of side a, whichforms the bases of the tetragonal unit cells, in terms of a non-primitive unitcell with a square base of side

√2 a with sides along the diagonal. This larger

non-primitive unit cell contains one extra lattice site at the center of the base

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and the top. When this base centered tetragonal structure is then stretchedalong one of its sides ( one of the diagonal sides of length

√2 a ), one obtains

the orthorhombic base centered lattice.

——————————————————————————————————

The monoclinic lattice system requires a minimum of one two-fold rotationaxis. Due to the conditions imposed by the two-fold rotation symmetry, it ispossible to choose α = γ = π

2 6= β. The monoclinic systems is compatiblewith the point groups C2, Cs and C2h.

3.5.5 Monoclinic Bravais Lattice.

The monoclinic Bravais lattice is obtained from the orthorhombic Bravais lat-tices by distorting the rectangular base perpendicular to the c axis into a par-allelogram. The base is a parallelogram, and the two basal lattice vectors areperpendicular to the c axis.

The simple monoclinic lattice (P) has a two-fold axis parallel to the c axis.The rotational group is C2. If a horizontal mirror plane is added to C2, thenone finds that the most symmetric monoclinic point group is C2h or 2/m whichhas four elements.

There are two types of monoclinic Bravais lattices: the simple monoclinic(P) and the body centered monoclinic Bravais lattice (I). The two monoclinicBravais lattices correspond to the two tetragonal Bravais lattices. The four or-thorhombic lattices collapse onto two lattices in the tetragonal and monoclinicsystems, as the centered square net is not distinct from a square net. Likewise,the centered parallelogram is not distinct from a parallelogram.

——————————————————————————————————

The groups C1 and Ci impose no specific restrictions on the lattice. This isthe triclinic lattice system.

3.5.6 Triclinic Bravais Lattice.

The triclinic Bravais Lattice is obtained from the monoclinic lattice by tiltingthe c axis so that it is no longer orthogonal to the base. There is only the simpletriclinic Bravais Lattice (P). The three axes are not orthogonal and the sidesare all different.

Apart from inversion, which is required by the periodic translational invari-ance of the lattice, the triclinic lattice has no special symmetry elements. The

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point group of highest symmetry is Ci or 1 which has two elements.

——————————————————————————————————

The presence of only one three-fold axes, either C3 (3) or S6 (3), producesthe trigonal system. There are two types of trigonal system. In one of thetrigonal systems, a primitive unit cell may be chosen with a = b = c andα = β = γ such that the three-fold axes is along the body diagonal. Theother trigonal system has a = b 6= c and α = β = π

2 and γ = 2 π3 . This

later system is denoted as the hexagonal system.

3.5.7 Trigonal Bravais Lattice.

The Trigonal Bravais Lattice is a deformation of the cube produced by stretch-ing it along the body diagonal. The lengths of the sides remain the same andthe three angles between the sides are all identical. There is only one trigonalBravais lattice. The point symmetry group is D3h or 62m with 12 symmetryoperations.

The body centered cubic and face centered cubic Bravais lattices can be con-sidered to be special cases of the trigonal lattice. For these cubic systems, thesides of the primitive unit cells are all equal and the angles are 109.47 degreesfor the b.c.c. structure and 60 degrees for the f.c.c. structure.

The trigonal unit cell contains two equilateral triangles. In the trigonal lat-tice the equilateral triangles form hexagonal nets. The difference between thetrigonal lattice and the hexagonal lattice is merely due to the different stackingof the hexagonal planes.

——————————————————————————————————

The presence of either a six-fold axes C6 (6) or a rotation inversion axesS3 (6), indicates that the system is hexagonal. The hexagonal unit cell hasa = b 6= c and α = β = π

2 and γ = 2 π3 .

3.5.8 Hexagonal Bravais Lattice.

The hexagonal Bravais lattice has a unit cell in which the base has sides ofequal length, inclined at an angle of 2 π

3 with respect to each other. The c axisis perpendicular to the base.

The hexagonal system has a point group D6h or 6/mmm which has 24 sym-metry elements.

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There is only one Hexagonal Bravais Lattice. The primitive unit cells arerhombic prisms which can be stacked to build the hexagonal non-primitive unitcell. The six-fold rotational symmetry of the hexagonal Bravais lattice is mostevident from the non-primitive unit cell.

The primitive lattice vector are given in terms of Cartesian coordinates by

a1 = a ex

a2 =a

2

(ex +

√3 ey

)a3 = c ez (34)

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In summary the following structures were found:

Cubic. 3 a = b = c α = β = γ = π2

Tetragonal. 2 a = b 6= c α = β = γ = π2

Orthorhombic. 4 a 6= b 6= c α = β = γ = π2

Monoclinic. 2 a 6= b 6= c α = β = π2 6= γ

Triclinic. 1 a 6= b 6= c α 6= β 6= γ

Trigonal. 1 a = b = c α = β = γ < 2 π3 6= π

2

Hexagonal. 1 a = b 6= c α = β = π2 , γ = 2 π

3

This completes the discussion of the set of fourteen Bravais lattices. In orderto specify crystal structures, it is necessary to associate a basis along with theunderlying Bravais lattice. The addition of a basis can reduce the symmetry ofthe crystal from the symmetry of the Bravais lattices. This results in thirty twopoint groups, and by adjoining the translations and combined operations, onefinds the two hundred and thirty space groups.

——————————————————————————————————

3.5.9 Exercise 2

Form a table of the number of the n-th nearest neighbors and the distancesto the n-th neighbors for the face centered cubic (f.c.c.), body centered cubic(b.c.c.) and simple cubic (s.c.) lattices, for n = 1, n = 2 and n = 3.

——————————————————————————————————

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Having just used symmetry to enumerate all the possible Bravais lattices,we shall now discuss the possible symmetries of crystals. Due to the addition ofthe basis, the point group symmetry of a crystal can be different from the pointgroup symmetry of the Bravais lattice.

3.6 Point Groups

The addition of a basis to a lattice can result in a reduction of the symmetryof the point group. Here the point groups are enumerated according to theBravais Lattice types and by the Schoenflies designation followed by the appro-priate (International) symbol.

The cubic system with a basis can have the point symmetry group of eitherOh (m3m), O (43), Td (43m), Th (m3) or T (23).

The tetragonal system can have point group symmetry of D4h (4/mmm),D4 (42), C4v (4mmm), C4h (4/m) or C4 (4).

The orthorhombic system can have point group symmetry of either D2h

(mmm), D2 (222) or C2v (2mm).

The monoclinic system can exist with point group symmetry of either C2h

(2/m), C2 (2) and Cs (m), the group which only consists of the identity andthe inversion operation.

The triclinic system only contains C1 (1) and Cs (m).

The trigonal system has the point groups D3h (62m), D3 (32), C3v (3m), S6

(3), or C3 (3).

The hexagonal system has the point groups D6h (6/mmm), D6 (62), C6v

(6mm), C6h (6/m), or C6 (6).

There are four remaining groups. The groups C3h (6) and D3d (3m) whichare usually included in the hexagonal system. Finally, there are the groups S4

(4) and D2d (42m) which are included with the tetragonal systems.

This completes the enumeration of the 32 point groups.

3.7 Space Groups

On combining the point group symmetry operations with lattice translations,one can generate 230 space groups. Often, the space group is composed fromsymmetry operations of the point group and symmetry operations that are

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translations by the vectors of the direct lattice. These space groups are calledsymmorphic groups. Lattices with symmorphic space groups can be constructedby attaching basis with the various point group symmetries on the various Bra-vais Lattices. For example, the 5 cubic point groups can be placed on the threecubic Bravais Lattices, yielding 15 cubic space groups. Likewise, 7 tetragonalgroups can be placed on the two tetragonal Bravais Lattices, yielding 14 tetrag-onal space groups. This process only leads to 61 different space groups. In theother cases, the space groups contain two new types of symmetry operationsthat cannot be compounded from translations by Bravais lattice vectors andoperations contained in the point groups. These groups are non-symmorphic.The new types of symmetry operations occur when there is a special relationbetween the basis dimensions and the size of the Bravais lattice. These newsymmetry elements include :-

Screw Axes. A screw operation is a translation by a vector, not in theBravais lattice, which is followed by a rotation about the axis defined by thetranslation vector. A screw symmetry is denoted by nm, where n represents therotations 2 π

n , where n = 2 , 3 , 4 , 6 and m represents the number of trans-lations by lattice vectors which produce one complete rotation by 2 π. Thus, nscrew operations, each producing a rotation of 2 π

n , produce a translation of mlattice spacings.

Glide Planes. A glide operation is composed of a translation by a vector,not in the Bravais lattice, which is followed by a reflection in a plane containingthe translation vector. Glide planes are denoted by a, b, c (according to whetherthe translation is along the a, b and c axis), or n and d (the diagonal or diamondglide which are special cases involving translations along more than one axis).

The hexagonal close-packed lattice structure has both of these types of non-symmorphic symmetry operations. The hexagonal close-packed structure can bedescribed by a three-dimensional unit cell which contains a centered hexagonalbase, and which has an identical centered hexagonal top located at vertical dis-tance c directly above the base. If one considers the base hexagon to be formedby six equilateral triangles, then there are lattice sites at the vertex of eachtriangle. These lattice sites form a triangular net in the basal plane and there isa similar triangular net in the upper plane. These lattice sites are designated asthe A sites. There is a second net of triangles at a distance c

2 vertically over thebase. The centers of the mid-plane equilateral triangles are located directly overthe (central) lattice sites of the base. There are two possible orientations forthese triangles. On choosing any one orientation, the set of lattice sites on thismid-plane are located such that they lie directly over the centers of every otherequilateral triangle in the base. These mid-plane lattice sites are designated asthe B sites.

Consider a line, parallel to the c axis. The line is equidistant between twoneighboring B lattice sites and is equidistant to the two A lattice sites that form

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the section of the perimeter of the basal hexagon which is parallel to the lineconnecting the above two B lattice sites. Viewed from the c axis, the verticalline passes through the center of the rectangle formed by the two A and twoB lattice sites. This line is the screw axis. The screw operation consists ofa translation by c

2 followed by rotation of π, and brings the A hexagons intocoincidence with the sites of the B hexagons.

The glide planes can also be found by considering the projection of the lat-tice along the c axis. A line can be constructed which connects any two of thethree B sites inside the hexagonal unit cell. Form a parallel line connecting apair of neighboring A sites that forms part of the perimeter of the hexagonalbase. Since this line is on the perimeter of the unit cell, it is equivalent to theparallel line segment connecting A sites at the opposite boundary. Consider thepair of parallel lines, one which connects the B sites, and the other which isthe closest line segment that connects the A sites on the perimeter of the basehexagon. The projection of the glide plane along the c axis is parallel to andequidistant from the above pair of lines. The glide operation is a translation byc2 along the c axis followed by a reflection in the plane.

There are two different systems of nomenclature for space groups, one dueto Schoenflies and the other is due to Hermann and Mauguin. The Hermann- Maugin space group nomenclature consists of a letter P , I , F , R , Cwhich describes the Bravais Lattice type, followed by a statement of the essen-tial symmetry elements that are present. Thus, for example, the space groupP63/mmc has a primitive (P ) hexagonal Bravais lattice with point group sym-metry 6/mmm. Another example is given by the space group Pba2 whichrepresents a primitive (P ) orthorhombic Bravais Lattice and has a point groupof mm2 (the a and b glide planes being simple mirror planes in point groupsymmetry).

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3.8 Crystal Structures with Bases.

Crystal structures are specified by giving the basis and the Bravais Lattice. Thebasis is specified by the positions of and types of atoms in the unit cell.

Sometimes it is also useful to specify the local coordination polyhedra aroundeach inequivalent site in the lattice. This provides information about the localenvironment of the atom which is important for bonding. Small deformationsin the positions of the atoms can lower the symmetry of a crystal structure,but usually does not affect the connectivity or topology of the atoms. There-fore, slight deformations of the local environment are often specified by thesame local coordination polyhedra. The local coordination polyhedra have beenenumerated by W. B. Jensen, (The Structures of Binary Compounds, NorthHolland publishers (1988)) and by Villars and Daams (Journal of Alloys andCompounds, 197, 177 (1993)).

3.8.1 Diamond Structure

The diamond lattice is formed by the carbon atoms in a diamond crystal. Thestructure is cubic, and has the space group Fd3m. The underlying Bravaislattice is the face centered cubic lattice, and has a two atom basis. In the dia-mond structure, both atoms are identical. They are located at the sites of theBravais lattice (0, 0, 0) and at a second site displaced by a distance a( 1

4 ,14 ,

14 )

in terms of the Cartesian coordinates of the conventional unit cell. There arefour lattice points corresponding to the sites of the conventional f.c.c. unit cell.There are also four interior points which are displaced from the Bravais latticepoints by the basis vector a( 1

4 ,14 ,

14 ). Thus, the diamond structure consists of

two interpenetrating face centered cubic lattices with atoms on each lattice site.Diamond possesses a center of inversion located half way between the originsof the two f.c.c. lattices. This is a glide-like inversion operation. The center ofinversion is located at a( 1

8 ,18 ,

18 ). When this is chosen as the origin, the crystal

is symmetric under the transformation r → − r.

Each atom is covalently bonded to four other atoms. The four neighbor-ing atoms form a tetrahedron centered on each atom. The tetrahedra centeredon the two inequivalent lattice sites have different orientations. The diamondlattice is most stable for compounds in which the bonds are highly directional.Directional covalent bonding is often found in the elements of column IV of theperiodic table. In particular, Carbon, Silicon and Germanium can crystallize inthe diamond structure. The great strength of diamond is a consequence of thethree-dimensional network of strong covalent bonds. The diamond structure isrelatively open as the packing fraction is only 0.34.

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3.8.2 Exercise 3

Find the angles between the tetrahedral bonds of diamond.

——————————————————————————————————

3.8.3 Graphite Structure

Graphite is the stable form of carbon. Graphite has a hexagonal unit cell andhas the space group P63/mmc. The primitive lattice vectors may be representedby

a1 =√

3 a ex

a2 =√

32

a ex +32a ey

a3 = c ez (35)

where a is the length of the side of the hexagon. The atoms are located at [0, 0, z]and [0, 0, 1

2 + z] where z ≈ 0, and the coordinates are given in terms of theprimitive lattice vectors. Another two atoms are located at the positions [ 23 ,

23 , z]

and [ 13 ,13 ,

12 + z], where z ≈ 0. The structure is formed in layers, in which

each atom is bonded to three other atoms, thereby forming a two-dimensionalhexagonal network. The central site of the two-dimensional hexagonal ring isopen. The stacking sequence of the layers just corresponds to a translation ofone layer by [13 ,

13 ,

12 ] with respect to the other, such that one C atom lies above

the hexagonal hollow in the layer below. The layers are relatively far apart,and as is expected, there is only weak van der Waals bonding between the lay-ers. This structure explains the cleavage and other characteristic properties ofgraphite.

Carbon may crystallize into either as a diamond lattice or as graphite, underdifferent conditions. This is an example of polymorphism which is quite commonamong the elements. Diamonds are not forever as they actually are an unstableform of C under ambient conditions, although the rate of transformation to thestable form (graphite) is exceedingly slow.

Boron and Nitrogen, which occur on either side of Carbon in the periodictable, form compounds which have properties that are strikingly similar to Car-bon. The Boron and Nitrogen atoms can be bonded in either planar structureslike graphite, or tetrahedral structures, like diamond. The tetrahedral bondedBoron - Nitrogen materials have extremely high melting points and hardness,and have great importance in materials engineering.

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3.8.4 Exercise 4

There are two forms of graphite. The most common form is hexagonal graphite,which has a stacking sequence A − B − A − B. The other form of graphite isbased on an f.c.c. form with a stacking sequence A−B−C−A−B−C. Describethe primitive unit cells for the two forms of graphite. How many atoms are inthe primitive unit cells of graphite?

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3.8.5 Hexagonal Close-Packed Structure

The hexagonal close-packed structure has hexagonal symmetry, the space groupis P63/mmc. It is composed of the hexagonal Bravais lattice, and has a basiscomposed of two atoms. The two identical atoms are positioned at [0, 0, 0]which is at the vertex of the primitive lattice cell, and has the other atomlocated at [ 13 ,

13 ,

12 ] as expressed in terms of the primitive lattice vectors. (The

square brackets indicate that the direction in the direct lattice are specified withrespect to the primitive lattice vectors.) The primitive lattice vectors are

a1 = a ex

a2 =a

2

(ex +

√3 ey

)a3 = c ez (36)

Thus, the hexagonal close-packed structure has a basis of two atoms one atr1 = (0, 0, 0) and the other at

r2 =13

(a1 + a2

)+

12a3

=a

2ex +

a

2√

3ey +

c

2ez (37)

Since a1 and a2 are inclined at an angle π3 , the structure can be considered to be

formed by two interpenetrating simple hexagonal Bravais lattices. Alternately,the structure may be viewed as being formed by stacking two-dimensional tri-angular lattices above one another, with a separation between the layers of halfthe height of the unit cell. Each atom has 12 nearest neighbors: six within thehexagonal plane and three in each of the planes above and below the atom.

The name hexagonal close-packed comes from thinking of this structure asbeing formed from hard spheres of radius and forming a close-packed hexagonallayer. The second layer is formed by stacking a second hexagonal layer of atomsabove the first. However, the center of the second layer of atoms are positionedabove the dimples in the first layer. There are two sets dimples of dimplesbetween the atoms, so there are two different choices for placing the second

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layer of atoms. The third layer is stacked such that the centers of the atomsare directly above the centers of the atoms of the first layer, and the fourth isstacked directly over the second layer, etc. Thus, there are two interpenetratinghexagonal lattices displaced by

13a1 +

13a2 +

12a3 (38)

or [ 13 ,13 ,

12 ].

There are a total of twelve nearest neighbor atoms which are distributed as6 neighbors in the plane, 3 in the plane above, and 3 in the plane below. Thisgives a total of 12 nearest neighbor atoms.

On assuming a radius of the atomic spheres to be r, the lattice constantssatisfy a = b = 2 r and c = 4

√2√3r. This yields the hexagonal close-packed

structure, and has the particular ratio of the c to the a axis lengths of

c

a=

√83

= 1.633 (39)

This is the ideal c to a ratio. Hexagonal close-packed systems with the idealratio have a packing fraction of 0.74. As atoms are not hard spheres, there is noreason for this value to be found in naturally occurring crystals, and deviationsfrom the ideal value are found most frequently. Only He has the ideal c to aratio.

The most frequently occurring structures are the close-packed structures.These are the hexagonal close-packed, face centered cubic and body centeredcubic structures, which have packing fractions of 0.74, 0.74 and 0.68, respec-tively. Both simple and transition metals frequently form in the hexagonalclose-packed structure, or other close-packed structures.

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3.8.6 Exercise 5

Show that the ca ratio for an ideal hexagonal close-packed lattice structure is

c

a=(

83

) 12

(40)

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3.8.7 Exercise 6

Na transforms from b.c.c. to h.c.p. at 23 K via a Martensitic transition. Onassuming that the density remains constant and the h.c.p. structure is ideal,find the h.c.p. lattice constant a in terms of the b.c.c. value a′.

——————————————————————————————————

3.8.8 Other Close-Packed Structures

One can form other close-packed structures by altering the sequence of stackingof the close-packed layers. The hexagonal close packed can be characterizedby the repeated stacking sequence A - B - A - B etc. That is, the atoms inthe planes above and below the triangular lattice have centers directly over thedimples and each other, thereby creating a two layer unit cell.

Another stacking sequence is given by A - B - C in which the unit cellconsists of three layers. The A and C layers have the atoms centered on the twoinequivalent sets of triangular dimples of the B layer. This close-packed stackingcorresponds to the face centered cubic lattice. The packing fraction of the facecentered cubic lattice and hexagonal close-packed lattice are identical. Thetriangular close-packed nets are the planes perpendicular to the body diagonalof the conventional f.c.c. unit cell. There are two such planes which pass throughthe conventional unit cell and two further planes that each just graze one vertexof the unit cell. The intercepts of the planes with the conventional (Cartesian)axes are (1, 0, 0), (0, 1, 0) and (0, 0, 1). The next plane has intercepts (2, 0, 0),(0, 2, 0) and (0, 0, 2). The sets of planes are known as 1, 1, 1 planes and havetriangular arrays of atoms, where the sides of the triangle side has length a√

2.

The normal to the planes are in the direction [1, 1, 1] i.e.

n =1√3

(ex + ey + ez

)(41)

where e are the orthogonal unit vectors of the conventional cell. The equationsof the planes are (

r − m a ex

). n = 0 (42)

where m is an integer that labels the plane by the intercept with the x axis.The quantity m is related to the perpendicular distance, s, between the planeand the origin through

s = ma√3

(43)

for integer m.

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It is convenient to introduce three new orthogonal unit vectors to describethe positions of the atoms in the planes. The first is n the normal to the planes

n =1√3

(ex + ey + ez

)(44)

The other vectors e1 and e2 are chosen to be vectors in the planes. These forma new set of Cartesian non-primitive lattice vectors which are defined by

e1 =1√2

(ex − ey

)(45)

which corresponds to the face diagonal of the conventional unit cell that lies inthe triangular plane and

e2 =1√6

(ex + ey − 2 ez

)(46)

which is the ”lateral” direction in the triangular plane. The lateral displace-ments of atoms between one triangular plane, say the plane which passes throughthe atom at ( 1

2 ,12 , 0)), and the atoms on the next plane (centered on the origin

(0, 0, 0)) can be written as

∆r =a

2

(ex + ey

)− n

a√3

=a

6

(ex + ey − 2 ez

)=

1√3

a√2e2 (47)

This can be re-written as23

√3

2a√2e2 (48)

as a√2

is the triangular lattice constant and√

32

a√2

is the height of the triangle.Thus, the atoms in consecutive planes are displaced ”laterally” by 0, 2

3 , and43 and then repeats. The resulting structure has layers which have a stackingsequence A−B − C −A−B − C etc.

There are other possible stacking sequences, with longer periodicities. Theearlier lanthanides and late actinides have a stacking sequence A - B - A - Cwith four layers per unit cell, however, the Sm structure only repeats itself af-ter nine layers. The longest known periodicity is 594 layers which is found ina polytype of SiC. The long-ranged crystallographic order is not due to long-ranged forces, but is caused by spiral steps caused by dislocations in the growthnucleus. There is also the possibility of random stacking sequences.

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3.8.9 Sodium Chloride Structure

The Sodium Chloride or NaCl structure is cubic. The space group is Fm3m.It has an ordered array of Na and Cl ions located on the sites of a simple cu-bic lattice of linear dimension a

2 . Each type of ion is surrounded by six ionsof the opposite charge, located at a distance a

2 away. The twelve next nearestneighbors have like charge and are located at a distance 1√

2a away along the

face diagonals of the cubic unit cell. There are four units of NaCl in the unitcell. The structure may be most efficiently visualized as having the Na+ ionslocated on the sites of a face centered cubic lattice with vertices at (0, 0, 0) andthe Cl− ions are located on a face centered cubic lattice with vertices at thecenter of the cubic unit cell ( 1

2 ,12 ,

12 ).

The Sodium Chloride structure is favored by many ionic compounds. In thisstructure, the electrostatic interactions are balanced by the short ranged repul-sive interactions due to the finite size of the ions. The short ranged repulsionsare due to the Pauli exclusion principle. The sizes of the ions are importantin determining the stability of this structure. If the ions of opposite charge areenvisaged as just touching, then the ionic radii must satisfy the equality

a = 2[r(Na+) + r(Cl−)

](49)

Ions of the same type are closest along the face diagonals, so if they do nottouch, the lattice constant satisfies the inequality

1√2a > 2 r(Cl−) (50)

Combining the above two equations yields an inequality for the ratio of the ionicradii of the ions

r(Cl−)r(Na+)

≤ 1 +√

2 (51)

If this inequality is not obeyed, the Pauli forces render the structure unstable.

Examples of materials that form in the NaCl structure are the alkali halidesmade from the alkaline elements Li, Na, K, Rb or Cs with a halide element F ,Cl Br or I. Alternatively, one can go to the next columns of the periodic tableand combine Mg, Ca, Sr or Ba with a chalcogen O, S, Se or Te to form theNaCl structure.

3.8.10 Cesium Chloride Structure

The ionic compound Cesium Chloride or CsCl has a cubic structure. The spacegroup is Pm3m. The Cs+ ion is located at (0, 0, 0) and the Cl− ion at the bodycenter of the cube (1

2 ,12 ,

12 ). Thus, the CsCl structure resembles a body centered

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cubic structure in which one type of atom is at the simple cubic sites and theother type of atom is at the body center. Each ion is surrounded by eight atomsof opposite charge located at a distance

√3

2 a away, which corresponds to halfthe length of the body diagonal of the cube. Each atom has six neighbors ofsimilar charge located a distance a away. The ratio of the ionic radii requiredfor this structure to be possible is

r(Cl−)r(Cs+)

≤ (√

3 + 1 )2

(52)

If the radii ratio is greater than 1.366, but less than 2.42, ionic compounds pre-fer the NaCl structure.

Examples of compounds that form the CsCl structure are the Cs halides,T l halides, CuZn (beta brass), CuPd, AgMg and LiHg.

Linus Pauling has produced a set of empirical rules which determine thecoordination numbers in terms of the ionic radii of the ions. If one assumes thatthe anion adopts the cubic close-packed structure (f.c.c.), there are three types ofholes between the close-packed spheres and each type of hole has a different size.It is assumed that the cations fit into one set of holes. The central site of theconventional f.c.c. unit cell is surrounded by an octahedron and, therefore, hasa coordination number of 6. There are also tetrahedral holes with coordinationnumber 4. The tetrahedral holes are located near the 8 corners of the f.c.c.cube, and the vertices of the tetrahedra are located at the corner and the threeneighboring face centers. The tetrahedral holes are best seen by consideringan octant of the f.c.c. cube. The tetrahedral hole site is at the center of theoctant, and the four vertices of the tetrahedron are located at four of the octantscorners. The are 12 trigonal holes which are located near the 8 vertices of theconventional unit cell. The trigonal sites lie in the plane formed by the vertexand any two of the closest face centers. The radius ratio rule suggests that thestructure is determined by maximizing the coordination numbers while keepingions of opposite charge in contact. This procedure seems likely to maximize theelectrostatic attraction energy. By considering the geometry of the holes, oneexpects that certain structures will be stable for different values of the radiusratio

rr =r(X−)r(R+)

(53)

For the tetragonal sites, by considering the body diagonal of the octant, oneexpects that

2[r(X−) + r(R+)

]=

√3

2a (54)

and by considering the face diagonal

2 r(X−) <a√2

(55)

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Hence, we find the tetragonal hole has the limiting radius ratio of

r(X−)r(R+)

> 2( √

32

+ 1)

(56)

In particular, the radius ratio rules suggest that the range of radii ratios wherethe various configurations are stable are given by

6.45 > rr > 4.45 trigonal 34.45 > rr > 2.41 tetrahedral 42.41 > rr > 1.37 octahedral 6

(57)

If the atoms have comparable sizes, then it is necessary to consider more openstructures with higher coordination numbers, such as simple cubic. For the sim-ple cubic structure, the coordination number is 8 hole and the hole size is larger1.37 > rr. Thus, since rr ∼ 1.8 for Na and Cl, it fits the radius ratio rulesas being octahedrally coordinated, like in the NaCl structure. On the otherhand, for Cs and Cl where the ions have comparable sizes, the radius ratio isrr ∼ 1.07 which is compatible with the cubic hole structure found in CsCl.

3.8.11 Fluorite Structure

Fluorite or CaF2 has a cubic structure. The space group is Fm3m. Ioniccompounds of the form RX2, in which the ratio of the ionic radii r satisfy theinequality

r(X−)r(R2+)

≤ (√

3 + 1 )2

(58)

can form the fluorite structure. The unit cell has four Ca2+ ions, one at theorigin and the others are located at the face centers of the cube. The eightF− ions are interior to the cube. The F− ions form simple cubes which areconcentric with the unit cells, but the simple cubes have only half the latticespacing of the unit cell. Alternatively, the eight F− ions can be considered to lieon two interpenetrating f.c.c. lattices with origins ( 3

4 ,14 ,

14 ) and ( 3

4 ,34 ,

34 ). Each

F anion occupies a site at the center of a tetrahedron formed by the Ca cations.

Materials, such as LiO2, form an anti-fluorite structure. The anti-fluoritestructure is the same as the fluorite structure except that the positions of theanions and cations are revered. The O anions are in the f.c.c. positions and theLi cations form a simple cubic array.

3.8.12 The Copper Three Gold Structure

The Cu3Au structure is cubic, and has the space group Pm3m. The Bravaislattice corresponds to a primitive cubic structure. There are 3 Cu atoms and oneAu per unit cell. All the atoms are located on the sites of a face centered cubic

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unit cell. The Au atom can be envisaged as being positioned on the corners ofthe cube, whereas the 3 Cu atoms sit on the centers of the faces of the cube,forming octahedra. Thus, the basis of the structure consists of the position ofthe Au atom

r0 = 0 (59)

and the three Cu atoms are located at

r1 =a

2( ey + ez )

r2 =a

2( ex + ez )

r3 =a

2( ex + ey ) (60)

The Au atoms have 12 Cu nearest neighbors located at a distance a√2, whereas

the Cu atoms only have 4 Au nearest neighbors.

Other compounds with the Cu3Au structure are Ni3Al, TiP t3 and themetastable compound Al3Li.

3.8.13 Rutile Structure

The structure possessed by rutile, TiO2, by cassiterite, SnO2 and by numer-ous other substances with small cations, is tetragonal. The space group isP42/mnm. The Ti4+ ions occupy positions : (0, 0, 0) ; ( 1

2 ,12 ,

12 ) while the O2−

ions occupy the four positions ± (x, x, 0) ; ± ( 12 + x, 1

2 − x, 12 ) where x ≈ 3

10 .Thus, the titanium atoms occupy the sites of a body centered tetragonal lattice.The oxygen atoms lie on lines which are oriented along one set of face diagonalsof the base. The atoms are also located on horizontal lines through the bodycenters, and are orthogonal to the lines in the base. The titanium ion is sur-rounded by six O atoms which form a slightly distorted octahedron.

3.8.14 Zinc Blende Structure

Zinc Blende structure or ZnS is cubic. This is also known as the Sphaleritestructure. The space group is F43m. The Zn2+ ions are positioned at (0, 0, 0)and the face centers of the cube. The S2− are positioned on an interpenetratingface centered cubic lattice with origin ( 1

4 ,14 ,

14 ). There are four units of ZnS in

the unit cell. The Zinc Blende structure is related to the diamond structure,except that Zinc Blende involves two different types of atoms. Each atom in ZnSis surrounded by a regular tetrahedron of atoms of the opposite type. Unlikediamond, Zinc Blende has no center of inversion, as the diamond inversionoperator interchanges the two different types of atoms. The radius ratio rulessuggest that this structure will be adopted whenever

2( √

32

+ 1)

> rr > 1 +√

2 (61)

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The Zinc Blende structure is often found for binary compounds formed frompairs of elements from either the II - VI columns, III - V columns or the I - VIIcolumns of the periodic table.

3.8.15 Zincite Structure

Zincite, ZnO, has a hexagonal structure. This structure is also known as theWurtzite structure. The space group is P63mc. The primitive lattice vectorsare given by

a1 = a ex

a2 =a

2( ex +

√3 ey )

a3 = z ez (62)

The Zn and O atoms occupy the positions [ 0, 0, z ]; [ 23 ,

23 ,

12 + z ] where

z = 0 for Zn and about z ≈ 38 for O. Since ZnS also is found in this form

above 1300 K, it is not surprising that Zincite structure has a local coordina-tion similar to that of the low-temperature Zinc Blende structure. Each atomis surrounded by a tetrahedron of atoms of the opposite type. The tetrahedraform continuous interconnected networks. However, symmetry does not requirethat the tetrahedra are regular.

The cubic Zinc Blende and the hexagonal Wurtzite structures are closelyrelated. They merely differ by the stacking sequence of the Zn (S) close-packedplanes. The structure consists of alternate close-packed planes which either con-tain only Zn or only S ions. The set of planes form layers consisting of a pairof planes. In a layer, the Zn atoms in one plane and the S atoms in the otherplane are bonded by vertical tetrahedral bonds. The remaining three tetrahe-dral bonds join the atoms in the successive layers. Due to the orientation of theinter-layer tetrahedral bonds, successive pairs of planes are displaced horizon-tally. Thus, the successive sets of vertical bonds are displaced horizontally.

In the cubic Zinc Blende sequence, the tetrahedra of the S atom bonds havethe same rotational orientation in each layer, so that each S layer is displaced inthe same direction. The net horizontal displacement produced in three verticalS layers is equal to the periodicity in the direction of the displacement. Thiscan be considered as having a stacking sequence A - B - C which repeats.

In the hexagonal Wurtzite sequence, the tetrahedra of bonds are rotated byπ between successive S layers. Thus, the horizontal displacement that occursbetween one S layer and the next are cancelled by the opposite displacementthat occurs by going to the very next S layer. This stacking sequence is A - Bwhich repeats.

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3.8.16 The Perovskite Structure

The perovskite structure, as exemplified by BaTiO3, is cubic at high tempera-tures but becomes slightly tetragonal on cooling below a ferro-electric transitiontemperature. The cubic structure has the space group Pm3m. The structure iscomposed of the Ti atoms positioned on the simple cubic lattice sites (0, 0, 0),and the Ba atoms positioned at the body center sites ( 1

2 ,12 ,

12 ). The three O

atoms are located at the mid-points of the edges of the cube, i.e. at (0, 0, 12 ),

(0, 12 , 0) and ( 1

2 , 0, 0). An alternate representation of the unit cell is found bycentering the lattice on the Ba ions, by translating the origin via 1

2 (1, 1, 1). Inthis representation, the Ti atoms are located at the body centers, and the Oatoms lies on the face centers. The TiO2 form a set of parallel planes sepa-rated by planes of BaO. Each Ti atom is surrounded by an octahedron of Oatoms, which have corners which are shared with the octahedron surroundingthe neighboring Ti atoms.

——————————————————————————————————

3.8.17 Exercise 7

The density of the face centered cubic structure is highest, body centered cubicis the next largest, followed by simple cubic and then diamond has the lowestdensity. This correlates with the coordination numbers. The coordination num-ber is defined to be the number of nearest neighbors. The coordination numbersare 12 for the f.c.c. lattice, 8 for b.c.c., 6 for s.c. and 4 for diamond. Assumethat the atoms are hard spheres that just touch. Find the packing fraction ordensity of these materials.

——————————————————————————————————

3.9 Lattice Planes

A Bravais lattice plane, by definition, passes through three non-collinear Bravaislattice points. Since these points are connected by combinations of multiplesof the primitive lattice vectors, and due to the periodic translational symmetryof the lattice, the lattice planes must contain an infinite number of lattice points.

Given one such lattice plane, there exists a family consisting of an infiniteset of parallel lattice planes with the same normal. One such lattice plane mustpass through each Bravais lattice point, since the lattice viewed from any latticepoint is identical to the lattice when viewed from any other lattice point. Thus,the family of parallel planes contain all the points of the Bravais lattice.

Each member of the set of lattice planes must intersect the axis given bythe primitive lattice vectors a1, a2 and a3. The planes need not intersect any

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particular axes at a lattice point, however, every lattice point on the three axeswill have one member of the family pass through it. In particular, one planemust pass through the origin O.

Each plane is uniquely specified by the three intercepts of the plane with theaxes formed by three primitive lattice vectors directed from the origin to theBravais lattice points a1, a2 and a3. The intercepts x1, x2 and x3 are measuredin units of the length of the primitive lattice vectors. That is, the intercepts arex1 a1, x2 a2 and x3 a3.

The three points of intersection between one lattice plane with the threeprimitive axes can be represented as κ [ 1

h1, 0, 0], κ [0, 1

h2, 0] and κ [0, 0, 1

h3],

where κ is a positive or negative integer, and (h1, h2, h3) are also positive ornegative integers. The integers (h1, h2, h3) are chosen such that they have nocommon factors. The index κ serves to distinguish between the different mem-bers of the same family of planes. The plane that passes through the origin hasκ = 0, whereas the plane that passes next closest to the origin has κ = 1.The planes that are at successively further distances from the origin have largermagnitudes of κ.

The indices (h1, h2, h3) are found by locating the intercepts of the plane withthe three primitive axes, say x1 a1, x2 a2 and x3 a3, inverting the intercepts1x1, 1

x2, 1

x3, and then finding the smallest three integers which have the same

ratio1x1

:1x2

:1x3

= h1 : h2 : h3 (63)

The set of integers (h1, h2, h3) are enclosed in round brackets and denote theMiller indices of the plane. A negative valued integer, such as − h1, is denotedby an overbar such as h1.

The Miller indices label the direction of the normal to the family of planes.Since the vectors between pairs of intercepts lay in the plane, the three vectors

1h1

a1 − 1h2

a2

1h2

a2 − 1h3

a3

1h3

a3 − 1h1

a1 (64)

are parallel to the plane. Any two of these vectors span the plane, so the thirdvector is not independent. The normal to the plane is parallel to the vectorproduct of any two non-collinear vectors in the plane

n ∝ κ2

(1h1

a1 − 1h2

a2

)∧(

1h2

a2 − 1h3

a3

)

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=κ2

h1 h2 h3

(h3 a1 ∧ a2 + h2 a3 ∧ a1 + h1 a2 ∧ a3

)(65)

Thus, the direction of the normal to the plane is given in terms of the compo-nents hi in the three directions defined by aj ∧ ak. The three vectors have thesame directions as the primitive ”reciprocal lattice vectors”.

The primitive reciprocal lattice vectors are defined by

b1 = 2 πa2 ∧ a3

a1 . ( a2 ∧ a3 )(66)

and cyclic permutations of the set (1, 2, 3). These primitive reciprocal latticevectors are, in general, not orthogonal. The normal to the plane is then givenby the direction of the reciprocal lattice vector Bh

Bh = h1 b1 + h2 b2 + h3 b3 (67)

where (h1, h2, h3) are the Miller indices. The length of this reciprocal latticevector is defined as

| Bh |2 =(h1 b1 + h2 b2 + h3 b3

)2

=(

2 πdh

)2

(68)

This is seen through the following consideration: The equation for the points ron the plane which intercept the primitive lattice vectors ai at distances xi = κ

hi

is given by

r . Bh =κ

h1a1 . Bh

h1a1 .

(h1 b1

)= 2 π κ (69)

The minimum distance, s, between the origin and the plane is given by

s = r .Bh

| Bh |

=(dh

2 π

)r . Bh

= κ dh (70)

Thus, it is found that the spacing between successive planes in the family isgiven by s = dh, and the planes are equidistant.

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Sets of families of planes that are equivalent in a given crystal structureare denoted by h, k, l. For example, in a cubic crystal the families of planes(1, 0, 0), (0, 1, 0) and (0, 0, 1) are equivalent and are denoted by 1, 0, 0.

A direction of a vector in the direct lattice is specified by three integers insquare brackets [n1, n2, n3] and specify a vector

n1 a1 + n2 a2 + n3 a3 (71)

A negative value for a component is also denoted by an overbar. The set of direc-tions which are equivalent for a crystal structure are denoted by < n1, n2, n3 >.

——————————————————————————————————

3.9.1 Exercise 8

Consider the planes (1, 0, 0) and (0, 0, 1) for a f.c.c. lattice with axes describedby the conventional unit cell. What are the indices of the planes when referredto the primitive axes?

——————————————————————————————————

3.9.2 Exercise 9

Show that the angles α1 ( 6 a2 , a3 ), α2 ( 6 a3 , a1 ) and α3 ( 6 a1 , a2 )between the three primitive lattice vectors of the direct lattice, ai, are related tothe angles between the three primitive lattice vectors of the reciprocal lattice,bi, β1 ( 6 b2 , b3 ), β2 ( 6 b3 , b1 ) and β3 ( 6 b1 , b2 ) via

cos α1 =cos β2 cos β3 − cos β1

| sin β2 sin β3 |(72)

and also find the inverse relation.

——————————————————————————————————

3.10 Quasi-Crystals

Quasi-crystals have symmetries intermediate between a crystal and a liquid.Quasi-crystals are usually intermetallic alloys. The quasi-crystal is space filling,but unlike a regular Bravais lattice, does not have just one unit cell. Thesedifferent ”unit cells” are stacked in a way such that there is no long-rangedpositional order, but nevertheless retain orientational order. The absence oflong-ranged positional order lifts the restriction on the symmetry of the lattice

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but puts a restriction on the vectors that describe the ”unit cells”. For example,an Al −Mn quasi-crystal (Schechtman, Blech, Gratais and Cahn, Phys. Rev.Lett. 53, 1951 (1984)) has icosahedral symmetry, with two, three and five-foldaxes. The structure is made from blocks consisting of a central Mn atom sur-rounded by 12 Al atoms arranged at the corners of an icosahedron. This typeof icosahedral structure often the arrangement of 13 atoms which has the lowestenergy (F.C. Frank, Proc. Roy. Soc. London, 215, 43 (1952)). The icosahedraare stacked together with the same orientation. The voids are formed with thesecond structural unit. The five-fold symmetry of the icosahedra is not allowedfor a regular Bravais Lattice. The five-fold point group symmetry imposes a re-striction on the lengths of the ”lattice vectors” of a quasi-crystal to have certainirrational ratios. Thus, the reciprocal lattice contains reciprocal lattice vectorsof arbitrary small magnitude which show up as an extremely high density ofBragg reflections (Levine and Steinhart, Phys. Rev. Lett. 53, 2477 (1984)).

A way of obtaining quasi-crystal structures is by projecting a periodic Bra-vais lattice structure in higher dimensions (six or more) onto three dimensions(P. Kramer and R. Neri, Acta. Crystallogr. Sec. A 40, 580 (1984)). To il-lustrate this, consider a square two-dimensional lattice, with lattice constant a.On any unit cell, construct two parallel lines with slope tan θ passing throughopposite corners. The equations of the lower line is given by

y = x tan θ (73)

and the upper line is determined by

y = a + ( x + a ) tan θ (74)

For rational values of the slope, tan θ = pq , the lattice points cross the line

periodically, with repeat distance q a along the x direction and have periodicityp a along the y direction. Lines with irrational values of the slope cannot crossmore than one lattice point and, therefore, do not have periodic long-rangedorder. The points (na,ma) contained in the area between the two lines satisfythe inequality

1 + ( n + 1 ) tan θ > m > n tan θ (75)

Project the lattice points contained within the strip onto one of the lines. Thedistance s along the lower line is given by

s = n a cos θ + m a sin θ (76)

wherem =

∑m′

m′ Θ(1 + (n+ 1) tan θ −m′) Θ(m′ − n tan θ) (77)

For irrational values of the slope, the resulting array of points is a quasi-periodicarray. The spacing between consecutive points of the quasi-periodic array is ei-ther given by cos θ or sin θ. The spacings are not distributed periodically, but

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nevertheless, are distributed according to some irregular or more complex pat-tern. If the slopes of the line are equal to 1

2 (√

5 − 1 ) the array of projectedpoints is a Fibonacci series. For a Fibonacci series of numbers, the first termcan be chosen in any way but the next term is given by the sum of the precedingtwo numbers, i.e., Fn+1 = Fn + Fn−1. Thus, both series 1 , 1 , 2 , 3 , 5, 8 , 13 etc. or 3 , 3 , 6 , 9 , 15 etc. are Fibonacci series. The golden mean12 (

√5 + 1) is the limit of the ratio of the successive terms. In our example,

the sequences of spacings is given by s c s c c s c s c . . .. The first element ofthe Fibonacci series is s the second element is c the third element comprises ofs c, the next element is c s c, which is followed by s c c s c etc. If this typeof analysis is applied to high dimensional Bravais lattices, one can find threedimensional quasi-crystal structures with five-fold symmetry.

A five-fold symmetry is also found when tiling a two-dimensional plane withtwo types of tiles, both having the same length of edge s, but with angles ofπ5 or 2 π

5 . The ”diameter” to side ratios of these two types of tiles satisfyds = s

d′ =√

5 − 12 . The sides of the tiles are marked and the tiles are adjoined

so that the markings match (Gardner, Scientific American, 236, 110 (1977)).The result is a tiling without long-ranged periodic order, although every finitearea segment repeats an infinite number of times in the plane. These types oftilings are known as Penrose tilings. The Penrose tiling has long-ranged orien-tational order, as can be seen by decorating each tile with lines. The lines onthe tiles join up to form five sets of parallel lines (Ammann lines). The five setsof lines make an angle of 2 π

5 with respect to each other. The spacing betweenthe successive members of a set form a Fibonacci series.

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4 Structure Determination

Structure can be determined by experiments in which beams of particles arescattered from the structure. Elastic scattering experiments are usually pre-ferred as the underlying lattices are not dynamically deformed by the process.In order that the results be easily interpretable in terms of the structure, it isnecessary that the wave length associated with the beam of particles should havethe same order of magnitude as the spacing between atoms in the structure andsecondly, the beam of particles should only interact weakly with the structure.The first condition allows for a clear resolution of diffraction peaks caused bythe atomic structure. The second condition ensures that the beam is scatteredprimarily in the bulk or interior of the material, and not just the surface. Italso allows for an easy interpretation of the data via second order perturbationtheory.

4.1 X Ray Scattering

X-rays are usually used in the determination of the atomic structure of solids.The strength of the interaction is measured by the deviation of the dielectricconstant from its vacuum value ( 1 ). At energies of about 10 keV, the wavelength of the x-rays λ is ∼ 10−10 m, and at these high energies the refractiveindex is almost unity.

In x-ray diffraction, the x-rays are elastically scattered from the charge den-sity of the electrons. The formal theory of x-ray scattering shows that theintensity of the reflected waves is given by the Fourier Transform of the electrondensity - density correlation function. For a solid which possesses long-rangedorder, the resulting expression for the intensity can be simplified down to involvethe square of the Fourier transform of the electron density. In order to eluci-date the role of the Bravais lattice and the coherent nature of x-ray scattering,the atoms shall first be considered to be point like objects. Later, the spatialdistribution of the electrons around the nuclei shall be re-introduced.

4.1.1 The Bragg conditions

Bragg considered the specular reflection of a beam of x-rays from successiveplanes of atoms separated by distances d. If the angle between the x-rays andthe planes (not the normal to the plane) is θ

2 , then the difference in optical pathlengths for a beam specularly reflected at the lower of two consecutive layers is

2 d sinθ

2(78)

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In this expression, θ is the scattering angle of the particles in the beam. Thereflected beams superimpose with a phase difference of

4 πd

λsin

θ

2(79)

and constructive interference occurs whenever

n λ = 2 d sinθ

2(80)

This is Bragg’s law. The value of n is called the order of the Bragg reflection.Since the successive planes are equi-spaced, the scattering for an entire familyof planes is constructive when the scattering from two neighboring planes in thefamily is constructive. Since there are a large number of planes in a family, andsince the solid is almost transparent to x-rays, the scattering amplitude fromeach member of the family adds coherently giving rise to a very high intensityof the scattered beam whenever Bragg’s condition is satisfied.

In the application of Bragg’s law to x-ray scattering, not only must one con-sider the different coherent scattering conditions from a single family of planes,but one must also consider scattering from the different families of planes in thesolid. Different families of planes of atoms in a solid have different orientations.Since a plane of every family passes through each lattice point, the differentorientations may have different spacings between members of the families ofplanes, so d can vary from family to family. The different Bragg reflections areusually indexed by the Miller indices (m1,m2,m3) of the planes that they arereflected from.

4.1.2 The Laue conditions

Laue’s condition is more general than that of Bragg. The Laue condition isderived by considering scattering from the basis atoms in each of the primitiveunit cells in the solid. The individual cells scatter the x-rays almost isotropi-cally, however, the scattering in a specific direction will only be coherent at wavelengths for which the scattered waves from each unit cell add constructively.

The wave vector of the incident beam is expressed as k, where

k =2 πλ

e (81)

and the reflected wave has wave vector k′

k′ =2 πλ

e′ (82)

where e and e′ are two unit vectors. Let us consider two scattering centersseparated by a vector displacement d. Then, the difference in optical path length

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for x-rays scattered from one atom is composed of two non-equal segments

d cosθ

2= d . e

d cosθ′

2= d . e′ (83)

The optical path difference between the two waves is given by the difference

d cosθ

2− d cos

θ′

2= d . ( e − e′ ) (84)

Thus, constructive interference of the scattered waves from two unit cells occurswhenever

d .

(e − e′

)= m λ (85)

holds for integer m. This condition can be re-expressed in terms of the wavevectors of the incident and scattered x-rays as

d .

(k − k′

)= 2 π m (86)

If this condition is fulfilled for the set of vectors d that are all the Bravais Latticevectors R, one finds the Laue condition for coherent scattering

R .

(k − k′

)= 2 π m (87)

or alternatively

exp[i

(k − k′

). R

]= 1 (88)

If this condition is satisfied for all R in a solid with N unit cells, constructiveinterference will occur between all pairs of unit cells, giving rise to coherentscattering. The cross-section will have N2 such contributions, and the scatteredwave will be extremely intense.

If the scattering vector q is defined as

q = k − k′ (89)

the Laue condition is satisfied for the special set of q values, Q which satisfy

exp[i Q . R

]= 1 ∀ R (90)

These special q values can be used to obtain the k values at which the reflectionwill occur. The expression for the momentum transfer is

k′ = k − Q (91)

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which can be squared to yield

k′2 = k2 − 2 k . Q + Q2 (92)

This equation may be combined with the condition for elastic scattering | k | =| k′ |, to result in a condition on the incident k values for coherent scattering ofthe form

Q2 = 2 k . Q (93)

Thus, k will satisfy the Laue condition for coherent scattering when the compo-nent of k along Q bisects Q. Thus, the projection of k along Q must be equal tohalf the length of Q. The incident wave vector k must lie on the plane bisectingthe origin and Q, which is called the Bragg plane.

The Laue condition is satisfied if Q . R = 2 π m for all lattice vectors R.In particular, if the Laue condition is satisfied, one can choose R to be any oneof three primitive lattice vectors. The three choices of primitive lattice vectorsyields the three equations,

a1 . Q = 2 π m1

a2 . Q = 2 π m2

a3 . Q = 2 π m3

(94)

Since any lattice vector R can be expressed as integer multiples of the primitivelattice vectors, these three Laue equations are equivalent to the Laue condition.The three Laue equations have a geometrical interpretation. Namely, Q lies ona cone around the direction of a1 with projection 2 π m1. Similarly, Q alsolies on a certain cone around a2, and also on a cone around a3. Thus, Q mustlie on the common intersection of the three cones. This is a severe constraint:the values of k for which this is satisfied can only be found by systematicallysweeping the magnitude of k or by rotating the direction of k which is equivalentto systematically re-orienting the crystal.

However, once Qivalues have been found which satisfy the Laue conditions,

other Q values can be found which are integral multiples of the initial Qi’s.

General considerations show that there are three basis vectors bi which can beused to construct the general Q.

4.1.3 Equivalence of the Bragg and Laue conditions

Since a plane belonging to each family of planes passes through each latticepoint, it is obvious that the Laue condition is equivalent to the Bragg condition.

Let Q = k − k′ be a scattering wave vector such that Q . R = 2 π m

for all lattice vectors R. As k and k′ have the same magnitude, they make the

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same angle θ2 with the Bragg plane.

Due to the elastic scattering condition one has |Q| = 2 k sin θ2 and if the

scattering is coherent then the magnitude of Q can be written as |Q| = 2 π nd ,

where n is the order of the reflection and d is a distance characteristic of thelattice. Combining the elastic and Laue conditions, one has

k sinθ

2=

π n

d

2 d sinθ

2= n λ (95)

Thus, the Laue diffraction peak associated with the change in k given byk − k′ = Q, just corresponds to a Bragg reflection by an effective familyof planes which have Q as their normal.

The order n of the Bragg reflection just corresponds to the magnitude of| Q | divided by 2 π

d , where d is the separation of a family of planes.

4.1.4 The Ewald Construction

Since the Laue condition is very restrictive, the vectors k which produce coher-ent scattering are relatively few and far between. The Ewald construction (P.P.Ewald, Z. Krist. 56, 129 (1921)) provides a convenient way of visualizing howthe Laue condition may be fulfilled.

The incident wave vector wave k is centered on the origin O. A sphere ofradius k′ ( = k ) is constructed which is centered on the tip of k. This is theEwald sphere. The scattered wave vectors have the magnitude k′ and may berepresented by vectors k′ directed from points on the sphere’s surface directedto the center of the sphere. The scattering wave vectors q = k − k′ aredirected from the origin towards the points on the surface of the sphere.

Since the wave vectors Q which are solutions of

Q . R = 2 π m (96)

form a lattice of points (the reciprocal lattice) including Q = 0, a latticepoint has to be centered on the origin. This lattice is indexed by three integers(m1,m2,m3) corresponding to the components along three primitive (recipro-cal) lattice vectors. When a second point of the lattice of Q points resides on thesurface of the Ewald sphere, say at − k′, it produces a Bragg reflected beam. Inthis case, the Laue condition is satisfied and the incident beam will be Bragg re-flected at this k′ value. In general, it is expected that the Ewald sphere will nothave a second lattice point on the surface. When Bragg reflections occur, theyare indexed by the integers (m1,m2,m3) which describe the family of planes

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associated with the momentum transfer Q.

4.1.5 X-ray Techniques

There are various techniques which can be used to obtain diffracted beams.

In the Laue Method, a beam of x-rays with a continuum of wave lengthsin the range between λ0 and λ1 is used, and the incident beam has a fixeddirection. Thus, it is only appropriate to use this method for a single crystal,as a polycrystalline sample would correspond to an average over the relativeorientation with the incident beam. In the Laue method, the continuous wave-length of the beam broadens the surface of the Ewald sphere into a finite volumeenclosed between two Ewald spheres with the limiting wave lengths. For a largeenough mismatch between the wave length of the interior Ewald sphere λ0 andthe exterior sphere λ1 , it is quite likely that at least one Bragg reflection willoccur. This method provides the simplest method for orienting a single crystalrelative to the direction of the incident beam. If the incident beam is along adirection of high symmetry of the lattice of Q points, the pattern of reflectedbeams should exhibit the same symmetry. It should be noted that the x-raypattern will always show a center of symmetry, even if the crystal does not haveone. This discovery is due to Friedel.

The Rotating Crystal Method uses a monochromatic beam of x rays, andin the experiment, the relative direction of the incident beam and the crystal isvaried. If one considers the lattice of points Q as being fixed, then the Ewaldsphere rotates around the origin and, for large enough k, will sweep some latticepoints through the surface of the sphere. This experiment produces a set ofBragg reflected beams that are recorded on a photographic film. In practice,the crystal is rotated about a crystallographic axis, say a1, while the incidentbeam has a fixed direction perpendicular to a1. The photographic film is bentinto a cylinder with an axis which is chosen to coincide with the axis of rotationof the crystal. Since the incident beam is perpendicular to the rotation axis, thenthe Bragg reflected beams occur within cones of fixed angle. That is, the b2 andb3 components of the lattice of Q points form planes which are perpendicularto a1. Therefore, under the rotation these two components of the Q vectorsare rotated in the planes. However, the components of Q parallel to a1 remaininvariant and are governed by m1, since

Q . a1 = 2 π m1 (97)

Furthermore, since k and a1 are perpendicular

k′ . a1 = − Q . a1 (98)

and the reflected beams produce a series of Bragg spots which exist in ringswrapped around the photographic film cylinder. Each ring corresponds to a

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different value of m1. Direct observation of the angle between k′ and a1 allowsthe magnitude of a1 to be obtained with ease.

The Debye-Scherrer Method uses a polycrystalline or powdered sample.Each grain of the sample has a random orientation, therefore, this method isequivalent to the rotating crystal method in which the sample is rotated overall possible orientations. Each reciprocal lattice point will generate a sphere ofradius equal to the magnitude of the reciprocal lattice vector. If this sphericalshell of reciprocal lattice vectors intersects with the Ewald sphere, it producesBragg reflections. Each lattice vector with length less than 2 k will produce acone of Bragg reflections, with an angle θ relative to the un-scattered beam. Themagnitude of the reciprocal lattice vector is given by Q = 2 k sin θ

2 . Thus,a measurement of θ will give the lengths of the smallest reciprocal lattice vectors.

These methods can be used to determine the reciprocal lattice vectors and,hence, the Bravais lattice associated with the crystal. In order to completelydetermine the crystal structure, one must determine the basis. This can be doneby examining the structure and form factors.

——————————————————————————————————

4.1.6 Exercise 10

Fleming and co-workers describe the structure of various alkaline metal C60

compounds in their Nature article, Nature 352, 701, 1991.

In figure (2.a) of the paper they indicate an f.c.c. structure for the solid.Indicate the conventional axes on their unit cell.

If a powder x -ray diffraction experiment is performed on Rb doped C60 withx-rays of wavelength λ = 0.9 A, for the dopings 3, 4 and 6 in the paper, whatare the angles 2 θ for the first 5 diffraction peaks for the observed structures?

——————————————————————————————————

4.1.7 The Structure and Form Factors

If the lattice has a basis, the scattered wave from each unit cell must be com-posed from the scattered waves from each atom in the basis. This means thatthe scattering from each type of atom in the basis must be determined and thensuperimposed to find the scattered wave. The scattering from the electron den-sity of each atom can be expressed in terms of the form factor. The form factorsfor an atom in a solid differ only slightly from the form factors of isolated atoms,and are mainly determined by the atomic charge number Z. Although there

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are differences due to the bonding, the form factors are determined by all theelectrons, and not just those involved in bonding. The form factor of the j-thatom in the basis is denoted by Fj(q). It is conventional to use a scale such thatthe forward scattering θ = 0 atomic form factor equals the number of electronsin the atom. Since the coherent scattering is restricted to scattering vectors Qthat satisfy the Laue condition, the form factor only needs to be evaluated atthese values of Q. The amplitude of the scattered wave from the atoms in thebasis of the unit cell can be expressed in terms of the structure factor S(Q)which is given by

S(Q) =∑

j

exp[i Q . rj

]Fj(Q) (99)

This is just the component of Fourier Transform of the electron density fromone unit cell. The intensity of the Bragg peaks is proportional to the factor

| S(Q) |2 (100)

The Q dependence of the intensity can be used to determine the basis of thecrystal. Unfortunately, since only the modulus of S(Q) can be found from ex-periment and not its phase, indirect methods have to be used to discover thecrystal structure. However, if the crystal is centro-symmetric, then if there isan atom at the basis point rj there is another atom of the same type at − rj

and S(Q) is purely real. The phase problem just simplifies to the question asto whether S(Q) is positive or negative.

If the basis of a crystal structure is mono-atomic, the atomic form factorcan be factorized out, and the amplitude of the scattered wave is partiallydetermined by the geometric structure factor

SG(Q) =∑

j

exp[i Q . rj

](101)

The geometric structure factor expresses the interference between identical atomsin the basis. The intensity of the Bragg peak is still determined by the productof the modulus of the form factor with the modulus of the geometric structurefactor. The vanishing or variation of the Bragg peak intensities due to interfer-ence can be used to determine the positions of the basis atoms.

An example of the ambiguity imposed by the non-measurability of the phaseof the Structure Factor is given by Friedel’s law, for non-centrosymmetric crys-tals. The structure factor S(Q) is a complex number, and can be written as

S(Q) = A + i B (102)

For each Q that satisfies the Laue condition, there is a vector −Q which cor-responds to the negative integer − m. The structure factor S(−Q) is just the

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complex conjugate of S(Q)

S(−Q) = A − i B (103)

Since the structure factor for both the vectors Q and −Q have the same magni-tude, the Bragg peaks have the same intensity. Thus, the diffraction pattern hasa center of inversion symmetry, even if the crystal structure does not. Excep-tions to Friedel’s law only occur if the crystal has anomalous dispersion. Thishappens when the x-rays are highly absorbed by the crystal.

Face Centered Cubic Lattice.

The face centered cubic lattice can be represented in terms of a simple cubiclattice with a four atom basis. The scattering from this lattice can be expressedin terms of the Laue condition for the simple cubic lattice, but modulated by thegeometric structure factor. The four atom basis of the non-primitive (conven-tional) unit cell of the face centered cubic lattice consists of the atomic positions

r1 = 0

r2 =a

2

(ex + ey

)r3 =

a

2

(ez + ex

)r4 =

a

2

(ez + ey

)(104)

The Bragg vectors for the conventional simple cubic cell are easily found to be

bx =(

2 πa

)ex

by =(

2 πa

)ey

bz =(

2 πa

)ez (105)

so a general simple cubic Bragg scattering vector is given by

Q =(

2 πa

) (m1 ex + m2 ey + m3 ez

)(106)

The geometric structure factor for the conventional f.c.c. unit cell is found tobe

SG(Q) =∑

j

exp[i Q . rj

]

=

[1 + exp

[+ i π ( m1 + m2 )

]+

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+ exp[

+ i π ( m1 + m3 )]

+ exp[

+ i π ( m2 + m3 )] ]

(107)

When evaluated at the Bragg vectors, the geometric structure factor adds co-herently

SG(Q) = 4 (108)

if the integers (m1,m2,m3) are either all even or are all odd. The geometricstructure factor interferes destructively

SG(Q) = 0 (109)

if only one integer is different from the other two. That is, if one integer is ei-ther even or odd, while the other two, respectively are odd or even, then SG(Q)vanishes. Thus, the f.c.c. lattice has the same pattern of Bragg reflections asthe simple cubic lattice, but has missing Bragg spots. The resulting lattice ofBragg spots is cubic with twice the dimensions (in q space) but has missingBragg spots at the mid points of the edges and at the face centers. Thus, it isfound that the diffraction pattern has the form of a body centered cubic lattice.

The Body Centered Cubic Lattice.

The body centered cubic lattice can be viewed as a simple cubic lattice witha two atom basis

r0 = 0

r1 =a

2

(ex + ey + ez

)(110)

Then, the geometric structure factor for the conventional b.c.c. unit cell is just

SG(Q) = 1 + exp[i Q .

a

2( ex + ey + ez )

]= 1 + exp

[ia

2( Qx + Qy + Qz )

](111)

Now the Bragg vectors for the simple cubic structure are just

Q =2 πa

( m1 ex + m2 ey + m3 ez ) (112)

therefore, at these Q values the geometric structure factor simplifies to

SG(Q) = 1 + exp[i π ( m1 + m2 + m3 )

]= 1 +

(− 1

)( m1 + m2 + m3 )

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= 2 for ( m1 + m2 + m3 ) even

= 0 for ( m1 + m2 + m3 ) odd

(113)

Thus, the body centered cubic lattice has Bragg spots that form a cubic lattice.However, the intensity of the odd indexed Bragg spots vanish, leading to a facecentered cubic lattice of Bragg spots.

The Diamond Lattice.

The diamond lattice is an f.c.c. lattice with a two atom basis

r0 = 0

r1 =a

4

(ex + ey + ez

)(114)

where the conventional f.c.c. unit cell has linear dimension a.

From the discussion of scattering from an f.c.c. lattice, one finds that theQ vectors of the Bragg spots can be expressed in terms of the set of primitivevectors for the b.c.c. lattice

Q =∑

i

mi bi (115)

The primitive vectors are given by

b1 =2 πa

(ey + ez − ex

)b2 =

2 πa

(ez + ex − ey

)b3 =

2 πa

(ex + ey − ez

)(116)

The geometric structure factor of the diamond lattice, relative to the lattice ofBragg spots of the real space f.c.c. lattice, is given by

SG(Q) = 1 + exp[iπ

2( m1 + m2 + m3 )

](117)

From this it is found that the geometric structure factor not only gives riseto extinctions but also modulates the intensity of the non-zero Bragg spots,according to the rule

SG(Q) = 2 for ( m1 + m2 + m3 ) 2 × even

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SG(Q) = 0 for ( m1 + m2 + m3 ) 2 × odd

SG(Q) = 1 ± i for ( m1 + m2 + m3 ) odd

(118)

As the f.c.c. lattice has Bragg spots arranged on a b.c.c. lattice, it is con-venient to transform the Bragg vectors into the coordinates system used for aconventional b.c.c. unit cell

Q =4 πa

[ex

(12

( m1 + m2 + m3 ) − m1

)+ ey

(12

( m1 + m2 + m3 ) − m2

)+ ez

(12

( m1 + m2 + m3 ) − m3

) ](119)

The rule for the modulation of intensities is expressed directly in terms of thequantity ∑

i

Qi a

4 π=

12

( m1 + m2 + m3 ) (120)

Thus, one can describe the system of Bragg spots as residing on a b.c.c. latticewith cubic cell of side 4 π

a . The b.c.c. lattice can be re-interpreted in termsof two interpenetrating simple cubic lattices. Thus, the Bragg spots with non-equal intensities reside on two interpenetrating simple cubic lattices of side 4 π

a .The length scale is twice as large as the reciprocal lattice spacing of the (simplecubic) lattice constructed from the conventional unit cell.

One simple cubic lattice contains the origin Q = 0, and the Bragg spotshave integer coefficients for the unit vectors ex, ey and ez. This means that( m1 + m2 + m3 ) is even for this simple cubic lattice. On dividing by a factorof 2, the resulting number is odd and even at consecutive lattice points. When( m1 + m2 + m3 )/2 is an even integer, S = 2 and the intensities are finite.However, when ( m1 + m2 + m3 )/2 is odd then S = 0 so the intensities arevanishing. Thus, the non-zero intensities on this simple cubic reciprocal latticeactually forms a face centered cubic reciprocal lattice.

The second interpenetrating simple cubic lattice has Bragg points with half(odd) integer coefficients for the unit vectors ex, ey and ez. This means thatthe sum ( m1 + m2 + m3 ) is odd for this simple cubic lattice. These latticepoints are the body center points of the underlying b.c.c. lattice. The geometricstructure factor is simply SG(Q) = 1 ± i and thus, the Bragg spots on this

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simple cubic lattice all have the same intensities.

Extinctions due to Glide Planes and Screw Axes.

Consider a solid with a glide plane, along the ez axis perpendicular to the ey

axis. Thus, if there is an atom at (x, y, z) in units of the lattice parameters, thereis an equivalent atom at (x, y, z + 1

2 ). The pairs of basis atoms each contributea term

SG(Q) = exp[

2 π i ( x m1 + y m2 + z m3 )]

+ exp[

2 π i ( x m1 − y m2 + ( z +12

)m3 )]

(121)to the geometric structure factor. One can see that for the special case m2 = 0the structure factor is composed of terms with the form

SG(Q) = exp[

2 π i ( x m1 + z m3 )] (

1 + exp[π i m3

] )= exp

[2 π i ( x m1 + z m3 )

] (1 + ( − 1 )m3

)= 0 if m3 is odd

= 2 exp[

2 π i ( x m1 + z m3 )]

if m3 is even

(122)

Thus, reflections of the type (m1, 0,m3) will be missing unless m3 is an evennumber.

Similar extinctions occur for screw axes. Consider a two-fold screw axisparallel to ey. The equivalent positions are (x, y, z) and (x, 1

2 + y, z). Thus, thestructure factor for m1 = 0 and m3 = 0 is made up of contributions with theform

SG(Q) = exp[

2 π i y m2

] (1 + ( − 1 )m2

)= 0 if m2 is odd

= 2 exp[

2 π i y m2

]if m2 is even

(123)

Thus, reflections of the type (0,m2, 0) will be missing unless m2 is an even in-teger.

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——————————————————————————————————

4.1.8 Exercise 11

Experiments on solid AxC60 show that the C60 molecules are located on a facecentered cubic lattice with lattice spacing a = 14.11 A, and that the (2, 0, 0)x-ray diffraction peak is very weak when compared to the (1, 1, 1) Bragg peak.Fleming et. al. Nature 352, 701 (1991). Calculate the structure factor for thesereflections in an approximation which assumes that the electron distribution ofeach fullerene molecule is uniformly spread over a spherical shell of radius 3.5 A.

——————————————————————————————————

4.1.9 Exercise 12

The Hendriks-Teller model for x-ray diffraction from a disordered system con-siders a one-dimensional line of molecules. The probability that a pair of atomsis separated by a distance a is given by p and the probability that they areseparated by a + da is given by 1 − p. The random system has an infiniteunit cell. Calculate the average geometric structure factor for this model, andshow that

SG(Q) =p ( 1 − p )

[1 − cosQ da

]1− p(1− p)− p cosQa− (1− p) cos[ Q(a+ da) ] + p(1− p) cosQda

(124)In a scattering measurement on a random system, one measures the average of| SG(Q) |2. Determine the relation between SG(Q) and | SG(Q) |2 and describethe results of a scattering measurement on this one-dimensional system.

——————————————————————————————————

Polyatomic Crystals.

For a polyatomic crystal the structure factor has both the geometric contri-bution and the contribution from the atomic form factors of the basis atoms

S(Q) =∑

j

exp[i Q . rj

]Fj(Q) (125)

The atomic form factor Fj(Q) is determined by the internal structure of theatom that occupies the position rj in the basis.

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The atomic form factor is normalized to the electronic charge of the atom.For a single atom, the form factor is given by

F (Q) =∫

d3r ρ(r) exp[− i Q . r

](126)

where ρ(r) is the atomic electron density. If the charge density is sphericallysymmetric, then the form factor can be reduced to a radial integral

F (Q) = 2 π∫ ∞

0

dr r2 ρ(r)∫ 1

−1

d cos θ exp[− i Q r cos θ

]= 2 π

∫ ∞

0

dr r2 ρ(r)2 sin Q r

Q r

= 4 π∫ ∞

0

dr r2 ρ(r)sin Q r

Q r(127)

For forward scattering, Q = 0, the form factor reduces to

F (0) = 4 π∫ ∞

0

dr r2 ρ(r)

= Z (128)

where Z is the atomic number. Typically F (Q) decreases monotonically withincreasing Q, falling off as a power of 1

Q2 for large Q.

——————————————————————————————————

4.1.10 Exercise 13

Calculate the x-ray scattering intensities for the following close-packed structureformed by stacking hexagonal layers, in the following sequences:

(a) The sequence ABAB... (the h.c.p. sequence).

(b) The sequence ABCABC... (the f.c.c. sequence).

(c) The random sequence in which all the consecutive layers are different,but given one layer (say A), there is an equal probability that it will be followedby either one of the two other layers.

——————————————————————————————————

4.1.11 Exercise 14

Find the atomic form factor for the hydrogen atom, using the electron density

ρ(r) =1

π a3exp

[− 2 r

a

](129)

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where a is the Bohr radius.

——————————————————————————————————

Sodium Chloride.

An example of a diatomic crystal with a basis is provided by NaCl. This hasa face centered cubic lattice and has Na+ ions at the positions (0, 0, 0), ( 1

2 ,12 , 0),

( 12 , 0,

12 ) and (0, 1

2 ,12 ). The Cl− ions reside at (1

2 , 0, 0), (0, 12 , 0), (0, 0, 1

2 ) and( 12 ,

12 ,

12 ). The structure can be viewed as a simple cubic lattice with a six atom

basis. In this case, we can use the simple cubic representation of the Braggvectors Q. Thus, the structure factor is given by

S(Q) = FNa(Q)(

1 + exp[i π ( m1 + m2 )

]+ exp

[i π ( m2 + m3 )

]+ exp

[i π ( m3 + m1 )

] )+ FCl(Q)

(exp

[i π m1

]+ exp

[i π m2

]+ exp

[i π m3

]+ exp

[i π ( m1 + m2 + m3 )

] )(130)

As exp[ i π m ] = ( − 1 )m the structure factor can be factorized as

S(Q) =(FNa(Q) +

(− 1

)m1

FCl(Q))

×[

1 +(− 1

)(m1+m2)

+(− 1

)(m2+m3)

+(− 1

)(m3+m1) ](131)

The structure factor is 0 unless the indices are either all odd or all even. This ischaracteristic of face centering. The intensities of the Bragg spots with all evenindices and all odd indices are different as the atomic form factors either add orsubtract.

——————————————————————————————————

4.1.12 Exercise 15

Potassium Chloride has the same structure as NaCl. However, K+ and Cl− areiso-electronic and so have very similar structure factors. Determine the indices(m1,m2,m3) of the allowed Bragg reflections.

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——————————————————————————————————

4.1.13 Exercise 16

Calculate the structure factor for the zincblende structure. The zincblendestructure is a face centered cubic lattice of side a, with a positively charged ionat the origin and a negatively charged ion at a

4 ( ex + ey + ez ).

——————————————————————————————————

Since the differences between the atomic form factors show up in the exper-imentally observed structure factor of compounds, it is possible to distinguishbetween ordered binary compounds and binary compounds with site disorder.The order-disorder transition in Cu3Au has been observed by x-ray scattering.At high temperatures, the atoms in this material are randomly distributed oneatom on each site of an f.c.c. lattice. However, there is a transition between thedisordered phase, which occurs above a critical temperature of Tc ≈ 660 K,to an ordered phase at lower temperatures. In the completely disordered phase,the structure factor is that pertaining to an f.c.c. crystal, in which the formfactor is replaced by the statistically averaged value

Fav(Q) =34FCu(Q) +

14FAu(Q) (132)

Thus, at high temperatures, the structure factor is given by

S(Q) = Fav(Q)(

1 + exp[i π ( m1 + m2 )

]+ exp

[i π ( m2 + m3 )

]+ exp

[i π ( m3 + m1 )

] )(133)

Hence, the peaks have intensity of either 16 | Fav(Q) |2 or zero depending onwhether the indices are all even or all odd, or whether they are mixed. In theordered phase, the Cu atoms reside on the face center sites and the Au on thevertices of the cubes. In this phase, ”super-lattice” peaks appear in the spectrafor mixed indices. For the completely ordered phase, the structure factor isgiven by

S(Q) = FAu(Q) + FCu(Q)(

exp[i π ( m1 + m2 )

]+ exp

[i π ( m2 + m3 )

]+ exp

[i π ( m3 + m1 )

] )(134)

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The ”super-lattice” peaks occur for mixed indices. The relative intensity of the”super-lattice” peaks are approximately given by

I(1, 0, 0)I(2, 0, 0)

∼(FAu(0) − FCu(0)FAu(0) + 3 FCu(0)

)2

(135)

which leads to a relative intensity of about 0.09. Since the x-ray form factorsare FCu(0) = 29 and FZn(0) = 30, the relative intensity of the ”super-lattice”peaks of CuZn, or beta brass, are of the order of 0.0003. Thus, the super-latticepeaks are difficult to observe in x-ray scattering. However, the order-disordertransition in CuZn is easily observable by neutron diffraction.

At very low temperatures, CuZn exists as an ordered compound of the CuCltype. The structure consists of two interpenetrating simple cubic sub-latticeswhich have a relative displacement of [ 12 ,

12 ,

12 ]. The Cu atoms occupy the sites of

one sub-lattice, say the A sub-lattice, and the Zn atoms are located on the othersub-lattice, say the B sub-lattice. For an infinite solid the A and B sub-latticesare equivalent, thus, the compound may also form with the Cu atoms on the Bsub-lattice and the Zn atoms on the A sub-lattice. At temperatures above theorder-disorder transition temperature, the material exists in a disordered phasein which the Cu and Zn atoms are randomly positioned on the sites of the A andB sub-lattices. At the transition temperature, a phase transition occurs betweenthe high temperature disordered phase and the low-temperature ordered phase.The order parameter for the phase transition is given by the scalar quantity,φ(T ), where

φ(T ) = n(Cu)A − n(Cu)B

= n(Zn)B − n(Zn)A (136)

where n(Cu)A and n(Cu)B are, respectively, the number of Cu atoms on the Aand B sub-lattices. The second line follows from the fact that an atom of onetype or the other exists at each site. In particular, if the total number of sitesis 2 N , the numbers of Zn atoms at the sites of the A and B sub-lattices are,respectively, given by

n(Zn)A = N − n(Cu)A

n(Zn)B = N − n(Cu)B (137)

Above the transition temperature, the Cu atoms are equally probable to befound on the A and B sublattices and so the order parameter is zero, φ = 0.Below the transition temperature, Tc ≈ 741 K, the order parameter has anon-zero magnitude φ0(T ) which is temperature dependent, and has either apositive or negative sign depending on whether the Cu atoms spontaneouslyselect to occupy the A or B sites, φ(T ) = ± φ0(T ). In the ordered state, thetemperature dependence of the order parameter is given by

φ0(T ) ∝ ( Tc − T )β (138)

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where β ≈ 0.32. As the Hamiltonian is symmetric under interchange of the Aand B sub-lattices, this order-disorder transition provides an example of spon-taneous symmetry breaking.

——————————————————————————————————

4.1.14 Exercise 17

Express the inelastic x-ray scattering intensity for CuZn in terms of the atomicform factors FCu(Q), FZn(Q), and the order parameter φ(T ). Assume that thedeviations of the site occupancies from the average values at different sites areun-correlated.

——————————————————————————————————

4.2 Neutron Diffraction

Elastic neutron scattering from the nuclei of a solid involves the change in themomentum of the neutron from the initial value h k to the final value h k′ of

q = k − k′ (139)

Conservation of momentum requires that the transferred momentum must beequal to a momentum component of the interaction potential. This momentumis ultimately transferred to the solid. Experimentally accessible ranges of q forneutrons are in the range of 0.01 < q < 30 A, which covers the range that isuseful to determine crystalline structures.

The interaction between the neutron and one nucleus is short ranged andcan be modelled by a point contact interaction,

Hint =2 π h2

mnb δ3( r − R ) (140)

where b is the scattering amplitude of the order of 10−14 m. The differentialscattering cross-section represents the number of particles scattered into solidangle dΩ per incident flux. The differential scattering cross-section for onenucleus is assumed to be isotropic and given by

dΩ= | b |2 (141)

Hence, the total cross-section is given by

σ = 4 π | b |2 (142)

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For a crystalline lattice of nuclei, as it shall be shown, the scattering cross-section is given by

dΩ=∑i,j

exp[i q . ( Ri − Rj )

]b∗i bj (143)

where bi is the scattering amplitude from the i-th nucleus. The value of bidepends on what isotope exists at the lattice site and also on the direction ofnuclear spins.

In general, the different isotopes are randomly distributed so they must beaveraged over. Thus, bi and bj are independent or uncorrelated if they belongto different sites, and the average for i 6= j is given by the product of theaverages

b∗i bj = b∗i bj = | b |2 (144)

while, if i = j, one has the average of the squared amplitude

b∗i bi = | b |2 (145)

In general, the average has the form

b∗i bj = | b |2 + δi,j

(| b |2 − | b |2

)(146)

The scattering cross-section can be written as the sum of two parts, a coherentpart where i 6= j and an incoherent part which has i = j.

The coherent cross-section is given by

dΩ=∑i,j

exp[i q . ( Ri − Rj )

]| b |2 (147)

For coherent scattering from every nuclei in the solid, the momentum transfermust satisfy the Laue condition and so q must be equal to Q, where Q satisfies

Q . R = 2 π m (148)

for all lattice vectors R and m is any integer. When this condition is satisfied,the scattering produces Bragg reflections similar to those observed in x-ray scat-tering. When the Bragg scattering condition is satisfied the coherent scatteringhas an intensity proportional to N2.

The incoherent scattering cross-section comes from the terms with i = jand is given by

dΩ= N

(| b |

2− | b |2

)(149)

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The incoherent scattering is proportional to the number of nuclei N and is in-dependent of the direction of q. It is obvious that the coherent and incoherentcontributions are profoundly different. Only the coherent contribution can beutilized to determine the crystalline structure.

4.3 Theory of the Differential Scattering Cross-section

By definition, the differential scattering cross-section dσdΩ is the ratio of the

number of particles scattered dNscatt (per unit time) into a solid angle dΩ =sin θ dθ dϕ to the incident flux of particles F (number of particles crossing unitarea per unit time) times the solid angle element

dNscatt = Fdσ

dΩdΩ (150)

Consider a beam of particles collimated to have a momentum k that fallsincident on a crystal. The particles are assumed to interact with either theelectrons or nuclei of the solid. An example is x-ray diffraction, in which thebeam of photons interacts elastically with the electron density, or alternativelyin neutron diffraction experiments the beam of neutrons interacts, via shortranged nuclear forces, with the nuclei of the solid.

The interaction Hamiltonian between a particle in the beam and the relevantparticles of the solid can be represented as the sum of single particle interactions

Hint =∑

j

Vj(r − rj) (151)

Here, r represents the position of the beam particle and rj is the position of thej-th particle in the solid.

For x-rays in which the energy of the photon is in the keV range, the photonenergy is much greater than the electronic energy scale. This has the effectthat only certain terms of the interaction Hamiltonian between the x-rays andthe electron need be considered. The non-relativistic form of the interactionbetween the electromagnetic field represented by a vector potential A(r, t) andparticles of charge q and mass m is given by

Hint = −∑

j

[q

2 m c

(p

j. A(rj , t) +A(rj , t) . pj

)− q2

2 m c2A(rj , t) . A(rj , t)

](152)

where rj and pj

are the position and momentum of the j-th particle. The firstpair of terms involve processes in which a single photon is absorbed or emitted,whereas the last term involves the interaction of two photons with the chargedparticle. To calculate the cross-section for light scattering, one needs to considerterms of fourth order in the vector potential A(r, t), as both the initial and final

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states each involve a photon. In principle, this requires including the first pairof terms in fourth order as well as the last term in second order. However, as thefourth order processes involve intermediate states in which a very high energyphoton has either been absorbed or emitted, the energy denominator involvingthe intermediate state is large. Thus, these contributions can safely be ignoredand only the last term in the interaction need be considered explicitly in thecalculation of the elastic scattering cross section. Thus, in this approximation,the x-rays couple to the density of the charged particles, ρ(r) =

∑j δ( r − rj ).

For electrons, the coupling constant is proportional to the length

e2

2 me c2=

e2

h c

h

me c∼ 10−15 m (153)

which involves the fine structure constant and the Compton wave length. Theresulting length scale is the so-called classical radius of the electron.

4.3.1 Time Dependent Perturbation Theory

The incident beam has the asymptotic form of a momentum eigenstate witheigenvalue h k

Ψk(r, t) =(

1V

) 12

exp[

+ i k . r

]exp

[− i ωk t

](154)

The time independent part of the asymptotic initial state will be denoted by| k > in Dirac notation. The scattered wave at the detector has an asymptoticform of a momentum eigenstate | k′ > with momentum eigenvalue h k′.

The matrix elements of the interaction potential are given as

< k | Hint | k′ > =1V

∑j

∫V

d3r exp[− i k . r

]Vj(r − rj) exp

[+ i k′ . r

]

=1V

∑j

∫V

d3R′ exp[− i

(k − k′

). R′

]Vj(R′) exp

[− i

(k − k′

). rj

](155)

where R′ = r − rj . The integration over R′ yields the Fourier transform ofthe interaction potential between the scattered particle and the j-th atom

Vj(q) =∫

V

d3R′ exp[− i q . R′

]Vj(R′)

≈∫d3R′ exp

[− i q . R′

]Vj(R′) (156)

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Given one incident particle in the state | Ψk (t) >, which is initially in anenergy eigenstate | k > before the interaction Hint is turned on adiabaticallyat t → − ∞, the state of this particle evolves according to the Schrodingerequation

i h∂

∂t| Ψk (t) > =

(H0 + Hint

)| Ψk (t) > (157)

As the interaction is weak, the Schrodinger equation can be solved perturba-tively using the interaction representation. In the interaction representation thestates are transformed through a unitary operator in a manner such that

| Ψk (t) > = exp[

+i

hH0 t

]| Ψk (t) > (158)

This unitary transformation would make the eigenstate of the non-interactingparticle time independent. However, the presence of a non-zero interaction termleads to the time dependent equation of motion

i h∂

∂t| Ψk (t) > = ˆHint(t) | Ψk (t) > (159)

where the new interaction operator is time dependent and is given by

ˆHint(t) = exp[

+i

hH0 t

]Hint exp

[− i

hH0 t

](160)

The equation of motion in the interaction representation can be solved by iter-ation. The equation is integrated to yield

| Ψk (t) > = | k > − i

h

∫ t

−∞dt′ ˆHint(t′) | Ψk (t′) > (161)

On iterating once, it is found that the state is given to first order in the inter-action by

| Ψk (t) > = | k > − i

h

∫ t

−∞dt′ ˆHint(t′) | k > + . . . (162)

This shows that, if wave function is started in an initial state which is an en-ergy eigenstate of the unperturbed Hamiltonian, the time evolution caused bythe interaction will admix other states into the wave function. In this sense,the particle described by the wave function may be considered as undergoingtransitions between the unperturbed energy eigenstates.

4.3.2 The Fermi-Golden Rule

The rate at which the particle makes a transition from the initial state | k >to state | k′ >, due to the effect of Hint, is given in second order perturbation

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theory by the Fermi-Golden rule. The probability that the system has made atransition at time t is given by the squared modulus of the transition amplitude

< k′ | Ψk(t) > (163)

However, it is more convenient to calculate the probability based on the matrixelements evaluated in the interaction representation

< k′ | Ψk(t) > (164)

These two quantities are equivalent, as they are simply related via

< k′ | Ψk(t) > = exp[− i

hE(k′) t

]< k′ | Ψk(t) > (165)

and the phase factor cancels out in the squared modulus.

To first order in the perturbation, the transition amplitude is given by

< k′ | Ψk(t) > = − i

h

∫ t

−∞dt′ < k′ | ˆHint(t′) | k >

= − i

h

∫ t

−∞dt′ exp

[i

h( E(k′) − E(k) − i η ) t′

]< k′ | Hint | k >

(166)

where E(k) and E(k′) are the unperturbed (non-interacting) energies of theinitial and final states of the beam particles. The factor η corresponds to adia-batically switching on the interaction at t′ → − ∞. The probability that thetransition has occurred at time t is given by

1h2

∣∣∣∣ ∫ t

−∞dt′ exp

[i

h( E(k′) − E(k) − i η ) t′

]< k′ | Hint | k >

∣∣∣∣2(167)

The rate at which the transition occurs is given by the time derivative of thetransition probability

P (k → k′, t) =1h2

∂t

∣∣∣∣ ∫ t

−∞dt′ exp

[i

h( E(k′) − E(k) − i η ) t′

]< k′ | Hint | k >

∣∣∣∣2(168)

The transition rate is evaluated as

P (k → k′, t) = | < k′ | Hint | k > |2 ∂

∂t

( exp[

2 η th

]( E(k′) − E(k) )2 + η2

)=

2h| < k′ | Hint | k > |2 exp

[2 η th

( E(k′) − E(k) )2 + η2

(169)

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Then, in the limit η → 0, the transition rate becomes time independent andenergy dependent terms reduce to π times an energy conserving delta functionsince

limη → 0

η

( E(k′) − E(k) )2 + η2= π δ( E(k′) − E(k) ) (170)

Hence, we have obtained the Fermi-Golden rule

limη → 0

P (k → k′, t) =2 πh

| < k′ | Hint | k > |2 δ( E(k′) − E(k) )

(171)

This expression represents the probability per unit time for a transition to oc-cur from the initial state to a very specific final state, with a precisely knownk′ that exactly conserves energy. As the rate contains a dirac delta function itis necessary, for the rate to be mathematically meaningful, to introduce a dis-tribution of final states. Thus, one must sum over all states with k′ in the solidangle subtended by the detector, irrespective of the magnitude of k′. Thus, thedirac delta function is to be replaced by the density of final states with energyE = E(k) which are travelling in the direction dΩ.

The probability that a particle makes the transition from state k to stateswith final momentum in a solid angle dΩ distributed around k′, per unit time,is given by summing over the number of allowed final states

P (k → dΩ) =V

( 2 π )3

∫ ∞

0

dk′ k′2 dΩ2 πh

| < k′ | Hint | k > |2 δ( E − E(k′) )

=2 πh

| < k′ | Hint | k > |2 ρdΩ(E, k′) (172)

where ρdω(E, k′) is the density of final scattering states per unit energy range,defined as

ρdω(E, k′) =V

( 2 π )3dΩ

∫ ∞

0

dk′ k′2 δ( E − E(k′) ) (173)

The matrix elements of the interaction operator are to be evaluated with k′ thathave the magnitude of k and are headed in direction dΩ.

4.3.3 The Elastic Scattering Cross-Section

The scattering cross-section is defined by(dσ

)dΩ = P (k → dΩ) / F (174)

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where the incident flux F is the density of particles (which is one per unitvolume, i.e. 1

V ) times the velocity. For massive particles the velocity is just h km .

Thus, for particles of mass mn, the flux is given by

F =h k

mn V(175)

On changing the variable of integration from dk′ to dE′, the density of finalstates is evaluated by integrating over the energy conserving delta function

ρdΩ(E, k′) =V

( 2 π )3

∫ ∞

0

dE′ dΩdk′ k′2

dE′δ( E − E(k′) )

=V

( 2 π )3dΩ

dk′ k′2

dE′(176)

where the magnitude of k′ is determined by the the solution of E = E(k′),hence k′ = k. For massive particles, one has the energy momentum relation

dE′ =h2 k′

mndk′ (177)

and so, the density of final states can be written as

ρdΩ(E, k′) =V

( 2 π )3mn k′

h2 dΩ (178)

On inserting the Fermi-golden rule expression for P (k → dΩ)

P (k → dΩ) =2 πh

| < k′ | Hint | k > |2 ρdΩ(E, k′) (179)

the final density of states ρdΩ(E, k) and the flux F into the expression eqn(174)for the scattering cross-section, one finds that the elastic scattering cross-sectionfor massive particles such as neutrons, is calculated as

dΩ=

(V mn

2 π h2

)2 ∣∣∣∣ ∫V

d3r Ψ∗k′(r) Hint(r) Ψk(r)

∣∣∣∣2=

(mn

2 π h2

)2 ∑j,j′

Vj(q) V ∗j′(q) exp[− i q .

(Rj − Rj′

) ](180)

where q is the scattering vector

k − k′ = q (181)

The magnitude of the scattering vector is related to the scattering angle θ via

q = 2 k sinθ

2(182)

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On substituting the point contact interaction appropriate for nuclear scattering,and noting that the Fourier transform of the delta function is q independent,one finds the expression for the Fourier component of the potential

Vj(q) =2 π h2

mnbj (183)

Substituting for Vj(q), in the above expression for the cross-section, yields theformulae for the elastic neutron scattering cross-section

dΩ=∑j,j′

bj b∗j′ exp

[− i q .

(Rj − Rj′

) ](184)

previously discussed.

For massless particles such as photons, the incident flux is just

F =c

V(185)

if the incident vector potential is

A(r, t) = eα c

√h

2 ω Vexp

[i ( k . r − ω t )

]+ c.c. (186)

With this normalization, the vector potential represents one incident photonper volume V , with frequency ω and incident polarization eα. The density offinal states (for polarization eβ) is just

ρdΩ(E, k′) =V

( 2 π )3k′2

h cdΩ (187)

Thus, it is found that the cross-section for elastic x-ray scattering is simplygiven by

dΩ=

2 πh2 c

V 2 ω2

( 2 π c )3

(e2

2 me c2

)2 ∣∣∣∣ ∫V

d3r A∗k′(r) . ρ(r) Ak(r)∣∣∣∣2

=∣∣∣∣ eα . eβ

∣∣∣∣2 ( e2

4 π me c2

)2 ∑j,j′

S(q) S∗(q) exp[− i q .

(Rj − Rj′

) ](188)

where the structure factor S(q) is the contribution of a unit cell to the Fouriertransform of the electron density. The vectors Ri are the lattice vectors. Thus,the factors of V and ω cancel, leading to a scattering cross-section that onlydepends on the Fourier transform of the electronic density and has a couplingconstant which is the square of the classical radius of the electron

r2e =(

e2

4 π me c2

)2

(189)

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From the form of this coupling constant, it can be seen that the scattering ofx-rays from the density of charged nuclei is entirely negligible compared withthe scattering from the electron density.

4.3.4 The Condition for Coherent Scattering

Consider scattering from a crystal which has a mono-atomic basis and has afinite spatial extent. In this case, the subscript on the atomic potential can bedropped, and the summation over j and j′ run over all the lattice sites. Forconvenience, it shall be assumed that the crystal has the same shape as theprimitive unit cell but has overall dimensions ( N1 − 1 ) a1, ( N2 − 1 ) a2

and ( N3 − 1 ) a3 along the various primitive lattice directions. The solid,therefore, contains a total of N1 N2 N3 primitive unit cells, and as the basisconsists of one atom, the solid contains a total of N = N1 N2 N3 atoms.

The summation over Rj , in the scattering cross-section can be performed byexpressing the general reciprocal lattice vector in terms of the primitive latticevectors,∑

j

exp[i q . Rj

]=

∑n1,n2,n3

exp[i n1 q . a1

]exp

[i n2 q . a2

]exp

[i n3 q . a3

](190)

The sums over n1 runs from 0 to N1−1, and similarly for n2 and n3. This givesthe products of three factors, each of the form

n1=N1−1∑n1=0

exp[i n1 a . a1

]=

1 − exp[i N1 q . a1

]1 − exp

[i q . a1

]= exp

[+ i

( N1 − 1 )2

q . a1

×

(exp

[i N1

2 q . a1

]− exp

[− i N1

2 q . a1

] )exp

[i 1

2 q . a1

]− exp

[− i 1

2 q . a1

]

= exp[

+ i( N1 − 1 )

2q . a1

] (sin N1

2 q . a1

sin 12 q . a1

)(191)

This function exhibits the effect of the constructive and destructive interferencebetween the scattered waves emanating from the various atoms forming the

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solid. The numerator of the function falls to zero at

q . a1 =2 m π

N1(192)

for general integer values of m. The numerator has maximum magnitude at

q . a1 =( 2 m + 1 ) π

N1(193)

The overall q dependence is dominated by the denominator which falls to zerowhen q . a1 = 2 m π, for integer m. At these special q values, the function hasto be evaluated by l’hopital’s rule and has the limiting value of N1. This occurssince, for these q values, the exponential phase factors are all in phase (andequal to unity) and so the sum over the N1 terms simply yields N1. Thus, thescattering cross-section is proportional to the product of the modulus square ofthree of these factors

dΩ= r2e | F (q) |2 ×

×

(sin N1

2 q . a1

sin 12 q . a1

)2 (sin N2

2 q . a2

sin 12 q . a2

)2 (sin N3

2 q . a3

sin 12 q . a3

)2

(194)

Since for a macroscopic solid the numbers N1, N2 and N3 are of the order of107, the three factors rapidly vary with the magnitudes of q . ai. The maximaoccur when the three conditions

q . a1 = 2 π m1

q . a2 = 2 π m2

q . a3 = 2 π m3

(195)

are satisfied. These special values of q are denoted by Q. In this case, one findsthat the scattering cross-section is simply proportional to

dΩ∼ r2e | F (Q) |2 N2 (196)

which is just equal to the square of the number of atoms in the solid. Thecoherent scattering from an ordered solid should be contrasted with incoherentscattering from the atoms of a gas. Due to the positional disorder in the gas, thephase factors may be considered to be random. The net scattering intensity forscattering of a gas of N atoms is then approximately equal to just N times thescattering intensity for an isolated atom. The coherent scattering from atoms ina solid possessing long-ranged order is a factor of N2 larger than the scattering

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intensity for an isolated atom. In summary, the condition that there is completeconstructive interference between all the atoms in the solid is given by

exp[i Q . Ri

]= 1 ∀ i (197)

The intensity of the scattered beam is exceptionally large at these special valuesQ, compared with all other q values. Thus, coherent scattering is the dominantfeature of diffraction from crystalline solids but occurs only infrequently, as itonly occurs when the scattered wave length and scattering angle satisfy theabove stringent condition. These special values of Q are the lattice vectors ofthe reciprocal lattice.

——————————————————————————————————

4.3.5 Exercise 18

Consider a sample with N unit cells arranged in M micro-crystals that areoriented parallel with respect to each other, but their positions are random.Calculate the width and height of the Bragg peak.

——————————————————————————————————

4.3.6 Exercise 19

At finite temperatures, the atoms of a crystal undergo thermal vibrations. Dueto the vibrations, the intensity of the Bragg peaks are reduced by a Debye-Waller factor which involves the spectrum of lattice vibrations. However, thissituation can be approximately modelled by assuming that each atom undergoesa small random displacement δR from its equilibrium position R. Assume thatthe displacements are small compared with the separation between neighbor-ing atoms, | δR | a, and are Gaussian distributed. Also assume that thedisplacements of different atoms are entirely uncorrelated δi,R δj,R′ = 0 forR 6= R′ . Calculate the diffraction peak intensity, and show that the largestreduction occurs for large Q values.

——————————————————————————————————

4.3.7 Exercise 20

Evaluate the effect of a significant number of thermally induced vacancies (miss-ing atoms) in the elastic scattering cross-section from a crystal.

——————————————————————————————————

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4.3.8 Anti-Domain Phase Boundaries

The order-disorder transition usually starts at several nucleation centers in acrystal. For CuZn the underlying CsCl lattice can be divided into two inter-penetrating simple cubic sub-lattices: the A and B sub-lattice. In several re-gions, the nucleation may start with the Cu atoms condensing on the A sub-lattice, whereas the nucleation may occur in other regions where the Cu atomscondense on the B sub-lattices. These distinct domains of nucleation grow andspread through the crystal until they meet and the entire crystal is ordered.The interfaces of the different domains meet at anti-domain phase boundariesat which there is a mismatch of the long-ranged ordering of the atoms. Dueto the mismatch, two planes containing similar atoms form the anti-domainphase boundary. The effect of anti-domain phase boundaries is to smear outthe ”super-lattice” Bragg peaks. This can be seen by considering the amplitudeof the scattered x-rays as a superposition of the scattering from the variousdomains. For simplicity, let us consider the scattering from two domains ofidentical shape and size. If the scattering amplitude from one domain is de-noted by A1(q) and the scattering from the second domain is denoted by A2(q)then, as the scattering amplitudes are additive, one obtains

A(q) = A1(q) + A2(q) (198)

where

A2(q) = exp[i q . δR

]A1(q) (199)

δR is the vector displacements of the origins of the two domains. The scatteringamplitude A1(q) is given by

A1(q) ∝

(sin N1 qx a

2

sin qx a2

) (sin N2 qy a

2

sin qy a2

) (sin N3 qz a

2

sin qz a2

)(200)

For a domain wall in the y − z plane, the displacement between the two Cusub-lattices is given by

δR = ( N1 +12

) a ex +a

2ey +

a

2ez (201)

Hence, for a CsCl-type structure and if q is close to Q, the total scatteringamplitude is given by the expression

A(q) ∼ A1(q)(

1 + ( − 1 )m1+m2+m3 exp[i N1 qz a

] )(202)

The total intensity of the scattered wave is proportional to

I(q) ∝ 2 | A1(q) |2(

1 + ( − 1 )m1+m2+m3 cosN1 qx a

)(203)

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Thus, if m1 +m2 +m3 is even the intensity is modulated by the factor

4 cos2N1 qx a

2(204)

whereas, if m1 +m2 +m3 is odd the intensity is modulated by the factor

4 sin2 N1 qx a

2(205)

This factor is due to the interference of the scattering from the two domains.The destructive interference causes an exact cancellation of the intensity at theexact Bragg wave vector, at odd m1 +m2 +m3. However, for qx slightly off-theBragg position

qx =2 πa

m1 + δqx

δqx ∼ π

N1 a(206)

the scattered intensity is finite and large. That is, the single anti-domain phaseboundary between identically domains of identical shapes and sizes produces ahole in the Bragg peak with odd m1 +m2 +m3.

For a crystal with a CuCl type structure which contains several anti-domainphases, one expects there to be three sets of anti-domain phase boundaries andone expects that each domain has a different size. On averaging over the distri-bution of domains, one expects the small oscillations in the scattered intensityfrom the single domain S1(q) to be washed out. Furthermore, one expects thatthe intensities of the ”super-lattice” peaks to be smeared out in q space.

4.3.9 Exercise 21

Consider the scattering produced by a CuCl type material, with anti-domainwalls. For simplicity, only consider the component of the scattering amplitudeassociated with a single primitive lattice vector. Let p be the probability of notcrossing a domain wall on traversing one step a along a primitive lattice vector,and q is the probability of crossing a domain wall, where q ∼ 1

N1. Show that

the average scattered intensity, near the ”super-lattice” peaks, is proportionalto the factor

| A(qx) |2 ∝ N1 +N1∑

m1=1

( N1 − m1 ) ( p − q )m1 2 cosm1 qx a

=2 N1 p q + ( p2 − q2 )

2 ( q2 + ( p2 − q2 ) sin2 qxa2 )

− ( p2 − q2 ) q2

( q2 + ( p2 − q2 ) sin2 qxa2 )2

+ O

(( p − q )N1+1

)(207)

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Hence, show that the intensity of the ”super-lattice” Bragg peaks are dimin-ished and acquire low amplitude tails.

4.4 Elastic Scattering from Quasi-Crystals

The scattering intensity from three-dimensional quasi-crystals show ten-fold,six-fold and five-fold symmetric diffraction patterns which can be understoodas arising from a space of six or more dimensions. Icosahedral symmetry can befound in a six dimensional hyper-cubic lattice. An icosahedron has 20 identicalfaces made of equilateral triangles. Five of the faces meet at each of the 12vertices of the icosahedron, which is responsible for the five-fold symmetry.

The x-ray scattering amplitude A(q) from a one-dimensional quasi-crystalcan be found by a projection from a two-dimensional lattice. The amplitude isa linear superposition from the scattered amplitudes from the sites sn, where

sn = n a cos θ + m′ a sin θ (208)

and where the points (na,m′a) are restricted to lie in a two-dimensional strip.The amplitude is given by

A(q) =∑

n

exp[i q sn

]=

∑n,m′

exp[i q a ( cos θ n + sin θ m′ )

]

=∑n,m

exp[i q a ( cos θ n + sin θ m )

]Θ(1 + (n+ 1) tan θ −m) Θ(m− n tan θ)

(209)

This can be expressed as an integral over a two-dimensional space

A(q) =∑n,m

exp[i q a ( cos θ n + sin θ m )

]Θ(1 + (n+ 1) tan θ −m) Θ(m− n tan θ)

=∫dx

∫dy exp

[i q ( cos θ x + sin θ y )

] ∑n,m

δ(x− na) δ(y −ma)×

× Θ(a+ (x+ a) tan θ − y) Θ(y − x tan θ)

This is a two-dimensional Fourier transform

A(q) =∫d2r exp

[i q . r

] ∑n,m

δ(x− na) δ(y −ma)×

× Θ(a+ (x+ a) tan θ − y) Θ(y − x tan θ)(210)

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which is to be evaluated on the one-dimensional line

q = q ( cos θ , sin θ ) (211)

The two-dimensional Fourier transform is recognized as the Fourier transformof a product

A(q) =∫d2r exp

[i q . r

]B(r) C(r) (212)

where B(r) is non-zero on the sites of a two dimensional array

B(r) =∑m,n

δ(x− na) δ(y −ma) (213)

and the function C(r) projects onto a two-dimensional strip

C(r) = Θ(a+ (x+ a) tan θ − y) Θ(y − x tan θ) (214)

This can be evaluated using the convolution theorem as the convolution of theproduct of Fourier Transforms

A(q) =∫

d2q′

( 2 π )2B(q − q′) C(q′) (215)

The function B(q) is the scattering amplitude from the two-dimensional lattice

B(q) =∑n,m

exp[i ( qx n a + qy m a )

](216)

while the function C(q′) is evaluated as

C(q′) =∫

dx exp[i ( q′x + q′y tan θ ) x

] (exp[ i q′y a (1 + tan θ) ] − 1

i q′y

)

= ( 2 π ) δ( q′x + q′y tan θ )

(exp[ i q′y a (1 + tan θ) ] − 1

i q′y

)(217)

The scattering amplitude for the two-dimensional lattice is only non-zero at thetwo dimensional reciprocal lattice vectors q = Q. Thus, the scattering fromthe the two-dimensional lattice is represented by the factor

B(q) =(

2 πa

)2 ∑Q

δ2(q −Q) (218)

Hence, we find that the amplitude in the two-dimensional space is given by

A(q) =1a2

∑Q

C(q −Q)

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=2 πa2

∑Q

δ( qx − Qx + ( qy − Qy ) tan θ ) ×

×

(exp[ i ( qy − Qy ) a (1 + tan θ) ] − 1

i ( qy − Qy )

)(219)

Evaluating this on the line in q space yields the amplitude for scattering fromthe one-dimensional quasi-crystal

A(q) = 2 π∑Q

δ(q a

cos θ− Qx a − Qy a tan θ ) ×

×

(exp[ i ( q sin θ − Qy ) a (1 + tan θ) ] − 1

i ( q a sin θ − Qy a )

)= 2 π

∑Q

δ( q a − Qx a cos θ − Qy a sin θ ) ×

×

(exp[ i ( Qx sin θ − Qy cos θ ) a (cos θ + sin θ) ] − 1

i ( Qx a sin θ − Qy a cos θ )

)(220)

This has delta function like peaks at the wave vectors given by

q a = 2 π ( m1 cos θ + m2 sin θ ) (221)

where m1 and m2 are integers. The intensities of the peaks are proportional to

| A(q) |2 ∝sin2

(π ( m1 sin θ − m2 cos θ ) ( cos θ + sin θ )

)( m1 sin θ − m2 cos θ )2

(222)

Thus, the inelastic scattering spectra consists of a dense set of sharp peaks, butwith varying intensities. The intensities are large when the ratios of m2 and m1

are close to the value tan θ.

4.5 Elastic Scattering from a Fluid

The structure of a fluid, as expressed by the pair correlation function, can beinferred from elastic scattering experiments. The intensity of a beam of particlesscattered from a liquid can be considered as analogous to the scattering from asolid with an infinite unit cell. First, we shall consider the atoms of the fluidas static point particles. The amplitude of the beams scattered from each atomadd, giving a total amplitude which is proportional to

S(q) =∑

j

exp[i q . rj

]

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=∫

d3r exp[i q . r

] ∑j

δ3(r − rj)

(223)

The scattering intensity is given by the square of the scattered amplitude

I(q) ∝ | S(q) |2

=∑i,j

exp[

+ i q . ri

]exp

[− i q . rj

]

=∫

d3r

∫d3r′ exp

[i q . ( r − r′ )

] ∑i,j

δ3(r − ri) δ3(r′ − rj)

(224)

On considering the long time average of the atomic positions, one obtains

I(q) ∝∫

d3r

∫d3r′ exp

[i q . ( r − r′ )

] ∑i,j

δ3(r − ri) δ3(r′ − rj)

=∫

d3r

∫d3r′ exp

[i q . ( r − r′ )

]C(r, r′)

(225)

The scattering intensity can be expressed in terms of the radial distributionfunction g(r), since

C(r − r′) = δ3(r − r′) ρ(0) + g(r − r′) (226)

Hence, the

I(q) ∝∫

d3r ρ(0) +∫d3r

∫d3r′ exp

[i q . ( r − r′ )

]g(r − r′)

= N + V

∫d3r exp

[i q . r

]g(r)

(227)

However, the integral over g(r) can be split into two parts

I(q) ∝ N + V

∫d3r exp

[i q . r

]ρ(0)

2+ V

∫d3r exp

[i q . r

]( g(r) − ρ(0)

2)

= N + V ( 2 π )3 ρ(0)2δ3(q) + V

∫d3r exp

[i q . r

]( g(r) − ρ(0)

2)

= N + N2 ( 2 π )3

Vδ3(q) + V

∫d3r exp

[i q . r

]( g(r) − ρ(0)

2)

= N + N2 ( 2 π )3

Vδ3(q) + V

4 πq

∫ ∞

0

dr r sin q r ( g(r) − ρ(0)2

)

(228)

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The first term represents the incoherent scattering. The second term representscoherent forward scattering. The integral in the last term is convergent andyields non-trivial information about the structure of the fluid.

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5 The Reciprocal Lattice

The reciprocal lattice vectors play an important role in describing the propertiesof a solid that has periodic translational invariance. Any property of the solid,whether scalar, vector or tensor, should have the same periodic translationalinvariance as the potential due to the charged nuclei. This means that, dueto the translational invariance, the physical property only needs to be specifiedin a finite volume, and this volume can then be periodically continued over allspace. The vectors of the reciprocal lattice play an important and special rolein the Fourier transform of the physical quantity.

The Reciprocal Lattice Vectors have dimensions of inverse distance and aredefined in terms of the direct primitive lattice vectors a1, a2 and a3. Theprimitive reciprocal lattice vectors, b(i), are defined via the scalar product

ai . b(j) = 2 π δj

i (229)

where the Kronecker delta function δji has the value 1 if i = j and is zero

if i 6= j. Thus, the primitive reciprocal lattice vectors are orthogonal to twoprimitive lattice vectors of the direct lattice.

The primitive reciprocal lattice vectors can be constructed via

b(1) = 2 πa2 ∧ a3

a1 . ( a2 ∧ a3 )

b(2) = 2 πa3 ∧ a1

a1 . ( a2 ∧ a3 )

b(3) = 2 πa1 ∧ a2

a1 . ( a2 ∧ a3 )(230)

where the last two expressions are found from the first by cyclic permutation ofthe labels (1, 2, 3). The denominator is just the volume of the primitive unit cell.

The reciprocal lattice consists of the points given by the set of vectors Qwhere

Q = m1 b(1) + m2 b

(2) + m3 b(3) (231)

and (m1,m2,m3) are integers. This set of vectors are the reciprocal lattice vec-tors. The reciprocal lattice vectors denote directions in the reciprocal lattice orare the normals to a set of planes in the direct lattice. In the latter case, asit shall be seen, the numbers (m1,m2,m3) are equivalent to Miller indices and,hence, are enclosed in round brackets.

——————————————————————————————————

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5.0.1 Exercise 22

Find the volume of the primitive unit cell of the reciprocal lattice.

——————————————————————————————————

5.1 The Reciprocal Lattice as a Dual Lattice

The reciprocal lattice vectors can be considered to be the duals of the directlattice vectors. This relation can be seen by expressing the primitive latticevectors aj in terms of the primitive reciprocal lattice vectors bi, via

aj =1

2 π

∑i

gj,i b(i) (232)

The quantity gi,j is given by the metric, since

aj . ak =1

2 π

∑i

gj,i b(i) . ak (233)

and sinceb(i) . ak = 2 π δi

k (234)

one hasgj,k = aj . ak (235)

Hence, gj,k is the metric tensor. The metric tensor expresses the length s of avector r in terms of its components xi along the basis vectors ai. That is, if

r =∑

i

xi ai (236)

then, for a constant metric, the length is given in terms of the components via

s2 =∑i,j

gi,j xi xj (237)

The metric tensor, when evaluated in terms of the parameters of the primitiveunit cell, is given by the matrix

( gi,j ) =

a21 a1a2 cosα3 a1a3 cosα2

a1a2 cosα3 a22 a2a3 cosα1

a1a3 cosα2 a2a3 cosα1 a23

(238)

The inverse transform is given by

b(i) = 2 π∑

k

gi,k ak (239)

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where the quantity gi,k is identified as the metric for the dual vectors. Since

aj =1

2 π

∑i

gj,i b(i)

=∑

i

gj,i

∑k

gi,k ak (240)

and asaj =

∑k

δkj ak (241)

one infers thatδkj =

∑i

gj,i gi,k (242)

Hence, the metric tensor is the inverse of the metric tensor for the dual vectors.

The volume of the unit cell, Vc, is given by

V 2c = det ( gi,j ) (243)

or

V 2c = a2

1 a22 a

23

(1− cos2 α1 − cos2 α2 − cos2 α3 + 2 cosα1 cosα2 cosα3

)(244)

The dual metric tensor is given by the inverse of the metric tensor, this isevaluated as the matrix

(gi,j)

=

a22a2

3(1−cos2 α1)V 2

c

a23a1a2(cos α1 cos α2−cos α3)

V 2c

a22a1a3(cos α1 cos α3−cos α2)

V 2c

a23a1a2(cos α1 cos α2−cos α3)

V 2c

a21a2

3(1−cos2 α2)V 2

c

a21a2a3(cos α2 cos α3−cos α1)

V 2c

a22a1a3(cos α1 cos α3−cos α2)

V 2c

a21a2a3(cos α2 cos α3−cos α1)

V 2c

a21a2

2(1−cos2 α3)V 2

c

(245)

This dual metric is also defined as

bi . bj = ( 2 π )2 gi,j (246)

From this, one can immediately find that the length of the reciprocal latticevectors are given by

b1 = 2 πa2 a3

Vc| sinα1 | (247)

etc., and the angle β3 between b(1) and b(2) is given by

cosβ3 =( cosα1 cosα2 − cosα3 )

| sinα1 sinα2 |(248)

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etc. On using the inverse transformation, the reciprocal lattice vectors are givenin terms of the primitive direct lattice vectors by

b(1) = 2πa21a

22a

23

V 2c

(a1

(1− cos2 α1)a21

+ a2

(cosα1 cosα2 − cosα3)a1a2

+ a3

(cosα1 cosα3 − cosα2)a1a3

)b(2) = 2π

a21a

22a

23

V 2c

(a1

(cosα1 cosα2 − cosα3)a1a2

+ a2

(1− cos2 α2)a22

+ a3

(cosα2 cosα3 − cosα1)a2a3

)b(3) = 2π

a21a

22a

23

V 2c

(a1

(cosα1 cosα3 − cosα2)a1a3

+ a2

(cosα2 cosα3 − cosα1)a2a3

+ a3

(1− cos2 α3)a23

)(249)

These expressions are equivalent to the expression in terms of the vector product,and they also satisfy the definitions of the primitive reciprocal lattice vectors

ai . b(j) = 2 π δj

i (250)

Any vector of the direct Bravais Lattice can be expressed as

R = n1 a1 + n2 a2 + n3 a3 (251)

A reciprocal lattice vector Q can also be written as

Q = m1 b(1) + m2 b

(2) + m3 b(3) (252)

where (m1,m2,m3) are integers. Any vector k in the reciprocal lattice can berepresented as a superposition of the reciprocal lattice vectors

k = µ1 b(1) + µ2 b

(2) + µ3 b(3) (253)

where the µi are non-integer. Thus, the scalar product of an arbitrary vector kin the reciprocal lattice and a Bravais Lattice vector R is evaluated as

k . R = 2 π(µ1 n1 + µ2 n2 + µ3 n3

)(254)

If k is a reciprocal lattice vector Q then the set of µi’s take on integer valuesmi, so that the scalar product reduces to

Q . R = 2 π(m1 n1 + m2 n2 + m3 n3

)(255)

As the sum of the products of integers is still an integer ( say M ), the Lauecondition can be expressed as

Q . R = 2 π M (256)

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for all R. Thus, the Reciprocal Lattice vectors satisfy the Laue condition. Thisrequirement is equivalent to the condition that the exponential phase factorgiven by

exp[i Q . R

]= 1 (257)

is unity for all Bravais Lattice vectors R.

The vectors Q form a Bravais Lattice in which the primitive lattice vectorscan be expressed in terms of the vectors b(i). Also, the reciprocal lattice of areciprocal lattice is the original direct lattice.

——————————————————————————————————

5.1.1 Exercise 23

Determine the primitive lattice vectors of the lattice that is reciprocal to the re-ciprocal lattice. How are they related to the vectors of the original direct lattice?

——————————————————————————————————

5.2 Examples of Reciprocal Lattices

Now some examples of reciprocal lattices are examined.

5.2.1 The Simple Cubic Reciprocal Lattice

In terms of Cartesian coordinates, the lattice vectors of the simple cubic directlattice are

a1 = a ex

a2 = a ey

a3 = a ez (258)

The reciprocal lattice vectors are determined to be

b(1) =2 πa

ex

b(2) =2 πa

ey

b(3) =2 πa

ez (259)

These are three orthogonal vectors which are oriented parallel to the directlattice vectors. The reciprocal lattice of the simple cubic direct lattice is also

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simple cubic.

5.2.2 The Body Centered Cubic Reciprocal Lattice

In terms of Cartesian coordinates, the primitive lattice vectors of the bodycentered cubic direct lattice are

a1 =a

2

(ex + ey − ez

)a2 =

a

2

(− ex + ey + ez

)a3 =

a

2

(ex − ey + ez

)(260)

The volume of the unit cell is Vc = | a1 . ( a2 ∧ a3 ) | = a3

2 .

The reciprocal lattice vectors are determined to be

b(1) =2 πa

(ex + ey

)b(2) =

2 πa

(ey + ez

)b(3) =

2 πa

(ex + ez

)(261)

The three reciprocal lattice vectors span the three-dimensional reciprocal lattice,but have different orientations from the direct lattice vectors. The reciprocallattice has cubic symmetry as can be seen by combining the three reciprocallattice vectors ( adding any two and subtracting the third ) to yield three or-thogonal vectors of equal magnitude. The reciprocal lattice of the body centeredcubic direct lattice is face centered cubic, with a conventional cell of side 4 π

a .

5.2.3 The Face Centered Cubic Reciprocal Lattice

In terms of Cartesian coordinates, the primitive lattice vectors of the face cen-tered cubic direct lattice are

a1 =a

2

(ex + ey

)a2 =

a

2

(ex + ez

)a3 =

a

2

(ey + ez

)(262)

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The reciprocal lattice vectors are determined to be

b(1) =2 πa

(ex + ey − ez

)b(2) =

2 πa

(ex − ey + ez

)b(3) =

2 πa

(− ex + ey + ez

)(263)

These are three non co-planar vectors, but have different orientations from thedirect lattice vectors. The reciprocal lattice has cubic symmetry. This can beseen by combining pairs of reciprocal lattice vectors, which yields three orthogo-nal vectors of equal magnitude. The reciprocal lattice of the face centered cubicdirect lattice is body centered cubic, with a conventional unit cell of side 4 π

a .

5.2.4 The Hexagonal Reciprocal Lattice

The hexagonal lattice has lattice vectors

a1 =a

2

( √3 ex + ey

)a2 =

a

2

(−√

3 ex + ey

)a3 = c ez (264)

The volume of the primitive unit cell is

Vc =√

32

a2 c (265)

The primitive reciprocal lattice vectors are

b(1) =2 πa

(1√3ex + ey

)b(2) =

2 πa

(− 1√

3ex + ey

)b(3) =

2 πc

ez (266)

Thus, the reciprocal lattice of the hexagonal lattice is its own reciprocal lattice,but is rotated about the z axis.

——————————————————————————————————

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5.2.5 Exercise 24

A trigonal lattice is defined by three primitive lattice vectors a1, a2 and a3, allof equal length a and an angle θ between any pair of these lattice vectors isa constant. Show that the three vectors a1 = [m,n, p], a2 = [p,m, n] anda3 = [n, p,m], referenced to an orthonormal basis represent a trigonal lattice.Prove that the reciprocal lattice of a trigonal lattice is another trigonal lattice.

——————————————————————————————————

5.3 The Brillouin Zones

The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. Thatis, the first Brillouin zone is a volume of a unit cell in the reciprocal lattice. Thiscell is found by first connecting a central reciprocal lattice point O to all theother reciprocal lattice points via the reciprocal lattice vectors Q

i. Secondly,

these connecting lines are bisected by planes. The equations for the set of theseplanes are given by (

k − 12Q

i

). Q

i= 0 (267)

for each i. The smallest volume around the origin O enclosed by these planesis the first Brillouin zone. That is, the first Brillouin zone consists of all theregions of space that can be reached from O without crossing any of the planes.

The regions of the entire reciprocal lattice can be partitioned off into Bril-louin zones of higher order. The planes defined by eqn(267) form a set ofboundaries for the set of Brillouin zones. The n-th order Brillouin zone consistsof the regions of k space that is accessed from the origin by crossing a minimumof n− 1 boundaries.

Although the n-th order Brillouin zone exists in the form of isolated regionsof k space, these regions can be brought together to make a contiguous volumeby translating the isolated regions through appropriately chosen reciprocal lat-tice vectors Q

i.

5.3.1 The Simple Cubic Brillouin Zone

The first Brillouin zone of the simple cubic direct lattice is a simple cube cen-tered at the origin O. The sides of the cube are 2 π

a and the Brillouin zone hasa volume of ( 2 π

a )3 which, when given in terms of the volume of the unit cell ofthe direct lattice, is equal to 8 π3

Vc.

Points of high symmetry are usually given special names. Points interior tothe first Brillouin zone are designated by Greek letters and those on the surface

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are designated by Roman letters. The center of the zone (0, 0, 0) is denoted byΓ, the vertex of the cube 2 π

a ( 12 ,

12 ,

12 ) is called R. The center of the x face lo-

cated at 2 πa ( 1

2 , 0, 0) is called X, and the mid-points of the edges at 2 πa ( 1

2 ,12 , 0)

are denoted by M .

Points on high symmetry lines are also given special designations. The pointsbetween M and X are denoted by Z. The points on the lines between R andX are denoted by S, the points on the lines between R and M are denoted byT . The points on high symmetry lines in the interior have the following desig-nations: the points between Γ and M are denoted by Σ, the points between Γand X by ∆, the points on lines between Γ and R are denoted by Λ.

5.3.2 The Body Centered Cubic Brillouin Zone

The first Brillouin zone for the body centered direct lattice is a dodecahedralrhombohedron. The cell is centered at the origin Γ = (0, 0, 0). The vertices arelocated either on the positive or negative Cartesian axes at H = 2 π

a (1, 0, 0)or at diagonal points P = 2 π

a ( 12 ,

12 ,

12 ). The centers of the faces are denoted

by N = 2 πa ( 1

2 ,12 , 0).

Points on the high symmetry lines joining P and H are denoted by F . Otherspecial points are: G which are on the high symmetry line between N and H, orD between P and N . The names of interior points on high symmetry lines areΣ which are intermediate between Γ and N , ∆ which are intermediate betweenΓ and H, and Λ which are intermediate between Γ and P .

5.3.3 The Face Centered Cubic Brillouin Zone

The Brillouin zone for the face centered cubic has twenty four vertices atW = 2 π

a (1, 12 , 0) and has six square faces which have centers on the Carte-

sian axes. The centers of the square faces are denoted by X and are located at2 πa (0, 1, 0). These squares are connected to eight hexagonal faces with centers

at the L points L = 2πa ( 1

2 ,12 ,

12 ). The mid-points of the edges joining two

hexagonal faces are at 2 πa ( 3

4 ,34 , 0), and are denoted by K. The mid-points of

the edges between the square and hexagonal faces are denoted by U .

The points on the lines between X and U contained on the square faces aredenoted by S while those between X and W are denoted by Z. The points onthe high symmetry lines between L and W on the hexagonal faces are denotedby Q. The points on the high symmetry lines between Γ and K are denoted byΣ: the points on the lines between Γ and X are denoted by ∆, and the pointson the line running through Γ and L are known as Λ.

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5.3.4 The Hexagonal Brillouin Zone

The Brillouin zone for the hexagonal lattice is hexagonal. The upper and lowerfaces are hexagons. The hexagonal face centers are at A = 2 π

a (0, 0, a2 c ). The

vertices are at the H points, H = 2 πa ( 1√

3, 1

3 ,a

2 c ). The centers of the verticalrectangular faces are denoted by M and M = 2 π

a ( 1√3, 0, 0). The mid-points

of the horizontal edges are denoted by L and L = 2 πa ( 1√

3, 0, a

2 c ).

Some of the interior high symmetry points are: Γ the zone center, Σ whichare located on the high symmetry lines Γ M , ∆ are the points on the lines Γ A.

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6 Electrons

The types of states of single electrons in the potentials produced by the crys-talline lattice are discussed in the next three chapters. For simplicity, we shallfirst implicitly assume that the effect of the Coulomb interactions between elec-trons can be neglected. The neglect of electron - electron interactions is un-justified, as can be seen by considering the electrical neutrality of solids. Thecondition of electrical neutrality leads to the electron charge density being com-parable with the charge density due to the lattice of nuclei or ions. Thus, thestrength of the interactions between the electrons is expected to be comparableto the strength of the potential due to the nuclei. A simple order of magni-tude estimate, based upon the typical linear dimensions of a unit cell a0 ∼ 2Angstroms, leads to the average value of e2

r ∼ 3 eV for both these interactionenergies. Nevertheless, as a discussion of the effect of pseudo-potentials reveals,for most metals, the effect of the periodic potential of the lattice may be consid-ered as small. The small value of the effective potential (or pseudo-potential)leads to a useful approximation namely, that of nearly free electrons. The effectof the finite strength of electron-electron interactions is a more complex issue,and is not yet fully understood. The density functional method in principleprovides a method of evaluating the ground state electron density including theeffect of electron-electron interactions. However, the density functional methoddoes not describe the excited states. The effect of the electron-electron interac-tions is that of disturbing the electron density around any excited electron. Onassuming that the interactions can be treated as a small perturbation, it can beshown that most of the effects of electron-electron interactions on the low en-ergy excited electrons merely involve the dressing of the single excited electronthereby, forming a quasi-particle excitation. That is, the effect of the electronexcitation of the surrounding gas of electrons can be absorbed as renormaliza-tions of the properties of the single-electron excitations. This feature can leadto the low-temperature properties of the electronic system being determined bythe gas of quasi-particles, which has the same form as a non-interacting gasof electrons. Systems where this simplification occurs are known as LandauFermi-liquids. The effect of electron-electron interactions will be delayed to alater chapter.

7 Electronic States

In describing electronic states in metals first, the nature of the many-electronwave function and its decomposition into the sum of anti-symmetric productsof one-electron wave functions shall be described. Then, the general propertiesof the one-electron basis wave functions shall be discussed. The one-electronwave functions or Bloch functions, are taken to be eigenfunctions of a suitablenon-interacting Hamiltonian in which the potential has the periodicity of theunderlying Bravais Lattice.

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7.1 Many Electron Wave Functions

The energy of the electrons in a solid can be written as the sum of the kineticenergies, the ionic potential energy acting on the individual electrons, and theinteraction potential between pairs of electrons. Thus, for a system with Ne

electrons, the Hamiltonian can be written as

H =i=Ne∑i=1

(− h2

2 m∇2

i + Vions(ri))

+12

∑i 6=j

e2

| ri − rj |(268)

where ri denotes the position of the i-th electron, Vions is the potential due tothe lattice of ions, and the last term is the pair-wise interaction between theelectrons. This Hamiltonian can be separated into two sets of terms,

H = H0 + Hint (269)

where

H0 =i=Ne∑i=1

(− h2

2 m∇2

i + Vions(ri))

(270)

is the sum of one-body Hamiltonians acting on the individual electrons, and theinteraction term is given by the sum of two body terms

Hint =12

∑i 6=j

e2

| ri − rj |(271)

Since electrons are indistinguishable, the Hamiltonian must be symmetric un-der all permutations of the indices i labelling the electrons. Also, the modulussquared wave function must be invariant under all possible permutations of theelectron labels. An arbitrary permutation of the labels can be built up throughsequentially permuting pairs of labels.

The permutation operator Pi,j is defined as the operator which interchangesthe indices i and j labelling a pair of otherwise indistinguishable particles. Thus,if

Ψ(r1, . . . ri, . . . rj , . . . rNe)

for an arbitrary Ne particle wave function, the permutation operator can bedefined as

Pi,j Ψ(r1, . . . ri, . . . rj , . . . , rNe) = Ψ(r1, . . . rj , . . . ri, . . . rNe

)(272)

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Since the Hamiltonian is symmetric under interchange of the indices i and jlabelling any two identical particles, the permutation operators commute withthe Hamiltonian

[ Pi,j , H ] = 0 (273)

Likewise, the permutation operators must also commute with any physical op-erator A

[ Pi,j , A ] = 0 (274)

otherwise measurements of the quantity A could lead to the particles being dis-tinguished.

Since the Hamiltonian commutes with all the permutation operators, one canfind simultaneous eigenstates of the Hamiltonian H and all the permutationoperators Pi,j . The energy eigenstates Ψ corresponding to physical states ofindistinguishable particles must satisfy the equations

H Ψ = E ΨPi,j Ψ = pi,j Ψ (275)

where pi,j are the eigenvalues of the permutation operators Pi,j . As permutatingthe same pair of particle indices twice always reproduces the initial wavefunc-tion, one has

Pi,j2 = I (276)

where I is the identity operator. Thus, the eigenstates of the permutationoperators satisfy the two equations

Pi,j2 Ψ = p2

i,j Ψ= Ψ (277)

Hence, the eigenvalues of the permutation operators must satisfy

pi,j2 = 1 (278)

orpi,j = ± 1 (279)

Thus, the Ne particle wave functions have the property that, under any permu-tation of a single pair of identical particles which are labelled by i and j, theun-permuted and permuted wave functions are related by

Ψ(r1, . . . rj , . . . ri, . . . rNe) = ± Ψ(r1, . . . ri, . . . rj , . . . rNe

)(280)

The upper sign holds for boson particles and the lower sign holds for fermions.Also, since pi,j is a constant of motion, the nature of the particles does notchange with respect to time. Electrons are fermions and, thus, the wave func-tion must always be anti-symmetric with respect to the interchange of any pair

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of electron labels. Furthermore, the modulus square of the many-electron wavefunction must be invariant under all possible permutations of the electron labels.

The energy eigenfunction for the system of Ne electrons can be written as

Ψ(r1, r2, . . . rNe) (281)

The many-electron energy eigenstates Ψ usually cannot be found exactly. How-ever, they can be expressed in terms of a superposition formed from a com-plete set of many-electron eigenfunctions Φα1,α2,...,αNe

(r1, r2, . . . , rNe) of the

one-particle Hamiltonian H0. The subscript αi represents the complete set ofquantum numbers (including spin) which completely describes the state of asingle fermion state.

H0 Φα1,α2,...αNe(r1, r2, . . . rNe

) = E0 Φα1,α2,...αNe(r1, r2, . . . rNe

)(282)

This many-electron eigenfunction is interpreted as representing the state inwhich the Ne electrons are distributed in the set of single-electron states withthe specific quantum numbers α1, α2, . . . αNe

. The basis states are orthonormaland so satisfy the relations

Ne∏j=1

(∫V

d3rj

)Φ∗β1,β2,...(r1, . . . , rNe

) Φα1,α2,...(r1, . . . , rNe) = δα1,β1 δα2,β2 . . .

(283)where we have assumed that the sets of single-electron eigenvalues have beenordered. Since the basis is complete, the exact many-body eigenstates of thefull Hamiltonian H can be written as a linear superposition of the complete setof basis functions

Ψ(r1, r2, . . . , rNe) =

∑α1,α2,...,αNe

Cα1,α2,...αNeΦα1,α2,...,αNe

(r1, r2, . . . rNe)

(284)where the sum over the set of αi runs over all possible distributions of the Ne

electrons in the set of all the single-electron states. The coefficients Cα1,α2,...,αNe

have to be determined. The coefficients represent the probability amplitudesthat electrons occupy the set of single-electron states labelled by α1, α2, . . . , αNe

.

The set of many-electron basis functions Φα1,α2,...αNe(r1, r2, . . . rNe

) can beexpressed directly in terms of the one-electron wave functions φα(r). First,note that the non-interacting Hamiltonian H0 can be decomposed as the sumof Hamiltonians which only act on the individual electrons

H0 =i=Ne∑i=1

Hi (285)

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where the one-particle non-interacting Hamiltonian is given by

Hi = − h2

2 m∇2

i + Vions(ri) (286)

This one-particle Hamiltonian has eigenstates, φβ(ri), which satisfy the eigen-value equation

Hi φβ(ri) = Eβ φβ(ri) (287)

The many-particle non-interacting Hamiltonian H0 has eigenfunctions whichare the products of Ne one-particle eigenfunctions φβ(r)

χ(r1, α1; r2, α2; . . . rNe, αNe

) = φα1(r1) φα2(r2) . . . φαNe(rNe

) (288)

and the non-interacting energy eigenvalue E0 for the many-particle state is givenas the sum of the one-electron energy eigenvalues Eαi

that are occupied by theelectrons

E0 =i=Ne∑i=1

Eαi(289)

However, the wave functions χ(r1, α1; r2, α2; . . . rNe, αNe

) do not represent phys-ical wave functions since each of the single particle states with quantum numbersα1, α2, . . . αNe

are occupied by the respective electron labelled by r1, r2, . . . rNe

and, hence, the electrons have been labelled. As the electrons are indistinguish-able, it is impermissible to distinguish them by this type of labelling. Thus,physical wave functions should contain terms which are related by all the pos-sible relabelling of the indices of the particles.

Electrons are fermions and, therefore, they have wave functions which areanti-symmetric under the interchange of any pair of particles. The proper ba-sis set of the many-electron wave function Φ must correspond to all possiblepermutations of the single-particle indices. The proper anti-symmetrized wavefunction Φα1,α2,...,αNe

is given by the Slater determinant

Φα1...αNe= 1√

Ne!

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

φα1(r1) φα1(r2) . . . φα1(ri) . . . φα1(rNe)

φα2(r1) φα2(r2) . . . φα2(ri) . . . φα2(rNe)

......

......

φαi(r1) φαi

(r2) . . . φαi(ri) . . . φαi

(rNe)

......

......

φαNe(r1) φαNe

(r2) . . . φαNe(ri) . . . φαNe

(rNe)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣The normalization is 1√

Ne!as there are Ne! terms in the determinant, corre-

sponding to the Ne! permutations of the electron labels.

The anti-symmetric wave function has the property that if there are two ormore particles in the same one-particle eigenstate, say αi = αj , then the wavefunction vanishes. This can be seen by noting that two rows of the determinant

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are identical and, hence, the determinant vanishes. As the wave function van-ishes if two or more electrons occupy the same single particle eigenstate, there isno state in which a one-particle eigenstate is occupied by two or more electrons.The anti-symmetric nature of the fermion wave function directly leads to thePauli exclusion principle. The Pauli exclusion principle can be stated as ”nounique single particle state can be occupied by two or more electrons.” For elec-trons which have spin one half, a single particle state is uniquely specified onlyif the spin quantum number is also specified. The single particle wave functionφα(r) should be supplemented by the spinor χσ. That is, the single-electronwave function should be replaced by the product

φα(r) → φα(r) χσ (290)

where χσ is a spinor or a normalized two component column vector. The spinindex σ can be considered to label an eigenstate of a component of an arbitrarysingle-electron spin operator, and the label σ should be considered as analogousto the single particle eigenvalue α. A complete set of labels for the single-electron state are given by α and σ. An arbitrary spinor χσ can be decomposedas the linear superposition of two basis spinors χ±

χσ =∑±

γ± χ± (291)

where the normalization condition is given by∑±

| γ± |2 = 1 (292)

The two basis spinors χ± are usually denoted by the two component columnvectors

χ+ =(

10

)(293)

corresponding to an eigenstate of the Pauli matrix σz with spin up and

χ− =(

01

)(294)

corresponding to the eigenstate with spin down. Thus, the arbitrary state canbe written as

χσ =(γ+

γ−

)(295)

In this representation, the two components of an arbitrary spinor, χσ, representthe internal degree of freedom of the spin and, thus, are analogous to the de-gree of freedom represented by r in the position representation. The complexconjugate wave function should be replaced by

φ∗α′(r) → χTσ′ φ

∗α′(r) (296)

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which contains χTσ′ which is the complex conjugated transpose of the spinor

states given by the two dimensional row matrices

χTσ′ =

(γ′+

∗ γ′−∗ ) (297)

In the situations where the electron spin has to be explicitly considered, thesereplacements lead to the inner product of two one-electron states not only in-volving the integration of the product φ∗α′(r) φα(r) over the electron’s position rbut also automatically involves evaluating the matrix elements of the individualelectron’s row spinor state χT

σ′ with the column spinor state χσ.

The probability density ρ(r) for finding an electron at position r can be ob-tained from the matrix elements of the many-electron wave function Ψ(r1; r2, . . . rNe

)with the one-electron density operator ρ. The one-electron density operator isgiven by a dirac delta function

ρ(r) =i=Ne∑i=1

δ3( r − ri ) (298)

The density ρ(r) is evaluated as

ρ(r) = Ne

∫V

d3r1

∫V

d3r2 . . .

∫V

d3rNeδ( r − r1 ) | Ψ(r1; r2, . . . rNe

) |2

(299)Thus, the trace over the positions particles can be evaluated by integrating overall but one of the particles positions

ρ(r) = Ne

∫V

d3r2

∫V

d3r3 . . .

∫V

d3rNe| Ψ(r; r2, . . . rNe

) |2 (300)

The matrix elements of the spin states has also to be taken. The resulting elec-tron density is normalized to Ne.

The probability density for finding an electron at position r and anotherelectron at r′ is a correlation function ρ(r, r′) which is given by the matrixelements of the operator

ρ(r, r′) =∑

i

∑j 6=i

δ3( r − ri ) δ3( r′ − rj ) (301)

The resulting expression for the two-particle density is found by integrating overthe positions of all the electrons except two

ρ2(r, r′) = Ne ( Ne − 1 )∫

V

d3r3

∫V

d3r4 . . .

∫V

d3rNe| Ψ(r; r′; . . . rNe

) |2

(302)This two-particle density correlation function is normalized to twice the numberof pairs of electrons, Ne ( Ne − 1 ).

——————————————————————————————————

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7.1.1 Exercise 25

Evaluate the single-particle density and two-particle density correlation functionfor a many-particle basis wave function Φα1,α2,...αNe

given by a single Slater de-terminant of single-particle wave functions φα(r).

——————————————————————————————————

The properties of the single-electron wave functions φα(ri), that are to beused in forming the many-particle basis functions Φα1,α2...αNe

(r1, r2, . . . rNe) as

Slater determinants, are discussed in the next chapter. In the following, theelectron labels i in the one-electron wave functions are omitted.

7.2 Bloch’s Theorem

Bloch’s theorem describes the properties of the one-electron states φα(r) whichare eigenstates of the one-electron Hamiltonian with a periodic potential. Anelectron in the solid experiences a periodic potential that has the periodicityof the underlying lattice of ions. In particular, the potential is invariant undertranslation through any Bravais lattice vector Ri

Vions(r −Ri) = Vions(r) (303)

General properties of the solution of the Schrodinger equation for a singleelectron in a solid can be found from the periodicity of Vions(r). If the electron-electron interactions are neglected, the independent electrons obey the one-particle Schrodinger equation with the periodic potential,

H φα(r) =(− h2

2 m∇2 + Vions(r)

)φα(r) = Eα φα(r) (304)

For an infinite solid, the physically acceptable solutions of this equation areknown as the Bloch wave functions. The energies of the Bloch states are usuallylabelled by two quantum numbers n and k, instead of by α. The one-dimensionalcase, where the values of k were restricted to real values, was investigated byKramers (H.A. Kramers, Physica 2, 483 (1935)).

Bloch’s theorem applies to the eigenstates of the one-particle Hamiltonian,

H =(− h2

2 m∇2 + Vions(r)

)(305)

in which the potential has the symmetry

Vion(r −Ri) = Vions(r) (306)

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for all lattice vectors Ri in the Bravais Lattice. Bloch theorem states that theeigenfunctions can be found in the form

φn,k(r) = exp[i k . r

]un,k(r) (307)

where the function un,k is invariant under the translation through any BravaisLattice vector

un,k(r −Ri) = un,k(r) (308)

Bloch’s theorem asserts that the periodic translational symmetry manifestsitself in the transformation of the wave function

φn,k(r −Ri) = exp[− i k . Ri

]φn,k(r) (309)

Thus, a translation of the wave function through a reciprocal lattice vector onlyshows up through the presence of an exponential factor. Furthermore, if thewave vector k is real, then the electron density for the Bloch state is identicalfor each unit cell in the crystal. This prevents the wave function from divergingat the boundaries of the solid.

The proof of Bloch’s theorem is based on the consideration of the translationoperator TR which, when acting on an arbitrary function f(r), has the effect oftranslating it through a Bravais lattice vector R

TR f(r) = f(r −R) (310)

This translation operator can be applied to the wave function H φ(r) whichyields

TR H φ(r) = H(r −R) φ(r −R)

= H(r) φ(r −R)= H TR φ(r) (311)

Thus, the Hamiltonian commutes with the translation operator which producesa translation through a Bravais lattice vector,

[ H , TR ] = 0 (312)

This means that it is possible to find simultaneous eigenstates of both TR and H.

Furthermore, the translation operators corresponding to translations throughdifferent lattice vectors commute. This can be shown by successive translationsTR and TR′ , which yields

TR TR′ φ(r) = φ(r −R′ −R)

φ(r −R−R′) = TR′ TR φ(r) (313)

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Thus, the translation operators commute

[ TR , TR′ ] = 0 (314)

This proves that the wave functions can be chosen to be simultaneous eigenstatesof the Hamiltonian and all the translation operators that produce translationsthrough Bravais lattice vectors. The Bloch functions are chosen such that theysatisfy

H φ(r) = E φ(r)

TR φ(r) = c(R) φ(r) (315)

and, thus, are the simultaneous eigenstates of H and all the TR.

The translation operators can be compounded as

TR′ TR φ(r) = φ(r −R−R′)

= TR+R′ φ(r) (316)

When two translation operators are successively applied to the simultaneouseigenfunctions of the translation operators, it may be re-interpreted in terms ofthe compound translation

TR′ TR φ(r) = c(R′) c(R) φ(r)

= TR+R′ φ(r) = c(R+R′) φ(r) (317)

This shows that the products of two eigenvalues of different translation operatorsgives the eigenvalue of the compound translation

c(R′) c(R) = c(R+R′) (318)

Since a general Bravais lattice vector can be expressed as the sum

R = n1 a1 + n2 a2 + n3 a3 (319)

where (n1, n2, n3) are integers, a general eigenvalue can be decomposed in termsof products

c(R) = c(a1)n1 c(a2)

n2 c(a3)n3 (320)

Hence, on defining

c(a1) = exp[− i 2 π x1

]

c(a2) = exp[− i 2 π x2

]

c(a3) = exp[− i 2 π x3

](321)

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one can define a vector k via

k =(x1 b

(1) + x2 b(2) + x3 b

(3)

)(322)

With these definitions, the eigenvalue of the translation operator can be ex-pressed in terms of the k vector as

c(R) = exp[− i k . R

](323)

Thus, the eigenvalue equation for the translation operator is expressed as

TR φ(r) = φ(r −R)= c(R) φ(r)

= exp[− i k . R

]φ(r) (324)

which completes the proof of Bloch’s theorem.

The wave functions which are simultaneous eigenfunctions of the energy andthe periodic translation operators are the Bloch functions. The Bloch functions,φn,k(r), are labelled by the translation quantum number k and a quantumnumber n that pertains to the single particle energy eigenvalue En,k. It shouldbe noted that Bloch’s theorem does not guarantee that the quantity k is real.Since k is the quantum number associated with the eigenvalue of the operatorwhich translates through a Bravais lattice vector

TR φn,k(r) = φn,k(r −R)

= exp[− i k . R

]φn,k(r) (325)

then it should be clear that as

exp[i Q . R

]= 1 (326)

the eigenvalue labelled by k is identical to the eigenvalue labelled by k+Q. Thismeans that the two wave vectors can be identified, i.e., k + Q ≡ k. Thus, theBloch wave vector when translated through a reciprocal lattice vector Q leadsto an equivalent wave vector. Furthermore, if the convention

φn,k+Q(r) = φn,k(r) (327)

is adopted, then the eigenvalues must be related through

En,k+Q = En,k (328)

Thus, if k is real, any k value can be restricted to lie within one unit cell ofreciprocal space (F. Bloch, Zeit. fur Physik, 52, 555 (1928)).

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7.3 Boundary Conditions

Bloch’s theorem does not ensure that the wave vector k is real. In fact, forsurface states or impurity states, k may become imaginary. However, for bulkstates the wave vector is real, as can be ascertained by applying appropriateboundary conditions.

Consider a crystalline solid of finite size which has the same shape as theprimitive unit cell of the Bravais Lattice but with dimensions L1 = N1 | a1 |,L2 = N2 | a2 | and L3 = N3 | a3 | along the three primitive axes. The solidthen contains N = N1 N2 N3 lattice points.

Born-von Karman or periodic boundary conditions are imposed on the wavefunction

φn,k(r −Ni ai) = φn,k(r) for i = 1 , 2 or 3 . (329)

The periodic boundary conditions ensure that the electronic states are homoge-neous bulk states and are unmodified in the vicinity of the surface of the solid.Application of Bloch’s theorem yields the condition

φn,k(r) = φn,k(r −Ni ai)

= exp[− i Ni k . ai

]φn,k(r) for i = 1 , 2 or 3

(330)

Thus, the periodic boundary conditions are fulfilled if the wave vectors k satisfythe conditions

exp[− i Ni k . ai

]= 1 (331)

Since k can be written in terms of the primitive reciprocal lattice vectors, b(i),via

k =i=3∑i=1

xi b(i) (332)

and as ai . b(j) = 2 π δj

i , then the periodic boundary conditions require that

exp[− i 2 π Ni xi

]= 1 for i = 1 , 2 or 3

(333)

Thus, the components xi must be in the form of ratios

xi =mi

Ni(334)

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where mi are integers. This proves that the general Bloch wave vector k is areal vector, and the k vectors have the general form

k =i=3∑i=1

mi

Nib(i) (335)

Since Ni 1, the k vectors form a dense set of points in reciprocal space.

The properties of a solid can be expressed in terms of summations over theelectronic states. Since each state can be expressed in terms of the discretek quantum number, the summation are over a dense set of k vectors. Thesummation over a dense set of k vectors can be represented in terms of anintegral over the energy, weighted by the density of states. From the form of k,the volume of k space per allowed k value is

∆3k =b(1)

N1.

(b(2)

N2∧ b(3)

N3

)=

1N

b(1) .

(b(2) ∧ b(3)

)(336)

As the volume of the Brillouin zone is given by

b(1) .

(b(2) ∧ b(3)

)(337)

the volume of one state is 1N times the volume of the Brillouin zone. This implies

that the number of allowed k values within the Brillouin zone is equal to thenumber of unit cells in the crystal. The volume ∆3k associated with a Blochstate is given by

∆3k =1N

( 2 π )3

a1 . ( a2 ∧ a3 )

=1N

( 2 π )3

Vc(338)

Now, since the volume of the solid V is N times the volume of the cell Vc,

V = N Vc (339)

then the volume of k space associated with each Bloch state is

∆3k =( 2 π )3

V(340)

Hence, in the continuum limit, the number of one-electron states (per spin) inan infinitesimal volume of phase d3k is given by

V

( 2 π )3d3k (341)

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7.4 Plane Wave Expansion of Bloch Functions

Any function obeying Born-von Karman boundary conditions can be expandedas a Fourier series. This implies that the Bloch functions can also be expandedas

φn,k(r) =∑

q

Cq exp[i q . r

](342)

where the wave vectors q are to be related to k. From Bloch’s theorem, theBloch functions can also expressed as

φn,k(r) = exp[i k . r

]un,k(r) (343)

Since un,k(r) has periodic translational invariance, it only contains reciprocallattice vectors Q. The Fourier series expansion of the periodic function is

un,k(r) =∑Q

un,k(Q) exp[i Q . r

](344)

and the inverse transform is given by the integral

un,k(Q) =1V

∫V

d3r un,k(r) exp[− i Q . r

](345)

On comparing the above two forms for the Bloch functions, one has

φn,k(r) =∑

q

Cq exp[i q . r

]

=∑Q

un,k(Q) exp[i ( k + Q ) . r

](346)

Thus, the allowed q values in the Bloch wave functions are equal to k, modulo areciprocal lattice vector. Furthermore, the Cq are equal to the Fourier compo-nents un,k(Q). Next, it shall be shown how the Cq can be determined directlyfrom the Schrodinger equation which contains the periodic potential Vions(r).

The Bloch functions can be found by solving the Schrodinger equation wherethe Hamiltonian contains the periodic potential Vions(r). The periodic potentialalso has a Fourier series expansion

Vions(r) =∑Q

Vions(Q) exp[i Q . r

](347)

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and the inverse transform is given by the integral

Vions(Q) =1V

∫V

d3r Vions(r) exp[− i Q . r

](348)

Furthermore, since Vions(r) is real, the Fourier transform of the potential hasthe symmetry

Vions(−Q) = V ∗ions(Q) (349)

This follows from taking the complex conjugate of the Fourier series expansionof Vions(r). A second condition on the Fourier expansion coefficients existsfor crystals which have an inversion symmetry around a suitable origin. Theinversion symmetry implies that the potential is symmetric

Vions(r) = Vions(−r) (350)

and this implies that the Fourier transform of the potential has the property

Vions(Q) = Vions(−Q) = V ∗ions(Q) (351)

The expansion coefficients Cq in the Bloch function are found by substitutingthe Fourier series into the energy eigenvalue equation. The kinetic energy termis evaluated from

p2

2 mφn,k(r) = − h2

2 m∇2 φn,k(r)

=∑

q

h2 q2

2 mCq exp

[i q . r

](352)

The potential term in the energy eigenvalue equation has the form of a convo-lution when expressed in terms of the Fourier Transforms

Vions(r) φn,k(r) =∑q′

∑Q′

Vions(Q′) Cq′ exp[i

(q′ + Q′

). r

](353)

The form of the energy eigenvalue equation is simplified if q′ is expressed asq′ = q − Q′, so

Vions(r) φn,k(r) =∑

q

∑Q′

Vions(Q′) Cq−Q′ exp[i q . r

](354)

Then, the energy eigenvalue equation takes the form

∑q

( [h2 q2

2 m− E

]Cq +

∑Q′

Vions(Q′) Cq−Q′

)exp

[i q . r

]= 0 (355)

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The wave vectors q are expressed as q = k − Q so that k is always locatedwithin the first Brillouin zone. On equating the coefficients of the plane waveswith zero, one finds the matrix eigenvalue equation(

h2 ( k − Q )2

2 m− E

)Ck−Q +

∑Q′

Vions(Q′) Ck−Q−Q′ = 0 (356)

The reciprocal lattice vector is transformed as Q′ → Q” = Q′ + Q in thesecond term, leading to an infinite set of coupled equations(

h2 ( k − Q )2

2 m− E

)Ck−Q +

∑Q”

Vions(Q”−Q) Ck−Q” = 0

(357)

Thus, because of the periodicity of the potential, the Bloch functions only con-tain Fourier components q that are connected to k via reciprocal lattice vectors.For fixed k, the set of equations couple Ck to all the Ck−Q via the Fouriercomponent of the potential Vions(Q). In principle, the set of infinite coupledalgebraic equations (357) could be used to find the coefficients Ck−Q and theeigenvalue En,k. The Bloch function is expressed in terms of the coefficientsCk−Q as

φn,k(r) =∑Q

Ck−Q exp[i ( k − Q ) . r

]

= exp[

+ i k . r

] ∑Q

Ck−Q exp[− i Q . r

](358)

Using this, the Bloch function can be expressed in terms of the periodic functionun,k(r) via

un,k(r) =∑Q

Ck−Q exp[− i Q . r

](359)

In order to make this approach tractable, it is necessary to truncate the infiniteset of coupled equations (357) to a finite set. However, if this set of equations aretruncated, it would require approximately 103 to 106 plane wave componentsbefore convergence is attained in three dimensions. Therefore, other methodsare frequently used.

7.5 The Bloch Wave Vector

The Bloch wave vector k plays a role similar to that of the momentum of afree electron. In fact, it reduces to the momentum quantum number in the

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limit Vions(r) → 0. However, for a non-zero crystal potential, k is not equalto the eigenvalue of the electron momentum p = − i h ∇ since it differsby amounts that are determined by the reciprocal lattice vectors Q and thecoefficients Ck+Q. That is,

p φn,k(r) = h k φn,k(r) − i h exp[i k . r

]∇ un,k(r) (360)

Thus, h k is known as the crystal momentum.

The crystal momentum can always be chosen to be in the first Brillouin zoneby making the transformation

k = k′ + Q (361)

On substituting this relation, the Bloch function is re-written as

exp[i k . r

]un,k(r) = exp

[i k′ . r

] (exp

[i Q . r

]un,k(r)

)= exp

[i k′ . r

]un,k(r) (362)

where

un,k(r) = exp[i Q . r

]un,k(r) (363)

is identified as a periodic function of the type that is used in Bloch’s theorem.The new function un,k(r) transforms like un,k(r) since it has the periodicity ofthe Bravais Lattice as

exp[i Q . R

]= 1 (364)

Due to the periodic translational symmetry, the eigenvalue problem can bereduced to finding a solution for the periodic function un,k(r) in a single cell ofthe lattice. The total number of energy eigenfunctions must correspond to thenumber of electron states originating from each atom in the crystal, and theremay be many basis atoms in the unit cell. As an isolated atom is expectedto have an infinite number of excited levels, and as the number of different kpoints in the Brillouin zone is equal to the number of primitive cells in thecrystal, there must be infinitely many energy eigenfunctions with fixed k. Thedifferent one-electron states with fixed k are distinguished by the index n. Theenergy En,k is a continuous function of k, forming energy bands. This is seen byexamination of the eigenvalue equation, when the Bloch functions are expressedas

φn,k(r) = exp[i k . r

]un,k(r) (365)

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This procedure leads to the energy eigenvalue equation

Hk un,k(r) =[

h2

2 m

(− i ∇ + k

)2

+ Vions(r)]un,k(r)

= En,k un,k(r) (366)

Due to the Born-von Karman boundary conditions, each energy band in theBrillouin zone contains N different states. The different k values are not partof a continuum but form a discrete dense set of points. The energy eigenvaluesEn,k, therefore, although a continuous function of k, only exist at the finite setof points.

7.6 The Density of States

A physical quantity A may be expressed in terms of the quantities An,k associ-ated with the individual electrons in each of the occupied Bloch states (n, k) inthe solid. That is, the quantity A is given by

A = 2∑n,k

An,k (367)

where the sum runs over each level (n, k) that is occupied by an electron. Thefactor 2 originates from the spin degeneracy. Since the different k states aredense and uniformly distributed in the Brillouin zone, the summation may berepresented by an integration. The volume ∆3k of phase space associated witha Bloch state is given by

∆3k =( 2 π )3

V(368)

The quantity A is expressed as the integral

A = 2V

( 2 π )3∑

n

∫En,k <EF

d3k An,k (369)

where the integration over k runs over the volume of occupied states in the firstBrillouin zone. Thus, for the partially filled bands the integration runs overa volume of k space enclosed by a surface of constant energy EF , and for thecompletely filled bands it runs over the entire Brillouin zone.

The integration over k space may be converted into an integral over theenergy E, by introducing the one-electron density of states ρ(E). The densityof states per spin is defined by the integration over the dirac delta function

ρ(E) = V∑

n

∫d3k

( 2 π )3δ( E − En,k ) (370)

If the quantity An,k only depends on (n, k) through En,k, then

An,k = A(E) (371)

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so the quantity A can be represented as an integral over the density of states

A = 2∫ EF

−∞dE ρ(E) A(E) (372)

The density of states ρ(E) can be calculated by noting that the infinitesimalintegral

∫ E+∆E

Eρ(E) dE ∼ ρ(E) ∆E is the number of states in the energy

range between E and E + ∆E, or the allowed number of k values between Eand E + ∆E in each of the energy bands. Thus, on integrating over an energyrange ∆E and using the definition of the density of states in terms of the diracdelta function, one finds

ρ(E) ∆E ∼ V∑

n

∫ E+∆E

E

dE

∫d3k

( 2 π )3δ( E − En,k )

=V

( 2 π )3∑

n

∫d3k

(Θ(E + ∆E − En,k) − Θ(E − En,k)

)(373)

where Θ(x) is the Heaviside step function. Thus, the density of states is ex-pressed by an integral over a volume of k space enclosed by surfaces of constantenergy E and E + ∆E. Furthermore, since ∆E is an infinitesimal quantity,∆E can be expressed in terms of the perpendicular distance between the twosurfaces of constant energy.

Let Sn(E) be the surface En,k = E lying within the primitive cell and letδk(k) be the perpendicular distance between the surfaces Sn(E) and Sn(E +dE) at point k. Then, as Sn(E) is a surface of constant E and ∇ En,k isperpendicular to that surface

E + ∆E = E + | ∇ En,k | δk(k)

δk(k) =∆E

| ∇ En,k |(374)

Hence, the density of states can be expressed as an integral over a surface ofconstant energy

ρ(E) = V∑

n

∫Sn(E)

d2S

( 2 π )31

| ∇ En,k |(375)

This gives an explicit relation between the density of states and the band struc-ture.

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Since En,k is periodic, it is bounded from above and below for each value ofn. This implies that there will be values of k in each Brillouin zone where thegroup velocity vanishes,

∇ En,k = 0 (376)

The band energy En,k must have at least one maximum and one minimum inthe Brillouin zone. At each of these k points, the integrand in ρ(E) diverges.Other divergences may be expected which originates from k points near theBrillouin zone boundary, where the dispersion relation is expected to have zeroslope. These divergences give rise to van Hove singularities in the density ofstates. L. van Hove provided a general discussion of these types of singularitiesusing the Morse index theorem (L. van Hove, Phys. Rev. 89, 1189 (1953), alsosee the discussion by H.P. Rosenstock, Phys. Rev. 97, 290 (1955)).

In three dimensions these singularities are integrable. That is, the integra-tion over the surface area yield a finite value for ρ(E). In the three-dimensionalcase the divergences show up in the slopes of the density of states ∂ρ(E)

∂E , andare the van Hove singularities. The van Hove singularities at the density ofstates occur at the values of E where ∇ En,k vanishes at some points of thesurface Sn(E). Typical van Hove singularities occur at the band edges wherethe density of states varies as

√| E | . Although the density of states ρ(E)

at van Hove singularities does not diverge in three dimensions, the derivativesdiverge and can give rise to anomalies in thermodynamics as can be seen byexamining the Sommerfeld expansion.

In low dimensional systems, the divergence can show up directly as a diver-gence in the density of states.

——————————————————————————————————

7.6.1 Exercise 26

The energy dispersion relation at a van Hove singularity has a zero gradient. Inthe vicinity of the van Hove singularity, the d-dimensional dispersion relationcan be written as

Ek = E0 + E1

i=d∑i=1

αi k2i a

2i (377)

where the coefficients αi determine whether the extremum is a maximum, min-imum or saddle point. The coefficients are given by

αi = ± 1 (378)

Characterize the different types of van Hove singularities in the density of statesand sketch the energy dependence in the vicinity of the singularity for d = 1, 2

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and d = 3.

——————————————————————————————————

7.7 The Fermi-Surface

The ground state of the electronic system has the lowest possible energy. Fornon-interacting electrons, the electrons occupy the lowest possible eigenvalues.However, the distribution of electrons must satisfy the restriction imposed bythe Pauli exclusion principle, which states that no uniquely specified electronstate can be occupied by more than one electron. This means that a spin de-generate state cannot be occupied by more than two electrons, one for each spinvalue. Thus, the ground state of H0 is represented by a Slater determinant wavefunction in which two electrons are placed in the lowest energy eigenstate, andtwo in the successively next lowest states, until all the Ne electrons have beenplaced in states. In the following, the convention is adopted that the electronswhich are associated with the states (n, k) have k restricted to be within thefirst Brillouin zone.

Two different types of ground states result.

Insulators.

In insulators, a number of bands are completely filled and all other bandsare completely empty. No band is partially filled. In this case, there must existan energy interval which separates the lowest unoccupied band state and thehighest occupied band state. The density of states must be zero in this energyinterval. The width of the interval, where ρ(E) = 0, is the threshold energyrequired to excite an electron from an occupied to an unoccupied state. Thisenergy interval is defined to be the band gap. In an insulator, the chemical po-tential µ falls in the band gap. An insulating state can only occur if the numberof electrons Ne is equal to an even number times the number of primitive unitcells N in the direct lattice. This is because each band can be occupied by 2 Nelectrons. For example, C being tetravalent when it crystallizes in the diamondstructure is insulating, and has a band gap of over 5 eV. The elements Si andGe are also insulating, but have smaller band gaps which are 1.1 eV and 0.67eV, respectively.

Metals.

A number of bands may be partially filled. In this case, the highest occu-pied Bloch states have an energy EF which lies within the range of one or morebands. This case corresponds to a metal, in which the one-electron density ofstates at EF is non-zero, ρ(EF ) 6= 0. Systems with an odd number of electronsper unit cell should be metallic, such as the simple mono-valent metals like Na

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or K. However, systems with two electrons per unit cell can be metallic. Forexample, divalent Mg is metallic. Mg crystallizes in the hexagonal close-packedsystem and, hence, has four electrons per unit cell. The small distance betweenthe atoms is responsible for the large dispersion of the bands which allows thebands to overlap. The overlapping of the bands leads to divalent Mg beingmetallic.

For each partially filled band, there will be a surface in the three-dimensionalk space which separates the occupied from the unoccupied states. The set of allsuch surfaces forms the Fermi-surface. The Fermi-surface is determined by theequation

En,k = EF (379)

Since En,k is periodic in the reciprocal lattice, the Fermi-surface may either berepresented within the full periodic reciprocal lattice or in a single unit cell ofthe reciprocal lattice. If the full reciprocal lattice is used, the Fermi-surface isrepresented in the extended zone scheme. If the Fermi-surface is representedwithin a single primitive unit cell of the reciprocal lattice, it is represented in areduced zone scheme.

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8 Approximate Models

Some of the earlier approaches to electronic structure of solids will be discussedin this chapter. These methods are not in common use, and are not reliablemethods for calculating electronic structures. These older methods also ne-glect the effect of electron-electron interactions. By contrast, the most commonmethod in use today is based on the Density Functional approach of Kohn andSham, which is quantitatively reliable and includes the effect of electron-electroninteractions. Nevertheless, the older methods were important in the develop-ment of the subject and yield important insights into the results of electronicstructure calculations.

8.1 The Nearly Free Electron Model

In the nearly free electron approach to electronic structure calculations, one as-sumes that the periodic potential due to the lattice is small. This assumption isnot justified, apriori, as the potential is of the order of 10 eV. However, the effectof the potential can be much smaller than this estimate, and for these cases, thenearly free electron model gives results which can be used to phenomenologicallydescribe metals found in groups I, II, III, and IV of the periodic table. Thesematerials have an atomic structure which consists of s or p electrons outside aclosed shell configuration.

The nearly free model works for two main reasons:-

(i) The region in which the electron - ion interaction is strongest is in thevicinity of the ion. However, since this region is occupied by the core electronsand the Pauli principle forbids the conduction electrons to enter this region, theeffective potential is weak.

(ii) In the region of space where the conduction electrons reside, the motionof other conduction electrons effectively screen the potential.

Since in the nearly free electron approximation the effective potential is as-sumed to be small, perturbation theory may be used.

8.1.1 Perturbation Theory

The wave function for an electron in a Bloch state with wave vector k is givenby

φk(r) =∑Q

Ck−Q exp[i ( k − Q ) . r

](380)

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where Q are reciprocal lattice vectors and the coefficients Ck have to be deter-mined. The coefficients satisfy the set of coupled algebraic equations[

h2

2 m( k − Q )2 − E

]Ck−Q +

∑Q′

Vions(Q−Q′) Ck−Q′ = 0 (381)

where the sum runs over all the reciprocal lattice vectors Q′. For fixed k, thereis an equation for each Q value. The solutions of this equation for fixed k arelabelled by n.

If one neglects the potential due to the lattice, one obtains the empty latticeapproximation. This is the result of the zero-th order perturbation theory. Tozero-th order in the perturbing potential Vions, the set of equations reduce to[

E(0)k − Q − E

]Ck−Q = 0 (382)

where the zero-th order energy eigenvalues are given by

E(0)k − Q =

h2

2 m( k − Q )2 (383)

and the zero-th order energy eigenfunctions are

φ(0)k (r) =

1√V

exp[i ( k − Q ) . r

](384)

If, for a given k, the energies associated with the set of reciprocal lattice vectorsQ

1, . . . , Q

mare degenerate,

E(0)k − Q

1= E

(0)k − Q

2= . . . = E

(0)k − Q

m

(385)

then φ(0)k (r) can be made of any linear combination of the functions exp[ i ( k −

Q ) . r ].

The type of perturbation theory that is appropriate depends on whether thezero-th order eigenvalues are degenerate or not.

8.1.2 Non-Degenerate Perturbation Theory

Non-degenerate perturbation theory can be used when the energy separationsbetween the level under consideration, E(0)

k − Q1

, and all other zero-th ordereigenvalues are large compared with the magnitude of the potential

| E(0)k − Q

1− E

(0)k − Q | | Vions(Q1

−Q) | (386)

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for fixed k and all Q 6= Q1. This corresponds to the non-degenerate case.

We shall evaluate the one-electron energy eigenvalue to second order in Vions

but first, we need to consider the first order correction to the energy and wavefunction of the state under consideration which, to zero-th order, has momentumk − Q

1. The amplitude corresponding to the plane wave component with this

momentum satisfies the secular equation[E

(0)k − Q

1− E

]Ck−Q

1+∑Q′

Vions(Q1−Q′) Ck−Q′ = 0 (387)

This shall be used to obtain the energy E and the coefficient Ck−Q1

to firstorder in Vions. The term involving the summation is explicitly of the order ofVions, so the coefficients Ck−Q in this term only need to be calculated to zero-thorder in the Vions. Only one coefficient is non-zero to zero-th order in Vions,since

C(0)k−Q = 0 ∀ Q 6= Q

1(388)

Thus, to first order in Vions, only one term survives in the summation and thecoefficient Ck−Q

1satisfies the eigenvalue equation[E − E

(0)k − Q

1

]Ck−Q

1= Vions(0) C(0)

k−Q1

(389)

This equation determines the energy eigenvalue E(1) to first order in Vions. Sincethe energy shift is to be calculated to first order in Vions, the coefficient Ck−Q

1

can be substituted by its zero-th order value C(0)k−Q

1. This procedure yields the

first order approximation for the energy eigenvalue

E(1) = E(0)k−Q

1+ Vions(0) (390)

This only yields a constant shift in the zero-th order energy eigenvalues whichcan be absorbed into the definition of the reference energy. It is also seenfrom eqn(389) that, to first order, the change in the coefficient Ck−Q

1remains

undetermined, so we may set

C(1)k−Q

1= C

(0)k−Q

1(391)

This is seen by substituting the first-order expression for E − E(0)k−Q

1into

eqn(389). In the following discussion, we shall neglect the effect of the averagepotential V (0)ions.

The coefficients of the other plane wave components of the Bloch functionsatisfy [

E(0)k − Q − E

]Ck−Q +

∑Q′

Vions(Q−Q′) Ck−Q′ = 0 (392)

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This is used to obtain the coefficients C(1)k−Q to first order in Vions. Since the

summand is explicitly of first order in Vions, then the coefficients Ck−Q′ need

only be considered to zero-th order. However, only C(0)k−Q

1is non-zero in this

order so,

C(1)k−Q =

Vions(Q − Q1)

E − E(0)k − Q

C(0)k−Q

1(393)

The coefficients C(1)k−Q and C

(0)k−Q

1completely determine the energy eigenfunc-

tion to first order in Vions.

The energy eigenvalue can now be found to second order in Vions usingthe wave function that have just been calculated to first order in Vions. Onsubstituting the expression for C(1)

k−Q, eqn(393), into the secular equation whichdetermines Ck−Q

1, eqn(387), one finds(

E − E(0)k − Q

1

)Ck−Q

1=∑Q

| Vions(Q1− Q) |2

( E − E(0)k − Q )

C(0)k−Q

1(394)

Since both the energy and wave function are unchanged to first order in Vions,the lowest order non-zero contribution to the term on the left hand side is foundwhen Ck−Q

1is evaluated in zero-th order and E is evaluated to second order.

Thus, to second order in Vions, the energy eigenvalue E is given by the solutionof (

E − E(0)k − Q

1

)=∑Q

| Vions(Q1− Q) |2

( E − E(0)k − Q )

(395)

or, since the eigenvalue E is approximately equal to E(0)k − Q

1, the energy eigen-

value is given by

E = E(0)k − Q

1+∑Q

| Vions(Q1− Q) |2

( E(0)k − Q

1− E

(0)k − Q )

(396)

This relation shows that weakly perturbed non-degenerate bands repel eachother. For example, if

E(0)k − Q > E

(0)k − Q

1(397)

then the second order contribution is negative and E is reduced further belowE

(0)k − Q

1. On the other hand, if

E(0)k − Q < E

(0)k − Q

1(398)

then the second order contribution is positive and E is increased further aboveE

(0)k − Q

1. Hence, the leading order effect of the perturbation increases the sep-

aration between the energy bands.

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8.1.3 Degenerate Perturbation Theory

The most important effect of the potential occurs when a pair of the free electroneigenvalues are within Vions of each other, but are far from all other eigenvalues.Under these conditions, the eigenvalues are almost doubly degenerate and onecan use degenerate perturbation theory to couple these energy levels.

In this case, the set of equations can be truncated to only two non-zero Ccoefficients. These two coefficients satisfy the pair of equations

( E − E(0)k − Q

1) Ck−Q

1= Vions(Q2

− Q1) Ck−Q

2(399)

and( E − E

(0)k − Q

2) Ck−Q

2= Vions(Q1

− Q2) Ck−Q

1(400)

which can be combined to yield the quadratic equation for E

( E − E(0)k − Q

1) ( E − E

(0)k − Q

2) = | Vions(Q1

− Q2) |2 (401)

This quadratic equation has the solution for the energy eigenvalue

E =( E

(0)k − Q

1+ E

(0)k − Q

2

2

√√√√(E(0)k − Q

1− E

(0)k − Q

2

2

)2

+ | Vions(Q1− Q

2) |2

(402)Whenever the Bloch wave vector k takes on special values such that unperturbedbands cross

E(0)k − Q

1= E

(0)k − Q

2(403)

the energy bands simplify to yield the two branches

E = E(0)k − Q

1± | Vions(Q1

− Q2) | (404)

If the unperturbed bands cross, the non-zero potential produces a splitting of2 | Vions(Q1

− Q2) |. This result is consistent with that previously found by

using non-degenerate perturbation theory.

The avoided crossings of the bands are expected to occur whenever

E(0)k − Q

1∼ E

(0)k − Q

2(405)

This gives rise to a specific condition on the wave vectors. For convenience ofnotation, let q = k − Q

1so that this criterion takes the form

E(0)q = E

(0)q − Q” (406)

for some reciprocal lattice vector Q” 6= 0. This requires that vector q lies onthe Bragg plane bisecting Q”, as this condition reduces to

Q”2 = 2 q . Q” (407)

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The vector q − Q” lies on a second Bragg plane. Thus, the geometric signifi-cance of the condition for the degeneracy of the unperturbed bands, is that theelectronic states satisfy the condition for Bragg scattering.

The origin of the gaps can be easily understood from consideration of thewave functions. When q lies on a single Bragg plane, then the energy eigenvaluesare simply given by

E = E(0)q ± | Vions(Q”) | (408)

The coefficients corresponding to these energies are found from the two coupledequations. In this case, where the unperturbed bands cross, the coefficients arerelated via

Cq = ± sign

(Vions(Q”)

)Cq−Q” (409)

which produces two standing wave solutions. If Vions(Q”) > 0, then the pairof states are the anti-bonding state

| φ+q (r) |2 ∝ cos2

(Q” . r

2

)E+ = E(0)

q + | Vions(Q”) | (410)

and the bonding state

| φ−q (r) |2 ∝ sin2

(Q” . r

2

)E− = E(0)

q − | Vions(Q”) | (411)

On the other hand, if Vions(Q”) < 0, then the situation is reversed, and theanti-bonding state is given by

| φ+q (r) |2 ∝ sin2

(Q” . r

2

)E+ = E(0)

q + | Vions(Q”) | (412)

while the bonding state is given by the other form

| φ−q (r) |2 ∝ cos2(Q” . r

2

)E− = E(0)

q − | Vions(Q”) | (413)

In this context, the wave function

φpq(r) ∝ sin

(Q” . r

2

)(414)

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is called p-like as it vanishes at the lattice points, whereas

φsq(r) ∝ cos

(Q” . r

2

)(415)

is called s-like as it is non-vanishing at the positions of the ions, r = R. Theorigin of the gap between the two branches is seen through examination of theaverage potential energy of the s and p like wave functions∫

V

d3r Vions(r) | φs,pq (r) |2 (416)

The s-like electrons congregate at the position of the ions where the potential islower, and the p-like electrons congregate between the ions where the potentialis higher. For an attractive interaction Vions(r) < 0, this leads to φs

q(r) havinga lower energy than φp

q(r), ( when Vions(Q”) < 0 ).

The Bragg planes have other significance as can be inferred from the gradientof the energy

E± =( E

(0)q + E

(0)q − Q”

2

√√√√(E(0)q − E

(0)q − Q”

2

)2

+ | Vions(Q”) |2

(417)which is found as

∇q E± =h2

m

[ (q −

Q”2

)±Q”2

(E

(0)q − E

(0)q − Q”

)√(

E(0)q − E

(0)q − Q”

)2

+ 4 | Vions(Q”) |2

]

(418)On the Bragg plane, one has

E(0)q = E

(0)q − Q” (419)

therefore, the second term in the expression for the gradient drops out on theseplanes. Thus, the gradient of the energy of the mixed bands is given by

∇q E± =h2

m

(q −

Q”2

)(420)

and, as q is on the Bragg plane, the vector q − Q”

2 is parallel to the plane andso is the gradient. The gradient of the energy is perpendicular to surfaces ofconstant energy and so, the constant energy surfaces are usually perpendicularto the Bragg planes at their points of intersection.

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Generally, the vanishing of the normal component of the gradient at theBrillouin zone boundary is not dependent on the validity of the nearly freeelectron approximation, but is a consequence of symmetry. Consider the casein which there is a mirror plane symmetry, σ. The mirror plane is assumed torun through the origin of the Brillouin zone and is parallel to the Brillouin zoneboundary under consideration. Then, the normal component of the gradient isdefined as

Q . ∇k Ek

∣∣∣∣ = limδ → 0

[Ek+δQ − Ek−δQ

2 δ

](421)

However, since the point k is equivalent to the point k −Q, one has

Ek+δQ = E−Q+k−δQ (422)

and, as there exists a mirror plane σ through the origin and perpendicular toQ, one also has

E−Q+k−δQ = EQ+σk+δQ (423)

Noting that as k is on the Bragg plane, k ≡ Q + σk, and substituting theabove equality into the definition, one finds that the normal component of thegradient vanishes at the Brillouin zone boundary

Q . ∇k Ek

∣∣∣∣ = 0 (424)

Thus, at the Brillouin zone boundary, either the normal component of the gra-dient vanishes or the gradient does not exist, i.e. there might be a cusp. Thepresence of other types of symmetry can give rise to similar conclusions.

8.1.4 Empty Lattice Approximation Band Structure

Since the nearly free electron approximation deviates only slightly from the freeelectron approximation, the gross features of the band structure can be foundusing the empty lattice approximation.

Since the Brillouin zone is a three-dimensional object and is highly sym-metric, it is only necessary to specify the bands within an irreducible wedge.Once the bands are specified within the wedge, then by use of symmetry, thebands are completely known throughout the Brillouin zone. Since it is difficultto represent the energy dispersion relations in a three-dimensional volume ofreciprocal space, it is customary to specify the dispersion relations on the linesdefining the boundaries of the irreducible wedge. These lines have high symme-tries.

Consider the case of an f.c.c. Bravais Lattice, and consider the bands withinthe first Brillouin zone. The high symmetry points are marked by special letters.

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Γ ≡ (0, 0, 0) ≡ (0, 0, 0)

K ≡ 3 π2 a (1, 1, 0) ≡ 2 π

a ( 34 ,

34 , 0)

W ≡ πa (2, 1, 0) ≡ 2 π

a (1, 12 , 0)

X ≡ 2 πa (1, 0, 0) ≡ 2 π

a (1, 0, 0)

L ≡ πa (1, 1, 1) ≡ 2 π

a ( 12 ,

12 ,

12 )

and in units of 2 πa correspond to

Γ ≡ (0, 0, 0) The zone center

K ≡ ( 34 ,

34 , 0) The hexagonal edge center

X ≡ (1, 0, 0) The diamond face center

W ≡ (1, 12 , 0) The corner

L ≡ ( 12 ,

12 ,

12 ) The hexagonal face center

The electron bands are usually plotted against k along the high symmetrydirections

Γ → X → W → L → Γ → K → X

The length of these linear segments ( in units of 2 πa ) are given by

1 12

1√2

√3

23√

24

√104

The band energies in the empty lattice approximation can be plotted alongthese axes in units of E0 where

E0 =h2

2 m

(4 π2

a2

)(425)

and the components of the reduced wave vectors ki where

ki =ki a

2 π(426)

The energy of the various bands can be constructed from the various E0k−Q.

The first band that is considered is simply E0k, where Q = (0, 0, 0) thus,

E0k

E0=(k2

x + k2y + k2

z

)(427)

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which for Γ → X is just

= k2x for 0 ≤ kx ≤ 1 (428)

For X → W this band dispersion is given

= 1 + k2y for 0 ≤ ky ≤ 1

2(429)

For W → L this band is described by

=14

+ k2x + ( 1 − kx )2 for

12≤ kx ≤ 1 (430)

For L → Γ this dispersion is given as

= 3 k2x for 0 ≤ kx ≤ 1

2(431)

For Γ → K this band takes the form

= 2 k2x for 0 ≤ kx ≤ 3

4(432)

The last segment is given by K → X in which the band takes the form

= k2x + 9 ( 1 − kx )2 for

34≤ kx ≤ 1 (433)

The next band to be considered is simply E0k−Q, where Q = 4 π

a (1, 0, 0)thus,

E0k−Q

E0=(

( kx − 2 )2 + k2y + k2

z

)(434)

which for Γ → X is just

= ( kx − 2 )2 for 0 ≤ kx ≤ 1 (435)

For X → W this band dispersion is given

= 1 + k2y for 0 ≤ ky ≤ 1

2(436)

For W → L this band is described by

=14

+ ( kx − 2 )2 + ( 1 − kx )2 for12≤ kx ≤ 1 (437)

For L → Γ this dispersion is given as

= ( kx − 2 )2 + 2 k2x for 0 ≤ kx ≤ 1

2(438)

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For Γ → K this band takes the form

= ( kx − 2 )2 + k2x for 0 ≤ kx ≤ 3

4(439)

The last segment is given by K → X in which the band takes the form

= ( kx − 2 )2 + 9 ( 1 − kx )2 for34≤ kx ≤ 1 (440)

The next band is E0k−Q, where Q = 4 π

a ( 12 ,

12 ,

12 ) thus,

E0k−Q

E0=(

( kx − 1 )2 + ( ky − 1 )2 + ( kz − 1 )2)

(441)

which for Γ → X is just

= 2 + ( kx − 1 )2 for 0 ≤ kx ≤ 1 (442)

For X → W this band dispersion is given

= 1 + ( ky − 1 )2 for 0 ≤ ky ≤ 12

(443)

For W → L this band is described by

=14

+ ( kx − 1 )2 + k2x for

12≤ kx ≤ 1 (444)

For L → Γ this dispersion is given as

= 3 ( kx − 1 )2 for 0 ≤ kx ≤ 12

(445)

For Γ → K this band takes the form

= 1 + 2 ( kx − 1 )2 for 0 ≤ kx ≤ 34

(446)

The last segment is given by K → X in which the band takes the form

= 1 + ( kx − 1 )2 + ( 2 − 3 kx )2 for34≤ kx ≤ 1 (447)

The next band is E0k−Q, where Q = 4 π

a ( 12 ,−

12 ,

12 ) thus,

E0k−Q

E0=(

( kx − 1 )2 + ( ky + 1 )2 + ( kz − 1 )2)

(448)

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which for Γ → X is just

= 2 + ( kx − 1 )2 for 0 ≤ kx ≤ 1 (449)

For X → W this band dispersion is given

= 1 + ( ky + 1 )2 for 0 ≤ ky ≤ 12

(450)

For W → L this band is described by

=94

+ ( kx − 1 )2 + k2x for

12≤ kx ≤ 1 (451)

For L → Γ this dispersion is given as

= 2 ( kx − 1 )2 + ( kx + 1 )2 for 0 ≤ kx ≤ 12

(452)

For Γ → K this band takes the form

= 1 + ( kx − 1 )2 + ( kx + 1 )2 for 0 ≤ kx ≤ 34

(453)

The last segment is given by K → X in which the band takes the form

= 1 + ( kx − 1 )2 + ( 4 − 3 kx )2 for34≤ kx ≤ 1 (454)

It is seen that some branches of these bands are highly degenerate. WhenVions 6= 0, the degeneracy of the various branches may be lifted. Group theorycan be used to determine whether or not the potential lifts the degeneracy ofthe branches.

Thus, even in the empty lattice approximation, the method of plotting bandsshows a great deal of structure. The real structure is actually inherent in theBragg planes which generally can be associated with an ”energy gap” in the dis-persion relations. The ”gap” may or may not extend across the entire Brillouinzone. A gap only appears in the density of states if the ”gap” extends across theentire Brillouin zone. The nearly free electron approximation has been workedout in detail for Al by B. Segall, Physical Review 124, 1797 (1961).

For the b.c.c. lattice, the reciprocal lattice vectors are

b1 =12

4 πa

( ex + ey )

b2 =12

4 πa

( ex + ez )

b3 =12

4 πa

( ey + ez ) (455)

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The Cartesian coordinates of the high symmetry points are

Γ ≡ (0, 0, 0)

H ≡ (1, 0, 0)

N ≡ ( 12 ,

12 , 0)

P ≡ ( 12 ,

12 ,

12 )

in units of 2 πa .

——————————————————————————————————

8.1.5 Exercise 27

Derive the lowest energy bands of a b.c.c. lattice in the empty lattice approx-imation. Plot the dispersion along the high symmetry directions (Γ → H →N → P → Γ → N).

——————————————————————————————————

8.1.6 Degeneracies of the Bloch States

The degeneracies of the bands at various points in the Brillouin zone, found inthe empty lattice approximation, can be raised by the crystalline potential. Thecharacter and degeneracies of the bands at symmetry points can be ascertainedby the use of group theory (L.P. Bouckaert, R. Smoluchowski and E. Wigner,Phys. Rev. 50, 58 (1936)).

Given a Bloch function φn,k(r), one can apply a general point group sym-metry operator O(Aj) to the Bloch function, thereby, transforming it into theBloch function corresponding to the wave vector Aj k

O(Aj) φn,k(r) = φn,Ajk(r) (456)

This is proved by considering the combined operation consisting of the pointgroup operation O(Aj) followed by a translation through a Bravais lattice vectorR. The effect of the combined operation is evaluated as

T (R) O(Aj) φn,k(r) = T (R) φn,k(A−1j r)

= exp[− i k . A−1

j R

]φn,k(A−1

j r) (457)

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where the second line follows from Bloch’s theorem. However, we note thatthe scalar product remains invariant if both vectors are transformed. We shalltransform the vectors k and ( A−1

j R ) by Aj . Hence, as

k . ( A−1j R ) = ( Aj k ) . ( Aj R

−1j R )

= ( Aj k ) . R (458)

we find that

T (R) Aj φn,k(r) = exp[− i ( Aj k ) . R

]φn,k(A−1

j r)

= exp[− i ( Aj k ) . R

]O(Aj) φn,k(r)

(459)

Since the quantity

exp[− i ( Aj k ) . R

](460)

is the eigenvalue of the translation operator T (R), the Bloch wave vector ofthe function O(Aj) φn,k(r) is Aj k. As this is an energy eigenfunction, thetransformed function is a Bloch function. That is,

φn,Ajk(r) = O(Aj) φn,k(r) (461)

Since the point group symmetry operations commute with the Hamiltonian,

[ H , O(Aj) ] = 0 (462)

the Bloch states O(Aj) φn,k(r) all have the same energy En,k.

A basis set can be constructed by repeated application of the point groupsymmetry operators on the Bloch functions. The same vector k cannot appearin distinct bases created from a Bloch function since the symmetry operationsform a group. This means that two such bases are either identical or have nowave vector k in common. A basis created from the Bloch function φn,k(r) inthis fashion may be either reducible or irreducible.

An irreducible basis can be constructed by selecting an appropriate subsetof Bloch functions from the above basis set. If one considers the set of wavevectors Ai k, then certain of these points may be equivalent in that

Ai k = Aj k + Q (463)

where Q is a reciprocal lattice vector. The star of k is the set of all the inequiv-alent wave vectors Ai k. More precisely, the star of the wave vector k consistsof the set of all mutually inequivalent wave vectors Ai k, where Ai ranges over

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all the operations of the point group. Since none of the Bloch wave vectors inthe star are equivalent, the corresponding Bloch functions are all linearly inde-pendent. Hence, the Bloch functions of the star may be used to construct anirreducible basis.

The group of the k vector consists of all symmetry operations which, whenacting on k, lead to an equivalent point. That is, the symmetry operations ofthe group of the k vector satisfy

Aj k = k + Q (464)

where Q is a reciprocal lattice vector. As an example, the groups of the k vec-tors for the points Γ, Z,M,A of the simple tetragonal lattice coincide with theD4h point group of the tetragonal lattice itself. The groups of the k vectors atX and R are D2h, and D2h is a subgroup of D4h. In general, the group of thek vector of the Γ point will always coincide with the point group of the crystal.The group of the k vector has irreducible representations, and these are calledthe small representations.

The basis functions of the star of k can be symmetrized with respect tothe small representations. The symmetrization can be performed by using theprojection method. Although the groups of the wave vector in the star may bedifferent, the small representation of any one can be chosen for the symmetriza-tion process. After the symmetrization, the resulting basis functions form anirreducible representation of the space group. Each basis function of the smallrepresentation only corresponds to exactly one wave vector in the star and theequivalent wave vectors. The basis functions corresponding to the different ir-reducible representations are orthogonal.

The irreducible representations of the space group constructed from theBloch functions are fully determined by the star of the k vector and the smallrepresentation. The basis functions forming the irreducible representation ofthe space group constructed from the Bloch state φn,k(r) are eigenstates of H0

with energy En,k. Barring accidental degeneracies, the degeneracy of this eigen-value is equal to the dimension of its irreducible representation. As k varies inthe Brillouin zone, the eigenvalue En,k and the corresponding basis functionsvary continuously. The group of the k vector also varies as k varies. Wheneverthe dimension of the small representation corresponding to the basis functionφn,k(r) changes, the degeneracy of En,k changes. This may signify that at thesepoints different bands cross or merge together.

Alternatively, at k there are a vast number of bands each corresponding toa different small representation. The degeneracy of each band is given by thedimension of the corresponding small representation. If an irreducible represen-tation of the group of the wave vector k can be decomposed into the irreduciblerepresentations of the group of k0, then on varying k to k0, the branch will

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split into sub-levels. The degeneracies of the sub-levels are determined by thedimensions of the irreducible representations contained in the decomposition.

——————————————————————————————————

As an example, consider the nearly free electron bands of zinc blende. Thematerial has tetrahedral point group symmetry, Td. The point group containstwenty four elements in five equivalence classes. One class consists of the iden-tity E. There is a class of eight C3 operations, which contain the rotation C3

and the inverse rotation C−13 about the four axes [1, 1, 1], [1, 1, 1], [1, 1, 1] and

[1, 1, 1]. There is a class consisting of three C2 operations around the [1, 0, 0],[0, 1, 0] and [0, 0, 1]. There is a class consisting of six S4 operations around the[1, 0, 0], [0, 1, 0] and [0, 0, 1] axes. Finally, there is a group consisting of six σ op-erations which are reflections in the six planes (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 0),(1, 0, 1) and (0, 1, 1). Therefore, the group has five irreducible representations.The character table is given by

Td E C2(3) C3(8) (S4)(6) σ(6)Γ1 1 1 1 1 1Γ2 1 1 1 - 1 - 1Γ3 2 2 - 1 0 0Γ4 3 - 1 0 1 - 1Γ5 3 - 1 0 - 1 1

Let us consider the band structure along the high symmetry directions[1, 1, 1] and [1, 0, 0] directions.

At the Γ point the group of the k vector coincides with the point group ofthe crystal. Since the nearly free electron approximation for the Bloch wavefunction for k = 0 is a constant, it is a basis for the Γ1 representation. Thus,the level is non-degenerate.

At a general point along the eight [1, 1, 1] directions, the group of the kvector is C3v and contains six elements in three classes. The are the identityE, a class consisting of the rotation C3 about the [1, 1, 1] axis and its inverseC−1

3 , and three reflections σ in the three equivalent (1, 1, 0) planes containingthe [1, 1, 1] axis. The character table is given by

C3v E C3(2) σ(3)Λ1 1 1 1Λ2 1 1 - 1Λ3 2 - 1 0

Thus, the branches along the Λ axis are either singly or doubly degenerate,when the crystalline potential is introduced. The branch which emanates from

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k = 0 with energy E = h2

2 m k2 belongs to the Λ1 representation as this iscompatible with the Γ1 representation.

At the end point L where k = πa (1, 1, 1), the symmetry operations are iden-

tical to those of Λ. In the free electron approximation, the state at L is doublydegenerate (ignoring spin) since the wave vectors π

a (1, 1, 1) and − πa (1, 1, 1)

differ by a reciprocal lattice vector Q = 2 πa (1, 1, 1). Using the compatibility

relations, one can show that the next highest band has Λ1 symmetry. These twolevels are accidentally degenerate, since they are not partner basis functions ofa multi-dimensional irreducible representation. Therefore, the degeneracy maybe lifted by the presence of a crystalline potential V (Q).

On continuing along the band with Λ1 symmetry, one reaches the pointk = 2 π

a (1, 1, 1). Since the primitive lattice vectors of the f.c.c. lattice are ofthe form

b1 =2 πa

(−1, 1, 1)

b2 =2 πa

(1,−1, 1)

b3 =2 πa

(1, 1,−1) (465)

thenQ = b1 + b2 + b3 is equal to 2 πa (1, 1, 1). Thus, the point k = 2 π

a (1, 1, 1)is equivalent to the Γ point. The star consists of just one wave vector. At thispoint, the eight nearly free electron bands corresponding to

φkj(r) ∼ exp

[i

2 πa

( ± x ± y ± z )]

(466)

are degenerate. They form the basis of an eight-dimensional representationwhich is reducible. In this representation, a symmetry transformation A isrepresented by the 8 × 8 matrices, D(A), which are constructed according tothe prescription

O(A) φki(r) = φki

(A−1r)

=∑

j

φkj(r) D(A)j,i (467)

The characters of this eight-dimensional representation are given by the traceof the 8 × 8 matrices and, therefore, the character of an operation is just thenumber of wave functions that are unchanged by the transformation.

Class Transformation χE x, y, z 8

C2(3) x, y, z 0C3(8) y, z, x 2S4(6) x, z, y 0σ(6) y, x, z 4

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This eight-dimensional representation, Γ, is reduced into the irreducible repre-sentations, Γµ, via

Γ =∑

µ

aµ Γµ (468)

The decomposition can be found from considering the characters. The charac-ters of a symmetry operation A, χ(A), is decomposed into the characters of theirreducible representations, χµ(A), via

χ(A) =∑

µ

aµ χµ(A) (469)

The multiplicity aµ can be found from the orthogonality relation∑i

gi χ(Ai) χµ(Ai) = g aµ (470)

where the sum over i runs over all the equivalence classes of the group, and gi

is the number of symmetry elements in the i-th equivalence class, and g is theorder of the group. This procedure leads to the decomposition

χ(Ai) = 2 χΓ1(Ai) + 2 χΓ4(Ai) (471)

Thus, the eight plane wave basis can be symmetrized into two sets of basisfunctions of Γ1 symmetry and two three-dimensional sets of basis functionsof Γ4 symmetry. The symmetrization process is performed by the use of theprojection method. A projector, Pµ which projects the functions on to anirreducible set of basis functions, is constructed from the symmetry operationsO(A) and the characters of the operations via

Pµ =nµ

g

∑A

χµ(A) O(A) (472)

In this, nµ is the dimension of the µ-th irreducible representation, i.e., nµ =χµ(E). When the projector acts on an arbitrary combination of functions withequivalent wave vectors k, φk(r), it produces a basis function, φµ

k(r) for the µ-thirreducible representation

Oµ φk(r) = φµk(r) (473)

In this way, one can construct the set of symmetrized basis functions:

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Representation Basis functionsΓ1

cos 2πxa cos 2πy

a cos 2πza

Γ1

sin 2πxa sin 2πy

a sin 2πza

Γ4

cos 2πxa sin 2πy

a sin 2πza

sin 2πxa cos 2πy

a sin 2πza

sin 2πxa sin 2πy

a cos 2πza

Γ4

sin 2πxa cos 2πy

a cos 2πza

cos 2πxa sin 2πy

a cos 2πza

cos 2πxa cos 2πy

a sin 2πza

In this basis, all the matrices D(A) representing the symmetry operators Ahave the same block diagonal form. The matrices contains two one-dimensionalblocks and two three-dimensional blocks. Thus, these levels may be split by theapplication of a potential, however, the degeneracies cannot be completely lifted.

Along the X direction, the wave vectors are of the form (k, 0, 0) where0 < k < 2 π

a . The group of k is C2v. It has four elements in four classes:the identity E, a two-fold rotation about the [1, 0, 0] axis, and the two diagonalmirror planes σd and σ′d. The character table is given by

C2v E C24 σd σ′d

∆1 1 1 1 1∆2 1 1 - 1 - 1∆3 1 - 1 1 - 1∆4 1 - 1 - 1 1

Therefore, along this direction, all the irreducible representations are one-dimensional. The symmetry of the wave function emanating from (0, 0, 0) belongto ∆1 since this is the only irreducible representation compatible with Γ1. Thisbranch continues up to the X point. The point 2 π

a (1, 0, 0) is equivalent to thepoint − 2 π

a (1, 0, 0), as they are related via the Q vector Q = b2 + b3. Atthe X point, the lowest energy level in the nearly free electron approximationis doubly degenerate.

The group of the k vector at the X point is D2d and consists of eight el-ements arranged in five classes. These are the identity E, a two fold rotationabout the x axis C2

4 , a class of two elements which are the two-fold rotations C2

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about the y and z axis, and two S4 operations about the x axis, and a class oftwo diagonal reflections σd on the (0, 1, 1) and the (0, 1, 1) planes. Thus thereare five irreducible representations. The character table is given by

D2d E C24 (1) C2(2) S4(2) σd(2)

X1 1 1 1 1 1X2 1 1 1 - 1 - 1X3 1 1 - 1 - 1 1X4 1 1 - 1 1 - 1X5 2 - 2 0 0 0

At the X point, the wave functions of the two-fold degenerate energy levels, E0,found in the nearly free electron approximation belong to the one-dimensionalX1 and X3 irreducible representations. This degeneracy may be raised by thepotential.

On continuing along the X direction, one reaches the point (2, 0, 0). The sixk points (±2, 0, 0), (0,±2, 0) and (0, 0,±2) are all equivalent to the zone center.The group of the wave vector is Td. The six wave functions

φk(r) = exp[± i

4 πa

x

]φk(r) = exp

[± i

4 πa

y

]φk(r) = exp

[± i

4 πa

z

](474)

can be used as a basis for a six-dimensional representation. In this representa-tion, the characters of the symmetry operations are given by:

Class Transformation χE x, y, z 6

C2(3) x, y, z 2C3(8) y, z, x 0S4(6) x, z, y 0σ(6) y, x, z 2

This representation is degenerate and can be decomposed via

Γ =∑

µ

aµ Γµ (475)

The multiplicities aµ are calculated from∑i

gi χ(Ai) χµ(Ai) = g aµ (476)

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which leads to the decomposition

Γ = Γ1 + Γ3 + Γ4 (477)

into a one-dimensional, a two-dimensional and a three-dimensional irreduciblerepresentation. The basis functions can be symmetrized using the projectionmethod. The basis functions for the small representations are

Representation Basis functionsΓ1

cos 4πxa + cos 4πy

a + cos 4πza

Γ3

cos 4πya − cos 4πz

a

2 cos 4πxa − cos 4πy

a − cos 4πza

Γ4

sin 4πxa

sin 4πya

sin 4πza

Hence, the six-fold degenerate energy level E0 = h2

2m ( 4 πa )2 may have the

degeneracy lifted by V (Q).

——————————————————————————————————

8.1.7 Exercise 28

Using the symmetrized wave functions at k = ( 2 πa ) (1, 1, 1) in the nearly free

electron model for Zn blende

φΓ1 =

√8a3

cos2 π xa

cos2 π ya

cos2 π za

φΓ4(x) =

√8a3

sin2 π xa

cos2 π ya

cos2 π za

φΓ4(y) =

√8a3

cos2 π xa

sin2 π ya

cos2 π za

φΓ4(z) =

√8a3

cos2 π xa

cos2 π ya

sin2 π za

(478)

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Show that the matrix elements of the momentum operator between the Γ1 andΓ4 basis functions are given by

| < Γ1 | px | Γ4(x) > |2 = | < Γ1 | py | Γ4(y) > |2 = | < Γ1 | pz | Γ4(z) > |2 =(

2 π ha

)2

(479)while all other matrix elements are zero.

——————————————————————————————————

8.1.8 Brillouin Zone Boundaries

The Brillouin zone boundaries play an important role in the understandingof Fermi-surfaces. In the empty lattice approximation, the Fermi-surface is asphere when represented in the extended zone scheme. The nearly free electronapproximation introduces a distortion to the sphere which is most marked nearthe Brillouin zone boundaries.

In general, if the spherical Fermi-surface crosses a Bragg plane, then thesphere may distort. In particular, the constant energy surface should be per-pendicular to the Bragg plane at the line where they intersect. Due to theappearance of the potential Vions(Q) in the expression for the Bloch energynear the Bragg plane, and also due to the accompanying band splitting, the cir-cles of intersection of the constant energy surfaces (corresponding to EF ) withthe Bragg plane do not match up. This is necessary since the distortion of theFermi-surface must conserve the volume enclosed. This volume is equal to thevolume enclosed by the spherical Fermi-surface of the empty lattice approxima-tion.

The Fermi-surface in the reduced Brillouin zone scheme can be constructedfrom the Fermi-surface in the extended zone scheme. This is done by translatingthe disjoint pieces of the Fermi-surface in the higher order zones by reciprocallattice vectors, so that the pieces fit back into the first Brillouin zone.

The first Brillouin zone is the Wigner-Seitz unit cell of the reciprocal lattice.It encloses the set of points that are closer to Q = 0 than they are to any otherreciprocal lattice vector Q 6= 0. This can be restated as, the first Brillouinzone consists of the volume in the reciprocal lattice which can be accessed fromthe origin without crossing a Bragg plane.

The second Brillouin zone is the volume that can be reached from the firstBrillouin zone by crossing only one Bragg plane.

Likewise, the (n + 1)-th Brillouin zone consists of the points, not in the(n − 1)-th zone, that can be reached from the n-th zone by crossing only one

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Bragg plane. Alternatively, the n-th Brillouin zone is the volume that can onlybe reached from the origin by crossing a minimum of (n− 1) Bragg planes.

The Fermi-surface is constructed by:

(i) Drawing the free electron sphere.

(ii) Distorting the sphere at the Bragg planes.

(iii) For each of the n Brillouin zones, take the portions of the surface inthe n-th zone and translate them by reciprocal lattice vectors so that they laywithin the first Brillouin zone. The resulting surface is the branch of the Fermi-surface assigned to the n-th band in the repeated zone scheme.

The Hume-Rothery rules provide a correlation of crystal structure with thenumber of electrons per unit cell, or band filling. It is an empirical rule whichonly applies to alloys of noble metals, such as Cu, Ag and Au, with s-p elementssuch as Zn, Al, Si, and Ge. If it is assumed that the noble metals have oneelectron outside the closed d shell, then the alloys have an f.c.c. phase foran average number of electrons per atom up to 1.38, while the b.c.c. phaseis stable for band-fillings between 1.38 and 1.48. In the f.c.c. structure, thesmallest vectors from the zone center to each face of the Brillouin zone have theform 1

22 πa (1, 1, 0), whereas for the f.c.c. lattice these vectors are of the form

12

2 πa (1, 1, 1). Therefore, the radius of the Fermi-sphere, kF , at which it first

makes contact with the Brillouin zone boundary is given by

kF =√

afor f.c.c.

kF =√

afor b.c.c. (480)

When the Fermi-sphere first makes contact with the zone boundary, the occu-pied band is depressed by V (Q) resulting in an energy lowering which stabilizesthe structure. In the free electron approximation, the number of electrons perprimitive unit cell, n, is given by

n = 2V

N ( 2 π )34 π3

k3F (481)

where

V

N=

a3

4for f.c.c.

V

N=

a3

2for b.c.c. (482)

Thus, one finds that the critical number n is given by√

3π4 = 1.36 for the f.c.c.

and√

2 π3 = 1.48 for the b.c.c. lattices.

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8.1.9 The Geometric Structure Factor

The potential Vions(r) is a periodic function and can be defined in terms of theionic potentials, Vatom, the lattice vectors R, and the basis vectors rj , via

Vions(r) =∑R

∑j

Vatom(r −R− rj) (483)

The evaluation of the Fourier Transform of the potential can be reduced to anevaluation of the Fourier Transform in one unit cell of the lattice as

Vions(Q) =1V

∫V

d3r exp[− i Q . r

] ∑R,j

Vatom(r −R− rj)

=1V

∫V

d3r∑R,j

exp[− i Q . ( r − R )

]Vatom(r −R− rj)

=1V

∫V ′

d3r′∑R,j

exp[− i Q . r′

]Vatom(r′ − rj)

(484)

where we have used the Laue condition

exp[i Q . R

]= 1 (485)

and the transformation r′ = r − R. Furthermore, since the Bravais Lat-tice vectors do not explicitly appear in the summand, the sum over R merelyproduces a factor of N ∑

R

≡ N (486)

one has

Vions(Q) =N

V

∫V

d3r exp[− i Q . r

] ∑j

Vatom(r − rj)

=N

V

∑j

exp[

+ i Q . rj

] ∫V

d3r” exp[− i Q . r”

]Vatom(r”)

=N

VS(Q) Vatom(Q) (487)

where S(Q) is the geometric structure factor associated with the basis and theother factor is the Fourier transform of the ionic potential

Vatom(Q) =∫

V

d3r exp[− i Q . r

]Vatom(r) (488)

Thus, when the geometric structure factor vanishes, the Fourier component ofthe lattice potential also vanishes and then the lowest order splitting at the

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Bragg plane also vanishes. An example of this is given by the hexagonal close-packed lattice.

The unit cell of the reciprocal lattice of the (direct space) hexagonal closedpacked lattice is a hexagonal prism. There are two hexagonal planes whichhave normals pointing along the positive and negative z axis. These are Braggplanes. The structure factor vanishes for all q values on the hexagonal top andbottom of the prism. The structure factor can be evaluated as

S(Q) = 1 + exp[i π (

23m1 +

43m2 + m3 )

](489)

which vanishes when m1 = m2 = 0 and m3 = ± 1, corresponding to q layingon the Bragg planes. The vanishing of the structure factor at these particularBragg planes is a consequence of a glide symmetry. In fact, group theory showsthat the splitting on these planes is rigorously zero in the absence of spin-orbitcoupling (C. Herring, Phys. Rev. 52, 361 (1937)).

Since the gaps vanish on some faces of the Brillouin zones, it is sometimeshelpful to define a set of zones, the Jones zones, which are separated by planesin which gaps do occur.

The spin-orbit interaction can lead to the re-occurrence of small gaps in thebands (M.H. Cohen and L. Falicov, Phys. Rev. Letts. 5, 544 (1960)). The spin-orbit interaction is a relativistic effect, which appears as low order correction tothe non-relativistic limit of the Dirac equation. For a particle of charge q in thepresence of a scalar and vector potential (φ,A), this process yields the singleparticle Hamiltonian in the form

H = m c2 +1

2 m

(( p − q

cA ) . σ

)2

+ q φ

− 18 m3 c

p4 +q h

4 m2 c2σ .

[∇ φ ∧ ( p − q

cA )

]+

q h3

8 m2 c2∇2φ

(490)

The first line, apart from the rest energy, coincides with the non-relativisticPauli Hamiltonian

HP =1

2 m

(( p − q

cA ) . σ

)2

+ σ0 q φ (491)

which, together with the identity(σ . a

) (σ . b

)= σ0

(a . b

)+ i σ .

(a ∧ b

)(492)

leads to

HP =[

12 m

σ0

(− i h ∇ − q

cA

)2

+ σ0 q φ

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− h q

2 m cσ .

(∇ ∧ A + A ∧ ∇

) ](493)

which, since

∇ ∧ A Ψ(r) = Ψ(r)(∇ ∧ A

)− A ∧ ∇ Ψ(r) (494)

and as B = ∇ ∧ A, yields the non-relativistic Pauli Hamiltonian includingthe anomalous Zeeman interaction

HP =[

12 m

σ0

(p − q

cA

)2

− h q

2 m cσ . B + σ0 q φ

](495)

Thus, all the terms in the first line of equation (490) are found in the non-relativistic theory whereas the terms in the second line represent interactions,Hrel, which have a relativistic origin. The relativistic terms are given by

Hrel = − 18 m3 c

p4 +q h

4 m2 c2σ .

[∇ φ ∧ ( p − q

cA )

]+

q h3

8 m2 c2∇2φ

(496)

The first term which is proportional to p4 represents a relativistic correctionto the kinetic energy. The next term is the spin-orbit interaction which can beinterpreted as being caused by the interaction of the spin with the magnetic fieldproduced by the electron’s own orbital motion. The last term is the Darwinterm, which is often discussed as an interaction with a classical electron offinite spatial extent. Thus, the spin-orbit interaction for an electron is trulya relativistic effect and, unlike the other relativistic corrections, is not verysymmetric. It is given by the pseudo-scalar interaction

− q h

4 m c2σ . ( v ∧ E ) (497)

Due to its reduced symmetry, the spin-orbit interaction raises the degeneracyof the bands at high symmetry points in k space (R.J. Elliott, Phys. Rev. 96,280 (1954)), such as those on the hexagonal faces of the h.c.p. Brillouin zone.

——————————————————————————————————

8.1.10 Exercise 29

The effect of the Bragg planes on the density of states can be calculated fromthe nearly free electron model. For simplicity, consider the effect of one Braggplane. The Bloch wave vector k is resolved into components parallel, k‖, andperpendicular, k⊥, to the reciprocal lattice vector Q

k = k⊥ + k‖ (498)

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The energy of the two bands can be written as

Ek,± =h2

2 mk2⊥ + ∆E±(k‖) (499)

where

∆E±(k‖) =h2

2 m

[k2‖ +

12

(Q2 − 2 k‖ Q

) ]

±

( [h2

4 m

(Q2 − 2 k‖ Q

) ]2+ | V (Q) |2

) 12

(500)

describes the splitting of the two bands. (Note that the band energies are notperiodic in k‖. This is a consequence of our artificial assumption that there isonly one Bragg plane.) For each band, the density of state per spin is

ρ±(E) =V

( 2 π )3

∫d3 k δ( E − Ek,± ) (501)

Show that the density of states is given by(2 mh2

)V

4 π2

(kmax‖(E) − kmin‖(E)

)(502)

where E = ∆E±(km‖) defines the maximum and minimum value of k‖.

Show that, if the constant energy surface cuts the zone, i.e.,

E0Q

2

− | V (Q) | ≤ E ≤ E0Q

2

+ | V (Q) | (503)

then for the lower band one has

kmax‖(E) =Q

2(504)

and

kmin‖(E) = −√

2 m E

h2 + O(|V (Q)|2) (505)

for E > 0.

Show that

ρ+(E) =V

4 π2

(m

h2

) (kmax‖(E) − Q

2

)for E ≥ E0

Q

2

+ | V (Q) | (506)

Show that the energy derivative of the density of states, ∂ρ∂E , is singular at

the energiesE = E0

Q

2

± | V (Q) | (507)

——————————————————————————————————

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8.1.11 Exercise 30

Consider the point W on the Brillouin zone boundary of an f.c.c. crystal. ThreeBragg planes meet at W. The k value at W is

kW =(

2 πa

)(1,

12, 0) (508)

The three planes are the (2, 0, 0), (1, 1, 1) and (1, 1, 1) planes. The four freeelectron energies are

E01 =

h2

2 mk2

E02 =

h2

2 m

(k − 2 π

a(1, 1, 1)

)2

E03 =

h2

2 m

(k − 2 π

a(1, 1, 1)

)2

E04 =

h2

2 m

(k − 2 π

a(2, 0, 0)

)2

(509)

These four energies are degenerate at W and are equal to E0W = h2

2 m k2W .

Show that near W, the first order energies are given by the solutions of∣∣∣∣∣∣∣∣∣∣∣∣∣∣

E01 − E V1 V1 V2

V1 E02 − E V2 V1

V1 V2 E03 − E V1

V2 V1 V1 E04 − E

∣∣∣∣∣∣∣∣∣∣∣∣∣∣= 0

where V2 = V (2, 0, 0) and V1 = V (1, 1, 1) = V (1, 1, 1), and that at Wthe roots are

E = E0W − V2 doubly degenerate

E = E0W + V2 ± 2 V1 singly degenerate (510)

Two Bragg planes meet at the point U, which corresponds to the k value

kU =(

2 πa

)(1,

14,14) (511)

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Show that at the U point the band energies are given by

E = E0U − V2

E = E0U +

V2

2± 1

2

√V 2

2 + 8 V 21 (512)

where

E0U =

h2

2 mk2

U (513)

is the free electron energy at point U.

——————————————————————————————————

8.1.12 Exercise 31

Consider a nearly free electron band structure near a Bragg plane. Let

k =Q

2+ q (514)

and resolve q into the components q‖ and q⊥ parallel and perpendicular to the

Bragg planeQ

2 . Then, the energy bands are given by

E = E0Q

2

+h2

2 mq2 ±

(4 E0

Q

2

h2

2 mq2‖ + | V (Q) |2

) 12

(515)

It is convenient to express the Fermi-energy µ in terms of the energy of thelower band at the Bragg plane

µ = E0Q

2

− | V (Q) | + ∆ (516)

Show that when 2 V (Q) > ∆ > 0, then the Fermi-surface is only composedof states in the lower Bloch band. Furthermore, show that the Fermi-surfaceintersects the Bragg plane in a circle of radius ρ where

ρ =

√2 m ∆h2 (517)

Show that, if ∆ > 2 | V (Q) |, the Fermi-surface cuts the Bragg plane intwo circles of radius ρ1 and ρ2 such that the area between them is

π

(ρ21 − ρ2

2

)=

4 π mh2 | V (Q) | (518)

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This area is measurable through de Haas - van Alphen experiments.

——————————————————————————————————

8.1.13 Exercise 32

In a weak periodic potential the Bloch states in the vicinity of a Bragg planecan be approximated in terms of two plane waves.

Let k be a wave vector with polar coordinates (θ, ϕ) in which the z axisis taken to be the direction Q of the reciprocal lattice vector that defines theBragg plane.

(i) If E < h2

2 m

Q

2

2

show that to order V (Q)2 the surface of energy E isgiven by

k(θ, ϕ) =

√2 m E

h2

(1 + δ(θ)

)(519)

where

δ(θ) =m

| V (Q) |2

E

h2 Q2 − 2 h Q cos θ√

2 m E(520)

(ii) Show that | V (Q) |2 results in a shift of the Fermi-energy given by

∆µ = µ − µ0 (521)

where

∆µ = − 18| V (Q) |2

µ0

(2 kF

Q

)ln∣∣∣∣ Q + 2 kF

Q − 2 kF

∣∣∣∣ (522)

——————————————————————————————————

8.1.14 Exercise 33

Consider an energy E which lies within the gap between the upper and lowerbands at point k on the Bragg plane which is defined by the reciprocal latticevector Q. Let

k =Q

2+ q (523)

(i) Find an expression for the imaginary part of k for E within the gap.

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(ii) Show that for E at the center of the gap, the imaginary part of k satisfies

(Im k

)2

= − Q2

√ (Q2

2

)2

+ | 2 mh2 V (Q) |2 (524)

Thus, on solving for k given E, there is a range of Im k when Re k = Q2 .

Complex wave vectors are important for the theory of Zener tunnelling be-tween two bands, caused by strong electric fields. Complex wave vectors alsooccur in the description of states that are localized near surfaces.

——————————————————————————————————

8.2 The Pseudo-Potential Method

The failure of the nearly free electron model is primarily due to the large valuesof the potentials, V (Q), calculated from first principles, and the small valuesof the experimentally observed splittings between the bands. Due to the largevalue of the lattice potential, if the wave functions are expanded terms of planewaves very many plane waves ( of the order of 106 ) are needed to obtain conver-gence. Furthermore, band structure calculations with the exact lattice potentialare expected to reproduce the entire set of wave functions ranging from the corewave functions located within the ions, up to the valence and/or conductionwave functions. Since the core electrons are very localized and almost atomic, alarge number of plane waves are needed for an accurate calculation of the corewave functions. Large numbers of plane waves are also needed to calculate thevalence band wave functions. The need for a large number of Fourier compo-nents to calculate the valence band wave functions can be understood by theconsideration of the fact that the conduction or valence band states have to beorthogonal to the wave functions of the core electrons. Thus, the conductionelectrons should have wave functions that exhibit rapid oscillations in the vicin-ity of the ion cores. Historically, there have been many methods which wereused to avoid the need to use many plane waves. The methods used range fromorthogonalized plane waves, augmented plane waves and pseudo-potentials. Allthese methods have some common features, namely the feature of producingwave functions that require fewer plane wave components in the expansion and,thereby, increase the rate of convergence, and concomitantly diminish the effectof the ionic potential. The pseudo-potential method provides a first principlesway of explicitly finding a smaller effective potential.

The electrons in the valence band move in a periodic potential Vions(r) pro-vided by the ions. The ionic potential already includes a partial screening ofthe nuclear potential by the ion core electrons.

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The valence band Bloch functions φvk,n(r) undergo many oscillations in the

region of the core as they must be orthogonal to the core electron wave functionsφc

k,α(r). In the Dirac notation, the orthogonality condition is expressed as

< φvk,n | φc

k,α > = 0 (525)

The valence band Bloch function can be expressed in terms of a smooth function

ψvk,n(r) (526)

that doesn’t contain the oscillations that orthogonalize the Bloch state, | φvk,n >,

with the core wave states. The smooth function is known as the pseudo-wavefunction. The pseudo-wave function is related to the valence band Bloch func-tion by the definition

| φvk,n > = | ψv

k,n > −∑α

| φck,α > < φc

k,α | ψvk,n > (527)

This definition automatically ensures the othornomality of the core states withthe valence band states without placing any restriction on the form of thepseudo-wave function. The basic idea behind pseudo-potential theory is thatthe smooth pseudo-wave function represents the electronic wave function in theregion between the cores, and may be expressed in terms of only a few planewave components (J.C. Phillips and L. Kleinman, Phys. Rev. 116, 287 (1959)).

Since the Bloch state, | φvk,n >, satisfies the one-particle Schrodinger equa-

tionH | φv

k,n > = Evk,n | φv

k,n > (528)

one finds that the smooth function satisfies

H | ψvk,n > −

∑α

Ecα | φc

k,α > < φck,α | ψv

k,n > =

= Evk,n

(| ψv

k,n > −∑α

| φck,α > < φc

k,α | ψvk,n >

)(529)

This equation can be re-arranged to yield an eigenvalue equation for the (un-known) smooth function, which has the same energy eigenvalues as the exacteigenfunction. The rearranged equation has the form(

H + V (Evk,n)

)| ψv

k,n > = Evk,n | ψv

k,n > (530)

where

V (Evk,n) =

∑α

(Ev

k,n − Eck,α

)| φc

k,α > < φck,α | (531)

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is a non-local and energy dependent contribution to the potential. The impor-tant point is that this potential may be regarded as being positive and, therefore,counteracts the effect of the large negative potential due to the ions. This canbe seen by taking the expectation value of the energy dependent potential inany arbitrary state | Ψ >

< Ψ | V (Evk,n) | Ψ > =

∑α

(Ev

k,n − Eck,α

)| < Ψ | φc

k,α > |2 (532)

and as the valence electrons have a higher energy than the core electrons, Evk,n >

Eck,α, one finds

< Ψ | V (Evk,n) | Ψ > ≥ 0 (533)

Thus, the potential operator is effectively positive as it increases the expectationvalue of the energy for an arbitrary state.

The operator V is non-local. This can be seen by considering the action of Von an arbitrary wave function Ψ(r). The operator has the effect of transformingthe state through

V (Evk,n) Ψ(r) =

∑α

(Ev

k,n − Eck,α

) ∫V

d3r′ φ∗ck,α(r′) Ψ(r′) φck,α(r) (534)

Thus, the operator when acting on the wave function at position r changes theposition to r′.

If the original one-particle Schrodinger equation for φvk,n(r) has the form(

− h2

2 m∇2 + Vions(r)

)φv

k,n(r) = Evk,n φv

k,n(r) (535)

then the Schrodinger equation for the smooth function ψvk,n(r) has the form(

− h2

2 m∇2 + Vions(r) + V (Ev

k,n))ψv

k,n(r) = Evk,n ψv

k,n(r) (536)

The Schrodinger equation for the smooth wave function has exactly the sameenergy eigenvalues as the original potential. The pseudo-potential is defined as

Vpseudo = Vions(r) + V (Evk,n) (537)

and, as has been shown, the effect of the pseudo potential is much weaker thanthat of Vions(r). Also as the eigenstate ψv

k,n(r) is a smooth function it can beexpanded in terms of a few planes waves

ψvk,n(r) =

∑Q

Ck−Q exp[i ( k − Q ) . r

](538)

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Thus, the pseudo-potential may be treated as a weak perturbation and givesresults very similar to those of the nearly free electron model.

There are many different forms that the pseudo-potential can take (B.J.Austin, V. Heine and L.J. Sham, Phys. Rev. 127, 276 (1962)). The non-localpseudo-potential can be approximated by a local potential and, as its energydependence is weak, Ev can be set to zero in the pseudo-potential. In thisapproximation, the pseudo-potential is almost zero within the core. This isa result of the so-called cancellation theorem (M. Cohen and V. Heine, Phys.Rev. 122, 1821 (1961)). The cancellation theorem can be found from classicalconsiderations. Classically, the gain in kinetic energy of a conduction electronas it enters the core region is equal to the potential energy. As the oscillationsin φc

k,α(r) give rise to the kinetic energy of the electron in the core region,one expects the pseudo-potential to cancel in the core region. Therefore, thepseudo-potential follows the ionic core potential for distances larger than theionic core radius Rc, at which point the attractive potential almost shuts off.The empty core approximation to the atomic pseudo-potential (N.W. Ashcroft,Phys. Letts. 23, 48 (1966)) is given by

Vpseudo(r) = − Z e2

rfor r > Rc

Vpseudo(r) = 0 for r < Rc

(539)

Basically, this is a reflection of the fact that the valence electrons do not probethe region of the cores as this region is already occupied by the core electronsand the Pauli exclusion principle forbids the overlap of states.

The Fourier transform of the local pseudo-potential is a smooth function ofthe wave vector q.

Vpseudo(q) = − 4 π Z e2

q2cos q Rc (540)

Only the values of Vpseudo(q) at the reciprocal lattice vectors Q are physi-cally important and most of these are small. When one includes the effect ofthe screening electron clouds, the pseudo-potential is replaced by the screenedpseudo-potential

Vpseudo(r) = − Z e2

rexp

[− kTF r

]for r > Rc

Vpseudo(r) = 0 for r < Rc (541)

The Fourier transform of the screened pseudo-potential is given by

Vpseudo(q) = − 4 π Z e2

q2 + k2TF

cos q Rc (542)

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which is weakened with respect to the original potential.

8.2.1 The Scattering Approach

The pseudo-potential is a potential that gives the same eigenvalues as Vions(r),for the valence electron states. The pseudo-potential may be obtained fromscattering theory.

Consider a single ionic scattering center with a spherically symmetric poten-tial V (r) which is zero for r > R. Then for r > R, the radial wave functionhas the asymptotic form

Rl(r, E) = Cl

[jl(kr) − tan δl ηl(kr)

](543)

where

E =h2 k2

2 m(544)

and jj(x) and ηl(x) are the spherical Bessel and Neumann functions. The coef-ficients Cl and the phase shifts δl(E) are obtained by matching the asymptoticform to the solution at some large distance r = R. The exact logarithmicderivative of Rl(r, E) at r = R can be defined as

Ll(E) =R′l(R,E)Rl(R,E)

(545)

The matching condition of the logarithmic derivative of the asymptotic formwith the logarithmic derivative of the wave function at r = R leads to theequation

tan δl(E) =jl(kR) Ll(E) − k j′l(kR)ηl(kR) Ll(E) − k η′l(kR)

(546)

The phase shifts δl(E) determine the scattering amplitude f(θ, E) for a particleof energy E to be scattered through an angle θ. Partial wave analysis yields therelation

f(θ, E) =1

2 i k

∑l

( 2 l + 1 )(

exp[

2 i δl

]− 1

)Pl(cos θ) (547)

The scattering amplitude only depends on the phase shift modulo π. The phaseshift can always be restricted to the range − π

2 to + π2 by defining

δl = nl π + ∆l (548)

where nl is an integer chosen such that the

| ∆l | <π

2(549)

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The value of nl denotes the number of the oscillations in the radial wave func-tion Rl(r, E). The (truncated) phase shifts ∆l produce the same scatteringamplitude as the original phase shift δl(E).

The atomic pseudo-potential is defined as any potential in which the com-plete phase shifts are the truncated phase shifts ∆l and, thus, gives rise to thesame scattering amplitude, but does not produce any bound states (accordingto Levinson’s theorem). The pseudo-radial wave functions Rl(r, E) have nonodes and, thus, have no rapid oscillations. Therefore, the pseudo-radial wavefunction can be represented in terms of a finite superposition of plane waves oflong wave length. The pseudo-potential actually only depends on the functionLl(E). From the knowledge of logarithmic derivative, Ll(E), one can constructthe pseudo-potential. One method has been proposed by Ziman and Lloyd.

8.2.2 The Ziman-Lloyd Pseudo-potential

Ziman and Lloyd independently proposed a pseudo-potential which is local in rand is zero everywhere except on the surface of a shell of radius R. The potentialoperator, V ZL, is written as

V ZL =∑

l

Bl(E) δ( r − R ) Pl (550)

where Pl projects onto the states with angular momentum l (J.M. Ziman, Proc.Phys. Soc. (London) 86, 337 (1965), P. Lloyd, Proc. Phys. Soc. (London),86, 825 (1965)). Inside the sphere the potential is zero and so the radial wavefunction is just proportional to jl(kr), since the Neumann function is excludeddue to the boundary condition at r = 0. The amplitude Bl(E) is chosen soas to give the proper asymptotic properties of the wave function of the truepotential V , for r > R.

The pseudo-radial wave functions satisfy the radial Schrodinger equation,given by

− h2

2 m1r2

∂r

(r2

∂rRl

)+[h2 l ( l + 1 )

2 m r2+ V ZL(r)

]Rl(r) = E Rl(r)

(551)The derivative of the pseudo-radial wave function is found by integrating theRadial Schrodinger equation over the shell at r = R

− h2

2 m∂

∂rRl(r)

∣∣∣∣R+

R−

+ Bl(E) Rl(R) = 0 (552)

The pseudo-wave function is matched with the true wave function at the radiusr = R+. The matching condition determines the function Bl(E) in the pseudo-potential in terms of the logarithmic derivative of the true wave function, Ll(E).

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Thus, the coefficient Bl(E) is related to Ll(E) via

Ll(E) − kj′l(kR)jl(kR)

=2 mh2 Bl(E) (553)

Therefore, the Bl(E), for different l, are determined in terms of the exact valueof logarithmic derivatives. The projection operator is simply given as

Pl =∑m

| l,m > < l,m | (554)

which also gives rise to the non-locality of the pseudo-potential operator. Thepseudo-potential for the solid can be constructed as a superposition of thepseudo-potentials of the ions.

It should be noted that the pseudo-potential only cancels for states of angu-lar momentum l if there are core states with angular momentum l otherwise, theelectrons experience the full potential. Thus, in C the 2s electron experience thecancelled pseudo-potential but the 2p electrons interact with the full potential.The 2p electrons are relatively tightly bound compared with the 2s. Thus, thes → p promotion energy is lower than in the other group IV elements Si, Ge,Sn and Pb. This allows C to easily form the tetrahedrally directed sp3 valencebonds. Similarly, in the 3d transition metals, the 3d electrons are tightly boundcompared with the 4d or 5d electrons in the second and third series. Thus, the3d electrons form tightly bound narrow bands, and pseudo-potential theory isinappropriate.

In summary, the pseudo-potentials can be created from first principles andthen, if the pseudo-potential is weak enough, the nearly free electron model canbe used to obtain the results for the valence bands of real solids.

——————————————————————————————————

8.2.3 Exercise 34

An electron outside a hydrogen atom with a 1s core state is treated by thepseudo-potential method. Calculate the Bloch wave function for an electronwhich has a pseudo-wave function that can be approximated by a single planewave. Discuss whether this function is appropriate to represent a 2s wave func-tion. Evaluate the magnitude of the pseudo-potential, for low energy electronstates.

——————————————————————————————————

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8.3 The Tight-Binding Model

The tight-binding method is appropriate to the situation in which the electrondensity in a solid can be considered to be mainly a superposition of the densi-ties of the individual atoms (J.C. Slater and G.F. Koster, Phys. Rev 94, 1498(1954)). However, the tight-binding method does produce slight corrections tothe atomic densities. It should be a good approximation for the inner coreorbitals where the ratio of the radius of the atomic orbit to the inter-atomicseparation is small.

Consider a lattice with a mono-atomic basis. The Hamiltonian for a singleion centered at 0 is H0 and has eigenstates | φm > defined by the eigenvalueequation

H0 | φm > = Em | φm > (555)

The periodic potential of the ions can be written as the sum of the potentialfrom the ion at site 0, V0, and the potential due to all other ions in the crystallinelattice ∆V

Vions = V0 + ∆V (556)

Thus, the Hamiltonian is written as the sum of a single ion Hamiltonian andthe potential due to the rest of the ions

H = H0 + ∆V (557)

In the tight-binding method it is convenient to define Wannier functions, φn, asa transform of the Bloch functions

φk,n(r) =∑R

exp[i k . R

]φn( r − R ) (558)

Thus, the Wannier functions are centered around the different lattice points R.The Wannier states are almost localized states and are composed of a linearsuperposition of the atomic states

| φn > =∑m

bn,m | φm > (559)

The band structure is found from the energy eigenvalue equation for the Blochwave functions

H | φk,n > = Ek,n | φk,n > (560)

or (H0 + ∆V

)| φk,n > = Ek,n | φk,n > (561)

This energy eigenvalue equation is projected onto the atomic wave function| φm > located at O leading to

< φm | H | φk,n > = < φm |(H0 + ∆V

)| φk,n >

= Ek,n < φm | φk,n > (562)

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However, the state | φm > is an eigenstate of the atomic Hamiltonian H0 andso the overlap is given by

< φm | H0 | φk,n > = Em < φm | φk,n >

(563)

On substituting this relation into the matrix elements of the eigenvalue equation,the equation reduces to

( Ek,n − Em ) < φm | φk,n > = < φm | ∆V | φk,n >

(564)

The Bloch wave function can be expressed in terms of the Wannier functions,and then the Wannier functions are expressed in terms of the atomic wavefunctions via

φk,n(r) =∑R

exp[i k . R

]φn( r − R )

=∑R,m′

bn,m′ exp[i k . R

]φm′( r − R )

(565)

The overlap of the Bloch functions and the atomic wave function is expressedas the sum of the overlap of atomic wave functions at the same site and theoverlaps of atomic wave functions centered at different sites

< φm | φk,n > =∑m′

δm,m′ bn,m′ +

+∑m′

∑R 6=0

bn,m′ exp[i k . R

] ∫d3r φ∗m(r) φm′(r −R)

(566)

Substituting this into the energy eigenvalue equation, one obtains the equation∑m′

(Ek,n − Em

)δm,m′ bn,m′ +

+(Ek,n − Em

) ∑m′,R 6=0

bn,m′ exp[i k . R

] ∫d3r φ∗m(r) φm′(r −R)

=∑m′

bn,m′

∫d3r φ∗m(r) ∆V (r) φm′(r) +

+∑

m′,R 6=0

bn,m′ exp[i k . R

] ∫d3r φ∗m(r) ∆V (r) φm′(r −R)

(567)

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The first term on the left side involves the overlap of two atomic wave functionboth centered at site 0. These atomic wave functions are part of an orthonormalset of eigenfunctions. The second term on the left hand side involves the overlapof atomic wave functions at site 0 and site R, and may be expected to beexponentially smaller than the first term.

1 ∣∣∣∣ ∫ d3r φ∗m(r) φm′(r −R)

∣∣∣∣ (568)

The two terms on the right both involve the potential ∆V and the atomic wavefunction φm(r) located at site 0. The first term on the right hand site involvesthe effect of the potential due to the other ions on the central atom. This termrepresents the effect of the crystalline electric field on the atomic levels. The re-maining term represents the delocalization of the electrons. The magnitudes ofthe coefficients bn,m that appear in the expansion of the Wannier state cruciallydepend on the ratios of the overlap integrals to the energy difference Ek,n − Em.Generally, this allows one to approximate the Wannier functions by retainingonly a finite number of atomic wave functions in their expansions. That is, theexpansion of the Wannier function is truncated by only considering atomic wavefunctions that have energies close to the energy of the Bloch state.

The set of equations can be solved approximately by considering the spatialdependence.

If one assumes that the potential ∆V is non-zero only in the range whereφm(r) is negligibly small, both terms on the right hand side will be approxi-mately zero. Thus, in a first order and very crude approximation, it is foundthat Ek,n = Em.

On keeping the two center and three center integrals in which R is limitedto a few neighbor sites to O, and to atomic states with a few energies close toEm, the set of equations truncate into a finite set. These can be solved to yieldthe Bloch state energies and the Bloch wave functions.

In general, the band widths are linearly related to the overlap matrix ele-ments, γi,j , where

γi,j(R) = −∫

d3r φ∗i (r) ∆V (r) φj(r −R) (569)

in which φj are atomic wave functions and R represent atomic positions relativeto the central atom 0. The band widths increase with the increase in the ratio ofthe spatial extent of φi(r) to the typical separation R. Thus, bands with largebinding energies which tend to have wave functions with small spatial extentsform narrow bands while the higher energy bands have broader band widths.

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The overlap integrals are conventionally expressed in terms of the angularmomentum quantum numbers (l,m) of the atomic wave functions that are quan-tized along the axis joining the atoms. The matrix elements are non-negligibleonly if the z-component of the angular momentum satisfies a selection rule. Thenon-zero overlap matrix elements are then characterized by m. In analogy tothe atomic wave functions, the type of bonding is labelled by the greek lettersσ, π and δ respectively, corresponding to m = 0, m = ± 1 and m = ± 2.The overlap integrals corresponding to ssσ and ppπ bonds are negative, as thelobes of the wave function with the same sign overlap the negative crystal fieldpotential. The ppσ bonds are positive at large to intermediate separations aslobes of opposite sign overlap the negative potential, but become negative atsmall values of R where the overlap of lobes with the same sign start to domi-nate. The spσ overlap is an odd function of R and vanishes for zero separationR = 0 as the different atomic wave functions are orthogonal. The sign of thespσ overlap depends on the ordering of the s and p orbitals along the axis. Thespσ bond is positive if lobes of different sign overlap and is negative if lobes ofthe same sign overlap.

The Helmholtz-Wolfsberg approximation consists of replacing the value ofthe potential ∆V by a constant. The magnitude of the potential is factorized outof the integral. Therefore, the overlap integrals merely depend on the displacedatomic wave functions, i and j. The overlap integrals are then written as

γi,j(R) = − ∆V ti,j(R) (570)

The overlap between hydrogen-like 1s wave functions

φ1s(r) =

√κ3

πexp

[− κ r

](571)

can be evaluated from the Fourier transformed wave function

φ1s(q) =

√κ3

π

8 π κ( q2 + κ2 )2

(572)

The overlap of two wave functions, with a relative displacement R, can beevaluated via the convolution theorem∫

d3r φ∗1s(r) φ1s(r−R) =∫

d3q

( 2 π )3φ1s(−q) φ1s(q) exp

[i q . R

](573)

with the result that

t1s,1s,σ = −(

1 + κ R +13κ2 R2

)exp

[− κ R

](574)

On using the hydrogenic-like 2s and 2p wave functions,

φ2s(r) =

√κ3

π

(1 − κ r

)exp

[− κ r

]

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φ2p,0(r) =

√κ3

πcos θ κ r exp

[− κ r

]φ2p,±1(r) =

√κ3

2 πsin θ exp

[± i ϕ

]κ r exp

[− κ r

](575)

one finds that the Fourier transform of the 2s and 2p wave functions are givenby

φ2s(q) =√κ3

32 π κ ( q2 − κ2 )( κ2 + q2 )3

Y 00 (θq, ϕq)

φ2p,0(q) = i

√κ5

364 π κ q

( κ2 + q2 )3Y 0

1 (θq, ϕq)

φ2p,±1(q) = − i

√κ5

364 π κ q

( κ2 + q2 )3Y ±1

1 (θq, ϕq) (576)

where the dependence on the direction of q is expressed through the factorsY m

l (θq, ϕq). The functions Y ml (θ, ϕ) are the spherical harmonics. On using the

convolution theorem, the approximate overlap integrals are evaluated as

t2s,2s,σ = −(

1 + κ R +13κ2 R2 +

115

κ4 R4

)exp

[− κ R

]

t2s,2p,σ =(

1330

κ3 R3

)exp

[− κ R

]

t2p,2p,σ = −(

1 + κ R +15κ2 R2 − 2

15κ3 R3 − 1

15κ4 R4

)exp

[− κ R

]

t2p,2p,π = −(

1 + κ R +25κ2 R2 +

115

κ3 R3

)exp

[− κ R

](577)

where κ determines the spatial extent of the wave function and R is the inter-atomic separation. Typically for a material such as C, the relative strength ofthe bonds are given by the ratios at the radius R where the bonding saturates.Typical values of the relative strengths are given by

t2s,2s,σ : t2s,2p,σ : t2p,2p,σ : t2p,2p,π = − 1 : 1 : 0.75 : − 0.49 (578)

The structure of tight-binding d bands can be found by expressing the Blochfunctions in terms of five atomic d wave functions that correspond to the differ-ent eigenvalues of the z component of the orbital angular momentummz = ± 2,mz = ± 1 and mz = 0. If mz is quantized along the axis between two atoms,

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the tight-binding overlap integrals between these sets of states are denoted,respectively, by td,d,δ, td,d,π and td,d,σ. The matrix elements for arbitrary orien-tations are tabulated in the article of Slater and Koster (1954). Representativeratios of the strengths of the td,d,δ, td,d,π and td,d,σ bonds are given by

td,d,δ : td,d,π : td,d,σ = − 6 : 4 : − 1 (579)

In general, the tight-binding bands obtained by considering d bands alone ishighly inaccurate. Usually, a broad s band crosses the narrow set of d bands.This degeneracy is lifted as the d and s bands hybridize strongly (V. Heine,Phys. Rev. 153, 673 (1967)).

The Bloch functions are constructed out of localized atomic levels with equalamplitude, but only involves the phase exp[ i k . R ]. Thus, the electrons areequally likely to be found in any atomic cell of the crystal. Also, Re φk,n

shows that the atomic structure is modulated by the sinusoidal variation ofexp[ i k . R ]. Since the mean velocity is given by

v(k) =1h∇Ek 6= 0 (580)

then the electrons have a non-zero velocity and will be able to move through-out the crystal. The non-zero velocity is due to the coherent tunnelling of theelectron between the atoms.

For a lattice with a basis, the Bloch wave function is given

φk(r) =∑R

exp[ i k . R ]∑j,m

aj,m φm(r − rj −R) (581)

where rj are the positions of the basis atoms and aj,m are the amplitudes ofthe orbitals on the j-th basis atom. The equation for the Bloch function has astructure in which the basis atoms in each unit cell can be viewed as formingmolecules. These molecular wave functions in each lattice cell are then com-bined via the tight-binding method.

8.3.1 Tight-Binding s Band Metal

For a simple s-band metal the Wannier state | φn > can be approximated bythe atomic s wave function. As this s wave function is non-degenerate, one has

| φ1 > ≈ | φs > (582)

or bs = 1. All other coefficients are set to zero, corresponding to the assumptionthat the energy of the s band, Es, is well separated from the energies of theother bands. This is probably a good assumption for the 1s band which is often

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regarded as forming part of the core of the ions. The energy eigenvalue equationtruncates to(

Es,k − Es

) (1 +

∑R 6=0

exp[i k . R

] ∫d3r φ∗s(r) φs(r −R)

)

= < φs | ∆V | φs > +∑R 6=0

exp[i k . R

] ∫d3r φ∗s(r) ∆V (r) φs(r −R)

(583)

The overlap between the atomic wave functions on different sites is defined tobe a function α(R) through∫

d3r φ∗s(r) φs(r −R) = α(R) (584)

The matrix elements of the atomic functions centered at 0 with the tail of thepotential, ∆V , is defined to be β where

< φs | ∆V | φs > = − β (585)

and the matrix elements of the atomic functions centered at 0 and R with thetail of the potential is defined to be γ(R) through∫

d3r φ∗s(r) ∆V (r) φs(r −R) = − γ(R) (586)

The dispersion relation can be expressed in terms of these three functions via

Es,k = Es −

( β +∑

R 6=0 γ(R) exp[i k . R

]1 +

∑R 6=0 α(R) exp

[i k . R

] ) (587)

Since γ(R) = γ(−R) and α(R) = α(−R) the dispersion relation E1,k is aneven periodic function of k. For bonding only to the nearest neighbors, the sumsover R are truncated to run only over the nearest neighbors.

For the f.c.c. structure the dispersion relation becomes

Es,k = Es −

(β + γ(k)1 + α(k)

)(588)

where

γ(k) = 4 γ(

coskx a

2cos

ky a

2+ cos

kx a

2cos

kz a

2+ cos

ky a

2cos

kz a

2

)(589)

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and

α(k) = 4 α(

coskx a

2cos

ky a

2+ cos

kx a

2cos

kz a

2+ cos

ky a

2cos

kz a

2

)(590)

Usually α is neglected as it is small. The tight-binding bands are off-set fromEs by an energy β due to the tail of the potential of all other atoms at O,

β = − < φs | ∆V | φs > (591)

The band width is governed by the overlap of the central atom’s wave functionwith the nearest neighbor atomic wave function. This overlap, γ, is evaluatedfrom

γ = −∫

d3r φ∗s(r) ∆V (r) φs(r −Rnn) (592)

The band width for the f.c.c. lattice is 12 γ.

For small | k | a one can expand the dispersion relation in powers of k

E1,k = Es − β − 12 γ + γ k2 a2 (593)

which is independent of the direction of k near k = 0. Thus, the constantenergy surfaces are spherical around k = 0.

The gradient of the energy has a component perpendicular to the squareface of the Brillouin zone (the face containing the X point) that is given by

∂Ek

∂kx= 2 a γ sin

kx a

2

(cos

ky a

2+ cos

kz a

2

)(594)

Thus, if E1,k is plotted along any line in k space which is perpendicular to thesquare face, it crosses with zero slope.

The points on the hexagonal face satisfy the equation

kx + ky + kz =3 πa

=32

(2 πa

)(595)

Since there is no plane of symmetry parallel to the hexagonal face, the energyplotted along any line perpendicular to the hexagonal face is not required tocross with zero slope,

∇ E1,k . e ∝ sinkx a

2

(cos

ky a

2+ cos

kz a

2

)+ sin

ky a

2

(cos

kx a

2+ cos

kz a

2

)+ sin

kz a

2

(cos

kx a

2+ cos

ky a

2

)(596)

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This only vanishes along the lines joining L (12 ,

12 ,

12 ) to the vertices W (1, 1

2 , 0).

For degenerate levels such as p or d levels, the tight-binding method leadsto a N × N secular equation where N is the orbital degeneracy.

For heavy elements, spin-orbit coupling should be included. In this case, thepotential ∆V should have a spin dependent contribution. The spin-orbit cou-pling breaks the spin degeneracy and increases the size of the secular equationby a factor of 2 (J. Friedel, P. Lenghart and G. Leman, J. Phys Chem. Solids25, 781 (1964)).

——————————————————————————————————

8.3.2 Exercise 35

Consider two p orbitals, one located at the origin and another at the pointR (cos θx, cos θy, cos θz), where R is the separation between the two ions andthe cos θ are the direction cosines of the displacements. The overlap parametersfor the orbitals φi(r) and φj(r) are defined by

γi,j(R) = −∫

d3r φ∗i (r) ∆V (r) φj(r −R) (597)

Show that the overlap parameters are given by

γx,x = −(tppσ cos2 θx + tppπ sin2 θx

)γx,y = −

(tppσ − tppπ

)cos θx cos θy (598)

Thus, the tight-binding parameters not only depend on the distance, R, butalso depend on the direction.

——————————————————————————————————

8.3.3 Exercise 36

Consider the p bands in a cubic crystal, which have the p wave functions

φpx(r) = x f(r)φp

y(r) = y f(r)φp

z(r) = z f(r) (599)

where f(r) is a spherically symmetric function. The energies of the three pbands are found from the secular equation∣∣∣∣ ( Ek − Ep

)δi,j + βi,j + γi,j(k)

∣∣∣∣ = 0 (600)

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and

γi,j(k) =∑R

exp[i k . R

]γi,j(R) (601)

andγi,j(R) = −

∫d3r φ∗i (r) ∆V (r) φj(r −R) (602)

andβi,j = γi,j(0) (603)

Show that, using cubic symmetry,

βx,x = βy,y = βz,z = β (604)

and all other overlap matrix elements are zero

βx,y = βy,z = βx,z = 0 (605)

Assuming that only the nearest neighbor overlaps γi,j(R) are non-zero, showthat for a simple cubic lattice γi,j(k) are diagonal in i and j. Hence, the px, py

and pz wave functions generate three independent bands

Ex,k = Ep + 2 tppσ cos kxa + 2 tppπ ( cos kya + cos kza )Ey,k = Ep + 2 tppσ cos kya + 2 tppπ ( cos kxa + cos kza )Ez,k = Ep + 2 tppσ cos kza + 2 tppπ ( cos kxa + cos kya )

(606)

The relative values of these parameters can be estimated from first princi-ples calculations of bulk silicon, where the ratios were found to be given bytppσ : tppπ = 3.98 : − 1 .

——————————————————————————————————

8.3.4 Exercise 37

Consider the p bands in a face-centered cubic lattice with nearest neighborhopping γi,j(R). Show that the system is described by a 3 × 3 secular equationwhich is expressed in terms of four integrals

0 =

∣∣∣∣∣∣E − E0

k + M0x − M1

z − M1y

− M1z E − E0

k + M0y − M1

x

− M1y − M1

x E − E0k + M0

z

∣∣∣∣∣∣ (607)

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where the functions M0i and M1

i are given by

M0x = 4 γ0 cos

ky a

2cos

kz a

2

M1x = 4 γ1 sin

ky a

2sin

kz a

2(608)

and cyclic permutations. The energy E0k is given by

E0,k = Ep − β − 4 γ2

(cos

ky a

2cos

kz a

2+ cos

kx a

2cos

kz a

2+ cos

kx a

2cos

ky a

2

)(609)

Evaluate the integrals in terms of the overlap of atomic wave functions by us-ing the Helmholtz-Wolfsberg approximation. Also show that the three energybands are degenerate at the Γ point, and that when k is directed along the cubeaxis (Γ X) or the cube diagonal (Γ L), two bands are degenerate.

——————————————————————————————————

8.3.5 Exercise 38

The parent compound of the doped high temperature superconductors is La2CuO4

which has the Perovskite structure. In this structure, the CuO2 atoms formplanes. Each Cu atom is surrounded by an octahedra of O atoms of which fouratoms are in the plane. The in-plane Cu − O bonds can serve to define thex and y axes. The O atoms that have the Cu−O bonds parallel to the x axisare denoted as Ox, whereas the other O atoms are denoted by Oy. In this coor-dinate system, the appropriate basis orbitals are the Cu dx2−y2 orbitals, whilethe only Ox states which mix with the Cu states are the px states and the onlyOy states that mix with the Cu are the py states.

Using the tight-binding form of the Bloch wave function

φk =∑R

exp[i k . R

] (a φd

x2−y2(r) + bx φpx(r− a

2ex) + by φ

py(r− a

2ey)

)(610)

find the energy bands for the CuO2 planes.

——————————————————————————————————

8.3.6 Exercise 39

Evaluate the tight-binding density of states for the s states of a simple hyper-cubic lattice in d = 1, d = 2, d = 3, d = 4, in which only the nearestneighbor hopping matrix elements t are retained. Calculate the form of the

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density of states when d → ∞.

——————————————————————————————————

8.3.7 Exercise 40

Consider the tight-binding density of states for s states on a tetragonal latticewhere the overlap in the c direction is t′ and the overlap in either the a or bdirection is t. Assume that t t′. Examine the form of the Fermi-surfacewhen the band is nearly half-filled. Evaluate the density of states.

——————————————————————————————————

8.3.8 Wannier Functions

Consider the position r to have a fixed value. The Bloch functions can bewritten as

φk,n(r) =∑R

exp[i k . R

]fn(r,R) (611)

The Bloch function φk for fixed r is periodic in k, with periodicity given by theprimitive reciprocal lattice vectors Q. Clearly

φk+Q,n =∑R

exp[i ( k + Q ) . R

]fn(r,R)

=∑R

exp[i k . R

]fn(r,R)

= φk,n (612)

since Q and R satisfies the Laue condition. Thus, the Bloch functions areperiodic functions in k space. The Fourier coefficients, fn(r,R), that appear inthe k space Fourier expansion can be found from the inversion formulae

fn(r,R) =1Ωc

∫Ωc

d3k′ exp[− i k′ . R

]φk′,n(r) (613)

where the integration volume Ωc is the volume of one cell of the reciprocal lattice.

The simultaneous transformations r → r − R0 and R → R − R0 leavefn(r,R) unchanged

fn(r,R) = fn(r −R0, R−R0) (614)

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This is proved by considering the effect of the transformation r → r − R0 onthe definition of the functions fn(r,R)

φk,n(r) =∑R′

exp[i k . R′

]fn(r,R′) (615)

Applying the transformation on the Bloch function yields

φk,n(r −R0) =∑R′

exp[i k . R′

]fn(r −R0, R

′) (616)

and then, on transforming the sum over R′ as R′ = R − R0, one has

φk,n(r −R0) =∑R

exp[i k . (R−R0)

]fn(r −R0, R−R0) (617)

On comparing the above expression with the result of Bloch’s theorem

φk,n(r −R0) = exp[− i k . R0

]∑R

exp[i k . R

]fn(r,R) (618)

one recovers the symmetry relation

fn(r,R) = fn(r −R0, R−R0) (619)

Using the above symmetry of f(r,R) under a translation R0, and on choosingR0 = R one finds

fn(r,R) = fn(r −R, 0) = φn(r −R) (620)

which shows that the function only depends on the difference r − R. Hence, ithas been shown that the Bloch function can be expressed as

φk,n(r) =∑R

exp[i k . R

]φn(r −R) (621)

where φn(r) are the Wannier functions (G. Wannier, Phys. Rev. 52, 191 (1947)).The Wannier functions at different sites are orthogonal. Thus, as they are lin-early related to the Bloch wave functions φk,n(r), the set of Wannier functionsform a complete orthogonal set.

The Wannier functions are given in terms of the Bloch functions via

φn(r −R) =1Ωc

∫Ωc

d3k exp[− i k . R

]φk,n(r) (622)

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The Wannier functions are localized around the site R, as can be seen by sub-stituting the expression for the Bloch functions in the above equation

φn(r −R) =1Ωc

∫Ωc

d3k exp[

+ i k . ( r − R )]un,k(r) (623)

The phase factor in the integral over d3k has the effect of localizing the Blochfunction around r = R, as at this r value, the phase of the integral is stationary.The integral is easy to evaluate for free electrons for which un(r) = 1. TheWannier functions appropriate to free electrons in an orthorhombic lattice aregiven by

φn(r) =( sin [ π x

ax]

π xax

) ( sin [ π yay

]π yay

) ( sin [ π zaz

]π zaz

)(624)

which have amplitudes that decay algebraically outside the unit cell. This alge-braic decay is found only for bands with infinite width. Bands that have allowedenergies that are separated by forbidden ranges of E of finite width have Wan-nier functions that decay exponentially. Furthermore, the rate of exponentialdecay is dependent on the band width (W. Kohn Phys. Rev. 115 (1959), E.I.Blount, Solid State Physics, Vol 13, Acad. Press, (1962)).

——————————————————————————————————

8.3.9 Exercise 41

Prove that the Wannier functions centered on different lattice sites are orthog-onal ∫

d3r φ∗n′(r −R′) φn(r −R) ∝ δn′,n δR′,R (625)

Also show that the Wannier functions are normalized to unity∫d3r | φn(r) |2 = 1 (626)

——————————————————————————————————

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9 Electron-Electron Interactions

In the last chapter, the effects of interactions between electrons were neglectedin the calculation of the energies of single-electron excitations and the single-electron wave functions. The neglect of the effects of electron-electron interac-tions is certainly not justifiable from considerations of the relative strength ofthe effect of the Coulomb interactions with the potential due to the lattice ofnuclei compared with the electron-electron interactions. However, due to thePauli exclusion principle, the lowest energy excitations of an interacting elec-tron gas can be put into a one to one correspondence with the excitations ofa non-interacting gas of fermions. The effects of electron-electron interactionsare weak for low energy excitations and this leads to the concept of treating theinteracting electron system as a Landau Fermi Liquid.

9.1 The Landau Fermi Liquid

The Pauli exclusion principle plays an important role in reducing the effect ofelectron-electron interactions. A important result of this blocking principle isthat the low energy excitations of an electron gas behave very similarly to thatof a non-interacting electron gas. This allows one to consider the low energyexcitations as quasi-particles, which have a one to one correspondence with theexcitations of a non-interacting electron gas. This is the basis of the Landautheory of Fermi-liquids.

An important step in deriving the Landau theory was proved by J.M. Lut-tinger, who showed that electrons with energies close to the Fermi-energy havescattering rates that vanish as the energy approaches the Fermi-energy, to allorders in the electron-electron interaction. This can already be be seen fromthe lowest order calculation of the lifetime of an electron in a Bloch state due toelectron-electron interactions. Although, a rigorous derivation of Fermi Liquidtheory must consider processes of all order in the electron-electron interaction,we shall only consider the lowest order processes. Consider the lowest orderprocess, in which an electron, initially in a state k above the Fermi-surface, isscattered to a state k − q. In this scattering processes a second electron isexcited from an initial state k′ below the Fermi-surface to a state k′ + q abovethe Fermi-surface. This process conserves momentum and will conserve energyif

h2 k2

2 m−

h2 ( k − q )2

2 m=

h2 ( k′ + q )2

2 m− h2 k′2

2 m(627)

or( k − k′ ) . q = q2 (628)

For fixed k and k′ this is an equation of a sphere of diameter | k − k′ |,centered on ( k − k′ )

2 . Thus, q ranges from 0 to k − k′, and conservation of

energy ensures that k − q lies on a sphere of radius | ( k − k′ )2 |, centered

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at ( k + k′ )2 , passing through k and k′. However, since k − q must be above

the Fermi-surface there are additional restrictions due to the Pauli exclusionprinciple, namely

| k − q | ≥ kF (629)

and| k′ + q | ≥ kF (630)

Thus, only a segment of the surface of this sphere represents final states of thepossible processes. This segment becomes small as k approaches kF . In thelimit | k | → kF this segment tends to a circle in the plane of intersection ofthe sphere and the Fermi-surface, unless of course k = − k′. The net result isthat the phase space available for the scattering process vanishes as k → kF ,and the scattering rate vanishes (J.J. Quinn and R.A. Ferrell, Phys. Rev. 112,812 (1958)).

9.1.1 The Scattering Rate

The scattering rate can be evaluated from Fermi’s Golden rule

1τk

=2 πh

∑q

(4 π e2

q2 + k2TF

)2 ∑k′<kF

m

h2 δ( ( k − k′ ) . q − q2 )

(631)

The sum over k′ is performed where k′ lies within the Fermi-sphere.

Thus, the quasi-particle scattering rate vanishes as Ek → µ at zero tem-perature. At finite temperatures the quasi-particle scattering rate at the Fermi-energy varies as ( kB T )2 (E. Abrahams, Phys. Rev. 95, 834 (1954)). Thequasi-particle concept remains valid in the limit Ek → µ and T → 0.

9.1.2 The Quasi-Particle Energy

The quasi-particle excitation energy Ek is affected by the interaction with theother electrons in the system. The manner in which this change in energyoccurs system can be estimated from perturbation theory. To second order inthe perturbation, the energy of the state with an additional electron in state kis given by

E+k = E0

k +∑

|kn|<kF

E0kn

+ < k∏

|kn|<kF

kn | Hint | k∏

|kn|<kF

kn >

+∑

q

∑|km|<kF

∣∣∣∣ < k∏|k

n|<kF

kn | Hint | k − q km + q∏|k

n|<kF ,n 6=m kn >

∣∣∣∣2E0

k + E0k

m− E0

k−q − E0k

m+q

(632)

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To second order in the interaction, the ground state energy is given by

Egs =∑

|kn|<kF

E0kn

+ <∏

|kn|<kF

kn | Hint |∏

|kn|<kF

kn >

+∑

q

∑m,m′

∣∣∣∣ < ∏|k

n|<kF

kn | Hint | km′ − q km + q∏|k

n|<kF ,n 6=m,m′ kn >

∣∣∣∣2E0

km′

+ E0k

m− E0

km′−q − E0

km

+q

(633)

The excitation energy for adding an electron to state k is defined by

Eexck = E+

k − Egs (634)

To this order, the excitation energy is expressed in terms of two-particle statesas

Eexck = E0

k +∑

|kn|<kF

< k kn | Hint | k kn >

+∑

|k−q|>kF

∑|k

m|<kF

∣∣∣∣ < k km | Hint | k − q km + q >

∣∣∣∣2E0

k + E0k

m− E0

k−q − E0k

m+q

−∑

|k+q|<kF

∑|k

m|<kF

∣∣∣∣ < k + q km | Hint | k km + q >

∣∣∣∣2E0

k+q + E0k

m− E0

k − E0k

m+q

(635)

The terms first order in the interaction represent the interaction of the particlewith the average density due to the other electrons. The last two terms are sec-ond order terms. The first of this pair represents the scattering of the electronfrom the state k from an electron km in the Fermi-sea, to final states k− q andkm + q above the Fermi-sea. The last term represents a subtraction, as thisrepresents a scattering process for a pair of electrons that initially are below theFermi-surface which is forbidden by the Pauli exclusion principle as the state kis occupied by an electron. The k independent terms are absorbed into a shiftof the Fermi-energy.

This excitation represents the excitation energy for adding an electron tostate k. However, the many-body state consists of a linear superposition ofsingle-electron states and states where the added electron is dressed by electron-hole pairs. The quasi-particle weight Z−1(k) is defined as the fraction of theinitial bare electron contained in the quasi-particle. To lowest order in Hint, the

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quasi-particle weight or wave function renormalization is calculated as

Z(k) = 1 +∑

|k−q|>kF

∑|k

m|<kF

∣∣∣∣ < k km | Hint | k − q km + q >

∣∣∣∣2( E0

k + E0k

m− E0

k−q − E0k

m+q )2

+∑

|k+q|<kF

∑|km|<kF

∣∣∣∣ < k + q km | Hint | k km + q >

∣∣∣∣2( E0

k+q + E0k

m− E0

k − E0k

m+q )2

(636)

which is greater than unity. Thus, the fraction of the bare electron in the quasi-particle state is always less than unity. This conclusion remains valid to all or-ders of perturbation theory, if the Fermi Liquid phase is stable. When |k| crosseskF , the quasi-particle changes from a quasi-particle to a quasi-hole. At zerotemperature due to the vanishing of the quasi-particle scattering rate, the dis-tribution of the number of bare particles has a discontinuity at the Fermi-energyof Z(k)−1. This discontinuity is small compared with the discontinuity for non-interacting electrons which is completely contained in the Fermi-function. Thus,the concept of a Fermi-surface remains well defined for interacting electron sys-tems.

The quasi-particle weight has the effect that the excitation energy for a singlequasi-particle is given by the expression

Eqp(k) =Eexc

k

Z(k)(637)

In addition to the shift in the excitation energy, the quasi-particle excitationenergy is reduced by Z(k) and these two effects combine to yield an reductionof the dispersion. The reduced dispersion is interpreted in terms of an increase inthe effective mass of the quasi-particle. The density of single-electron excitationsis given by the quasi-particle contribution

ρqp(E) =∑

k

Z(k)−1 δ

(E − Eqp(k)

)(638)

where E is the excitation energy relative to the Fermi-energy. Due to quasi-particle weight factor, the single-electron density of states is narrowed and peaksup near the Fermi-energy. As the quasi-particles obey Fermi-Dirac statistics,the quasi-particles can give rise to an enhancement of the coefficient of the linearT term in the low-temperature electronic specific heat.

Despite the apparent simplicity of the Fermi Liquid picture, it is exceed-ingly difficult to quantitatively derive the Fermi Liquid description appropriate

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to a specific microscopic Hamiltonian. Since the perturbation due to electron-electron interaction is long-ranged, there are divergent terms in the perturbationexpansion. The divergent terms first appear in the expansion taken to secondorder. The divergent terms can actually be re-summed to yield finite results.The re-summations are made possible by the fact that the long-ranged Coulombinteraction in a metal is screened by the other electrons. The screening processesinvolves the Coulomb interaction to infinite order. By taking into account thescreening of the long-ranged Coulomb interaction, the divergent terms can besummed to infinite order leading to finite results. That is, the divergence asso-ciated with any term can be eliminated by combining it with a subset of otherdivergent terms. However, the re-summation of all the terms in the perturbationexpansion presents a serious challenge and so approximations have been devel-oped. These approximations involve the summation of infinite subsets of theterms that appear in the perturbation expansion. One such approximation is theHartree-Fock approximation. The Hartree-Fock approximation is self-consistentfirst order perturbation theory in that it just consists of the first order termsin the perturbation expansion. However, in these terms, all the wave functionsare calculated self-consistently by taking the first order processes into account.

——————————————————————————————————

9.1.3 Exercise 42

Using a perturbation expansion, find the energy of a free electron gas to firstorder in the electron-electron interaction.

——————————————————————————————————

9.2 The Hartree-Fock Approximation

The Hartree-Fock approximation consists of writing the many-electron wavefunction as a single Slater determinant, much the same way as for independentor non-interacting electrons. This should be contrasted with the exact wavefunction which is expected to be composed of a linear superposition of Slaterdeterminants. The Hartree-Fock approximation, therefore, involves finding thebest one-electron basis functions that takes the average effect of electron-electroninteractions into account (D.R. Hartree, Proc. Camb. Phil. Soc., 24, 89,1928)).

The Hartree-Fock approximation can be expressed in terms of the Rayleigh-Ritz variational principle (V.A. Fock, Zeit. fur Physik, 61, 126, (1930)), in whichthe many-particle wave function is written as a single Slater determinant (J.C.

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Slater, Phys. Rev. 35, 210 (1930)). The Hamiltonian operator is expressed as

H =∑

i

[p2

i

2 m+ Vions(ri)

]+

12

∑i 6= j

e2

| ri − rj |(639)

The expectation value of the Hamiltonian in terms of a Slater determinant Φ ofa complete set unspecified single-electron wave functions φα,σ(r) is given by

H =i=Ne∏i=1

[ ∫d3ri

]Φ∗α1, . . . αNe

(r1, . . . rNe) H Φα1, . . . αNe

(r1, . . . rNe)

(640)

The expectation value of the energy is evaluated as

E =∑α

∫d3r φ∗α(r)

[− h2

2 m∇2 + Vions(r)

]φα(r)

+12

∑α,β

∫d3r

∫d3r′ φ∗α(r) φ∗β(r′)

e2

| r − r′ |φβ(r′) φα(r)

− 12

∑α,β

∫d3r

∫d3r′ φ∗α(r) φ∗β(r′)

e2

| r − r′ |φα(r′) φβ(r)

(641)

where the sums over α and β run over all the single particle quantum numberslabelling the Slater determinant Φ. The first term just represents the sum ofone-particle energies of the electrons. The second term represents the interactionenergy between an electron and the average charge density of all the electrons.The last term is the exchange term; it arises due to the Coulomb interactionand the anti-symmetry of the many-electron wave function. The spin indiceshave been suppressed in the expression for the energy. The quantum number αneeds to be supplemented by the spin quantum number σ to uniquely specifythe state and φα(r) → φα(r) χσ. Therefore, in the matrix elements there isnot only an integration over r, but also the matrix elements of the spin stateshas to be evaluated.

The single-electron wave functions are to be chosen such that they minimizethe energy, subject to the constraint that they remain normalized to unity.Hence, subject to this condition, the single-electron wave functions are chosensuch that the first order variation of the energy is identically equal to zero.The minimization is performed by using the Lagrange method of undeterminedmultipliers. First, one forms the functional Ω which is the average value of theHamiltonian minus the Ne constraints that ensure that the one-electron wavefunctions are normalized to unity. The functional Ω is given by

Ω =i=Ne∏i=1

[ ∫d3ri

]Φ∗α1, . . . αNe

(r1, . . . rNe) H Φα1, . . . αNe

(r1, . . . rNe)

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−i=Ne∑i=1

λαi

( ∫d3ri φ

∗αi

(ri) φαi(ri) − 1

)(642)

where the λα are the undetermined multipliers. Since φα is an arbitrary complexfunction, the real and imaginary parts are independent. Instead of working withthe real and imaginary parts, we shall consider the function φα and its complexconjugate φ∗α as being independent. The second step of the Lagrange methodconsists of considering the effect of varying the set of φ∗α. The deviation of thevariational functions φ∗α(r) from the extremal function, φ∗HF,α(r), are denotedby δφ∗α, i.e.,

φ∗α(r) = φ∗HF,α(r) + δφ∗α(r) (643)

To first order in the deviation δφ∗α(r), the expectation value of the functional Ωchanges to first order in δφ∗α by an amount δΩ. The change δΩ is evaluated as

δΩ =∑α

∫d3r δφ∗α(r)

[− h2

2 m∇2 + Vions(r) − λα

]φHF,α(r)

+∑α,β

∫d3r

∫d3r′ δφ∗α(r) φ∗HF,β(r′)

e2

| r − r′ |φHF,β(r′) φHF,α(r)

−∑α,β

∫d3r

∫d3r′ φ∗HF,β(r) δφ∗α(r′)

e2

| r − r′ |φHF,β(r′) φHF,α(r)

(644)

The expression for δΩ must vanish identically for any of the independent andarbitrary variations δφ∗α(r), if the Hartree-Fock wave functions φHF,α(r) mini-mize the average energy. In order for this to be true, for each value of α, thecoefficient of δφ∗α(r) must vanish identically. After interchanging the variablesr and r′ in the last term, one finds that the normalized Hartree-Fock wavefunctions must satisfy the set of equations

0 =[− h2

2 m∇2 + Vions(r) − λα

]φHF,α(r)

+∑

β

∫d3r′

(φ∗HF,β(r′)

e2

| r − r′ |φHF,β(r′)

)φHF,α(r)

−∑

β

∫d3r′

(φ∗HF,β(r′)

e2

| r − r′ |φHF,α(r′)

)φHF,β(r)

(645)

in order to minimize the energy. To simplify further analysis, we shall explicitlydisplay the spin dependence by writing

φHF,α(r) = ψα(r) χσ

φHF,β(r) = ψβ(r) χσ′ (646)

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This notation recognizes that the spatial component of the wave function, ψα(r),depends on all the quantum numbers represented by α, including the spin quan-tum number, as in the un-restricted Hartree-Fock approximation. The Hartree-Fock equations are re-written as

0 =[− h2

2 m∇2 + Vions(r) − λα

]ψα(r) χσ

+∑

β

∫d3r′

(χT

σ′ ψ∗β(r′)

e2

| r − r′ |ψβ(r′) χσ′

)ψα(r) χσ

−∑

β

∫d3r′

(χT

σ′ ψ∗β(r′)

e2

| r − r′ |ψα(r′) χσ

)ψβ(r) χσ′

(647)

In the inner product, the integrations over the position r′ of the spatial com-ponent of the wave function is combined with the matrix elements of the spinwave functions. The spin matrix elements are given by

χTσ′ χσ = δσ′,σ (648)

Since the Coulomb interaction is spin independent, that last term contains aKronecker delta function that is non-vanishing only when σ = σ′. The set ofHartree-Fock equations are eigenvalue equations for a non-local linear operator

0 =[− h2

2 m∇2 + Vions(r) − λα

]ψα(r)

+∑

β

∫d3r′

(ψ∗β(r′)

e2

| r − r′ |ψβ(r′)

)ψα(r)

−∑

β

δσ′,σ

∫d3r′

(ψ∗β(r′)

e2

| r − r′ |ψβ(r)

)ψα(r′)

(649)

There is one such equation for each value of α. In solving the above equationsfor ψα(r), one should consider the functions ψβ(r) as known quantities. Inthis case, the eigenvalue equations are linear in the eigenfunctions, ψα, and theundetermined multipliers, λα, are the eigenvalues. The term proportional to

Vdirect(r) =∑

β

∫d3r′

e2 | ψβ(r′) |2

| r − r′ |(650)

represents a contribution to the potential from the average electrostatic potentialdue to the all electrons in the system. This potential includes the contributionfrom an the electron in state α. This potential is independent of the spinstates of the electrons, and is called the direct interaction. The last term inthe Hartree-Fock equation is non-local, as it relates the unknown eigenfunction

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ψα(r) to the weighted average of the unknown eigenfunction at other points inspace, ψα(r′). The non-local potential represented by

V σexch(r, r′) = −

∑β

δσ,σ′ ψ∗β(r′)

e2

| r − r′ |ψβ(r) (651)

is called the exchange interaction. Since the Coulomb interaction is spin inde-pendent, the matrix elements in the non-local exchange potential are non-zeroonly if the spin of state α is identical to the spin of state β. If the spins are anti-parallel, the exchange term is zero. Thus, the exchange term is spin dependent.

With this notation, the Hartree-Fock equations can be written as[− h2

2 m∇2 + Vion(r) + Vdirect(r)

]ψα(r) +

∫d3r′ V σ

exch(r, r′) ψα(r′) = λα ψα(r)

(652)These sets of equation can be solved iteratively. Using approximations for thedirect and exchange potentials, one can solve the equations to find a set of wavefunctions which are approximations for the ψα(r). These approximate wavefunctions are then used to construct new approximations for the direct and ex-change potentials. The procedure is repeated until self-consistency is achieved.The contributions to the direct and exchange potentials, arising from the statewhere β = α, exactly cancel in the non-local operator. Therefore, there are noself interaction terms in the Hartree-Fock approximation. The cancellation ofthe self interaction has the effect that the linear potential operator is the samefor all the single-electron wave functions.

The Hartree-Fock approximation can be solved exactly for the free electrongas in which the potential of the lattice of ions is replaced by a constant value.This (unrealistic) uniform potential is of special importance, since the solutionis often used as a starting point to discussing the electronic structure of a non-uniform electron gas. Specifically, the most common method of determiningelectronic structure, the local density functional method, utilizes the expressionfor the ground state energy of the uniform electron gas.

9.2.1 The Free Electron Gas.

The Hamiltonian for the free electron gas is invariant under all translations and,as long as the translational symmetry is not spontaneously broken, the Hartree-Fock eigenstates should be simultaneous eigenstates of the momentum operator.Thus, the Hartree-Fock equations for a uniform potential Vions = V0 shouldhave the eigenfunctions

ψk,σ(r) =1√V

exp[i k . r

]χσ (653)

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where V is the volume of the crystal. It can be seen that this is true by substi-tuting the wave functions into the Hartree-Fock eigenvalue equations.

The charge density due to the electrons is a constant, and this combines withthe uniform charge density from the background gas of ions. Due to chargeneutrality, the resulting net direct Coulomb potential from the total chargedensity vanishes

Vions(r) + Vdirect(r) = 0 (654)

In order to evaluate the exchange potential, one has to perform the sumover values of k′, σ′. The sum over k′, σ′ only runs over the occupied states. Weshall assume that the Hartree-Fock state does not spontaneously break the spinrotational symmetry and lead to magnetism. Likewise, we shall also assume thatthe Hartree-Fock solution does not break translational invariance. Magneticsolutions which also break translation invariance have been found by Overhauser(A.W. Overhauser, Phys. Rev. Letts. 4, 462 (1960), Phys. Rev. 128, 1437(1962)) and also by Kohn and Nettel (W. Kohn and S.J. Nettel, Phys. Rev.Letts. 5, 8 (1960)). In the non-magnetic translationally invariant case, theHartree-Fock states are spin degenerate, and the one-particle states are filledaccording to the magnitude of the kinetic energy. All the one-particle stateslabelled by (k, σ), where k is contained inside a sphere of radius kF , are filledwith electrons. The spin-dependent exchange term is evaluated as

Vexch(r, r′) = − 1V

e2

| r − r′ |∑

|k′|≤kF , σ′

δσ,σ′ exp[i k′ . ( r − r′ )

](655)

The exchange potential also has translational invariance, and so it is possiblethat plane waves are eigenfunctions of the Hartree-Fock equations. The ex-change potential is evaluated from

Vexch(r, r′) = − 1( 2 π )3

∫ 2 π

0

∫ π

0

dθ sin θ

×∫ kF

0

dk′ k′2 exp[i k′ . ( r − r′ )

]e2

| r − r′ |

= − 1( 2 π )3

2 π e2

×∫ kF

0

dk′ k′

( exp[

+ i k′ | r − r′ |]− exp

[− i k′ | r − r′ |

]i | r − r′ |2

)(656)

The integration over k′ can be performed with the aid of an identity obtained

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by differentiating the expression∫ 1

0

dx cos α x =sin α

α(657)

with respect to α. That is,∫ 1

0

dx x sin α x =(

sin α

α2− cos α

α

)(658)

The resulting expression for the exchange potential is

Vexch(r, r′) = − e2 k4F

2 π2

(sin kF | r − r′ |( kF | r − r′ | )4

− cos kF | r − r′ |( kF | r − r′ | )3

)(659)

The long-ranged oscillatory behavior of the exchange potential is due to thesharp cut off of the integration at kF . This cut off occurs as the Fermi-wavevector kF is the largest wave vector associated with the occupied one-electronstates.

The contribution of the exchange potential to the energy eigenvalue λk canbe found from∫

d3r′ Vexch(r, r′) ψk(r′) =1√V

exp[i k . r

] ∫d3r′ exp

[i k . ( r′ − r )

]Vexch(r, r′)

(660)

Thus, the contribution of the eigenvalue stemming from exchange potential isjust the Fourier transform of the exchange term, Vexch(k),

Vexch(k) = − e2 k4F

2 π2

∫d3R exp

[i k . R

] (sin kF R

( kF R )4− cos kF R

( kF R )3

)(661)

which can be evaluated directly. An alternate method involves using the con-volution theorem, in which case the expression

Vexch(k) = − V

( 2 π )3

∫d3r′

e2

| r − r′ |

×∫|k′|≤kF

d3k′ ψ∗k′(r′) ψk′(r) exp

[i k . ( r′ − r )

](662)

can be used. The plane wave nature of the eigenfunctions can be utilized towrite the expression as

Vexch(k) = − V

( 2 π )3

∫d3r′

e2

| r − r′ |×

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×∫|k′|≤kF

d3k′ | ψk′(r′) |2 exp[i ( k′ − k ) . ( r − r′ )

](663)

The electron density, per spin, arising from state k is just | ψk′(r′) |2 = 1V

for | k′ | ≤ kF . Since this is independent of r′, the exchange contribution tothe eigenvalue involves the Fourier Transform of the Coulomb potential. TheFourier transform of the exchange potential is found as

= − 1( 2 π )3

∫d3r′

∫|k′|≤kF

d3k′e2

| r − r′ |exp

[i ( k′ − k ) . ( r − r′ )

](664)

Hence, the expression for the exchange contribution to the eigenvalue λk is givenby

Vexch(k) = − 1( 2 π )3

∫|k′|≤kF

d3k′4 π e2

| k − k′ |2

= − e2

π k

∫ kF

0

dk′ k′ ln| k + k′ || k − k′ |

(665)

The integral can be evaluated as

Vexch(k) = − 2 e2

πkF F

(k

kF

)(666)

where

F (x) =12

+1 − x2

4 xln| 1 + x || 1 − x |

(667)

At k = 0, the function F (0) is unity. At k = kF , the function falls to thevalue F (1) = 1

2 and has a logarithmic singularity in the slope. This singularityin the slope is due to the long-ranged nature of the Coulomb interaction ( 4 π

k2

). The function F (x) falls to zero in the limit limx → ∞ F (x) → 0. Thus, theeigenvalue λk is given by

λk =h2 k2

2 m− 2 e2

πkF F

(k

kF

)(668)

The total energy of the electron system is given by the sum of the kineticenergy and the exchange energy

EHF = 2∑

k

h2 k2

2 m

−∑k,k′

∫d3r

∫d3r′ ψ∗k(r) ψ∗k′(r

′)e2

| r − r′ |ψk(r′) ψk′(r)

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=∑

k

(h2 k2

2 m+ λk

)(669)

where the summations are restricted to the values of k and k′ which are withinthe Fermi-sphere. The Hartree-Fock energy can be re-expressed as

EHF = 2∑

k ≤ kF

h2 k2

2 m

− 2 e2

πkF

∑k ≤ kF

[12

+(k2

F − k2

4 k kF

)ln| kF + k || kF − k |

](670)

The summations over k can be evaluated by transforming them into integrals

EHF = 24 π V

( 2 π )3

∫k ≤ kF

dk k2 h2 k2

2 m

− 2 e2

πkF

4 π V( 2 π )3

∫k ≤ kF

dk k2

[12

+(k2

F − k2

4 k kF

)ln| kF + k || kF − k |

]=

V

π2

h2 k5F

10 m− V e2

π3k4

F

[13− 1

12

](671)

The number of electrons, per spin, Ne

2 is given by

Ne

2=

V

8 π3

4 π3

k3F (672)

Using this, the Hartree-Fock approximation for the cohesive energy of the freeelectron gas can be expressed as

EHF = Ne

[35h2 k2

F

2 m− 3

4e2

πkF

](673)

An alternative expression is given by introducing a characteristic dimension, orradius rs, such that there exists one electron in a sphere of radius rs a0, wherea0 is the Bohr radius ( a0 = h2

m e2 ). Then, the uniform electron density, ρ, isgiven by the equivalent expressions

=4 π3

a30 r

3s

=3 π2

k3F

(674)

Thus, the magnitude of the Fermi-wave vector kF is given by

kF =(

9 π4

) 13 1rs a0

(675)

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and so the electronic energy is expressed as

EHF

Ne=

3 h2

10 m a20

(9 π4

) 23 1r2s

− 3 e2

4 π a0

(9 π4

) 13 1rs

=e2

2 a0

(9 π4

) 13[

35

(9 π4

) 13 1r2s− 3

2 π1rs

]=

2.21r2s

− 0.9163rs

Rydbergs (676)

where 1 Rydberg = e2

2 a0. The Hartree-Fock energy has a minimum at the

rs value given by rs ∼ 4.8 and has a cohesive energy of about 0.1 Rydbergs.Typical materials have spatially varying densities, hence, the local value of rsalso varies. For a hydrogen-like atom, the ground state density is given by

ρ(r) =Z3

π a30

exp[− 2 Z r

a0

](677)

Therefore, typical values of rs are given by the density at the nuclear position

rs =( 34 )

13

Z

=0.9086Z

(678)

and at the first Bohr radius r = Z a0

rs =( 34 )

13 e

23

Z

=1.7696Z

(679)

Since for metals the density of electrons corresponds to rs values in the rangeof 2 to 5, the exchange term is of similar magnitude to the kinetic energy term.The Hartree-Fock approximation indicates that the cohesive energy is largestfor low density metals, i.e., those with rs ∼ 5.

In the particular case of the free electron gas where the lattice potential iszero, the Hartree-Fock approximation coincides with second order perturbationtheory. If higher order terms are included (M. Gell-Mann and K. Brueckner,Phys. Rev. 106, 347, (1957), W.J. Carr and A.A. Maradudin, Phys. Rev. A133, 371 (1964)), one obtains the expression for the energy per electron

E

Ne=[

2.21r2s

− 0.9163rs

+ 0.06218 ln rs − 0.094 + O(rs)]

(680)

in units of e2

2 a0. The energy is a form of an expansion in rs, valid for rs < 1.

Thus, the Hartree-Fock result can be thought of as an approximation which

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reproduces the high density limit ( small rs limit ) correctly. The other terms inthe expression are due to electron correlations. A completely different behavioris expected to occur in the low density limit. In reducing the density from thehigh density metallic limit to the low density limit, the system is expected toundergo a transition to a Wigner crystal phase (E.P. Wigner, Phys. Rev. 46,1002 (1934)). In a Wigner crystal, the electrons are expected to localize in ab.c.c. structure. The total energy is expected to be dominated by the elec-trostatic interaction and the energies of the vibrations of the electronic lattice(W.J. Carr, R.A. Coldwell-Horsfall, and A.E. Fein, Phys. Rev. 124, 747 (1961)).The energy of the Wigner crystalline phase is given by

E

Ne=

e2

2 a0

(− 1.792

rs+

2.65

r32s

− 0.73r2s

+ . . .

)(681)

for rs 1.

The electronic wave functions described by a Slater determinant are notdevoid of correlations. The correlations are a result of the Pauli exclusion prin-ciple. The two-particle density-density correlation function for a single Slaterdeterminant can be written as

ρ2(r, r′) =∑α,β

12| φα(r) φβ(r′) − φβ(r) φα(r′) |2

=∑α

| φα(r) |2∑

β

| φβ(r′) |2 −∑α,β

φ∗α(r) φβ(r) φ∗β(r′) φα(r′)

(682)

On making the spin dependence explicit, by writing

φα(r) = ψα(r) χσ

φβ(r) = ψβ(r) χσ′ (683)

one finds that the two-particle density-density correlation function is given by

ρ2(r, r′) =∑α

| ψα(r) |2∑

β

| ψβ(r′) |2 −∑α,β

δσ,σ′ ψ∗α(r) ψα(r′) ψ∗β(r′) ψβ(r)

= ρ(r) ρ(r′) −∑

σ

Gσ(r′, r) Gσ(r, r′) (684)

where Gσ is given by a sum over the single-particle states labelled by α whichhave the spin quantum number σ

Gσ(r, r′) =∑α

ψ∗α(r′) ψα(r) (685)

The last term in the two-particle density-density correlation function is the ex-change term. The exchange term originates from pairs of electrons with parallel

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spins. In the Hartree-Fock approximation for the free electron gas, the exchangecontribution to the two-particle density-density correlation function ρ2(r, r′) isexpressed in terms of the factors

Gσ(r, r′) =∑

|k′| < kF

ψ∗k′(r′) ψk′(r)

=1V

∑|k′| < kF

exp[i k′ . ( r − r′ )

]

=k3

F

2 π2

[sin kF | r − r′ |( kF | r − r′ | )3

− cos kF | r − r′ |( kF | r − r′ | )2

](686)

where the summation is over the Fermi-sphere. The density-density correlationfunction shows a hole in the density of parallel spin electron around the electronand vanishes as | r − r′ | → 0, as expected from the Pauli exclusion principle.The exchange potential has a similar form and can be thought of arising from adeficiency in the density of parallel spins around an electron at r. The Hartree-Fock approximation is deficient in that it does not include a similar correlationhole between electrons with anti-parallel spins.

In the Hartree-Fock approximation, the energies of the excited states aregiven by Koopmans’ theorem (T.A. Koopmans, Physica 1, 104 (1933)). Thatis, the energy for adding or removing an electron from the system is given bythe eigenvalue λk, if the other one-electron states in the many-particle Slaterdeterminant are not changed or that the other electrons in the ground state arenot re-arranged. Thus, in the Hartree-Fock approximation, the quasi-particlesenergies are given by

Eqp(k) = λk (687)

The quasi-particle density of states, per spin, is given by

ρqp(E) =∑

k

δ( E − Eqp(k) )

=V

2 π2

∫ kF

0

dk k2 δ( E − Eqp(k) )

=V

2 π2k2

(dEqp(k)dk

)−1∣∣∣∣k(E)

(688)

where k(E) is the value of k that satisfies the equation

Eqp(k) = E (689)

From the above, one sees that at the Fermi-energy defined by

EF = Eqp(kF ) (690)

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the quasi-particle density of states is zero since

dEqp(k)dk

=h2 k

m− e2

π

kF

k

+e2

π

( k2F + k2 )2 k2

ln| kF + k || kF − k |

(691)

which diverges logarithmically at k(EF ) = kF . Thus, the Hartree-Fock ap-proximation for the free electron gas is of limited utility in discussing propertiesof real metals. This is caused by the divergent slope of the one-electron eigen-values near the Fermi-surface. This spurious divergence caused by the neglectof screening, results in the one-electron density of states falling to zero just atthe Fermi-energy.

——————————————————————————————————

9.2.2 Exercise 43

Show, using perturbation theory, that the second order correction to the energyof a free electron gas is given by

∆E(2) = − m

h2

∑k,k′,q

(4 π e2

q2 V

)2 1q . ( k − k′ + q )

(692)

where k < kF , k′ < kF , | k + q | > kF and | k′ − q | > kF . Since thisintegral is dominated by the region q → 0, the value of k ∼ kF and k′ ∼ kF .Show that the contribution to ∆E(2) is proportional to∫

d3q

q3= 4 π

∫dq

q

= 4 π ln q (693)

and, thus, diverges for q → 0.

Simple second order perturbation theory does not work for the free electrongas. None the less, perturbation theory can be applied by using more elaboratetechniques which take into account the screening of the Coulomb interaction.

——————————————————————————————————

9.3 The Density Functional Method

The density functional method provides an exact method for calculating theelectron density and ground state energy for interacting electrons in the pres-ence of a crystalline potential. As such, it can be used to determine the stability

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of various lattice structures. It can also be used to determine ground state prop-erties or static properties of the electronic systems such as those provided byelastic scattering experiments. It is based on the Hohenberg and Kohn Theorem(P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)).

The Hohenberg-Kohn theorem considers the form of a many-electron Hamil-tonian in which the form of the Coulomb interaction term between pairs of elec-trons, Vint(r, r′), is known, but in which the one-particle potential due to theion cores is considered to be an external potential. Thus, the external potentialVext(r) varies between one crystal structure and the next. The Hamiltonian iswritten as the sum of the kinetic energy of the electrons and the interaction andexternal potentials acting on each electron.

The Hohenberg-Kohn theorem associates every non-degenerate many-bodyground state wave function with a unique external potential. A second mapexists between the many-body ground state wave function and the ground stateelectron density. Therefore, the expectation value of any ground state propertycan be expressed in terms of a unique functional of the ground state density.Having established this, the ground state properties and electron density canthen be evaluated from the Rayleigh-Ritz variational principle for the groundstate energy in which the electron density is the function to be varied.

This leads to a knowledge that, if one can construct the unique energy func-tional which contains the external potential due to the lattice potential, thenone can find the ground state energy and electron density. This functional isnot known, however, it is customary to make the local density approximation.In this approximation, an un-testable assumption is made about the electron-electron interactions in a non-uniform electron gas. The method also generateseigenvalues which are interpreted in terms of the energies of independent Blochelectrons. The energy dispersion relations generated this way do show a markedsimilarity with the experimentally determined bands of simple metals.

The basis for density functional theory is provided by a theorem provedby Hohenberg and Kohn (P. Hohenberg and W. Kohn, Phys. Rev. 136 B864(1964)).

9.3.1 Hohenberg-Kohn Theorem

The Hohenberg-Kohn theorem first assures us that the electron density in asolid, ρ(r), uniquely specifies the electrostatic interaction potential between theNe electrons and the ionic lattice. Thus, the density ρ(r) can be used as thebasic variable. Furthermore, the energy can be expressed as a unique functionalof the energy density involving the potential due to the ionic lattice. This estab-lishes a variational principle which can be used to calculate the electron density,ρ(r), and the total energy of the electronic system.

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First, it shall be assumed that Vions(r) is not uniquely specified if ρ(r) isgiven. That is, it is assumed that there exists at least two potentials V and V ′

which give rise to the same ground state electron density. These potentials arerelated to the exact ground state many-particle wave functions via the energyeigenvalue equations,

H Ψ(r1, . . . rNe) = E Ψ(r1, . . . rNe

) (694)

andH ′ Ψ′(r1, . . . rNe

) = E′ Ψ′(r1, . . . rNe) (695)

From the Rayleigh-Ritz variational principle, one finds that the primed wavefunction Ψ′(r1, . . . rNe

) provides an upper bound to the ground state energyof the unprimed Hamiltonian H,

E =i=Ne∏i=1

[ ∫d3ri

]Ψ∗(r1, . . . rNe

) H Ψ(r1, . . . rNe)

E <

i=Ne∏i=1

[ ∫d3ri

]Ψ′∗(r1, . . . rNe

) H Ψ′(r1 . . . rNe) (696)

However, as the primed and unprimed Hamiltonian are related through

H = H ′ + V − V ′ (697)

and as Ψ′ is the ground state of H ′ with energy eigenvalue E′, the energiessatisfy an inequality

E < E′ +∫d3r ρ(r) ( V (r) − V ′(r) ) (698)

However, by similar reasoning, it can also be shown that the energies also satisfythe inequality

E′ < E +∫d3r ρ(r) ( V ′(r) − V (r) ) (699)

where the prime and unprimed quantities are interchanged. The assumptionthat the ground state densities of the primed and unprimed Hamiltonian areequal has been used. Adding these two inequalities leads to an inconsistency

E + E′ < E + E′ (700)

Therefore, the assumption that the same ground state density can be found fortwo different potentials is false. Furthermore, the potentials can, at most, onlydiffer by a constant V ′(r) − V (r). Thus, the ground state electron density ρ(r)must correspond to a unique V (r). This means that the electron density, ρ(r),can be taken to be the principal variable.

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9.3.2 Functionals and Functional Derivatives

As a mathematical prelude, we shall define functionals and functional deriva-tives.

A functional is a generalization of a function. A function f(r) can be definedas a mapping which maps each point in space, r, to a number. The value ofthe number depends on the position of the point. The functional is similar inthat it maps a scalar function onto a number. The value of the functional, F [ρ],depends upon the function ρ(r), i.e., the values of the function ρ at each pointin space. Functionals are usually expressed in terms of integrals over space,usually as multiple integrals.

A simple example of a functional is given by the number of electrons Ne[ρ],which is a functional of the density. The number of electrons is given by

Ne[ρ] =∫

d3r ρ(r) (701)

It is a functional as different densities may correspond to different number ofparticles i.e., Na has a different density than Li and they have different numbersof electrons.

The classical Coulomb energy is a more interesting functional. The Coulombenergy is defined as the pairwise sum of interactions

ECoul[ρ] =e2

2

∫d3r

∫d3r′

ρ(r) ρ(r′)| r − r′ |

(702)

This yields a number which is the value of the energy, and this number dependson the density at all points of space.

Given a functional F [ρ], one can define a functional derivative. The defini-tion of the functional derivative is similar to the definition of a derivative of afunction. However, instead of defining the derivative in terms of the differenceof the function at two nearby points, one defines the functional derivative interms of the difference of the functional for two functions that are close. Forexample, an arbitrary family of functions, ρ′(r), can be defined in terms of afixed function ρ(r) and an arbitrary deviation δρ(r) via

ρ′(r) = ρ(r) + λ δρ(r) (703)

The scale factor λ varies from unity to zero continuously. When λ = 1, thisrelation defines the shape of the deviation δρ(r). If λ is changed continuouslyto zero, the differences between the function ρ′ and the fixed function ρ van-ish. The shape of the deviation λ δρ(r) is arbitrary and does not change, onlythe magnitude of the deviation is changing. The functional derivative can beexpressed in terms of the limit of the difference of the functional evaluated at

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these two functions. If one assumes that one may Taylor expand the functionalin powers of λ, one has

F [ρ′] = F [ρ] + λ δ1F [ρ, δρ] +12λ2 δ2F [ρ, δρ] + . . . (704)

since the differences now depend on two functions ρ and δρ. If one defines theterms of first order in λ to have the form

δ1F [ρ, δρ] =∫

d3r δρ(r)δF [ρ]δρ(r)

(705)

then the quantityδF [ρ]δρ(r)

(706)

is independent of the shape of the deviation, and is defined to be the first orderfunctional derivative. Sometimes a functional may depend on the higher orderderivatives of ρ i.e.,

F [ρ] =∫

d3r f(r, ρ,∇ρ) (707)

In this case, one can define a functional derivative in terms of the partial deriva-tives,

∂f

∂ρ(708)

and the vector quantity∂f

∂∇ρ(709)

etc., where the functions ρ and ∇ρ etc. are treated as independent variables.This yields the first order variation as

δ1F [ρ, δρ] =∫

d3r

(δρ

∂f

∂ρ+ ∇δρ . ∂f

∂∇ρ

)(710)

If the functions ρ satisfy appropriate conditions at the boundaries of the inte-gration, the equation can be integrated by parts to eliminate the term∫

d3r ( ∇δρ ) .∂f

∂∇ρ(711)

In this case, the first order functional derivative is evaluated as

δF [ρ]δρ(r)

=∂f

∂ρ− ∇ .

∂f

∂∇ρ(712)

The extension to functionals containing higher order derivatives is quite straight-forward.

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An alternative method of evaluating functional derivatives is based on theobservation that the functional derivative is independent of the variation δρ.Since δρ is arbitrary, one may choose δρ to have any particular form. Theparticular variation of the form of a dirac delta function proves to be a usefulchoice

δρ(r) = δ3(r − r′) (713)

since, for this particular choice, the value of δF 1[ρ, δ3(r − r′)] is given by

δ1F [ρ, δ3(r − r′)] =δF [ρ]δρ(r′)

(714)

An example of the first order functional derivative is given by the functionalderivative of the Coulomb energy

δECoul[ρ]δρ(r1)

=e2

2

∫d3r

ρ(r)| r − r1 |

+e2

2

∫d3r′

ρ(r′)| r′ − r1 |

= e2∫

d3rρ(r)

| r − r1 |(715)

In obtaining the second line, we have relabelled the variable of integration. Thefirst order functional derivative of the mono-nomial functional

Fn[ρ] =∫

d3r ρ(r)n (716)

is simply evaluated asδFn[ρ]δρ(r1)

= n ρ(r1)n−1 (717)

The delta function method also proves useful for evaluating functional deriva-tives of higher orders.

The first order functional derivative is often encountered in variational prin-ciples. In a variational principle, there exists a function ρ(r) which yields anextremal value of the functional. That is, if the functional is changed by anarbitrary small variation λδρ away from the extremal function, the functionaldoes not change. On regarding the functional F [ρ+λδρ] as a function of λ, theextremal condition is equivalent to

∂λF [ρ+ λδρ]

∣∣∣∣λ=0

= 0 (718)

since the value of the functional does not change to order λ as λ approacheszero. This equation is satisfied for an arbitrary shape δρ(r), if the functionalderivative is identically zero

δF [ρ]δρ(r)

= 0 (719)

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for all r. The extremal function ρ(r) must satisfy this extremal condition for allr. Often, the extremal condition provides an integro-differential equation thatcan be used to uniquely determine ρ(r). The above condition only guaranteesthat F [ρ] is an extremal. In order that the functional F [ρ] is minimized, werequire that

δ2F [ρ, δρ] > 0 (720)

for every δρ.

The second order functional derivative is defined via

δ2F [ρ, δρ] =∫d3r

∫d3r′ δρ(r) δρ(r′)

δ2F [ρ]δρ(r) δρ(r′)

(721)

On using the choiceδρ(r) = δ3(r − r1) (722)

for the deviation centered at r′ in the first derivative and the choice

δρ′(r′) = δ3(r′ − r2) (723)

when differentiating the second time, one obtains

δ2F [ρ, δρ, δρ′] =δ2F [ρ]

δρ(r1) δρ(r2)(724)

As an example, the second order functional derivative of the Coulomb energy isfound to be

δ2ECoul[ρ]δρ(r1) δρ(r2)

=e2

| r1 − r2 |(725)

A second example is provided by the functional derivative of the mono-nomial

Fn[ρ] =∫

d3r ρ(r)n (726)

for real φ. For this functional, the second order functional derivative has theform

δ2Fn[ρ]δρ(r1) δρ(r2)

= δ3(r1 − r2) n ( n − 1 ) ρ(r1)n−2 (727)

etc.

9.3.3 The Variational Principle

Hohenberg and Kohn defined an energy functional of the electron density

E[ρ] = F [ρ] +∫

d3r Vions(r) ρ(r) (728)

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in which the energy functional F [ρ] depends on the kinetic energy T given by

T = − h2

2 m

Ne∑i=1

∇2i (729)

and the electron-electron interaction energy, Vint, given by

Vint =12

∑i 6=j

e2

| ri − rj |(730)

The functional F [ρ] can be evaluated as

F [ρ] =i=Ne∏i=1

[ ∫d3ri

]Ψ∗(r1, . . . rNe

)(T + Vint

)Ψ(r1, . . . rNe

)

(731)

The functional F [ρ] is a universal functional of ρ, as the functional F [Ψ] is auniversal functional of Ψ. Furthermore, as will be shown, the energy of theelectronic system E is given by the minimum value of the functional E[ρ] whereρ(r) is the correct ground state density of an Ne electron system associated withthe lattice potential Vions(r). In fact, E is the minimum value of E[ρ] evaluatedfor the set of functions, ρ(r), which correspond to the Ne-electron ground statedensities of arbitrary potentials. Such densities are known as V -representabledensities. Not all densities are V -representable.

Let ρ′(r) 6= ρ(r) be an arbitrary density associated with some many-bodywave function Ψ′ 6= Ψ, that is not the ground state of our system. The groundstate energy is defined by

E =i=Ne∏i=1

[ ∫d3ri

]Ψ∗(r1, . . . rNe

) H Ψ(r1, . . . rNe) = E[ρ]

(732)

The Rayleigh-Ritz variational principle asserts that the expectation values ofthe Hamiltonian satisfies the inequality

i=Ne∏i=1

[ ∫d3ri

]Ψ∗(r1, . . . rNe

) H Ψ(r1, . . . rNe)

≤i=Ne∏i=1

[ ∫d3ri

]Ψ′∗(r1, . . . rNe

) H Ψ′(r1, . . . rNe) (733)

and soE[ρ] ≤ E[ρ′] (734)

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This establishes the minimum principle for the energy functional

δE[ρ]δρ(r)

= 0 (735)

subject to the constraint that the total number of electrons are fixed∫d3r ρ(r) = Ne (736)

The condition that ρ is V -representable may be replaced by a less stringentcondition of N representable (M. Levy, Proc. Nat. Acad. Sci. 76, 6062 (1979)),which only requires

ρ(r) > 0∫d3r ρ(r) = Ne∫

d3r | ∇ ρ12 (r) |2 < ∞ (737)

Having established the existence of the variational function, the precise form ofthe functional remains to be determined.

9.3.4 The Electrostatic Terms

Hohenberg and Kohn suggest that one should separate the long-ranged classicalCoulomb energy of the electrons from the functional F [ρ]. This term representsthe average Coulomb interaction with the electrons in the system and, therefore,represents the Hartree terms. That is, the energy functional representing thekinetic and electron-electron interaction energies is written as

F [ρ] =12

∫d3r

∫d3r′

ρ(r) ρ(r′)| r − r′ |

+ G[ρ] (738)

The total energy functional is given by

E[ρ] =∫

d3r Vions(r) ρ(r) +e2

2

∫d3r

∫d3r′

ρ(r) ρ(r′)| r − r′ |

+ G[ρ]

(739)

The electrostatic potential φes(r) is given by the sum of the potential due tothe lattice of ions and the electron-electron interaction

− | e | φes(r) = Vions(r) + e2∫

d3r′ρ(r′)

| r − r′ |(740)

This potential may be obtained directly from Poisson’s equation from the densityof the ions and electrons

− ∇2 φes(r) = 4 π | e |(Z ρions(r) − ρ(r)

)(741)

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where | e | is the magnitude of the charge on the electron. The electrostaticpotential determines the chemical potential through the variational procedure.The energy functional is minimized w.r.t variations of ρ(r) subject to the con-straint that the density is normalized to Ne. This is performed by using La-grange’s method of undetermined multipliers. The method consists of construct-ing the functional Ω[ρ] as

Ω[ρ] = E[ρ] − µ

( ∫d3r ρ(r) − Ne

)(742)

Then on writing ρ′(r) = ρ(r) + λ δρ(r) and Taylor expanding in λ one has

Ω[ρ′] = Ω[ρ] + λ

∫d3r δρ(r)

δΩ[ρ]δρ(r)

+λ2

2

∫d3r

∫d3r′ δρ(r) δρ(r′)

δ2Ω[ρ]δρ(r) δρ(r′)

+ . . . (743)

The extremal conditionδΩ[ρ]δρ(r)

= 0 (744)

becomesδE[ρ]δρ(r)

= µ (745)

The first order functional derivative of E is evaluated from the Taylor expansionby retaining the terms of first order in λ. The first order term in E, δE1, isevaluated as

δE1 =∫

d3r δρ(r) Vions(r)

+e2

2

∫d3r

∫d3r′

(δρ(r)

ρ(r′)| r − r′ |

+ δρ(r′)ρ(r)

| r − r′ |

)+∫

d3r δρ(r)δG[ρ]δρ(r)

(746)

On interchanging the variables of integration r and r′ in the second part of theCoulomb term and combining it with the first, one obtains

δE1 =∫

d3r δρ(r)(Vions(r) + e2

∫d3r′

ρ(r′)| r − r′ |

+δG[ρ]δρ(r)

)(747)

Since the first two terms are identified with the electrostatic potential, the func-tional derivative is given by

δE[ρ]δρ(r)

= Vions(r) + e2∫

d3r′ρ(r′)

| r − r′ |

= − | e | φes(r) +δG[ρ]δρ(r)

(748)

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Hence, Ω is minimized if ρ satisfies the equation

− | e | φes(r) +δG[ρ]δρ(r)

= µ (749)

For large Ne, µ is equal to the chemical potential given by

µ =∂E

∂Ne(750)

9.3.5 The Kohn-Sham Equations

The Kohn-Sham equations provide a formal correspondence between the many-body problem and an effective (non-interacting) one-body problem (W. Kohn,L.J. Sham, Phys. Rev. 140, A1133 (1965)). This allows the kinetic energy termin the energy functional to be determined.

The kinetic energy functional T [ρ] can be defined via

T [ρ] =i=Ne∏i=1

[ ∫d3ri

]Ψ∗(r1, . . . rNe

) T Ψ(r1, . . . rNe) (751)

so the non-electrostatic contribution to the energy functional may be written asthe sum

G[ρ] = T [ρ] + Exc[ρ] (752)

which defines the exchange and correlation functional Exc[ρ]. The variationalprinciple for the density functional gives

− | e | φes(r) +δExc[ρ]δρ(r)

+δT [ρ]δρ(r)

= µ (753)

Thus, the quantity

− | e | φes(r) +δExc[ρ]δρ(r)

(754)

plays the role of an effective potential, Veff [ρ, r], which not only depends on r,but is also a functional of ρ. The effective potential is given by

Veff [ρ, r] = − | e | φes(r) +δExc[ρ]δρ(r)

(755)

Thus, minimizing the energy functional entails solving the equation

Veff [ρ, r] +δT [ρ]δρ(r)

= µ (756)

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Formally, this is equivalent to solving for the ground state of a (non-interacting)problem with the energy functional given by

Es[ρ] = T [ρ] +∫d3r Vs(r) ρ(r) (757)

in which the electron-electron interaction terms are absent. The variationalprocedure leads to

Vs(r) +δT [ρ]δρ(r)

= µ (758)

Since the particles are non-interacting, this equation is solved by exactly findingthe single-particle wave functions φs,α which make up the single Slater determi-nant that represents the non-interacting ground state. The set of single-particlewave functions are given as the solutions of the eigenvalue equation[

− h2

2 m∇2 + Vs(r)

]φs,α(r) = Es,α φs,α(r) (759)

and then the electron density is given by

ρ(r) =i=Ne∑i=1

| φs,αi(r) |2 (760)

By analogy, one can find the solution of the effective one-body eigenvalue equa-tion [

− h2

2 m∇2 + Veff [ρ, r]

]φeff,α(r) = λeff,α φeff,α(r) (761)

and the electron density is given by

ρ(r) =i=Ne∑i=1

| φeff,αi(r) |2 (762)

The value of the kinetic energy functional for this effective one-body problemcan be found from the eigenvalues λeff,αi

by

T [ρ] =Ne∑i=1

λeff,αi−

∫d3r Veff [ρ, r] ρ(r) (763)

Thus, one also has to minimize the sum of the effective one-body eigenvalues

Ne∑i=1

λeff,αi (764)

This shows that the Kohn-Sham equations provide a method of obtaining thekinetic energy functional and also minimizes the energy functional. Although

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Kohn-Sham eigenvalues λeff,α are often used to describe electron excitation en-ergies, they have no physical meaning. In general, the method only provides theground state energy and ground state electron density. However, there is a den-sity functional analogue of Koopmans’ theorem: the eigenvalue of the highestoccupied effective single-particle level is the Fermi-energy. All the non-trivialinformation about the many-body ground state is contained in the exchange andcorrelation function. This is usually approximated in an uncontrolled fashionby using the local density functional approximation.

9.3.6 The Local Density Approximation

In the Kohn-Sham equations, the remaining unknown function is the exchangeand correlation functional Exc[ρ]. This contains the information about themany-body interactions. The Local Density Approximation is motivated byan assumption namely, that this functional can be represented as an integralover all space of a function of ρ. This assumes that the functional has no non-local terms, or equivalently, that the non-local terms in the density functionalcan be expanded in powers of the gradient. This expansion could be justifiable ifthe density ρ(r) was slowly varying in space. The first few terms of the gradientexpansion of the exchange-correlation energy would be

Exc[ρ] =∫d3r

[Exc0(ρ(r)) + Exc2(ρ(r)) | ∇ ρ(r) |2 + . . .

](765)

where the coefficients Exc0(ρ(r)) and Exc2(ρ(r)) are ordinary functions of thedensity. The local density approximation neglects the gradient terms and usesthe same form of the exchange-correlation function Exc0(ρ(r)) as it pertains tothe free electron gas. In the free electron gas, the electron density ρ is inde-pendent of r. However, in the local density approximation, the uniform densityappearing in the expressions for the uniform electron gas is replaced by similarexpressions but which depend upon the local electron density.

The exchange and correlation terms from the local density approximationare taken from the free electron gas. The energy of the free electron gas iswritten as

E = Ne

[35h2 k2

F

2 m− 3

4e2

πkF − 0.0311

m e4

h2 ln kF + O(1)]

(766)

where the first term is due to the kinetic energy, and the second term is the ex-change energy. The final term is the leading term in the high density expansionof the electron correlation energy, as evaluated by Gell-Mann and Brueckner(M. Gell-Mann and K. Brueckner, Phys. Rev. 106, 364 (1957)). For the freeelectron gas, the electrostatic interaction energy between the electrons and thesmeared out lattice of ions cancels identically with the Hartree term. To obtain

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the exchange-correlation energy, the kinetic energy term is omitted to find

Exc = Ne

[− 3

4e2

πkF − 0.0311

m e4

h2 ln kF + O(1)]

(767)

Combining this together with the two relations

kF =(

3 π2 ρ

) 13

(768)

andNe = V ρ (769)

the exchange-correlation term can be expressed as

Exc = V ρ

[− 3

4e2

π

(3 π2 ρ

) 13

− 0.0104m e4

h2 ln ρ + O(1)]

(770)

The exchange-correlation energy in the local density approximation is simplygiven by

Exc[ρ] =∫

d3r ρ(r)[− 3

4e2

π

(3 π2

) 13

ρ(r)13 − 0.0104

m e4

h2 ln ρ(r) +O(1)]

(771)

Since the effective potential is given by the sum

Veff [ρ, r] = − e φ(r) +δG[ρ]δρ(r)

(772)

the local density approximation for the exchange and correlation energy func-tional contributes a term to the potential in the Kohn-Sham equations of

Vxc[ρ] = − e2

π

(3 π2 ρ(r)

) 13

− 0.0104m e4

h2 ln ρ(r) + O(1) (773)

which adds to the electrostatic potential. The first term comes from the ex-change interaction, and has the form that was originally proposed by J.C. Slaterbut has a different coefficient (J.C. Slater, Phys. Rev. 81, 385 (1951)). Thehigher order terms come from the correlation energy. In practice, the form ofthe exchange-correlation energy that is used as an input to the local densityapproximation is a form which interpolates between the high density limit andthe low density limit. As the density is reduced, the electrons are expected toundergo a phase transition and form a Wigner crystal. Since the energy is ex-pected to be a non-analytic function at the phase transition, the interpolation isof doubtful utility. It seems more appropriate to use the results of Monte Carlocalculations for the correlation energy of the homogenous electron gas (D.M.

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Ceperley and B.J. Alder, Phys. Rev. Lett. 45, 566 (1980)).

The local density functional approximation has been used to successfully de-scribe many different materials, and fails miserably for some others. Attemptsto justify this expression based on the gradient expansion have failed. Basically,the electron density varies too rapidly for the gradient expansion to be useful.

9.4 Static Screening

The response of an electronic system to a static or time independent externalpotential is quite remarkable in a metal. In a metal, the static external potentialis screened out by the electron response. The screening is characterized by thedielectric constant. Classically, the total electrostatic potential φes(r) is relatedto the charge density through Poisson’s equation. In the absence of the externalpotential, Poisson’s equation is written as

− ∇2 φes(r) = 4 π | e |(Z ρions(r) − ρ(r)

)(774)

For a free electron gas, the charge density for the electrons exactly cancels thecontributions from the smeared out charges of the ions. The correspondingpotential is constant, and the reference value φes(r) may be set to be zero.It is expected that a positive external charge with density ρext(r) will inducea change in the electronic density ρind(r). The external charge produces theexternal potential which is defined by the Poisson equation

− ∇2 φext(r) = 4 π | e | ρext(r) (775)

The total potential φes(r) satisfies the Poisson equation

− ∇2 φes(r) = 4 π | e |(ρext(r) − ρind(r)

)(776)

where ρext is assumed to have a positive charge, and the induced electron densityρind is associated with a negative charge. The external potential is related tothe total potential via the dielectric constant through the non-local relation

φext(r) =∫

V

d3r′ ε(r, r′) φes(r′) (777)

In a spatially homogeneous system, the dielectric constant is translationallyinvariant and, therefore, only depends upon the difference r − r′. In this case,the linear response relation is expressed as a convolution

φext(r) =∫

V

d3r′ ε(r − r′) φes(r′) (778)

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This non-local relation, which is valid for homogeneous systems, is simpler afterit has been Fourier transformed. The Fourier transform of φext(r) is defined by

φext(q) =1V

∫V

d3r φext(r) exp[− i q . r

](779)

and the Fourier transform of the dielectric constant is defined by

ε(q) =∫

V

d3r ε(r) exp[− i q . r

](780)

Hence, the Fourier transform of the convolution is just the product of the re-spective Fourier transforms. Thus, the relation becomes

φext(q) = ε(q) φes(q) (781)

Hence, the total potential is reduced by the dielectric constant

φes(q) =φext(q)ε(q)

(782)

The Fourier Transform of the Poisson equations yield

q2 φext(q) = 4 π | e | ρext(q) (783)

and

q2 φes(q) = 4 π | e |(ρext(q) − ρind(q)

)(784)

On using the first equation to eliminate ρext(q) in the second, one obtains

q2 φes(q) = q2 φext(q) − 4 π | e | ρind(q) (785)

Taking the induced charge density term to the other side of the equation pro-duces

q2 φext(q) = q2 φes(q) + 4 π | e | ρind(q) (786)

The definition of the dielectric constant, ε(q), can be used to yield the relation

ε(q) = 1 +4 π | e |q2

ρind(q)φes(q)

(787)

On expressing the total scalar potential as a potential energy term acting onthe electrons

V (q) = − | e | φes(q) (788)

and defining the response function χ(q) as the ratio of the induced density tothe potential

χ(q) =ρind(q)V (q)

(789)

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one finds that the expression for the dielectric constant reduces to

ε(q) = 1 − 4 π | e |2

q2χ(q) (790)

Thus, the dielectric constant is related to the response of the charge densityto the total potential. This response function can be calculated via differenttechniques. However, in making approximations, it is imperative that only theresponse to the total field is approximated and not the response to the externalfield. In a metal, it is all the electrons that take part in screening an exter-nal charge. If each electron were to react independently to screen the externalcharge, the external charge density would be over-screened by a factor of Ne

as each electron by itself could neutralize a charge of | e |. The simplest ap-proximate theory of the system’s response to the total field is given by theThomas-Fermi approximation. The Thomas-Fermi theory pre-dates linear re-sponse theory and density functional theory. A more accurate approximationfor weak potentials is based on linear response theory.

The above derivation has the following drawbacks: First, the use of Poisson’sequation only treats the classical direct Coulomb interactions between aggre-gates of electrons, neglecting the effect of the exchange interactions. Second,the assumption of spatial homogeneity neglects the effect of Umklapp interac-tions in a solid. This neglect produces simple algebraic coupled equations. Theinclusion of Umklapp scattering produces an infinite set of coupled equationswhich has no known analytic solution.

9.4.1 The Thomas-Fermi Approximation

The Thomas-Fermi approximation is based on the assumption that the potentialis slowly varying. The energy of a Bloch state is given by

h2 k2

2 m− | e | φes(r) (791)

The momentum of the highest occupied energy is r dependent kF (r) and isgiven by

h2 k2F (r)

2 m− | e | φes(r) = µ (792)

Thus, the electron density at position r is expressed in terms of a local Fermi-wave vector

ρ(r) = 21

8 π3

4 π k3F (r)

3(793)

On expressing the Fermi-wave vector in terms of the chemical potential and theelectrostatic potential, the total density becomes

ρ(r) =1

3 π2

(2 mh2

) 32(

µ + | e | φes(r)) 3

2

(794)

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The induced density is given in terms of the electrostatic potential via

ρind(r) =1

3 π2

(2 mh2

) 32[ (

µ + | e | φes(r)) 3

2

−(µ

) 32]

(795)

This is the basis of the Thomas-Fermi Theory. On assuming that φes(r) is smallcompared with µ, the equation can be linearized yielding

ρind(r) = | e |(∂ρ0

∂µ

)φes(r) (796)

Thus, the Thomas-Fermi response function is given by

χTF = −(∂ρ0

∂µ

)= − 1

2 π2

(2 mh2

) 32

µ12

= − m kF

π2 h2 (797)

This leads to the Thomas-Fermi approximation for the dielectric constant

ε(q) = 1 +4 π e2

q2

(∂ρ0

∂µ

)= 1 +

k2TF

q2(798)

The Thomas-Fermi wave vector is given in terms of the Fermi-wave vector by

k2TF =

4 m e2

π h2 kF

=4π

kF

a0(799)

and by the alternate expression

kTF

kF=

√4

π kF a0

=(

163 π2

) 13

r12s

= 0.8145 r12s (800)

Thus, kTF is of the order of kF in a metal, and depends on the density ofmobile electrons available to perform screening. This means that the externalpotential or charge is screened over distances of the order of k−1

TF ∼ 1 Angstrom.

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This can be most clearly seen by applying the Thomas-Fermi approximationto the screening of a point charge Z e in a metal. The charged particle is locatedat the origin. From the Fourier transform of Poisson’s equation, the externalpotential is given by

φext(q) =4 π Z | e |

q2(801)

The total potential is given by

φes(q) =φext(q)ε(q)

=4 π Z | e |q2 + k2

TF

(802)

which no longer possesses the long-ranged divergence as k → 0. On perform-ing the inverse Fourier transform, thereby transforming the potential back intodirect space, one has

φes(r) =Z | e |r

exp[− kTF r

](803)

Thus, the charged impurity is exponentially screened over a distance k−1TF . The

induced charge density is given by

ρind(r) =Z e2

r

(∂ρ0

∂µ

)exp

[− kTF r

]=

Z

r

(k2

TF

4 π

)exp

[− kTF r

](804)

On integrating this over all space, one finds that the screening in a metal isperfect in that the total number of electrons in the induced density is equal toZ.

The Thomas-Fermi approximation is deficient. For isolated atoms, it can beshown that the Thomas-Fermi approximation breaks down as it predicts thatthe electron density at the nuclear position is infinite (L.D. Landau and E.M.Lifshitz, Quantum Mechanics), i.e.,

limr → 0

ρ(r) ∼ r−32 (805)

The Thomas-Fermi approximation cannot describe negative ions. That is, inthe Thomas-Fermi approximation, the number of electrons must always be lessthan the nuclear charge. Furthermore, the Thomas-Fermi method also precludesthe binding of neutral atoms into molecules (N.L. Balazs, Phys. Rev. 156, 42(1967)). The Thomas-Fermi method is deficient as it assumes that the poten-tial is slowly varying in space compared to the distance over which the electronsadjust to the potential. Therefore, the Thomas-Fermi method assumes that a

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local approximation for the kinetic energy is valid. This is not the case for mostsimple metals, where the potential due to the ions varies over distances of theorder of Angstroms.

9.4.2 Linear Response Theory

Linear response theory describes the response of a system to a weak perturbingpotential. In such cases, the response is approximately linear in the perturba-tion, so perturbation theory may be used. The effect of a perturbing potentialδV (r) on the electronic system is considered. The effect of this one-body po-tential on the one-body Bloch functions φn,k(r), is examined via perturbationtheory. To first order in the perturbation, the one-electron eigenfunctions arealtered. The one-electron eigenfunctions are no longer Bloch functions, but aregiven by

ψn,k(r) = φn,k(r) +∑

n′,k′ 6=n,k

Mn′,k′;n,k

En,k − En′,k′φn′,k′(r) (806)

where the Mn′,k′;n,k is the matrix element of the perturbing potential betweentwo Bloch functions,

Mn′,k′;n,k =∫

d3r′ φ∗n′,k′(r′) δV (r′) φn,k(r′) (807)

The induced change in the electron density, to first order in δV (r), is found as

ρind(r) =∑n,k,σ

[ ∑n′,k′ 6=n,k

φ∗n,k(r)Mn′,k′;n,k

En,k − En′,k′φn′,k′(r) + c.c.

](808)

where the summation over n, k runs over all the occupied states and c.c. denotesthe complex conjugated term. Thus, the response is not local in the perturbationbut is non-local. The response is expressed in the form

ρind(r) =∫

d3r′ χ(r, r′) δV (r′) (809)

The response function χ(r, r′) is given by the expression

χ(r, r′) =∑n,k,σ

[φ∗n,k(r) φn,k(r′)

∑n′,k′ 6=n,k

φ∗n′,k′(r′) φn′,k′(r)

En,k − En′,k′+ c.c.

](810)

where the summation over n, k, σ runs over all one-electron states that were oc-cupied before the perturbation was turned on. Due to the Pauli exclusion prin-ciple, the summation over n′, k′ is restricted to the unoccupied states. The ex-pression for the response is expected to be modified by the presence of electron-electron interactions.

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The expression for the non-interacting response can easily be evaluated forfree electrons. First, the variables k′ and k are interchanged in the complexconjugate term, and then, due to a cancellation between the two terms, therange of one integration in each term is extended over all momentum space.Once again, the variables k′ and k are interchanged in the second term, to yield

χ(r, r′) =4 mh2

∫|k|≤kF

d3k

( 2 π )3

∫d3k′

( 2 π )3

[exp

[i ( k′ − k ) . ( r − r′ )

]

+ c.c.

] (1

k2 − k′2

)(811)

where the integration over k′ runs over all space. As the Hamiltonian possessestranslational invariance, the response function only depends on the vector R =r − r′. Thus, for the homogeneous electron gas, the real space linear responserelation is in the form of a convolution. The integrations over the directions ofk and k′ can be evaluated by standard means. The range of integration overthe magnitude of k′ can be extended between − ∞ and + ∞ and evaluated bymeans of contour integration which leads to

χ(r, r′) = − 2 mh2

2( 2 π )3

∫ kF

0

dk ksin 2 k | r − r′ || r − r′ |2

(812)

The resulting expression is

χ(r, r′) =2 mh2

1π3

k4F

[cos 2 kF | r − r′ |( 2 kF | r − r′ | )3

− sin 2 kF | r − r′ |( 2 kF | r − r′ | )4

](813)

This is the response to a delta function perturbation at the origin. This deltafunction perturbation requires the electron gas to adjust at very short wavelengths. Instead of having the exponential decay as predicted by the Thomas-Fermi approximation, the response only decays algebraically, with characteristicoscillations determined by the wave vector 2 kF due to the sharp cut off at theFermi-surface. That is, 2 kF is the largest wave vector available for a zero en-ergy density fluctuation in which an electron is excited from just below to justabove the Fermi-surface. The oscillations in the density that occur in responseto a potential are known as Friedel oscillations.

It is more convenient to consider the Fourier transform of the response func-tion

χ(q) =∫

V

d3r χ(r) exp[− i q . r

](814)

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The response function χ(q) is evaluated from

χ(q) = 22 mh2

1V

∑k<kF

[1

k2 − ( k + q )2+

1k2 − ( k − q )2

](815)

where the summation over k runs over the occupied states within the Fermi-sphere. The summation can be replaced by an integration

χ(q) = − 22 mh2

14 π2

∫ kF

0

dk k2

∫ +1

−1

d cos θ[

1q2 + 2 k q cos θ

+1

q2 − 2 k q cos θ

]= − 2

2 mh2 q

14 π2

∫ kF

0

dk k ln| q + 2 k || q − 2 k |

(816)

The response is given explicitly by

χ(q) = −(m kF

h2 π2

) [12

+4 k2

F − q2

8 q kFln| 2 kF + q || 2 kF − q |

](817)

This is the Lindhard function for the free electron gas (J. Lindhard, Kgl. DanskeVidenskab. Selskab. Mat. Fys. Medd. 28, 8 (1954)). The Lindhard functionreduces to the value of the corresponding Thomas-Fermi response function atq = 0, which is

χTF = − k2TF

4 π e2(818)

Thus, for very slowly varying potentials, the response of the free electron gasis identical to the response function found using the Thomas-Fermi approxima-tion. The magnitude of the Lindhard function drops with increasing q, fallingto half the q = 0 value at q = 2 kF . At this point, the slope has a weaklogarithmic singularity. The electron gas is ineffective in screening the appliedpotential for q ≥ 2 kF as 2 kF corresponds to the largest wave vector at whichelectrons on the spherical Fermi-surface can readjust.

9.4.3 Density Functional Response Function

The change in the electron density ρind(r) due to an external potential, φext(r),in which electron-electron interactions are included can be obtained from densityfunctional theory. The relation between the induced density and the externalpotential is given by the screened response function

ρind(r) = −∫

d3r′ χs(r − r′) | e | φext(r′) (819)

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Density functional theory yields an effective potential which contains the effectof the electron-electron interactions

| e | φeff (r) = | e | φext(r) −∫

d3r′ ρind(r′)

[| e |2

| r − r′ |+

δ2Exc

δρ(r) δρ(r′)

](820)

The relation between the induced electron density and the effective potential isgiven by

ρind(r) = −∫

d3r′ χ0(r − r′) | e | φeff (r′) (821)

where χ0(r−r′) is the Lindhard response function for non-interacting electrons.

The response function, including the effects of the electron-electron interac-tions, can be found by Fourier transforming the above set of equations. Thus,the full response function is given by

ρind(q) = − χs(q) | e | φext(q) (822)

and the non-interacting response function is given by

ρind(q) = − χ0(q) | e | φeff (q) (823)

The relationship between the effective and external potential is given by

φeff (q) = φext(q) +(− 4 π | e |

q2+

π | e |k2

TF

Γxc(q))ρind(q) (824)

This equation can be solved for χs(q) in terms of the non-interacting responsefunction χ0(q).

χs(q) =χ0(q)

1 − | e |2(

4 πq2 − π

k2T F

Γxc(q))χ0(q)

(825)

The dielectric constant ε(q) is given by

1ε(q)

=φes(q)φext(q)

1ε(q)

= 1 − 4 π | e |q2

ρind(q)φext(q)

1ε(q)

= 1 +4 π e2

q2χs(q) (826)

The exchange contribution to Γxc(q) is given in the limit q → 0 by

Γxc(q) =[

1 +59

(q

2 kF

)2

+73225

(q

2 kF

)4

+ . . .

](827)

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It is noted that if the effect of the exchange-correlation terms to the screeningcould be dropped, then the dielectric constant is approximated by

1ε(q)

≈ 1 +4 π e2

q2χ0(q)ε(q)

(828)

which is consistent with the result for free-electrons using the Lindhard approx-imation for the response to the total field, and treating the total scalar potentialclassically via Poisson’s equation. In obtaining this approximate result, it wasnecessary to calculate the response of the system to the external potential byincluding processes, to all orders in e2, in which the electron gas is polarized.That is, the electron gas is polarized by the external potential and then the re-sulting polarization and the external potential are screened by the electron gas,ad infinitum. This infinite regression is necessary for the external charge to becompletely screened at large distances, and is a consequence of the long-rangednature of the Coulomb interaction limq → 0

4 π e2

q2 → ∞. This re-emphasizesthe importance of only making approximations in the response to the total po-tential χ and not in the response to the external potential χs.

The response of the electronic system to an applied potential can be usedto examine the stability of a structure. The electronic energy change due tothe perturbation consists of the potential energy of interaction between theions and the electron gas, as well as the change induced into the energy ofelectron-electron repulsions. All of these energies can be expressed in terms ofthe induced charge density.

——————————————————————————————————

9.4.4 Exercise 44

Calculate the Lindhard function for a free electron gas E0k = h2 k2

2 m in d =1, d = 2 and d = 3 dimensions, at zero temperature.

——————————————————————————————————

9.4.5 Exercise 45

Consider the Lindhard function for a tight-binding non-degenerate s band on ahyper-cubic lattice with the dispersion relation

Ek = E0 − 2 ti=d∑i=1

cos ki a (829)

Show that the response function at the corner of the Brillouin zone q =πa (1, 1, 1, ., ., .) diverges as the number of electrons in the band approaches one

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per site.

——————————————————————————————————

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10 Stability of Structures

In this chapter, the structural stability of a metal is discussed. The total energyof the metal will be expressed in terms of the energy for a uniform electron gas,and the interaction with the periodic structure will be treated as a perturbation.

10.1 Momentum Space Representation

In the uniform electron gas, the electro-static energy between pairs of electronsand also between the particles forming the background positive charge exactlycancels with the interaction between the electrons and the positive charges.When the periodic potential is introduced as a perturbation, the change in thetotal energy can be expressed in terms of the change in the one-electron eigen-values. However, the inclusion of the Coulomb interaction between the latticeand the electrons will also require that the contributions from electron-electronand ion-ion interaction be explicitly reconsidered in the calculation of the totalenergy.

The energy of a one-electron Bloch state, calculated to second in the poten-tial due to the ionic lattice, can be expressed in terms of the one-electron energyeigenvalues for a free electron gas as

En,k =h2 k2

2 m+ Vions(k, k) +

2 mh2

∑k 6=k′

| Vions(k′, k) |2

k2 − k′2(830)

The zero-th order and first order terms in this energy are independent of thelattice structure of the ionic potential. This can be seen by examining the matrixelements

Vions(k′, k) =1V

∫d3r Vions(r) exp

[i ( k − k′ ) . r

](831)

which is just the average potential when k = k′. The sum over the energiesof all the occupied Bloch states, (k, σ), contribute to the total energy of thesolid. The first order contribution from Vions(k, k), like the kinetic energy ofthe free electron gas, does not depend on the structure. These terms combineto produce a volume-dependent contribution to the solid’s total energy.

The other volume-dependent contribution to the total energy of the solidoriginates from the electron-electron interactions and the ion-ion interactions.It is convenient to combine these terms with the energy of the zero-th orderelectron-ion interaction, due to the exact cancellation for the uniform electrongas. This combination is the total electrostatic interaction. It can be evaluatedin the approximation that the Coulomb interactions between different Wigner-Seitz cells are totally screened (E. Wigner and F. Seitz, Phys. Rev. 43, 804(1933), Phys. Rev. 45, 509 (1934)). This means that the ion-ion interactions

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need not be considered explicitly. The electrostatic contribution to the energyis then written as

Ees =∫

d3r Vions(r) ρ(r) +e2

2

∫d3r

∫d3r′

ρ(r) ρ(r′)| r − r′ |

(832)

To lowest order in the structure, the electrostatic contribution to the totalenergy can be evaluated by considering the Wigner-Seitz unit cell to be sphericalwith radius RWS . The electron density is given by

ρ =3 Z

4 π R3WS

(833)

For the uniform density, the electron-electron repulsion term is evaluated as

Ees =∫

d3r Vions(r) ρ(r) +35Z2 e2

RWS(834)

For the free-electron approximation for the kinetic energy to be valid, the elec-trostatic contribution from the ions should be calculated using the pseudo-potential. We shall use the Ashcroft empty core approximation for the ionicpseudo-potential. Inside the Wigner-Seitz cell, the pseudo-potential reduces tothat of an isolated atom

Vatom(r) = − Z e2

rfor r ≥ Rc

= 0 for r ≤ Rc (835)

where Rc is the radius of the ionic core. Hence, for a structureless metal, theelectrostatic terms can be expressed as

Ees = − 32Z2 e2

RWS

[1 −

(Rc

RWS

)2]

+35Z2 e2

RWS(836)

The potential terms inversely proportional to the Wigner-Seitz radius can becombined as − 9

10Z2 e2

RW S. The coefficient α = 9

10 is the Madelung constant fora solid composed of spherical unit cells. In general, the Madelung constant willdepend slightly on the structure of the lattice.

For a solid with structure, the electrostatic energy can be expressed as thesum

E = EM + Ec (837)

where EM is the Madelung energy and Ec is the core energy. The Madelungenergy is the electrostatic energy due to point charges immersed in a neutralizinguniform distribution of electrons. The Madelung energy is given by

EM = − αZ2 e2

RWS(838)

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where α is the structure-dependent Madelung constant. The Madelung con-stants are evaluated as

Structure α

b.c.c. 0.89593f.c.c. 0.89587h.c.p. 0.89584

simple hexagonal 0.88732simple cubic 0.88006

The Madelung energy is seen to increase as the symmetry is lowered. Theremaining contribution to the electrostatic energy is defined to be the coreenergy. The core energy is given by

Ec =32Z2 e2

RWS

(Rc

RWS

)2

(839)

and, as it is the electrostatic energy associated with the spherical pseudo-potential core, is not dependent on the solid’s structure.

The largest structural-dependent contribution to the energy originates fromthe second order terms of the Bloch energies in the electron-ion interaction

E(2)n,k =

2 mh2

∑k 6=k′

| Vions(k′, k) |2

k2 − k′2(840)

On summing over all the occupied Bloch states ( | k | < kF ) and both spinvalues σ, one obtains a contribution E2 to the total energy of

E2 =2 mh2

∑|k|<kF ,σ

∑k 6=k′

| Vions(k′, k) |2

k2 − k′2(841)

In the free electron basis, the matrix elements of the electron-ion interaction,Vions(k′, k), only depends on the momentum difference q = k′ − k.

Vions(k′, k) =1V

∫d3r Vions(r) exp

[i ( k − k′ ) . r

](842)

The potential due to the lattice can be written as the sum of the individualpotentials from the atoms. The basis position of the j-th atom in the unit cellis denoted by rj and the Bravais lattice vector is denoted by Ri. Thus, thepotential for the lattice of ions is given by

Vions(r) =∑i,j

Vj(r −Ri − rj) (843)

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The matrix elements are then given by

Vions(q) =1V

∫d3r

∑i,j

Vj(r −Ri − rj) exp[− i q . r

]

=1V

∑i,j

∫d3r exp

[− i q . Ri

]Vj(r −Ri − rj) exp

[− i q . ( r −Ri )

](844)

This can be expressed as

Vions(q) =∑

i

exp[− i q . Ri

] ∑j

exp[− i q . rj

]Vj(q)

(845)

where Vj(q) is related to the Fourier transform of the potential from the j-thatom of the basis

Vj(q) =∫

d3r Vj(r) exp[− i q . r

](846)

For simplicity, a crystal with a mono-atomic basis is considered. The matrixelements are only non-zero when q is a reciprocal lattice vector Q. The matrixcan be expressed in terms of the structure factor S(Q), via

Vions(Q) =N

VS(Q) V0(Q) (847)

The structure dependence of the total electronic energy is contained in thesecond order contribution

E2 =N2

V 2

∑k<kF ,σ

2 mh2

∑Q6=0

| S(Q) |2 | V0(Q) |2

k2 − ( k + Q )2(848)

where the sum over k, σ runs over the occupied states ( k < kF ), and theterm with Q = 0 is omitted. On interchanging the order of the summationsover k and Q, one finds that the second order term can be expressed in termsof the Lindhard function χ(q),

E2 =N2

V 2

∑Q6=0

| S(Q) |2 | V0(Q) |2∑k,σ

fk

Ek − Ek+Q

=12N2

V 2

∑Q6=0

| S(Q) |2 | V0(Q) |2∑k,σ

fk − fk+Q

Ek − Ek+Q

=12N2

V

∑Q6=0

| S(Q) |2 | V0(Q) |2 χ(Q) (849)

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The summation over q is limited to the reciprocal lattice vectors Q. Therefore,it depends on the lattice structure through the structure factors | S(Q) |2, andon the electron density through the factors χ(q), and the nature of the ionsthrough V0(q). The latter is often expressed in terms of the Thomas-Fermiscreened pseudo-potential

V0(q) = − 4 π Z e2cos q Rc

q2 + k2TF

(850)

where Rc is the radius of the ionic core. The potential has a node at q0 Rc = π2 .

The structural part of the electronic energy depends sensitively on the positionof the node q0 with respect to the smallest reciprocal lattice vectors Q. Recip-rocal lattice vectors close to a node q0 contribute little to the cohesive energy.The system may lower its structural energy, if Q moves away from q0 withoutcausing a change in the volume-dependent contribution to the energy. Recip-rocal lattice vectors greater than 2 kF contribute little as the response of theelectron gas is negligible.

In addition to these terms, there is a structural contribution arising fromthe electron-electron interactions which comes from the induced change in theelectron density

E2 es = − 12

∑q

ρ∗ind(q)4 π e2 V

q2ρind(q) (851)

This term occurs since the effect of electron-electron interactions have beendouble counted. On noting that the ionic potential only has non-zero Fouriercomponents at q = Q, and that

ρind(Q) = χ(Q) Vions(Q) (852)

one can combine this with the contribution from the Bloch energies. The factor4 πq2 χ(q) is related to the dielectric constant ε(q) through

ε(q) = 1 − 4 π e2

q2χ(q) (853)

The two second order terms can be combined to yield the dominant contributionto the structural energy

Estructural =12N2

V

∑Q6=0

| S(Q) |2 | V0(Q) |2 χ(Q) ε(Q) (854)

Since both pseudo-potential terms V0(Q) include screening, the explicit factorof ε(q) cancels with one factor of ε(q) in the denominators. Thus, the structuralenergy is only screened by one factor of the dielectric constant. The magni-tude of the structural energy is quite small. The maximum magnitude of the

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pseudo-potential is Z e2

Rcwhich may be as small as 1

2 eV. The magnitude of χis given by the inverse of the Fermi-energy which is typically 5 eV. Thus, thestructural energy is of the order of milli-Rydbergs. Since the structure factorvanishes unless q = Q, the structural energy depends on the screened potentialonly at the reciprocal lattice vectors. Note that the pseudo-potential containsnodes at the wave vectors q0 = n 2 π

Rc. The structural energy is composed of

negative contributions, but the contributions from the reciprocal lattice vectorswhich are close to the nodes, contribute little to the stability of the structure.In fact, reciprocal lattice vectors at the nodes would correspond to the specialcase in which the band gap at the appropriate Brillouin zone boundary is zero.Usually, the opening of a band gap at a Brillouin zone boundary in a conductionband can result in an increased stability of the structure. The electronic statesbelow the ”band gap” are depressed and, if occupied, result in a lowering of thesolid’s energy. However, the states above the ”band gap,” if empty, are raisedbut don’t contribute to the solid’s energy.

Al is f.c.c. and the reciprocal lattice vectors (1, 1, 1) and (2, 0, 0) are bothlarger than q0. On moving down the column of the periodic table from Al toGa and then In, the ratios of Q/q0 are reduced.

Al Ga In

Q(1, 1, 1)/q0 1.04 0.94 0.93

Q(2, 0, 0)/q0 1.20 1.09 1.08

As the Q vector for (2, 0, 0) approaches q0 in In, there is a loss in structuralstability and the series undergoes a transition from the f.c.c. to a tetragonalstructure (V. Heine and D.L. Weaire, Solid State Physics, 24, 1 (1970)). Whenthis transition occurs, the set of equivalent f.c.c. reciprocal lattice vectors thathave Q

q0∼ 1, split. In the tetragonal structure, as the structure is sheared, the

reciprocal lattice vectors undergo different changes. Some values of Qq0

move tohigher values while others move to lower values. This type of transformationleaves the atomic volume unchanged, but as all the ”band gaps” V (Q) increase,the transition lowers the energy of the structure. This structural transition oc-curs when the lowering of the electronic energy outweighs the increase in theMadelung energy.

10.2 Real Space Representation

The dominant electronic structural energy is given by a sum over all Q of

Estructural =12N2

V

∑Q6=0

| S(Q) |2 | V0(Q) |2 χ(Q) ε(Q) (855)

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where S(Q) is the structure factor evaluated at a reciprocal lattice vector. Thiscan be written as a sum over all vectors q, by using the Laue identity

N2∑Q

δq,Q =∑

R 6=R′

exp[i q . ( R − R′ )

](856)

where R and R′ are Bravais Lattice vectors. The structural energy then takesthe form

Estructural =1

2 V

∑q 6=0

∑R 6=R′

exp[i q . ( R − R′ )

]| S(q) |2 θ(q) (857)

where θ(q) is defined to be

θ(q) =1V| V0(q) |2 χ(q) ε(q) (858)

It should be noted that in this approximation, θ(q) is independent of the direc-tion of q. The product of the structure factors can be written as

| S(q) |2 =∑i 6=j

exp[i q . ( ri − rj )

](859)

Thus, on denoting the position of the atoms by Rj = R + rj , one has

Estructural =12

∑i 6=j

∑q 6=0

exp[i q . ( Ri − Rj )

]θ(q) (860)

The Fourier transform of θ(q) is defined as

θ(Ri,j) =∑

q

θ(q) exp[i q . Ri,j

](861)

where the vector Ri,j denotes the relative position of the two atoms. On chang-ing the sum to an integration, θ(Ri,j) is evaluated as

θ(Ri,j) =V

( 2 π )3

∫d3q θ(q) exp

[i q . Ri,j

]=

2 π V( 2 π )3

∫ ∞

0

dq q2∫ 1

−1

d cos θ θ(q) exp[i q Ri,j cos θ

]

=V

( 2 π )2

∫ ∞

0

dq q2 θ(q)

( exp[i q Ri,j

]− exp

[− i q Ri,j

]i q Ri,j

)

=V

2 π2

∫ ∞

0

dq q2 θ(q)(

sin q Ri,j

q Ri,j

)(862)

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Thus, the electronic contribution to the structural energy has the real spacerepresentation

Estructural =12

∑i 6=j

θ(Ri,j) (863)

The Madelung energy, which is the sum over the interaction energies of the ions

EMadelung =12

∑i 6=j

Z2 e2

| Ri,j |(864)

should also be added to the structural energy. Thus, the total structural energycan be expressed in terms of the sum of pair potentials Θ(R), where

Θ(R) =Z2 e2

R+ θ(R) (865)

The pair potential represents the interaction between a pair of bare ions in thesolid plus the effect of the screening clouds. The pair potential does not de-scribe the volume dependence of the energy of the solid, but only the structure-dependent contribution to the energy. The pair potential can be expressed as

Θ(R) =Z2 e2

R+

V

( 2 π2 )

∫ ∞

0

dq q2(

sin q Rq R

)θ(q) (866)

The first and second term can be combined to yield the interaction between anion and a screened ion. This can be seen by expressing the potential in termsof a dimensionless function V (q) defined by

V0(q) =4 π Z e2

q2 ε(q)V (q) (867)

Thus, the interaction can be expressed as

Θ(R) =Z2 e2

R

[1 +

∫ ∞

0

dq

(sin q Rq

)4 π e2

q2χ(q)ε(q)

| V (q) |2]

=Z2 e2

R

[1 +

∫ ∞

0

dq

(sin q Rq

)1 − ε(q)ε(q)

| V (q) |2]

=Z2 e2

R

[1 − 2

π

∫ ∞

0

dq

(sin q Rq

)| V (q) |2

]

+Z2 e2

R

[2π

∫ ∞

0

dq

(sin q Rq

)1ε(q)

| V (q) |2]

(868)

The integral in the first term can be evaluated with the calculus of residues, andis evaluated in terms of the pole at q = 0. Since V (0) = 1 and as R > Rc, the

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integral is equal to unity. Therefore, the first term cancels identically. Hence,the interaction energy between a bare ion and a screened ion is given by theexpression

Θ(R) =Z2 e2

R

[2π

∫ ∞

0

dq

(sin q Rq

)1ε(q)

| V (q) |2]

(869)

The long-ranged nature of the Coulomb interaction between the bare ions hasbeen completely eliminated due to the screening. The very weak logarithmicsingularity at q = 2 kF leads to Friedel oscillations in the potential at asymp-totically large distances R

Θ(R) = Acos 2 kF R

R3(870)

However, at intermediate distances, the pair potential can be approximatelyexpressed as the sum of three (damped) oscillatory terms (D.G. Pettifor andM.A. Ward, Solid. State. Commun. 49, 291 (1984))

Θ(R) =Z2 e2

R

3∑n=1

Bn cos(αn 2 kF R + φn

)exp

[− βn kF R

](871)

where the phase shift depends on the ionic core radius Rc and the electron den-sity. This form is obtained as a result of approximating the Lindhard functionχ(q) by a ratio of polynomials (Pade approximation). The integration over qcan be performed via contour integration. The pairs of complex poles in theintegrand produce the terms which have a damped oscillatory dependence onR. The fit parameters for Na are given by:

Nan 1 2 3αn 0.291 0.715 0.958βn 0.897 0.641 0.271Bn 1.961 0.806 0.023φn

π 1.706 1.250 1.005

while for Mg the interaction is specified by

Mgn 1 2 3αn 0.224 0.664 0.958βn 0.834 0.675 0.277Bn 5.204 1.313 0.033φn

π 1.599 0.932 0.499

and for Al one has

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Aln 1 2 3αn 0.156 0.644 0.958βn 0.793 0.698 0.279Bn 7.954 1.275 0.030φn

π 1.559 0.832 0.431

The contributions to the pair potential are arranged in order of increasing rangei.e., they are arranged in order of decreasing βn. The Z dependence of the phaseshifts determine the position of the minima of the pair potential. This pair po-tential, although it only has a magnitude of about 10−2 eV, dominates thestructural energy.

neighbor shell number 1 2 3 4 5b.c.c.

number of neighbors 8 6 12 24 8neighbor distance

√3

2 1√

2√

112

√3

f.c.c.number of neighbors 12 6 24 12 24neighbor distance

√2

2 1√

62

√2

√102

h.c.p.number of neighbors 12 6 2 18 12neighbor distance

√2

2 1 2√3

√6

2

√11√6

The energy difference between the f.c.c. and h.c.p. structures are determinedby the third, fourth and fifth nearest neighbors, as the number and positionsof the nearest and next nearest neighbors are the same. Hence, the relativestability of this pair of structures is determined by the reasonably long distancebehavior of the pair potential. The form of the pair potential can be used todescribe the relative stability of the h.c.p. and f.c.c structures of Na, Mg andAl (A.K. McMahan and J.R. Moriarty, Phys. Rev. B 27, 3235 (1983)). Atambient pressure, Na and Mg are h.c.p. and Al is f.c.c.. The f.c.c. form ofMg is unstable due to a repulsive contribution from the pair potentials betweenthe (12) fourth nearest neighbor pairs. The h.c.p. form of Al is unstable dueto a repulsive contribution from the pair potentials between the (12) fifth near-est neighbor pairs. This trend is understood as almost entirely being due tothe long-ranged component of the pair potential. Basically, as the value of Zincreases, when going across the column from Na to Al, the phase shift of thelong-ranged interaction decreases. This means that the oscillations in the pairpotential move out to larger distances. This causes the changes in the pair po-tential at the positions of the fourth or fifth nearest neighbors.

Under pressure, these materials are predicted to transform to a b.c.c. phase.The phase shift of the long-ranged component decreases monotonically withincreasing Rc

rs, which corresponds to increasing pressure. The change in the

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phase shifts moves the oscillations in the pair potentials to distances larger dis-tances than the neighbor distances. This shows that as the pressure is increased,one may expect the energy differences between the h.c.p. and f.c.c. phases tooscillate. The energy differences between the b.c.c. and close-packed phasesoriginate from the combined (14) first and second nearest neighbors in b.c.c.and the (12) nearest neighbors of the close-packed structures. The separationsof the neighbors in the b.c.c. structure should be scaled by a factor of 2−

13 to

yield the same electron density as the close-packed structures. After this scal-ing, it is found that the nearest neighbor distances in the close-packed structuresare intermediate between the nearest neighbor and the next nearest neighbordistances of the b.c.c. structure. On decreasing the phase shift, one may ex-pect to see the b.c.c. phase become unstable to a close-packed phase when the(8) nearest neighbors experience the hard core repulsive potential. On furtherdecreasing the phase shift, the (12) neighbors of the close-packed phase will ex-perience the same hard core potential at which point, the b.c.c. becomes stableagain. This region of stability of the b.c.c. structure will remain until the (8)next nearest neighbors are compressed to distances where the pair potential hasthe form of a hard core repulsion.

These and similar considerations illuminate the origins of the stability ofdifferent structures, which are hard to extract from other methods, as the struc-tural energy typically amounts to only 1% of the cohesive energy of a solid. Ingeneral, the cohesive energy of the solid will also involve three and four-atominteractions etc., in addition to the pair potential. To obtain a more accuratedescription of structural stability, it is necessary to utilize density functionalcalculations.

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11 Metals

In a metal with Ne electrons, the state with minimum energy has the Ne low-est one-electron energy eigenvalue states filled with one electron per state (perspin) in accordance with the Pauli exclusion principle. In a metal, the highestoccupied and the lowest unoccupied state have energies which only differ by aninfinitesimal amount. This energy is called the Fermi-energy, εF . Thus, theone-electron states have occupation numbers distributed according to the law

f(ε) = 1 if ε < εF

f(ε) = 0 if ε > εF (872)

The number of electrons in a solid Ne is dictated by charge neutrality to be equalto the number of nuclear charges N Z. At finite temperatures, the electronoccupation numbers are statistically distributed according to the Fermi-Diracdistribution function

f(ε) =1

1 + exp[β ( ε − µ )

] (873)

where β−1 = kB T is the inverse temperature. The Fermi-Dirac distributionrepresents the probability that a state with energy ε is occupied. Due to thePauli exclusion principle, the distribution also represents the average occupationof the level with energy ε. The value of the chemical potential coincides withthe Fermi-energy at zero temperature µ(0) = εF . Since the solid remainscharge neutral at finite temperatures, the chemical potential is determined bythe condition that the solid contains Ne electrons. For a solid with a density ofstates given by ρ(ε), per spin, the total number of electron is given by

2∫ +∞

−∞dε ρ(ε) f(ε) = Ne (874)

which is an implicit equation for µ. The factor of two represents the number ofdifferent spin polarizations of the electron.

11.1 Thermodynamics

Due to the Pauli exclusion principle, the density of states at the Fermi-energycan often be inferred from measurements of the thermodynamic properties of ametal. As the characteristic energy scale for the electronic properties is of theorder of eV, and room temperature is of the order of 25 meV, the thermody-namic properties can usually be evaluated in the asymptotic low-temperatureexpansion first investigated by Sommerfeld (A. Sommerfeld, Zeit. fur Physik,47, 1 (1928)). The low-temperature Sommerfeld expansion of the electronicspecific heat, for non-interacting electrons, shall be examined.

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11.1.1 The Sommerfeld Expansion

The total energy of the solid can be expressed as an integral

E = 2∫ +∞

−∞dε ρ(ε) ε f(ε) (875)

Integrals of this type can be evaluated by expressing them in terms of the zerotemperature limit of the distribution and small deviations about this limit.

E = 2∫ µ

−∞dε ρ(ε) ε

+ 2∫ µ

−∞dε ρ(ε) ε

[f(ε) − 1

]+ 2

∫ +∞

µ

dε ρ(ε) ε f(ε) (876)

The variable of integration in the terms involving the Fermi-function is changedfrom ε to the dimensionless variable x defined by

ε = µ + kB T x (877)

The Fermi-function becomes

f(µ + kB T x) =1

1 + exp x(878)

Thus, the integral becomes

E = 2∫ µ

−∞dε ρ(ε) ε

+ 2 kB T

∫ 0

−∞dx ρ(µ+ kBTx) ( µ+ kBTx )

[f(µ+ kBTx) − 1

]+ 2 kB T

∫ +∞

0

dx ρ(µ+ kBTx) ( µ+ kBTx ) f(µ+ kBTx)

(879)

The integral over the negative range of x is re-expressed in terms of the newvariable y where

y = − x (880)

Thus, the energy is expressed as

E = 2∫ µ

−∞dε ρ(ε) ε

+ 2 kB T

∫ ∞

0

dy ρ(µ− kBTy) ( µ− kBTy )[f(µ− kBTy) − 1

]+ 2 kB T

∫ +∞

0

dx ρ(µ+ kBTx) ( µ+ kBTx ) f(µ+ kBTx)

(881)

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However, the Fermi-function satisfies the relation

1 − f( µ− kBTy ) = f( µ+ kBTy ) (882)

or equivalently

1 − 11 + exp[ − y ]

=1

1 + exp[ y ](883)

On setting y back to x, one finds

E = 2∫ µ

−∞dε ρ(ε) ε

+ 2 kB T

∫ +∞

0

dx1

1 + exp x×

×

[ρ(µ+ kBTx) ( µ+ kBTx ) − ρ(µ− kBTx) ( µ− kBTx )

](884)

The terms within the square brackets can be Taylor expanded in powers ofkB T x, and the integration over x can be performed. Due to the presence ofthe Fermi-function, the integrals converge. One then has an expansion which iseffectively expressed in powers of kB T / µ. Thus, the energy is expressed as

E = 2∫ µ

−∞dε ρ(ε) ε

+ 4 kB T

∫ ∞

0

dx

∞∑n=0

x(2n+1)

1 + expx( kBT )2n+1

(2n+ 1)!

(∂

∂µ

)(2n+1)[µ ρ(µ)

] (885)

The integrals over x are evaluated as∫ ∞

0

dxxn

1 + expx=

∫ ∞

0

dx xn∞∑

l=1

( − 1 )l+1 exp[− l x

]

= n!∞∑

l=1

( − 1 )l+1

ln+1(886)

which are finite for n ≥ 1. Furthermore, the summation can be expressed interms of the Riemann ζ functions defined by

ζ(m) =∞∑

l=1

1lm

(887)

Using this, one finds that∞∑

l=1

( − 1 )l+1

l2n+2=(

22n+1 − 122n+1

)ζ(2n+ 2) (888)

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The Riemann zeta functions have special values

ζ(2) =π2

6

ζ(4) =π4

90(889)

Thus, the Sommerfeld expansion for the total electronic energy only involveseven powers of T 2, that is,

E = 2∫ µ

−∞dε ρ(ε) ε

+ 4 ( kB T )2∞∑

n=0

(22n+1 − 1

22n+1

)ζ(2n+ 2) ( kBT )2n

(∂

∂µ

)(2n+1)[µ ρ(µ)

] (890)

The coefficients may be evaluated in terms of the Riemann ζ functions.

Although the expansion contains an explicit temperature dependence, thereis an implicit temperature dependence in the chemical potential µ. This tem-perature dependence can be found from the equation

Ne = 2∫ +∞

−∞dε ρ(ε) f(ε) (891)

which also can be expanded in powers of T 2 as

Ne = 2∫ µ

−∞dε ρ(ε)

+ 4 ( kB T )2∞∑

n=0

(22n+1 − 1

22n+1

)ζ(2n+ 2) ( kBT )2n

(∂

∂µ

)(2n+1)[ρ(µ)

] (892)

Since Ne is temperature independent, in principle, the series expansion can beinverted to yield µ in powers of T .

11.1.2 The Specific Heat Capacity

The electronic contribution of the heat capacity, for non-interacting electrons,can be expressed as

CNe(T ) = T

(∂S

∂T

)Ne

=(∂E

∂T

)Ne

(893)

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as the solid remains electrically neutral. Using the Sommerfeld expansion of theenergy, the specific heat can can be expressed as the sum of the specific heat atconstant µ and a term depending on the temperature derivative of µ at constantNe.

CNe= 4 k2

BT∞∑

n=0

(n+ 1)(

22n+1 − 122n

)ζ(2n+ 2) ( kBT )2n

(∂

∂µ

)(2n+1)[µ ρ(µ)

]

+(∂µ

∂T

)Ne

[2 µ ρ(µ)

+ 4 k2BT

2∞∑

n=0

(22n+1 − 1

22n+1

)ζ(2n+ 2) (kBT )2n

(∂

∂µ

)(2n+2)[µ ρ(µ)

] ](894)

In the above expression, µ is to be expanded in powers of T about its zero tem-perature value µ = εF . The temperature derivative of the chemical potentialcan be evaluated from the temperature derivative of the equation for the fixednumber of electrons Ne,

0 = 4 k2BT

∞∑n=0

(n+ 1)(

22n+1 − 122n

)ζ(2n+ 2) ( kBT )2n

(∂

∂µ

)(2n+1)

ρ(µ)

+(∂µ

∂T

)Ne

[2 ρ(µ)

+ 4 k2BT

2∞∑

n=0

(22n+1 − 1

22n+1

)ζ(2n+ 2) ( kBT )2n

(∂

∂µ

)(2n+2)

ρ(µ)

](895)

This equation yields the temperature dependence of µ which can be substitutedback into the expression for the temperature dependence of CN . This yields theleading term in the low-temperature expansion for the electronic-specific heatof non-interacting electrons as

CN = k2B T 4 ζ(2) ρ(µ) + O(k4

B T 3)

= k2B T

2 π2

3ρ(µ) + O(k4

B T 3) (896)

The coefficient of the linear term is proportional to the density of states, perspin, at the Fermi-energy. The result is understood by noting that the Pauliexclusion principle prevents electrons from being thermally excited, unless theyare within kB T of the Fermi-energy. There are ρ(µ) kB T such electrons, andeach electron contributes kB to the specific heat. Thus, the low-temperaturespecific heat is of the order of k2

B T ρ(µ). The inclusion of electron-electroninteraction changes this result, and in a Fermi-liquid, may increase the coefficient

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of T . The low-temperature specific heat is enhanced, due to the enhancement ofthe quasi-particle masses. This can be demonstrated by a simplified calculationin which the quasi-particle weight is assumed to be independent of k. Since thequasi-particle width in the vicinity of the Fermi-energy is negligible, one hasthe relationship between the quasi-particle density of states and the density ofstates for non-interacting electrons given by

ρqp(E) =∑

k

δ

(Z(k) E − Ek + µ

)

=∑

k

1Z(k)

δ

(E −

( Ek − µ )Zk

)

=∑

k

1Z(k)

δ

(E − Eqp(k)

)(897)

Also, the quasi-particle density of states at the Fermi-energy is un-renormalizedas

ρqp(0) =∑

k

δ

(µ − Ek

)= ρ(µ) (898)

The γ term in the low-temperature specific heat is calculated from the quasi-particle entropy S defined in terms of the quasi-particle occupation numbersnqp

k by

S = − kB

∑σ,k

[nqp

k ln nqpk + ( 1 − nqp

k ) ln( 1 − nqpk )

]

= − 2 kB

∫ ∞

−∞dE Z ρqp(E)

[f(E) ln f(E) + ( 1 − f(E) ) ln( 1 − f(E) )

](899)

Thus, in this approximation, the coefficient of the linear T term is given by

γ = limT → 0

CN

T

=(∂S

∂T

)Ne

= k2B Z

2 π2

3ρ(µ) (900)

In the more general case, the specific heat coefficient is enhanced through a kweighted average of the quasi-particle mass enhancement Zk. For materials likeCeCu6, CeCu2Si2, CeAl3 and UBe13, the value of the γ coefficients are ex-tremely large, of the order of 1 J / mole of f ion / K2, which is 1000 times larger

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than Cu. The quasi-particle mass enhancements are inferred by comparison toL.D.A. electronic density of states calculations and are about 10 to 30. Theenhancement is assumed to be due to the strong electron-electron interactions,which the L.D.A. fails to take into account.

——————————————————————————————————

11.1.3 Exercise 46

Calculate the next to leading order term in the low-temperature electronic-specific heat.

——————————————————————————————————

11.1.4 Exercise 47

CeNiSn is thought to be a zero-gap semiconductor with a V shaped density ofstates. The density of states near the Fermi-level is approximated by

ρ(ε) = α0 ε for ε > 0ρ(ε) = − α1 ε for ε < 0 (901)

where α0 and α1 are positive numbers. Find the leading temperature depen-dence of the low-temperature specific-heat.

——————————————————————————————————

11.1.5 Pauli Paramagnetism

In the absence of spin-orbit scattering effects, the susceptibility of a metal canbe decomposed into two contributions; the susceptibility due to the spins of theelectrons, and the susceptibility due to the electrons orbital motion. The spinsusceptibility for non-interacting electrons gives rise to the Pauli-paramagneticsusceptibility which is positive, and is temperature independent at sufficientlylow temperatures. The susceptibility due to the orbital motion has a negativesign and, therefore, yields the Landau-diamagnetic susceptibility.

The magnetization due to the electronic spins can be calculated from

Mz = −(∂Ω∂Hz

)(902)

where the grand canonical potential is given by

Ω = − kB T∑α

ln

(1 + exp

[− β ( Eα − µ )

] )(903)

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and where the sum over α runs over the quantum numbers of the single particlestates including the spin. The applied magnetic field Hz couples to the quantumnumber corresponding to the z component of the spin of the electron, σ, via theZeeman energy

HZeeman = − g | e |2 me c

Hz Sz

= − µB Hz σ (904)

where the spin angular momentum is given by S = h2 σ and the gyromagnetic

ratio g = 2 originates from the Dirac or Pauli equation. The quantity µB isthe Bohr magneton and is given in terms of the electron’s charge and mass by

µB =| e | h2 me c

(905)

The energy of a particle can then be written as

Eσ(k) = E(k) − µB σ Hz (906)

where σ is the eigenvalue of the Pauli spin matrix σz. The density of states, perspin, in the absence of the field is defined as

ρ(ε) =∑

k

δ( ε − E(k) ) (907)

Thus, in the presence of a field, one has the spin dependent density of states

ρσ(ε) = ρ(ε+ µBσHz) (908)

The grand canonical potential can be expressed as an integral over the densityof states

Ω = − kB T

∫ ∞

−∞dε∑

σ

ρ(ε+ µBσHz) ln

(1 + exp

[− β ( ε− µ )

] )

= − kB T

∫ ∞

−∞dE

∑σ

ρ(E) ln

(1 + exp

[− β ( E − µBσHz − µ )

] )(909)

where the variable of integration has been changed in the last line. The summa-tion over σ runs over the values ± 1. The spin contribution to the magnetizationinduced by the applied field is given by

Mz = µB

∫ ∞

−∞dE

∑σ

σ ρ(E)1

1 + exp[β ( E − µBσHz − µ )

]= µB

∫ ∞

−∞dE

∑σ

σ ρ(E) f( E − µBσHz )

= µB

(Ne(σ = 1) − Ne(σ = − 1)

)(910)

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The magnetization due to the spins is just proportional to the number of up-spinelectrons minus the down-spin electrons. The spin susceptibility is given by

χzzp (T,Hz) =

(∂Mz

∂Hz

)(911)

and is given by

χp(T,Hz) = − µ2B

∫ ∞

−∞dE

∑σ

σ2 ρ(E)∂

∂Ef( E − µBσHz )

(912)

It is usual to measure the susceptibility at zero field. Since the derivative of theFermi-function is peaked around the chemical potential, only electrons withinkB T of the Fermi-energy contribute to the Pauli-susceptibility. At sufficientlylow temperatures, one may use the approximation

− ∂

∂Ef = δ( E − εF ) (913)

so that the zero temperature value of the Pauli-susceptibility is evaluated as

χp(0) = 2 µ2B ρ(εF ) (914)

which is inversely proportional to the free electron mass, and is also proportionalto the density of states at the Fermi-energy. The finite temperature susceptibil-ity can be evaluated by integration by parts, to obtain

χp(T ) = 2 µ2B

∫ ∞

−∞dE f(E)

∂Eρ(E) (915)

The zero field spin susceptibility can then be obtained via the Sommerfeld ex-pansion

χp(T ) = 2 µ2B

[ρ(µ) + 2 k2

BT2

∞∑n=0

(22n+1 − 1

22n+1

)ζ(2n+2) (kBT )2n

(∂

∂µ

)(2n+2)

ρ(µ)

](916)

Thus, the spin susceptibility has the form of a power series in T 2. The temper-ature dependence of the chemical potential can be found from the equation forNe. The leading change in the chemical potential ∆µ due to T is given by

∆µ = − k2BT

2 π2

6

(∂ρ(εF )

∂εF

ρ(εF )

)+ O( k4

BT4 ) (917)

The temperature dependence of the chemical potential depends on the logarith-mic derivative of the density of states, such that it moves away from the region

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of high density of states to keep the number of electrons fixed. This leads tothe leading temperature dependence of the Pauli susceptibility being given by

χp(T ) = 2 µ2B

[ρ(εF ) +

π2

6k2

BT2

(∂2ρ(εF )∂ε2F

−( ∂ρ(εF )

∂εF)2

ρ(εF )

)+ O( k4

BT4 )

](918)

The temperature dependence gives information about the derivatives of the den-sity of states.

The coefficient γ of the linear T term in the low-temperature specific-heatand the zero temperature susceptibility are proportional to the density of statesat the Fermi-energy. The susceptibility and specific heat can be used to definethe dimensionless ratio

limT → 0

T χp(T )C(T )

=χp(0)γ

(919)

This ratio is known as the Sommerfeld ratio. For free electrons, this ratio hasthe value

limT → 0

T χp(T )C(T )

=3 µ2

B

π2 k2B

(920)

The effect of electron-electron interactions can change this ratio, as they mayaffect the susceptibility in a different manner than the specific heat. The Stonermodel, discussed in the chapter on magnetism, shows that the effect of electron-electron interactions can produce a large enhancement of the paramagnetic sus-ceptibility for electron systems close to a ferromagnetic instability. Thus, near aferromagnetic instability, the Sommerfeld ratio is expected to be large. However,for heavy fermion materials where both C(T )/T and χp(0) are highly enhanced,the value of the Sommerfeld ratio is very close to that of non-interacting elec-trons.

——————————————————————————————————

11.1.6 Exercise 48

Determine the field dependence of the low-temperature Pauli susceptibility.

——————————————————————————————————

11.1.7 Exercise 49

Determine the high temperature form of the Pauli susceptibility.

——————————————————————————————————

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11.1.8 Landau Diamagnetism

Free electrons in a magnetic field aligned along the z axis have quantized energiesgiven by

Ekz,n =h2 k2

z

2 m+(n +

12

)h ωc (921)

where

ωc =| e | Hz

m c(922)

is the cyclotron frequency and n is a positive integer. For a cubic environmentof linear dimension L, the value of kz is given by

kz =2 πL

nz (923)

The Landau levels have their orbits in the x − y plane quantized and have alevel spacing of h ωc. Each Landau level is highly degenerate. The degeneracyD, or number of electrons with a given n and kz, can be found as the ratio ofthe area of the sample divided by the area enclosed by the classical orbit

D =L2

2 π r2c(924)

where rc is the radius of the classical orbit. This radius can be obtained byequating the field energy with the zero point energy of the Landau level

m

2ω2

c r2c =

12h ωc (925)

Thus, the degeneracy is given by

D =L2

2 π hm ωc

D =| e | L2

h cHz (926)

Since Hz ∼ 1 kG, a typical value of the degeneracy is of the order ofD ∼ 1010. These levels can be treated semi-classically as there are an enormousnumber of Landau levels in an energy interval. The number of occupied Landaulevels is given by the Fermi-energy µ divided by h ωc,

µ

h ωc=

µ| e | hm c Hz

(927)

The numerical constant has the value

| e | hm c

∼ 1.16 × 10−8 eV / G (928)

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so, with µ ∼ 1 eV and Hz ∼ 104 G, one finds that the number of occupiedLandau levels is approximately given by

µ

h ωc∼ 104 (929)

11.1.9 Landau Level Quantization

The Hamiltonian of a free electron in a magnetic field is given by

H =(p +

| e |c

A

)2

/ ( 2 m ) (930)

Using the gauge A = (0,Hzx, 0) appropriate for a field along the z axis, thenthe Schrodinger equation takes the form

− h2

2 m∇2φ − i

| e | Hz

m cx

(∂φ

∂y

)+

e2 H2z

2 m c2x2 φ = E φ (931)

This can be solved by the substitution

φ(r) = f(x) exp[i ( ky y + kz z )

](932)

so that f(x) satisfies

− h2

2 m

(∂2f

∂x2

)+

[ (h ky +

| e | Hz

cx

)2 12 m

−(E − h2 k2

z

2 m

) ]f(x) = 0

(933)which is recognized as the equation for the harmonic oscillator with energyeigenvalue

E − h2 k2z

2 m=(n +

12

)h ωc (934)

where

ω2c =

e2 H2z

m2 c2(935)

That is, the motion in the plane perpendicular to the field, Hz, is quantizedinto Landau levels (L.D. Landau, Zeit. fur Physik, 64, 629 (1930)). The energyspacing between the levels is given by h ωc, where

ωc =| e | Hz

m c(936)

and the orbit is centered around the position

x0 = − h ky c

| e | Hz(937)

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The momentum dependence of the position x0 has a classical analogy. Thecenter of the classical orbit is determined by its initial velocity vy via

vy = ωc x0 (938)

so the center of the quantum orbit is determined by py. The energy of theLandau orbit is given by

Ekz,n =h2 k2

z

2 m+(n +

12

)h ωc (939)

The degeneracy of the n-th level must correspond to the number of kx, ky valuesfor Hz = 0 that collapse onto the Landau levels as Hz is increased.

The degeneracy can be enumerated in the case of periodic boundary condi-tions,

φ(x, y, z) = φ(x, Ly − y, z) (940)

The periodic boundary conditions imply that

exp[i ky Ly

]= 1 (941)

or

ky =2 πLy

ny (942)

The x dependent factor of the wave function f(x) is centered at x0 where

x0 = − h ky c

| e | Hz(943)

For a sample of width Lx, one must have Lx > x0 > 0, so one has the equality

| e | Hz Lx

h c> − ky > 0 (944)

The degeneracy, D, is the number of quantized ky values that satisfy this in-equality. The degeneracy is found to be

D =| e | Hz Lx

h c/

2 πLy

=| e | Hz Lx Ly

2 π h c(945)

independent of n. Thus, the degeneracy D of every Landau is given by

D =| e | Hz Lx Ly

2 π h c(946)

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The degeneracy can be expressed in terms of the amplitude of the oscillationsin the x direction, which is defined as the length scale that determines theexponential fall off of the ground state wave function

rc =√

h

m ωc(947)

The degeneracy of the Landau levels can also be expressed as

D =Lx Ly

2 π r2c(948)

as previously found from classical considerations. The quantization of the or-bital motion, in the presence of a periodic potential, has been considered byRauh (A. Rauh, Phys. Stat. Solidi, B 65, K131 (1974), A. Rauh, Phys. Stat.Solidi, B 69, K9 (1975)) and by Harper (P.G. Harper, Ph.D. Thesis, Univer-sity of Birmingham (1954), P.G. Harper, Proc. Phys. Soc. London, A 68, 874(1955)). These authors have shown that the periodic potential causes the Lan-dau levels to be broadened or split.

11.1.10 The Diamagnetic Susceptibility

The diamagnetic susceptibility is determined from the field dependence of thegrand canonical potential, Ω,

Ω = 2D Lz

2 π h

∫ ∞

−∞dkz

∑n

(h ωc ( n+

12

) +h2 k2

z

2 m− µ

)Θ(µ−h ωc ( n+

12

)− h2 k2z

2 m

)(949)

On integrating over kz, one finds

Ω = −83D Lz

2 π

(2 mh2

) 12

µhωc

− 12∑

n=0

(µ − h ωc ( n +

12

)) 3

2

(950)

or

Ω = −23

V

( π2 h )m ωc

(2 mh2

) 12

µhωc

− 12∑

n=0

(µ − h ωc ( n +

12

)) 3

2

(951)

The summation over n can be performed using the Euler-MacLaurin formula

n=N∑n=0

F (n) =∫ N

0

dx F (x) +12

( F (0) + F (N) ) +112

( F ′(N)− F ′(0) ) + . . .

(952)

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This produces the leading order field dependence of the grand canonical poten-tial, given by

Ω = − 23

V

( π2 h2 )m

(2 mh2

) 12[

25µ

52 − 1

16h2 ω2

c µ12 + . . .

](953)

The diamagnetic susceptibility is given by the second derivative with respect tothe applied field

χd = −(∂2Ω∂H2

z

)= − µ2

B

V m

3 π2 h2

(2 mh2

) 12

µ12 (954)

where we have expressed the orbital magnetic moment in terms of the (orbital)Bohr magneton

µB =| e | h2 m c

(955)

The diamagnetic susceptibility χd can be compared with the Pauli paramagneticsusceptibility χp. For free electrons, the Pauli susceptibility is given by

χp = 2 µ2B ρ(µ)

= 2 µ2B

V

2 π2

m kF

h2

= µ2B

V m

π2 h2

(2 mh2

) 12

µ12 (956)

Hence, the spin and orbital susceptibilities are related via

χd = − 13χp (957)

Thus, the Landau diamagnetic susceptibility is negative and has a magnitudewhich, for free electrons, is just one third of the Pauli paramagnetic susceptibility(L.D. Landau, Zeit. fur Physik, 64, 629 (1930)). The diamagnetism results fromthe quantized orbital angular momentum of the electrons. The value of µB inthe diamagnetic susceptibility is given by the band mass m∗, whereas the factorof µB in the Pauli susceptibility is defined in terms of the mass of the electronin vacuum me. In systems such as Bismuth, in which the band mass is smallerthan the free electron mass, the diamagnetic susceptibility is larger by a factorof

χd

χp= − 1

3

(me

m∗

)2

(958)

and the diamagnetic susceptibility can be larger than the Pauli susceptibility.The susceptibility of Bismuth is negative.

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In the presence of spin-orbit coupling, the orbital angular momenta are cou-pled with the spin angular momenta. As a result, the components of the totalsusceptibility are coupled. The manner in which the total angular momentumcouples to the field is described by the g factor.

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11.2 Transport Properties

11.2.1 Electrical Conductivity

The electrical conductivity of a normal metal is considered. The applicationof an electromagnetic field will produce an acceleration of the electrons in themetal. This implies that the distribution of the electrons in phase space willbecome time dependent, and in particular the Fermi-surface will be subject toa time dependent distortion. However, the phenomenon of electrical transportin metals is usually a steady state process, in that the electric current density jproduced by a static electric field E is time independent and obeys Ohm’s law

j = σ E (959)

where σ is the electrical conductivity. This steady state is established by scatter-ing processes that dynamically balances the time dependent changes producedby the electric field. That is, once the steady state has been established, theacceleration of the electrons produced by the electric field is balanced by scat-tering processes that are responsible for equilibration.

Since Ohm’s law holds almost universally, without requiring any noticeablenon-linear terms in E to describe the current density, it is safe to assume thatthe current density can be calculated by only considering the first order terms inthe electro-magnetic field. The validity of this assumption can be related to thesmallness of the ratio of λ | e | E

µ where λ is the mean free path, E the strengthof the applied field and µ the Fermi-energy. This has the consequence that theFermi-surface in the steady state where the field is present is only weakly per-turbed from the Fermi-surface with zero field. A number of different approachesto the calculation of the electrical conductivity will be described. For simplic-ity, only the zero temperature limit of the conductivity shall be calculated. Thedominant scattering process for the conductivity in this temperature range isscattering by static impurities.

11.2.2 Scattering by Static Defects

The electrical conductivity will be calculated in which the scattering is due toa small concentration of randomly distributed impurities. The potential dueto the distribution of impurities located at positions rj , each with a potentialVimp(r) is given by

V (r) =∑

j

Vimp(r − rj) (960)

This produces elastic scattering of electrons between Bloch states of differentwave vectors. The transition rate in which an electron is scattered from thestate with Bloch wave vector k to a state with Bloch wave vector k′ is denotedby 1

τ(k→k′) . If the strength of the scattering potential is weak enough, the

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transition rate can be calculated from Fermi’s golden rule as

1τ(k → k′)

=2 πh

∣∣∣∣ < k | V | k′ >∣∣∣∣2 δ( E(k) − E(k′) )

=2 πh

1V 2

∑i,j

exp[i (k − k′) . (ri − rj)

] ∣∣∣∣ Vimp(k − k′)∣∣∣∣2 δ( E(k) − E(k′) )

(961)

where the delta function expresses the restriction imposed by energy conserva-tion in the elastic impurity scattering processes. As usual, the presence of thedelta function requires that the transition probability is calculated by integrat-ing over the momentum of the final state. As the positions of the impuritiesare distributed randomly, the scattering rate shall be configurational averaged.The configurational average of any function is obtained by integrating over thepositions of the impurities

F =∏j

[1V

∫d3rj F (rj)

](962)

The configurational average of the scattering rate is evaluated as

1τ(k → k′)

= =2 πh

1V 2

∑j

∣∣∣∣ Vimp(k − k′)∣∣∣∣2 δ( E(k) − E(k′) ) (963)

where only the term with i = j survives. The conductivity can be calculatedfrom the steady state distribution function of the electrons, in which the scat-tering rate dynamically balances the effects of the electric field. This is found,in the quasi-classical approximation, from the Boltzmann equation.

The Boltzmann Equation.

The distribution of electrons in phase space at time t, f(k, r, t), is deter-mined by the Boltzmann equation. The Boltzmann equation can be found beexamining the increase in an infinitesimally region of phase space that occursduring a time interval dt. The number of electrons in the infinitesimal volumed3k d3r located at the point k, r at time t is

f(k, r, t) d3k d3r (964)

The increase in the number of electrons in this volume that occurs in timeinterval dt is given by(f(k, r, t+ dt) − f(k, r, t)

)d3k d3r =

∂tf(k, r, t) d3k d3r dt + O(dt2)

(965)

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This increase can be attributed to changes caused by the regular or deterministicmotion of the electrons in the applied field, and partly due to the irregularmotion caused by the scattering. The appropriate time scale for the changes inthe distribution function due to the applied fields is assumed to be much longerthan the time interval in which the collisions occur. The deterministic motionof the electrons trajectories in phase space results in a change in the numberof electrons in the volume d3k d3r. The increase due to these slow time scalemotions is equal to the number of electrons entering the six-dimensional volumethrough its surfaces in the time interval dt minus the number of electrons leavingthe volume. This is given by

∆ f(k, r, t) d3k d3r = − dt

[∇ .

(r f(k, r, t)

)+ ∇k .

(k f(k, r, t)

) ](966)

and the slow rates of change in position and momentum of the electrons isdetermined via

r =h k

m

h k = − | e | Em

(967)

Hence, the deterministic changes are found as

∆ f(k, r, t) d3k d3r = −[∇ .

(h k

mf(k, r, t)

)− ∇k .

(| e | Em h

f(k, r, t)) ]

d3k d3r dt

(968)

This involves the sum of two terms, one coming from the change of the electronsmomentum and the other from the change in the electrons position. The twogradients in this expression can be evaluated, each gradient yields two terms.One term of each pair involves a gradient of the distribution function, while theother only involves the distribution function itself. One term, originating fromthe change in the electrons position involves the spatial variation of the velocity.From Hamilton’s equations of motion it can be shown that the coefficient of thisterm is equal to the second derivative of the Hamiltonian,

∇ . r = ∇ . ∇p H (969)

while the similar term originating from the change in particles momentum isjust equal to the negative of the second derivative

∇p . p = − ∇ . ∇p H (970)

Since the Hamiltonian is ana analytic function these terms are equal magnitudeand of opposite sign. Thus, these terms cancel yielding only

∆ f(k, r, t) d3k d3r =

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= −[h k

m. ∇

(f(k, r, t)

)−(| e | Em h

. ∇k f(k, r, t)) ]

d3k d3r dt

(971)

The remaining contribution to the change in number of electrons per unit timeoccurs from the rapid irregular motion caused by the impurity scattering. Thenet increase is due to the excess in scattering of electrons from occupied statesat (k′, r) into an unoccupied state (k, r) over the rate of scattering out of state(k, r) into the unoccupied states at (k′, r). The restriction imposed by the Pauliexclusion principle, is that the state to which the electron is scattered intoshould be unoccupied in the initial state. This restriction is incorporated byintroducing the probability that a state (k, r) is unoccupied, through the factor( 1 − f(k, r, t) ).

∆ f(k, r, t) d3k d3r =∑k′

[1

τ(k′ → k)f(k′, r, t) ( 1 − f(k, r, t) )

− 1τ(k → k′)

f(k, r, t) ( 1 − f(k′, r, t) )

]d3k d3r dt (972)

On equating these three terms, cancelling common factors of d3k d3r dt oneobtains the Boltzmann equation

∂tf(k, r, t) = −

[∇ .

(h k

mf(k, r, t)

)− ∇k .

(| e | Em h

f(k, r, t)) ]

+ I

[f(k, r, t)

](973)

where the functional I[f]

is the collision integral and is given by

I

[f(k, r, t)

]=∑k′

[1

τ(k′ → k)f(k′, r, t) ( 1 − f(k, r, t) )

− 1τ(k → k′)

f(k, r, t) ( 1 − f(k′, r, t) )

](974)

Thus, the Boltzmann equation can be written as the equality of a total derivativeobtained from the regular motion and the collision integral which represents thescattering processes

d

dtf(k, r, t) = I

[f(k, r, t)

](975)

Since1

τ(k → k′)=

1τ(k′ → k)

(976)

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the collision integral can be simplified to yield

I

[f(k, r, t)

]=∑k′

[1

τ(k → k′)

(f(k′, r, t)− f(k, r, t)

) ](977)

Due to time reversal invariance of the scattering rates and conservation of en-ergy, the collision integral vanishes in the equilibrium state. In equilibrium,the distribution function is time independent and uniform in space. The dis-tribution function, therefore, only depends on k in a non-trivial manner, andcan be written in terms of the Fermi-function f0(k). Thus, in this case, thedistributions are related via

f(k, r, t) =1Vf0(k) (978)

The equilibrium distribution function f0(k) is only a function of the energyE(k). The condition of conservation of energy which occurs implicity in thescattering rate requires f0(k) = f0(k′). Hence, in equilibrium the collisionintegral vanishes.

In the steady state produced by the application of an electric field, theelectron density will be time independent and uniform throughout the metal,and so the temporal and spatial dependence of f(k, r, t) can still be neglected.In this case, the distribution function in momentum space is still related to thedistribution function in phase space via

f(k, r, t) =1Vf(k) (979)

where f(k) is the non-equilibrium distribution describing the steady state.

The Solution of the Boltzmann equation.

Since the electron distribution in the steady state conduction of electrons isclose to equilibrium one may look for solutions, for f(k) close to the equilibriumFermi-Dirac distribution function. Thus, solutions of the form can be sought

f(k) = f0(k) + Φ(k)∂f0(k)∂E(k)

(980)

where Φ is an unknown function, with dimensions of energy. It is to be shownthat Φ(k) is determined by the electric field and small compared with the Fermi-energy µ. The above ansatz for the non-equilibrium distribution function ismotivated by the notion that the term proportional to Φ occurs from a Taylorexpansion of the steady state distribution function. In other words, the varia-tion of Φ with k occurs from the distortion of the Fermi-surface in the steadystate.

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If the above ansatz for the steady state distribution is substituted into theBoltzmann equation one obtains

− ∇k .

(| e | Em h

f(k, r, t))

= I

[Φ(k)

∂f0(E(k))∂E(k)

](981)

This shows that the energy Φ has a leading term which is proportional to thefirst power of the electric field. However, in order to obtain a current thatsatisfies Ohm’s law, only the terms in Φ terms linear in E need to be calculated.Therefore, the Boltzmann equation can be linearized by dropping the term thatinvolves the electric field and Φ, since this is second order in the effect of thefield. The linearized Boltzmann equation can be solved by noticing that thecollision integral is equal to the source term which is proportional to the scalarproduct ( k . E ). Hence, it is reasonable to assume that Φ(k) has a similarform

Φ(k) = A(E(k)) ( k . E ) (982)

where A(E) is an unknown function of the energy, or other constants of motion.Due to conservation of energy, the unknown coefficient can be factored out ofthe collision integral, as can be the factor of ∂f0

∂E since both are only functionsof the energy. It remains to evaluate an integral of the form∫

d3k′ δ( E(k) − E(k′) ) | Vimp(k − k′) |2(

( k′ − k ) . E)

(983)

The integration over k′ can be performed by first integrating over the magnitudeof q = k − k′. On using the property of the energy conserving delta function,

δ( E(k) − E(k′) ) =2 mh2 δ( q2 − 2 k q cos θ′ ) (984)

this sets the magnitude of q = 2 k cos θ′, where the direction of k was chosenas the polar axis. For simplicity it shall be assumed that the impurity potentialis short ranged, so that the dependence of V (q) on q is relatively unimportant.The integration over the factor of q . E can easily be evaluated, and the resultcan be shown to be proportional to just ( k . E ) . That is, on expressing thescalar product as

q . E = ( q sin θ′ cosφ′ Ex + q sin θ′ sinφ′ Ey + q cos θ′ Ez ) (985)

on integrating over the azimuthal angle φ′

dΩ′ = dθ′ sin θ′ dφ′ (986)

the terms proportional to Ex and Ey vanish. The integration over the polarangle θ′ produces a factor

8 π2 mh2 k2

∫ 1

−1

d cos θ′ cos3 θ′ | V ( 2 k cos θ′ ) |2 Ez (987)

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This yields the result

= 4 π2 mh2 k ( k . E ) 2

∫ 1

−1

d cos θ′ cos3 θ′ | V ( 2 k cos θ′ ) |2 (988)

which can be expressed as an integral over the scattering angle θ = π − 2 θ′

= 2 π2 mh2 k ( k . E )

∫ 1

−1

d cos θ ( 1 − cos θ ) | V ( 2 k sinθ

2) |2 (989)

On identifying the non-equilibrium part of the distribution function with

Φ(k)∂f0(k)∂E(k)

= ( k . E ) A(E)∂f0(k)∂E(k)

(990)

yields the solution for the non-equilibrium contribution of the distribution func-tion as

Φ(k) = + τtr(k)| e |h

(E . ∇k E(k)

)(991)

Thus, Φ is proportional to the energy change of the electron produced by theelectric field in the interval between scattering events. In the above expression,the term

1τtr(k)

= c2 πh

∑k′

δ( E(k) − E(k′)) | V (k − k′) |2(

1 − cos θ)

(992)

is identified as the transport scattering rate, in which c is the concentration ofimpurities. The transport scattering rate has the form of the rate for scatteringout of the state k but has an extra factor of ( 1 − cos θ ). In the quantum for-mulation of transport this factor appears as a vertex correction. Basically, theelectrical current is related to the momentum of the electrons in the directionof the applied field. Forward scattering processes do not result in a reductionof the momentum and, therefore, leave the current unaffected. The transportscattering rate involves a factor of ( 1 − cos θ ) where θ is the scattering angle.This factor represents the relative importance of large angle scattering in thereduction of the total current.

The Current Density.

The current density can be obtained directly from the expression.

j = − 2 | e | 1V

∑k

1h

∂E(k)∂k

f(k)

= − 2 | e | 1V

∑k

1h

∂E(k)∂k

(f0(k) + Φ(k)

∂f0(k)∂E(k)

)(993)

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where the factor of 2 represents the sum over the electron spins. On viewing theelectron distribution function as the first two terms in a Taylor expansion, theelectron distribution function can be described by an occupied Fermi-volumewhich has been displaced from the equilibrium position in the direction of theapplied field. The displacement of the Fermi-volume produces the average cur-rent in the direction of the field. The first term represents the current that isexpected to flow in the equilibrium state. This term is zero, as can be seen byusing the symmetry of the energy E(k) = E(−k) in the Fermi-function. Dueto the presence of the velocity vector 1

h ∇kE(k), it can be seen that the currentproduced by an electron of momentum k identically cancels with the currentproduced by an electron of momentum −k.

Thus, the non-zero component of the current originates from the non-equilibriumpart of the distribution function. This can only be evaluated once the Blochenergies are given. The current is given by

j = − 2e2

h2

∑k

τ(k)tr ∇kE(k)(∇kE(k) . E

) (∂f0(k)∂E(k)

)(994)

On recognizing the zero temperature property of the Fermi-function

−(∂f0(k)∂E(k)

)= δ( E(k) − µ ) (995)

it is seen that the electrical current is carried by electrons in a narrow energyshell around the Fermi-surface. On using the symmetry properties of the integralone finds that only the diagonal component of the conductivity tensor is non-zero and is given by

σα,β = − 2 δα,β

3e2

h2

∑k

τ(k)tr | ∇kE(k) |2(∂f0(k)∂E(k)

)(996)

For free electron bands, the conductivity tensor is evaluated as

σα,β = δα,βρ e2 τtrm

(997)

where ρ is the density of electrons, m is the mass of the electrons and τtr is theFermi-surface average of the transport scattering rate τtr(k).

——————————————————————————————————

11.2.3 Exercise 50

Determine the conductivity tensor σα,β(q, ω) which relates the Fourier compo-nent of a current density jα(q, ω) to a time and spatially varying applied electric

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field with a Fourier amplitude Eβ(q, ω) via Ohm’s law

jα(q, ω) =∑

β

σα,β(q, ω) Eβ(q, ω) (998)

Assume that ω is negligibly small compared with the Fermi-energy so that thescattering rate can be evaluated on the Fermi-surface.

The above result should show that in the zero frequency limit ω → 0the q = 0 conductivity is purely real and given by the standard expressionσα,β(0, 0) = δα,β

ρ e2 τtr

m , and decreases for increasing ω. The frequency widthof the Drude peak is given by the scattering rate 1

τtr.

——————————————————————————————————

11.2.4 The Hall Effect and Magneto-resistance.

The Hall effect occurs when an electrical current is flowing in a sample and amagnetic field is applied in a direction transverse to the direction of the cur-rent density. Consider a sample in the form of a rectangular prism, with axesparallel to the axes of a Cartesian coordinate system. The magnetic field is ap-plied along the z direction and a current flows along the y direction. The Halleffect concerns the appearance of a voltage (the Hall voltage) across a samplein the x direction. The Hall voltage appears in order to balance the Lorentzforce produced by the motion of the charged particles in the magnetic field.The initial current flow in the x direction sets up a net charge imbalance acrossthe sample in accordance with the continuity equation. The build up of staticcharge produces the Hall voltage. In the steady state, the Hall voltage balancesthe Lorentz force opposing the further build up of static charge. The sign of theHall voltage is an indicator of the sign of the current carrying particles.

The Hall Electric field is given by

Ex ex = + vy Bz ey ∧ ez (999)

The Hall voltage VH is related to the electric field and the width of the sampledx via

VH = − Ex dx = − vy Bz dx

= − jy Bz dx

ρ q

(1000)

Hence, measurement of the Hall voltage VH and jy, together with the magnitudeof the applied field Bz, determines the carrier density ρ and the charge q. This

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is embodied in the definition of the Hall constant, RH

RH =Ey

jx Hz(1001)

which for semi-classical free carriers of charge q and density ρ is evaluated as

RH =1ρ q

(1002)

In other geometries, one notices that the current will flow in a direction otherthan parallel to the applied field. The conductivity tensor will not be diagonal,as will the resistivity tensor. The dependence of the resistivity on the magneticfield is known as magneto-resistance. The phenomenon of transport in a mag-netic field can be quite generally addressed from knowledge of the conductivitytensor in an applied magnetic field. This can be calculated using the Boltzmannequation approach.

The Boltzmann Equation.

The Boltzmann equation for the steady state distribution f(p), in the pres-ence of static electric and magnetic fields, can be expressed as

− | e |(E + v ∧ B

). ∇p f(p) = I

[f(p)

](1003)

Since only a solution for f(p) is sought which contain terms linear in the electricfield E, the equation can be linearized by making the substitution f(p) → f0(p)but only in the term explicitly proportional to E.

− | e | E . ∇p f0(p) − | e | ( v ∧ B ) . ∇p f(p) = I

[f(p)

](1004)

The substitution of the zero field equilibrium distribution function f0(p) in theterm proportional to f0(p) without any magnetic field corrections is consistentwith the equilibrium in the presence of a static magnet field. This can be seenby examining the limit E = 0, where the Boltzmann equation reduces to

− | e | ( v ∧ B ) . ∇p f(p) = I

[f(p)

](1005)

which has the solution f(p) = f0(p) since in this case the collision integralvanishes and the remaining term is also zero as

( v ∧ B ) . ∇p f0(p) = ( v ∧ B ) . ∇p E(p)∂f0(p)∂E(p)

= ( v ∧ B ) . v∂f0(p)∂E(p)

= 0

(1006)

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since due to the vector identity

( A ∧ B ) . A = 0 (1007)

the scalar product vanishes. This is just a consequence of the fact that a mag-netic field does not change the particles energy. The observation can be usedto simplify the Boltzmann equation as, has been seen, the magnetic force termonly acts on the deviation from equilibrium.

The ansatz for the steady state distribution function is

f(p) = f0(p) + Φ(p)∂f0(p)∂E(p)

= f0(p) + v . C∂f0(p)∂E(p)

(1008)

where C is an unknown vector function. It shall be shown that the vector Cis independent of p. In this case, the collision integral simplifies to the casethat was previously considered. Namely, the collision integral reduces to thetransport scattering rate times the non-equilibrium part of the steady statedistribution function. On cancelling the common factor involving the derivativeof the Fermi-function, and using

∇p . v =1m∗ (1009)

one finds

| e | E . v +| e |m∗ c

( v ∧ B ) . C =1τtr

( v . C ) (1010)

The solution of this equation is independent of v, hence C is a constant vector.This can be seen explicitly by substituting the identity

( v ∧ B ) . C = ( B ∧ C ) . v (1011)

back into the Boltzmann equation. The resulting equation can be solved for allv if C satisfies the algebraic vector equation

| e | E +(| e |m∗ c

)B ∧ C =

1τtr

C (1012)

To solve the above algebraic equation it is convenient to change variables

ωc =(| e |m∗ c

)B (1013)

This shall be solved by finding the components parallel and transverse to B.

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If the scalar product of the algebraic equation is formed with ωc and onrecognizing that ωc . ( ωc ∧ C ) = 0 one finds that the component of Cparallel to the magnetic field is given by

ωc . C = | e | τtr ωc . E (1014)

The transverse component of C can be obtained by taking the vector productof the algebraic equation with ω. This results in the equation

ωc ∧ ( | e | E ) + ωc ∧ ( ωc ∧ C ) =1τtr

ωc ∧ C (1015)

butωc ∧ ( ωc ∧ C ) = ωc ( ωc . C ) − ω2

c C (1016)

so one recovers the relation between the transverse component and C from

ωc ∧ ( | e | E ) + ωc ( ωc . C ) − ω2c C =

1τtr

ωc ∧ C (1017)

by eliminating the longitudinal component. The resulting relation is found as

ωc ∧ ( | e | E ) + ωc ( | e | τtr ωc . E ) − ω2c C =

1τtr

ωc ∧ C (1018)

The transverse component can be substituted back into the original algebraicequation to find the complete expression for C.

| e |(ωc ∧ E + τ ωc ( ωc . E )

)− ω2

c C =1τ2

C − 1τ| e | E (1019)

Therefore, C is given by the constant vector(1 + ω2

c τ2

)C = τ | e |

(E + τ2 ( ωc . E ) ωc + τ ( ωc ∧ E )

)(1020)

which only depends upon E and B but not on the momentum p. This leads tothe explicit expression for the non-equilibrium distribution function of

f(k) = f0(k)

+τ | e |

1 + ω2c τ

2

(v . E + τ2 ( v . ωc ) ( ωc . E ) + τ ( v ∧ ωc ) . E

)∂f0(k)∂E(k)(1021)

The deviation from equilibrium can be interpreted in terms of an anisotropicdisplacement of the Fermi-function involving the work done by the electric field

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on the electron in the time interval between scattering events.

The Conductivity Tensor.

The average value of the current density is given by

j = − | e | 2V

∑k

v(k) f(k) (1022)

where f(k) is the steady state distribution function, and the factor of 2 rep-resents the summation over the electrons spin. On substituting for the steadystate distribution function, and noting that because of the symmetry k → − kin the equilibrium distribution function, no current flows in the absence of theelectric field. The current density j is linear in the magnitude of the electricfield E, and is given by

j = 2 | e2 | τ

1 + ω2c τ

2

∫d3k

( 2 π )3v

(− ∂f0

∂E

×

[( v . E ) + τ2 ( v . ωc ) ( ωc . E ) + ( v ∧ ωc ) . E

](1023)

Thus, the conductivity tensor is recovered in dyadic form as

σ = 2 | e2 | τ

1 + ω2c τ

2

∫d3k

( 2 π )3

(− ∂f0

∂E

×

[v v + τ2 ( v . ωc ) v ωc + τ v ( v ∧ ωc )

](1024)

Furthermore, if E(k) is assumed to be spherically symmetric, one finds that thecomponents of the tensor can be expressed as

σα,β = e2∫

dE ρ(E)(− ∂f0∂E

)v2

1 + ω2 τ2

[δα,β + τ2ωα ωβ ± ( 1−δα,β ) τ ωγ

](1025)

where in the off-diagonal term the convention is introduced such that γ is chosensuch that (α, β, γ) corresponds to a permutation of (x, y, z). Since the densityof states per unit volume, ρ(E), is proportional to E

32 , the conductivity tensor

can be evaluated, by integration by parts, to yield

σα,β =ρ e2

m

τ

1 + ω2 τ2

[δα,β + τ2ωα ωβ ± ( 1 − δα,β ) τ ωγ

](1026)

where the ± sign is taken to be positive when (α, β, γ) are an odd permutation of(x, y, z) and is negative when (α, β, γ) are an even permutation of (x, y, z). Thus,

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if the field is applied along the z direction it is found the diagonal componentsof the conductivity tensor are given by

σx,x = σy,y =ρ e2

m

τ

1 + ω2c τ

2

σz,z =ρ e2

mτ (1027)

The non-zero off-diagonal terms are found as

σy,x = − σx,y =ρ e2

m

ωc τ2

1 + ω2c τ

2(1028)

Thus, for the diagonal component of the conductivity tensor are anisotropic.The component parallel to the field is constant while the other two componentsdecrease like ω−2

c in high fields. The off diagonal components are zero at zerofield, but increase linearly with the field for small ωc but then decreases like ω−1

c

at high fields.

A useful representation of the conductivity is through the Hall angle. Forexample, if one applies the magnetic field along the z direction and then anelectric field along the x direction Ex 6= 0, then the current will have an x andy component that can be characterized by a complex number z

z = Jx + i Jy

= σ0 Ex

(1 − i ωc τ

1 + ω2c τ

2

)= σ0

11 + i ωc τ

(1029)

This complex number z lies on a semi-circle of radius σ02 Ex centered on the

point (σ02 Ex, 0), as

z − σ0

2Ex =

σ0

2Ex

(1 − i ωc τ

1 + i ωc τ

)(1030)

and the modulus is just given by

| z − σ0

2Ex | =

σ0

2Ex (1031)

Thus, the number z lies on a semi-circle of radius σ02 Ex passing through the

origin. The Hall angle ΨH is defined as the angle between the line subtendedfrom the point z to the origin and the Jx axis. Thus

tan ΨH =Jy

Jx(1032)

and from the Boltzmann equation analysis of the magneto-conductivity

ΨH = tan−1 ωc τ (1033)

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Thus, from knowledge of σ0 and E one can find z and, thence, J .

The resistivity tensor ρi,j is obtained from the conductivity tensor σi,j byinverting the relation

Ji =∑

j

σi,j Ej (1034)

to obtainEi =

∑j

ρi,j Jj (1035)

The resistivity tensor is found as

ρ =

ρ0 ρ0 ωc τ 0− ρ0 ωc τ ρ0 0

0 0 ρ0

Thus, for the free electron model the diagonal part of the resistivity tensor is

completely unaffected by the field. There is neither a longitudinal or transversemagneto-resistance.

However, as Hz increases, the transverse component of the electric field Ey

increases. This is the Hall field. The Hall field is given by

Ey = ωc τ ρ0 Jx =Jx Hz

ρ | e |(1036)

Thus, the Hall resistivity is

ρyx =Ey

Jx

=Hz

ρ | e |(1037)

Thus, the Hall constant RH is given by

RH =EY

Hz Jx=

1ρ | e |

(1038)

The magneto-resistivity is usually classified as being longitudinal or trans-verse. The longitudinal magneto resistance is the change in the resistivity tensorρz,z due to the application of a magnetic field along the z direction. The trans-verse magneto-resistance is given by the change in ρx,x or ρy,y due to afield Hz.The longitudinal magneto-resistance is usually due to the dependence of thescattering rate on the magnetic field, whereas the transverse magneto-resistanceis due to the action of the Lorentz force.

The general features of the magneto-resistance are:-

(i) for low fields, Hz such that ωc τ < 1 then

∆ρx,x = ρx,x(Hz) − ρx,x(0) ∝ H2z (1039)

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(ii) There is an electric field Ey transverse to Jx and Hz, which has a mag-nitude proportional to Hz.

(iii) For large fields ρx,x(Hz) may either continue to increase with H2z or

saturate.

(iv) For a set of samples all which have different residual resistivities ρzz(T =0,Hz = 0), then the transverse magneto resistance usually satisfies Koehler’srule

∆ρx,x(Hz)ρx,x(T = 0,Hz = 0)

= F

(Hz

ρx,x(0, 0)

)(1040)

Basically, Koehler’s law expresses the fact that ρ(Hz) only depends on Hz

through the combination ωc τ and that ∆ρx,x and ρzz(T = 0,Hz = 0) areboth proportional to τ−1.

The standard form of the relationship between E and J is expressed as avector equation

E = ρ0 J + a ( J ∧ H ) + b H2 J

+ c ( J . H ) H + d T J (1041)

where T is a tensor which only has diagonal components that, when referred tothe crystalline axes, are (H2

x , H2y , H

2z ). That is T is the matrix

H2x 0 0

0 H2y 0

0 0 H2z

The five unknown quantities may be determined by five experiments.

(1) When J and H are parallel to the x axis one has the longitudinalmagneto-resistance given by

ρx,x = ρ0 + ( b + c + d ) H2 (1042)

(2) When J ‖ x ,H ‖ y then

ρx,x = ρ0 + b H2 (1043)

which is the transverse magneto-resistance.(3) With J ‖ (1, 1, 0) i.e.

J =J√2

(1, 1, 0) (1044)

andH =

H√2

(1, 1, 0) (1045)

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one has a different longitudinal magneto-resistance

∆ρ = ( b + c +d

2) H2 (1046)

(4) When H = H (0, 0, 1) then a second transverse magneto-resistance is foundas

∆ ρ = b H2 (1047)

but when H = H√2

(1,−1, 0) then the magneto-resistance is found as

∆ ρ = ( b +d

2) H2 (1048)

(5) The constant a makes no contribution to the magneto-resistance, but isfound from the Hall effect. If H is transverse to J then the Hall effect is onlydetermined by a alone, and is isotropic.

The magneto-resistance is usually negative except for cases where the scat-tering is of magnetic origin, such as disorder with spin - orbit coupling or fromKondo scattering by magnetic impurities in metals.

11.2.5 Multi-band Models

The transverse magneto-resistance for a multi-band model is non-trivial, unlikethe one band free electron model. The resistance can be obtained from thecurrent field diagram, in which the currents originating from the various sheetsof the Fermi-surface are considered separately.

For example, a two band model with positive and negative charge carriersproduces two components of the current J+ and J− by virtue of their responsesσ+, σ− in response to the electric field Ex. On assuming that the carriers havethe same Hall angles ΨH , then the total current is found as

Jx =(σ+ + σ−

2

)Ex ( 1 + cos 2 ΨH ) (1049)

and

Jy =(σ+ − σ−

2

)Ex sin 2 ΨH (1050)

Thus,

σx,x = σ0 cos2 ΨH

σy,x = σ0

(ρ+ − ρ−ρ+ + ρ−

)sin ΨH cos ΨH

(1051)

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Since the conductivity tensor is anisotropic and given by

σ =

σx,x σx,y 0− σy,x σx,x 0

0 0 σz,z

then the transverse magneto resistivity can be found from

ρx,x =σx,x σz,z

σz,z ( σ2x,x + σ2

x,y )

=σx,x

( σ2x,x + σ2

x,y )

=1σ0

cos2 ΨH

( cos4 ΨH +(

ρ+ − ρ−ρ+ + ρ−

)2

sin2 ΨH cos2 ΨH )

=1σ0

1

( cos2 ΨH +(

ρ+ − ρ−ρ+ + ρ−

)2

sin2 ΨH )

=1σ0

sec2 ΨH

1 +(

ρ+ − ρ−ρ+ + ρ−

)2

ω2c τ

2

(1052)

since tanΨh = ωc τ .

Furthermore, as

sec2 ΨH = 1 + tan2 ΨH

= 1 + ω2c τ

2 (1053)

then

ρx,x =1σ0

( 1 + ω2c τ

2 )

1 +(

ρ+ − ρ−ρ+ + ρ−

)2

ω2c τ

2

(1054)

This saturates if | ρ+ − ρ− | > 0 and increases indefinitely for a compensatedmetal ρ+ = ρ−. Basically, the positive magneto-resistance occurs because theLorentz force produces a transverse component of the current in each sheet ofthe Fermi-surface. The Lorentz force, the acting on these transverse currentsthen produces a shift of the Fermi-surface opposite to the shift produced by theelectric field.

A similar analysis can be performed on the Hall coefficient

RH =E⊥H J

(1055)

The value of E⊥ is the component of the field perpendicular to the current.This is found from the angle θ between J and E

cos θ =Jx

J

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sin θ =Jy

J(1056)

Thus

RH =E sin θH J

=E Jy

H J2

=1

σ0 H

(ρ+ − ρ−ρ+ + ρ−

)tan ΨH

cos2 ΨH

1 +(

ρ+ − ρ−ρ+ + ρ−

)2

tan2 ΨH

=ωc τ

σ0 H

(ρ+ − ρ−ρ+ + ρ−

)( 1 + ω2

c τ2 )

1 +(

ρ+ − ρ−ρ+ + ρ−

)2

tan2 ΨH

=1

| e | ( ρ+ − ρ− )

(ρ+ − ρ−ρ+ + ρ−

)2

( 1 + ω2c τ

2 )

1 +(

ρ+ − ρ−ρ+ + ρ−

)2

tan2 ΨH

(1057)

The Hall coefficient saturates to

RH → 1| e | ( ρ+ − ρ− )

(1058)

for large magnetic fields.

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11.3 Electromagnetic Properties of Metals

Maxwell’s equations relate the electromagnetic field to charges and currentsources ρ(r; t) and j(r; t). Maxwell’s equations can be formulated as

∇ . E(r; t) = 4 π ρe(r; t)∇ . B(r; t) = 0

∇ ∧ B(r; t) =4 πc

j(r; t) +1c

∂E(r; t)∂t

∇ ∧ E(r; t) = − 1c

∂B(r; t)∂t

(1059)

where E andB represent the microscopic electric and magnetic fields, and ρe andj are the microscopic charge and current densities. These are eight equationsfor the six unknown quantities. The six unknown quantities are the componentsof E and B.

The sourceless equations have a formal solution in terms of a scalar potentialφ and a vector potential A, which are related to the electric field E and magneticfield B via

E = − ∇ φ − 1c

∂A

∂tB = ∇ ∧ A (1060)

The solutions for the potentials are not unique, as the gauge transformations

A → A′ = A + ∇ Λ (1061)

andφ → φ′ = φ − 1

c

∂Λ∂t

(1062)

yield new scalar and vector potentials, (A′, φ′), that produce the same physicalE and B fields as the original potentials (A,φ). The four quantities φ and Asatisfy the four source equations

∇2 φ +1c

∂t

(∇ . A

)= − 4 π ρe (1063)

and

∇2 A − 1c2

∂2A

∂t2− ∇

(∇ . A +

1c

∂φ

∂t

)= − 4 π

cj (1064)

These equations are usually simplified by choosing a gauge condition. The gaugeconditions which are usually chosen are either the Coulomb Gauge

∇ . A = 0 (1065)

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or the Lorentz Gauge

∇ . A +1c

∂φ

∂t= 0 (1066)

The Lorentz gauge has the advantage that it is explicitly covariant under Lorentztransformations. The Coulomb gauge, also known as the transverse gauge orradiation gauge, is quite convenient for non-relativistic problems in that it sep-arates out the effect of radiation from electrostatics.

The space and time Fourier transform of the charge density is defined as

ρe(q, ω) =1V

∫d3r

∫dt exp

[− i ( q . r − ω t )

]ρe(r; t) (1067)

On Fourier transforming the Source equations with respect to space and time,one has

− q2 φ(q, ω) +ω

cq . A(q, ω) = − 4 π ρe(q, ω)[

− q2 +ω2

c2

]A(q, ω) + q

(q . A(q, ω) − ω

cφ(q, ω)

)= − 4 π

cj(q, ω)

(1068)

In the wave-vector and frequency domain, the Coulomb gauge condition is ex-pressed as

q . A(q, ω) = 0 (1069)

which shows that the vector potential is transverse to the direction of q. In thetransverse gauge, the equation for the vector potential reduces to[

− q2 +ω2

c2

]A(q, ω) − q

cφ(q, ω)

)= − 4 π

cj(q, ω)

(1070)

The first term is transverse and the second term is longitudinal. Thus, thecurrent can also be divided into a longitudinal term

jL(q, ω) = q

(q . j(q, ω)

)(1071)

and a transverse term

jT(q, ω) = j(q, ω) − q

(q . j(q, ω)

)(1072)

Thus, the second non-trivial Maxwell equation separates into the transverseequation [

− q2 +ω2

c2

]A(q, ω) = − 4 π

cj

T(q, ω) (1073)

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and the longitudinal equation

− qω

cφ(q, ω) = − 4 π

cj

L(q, ω) (1074)

In the Coulomb gauge, the other non-trivial Maxwell equation relates the chargedensity to the scalar potential via

− q2 φ(q, ω) = − 4 π ρe(q, ω) (1075)

This is just Poisson’s equation, and it has the solution

φ(q, ω) =4 πq2

ρe(q, ω) (1076)

which is equivalent to Coulomb’s law. When Fourier transformed with respect tospace and time, Poisson’s equation yields an instantaneous relation between thecharge density and the scalar potential in the form of Coulomb’s law. Althoughthis is an instantaneous relation, the signals transmitted by the electromagneticfield still travel with speed c and are also causal. This is because, in the Coulombgauge, the retardation effects are contained in the vector potential. Poisson’sequation actually has the same content as the longitudinal equation, as canbe seen by examining the continuity equation which expresses conservation ofcharge

∂ρe

∂t+ ∇ . j = 0 (1077)

The continuity equation can be Fourier transformed to yield

− ω ρe(q, ω) + q . j(q, ω) = 0 (1078)

This shows that the fluctuations in the charge density are related to the longi-tudinal current. On solving the continuity condition, one finds that the longi-tudinal current is given by

jL(q, ω) = q

ω

qρe(q, ω) (1079)

On substituting the above expression for the longitudinal current into the lon-gitudinal equation, one finds

− qω

cφ(q, ω) = − 4 π

cqω

qρe(q, ω) (1080)

On cancelling the factors of ω/c and q, one obtains Poisson’s equation. Thisproves that the longitudinal equation is equivalent to Poisson’s equation. Wehave also found that the longitudinal current can be expressed in the forms

jL(q, ω) = q

ω

qρe(q, ω)

=q ω

4 πφ(q, ω) (1081)

so the longitudinal current can be viewed as being produced either by the chargedensity or by the scalar potential.

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11.3.1 The Longitudinal Response

The currents and charge densities are usually broken down into the externalcontributions and the induced contribution, via

j(q, ω) = jind

(q, ω) + jext

(q, ω)ρe(q, ω) = ρe ind(q, ω) + ρe ext(q, ω) (1082)

The external scalar potential is given in terms of the external charge density viaPoisson’s equation

− q2 φext(q, ω) = − 4 π ρe ext(q, ω) (1083)

The frequency and wave vector dependent dielectric constant for a homogeneousmedium, ε(q, ω), is defined by the ratio

ε(q, ω) =φext(q, ω)φ(q, ω)

(1084)

The dielectric constant describes the screening of the external potential by lon-gitudinal or charge density fluctuations. The dielectric constant is related tothe longitudinal conductivity. This can be seen by combining the relation

jL(q, ω) =

q ω

4 πφ(q, ω) (1085)

with the expression for the induced component of the longitudinal current

jL(q, ω)ind =

q ω

4 π

(φ(q, ω) − φ(q, ω)ext

)(1086)

Hence, on using the definition of the frequency dependent dielectric constant,one obtains

jL(q, ω)ind =

q ω

4 π

(1 − ε(q, ω)

)φ(q, ω) (1087)

The total scalar potential φ(q, ω) can be related to the the longitudinal electricfield, EL(q, ω), since the electric field can be written as the sum of the timedependence of the vector potential and the gradient of the scalar potential

E(q, ω) =i ω

cA(q, ω) − i q φ(q, ω) (1088)

If the longitudinal part of the electric field is identified as

EL(q, ω) = − i q φ(q, ω) (1089)

then one obtains the relation between the longitudinal current and the longitu-dinal electric field

jL(q, ω)ind =

i ω

4 π

(1 − ε(q, ω)

)EL(q, ω) (1090)

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Hence, as the longitudinal conductivity σL is defined by the relation

jL(q, ω)ind = σL(q, ω) EL(q, ω)

(1091)

one finds that the conductivity and the dielectric constant are related through

σL(q, ω) =i ω

4 π

(1 − ε(q, ω)

)(1092)

The frequency dependent dielectric constant can be expressed in terms of theresponse of the charge density due to the potential

ε(q, ω) =φext(q, ω)φ(q, ω)

ε(q, ω) =φ(q, ω) − φind(q, ω)

φ(q, ω)

ε(q, ω) = 1 − 4 πq2

ρe ind(q, ω)φ(q, ω)

(1093)

The charge density is related to the electron density via a factor of the electron’scharge

ρe ind(q, ω) = − | e | ρind(q, ω) (1094)

and the scalar potential acting on the electrons produces the potential δV (q, ω)where

δV (q, ω) = − | e | φ(q, ω) (1095)

Thus, the frequency dependent dielectric constant may be written as

ε(q, ω) = 1 − 4 π e2

q2ρind(q, ω)δV (q, ω)

= 1 − 4 π e2

q2χ(q, ω) (1096)

(H. Ehrenreich and M.H. Cohen, Phys. Rev. 115, 786 (1959)) where we haveused the definition of the frequency dependent response function χ(q, ω). Thefrequency dependent response function is defined by

χ(q, ω) =ρind(q, ω)δV (q, ω)

(1097)

The real space and time form of the linear response relation can be found byre-writing this relation as

ρind(q, ω) = χ(q, ω) δV (q, ω) (1098)

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and then performing the inverse Fourier transform. The real space and timeform of the linear response relation is in the form of a convolution

ρind(r, t) =∫

d3r′∫ ∞

−∞dt′ χ(r − r′; t− t′) δV (r′, t′) (1099)

The dependence of the response function on r − r′ is a direct consequence ofour assumption that space is homogeneous. As the response function relatesthe cause and effect in a linear fashion, the response function can be calculatedperturbatively. The induced electron density is found, in real space and time, bytreating the time dependent potential as a perturbation. The resulting causal,non-local relation is then Fourier transformed with respect to space and time.This procedure is a generalization of our previous treatment of static screening.

The expectation value of the electron density operator ρ(r) at time t, is cal-culated in a state that has evolved from the ground states due to the interaction.The electron density operator is given by

ρ(r) =∑

i

δ3(r − ri

)(1100)

and the time dependent perturbation is

Hint(t) =∫

d3r′ ρ(r′, t) δV (r′, t) (1101)

The expectation value of the electron density is to be evaluated in the interactionrepresentation. The expectation value of the density is given by

ρ(r, t) = < Ψint(t) | ρint(r, t) | Ψint(t) > (1102)

where the state and operators are expressed in the interaction representation.In the interaction representation the operators evolve with respect to time underthe influence of the unperturbed Hamiltonian H0, and are given by

ρint(r, t) = exp[

+i t

hH0

]ρ(r) exp

[− i t

hH0

](1103)

In the interaction representation, the state evolves under the influence of theinteraction Hint(t). To first order in the perturbation, the ground state is givenby

| Ψint(t) > =

[1 − i

h

∫ t

−∞dt′ Hint(t′) + . . .

]| Ψ0 > (1104)

where | Ψ0 > is the initial ground state eigenfunction of H0. The inducedelectron density is defined as

ρind(r, t) = < Ψint(t) | ρint(r, t) | Ψint(t) > − < Ψ0 | ρint(r, t) | Ψ0 >

(1105)

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The second term is time independent, as it is the expectation value in the groundstate of the time independent Hamiltonian H0. On substituting the expressionfor the perturbed wave function, one finds a linear relationship between theinduced density and the perturbing potential

ρind(r, t) = − i

h

∫ t

−∞dt′ < Ψ0 |

[ρint(r, t) , Hint(t′)

]| Ψ0 >

= − i

h

∫ t

−∞dt′

∫d3r′ < Ψ0 |

[ρint(r, t) , ρint(r′, t′)

]| Ψ0 > δV (r′, t′)

=∫ +∞

−∞dt′

∫d3r′ χ(r, r′; t− t′) δV (r′, t′) (1106)

This is a causal relation in which the response function is identified as

χ(r, r′; t− t′) = − i

h< Ψ0 |

[ρint(r, t) , ρint(r′, t′)

]| Ψ0 > Θ( t − t′ )

(1107)where Θ(t) is the Heaviside step function. Thus, the response function is a twotime correlation function, which involves the ground state expectation valueof the commutator of the density operators at different positions and differ-ent times. Due to the time homogeneity of the ground state, the correlationfunction only depends on the difference of the two times. For a spatially ho-mogeneous system, the correlation function only depends on the difference r−r′.

The expression can be evaluated by using the completeness relation∑n

| Ψn > < Ψn | = I (1108)

On inserting a complete set of states between the density operators, one obtains

χ(r, r′; t− t′) = − i

h

∑n

[< Ψ0 | ρint(r, t) | Ψn > < Ψn | ρint(r′, t′) | Ψ0 >

− < Ψ0 | ρint(r′, t′) | Ψn > < Ψn | ρint(r, t) | Ψ0 >

]Θ( t − t′ )

(1109)

On expressing the time dependence of the operators in terms of the eigenvaluesof the unperturbed Hamiltonian, H0, the response function reduces to

= − i

h

∑n

exp[

+i

h(t− t′)(E0 − En)

]< Ψ0 | ρ(r) | Ψn > < Ψn | ρ(r′) | Ψ0 >

+i

h

∑n

exp[− i

h(t− t′)(E0 − En)

]< Ψ0 | ρ(r′) | Ψn > < Ψn | ρ(r) | Ψ0 >

(1110)

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for t − t′ > 0, and is zero otherwise In the above expression for the responsefunction, the density operators are no longer time dependent.

Up to this point, our analysis has been completely general. To illustrate thestructure of the response function, we shall now make the assumption that theelectrons are non-interacting. The ground state | Ψ0 > and the excited states| Ψn > can represented by single Slater determinants, composed of the set ofone-electron energy eigenfunctions φαj

(rj); j ∈ 1, 2, . . . Ne and φβj(rj); j ∈

1, 2, . . . Ne, respectively. The matrix elements of the one-electron operatorρ(r) are non-zero only if the set of quantum numbers αj ; j ∈ 1, 2, . . . Ne andβj ; j ∈ 1, 2, . . . Ne only differ by at most one element, say the i-th value. Thus,we may permute the indices in the set βj until one has

αi 6= βi (1111)

andαj = βj ∀ j 6= i (1112)

In this case, the matrix elements < Ψ0 | ρ(r) | Ψn > are trivially evaluatedas

< Ψ0 | ρ(r) | Ψn > =∫

d3ri φ∗αi

(ri) δ3( r − ri ) φβi(ri)

= φ∗αi(r) φβi(r) (1113)

The matrix element is only non zero if the spin state of α is identical to the spinstate of β, so the spin quantum number is conserved. In the above expression,the single electron state αi is occupied in the initial state |Ψ0 > and unoccupiedin the final state | Ψn > and the single electron state βi is unoccupied in theinitial state | Ψ0 > and occupied in the final state | Ψn > . All the othersingle-electron quantum numbers in | Ψ0 > and | Ψn > are unchanged, i.e.,αj = βj for ∀ j 6= i. Furthermore, the Pauli exclusion principle requires thatβi 6= βj . This shows that the final states of the non-interacting many-electronsystem are obtained by exciting a single electron from the state αi to the stateβi. For non-interacting electrons, the excitation energy En − E0 is simplygiven by the difference in the single-electron energy eigenvalues

En − E0 = Eβi− Eαi

(1114)

Thus, the response function is simply given by

χ(r, r′; t) = − i

h

∑α,β

exp[

+i

ht ( Eα − Eβ )

]φ∗α(r) φβ(r) φ∗β(r′) φα(r′)

+i

h

∑α,β

exp[− i

ht ( Eα − Eβ )

]φα(r) φ∗β(r) φβ(r′) φ∗α(r′)

(1115)

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for t > 0. The sum over α is restricted to run over the single particle quantumnumbers that are occupied in the ground state, and β runs over the quantumnumbers that are unoccupied in the ground state. The spin quantum numberis conserved, that is σα = σβ .

On evaluating the response function for free electrons, summing over spinstates and using the Bloch state energy eigenvalues, one finds

= − 2 ih V 2

∑|k|<kF

∑|k′|>kF

exp[

+i t

h(Ek − Ek′)

]exp

[− i

h(k − k′) . (r − r′)

]

+2 ih V 2

∑|k|<kF

∑|k′|>kF

exp[− i t

h(Ek − Ek′)

]exp

[+i

h(k − k′) . (r − r′)

](1116)

for t > 0. Since the free electron gas is homogeneous, the response functiononly depends on the distance between the perturbation and the response r− r′.On Fourier transforming the response function with respect to space and timeone obtains χ(q, ω) as

χ(q, ω) =∫ +∞

−∞dt

∫d3r exp

[− i ( q . r − ω t )

]χ(r; t) (1117)

Since the response function contains the Heaviside step function Θ(t), the inte-gral over t can be evaluated in the interval ∞ > t ≥ 0. The integral over tconverges faster if ω is analytically continued into the upper half complex planeto ω → z = ω + i δ. The factor of exp [ − δ t ] damps out the oscillationsin the integrand as t → ∞. Thus, one finds that in the (q, ω) domain theresponse function is complex and is given by the expression

χ(q, ω + iδ) =2V

∑|k|<kF |k+q|>kF

[1

h ω + i δ + Ek − Ek+q

]

− 2V

∑|k|>kF |k+q|<kF

[1

h ω + i δ + Ek − Ek+q

](1118)

The restrictions on the summation over k can be simplified. To see this, weshall introduce a function fk which behaves like the T → 0 limit of the Fermi-function. The function is defined by

fk = 1 for Ek < EF (1119)

andfk = 0 for Ek > EF (1120)

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The response function can then be written as the sum over all k as

χ(q, ω + iδ) =2V

∑k

[fk ( 1 − fk+q )

h ω + i δ + Ek − Ek+q

]

− 2V

∑k

[fk+q ( 1 − fk )

h ω + i δ + Ek − Ek+q

]

=2V

∑k

[fk − fk+q

h ω + i δ + Ek − Ek+q

](1121)

In the last line, it is seen that the factors which explicitly enforce the Pauli-exclusion principle cancel. For ω just above the real axis, i.e in the limit δ → 0,the imaginary part of the response function is found as

Im χ(q, ω + iδ) = − 2 πV

∑|k|<kF

δ

(h ω + Ek − Ek+q

)

+2 πV

∑|k+q|<kF

δ

(h ω + Ek − Ek+q

)(1122)

From this analysis, one can see that for positive ω the imaginary part of χ(q, ω)is non-zero in for the region of (ω, q) phase space, where

h

2 m( − 2 kF q + q2 ) < ω <

h

2 m( + 2 kF q + q2 ) (1123)

It is only in this region that the argument of the first delta function in Im χ(q, ω)has a solution

h

mk . q = ω − h

2 mq2 (1124)

with k < kF . These conditions divide (q, ω) space into non-overlapping regions.

For completeness, the complete expressions for the real and imaginary partsof the Lindhard dielectric function at finite frequencies are given (J. Lindhard,Kgl. Danske Videnskab. Selskab. Mat. Fys. Medd. 28, 8 (1954)). The realpart is given by

Re

[ε(q, ω)

]= 1 +

k2TF

2 q2

[1 +

kF

2 q

(1 − (2 m ω − h q2)2

4 h2 q2 k2F

× ln

∣∣∣∣∣2 m ω − 2 h q kF − h q2

2 m ω + 2 h q kF − h q2

∣∣∣∣∣+(

1 − (2 m ω + h q2)2

4 h2 q2 k2F

)ln

∣∣∣∣∣2 m ω + 2 h q kF + h q2

2 m ω − 2 h q kF + h q2

∣∣∣∣∣ ]

(1125)

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and the imaginary part is given by

Im

[ε(q, ω + iδ)

]=

π

2k2

TF

q2m ω

h q kF2 m ω < 2 h q kF − h q2

(1126)

Im

[ε(q, ω + iδ)

]=

π

4k2

TF

q2kF

q

[1 − (2 m ω − h q2)2

4 h2 q2 k2F

]2 h q kF − h q2 < 2 m ω < 2 h q kF + h q2 (1127)

and

Im

[ε(q, ω + iδ)

]= 0 2 h q kF + h q2 < 2 m ω (1128)

The real part is an even function of ω and the imaginary part is an odd functionof ω. For ω = 0 the response function reduces to the real static responsefunction calculated previously. For | ω | > h

2 m ( 2 kF q + q2 ) the imaginarypart of the function vanishes, as the denominator never vanishes for any k valuein the range of integration. In this region of q and ω there are no poles, therefore,the real part of the response function χ(q, ω) can be expanded in powers of q2.To the order of q4, one finds

Re χ(q, ω) = +k3

F

3 π2

q2

m ω2

[1 +

35

(h kF q

m ω

)2

+ . . .

](1129)

Thus, for high frequencies such that ω q h kF

m the dielectric constant canbe approximated by

ε(q, ω) = 1 − 4 π ρ e2

m ω2

[1 +

35

(h kF q

m ω

)2

+ . . .

]

= 1 −ω2

p

ω2

[1 +

35

(h kF q

m ω

)2

+ . . .

](1130)

where the expression for the electron density ρ

ρ = 2k3

F

6 π2(1131)

has been used, and the plasmon frequency ωp has been defined via

ω2p =

4 π ρ e2

m(1132)

Thus, the dielectric constant has zeros at the frequencies ω = ωp(q), where

ω2p(q) = ω2

p

[1 +

35

(h kF q

m ωp(q)

)2

+ . . .

](1133)

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If the external potential is zero, φext(q, ωp(q)) = 0, and the total potential isnon-zero φ(q, ωp(q)) 6= 0, then the real and imaginary parts of the dielectricconstant must vanish, ε(q, ωp(q)) = 0, as

ε(q, ωp(q)) φ(q, ωp(q)) = φext(q, ωp(q))ε(q, ωp(q)) φ(q, ωp(q)) = 0 (1134)

In this case, when the total potential inside the solid, φ(q, ωp(q)) is non-zero,the induced density and current fluctuations must be finite. These longitudinalcollective charge oscillations excitations are plasmons. A typical energy rangefor the plasmon energy, h ωp, in metals ranges from the low values of 3.72 eVfound in K, 5.71 eV found in Na, to values as high as 15.8 eV found in Al. Thedielectric materials Si, Ge etc. also have plasmon energies of the order 16 eV.

One may enquire as to the nature of the excitations at larger q values, suchthat the phase velocity of the plasmons becomes greater than the Fermi-velocityvF = h kF

m . At a critical value of q the denominator of the response functionmay vanish, so the response function acquires a sizeable imaginary part. Theplasmon excitations merge with a continuum of particle hole excitations whichhave excitation energies given by

h ω(q, k) = Ek+q − Ek (1135)

for k < kF . The edges of the continuum stretch from h2

2 m ( 2 kF q + q2 )to h2

2 m ( − 2 kF q + q2 ). When the plasmon merges into the continuum itundergoes significant broadening. This sort of damping is called Landau damp-ing. Landau damping can also be viewed classically, in terms of electrons surfriding the waves in the potential field. Imagine a wave with phase velocity ω

q ispropagating through an electron gas, and consider the electrons with velocityis almost parallel and close to the phase velocity of the wave. In the frame ofreference travelling with the wave, the electron is at rest and experiences an es-sentially time independent electric field. The electric field continuously transfersenergy from the wave to the electrons that have the same velocity. If there isa slight mismatch in the velocities, electrons with lower velocity than the wavedraw energy from the wave and accelerate, whereas electrons that are movingfaster lose energy and slow down. This has the consequence that the rate ofenergy loss of the wave is proportional to the derivative of the electron velocitydistribution, evaluated at the wave’s phase velocity.

11.3.2 Electron Scattering Experiments

The longitudinal excitations of the electrons in a metal can be probed by scat-tering of a beam of charged particles or fast electrons. The coupling takes placevia the Coulomb interaction

Hint =∑

i

e2

| r − ri |(1136)

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where ri labels the positions of the electrons in the plasma and r is the posi-tion of the incoming high energy electron. If the incident beam is composedof electrons which have high energies, the beam electrons can be considered tobe as classical and are, therefore, distinguishable. This ignores the possibilityof exchange interactions with the electrons in the metal. Analysis of the Mottscattering formula for electrons also shows that the neglect of the exchangescattering is an excellent approximation for scattering through small angle scat-tering. Therefore, we shall consider the charged particles in the beam as beingdistinguishable from the electrons in the solid.

The rate at which a charged particle is scattered inelastically from state kwith energy E(k) to state k′ with energy E(k′) is given by

1τ(k → k′)

=2 πh

∣∣∣∣ < k′ Ψn | Hint | k Ψ0 >

∣∣∣∣2 δ( En + E(k′)− E0 − E(k))

(1137)where | Ψ0 > and E0 are the ground state wave function and ground stateenergy of the solid. The final state wave function and energy is given by | Ψn >and En. The momentum and energy loss of the charged particle are defined tobe

h q = h k − h k′

h ω = E(k) − E(k′) (1138)

On performing the integral over the position of the fast charged particle one has

1τ(k → k′)

=2 πh

(4 π e2

q2 V

)2 ∣∣∣∣ < Ψn |∑

i

exp[ i q . ri ] | Ψ0 >

∣∣∣∣2 ×

δ

(h ω + E0 − En

)(1139)

The energy conserving delta function can be replaced by an integral over timeby using the identity

δ

(h ω + E0 − En

)=∫ ∞

−∞

dt

2 π hexp[ i ω t ] exp

[it

h( E0 − En )

](1140)

The energy eigenvalues in the exponential time evolution factors can be replacedby the general time evolution operators involving the unperturbed Hamiltonianoperator H0,

1τ(k → k′)

=2 πh2

(4 π e2

q2 V

)2 ∫ ∞

−∞

dt

2 πexp[ i ω t ] < Ψn |

∑i

exp[ i q . ri ] | Ψ0 > ×

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< Ψ0 |∑

i

exp[ iH0 t

h] exp[ − i q . ri ] exp[ − i

H0 t

h] | Ψn >

(1141)

The factor involving ri can be expressed as the Fourier transform of the electrondensity operator

1V

∑i

exp[ − i q . ri ] =1V

∫d3r exp[ − i q . r ]

∑i

δ3(r − ri

)=

1V

∫d3r exp[ − i q . r ] ρ(r)

= ρq (1142)

Thus, on combining the above expressions, the inelastic scattering rate is foundas

1τ(k → k′)

=2 πh2

(4 π e2

q2

)2 ∫ ∞

−∞

dt

2 πexp[ i ω t ] < Ψn | ρ−q | Ψ0 > ×

< Ψ0 | exp[ iH0 t

h] ρq exp[ − i

H0 t

h] | Ψn >

=2 πh2

(4 π e2

q2

)2 ∫ ∞

−∞

dt

2 πexp[ i ω t ] < Ψ0 | ρq(t) | Ψn > < Ψn | ρ−q(0) | Ψ0 >

(1143)

where the density operator is evaluated in the interaction representation. If thefinal state of the solid | Ψn > is not measured, there is a distribution of possiblefinal states of the solid. If only the final state of the charged particle is measuredand the final state of the solid is not measured, the index n corresponding tothe different possible final states must be summed over

1τ(k → k′)

=∑

n

2 πh2

(4 π e2

q2

)2 ∫ ∞

−∞

dt

2 πexp[ i ω t ] ×

< Ψ0 | ρq(t) | Ψn > < Ψn | ρ−q(0) | Ψ0 >

(1144)

The sum over the final states can be evaluated using the completeness relation,which leads to the result

1τ(k → k′)

=2 πh2

(4 π e2

q2

)2 ∫ ∞

−∞

dt

2 πexp[ i ω t ] < Ψ0 | ρq(t) ρ−q(0) | Ψ0 >

(1145)

The factor of q−4 shows the scattering process is dominated by small momentumtransfers. The density-density correlation function, S(q, ω), is defined via

S(q, ω) = V 2

∫ ∞

−∞

dt

2 π hexp[ i ω t ] < Ψ0 | ρq(t) ρ−q(0) | Ψ0 > (1146)

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On substituting this relation into the scattering rate we obtain the result

1τ(k → k′)

=2 πh

(4 π e2

q2 V

)2

S(q, ω) (1147)

Thus, it is seen that the long wavelength electron density fluctuations are mainlyresponsible for scattering the incident charged particle. For non-interactingelectrons, S(q, ω) is evaluated as

S(q, ω) = 2∑

|k|<kF |k+q|>kF

δ

(h ω + Ek − Ek+q

)(1148)

where the summation over k is over the filled Fermi-sphere, subject to the re-striction that the final state be allowed by the Pauli exclusion principle.

The inelastic scattering cross-section can be evaluated in terms of the scat-tering rate, and is found to be given by the expression

d2σ

dΩ dω=

k′

k

(2 mq e

2

h2 q2

)2

h S(q, ω) (1149)

where mq is the mass of the charged particle. Thus, in the Born approximationthe scattering cross-section is directly related to the density-density correlationfunction. This type of correlation function was first introduced by van Hove inthe context of neutron scattering (L. van Hove, Phys. Rev. 95, 249 (1954)).

The Fluctuation-Dissipation Theorem (H.B. Callen and T.A. Welton, Phys.Rev. 83, 34 (1951)) relates the spectrum of electron density fluctuations to theimaginary part of the dielectric constant. At finite temperatures this relationhas the form

S(q, ω) =q2 V

4 π2 e2

[1

exp[ − β h ω ] − 1

]Im

[1

ε(q, ω + iδ)

](1150)

The relation between S(q, ω) and the inverse dielectric constant can be seenthrough the following classical argument. The power, per unit volume, dissi-pated by the electromagnetic field of the charged particle is given by

P (r, t) =1

4 πE .

∂D

∂t(1151)

For a negatively charged particle travelling with velocity v(t), the displacementfield D(r, t) is the experimentally controllable quantity and is given by theexpression

D(r, t) = − ∇

[− | e || r − r(t) |

](1152)

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On Fourier transformingD(r, t) with respect to space and time one findsD(q, ω).However, D(q, ω) is related to the Fourier transform of the electric field E(q, ω)via a factor of the dielectric constant

E(q, ω) =D(q, ω)

ε(q, ω + iδ)(1153)

On Fourier transforming the expression for the power density, P (r, t), withrespect to r and t, one finds P (q, ω) to be given by

P (q, ω) = − ω

8 πIm

[1

ε(q, ω + iδ)

] ∣∣∣∣ D(q, ω)∣∣∣∣2

8 π

[Im ε(q, ω + iδ)| ε(q, ω + iδ) |2

] ∣∣∣∣ D(q, ω)∣∣∣∣2

8 π

[Im ε(q, ω + iδ)

( Re ε(q, ω) )2 + ( Im ε(q, ω + iδ) )2

] ∣∣∣∣ D(q, ω)∣∣∣∣2

(1154)

This result implies that the zero in the real part of ε(q, ω) should show up as adelta function peak in the power loss.

——————————————————————————————————

11.3.3 Exercise 51

Use linear response theory to express the change in the electron density inducedby an external charge. Hence, express the inverse of the dielectric constant interms of the exact eigenstates and energy eigenvalues of the interacting many-electron system. Use the resulting expression to find the T = 0 form of thefluctuation-dissipation theorem (P. Nozieres and D. Pines, Nuovo Cimento, 9,470 (1958)).

——————————————————————————————————

Solution

In the Coulomb gauge, the Fourier transform of the external charge densityρext(r, t) is related to the external potential via Poisson’s theorem

− q2 φext(q, ω) = − 4 π | e | ρext(q, ω) (1155)

The total field is related to the external charge density and the induced chargedensity via

− q2 φ(q, ω) = − 4 π | e | ρext(q, ω) − 4 π | e | ρind(q, ω) (1156)

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The dielectric constant is defined as

1ε(q, ω)

=φ(q, ω)φext(q, ω)

(1157)

which can be expressed as

1ε(q, ω)

= 1 +ρind(q, ω)ρext(q, ω)

(1158)

Hence, the linear response relation can be expressed as

ρind(q, ω) =(

1ε(q, ω)

− 1)ρext(q, ω)

=(

1ε(q, ω)

− 1)

q2

4 π | e |φext(q, ω)

=(

1 − 1ε(q, ω)

)q2

4 π e2δV (q, ω) (1159)

However, the interaction operator is given by

Hint(r, t) =∫

d3r ρ(r) δV (r, t) (1160)

where the electron density is given by

ρ(r) =∑

i

δ3(r − ri) (1161)

The induced electron density is evaluated using linear perturbation theory, inthe interaction representation. In the absence of the perturbation, the groundstate is denoted by | Ψ0 > . The perturbation is turned on adiabatically, andthe ground state evolves to the state | Φ0(t) > which, to first order in theinteraction, is given by

| Φ0(t) > =(

1 − i

h

∫ t

−∞dt′ Hint(t′)

)| Ψ0 > (1162)

The induced electron density ρind(r, t) is then given by the expectation value ofthe commutator

ρind(r, t) = − i

h

∫ t

−∞dt′ < Ψ0 | [ ρ(r, t) , Hint(t′) ] | Ψ0 >

= − i

h

∫ t

−∞dt′

∫d3r′ < Ψ0 | [ ρ(r, t) , ρ(r′, t′) ] | Ψ0 > δV (r′, t′)

=∫ ∞

−∞dt′

∫d3r′ χ(r − r′, t− t′) δV (r′, t′) (1163)

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The material is assumed to be homogeneous, therefore, the response functionis only a function of the spatial separation r − r′. Furthermore, since H0 isindependent of time, the response function is only a function of t − t′. OnFourier transforming this equation with respect to space and time, one finds

ρind(q, ω) = χ(q, ω) δV (q, ω) (1164)

where

χ(q, ω) = V∑

n

[ | < Ψ0 | ρq | Ψn > |2

h ω + i δ + En − E0

+| < Ψ0 | ρq | Ψn > |2

− h ω − i δ + En − E0

](1165)

The imaginary part of the response function is found as

Im χ(q, ω) = − π V∑

n

| < Ψ0 | ρq |Ψn > |2[δ( h ω + En − E0 )− δ( h ω − En + E0 )

](1166)

Thus, the zero temperature limit of the fluctuation-dissipation theorem has theform

Im1

ε(q, ω)=

4 π2 e2 V

q2

∑n

| < Ψ0 | ρq | Ψn > |2[δ(hω + En − E0)− δ(hω − En + E0)

]=

4 π2 e2

q2 V

[S(q,−ω) − S(q, ω)

](1167)

The first term is only non-zero if 0 > ω, and the second term is only non-zeroin the range ω > 0.

——————————————————————————————————

11.3.4 Exercise 52

Show, using classical electromagnetic theory, that the power loss spectrum of aparticle with charge e moving with velocity v, due to plasmons can be expressedas

P (ω) = − 2 e2

π vIm

ε(ω + iδ)

]lnq0 v

ω(1168)

Assume that the dielectric constant is independent of q, for q < q0 where h q0is the maximum momentum transfer.

——————————————————————————————————

Solution 52

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The average power P dissipated by the charged particle can be expressed asthe limit τ → ∞

P =1τ

∫ ∞

0

dt P (t) exp[− t

τ

]=

∫ ∞

0

dt

∫d3r P (r, t) exp

[− t

τ

]=

14 π τ

∫ ∞

0

dt

∫d3r E(r, t)

∂tD(r, t) exp

[− t

τ

](1169)

where we have inserted an exponential convergence factor. The convergencefactor will be absorbed in the displacement and electric fields. The Fouriertransform is expressed as

D(q, ω) =1V

∫d3r

∫ ∞

−∞dt D(r, t) exp

[− i ( q . r − ω t )

](1170)

and the inverse Fourier transformation is given by

D(r, t) =V

( 2 π )3

∫d3q

∫dω

2 πD(q, ω) exp

[+ i ( q . r − ω t )

](1171)

On inserting the expressions for the inverse Fourier transforms into the expres-sion for the average power loss, one finds

P = − i V 2

2 τ ( 2 π )

∫ ∞

−∞

2 π

∫d3q

( 2 π )3E(−q,−ω) ω D(q, ω)

=i V 2

2 τ ( 2 π )

∫ ∞

−∞

2 π

∫d3q

( 2 π )3D(q, ω)

ε(q, ω + i2τ )

ω D∗(q, ω)

=i V 2

2 τ ( 2 π )

∫ ∞

−∞

2 π

∫d3q

( 2 π )3ω

ε(q, ω + i2τ )

| D(q, ω) |2(1172)

in the limit τ → ∞. The Fourier transform of the displacement field is givenby

D(q, ω) =1V

∫d3r

∫ ∞

0

dt exp[− i ( q . r − ω t ) − t

2 τ

]∇ | e |

| r − v t |

= −4 π i q | e |

V q2

∫ ∞

0

dt exp[i ( ω − q . v ) t − t

2 τ

]= −

4 π i q | e |V q2

2 τ1 − i ( ω − q . v ) 2 τ

(1173)

Hence,

P =4 e2

2 τ ( 2 π )3

∫ ∞

−∞dω

∫d3q

q2i ω

ε(q, ω + i2τ )

11

4 τ2 + ( ω − q . v )2

(1174)

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which in the limit τ → ∞ reduces to

P =2 e2

( 2 π )2

∫ ∞

−∞dω

∫d3q

q2i ω

ε(q, ω + iδ)δ( ω − q . v ) (1175)

This yields the expression

P =2 e2

( 2 π )

∫ ∞

−∞dω

∫dq

q v

i ω

ε(q, ω + iδ)

[θ( ω + q v ) − θ( ω − q v )

](1176)

Hence, on assuming that the dielectric constant is independent of q, from q = 0to an upper cut off q = q0 one obtains the result

P = − e2

π v

∫ ∞

−∞dω

i ω

ε(ω + iδ)lnq0 v

ω(1177)

The power loss spectrum, P (ω), is defined in terms of an integral over positivefrequencies

P =∫ ∞

0

dω P (ω) (1178)

On using the symmetry properties of the dielectric constant under the transfor-mation ω → − ω, one finds that the contribution from real part of the inversedielectric constant vanishes. Hence, one obtains the final result for P (ω)

P (ω) = − 2 e2

π vIm

ε(ω + iδ)

]lnq0 v

ω(1179)

——————————————————————————————————

In a typical experiment, monochromatic electron beams with energies E(k)of the order of keV fall incident on thin films, and the energies of the scatteredelectrons, E(k′) are analyzed (L. Marton, J.A. Simpson, H.A. Fowler and N.Swanson, Phys. Rev. 126, 182 (1962)). Experimentally, it is found that the fastelectron loses energy in almost exact multiples of h ωp. That is, the energy lossspectrum shows peaks separated by energies which are multiples of h ωp. Theabove analysis predicts a single pole near ω = ωp. The discrepancy is causedby the use of the Born Approximation, which neglects the effects of multiplescattering. The experiments are usually analyzed by fitting the intensities ofthe peaks to a Poisson distribution

In =1n!

(L

λ

)n

exp[− L

λ

]I0 (1180)

where In is the intensity of the n-th plasmon peak, L is the sample thicknessand λ is the mean free path. The mean free path is then compared with the

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theoretically derived inelastic scattering cross-section.

The mean free path can be estimated from the scattering cross-section. Oneexpressing the density-density correlation function in terms of the imaginarypart of the inverse of the dielectric constant one finds

d2σ

dΩ dω= − k′

k

(mq e

π h2 q

)2

h V Im

[1

ε(q, ω + iδ)

](1181)

For the frequency range

2 m ω > 2 h q kF + q2 (1182)

the imaginary part of χ(q, ω + iδ) vanishes as δ → 0. Therefore, in this limit,the imaginary part of the dielectric constant also vanishes

Im ε(q, ω + iδ) → κ δ (1183)

for some value finite of κ. Hence, in this limit one has

Im

[1

ε(q, ω + iδ)

]= − π δ( Re ε(q, ω) ) (1184)

Furthermore, in this region of (ω, q) space one has the approximate expression

ε(q, ω) = 1−ω2

p(q)ω2

(1185)

where the plasmon dispersion relation is given by

ω2p(q) =

(ω2

p +35q2 v2

F + . . .

)(1186)

and the plasmon frequency by

ω2p =

4 π ρ e2

m(1187)

Thus, one finds the single (plasmon) pole approximation for the inverse dielectricconstant

Im

[1

ε(q, ω + iδ)

]= − π δ

(ω2 − ω2

p(q)ω2

)= − π

ω2

ω + ωp(q)δ( ω − ωp(q) )

= − π

2ωp(q) δ( ω − ωp(q) ) (1188)

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for positive ω. The plasmon contribution to the differential scattering cross-section, is found by integrating over the energy loss ω and is given by

dΩ=

k′

k

2 m2q ρ V

m h ωp

(e2

h q

)2

(1189)

Thus, the scattering cross-section is directly proportional to the number of elec-trons in the solid. In deriving the differential scattering cross-section, we haveneglected the q dependence of the plasmon dispersion relation. The total plas-mon scattering cross-section is found by integrating the differential cross-sectionover the scattering angle θ. We note that energy and momentum conservationleads to the two conditions

q2 = k2 + k′2 − 2 k k′ cos θ

q2 = ( k − k′ )2 + 4 k k′ sin2 θ

2(1190)

and

( k − k′ ) ( k + k′ ) =2 mq

hωp

( k − k′ ) =2 mq ωp

h ( k + k′ )(1191)

Foe small scattering angles, θ 1, these can be combined to yield

q2 ≈ k2

(4 sin2 θ

2+

h2ω2p

4 E(k)2

)≈ k2

(θ2 + θ20

)(1192)

Hence, one has

dΩ≈ k′

k

2 m2q ρ V

m h ωp

e4

h2 k2 ( θ2 + θ20 )

≈ mq e4 ρ V

m h ωp E(k) ( θ2 + θ20 )(1193)

On integrating over the solid angle dΩ, but restricting the range of θ from zeroto a maximum momentum transfer given by 2 kF ∼ θm k, one finds the totalcross-section, σ, for plasmon scattering is given by

σ ≈ 2 πe4 ρ V

h ωp E(k)mq

mlnθm

θ0(1194)

The mean free path, λ, is then found by noting that a trajectory of cross-sectional area σ covers a volume λ σ between consecutive collisions, which must

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equal V the volume of the solid. This leads to the mean free path being givenby

λ−1 ≈ 2 πe4 ρ

h ωp E(k)mq

mlnθm

θ0(1195)

Thus, the mean free path depends linearly on the kinetic energy of the incidentelectron. This value has been found to track the mean free path obtained byfitting the measured intensities of the multi-plasmon peaks.

11.3.5 The Transverse Response

In the Coulomb or radiation gauge, the vector potential describes the transverseelectromagnetic field. It satisfies the equation(

− q2 +ω2

c2

)A(q, ω) = − 4 π

cj

T(q, ω) (1196)

The situation in which there are no transverse external currents impressed onthe system, j

T ext(q, ω) = 0 is considered. Thus, one obtains the microscopic

equation (− q2 +

ω2

c2

)A(q, ω) = − 4 π

cj

T ind(q, ω) (1197)

Ohm’s law can be expressed in the form

jT ind

(q, ω) = σT (q, ω) ET (q, ω) (1198)

where σT is the transverse conductivity and the total transverse electric field isgiven by

ET (q, ω) = iω

cA(q, ω) (1199)

This leads to(− q2 +

ω2

c2+ i

4 π ωc2

σT (q, ω))A(q, ω) = 0 (1200)

The transverse dielectric function is identified in terms of the optical conduc-tivity

εT (q, ω) = 1 +4 π iω

σT (q, ω) (1201)

The photon dispersion relation can be re-written as(ω

c

)2

εT (q, ω) = q2 (1202)

If εT (q, ω) > 0, then there are undamped transverse electromagnetic waves.Otherwise, q would be complex which implies that ET (r; t) is attenuated as itenters into the sample. In other words, if Im ε(q, ω) = 0 and Re ε(q, ω) > 0,

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the material is transparent to transverse electromagnetic waves. The dispersionrelation is given by

εT (q, ω) =(c q

ω

)2

(1203)

Thus, the transverse excitations have a completely different character to thelongitudinal excitations, specially at large q. As q → 0, one expects thatσL(q, ω) → σT (q, ω) since electrons cannot differentiate between transverseand longitudinal waves in this limit. In this limit the conductivity may bemodelled by the complex Drude expression

σ(0, ω) =ρ e2 τ

m

11 − i ω τ

(1204)

which leads to the dielectric constant being given by

ε(0, ω) = 1 − 4 π e2 ρm ω2

11 + i

ω τ

(1205)

This approximate equality between the longitudinal and transverse dielectricconstants implies that the plasmon frequency also sets the frequency scale forthe interaction of photons with a metal. The dispersion relation for transverseradiation becomes

ε(0, ω) ω2 = c2 q2 (1206)

which is given by

ω2 − ω2p = c2 q2 (1207)

Thus, for ω < ωp, the wave will be reflected from a metal. For a typical metal,where ρ ∼ 1022 electrons/cm3, a typical plasmon frequency is 1015 sec−1. Thistypical frequency corresponds to a typical wave length of light in vacuum ofλp ∼ 10−7m. Incident light with longer wave length will be reflected from themetal. Hence, as εL(0, ω) = εT (0, ω) which implies that optical experimentsthat measure εT (q, ω) produce similar information to characteristic energy lossexperiments that determine εL(q, ω).

The transverse conductivity may be evaluated directly by linear responsetheory. The vector potential couples to the electrons via the interaction

Hint =| e |

2 m c

∑i

[ (p

i. A(ri, t) + A(ri, t) . pi

)+| e |c

A(ri, t)2

](1208)

The interaction contains a paramagnetic contribution that involves a coupling tothe momentum density and a diamagnetic contribution that involves a couplingwith the density of the charged electrons. The transverse current density j(r, t)

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is the mechanical current density e v and is given by the sum of a paramagneticcurrent j

pand a diamagnetic current j

d

j(r, t) = jp(r, t) + j

d(r, t) (1209)

where the paramagnetic current is given by the symmetric operator

jp(r) = − | e |

2 m

∑i

[δ3(r − ri) pi

+ piδ3(r − ri)

](1210)

and the diamagnetic current is given by

jd(r) = − | e |2

m c

∑i

δ3(r − ri) A(ri) (1211)

To linear order in the vector potential, the interaction can be written as

H1int = −1

c

∫d3r j

p(r) . A(r) (1212)

The induced paramagnetic current density is then found from linear responsetheory, in which the ground state is evaluated to first order in the perturbinginteraction H1

int. The components of the induced paramagnetic current aregiven as a causal convolution of a paramagnetic current - paramagnetic currenttensor correlation function and the components of the total vector potential.

jαind p(r, t) =

∑β

∫ +∞

−∞dt′

∫d3r′

1cRα,β

j,j (r, r′, t− t′) Aβ(r′, t′) (1213)

where the paramagnetic response function is given by the ground state expec-tation value

Rα,βj,j (r, r′, t− t′) = +

i

h< Ψ0 |

[jαp (r, t) , jβ

p (r′, t′)]| Ψ0 > Θ( t − t′ )

(1214)

This is known as the Kubo formula for the conductivity (R. Kubo, J. Phys. Soc.Jpn. 12, 570 (1957)). The structure of the Kubo formula for the response R issimilar to that of the longitudinal response function χ. They both involve theexpectation value of a retarded two time commutator. However, the Lindhardfunction involves the commutator of the density operator and the Kubo formulainvolves the current operator.

On Fourier transforming the non-local relation between jp

and A with re-spect to space and time, one has

jαind p(q, ω) =

∑β

1cRα,β

j,j (q, ω) Aβ(q, ω) (1215)

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Hence, to linear order in the vector potential, the total transverse current isgiven by

jαind(q, ω) =

∑β

[1cRα,β

j,j (q, ω) − δα,β| e |2

m cρ0

]Aβ(q, ω) (1216)

where it is assumed that the electron density is uniform and is given by ρ0. Thetransverse conductivity is then found with the aid of the relation between thetransverse electric field and the vector potential

ET (q, ω) = iω

cA(q, ω) (1217)

as

σα,βT (q, ω) =

1i ω

[Rα,β

j,j (q, ω) − δα,β| e |2

mρ0

](1218)

The conductivity should be evaluated using a microscopic theory, and has a realpart and an imaginary part that are connected by causality. The conductivitydetermines the material’s properties and how transverse electromagnetic radia-tion or light interacts with the electrons in the metal.

The energy loss due to a longitudinal field is related to the inverse of thedielectric constant, but the energy loss of a transverse field is related to theconductivity or the imaginary part of the dielectric constant. This can be seenfrom the expression for the time averaged dissipated power density

P (q, ω) =ω

4 πIm ε(q, ω + iδ) | ET (q, ω) |2

=ω3

4 π c2Im ε(q, ω + iδ) | A(q, ω) |2 (1219)

As the imaginary part of the dielectric constant is related to the real part of theconductivity,

Im ε(q, ω + iδ) =(

4 πω

)Re σ(q, ω) (1220)

the absorption of light measures the conductivity.

11.3.6 Optical Experiments

The optical conductivity can be measured in optical absorption and reflectionexperiments. The wave vector of light in the medium is given by the complexnumber

q =ω

c

(1 +

4 π i σ(ω)ω

) 12

c

(n + i κ

)(1221)

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and this has the effect that intensity of light is exponentially attenuated as itpasses through the material

E(r, t) = E0 exp[i ω (

n z

c− t )

]exp

[− κ ω z

c

](1222)

Experiments measure the absorption coefficient η which is the fraction of lightabsorbed in passing through unit thickness of the material

η =Re(j . E

)n | E |2

= 2κ ω

c(1223)

Another, experimental method (ellipsometry) measures the reflectance of light.This involves measuring the ratio of the reflected to the incident intensities, andgives rise to the real reflection coefficient. At oblique incidence, with angle ofincidence θ one distinguishes between s and p polarized light. The s polarizedlight has the polarization perpendicular to the plane of incidence and the p po-larized light has polarization parallel to the plane of incidence. The reflectancesare given in terms of the complex refractive index n = n + i κ, via the Fresnelformulas

Rs(θ) =∣∣∣∣ cos θ − ( n2 − sin2 θ )

12

cos θ + ( n2 − sin2 θ )12

∣∣∣∣ (1224)

and

Rp(θ) =∣∣∣∣ n2 cos θ − ( n2 − sin2 θ )

12

n2 cos θ + ( n2 − sin2 θ )12

∣∣∣∣ (1225)

The complex refractive index can then be inferred from measurements of Rs(θ)and Rp(θ). However, it is usual to infer the real part from the imaginary partvia the Kramers-Kronig relation (H.A. Kramers, Nature 117, 775 (1926), R. deL. Kronig, J. Opt. Soc. Am., 12 547 (1926)).

11.3.7 Kramers-Kronig Relation

Causality requires that the frequency be continued in the upper half complexplane ω + i δ in the response functions. This has the consequence that theresponse function is analytic in the upper half complex plane. Also, it is requiredthat the integrand vanishes over a semi-circular contour at infinity in the upperhalf complex plane. With these restrictions one can consider the Cauchy integral

( ε(q, ω + iδ) − 1 ) =1

2 π i

∫c

ε(q, z) − 1z − ω − i δ

dz (1226)

where the contour is taken around the point z = ω + iδ. If ε(q, z) does nothave a pole at z = 0, the contour of integration can be deformed to the realaxis and an infinite semi-circular contour in the upper half complex plane. Inthis case, one finds

ε(q, ω + iδ) − 1 =1π i

Pr

∫ +∞

−∞

ε(q, z) − 1z − ω

dz (1227)

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in which the contribution of the small semi-circle around the pole at z = ω + i δhas been cancelled out. On writing

ε(q, z) = Re ε(q, z) + i Im ε(q, z) (1228)

one finds

Re ε(q, ω + iδ) − 1 =1πPr

∫ +∞

−∞

Im ε(q, z)z − ω

dz (1229)

and

Im ε(q, ω + iδ) = − 1πPr

∫ +∞

−∞

Re ε(q, z) − 1z − ω

dz (1230)

These relations can be recast in the form

Re ε(q, ω + iδ) − 1 =2πPr

∫ ∞

0

z Im ε(q, z)z2 − ω2

dz (1231)

and

Im ε(q, ω + iδ) = − 2 ωπ

Pr

∫ ∞

0

Re ε(q, z)z2 − ω2

dz (1232)

These are the Kramers-Kronig relations (H.A. Kramers, Nature 117, 775 (1926),R. de L. Kronig, J. Opt. Soc. Am., 12 547 (1926)). They can be used to analyzeexperimental data or as consistency checks.

——————————————————————————————————

11.3.8 Exercise 53

Derive the form of the Kramers-Kronig relation for the imaginary part of thedielectric constant ε(q, ω)

Im ε(q, ω) =4 π σ(q, 0)

ω− 2 ω

πPr

∫ ∞

0

dzRe ε(q, z)z2 − ω2

(1233)

valid for a material which has a finite d.c. conductivity σ(q, 0).

——————————————————————————————————

Another sum rule, the optical sum rule is stated as∫ ∞

0

dω ω Im

[εT (q, ω + iδ)

]=

π

2ω2

p (1234)

The optical sum rule can be derived by exact methods. However, it can also beproved by noting that at high frequencies

limω → ∞

εT (q, ω) = 1 −ω2

p

ω2(1235)

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On expressing the imaginary part of the dielectric constant in terms of the realpart, one can verify the sum rule using contour integration. A more usual formof the optical sum rule is stated in terms of a sum rule for the conductivity∫ ∞

0

dω σ(0, ω) =π

2ρ e2

m(1236)

where ρ is the electron density. Kramers-Kronig relations and sum rules can beestablished for a variety of response functions (P.C. Martin, Phys. Rev. 161,143 (1967)). Since the inverse dielectric constant is the longitudinal responsefunction, 1/ε(q, ω) − 1 also satisfies a Kramers-Kronig relation.

——————————————————————————————————

11.3.9 Exercise 54

The n-th moment of the imaginary part of the dielectric constant is defined byMn

Mn =∫ ∞

0

dω ωn Im ε(q, ω + iδ) (1237)

Show that M1 is given by

M1 = 2 π2 e2 ρ

m(1238)

and that M−1 is given by

M−1 =π

2

[ε(q, 0) − 1

](1239)

——————————————————————————————————

11.3.10 The Drude Conductivity

Metals have a large conductivity, and as a result, electromagnetic fields onlypenetrate a small distance into the metal before the energy of the field is ab-sorbed and dissipated as Joule heating. For low frequencies, or slowly spatiallyvarying fields, the penetration depth δ can be calculated from Maxwell’s equa-tions using the frequency dependent Drude electrical conductivity. The Drudeconductivity is calculated by assuming that the photon has a long wavelength,therefore q ≈ 0. On assuming that the medium is homogeneous and isotropic,one finds that the conductivity tensor is diagonal

σα,β(ω) = δα,β e2 ρ τ

m

11 − i ω τ

(1240)

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The Drude formula for the conductivity can be obtained directly from Kuboformulae, in the case of a non-interacting electrons. On expressing the Kuboformulae in terms of the completes set of exact eigenstates of the many-particleHamiltonian H0

H0 | Ψn > = En | Ψn > (1241)

for t > 0, one finds

Rα,β(r, r′, t) =i

h

∑n

< Ψ0 | jαp (r) | Ψn > < Ψn | jβ(r′) | Ψ0 > exp

[+ i

t

h(E0 − En)

]− i

h

∑n

< Ψ0 | jβp (r′) | Ψn > < Ψn | jα(r) | Ψ0 > exp

[− i

t

h(E0 − En)

](1242)

On Fourier transforming the Kubo formula with respect to time, one obtains

Rα,β(r, r′, ω) = −∑

n

< Ψ0 | jαp (r) | Ψn > < Ψn | jβ(r′) | Ψ0 >

h ω + i η + E0 − En

+∑

n

< Ψ0 | jβp (r′) | Ψn > < Ψn | jα(r) | Ψ0 >

h ω + i η + En − E0

(1243)

where the convergence factor η is to be assigned a physical meaning. Thisexpression is to be evaluated for non-interacting electrons, in which case thestates | Ψn > can be taken to be Slater determinants. The matrix elementsof the current density operators can be expressed in terms of the one-electronwave functions φγ(r) and φγ′(r) via

< Ψn | jα(r) | Ψ0 > = − | e | h2 i m

[φ∗γ′(r) ∇αφγ(r) − ∇αφ∗γ′(r) φγ(r)

]= − | e | h

mIm

[φ∗γ′(r) ∇αφγ(r)

](1244)

where the electron in the state labelled by the one-electron quantum number γ isthe excited to the state with quantum number γ′ in the final state. The energydifference between the initial and final states is given by the energy differencebetween the initial and final energies of the excited electron

En − E0 = Eγ′ − Eγ (1245)

Thus, one has

Rα,β(r, r′, ω) = − e2 h2

m2

∑γ,γ′

Im

[φ∗γ(r) ∇αφγ′(r)

]Im

[φ∗γ′(r

′) ∇βφγ(r′)]

h ω + i η + Eγ − Eγ′

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+e2 h2

m2

∑γ,γ′

Im

[φ∗γ(r′) ∇βφγ′(r′)

]Im

[φ∗γ′(r) ∇αφγ(r)

]h ω + i η + Eγ′ − Eγ

(1246)

where Eγ < µ and Eγ′ > µ.

The conductivity response function will be evaluated for free-electrons. Oninserting the single electron wave functions, the response function is found as

Rα,β(r, r′, ω) = − e2 h2

4 m2 V 2

∑k,σ;k′

(kα + k′α) (kβ + k′β)exp

[− i

h (k − k′) . (r − r′)]

h ω + i η + Ek − Ek′

+e2 h2

4 m2 V 2

∑k,σ;k′

(kα + k′α) (kβ + k′β)exp

[− i

h (k − k′) . (r − r′)]

h ω + i η − Ek + Ek′

(1247)

where kα and kβ denote the α and β components of the vector k. The summationover k, σ runs over the occupied states k < kF whereas the sum over k′ runsover the unoccupied states with the spin σ but with k′ > kF . The initialand final state spin quantum numbers are identical. On Fourier transformingwith respect to the space variable, and re-arranging the summation index in thesecond term, one finds

Rα,β(q, ω) = − e2 h2

4 m2 V

∑k,σ

(2kα + qα) (2kβ + qβ)f(Ek) − f(Ek+q)

h ω + i η + Ek − Ek+q

(1248)

In this expression, the effect of the Pauli-exclusion principle is automaticallyaccounted for.

Due to the large magnitude of c, for fixed ω, this expression can be evaluatedto leading order in q. In this case, as space is isotropic, the response functionis also isotropic. That is, the response function is diagonal in the indices α andβ and the diagonal components have equal magnitudes. Hence, the diagonalcomponents can be evaluated from the relation

Rα,α(q, ω) =13

3∑β=1

Rβ,β(q, ω) (1249)

The response function can be expressed as

Rα,β(q, ω) = − δα,β 2 e2

3 m V

∑k,σ

Ek

f(Ek) − f(Ek+q)

h ω + i η + Ek − Ek+q

(1250)

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On Taylor expanding the Fermi-function f(Ek+q) in powers of (Ek+q − Ek)one has

Rα,β(q, ω) = − δα,β 2 e2

3 m V

∑k,σ

Ek

(∂f

∂Ek

) ( Ek − Ek+q )

h ω + i η + Ek − Ek+q

= − δα,β 2 e2

3 m V

∑k,σ

Ek

(∂f

∂Ek

) [1 − h ω + i η

h ω + i η + Ek − Ek+q

](1251)

The first term can be evaluated through integration by parts

− 23

∑σ,k

Ek

(∂f

∂Ek

)= − 2

3

∑σ

∫ ∞

0

dE E ρ(E)(∂f

∂E

)

=23

∑σ

∫ ∞

0

dE f(E)∂

∂E

(E ρ(E)

)(1252)

since the boundary terms vanish. Furthermore, since ρ(E) ∝√E, this term

is evaluated as

− 23

∑σ,k

Ek

(∂f

∂Ek

)=

∑σ

∫ ∞

0

dE f(E) ρ(E)

= Ne (1253)

as the factor of 23 cancels with the factor of 3

2 from the derivative. Due to thissimplification, the response function is given by

Rα,α(q, ω) =ρ0 e

2

m+

2 e2

3 m V

∑k,σ

Ek

(∂f

∂Ek

)h ω + i η

h ω + i η + Ek − Ek+q

(1254)

On substituting this expression into the conductivity, one finds the first termcancels with the diamagnetic current. This cancellation is responsible for pro-hibiting current flow occurring in a metal as a response to an applied magneticfield. In other words, a normal metal does not superconduct due to the cancel-lation of the diamagnetic current.

The conductivity is simply given by

σα,β(q, ω) = δα,β 2 e2

3 i m V

∑k,σ

Ek

(∂f

∂Ek

)h

h ω + i η + Ek − Ek+q

(1255)

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The derivative of the Fermi-function is only non-zero in the vicinity of theFermi-energy. In the limit, T → 0, the derivative may be expressed as

− ∂f

∂E= δ( E − EF ) (1256)

The appearance of the derivative of the Fermi-function in the expression for theconductivity is a consequence of the Pauli-exclusion principle. Only electronsclose to the Fermi-energy can absorb relatively small amounts of energy andbe excited to unoccupied single-electron states and, hence, carry current. Aphenomenological relaxation time τ can be defined via

h

τ= η (1257)

The relaxation time τ can be thought of as the lifetime of the current carryinghole or the current carrying excited electron. This lifetime must be causedby a scattering mechanism. In more rigorous treatments of the conductivity,the scattering rate is calculated using microscopic descriptions of the scatteringinteraction. When expressed in terms of the relaxation rate, the conductivitybecomes

σα,β(q, ω) = − δα,β e2 τ

m V

∑k,σ

23

Ek

(∂f

∂Ek

)1 − i τ ( ω − q . v(k) )

(1258)

In the limit, q → 0, one recovers the Drude approximation for the conductivity

σα,β(ω) = δα,β ρ e2 τ

m

11 − i τ ω

(1259)

where ρ is the electron density. The Drude conductivity is purely real in thed.c. limit, and is given by

σα,β(0) = δα,β ρ e2 τ

m(1260)

and at finite frequencies has a real part that decays like ω−2

Re σα,β(ω) = δα,β ρ e2τ

m

11 + ω2 τ2

(1261)

Thus, the Drude conductivity has a peak at zero frequency and the width ofthe peak is determined by the relaxation time. The integrated strength of thelow energy Drude peak in the conductivity is given by∫ ∞

0

dω Re

[σα,β(ω)

]= δα,β π ρ e2

2 m(1262)

Hence, the intensity of the Drude peak provides a measure of the number ofconduction electrons in a system of non-interacting electrons. For a metal with

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interacting electrons, the Drude peak, when integrated over a low frequencyrange yields an estimate of the quasi-particle weight.

——————————————————————————————————

11.3.11 Exercise 55

Show that microwaves, with low frequency frequency ω, satisfy the equation

− ∇2 E(r, ω) =4 π i σ(ω) ω

c2E(r, ω) (1263)

where σ(ω) is the diagonal component of the conductivity tensor. Solve thisequation for the electric field and hence calculate the classical skin depth δ.The classical skin depth is defined as the distance δ that an electric field pene-trates into a metal before being attenuated.

——————————————————————————————————

The analysis of Exercise 55 is only valid if the electric field vary slowly overdistances of the order of a mean free path λ. The analysis is only valid for lowfrequencies and dirty metals. However, for good metals, E(r, ω) varies rapidlyin space. This regime corresponds to the anomalous skin effect (A.B. Pippard,Proc. Roy. Soc. A, 191, 385 (1947), A.B. Pippard, Proc. Roy. Soc. A, 224273 (1954)). Since the electrons do not respond to the field instantaneously andlocally, the retarded and non-local response function ought to be used. In thiscase, one should solve Maxwell’s equations by solving for the Fourier compo-nents of the fields E(q, ω) and B(q, ω) and by using an approximate expressionfor the conductivity tensor in which both the wave vector and frequency depen-dence are kept (D.C. Mattis and G. Dresselhaus, Phys. Rev. 111, 403 (1958)).This procedure is crucial for the discussion of the anomalous skin effect.

——————————————————————————————————

11.3.12 Exercise 56

The conductivity tensor can be expressed as an integral over the Fermi-surface,

σα,β(q, ω) =∫

d2S

| h v(k) |τ vα(k) vβ(k)

1 − i τ ( ω − q . v(k) )(1264)

Consider a clean material with a sufficiently long mean free path λ such thatqz λ 1 for fixed qz. Show that the transverse conductivity σx,x(qz ez, ω) isgiven by the approximate expression

σx,x(qz ez, 0) =3π4

σ0

| qz | λ(1265)

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(G.E.H. Reuter and E.H. Sondheimer, Proc. Roy. Soc. A195, 336 (1948))

——————————————————————————————————

11.3.13 The Anomalous Skin Effect

For clean materials with large mean free-paths λ, the penetration of an electricfield into a metal is governed by the anomalous skin effect. In the low frequencylimit, the electric field Ex(z, ω) is governed by

∂2Ex

∂z2+

ω2

c2Ex = − 4 π i ω

c2jx (1266)

where the surface of the material is the z = 0 plane. we shall assume thatelectrons are specularly reflected from the surface. This boundary conditioncan be understood by imagining that the surface of the metal demarcates theboundary between two identical solids. One solid represents an extension ofthe actual material generated by mirror symmetry. The condition of specularreflection amounts to assuming that the electrons and fields in the mirror imagesolid behave in the same way as in the actual solid. This leads to the boundarycondition for the electric field being given by(

∂Ex

∂z

)∣∣∣∣z=−ε

= −(∂Ex

∂z

)∣∣∣∣z=ε

(1267)

Therefore, on subsuming the boundary condition in the equation of motion forthe field, one has

∂2Ex

∂z2+

ω2

c2Ex = − 4 π i ω

c2jx + 2 δ(z)

(∂Ex

∂z

)∣∣∣∣z=0

(1268)

Hence, on Fourier transforming with respect to z and using Ohm’s law

jx(qz, ω) = σx,x(qz) Ex(qz, ω) (1269)

one obtains the solution

Ex(qz, ω) =2(

∂Ex

∂z

)∣∣∣∣z=0

ω2

c2 − q2z + 4 π i ωc2 σx,x(qz, ω)

(1270)

In the limit of extremely long mean free path λ → ∞, the conductivitysimplifies to

σx,x(qz, 0) =3 π σ(0, 0)4 | qz | λ

(1271)

(G.E.H. Reuter and E.H. Sondheimer, Proc. Roy. Soc. A195, 336 (1948)). Thespatial dependence of the electric field is given by the inverse Fourier transform,

Ex(z, ω) =(∂Ex

∂z

)∣∣∣∣z=0

∫ ∞

−∞

dqzπ

exp[− i qz z

]ω2

c2 − q2z + 3 π2 i ωc2 λ | qz | σ(0, 0)

(1272)

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On defining δ via

δ−1 =(

3 π2 ω σ(0, 0)c2 λ

) 13

(1273)

one has

Ex(z, ω) =(∂Ex

∂z

)∣∣∣∣z=0

∫ ∞

−∞

dqzπ

| qz | exp[− i qz z

]ω2

c2 | qz | − | q3z | + i δ−3

= − 2 i δ(∂Ex

∂z

)∣∣∣∣z=0

∫ ∞

0

dx

π

x cos( x zδ )

1 + i x3 − i ω2 δ2

c2 x

(1274)

For low frequencies, the decay of the electric field is governed by δ the anoma-lous skin depth.

At the surface, the value of the field is given by

Ex(0, ω) = − 2 i δ(∂Ex

∂z

)∣∣∣∣z=0

∫ ∞

0

dx

π

x

1 + i x3 − i ω2 δ2

c2 x

≈ − 2 δ3

(∂Ex

∂z

)∣∣∣∣z=0

( 1 +i√3

) (1275)

Far from the surface, the field has an exponential decay

Ex(z, ω) ∼(δ

z

)2

exp[− z

δ

](1276)

which decays over a distance δ.

This result for the anomalous skin depth δ was first obtained by Pippard,upto a numerical factor (A.B. Pippard, Proc. Roy. Soc. A, 191, 385 (1947),A.B. Pippard, Proc. Roy. Soc. A, 224 273 (1954)). Pippard noted that only thefraction of the electrons δ

λ close to the surface may participate in the screeningprocess. That is, only the electrons moving parallel to the surface are stronglyeffected by the electric field. The electrons that remain within the penetrationdepth δ before being scattered, subtend an angle of

dθ ≈ δ

λ(1277)

The number of the electrons which are capable of responding to the field isproportional to the solid angle dΩ,

dΩ = 2 π sin θ dθ ∼ 2 π dθ (1278)

since θ ≈ π2 . Hence, the effective electron density, ρeff is given by

ρeff ≈ ρδ

λ(1279)

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where ρ is the uniform electron density. The implies that conductivity parallelto the surface should be reduced by the factor δ

λ . Thus, on applying the analysisof the classical skin effect, one recovers the relation

δ−2 ∼ 4 π ωc2

σ(0)δ

λ(1280)

Hence, one has Pippard’s relation

δ−1 ∼(

4 π ω σ(0)c2 λ

) 13

(1281)

which only differs by a numerical factor from the previously given expressionfor the skin depth.

11.3.14 Inter-Band Transitions

The absorption of a photon of wave vector q may cause an electron to make atransition between the initially occupied state with Bloch wave vector k to a finalstate with wave vector k+ q. Since the wave vector of light q is small comparedwith kF , for a given ω, the final state must be in a different band and must beempty. These are inter-band transitions. Materials with small inter-band gapscan have large dielectric constants. The inter-band contribution to the dielectricconstant can be obtained, by neglecting q, thereby producing vertical transitionbetween the different Bloch bands. The imaginary part of the dielectric constantdue to inter-band transitions can be written as

Im

[ε(ω+iδ)

]= 8 π2 h2

(e

m ω

)2 ∫d3k

( 2 π )3

∣∣∣∣ eα . Mk

∣∣∣∣2 δ( Ec,k − Ev,k − h ω )

(1282)The matrix elements for the inter-band transitions are given by

eα . Mk = eα .

∫d3r exp

[− i k . r

]uv,k(r) ∇ exp

[+ i k . r

]uc,k(r)

(1283)

where un,k are the periodic functions of r in the Bloch functions. The sum overk can transformed into an integral

Im

[ε(ω+iδ)

]= 8 π2 h2

(e

m ω

)2 ∫d2S

( 2 π )3

∣∣∣∣ eα . Mk

∣∣∣∣2∣∣∣∣ ∇ ( Ec,k − Ev,k )∣∣∣∣Ec−Ev=hω

(1284)

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where d2S represents an element of the surface in k space defined by the equationh ω = Ec,k − Ev,k. The quantity

J(ω) =∫

d2S

( 2 π )3

∣∣∣∣ eα . Mk

∣∣∣∣2∣∣∣∣ ∇k ( Ec,k − Ev,k )∣∣∣∣Ec−Ev=hω

(1285)

is known as the joint density of states. The joint density of states varies rapidlywith respect to ω at the critical points, at which

∇k ( Ec,k − Ev,k )∣∣∣∣Ec−Ev=hω

= 0 (1286)

The inter-band transitions produces a broad continuum in the absorption spec-trum, and only the van Hove singularities may be uniquely identified in thespectrum. The analytic behavior of the dielectric constant near a singularitymay be obtained by Taylor expanding about the critical point.

When other processes such as electron-phonon scattering are considered,second order time dependent perturbation theory describes indirect transitions.In this case, a phonon may be absorbed or emitted by the lattice while thephoton is being absorbed. The emission or absorption of the phonon introducesa change of momentum q. Conservation of momentum leads to the momentaof the initial and final state of the electron being related via q = k′ − k.Since the energy of the phonon is usually negligible compared with the energyof the photon, the energy of the absorbed photon is approximately given by theenergy difference of the electron’s initial and final states

h ω ≈ Ec,k − Ev,k′ (1287)

Thus, since q varies continuously, the indirect inter-band transitions have a con-tinuous spectrum. The threshold energy for the inter-band transition is close tothe minimum value of Ec,k − Ev,k′ , for all possible values of k and k′. If theminimum value of Ec,k − Ev,k′ occurs for k = k′ the band gap is known asa direct band gap, whereas if k 6= k′ the band gap is called an indirect band gap.

11.4 Measuring the Fermi-Surface

The Fermi-surface determines most of the thermodynamic, transport and opticalproperties of a solid. The geometry of the Fermi-surface can be determinedexperimentally, through a variety of techniques. The most powerful of thesetechniques is the measurement of de Haas - van Alphen oscillations. The de Haas- van Alphen effect is manifested as an oscillatory behavior of the magnetization(W.J. de Haas and P.M. van Alphen, Proc. Amsterdam, Acad. 33, 1106 (1930)).

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The magnetization is periodic in the inverse of the applied magnetic field 1H .

Onsager pointed out that the period in 1H is given by the expression

∆(

1H

)= 2 π

| e |h c

Ae (1288)

where Ae is the extremal cross-sectional area of the Fermi-surface in the planeperpendicular to the direction of the applied field H (L. Onsager, Phil. Mag. 45,1006 (1952)). By observing the period of oscillations for the different directionsof the applied field, one can measure the extremal areas for each direction. Thisinformation can then be used to reconstruct the three-dimensional Fermi-surface(D. Schoenberg, Proc. Roy. Soc. A 170, 341 (1939)). First, some properties ofthe electron orbits in an applied field will be examined, then the experimentalmethods used in the determination of the Fermi-surface will be described.

11.4.1 Semi-Classical Orbits

In the classical approximation, the Hamilton equations of motion are given bythe pair of equations

r = v(k) =1h∇k Ek (1289)

and

h k = − | e |c

v(k) ∧ H (1290)

From these one finds that k changes in a manner such that it remains on theconstant energy surfaces. This is found by observing that, from the above equa-tion, k is perpendicular to ∇kEk. Also since k is perpendicular to H, the kspace orbits are a section of the constant energy surfaces with normal alongthe z axis. That is the orbits traverse the constant energy surfaces, but kz is aconstant.

The real space orbits are perpendicular to the k space orbits. To show this,we shall first prove that k is perpendicular to r. On taking the vector productof the equation of motion with the vector H, one finds

h H ∧ k = − | e |c

H ∧(v(k) ∧ H

)(1291)

The component of the velocity perpendicular to the applied field is given by

r⊥ = r − H

(r . H

)= H ∧

(r ∧ H

)(1292)

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where H is the unit vector in the direction of H. Hence,

r⊥ = − c h

| e | HzH ∧ k (1293)

Thus, on integrating this one finds that the displacement ∆r⊥ is given in termsof the displacement in k space through

∆r⊥ = − c h

| e | HzH ∧ ∆k (1294)

Thus, the real space orbit is perpendicular to the k space orbit and is scaled bya factor of c h

e Hz.

The period T at which the orbit is traversed is given by the integral overone orbit

T =∮

dk

k(1295)

The rate of change of k is given by the Lorentz Force

k =| e |h2 c

∣∣∣∣ ∇kE ∧ H

∣∣∣∣=

| e |h2 c

Hz

∣∣∣∣ ∇kE⊥

∣∣∣∣ (1296)

where ∇kE⊥ is the component of the gradient perpendicular to H, i.e., theprojection in the plane of the orbit. Thus,

T =∮

dk∣∣∣∣ ∇kE⊥

∣∣∣∣h2c

| e |1Hz

(1297)

If semi-classical quantization considerations are applied, then the energy ofthe orbits become quantized as do the orbits themselves. The area enclosed bythe orbits are related to the energy, and so the areas are also expected to bequantized. This shall be shown by two methods, in the first the quantizationcondition is imposed through the energy - time uncertainty relation, and thesecond method will utilize the Bohr-Sommerfeld quantization condition.

Quantization Using Energy - Time Uncertainty.

The relationship between the energy and the areas enclosed by the k spaceorbits can be found from consideration of two classical orbits, one with energyE and another with energy E + ∆E where both orbits are in the same kz

plane. Then, let ∆k be the minimum distance between these two orbits. Thevalue of ∆k is related to ∆E via

∆E =∣∣∣∣ ∇kE⊥

∣∣∣∣ ∆k (1298)

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This relation can be substituted into the expression for the period to yield

T =h2c

| e | Hz

1∆E

∮∆k dk (1299)

However, the area between the two successive orbits is given by the integral

∆A =∮

∆k dk (1300)

Thus, the period can be expressed as

T =h2c

| e | Hz

∆A∆E

(1301)

The orbits can be quantized through the energy uncertainty relation

En+1 − En =h

T

=| e | Hz

h c

∣∣∣∣ ∆E∆A

∣∣∣∣ (1302)

Furthermore, as∆E∆A

=En+1 − En

An+1 − An(1303)

one can cancel a factor of ∆E to find that the area enclosed between consecutiveLandau orbits is quantized

An+1 − An =2 π | e |h c

Hz (1304)

This difference equation can be solved to yield the area enclosed by the n-thLandau orbital as

An =(n + λ

)2 π | e |h c

Hz (1305)

where λ is a constant, independent of n. Thus, the area of a Landau orbit ink space is related to n and the applied field Hz, through the Onsager equation(L. Onsager, Phil. Mag. 43, 1006, (1952)).

Bohr-Sommerfeld Quantization.

An alternate derivation of the Onsager equation follows from the Bohr-Sommerfeld quantization condition∮

p . dr = 2 π h(n +

12

)(1306)

The mechanical momentum is given by

p = h k − | e |c

A (1307)

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The integral is evaluated over an orbit in the x − y plane perpendicular to H.The orbit is obtained from the equation of motion with the Lorentz Force Law

h k = − | e |c

r ∧ H (1308)

The equation of motion can be integrated with respect to time, to yield

h k = − | e |c

r ∧ H (1309)

Thus, the Bohr-Sommerfeld quantization condition reduces to

− | e |c

∮ (r ∧ H + A

). dr = 2 π h

(n +

12

)=

| e |c

(H .

∮dr ∧ r −

∮dr . A

)(1310)

However, the integral ∮dr ∧ r = 2 Ar ez (1311)

is just twice the area enclosed by the real space orbit, and the integral of thevector potential around the loop is given by∮

dr . A = Φ (1312)

where Φ = Ar Hz is the flux enclosed by the orbit. Hence, the magnitude ofthe area of the orbit, Ar, in real space is quantized and is given by

Ar =(n +

12

)2 π h c| e | Hz

(1313)

Since the real space and momentum space orbitals are related via

∆r =h c

| e | Hz∆k (1314)

one can scale the areas of the real and momentum space orbits. Thus, onerecovers the Onsager formulae for the area of the momentum space orbit

An =(n +

12

)2 π | e |h c

Hz (1315)

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11.4.2 de Haas - van Alphen Oscillations

Given a solid with Hz = 0, surfaces of constant energy do not intersect whenplotted in k space. The consecutive constant energy surfaces, corresponding tothe different allowed values of energy, completely fill momentum space. Thestates on the surfaces which have energy less than µ will be occupied, andthose with energy greater than µ are empty. On applying a magnetic field, Hz,the momentum perpendicular to the field is no longer a constant of motion,but kz is constant. However, as time evolves, an orbit never leaves its surfaceof constant energy. The magnetic field quantizes the orbits. In momentumspace, the allowed orbits form a nested set of discrete Landau tubes. Orbits inthe regions between the tubes are forbidden. For a general Fermi-surface of athree-dimensional crystal, the intersection of the constant energy surfaces witha plane of fixed kz need not be circular, so that the Landau tubes need not havecylindrical cross-sections. However, for free electrons, the zero field constantenergy surfaces are spherical and the Landau tubes are cylindrical. The radiusof the tubes is determined by the energy of the the x-y motion, while the heightis determined by the component of the kinetic energy due to motion in the z-direction. For free electrons, an occupied orbit is specified by kz and n. Theenergy of the free-electron orbit is given by

En,kz = ( n +12

) h ωc +h2 k2

z

2 m(1316)

where

ωc =| e | Hz

m c(1317)

The orbit maps out a circle of area

An =2 π | e |h c

Hz

(n +

12

)(1318)

so the orbits will consist of concentric circles. On varying kz but holding n fixed,the consecutive orbits will map out a tube in k space. The occupied portions ofk space will lie on portions of a series of tubes. These portions will be containedin a volume similar to the volume of the Fermi-surface, when Hz = 0. Thebounding volume must reduce to the volume enclosed by the Fermi-surface whenthe field is decreased. For fields of the order of H ∼ 1 kG, the Fermi-surfacecuts about 103 such tubes, so the quasi-classical approximation can be expectedto be valid.

As the field increases, the cross-sectional area enclosed by the tubes alsoincreases, as does the number of electrons held by the tubes. The extremaltube may cross the zero field Fermi-surface ( H = 0 ) at which point theelectrons in the tube will be entirely transferred into the tubes with lower nvalues. The changing structure gives rise to a loss of tubes from the occupiedFermi-volume when the field changes by amounts ∆H. Thus, if at some value

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of kz, the occupied Landau tube with the largest area has the largest value ngiven by the extremal area of the Fermi-surface A(kF )

2 π | e |h c

Hz ( n +12

) ∼ π k2F (kz) = A(kF ) (1319)

then, on changing Hz to Hz + ∆H the tube becomes unoccupied so the largestLandau tube changes from n to n− 1. This occurs when

( n − 12

) ( Hz + ∆H ) = ( n +12

) Hz

(1320)

Thus, the extremal orbit crosses the Fermi-surface when Hz is increased by

n ∆H ≈ Hz (1321)

which can be used to eliminate n and relate ∆HHz

to the momentum space areaof the extremal orbit.

A(kF ) = π k2F (kz) =

Hz

∆H2 π | e |h c

Hz (1322)

Thus, n decreases by unity at fields given by

− ∆HH2

= − 2 π | e |h c

1A(kz)

(1323)

In other words, ∆n changes by − 1 with increasing ∆(

1Hz

). The non-

monotonic variation of the occupancy of the extremal orbits or tubes gives riseto oscillations in the Free energy as Hz is varied. This can also be seen fromexamination of the density of states, per spin polarization, for free electrons

ρ(E) = D∑

n

Lz

∫ ∞

−∞

dkz

2 πδ( E − n h ωc −

h2 k2z

2 m)

=Lx Ly Lz

4 π2 hm ωc

∑n

∫ ∞

0

dkz δ( E − n h ωc −h2 k2

z

2 m)

=Lx Ly Lz

4 π2 h2 m2 ωc

∑n

θ( E − n h ωc )√2 m ( E − n h ωc )

(1324)

where D is the degeneracy of a Landau orbital. The degeneracy is given by theratio of the cross-section of the crystal to the real space area enclosed betweenthe Landau orbits

D =Lx Ly

∆Ar

=Lx Ly

2 π hm ωc (1325)

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which increases with increasing field. The density of states has equally spacedsquare root singularities determined by the energies of the Landau levels, butyet still roughly follows the zero field density of states. On changing the fieldthe spacing between the singularities increases. This means that, as the field isincreased, successive singularities may cross the Fermi-energy, and give rise tooscillations in physical properties.

Physical properties are expressible as averages which are weighted by theproduct of the Fermi-function and the density of states. For zero spin-orbitcoupling, the average of A is given by

A =∑

σ

∫ +∞

−∞dE f(E) ρ( E − µB σ Hz ) Aσ(E) (1326)

in which the electronic density of states is spin split by the Zeeman field. Thissplitting is comparable with the effect of h ωc. Increasing the field will produceregular oscillations in the integrand which will show up in A. Due to the thermalsmearing manifested by the Fermi-function, the oscillations of A as a functionof 1

Hzcan only be seen at sufficiently low temperatures such that

kB T h ωc (1327)

If this condition is not satisfied, the Fermi-function becomes broad and washesout the peaks in the integrand near µ. As

| e | hm c kB

∼ 1.34 × 10−4 k/G (1328)

it is found that, for a typical field of H = 10 kG, the oscillations will only beappreciable below T ∼ 2 K.

——————————————————————————————————

11.4.3 Exercise 57

A non-uniformity of the magnetic field in a de Haas - van Alphen experimentmay cause the oscillations in Mz to be washed out. Calculate the field derivativeof the electron energy

∂En,k

∂H(1329)

for an extremal orbit. Determine the maximum allowed variation of Hz that isallowable for the oscillations to still be observed. Show that it is given by δH,where

δH

H2z

<2 π | e |h c A

(1330)

and A is the area of the extremal orbit.

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——————————————————————————————————

11.4.4 The Lifshitz-Kosevich Formulae

The de Haas - van Alphen Oscillations in the magnetization M can be foundfrom the grand canonical potential Ω

Ω = − kB T∑α

ln(

1 + exp[− β ( Eα − µ )

] )(1331)

where the sum over α runs over all the one-electron states.

The Lifshitz-Kosevich formulae describes the oscillatory parts of M (I.M.Lifshitz and A.M. Kosevich, Sov. Phys. J.E.T.P. 2, 636 (1956)). This shall beexamined in the T → 0 limit. In the limit T → 0 one has

limT → 0

Ω =∑α

(Eα − µ

)Θ( µ − Eα ) (1332)

where Θ(x) is the Heaviside step function. Also, the total number of electronsis given by

Ne =∑α

Θ( µ − Eα ) (1333)

The dispersion relation for free electrons in an applied field is given by

Eα =h2 k2

z

2 m+(n +

12

)h ωc − µB Hz σ (1334)

so

Ω =| e | Hz V

4 π2 c h

∑σ

n=∞∑n=0

∫ ∞

−∞dkz

(h2 k2

z

2 m+(n +

12

)h ωc − µB Hz σ − µ

)× Θ

(µ − h2 k2

z

2 m− ( n +

12

) h ωc + µB Hz σ

)(1335)

For fixed n the step function has the effect that the kz integration is limited tothe range of kz values, kz(σ, n) > kz > − kz(σ, n) , where

h2 kz(σ, n)2

2 m= µ −

(n +

12

)h ωc + µB Hz σ (1336)

The kz integration can be performed yielding

Ω = − 43| e | Hz V

4 π2 c h

(2 mh2

) 12 ∑

σ

n=∞∑n=0

(µ −

(n +

12

)h ωc + µB Hz σ

) 32

× Θ(µ − ( n +

12

) h ωc + µB Hz σ

)(1337)

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Thus, the summation over n only runs over a finite range of values, where nruns from 0 to n+, where n+ denotes the integer part of

n+ =µ + µB Hz σ

h ωc− 1

2(1338)

Hence,

Ω = − 43| e | Hz V

4 π2 c h

(2 mh2

) 12 ∑

σ

n=n+∑n=0

(µ −

(n +

12

)h ωc + µB Hz σ

) 32

(1339)

The thermodynamic potential shows oscillatory behavior as H increases, sincewhen µ

h ωcchanges by an integer the upper limit of the summation over n also

changes by an integer.

In order to make the oscillatory nature of the summation more explicit, aperiodic function β(x) is introduced. The periodic function is defined as

β(x) =n=+∞∑n=−∞

δ

(x − ( n +

12

))

(1340)

The summation over n in the thermodynamic potential can be expressed interms of an integral over β(x) via

Ω = − 43| e | Hz V

4 π2 c h

(2 mh2

) 12 ∑

σ

∫ x+

0

dx β(x)

(µ − x h ωc + µB Hz σ

) 32

(1341)

where the upper limit of integration is given by

x+ =µ

h ωc+

m c

| e | hµB σ (1342)

However, as

µB =| e | h2 m c

(1343)

the upper limit of integration becomes

x+ =µ

h ωc+

σ

2(1344)

in which the mass of the electron has cancelled in the second term. In general,the spin splitting term will depend on the ratio of the mass of the electron tothe band mass.

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On Fourier analyzing β(x) one has

β(x) = 1 + 2∞∑

p=1

cos 2πp ( x − 12

) (1345)

which on substituting into the expression for Ω yields the expression

Ω = − 43| e | Hz V

4 π2 c h

(2 mh2

) 12 ∑

σ

[2

5 h ωc

(µ + µB σ Hz

) 52

+ 2∞∑

p=1

∫ x+

0

dx cos 2πp ( x − 12

)

(µ − x h ωc + µB Hz σ

) 32]

(1346)

The first term is non-oscillatory. The term containing the summation producesthe oscillatory terms. The second term can be evaluated by integration by parts

Ip =∫ x+

0

dx cos 2πp ( x − 12

)(

µ

h ωc− x +

µB Hz σ

h ωc

) 32

=∫ x+

0

dx

2 π pd

dx

(sin 2πp ( x − 1

2)) (

x+ − x

) 32

=32

∫ x+

0

dx

2 π psin 2πp ( x − 1

2)(x+ − x

) 12

(1347)

since the boundary term vanishes. Integrating by parts once again

Ip = − 38 π2 p2

∫ x+

0

dxd

dx

(cos 2πp ( x − 1

2)) (

x+ − x

) 12

=3

8 π2 p2

[x

12+ cosπp − 1

2

∫ x+

0

dx

(cos 2πp ( x − 1

2)) (

x+ − x

)− 12]

(1348)

Changing variables from x to u, where

2 p(x+ − x

)=

u2

2(1349)

sodx = − 1

2 pu du (1350)

Then, the integration becomes

Ip =3

8 π2 p2

[x

12+ cosπp − 1√

4 p

∫ u0

0

du

(cos

π

2( u2

0 − u2 − 2 p )) ]

(1351)

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The cosine term can be decomposed as

cos (π u2

2+ φ ) = cos

π u2

2cosφ − sin

π u2

2sinφ (1352)

so one has the integrals

C(x) =∫ x

0

du cosπ u2

2

S(x) =∫ x

0

du sinπ u2

2(1353)

which for large x have the limits

C(∞) = S(∞) =12

(1354)

Thus, the integral is evaluated as

Ip =3

8 π2 p2

[x

12+ cosπp +

− 1√4 p

(C(u0) cos

π

2( u2

0 − 2 p ) + S(u0) sinπ

2( u2

0 − 2 p )) ]

∼ 38 π2 p2

[x

12+ cosπp − 1√

8 pcos

π

2( u2

0 − 2 p − 12

)

](1355)

Thus, the oscillatory part of the grand canonical potential ∆Ω is

∆Ω ∼(h ωc

) 32 | e | Hz V

4 π4 c h

(2 mh2

) 12

×

×∑

σ

∞∑p=1

2( 2 p )

52

cos(

2πp(

µ

h ωc+σ − 1

2

)− π

4

)(1356)

This depends on the ratio of the extremal cross-section of the zero field Fermi-surface, AF = π k2

F , and the difference in areas of the Landau orbits inmomentum space, ∆A = 2π | e |

h c Hz,

∆Ω ∼(h ωc

) 32 | e | Hz V

4 π4 c h

(2 mh2

) 12

×

×∑

σ

∞∑p=1

2( 2 p )

52

cos(

2πp(

c h π k2F

2 π | e | Hz+σ

2− 1

2

)− π

4

)

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∆Ω ∼(h ωc

) 32 | e | Hz V

4 π4 c h

(2 mh2

) 12

×

×∑

σ

∞∑p=1

2( 2 p )

52

cos(

2πp(AF

∆A+σ − 1

2

)− π

4

)(1357)

Thus, oscillations in the grand canonical potential occur when the number ofLandau orbits inside the extremal cross-sectional area change. The oscillationsoccur in the magnetization Mz as it is related to the grand canonical potentialΩ via

Mz = −(∂Ω∂Hz

)(1358)

Thus, the magnetization also has oscillations that are periodic in 1Hz

. Further-more, for a free electron gas the extremal area of the Fermi-surface is just π k2

F ,so the period of oscillations is proportional to the extremal cross-sectional areaof the zero field Fermi-sphere. In addition to the fundamental oscillations, thereare also higher harmonics which can be observed in experiments. For the moregeneral situation, where the Fermi-surface is non-spherical different extremalcross-sections will be observed when the magnetic field is applied in differentdirections. This can be used to map out the Fermi-surface.

The Lifshitz-Kosevich formulae, valid at finite temperatures, is

∆Ω =(| e | Hz

2 π h c

) 32 kB T V

( 2 π )12

∑σ

∑p

1p

32

exp[− π p

ωc τ

]sinh 2π2p kBT

h ωc

× cosπp cos[

2πp(AF

∆A+

σ m∗

2 me

)− π

4

](1359)

On performing the sum over the spin polarizations, one obtains the result

∆Ω =(| e | Hz

2 π h c

) 32 2 kB T V

( 2 π )12

∑p

1p

32

exp[− π p

ωc τ

]sinh 2π2p kBT

h ωc

× cosπp(

cosπpm∗

me

)cos[

2πp(AF

∆A

)− π

4

](1360)

The splitting between the up-spin and down-spin bands has modified the rel-ative phase of the higher harmonics in the oscillations. This can be used to

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extract the ratio of the band mass of the electron to the electron mass in vac-uum. For systems which are on the verge of ferromagnetism, the spin splittingfactor should be enhanced by including the effective field on the spins due to theinteractions with the other electrons. This formula also includes the exponentialdamping of the oscillations due to T through the thermal smearing of the Fermi-surface and also has an exponential damping term depending on the rate forelastic scattering of the impurities 1

τ . Both these effects reduce the amplitudeof the de Haas - van Alphen oscillations (R.B. Dingle, Proc. Roy. Soc. A 211,257 (1952)). The oscillations can only be seen at low temperatures T < 1 Kand for samples of high purity, as indicated by small residual resistances. Theoscillations are only seen in materials where the zero temperature limit of theresistivity ρ(0) is less than 1 µΩ cm. The term involving the lifetime comes fromthe width of the quasi-particle spectrum, and should also be accompanied withthe change in quasi-particle energy due to interactions. Therefore, the increasein the quasi-particle mass can also be extracted from the amplitude of the deHaas - van Alphen oscillations. However, the amplitude of the heavier massbands are small compared with the light quasi-particle bands. In the heavyfermion materials such as CeCu6 and UPt3 quasi-particle masses of about 200free electron masses have been observed in de Haas - van Alphen experiments.

11.4.5 Other Fermi-Surface Probes

There are many other probes of the Fermi-surface, these include the attenuationof sound waves. Consider sound waves in a crystal propagating perpendicular tothe direction of the applied magnetic field and having a transverse polarizationthat is also perpendicular to Hz. The motion of the ions is accompanied by anelectric field of the same frequency, wave vector and polarization. The electronsinteract with the sound wave through the electric field. If the wave length ofthe mean free path is sufficiently long, the attenuation of the sound waves canbe used to determine the Fermi-surface.

The electrons follow real space orbits which have projections in the planeperpendicular toHz, which are just cross-sections of the constant energy surfacesin momentum space, but are scaled by h c

| e | Hzand rotated by π

2 . As velocitiesof the ions are much smaller than the electrons velocities, the electric field maybe considered to be static. If the phonon wave vector q is comparable to theradius of the real space orbit, or more precisely the diameter of the orbit inthe direction of q, then the electric field can significantly perturb the electronsmotion. This strongly depends on the mismatch between q−1 and the diameterof the orbit. When the radius of rq the orbit is such that

2 rq =λ

2(1361)

then the electron may be accelerated tangentially by the electric field at bothextremities of the orbit. The coupling is coherent over the electron’s orbit and

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the coupling is strong. When2 rq = λ (1362)

the electron is sequentially accelerated and decelerated by the field. The cou-pling is out of phase on the different segments of the electron’s orbit so that theresulting coupling is weak. In general the condition for strong coupling is thatof constructive interference

2 rq = ( n +12

) λ (1363)

and weak coupling occurs when the interference is destructive

2 rq = n λ (1364)

The period differs slightly from the asymptotic large n variation just described.Assume that the projection of the trajectory on the plane perpendicular to theapplied field Hz is circular. The energy transfer between the electron and theelectronic wave in one orbit is given by∫ 2π

ωc

0

dt E(r, t) . v(t) (1365)

The electric field is assumed to be polarized along the y direction, and q isdirected along the x direction. Since the orbit in momentum space is rotatedby π

2 with respect to the real space orbit one has

vy(t) = vF sin ωc t (1366)

andx(t) = rq sin ωc t (1367)

Thus, the energy transfer in a period is evaluated as

Ey vF

∫ 2πωc

0

dt exp[i q rq sin ωc t

]sin ωc t

= Ey vF

(2πωc

)J1( q rq ) (1368)

Thus, the resonances occur for phonon wave lengths which match the maximaof the Bessel function J1(x). Only electrons near the Fermi-surface can absorbenergy from the sound wave. The Pauli exclusion principle forbids electrons inother states to undergo low energy excitations, since the slightly higher energystates are already occupied.

The electrons with the extremal diameter on the Fermi-surface are morenumerous and, therefore, play a dominant role in the attenuation process. Thus,the sound wave may display an approximately periodic variation in λ where theasymptotic period is determined by

∆(

)=

12 rq

(1369)

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By variation of q, and H one can map out the Fermi-surface.

11.4.6 Cyclotron Resonances

This method requires the application of an microwave electric field at the sur-face of a metal. The field is attenuated as it penetrates into the metal, and isonly appreciable with a skin depth δ from the surface. Since the field does notpenetrate the bulk, electrons can only pick up energy from the field when theyare within the skin depth of the surface.

A static (d.c.) magnetic field is applied parallel to the surface say in the xdirection, so that the electrons undergo spiral orbits in real space. The veloc-ity vx remains constant, but the electrons undergo circular motion in the y-zplane. It is only necessary to consider the electrons that travel in spirals thatare close and parallel to the surface as it is these electron couple that to themicrowave field. The size of the orbit and the electron’s mean free path λ shouldbe much larger than the skin depth δ. This holds true when the cyclotron fre-quency ωc is large, and for microwave frequencies ω where the anomalous skindepth phenomenon occurs. The condition of a long mean free path and largecyclotron frequency is necessary for the electrons to undergo well defined spirals,so ωc τ 1.

The electrons pick up energy from the field only if they are within δ of thesurface. The electrons in the spiral orbits only experience the electric field eachtime they enter the surface region. They enter the skin depth periodically, withperiod

TH =2 πωc

(1370)

which is the period of the cyclotron motion. In general, the period is given by

TH =h2 c

| e | Hx

(∂A

∂E

)(1371)

The electron will experience an E field with the same phase, if the appliedfield has completed an integral number of oscillations during each cyclotronperiod. That is

TH = n TE = n2 πω

(1372)

Hence, this requires that the frequency of the cyclotron orbit match with thefrequency of the a.c. electric field

ω = n ωc (1373)

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so that the a.c. field resonates with the electronic motion in the uniform field.This condition can be written as

1Hx

= 2 π | E | h2 c ωn(

∂A∂E

) (1374)

The factor

mc =h2

2 π

(∂A

∂E

)(1375)

is known as the cyclotron mass. For free electrons, A = π k2 and E = h2 k2

2 mso the cyclotron mass coincides with the electron mass

mc = m (1376)

If the microwave absorption is plotted versus 1Hx

a series of uniformly spacedresonance peaks should be found.

The calculation of the absorption is the simplest in the case when the wavelength of the electromagnetic field λ is much larger than the cyclotron orbit andωc τ 1.

The geometry shall be considered where a surface has a normal in the zdirection and a d.c. magnetic field is applied parallel to the surface in the xdirection

H = ex Hx (1377)

and the a.c. electric field is in the y direction

E = ey Ey exp[i ( q z − ω t )

](1378)

The electron in its orbit experiences a rapidly alternating electric field. For mostvalues of z the contributions cancel. The cancellation only fails at the extremalvalues of z where the velocity lies in the plane z = const. .

To the zero-th order approximation, the z component of the electron’s posi-tion can be expressed as

z(t) = z0 +v

ωcsinωc t (1379)

The total change in momentum of the electron due to the oscillating field, inone period, can be calculated in the semi-classical approximation. The changeof momentum and, hence, the current will be in the direction of the a.c. field.The impulse imparted to the electron is given by the integral of the electric field

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evaluated at the electron’s position

− | e |∫ t+TH

t

dt′ Ey exp[i

(q z0 + q

v

ωcsinωc t

′ − ω t′) ]

∼ − | e | Ey2 πωc

exp[i

(q z0 − ω t

) ]I ω

ωc

(q v

ωc

)(1380)

where In(x) is the modified Bessel function of order n. For large q vωc

, this hasthe asymptotic form

∼ − | e | Ey

(2 πq v ωc

) 12

exp[i

(q z0 − ω t − π

4

) ](1381)

Due to the phase differences around the orbit, only a fraction(2 π ωc

q v

) 12

(1382)

of the orbit contributes to the integral. The energy gain of the electron in onetraversal is given by

v h δky = − | e | Ey

(2 π vq ωc

) 12

exp[i

(q z0 − ω t − π

4

) ](1383)

The previous traversal caused a similar displacement, but with t → t − 2 πωc

.However, only the fraction

exp[− 2 π

ωc τ

](1384)

of electrons survive traversing one cyclotron orbit without scattering. As theseall have a similar form, one can obtain the average energy displacement experi-enced by an electron between one scattering and the next

v h ∆ky = − | e | Ey

(2 π vq ωc

) 12

exp[i

(q z0 − ω t − π

4

) ]F (1385)

The factor F reflects the sum of the probabilities that the electron survive norbits without scattering. The value of F is given by

F =∞∑

n=0

exp[− 2 π n

ωc τ( 1 + i ω τ )

]

=1

1 − exp[− 2 π

ωc τ ( 1 + i ω τ )] (1386)

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It is the imaginary part of the probability for survival that causes the resonancesin the surface impedance.

To calculate the current at a depth z0 one must examine the orbits on theFermi-surface. The orbits circulate around the Fermi-surface in sections thatare perpendicular to the d.c. field. Thus, the orbits are in the y− z plane. Theportions of the Fermi-surface orbits which contribute most to the current arethose in which the electrons are moving parallel to the surface. These portionsof the orbits are those where kz = 0, and form the effective zone. An electronon the effective zone has kz = 0 and is, thus, moving at the extreme of its orbit.Due to the effect of the a.c. field this orbit has been displaced by a distance∆ky from the orbit in which the a.c. field has been turned off.

The total current from the orbits in a section of width dkx, around kx, canbe calculated by considering the contribution of the electrons around the orbit.Due to the phase differences around the orbits only those within a distance

kx

(2 π ωc

q v

) 12

of the effective zone contribute. These orbits are displaced from

their equilibrium positions by an amount ∆ky. Only the displacements fromequilibrium contribute to the current. The contribution to the current densityis

δJy = − | e | 28 π3

dkx kx

(2 π ωc

q v

) 12

exp[iπ

4

]∆ky v

=e2

2 π2dkx kx

(1h q

)F Ey (1387)

where the phases of π4 cancel. The integration over dkx can be converted into

an integral over the effective zone via φ. If the effective mass and cyclotronfrequency are constant over the Fermi-surface, then F is also constant. Theconductivity σ is proportional to F .

The surface impedance is defined as

Z(ω) =4 π i ω Ex

∂Ex

∂x

∼ ( 1 − i )(

2 π ωσ

) 12

∼ F−12 (1388)

The surface impedance has oscillations with varying field Hx. It should be ob-vious from the above discussion that the oscillations provide information on ω

ωc,

but the dominant contribution occurs from the extremal parts of the line ofintersection of the Fermi-surface with the plane kz = 0. Thus, the cyclotronresonance can be used to study points on the Fermi-surface.

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11.5 The Quantum Hall Effect

The Quantum Hall Effect is found in two-dimensional electron systems, in whichan electric field is applied perpendicular to the plane where the electrons areconfined. Experimentally, the electrons can be confined to a two-dimensionalsheet in a metal oxide semiconductor field effect transistor. The applicationof a strong electric field to the surface of a semiconductor may pull down theconduction band at the surface of the semiconductor. If the energy of these fieldinduced surface states is less than the Fermi-energy of the metal, electrons willtunnel across the insulating oxide barrier and occupy them. After equilibriumhas been established, the electrons at the surface of the semiconductor will forma two-dimensional electron gas.

11.5.1 The Integer Quantum Hall Effect

The Integer Quantum Hall Effect can be understood entirely within the frame-work of non-interacting electrons. The calculation of the Hall coefficient canbe performed using the Kubo formula. However, in applying the Kubo for-mulae, one must recognize that the vector potential has two components: ana.c. component responsible for producing the applied electric field and a secondcomponent which produces the static magnetic field. The usual derivation onlytakes the a.c. component of the vector potential into account as a perturbation.The d.c. component of the vector potential must be added to the paramagneticcurrent operator, yielding the electron velocity operator appropriate for the sit-uation where the weakly perturbing electric field is zero.

The application of a static magnetic field Bz perpendicular to the surfacewill quantize the motion of the electrons parallel to the surface. The motionparallel to the surface is quantized into Landau orbits, and the energy eigenvalueequation reduces to the energy eigenvalue equation of simple harmonic motion.Due to the confinement in the direction perpendicular to the surface, kz will notbe a good quantum number, and the perpendicular component of the energywill form highly degenerate discrete levels εn. For large enough fields only thelowest level ε0 will be occupied. On choosing a particular asymmetric gauge forthe vector potential,

A(r) = + ey Bz x (1389)

the Hamiltonian for the two-dimensional motion in the x − y plane is given by

H =p2

x

2 m+

m ω2c

2

(x − X

)2

(1390)

where the cyclotron frequency ωc is given by

ωc =| e | Bz

m c(1391)

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The operator X is given in terms of the y component of the momentum operatorpy. Since the Hamiltonian is independent of y, both py and X can be taken asconstants of motion. The momentum component px is canonically conjugate tothe x component of the particle’s position relative to the center of the orbit

x − X = x +py c

| e | Bz(1392)

The energy eigenvalues of the shifted Harmonic oscillator are given by

Eν,0 = ε0 + h ωc ( ν +12

) (1393)

where ν is the quantum number for the Landau levels. The Landau levels areindependent of ky and, therefore, are degenerate. Since the values of ky arequantized via

ky =2 π ny

Ly(1394)

for a surface of length Ly, then the possible values of ky are limited by therestriction

Lx > X > 0 (1395)

which yields the total number of degenerate states as

D = Lx LyBz | e |2 π h c

=Φ | e |2 π h c

(1396)

where Φ is the total magnetic flux passing through the sample. The fundamentalflux quantum Φ0 is defined as the quantity

Φ0 =2 π h c| e |

(1397)

The density of states can be approximately expressed as a discrete set of deltafunctions

ρ(E) =| e | Bz Lx Ly

h c

∑ν

δ

(E − ε0 − h ωc ( ν +

12

))

=m ωc Lx Ly

2 π h

∑ν

δ

(E − ε0 − h ωc ( ν +

12

))

(1398)

The weight associated with each delta function corresponds to the degeneracyof each Landau level. On defining the cyclotron radius rc as

rc =

√h c

| e | Bz(1399)

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one finds that the relative position operator can be expressed as

x − X =rc√2

(a†ky

+ aky

)(1400)

and from the Heisenberg equation of motion for x one finds that the x componentof the velocity is given by

vx =1i h

[x , H

]= i

rc ωc√2

(a†ky

− aky

)(1401)

as py can be taken to be diagonal. The y component of the velocity is foundfrom the Heisenberg equation of motion for y and is given by

vy =1i h

[y , H

]= ωc ( x − X ) (1402)

since the commutator[y , X

]= − i h c

| e | Bz. On substituting for the x

component of the displacement from the center of the orbit

vy =rc ωc√

2( a†ky

+ aky ) (1403)

Thus, as the velocity operators are non-diagonal, in the quantized Landau levelindices, the Landau orbitals do not carry a net current.

The Kubo formula can be expressed in terms of the electron velocity op-erators, which includes the diamagnetic current contributions from the staticmagnetic field. On using the form of the Kubo formula for the conductivitytensor, per unit area, appropriate for single particle excitations

σα,β(ω) =i e2

ω + i η

1Lx Ly

[ ∑ν,ν′,ky

< ν, ky | vα | ν′, ky > < ν′, ky | vβ | ν, ky > ×

× f(Eν) − f(Eν′)h ω + i η + h ωc ( ν − ν′ )

−∑ν,ky

f(Eν)m

δα,β

](1404)

one finds that the diagonal component is zero

Re σx,x(0) = 0 (1405)

but, nevertheless, the off-diagonal term is finite and quantized

Re σx,y(0) =e2

2 π h( n + 1 ) (1406)

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where n is the quantum number for the highest occupied Landau orbital. Asthe field is changed, the peaks in the density of states associated with the Lan-dau levels sweep through the Fermi-level. The Hall resistivity should, therefore,show a set of steps as the applied field is increased. This phenomenon is theinteger quantum Hall effect.

Experimentally, it is found that the steps in σx,y(0) are not discontinuous,but instead show a finite slope in the transition region. Furthermore, the di-agonal component of the resistivity is non-zero, but shows spikes for fields inthe region where the transition between successive plateaus occur. This phe-nomenon is associated with impurity scattering. The effect of impurities is tobroaden the set of delta function peaks in the density of states into a set ofGaussians. This allows the transition between the plateaus to be continuous.In fact, if all the states contributed to the conductivity, the steps of the stair-case would be smeared out into a straight line, just like in the Drude theory forthree-dimensional metals. Fortunately, the states in the tails of each Gaussianare localized, as the deviation from the ideal Landau level energy indicates thatthese states experience a larger impurity potential than average. The large po-tential acts to localize the electrons in the states with energies in the Gaussiantail and so do not contribute to the Hall conductivity. In fact, in two dimen-sions with zero field, one can show that all the states are localized in an infinitesample. However, the samples are finite and have edges. The edges have ex-tended states that carry current. The edge states can be understood in analogywith the classical motion where there are skipping orbits, in which the cyclotronorbits are reflected at the edges. The classical skipping orbits would produceoppositely directed currents at pairs of edges. Quantum mechanically, the bulkstates do not contribute to the current since the velocity operator is given by

vy =py

me− q

me cAy

= ωc ( x − X ) (1407)

and the probability density for the shifted harmonic oscillator is symmetricallypeaked about X. Since the current carried by the state is given by an integralwhich is almost anti-symmetric

jky,ν =q ωc

me

∫ L

0

dx | φν( x − X ) |2 ( x − X ) (1408)

it vanishes. On the other hand for X close to the boundary, say at X = 0,then the wave function must vanish at the boundary for a hard core potentialand so the current is given by

jky,ν =q ωc

me

∫ L

0

dx | φν( x ) |2 x (1409)

Since the wave function is cut off at x = 0, the integral is positive and theedge state carries current. The other edge state carries an oppositely directed

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current. The presence of the confining potential also lifts the degeneracy of thestates in the Landau levels, by increasing the energy of the states close to theboundary. As the wave function of the odd order excited state Landau level ofthe homogeneous system vanish at x = X, one finds that the energy of theLandau levels with hard wall confining potentials increases from ( ν + 1

2 ) h ωc

to ( 2 ν + 32 ) h ωc as X → Lx. The increase in the Hall resistivity only occurs

when the Fermi=level sweeps through the itinerant or delocalized portions ofthe density of states.

As the above calculations completely neglects the effect of impurities andlocalization, Laughlin proposed a gauge theoretic argument which overcomesthese shortcomings (R.B. Laughlin, Phys. Rev. B 23 5632, (1981)). Laughlinenvisaged an experiment in which the two-dimensional sample is in the form ofthe surface of a hollow cylinder of radius R. The axis of the cylinder is takento be along the z direction. A uniform magnetic field Br is arranged to flowthrough the sample in a radial direction. Locally, this field is perpendicularto the plane in which the electrons are confined. A second field is arranged tothread through the cylinder parallel to its axis, but is entirely contained insidethe hollow and falls to zero at r = R. This field does not affect the motion ofthe electrons directly since it is zero inside the sample, however, the associatedflux threading through the cylinder Φ does lead to a finite vector potential AΦ

which satisfies∇ ∧ AΦ = 0 (1410)

which, therefore, can be written in the form

AΦ = ∇ Λ (1411)

This vector potential does cause the electronic wave functions to acquire anAharonov-Bohm phase factor of

exp[iq c

hΛ]

(1412)

For the vector potential

AΦ =Φ

2 π reϕ (1413)

one finds thatΛ =

Φ2 π

ϕ (1414)

and, hence, the phase factor is given by

exp[iq c Φ2 π h

ϕ

](1415)

Thus, on traversing a singly connected path around the cylinders axis, theAharonov-Bohm flux changes the extended state wave function by a factor of

exp[iq c Φ2 π h

2 π]

(1416)

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Since the fundamental flux quantum Φ0 is defined by

Φ0 =2 π hq c

(1417)

this Aharonov-Bohm factor can be written as

exp[i

ΦΦ0

2 π]

(1418)

Thus, if Φ is an integer multiple of Φ0, i.e.

Φ = µ Φ0 (1419)

the extended wave functions are single valued.

The presence of the perpendicular field Br within the sample quantizes themotion into Landau levels. These states may either be localized or may beextended throughout the sample. The vector potential at position z is given by

Br z +Φ

2 π R= µ Φ0 (1420)

which, according to the flux quantization condition, must be an integer multipleof Φ0 for the phase of an extended wave function to be single valued. If Φis adiabatically increased by Φ0, then the maximum the extended states in aLandau level must be translated along the z axis by amounts ∆z

∆z = − Φ0

2 π R Br

= − LΦ0

Φ(1421)

The phase of the localized states can shift by arbitrary amounts. The presenceof the gap forbids excitation of electrons to states in the higher Landau levels.Since, in the pure systems, the fully occupied Landau level contains a number

m =| ∆z |L

=ΦΦ0

(1422)

of electrons, the adiabatic change of Φ results in the transfer of electrons be-tween neighboring extended states. Hence, in the dirty system, all the delocal-ized electrons in the Landau level are translated along the z direction by onespacing, skipping over the localized states. The net result is that one electron istranslated across the entire length of the sample. In the absence of an appliedelectric field, the initial and final states have the same energy. Thus, by gaugeinvariance, adding Φ0 maps the system back on itself. However, if there is anelectric field Ez across the length of the cylinder, this process requires an energychange of

∆E = q Ez L (1423)

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The current around the cylinder Iϕ, from all the electrons in a single Landaulevel, is given by

Iϕ = − c∂E

∂Φ(1424)

which leads to a current density

jϕ = − c

L

q Ez L

Φ0

= − q2

2 π hEz (1425)

Hence, on summing over all occupied Landau levels, one has

jϕ = − q2

2 π hn Ez (1426)

In this n is the number of completely occupied Landau levels with extendedstates and the Fermi-energy is in a mobility gap. The Hall conductivity is givenby

σH = − jyEz

=q2

2 π hn (1427)

Hence, as long as there are Landau levels with extended states, there is an in-teger quantum Hall effect.

The integer quantum Hall effect was measured experimentally by von Klitz-ing in 1980. (K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Letts.494 (1980)). The steps can only be discerned in very clean samples. At muchhigher fields, where only the lowest Landau level should be occupied, Gossard,Stormer, and Tsuei discovered a similar type of effect which is known as thefractional quantum Hall effect (D.C. Tsuei, H.L. Stormer and A.C. Gossard,Phys. Rev. Letts. 48, 1559 (1982)). This phenomenon involves the effect of theCoulomb repulsion between electrons in the Landau levels. Laughlin showedthat the energy of the interacting electron states can be minimized by allowingthe electrons to form a ground state with a different symmetry from the bulk(R.B. Laughlin, Phys. Rev. Letts. 50, 1395 (1983)).

——————————————————————————————————

11.5.2 Exercise 58

Evaluate the Kubo formula for the real part of the diagonal and off diagonalquantum Hall conductivities by first taking the limit ω → 0 and then takingthe limit ε → 0. Also estimate the effects of introducing scattering due torandom impurities. The effect of the scattering lifetime can be introduced byincluding imaginary parts of the energies of the occupied and unoccupied singleparticle states of the form ± i h

2 τ . Choose the signs to ensure that the wave

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functions for the excited states (with an electron - hole pair) will decay to theground state after a time τ . Compare your result with the conductivities ob-tained for the three-dimensional Drude model.

——————————————————————————————————

11.5.3 The Fractional Quantum Hall Effect

Consider a particle of mass me, confined to move in the x − y plane with auniform magnetic field in the z direction. Using the circularly symmetric gauge,the energy eigenstates are also eigenstates of angular momentum. The singleparticle wave functions describing the states in the lowest Landau level withangular momentum m can be written as

φm(rj) =

√2

π m!

(xj + i yj

2 ξ

)m

exp[−

x2j + y2

j

4 ξ2

](1428)

where the length ξ is given by

ξ =

√h c

q Bz(1429)

The probability density for finding a particle has a peak which form a circlearound the origin. The radius of the circle depends on m.

The many-particle ground state wave function corresponding to the lowestLandau level is constructed as a Slater determinant from the states of differentm. The spins of the electrons are assumed to be fully polarized by the appliedfield. The N particle wave function is

Ψ ∼∏i>j

(( xi − xj ) + i ( yi − yj )

) Ne∏k=1

exp[− x2

k + y2k

4 ξ2

](1430)

This satisfies the Pauli exclusion principle as the wave function vanishes linearlyas ri → rj . The linear vanishing is a signature that the a pair of particles arein a state of relative angular momentum m = 1, together with contributionsfrom states of higher angular momentum. This can be seen by expressing theprefactor as a van der Monde determinant

∏i>j

(zi − zj

)=

∣∣∣∣∣∣∣∣∣z01 z1

1 z21 . . . zNe−1

1

z02 z1

2 z22 . . . zNe−1

2...

...z0Ne

z1Ne

z2Ne

. . . zNe−1Ne

∣∣∣∣∣∣∣∣∣ (1431)

Hence, the Ne electrons occupy the zero-th Landau levels single particle stateswith all the angular momentum quantum numbers in the range between m =

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0 and m = Ne − 1. The wave function is an eigenfunction of the totalangular momentum. The total angular momentum about the origin is Mz =Ne ( Ne − 1 )

2 h. Since each m value is occupied, this state corresponds to auniform particle density of

ρ =1

2 π ξ2(1432)

particles per unit area. Hence, this many-particle state corresponds to the com-pletely filled lowest Landau level.

For larger Bz the lowest Landau level is only partially filled. The wavefunction which minimizes the interactions between pairs of particles is given bythe Laughlin trial wave function

Ψp ∼∏i>j

(( xi − xj ) + i ( yi − yj )

)p Ne∏k=1

exp[− x2

k + y2k

4 ξ2

](1433)

for odd integers p. The higher power of p has the effect of minimizing theinteractions between particles, since the square of the wave function vanisheslike a power law with power 2p instead of quadratically. This is a consequenceof the pairs of particles being in states with relative angular momentum p. TheLaughlin state is also an eigenstate of total angular momentum with eigenvalueMz = Ne ( Ne − 1 )

2 p h. Since the linear superposition contains 1 particle forevery p values of m, this state corresponds to a uniform particle density of

ρ =1

2 π ξ2 p(1434)

particles per unit area. The filling factor, ν is defined as

ν = NeΦ0

Φ=

ρ Φ0

Bz(1435)

This state corresponds to a state with the fractional filling, ν = 1p , of the

lowest Landau level. The energy of the Laughlin ground state is given by theCoulomb interaction energy. Since the Coulomb potential is a central potentialit conserves the relative angular momentum. In the Laughlin state, the energyis evaluated as

Eg

Ne= − 0.78213

√p

(1 − 0.211

p0.74+

0.012p1.7

)e2

rc(1436)

where

rc =√

h

me ωc(1437)

The energy per particle, for small p, is lower than any other candidate state byan amount determined by the Coulomb interaction between states with angularmomentum m < p.

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11.5.4 Quasi-Particle Excitations

The quasi-particle excitations of the Laughlin state, like the quasi-particle ex-citations of a completely filled Landau level, can be obtained from consideringthe effect of adding a number of flux quanta, Φ, passing through the center ofthe system. Although the magnetic field generating the extra flux does not acton the electrons at zero, it does add an Aharonov-Bohm phase to the system.First we shall consider the filled nr = 0 Landau level.

The single particle wave function experiences a vector potential of the form

A =[Bz r

2+

Φ2 π r

]eϕ (1438)

where Bz is the uniform field and Φ is the Aharonov-Bohm flux. Since the singleparticle energy eigenstates satisfy[− h2

2 me

1r

∂r

(r∂

∂r

)+

12 me

(− i h

r

∂ϕ− q

c(Bz r

2+

Φ2 π r

))2

− E

]φ(r, ϕ) = 0

(1439)Then, with the ansatz

φ(r, ϕ) =1√2 π

exp[

+ i µ ϕ

]R(r) (1440)

one finds that the radial wave function is given by the solution of[− h2

2 me

1r

∂r

(r∂

∂r

)+

h2

2 me r2

(µ− q Φ

c 2 π h− q Bz r

2

2 h c

)2

− E

]R(r) = 0

(1441)Hence, the solutions for the lowest Landau level are of the form

φ(r, ϕ) ∼ exp[i

q Φc 2 π h

ϕ

]exp

[i ν ϕ

]rν exp

[− r2

4 ξ2

](1442)

whereν = µ − q Φ

c 2 π h(1443)

Since the wave function is single valued µ must be an integer, say m. Thus, onincreasing ν the particles move away from the origin. On subtracting one fluxquantum, Φ0, through the center of the loop, where

Φ0 =c 2 π h

q(1444)

then the degenerate eigenfunctions transform into themselves, m → m + 1. Ifthe Landau level had been completely filled, then one particle has been pushedto the edge of the system and a hole has been created in the m = 0 orbit. This

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is the quasi-hole excitation in the filled Landau level.

The wave function of the Laughlin state, when a quasi-hole has been added isgiven by similar considerations. The insertion of a flux quanta produces an extraAharonov-Bohm phase. The requirement that the wave function is single valuedrestricts µ to be integer, m. The Laughlin state in which the flux is decreased byone flux quantum Φ0 has m shifted by m → m + 1 which creates a quasi-holeat the origin. The many-particle wave function with a quasi-hole at the originis given by the expression

Ψ+p ∼

∏i

( xi + i yi )∏i>j

(( xi − xj ) + i ( yi − yj )

)p Ne∏k=1

exp[− x2

k + y2k

4 ξ2

](1445)

and the wave function with a quasi-hole at r0 is given by

Ψ+p ∼

Ne∏i=1

(xi − x0 + i ( yi−y0 )

) ∏i>j

(( xi − xj ) + i ( yi − yj )

)p Ne∏k=1

exp[− x2

k + y2k

4 ξ2

](1446)

where one flux quanta has also been removed from r0. This state has angularmomentum of

Mz =Ne ( Ne − 1 )

2p h + Ne h (1447)

as there is now an extra zero at point r0. Due to the zero, the charge densityof this state is depleted around r0. The charge deficiency is smaller than thataround the position of any electron by a factor of 1

p . Hence, the quasi-hole hascharge − q

p . Alternatively, one may notice that by adding p quasi-holes at thesame point and then add an electron there, one just obtains the Laughlin wavefunction with one more electron. Hence, p quasi-holes are neutralized by anextra electron. The operator creating a quasi-hole can be written just as

Sp ∼Ne∏i=1

(xi − x0 + i ( yi − y0 )

)(1448)

since it just adds zeros to the wave function.

Creating a quasi-particle is a little more complicated, as adding a flux quan-tum results in the transformation m → m − 1. Hence, the circles contract tothe origin, but the state with m = 0 is already filled in the initial state. Thismust be lifted to the next Landau level, nr = 1. An operator S†p which addsa flux quantum at r0, creating a quasi-particle, can be written just as

S†p ∼Ne∏i=1

(∂

∂xi− i

∂yi− x0 − i y0

ξ2

)(1449)

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where this operator only acts on the polynomial part of the wave function andnot the exponential part. It reduces the angular momentum of each single par-ticle state by one unit of h and sends the particle at r0 into the higher Landaulevels. This activation process ensures that the quasi-particle excitation spectrahas a gap.

Since each quasi-particle of charge qp is attached to one flux quantum, the

statistics are neither fermion nor boson. Two quasi-particles are exchanged inthe process, whereby one quasi-particle is rotated by π in a semi-circle centeredon the other fixed quasi-particle and then the two particles are translated inthe same direction along the diameter. This process results in an interchangeof the electrons between their initial position states. The phase of the wavefunction changes in this permutation. The rotation of the quasi-particle throughπ around a flux tube produces an Aharonov-Bohm phase of

πq

p

Φ0

2 π c h=

π

p(1450)

since the quasi-particle has a fractional charge. Thus, the quasi-particles havefractional statistics.

These types of fractional or anyon statistics is only possible in two or lessdimensions. If the permutation of two particles produces a phase difference of π

pthen the reverse permutation process must yield a phase change of − π

p . In twodimensions, the permutation process and the reversed process are distinguish-able. However, if the two-dimensional process is embedded in three dimensionsthe processes are no longer distinct. On rotating the plane by in which the par-ticles are contained in by π, the interchange becomes equivalent to the reverseinterchange. Hence,

exp[

+ iπ

p

]= exp

[− i

π

p

](1451)

which yields p = 1.

11.5.5 Skyrmions

Although, we have considered the effect of extremely high magnetic fields, theelectronic spin system is not completely polarized, as we have been assuming.The reasons for the relatively weak coupling between the electronic spin and themagnetic field, compared with the coupling of the orbital motion to the field ismainly due to the small band mass of the electron. The strength of the orbitalcoupling is determined by the quantity

q

2 m∗ c(1452)

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where m∗ is the band mass, which in GaAs has the value of m∗ ≈ 0.07 me

where me is the free electron mass. The strength of the spin coupling is givenby

g q

me c(1453)

where g is the gyro-magnetic ratio g ≈ 2. This is further reduced by thestrong spin-orbit coupling in GaAs to a value given by the Lande gl factor,gL ≈ 0.45. The spin directions are, therefore, determined via the exchangeparts of the Coulomb interaction. The magnitude of the Coulomb interactionis given by

q2

ε ξ(1454)

where ε ∼ 12 and ξ is the magnetic length. For fields of order B = 10 Teslathis is the same order of magnitude as h ωc.

The spin magnetization M contributes to the effective magnetic field

Beff = B + 4 π M (1455)

Hence, if we consider excitations in the spin system the magnetization will bespatially varying, and so will the effective field. A variation in the effective fieldwill result in a local change in the filling factor. The system will respond tothe change in the filling factor by transferring charge. Thus, spin and chargeexcitations are coupled. The lowest energy coupled spin-charge excitations areskyrmions, not the Laughlin quasi-particles.

Consider an electron with spin S moving in the exchange field of the otherfixed electrons. The spin degree of freedom is governed by the effective ZeemanHamiltonian

Hint = − g µB Beff (r) . S (1456)

In the lowest energy state, the electron aligns its spin with the static effectivemagnetic field. If the electron is moved around a closed contour, the spin willremain aligned with the local magnetic field all along the contour. However,the spin wave function does not return to its initial value but instead acquires aphase, the Berry phase. The Berry phase is related to the solid angle enclosedby the spins trajectory, as mapped onto the unit sphere in spin space. The solidangle Ω traced out by the spin when completing the contour is given by

Ω =∮

dϕ ( 1 − cos θ ) (1457)

where the spin direction is specified by the polar coordinates (θ, ϕ). After thecontour is traversed the spin wave function acquires an extra phase is S Ω

h .

The Berry phase can be illustrated by considering a spin one half in a mag-netic field of constant magnitude oriented along the direction (θ, ϕ). In this

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case, the Zeeman Hamiltonian is given by

HZ = − µB ( B . σ ) (1458)

which can be expressed as

HZ = − µB B

(cos θ sin θ exp[− i ϕ ]

sin θ exp[ + i ϕ ] − cos θ

)(1459)

which for fixed (θ, ϕ) has an eigenstate of HZ

χ+ =(

cos θ2

sin θ2 exp[ + i ϕ ]

)(1460)

which has the eigenvalueE0 = − µB B (1461)

Thus, in this state the spin is aligned parallel to the applied field. For a staticfield one has the time dependent wave function given by

χ+(t) =(

cos θ2

sin θ2 exp[ + i ϕ ]

)exp

[+ i

µB B

ht

](1462)

where the time dependence is given purely by the exponential phase factor.

If the direction of the field (θ(t), ϕ(t)) is changed very slowly, one expects thespin will adiabatically follow the field direction. That is, if the field is rotatedsufficiently slowly, one does not expect the spin to make a transition to thestate with energy E = + µB B where the spin is aligned anti-parallel to thefield. However, the wave function may acquire a phase which is different fromthe time and energy dependent phase factor expected for a static field. Thisextra phase is the Berry phase δ, and can be calculated from the Schrodingerequation

i h∂

∂t

(α(t)β(t)

)= − µB B

(cos θ(t) sin θ(t) exp[− i ϕ(t) ]

sin θ(t) exp[ + i ϕ(t) ] − cos θ(t)

) (α(t)β(t)

)(1463)

We shall assume that the wave function takes the adiabatic form(α(t)β(t)

)=

(cos θ(t)

2

sin θ(t)2 exp[ + i ϕ(t) ]

)exp

[+ i

(µB B

ht − δ(t)

) ](1464)

which instantaneously follows the direction of the field but is also modified by theinclusion of the Berry phase. On substituting this ansatz into the Schrodingerequation, one finds that the non-adiabatic terms satisfy

− ∂δ

∂t

(cos θ(t)

2

sin θ(t)2 exp[ + i ϕ(t) ]

)+

∂ϕ

∂t

(0

sin θ(t)2 exp[ + i ϕ(t) ]

)

=i

2∂θ

∂t

(− sin θ(t)

2

cos θ(t)2 exp[ + i ϕ(t) ]

)(1465)

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The above equation is projected onto the adiabatic state by multiplying it bythe row matrix (

cos θ(t)2 sin θ(t)

2 exp[ − i ϕ(t) ])

(1466)

One finds that the derivative of θ w.r.t. t cancels and that the equation simplifiesto

− ∂δ

∂t+

∂ϕ

∂tsin2 θ

2= 0 (1467)

Hence, the Berry phase is given by integrating w.r.t. to t,

δ(t) =∫ t

0

dt′∂ϕ

∂t′sin2 θ(t

′)2

=∫

dϕ sin2 θ

2

=12

∫dϕ ( 1 − cos θ ) (1468)

On completing one orbit in spin space, the extra phase is given by

δ =Ω2

(1469)

as was claimed.

Thus, an inhomogeneous effective field on the spin introduces an extra phaseof Ω

2 in the wave function of the electron which is dragged around a contour.This extra phase has the same effect as if the contour contains an additionalcontribution to the magnetic flux of

∆Φ =Ω2

Φ0

2 π(1470)

since encircling a flux quanta Φ0 produces a phase change of 2 π. Furthermore,as the effective flux enclosed in the region is increased by ∆Φ, and the fillingfraction ν is constant

ν = ∆NΦ0

∆Φ(1471)

Hence, the contour encloses an extra charge

∆Q = q ∆N

= ν q∆ΦΦ0

= ν qΩ

4 π(1472)

Thus, the extra charge is determined by the Berry phase and the filling fractionν, also the spin and charge excitations are coupled.

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Due to the coupling of spin and charge, a localized spin-flip excitation of afully polarized ground state introduces a non-uniform charge density. Considera skyrmion excitation in the fully filled lowest Landau level. The ground statewave function is written in second quantized form as

| Ψ0 > =∏m

a†m,↑ | 0 > (1473)

The creation of a charged spin-flip excitation at the origin requires adding adown spin electron in m = 0. However, to allow for the spin excitation to havea finite spatial extent and the charge density to re-adjust, the wave functionneeds to be able to reduce the charge density and net spin at the origin byredistributing them on neighboring shells. The state is also an eigenstate oftotal angular momentum Jz. Thus, to a first approximation the excited statewave function can be written as

| Ψ+ > ≈(v0 + u0 a

†1,↓ a0,↑

)a†0,↓

Ne∏m=0

a†m,↑ | 0 >

≈(v0 a

†0↑ + u0 a

†1,↓

)a†0,↓

Ne∏m6=0

a†m,↑ | 0 >

(1474)

where v0 and u0 are variational parameters which, since the wave function isnormalized, must satisfy

| u0 |2 + | v0 |2 = 1 (1475)

Iterating this process leads to the skyrmion wave function

| Ψ+ > =Ne∏

m=0

(vm a†m↑ + um a†m+1,↓

)a†0,↓ | 0 >

(1476)

where one expects that as m → N then | vm | → 1 and | um | → 0. Thisis a variational wave function for the excited state, and the parameters vm andum are to be determined by minimizing the expectation value of H.

The Hamiltonian can be approximated by

H =∑m,σ

εm,σ a†m,σ am,σ +

12!

∑m,m′

Vm,m′ a†m,↑ a†m′+1,↓ am′,↑ am+1,↓ (1477)

which is a simplified version of the skyrmion Hamiltonian. On using the relations

< Ψ+ | a†m,↑ am,↑ | Ψ+ > = | vm |2

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< Ψ+ | a†m,↑ am+1,↓ | Ψ+ > = v∗m um

< Ψ+ | a†m+1,↓ am,↑ | Ψ+ > = u∗m vm

< Ψ+ | a†m+1,↓ am+1,↓ | Ψ+ > = | um |2

(1478)

the expectation value of the Hamiltonian is found as

< Ψ+ | H | Ψ+ > =∑m

(εm,↑ | vm |2 + εm+1,↓ | um |2

)+

12!

∑m,m′

Vm,m′ v∗m um u∗m′ vm′

(1479)

The energy of this excited state is to be minimized w.r.t. um and vm subject tothe constraint

| vm |2 + | um |2 = 1 (1480)

The minimization is performed using Lagrange’s method of undetermined mul-tipliers, λm. The minimization results in the set of equations

( εm,↑ − λm ) v∗m +12u∗m

∑m′

Vm,m′ um′ v∗m′ = 0 (1481)

and

( εm+1,↓ − λm ) u∗m +12v∗m

∑m′

Vm,m′ vm′ u∗m′ = 0 (1482)

These sets of equations can be solved to yield the undetermined multipliers

λm =(εm↑ + εm+1,↓

2

√ (εm↑ − εm+1,↓

2

)2

+ | ∆m |2 (1483)

where we have defined the parameter

∆m =12!

∑m′

Vm,m′ v∗m′ um′ (1484)

which we expect will decrease with increasing m. The factors | vm |2 and | um |2are then found as

| vm |2 =12

(1 ∓ εm,↑ − εm+1,↓√

( εm↑ − εm+1,↓)2 + 4 | ∆m |2

)| um |2 =

12

(1 ± εm,↑ − εm+1,↓√

( εm↑ − εm+1,↓)2 + 4 | ∆m |2

)(1485)

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Far from the center of the spin flip excitation one expects that the spins will bepolarized parallel to the field and the ground state will be recovered. Hence, weshall use the upper signs. The equations (1484) and (1485) have to be solvedself-consistently for ∆m. We shall use a real solution for the gap. These arecombined to yield the ”gap” equation

∆m =12

∑m′

Vm,m′∆m′√

( εm′↑ − εm′+1,↓)2 + 4 | ∆m′ |2(1486)

The solution uniquely determines the wave function up to an undeterminedphase. We note that as the Zeeman splitting increases, the magnitude of ∆m

decreases.

Once the wave function has been determined, one can examine the spindistribution. The direction of the spin at the point (r, ϕ) is defined as thedirection along which the spin density operator is maximum. The spin densityoperator, projected along a unit vector η in an arbitrary direction (θ′, ϕ′) insecond quantized form is given by

( η . σ )(r) = Ψ†(r) ( η . σ ) Ψ(r) (1487)

However, the field creation and annihilation operators are given by

Ψ†(r) =∑m,α

a†m,α φ∗m(r) χ†α

Ψ(r) =∑m′,β

am′,β φm′(r) χβ

(1488)

Hence, one finds the component of the spin density operator in the form

( η . σ )(r) =∑

m,m′

∑α,β

φ∗m(r) ( η . σ )α,β φm′(r) a†m,α am′,β

=∑

m,m′

φ∗m(r) φm′(r) sin θ′(a†m,↑ am′,↓ exp[ − i ϕ′ ] + a†m,↓ am′,↑ exp[ + i ϕ′ ]

)

+∑

m,m′

φ∗m(r) φm′(r) cos θ′(a†m,↑ am′,↑ − a†m,↓ am′,↓

)(1489)

On taking the expectation value of the spin density operator in the skyrmionstate one finds

< Ψ+ | ( η . σ )(r) | Ψ+ > =∑m

sin θ′ exp[ − i ϕ′ ] φ∗m(r) φm+1(r) um v∗m

+∑m

sin θ′ exp[ + i ϕ′ ] φ∗m+1(r) φm(r) vm u∗m

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+∑m

cos θ′(| φm(r) |2 | vm |2 − | φm+1(r) |2 | um |2

)(1490)

The wave functions can be expressed in planar polar coordinates as

φm(r, ϕ) =1√2 π

exp[

+ i m ϕ

]Rm(r) (1491)

where Rm(r) is real. Hence, we have

< Ψ+ | ( η . σ )(r) | Ψ+ > =∑m

sin θ′ exp[ − i ( ϕ′ − ϕ ) ] Rm(r) Rm+1(r) um v∗m

+∑m

sin θ′ exp[ + i ( ϕ′ − ϕ ) ] Rm+1(r) Rm(r) vm u∗m

+∑m

cos θ′(R2

m(r) | vm |2 − R2m+1(r) | um |2

)(1492)

On maximizing w.r.t ϕ′ one finds

exp[− 2 i ( ϕ′ − ϕ ) ]∑m

Rm(r)Rm+1(r) um v∗m =∑m

Rm+1(r)Rm(r) vm u∗m

(1493)Hence, ϕ′ = ϕ, that is, the in-plane component of the spin is directed radiallyoutwards. On substituting this relation into the wave function, one finds thatthe spin density along this direction simplifies to

< Ψ+ | ( η . σ )(r) | Ψ+ > = sin θ′∑m

Rm+1(r) Rm(r) ( vm u∗m + v∗m um )

+ cos θ′∑m

(| vm |2 R2

m(r) − | um |2 R2m+1(r)

)(1494)

The out of plane component of the spin is determined by θ′. This is found frommaximizing w.r.t. θ′, and leads to

tan θ′ =∑

m( vm u∗m + v∗m um ) Rm+1(r) Rm(r)∑m′

(| vm′ |2 R2

m′(r) − | um′ |2 R2m′+1(r)

) (1495)

At large distances r from the origin, the wave functions are dominated by arange of m values around the value given by

r2 = 2 m ξ2 (1496)

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In this case, one has Rm+1(r) ∼ r2√

m ξRm(r), hence, the out of plane angle

at the distance r from the origin is governed by

tanθ′

2∼ r

2√m ξ

∣∣∣∣ um

vm

∣∣∣∣ (1497)

Since, the ratio decreases with increasing m, the spin direction varies fromθ′ = π at the origin to θ′ = 0 as r → ∞. The texture can be expressedempirically as

tanθ′

2=

λ

r(1498)

where λ expresses the size of the skyrmion. In fact the size of the skyrmion isdetermined by the m variation of um, or more explicitly on the ratio(

εm+1,↓ − εm,↑

∆m

)(1499)

The size of the skyrmion decreases as the magnitude of the Zeeman splittingincreases. This reflects the fact that the energy required to flip the spins in aregion of large spatial extent becomes prohibitively costly as the Zeeman inter-action is increased.

Skyrmions can also be created in the Laughlin state. The skyrmions havelower energy than the Laughlin quasi-particles for all values, of the gyromag-netic ratio g. The energy difference is largest for g → 0. However, as g → ∞the region over which the spin is varying is reduced, and the energy approachesthat of the Laughlin quasi-particle. In fact, in this limit, the skyrmion becomesidentical to the Laughlin quasi-particle.

11.5.6 Composite Fermions

The Laughlin wave function describes states with filling fractions 1p , where p

is odd. However, the sequence of fillings at which the fractional quantum Halleffect is observed is given by the expressions

ν =n

2 n r ± 1(1500)

andν = 1 − n

2 n r ± 1(1501)

where n and r are integers. These two filling fractions are related by approxi-mate electron-hole symmetry, in which occupations of only the lowest Landaulevel are considered.

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These other states can be expressed in terms of composite fermions. Thegeneral Laughlin state can be written as

Ψ2r+1 ∼∏i>j

(( xi − xj ) + i ( yi − yj )

)2r+1 Ne∏k=1

exp[− x2

k + y2k

4 ξ2

]

=∏i>j

(( xi − xj ) + i ( yi − yj )

)2r

×

×∏i>j

(( xi − xj ) + i ( yi − yj )

) Ne∏k=1

exp[− x2

k + y2k

4 ξ2

](1502)

This can be thought of as taking the state in which the lowest Landau level hasthe filling ν = 1 and attaching an even number 2 r flux quanta to each particle.Then, the statistics of the composite particle (composed of the 2 r flux quantaand the electron) will be that of an electron (π) plus a multiple of π for eachflux quantum. Hence, the total exchange phase of the composite particle wavefunction will be

π + 2 r π = ( 2 r + 1 ) π (1503)

and, thus, the composite particle is a fermion.

Consider the state with the general filling factor

ν =n

2 r n ± 1(1504)

If the electrons are attached to 2 r flux tubes, one has composite fermions. Thishas the effect of reducing the magnetic field from Φ to a value Φ∗ given by

Φ∗ = Φ − 2 r Ne Φ0 (1505)

This effective free flux can be either positive or negative. The fractional quantumHall effect of electrons with filling factor ν can be related to the fractionalquantum Hall effect of composite fermions with filling factor ν∗. The relationbetween ν and ν∗ is found by first inverting the definitions

ν =Ne Φ0

Φ

ν∗ =Ne Φ0

Φ∗(1506)

in which Φ∗ and, therefore, ν∗ are assumed to be positive. Then, substitutingΦ∗ and Φ into the relation given by eqn(1505) yields

1ν∗

=1ν− 2 r (1507)

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orν =

ν∗

2 ν∗ r + 1(1508)

Hence, the fractional quantum Hall effect at general fillings can be related to theinteger quantum Hall effect, with integer fillings ν∗ for composite fermions with2 r flux tubes attached to each electron. The expression for the filling fractionswith the minus sign in the denominator can be obtained by considering negativevalues of Φ∗, for which a minus sign has to be inserted into the definition of ν∗

in order to keep the filling fraction positive.

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12 Insulators

The existence of band gaps is a natural consequence of Bloch’s theorem for pe-riodic crystals. However, the existence of band gaps is a much more universalphenomenon, for example it also appear in amorphous materials. In these cases,the existence of band gaps can be traced back to the energy gaps separating thediscrete bound state energies of isolated atoms. When the atoms are broughttogether to form a solid, each electron will be shared with all the atoms in acrystal like in a giant molecule of N atoms. The set of discrete energy lev-els from each of the N atoms, are nearly degenerate. The binding of the Natomic degenerate atomic states into a N molecular states will involve bonding/ anti-bonding splittings that raises the degeneracy. As the energy spread ofthe bonding anti-bonding states are fixed, the levels form a dense set of discretelevels, which can be approximated by a continuous energy band. The separationbetween consecutive bands is roughly determined by the energy separation ofthe discrete levels the isolated atom. Generally, the low energy bands have asmall energy spread, and a clear correspondence with the atomic levels can beestablished. However, the higher energy bands tend to have larger band widths,so the bands overlap and the correspondence with the atomic orbitals becomesmore obtuse.

An example is given by the ionic compound LiF , in which the Li ion losesan electron, and the F ion gains an electron in order that each ion only havecompletely filled atomic shells. The Li 1s and F 1s levels are completely filledand are well separated forming the core levels. The F 2s and 2p levels are alsooccupied, but have broader band widths and form the occupied valence bands.The unoccupied Li 2s and the unoccupied F 3s and 3p levels have large bandwidths which strongly overlap yielding a conduction band which has mixed char-acter. The density of states from the completely filled valence band states areseparated by an energy gap from the completely empty conduction band portionof the density of states. By definition, the Fermi-energy or chemical potentiallies somewhere in the energy gap. The existence of a gap in the density ofstates at the Fermi-energy is the characteristic feature that defines an insulatoror semiconductor.

The distinction between a semiconductor and insulator is only by the mag-nitude of the energy gap between the lowest unoccupied state and the highestoccupied state. In insulators this energy gap is large. In semiconductors thisenergy gap is small, so the electronic properties are determined by the electronicstates close to the bottom of the conduction band and the states close to thetop of the valence band. The density of states close to the band gap can usuallybe parameterized by a few quantities, such as the value of the band gap, andthe effective masses me and mh for the valence and conduction bands. Thisis true, since the discontinuities at the band edges are van Hove singularities.Due to the symmetry one can represent the single particle Bloch energies of the

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valence band and conduction band states as

Ec(p) = Eg +d∑

i=1

ai p2i

Ev(p) = −d∑

i=1

bi p2i (1509)

where the zero of energy was chosen to be at the top of the valence band. Usingthe definition of the effective mass as

1mα,β

=∂2 E(p)∂pα ∂pβ

(1510)

one finds a diagonal effective mass tensor mα,β , which is positive for the con-duction band and negative for the valence band.

There are two types of semiconductors that are frequently encountered. In-trinsic semiconductors and extrinsic semiconductors.

A Intrinsic Semiconductors.

These are pure semiconductors, where the density of states consists of acompletely filled valence band and a completely empty conduction band at T =), which are separated by a band gap Eg. At temperatures comparable to theband gap

Eg ∼ kB T (1511)

a finite number of electrons can be excited from valence band states to theconduction band. The thermal excited electrons in the conduction band areassociated with empty states in the valence band. For each conduction electronthere is one empty valence band state. A hole is defined as the absence of anelectron in a valence band state. Thus, at finite temperatures one has a finitenumber of electron - hole pairs. For materials such as sI or Ge the gap in thedensity of states is of the order of 1 to 0.5 eV. Thus, the number of thermallyactivated electron hole pairs is expected to be extremely small under ambientconditions.

B Extrinsic Semiconductors.

Extrinsic semiconductors are a type of semiconductor that contain impuri-ties. Semiconductors with impurities can have discrete atomic levels that haveenergies which are lower than the empty conduction band and higher than thefull valence band. That is the impurity level lie within the gap.

There are two types of extrinsic or impurity semiconductors.

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N type semiconductors have impurities with levels which at T = 0 would befilled with electrons. At higher temperatures the electrons can be excited fromthe levels into the conduction band. These types of impurities are called donors,since at high temperatures they donate electrons to the conduction band. Anexample of an N type semiconductor is given by semiconducting Si in whichAs impurities are substituted for some Si ions or alternatively semiconductingGe substitutionally doped with P impurities. These are examples of elementalsemiconductors from the IV column of the periodic table doped with impuritiestaken from the V column. Since the impurity ion has one more electron thanthe host material, the host bands are completely filled by taking four electronsfrom each impurity, but the impurity ion can still release one extra electron intothe conduction band.

For Si doped with a low concentration of As impurities, each As impuritycan be considered individually. The As atom contains 5 electrons, while theperfect valence band only contains 4 electrons per site. Thus, the As ion hasone extra electron which, according to the Pauli exclusion principle has to beplaced in states above the valence band. In the absence of the impurity potentialof the ionized As atom, this extra electron would go into the conduction bandand would behave very similarly to a free electron with mass me. However,one must consider the effect of the potential produced by the As+ ion and thespatial correlation that this imposes on the extra electron.

The As+ ions has a positive charge, which affects the free conduction elec-tron much the same way as the positive nuclear charge effects the electron in aH atom. The extra electron becomes bound to the donor atom. In the semi-conductor, the binding energy is extremely low and the radius of the orbit islarge. This can be seen by examining the potential produced by the As+ ion

V (r) = − e2

ε r(1512)

which is screened by the dielectric constant ε. The dielectric constant for si hasa value of about 70. The mass of the electron is the effective mass me. TheBohr radius of the donor orbital is given by

ad(n) =ε h2

me e2n2 (1513)

where n is the principal quantum number. The energy of the donor levels belowthe bottom of the conduction band are given by the expression

Ed(n) = − me e4

2 ε2 h2

1n2

(1514)

The lowest state of the donor level is occupied at T = 0, where the electronis located in an orbit of radius ad(1) ∼ 30 A and the energy of the donorlevel Ed(1) ' − 0.02 eV. Thus, there are shallow impurity levels just below

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the bottom of the conduction band, one of these set are occupied at T = 0K. These discrete levels can be represented by a set of delta functions in thedensity of states. For sufficiently large concentrations these impurities can berepresented in terms of a finite impurity band which appears inside the gap inthe density of states of the pure host material. Since the value of the gap iscomparable to room temperature, one expects that under ambient conditionsthere are a finite number of thermally activated conduction electrons availablefor carrying current.

P type semiconductors have impurities with levels that at T = 0 wouldbe empty of electrons. At higher temperatures, electrons from the filled valenceband will be excited into the empty impurity levels. These thermally excitedlevels will be localized on the impurities ( for small concentrations of impurities). However, the holes present in he valence band allows the valence electrons toconduct electricity and contribute to the properties of the semiconductor. Theimpurities in P type semiconductors are called acceptors as they accept electronsat finite temperatures. Examples of P type semiconductors are Ga impuritiesdoped substitutionally on the sites of a Si host, or Al impurities substitutedfor the atoms in a Ge crystal. These are examples of impurities from the IIIcolumn of the periodic table being substituted for the atoms in a semiconductorcomposed of an element from the IV column of the periodic table. In this casethe type III impurity provides only 3 electrons to the host conduction band,which at finite T contains one hole per Ga impurity.

The Ga impurity atom shares the electrons of the surrounding Si atom andbecomes negatively charged. The extra hole orbits around the negative ion pro-ducing acceptor levels that lie just above the top of the valence band. A finiteconcentration of acceptor levels is expected to produce a smeared out acceptorband just above the top of the valence band. Due to the smallness of the gapbetween the valence band and the acceptor levels, at room temperature an ap-preciable number of electrons can be excited from the valence band onto theacceptor levels.

12.1 Thermodynamics

The thermodynamic and transport properties of semiconductors are governedby the excitations in the filled valence band or empty conduction band, as thefully filled or empty bands are essentially inert. The excitations of electronson the localized impurity levels of extrinsic semiconductors, do not directlycontribute to physical properties. However, the electrons in the conductionband and the valence band are itinerant and do contribute. To develop a theoryof the properties of semiconductors it is convenient to focus attention on thefew unoccupied states of the valence band rather than the many filled states.This is achieved by reformulating the properties in terms of holes.

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12.1.1 Holes

A hole is an unoccupied state in an otherwise completely occupied valence band.The probability of finding a hole in a state of energy E, Ph(E) is given by theprobability that an electron is not occupying that state

Ph(E) = 1 − f(E) (1515)

Thus, as

Ph(E) = 1 − f(E) =1

1 + exp[− β (E − µ)

] (1516)

one finds that the probability of finding a hole in a state of energy E is alsogiven by the Fermi-function except that E → − E and µ → − µ. That isthe energy of the hole is the energy of a missing electron.

The momentum of a hole can be found as the momentum of a completelyoccupied band is zero, since inversion symmetry dictates that for each state withmomentum ke there is a state with momentum −ke with the same energy, andby assumption both are occupied. Then by definition the momentum of a holein ke is that of the full band with one electron missing

kh = 0 − ke (1517)

Thus, the momentum of a hole is opposite to that of the missing electron

kh = − ke (1518)

The charge on the electron is − | e |. The charge on the hole can be obtainedfrom the quasi-classical form of Newton’s laws applied to an electron in anelectric field E,

hdke

dt= − | e | E (1519)

As the momentum of a hole is given by

kh = − ke (1520)

one finds thathdkh

dt= + | e | E (1521)

Thus, the charge of the hole is just | e |. This is consistent with considerationsof electrical neutrality. In a solid in equilibrium, the charge of the nuclei is equalin magnitude to the charge of the electrons. Thus

| e |(Z Nn − Ne

)= 0 (1522)

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Thus, in equilibrium the number of electrons is related to the number of nucleivia Ne = Z Nn. Now if one electron is removed from the valence band andremoved from the solid, there is just one hole. The total charge of the one holestate is given by

Qh = | e |(Z Nn − ( Ne − 1 )

)(1523)

where the number of electrons is now Ne − 1 = Z Nn − 1. Hence,

Qh = | e | (1524)

Thus, the charge on the hole is minus the charge of the electron.

The velocity of a hole vh can be found by considering the electrical currentcarried by the electrons. The electrical current carried by a full band is zero.The electrical current carried by a hole is the electrical current carried by a fullband minus one electron.

jh

= | e | vh = 0 − ( − | e | ve ) (1525)

Thus,vh = ve (1526)

Thus, the velocity of the hole is the same as the velocity of the missing electron.

The above results indicate that the velocity of the hole is in the same di-rection as the velocity of the missing electron, but the momenta have oppositedirections. This implies that the sign of the hole mass should be opposite of themass of the missing electron. This can also be seen by considering the alternateform of the quasi-classical equation of motion

medve

dt= − | e | E (1527)

and sinceve = vh (1528)

and the charge of the hole is | e | one has

− medvh

dt= | e | E (1529)

Hence,mh = − me (1530)

the mass of the hole is the negative of the mass of the missing electron.

The most important number for the thermodynamic and transport proper-ties of a semiconductor is the number of charge carriers. This can be obtainedfrom analysis of the appropriate semiconductor.

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12.1.2 Intrinsic Semiconductors

The number of electrons thermally excited into the conduction band of an intrin-sic semiconductor can be calculated with knowledge of the chemical potentialµ(T ). Consider an intrinsic semiconductor like pure Si which has an energygap Eg of the order of 1 eV. The energy of the top of the valence band shall beset to zero. The energy of the hole is zero at the top of the valence band andincreases downwards, as the energy is that of the missing electron. This agreeswith the fact that mh < 0, as

∂2E

∂2k=

1mh

< 0 (1531)

The energy of the conduction electron is given by

E(k) = Eg +h2 k2

2 me(1532)

The chemical potential can be obtained from considerations of charge neutrality,which implies that the number of conduction electrons is equal to the numberof holes in the valence band. The number of electrons can be calculated fromthe electron density of states

ρ(E) =V

2 π2

(2 me

h2

) 32

( E − Eg )12 for E > Eg (1533)

The number of electrons in the conduction band, Nc is given by

Nc =∫ ∞

Eg

dE ρ(E) f(E) (1534)

The Fermi-function can be expressed in terms of ( E − Eg ) as

f(E) =1

exp[β ( E − Eg )

]exp

[− β ( µ − Eg )

]+ 1

(1535)

If it is assumed that µ = Eg

2 then, as the lowest conduction band state is atE = Eg, the Fermi-function for the conduction electrons is almost classicalsince both exponents are positive. Thus, with the assumption

f(E) ∼ exp[− β ( E − µ )

](1536)

As shall be shown later the above assumption is valid. The number of thermallyactivated conduction electrons can now be obtained by evaluating the integralover the classical Boltzmann distribution by changing variables from E to the

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dimensionless variable x = β ( E − Eg ). Then

Nc ∼ V

2 π2

(2 me kB T

h2

) 32

exp[− β ( Eg − µ )

] ∫ ∞

0

dx x12 exp

[− x

]= 2 V

(2 π me kB T

h2

) 32

exp[− β ( Eg − µ )

](1537)

which is usually expressed in terms of the thermally De Broglie wavelength ofthe conduction electrons λe defined by

λe =h√

2 π me kB T(1538)

as

Ne = 2V

λ3e

exp[− β ( Eg − µ )

](1539)

The number of thermally excited conduction electrons depends exponentiallyon the unknown quantity µ.

The number of holes in the valence band can be found from a similar cal-culation. The occupation number for the holes in the valence band is givenby

Ph(E) = 1 − f(E)

=1

1 + exp[− β (E − µ)

]which is the Fermi-function except that E → − E and µ → − µ. The densityof states for the holes in the valence band is given by

ρ(E) =V

2 π2

(− 2 mh

h2

) 32

( − E )12 for E < 0 (1540)

Since the value of µ is assumed to be positive and of the order of Eg

2 the Fermi-function can also be treated classically. Thus, the number of holes Nv in thevalence band is given by the integral

Nv =∫ 0

−∞dE Ph(E) ρ(E)

=V

2 π2

(− 2 mh kB T

h2

) 32∫ 0

−∞dE

( − E )12

exp[− β ( E − µ )

]+ 1

∼ 2 V(

− 2 π mh kB T

h2

) 32

exp[− β µ

](1541)

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which also depends on µ.

Since in an intrinsic semiconductor the electrons and holes are created inpairs one has

Nc = Nv (1542)

Therefore, the equation for the unknown variable µ is given by

exp[

2 β µ]

=(− mh

me

) 32

exp[β Eg

](1543)

or on taking the logarithm one has

µ =Eg

2+

34kB T ln

∣∣∣∣ − mh

me

∣∣∣∣ (1544)

Hence, at T = 0, the chemical potential lies half way in the gap and onlyacquires a temperature dependence if there is an asymmetry in the magnitudeof the conduction band density of states and the valence band density in thevicinity of the band edge. This result justifies the previous assumption that thedistribution functions can be treated classically. The result also shows that thenumber of thermally activated electrons or holes crucially depends on the gapvia the exponential factor

exp[− β

Eg

2

](1545)

and is extremely small at room temperature for a semiconductor with a gapthat has a magnitude of the order of electron volts.

12.1.3 Extrinsic Semiconductors

In an intrinsic semiconductor the gap in the host material is usually much largerthan the gap between the impurity levels and the band edges. Therefore, underambient temperatures one can neglect one of the bands, as the number of ther-mally excited electrons or holes in that band is small.

For concreteness, consider an N type semiconductor. The bottom of theconduction band is defined as the reference energy E = 0 and the energy ofthe donor levels is defined as − Ed. Let Nd be the number of donor atoms,and nd be the number of electrons remaining in the donor atoms, and nc be thenumber of electrons thermally excited to the conduction band. The importantphysical properties are determined by the number of conduction electrons. Thiscan be calculated from the free energy F .

The free energy F is given in terms of the energy and entropy via the relation

F = E − T S (1546)

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where the total energy isE = − nd Ed (1547)

as the energy of the thermally activated conduction electrons E ∼ 0 as theyoccupy states at the bottom of the band. The entropy S is given by

S = kB lnΩ (1548)

where Ω is the number of possible states of the system. This is just the numberof ways of distributing nd electrons in the 2 Nd states of the donor atoms. Itis assumed that the donor atoms only have a two-fold spin degeneracy, andthat there is no interaction energy between two electrons occupying the twospin states on the same donor atom. With these assumptions, the number ofaccessible states is given by

Ω =( 2 Nd )!

nd! ( 2 Nd − nd )!(1549)

Hence, the Free energy can be calculated in the thermodynamic limit as

F = − Ed nd

− kB T

[2Nd ln 2Nd − nd lnnd − ( 2Nd − nd ) ln( 2Nd − nd )

](1550)

The chemical potential is found from minimizing the free energy with respectto nd

µ =∂F

∂nd

= − Ed − kB T

[− lnnd + ln( 2Nd − nd )

](1551)

Thus, on exponentiating

exp[β (µ + Ed )

]=

nd

2 Nd − nd(1552)

which leads to the number of electrons occupying the donor orbitals as

nd =2 Nd

exp[β ( − Ed − µ )

]+ 1

(1553)

which is just governed by the Fermi-function f(−Ed) and the number of orbitals2 Nd.

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The number of thermally excited conduction electrons is given by

nc = 2 V∫

d3k1

exp[β ( E(k) − µ )

]+ 1

= 2 V(

2 π me kB T

h2

) 32

exp[β µ

](1554)

The unknown chemical potential µ can be eliminated from the above two equa-tions and thereby a relation between nc and nd can be found

nc ( 2 Nd − nd )nd

= exp[− β Ed

]2 V

(2 π me kB T

h2

) 32

(1555)

This is the law of mass action for the dissociation reaction

nd → nc + ( Nd − nd ) (1556)

in which filled donors dissociate into conduction electrons and unoccupied donoratoms. This relation can be written as

nc ( 2 Nd − nd )nd

= exp[− β Ed

]2V

λ3(1557)

where λ is the thermal De-Broglie wavelength,

λ =h√

2 π me kB T(1558)

On using the condition of electrical neutrality

Nd = nd + nc (1559)

one can eliminate nd to find the number of conduction electrons as

nc ( Nd + nc )( Nd − nc )

= exp[− β Ed

]2 Vλ3

(1560)

This is a quadratic equation for nc which can be solved to yield the positiveroot

nc = − 12

(Nd +

2 Vλ3

exp[− β Ed

] )

+12

√(Nd +

2 Vλ3

exp[− β Ed

] )2

+8 Nd V

λ3exp

[− β Ed

](1561)

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With this expression for the number of conduction electrons one can solve for µfrom

µ = kB T ln[nc λ

3

2 V

](1562)

At sufficiently low temperatures when

λ3 Nd

2 V exp

[− β Ed

](1563)

one finds thatµ = − Ed (1564)

as at T = 0 the donor level is only partially occupied as there are 2 Nd orbitalsincluding spin and Nd electrons. At sufficiently high temperatures, the donorlevels are almost completely ionized and

nc ' Nd (1565)

and the chemical potential is found as

µ = kB T ln[Nd λ

3

2 V

](1566)

——————————————————————————————————

12.1.4 Exercise 59

Assume that if the donor levels are localized to such an extent that the coulombrepulsion between two opposite spin electrons in the same donor atom is ex-tremely large. This assumption makes the occurrence of doubly occupied donoratoms extremely improbable. Show that the number of accessible states is givenby

Ω = 2NdNd!

nd! ( Nd − nd )!(1567)

Hence, show that

nd =Nd

1 + 12 exp

[− β ( Ed + µ )

] (1568)

Thus, the interaction affects the statistics of the occupation numbers. Also showthat the law of mass action becomes

nc ( Nd − nd )nd

=V

λ3exp

[− β Ed

](1569)

——————————————————————————————————

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12.2 Transport Properties

Transport in doped semiconductors is mainly due to scattering from donor impu-rities. The potential due to the isolated donor impurities is a screened Coulombinteraction

V (r) = − Z e2

ε rexp

[− kTF r

](1570)

The transport scattering rate is given by

1τ(E(k))

=c

h

∫dk′ k′2 δ( E(k)− E(k′) )

∫d cos θ ( 1− cos θ )

(4 π Z e2

ε ( (k − k′ )2 + k2TF )

)2

(1571)where c is the impurity concentration and θ is the scattering angle. The inte-gration over the magnitude of k′ can be performed over the delta function. Theintegration over the angle can be performed as

1τ(E(k))

=2 m c

h3

1k3

∫d cos θ ( 1 − cos θ )

(4 π Z e2

ε ( 1 − cos θ + ( kT F

k )2 )

)2

(1572)Hence, τ(k) ∼ k3. The conductivity can be evaluated from the formulae

σx,x = − 2 e2

V

∑k

h2

m2k2

x τ(k)∂f(E(k))∂E(k)

=2 e2

V

∑k

h2

m2k2

x τ(k) β exp[− β ( E(k) − µ )

]

∼ β exp[β µ

] ∫dk k7 exp

[− β h2

2 mek2

](1573)

On changing the variable of integration from k to the dimensionless variable

x =h2 k2

2 me kB T(1574)

one finds that the temperature dependence of the conductivity is governed by

σx,x ∼ ( kB T )3 exp[β µ

](1575)

The conductivity is proportional to the electron density and the scattering time.For Coulomb scattering, the average scattering time is proportional to T

32 . On

the other hand, scattering from lattice vibrations gives rise to a conductivitythat is just proportional to exp[ β µ ].

12.3 Optical Properties

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13 Phonons

14 Harmonic Phonons

The Hamiltonian describing the motion of the ions can be formulated by as-suming that the ions move slowly compared with the electrons. Thus, at anyinstant of time, the electrons have relaxed into equilibrium positions and theions are frozen into their instantaneous positions.

It is assumed that the ions have definite mean equilibrium positions andthat the displacements of the ions from these equilibrium positions are small.First, the situation in which the crystal lattice can be described by a Bravaislattice, with a one atom basis is considered. In a later section, the effects of amulti-atom basis will be described. For the case of a one atom basis, the energyof the solid can be calculated in the which the ionic positions are displaced byamounts ui,

r(Ri) = Ri + ui (1576)

where ui is the deviation from the equilibrium position Ri. It is assumed thatthe total energy can be formulated as a constant plus pairwise interactions whichdepends on r(Ri) − r(Rj). The pair-wise interaction is given in terms of thepair-potentials Θ(r(Ri)− r(Rj)) via

V =12

∑i,j

Θ(r(Ri)− r(Rj))

=12

∑i,j

Θ(Ri −Rj + ui − uj) (1577)

The Hamiltonian governing the motion of the ions is just given by the sum ofthe ionic kinetic energies of the atoms of mass M and the pair-wise interactions

H =∑

i

P2

i

2 M+ V (1578)

The harmonic approximation assumes that ui are sufficiently small so that Hcan be expanded in powers of ui. The terms involving ui describe the changein energies of the ions due to the lattice vibrations. The expansion is

H =∑

i

P2

i

2 M+

12

∑i,j

Θ(Ri −Rj)

+12

∑i,j

(ui − uj) . ∇ Θ(Ri −Rj)

+12

12!

∑i,j

[(ui − uj) . ∇

]2Θ(Ri −Rj) + ....

(1579)

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However, since the Ri are equilibrium positions, and the total force on the atomlocated at Ri is given by ∑

j

∇ Θ(Ri −Rj) (1580)

the total force must vanish in equilibrium. Hence, the potential to second orderin u is just given by

V = Veq + Vharmonic (1581)

where the harmonic potential is given by

Vharmonic =14

∑i,j

∑µ,ν

(uµi − uµ

j ) Θνµ(Ri −Rj) (uν,i − uν,j) (1582)

In the above equation, the quantities Θνµ are defined in terms of the second

derivatives of the pair potential

Θνµ(Ri −Rj) = ∇µ ∇ν Θ(Ri −Rj + ui − uj)

∣∣∣∣u≡0

(1583)

The harmonic potential Vharmonic is usually expressed directly in terms of thedisplacements, and not their differences

Vharmonic =12

∑i,j

∑µ,ν

uµi D

νµ(Ri −Rj) uν,j (1584)

where the dynamical matrix is given by

Dνµ(Ri −Rj) = δRi,Rj

∑R”

Θνµ(Ri −R”) − Θν

µ(Ri −Rj) (1585)

In general, when the interactions are not just pairwise, the harmonic potentialcan still be defined via the second derivative of the total energy

Dνµ(R−R′) =

∂2E

∂uµi ∂uν,j

(1586)

For a solid with a mono-atomic Bravais lattice, the dynamical matrixDνµ(R−R′)

possesses the symmetry

Dνµ(R−R′) = Dµ

ν (R′ −R) (1587)

due to the analyticity of the pair potential. Also, one has

Dνµ(R−R′) = Dν

µ(R′ −R) (1588)

which follows as every Bravais Lattice has inversion symmetry. Due to transla-tional invariance, one has the sum rule∑

R

Dνµ(R) = 0 (1589)

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This follows from consideration of the absence of energy change due to a uniformdisplacement of the solid

r(Ri) = Ri + ∆ ∀ i (1590)

Under this displacement, the energy change of the solid is zero, and so

0 =12

∑i,j

∑µ,ν

∆µ Dνµ(Ri −Rj) ∆ν (1591)

for an arbitrarily chosen ∆µ.

Since ui,µ represents the displacement canonically conjugate to Pi,µ, themomentum and displacement operators satisfy the commutation relations

[ ui,µ , Pj,ν ] = i h δi,j δµ,ν (1592)

and[ ui,µ , uj,ν ] = [ Pi,µ , Pj,ν ] = 0 (1593)

To diagonalize the harmonic Hamiltonian, first a canonical transformationis performed to a representation in which the periodic translational invarianceof the lattice is explicit. The displacement is expressed as

ui =1√N

∑q

uq exp[i q . Ri

](1594)

and the momentum operator becomes

P i =1√N

∑q

pq

exp[i q . Ri

](1595)

where N = N1 N2 N3 is the number of unit cells in the crystal. On assumingthat the lattice displacements satisfy Born-von Karman boundary conditions,the momenta are quantized by

q =n1

N1b1 +

n2

N2b2 +

n3

N3b3 (1596)

where ni are integers such that 0 < ni < Ni. In the Fourier transformedbasis, the commutation relations become

[ uq,µ , pq′,ν ] = i h ∆(q + q′) δµ,ν (1597)

and[ uq,µ , uq′,ν ] = [ pq,µ , pq′,ν ] = 0 (1598)

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The quantity ∆(q) is the Dirac delta function, modulo reciprocal lattice vectors,and is defined via ∑

Ri

exp[i q . R

]= N ∆(q) (1599)

This is non-zero if q = Q is a reciprocal lattice vector, and is zero otherwise.In terms of the new coordinates the Hamiltonian becomes

H =∑

q

[ p†q,µ δµ,ν pq,ν

2 M+

12u†q,µ D

µν (q) uq,ν

](1600)

where D(q) is the Fourier Transform of the dynamical matrix

D(q) =∑R

D(R) exp[− i q . R

](1601)

Thus, the harmonic Hamiltonian is diagonal in the quantum number q.

The symmetries of the dynamical matrix D(R) can be used to show that

D(q) =∑i,j

D(Ri −Rj) exp[− i q . ( Ri − Rj )

]

=12

∑R

D(R)

(exp

[− i q . R

]+ exp

[+ i q . R

]− 2

)

= − 2∑R

D(R) sin2

(q . R

2

)(1602)

Thus, D(q) is a real symmetric matrix. Every real 3× 3 symmetric matrix hasthree real eigenvalues, thus, one may find three eigenfunctions

D(q) εα(q) = M ω2α(q) εα(q) (1603)

where εα(q) are the eigenvectors and the eigenvalues have been written asM ω2

α(q). It is necessary that the eigenvalues are positive for the lattice tobe stable.

The eigenvectors are usually normalized, such that

εα(q) . εβ(q) = δα,β (1604)

Since the eigenvectors form a complete orthonormal set, one may expand thedisplacements and the momentum operators in terms of the eigenvectors as

uq =∑α

Qαq εα(q) (1605)

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andp

q=∑α

Pαq εα(q) (1606)

This transformation diagonalizes the Hamiltonian in terms of the polarizationindex α. This can be seen as

u†q D(q) uq =∑α,β

Q† βq εβ(q) D(q) εα(q) Qα

q

=∑α,β

M ω2α(q) Q† β

q εβ(q) . εα(q) Qαq

=∑α

M ω2α(q) Q† α

q Qαq (1607)

and alsop†

q. p

q=∑α

P † αq Pα

q (1608)

Hence, the Hamiltonian is diagonal in the polarization indices α,

H =∑q,α

[P † α

q Pαq

2 M+

M ω2α(q)

2Q† α

q Qαq

](1609)

The Hamiltonian has the form of 3 N independent harmonic oscillators. Theeigenvalues of the Hamiltonian can be found by introducing boson annihilationand creation operators. The annihilation operator is defined by

aq,α =

√M ωα(q)

2 hQα

q + i

√1

2 h M ωα(q)Pα

q (1610)

and the creation operator is the Hermitean conjugate of the annihilation oper-ator

a†q,α =

√M ωα(q)

2 hQ† α

q − i

√1

2 h M ωα(q)P † α

q (1611)

The commutation relations for the boson operators can be calculated from thecommutation relations of Pα

q and Qαq , so

[ aq,α , a†q′,β ] = ∆(q − q′) δα,β (1612)

These are the usual commutation relations for boson creation and annihilationoperators. In second quantized form, the Hamiltonian is expressed as

H =∑q,α

h ωα(q)2

(a†q,α aq,α + aq,α a†q,α

)(1613)

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Due to the commutation relations, one can show that the operator a†q,α whenacting on an energy eigenstate produces an energy eigenstate in which the en-ergy eigenvalue is increased by an amount h ωα(q). Likewise, the annihilationoperator acting on an energy eigenstate produces an energy eigenstate with aneigenvalue which is lower by h ωα(q). If one assumes the existence of a groundstate of the oscillator | 0q,α > such that

aq,α | 0q,α > = 0 (1614)

then the energy eigenvalue of the ground state is just 12 h ωα(q). The excited

states can be found by the raising operator to be ( nq,α + 12 ) h ωα(q), where

nq,α is a positive integer. Thus, the Hamiltonian may be expressed as

H =∑q,α

h ωα(q)(nq,α +

12

)(1615)

This implies that each normal mode (q, α) has quantized excitations. Thesequantized lattice vibrations are known as phonons.

The completeness relation for the polarization vectors is just

εα,µ(q) . εα,ν(q) = δµ,ν (1616)

The original displacements and momenta can be expressed in terms of thephonon creation and annihilation operators via

ui =1√N

∑q,α

√h

2 M ωα(q)εα(q)

(aq,α + a†−q,α

)exp

[i q . Ri

](1617)

and

P i =i√N

∑q,α

√h M ωα(q)

2εα(q)

(a†−q,α − aq,α

)exp

[i q . Ri

](1618)

In the Heisenberg representation, the displacements and momentum operatorsbecome time dependent. The time dependence occurs through factors of

exp[± i ωα(q) t

](1619)

appearing along with the phonon creation and annihilation operators.

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Excited lattice vibration normal modes that resemble classical waves, in thatthey have a definite displacement and definite phase, cannot be energy eigen-states of the Hamiltonian as they are not eigenstates of the number operator.The classical lattice vibrations are best described in terms of coherent stateswhich are a superposition of states each containing large numbers of phonons.The time dependence contained in the exponential phase factors has the effectthat the displacements associated with each excited normal mode oscillate peri-odically with time. In general, the time dependent factors have the effect that,if the eigenvalues of the dynamical matrix are such that ω2

α(q) > 0, the variousexpectation values of the displacements can simply be represented as a sumof periodic oscillations. On the other hand, if ω2

α(q) < 0 the displacementshave unbounded exponential growth, the harmonic approximation fails and thelattice becomes unstable. The discussion will now be restricted to the case ofstable lattice structures.

14.1 Lattice with a Basis

When the lattice has a basis composed of p atoms, the analysis of the phononexcitations is similar. The displacements are labelled by the lattice vector indexi and also by an index l which labels the atoms in the basis. Also, the dynamicalmatrix becomes a 3 p× 3 p matrix and there are 3 p normal modes labelled byα. Thus, one has

uli =

1√N

∑q

∑α

Qαq εlα(q) exp

[i q . Ri

](1620)

The polarization vectors εlα(q) satisfy the generalized ortho-normality condition

l=p∑l=1

εlβ(q)∗ . εlα(q) Ml = δα,β (1621)

where Ml is the mass of the l-th atom in the basis.

14.2 A Sum Rule for the Dispersion Relations

The eigenvalue equation

D(q) εα(q) = M ω2α(q) εα(q) (1622)

can be related to the Fourier transform of the pair potential. The FourierTransform of the dynamical matrix is given by

D(q) =∑R

D(R) exp[− i q . R

](1623)

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Hence, one has∑R

D(R) exp[− i q . R

]εα(q) = M ω2

α(q) εα(q) (1624)

The dynamical operator is given in terms of the pair potential

Dνµ(Ri −Rj) = δRi,Rj

∑R”

∇µ ∇ν Θ(Ri −R”) − ∇µ ∇νΘ(Ri −Rj)

(1625)

and the Fourier Transform of the pair potential is given by

Θ(R) =∑

k

exp[i k . R

]Θ(k) (1626)

On substituting the expression for the Fourier transform of the pair potentialinto the expression for the Fourier transform of the dynamical matrix, one ob-tains

Dνµ(q) =

∑R

j

∑k

kµ kν Θ(k) exp

[i ( k − q ) . ( Ri − Rj )

]

−∑R”

kµ kν Θ(k) exp

[i k . ( Ri − R” )

](1627)

Thus, the phonon frequencies satisfy the eigenvalue equation

M ω2α(q) εα(q) =

∑j

∑k

k Θ(k) exp[i ( k − q ) . ( Ri − Rj )

]k . εα(q)

−∑

j

∑k

k Θ(k) exp[i k . ( Ri − Rj )

]k . εα(q)

(1628)

On making use of conservation of momentum modulo the reciprocal lattice vec-tors Q, one has

M ω2α(q) εα(q) = N

∑Q

( q +Q ) Θ(q +Q) ( q +Q ) . εα(q)

− N∑Q

Q Θ(Q) ( Q . εα(q) ) (1629)

It can be seen that the transverse modes only exist because of the periodicity ofthe lattice. That is, if only the Q = 0 reciprocal lattice vector were included

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in the sum, the eigenvectors would be longitudinal as ε(q) would be parallel toq. In this case,

M ω2α(q) εα(q) = N q Θ(q) q . εα(q) (1630)

which are the longitudinal sound modes. Hence, the transverse modes only ex-its because of the periodicity of the lattice. Furthermore, by letting q → 0one finds that the phonon frequencies M ω2

α(0) must vanish in this limit. Thisprovides an example of the Goldstone mode which occurs because the continu-ous translational symmetry of the Hamiltonian is spontaneously broken at thephase transition when the solid is formed. Goldstone’s theorem may be roughlystated as ”When a continuous symmetry of the Hamiltonian is spontaneouslybroken in a phase transition, a continuous branch of normal modes appearswhich extend to zero energy that dynamically restore the broken symmetry”.Goldstone’s theorem also depends on the condition that long range forces arenot present. If long range forces are present the symmetry restoring mode mayacquire a finite frequency at q = 0 through the Kibble-Higgs mechanism. Inthis case the resulting modes are called Higgs bosons.

The Sum Rule.

The Sum Rule is obtained by using the orthogonality relations for the po-larization vectors. On taking the scalar product of the eigenvalue equation forthe dispersion relation with an eigenvector and summing over the polarizations,one obtains

M∑α

ω2α(q) = N

∑α

∑Q

εα(q) . ( q +Q ) Θ(q +Q) ( q +Q ) . εα(q)

− N∑α

∑Q

εα(q) . ( Q ) Θ(Q) ( Q ) . εα(q)

(1631)

Utilizingεα,µ(q) . εα,ν(q) = δµ,ν (1632)

one finds the sum rule

M∑α

ω2α(q) = N

∑Q

Θ(q +Q) ( q +Q )2

− N∑Q

Θ(Q) ( Q )2 (1633)

for the phonon modes.

——————————————————————————————————

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14.2.1 Exercise 60

Show that if the pair potential is approximated by the non-screened Coulombpotential between the charged nuclei, one finds∑

α

ω2α(q) =

4 π N Z2 e2

M V(1634)

which defines the plasmon frequency for the ions. In the long wave length limit,one has the longitudinal plasmon mode and at most two transverse modes de-pending on the presence of long-ranged order. As the longitudinal plasmonmode saturates the sum rule at q = 0, the transverse modes if they exist mustbe acoustic.

——————————————————————————————————

14.3 The Nature of the Phonon Modes

The long wavelength form of the dynamical matrix can easily be calculated from

D(q) = − 2∑R

D(R) sin2

(q . R

2

)(1635)

as

D(q) = − 12

∑R

D(R)(q . R

)2

(1636)

This implies that ωα(q)2 ∝ q2 as q → 0, which gives rise to acoustic modes.These acoustic modes include the two Goldstone modes as well as the longitu-dinal sound mode, which can be considered as a density fluctuation similar tothe sound waves found in fluids. If the crystal is anisotropic the frequency orsound velocity may depend on the direction of propagation.

In an isotropic solid, there should be one longitudinal and two transversepolarizations (ε ‖ q) and (ε ⊥ q). In an anisotropic solid the relation betweenε and q is not so simple, except at high symmetry points. However, because thepolarization vectors are continuous functions of q, one may still use the termi-nology of longitudinal and transverse polarizations in the vicinity of the highsymmetry points.

The solid has 3 N p degrees of freedom, 3 N of which are tied up in theacoustic phonon branches. The other 3 ( p − 1 ) N modes appear as opticbranches.

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The phonon density of states is given by an integral over the first Brillouinzone, and a summation over the polarization index α.

ρ(ω) = V∑α

∫d3q

( 2 π )3δ( ω − ωα(q) ) (1637)

This can be written as an integral over a surface of constant ωα(q). This surfaceis denoted by Sα(ω) which consists of the points ω = ωα(q) where q is in thefirst Brillouin zone. This yields

ρ(ω) = V∑α

∫Sα(ω)

d2S

( 2 π )31

| ∇ωα(q) |(1638)

The van Hove singularities occur when the group velocity vanishes,

∇ωα(q) = 0 (1639)

The van Hove singularities are usually integrable in three dimensions, but stillgive rise to anomalous slopes or discontinuities in the derivatives of ρ(ω). Anexample of the van Hove singularities in the phonon modes is given by Al whichhas two transverse and one longitudinal contribution to the density of states.

——————————————————————————————————

14.3.1 Exercise 61

Consider a one-dimensional linear chain, with a unit cell composed of two atoms,one with mass M1 and the other with mass M2. The atoms interact with theirnearest neighbors via a harmonic force, with force constant γ. Find the phonondispersion relation.

——————————————————————————————————

14.3.2 Exercise 62

Consider a one-dimensional line of ions, with equal mass, but alternating charges,such that the charge on the p-th ion is

ep = e ( − 1 )p (1640)

Assume that the inter-atomic potential has two contributions:-

(A) A short ranged force between nearest neighbors with a force constantC1 = γ.

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(B) A Coulomb interaction between all the ions Cp = 2 ( − 1 )p e2

p3 a3

where a is the atomic spacing.

(i) Show that

ω(q)2

ω20

= sin2 q a

2+ σ

∞∑p=1

( − 1 )p

p3

(1 − cos q p a

)(1641)

where ω20 = 4 γ

M and σ = e2

γ a3 .

(ii) Show that ω2(q) becomes soft ω2(q) = 0 at q = πa if σ > 4

7 ξ(3).

(iii) Show that the speed of sound becomes imaginary if σ > 12 ln 2 .

Thus, ω2 goes to zero and the lattice becomes unstable for q in the interval(0, π) if σ lies in the range 0.475 < σ < 0.721.

——————————————————————————————————

14.3.3 Exercise 63

Consider a two-dimensional square lattice, with a mono-atomic basis. The atomshave mass M and interact with their nearest neighbors and next nearest neigh-bors through a harmonic force of strength γ1 and γ2, respectively. Calculatethe frequencies of the longitudinal and transverse phonons at q = (π

a ) (1, 0).

——————————————————————————————————

14.3.4 Exercise 64

(i) Show that the linear chain with nearest neighbor (harmonic) interactions hasa dispersion relation

ω(q) = ω0 | sinq a

2| (1642)

and that the density of states is given by

ρ(ω) =2π a

1√ω2

0 − ω2(1643)

which has a van Hove singularity at ω = ω0.

(ii) Show that in three dimensions the van Hove singularities near a maxi-mum of ωα(q) gives rise to a term in the density of states that varies as

ρ(ω) ∝√

ω20 − ω2 (1644)

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and, thus, has a singularity in the first derivative of the density of states withrespect to ω.

——————————————————————————————————

14.3.5 Exercise 65

(i) Show that if the wave vector q lies along a 3 , 4 or 6 fold axis, then onenormal mode is polarized along q and the other two modes are degenerate andpolarized perpendicular to q.

(ii) Show that if q lies in a plane of mirror symmetry, then one mode has apolarization perpendicular to q and the plane, and the other two modes havepolarizations within the plane.

(iii) Show that if q lies on a Bragg plane that is parallel to a plane of mirrorsymmetry, then one mode is polarized perpendicular to the Bragg plane, whilethe other two modes have polarizations laying within the plane.

——————————————————————————————————

14.3.6 Exercise 66

Consider an f.c.c. mono-atomic Bravais Lattice in which the atoms interact viaa nearest neighbor pair potential Θ.

(i) Show that the frequencies of the phonon modes are given by the eigen-values of a 3 × 3 matrix given by

D(q) =∑R

D(R) sin2

(q . R

2

) [A I + B R R

](1645)

where the sum over R runs over the 12 lattice sites closest to the site R = 0,and the constants A and B are given in terms of the pair potential and itsderivatives at the nearest neighbor separation d = a√

2via

A = 2 Θ′(d)/d (1646)

and

B = 2[

Θ”(d) − Θ′(d)/d]

(1647)

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(ii) Show that when q = (q, 0, 0) the longitudinal and transverse acousticphonon frequencies are given by

ωl(q) =

√8 A + 4 B

Msin

q a

4(1648)

and

ωt(q) =

√8 A + 2 B

Msin

q a

4(1649)

(iii) Find the frequencies when q = q√3

(1, 1, 1).

(iv) Show that when q = q√2

(1, 1, 0) that the degeneracy between thetransverse modes is lifted and the frequencies are given by

ωl(q) =

√8 A + 2 B

Msin2 q a

4√

2+

2 A + 2 BM

sin2 q a

2√

2(1650)

and the two transverse modes are

ω1t (q) =

√8 A + 4 B

Msin2 q a

4√

2+

2 AM

sin2 q a

2√

2(1651)

and

ω2t (q) =

√8 A + 2 B

Msin2 q a

4√

2+

2 AM

sin2 q a

2√

2(1652)

——————————————————————————————————

14.3.7 Exercise 67

Consider a phonon with wave vector along the axis of a cubic crystal. Thenconsider the sums in

D(q) =∑R

D(R) sin2

(q . R

2

)(1653)

be restricted to the sites in two planes perpendicular to q separated by a distanceq . R. In metals there exists a long-ranged interaction between the planes

D(q) =∑R

D(R) sin2

(q . R

2

)= A

sin 2kF q . R

2kF q . R(1654)

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where A is a constant.

(i) Find an expression for ω2(q) and∂ω2(q)

∂q .

(ii) Show that∂ω2(q)

∂q is infinite at q = 2 kF . The kink in the dispersionrelation at the Fermi-wave vector is the Kohn anomaly.

——————————————————————————————————

14.4 Thermodynamics

A harmonic lattice has an energy given by

Eharmonic = Ecl

∑q,α

h ωα(q)(nq,α +

12

)(1655)

where Ecl is the ground state energy of the lattice in the classical approximationand free energy F is defined in terms of the partition function as

Z = exp[− β F

]

= exp[− β Ecl

] ∏q,α

nq,α=∞∑nq,α=0

exp[− β h ωα(q) ( nq,α +

12

)]

= exp[− β Ecl

] ∏q,α

(exp[ − 1

2 β h ωα(q) ]1 − exp[ − β h ωα(q) ]

)(1656)

Thus, the free energy is given by

F = Ecl +∑q,α

h ωα(q)2

+ kBT∑q,α

ln(

1 − exp[− β h ωα(q)

] )(1657)

The pressure P is found from the thermodynamic relation

dF = dE − T dS − S dT

dF = − S dT − P dV (1658)

Hence, the pressure is given by

P = −(∂F

∂V

)T,N

= −(∂Ecl

∂V

)T,N

− 12

∑q,α

(∂hωα(q)∂V

)T,N

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−∑q,α

(∂hωα(q)∂V

)T,N

1

exp[β h ωα(q)

]− 1

(1659)

The first two terms are temperature independent, and the last term depends ontemperature through the average phonon occupation numbers. The pressure isonly temperature dependent if the phonon frequencies depend on the volume V .

The thermal volume expansion coefficient α is defined by

α =1V

(∂V

∂T

)P

(1660)

As the equation of state is a relation between pressure, temperature and volume

P = P (T, V ) (1661)

Thus, the infinitesimal derivatives are related by

dP =(∂P

∂T

)V

dT +(∂P

∂V

)T

dV (1662)

For a process at constant P , dP = 0, thus

(∂V

∂T

)P

= −

(∂P∂T

)V(

∂P∂V

)T

(1663)

The bulk modulus, B defined by

B = − V

(∂P

∂V

)T

(1664)

is finite as Ecl is expected to be volume-dependent. Hence, the denominator isfinite. The thermal expansion coefficient is non-zero, only if (∂P

∂T )V 6= 0. Usingthe harmonic approximation, the frequencies must be functions of the volumeV if the solid is to undergo thermal expansion.

The specific heat at constant pressure is different from the specific heat atconstant volume. This relation is found by relating the temperature derivative ofthe entropy with respect to temperature, at constant volume to the temperaturederivative of the entropy with respect to temperature, at constant pressure. Thisrelation is found by considering the infinitesimal change in entropy, with eitherthe change in volume or pressure

dS =(∂S

∂T

)V

dT +(∂S

∂V

)T

dV

=(∂S

∂T

)P

dT +(∂S

∂P

)T

dP (1665)

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Using the equation of state relating P and V

V = V (T, P ) (1666)

to find

dV =(∂V

∂T

)P

dT +(∂V

∂P

)T

dP (1667)

Thus, on combining the expression for dS in terms of dV and dT and theequation for dV results in the expression

dS =[ (

∂S

∂T

)V

+(∂S

∂V

)T

(∂V

∂T

)P

]dT +

(∂S

∂V

)T

(∂V

∂P

)T

dP

(1668)

Thus, one has the relation(∂S

∂T

)P

=(∂S

∂T

)V

+(∂S

∂V

)T

(∂V

∂T

)P

(1669)

A Maxwell relation can be used to eliminate(

∂S∂V

)T. The Maxwell relation comes

from the analyticity condition on a thermodynamic function with independentvariables (T, V ), which is F (T, V ). Hence, one has(

∂S

∂V

)T

=(∂P

∂T

)V

(1670)

and, thus,

T

(∂S

∂T

)P

= T

[ (∂S

∂T

)V

+(∂P

∂T

)V

(∂V

∂T

)P

]CP = CV + T

(∂P

∂T

)V

(∂V

∂T

)P

CP = CV − T

(∂P∂T

)2

V(∂P∂V

)T

(1671)

Thus, if there is to be a difference between CP and CV the phonon frequenciesmust be dependent on V .

14.4.1 The Specific Heat

The specific heat at constant volume can be found from the entropy of thephonon gas

S = kB

∑q,α

[ (N(ωα(q)) + 1

)ln(N(ωα(q)) + 1

)−N(ωα(q)) ln N(ωα(q))

](1672)

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The specific heat is given by

CV = T

(∂S

∂T

)V

= kB

∑q,α

∂N(ωα(q))∂T

ln

(N(ωα(q)) + 1N(ωα(q))

)

=∑q,α

∂N(ωα(q))∂T

h ωα(q)

= kB

∑q,α

(β h ωα(q)

)2

N(ωα(q))[N(ωα(q)) + 1

](1673)

This can be expressed as an integral over the phonon density of states ρ(ω) via

CV = kB

∫ ∞

0

dω ρ(ω)(β h ω

)2

N(ω)[N(ω) + 1

](1674)

Now some approximate models of the specific heat of the lattice vibrations shallbe examined.

14.4.2 The Einstein Model of a Solid

The Einstein model of a solid considers the phonons to have a constant frequencyω0, and is an approximate representation of the optic phonons. The phonondensity of states is given by

ρ(ω) = 3 N δ(ω − ω0) (1675)

where there are 3 modes per atom. The specific heat is given by

CV = 3 N kB

(β h ω0

)2

N(ω0)[N(ω0) + 1

](1676)

This vanishes exponentially at low temperatures, kB T h ω0 where

N(ω0) ≈ exp[− β h ω0

](1677)

and at high temperatures kB T h ω0

N(ω0) ≈ kB T

h ω0(1678)

so the specific heat saturates to yield the classical result

limT → ∞

CV → 3 N kB (1679)

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The Einstein model of the specific heat fails to describe the lattice contributionto low-temperature specific heat of a solid. This is because it fails to describethe low energy acoustic phonon excitations, which gives rise to a power lawtemperature variation. The Debye model of a solid provides an approximatedescription of the low-temperature specific heat of a solid.

14.4.3 The Debye Model of a Solid

The Debye model of a solid approximates the phonon density of states for thetwo transverse acoustic mode and longitudinal acoustic mode in a an isotropicsolid. The dispersion relations of the phonon modes are represented by thetwo-fold degenerate transverse mode

ωT (q) = vT q (1680)

and the singly degenerate longitudinal mode

ωL(q) = vL q (1681)

The transverse sound velocity, in general, will be different from the longitudinalsound velocity, vL 6= vT . There are 3N such phonon modes in the first Brillouinzone. The phonon density of states is given by the integral over a surface areain the first Brillouin zone

ρ(ω) =V

( 2 π )3∑α

∫d2Sα(ω)

(dωα

dq

)−1

(1682)

If the Brillouin zone is approximated as a sphere of radius qD, then the densityof states is given by

ρ(ω) =V

( 2 π )3∑α

4 π q2(dωα

dq

)−1

(1683)

for q < qD. Using the form of the dispersion relation, the density of states canbe re-written as

ρ(ω) =V

( 2 π2 )

∑α

ω2

v3α

Θ( vα qD − ω ) (1684)

This can be further approximated by requiring that the upper limit on thefrequency of all three phonon modes be set to the Debye frequency ωD. In thiscase the density of states is simply given by

ρD(ω) =V

( 2 π2 )

∑α

ω2

v3α

Θ( ωD − ω ) (1685)

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The value of the Debye frequency is determined by the condition∫ ωD

0

dω ρD(ω) = 3 N

=V

( 6 π2 )

∑α

ω3D

v3α

(1686)

Using this condition, the Debye density of states is written as

ρD(ω) = 9 Nω2

ω3D

Θ( ωD − ω ) (1687)

Thus, the Debye density of states varies as ω2 at low frequencies and has a cutoff at the maximum frequency ωD.

The temperature dependence of the specific heat of the Debye model is givenby

CV =9 N kB

ω3D

∫ ωD

0

dω ω2

(β h ω

)2

N(ω)[N(ω) + 1

](1688)

The asymptotic low-temperature variation of the specific heat can be found bychanging variable x = β hω. The specific heat can be written in the form

CV = 9 N kB

(kB T

h ωD

)3 ∫ xD

0

dx x4 exp[ x ](exp[ x ] − 1

)2 (1689)

where the upper limit of integration is given by xD = β hωD. At sufficientlylow temperatures, kB T h ωD, the upper limit may be set to infinity yielding

CV = 9 N kB

(kB T

h ωD

)3 ∫ ∞

0

dx x4 exp[ x ](exp[ x ] − 1

)2

=12 π4

5N kB

(kB T

h ωD

)3

(1690)

where the integral has been evaluated as 4 π4

15 . Thus, the low-temperature spe-cific heat varies as T 3 in agreement with experiment. The asymptotic hightemperature specific heat, kB T h ωD is found from

CV =9 N kB

ω3D

∫ ωD

0

dω ω2

(β h ω

)2

N(ω)[N(ω) + 1

](1691)

noting that the number of phonons is given by N(ω) = kB Th ω , so

CV =9 N kB

ω3D

∫ ωD

0

dω ω2

= 3 N kB (1692)

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which is the classical limit. Thus, the Debye approximation provides an interpo-lation between the low-temperature limit and the high temperature limit, whichis only governed by one parameter, the Debye temperature kB TD = h ωD.

——————————————————————————————————

14.4.4 Exercise 68

Evaluate the integral ∫ ∞

0

dx x4 exp[ x ](exp[ x ] − 1

)2 (1693)

needed in the low-temperature limit of the specific heat for the Debye model.

——————————————————————————————————

14.4.5 Exercise 69

Generalize the Debye model to a d-dimensional solid. Determine the high tem-perature and leading low-temperature variation of the specific heat due to latticevibrations.

——————————————————————————————————

14.4.6 Exercise 70

Show that the leading high temperature correction to the Dulong and Petitvalue of the specific heat due to lattice vibrations is given by

∆CV

CV= − 1

12

∫dω

(h ω

kB T

)2

ρ(ω)∫dω ρ(ω)

(1694)

Also evaluate the moment of the phonon density of states∫dω ω2 ρ(ω) (1695)

in terms of the pair potentials between the ions.

——————————————————————————————————

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14.4.7 Exercise 71

Numerically calculate the phonon density of states for a single phonon mode fora two-dimensional lattice with a dispersion

ω2(q) = ω20

(2 − cos qxa − cos qya

)(1696)

and hence obtain the temperature dependence of the specific heat. Comparethis with the numerical evaluation of an appropriate Debye model.

——————————————————————————————————

14.4.8 Lindemann Theory of Melting

Lindemann assumed that a lattice melts when the displacements due to latticevibrations becomes comparable to the lattice constants. Although this theorydoes not address the appropriate mechanism it does give the right order ofmagnitude for simple metals and transition metals. It is shall assumed thatmelting occurs at a critical value of the ratio

γc =u2

i

a2i

(1697)

This determines the estimated melting temperature through

γc =h

2 M N a2

∑q,α

2 nq,α + 1

ωα(q)(1698)

The right hand side can easily be evaluated in two limits, the zero temperaturelimit and the high temperature limit. At zero temperature, the relative meansquared displacement is given by

γ =h

2 M N a2

∑q,α

1vα q

=3 h

2 M N v a2

V

( 2 π )3

∫d3q

1q

=3 h

2 M N v a2

V

( 2 π )32 π q2D (1699)

Using the relation

N =V q3D6 π2

=3 V

4 π a3(1700)

one obtains

γ ∼ 12

(9

2 π2

) 13 h qDM v

∼ 0.4h qDM v

= 0.4kB TD

M v2(1701)

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At high temperatures, the relative mean squared displacement is given by

γ =h

2 M N a2

V

2 π2

∫ qD

0

dq q26 kB T

h v2 q2

=3

M a2

V

2 π2 N

kB T qDv2

=9

M a2 q2D

kB T

v2=(

36π2

) 13 kB T

M v2

∼ 1.54kB T

M v2= 1.54

h2 q2DM k2

B T 2D

kB T (1702)

The Lindemann criterion provides a relation between the Debye temperatureand the melting temperature. The experimental data for alkaline metals andtransition metals suggest that γc has a value of 1

16 , independent of the metal.Of course, one expects that anharmonic effects may become important for largedisplacements of the ions from their equilibrium positions.

Mermin-Wagner Theorem.

The Lindemann theory of melting may be extended to provide an example ofthe Mermin-Wagner theorem. The Mermin-Wagner theorem states that finitetemperature phase transitions in which a continuous symmetry is spontaneouslybroken cannot occur in lower than three dimensions (N.D. Mermin and H. Wag-ner, Phys. Rev. Letts. 17, 1133 (1966)). Basically, if such a transition occursthen there should be a branch of Goldstone modes that dynamically restoresthe spontaneously broken symmetry. These normal modes produce fluctuationsin the order parameter. In a periodic solid where continuous translational in-variance is broken, the Goldstone modes are the transverse sound waves. Thetransverse sound modes have dispersion relations of the form ω(q) = v q. Thefluctuations in the order parameter are the fluctuations in the choice of originof the lattice and, therefore, are just the fluctuations in positions of any oneion. In d dimensions, at finite temperatures, the fluctuations have contributionsfrom the region of small q which are proportional to

u2i ∼

∫ qD

0

ddqh

2 M ω(q)kB T

h ω(q)

∼∫ qD

0

dq qd−3 (1703)

where qD is a cut off due to the lattice. The integral diverges logarithmically ford = 2, indicating that the fluctuations in the equilibrium lattice positions willbe infinitely large, thereby preventing the solid from being formed. Likewise,for lower dimensions, such as one dimension, the integral will also diverge atthe lower limit. Therefore, no truly one-dimensional solid is stable against tem-perature induced fluctuations. An analysis of the zero point fluctuations alsorules out the possibility of a one-dimensional lattice forming in the limit of zero

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temperature.

For a harmonic solid, the phonon frequencies are independent of the volumeV . This can be seen by considering the energy of a solid which has expandedin the linear dimensions by an amount proportional to ε.

The energy of a harmonic solid with static displacements about the originalequilibrium position is given by the harmonic expression

E = Eeq +12

∑i,j

ui D(Ri −Rj) uj (1704)

Now consider the expanded lattice, in which the displacements are given by

ui = ε Ri + ui (1705)

Here ui are the new displacements from the new lattice sites of the lattice whichhas undergone an increase in volume of ( 1 + ε )3, through the application ofexternal forces. The expanded solid has an energy given by

E = Eeq +ε2

2

∑i,j

Ri D(Ri −Rj) Rj +12

∑i,j

ui D(Ri −Rj) uj

(1706)

The terms linear in ε vanish identically, as the total force on an ion must vanishin equilibrium. The total force is the sum of the internal forces opposing theexpansion and the applied external forces that result in the expansion. Sincethe dynamical matrix that governs the lattice displacements ui is unchanged,its eigenvalues which are the phonon frequencies are unchanged by expansion ofa harmonic solid.

Thermal expansion only occurs for an anharmonic lattice. Thermal expan-sion provides a measure of the volume dependence of the phonon frequencies∂ h ω∂ V or the anharmonicity.

14.4.9 Thermal Expansion

The coefficient of thermal expansion of an insulator can be evaluated from(∂V

∂T

)P

= −(∂P

∂T

)V

/ (∂P

∂V

)T

(1707)

where the pressure is found from P = −(

∂F∂V

)T

. Using the expression for the

free energy of the lattice one finds that the coefficient of thermal expansion can

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be written as

α = − 1B

∑q,α

(∂ h ωα(q)

∂V

) (∂N(ωα(q))

∂T

)(1708)

where B is the bulk modulus and N(ω) is the Bose-Einstein distribution func-tion.

The specific heat can be written as

CV =∑q,α

h ωα(q)(∂N(ωα(q))

∂T

)(1709)

On identifying the contributions from each normal mode, one can define aGruneisen parameter for each normal mode

γα(q) = − V

ωα(q)

(∂ ωα(q)∂V

)(1710)

which is a dimensionless ratio of α B VCV

. Thus,

γα(q) = −

(∂ lnωα(q)∂ lnV

)(1711)

The Gruneisen parameter for the entire solid can be expressed as a weightedaverage of the Gruneisen parameter of each normal mode

γ =

∑q,α γα(q) Cq,α∑

q,α Cq,α(1712)

with weights given by Cq,α. This is consistent with the definition of the Gruneisenparameter in terms of thermodynamic quantities

γ =α B V

CV(1713)

For most models γα(q) is roughly independent of T and is a constant.

γα(q) ∼ γ = −

(∂ lnωD

∂ lnV

)(1714)

Hence, as B is roughly T independent, the specific heat CV tracks the coefficientof thermal expansion α. A typical Gruneisen parameter has a magnitude of ∼1 or 2, and a slow temperature variation, which changes on the scale of TD.

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14.4.10 Thermal Expansion of Metals

For a metal, there is an additional contribution to the pressure from the elec-trons. The electronic contribution to the pressure is calculated as

Pel =23Eel

V

V(1715)

and as the electronic energy is temperature dependent, the electronic contribu-tion to the pressure is also temperature dependent. This gives an additionalcontribution to the rate of change of pressure with respect to temperature(

∂Pel

∂T

)=

23Cel

V (1716)

Hence, the coefficient of thermal expansion for a metal is determined from

(∂V

∂T

)P

= −

(∂P∂T

)V(

∂P∂V

)T

(1717)

Hence,

α =1B

(γ Clatt

V +23Cel

V

)(1718)

where 23 is the electronic Gruneisen parameter.

14.5 Anharmonicity

The anharmonic interactions give rise to the lifetime of phonons, provide tem-perature dependent corrections to the phonon dispersion relations. These mayusually be thought of as producing small corrections to the harmonic phonons,except when the systems is on the verge of a structural instability where theyplay an important role. The phonon modes are not the only excitations of thecrystalline lattice, there are also large amplitude excitations like dislocations.Although these excitations may have a large ( macroscopic ) spatial extent theydo not extend all through the crystal, like the phonon modes, and the deviationsof the atoms from the ideal equilibrium positions can be large, comparable tothe lattice spacing. If the lattice displacements in the dislocations were consid-ered to be made up of a superposition of coherent states for each phonon mode,in the absence of the anharmonic interactions, the distortions would disperseand the dislocations would lose their shape. The anharmonic interactions areresponsible for stabilizing these large amplitude, spatially localized, excitationsby balancing the effects of dispersion of the phonon modes. These excitationdo have macroscopically large excitation energies but they do also have macro-scopically large effects. In essence, these dislocations are non-linear excitations,

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like solitons, and play an extremely important role in determining the actualmechanical properties of any real solid.

——————————————————————————————————

14.5.1 Exercise 72

The full ionic potential of a mono-atomic Bravais Lattice has the form

Veq +12!

∑R,R′

∑µ,ν

uµ(R) Dµ ν(R−R′) uν(R′)

+13!

∑R,R′,R”

∑µ,ν,λ

uµ(R) uν(R′) uλ(R”) Dµ ν λ(R,R′, R”)

(1719)

where u(R) gives the displacement from the equilibrium position R.

(i) Show that if an expansion is made about the expanded lattice positionsdefined by

R = ( 1 + ε ) R (1720)

then the dynamical matrix is changed to

Dµ,ν(R−R′) = Dµ,ν(R−R′) + ε δDµ,ν(R−R′) (1721)

where the change in the dynamical matrix is given by

δDµ,ν(R−R′) =∑λ,R”

Dµ ν λ(R,R′, R”) Rλ” (1722)

(ii) Show that the Gruneisen parameter is given by

γα(q) =εα(q) δD(q) εα(q)

6 M ω2α(q)

(1723)

——————————————————————————————————

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15 Phonon Measurements

The spectrum of phonon excitations in a solid can be measured directly, viainelastic neutron scattering or Raman scattering of light.

15.1 Inelastic Neutron Scattering

The neutrons interact with the atomic nuclei by a very short ranged contactinteraction

Hint =∑

i

2 π h2

mnb δ3( r − r(Ri) ) (1724)

where r is the position of the neutron, and r(Ri) are the positions of the ions.The inelastic neutron scattering cross-section contains information about theground state and the all the excited states of the lattice. The various contribu-tions to the spectrum are analyzed by use of the conservation laws.

In inelastic neutron scattering experiments the incident neutron energy isgiven by

E =P 2

2 mn=

h2 k2

2 mn(1725)

and the final energy is given by

E′ =P ′2

2 mn=

h2 k′2

2 mn(1726)

The energy transfer from the neutron to the sample is given by

h ω = E − E′ (1727)

This energy is the energy given to the excited phonon modes

h ω =∑q,α

h ωα(q) ( n′q,α − nq,α ) (1728)

as found from conservation of energy.

Due to the periodic translational invariance of the crystal and the shortrange of the interaction, the momentum change of the neutron is given by

p − p′ = h k − h k′ =∑q,α

h q

(n′q,α − nq,α

)+ h Q (1729)

where Q is a reciprocal lattice vector. Thus, even if the scattering is elastic theneutron may still be diffracted. The use of the two conservation laws allows thedispersion relation ωα(q) to be determined.

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15.1.1 The Scattering Cross-Section

The inelastic scattering cross-section d2σdΩdω depends on the scattering geometry,

through the scattering angle θ and dΩ is the angle subtended by the detector tothe target. The inelastic neutron scattering cross-section is given by the Fermi-Golden rule expression. The Fermi-Golden rule is derived in the interactionrepresentation, and involves an integral over time of the expression with thematrix elements

1h2 Re

[ ∫ + ∞

−∞dt′ <

∏q,α

nα,q | < k | Hint | k′ > |∏q,α

n′α,q > ×

<∏q,α

n′α,q | < k′ | exp[

+i t′

hH0

]Hint exp

[− i t′

hH0

]| k > |

∏q,α

nα,q >

](1730)

The matrix elements involve the initial and final states each of which are prod-ucts of the neutron states k and the states of the lattice. The initial and finalstates of the lattice are represented by the number of quanta in each normalmode and respectively are written as |

∏q,α nα,q > and |

∏q,α n′α,q >. The

dependence on the states of the lattice are suppressed in the following. The in-tegration over t′ gives rise to an energy conserving delta function. This involvesthe matrix elements of the interaction between the neutron and the nuclei inthe solid, but unlike the elastic scattering cross-section, previously derived, thenuclei may be displaced from their equilibrium positions by ui according to

r(Ri) = Ri + ui (1731)

Hence, the interaction Hamiltonian is given by

Hint =∑

i

2 π h2

mnb δ3( r − Ri − ui ) (1732)

and the matrix elements between the initial and final states of the neutron,respectively, labelled by momentum k and k′, u,

< k′ | Hint | k > =∑

i

2 π h2

mnb exp

[i ( k − k′ ) . ( Ri + ui )

](1733)

If the displacements ui have sufficiently small magnitudes, compared with theneutron wave length, or the lattice spacing, the exponential term can be ex-panded as a series in powers of ui,

< k′ | Hint | k > ≈∑

i

2 π h2

mnb exp

[i ( k − k′ ) . Ri

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×

(1 + i ( k − k′ ) . ui + ....

)(1734)

The first term in the expansion has been previously considered in the discussionof neutron diffraction by a crystalline lattice, and gives rise to Bragg scattering.The next term gives rise to single phonon scattering, while the higher orderterms represent scattering from multi-phonon excitations. In the interactionrepresentation, the terms involving the lattice displacements depend on time,via

Hint(t′) = exp[

+i H0 t

h

]Hint(0) exp

[− i H0 t

h

](1735)

Thus, on expressing the lattice displacements in terms of the phonon modes as

ui(t′) =

1√N

∑q,α

√h

2 M ωα(q)εα(q) exp

[i q . Ri

×(aq,α exp[ − i ωα(q) t′ ] + a†−q,α exp[ + i ωα(q) t′ ]

)(1736)

where ωα(q) are the phonon frequencies. The integrals over t′ can be expressedin terms of energy conserving delta functions∫ ∞

−∞

dt′

2 π hexp

[i t′

h( E′ − E ± h ω )

]= δ( E′ − E ± h ω )

(1737)

On evaluating the matrix elements between the initial and final states of thelattice, the scattering cross-section is found as the sum of terms

d2 σ

dΩ dω∝ k

k′

(2 π h2

mnb

)2[ ∣∣∣∣ ∑

i

exp[i ( k − k′ ) . Ri

] ∣∣∣∣2 δ( E′ − E )

+1N

∑q,α

[n−q,α + 1

]h

2 M ωα(q)δ( E′ − E + h ωα(q) ) ×

×∣∣∣∣ ∑

i

exp[i ( k + q − k′ ) . Ri

]( k − k′ ) . εα(q)

∣∣∣∣2+

1N

∑q,α

nq,αh

2 M ωα(q)δ( E′ − E − h ωα(q) )×

×∣∣∣∣ ∑

i

exp[i ( k + q − k′ ) . Ri

]( k − k′ ) . εα(q)

∣∣∣∣2

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+ . . . . .

](1738)

The displacements have been expressed in terms of the normal modes, and nq,α

is just the number of phonons with momentum q and polarization α in the initialstate. On performing the sum over lattice vectors Ri, one finds the conditionfor conservation of crystal momentum,∑

i

exp[i ( k − k′ − q ) . Ri

]= N ∆

(k − k′ − q

)(1739)

modulo Q. Thus, the summation over q can be performed, leading to the secondand third terms involves the absorption or emission of a phonon of wave vectorq = (k−k′+Q) where Q is a reciprocal lattice vector. These terms are smallerthan the coherent Bragg terms by a factor of 1

N .

d2 σ

dΩ dω∝ k

k′

(2 π h2

mnb

)2[N2 ∆( k − k′ ) δ( E′ − E )

+ N∑q,α

∆( k − k′ − q ) δ( E′ − E + h ωα(q) ) ×

×[nq,α + 1

]h

2 M ωα(q)

∣∣∣∣ ( k − k′ ) . εα(q)∣∣∣∣2

+ N∑q,α

∆( k − k′ + q ) δ( E′ − E − h ωα(q) )×

× nq,αh

2 M ωα(q)

∣∣∣∣ ( k − k′ ) . εα(q)∣∣∣∣2

+ . . . . .

](1740)

The inelastic one phonon contributions are coherent as it involves the conserva-tion of momentum, but has an intensity that is only proportional to N .

The energy of the phonon is given by h ωq,α. The thermal average of thenumber of phonons nq,α is given by the Bose Einstein distribution function,which is a Boltzmann weighted average

N(ωα(q)) = nq,α

=1

Zq,α

n=∞∑n=0

n exp[ − β n h ωα(q) ]

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=1

Zq,α

exp[ − β h ωα(q) ](1 − exp[ − β h ωα(q) ]

)2

(1741)

However, the partition function for a single phonon mode Zq,α is given by

Zq,α =n=∞∑n=0

exp[ − β n h ωα(q) ]

=1

1 − exp[ − β h ωα(q) ]

(1742)

Hence, the Bose-Einstein distribution function is found to be

N(ωα(q)) =1

exp[ β h ωα(q) ] − 1(1743)

At low temperatures, the number of thermally activated bosons is small, there-fore, the inelastic scattering intensity for processes which lead to an increase inthe energy of the neutron, due to absorption of phonons is small. On the otherhand, the intensity of processes which involves the energy loss by the neutronbeam due to creation of individual phonons has an intensity governed by the1 + N(ωα(q)) which is almost unity at low temperatures. The rate for inelas-tic transitions of the incident neutrons obeys the principle of detailed balance.That is, although the neutron beam is not in equilibrium with the solid, thetransition rate is such that it drives the beam towards equilibrium. This canbe seen by inspection of the one phonon contribution to the spectrum. The onephonon absorption and emission spectrum is proportional to[

1 + N(ωα(q))]δ( E − E′ − h ωα(q) ) + N(ωα(q)) δ( E − E′ + h ωα(q) )

(1744)The first term represents a processes in which the neutron loses energy due tothe emission of a phonon, whereas the second term represents a processes inwhich the neutron gains energy due to the absorption of a phonon. The ratioof the rate at which the neutron beam gains energy to the rate at which theneutron beam loses energy is given by

W (E → E + h ω)W (E + h ω → E)

=N(ω)

[ N(ω) + 1 ]= exp

[− β h ω

](1745)

If equilibrium with the beam were to be established, the kinetic energy of theneutron beam would be distributed according to the Boltzmann formula

P (E) =1Z

exp[− β E

](1746)

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such that dynamic equilibrium would be established. In this case the totalnumber of transitions from E → E + h ω precisely equals the number oftransitions in the reverse direction E + h ω → E

P (E) W (E → E + h ω) = P (E + h ω) W (E + h ω → E) (1747)

However, the beam produced by the neutron source is not in equilibrium withthe sample, and would only equilibrate if the beam traverses an infinite pathlength through the sample.

In addition to the inelastic one phonon scattering cross-section, there aretwo second order terms in u2, which are cross-terms involving the term secondorder in u(t′) from the expansion of one factor

2 π h2

mnb exp

[i ( k − k′ ) . ( Ri + ui(t

′) )]

(1748)

and the leading zero-th order term from the other factor. This cross term isproportional to

− 12!

(( k − k′ ) . ui(t

′))2

. 1

= − 12!

1N

∑q,q′,α,α′

h

2 M√ωα(q) ωα′(q′)

(( k − k′ ) . εα(q)

×(

( k − k′ ) . εα′(q′))

exp[i ( q + q′ ) . Ri

×(aq,α exp[ − i ωα(q) t′ ] + a†−q,α exp[ + i ωα(q) t′ ]

×(aq′,α′ exp[ − i ωα′(q′) t′ ] + a†−q′,α′ exp[ + i ωα′(−q′) t′ ]

)(1749)

The expectation value of both the cross-terms

−(

12!

) (( k − k′ ) . ui(t

′))2

. 1 −(

12!

)1 .(

( k − k′ ) . ui(0))2

(1750)

is time independent and is given by

− 1N

∑q,α

h

2 M ωα(q)

(( k − k′ ) . εα(q)

)2 (2 nq,α + 1

)(1751)

Hence, it is seen that the second order contribution can be decomposed intoan elastic and an inelastic one phonon contribution. The elastic contribution,

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involves the scattering intensity from the nuclei, when they are displaced fromtheir equilibrium positions, either through the zero point fluctuations or throughthe effect of a thermal activated population of lattice vibrations. This secondorder contribution to the elastic scattering has a form similar to the intensityof the coherent Bragg peak, and cannot be distinguished from it through ex-periments. It is expected that inspection of all the even order terms in theexpansion should provide similar contributions, which will modify the intensityof the observed Bragg scattering peak. These contributions, due to the fluc-tuations of the nuclei from their equilibrium positions, give rise to a reductionintensity which is governed by the Debye Waller factor.

15.2 The Debye-Waller Factor

The above second order contribution can be combined with the leading orderterm, to give the first two terms of the expansion of the elastic scattering cross-section, W ,

W = 1 − 1N

∑q,α

h

2 M ωα(q)

(( k − k′ ) . εα(q)

)2 (2 N(ωα(q)) + 1

)+ . . .

(1752)

which can be exponentiated to yield the Debye-Waller factor

W = exp[− 1

N

∑q,α

h

2 M ωα(q)

(( k − k′ ) . εα(q)

)2

cothβ h ωα(q)

2

](1753)

The Debye-Waller factor reduces the intensity of the Bragg peak. The Debye-Waller factor also modifies the intensity of the Bragg peak in x-ray scattering.

The effects of the multi-phonon processes, of all order, can be ascertained byexamining the expectation value of the factor in the neutron scattering cross-section given by∫ ∞

−∞dt′ < k | Hint(0) | k′ > < k′ | Hint(t′) | k >

=(

2 π h2

mnb

)2 ∑i,j

exp[− i ( k − k′ ) . ( Ri − Rj )

×∫ ∞

−∞dt′ exp

[− i ( k − k′ ) . ui(0)

]exp

[+ i ( k − k′ ) . uj(t

′)]

(1754)

The energy conserving delta functions have been expressed as an integrals over

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t′ via,

δ( E′ − E ± h ω ) =∫ ∞

−∞

dt′

2 π hexp

[i t′

h( E′ − E ± h ω )

](1755)

and the energies in the exponentials can be expressed in terms of the non-interacting Hamiltonian. Using this identity, the scattering cross-section can berepresented as a Fourier Transform of the thermal average two time correlationfunction∫ ∞

−∞dt′ exp

[i t′

h( E − E′ )

]< | exp

[− i ( k − k′ ) . ui(0)

]exp

[+ i ( k − k′ ) . uj(t

′)]| >

(1756)For harmonic phonons, one can express this correlation function as

=∫ ∞

−∞dt′ exp

[i t′ ω

]< | exp

[− i ( k − k′ ) . ui(0)

]exp

[+ i ( k − k′ ) . uj(t

′)]| >

=∫ ∞

−∞dt′ exp

[i t′ ω

]exp

[− 1

2< |

(( k − k′ ) . ( ui(0) − uj(t

′) ))2

| >]

=∫ ∞

−∞dt′ exp

[i t′ ω

]exp

[− < |

(( k − k′ ) . ui(0)

) (( k − k′ ) . uj(t

′))| >

× exp[− 1

2< |

(( k − k′ ) . ui(0)

)2

| >]

exp[− 1

2< |

(( k − k′ ) . uj(t

′))2

| >]

(1757)

where h ω = E − E′ is the energy loss experienced by the neutron. The lasttwo factors are identified with the Debye-Waller factor, which is given by

W = exp[− < |

(( k − k′ ) . ui(0)

)2

| >]

= exp[− 1

N

∑q,α

h

2 M ωα(q)

(( k − k′ ) . εα(q)

)2

cothβ h ωα(q)

2

](1758)

The frequency dependent factor can be expanded in terms of the number ofphonons ∫ ∞

−∞dt′ exp

[i t′ ω

]exp

[− < |

(( k − k′ ) . ui(0)

) (( k − k′ ) . uj(t

′))| >

]=∫ ∞

−∞dt′ exp

[i t′ ω

] (1 − < |

(( k − k′ ) . ui(0)

) (( k − k′ ) . uj(t

′))| > + . . .

)(1759)

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Thus, all the contributions to the scattering cross-section are reduced in inten-sity by the Debye-Waller factor. The first term in the expansion is found to beproportional to δ(ω) and gives rise to the elastic scattering. The second term isjust the one phonon contribution to the scattering cross-section.

15.3 Single Phonon Scattering

The phonon dispersion relation can be inferred from a measurement of the singlephonon scattering peak. The scattering cross-section for processes in which asingle phonon is emitted have to satisfy the energy and momentum conservationlaws

h2 k2

2 mn=

h2 k′2

2 mn+ h ωα(q) (1760)

andk = k′ + q + Q (1761)

since ω(q) is periodic with a periodicity of the reciprocal lattice vectors

ωα(q) = ωα(q +Q) (1762)

One can combine the equations as

h2 k2

2 mn=

h2 k′2

2 mn+ h ωα(k − k′) (1763)

In the scattering experiments the beam of neutrons is generally collimated tohave a definite direction of the k vector, and also a definite initial energy. Fora given k, the solution of the above equation for the three components of k′,form a two-dimensional surface. For a detector placed in a particular scatteringdirection, the solution only exists at isolated points. On measuring the scat-tering cross-section at the various magnitudes of the final momentum k′ yieldsharp peaks in the spectrum. With knowledge of the magnitude of the finalmomentum k′, one can construct k′−k, and also E′−E and hence find h ωα(q)for the normal mode. By varying the direction of k′ and the magnitude of E,one can map out successive surfaces and hence obtain the dispersion relation.

Information about the polarization of the phonon modes can be obtainedfrom the dependence of the intensity on the scattering wave-vector k−k′ as thescattering cross-section is proportional to∣∣∣∣ ( k − k′ ) . εα(q)

∣∣∣∣2 (1764)

The width of the single phonon peak obtained in experiments have two ori-gins, one is the experimental resolution and another component is not resolution

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limited. The second component is due to the lifetime of the phonon τ , which ac-cording to the energy time uncertainty principle gives rise to an energy width ofhτ . The lifetime occurs because the phonons are scattered either by anharmonicprocesses or by electrons. The small magnitude of the width of the phononpeaks attests to the effectiveness of the harmonic approximation and the Born-Oppenheimer approximation.

15.4 Multi-Phonon Scattering

Processes in which two phonons are absorption or emitted satisfy the two con-servation laws

h2 k2

2 mn=

h2 k′2

2 mn± h ωα(q) ± h ωα′(q′) (1765)

andk = k′ ± q ± q′ + Q (1766)

Conservation of momentum can be used to express q′ in terms of q, this givesrise to the restriction

h2 k2

2 mn=

h2 k′2

2 mn± h ωα(q) ± h ωα′(k − k′ ± q) (1767)

Since there are six quantities k′ and q, which are undetermined. Even if thedirection of k′ is fixed there still remains three unknown quantities q, whichproduces a continuously varying final neutron energy. Hence, one obtains acontinuous spectrum. A similar analysis of the higher order multi-phonon pro-cesses also yields a continuous spectrum. Only the one phonon spectrum givesrise to a single peak.

Thus, in a general scattering experiment, with a specific scattering direction,the analysis of the scattered neutrons energy provides a spectrum which con-tains a continuous portion on superimposed with sharp peaks. The spectrummay show an elastic Bragg peak depending on the magnitude of k and θ, or ifthere are different isotopes one may observe incoherent nuclear scattering at zeroenergy transfers. The peaks of the one phonon scattering can be used to map outthe dispersion relations. This has been performed for f.c.c. lead. However, somebranches were not observed. The intensity of the one phonon absorption peak isproportional to the Bose-Einstein distribution function N(ωα(q)), whereas theone phonon emission process has intensity proportional to [ N(ωα(q)) + 1 ].Thus, it is usual to measure phonon emission at low temperatures.

——————————————————————————————————

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15.4.1 Exercise 73

(i) Find a graphical description of the conservation laws for the phonon emissionprocess.

(ii) Show that there is a minimum or threshold energy required for phononemission.

——————————————————————————————————

15.4.2 Exercise 74

(i) Evaluate the Debye-Waller factor for a one, two or three dimensional systemof acoustic phonons.

(ii) Determine the temperature dependence of the integrated intensity of thescattering cross-section, defined by

I(q) =∫ +∞

−∞dω

k′

k

d2 σ

dω dΩ(1768)

——————————————————————————————————

15.4.3 Exercise 75

Consider inelastic neutron scattering from a perfect fluid, described by theHamiltonian

H0 =∑

i

P2

i

2 M(1769)

Show that the inelastic scattering cross-section is proportional to

d2 σ

dω dΩ∝(

β M

2 π h2 q2

) 12

exp[− β M

2 h2 q2

(h ω − h2q2

2 M

)2 ](1770)

——————————————————————————————————

15.5 Raman and Brillouin Scattering of Light

Since the energy of visible light is of the order of eV and the energy of a typicalphonon is of the order of meV, ( 10−3 ) eV, it is not possible to observe thephonons by direct absorption or emission of light. However, it is possible toobserve the phonons in a solid via light scattering. Even though the scattering

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processes proceed via the same mechanism, the scattering from optical phononsis called Raman scattering and scattering from acoustic phonons is called Bril-louin scattering.

As in neutron scattering, the basic process may involve emission of phononsor absorption of phonons. The conservation laws for the one phonon absorptionor emission process

h ω′ = h ω ± h ωα(q) (1771)

and one phonon absorption or emission process

h k′ n = h k n ± h q + h Q (1772)

In these expressions (k, ω) and (k′, ω′) are, respectively, the momentum andenergy of the incident beam of photons and the scattered photons, and n is therefractive index of the media. It reflects the change in the wavelength of the lightas it enters the solid. The phonon absorption (+) gives rise to the Stoke’s siftedline, which has an intensity proportional to the number of activated phonons

∝ N(ωα(q)) (1773)

The phonon emission (−) gives rise to the anti-Stoke’s line with an intensityproportional to

∝ [ 1 + N(ωα(q)) ] (1774)

which has contributions from spontaneous and stimulated emission.

Since the phonon frequency is given by the Debye frequency h ωD ∼ 10−2

eV, which is small compared with a typical photon energy h c k n ∼ 1 eV, thechange in photon wave vector k − k′ is small. Thus, as far as the scattering isconcerned the scattering triangle is almost isosceles. The momentum transfer qis given by

| q | = 2 n k sinθ

2

= 2 n kω

csin

θ

2(1775)

Since the direction of k and k′ are known from the experimental geometry, thedirection of q can be inferred, if the small change in the photon energy, h ω, ismeasured.

For Brillouin scattering the phonon energy is given by

ωα(q) = vα q (1776)

where vα is the velocity of sound. The magnitude of the phonon’s momentumis given by

q =ωα(q)vα(q)

= 2ω n

csin

θ

2(1777)

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However, the energy of the acoustic phonon energy is equal to the change inphoton energy, ∆ ω,

ωα(q) = ∆ω (1778)

Thus, the velocity of the acoustic phonon is found as

vα(q) =∆ω2 ω

c

ncsc

θ

2(1779)

The experimentally determined spectra has the form of a strong un-scatteredlaser line, surrounded by a small Stoke’s line at higher frequencies, and a slightlymore intense anti-Stoke’s line at lower frequencies. The Stokes and anti-Stoke’sline are both separated from the main line by the same frequency shift ∆ω.

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16 Phonons in Metals

An alternate approach to the phonon dispersion in metals is based on a twocomponent plasma composed of electrons and ions. The approach starts byconsideration of a plasma composed of the positively charged ions, with chargeZ | e | and mass M . The plasma of ions support longitudinal charge densityoscillations which occur in the absence of an external potential. Since the totalpotential is related to the external scalar potential via

φ(q, ω) =φext(q, ω)ε(q, ω)

(1780)

and if φext(q, ω) = 0 one must have ε(q, ω) = 0 for φ(q, ω) 6= 0. In this case,one has an spontaneous density fluctuations and induced longitudinal current

jL(q, ω) =q ω

4 π

[φ(q, ω) − φext(q, ω)

]

=q ω

4 π

[1 − ε(q, ω)

]φ(q, ω)

(1781)

where Poisson’s equation and the continuity condition on the charge density havebeen used. Using the definition of the longitudinal conductivity one recovers therelation

ε(q, ω) = 1 − 4 π σ(ω)i ω

(1782)

which together with the Drude expression for the conductivity of a gas of ionsof charge Z | e | and mass M

σ(ω) =Z2 e2 ρions τ

M

11 − i ω τ

(1783)

This yields the Drude model for the dielectric constant of the ions, which forω 1

τ becomes

ε(q, ω) = 1 − 4 π Z e2 ρ

M ω2(1784)

where the density of ions is given in terms of the electron density ρ via

ρ

Z(1785)

The condition for plasmon oscillations is given by

ε(q, ω) = 0 (1786)

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The ionic plasmon frequency man be written in terms of the plasmon frequency

Ω2p =

Z m

Mω2

p (1787)

This corresponds to an unscreened phonon frequency. Since the factor Z mM ∼

14000 and h ωp ∼ 10 eV, the unscreened phonon frequency is approximately∼ 1

10 eV.

16.1 Screened Ionic Plasmons

The above model is inadequate as it neglects the effects of the conduction elec-trons. This effect of the electrons can be included by screening the Coulombinteractions between the charged nuclei

4 π Z2 e2

q2(1788)

with the dielectric constant of the electron gas. In the Thomas-Fermi approxi-mation this is given by

εeg(q, ω) = 1 +k2

TF

q2(1789)

Thus, within the Born-Oppenheimer approximation, one obtains the dielectricconstant as

ε(q, ω) = 1 − 4 π Z e2 ρ

M ( 1 + k2T F

q2 ) ω2(1790)

The screened ionic plasmons have frequencies given by

ε(q, ω) = 1 − 4 π Z e2 ρ

M ( 1 + k2T F

q2 ) ω2= 0 (1791)

Thus,

ω2 =Z m

Mω2

p

q2

q2 + k2TF

(1792)

This is the Bohm-Staver model of the phonon frequency. This model results ina linear dispersion relation ω(q) ≈ v q, where the velocity v is given by

v2 =Z m

M

ω2p

k2TF

(1793)

As the Thomas-Fermi wave vector is given in terms of the Fermi-wave vector by

4 π e2

k2TF

≈ h2 π

m kF(1794)

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and the electron density is given by

ρ =k3

F

3 π2(1795)

the velocity of sound is related to the Fermi-velocity vF = hm kF via

v2 =13Z m

Mv2

F (1796)

Thus, the velocity of sound v is reduced below the Fermi-velocity vF as mM ∼

10−3 − 10−5.

16.1.1 Kohn Anomalies

A more accurate treatment of the phonon frequency replaces the Thomas-Fermidielectric constant with the Lindhard expression

εeg(q, ω) = 1 − 4 π e2

q2

∫d3k

4 π3

f(E(k + q)) − f(E(k))E(k + q) − E(k) + h ω

(1797)

where f(x) is the Fermi-Dirac distribution function. This has singularities inthe derivative at q = 2 kF . These singularities correspond to the extremaldiameters of the Fermi-surface. Walter Kohn showed that these singularitiesshould appear in the phonon spectrum by producing kinks or infinities in thederivative (

∂ω

∂q

) ∣∣∣∣q=2kF

(1798)

16.2 Dielectric Constant of a Metal

The dielectric constant of a metal represents the process in which an externalcharge is screened by the combined effects of the electrons and the ions

φext(q, ω) = φ(q, ω) ε(q, ω) (1799)

A dielectric function can be defined for just the electrons, in which the totalpotential φ(q, ω) is produced as the response to a total external potential whichis external to the electron gas. That is the total external potential is consideredto be the sum of the applied external potential and the total potential due tothe ion charge density

φext(q, ω) + φions(q, ω) = φ(q, ω) εel(q, ω) (1800)

Analogously, a dielectric function can be defined for the ions as the responseof the ions to an external potential composed of the applied potential and theelectrons

φext(q, ω) + φel(q, ω) = φ(q, ω) εions(q, ω) (1801)

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This goes beyond the Born-Oppenheimer approximation. The total potential isgiven by the sum of the potentials due to the external, electron and ion charges

φ(q, ω) = φext(q, ω) + φions(q, ω) + φel(q, ω) (1802)

The dielectric constant of the metal is given in terms of the dielectric constant ofthe electrons and the dielectric constant of the ions, by adding the two equationsdefining the electronic and ionic dielectric constants(

εions(q, ω) + εel(q, ω))φ(q, ω) = φ(q, ω) + φext(q, ω) (1803)

Then with the definition of the total dielectric constant one has the relation

ε(q, ω) =(εions(q, ω) + εel(q, ω) − 1

)(1804)

The dielectric constant of the ions goes beyond the Born-Oppenheimer approxi-mation. It describes how the ions, alone, screen the potential due to the appliedpotential and the potential due to the electrons. As the dielectric constant dueto the ions alone is given by

εions(q, ω) = 1 −Ω2

p

ω2(1805)

and the electronic dielectric constant ( at low frequencies ) is given by theThomas-Fermi approximation

εel(q, ω) = 1 +k2

TF

q2(1806)

Hence, the low frequency dielectric constant is given by

ε(q, ω) = 1 +k2

TF

q2−

Ω2p

ω2(1807)

for ωp ω.

An alternate definition of the dielectric constant of the ions may be intro-duced, in which one considers the external potential to be first screened by theelectron gas. Secondly the resulting dressed external potential is screened bythe ions. That is instead of the electron gas screening the external potential ofthe ions and the applied potential

φ(q, ω) =φext(q, ω)εel(q, ω)

+φions(q, ω)εel(q, ω)

(1808)

one considers only the dressed external potential

φdressed(q, ω) =φext(q, ω)εel(q, ω)

(1809)

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It is this dressed external potential that is screened by the ions to produce thetotal potential. This relation defines the dressed dielectric constant of the ions

φ(q, ω) =φdressed(q, ω)εdressed

ions (q, ω)

=φext(q, ω)

εel(q, ω) εdressedions (q, ω)

(1810)

Hence, the electronic and dressed ionic dielectric constants are related to thedielectric constant via

ε(q, ω) = εel(q, ω) εdressedions (q, ω) (1811)

Combining this with the relation of the dielectric constant in terms of dielectricconstants of the electrons and ions

ε(q, ω) =(εions(q, ω) + εel(q, ω) − 1

)(1812)

the dressed ionic dielectric constant can be defined by

εdressedions (q, ω) =

1εel(q, ω)

(εions(q, ω) + εel(q, ω) − 1

)= 1 +

1εel(q, ω)

(εions(q, ω) − 1

)(1813)

The dressed dielectric constant is calculated as

εdressedions (q, ω) = 1 +

1

1 + k2T F

q2

(εions(q, ω) − 1

)

= 1 − 1

1 + k2T F

q2

(Ω2

p

ω2

)(1814)

This can be written in terms of the phonon dispersion relation ω(q)2

εdressedions (q, ω) = 1 −

ω(q)2

ω2(1815)

since the phonon oscillations occur when the dielectric constant vanishes

εdressedions (q, ω(q)) = 0 (1816)

By inspection of the dressed dielectric constant the phonon frequency is foundas

ω(q)2 =q2

q2 + k2TF

Ω2p (1817)

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The introduction of screening by the electron gas has reduced the frequency ofthe ionic density oscillations from the ionic plasmon frequency to a branch oflongitudinal acoustic phonons. The total dielectric constant, which is a productof the dressed dielectric constant and the Thomas-Fermi dielectric constant ofthe electron gas, can now be written in terms of the phonon frequencies as

1ε(q, ω)

=1

1 + k2T F

q2

1

1 − ω(q)2

ω2

=1

1 + k2T F

q2

ω2

ω2 − ω(q)2

(1818)

This is in agreement with the expression discussed earlier.

16.3 The Retarded Electron-Electron Interaction

Consider the screening of the Coulomb interaction between a pair of electronsvia the dielectric constant

4 πq2

→ 4 πε(q, ω) q2

=4 π

k2TF + q2

(1 +

ω(q)2

ω2 − ω(q)2

)(1819)

Thus, there is an additional contribution in the effective interaction due to thescreening by the ions. The ω dependence of the interaction is representative thatthe effective interaction is not instantaneous but instead is a retarded interac-tion. The effective interaction between a pair of electrons involves a momentumtransfer q = k − k′ and energy transfer h ω = E(k) − E(k′). The effectiveinteraction has the following limits

(i) This interaction reduces to the Thomas-Fermi screened electron-electroninteraction when the electron energy transfer is greater than the typical phononfrequency ωD ∼ Ωp. In this case, when ω > ωD, the phonon correction isunimportant.

(ii) The electron-electron interaction is strongly modified at low frequencies,where ω < ωD. The contribution from the phonons is large and of oppositesign to the direct Coulomb repulsion, and exactly cancels at ω = 0. Theimportant point, however, is that the retarded interaction is attractive at lowfrequencies. It exhibits the phenomenon of over-screening and can give rise tosuperconductivity.

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16.4 Phonon Renormalization of Quasi-Particles

The electron-phonon interaction can give rise to a change in the quasi-particledispersion relation. The Hartree-Fock contribution to the quasi-particle energyfrom the screened electron-electron interaction is

∆E(k) =∑k′

f(E(k′)) < k k′ | e2

| r − r′ || k k′ >

=∑k′

f(E(k′))∫

d3r

∫d3r′

e2

| r − r′ |

(1 − exp

[i ( k − k′ ) . ( r − r′ )

] )

= ∆EH −∑k′

f(E(k′))4 π e2

| k − k′ |2

(1820)

The first term is the Hartree term which is k independent and can be absorbedinto a shift of the chemical potential and the second term is the exchange termwhich depends on k. The exchange term affects the quasi-particle dispersionrelation. If the effect of phonon screening is included the exchange term becomes

−∑k′

f(E(k′))4 π e2

| k − k′ |2 + k2TF

[1 +

h2 ω(k − k′)2

( E(k) − E(k′) )2 − h2 ω(k − k′)2

](1821)

where the exchange interaction is Thomas-Fermi screened, and there is also aphonon contribution.

On utilizing the smallness of the Debye frequency with respect to the Fermi-energy, and integrating over the magnitude of k′, one can show that the changein energy due to the electron-phonon interaction is given by

= −∫

d2S′

8 π3

1h v(k)

4 π e2

| k − k′ |2 + k2TF

h ω(k − k′) ln∣∣∣∣µ− E(k)− hω(k − k′)µ− E(k) + hω(k − k′)

∣∣∣∣(1822)

where k′ lies on the Fermi-surface. Substitution of E(k) = µ immediatelydemonstrates that the value of the Fermi-energy µ and the shape of the Fermi-surface are unaltered by the coupling to the phonons, which in the approxima-tion under consideration is given by the Thomas-Fermi quasi-particle theory.Secondly, when the quasi-particle energy is within h ωD of µ, | E(k) − µ | <h ωD, the logarithmic term can be expanded in inverse powers of h ω. Then itis seen that the phonon contribution to the screening produces a change in thedispersion relation

E(k) − µ =ETF (k) − µ

1 + λ(1823)

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where λ is the wave function renormalization due to the phonons and is givenby

λ =∫

d2S′

8π3

1h v(k′)

4 π e2

| k − k′ |2 + k2TF

(1824)

This has the result that the quasi-particle velocity is given by

v(k) =1h∇ E(k)

=1

1 + λ

1h∇ ETF (k)

(1825)

Thus, the quasi-particle contribution to the density of states is enhanced by afactor of 1 + λ

ρ(µ) = ( 1 + λ ) ρTF (µ) (1826)

The coupling can be estimated via

λ <4 π e2

k2TF

∫d2S′

8 π3

1h v(k′)

(1827)

but

4 π e2

k2TF

=∂ρ

∂µ=

1ρ(µ)

=[ ∫

d2S′

4 π3 h v(k′)

]−1

(1828)

Hence, the phonon renormalization factor is usually less than unity

λ < 1 (1829)

Finally, the phonon corrections are negligible for electron energies far from theFermi-energy. For example when

| E(k) − µ | > h ωD (1830)

then the dispersion relation suffers only small corrections

E(k) − µ = ETF (k) − µ + O

(h ωD

E(k) − µ

)2

(1831)

Thus, there has to be a kink in the quasi-particle dispersion relation near theFermi-energy.

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16.5 Electron-Phonon Interactions

The effect of coupling with the phonons on the quasi-particle spectrum can beused to deduce the form of the electron-phonon interaction. The change in theground state energy of a metal due to the electron phonon interaction, Hint,can be estimated from second order perturbation theory as

∆2E =∑

i

| < Ψ0 | Hint | Ψm > |2

E0 − Em(1832)

It is assumed that the form of the electron - phonon interaction is dominated bythe first non-trivial term in the expansion of potential acting on the electronsin powers of the displacements of the ions

Hint =∑

i

ui . ∇RiVions(r) (1833)

Thus, the most important excitation process comes from excited states | Ψm >in which an electron has been scattered from state k to k − q and in also aphonon of wave vector q has been excited. Hence,

Em − E0 = E(k − q) + h ω(q) − E(k) (1834)

Thus, one can express the second order correction to the ground state energyin a phenomenological manner as

∆2E = −∑k,q

f(E(k)) ( 1 − f(E(k − q)) )| λq |2

E(k − q) + h ω(q) − E(k)

(1835)where f(x) is the Fermi-function. One can identify an effective electron - elec-tron interaction due to the phonons from the functional derivative of the energywith respect to the Fermi-functions

Veff (q) =δ2 ∆2E

δf(E(k)) δf(E(k − q))(1836)

Hence,

Veff (q) = −| λq |2

E(k) − E(k − q) − h ω(q)

−| λq |2

E(k − q) − E(k) − h ω(q)

= | λq |2[

2 h ω(q)

h2 ω(q)2 − ( E(k) − E(k − q) )2

](1837)

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On identifying the above effective potential with the phonon contribution tothe screened interaction between the electrons one obtains an expression for theeffective coupling constant | λq |2 as

| λq |2 =1V

4 π e2

q2 + k2TF

h ω(q)2

(1838)

For small q the coupling constant vanishes linearly with q, since

4 π e2

k2TF

=23µ

ρ(1839)

for q < kTF the coupling constant varies as

| λq |2 =µ

ρ V

h ω(q)3

=h ω(q) µ3 N Z

(1840)

16.6 Electrical Resistivity due to Phonon Scattering

The electron-phonon scattering contributes to the electrical resistivity. Thephonon gas acts as a source or sink for the electron momentum and, thus, theinteractions with the electron gas reduces the current flow and hence increasesthe resistivity. The electron-ion interaction is given by

Hions =∑R

V (r −R) (1841)

and as the position of the i-th ion can be written in terms of the equilibriumposition and a displacement

R = Ri + ui (1842)

The potential of the ions is expanded up to linear order in the lattice displace-ments ui

Hions =∑

i

[V (r −Ri) − ui . ∇R V (r −Ri) + . . .

](1843)

The first term represents the static lattice and the second term is the electronphonon interaction. The electron phonon interaction is given by

Hint = −∑

i

ui . ∇R V (r −Ri) (1844)

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Thus, the interaction produces scattering of the electrons between Bloch statesand, through ui involves the absorption or emission of phonons. The conditionof conservation of energy yields the selection rule

E(k) = E(k′) ± h ω(k − k′) (1845)

This is a single restriction leads to a two-dimensional surface of Bloch statewave vectors k′ that are allowed final states for the electron initially in Blochstate k. The momentum transfer for these processes is given by q = k − k′.The surface of allowed final states must be close to the surface of initial energyas h ω µ, hence, E(q) ∼ E(k − q). The scattering rate out of the statewith momentum k is given by

1τ(k → k′)

=

2 πh

∑α

| λαq |2 f(E(k))

(1 − f(E(k + q))

)

×

[N(ωα(q)) δ

(E(k) − E(k + q) + h ωα(q)

)

+(

1 + N(ωα(q)))δ

(E(k) − E(k + q) − h ωα(q)

) ](1846)

The rate for scattering into the momentum state k is given by

1τ(k′ → k)

=

2 πh

∑α

| λαq |2 f(E(k + q))

(1 − f(E(k))

)

×

[N(ωα(q)) δ

(E(k + q) − E(k) + h ωα(q)

)

+(

1 + N(ωα(q)))δ

(E(k + q) − E(k) − h ωα(q)

) ](1847)

The transport scattering rate is the rate for momentum change of an electronat the Fermi-surface is defined by

( k . E )1τf(E(k))

(1 − f(E(k))

)=∑k′

[( k . E )

1τ(k → k′)

− ( k′ . E )1

τ(k′ → k)

](1848)

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The rate for scattering out of state k will be transformed into a form comparableto the rate for scattering in. The rate for scattering out of momentum state kis re-written as

=2 πh

∑α

| λαq |2 f(E(k))

(1 − f(E(k + q))

)exp

[β ( E(k) − E(k + q) )

]

×

[ (1 + N(ωα(q))

(E(k) − E(k + q) + h ωα(q)

)

+ N(ωα(q)) δ(E(k) − E(k + q) − h ωα(q)

) ]

=2 πh

∑α

| λαq |2 f(E(k + q))

(1 − f(E(k))

)

×

[ (1 + N(ωα(q))

(E(k) − E(k + q) + h ωα(q)

)

+ N(ωα(q)) δ(E(k) − E(k + q) − h ωα(q)

) ](1849)

Thus, the transport scattering rate can be expressed as

( k . E )1τf(E(k))

(1 − f(E(k))

)=

(1850)

=2 πh

∑α, q

( q . E ) | λαq |2 f(E(k + q))

(1 − f(E(k))

)

×

[ (1 + N(ωα(q))

(E(k) − E(k + q) + h ωα(q)

)

+ N(ωα(q)) δ(E(k) − E(k + q) − h ωα(q)

) ](1851)

Furthermore, as

f(E(k + q))(

1 − f(E(k))) (

1 + N(ωα(q)))δ

(E(k) − E(k + q) + h ωα(q)

)= f(E(k))

(1 − f(E(k + q))

)N(ωα(q)) δ

(E(k) − E(k + q) + h ωα(q)

)(1852)

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the scattering rate can be expressed as

( k . E )1τf(E(k))

(1 − f(E(k))

)=

2 πh

∑α, q

( q . E ) | λαq |2 N(ωα(q))

×

[f(E(k))

(1 − f(E(k + q))

(E(k) − E(k + q) + h ωα(q)

)

+ f(E(k + q))(

1 − f(E(k)))δ

(E(k) − E(k + q) − h ωα(q)

) ]

=2 πh

∑α, q

( q . E ) | λαq |2 N(ωα(q)) N( − ωα(q))

×

[ (f(E(k + q)) − f(E(k))

(E(k) − E(k + q) + h ωα(q)

)

+(f(E(k)) − f(E(k + q))

(E(k) − E(k + q) − h ωα(q)

) ]

=2 πh

∑α, q

( q . E ) | λαq |2 N(ωα(q)) N( − ωα(q))

×

[ (f(E(k) + hωα(q)) − f(E(k))

(E(k) − E(k + q) + h ωα(q)

)

+(f(E(k)) − f(E(k)− hωα(q))

(E(k) − E(k + q) − h ωα(q)

) ](1853)

The summation over q is evaluated by transforming it into an integral

=2 πh

∑α, q

( q . E ) | λαq |2 N(ωα(q)) N( − ωα(q))

×

[ (f(E(k) + hωα(q)) − f(E(k))

(h2

m( k . q ) +

h2

2 mq2 − h ωα(q)

)

+(f(E(k)) − f(E(k)− hωα(q))

(h2

m( k . q ) +

h2

2 mq2 + h ωα(q)

) ](1854)

The integration over the direction of q is performed in spherical polar coordi-nates, in which the direction of k is fixed as the polar axis. The integral overthe azimuthal angles result in the factors of sinφ and cosφ in

( q . E ) = q cos θ Ez + q sin θ ( sinφ Ey + cosφ Ex ) (1855)

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vanishing. The sole surviving term, proportional to Ez, can then be written ina manner independent of the choice of axis as k . E which can be factored outof the integral

=2 πh

(2 m π

h2 k2) ( k . E )

∑α

∫dq q2 | λα

q |2 N(ωα(q)) N( − ωα(q))

×

[ (f(E(k) + hωα(q)) − f(E(k))

) ∫ 1

−1

d cos θ cos θ δ(

cos θ +q

2 k−

m ωα(q)h k q

)

+(f(E(k)) − f(E(k)− hωα(q))

) ∫ 1

−1

d cos θ cos θ δ(

cos θ +q

2 k+

m ωα(q)h k q

) ](1856)

On neglecting the term of order vα

vF, cancelling the factors of ( k . E ), and Taylor

expanding the Fermi-function factors in powers of the phonon frequencies, onefinds the transport scattering rate for electrons on the Fermi-surface is given by

1τf(E(k))

(1 − f(E(k))

)=

= − 2 πh

(2 m π

h2 k3)∑α

∫dq q3 | λα

q |2 N(ωα(q)) N( − ωα(q)) h ωα(q)(∂f(E(k))∂E(k)

)(1857)

On using (∂f(E(k))∂E(k)

)= − β f(E(k))

(1 − f(E(k))

)(1858)

one finds

= β (4 m π2

h3 k3)∑α

∫dq q3 | λα

q |2 N(ωα(q)) N( − ωα(q)) h ωα(q)

(1859)

The temperature dependence of the transport scattering rate can be evaluatedusing the Debye model for the phonons, and using a linear q dependence of|λα

q |2. The integral over q is evaluated through the substitution z = β h ωα(q)and ωα(q) = vα q to yield

1τ∝ T 5

∫ TDT

0

dzz5

( exp[z] − 1 ) ( 1 − exp[−z] )(1860)

For this temperature range, the number of thermally exited phonons is propor-tional to T 3. One would expect that the scattering rate would be proportional

1τ∝∫

dq q2 N(ω(q))

∝ T 3 (1861)

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However, as forward scattering is ineffective in transport properties, the trans-port scattering rate is proportional to the change in momentum along the di-rection of the electric field and therefore, is proportional to

( 1 − cos θ ) = 2 sin2 θ

2

≈ 12q2

k2F

(1862)

which produces an additional T 2 dependence. For low temperatures ( T < TD

) , the upper limit on the integration may be set to infinity yielding

σ(T )−1 ∝(T

TD

)5

(1863)

Thus, the combined effect of the factor ( 1 − cos θ ) and 1τ ∝ T 2 produces a

T 5 temperature dependence in the low-temperature resistivity.

At high temperatures ( T > TD ), the range of integration is less thanunity so the integrand may be expanded in powers of z yielding

σ(T )−1 ∝ T 5

∫ TDT

0

dz z3 = T T 4D

∝(T

TD

)(1864)

which is the result for the classical limit of the scattering. This can be consideredto arise merely as the number of thermally activated phonons is given by theclassical expression

N(ωα(q)) =kB T

h ωα(q)(1865)

The above results were first derived independently by Bloch and Gruneisen andthe resulting formula is known as the Bloch - Gruneisen resistivity due to phononscattering.

16.6.1 Umklapp Scattering

Umklapp processes may change the leading low-temperature variation of theresistivity. Umklapp scattering circumvents the factor of ( 1 − cos θ ) whichproduces the extra T 2 factor. When kF is close to the zone boundary, a small qvalue may couple the sheets of the Fermi-surface in neighboring Brillouin zones.These are the umklapp processes. They produce a large change in the electronvelocity ∆v, by a phonon induced Bragg reflection.

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16.6.2 Phonon Drag

The resistivity could decrease faster than T 5 if the system was relatively freeof defects and umklapp scattering could be neglected. This would occur if thephonons were allowed to equilibrate with the electronic system in its steadystate. The combined system of electrons and phonons should have a total mo-mentum, which is conserved in collisions. As a result, the phonon system wouldnot be able to momentum ( or current ) from the electron system as they drifttogether.

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17 Phonons in Semiconductors

17.1 Resistivity due to Phonon Scattering

The transport scattering rate in a semiconductor can be obtained from thecollision integral of the Boltzmann equation

I

[f(k)

]=∑

q

λ2q

[f(k) ( 1 − f(k + q) ) N(ω(q)) δ( E(k) − E(k + q) + h ω(q) )

− ( 1 − f(k) ) f(k + q) ( 1 + N(ω(q)) ) δ( E(k + q) − E(k) − h ω(q) )

](1866)

in which f() is the non-equilibrium distribution function. On linearizing aboutthe equilibrium Fermi-distribution function

f(k) = f0(k) + A ( k . E )∂f0(k)∂E

(1867)

yields the linearized collision integral.

Using the identity

( 1− f(E(k)) ) f((E(k)+hω(q)) =(

1− exp[β h ω(q)

] ) (f(E(k))− f(E(k)+hω(q))

)(1868)

one finds the result

I

[f(k)

]=

2 m A V

h2 kB T kexp

[− β ( E(k) − µ )

] ∫ 2k

0

dq

2 πq2 Ezλ

2q N(ω(q)) N(−ω(q))

×

[ (q

2 k+

m ω(q)h k q

) (1 − exp

[− β hω(q)

] )

−(

q

2 k−

m ω(q)h k q

) (1 − exp

[+ β hω(q)

] ) ](1869)

For low frequency acoustic phonons, the Bose-Einstein distribution can be ap-proximated by its high temperature form leading to the collision integral

I

[f(k)

]= A ( k . E )

(− ∂f(k)

∂E

)m V

k3

∫ 2k

0

dq

2 πq2 λ2

q

(− q

2 β h ω(q)

)(1870)

The transport scattering rate is found by factoring out the non-equilibrium partof the distribution function

1τ(E)

=m V

k3kB T

∫ 2k

0

dq

4 πq3

N

2 M ω(q)2q2 | V (q) |2

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=N V

4 π c2m

Mk | V (0) |2 kB T (1871)

Hence, the conductivity in a semiconductor, in which the scattering rate isdominated by phonon scattering, is given by

σx,x ∼ β2 exp[β µ

] ∫dk k3 exp[ − β E(k)

](1872)

Thus, the conductivity has a temperature dependence given by

σx,x ∼ exp[β µ

](1873)

Thus, as expected, the conductivity is still dominated by the number of carriers,but the conductivity has an additional T dependence of T−

32 above and beyond

the prefactor in the number of carriers.

17.2 Polarons

Electron-phonon coupling in semiconductors can be large. For low density ofcarriers, each carrier can cause a distortion of the lattice. The carrier and thesurrounding distortion forms an excitation which is known as a polaron. At lowtemperatures the polaron appears to have a large effective mass, as the motionof the carrier is hindered by the need to drag the surrounding lattice distortion.Thus, there is a low-temperature regime in which the conductivity is governedby the motion of the heavy quasi-particles with an extremely large and tem-perature dependent effective mass. At high temperatures, the conductivity isdominated by incoherent hopping processes, which are thermally assisted by thepresence of a thermal population of phonons.

17.3 Indirect Transitions

In a semiconductor, light can be absorbed in processes where by an electronis excited from the filled valence band into the empty conduction band. Theminimum energy of the photon must be greater than the band gap between theconduction and valence band density of states. Since the speed of light c isso large, the wave vector of the photon absorbed in a transition between twostates with energy difference of the scale of eV is extremely long. Thus, themomentum of the photon is negligible on the scale of the size of the Brillouinzone. This means that in a semiconductor, if only a photon and an electron areinvolved, momentum conservation only allows transitions in which the initialand final state of the electron have the same k value. This type of transition iscalled a direct transition. In some semiconductors the minimum of the conduc-tion band dispersion relation lies vertically above the maximum of the valenceband, and the band gap is called the direct gap. In this case, the threshold for

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direct absorption should coincide with the gap observed in the density of states.On the other hand, if the energetic separation between the maximum of the va-lence band and the minimum of the conduction band dispersion relations occurat different k values, then the threshold energy for the absorption of a photonin a direct transition should be greater than the separation inferred from justconsidering the density of states alone. This second type of semiconductor hastwo gaps, the indirect band gap inferred from the density of states and a directgap inferred for q = 0 transitions by consideration of the dispersion relations.

If the ions of the lattice are displaced from their equilibrium positions, simpleconservation of momentum arguments do not apply. In this case, it is possibleto have absorption at the indirect gap. At the threshold for indirect transitions,the absorption process involves the absorption or emission of a phonon withwave vector equal to the wave vector Q separating the valence band maximumto the conduction band minimum. The transition rate has to be calculated viasecond order perturbation theory, one power of the interaction involves the ab-sorption of one photon and the other power of the interaction involves eitherthe absorption or emission of one phonon.

The state of the joint system composed of an electron, phonons of wavevector Q and photons of frequency ω is denoted by | Ψ >. This state satisfiesthe equation of motion

i h∂

∂t| Ψ > =

(H0 + Hint

)| Ψ > (1874)

The state is decomposed in terms of eigenstates of H0, | φn > with energy En

| Ψ > =∑

n

Cn(t) exp[− i t

En

h

]| φn > (1875)

Then, one finds that the expansion coefficients Cn(t) satisfy the equation

i h∂

∂tCn(t) =

∑m

< φn| Hint | φm > exp[i t

Em − En

h

]Cm(t) (1876)

Since the system is initially in the ground state, then the state is subject to theinitial condition given by

Cn(0) = δn,0 (1877)

To first order one has

C1n(t) = − i

h

∫ t

0

dt′ < φn| Hint | φm > exp[i t′

E0 − En

h

](1878)

We assume the perturbation has no diagonal elements, therefore, C10 (t) = 0.

To second order one has

i h∂

∂tC2

n(t) =∑m6=0

< φn| Hint | φm > exp[i t

Em − En

h

]C1

m(t) (1879)

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18 Impurities and Disorder

If an isolated impurity is introduced into a solid, and the impurity has no lowenergy degrees of freedom which can be excited, then it can be treated as animpurity potential Vimp(r). Since the impurity breaks the periodic translationalinvariance of the solid, the impurity potential will scatter an electron betweenBloch states with different Bloch wave vectors. The non-zero matrix elementsof the potential can be written as∫

d3r φ∗k′(r) Vimp(r) φk(r) = < k′ | Vimp | k > (1880)

If the wave function, in the presence of an impurity, is written as the superpo-sition

ψα(r) =∑

k

Cα(k) φk(r) (1881)

the energy eigenvalue can be expressed as

( Eα − E(k) ) Cα(k) =∑k′

Cα(k′) < k | Vimp | k′ > (1882)

If the quantity∑

k′ Cα(k′) < k | Vimp | k′ > is well defined and non zerofunction of k, then there exist eigenvalues Eα between every consecutive pairsof E(k). For a large system, where E(k) are very closely spaced, the eigen-values form a continuum. These eigenstates correspond to weakly perturbedBloch states. On the other hand, if the potential is attractive, and there is aminimum value of E(k), below which there can be bound state with energies Eα.

The dependence of the bound state energy on the density of states of theordered material can be easily found, for the case where the potential has theproperty that the matrix elements are independent of k and k′. In this caseone can easily solve for the bound states. The above equations can be solvedwriting

γ =∑k′

Cα(k′) (1883)

Therefore, one has

Cα(k) =Vimp

Eα − E(k)γ (1884)

The above two equations leads to the self-consistency condition for the boundstate energy Eα

1 =∑

k

Vimp

Eα − E(k)(1885)

which shows that, for an attractive potential, there may be a critical value ofVimp needed for a bound state to form.

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A more powerful way of solving the same problem involves use of the one-particle resolvent Green’s function defined by the operator

G(z) = ( z − H )−1 (1886)

where z is a complex number. Since H is a Hermitean operator the matrixelements of the Green’s function can be expressed in terms of a sum of simplepoles at the energy eigenvalues. Since the poles of the Hamiltonian are composedof the discrete bound states at negative energies and a semi-continuous spectrumat positive energies, the Green’s function has a branch cut across the real axis,x = Re z,

< Ψ |(G(x−iε)− G(x+iε)

)| Φ >= 2 π i

∑n

< Ψ | En > < En | Φ > δ( x− En )

(1887)where | En > is the energy eigenstate corresponding to the energy eigenvalueEn.

The resolvent Green’s function can be obtained by expressing the Hamilto-nian in terms of the unperturbed Hamiltonian H0 and the interaction Hint,

H = H0 + Hint (1888)

Then, the Green’s function

G(z) =(z − H

)−1

=(z − H0 − Hint

)−1

(1889)

can be re-written as

G(z) =(z − H0

)−1

+(z − H0

)−1

Hint G(z) (1890)

which can be expressed in terms of the non-interacting resolvent Green’s func-tion, G0(z), as

G(z) = G0(z) + G0(z) Hint G(z) (1891)

The non-interacting resolvent Green’s function is easily evaluated in terms ofthe matrix elements between the eigenstates of H0.

< φn |1

z − H0

| φm > = < φn |1

z − Em| φm >

=< φn | φm >

z − Em

=δn,m

z − Em(1892)

which is diagonal. For simultaneous momentum eigenstates the non-interactingresolvent Green’s function only has the diagonal matrix elements

< k′ | 1z − H0

| k > =< k′ | k >

z − E(k)(1893)

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The interacting Green’s function can be expressed in terms of the T (z) matrixas

G(z) = G0(z) + G0(z) T (z) G0(z) (1894)

where the T-matrix is defined as

T (z) = Hint

(1 − G0(z) Hint

)−1

(1895)

Thus, the poles of the T-matrix are related to the poles of the Green’s function.As the matrix elements of Hint are independent of k, the matrix elements of theT-matrix between the Bloch states can be evaluated as

T (z) = Vimp

(1 −

∑k

Vimp

z − E(k)

)−1

(1896)

Since the imaginary part of the trace of the Green’s function is related to thedensity of state via

ρ0(E) = − 1π

∑k

Im < k | G0(E + iε) |k > (1897)

one can express the real part of the Greens’s function as an integral∑k

Vimp

z − E(k)=∫ ∞

0

dEVimp ρ0(E)z − E

(1898)

Thus, the imaginary part of the T-matrix is non-zero for z on the positive realaxis. The T-matrix has isolated poles outside the continuum at the negativeenergy z which are given by the solutions of

1Vimp

=∫ ∞

0

dEρ0(E)z − E

(1899)

The minimum value of the attractive potential Vimp that produces a boundstate strongly depends on the form of the density of states at the edge of thecontinuum. The critical value of Vimp denoted as Vc is given by the conditionz = 0

1Vc

= −∫ ∞

0

dEρ0(E)E

(1900)

Since ρ0(E) ∝ Ed−22 , the integral converges for three dimensions and higher,

but diverges for two and one dimensions. The bound states are exponentiallylocalized around the impurity site.

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18.1 Scattering By Impurities

The exact eigenstates of a Hamiltonian containing a scattering potential Vimp

satisfies the equation

H | Ψ > = ( H0 + Vimp ) | Ψ > = E | Ψ > (1901)

The can be re-expressed as an integral equation with an initial state given byincident momentum eigenstate | p > as

( E − H0 + i ε) | Ψ > = Vimp | Ψ > (1902)

has general solutions which are the superposition of the solutions of the homo-geneous equation and a particular solution of the inhomogeneous equation

| Ψ > = | k > +1

E − H0 + iεVimp | Ψ > (1903)

To ensure that | Ψ > − | k > is an outgoing wave ε must be chosen as apositive infinitesimal term. The asymptotic behavior of the outgoing equationcan be expressed as

Ψ(r) =1

( 2 π h )32

exp[i k . r

]

+∑k′

∫d3r′

( 2 π h )3

exp[i k′ . ( r − r′ )

]E − E(k′) + i ε

Vimp(r′) Ψ(r′)

(1904)

which has the well known solution

Ψ(r) =1

( 2 π h )32

exp[i k . r

]

− m

2 π

∫d3r′

( 2 π h )3

exp[i k | r − r′ |

]| r − r′ |

Vimp(r′) Ψ(r′)

(1905)

The asymptotic form can be expressed in terms of the phase shifts δl(k) via apartial wave analysis.

limr → ∞

Ψ(r) ∼∞∑

l=0

( 2 l + 1 )k r

il exp[i δl

]Pl(cos θ) sin( k r − lπ

2+ δl )

(1906)

Since the T-matrix has matrix elements which satisfy

< k′ | T | k > = < k′ | Vimp | Ψ > (1907)

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This leads to

m

2 π< k′ | T | k > =

∞∑l=0

Pl(cos θ)exp

[i 2 δl

]− 1

2 i k(1908)

and the differential scattering cross-section is given by

dΩ=∣∣∣∣ m

2 π< k′ | T | k >

∣∣∣∣2 (1909)

In the limit k → 0 only the s-wave phase shift δ0 is significant and one finds

m

2 πT =

exp[i 2 δ0

]− 1

2 i k(1910)

and the scattering cross-section is given by

dΩ=

sin2 δ0k2

(1911)

and the total cross-section σ is given by

σ =4 π sin2 δ0

k2(1912)

The density of states due to the impurity can be expressed in terms of the phaseshift δ0(k). The number of states is evaluated in a volume of radius R, and thewave function is required to vanish at r = R. Hence, the phase shift mustsatisfy the condition

k R + δ0(k) = n π (1913)

Since successive states satisfy this condition with consecutive integers n andn + 1 then the change in k between two states is given by

∆k ( R +∂δ0∂k

) = π (1914)

Thus, the number of states per k interval is

1∆k

=1π

(R +

dδ0(k)dk

)(1915)

On integrating this with respect to E one finds that the number of states dueto the impurity with energy less than E, N(E) is given by

N(E) =1πδ0(E) (1916)

The impurity density of states, per spin is given by

ρimp(E) =1π

(∂δ0∂E

)(1917)

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The condition for electrical neutrality for a charge Z | e | determines the phaseshift at the Fermi-energy, through Freidel’s sum rule

Z =2πδ0(µ) (1918)

For systems were the phase shift is rapidly varying at the Fermi-energy, there isa large impurity density of states. For resonant scattering, as Friedel has shown,there exists a virtual bound state at the Fermi-energy. This finite density ofstates at the Fermi-energy gives rise to an impurity contribution to the specificheat and the susceptibility.

18.2 Virtual Bound States

The virtual bound state can be envisaged as an ( almost ) localized level that hasa finite probability amplitude for transitions into the conduction band states.These virtual bound states are most frequently found for 3 d transition metalimpurities in metals or in mixed valent lanthanide element impurities in metals.In both these cases, the potential well has a large centrifugal barrier

Vl(r) =h2 l ( l + 1 )

2 m r2(1919)

which prevents the 3 d states from being filled until after the 4 s states are filledor, in the case of the lanthanide elements, the 4 f states remain unfilled untilafter the 6 s, 5 p and 5 d states are all occupied. When the nuclear potential isstrong enough, such that the 3 d or 4 f can be occupied in the ground state, theion localizes an electron within the centrifugal barrier in an inner ionic shell.For example, in the Ce atom, although the 4 f wave function is localized, inthat it has a spatial extent of 0.7 a.u. which lies inside the core like 5 s and 5 porbitals, its’ ionization energy is small and comparable to the ionization energyof the band like 6 s and 6 p orbitals. As the localized state is degenerate withthe conduction band states, there is a finite probability amplitude for tunnellingthrough the barrier. The virtual bound state describes an extended state that,through resonant scattering builds up a significant local character. The virtualbound state in a metal may be modelled by a Hamiltonian which is the sum ofthree terms

H = H0 + HV = Hc + Hd + HV (1920)where Hc describes the electrons in th conduction band, the Hamiltonian Hd

represents the (isolated) localized d level on the impurity and the term HV

describes the coupling. The conduction band is expressed in terms of the numberof conduction electrons in the Bloch states (k, σ) with dispersion relation E(k)through

Hc =∑k,σ

E(k) nk,σ

=∑k,σ

E(k) c†k,σ ck,σ (1921)

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where c†k,σ and ck,σ Likewise the energy for an electron in the localized d stateis given by the binding energy Ed times the number of d electrons of spin σ,

Hc =∑

σ

Ed nd,σ

=∑

σ

Ed d†σ dσ (1922)

where d†σ and dσ respectively create and annihilate an electron of spin σ inthe localized d state. The hybridization or coupling term is given by the spinconserving term

HV =1√N

∑k,σ

[V (k) c†k,σ dσ + V (k)∗ d†σ ck,σ

](1923)

The first term represents a process whereby an electron in the d orbital tun-nels into the conduction band, and the Hermitean conjugate term representsthe reverse process. It is assumed that the conduction band states have beenorthogonalized to the localized states, so that the conduction band fermion op-erators ant-commute with all the local operators.

The Resolvent Green’s function can be calculated from the expression

( z − H0 )1

z − H= HV

1z − H

(1924)

Evaluating the matrix elements of this equation between the eigenstates of H0

yields the coupled equations

( z − Ed ) < d | 1z − H

| d > =1√N

∑k

V (k) < d | 1z − H

| k >

(1925)

and

( z − E(k) ) < d | 1z − H

| k > =1√N

V (k)∗ < d | 1z − H

| d >

(1926)

These equations can be combined to yield the matrix elements of the resolventGreen’s functions as

Gd,d(z) = < d | 1z − H

| d >

=1

z − Ed − Σ(z)(1927)

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where the self-energy Σ(z) is given by

Σ(z) =1N

∑k

| V (k) |2

z − E(k)(1928)

the real part of the self energy can be interpreted as producing a renormalizationof the energy of the localized level Ed, and the imaginary part can be interpretedas giving rise to an width or lifetime τ such that

h

2 τ= Im Σ(Ed) (1929)

The conduction band Resolvent Green’s function is evaluated, from a similarset of coupled equations as

Gk,k′(z) = < k | 1z − H

| k′ >

=δk,k′

z − E(k)+

+1N

V (k)z − E(k)

Gdd(z)V (k′)∗

z − E(k′)(1930)

From these equations it can be seen that the density of states of the d level isgiven in terms of the imaginary part of Σ(E) via

ρd(E) =1πIm

[Gd,d(E − iε)

]=

Im Σ(E − iε)(E − Ed − Re Σ(E − iε)

)2

+(Im Σ(E − iε)

)2

(1931)

The impurity density of states is approximately in the form of a Lorentziancentered on Ed with a width given by Im Σ(E). The width is given by

ImΣ(E) =π

N

∑k

| V (k) |2 δ( E − E(k) )

≈ 1N

π | V |2 ρ(Ed) (1932)

which is related to the Fermi-Golden rule expression for the rate for the local-ized electron to tunnel into the conduction band density of states ρ(E). Thus,the virtual bound state can be interpreted in terms of a narrow band density ofstates which is connected to the extended conduction band states.

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18.3 Disorder

Give a distribution of impurities in a solid, the potential in the solid will benon-uniform. The thermodynamic properties of the solid can be expressed interms of the energy eigenvalues, or alternatively the poles of the Green’s func-tion. For a macroscopic sample, the exact distribution of impurities will not bemeasurable and the thermodynamic properties is expected to be representativeof all distributions of impurities. Therefore, the average value of a quantity canbe represented by averaging over all configurations of the impurities. It caneasily be shown that the configurational averaged density of states is given bythe discontinuity across the real axis of the configurational averaged resolventGreen’s function.

The Hamiltonian of a binary (A-B) alloy, with site disorder, may be repre-sented by

H = H0 + V (1933)

in which H0 describes the tight-binding bands of a pure metal with a dispersionrelation

E(k) = − td∑

i=1

cos ki ai (1934)

and the randomness appears as a shift of the binding energies of the atomicorbitals

V =∑R

ER | ψR > < ψR | (1935)

where ER can take on the values EA or EB depending on the type of atompresent at site R.

The average Green’s function is given by

G(z) =(z − H0 − V

)−1

(1936)

which can be expressed as

G(z) =(z − H0 − Σ(z)

)−1

(1937)

where the operator Σ(z) is complex and is known as a self-energy. Since the con-figurational averaged Green’s function has translational invariance, then so doesthe self-energy. It represents the effect of the randomly distributed impuritieson the eigenvalue spectrum. Due to the fluctuations in the random potential,the energy eigenvalues may form continua.

The averaged Green’s function can be calculated by expanding the Green’sfunction in powers of the potential and then performing the configurational av-erage. For strongly fluctuating potentials the resulting power series may be

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slowly convergent, or it may not even be convergent at all. Therefore, it may bepreferable to expand the Green’s function about the self energy. This procedureleads to the coherent potential approximation.

18.4 Coherent Potential Approximation

The potential difference between a specific potential due to the impurities andthe self energy can be expressed as

V (z) = V − Σ(z) (1938)

The resolvent Green’s function for this type of disordered impurity problem canbe expressed as

G(z) =(z − H0 − Σ(z) − V (z)

)−1

(1939)

which can be expressed in terms of the T-matrix via

G(z) = G(z) + G(z) T (z) G(z) (1940)

where the T-matrix is given by

T (z) = V (z)(

1 − G(z) V (z))−1

(1941)

On taking the configurational average one finds that the averaged T-matrixmust be zero

T (z) = V (z)(

1 − G(z) V (z))−1

= 0 (1942)

This equation can be used to obtain the self-energy.

For the A−B alloy the effective potential is

V (z) =∑R

( ER − Σ(z) ) | ψR > < ψR | (1943)

It is assumed that the concentration of A atoms is c and the concentration of Batoms is ( 1 − c ) and these are randomly distributed. It is also assumed thatthe T-matrix can be represented as a sum of single site T-matrices, in whichthe scattering is referenced to an appropriately chosen averaged medium. Thisis the single site approximation. The averaged T-matrix can be written as

T (z) = cEA − Σ(z)

1 −[EA − Σ(z)

]< R0 | G(z) | R0 >

+ ( 1 − c )EB − Σ(z)

1 −[EB − Σ(z)

]< R0 | G(z) | R0 >

(1944)

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The Coherent Potential Approximation (C.P.A.) sets

T (z) = 0 (1945)

The resulting equations are non-trivial to solve since the Green’s function in thedenominator is formed from a sum over the Bloch states and also involves theself-energy.

< R0 | G(z) | R0 > =1N

∑k

1z − Σ(z) − E(k)

= < R0 | G0( z − Σ(z) ) | R0 > (1946)

Nevertheless, this can be solved numerically or alternatively if the sum overBloch energies can be evaluated analytically, a analytic solution may be found.

The C.P.A. is expected to be valid in the limit of a dilute concentration ofimpurities 1 c, weak scattering t | EA − EB | and fortuitously in theatomic limit, where the single site approximation is exact. In general the C.P.A.may be only trusted to yield the density of states, and not transport properties.The density of states obtained from this method resembles a smeared versionof the weighted sums of the density of states of a solid composed of A atomsand the density of states composed of B atoms. For small magnitudes in thedifferences of the site energies, the two components overlap, but they separatefor large differences in the site energies. When the bands are split, the widthsof the component bands are drastically modified from the ideal superposition,reflecting the increasing separation between sites which decreases the tendencyto form bands. The effects of the impurity scattering is to produce a smearing,which washes out any structure such as van Hove singularities. Transport prop-erties crucially depend on the spatial extended nature of the energy eigenstates,which may be destroyed by long-ranged correlations in the random potentials.This type of phenomenon is completely absent in C.P.A., and can lead to theenergy eigenstates becoming localized.

18.5 Localization

The phenomenon of disorder induced localization is easiest to understand interms of states at the tail edge of a band. Just as one impurity with a suffi-ciently strong attractive potential may cause a bound state to form around it,a bound state may also be formed for a number of nearby atoms with weakerattractive interactions, in which case the bound state may be of larger spatialextent. In both cases, they will produce localized states below the density ofstates. A distribution in the spatial separation between impurity atoms, willsmear these discrete bound states. The localized states manifest themselvesas low energy tails to the density of states. As the strength of the disorder isincreased, the number of localized states in the tails of the density of states will

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increase. One surprising feature is that a sharp energy, the mobility edge, sep-arates the states that extend throughout the crystal from the localized states.The length scale over which the states on the localized side of the mobility edgeare extra-ordinarily long, and cannot be treated by perturbation methods butrequire renormalization group type of approaches.

On using the electron-hole symmetry for states at the top of the band, onediscovers that the states at the top edge of the band will also become localizeddue to disorder, and also have a mobility edge. On increasing the strength ofthe disorder, the mobility edges will move towards the middle of the bands.A disorder driven metal insulator transition will occur when the mobility edgecrosses the Fermi-energy. This type of transition is known as the Andersontransition. The effect of many-body interactions complicate the physics on themetallic side of the Anderson transition, where weak localization occurs andthere effects are most marked. On the insulating side of the transition, con-duction will be still possible but only due to the thermal excitation of electronsto the itinerant states above the mobility edge, or by thermally assisted tun-nelling processes. For sufficiently strong disorder all the states in the band willbecome localized. All states in one-dimensional and two-dimensional systemsmust become localized, for arbitrarily small strengths of disorder. However, thislocalization need not show up in experiments if the length scale over which thestates are localized is smaller than the sample size.

18.5.1 Anderson Model of Localization

In a doped semiconductor such as P doped Si, as the impurity concentrationis increased, it is expected that the energy levels of the isolated impurities willbroaden and form bands. For large concentrations, the impurity level wave func-tions are expected to overlap and become extended. Thus, it is expected thata metal insulator transition will occur as a function of impurity concentration.The metal insulator transition can be described by a tight-binding model of adisordered system

H =∑i,σ

εic†i,σ ci,σ −

∑i,j,σ

t c†i,σ cj,σ (1947)

where t are the nearest neighbor tight-binding hopping matrix elements and thesum over (i, j) are assumed to run over pairs of nearest neighbors lattice sites.The site energies εi are assumed to be random variables uniformly distributedover an energy width W

P (ε) =1W

for − W

2< ε <

W

2= 0 otherwise

(1948)

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The degree of disorder is measured by the dimensionless parameter W t.

For sufficiently large W/t the states are expected to all be localized. Thecritical value of (W/t)c is expected to be dependent on the dimensionality ofthe lattice. For W/t less than the critical value the states around the center ofthe tight-binding bands are extended, while states near the band edges are lo-calized. There are energies Ec, called mobility edges that separate the localizedand extended states. When the chemical potential µ crosses the mobility edge,the states at the Fermi-energy change their characters and a metal non-metaltransition occurs. This is known as the Anderson transition.

The wave functions corresponding to extended and localized states havedifferent characters. A wave function for the disordered solid can be expressedas a linear combination of atomic wave functions

ψ(r) =∑R

C(R) φ(r −R) (1949)

A delocalized wave function has an amplitude C(R) which does not decay tozero at large distances. A localized wave function is expected to decay to zerowith an exponential envelope

| C(R) | ∼ exp[− | R | /ξ

](1950)

The spatial extent of the envelope is given by the correlation length ξ. The cor-relation length is expected to depend on the energy E of the energy eigenstate.The correlation length is expected to diverge as E approaches the mobility edgeEc. In the Anderson transition, the density of states of the localized statesis expected to be continuous. Numerical studies show that the wave functionexhibits long-ranged fluctuations close to the critical value of W/t, and appearsto be self similar when viewed at all length scales.

18.5.2 Scaling Theories of Localization

Since numerical studies of Anderson localization are hampered by finite sizeeffects which tend to obscure the effect of localization, Licciardello and Thoulessintroduced a number g(L) which describes the sensitivity of energy eigenvalueson the boundary conditions, for a system with linear dimension L. This numberis defined as the ratio

g(L) =∆EδE

(1951)

where ∆E is the shift in energy levels that occurs when the boundary conditionson the wave function are changed from periodic to anti-periodic. The quantityδE is the mean spacing of the energy levels of the finite size sample. If the wave

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functions are exponentially localized, it is expected that

g(L) ∝ exp[− 2 L

ξ(E)

](1952)

On the other hand, if the wave functions are extended the energy shift due tothe different boundary conditions should be of the order of

h

τ(1953)

where τ is the time required for the electron to diffuse to the boundary ofthe sample. This diffusion time is essentially independent of L. The differentdependencies of g(L) on L provide a simple criterion in numerical studies as towhether the states are extended or localized. The quantity g(L) is also foundto be equal to a conductance

g(L) =G(L) he2

(1954)

and is related to the conductivity via

g(L) ∝ Ld−1 σ (1955)

The scaling theory of localization is based upon the length dependence ofg(L), (Abrahams, Anderson, Licciardello and Ramakrishnan). In a scale changeof a d-dimensional system with linear dimension L, the length scale L is changedto b L. It is expected that g(bL) is related to g(L) and the factor b, and nothingelse. This is summarized in the formula

g(bL) = f [b, g(L)] (1956)

where f(x) is a universal scaling function, which only depends on the dimen-sionality d of the lattice. On considering an infinitesimal scale change

b = 1 +dL

L(1957)

then one can introduce a scaling function

d ln g(L)d lnL

=∂f(b, g)/∂b

g(L)= β[g(L)] (1958)

The functional β[g] completely specifies the scaling property of the conductivityin disordered systems. It is assumed that β(g) is a smooth continuous functionof g.

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The asymptotic forms of β can be found in the limits g → 0 and g → ∞.In the strongly localized regime g → 0 where the wave function is exponentiallylocalized one finds that as

g(L) ∝ exp[− 2

L

ξ

](1959)

thenβ(g) = − 2

L

ξ= ln g (1960)

Thus, β(g) tends to − ∞ as g → 0. In the metallic limit g → ∞, then as σis finite and independent of L one has

β(g) = ( d − 2 ) (1961)

From this one finds that the system is localized for all spatial dimensionalitiesless or equal to two, d < 2, as β(g) is always negative and, thus, on increasingL g(L) scales to zero. Thus, in the limit of a large system, no matter howweak the randomness is, the states are always localized in two dimensions. Intwo dimensions the conductivity decreases with increasing L, logarithmicallyat small values of g and exponentially at large values of g. Furthermore, ford > 2, there is a critical value of gc such that for g > gc the system scalesto the metallic limit, β(g) = ( d − 2 ) since as L is increased and when β(g)is positive then limL → ∞ g(L) → ∞. For g < gc, on increasing L, β(g)is negative and g(L) scales to zero. From the scaling theory one can infer thedependence of conductivity on the concentration of impurities, c > c0 , closeto the metal insulator transition

σ = σ0 ( c − c0 )1 (1962)

where the exponent of unity can be exactly obtained via perturbation theory.

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19 Magnetic Impurities

19.1 Localized Magnetic Impurities in Metals

When transition metal or rare earth impurities are dissolved in simple metals,the electronic states on the impurities hybridize with the conduction band statesand form a Friedel virtual bound state. Since the impurity states are localizedthe Coulomb interaction U between two electrons occupying these states is largeand has to be taken into consideration. The Hamiltonian can be expressed as

H = H0 + Hint (1963)

where the Hamiltonian H0 represents the non-interacting virtual bound state

H0 =∑k,σ

E(k) c†k,σ ck,σ +∑

σ

Ed d†σ dσ

+∑k,σ

[V (k) c†k,σ dσ + V (k)∗ d†σ ck,σ

](1964)

and the Coulomb interaction U between a pair of electrons in the ( spin onlydegenerate ) impurity state is given by

Hint = U d†↑ d†↓ d↓ d↑ (1965)

This is the Anderson impurity Hamiltonian is exactly soluble using numericalrenormalization group or Bethe-Ansatz techniques. The mean field solution willbe outlined below.

19.2 Mean Field Approximation

The interaction term Hint can be represented in terms of fluctuations about theaverage value

∆nσ = d†σ dσ − < | d†σ dσ | > (1966)

and the average spin value

nσ = < | d†σ dσ | > (1967)

asHint = U ∆n↑ ∆n↓ + U

∑σ

∆nσ n−σ + U n↑ n↓ (1968)

In the mean field approximation, the term quadratic in the occupation numberfluctuations is neglected, yielding

HMF = U∑

σ

nσ n−σ − U n↑ n↓ (1969)

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The localized electrons experience an effective spin dependent binding energygiven by

Hd =∑

σ

( Ed + U n−σ ) d†σ dσ (1970)

where the average number of electrons in the localized level of spin sigma isfound as an integral over the density of states of the virtual bound state, whichis given by

nσ =∫ ∞

−∞dε f(ε) ρσ

d (ε) (1971)

where

ρσd (ε) =

1πIm

[1

ε − Ed − U n−σ − Σ(ε)

](1972)

The self energy can be represented by a constant imaginary part with value ∆and a small energy shift that can be absorbed into the definition of Ed. Hence,the spin dependent density of states can be approximated as

ρσd (ε) =

∆(ε − Ed − U n−σ

)2 + ∆2(1973)

Thus, at T = 0 one finds that

nσ =1π

cot−1

(Ed − µ + U n−σ

)(1974)

where as cot θ is defined on the interval 0 to π and runs between ∞ and − ∞,then cot−1 x runs over the range from π to 0. These two coupled equations haveto be solved self-consistently. This can be done by changing the variables

x =µ − Ed

y =U

∆(1975)

which are dimensionless measures of the position of the Fermi-energy relative tothe d level and the Coulomb interaction. The pair of self-consistency equationsbecome

cotπ n↑ = ( y n↓ − x )cotπ n↓ = ( y n↑ − x )

(1976)

Thee is a non-magnetic solution

n↑ = n↓ = n (1977)

This has a unique solution for 0 < n < 1 given by the solution of

cotπ n = ( y n − x ) (1978)

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corresponding to a partial occupation of the localized levels. In this case, thevirtual bound state does not posses a magnetic moment. However, if y is largethe equations have two magnetic solutions. These solutions only occur for suffi-ciently large values of y as can be seen by linearizing the self-consistency equa-tions in terms of the variable m defined by

nσ = n + σ m (1979)

On equating the first two terms in the expansion in m, one finds

cotπ n = ( y n − x )π

sin2 π n= y (1980)

These can be re-written as

x

y=

12 π

( θ − sin θ )

1y

=1

2 π( 1 − cos θ ) (1981)

where θ = 2 π n. This leads to the identification of line separating the areasof phase space in which the impurity is magnetic from the area in which theimpurity is non-magnetic. The tendency for magnetism is strongest when thed-d interaction U is large and when n is close to 1

2 , i.,e, when Ed and Ed + U arepositioned symmetrically about the Fermi-level. In this case the total numberof d electrons of both spins is almost unity. The non-magnetic solution occurswhen U is small or when the d level is either almost completely filled or almostcompletely empty.

For large y, magnetic solutions are described by

nσ = 1 − 1π y

n−σ =1

π ( y − x )(1982)

These solutions are doubly degenerate and correspond to the spin up and spindown states of the impurity. It is to be expected that the solution should have acontinuous symmetry with respect to the orientation of the impurity spin. How-ever, the spin rotational invariance has been specifically broken by the mean fieldapproximation through the choice of a specific quantization axis.

Thus, the mean field solution of the Anderson model contains magnetic andnon-magnetic solutions. The appearance of magnetic moments of transitionmetal impurities in metals can be interpreted in terms of the change of positionand width of the virtual bound state.

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19.2.1 The Atomic Limit

In the case, when the hybridization is set to zero, the d orbital is entirelylocalized. The local level is entirely decoupled from the conduction band andthe model is exactly soluble. The local d level can be described in terms of theeigenstates of the d number operator. The four basis states that correspondto the d level being unoccupied, with energy 0, two states which correspond tothe d state being occupied by one electron, with energy Ed and one state inwhich the d level is occupied by two electrons. This has energy 2 Ed + U .The excitation energy required to put an additional particle in the d shell is,therefore, either Ed or Ed + U depending on whether the initial d state of theimpurity is unoccupied or singly occupied. The ladder of excitations betweenthese four states are described by the four operators

d†σ ( 1 − d†−σ d−σ )

d†σ d†−σ d−σ

(1983)

The first two take the system from the non-degenerate vacuum state to the dou-bly degenerate singly occupied states, and the second pair of operators take thesystem from the doubly degenerate singly occupied state to the non-degeneratedoubly occupied state.

19.3 The Schrieffer-Wolf Transformation

If the local magnetic impurity has a narrow width and is almost completelyoccupied by one electron, then the Anderson Model can be mapped onto amodel of a localized magnetic moment by the Schrieffer-Wolf transformation.The zero-th order Hamiltonian can be considered to be the terms in which thehybridization is set to zero. Thus, for the present purposes one may write

H = H0 + HV (1984)

where H0 describes the ionic d states and the conduction band states.

H0 =∑k,σ

E(k) c†k,σ ck,σ +∑

σ

Ed d†σ dσ

+ U d†↑ d†↓ d↓ d↑

(1985)

The Hamiltonian HV is the hybridization which couples the local and conductionband states.

HV =∑k,σ

[V (k) c†k,σ dσ + V (k)∗ d†σ ck,σ

](1986)

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The Schrieffer-Wolf transformation is based on a canonical transformation whichacts on the operators A and is of the form

A′ = exp[

+ S

]A exp

[− S

](1987)

where S is an anti-Hermitean operator. That is the operator S satisfies

S† = − S (1988)

Thus, if the operator A is Hermitean then A′ is also Hermitean. The canonicaltransformation leads to the same expectation values if the states | Ψ > arealso transformed to

| Ψ′ > = exp[

+ S

]| Ψ > (1989)

In particular the eigenvalues of H ′ and H are identical. The Schrieffer-Wolftransformation S is chosen such that terms linear in the hybridization HV vanishin the transformed Hamiltonian H ′. This can only be achieved if S is assumedto be of the same order of HV . In this case, the transformed Hamiltonian can beexpanded in powers of HV and S. On retaining the terms up to second order,one finds

H ′ = H0 + HV + [ S , H0 ]

+ [ S , HV ] +12!

[ S , [ S , H0 ] ] + . . .

(1990)

The operator S is chosen such that the terms linear in V (k) vanish. Hence, itis required that S satisfies the linear equation

[ S , H0 ] = − HV (1991)

This is an operator equation, and S is determined if all its matrix elements areknown. This requires that a complete set of states be used. The simplest set ofcomplete sets correspond to the eigenstates of H0, | φn > with eigenvalues En.In this case, matrix elements are found as

< φm | S | φn > =< φm | HV | φn >

Em − En(1992)

Thus, the operator S connects states which differ through the presence of anadditional conduction electron and a deficiency of an electron in the local orbital,and vice versa. The energy denominators are of the form E(k) − Ed or E(k) −Ed − U depending on the state of occupation of the local level. Thus, the anti-Hermitean operator S can be expressed in terms of the four creation operators

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for the localized level. The operator S is found as

S =1√N

∑k,σ

[V (k)

c†k,σ dσ

E(k) − Ed

(1 − d†−σ d−σ

)

+ V (k)c†k,σ dσ

E(k) − Ed − U

(d†−σ d−σ

)− V (k)∗

d†σ ck,σ

E(k) − Ed

(1 − d†−σ d−σ

)− V (k)∗

d†σ ck,σ

E(k) − Ed − U

(d†−σ d−σ

) ](1993)

Having determined the operator S, the Hamiltonian to second order in V isgiven by

H ′ = H0 + [ S , HV ] +12!

[ S , [ S , H0 ] ] + . . .

= H0 +12!

[ S , HV ] + . . .

(1994)

The transformed Hamiltonian H ′ contains an interaction term whereby theconduction electrons are scattered from the different singly occupied states ofthe d impurity. On expressing the conduction band factors in terms of thematrix elements of the Pauli-spin matrices

σαk,k′ =

∑δ,γ

c†k,δ < δ | σα | γ > ck′,γ (1995)

and likewise for the local operators

Sαk,k′ =

∑δ,γ

d†δ < δ | σα | γ > dγ (1996)

one finds that in addition to a potential scattering term there is also an in-teraction between the components of the spin density operators. The spin-flipcontribution of the interaction is of the form

H ′spin−flip =

12

∑k,k′,σ

[ (1

E(k′) − Ed

)+(

1E(k′) − Ed − U

) ]

×(V (k) V (k′)∗ c†k′,−σ ck,σ d

†σ d−σ + V (k)∗ V (k′) c†k,σ ck′,−σ d

†−σ dσ

)(1997)

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To describe scattering of electrons close to the Fermi-energy one may set E(k) =E(k′), then the effective exchange interaction has the strength

Jk,k′ = Re

(V (k) V (k′)∗

) [1

Ed − E(k)+

1U + Ed − E(k)

](1998)

The total spin dependent part of the interaction is recognized as just involvingthe scalar product of the Fourier components of the two spin densities. For asingly occupied level, where Ed − µ is negative, the coefficient Jk,k′ also has anegative sign if U is sufficiently large, so that the energy is lowered whenever theexpectation values of both the spin density operators are anti-parallel. Thus,classically, the energy is lowered whenever the polarization produced by theconduction electron gas is anti-parallel to the spin of the local moment. Thistype of coupling is known as an anti-ferromagnetic interaction. The alternativetype of coupling occurs when the sign of J is positive, and the ferromagneticinteraction attempts to polarize the conduction electron spin density to be par-allel to the local spin density.

19.3.1 The Kondo Hamiltonian

The resulting Hamiltonian is the Kondo Hamiltonian, it contains an interac-tion between the localized magnetic moment and the spins of the conductionelectrons. The Hamiltonian can be expressed as

H = H0 + Hint (1999)

where H0 represents the Hamiltonian for the conduction electrons

H0 =∑k,σ

E(k) c†k,σ ck,σ (2000)

and the interaction is given by

Hint = − J S . σ(0) (2001)

where S is a local moment and σ(0) is the spin of the conduction electrons atthe position of the impurity spin. The components of the conduction electronspin is given in terms of matrix elements of the Pauli-spin matrices

σα(0) =1N

∑k,k′;γ,δ

c†k,δ < δ | σα | γ > ck′,γ (2002)

It is convenient to write the spin dependent interaction in terms of the spinraising and lowering operators for the local spin and the conduction electron

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spin density

S± = Sx ± i Sy

σ± = σx ± i σy

(2003)

with the aid of the identity

S . σ = Sz σz +12

( S+ σ− + S− σ+ ) (2004)

Hence, the interaction is written as

Hint = − J

N

12

∑k,k′

[Sz ( c†k,↑ ck′,↑ − c†k,↓ ck′,↓ )

+ S+ c†k,↓ ck′,↑ + S− c†k,↑ ck′,↓

](2005)

19.4 The Resistance Minimum

The Kondo effect results in a minimum in the resistivity of metals. The mini-mum in the resistivity is due to the increasing T 5 resistivity caused by electron-phonon scattering and a decreasing contribution from the impurity spin flipsscattering, which in an intermediate temperature regime follows a ln T varia-tion

ρ(T ) = ρ(0) + b T 5 − c ρ1 Jρ(µ) S ( S + 1 ) ln kBT ρ(µ) (2006)

where c is the concentration of impurities. Then, the resistivity shows a mini-mum at a concentration temperature

Tmin = (ρ1 J ρ(µ) S(S + 1)

5 b)

15 c

15 (2007)

in agreement with experimental findings.

The lnT term in the resistivity comes from scattering process to third orderin J . This can be seen by considering the T-matrix for non-spin flip scatteringof an up sin electron in second order. The T-matrix will be evaluated on theenergy shell E(k) = E(k′), and E will be set to the ground state energy. Tolowest order, the non-spin flip scattering matrix elements are given by

< k′ ↑ | T (1)(E + iε) |k ↑ > =J

NSz (2008)

whereas to second order one has four non-zero contributions, two contributionsfrom the spin flip part ( S± ) of the interactions and two contributions from

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the non spin flip part ( Sz ). The non spin flip part gives rise to a term inT (2)(E + iε) of

< k′ ↑ | T (2)zz (E + iε) |k ↑ > =

(J

N

)2 (Sz

2

)2 ∑k1,k2

< k′ ↑ | ( c†k1,↑ ck′1,↑ − c†k1,↓ ck′1,↓ )

× 1E − H0 + iε

( c†k2,↑ ck′2,↑ − c†k2,↓ ck′2,↓ ) | k ↑ >

(2009)

As only the spin up terms contribute to the scattering of the spin up electronthe term simplifies to yield

=(J

N

)2 (Sz

2

)2 ∑k1,k2

< k′ ↑ | c†k1,↑ ck′1,↑1

E − H0 + iεc†k2,↑ ck′2,↑ | k ↑ >

(2010)

This has two contributions, one which corresponds to k′ = k1 and k = k′2 andthe other with k′ = k2 and k = k′1. The sum of these terms are evaluated as

=(J

N

)2 (Sz

2

)2 ∑k2

1 − f(E(k2))E(k) − E(k2) + iε

−(J

N

)2 (Sz

2

)2 ∑k1

f(E(k1))E(k1) − E(k) + iε

=(J

N

)2 (Sz

2

)2 ∑k1

1E(k) − E(k1) + iε

(2011)

The singularity at E(k1) = E(k) yields a finite result when integrated over k1.Thus, there is no non-analytic behavior originating from the Sz terms in theinteraction, which is just of the order J2 ρ(µ) which is just a factor of J ρ(µ)smaller than the leading contribution to the T-matrix.

The two spin flip contributions to the T-matrix are given by

< k′ ↑ | T (2)+−(E + iε) | k ↑ > =

(J

2 N

)2 ∑k1,k2

< k′ ↑ | S+c†k1,↓ck′1,↑1

E − H0 + iεS−c†k2,↑ck′2,↓| k ↑ >

(2012)

and

< k′ ↑ | T (2)−+(E + iε) | k ↑ > =

(J

2 N

)2 ∑k1,k2

< k′ ↑ | S−c†k1,↑ck′1,↓1

E − H0 + iεS+c†k2,↓ck′2,↑| k ↑ >

(2013)

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respectively. These terms are calculated to be

< k′ ↑ | T (2)+−(E + iε) | k ↑ > =

(J

2 N

)2 ∑k1,k2

< k′ ↑ | c†k1,↓ck′1,↑S+ S−

E − H0 + iεc†k2,↑ck′2,↓| k ↑ >

=(

J

2 N

)2

S+ S−∑k2

f(E(k2)E(k) − E(k2) + iε

(2014)

and

< k′ ↑ | T (2)−+(E + iε) | k ↑ > =

(J

2 N

)2 ∑k1,k2

< k′ ↑ | c†k1,↑ck′1,↓S− S+

E − H0 + iεc†k2,↓ck′2,↑| k ↑ >

=(

J

2 N

)2

S− S+∑k1

1 − f(E(k1)E(k) − E(k1) + iε

(2015)

In this case, the two terms cannot be combined to give a result independent ofthe Fermi-function, as S+ and S− do not commute. In this case, one can usethe identities

S+ S− = S2 − ( Sz )2 + Sz

S− S+ = S2 − ( Sz )2 − Sz

(2016)

The terms proportional to S2 − ( Sz )2 combine to yield an analytic contributionto the T-matrix of

< k′ ↑ | T (2)sf (E + iε) | k ↑ > =

(J

2 N

)2

( S2 − ( Sz )2 )∑k1

1E(k) − E(k1) + iε

(2017)

whereas the remaining contribution is proportional to Sz and the integration isdivergent at E(k) = E(k1) but the integration is cut off by the Fermi-function.

< k′ ↑ | T (2)sf (E + iε) | k ↑ > =

(J

2 N

)2

Sz∑k1

2 f(E(k1)) − 1E(k) − E(k1) + iε

(2018)

At finite temperatures, either the Fermi-function acts as a cut-off for the sin-gularity when the scattered particle is on the Fermi-surface E(k) = µ, or ifthe scattered particle is off the Fermi-surface, the excitation energy acts as a

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cut off. In the latter case, the second order contribution to the real part of theT-matrix can be evaluated as

< k′ ↑ | T (2)sf (E + iε) | k ↑ > ∼

(J

2 N

)2

Sz ρ(µ) ln∣∣∣∣ ( E(k) − µ ) ρ(µ)

∣∣∣∣(2019)

which is divergent when E(k) approaches µ. Thus, this second order termcan be as large as the first order term which is also proportional to Sz. Thescattering rate which enters into the resistivity is proportional to the square ofthe T-matrix and is found as

=2πhρ(µ) J2 S ( S + 1 )

3

(1 − 4 J ρ(µ) ln

∣∣ kB T ρ(µ)∣∣∣∣ + . . .

)(2020)

which gives the logarithmically increasing resistivity for magnetic impurities insimple metals. Since, the logarithmic divergence is caused by spin flip scatteringin the intermediate states, the application of a field should suppress the Kondoeffect. The resistivity and the T-matrix do not diverge at T = 0. The logarith-mic dependence found in perturbation theory saturates when all the scatteringprocesses are taken into account. The leading order logarithmic coefficient ofeach term in the perturbation expansion series (in powers of J ρ(µ)) can be cal-culated by various means (Abrikosov). In the ferromagnetic case, where J > 0,the saturation occurs at a characteristic Kondo energy or Kondo temperatureTK given by

2 J ρ(µ) ln∣∣∣∣ kB TK ρ(µ)

∣∣∣∣ = − 1 (2021)

or

kB TK = ρ(µ)−1 exp[− 1

2 J ρ(µ)

](2022)

and all the results are finite.

For the case of anti-ferromagnetic coupling, the physics scales to a strongcoupling fixed point (Anderson) so the solution must be obtained by othermeans such as Bethe Ansatz (Andrei,Weigmann). The properties of the anti-ferromagnetic solution include the cross-over from a high temperature ( T > TK

) Curie susceptibility for the free impurity moments to a Pauli paramagneticsusceptibility for T < TK , and the specific heat originating from the impuritychanges from a constant value at high temperatures to a low-temperature formhaving a linear T dependence. This indicates that the magnetic moments of theimpurity are being removed and that at low temperatures, the properties arethose of a narrow virtual bound state of width kB TK located near the Fermi-energy. In fact analysis shows that the magnetic moments are being screenedby a compensating polarization of conduction electrons, and that the cloud and

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moment form a singlet bound state of binding energy TK . For T < TK theconduction electrons occupy the bound state and the moment is screened, forT > TK the bound state is thermally depopulated and the system exhibitsproperties of the free moments. From the perspective of the Anderson impu-rity model, the density of states that is found at high temperatures followsdirectly from Anderson’s picture of a spin split virtual bound state. However,as T decreases below TK , the density of states shows a sharp peak of widthkB TK growing in the vicinity of µ. In the low-temperature limit, the heightof the Abrikosov-Suhl peak saturates on the order of ( kB TK )−1. Thus, thelow-temperature properties can be directly understood in terms of the virtualbound state with a density of states which is very large ∝ T−1

K . The propertiesof this low-temperature Fermi-liquid were established by Nozieres.

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20 Collective Phenomenon

21 Itinerant Magnetism

21.1 Stoner Theory

The Stoner theory of itinerant magnetism examines the stability of a bandof electrons to Coulomb interactions (E.C. Stoner, Rep. Prog. in Phys. 1143 (1948)). The Hamiltonian is expressed as the sum of two terms, H0 thenon-interacting electrons in the Bloch states and Hint describing the Coulombrepulsion between the electrons

H = H0 + Hint (2023)

The Hamiltonian for the non-interacting electrons in the Bloch states is writtenas

H0 =∑k,σ

E(k) nk,σ (2024)

The interaction Hamiltonian is given by

Hint =U

2

∑i,σ

ni,σ ni,−σ (2025)

where U represents the short ranged Coulomb interaction between a pair ofelectrons occupying the orbitals on the i-th lattice site. The operator ni,σ cor-responds to the number of electrons of spin σ which occupy the i-th lattice site.It is assumed that the band is non-degenerate, therefore, there is only one or-bital per lattice site which due to the limitations of the Pauli exclusion principlecan only hold a maximum of two electrons.

The interaction is treated in the mean field approximation. First it shall beassumed that translational invariance holds, so that the orbitals in each unitcell have the same occupation numbers. Also the Hamiltonian is expanded inpowers of the fluctuation operator ∆ni,σ = ni,σ − nσ so that

Hint =U

2

∑i,σ

[∆ni,σ ∆ni,−σ + nσ ∆ni,−σ + n−σ ∆ni,σ + n−σ nσ

](2026)

and then the second order fluctuations are ignored. This leads to the interactionenergy being approximated in terms of single particle operators

Hint ≈U

2

∑i,σ

[ni,σ n−σ + ni,−σ nσ − n−σ nσ

]

=U

2

∑k,σ

[nk,σ n−σ + nk,−σ nσ − n−σ nσ

](2027)

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Thus, in the mean field approximation, the Hamiltonian is given by

HMF =∑k,σ

( E(k) + U n−σ ) nk,σ − NU

2

∑σ

n−σ nσ (2028)

The single particles have the spin dependent energy eigenvalues

Eσ(k) = E(k) + U n−σ (2029)

The magnetization is given by

Mz = g µB12

(n+ − n−

)= µB

∫ ∞

−∞dε f(ε)

(ρ(ε − U n−) − ρ(ε − U n+)

)(2030)

This equation has non-magnetic solutions with n+ = n− and may have fer-romagnetic solutions in which the number of up spin electrons is greater thanthe number of down spin electrons n+ 6= n−. In the ferromagnetic state, theStoner model predicts that the up-spin sub bands are rigidly shifted relativelyto the down spin bands by the exchange splitting which has a magnitude ofU (n+ − n− ). On increasing U from zero, the ferromagnetic solutions firstbecome stable when Mz ∼ 0, in which case the equations can be linearized toyield

( n+ − n− ) = U

(n+ − n−

) ∫ ∞

−∞dε f(ε)

∂ερ(ε) (2031)

The ferromagnetic state has the lowest energy when the self-consistency equa-tion is satisfied. The integral can be performed by integration by parts yielding

1 = U

∫ ∞

−∞dε f(ε)

∂ερ(ε)

= − U

∫ ∞

−∞dε ρ(ε)

∂εf(ε)

(2032)

At low temperatures, the derivative of the Fermi-function can be replaced by adelta function at the Fermi-energy.

− ∂

∂εf(ε) = δ(ε − µ) (2033)

This yields the Stoner criterion for ferromagnetism as

1 < U ρ(µ) (2034)

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where ρ(µ) is the density of states per spin at the Fermi-energy.

If the Stoner criterion is satisfied the paramagnetic state is unstable to theferromagnetic state, and a spontaneous magnetic moment Mz occurs at T = 0.The magnetization is given by the solution of the non-linear equation. The non-linear equation shows that the magnetization increases with increasing U , andsaturates to a value which is one Bohr magneton per electron, for low densitymaterials which the bands have a filling of less then one per atom. In systemswhich have bands that are more than half filled, the saturation magnetic mo-ment is equal to a Bohr magneton per unoccupied state. At finite temperatures,the value of the magnetization is reduced and disappears at a critical tempera-ture Tc. Unfortunately, Stoner theory does not predict reasonable values of thecritical temperatures.

In the paramagnetic state Stoner theory predicts that the susceptibilityshould be exchange enhanced over the non-interacting susceptibility χ0

p via

χp =χ0

p

1 − U ρ(µ)(2035)

For systems which are close to the ferromagnetic instability, the susceptibilityshould take on large values. This is the case for Pd in which the d band isalmost completely occupied.

——————————————————————————————————

21.1.1 Exercise 76

Determine the critical temperature Tc predicted by Stoner theory.

——————————————————————————————————

21.1.2 Exercise 77

Determine the paramagnetic susceptibility by using Stoner theory.

——————————————————————————————————

21.2 Linear Response Theory

The spatially varying magnetization MZ(r) of a paramagnetic system to a spa-tially varying applied magnetic field Hz(r) are related by the z − z component

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of the magnetic tensor susceptibility, via the linear relationship

Mz(r) =∫

d3r′ χz,z(r, r′) Hz(r′) (2036)

This is a special case of the more general relation

Mα(r) =∫

d3r′∑

β

χα,β(r, r′) Hβ(r′) (2037)

For translational invariant systems the expression for the response function isonly a function of the difference r − r′. Also for non magnetic systems, thatpossess spin rotational invariance, the susceptibility tensor is diagonal and thediagonal components are related via

χx,x(r − r′) = χy,y(r − r′) = χz,z(r − r′) (2038)

The relation between the magnetic response and the applied field becomes sim-pler, after Fourier transforming. The Fourier transform of the magnetization isdefined as

M(q) =∫d3r exp

[− i q . r

]M(r) (2039)

The Fourier transform of the magnetization is related to the Fourier transformof the applied field via

Mα(q) =∑

β

χα,β(q) Hβ(q) (2040)

The response function can be evaluated from perturbation theory, in which theZeeman interaction

HZeeman = −∫

d3r′ M(r′) . H(r′)

= −∫

d3q M(q) . H(q)

(2041)

is treated as a small perturbation.

For convenience, χz,z(q) shall be calculated by reducing it to a previouslyknown case. The change in density of electrons of spin σ, with Fourier com-ponent q, produced in response to an applied spin dependent potential. TheFourier component of the potential is given by

Vσ(q) = − gµB

2Hz(q) σ + U ρ−σ(q) (2042)

Thus, the charge density is given by two coupled equations

ρσ(q) = χ0(q)(− g µB

2Hz σ + U ρ−σ(q)

)(2043)

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for each spin polarization. In the above expression, χ0(q) is the Lindharddensity-density response function, per spin. On combining these equations onefinds, the z component of the magnetization produced by a magnetic field ap-plied along the z direction is

Mz(q) = − g µB

2χ0(q)

(g µB Hz(q) + U Mz(q)

)(2044)

Thus, it is found that the Pauli paramagnetic susceptibility

χz,zp (q) =

Mz(q)Hz(q)

= − g2 µ2B

42 χ0(q)

1 + U χ0(q)(2045)

It is usual to use re-write this expression in terms of the reduced non-interactingmagnetic susceptibility defined by

χz,z0 (q) = − 2 χ0(q) (2046)

instead of the density-density response function χ0(q). This yields the result

χz,zp (q) =

(g µB

2

)2 χz,z0 (q)

1 − U2 χz,z

0 (q)(2047)

Since the reduced non-interacting magnetic susceptibility is positive, and U ispositive, the paramagnetic susceptibility is enhanced for sufficiently small valuesof U .

21.3 Magnetic Instabilities

The reduced non-interacting susceptibility may have maxima at certain valuesof q, say Q, which are determined by the band structure and the occupancy ofthe non-interacting bands, through the non-interacting susceptibility χ0(q). Ifthe value of the non-interacting susceptibility at these maxima are finite, thenthe denominator of the Pauli-paramagnetic susceptibility may become small atthese q values for sufficiently small values of U . This has the effect that, forsmall U , the Pauli-paramagnetic susceptibility is enhanced at these Q values. IfU is increased further, there will be a critical value of U , Uc at which point thedenominator will fall to zero and the susceptibility at Q will become infinite.The divergence of the susceptibility at Q indicates that an infinitesimal appliedfield can produce a finite staggered magnetization Mz(Q). Although the anal-ysis was performed with the z axis being the axis of quantization, a similaranalysis could have been performed in which any arbitrarily chosen direction.The infinitesimal field may be produced by a spontaneously statistical fluctua-tion, and have an arbitrary direction. This field will force the system to order

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magnetically by having a finite M(Q) in the spontaneously chosen direction.The system, by spontaneously choosing a direction for the magnetization hasspontaneously broken the symmetry of the Hamiltonian.

The critical value of U , above which the paramagnetic state becomes unsta-ble to a state with a modulated spin density M(Q), is given by

1 =Uc

2χz,z

0 (Q) (2048)

If a non-interacting system is considered which has a maximum at Q = 0 thisreduces to the Stoner criterion for ferromagnetism as

limQ → 0

χz,z0 (Q) → 2 ρ(µ) (2049)

where ρ(µ) is the density of states, per spin, at the Fermi-energy. Thus, it isfound the critical value of U is given by the criterion

1 = Uc ρ(µ) (2050)

For values of U larger than the critical value, the paramagnetic state is unstableto the formation of a ferromagnetic state.

For values of U greater than Uc, the mean field analysis has to be modifiedto include the effect of the spontaneous magnetization. For a ferromagnet, theinteraction produces a rigid splitting between the up-spin bands and down spinbands by an amount ∆ = U ( nσ − n−σ ) called the exchange splitting. Forisotropic systems, the magnetic response will crucially depend on the directionof the applied field compared to that of the spontaneous magnetization. For aferromagnet, the longitudinal response ( produced by a field which parallel tothe spontaneous magnetization M ) will be finite, as this corresponds to pro-cesses which excites the system as it stretches the magnitude of M . However,the transverse response will be infinite as this corresponds to applying a fieldthat will rotate the direction of the spontaneous magnetization until it alignswith the applied field. As the system is isotropic, this can be achieved withoutany finite energy excitations. The zero energy excitations that uniformly rotatethe magnetization in a ferromagnet are the q = 0 Goldstone modes associ-ated with the spontaneously broken continuous spin rotational invariance of theHamiltonian.

In three-dimensional systems, with almost spherical Fermi-surfaces, the in-stability can only occur at 2 kF . This can lead to a spin density wave whichhas a periodicity which is incommensurate with the underlying lattice. In lowdimensional systems, such as two-dimensional and one-dimensional organic ma-terials, there can be large sheets of the Fermi-surface which can produce a largenon-interacting susceptibility at the Q value connecting these sheets. This sus-ceptibility coupled by an interaction can produce a spin density wave in which

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the magnetization is modulated with this wave vector.

For tight-binding bands which satisfy the perfect nesting condition

E(k +Q) = − E(k) (2051)

for some Q. The non-interacting susceptibility can be evaluated as an integralover the density of states

χz,z0 (Q) = 2

∑k

f(−E(k)) − f(E(k))2 E(k)

= 2∫ ∞

−∞dε ρ(ε)

1 − 2 f(ε)2 ε

(2052)

From this one can see that if the density of states ρ(0) is non-zero, the sus-ceptibility χ(Q) will diverge logarithmically or faster than logarithmically whenµ → 0. The divergence occurs when two large portions of the Fermi-surfaceare connected by the wave vector Q, which allows the system to rearrange theelectrons at the Fermi-surface by zero energy excitations involving a momen-tum change Q. Thus, in this case, there is no energy penalty to be incurredin producing a spin density wave M(Q). The perfect nesting condition occursfor Q = π

a (1, 1, 1) in non-degenerate tight-binding bands on a simple cubiclattice, where

E(k) = − 2 ti=3∑i=1

cos ki a (2053)

Since the bands are symmetric around ε = 0, the non-interacting susceptibilitydiverges for half filled bands. In this case, the critical value of U is zero. Hence,the paramagnetic state will become unstable to a state in which the magnetiza-tion exhibits spatial oscillations with wave vector Q, even with an infinitesimallysmall value of U . In real space, the staggered magnetization of this orderedstate is given by

M(r) = M∑

i

cos πria

(2054)

The staggered magnetization on the neighboring lattice sites is oppositely ori-ented, and is anti-ferromagnetically ordered. Since anti-ferromagnetic orderingwas first proposed by Louis Neel to describe classical magnets (L. Neel, Ann. dePhysique, 17, 64 (1932), Ann. de Physique 5, 256 (1936)), this type of orderingis known as Neel ordering. Unfortunately, the Neel state is not an exact groundstate for a quantum system. The occurrence of an anti-ferromagnetically orderedstate may be accompanied by a metal-insulator transition. This process was firstdiscussed by J.C. Slater. Physically, the appearance of anti-ferromagnetic ordercould result in a doubling of the size of the real space unit cell. The electronsof spin σ travelling in the solid experience a periodic potential which contains a

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contribution due to the interaction with electrons of opposite spin. The doublingof the size of the real space unit cell, produced by the magnetic order, resultsin the volume of the Brillouin zone being halved. The new periodicity causedby the sub-lattice magnetization shows up as a spin dependent contribution tothe potential, and may produce gaps in the electronic dispersion relations at thesurface of the new Brillouin zone. If the magnitude of the spin dependent po-tential is large enough, a gap may occur all around the Brillouin zone resultingin a gap in the density of states and hence the insulating state. Such insulat-ing anti-ferromagnetic states occur in undoped La2CuO4, which is the parentmaterial of some high temperature superconductors. Although the insulatinganti-ferromagnetic state does not have low energy electronic excitations, it doeshave low energy spin excitations in the form of Goldstone modes. These arespin waves, which have the dispersion relation ω = c q.

21.4 Spin Waves

The dynamical response of the magnetization to a time and spatially varyingapplied magnetic field of wave-vector q and frequency ω is given by the dynam-ical response χz,z

p (q;ω). The imaginary part of this response function yields thespectrum of magnetic excitations. The imaginary part of the reduced suscepti-bility can be measured indirectly by inelastic neutron scattering experiments, inwhich the neutrons spin interacts with the electronic spin density via a dipole-dipole interaction. A simple extension of our previous analysis shows that, inthe mean field approximation,

χz,zp (q;ω) =

(g µB

2

)2 χz,z0 (q, ω)

1 − U2 χz,z

0 (q;ω)(2055)

Let us examine the imaginary part of the response function for a paramagneticmetal, such as Pd, which is on the verge of an instability to a ferromagneticstate. Then,

Im

[χz,z

p (q;ω)]

=(g µB

2

)2 Im χz,z0 (q, ω)[

1 − U2 Re χz,z

0 (q;ω)]2

+[

U2 Im χz,z

0 (q;ω)]2

(2056)which on using the approximation

Re χz,z0 (q;ω) = 2 ρ(µ)

Im χz,z0 (q;ω) =

π

q vFρ(µ)

(2057)

for the Lindhard susceptibility, shows that the system exhibits a continuum ofquasi-elastic magnetic excitations. As the instability is approached the spec-trum is enhanced at low frequencies. These magnetic excitations are known as

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paramagnons. The lifetime of the paramagnon excitations and the frequencyof the excitations soften as the value of U is increased to the critical value Uc.Basically, this represents a slowing down of the rate at which a small regionof ferromagnetically aligned spins relax back to the equilibrium (paramagnetic)state. The existence of large amplitude paramagnon fluctuations not only mani-fest themselves in the inelastic neutron scattering cross-section which is directlyproportional to Im [ χα,β(q;ω) ] , an enhanced susceptibility but also lead tologarithmic enhancement of the linear T term in the electronic specific heat(Berk and Schrieffer, Doniach and Engelsberg), and enhancement in the T 2

term in the electrical resistivity (Lederer and Mills). These characteristics havebeen observed in metallic Pd (Schindler and Coles).

For values of U greater than Uc, the q = 0 response shows a sharp zeroenergy mode that represents the Goldstone mode of the system. In the ferro-magnetically ordered state, these excitations form a sharp (delta function like)branch of spin waves which stretch up from ω = 0 at q = 0. The transverseresponse functions are equal

χx,x(q;ω) = χy,y(q;ω) (2058)

The transverse response is expressed in terms of the spin flip response functioninvolving the spin raising and lowering operators

M±(q) = Mx(q) ± i My(q) (2059)

The spin flip response functions are

χ+,−(q;ω) = χ−,+(q;ω) = 2 χx,x(q;ω) (2060)

In the limit q → 0, the non-interacting transverse response is given by

limq → 0

χ+,−0 (q, ω) =

∆U

h ω + ∆(2061)

so the full transverse response function is given by

limq → 0

χ+,−(q, ω) = limq → 0

χ+,−0 (q, ω)

1 − U χ+,−0 (q, ω)

=∆

U h ω(2062)

Thus, in accordance with Goldstone’s theorem the instability has produced adynamic response with a zero-frequency pole. For small q and low frequenciesω < ∆, the spin waves have a dispersion relation of the form

ω = D q2 (2063)

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The spin waves of the ferromagnetic state become broader for larger (ω, q) valueswhere the spin wave branch enters and coexists with the continuum of (Stoner)spin-flip electron-hole excitations.

An anti-ferromagnet also has Goldstone modes, but unlike the ferromagnetthe order parameter (the sub-lattice magnetization) for the anti-ferromagnet isnot a constant of motion. This results in the dispersion of the Goldstone modesbeing linear in q, ω = c q, similar to the transverse sound waves in a crystallinesolid.

The above mean field type of analysis has shown that close to a magneticinstability, there will be large amplitude Gaussian fluctuations. This continuousspectrum of excitations is expected to soften as the instability is approached.The fluctuations are expected to be long-ranged and long-lived. However, theabove mean field analysis is expected to fail close to the transition, where criticalfluctuations should be taken into account. Unlike, most other phase transitions,the phase transition that has just been described occurs at T = 0. The criticalfluctuations are not thermally excited and cannot be treated classically but arezero point fluctuations associated with the existence of a quantum critical point.

21.5 The Heisenberg Model

The above model of itinerant magnetism is believed to be appropriate for tran-sition metals only involves one type of electrons. Another model is appropriatefor materials which contain two types of electrons, such as rare earth materials,in which the magnetic moments occur in the f states which are inner orbitalsburied deep inside the f ion and the interaction is mediated by the itinerantconduction electrons.

The spin localized at site Ri is denoted by Si. The spin at site i interactswith the conduction electrons spin σ at i via a local exchange interaction

Hint = − J∑

i

Si . σi (2064)

which acts like a localized magnetic field of 2g µB

J Si. This localized magneticfield polarizes the conduction electrons, producing a polarization at site j of(

2g µB

)J Si χ

z,z(Ri −Rj) (2065)

This polarization then interacts with the spin at site j via the interaction leadingto an oscillatory interaction between pairs of localized spins of the form

H = −∑i,j

J(Ri −Rj) Si . Sj

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= −∑i,j

(2

g µB

)2

J2 χz,z(Ri −Rj) Si . Si

(2066)

The oscillations are produced by the oscillations of the response function of theconduction electrons. The Fourier transform of J(R) shows oscillations at 2 kF .This interaction was discovered independently, by Ruderman and Kittel, Ka-suya and Yosida.

22 Localized Magnetism

The nearest neighbor Heisenberg exchange interaction couples spins localizedon adjacent lattice sites.

H = − J

2

∑R,δ

S(R+ δ) . S(R) (2067)

This interaction Hamiltonian can be derived from the model of itinerant mag-netism, for large U in the case when the bands are half filled. In this case, thereis a spin at each lattice site and the exchange between the spins is the anti-ferromagnetic super exchange interaction found by Anderson. The exchangeconstant is given in terms of the tight-binding matrix element t and the Coulombrepulsion via

J = − t2

U(2068)

The Heisenberg Hamiltonian can be expressed as

H = − J

2

∑R, δ

[Sz(R) Sz(R+δ) +

12

(S+(R) S−(R+δ) + S−(R) S+(R+δ)

) ](2069)

For a ferromagnetic exchange, J > 0, the ground state of a three-dimensionallattice of spins of magnitude S consists of parallel aligned spins. The directionof quantization is chosen as the direction of the magnetization. All the spinshave their z components of the spin maximized

| Ψg > = |∏R

( mR = S ) > (2070)

This state has a total magnetization proportional to N S, and is an eigenstateof the Hamiltonian, with energy Eg = − 3 N J S2 since the z component of

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the Hamiltonian is diagonal and the spin flip terms vanish as the effect of theraising operators acting on the fully polarized state

S+(R) | mR = S > = 0S+(R+ δ) | mR+δ = S > = 0

(2071)

are both zero. The excitations of this system are the spin waves. The excitedstate wave function corresponding to a single spin wave is given by

| q > =1√N

∑R

exp[− i q . R

]S−(R) | Ψg > (2072)

It corresponds to a state with total spin N S − 1 as one spin is flipped over.This state is a coherent superposition of all states with one spin flipped and hastotal momentum q. The excitation energy is found from

( H − Eg ) | q > = 2 J S(

3 − cos qxa − cos qya − cos qza)| q > (2073)

At long wave lengths, the spin wave excitation energy hω(q) is found as

h ω(q) = 2 J S

(3 − cos qxa − cos qya − cos qza

)∼ J S q2 a2

(2074)

which is the branch of collective Goldstone modes which restore the sponta-neously broken spin rotational invariance of the ferromagnet. The vanishingof the frequency at q = 0 is a general consequence that the total spin isa constant of motion, and in this limit the spin wave state just corresponds to

a reduced value of the total magnetization∑

R exp[− i q . R

]S−(R) → S−.

The above excitations of the spin system are small amplitude excitations thathave a close resemblance to the harmonic phonons of the crystalline lattice. Theeffects of the interactions could be expected to produce small anharmonic cor-rections to these excitations, providing them with a lifetime and a renormalizeddispersion relation. Not all the excitations can be expressed as small amplitudeexcitations, some systems have large amplitude soliton excitations that cannotbe treated by perturbation theory. However, the small amplitude excitationscan be adequately treated as harmonic modes, as can be seen from an analysisbased on the Holstein - Primakoff transformation.

22.1 Holstein - Primakoff Transformation

The Holstein-Primakoff transformation provides a representation of localizedspins, which enables the low-temperature properties of an ordered spin system

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to be analyzed in terms of boson operators. The technique is particularly usefulfor systems where the magnitude of the spin S is large S > 1. The Hamilto-nian can be expanded in terms of boson operators, providing a description of theground state, small amplitude spin fluctuations and the anharmonic interactionbetween them.

The Holstein - Primakoff transformation of the spins represents the effectof the spin operators by a function of bosons operators (T. Holstein and H.Primakoff, Phys. Rev. 58, 1098 (1940)). The components of the spin operatorsat the i-th site Sα

i can be defined by their action on the eigenstates of Szi

Szi | mi > = mi h | mi > (2075)

In particular the spin raising and lowering operators

S±i = Sxi ± i Sy

i (2076)

have the commutation relations with Sx[Sz

i , S±i

]= ± h S±i (2077)

This can be used to show that the operators S±i have the effect of raising andlowering the magnitude of the eigenvalue Sz by one unit of h

S±i | mi > =√

S ( S + 1 ) − mi ( mi ± 1 ) h | mi ± 1 > (2078)

The Holstein Primakoff transformation represents the (2 S + 1 ) basis states| mi > by the infinite number of basis states of a boson number operator a†i ai.The boson basis | ni > states are defined through

a†i ai | ni > = ni | ni > (2079)

The relation between the basis spin states and boson states is provided by theboson representation of the z component of the spin operator

Szi = S − a†i ai (2080)

Thus, the state where the spin is aligned completely along the z axis ( mi = S)is the state with boson occupation number of zero, and the state with the lowesteigenvalue of the spins z component (mi = − S) corresponds to the bosonstate where 2 S bosons are present. The states with higher number of bosonsare un-physical and must be projected out of the Hilbert space. The effect ofthe spin raising and lowering operators are similar to the boson annihilationand creation operators, within the space of physical states. The correspondencebetween the spin and raising and lowering operators and the boson operatorscan be made exact, by multiplying the boson creation and annihilation operatorswith a function that ensures that only the physical acceptable boson states form

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the Hilbert space of the spin system. This is achieved by the representation ofthe spin lowering operator

S−i = a†i

( √2 S − a†i ai

)(2081)

which projects out states with more than 2 S bosons present. The raisingoperator is given by the Hermitean conjugate

S+i =

( √2 S − a†i ai

)ai

(2082)

This transformation respects the spin commutation relations.

Given the nearest neighbor ferromagnetic ( J > 0 ) Heisenberg Hamiltonian

H = − J

2

∑R,δ

S(R) . S(R+ δ) (2083)

this can be expressed as

H = − J

2

∑R,δ

[Sz(R) Sz(R+ δ) +

12

(S+(R) S−(R+ δ) + S−(R) S+(R+ δ)

) ](2084)

On representing this in terms of the boson operators, and expanding in powersof 1

S one finds

H = − J

2

∑R,δ

[( S − a†R aR ) ( S − a†R+δ aR+δ ) + S

(a†R aR+δ + a†R+δ aR

) ](2085)

The terms of order S2 just represents the classical ferromagnetic ground state, inwhich all the spins are aligned. The terms of order S represent excitations fromthe ground state and can be put in diagonal form by expressing them in termsof the Fourier transformed boson operators. The spatial Fourier transform ofthe boson operators are defined as

aR =1√N

∑q

exp[i q . R

]aq (2086)

and the creation operator is the Hermitean conjugate

a†R =1√N

∑q

exp[− i q . R

]a†q (2087)

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Substitution of these, and performing the sums over the spatial index, yields anapproximate expression for the Hamiltonian. The expression is the sum of theground state energy and the energies of harmonic normal modes that representthe excitations of the spin waves from the ferromagnetic ground state

H = − J

2N Z S2 + J S

∑q

a†q aq

∑δ

(1 − cos q . δ

)(2088)

The terms of higher order in 1S yield quantum corrections to the ground state

energy, the spin wave energies and also produce anharmonic interactions be-tween the spin waves.

The thermally excited spin waves have the effect of reducing the magnetiza-tion from the fully saturated T = 0 value

Mz(T ) − N S =∑

q

N(ω(q)) (2089)

where N(ω) is the Bose - Einstein distribution function. Since the ferromagneticspin waves have a dispersion relation which is ω(q) ∼ q2 at small q, one findsthat the temperature induced change in the magnetization is given by

M(T ) − M(0) ∼(kB T

J S

) 32

(2090)

for a three-dimensional lattice at low temperatures. Likewise, the thermal av-erage value of the energy can be calculated as

E(T ) + N ZJ

2S2 =

∑q

h ω(q) N(ω(q))

E(T ) − E(0) ∼ N J S

(kB T

J S

) 52

(2091)

This should result in the low-temperature specific heat being proportional toT

32 for a long range ordered insulating ferromagnet.

22.2 Spin Rotational Invariance

In the limit of zero applied magnetic field, the Heisenberg exchange Hamiltonianis invariant under the simultaneous continuous rotation of all the spins. As thereis no preferred choice of z axis, the energy of the ferromagnetic state of a fullypolarized spin system with a total spin ST = N 1

2 has the same energyas the state where all the spins are oriented along the direction (θ, ϕ). The

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ferromagnetic state where the spins are fully polarized along the z axis is givenby

| 0 > =∏j

| mj =12> (2092)

The state where the polarization is rotated through (θ, ϕ) is given by

| θ, ϕ > =∏j

(cos

θ

2+ exp

[i ϕ

]sin

θ

2S−j

)| 0 > (2093)

This can be proved by representing the spin vector operator for one site σj interms of its component along the unit vector η in the direction (θ, ϕ)

η . σj = sin θ cosϕ σx j + sin θ sinϕ σy j + cos θ σz j (2094)

where σx j , σy j and σz j are the three Pauli spin matrices for the spin at sitej. Thus, the representation of the operators for the spins at site j is found as

cos θ sin θ exp[− i ϕ

]

sin θ exp[

+ i ϕ

]− cos θ

This spin operator has two eigenstates

| θ, ϕ + > = cosθ

2| + > + exp

[+ i ϕ

]sin

θ

2| − > (2095)

and

| θ, ϕ − > = cosθ

2| − > − exp

[− i ϕ

]sin

θ

2| + > (2096)

The un-rotated ferromagnetic state has all the spins in the | + > spinor stateand after rotation all the spins have maximal eigenvalue along the direction(θ, ϕ) and are in the | θ, ϕ + > spinor state. The rotated ferromagnetic statehas all the spins aligned in the same direction. This classical ferromagnetic statehas an infinitesimal overlap with the un-rotated ferromagnetic state, as

< 0 | θ, ϕ > =(

cosθ

2

)N

= exp[N ln cos

θ

2

](2097)

which vanishes in the limit N → ∞. The rotated state can be consideredto be a Bose-Einstein condensate of the q = 0 spin waves. For example onexpanding the rotated state in powers of exp[ i ϕ ] one finds

| θ, ϕ > =∞∑

n=0

A(n) exp[i n ϕ

]( S−Tot )n

n!| 0 >

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=∞∑

n=0

cos(N−n) θ

2sinn θ

2exp

[i n ϕ

]( S−Tot )n

n!| 0 >

(2098)

where the total spin operator is given by

S−Tot =∑

j

S−j (2099)

since ( S−j )2 ≡ 0. The number of q = 0 spin waves, n, are distributed withthe binomial probability

P (n) = C

(Nn

)cos2(N−n) θ

2sin2n θ

2(2100)

Since N is a macroscopic number, the distribution of spin flips is a sharp Gaus-sian distribution with a peak at nmax = N sin2 θ

2 , and a width given by∆n = N

12 sin θ

2 cos θ2 . The number of bosons in the q = 0 spin wave mode

is macroscopic and of the order N . This is a coherent representation, and it isthe quantum state that is closest to a classical state as possible.

The coherent state corresponds to the classical state in which the total spinis oriented along the direction (θ, ϕ). This can be established by examining thematrix elements of the spin operators. The un-normalized states with n spinwave present are found as

| n > =( S−tot )n

n!| 0 > (2101)

These states have the normalization

< n | n > = C

(Nn

)(2102)

The matrix elements of the total spin lowering operator between states withtotal numbers of q = 0 spin waves close to the maximum of the wave packetare found first by noting that

< n + 1 | S−Tot | n > =(N − n

)C

(Nn

)(2103)

since S− can only have a non zero effect on the N − n up spins and createsan extra down spin. The expectation value of the spin lowering operator in therotated state is found to be

< θ, ϕ | S−Tot | θ, ϕ > =∑

n

exp[− i ϕ

]A(n) A(n+ 1) < n + 1 | S−Tot | n >

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∼ 12N sin θ exp

[− i ϕ

] ∑n

A(n)2 C(Nn

)=

12N sin θ exp

[− i ϕ

](2104)

and likewise, the matrix elements of the spin raising operator are given by thecomplex conjugate. Likewise, the matrix elements of the Sz

Tot is given by

< n | SzTot | n > = (

N

2− n ) C

(Nn

)(2105)

which for n ∼ nmax = N sin2 θ2 yields

< n | SzTot | n > ∼ N cos θ C

(Nn

)(2106)

Thus, one has

< θ, ϕ | SzTot | θ, ϕ > =

∑n

A(n)2 < n | SzTot | n >

∼ N cos θ∑

n

A(n)2 C(Nn

)= N cos θ

(2107)

Thus, the total spin operator has matrix elements between the coherent statethat exactly corresponds to the classical vector.

Thus, the different classical ferromagnetic states are represented by coherentstates which are superpositions of states with arbitrary numbers of Goldstonemodes excited. The thermal average in a ferromagnetic state should yield a zeromagnetization for a system in the thermodynamic limit. The thermal averagehas to be taken, in the thermodynamic limit, in the presence of an arbitrarysmall magnetic field. In this case the different classical states have zero over-lap and can be considered as being in disjoint portions of Hilbert space. In thisquasi-static state the field may then be driven to zero leading to a non-vanishingvector order parameter.

——————————————————————————————————

22.2.1 Exercise 78

Determine the spin wave spectrum for an isotropic Heisenberg ferromagnet inthe presence of an applied magnetic field. Do the conditions of Goldstone’s the-orem apply, and what happens to the excitation energy of the q = 0 spin wave?

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——————————————————————————————————

22.3 Anti-ferromagnetic Spinwaves

One can perform a similar analysis for an anti-ferromagnet ( J < 0 ) in aNeel state. Neel ordering shall be considered on a crystal structure that can bedecomposed into two interpenetrating sub-lattices. The spins on one sub-lattice(the A sub-lattice sites) shall be oriented parallel to the z axis, and the spins onthe second sub-lattice (the B sub-lattice) are anti-parallel to the z axis. In orderfor the bosons to represent excitations, it is necessary to switch the directions ofthe spins on the B sub-lattice Sz

i → − Szi , Sx

i → Sxi and Sy

i → − Syi . This

is a proper rotation of π about the x axis so that the commutation relationsremain the same. The Holstein - Primakoff transformation for the operatorrepresenting the z component of the B spins is of the form

Szi = b†i bi − S (2108)

and the spin raising operators for the B spins are

S+i = b†i

( √2 S − b†i bi

)(2109)

which projects out states with more than 2 S bosons present. The loweringoperator is given by the Hermitean conjugate

S−i =( √

2 S − b†i bi

)bi

(2110)

for the B sub-lattice.

The Hamiltonian can be written as

H = − J∑i,j

[( S − a†i ai ) ( S − b†j bj ) − S ( a†i b

†j + ai bj )

](2111)

On defining the Fourier transformed operators

aq =

√2N

∑i

exp[− i q . Ri

]ai

bq =

√2N

∑j

exp[− i q . Rj

]bj

(2112)

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then one has

H = − N z J S2 + J S∑

q

[z ( a†q aq + b†−q b−q )

+∑

δ

exp[− i q . δ

]( a†q b

†−q + aq b−q )

](2113)

This form is still not diagonal, it is necessary to use the Bogoliubov canonicaltransformation

αq = exp[

+ S

]aq exp

[− S

]β−q = exp

[+ S

]b−q exp

[− S

](2114)

where the operator is given by

S =∑

q

θq

2

(b†−q a

†q − aq b−q

)(2115)

and θq has still to be determined. The transformation is evaluated as

αq = coshθ

2aq − sinh

θ

2b†−q

β−q = coshθ

2b−q − sinh

θ

2a†q

(2116)

in which θq is chosen so that the Hamiltonian is diagonal. The inverse transfor-mation is given by

aq = coshθ

2αq + sinh

θ

2β†−q

b−q = coshθ

2β−q + sinh

θ

2α†q

(2117)

This value of θq is found as

tanh θq = − 1z

∑δ

exp[− i q . δ

](2118)

The resulting approximate Hamiltonian again can be interpreted in terms of azero point energy and a sum of harmonic normal modes.

——————————————————————————————————

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22.3.1 Exercise 79

Find the approximate dispersion relation for spin waves of a Heisenberg anti-ferromagnet, J < 0, for spins of magnitude S. The Hamiltonian describesinteractions between nearest neighbor spins arranged on a simple cubic lattice,

H = − J∑R, δ

S(R) . S(R+ δ) (2119)

Assume that S is large so that the classical Neel state can be considered asbeing stable. Also calculate the zero point energy.

——————————————————————————————————

Since tanh θ → − 1 in the limit q → 0 then θ → − ∞ so both sinh θ2 and

cosh θ2 diverge. In one dimension the change in the sub-lattice magnetization

< ψ | a†i ai | ψ > diverges logarithmically. In two and three dimensions, thedivergence in

∑q < ψ | a†q aq | ψ > is integrable and converges. There is an

energy change relative to the nominal classical energy of the Neel state, givenby the sum of the zero point energies,

Eg = J z N S2

[1 +

1S

( 1 − Id )]

(2120)

where

Id =2N d

∑q

√√√√ d2 −( d∑

i=1

cos qi a)2

(2121)

The amplitude of the q = 0 spin wave is divergent, but can be neglected inthe limit N → ∞. As the q = 0 spin wave has zero frequency, one cancompose a state which is a superposition of the q = 0 spin wave excitations.The dynamics of the finite frequency spin wave excitations can be examined forfinite time scales before the zero energy amplitude wave packet of q = 0 spinwaves diverges re-orienting the sub-lattice magnetization.

23 Spin Glasses

Spin glasses are found when magnetic impurities are randomly distributed in ametal, such as Fe in a gold Au or Mn in Cu. Due to the random separationsbetween the moment carrying impurities, the R.K.K.Y. interaction between themagnetic moments are also randomly distributed and can take on both ferromag-netic and anti-ferromagnetic signs. The distribution of interactions prevents thelocal magnetic moments from forming a long-range ordered phase at low temper-atures. Nevertheless, the random spin system may freeze into a spin glass statebelow a critical temperature. At high temperatures the spins are disordered,

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and as the temperature is reduced, the spins which are most strongly interactingprogressively build up there correlations and freeze into clusters. The dynamicsof the spin clusters slow down as they grow, and at a critical temperature Tf

they lock into the spin glass phase. However, this may not be the lowest energystate, as in order to reach the ground state there may have to be large scalereorientation of the spin clusters. Thus, the spin glass state is not unique butinstead is highly degenerate. This occurs as a result of frustration. The conceptof frustration is illuminated by imagining that all the spins on the magnetic sitesare frozen in fixed directions, except one, then there is a high probability thatthe long-ranged interactions of the spin under consideration with the fixed spinsalmost average out to zero. At finite temperatures, the spin under considerationis almost degenerate with respect to the orientation of the spin as it leads to aninsignificant lowering of the energy of the spin glass state.

The experimental signatures of spin glass freezing are a plateau in the staticsusceptibility and a rounded peak in the specific heat. The susceptibility followsa Curie-Weiss law at high temperatures

χ(T ) = cµ2

B S ( S + 1 )3 kB ( T − Θ )

(2122)

where Θ is the strength of the resultant interaction on an individual spin. Foran R.K.K.Y. interaction, the Curie-Weiss temperature Θ should be proportionalto c. At lower temperatures where the spin freeze into clusters, the effectivemoment increases, reflecting the growth of the clusters. The peak in the specificheat, encloses an entropy which is a considerable fraction of

∆S = c kB ln (2S + 1) (2123)

where c is the concentration of magnetic impurities of spin S. This entropyrepresents the entropy of the spins gradually freezing into clusters, but does notcontain the entropy of the frustrated spins. This maximum disappears and isbroadened to higher temperatures as a magnetic field is applied. The effect ofthe applied field is to order the spins at higher temperatures. Crude estimatesindicate that about 70 percent of the spins are already ordered above Tf . Thetemperature dependence of the resistivity shows a sharp drop or knee at thespin glass freezing temperature. At this temperature, the majority of spin arefrozen in specific directions preventing the logarithmic increase with decreasingtemperature associated with spin flip scattering.

As the spin glass phase is not a ground state but is instead a highly de-generate meta-stable state the most unusual properties occur in the dynamicalproperties. The low field a.c. susceptibility shows a very sharp cusp at the spinglass freezing temperature. The cusp becomes rounded and the temperature ofthe peak diminishes as the a.c. frequency is lowered. The susceptibility satu-rates to a finite value at T = 0 which is roughly half the value of the cusp andhas a T 2 variation on the low-temperature side. The d.c. susceptibility shows a

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memory effect, in that, in field cooled samples ( H 6= 0 ) the susceptibility sat-urates at Tf , and the curve χ(T ) is reversible as it is also followed for increasingtemperature at fixed field. However, the zero field cooled sample ( H = 0 )shows a cusp at Tf and as the susceptibility is zero until the field is applied,by definition the temperature is only allowed to increase. However, the valueof the susceptibility increases with increasing measurement time. The magne-tization is slowly increasing as the spins slowly adjust to lower energy state inthe presence of the applied field. This is contrasted to the field cooled state inwhich the spins have already minimized the field energy before the temperatureis lowered and they are frozen into the spin glass state.

The spin glass freezing resembles a phase transition, but the nature of theorder is unclear as the spin glass state involves disorder and is a highly degener-ate meta-stable state. Likewise, the description of the low frequency dynamicsof the magnetization is complicated by the existence of long-ranged correlationsbetween large groups of spins. Since there is no well defined order parame-ter, there is no well defined low frequency Goldstone mode. Several importantsteps in the solution of the thermodynamics of the spin glass problem have beenundertaken, this includes the discovery of the nature of the order parameter,by Edwards and Anderson, the formulation of a model which is exactly solublemean field theory by Sherrington and Kirkpatrick. The Sherrington-Kirkpatrickmodel consists of an Ising interaction

H = −∑i,j

Ji−j Szi . S

zj (2124)

where Ji−j is a randomly distributed long-ranged interaction between the spins.The average value of Ji−j is zero

< Ji−j > = 0 (2125)

and the average value of the square is given by

< J2i−j > =

J2

N(2126)

The averaging over the randomly distributed interactions is not commutative.

23.1 Mean Field Theory

The simplest mean field approximation is based on a representation of the freeenergy, for a spin glass with long-ranged interactions between the Ising spinS = 1

2 , in which the exact value of the spin on a site i is replaced by thethermal averaged value mi. The mean field free energy F [mi] is given by

F [mi] = −∑i,j

Ji,j mi mj + kB T∑

i

[1 +mi

2ln

1 +mi

2+

1−mi

2ln

1−mi

2

](2127)

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where the exchange interactions jij are randomly distributed. On minimizingthe Free energy one finds, the mean field magnetization at every site. Above thespin glass freezing temperature, the average magnetization at each site is zero.Below the freezing temperature the spin on each site has a non-zero averagevalue, the direction and magnitude varies from site to site and is determined bythe non-trivial solution of

0 = 2∑

j

Ji,j mj +kB T

2ln∣∣∣∣1 +mi

1−mi

∣∣∣∣ (2128)

On linearizing in mi ( only valid for T ≥ Tf ) one obtains the eigenvalueequation which determines the spin glass freezing temperature

kB Tf mi = 2∑

j

Ji,j mj (2129)

as the largest eigenvalue of the random matrix Ji,j . This is solved by finding abasis λ that diagonalizes the matrix

Ji,j =∑

λ

Jλ < i | λ > < λ | j > (2130)

The basis gives the set of the spin configurations that the spins will be frozen intobelow the spin glass freezing temperature. In the limit N →∞, the eigenvaluesof the random exchange matrix are distributed according to a semi-circular law(Edwards and Jones)

ρ(Jλ) =1

2 π J2

√4 J2 − J2

λ (2131)

where, obviously, 2 J is the largest eigenvalue. The spin glass freezing temper-ature is determined as

kB Tf = 4 J (2132)

This mean field theory predicts a transition temperature which is a factor of 2too large. This is because the mean field theory needs to incorporate a self re-action term. Namely, the reaction term includes the effect of the central spin onthe neighbors back on itself, before the thermal averaging is performed (Thou-less, Anderson and Palmer).

23.2 The Sherrington-Kirkpatrick Solution.

The correct mean field solution for the Sherrington-Kirkpatrick model can beobtained in a systematic manner, starting from the partition function. Althoughthe average value of the partition function Z is easily evaluated, the averagevalue of the Free energy is difficult to evaluate. However, the logarithm of thefree energy can be evaluated with the aid of the mathematical identity

− β F = limn→0

(Zn − 1

n

)(2133)

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For finite integer n the configurational average over Ji−j can be evaluated lead-ing to an expression for the partition function for n replicas of the spin systemin which the replicas are interacting. The Gaussian averaged value of Zn isgiven by

Zn =∏i−j

( √N

2 π J2

∫dJi−j exp

[−

N J2i−j

2 J2

] )×

×(Trace exp

[− β

∑i−j

Ji−j Si Sj

] )n

= Trace∏i−j

( √N

2 π J2

∫dJi−j exp

[−

N J2i−j

2 J2−∑α

β Ji−j Sαi Sα

j

] )

Zn = Trace exp[

( β J )2

2 N

∑i,j

∑α β

Sαi Sα

j Sβi Sβ

j

](2134)

where α and β are the indices labelling members of the n different replicas. Thetrace can be evaluated for integer n and then the result can be extrapolated ton → 0. The spin glass order parameter is given by the correlation between thespins of different replicas

qα,β = < | Sαi Sβ

i | > (2135)

which becomes non zero below the freezing temperature. The Free energy isevaluated by re-writing the trace in terms of a Gaussian integral

Trace exp[

( β J )2

2 N

∑i,j

∑α β

Sαi Sα

j Sβi Sβ

j

]

= Trace∏α,β

( ∫dyα,β β J

√N√

2 πexp

[− ( β J )2

2

(N y2

α,β − 2 yα,β

∑i

Sαi Sβ

i

) ] )

=∏α,β

( ∫dyα,β β J

√N√

2 πexp

[− N

( β J )2

2y2

α,β

] )×

× exp

[N ln Trace exp

[ ∑α,β

( β J )2 yα,β Sα Sβ

] ](2136)

In this the thermodynamic limit N → ∞ and the limit n → 0 have beeninterchanged. Due to the long-ranged nature of the interaction, the trace isover a single spin replicated n times. In the thermodynamic limit N → ∞,this integral can be evaluated by steepest descents. The saddle point value ofyα,β is denoted by qα,β . For temperatures above Tf it is easy to show that the

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interaction part of the Free energy originates from the terms with α = β asthe off diagonal terms of qα,β are all equal and zero, and leads to

− β F = N ln 2 +N

2( β J )2 (2137)

Just below the spin glass freezing transition the off diagonal terms qα,β are allequal and finite. separating out the terms where α = β and replacing theintegral by the saddle point value

= exp[n N

( β J )2

2

(1 − q2 (n− 1)

) ]×

× exp

[N ln Trace exp

[ ∑α6=β

( β J )2 q Sα Sβ

] ]

= exp[n N

( β J )2

2

(1 − 2 q − q2 (n− 1)

) ]×

× exp

[N ln Trace exp

[ ∑α,β

( β J )2 q Sα Sβ

] ]

= exp[n N

( β J )2

2

(1 − 2 q − q2 (n− 1)

) ]×

× exp

[N ln Trace

∫dz√2 π

exp[− z2

2

]exp

[ ∑α

( β J ) z√

2 q Sα

] ]

= exp[n N

( β J )2

2

(1 − 2 q − q2 (n− 1)

) ]×

× exp

[N ln

∫dz√2 π

exp[− z2

2

]2n coshn

[( β J ) z

√2 q

] ](2138)

The saddle point value of q is found by differentiating with respect to q. Afteran integration by parts and then clearing away fractions, one obtains(

1 + q (n− 1)) ∫

dz√2 π

exp[− z2

2

]coshn

[( β J ) z

√2 q

]=∫

dz√2 π

exp[− z2

2

]coshn

[( β J ) z

√2 q

] (1 + (n− 1) tanh2

[( β J ) z

√2 q

] )(2139)

In the limit n → 0, the order parameter is given by the solution of the equation

q(T ) =∫ ∞

−∞

dz√2 π

exp[− z2

2

]tanh2 β J

√2 q(T ) z (2140)

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The temperature variation of the order parameter is given by

q(T ) =12

[1 −

(T

Tf

)2 ]for T < Tf

limT → 0

q(T ) = 1 −(

23 π

) 12 T

TF(2141)

The finite value of the order parameter produces the cusp in the susceptibilityand the low-temperature saturation, since one can show that

χ(T ) =g2 µ2

B

3 kB T

(1 − q(T )

)(2142)

Although the long-ranged model is exactly soluble in the mean field approxi-mation, it is only soluble for all temperatures below the freezing temperature ifthe symmetry between the different replicas are broken. Replica symmetry isspecific to interacting random systems (Almeida and Thouless), and the exactsolution of the mean field model involves repeated replica symmetry breaking(Parisi). This repeated replica symmetry breaking has the consequence thatthe dynamics of the low-temperature system are frozen and no longer consis-tent with the ergodic hypothesis.

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24 Magnetic Neutron Scattering

The excitations of the electronic system can be probed by inelastic neutron scat-tering experiments. These experiments provide information about the magneticcharacter of the excitations, due to the nature of the interaction.

24.1 The Inelastic Scattering Cross-Section

The neutron scattering occurs through the interaction with the magnetic mo-ments of the electronic system.

24.1.1 The Dipole-Dipole Interaction

A neutron has a magnetic moment given by

µn

= gn µn σn (2143)

where the neutrons gyromagnetic ratio is given by gn = 1.91 and interacts withthe magnetic moments of electrons via dipole-dipole interactions. The magneticfield produced by a single electron is a dipole field given by

H = ∇ ∧[ge µB σe ∧ r

| r |3

]− | e |

c

v ∧ r

| r |3(2144)

where r is the position of the field relative to the electron. The interactionbetween the neutron and the magnetic field is given by the Zeeman interaction

Hint = − gn µn σn .

[∇ ∧

(ge µB

σe ∧ r

| r |3

)− | e |

c

v ∧ r

| r |3

]

= gn µn

[σn . ∇ ∧

(ge µB

σe ∧ r

| r |3

)

− | e |2 me c

(p .

σn ∧ r

| r |3+

σn ∧ r

| r |3. p

) ](2145)

The first term is a classical dipole - dipole interaction and the second term is aspin - orbit interaction.

24.1.2 The Inelastic Scattering Cross-Section

The scattering cross-section of a neutron, from an initial state (k, σn) to a finalstate (k′, σ′n), in which the electron makes a transition from the initial state

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| φn > to the final state | φn′ > is given by

d2σ

dω dΩ= ( ge gn µn µB )2

k′

k

(V mn

2 π h2

)2 ∑n,n′

P (n)∣∣∣∣ < φn′ ; k′, σ′N | σN . ∇ ∧

(σe ∧ r

| r |3

)

− 12 h

(p .

σn ∧ r

| r |3+

σn ∧ r

| r |3. p

)| φn; k, σn >

∣∣∣∣2 δ( h ω + En − En′ )

(2146)

Here, the probability that the electronic system is in the initial state is rep-resented by P (n). The neutron’s energy loss h ω and the momentum loss orscattering vector are defined via

h ω = E(k) − E(k′)

h q = h k − h k′

(2147)

As the neutron states are momentum eigenstates, the matrix elements of theinteraction can be easily evaluated. The spin component of the magnetic inter-action is evaluated by considering the neutron component of the matrix elements

< k′ | ∇ ∧(σe ∧ r

| r |3

)| k >

= − < k′ | ∇ ∧(σe ∧ ∇ 1

| r |

)| k >

=1V

∫d3 rn exp

[+ i q . rn

]∇ ∧

(σe ∧ ∇ 1

| r |

)=

4 πV q2

(q ∧ ( σe ∧ q )

)exp

[+ i q . re

](2148)

This shows that the neutron only interacts with the component of the electron’sspin σ perpendicular to the scattering vector. Likewise, the orbital componentcan be evaluated as

< k′ | pe∧(σe ∧ r

| r |3

)| k > = − 4 π i

V q2

(σe ∧ ( q ∧ p

e))

exp[

+ i q . re

](2149)

Furthermore, the operator ( q ∧ pe

) commutes with exp[

+ i q . re

]as

q ∧ q ≡ 0. Hence, the neutron scattering cross-section from a multi-electronsystem can be written as

d2σ

dω dΩ= ( ge gn µn µB )2

k′

k

(2 mn

h2

)2 ∑n,n′

P (n) δ( h ω + En − En′ )

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×∣∣∣∣ < φn′ ;σ′n | σn .

∑e

(q ∧

(σe ∧ q

| q |2

)− i

h

q ∧ pe

| q |2

)exp

[+ i q . re

]| φn;σn >

∣∣∣∣2(2150)

Since the nuclear Bohr magneton has the value

µn =| e | h2 mp c

(2151)

the coupling constant can be simplified

2 mn

h2 gn ge µn µB =gn e2

me c2= re (2152)

to yield re, the classical radius of the electron. Thus, the scattering cross-sectioncan be written as

d2σ

dω dΩ= r2e

k′

kS(q;ω) (2153)

where the response function is given by

S(q;ω) =∑n,n′

P (n) δ( h ω + En − En′ )

×∣∣∣∣ < φn′ ;σ′n | σn .

∑e

(q ∧

(σe ∧ q

| q |2

)− i

h

q ∧ pe

| q |2

)exp

[+ i q . re

]| φn;σn >

∣∣∣∣2(2154)

This expression still depends on the polarization of the neutrons in the incidentbeam, and also on the polarization of the detector. Polarized neutron scatteringmeasurements reveal more information about the nature of the excitations ofa system. However, due to the reduction of the intensity of the incident beamcaused by the polarization process, and the concomitant need to compensatethe loss of intensity by increase the measurements time, it is more convenientto perform measurements with unpolarized beams. For an un-polarized beamof neutrons, the initial polarization must be averaged over. The averaging canbe performed with the aid of the identity∑

σn

12< σn | σα

n σβn | σn > = δα,β (2155)

which follows from the anti-symmetric nature of the Pauli spin matrices. Foran un-polarized beam of neutrons the response function reduces to

S(q;ω) =∑n,n′

P (n) δ( h ω + En − En′ )∑α,β

(δα,β − qα qβ

)

× < φn |∑

e

(σe +

i

h

q ∧ pe

| q |2

exp[− i q . re

]| φn′ >

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× < φn′ |∑

e

(σe − i

h

q ∧ pe

| q |2

exp[

+ i q . re

]| φn >

(2156)

where q is the unit vector in the direction of q. On defining the spin densityoperator Sα(q) via

Sα(q) =∑

e

(σe +

i

h

q ∧ pe

| q |2

exp[− i q . re

](2157)

then the response function can be expressed as a spin - spin correlation function

S(q;ω) =∑n,n′

P (n) δ( h ω + En − En′ ) ×

×∑α,β

(δα,β − qβ qβ

)< φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

(2158)

Thus, the inelastic neutron scattering measures the excitation energies of thesystem, with intensity governed by the matrix elements < φn | Sα(q) | φn′ >which filters out the excitations of a non-magnetic nature. Furthermore, thescattering only provides information about the magnetic excitations which havea component of the fluctuation perpendicular to the momentum transfer.

In the case where the spin density can be expressed in terms of the atomicspin density due to the unpaired spins in the partially filled shells, such asin transition metals or rare earths, it is convenient to introduce the magneticatomic ( ionic ) form factor F (q). For a mono-atomic Bravais lattice, this isachieved by decomposing the spin density in terms of the spin density from eachunit cell

Sα(q) =∑

e

(σe +

i

h

q ∧ pe

| q |2

exp[− i q . re

]=

∑R

exp[− i q . R

] ∑j

(σj +

i

h

q ∧ pj

| q |2

exp[− i q . rj

](2159)

Since the unpaired electrons couple together to give the ionic spin SR, theWigner - Eckert theorem can be used to express the spin density operator

Sα(q) =∑R

exp[− i q . R

]F (q) SR

(2160)

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The form factor F (q) is defined as the Fourier transform of the normalized spindensity for the ion. By definition

F (0) = 1 (2161)

In this case, the inelastic neutron scattering spectrum can be expressed as

d2σ

dω dΩ= r2e

k′

k| F (q) |2 S(q;ω) (2162)

where the spin - spin correlation function is expressed in terms of the local ionicspins SR. Of course, it is being implicity assumed that the magnetic scatteringcan be completely separated from the phonon scattering. Thus, the analysis hasignored the existence of phonon excitations, in the case of zero phonon excita-tions, the intensity of the magnetic scattering is expected to be reduced by theDebye Waller factor of the phonons.

24.2 Time Dependent Spin Correlation Functions

The spin components of the correlation function measured in a scattering ex-periment can be defined through the function Sα,β(q;ω) where

Sα,β(q;ω) =∑n,n′

P (n) δ( h ω + En − En′ ) ×

×

[< φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

](2163)

Using the expression for the energy conserving delta function as an integral overa time variable

δ( h ω + En − En′ ) =∫ ∞

−∞

dt

2 π hexp

[i

h( h ω + En − En′ ) t

](2164)

the spin - spin correlation function can be written as a Fourier transform of atime dependent correlation function.

Sα,β(q;ω) =1h

∑n,n′

P (n)∫ ∞

−∞

dt

2 πexp

[i ω t

]exp

[i

h( En − En′ ) t

]

×

[< φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

]

=∫ ∞

−∞

dt

2 πexp

[i ω t

]1h

∑n,n′

P (n) exp[i

h( En − En′ ) t

]

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×

[< φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

](2165)

On expressing the product of the phase factor and the matrix elements of theS(q) as an operator in the interaction representation

exp[i

h( En − En′ ) t

]< φn | Sα(q) | φn′ >

= < φn | exp[

+i

hH0 t

]Sα(q) exp

[− i

hH0 t

]| φn′ >

= < φn | Sα(q; t) | φn′ >

(2166)

Hence, the spin - spin correlation function is given by

Sα,β(q;ω) =∫ ∞

−∞

dt

2 πexp

[i ω t

×

[1h

∑n,n′

P (n) < φn | Sα(q; t) | φn′ > < φn′ | S†β(q; 0) | φn >

](2167)

The final states are a complete set of states, therefore, on using the completenessrelation, one finds

Sα,β(q;ω) =∫ ∞

−∞

dt

2 πexp

[i ω t

] [1h

∑n

P (n) < φn | Sα(q; t) S†β(q; 0) | φn >

]

=∫ ∞

−∞

dt

2 πexp

[i ω t

]1h

< | Sα(q; t) S†β(q; 0) | >

(2168)

This is the Fourier Transform with respect to time of the thermal averaged spin -spin correlation function. Furthermore, as the inverse spatial Fourier transformof the spin density operator and the Hermitean conjugate are given by

Sα(q) =1V

∫d3r exp

[− i q . r

]Sα(r)

S†α(q) =1V

∫d3r′ exp

[+ i q . r′

]Sα(r′)

(2169)

Using this, and the spatial homogeneity of the system, one finds that the inelas-tic neutron scattering spectrum is related to the spatial and temporal Fourier

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transform of the spin - spin correlation function

Sα,β(q;ω) =1V

∫ ∞

−∞

dt

2 π

∫d3r exp

[i ( ω t − q . r )

]1h

< | Sα(r; t) Sβ(0; 0) | >

(2170)

Thus, the inelastic neutron scattering probes the Fourier transform of the equi-librium spin correlation functions.

24.3 The Fluctuation Dissipation Theorem

The spin - spin correlation function satisfies the principle of detailed balance

Sα,β(q;ω) =∑n,n′

P (n) δ( h ω + En − En′ ) < φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

(2171)

If the equilibrium probability P (n) is given by the Boltzmann expression

P (n) =1Z

exp[− β En

](2172)

then spin-spin correlation function can be re-written as

Sα,β(q;ω) =∑n,n′

P (n) δ( h ω + En − En′ ) < φn′ | S†β(q) | φn > < φn | Sα(q) | φn′ >

= exp[β h ω

] ∑n,n′

P (n′) δ( En′ − En − h ω ) < φn′ | S†β(q) | φn > < φn | Sα(q) | φn′ >

= exp[β h ω

] ∑n,n′

P (n) δ( En − En′ − h ω ) < φn | S†β(q) | φn′ > < φn′ | Sα(q) | φn >

= exp[β h ω

]Sβ,α(−q;−ω)

(2173)

which is a statement of the principle of detailed balance. The correlation func-tion Sα,β(q;ω) is also related to the imaginary part of the magnetic susceptibilityχα,β(q;ω) via the fluctuation dissipation theorem.

The reduced dynamical magnetic susceptibility is given by the expression

χα,β(r, r′; t− t′) = − i

h< |

[Sα(r, t) , Sβ(r′, t′)

]| > Θ( t − t′ ) (2174)

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which can be expressed as

χα,β(r; t) = − i

h

∑n,n′

P (n) exp[i

h( En − En′ ) t

]< φn | Sα(r) | φn′ > < φn′ | Sβ(0) | φn > Θ( t )

+i

h

∑n,n′

P (n) exp[i

h( En′ − En ) t

]< φn | Sβ(0) | φn′ > < φn′ | Sα(r) | φn > Θ( t )

(2175)

The Fourier transform is defined as

χα,β(q;ω) =1V

∫ ∞

−∞

dt

2 π

∫d3r exp

[i ( ω t − q . r )

]χα,β(r; t)

(2176)

and is evaluated as

χα,β(q;ω) =1

2 π

∑n,n′

P (n)

[< φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

h ω + En − En′ + i δ

−< φn | S†β(q) | φn′ > < φn′ | Sα(q) | φn >

h ω + En′ − En + i δ

](2177)

The imaginary part of the dynamic susceptibility is given by

Im

[χα,β(q;ω)

]= − 1

2

∑n,n′

P (n) ×

×

[δ( h ω + En − En′ ) < φn | Sα(q) | φn′ > < φn′ | S†β(q) | φn >

− δ( h ω + En′ − En ) < φn | S†β(q) | φn′ > < φn′ | Sα(q) | φn >

](2178)

which can be written as

Im

[χα,β(q;ω + iδ)

]= − 1

2Sα,β(q;ω)

[1 − exp

[− β h ω

] ](2179)

or

Sα,β(q;ω) = 2(

1 + N(ω))Im

[χα,β(q;ω − iδ)

](2180)

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This is the fluctuation dissipation theorem, it relates the dynamical response ofthe system to an external perturbation to the naturally occurring excitations inthe system as measured by neutron scattering experiments.

24.4 Magnetic Scattering

The neutron scattering cross-section is given in terms of the components of thespin spin correlation function.

As can be seen by inspection from the Holstein-Primakoff representation ofthe spins and the spin waves, the spin correlation function is a non-linear func-tion of the spin wave creation operators. The inelastic scattering cross-sectioncan be expanded in powers of the number of spin waves. The lowest order termis time independent and corresponds to Bragg scattering.

24.4.1 Neutron Diffraction

The ω = 0 component of the inelastic scattering cross-section given by thelimit of

Sα,β(q;ω) =∫ ∞

−∞

dt

2 πexp

[i ω t

]1h

< | Sα(q; t) S†β(q; 0) | >

(2181)

diverges if the integrand does not decay rapidly as t → ∞. In this case thetime independent component of the spin - spin correlation function given by

limt → ∞

1h

< | Sα(q; t) S†β(q; 0) | > (2182)

produces a Bragg peak with finite intensity as

δ(ω) =∫ ∞

−∞

dt

2 πexp

[i ω t

](2183)

Thus, the intensity of the Bragg peak represents the static correlations. If theergodic hypothesis holds, then in the long time limit, the correlation functiondecouples into the product of two expectation values

limt → ∞

1h

< | Sα(q; t) S†β(q; 0) | >

= limt → ∞

1h

< | Sα(q; t) | > < | S†β(q; 0) | >

(2184)

and for a static system for which

< | Sα(q; t) | > = < | Sα(q; 0) | > (2185)

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then the Bragg peak has an intensity given by

= limt → ∞

1h

< | Sα(q; 0) | > < | S†β(q; 0) | > (2186)

For a paramagnetic system the (quasi-stationary) average value of the spinvector is zero

< | Sα(q; 0) | > = 0 (2187)

and, thus, there is no magnetic Bragg peak for a paramagnetic system. On theother hand, if there is long-ranged magnetic order with wave vectors Q alongcertain directions ( say α ), then

< | Sα(Q; 0) | > 6= 0 (2188)

then the magnetic Bragg peaks are non-zero. The temperature dependence ofthe intensity of the Bragg peaks provides a direct measure of the temperaturedependence of the magnetic order parameter. For a ferromagnet, the magneticBragg peaks coincide with the Bragg peaks due to the crystalline order, so nonew peaks emerge. The Bragg scattering cross-section is given by

dΩBragg= r2e

( 2 π N )2

V

∑Q

(1 − q2z

)δ( q − Q )

∣∣∣∣ < | Sz | >∣∣∣∣2 (2189)

For a small single domain single crystal the magnetic elastic scattering is ex-tremely anisotropic, the scattering should be zero for momentum transfers alongthe direction of the magnetization.

For anti-ferromagnetic or spin density wave order new Bragg peaks mayemerge at the vectors of the anti-ferromagnetic reciprocal lattice. Analysis ofthe anisotropy of the neutron scattering intensity for anisotropic single crystalsleads to the determination of the preferred directions of the magnetic moments.

——————————————————————————————————

24.4.2 Exercise 80

Evaluate the elastic scattering cross-section for a anti-ferromagnetic insulator,using the Holstein-Primakoff representation of the low energy spin wave exci-tations. Discuss the anisotropy and also the temperature dependence of theintensity of the Bragg peaks.

——————————————————————————————————

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24.4.3 Exercise 81

Design a neutron diffraction experiment that will determine if a system has aspiral spin density wave order, as opposed to a magnetic moment that is mod-ulated in intensity. How can the direction of spiral be determined?

——————————————————————————————————

24.4.4 Spin Wave Scattering

The spin wave excitations of an ordered magnet show up in the inelastic neutronscattering spectra. In a process whereby a single spin wave is emitted in thescattering, the energy difference between the initial state En and the final stateEn′ so that energy conservation leads to

En′ − En = h ωq (2190)

The matrix elements in the spin - spin correlation function can be evaluated as

< φn | Sα(q) | φn′ > = <∏q′

nq′ | Sα(q) |∏q”

n′q” > (2191)

where the number of spin waves in the initial state are related to the numberin the final state via

n′q′ = nq′ for q′ 6= q (2192)

and

n′q = nq + 1 (2193)

For a ferromagnet the matrix elements are evaluated as

< nq | Sz(q) | nq + 1 > = 0 (2194)

while the transverse matrix elements are

< nq | Sx(q) | nq + 1 > = < nq |12

( S+(q) + S−(q) ) | nq + 1 >

=12

√nq + 1

√2 S

(2195)

and

< nq | Sy(q) | nq + 1 > = < nq |1

2 i( S+(q) − S−(q) ) | nq + 1 >

= +1

2 i

√nq + 1

√2 S

(2196)

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Thus, the inelastic neutron scattering from the single spin wave excitations isgiven purely by the diagonal components of the transverse spin-spin correlationfunction, as the longitudinal components are zero. The off diagonal terms cancel.The cross-section for the spin wave emission process is given by

d2 σ

dω dΩ emit= r2e

( 2 π)2

V

N S

2

(1 + q2z

) ∑q′,Q

δ( h ω − h ω(q′) ) δ( q − q′ −Q )[

1 +N(ω(q′))]

(2197)Likewise, the absorption process has a scattering cross-section given by

d2 σ

dω dΩabs= r2e

( 2 π)2

V

N S

2

(1 + q2z

) ∑q′,Q

δ( h ω + h ω(q′) ) δ( q − q′ −Q )N(ω(q′))

(2198)These satisfy the principle of detailed balance, and give rise to a Stokes andanti-Stokes line in the spectrum of the scattered neutrons.

——————————————————————————————————

24.4.5 Exercise 82

Evaluate the two lowest order terms in the inelastic scattering cross-section foran anti-ferromagnetic insulator, using the Holstein-Primakoff representation ofthe low energy spin wave excitations. Discuss the differences between the spec-trum obtained from magnetic scattering and that found in measurements of thephonon excitations.

——————————————————————————————————

24.4.6 Critical Scattering

Just above the temperature where magnetic ordering occurs, the inelastic neu-tron scattering cross-section in the paramagnetic phase shows a softening orbuild up at low frequencies and becomes sharply peaked at q values close tothe magnetic Bragg vectors Q. Below the ordering temperature, the intensitytransforms into the Bragg peak. This phenomenon of the build up of inten-sity close to the Bragg peak is known as critical scattering. The Bragg peakis extracted from the inelastic scattering spectrum by extracting a delta func-tion δ(ω), i.e., the inelastic scattering cross-section is integrated over a smallwindow dω. On invoking the fluctuation dissipation theorem and then notingthat if, in the paramagnetic phase, the main portion of the scattering occurswith frequencies such that β ω 1, then one finds that by using the Kramers- Kronig relation, the intensity of the critical scattering is given by the static

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susceptibility. For q values close to the Bragg peak, the susceptibility varies as

∝ 1( q − Q )2 + ξ−2

(2199)

where ξ the correlation length, in the mean field approximation, is given by

ξ−1 = a

√Tc

6 ( T − Tc )(2200)

Thus, the critical scattering diverges as ( T − Tc )−1 as the transition temper-ature is approached.

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25 Superconductivity

The electrical resistivity ρ(T ) of metals at low temperatures is expected to bedescribed by the Drude model

ρ(T ) ∝ m

e2 τ(2201)

The resistivity vary with temperature according to

ρ(T ) ∼ ρ(0) + A T 2 + B T 5 (2202)

as the scattering rates for scattering from static impurities, electron-electronscattering and phonon scattering are expected to be additive. The resistivity ofa perfect metal, without impurities, may be expected to vanish at T = 0. How-ever, it was discovered by Kammerlingh Onnes that the resistivity of a metalmay become so small as to effectively vanish for all temperatures below a criticaltemperature Tc (H. Kammerlingh Onnes, Comm. Phys. Lab. Univ. Leiden,Nos. 119, 120, 122 (1911)). This indicates that the scattering mechanismssuddenly becomes ineffective for temperatures slightly below the critical tem-perature, where the metal seems to act like a perfect conductor. The resistivityis so small that persistent electrical currents have been observed to flow withoutattenuation for very long time periods. The decay time of a super-current infavorable materials is apparently not less than 10,000 years.

25.1 Experimental Manifestation

The first manifestation of superconductivity is zero resistance, below Tc. An-other manifestation of superconductivity was found by Meissner and Ochsen-feld, which is flux exclusion (W. Meissner and R. Ochsenfeld, Naturwiss. 21,787 (1933)). A superconductor excludes the magnetic induction field B fromits interior, irrespective of whether it was cooled from above Tc to below Tc inthe presence of an applied field, or whether the field is only applied when thetemperature is smaller than Tc. In other words the Meissner effect excludestime independent magnetic field solutions from inside the superconductor. TheMeissner effect distinguishes superconductivity from perfect conductivity, as astatic magnetic field can exist in perfect conductor.

The perfect conductor has the property that the current produced by anapplied electric field increases linearly with time. Therefore, a perfect conductorexcludes electric fields from within its bulk. Maxwell’s equations reduce to

− 1c

∂B

∂t= 0

∇ ∧ B =4 πc

j

∇ . B = 0(2203)

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Thus, a perfect conductor only excludes a time varying magnetic field, but nota static magnetic field.

The Meissner effect shows hat the magnetic induction inside a superconduc-tor is zero. However, the magnetic induction B can be expressed in terms ofthe applied field H and the magnetization M via

B = H + 4 π M (2204)

so B = 0 implies that

M = − 14 π

H (2205)

so that perfect diamagnetism implies that the susceptibility is given by

χ = − 14 π

(2206)

The perfect diamagnetism does not hold for arbitrarily large applied magneticfields. For fields larger than a critical magnetic field, the induction inside thesuperconductor becomes non-zero. For a type I superconductor, the appliedfield fully penetrates into the bulk of the superconductor above the critical fieldHc. The magnetization drops discontinuously to zero at Hc. The value of Hc

depends on temperature according to

Hc(T ) = Hc(0)(

1 − T 2

T 2c

)(2207)

For a type II superconductor, the induction first starts penetrating into thebulk at the lower critical field Hc1, For fields larger than the lower critical field,the magnetization deviates from linear relation associated with perfect diamag-netism. The magnitude of the magnetization is reduced as the applied field isincreased above Hc1. The magnetization falls to zero at the upper critical fieldHc2, at which point the applied field fully penetrates into the bulk.

The experimental observations of a drop in the resistivity and the Meissnereffect demonstrate that the transition to the superconducting state is a phasetransition as the properties are independent of the history of the sample. Fora type I superconductor, the bulk superconductivity is completely destroyed atHc(T ).

25.1.1 The London Equations

A phenomenological description of superconductivity was developed by the Lon-don brothers. Basically, this description is based on two phenomenological con-stitutive equations for the electromagnetic field and its relation to current anddensity. The first London equation is of the form

j(r, t) = − ns e2

m cA(r, t) (2208)

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which expresses the perfect conductivity of a superconductor. It relates themicroscopic current in the superconductor which screens the applied magneticfield. Here, ns is density of superfluid electrons. This is slightly different fromthe condition of perfect conductivity in a metal which would be a relation be-tween the time derivative of the current that flows in response to an electricfield. Here, it has been assumed that the electric field is transverse and the con-dition for perfect conductivity has been integrated with respect to time, therebyallowing constant currents to screen the static applied magnetic field. In orderthat the continuity equation be satisfied in a steady state, the condition that∇ . A(r, t) = 0, which defines the London gauge must be imposed.

The second London equation comes from Maxwell’s equations

∇ ∧ B(r, t) =j(r, t)c

+1c

∂tE(r, t) (2209)

and with the definitions

B(r, t) = ∇ ∧ A(r, t)

E(r, t) = − 1c

∂tA(r, t)

(2210)

one finds (∇ ∧ ∇ ∧ +

1c2

∂2

∂t2

)A(r, t) = − ns e

2

m c2A(r, t) (2211)

This is referred to as the second London equation. The quantity ns e2

m c2 has unitsof inverse length squared and is used to define the London penetration depthλL, via

ns e2

m c2=

1λ2

L

(2212)

The second London equation expresses the Meissner effect. Namely, thata superconductor excludes the magnetic induction field B from the bulk of itsvolume. However, the field does penetrate the region at the surface and extendsover a distance λL into the superconductor. This can be seen by examiningvarious cases in which a static applied magnetic field is produced near a super-conductor. The geometry is considered in which the applied field is parallel tothe surface.

Let the surface be the plane z = 0, which separates the superconductorz > 0 from the vacuum z < 0. The external field is applied in the x direction,so B = B0 x for z < 0. The vector potential inside the superconductor mustsatisfy the boundary condition Az(z = 0) = 0 as any current should be per-pendicular to the surface. The London gauge requires the non-zero components

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of the vector potential to be Ax and Ay. Thus, the vector potential must beparallel to the surface. The static solution for the vector potential that satisfiesthe boundary conditions on the current for the semi-infinite solid is

A(z) = A0 exp[− z

λL

](2213)

An additional boundary condition at z = 0 is that Bx should be continuous.Hence, as the equation for the magnetic induction simplifies to

Bx(z) = − ∂Ay(z)∂z

(2214)

one finds that the vector potential is directed parallel to the surface, but is alsoperpendicular to the applied field. The only non-zero component of the vectorpotential in the superconductor is found as

Ay(z) = + λL Bx(0) exp[− z

λL

]for z > 0 (2215)

London’s first equation then implies that a supercurrent, jy(z), flows in a regionnear the surface of the superconductor which, through Ampere’s law, producesmagnetic field that screens or cancels the applied field. The magnetic inductionand the supercurrent are non-zero in the superconductor only within a distanceof λL from the surface. Hence, λL is called the penetration depth.

25.1.2 Thermodynamics of the Superconducting State

The phase transition to a superconducting state, in zero field, is a second or-der phase transition. This can be seen by examining the specific heat whichexhibits a discontinuous jump at Tc. The absence of any latent heat impliesthat the entropy is continuous, and since the entropy is obtained as a first orderderivative of the Free energy the transition is not first order. The non-analyticbehavior of the specific heat, which is obtained from a second derivative of thefree energy defines the transition to be second order.

In the presence of a field the transition is first order. The thermodynamicrelations are derived from the Gibbs free energy G in which M plays the role ofthe volume V and the externally applied field H plays the role of the appliedpressure P . Then, G(T,H) has the infinitesimal change

dG = − S dT − M . dH (2216)

where S is the entropy and T is the temperature. Since G is continuous acrossthe phase boundary at (Hc(T ), T )

Gn(T,Hc(T )) = Gs(T,Hc(T )) (2217)

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which on taking an infinitesimal change in both T and H = Hc(T ) so as to stayon the phase boundary one finds

( Ss − Sn ) dT = ( Mn − Ms ) dHc(T ) (2218)

The magnetization in the normal state is negligibly small, but the supercon-ducting state is perfectly diamagnetic, so

Mn − Ms = +1

4 πHc(T ) (2219)

This shows that in the presence of an applied field the superconducting transi-tion involves a latent heat L given by

L = T ( Sn − Ss )

= − T

4 πHc

∂Hc

∂T(2220)

Thus, the transition is first order in the presence of an applied field.

In the limit that the critical field is reduced to zero, the transition tempera-ture T is reduced to the zero field value Tc, and the entropy becomes continuousat the transition. However, there is a change in slope at Tc, which can be foundby taking the derivative of Sn − Ss with respect to T , and then letting Hc → 0

∂T( Sn − Ss ) = − 1

4 π

[Hc

(∂2Hc

∂T 2

)+(∂Hc

∂T

)2]

= − 14 π

(∂Hc

∂T

)2

(2221)

The specific heat may show a discontinuity or jump at Tc that is a measure ofthe initial slope of the critical field

Cs − Cn =T

4 π

(∂Hc

∂T

)2

(2222)

The discontinuous jump in the (zero field) specific heat is a characteristic of amean field transition. For temperatures below Tc the specific heat is exponen-tially activated

Cv ∼ γ Tc exp[− ∆

kB T

](2223)

The activated exponential behavior of the specific heat suggests that there is anenergy gap in the excitation spectrum. The existence of a gap is confirmed bya threshold frequency for photon absorption by a superconductor. Above Tc,the absorption spectrum is continuous and photons of arbitrarily low frequencycan be absorbed by the metal. However, for temperatures below Tc, there isa minimum frequency above which photons can be absorbed. The threshold

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frequency is related to ∆.

In most superconductors, the interaction mechanism that is responsible forpairing is mediated by the electron-phonon coupling. This was first identifiedthrough the insight of Frohlich (H. Frohlich, Phys. Rev. 79, 845 (1950)), whopredicted that the superconducting transition temperature Tc should be pro-portional to the phonon frequency. Furthermore, as the square of the phononfrequency is inversely proportional to the mass of the ions M , the supercon-ducting transition temperature should depend upon the isotopic mass through

Tc ∝ M− 12 (2224)

This isotope effect was confirmed in later experiments by Maxwell (E. Maxwell,Phys. Rev. 78, 447 (1950), Phys. Rev. 79, 173 (1950)) and Reynolds et. al.(C.A. Reynolds, B. Serin, W.H. Wright and L.B. Nesbitt, Phys. Rev. 78, 487(1950)) on simple metals. However, in transition metals the exponent of theisotope effect is reduced and may become zero, and in α − U the exponent ispositive. The occurrence of a positive isotope effect does not necessarily signifythe existence of alternate pairing mechanisms, but can indicate the effect ofstrong electron-electron interactions.

25.2 The Cooper Problem

The electron-electron interaction in a metal, is attractive at low frequencies.The attractive interaction originates from the screening of the electrons by theions, but only occurs for energy transfers less than h ωD . The effective attrac-tion is retarded, and occurs due to the attraction of a second electron with theslowly evolving polarization of the lattice produced by the first electron. Coopershowed that two electrons which are close to the Fermi-energy, will bind intopairs whenever they experience an attractive interaction, no matter how weakthe interactions is.

Consider a pair of electrons of spin σ and σ′, excited above the Fermi-energy. Due to the interaction between the pair of particles, the center of massmomentum q will be a constant of motion, but not the relative motion. Thus,the wave function of the Cooper pair with total momentum q can be written as

| Ψq > =∑

k

C(k) | σ, k + q σ′,−k > (2225)

Due to the Pauli exclusion principle the single particle energies E(−k) and E(k+q) must both be above the Fermi-energy µ. The wave function is normalizedsuch that ∑

k

| C(k) |2 = 1 (2226)

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The wave function must be an energy eigenstate of the Hamiltonian H and,thus, satisfies

H | Ψq > = Eq | Ψq > (2227)

On projecting out C(k) using the orthogonality of the different momentumstates | σ, k + q σ′,−k > one finds the secular equation(

Eq − E(k) − E(k + q))C(k) = − 1

N

∑k′

V (k, k′) C(k′) (2228)

The attractive pairing potential V , ( − V < 0 ), scatters the pairs of electronsbetween states of different relative momentum. The summation over k′ is re-stricted to unoccupied Bloch states within h ωD the Fermi-surface, where theinteraction is attractive. The above equation has a solution for the amplitudeC(k) which is given by

C(k) =α(k)

Eq − E(k) − E(k + q)(2229)

where α is given by

α(k) = − 1N

∑k′

V (k, k′) C(k′)

(2230)

This equation can be solved analytically in the case where the potential is sepa-rable, such as the case where V is just a constant. In such cases, the summationover k′ can be performed to yield a result which is independent of k. For sim-plicity, the separable potential shall be assumed to have a magnitude of V whenboth k and k′ are within h ωD of the Fermi-surface. Then, α is independent ofk and

α =α

N

∑k′

V (k, k′)E(k′) + E(k′ + q) − Eq

(2231)

Thus, the energy eigenvalue is determined from the equation

1 =1N

∑k′

V (k, k′)E(k′) + E(k′ + q) − Eq

(2232)

For Cooper pairs with zero total momentum q = 0, this equation reduces to

1 = V

∫ µ + hωD

µ

dερ(ε)

2 ε − E(2233)

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The density of states ρ(ε) can be approximated by a constant ρ(µ), and theintegral can be inverted to give the energy eigenvalue as

E = 2 µ − 2 h ωD

exp[

2V ρ(µ)

]− 1

(2234)

This eigenvalue is less than the energy of the two independent electrons, thus,the electrons are bound together. It is concluded that, due to the sharp cut offof the integral at the Fermi-energy, the electron bind to form Cooper pairs nomatter how small the attractive interaction is. The binding energy is small andis a non-analytic function of the pairing potential V , that is, the binding energycannot be expanded as a power series in V .

In the case that the pairing potential is spin rotationally invariant, the totalspin of the pair S is a good quantum number. The pairing states can be cat-egorized by the value of their spin quantum number and the projection of thetotal spin along the z axis. On pairing two spin one half electrons, there are fourpossible state, a spin singlet state S = 0 and a spin triplet state S = 1 whichis three-fold degenerate. The four Cooper pair wave functions corresponding tothese states have to obey the Paul-exclusion principle and are written as

ψS=0(r1, r2) =∑

k

CS=0(k)12

(φk(r1) φ−k(r2) − φ−k(r1) φk(r2)

×(χ+ 1 χ− 2 + χ− 1 χ+ 2

)(2235)

for the spin singlet pairing. The three spin triplet pair wave functions are

ψS=1,m=1(r1, r2) =∑

k

CS=1(k) φk(r1) φ−k(r2) χ+ 1 χ+ 2

ψS=1,m=0(r1, r2) =∑

k

CS=1(k)12

(φk(r1) φ−k(r2) + φ−k(r1) φk(r2)

×(χ+ 1 χ− 2 − χ− 1 χ+ 2

)ψS=1,m=−1(r1, r2) =

∑k

CS=1(k) φk(r1) φ−k(r2) χ− 1 χ− 2

(2236)

Thus, for singlet pairing one must have

CS=0(k) = CS=0(−k) (2237)

which requires that, when expanded in spherical harmonics, the expansion onlycontains even components of orbital angular momentum. For triplet pairing one

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hasCS=1(k) = − CS=1(−k) (2238)

thus, the triplet pair can only be composed of odd values of orbital angularmomentum.

Most superconductors that have been found have singlet spin pairing andare in a state which is predominantly in a state of orbital angular momentuml = 2. The high Tc superconductor such as Sr doped La2CuO4 found by Bed-norz and Muller in 1986 (Tc = 35 K) or Y Ba2Cu3O7 (Tc = 90 K) are slightlyexceptional to this rule. These materials evolve from an anti-ferromagneticinsulator phase at zero doping, but as the doping increases they lose the an-tiferromagnetism and become metallic paramagnets. A superconducting phaseappear above a small critical doping concentration. The superconductivity isexceptional, not just in the magnitude of the transition temperature Tc but alsoin that the pairing is singlet, but with an appreciable admixture of a compo-nent with l = 2 in the pair. Due to this admixture the pairing in high Tc

superconductors is sometimes referred to as d wave pairing. In heavy fermionsuperconductors, such as CeCu2Si2, UBe13, UPt3 and URu2Si2, experimentalevidence exists that these materials do not show exponentially activated behav-ior characteristic of a gap. Instead the specific heat and susceptibility showpower law variations. This and the multiple superconducting transitions foundin UPt3 and Th doped UBe13 indicate that the gap is dominated by compo-nents with non-zero angular momentum.

It is customary to represent the wave function of the Cooper pair in terms ofrelative coordinates r = r1 − r2 and center of mass coordinates R = r1 + r2

2 .Thus, the Cooper pair wave function is written as

ψ(r1, r2) → ψ(r,R) (2239)

and as the pair usually is in a state with zero total momentum, q = 0, thecenter of mass dependence can be ignored.

The mean square radius of the Cooper pair wave function is given by

ξ2 =∫

d3r r2 | ψ(r) |2 (2240)

but

ψ(r) =∑

k

C(k) exp[i k . r

](2241)

Thus,

ξ2 =∫

d3r r2∑k,k′

C(k) C∗(k′) exp[i ( k − k′ ) . r

]

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=∑

k

| ∇k C(k) |2

=43

(h vF

2 µ − E

)2

=43

(vF

2 ωD

)2

exp[

4V ρ(µ)

](2242)

For a binding energy of order 10 K and a Fermi-velocity vF of the order of 106

m/sec one obtains a pair size ξ of order 104 Angstroms.

The standard weak coupling theory of superconductivity due to Bardeen,Cooper and Schrieffer, (B.C.S.), treats the Cooper pairing of all electrons closeto the Fermi-surface in a self-consistent manner.

25.3 Pairing Theory

25.3.1 The Pairing Interaction

The attractive pairing interaction can be obtained from the electron phononinteraction, via an appropriately chosen canonical transform. The energy of theelectron phonon system can be expressed as the sum

H = H0 + Hint (2243)

where the non-interacting Hamiltonian is given by

H0 =∑k,σ

ε(k) c†k,σ ck,σ +∑q,α

h ωα(q) a†q,α aq,α (2244)

and the interaction term is given by

Hint =∑k,σ

∑q,α

λq c†k+q,σ ck,σ

(aq,α + a†−q,α

)(2245)

The Hamiltonian will be transformed via

H ′ = exp[

+ S

]H exp

[− S

](2246)

where S is chosen in a way that will eliminate the interaction term ( at least toin first order ). The operator S can be thought of as being of the same order asHint. That is on expanding the transformed Hamiltonian in powers of S

H ′ = H0 + Hint +[S , H0

]+

12

[S ,

[S , H0

] ]+[S , Hint

]+ O

(H3

int

)(2247)

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and then S is chosen such that

Hint =[H0 , S

](2248)

Since this is an operator equation, this is solved for S by taking matrix elementsbetween a complete set. A convenient complete set is provided by the eigenstates| φm > of H0, which have energy eigenvalues Em. Thus, the matrix elementsof S are found from the algebraic equation

< φm | Hint | φn > = ( Em − En ) < φm | S | φn > (2249)

The complete set of energy eigenstates are energy eigenstates of the non inter-acting electron and phonon Hamiltonian. The non-zero matrix elements onlyoccur between states which involve a difference of unity in the occupation num-ber of one phonon mode ( either q or − q ) , and also a change of state of oneelectron ( k to k + q ). The anti-Hermitean operator S can be represented insecond quantized form as

S = −∑k,σ

∑q,α

c†k+q,σ ck,σ

[λq aq,α

ε(k + q) − ε(k) − h ω(q)+

λq a†−q,α

ε(k + q) − ε(k) + h ω(q)

](2250)

The transformed Hamiltonian contains the effects of the interaction only throughthe higher order terms

H ′ = H0 +12

[S , Hint

]+ O

(H3

int

)(2251)

On evaluating the commutation relation one finds a renormalization of the elec-tron dispersion relation of order | λq |2, and electron-electron interaction terms.The electron-electron interaction terms combine and can be written as

H ′int =

∑k,σ;k′,σ′

∑q,α

| λq |2 h ω(q)

( ε(k + q) − ε(k) )2 − h2 ω(q)2c†k+q,σ ck,σ c

†k′−q,σ′ ck′,σ′

(2252)Thus, for electrons within h ω(q) of the Fermi-energy there is an attractive inter-action between the electrons. As this interaction depends on the energy transferbetween the electrons, and the energy transfer corresponds to a frequency. Asthe interaction is frequency dependent, it corresponds to a retarded interac-tion. As the interaction is only attractive at sufficiently small frequencies theinteraction is only attractive after long time delays. In the B.C.S. theory thisinteraction is further approximated. The approximation consists of only retain-ing scattering between electrons of opposite momentum, as this maximizes thephase space of allowed final states. That is the momenta and spin are restrictedsuch that

k = − k′ (2253)

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and alsoσ = − σ′ (2254)

This produces a pairing between electrons of opposite spins.

The B.C.S. Hamiltonian is composed of the energy of electrons in Blochstates and an attractive interaction between the electrons mediated by thephonons. The B.C.S. Hamiltonian is written as

H =∑k,σ

ε(k) c†k,σ ck,σ −∑k,k′

V (k, k′) c†−k′,↓ c†k′,↑ ck,↑ c−k,↓ (2255)

25.3.2 The B.C.S. Variational State

The pairing theory of superconductivity considers the ground state to be a statewithin the grand canonical ensemble. That is the ground state is composed ofa linear superposition of states with different numbers of particles. If required,a ground state in the canonical ensemble can be found by projecting the B.C.S.ground state onto one with a fixed number of particles. The B.C.S. state ischosen variationally, by minimizing the energy.

The B.C.S. ground state is found from anti-symmetrizing the many-particlestate which is composed of product over wave vector k. For each wave vectork the Cooper pair ((k, ↑), (−k, ↓)) is occupied with probability amplitude u(k)and unoccupied with probability amplitude v(k). The probability amplitudesare often referred to as coherence factors.

| ΨBCS > =∏k

(v(k) + u(k) c†k,↑ c

†−k,↓

)| 0 > (2256)

The amplitudes satisfy the constraint

| u(k) |2 + | v(k) |2 = 1 (2257)

The normal state for non-interacting electrons just corresponds to the specialcase,

| u(k) |2 = Θ( µ − ε(k) ) (2258)

The functions u(k) and v(k) are variational parameters that are found be min-imizing the expectation value of the Hamiltonian, which includes the pairinginteraction.

The expectation value for the appropriate energy, in the B.C.S. state, isgiven by

< ΨBCS | ( H − µ N ) | ΨBCS > =∑

k

2 ( ε(k) − µ ) | u(k) |2

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−∑k,k′

V (k, k′) v(k)∗ u(k) u(k′)∗ v(k′)

(2259)

The term involving the double sum is eliminated by introducing a quantity

∆(k) =∑k′

V (k, k′) u(k′)∗ v(k′) (2260)

On minimizing the energy, subject to the constraint of conservation of proba-bility, with respect to u(k) and v(k)∗ one finds

0 =[

2 ( ε(k) − µ ) + λ

]u∗(k) − ∆(k) v∗(k)

0 = λ v(k) − ∆(k) u(k)(2261)

where λ is the Lagrange undetermined parameter. These equations can besolved to yield

| u(k) |2 =12

(1 − ε(k) − µ

Eqp(k)

)(2262)

and

| v(k) |2 =12

(1 +

ε(k) − µ

Eqp(k)

)(2263)

where we have defined

u(k) v∗(k) =∆(k)

2 Eqp(k)(2264)

The first two equations can be multiplied and equated to the modulus squaredof the third equation according to the identity

| u(k) |2 | v(k) |2 =(u(k) v∗(k)

)∗ (u(k) v∗(k)

)(2265)

The resulting expression can be solved for Eqp(k) to yield

Eqp(k) = +√

( ε(k) − µ )2 + | ∆(k) |2 (2266)

The factor | u(k) |2, is the probability of finding an electron of momentum kin the B.C.S. ground state and, therefore, is just n(k). Unlike a Fermi-liquid,where n(k) is discontinuous at the Fermi-surface with magnitude 1/Zk, in thesuperconductor the distribution drops smoothly to zero as k increases above kF .Thus, the concept of Fermi-surface is not well defined in the superconductingstate. The energy Eqp(k), relative to µ, turns out to be the energy required tocreate a quasi-particle of momentum k from the ground state. The quasi-particleis either of the form of an added electron or a hole. With the B.C.S. ground stateboth of these leave a single unpaired electron in an otherwise perfectly pairedB.C.S. state. The minimum energy required to create two quasi-particles, thatis two individual electrons, is just 2 ∆(kF ) .

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25.3.3 The Gap Equation

The ”energy gap” parameter satisfies the non-linear integral equation

∆(k) =∑k′

V (k, k′)∆∗(k′)2 E(k′)

(2267)

where V (k, k′) is the attractive pairing interaction mediated by the phonons.The interaction can be approximated by the attractive s-wave potential

V (k, k′) = V for | ε(k) − µ | < h ωD

V (k, k′) = 0 for | ε(k) − µ | > h ωD

(2268)

In this case one finds

∆(k) = ∆(0) for | ε(k) − µ | < h ωD

∆(k) = 0 for | ε(k) − µ | > h ωD

(2269)

where the gap in the quasi-particle dispersion relation at the Fermi-energy isgiven by the solution of

1 = V ρ(µ)∫ hωD

0

dε1√

ε2 + | ∆(0) |2

= V ρ(µ) sinh−1 h ωD

| ∆(0) |(2270)

which is solved as| ∆(0) | =

h ωD

sinh 1V ρ(µ)

(2271)

This gap 2 ∆(0) just corresponds to the energy required to break a Cooper pair.At finite temperatures, the superconducting gap satisfies the equation

∆(k) =∑k′

V (k, k′)∆(k′)

2 Eqp(k′)( 1 − 2 f(Eqp(k′)) )

=∑k′

V (k, k′)∆(k′)

2 Eqp(k′)tanh

βEqp(k′)2

(2272)

The tanh factor is a decreasing function for increasing temperature, therefore,for the equation to have a non-trivial solution the denominator has to decrease

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with increasing temperature. This can only happen if | ∆(T ) | decreases withincreasing temperature. For sufficiently high temperatures, the equation canreduces to

∆(T ) = ∆(T ) V ρ(µ) β hωD (2273)

which only has the trivial solution ∆(0) = 0. The critical temperature wherethe gap first vanishes ∆(Tc) = 0 is given by

1 = ρ(µ) V∫ hωD

0

dεtanh βc ε

2

2 ε

= ρ(µ) V∫ βhωD

2

0

dztanh zz

= ρ(µ) V(

lnβhωD

2−∫ ∞

0

dz ln z sech2z

)= ρ(µ) V

(lnβhωD

2− ln

π

4 exp γ

)(2274)

The superconducting gap decreases with increasing temperature and vanishesat a critical temperature Tc given by

kB Tc = 1.14 hωD exp[− 1

V ρ(µ)

](2275)

The critical temperature is proportional to M− 12 as expected from the isotope

effect. Above the critical temperature ∆(T ) = 0 and the B.C.S. state reducesto the normal state. Just below the critical temperature one has

∆(T )2 =8 π2

7 ξ(3)k2

B Tc ( Tc − T ) T → Tc (2276)

Thus, the order parameter has a typical mean field variation with an exponentof β = 1

2 close to Tc.

25.3.4 The Ground State Energy

The normal state is unstable to the B.C.S. state only if it has a higher energy.At T = 0 the stability can be seen be examining the energy

< ΨBCS | ( H − µ N ) | ΨBCS > =∑

k

[( ε(k) − µ ) − ( ε(k) − µ )2√

( ε(k) − µ )2 + | ∆(k) |2

]

− | ∆(0) |2

V(2277)

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The condensation energy is defined as the difference between the energy of thesuperconducting state and the normal state

∆E = < ΨBCS | ( H − µ N ) | ΨBCS > − 2∫ 0

−∞dε ε ρ(µ+ ε)

(2278)

The condensation energy is evaluated by writing the sum over k as an integralover the density of states.

∆E =∫ hω

0

dε ρ(µ+ ε)

[ε − ε2√

ε2 + | ∆(0) |2

]

+∫ 0

−hωD

dε ρ(µ+ ε)

[− ε − ε2√

ε2 + | ∆(0) |2

]

− | ∆(0) |2

V(2279)

Then the integral over states below the Fermi-energy, ε < 0, is transformed toan integral over positive ε. This leads to

∆E = 2∫ hωD

0

dε ρ(µ+ ε)

[ε − ε2√

ε2 + | ∆(0) |2

]

− | ∆(0) |2

V

= 2 ρ(µ)∫ hωD

0

[ε −

√ε2 + | ∆(0) |2

]

+ 2 ρ(µ)∫ hωD

0

dε| ∆(0) |2√

ε2 + | ∆(0) |2− | ∆(0) |2

V

(2280)

The integrals are evaluated as

∆E = − ρ(µ)| ∆(0) |2

2+| ∆(0) |2

V− | ∆(0) |2

V

= − ρ(µ)| ∆(0) |2

2(2281)

which shows that the condensation energy comes from the attractive potentialthat lowers the energy of the pair more than the increase in the potential energy

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caused by the confinement within the coherence length ξ. The net lowering canbe understood in terms of the quasi-particle dispersion relation. The electronswith energy within | ∆(0) | of µ have their energy lowered by an amount | ∆(0) |.The net lowering of energy is just the number of electrons ρ(µ) | ∆(0) | times| ∆(0) |. Therefore, the B.C.S. state has lower energy than the normal statewhen the gap is non-zero.

25.4 Quasi-Particles

The B.C.S. Hamiltonian can be solved for the quasi-particle excitations, in themean field approximation, by linearizing the pairing interaction terms. In anormal metal, the only allowed matrix elements are between initial and finalstates which have the same number of electrons. However, since for a super-conductor the average is to be evaluated in the B.C.S. ground state, matrixelements between operators with different numbers of pairs are non zero. Thesegive rise to the anomalous expectation values. For example, the anomalous ex-pectation value associated with adding a pair of electrons ((k′, ↑), (−k′, ↓)) tothe superconducting condensate is given by the probability amplitude

< ΨBCS | c†k′,↑ c†−k′,↓ | ΨBCS > = u(k′)∗ v(k′) (2282)

The linearized mean field Hamiltonian is given by

HMF − µ N =∑

k

(( ε(k) − µ ) c†k,↑ ck,↑ + ( ε(−k) − µ ) c†−k,↓ c−k,↓

)−

∑k,k′

V (k, k′) < ΨBCS | c†−k′,↓ c†k′,↑ | ΨBCS > ck,↑ c−k,↓

−∑k,k′

V (k, k′) c†−k′,↓ c†k′,↑ < ΨBCS | ck,↑ c−k,↓ | ΨBCS >

+∑k,k′

V (k, k′) < ΨBCS | c†−k′,↓ c†k′,↑ | ΨBCS > < ΨBCS | ck,↑ c−k,↓ | ΨBCS >

(2283)

The anomalous expectation value leads to a term in the Hamiltonian withstrength

∆(k) =∑k′

V (k, k′) u(k′)∗ v(k′) (2284)

which corresponds to a process in which two electrons ((k, ↑), (−k, ↓)) are ab-sorbed into the condensate. The mean field Hamiltonian also contains the Her-mitean conjugate which represents the reverse process in which two electronsare emitted from the condensate.

HMF − µ N =∑

k

(( ε(k) − µ ) c†k,↑ ck,↑ + ( ε(−k) − µ ) c†−k,↓ c−k,↓

)

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−∑

k

[∆(k) ck,↑ c−k,↓ + c†−k,↓ c

†k,↑ ∆(k)∗

]+| ∆(0) |2

V

(2285)

In the absence of an electromagnetic field, the order parameter ∆(k) can bechosen to be real. The mean field Hamiltonian involves terms in which the con-densate emits or absorbs two electrons. This is reminiscent of the treatment ofanti-ferromagnetic spin waves, using the method of Holstein and Primakoff, ex-cept here the Hamiltonian involves fermions rather than bosons. The quadraticHamiltonian can be diagonalized by means of a canonical transformation.

We shall define two new fermion operators via the transformation

αk = exp[

+ S

]ck,↑ exp

[− S

](2286)

and

β†k = exp[

+ S

]c†−k,↓ exp

[− S

](2287)

where S is an anti-Hermitean operator, S† = − S. The energy eigenvalues ofthe Hamiltonian can be found directly from the transformed Hamiltonian

H ′MF = exp

[+ S

]HMF exp

[− S

](2288)

as they have the same eigenvalues and the eigenstates are related via

| φ′n > = exp[

+ S

]| φn > (2289)

The operator S is chosen to be of the form

S =∑

k

θk

(c†k,↑ c

†−k,↓ − c−k,↓ ck,↑

)(2290)

Explicitly, the transformation yields

αk = ck,↑ cos θk − c†−k,↓ sin θk

β†k = c†−k,↓ cos θk + ck,↑ sin θk

(2291)

Rather than working with the transformed Hamiltonian, we shall express theoriginal Hamiltonian in terms of the transformed operators. Hence, we shallrequire the inverse transformation which expresses the original electron andholes operators in terms of the new quasi-particles. The inverse transformation

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is expressed in terms of the transformation matrix but with θk → − θk so onehas

ck,↑ = αk cos θk + β†k sin θk

c†−k,↓ = β†k cos θk − αk sin θk

(2292)

The mean field Hamiltonian is expressed in terms of the new operators and θk ischosen so that the terms that are not represented in terms of the quasi-particlenumber operators vanish. The normal terms in the Hamiltonian are found as∑

k

(( ε(k) − µ ) c†k,↑ ck,↑ + ( ε(−k) − µ ) c†−k,↓ c−k,↓

)

=∑

k

( ε(k) − µ )

[sin2 θk

(αk α

†k + βk β

†k

)+ cos2 θk

(α†k αk + β†k βk

) ]

+∑

k

( ε(k) − µ ) sin 2θk

(α†k β

†k + βk αk

)(2293)

The anomalous terms are evaluated as

−∑

k

(∆(k) ck,↑ c−k,↓ + c†−k,↓ c

†k,↑ ∆(k)

)

= −∑

k

Re

[∆(k)

]sin 2θk

(β†k βk − αk α

†k

)

+∑

k

Re

[∆(k)

]cos 2θk

(α†k β

†k + βk αk

)(2294)

The off diagonal terms can be made to vanish by choosing

tan 2θk = −Re[

∆(k)](

ε(k) − µ) (2295)

Thus, θk decreases from a value less than π4 to less than − π

4 as k varies fromhωD below µ to hωD above µ. After this value has been chosen, the Hamiltonianis expressed as the sum of a constant and terms involving the number operatorsof the α and β quasi-particles

HMF = E0 +∑

k

Eqp(k)(α†k αk + β†k βk

)(2296)

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This procedure shows that the excitations are quasi-particles as they are stillfermions. Furthermore, these quasi-particles have excitation energies which havethe dispersion relation

Eqp(k) = +√

( ε(k) − µ )2 + | ∆(k) |2 (2297)

The canonical transformation shows that the quasi-particles are part electronand part hole like. Basically, this is a consequence that the quasi-particle ex-citation consists of a single unpaired electron (k, σ), in the presence of thecondensate. This specific state can be produced from the ground state, eitherby adding the electron (k, σ) to the system or by breaking a Cooper pair byremoving the partner electron (−k,−σ). We note that the quasi-particles areeigenstates of the spin operator. The α quasi-particle is a spin up excitationas it is composed of an up spin electron and down spin hole, whereas the βquasi-particle is a spin down excitation. From the dispersion relation, one findsthat the B.C.S. superconductor is actually characterized by the presence of agap in the excitation spectrum. That is, there is a minimum excitation energy2 | ∆(kF ) | corresponding to breaking a Cooper pair and producing two inde-pendent quasi-particles.

——————————————————————————————————

25.4.1 Exercise 83

Evaluate the constant term in the mean field B.C.S. Hamiltonian. Show thatthe variational B.C.S. ground state is the lowest energy state of the mean fieldHamiltonian by showing that the quasi-particle destruction operators annihilatethe B.C.S. state

αk | ΨBCS > = 0

βk | ΨBCS > = 0(2298)

——————————————————————————————————

25.5 Thermodynamics

Since the quasi-particles are fermions, the entropy S due to the gas of quasi-particles is given by the formulae

S = − 2 kB

∑k

[( 1− f(Eqp(k)) ) ln[ 1− f(Eqp(k)) ] + f(Eqp(k)) ln[ f(Eqp(k)) ]

](2299)

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By the usual procedure of minimizing the grand canonical potential Ω withrespect to the distribution f(Eqp(k)) , one can show that the non-interactingquasi-particles are distributed according to the Fermi-Dirac distribution func-tion. Therefore, the quasi-particle contribution to the specific heat is just givenby

Cqp(T ) = − 2T

∫ +∞

−∞dE ρqp(E)

(E2 − T

2∂∆(T )2

∂T

) (∂f

∂E

)(2300)

which involves the average of the temperature derivative of the square of thequasi-particle energy at µ, and the quasi-particle density of states

ρqp(E) =∑

k

δ

(E − Eqp(k)

)(2301)

Since, in the mean field approximation, the square of the gap has a finite slopefor T just below Tc and is zero above,

∆(T )2 ∼ ∆(0)2(

1 − T

Tc

)Θ( Tc − T ) (2302)

the specific heat has a discontinuity at Tc. In B.C.S. theory, the magnitudeof the specific heat jump has the value given by, 3.03 ∆2(0) ρ(µ) / Tc. Thus,the value of the specific heat jump found in weak coupling B.C.S. theory whennormalized to the normal state specific heat is given by

∆C(Tc)C(Tc)

=Cs − Cn

Cn

=12

7 ξ(3)= 1.43

(2303)

This ratio is a measure of the quantity

12 k2

B Tc

(∂∆2(T )∂T

) ∣∣∣∣Tc

∼(

∆(0)kB Tc

)2

(2304)

The values of the specific heat jumps for strong coupling materials tend to behigher than the B.C.S. value, for example the normalized jump for Pb is as largeas 2.71. This trend is understood as being due to inelastic scattering processeswhich tend to suppress Tc more than ∆(0). The heavy fermion superconductorsshow that the normalized specific heat discontinuities are significantly smallerthan the B.C.S. ratio.

Low Temperatures.

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The gap in the quasi-particle density of states could be expected to show upin an activated exponential dependence of the low-temperature electronic spe-cific heat, for T Tc. For these temperatures the order parameter is expectedto have saturated, and so if one considers the Fermi-liquid as being well formedthen the quasi-particle contribution is given by

Cqp(T ) = − 2T

∫ +∞

−∞dE ρqp(E) E2

(∂f

∂E

)(2305)

The B.C.S. quasi-particle density of states is evaluated as

ρqp(E) =∑

k

δ

(E − Eqp(k)

)

=∫ ∞

−∞dε ρ(ε) δ

(E −

√( ε − µ )2 + ∆(T )2

)∼ ρ(µ)

| E || ε − µ |

= ρ(µ)| E |√

E2 − ∆(T )2for | E | > ∆(T )

(2306)

In evaluating the B.C.S. density of states, the conduction band electron densityof states has been approximated by a constant value. The resulting B.C.S.quasi-particle density of states has a gap of magnitude 2 ∆(T ) around theFermi-energy. This yield an exponentially activated behavior of the specificheat,

Cqp(T ) ∼ 9.17 γ Tc exp[− ∆(0)

kB T

](2307)

in B.C.S. theory.

25.6 Perfect Conductivity

The current is composed of the sum of a paramagnetic current and a diamagneticcurrent. The paramagnetic current can be evaluated from the Kubo formula.The paramagnetic current is expressed as

jp(q;ω) =

e2 h

4 m2 c

1V

∑k

( 2 k − q )(

( 2 k − q ) . A)×

×(

f(E(k)) − f(E(k − q))E(k − q) − E(k) + h ω

+f(E(k − q)) − f(E(k))E(k) − E(k − q) + h ω

)(2308)

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In this expression E(k) is the quasi-particle energy in the superconductor. Inthe static limit with uniform fields, ( ω → 0, q → 0 ), the paramagneticcurrent reduces to

jp(0; 0) = 2

e2 h

m2 c

1V

∑k

k ( k . A )(− ∂f(E(k))

∂E

)(2309)

The total current is found by combining the paramagnetic current with thediamagnetic current

j(0; 0) = 2e2 h

m2 c

1V

∑k

k ( k . A )(− ∂f(E(k))

∂E

)− ρ e2

m cA

= 2e2 h

m2 c

16 π2

∫dk k4 A

(− ∂f(E(k))

∂E

)− ρ e2

m cA

=(e2 h

m2 c

13 π2

∫dk k4

(− ∂f(E(k))

∂E

)− ρ e2

m c

)A

(2310)

In the normal state, where the gap in E(k) vanishes the derivative of the Fermi-function can be approximated as(

− ∂f(E(k))∂E

)= δ( E(k) − µ ) (2311)

which leads to the vanishing response as

2 mh2 k6

F

µ k4F = 1 (2312)

Thus, in the normal state current does not flow in response to a static vectorpotential. However, in the superconducting state the total current is given by

j = − ρ e2

m cA

[1 − 2 µ

k5F

∫dk k4

(− ∂f(E(k))

∂E

) ](2313)

and as there is a gap on the Fermi-surface, the derivative of the Fermi-functionis always exponentially small. Because of the finite superconducting gap, thesecond term is small and the cancellation does not occur. In the superconductingstate, this reduces to the London equation

j = − ρ e2

m cA (2314)

This shows that a current will flow in a superconductor in response to a staticvector potential, that is the current will screen an applied magnetic field. Thisleads to the Meissner effect.

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25.7 The Meissner Effect

In the superconducting state the susceptibility is expected to be dominated bythe diamagnetic susceptibility produced by the supercurrent shielding the exter-nal field. The Pauli spin susceptibility will also be modified by the superconduc-tivity, and provide information about the pairing. The zero field susceptibilityis defined as a derivative of the magnetization, χs(T ) = (∂M

∂H ). The magneti-zation, produced by the electronic spins aligning with a magnetic field appliedalong the z axis, is given by

Mz =(g µB

2

) ∑k

[f(E↑(k)) − f(E↓(k))

](2315)

which is given in terms of the Fermi-distribution for quasi-particles with spin σand quasi-particle energy Eσ(k).

For singlet pairing the magnetic field couples to the spins of the quasi-particles via the Zeeman energies and, as can be seen from inspection of thematrix only the time reversal partners pair. The quasi-particles consist of bro-ken pairs, i.e. electrons of spin σ and holes of spin − σ. Since a down spin holehas the same Zeeman energy as an up spin electron, the quasi-particle energiesdepend on field through

Eσ(k) = EH=0(k) −(g µB σ H

2

)(2316)

and so the spin susceptibility takes the usual form

χs(T ) = − 2(g µB

2

)2 ∫ +∞

−∞dE ρqp(E)

(∂f

∂E

)(2317)

which involves the B.C.S. quasi-particle density of states. The Pauli suscepti-bility tends to zero as T → 0 in an exponentially activated way

χp(T ) ∼ exp[− ∆(0)

T

](2318)

The exponential vanishing of the spin susceptibility occurs as the electrons formsinglet pairs in the ground state, and the finite spin moment is caused by ther-mal population of quasi-particles.

Thus, in the spin singlet phases, the spin susceptibility could be expectedto vanish as T → 0. However, spin-orbit coupling will produce a residualsusceptibility that depends on the ratio of the superconducting coherence length,ξ0 to the mean free path due to spin-orbit scattering, lso. In the presenceof spin-orbit coupling, the spin is no longer a good quantum number for thesingle particle eigenstates, and the spin up and spin down states are mixed. In

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the limit that the strength of the spin-orbit coupling λ−→L .−→S is so large that

λ ∆0, the average value of σz for a single particle state tends to zero.The spin susceptibility is, therefore, reduced. The scattering has the effect thata significant contribution to the normal state χ(T ) comes from single particlestates separated by an energy of the order of the spin-orbit scattering rate,which is by our assumption greater than ∆. As an opening up of a gap at theFermi-energy is not expected to change the contribution of these higher energystates, one finds that the susceptibility in the superconducting state can remaincomparable in magnitude to the normal state value. According to Anderson,the normalized susceptibility should have the two limits,

χs(0)χn

= 1 − 2 lso

π ξ0(2319)

for strong spin-orbit scattering and for weak spin-orbit scattering one has

χs(0)χn

=π ξ06 lso

(2320)

Hence, a partial Meissner effect at T = 0 can be found in a conventionalsuperconductor.

26 Landau-Ginsberg Theory

Superconductors can be divided into two categories, which depend on theirmacroscopic characteristics when an applied magnetic field is present. The clas-sification is based on the length scale over which the magnetic field is screenedλL relative to the length scale over which the superconducting order parameterchanges. The latter length is given by the spatial extent of The Cooper pairwave function or coherence length ξ

ξ =h vf

π ∆(T )(2321)

Type I Superconductors.

In simple (non-transition) metals the penetration depth is small, e.g., λ ∼300 A for Al while the coherence length is large ξ ∼ 1 × 104 A as vF is large.Materials where

κ =λ

ξ<

1√2

(2322)

are type I superconductors.

Type II Superconductors.

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For transition metals, rare earths and intermetallic compounds, where theband mass is very large, λL is very large ( ∼ 2000 A for V3Ga ) and as theFermi-velocity is small ( vF ∼ 104 m /s ) then as Tc and ∆(0) are high xi issmall ( ∼ 50 A ). Materials where

κ =λ

ξ>

1√2

(2323)

are type II superconductors.

Since λL and κ diverge the same way at Tc the dimensionless ratio κ is ap-proximately temperature independent.

If a magnetic field H < Hc is applied to a small superconductor, the fieldis excluded from the superconductor, but if H > Hc the field will penetrateand the superconductor will undergo a transition to the normal state. If a fieldis applied normal to the surface of a large superconducting slab then, because∇ . B = 0, the field has to penetrate the slab.

In a type I superconductor the magnetic field will concentrate into regionswhere

| B | = Hc (2324)

which are normal and regions where

B = 0 (2325)

which are superconducting. The condensation energy density in the supercon-ducting state is H2

c

8 π and the diamagnetic energy of the normal state is also H2c

8 π .These regions are separated by a domain wall which has positive energy. Theenergetic cost of forming a domain wall of area A can be estimated as

E

A∼ ξ

H2c

8 π− λ

H2c

8 π(2326)

where the term ξH2

c

8 π is the energetic cost of setting the order parameter to

zero, and the diamagnetic energy is reduced by λH2

c

8 π . Because of this pos-itive domain wall energy in a type I superconductor, the number of domainsand domain walls will be minimized. The domain pattern will have a scale ofsubdivision which is intermediate between ξ and the sample size.

In type II superconductors, a similar separation occurs, but as the domainwall energy is negative, the superconductor will break up into as many normalregions as possible. These normal regions have the form of magnetic flux car-rying tubes that thread the sample, which are known as vortices. Each vortexcarries a minimum amount of flux Φ0, the flux is quantized in units of Φ0.

Φ0 =h c

2 e= 2.07 × 10−7 Gauss cm2 (2327)

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The vortices first enter the superconductor at a critical field Hc1. The vorticesform a triangular lattice of vortices. The superconductor becomes saturatedwith vortices when it becomes completely normal at an upper critical field Hc2.

The magnetization M is linear in field up to Hc1 with susceptibility − 14 π .

At Hc1 the magnitude of the magnetization has a cusp and the magnitude fallsto zero at Hc2.

506