Frederick Seitz - Solid State Physics, Vol.26

Contributors to This Volume

J. 0. Dimmock

Maurice Glicksman

I. S . Zheludev

J. M. Ziman

SOLID STATE PHYSICS

Advances in Research and Applications

Editors

HENRY EHRENREICH

Division of Engineering and Applied Physics Harvard University, Cambridge, Massachusetts

FREDERICK SEITZ

The Rockefeller Univer&y, New York, New York

DAVID TURNBULL

Division of Engineering and Applied Physics Harvard University, Cambrzilge, Massachusetts

VOLUME 26

197 1

@ ACADEMIC PRESS NEW YORK AND LONDON

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Contributors to Volume 26

J. 0. DIMMOCK, Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts

MAURICE GLICKSMAN, Brown University, Providence, Rhode Island

I. 5. ZHELUDEV,* Institute of Crystallography, Academy of Sciences, Moscow, USSR

J. M. ZIMAN, H. H. Wills Physics Laboratory, University of Bristol, Bristol, England

* Temporary address: Interiiationel Atomic Energy Ageqcy, Vienna, Aust.ria. vii

Preface

Of the four articles in this volume, the first two deal with band theory. In some sense they complement those concerned with the pseudopotential method which take up Volume 24 of. this series. Ziman has succeeded admirably in dissecting the essential physics underlying the single particle description of the electronic energy level structure of solids out of the surrounding algebraic complications. His article emphasizes that a unified view has at last emerged from the large variety of approaches to this complicated problem that has been developed over the years. This contribution will prove valuable both to those interested in a didactic pres- entation of the subject and as a handy reference for experts concerned with specific formal results.

Dimmocks article focuses on the results of the large body of band structure calculations based on the so-called APW and KKR methods that have now accumulated. The formal preliminaries duplicate a few of the points made in Zimans article. This slight redundancy, however, is offset by the advantage that the article is self-contained. The results presented illustrate dramatically the degree of success band theory has achieved in explaining a variety of experiments in a large range of crystalline solids, some of them of considerable complexity. In addition to ex- tensive calculations of band structure, the reader may also find the bibliog- raphy of APW and KKR calculations given at the end of the article to be of considerable value.

The subject of plasmas in solids has not been treated explicitly in this series since the appearance of the article by Pines in Volume 1. The contribution by Glicksman is the first of a group on various aspects of plasmas which are to appear soon in this series. It presents a comprehensive review of experiment and theory for plasmas in metals and semiconductors.

The final article in this volume by Zheludev treats the crystallographic features of phase transitions in ferroelectrics and antiferroelectrics. This topic also was discussed by Kanzig in his review of ferroelectrics and antiferroelectrics which appeared in Volume 4 of this series.

Supplementary Monographs

Supplement 1: T. P. DAS AND E. L. HAHN Nuclear Quadrupole Resonance Spectroscopy, 1958

Supplement 2 : WILLIAM Low Paramagnetic Resonance in Solids, 1960

Supplement 3: A. A. MARADUDIN, E. W. MONTROLL, AND G. H. WEISS Theory of Lattice Dynamics in the Haxmonic Approximation, 1963

Supplement 4: ALBERT C. BEER Galvanomagnetic Effects in Semiconductors, 1963

Supplement 5: R. S. KNOX Theory of Excitons, 1963

Supplement 6: S. AMELINCKX The Direct Observation of Dislocations, 1964

Supplement 7: J. W. CORBETT Electron Radiation Damage in Semiconductors and Metals, 1966

Supplement 8: JORDAN J. MARKHAM F-Centers in Alkali Halides, 1966

Supplement 9: ESTHER M. CONWELL High Field Transport in Semiconductors, 1967

Supplement 10: C. B. DUKE Tunneling in Solids, 1969

Supplement 11 : MANUEL CARDONA Optical Modulation Spectroscopy of Solids, 1969

X v i

The simple believeth every word: but the prudent man looketh well to his going.-Proverbs 14:15

The Calculation of Bloch Functions

J. M. Z I M ~ H. H. Wills Physics Laboratory, University of Brislol, Bristol, England

I. The Basic Problem of Solid State Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Technique, Art, or Science?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Lattice Potential.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Relativistic Hamiltonians.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11. Mathematical Insights and Physical Concepts. . . . . . . . . . . . . . . . . . 4. The LCAO Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The Cellular Method .............................................. 22 6. The NFE Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . 27 7. d Bands and Resonanc ............................................. 33

. 39 8. Orthogonalized Plane Waves.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

. . . . . 59 69 69 75 81 84 92 99

1 1 4

1 3

111. Pseudopotentials.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9. Analytic Pseudopotentials. . . . . . . . . .................... . . . . . 47 10. Model Pseudopotentials. . . . . . . . . . . ....................

IV. Augmented Plane Waves and Greenians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Augmented Plane Waves.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. The Greenian (KKR)Method . . . . . . . . .............................. 13. The Greenian in Reciprocal Lattice Representation.. . . . . . . . . . . . . . . . . . . 14. The KKR Method in APW Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. The KKR Method and the Model Hamiltonian.. . . . . . . . . . . . . . . . . . . . . .

V. Conclusion.. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. The Basic Problem of Solid State Theory

1. TECHNIQUE, ART, OR SCIENCE?

It is now some 40 years since Felix Blochl first discussed the eigenfunctions of the Schrodinger equation for an electron in a crystal. The theorems proved at that time are the foundation of all subsequent work on the quantum theory of solids: It is impossible to understand the behavior

F. Bloch, 2. Phys. 62, 555 (1928).

1

2 J. M. ZIMAN

of a metal, semiconductor, or insulator without a good knowledge of its electronic band structure. The direct computation of Bloch functions, or their investigation by inference from experiment, is therefore one of the major industries of solid state physics.

This subject cannot be explored in all its ramifications. We should need to review transport properties, the study of the Fermi surface, optical phenomena, lattice dynamics, cohesion, alloying, ferromagnetism, nuclear magnetic resonance, Mossbauer spectra, positron annihilation, etc., for these are all directly dependent upon the electronic structure. We shall concern ourselves solely with the problem of finding &(k) and h ( r ) to satisfy

where the potential is given and satisfies the periodicity condition

U(r + 2) = U(r) for any translation vector 2 of a given perfect crystal lattice. No attempt will be made to catalog the results of such calculations for particular elements and compounds nor shall we consider the evidence upon which it may be claimed that such a computed band structure is in agreement with experiment. Somewhat artificially, we shall also truncate the discussion of the most important recent development in the theory of metals-the concept of a pseudopotential-which arose from the band-structure problem but which has found its most useful applications in transport theory, lattice dynamics, alloy stability, etc.2

Even within these boundaries, the subject really demands a treatise. It would take us into the vast literature of the many-body problem of quantum mechanicsa to give a systematic account of the procedure for replacing the electron-electron interactions by a self-consistent potential U(r) within which we need only solve a one-electron equation (1.1). Another large topic is the group theoretical analysis of Bloch functions at points of high symmetry in the Brillouin zone, where they transform under the various irreducible representations of the corresponding point group operations. This algebraic theory is of great practical value in the computation of eigenfunctions and eigenvalues by reducing the order of the

* J. M. Ziman, Aduan. Phys. 13, 89-138 (1964); W. A. Harrison, Pseudopotentials in the Theory of Metals. Benjamin, New York, 1966; V. Heine, in The Physics of Metals (J. M. Ziman, ed.), pp. 1-61. Cambridge Univ. Press, London and New York, 1969.

* L. Hedin and S. Lundqvist, Solid State Phys. 28, 2-183 (1969); N. Hf March, W. H. Young, and S. Sampanthar, The Many-Body Problem in Quantum Mechanics. Cambridge Univ. Press, London and New York, 1967.

THE CALCULATION OF BLOCH FUNCTIONS 3

matrices to be diagonalized, but is well documented4 and is sufficiently self-contained and independent of the rest of the subject to be treated as a topic in its own right. We also take for granted the basic geometrical theory of lattices, reciprocal lattices, WignerSeitz cells, Brillouin zones, cyclic boundary conditions, constant energy surfaces, the Fermi level, band gaps, multiply connected Fermi surfaces, extended and reduced zone schemes, etc. which may be found in any standard textbook of the theory of s01ids.~

As in most active branches of science, the attack upon the Bloch problem has developed along many different lines, with many different tactical objectives. A primary purpose of this review is to show the progress we have made, in the past decade, in reconciling and reuniting these apparently conflicting points of view and distinct methods. We shall not be concerned so much with the details of algebra, but with the underlying mathematical and physical principles, the implicit assumptions that are made, and the hidden identities in the analysis.

A field such as this has its own history, and its own philosophies. To many of its practitioners, the calculation of band structures& a problem of computational technique. It is simply a matter of finding a workable procedure that will generate solutions of the equations to any desired degree of numerical accuracy. Historically, we may distinguish an eotechnic phase, during which various methods-cellular, orthogonalized plane wave, augmented plane wave, Green Function, etc.-were invented, but gave only limited results because of the enormous labor of hand calculation. The palaeotechnic era began in the ~ O S , with the advent of electronic computers. Very elaborate programs were constructed, and set going, for various materials, but the outcome was often disappointing. Generally speaking, the different methods appeared to converge satisfactorily to much the same numerical results, but these were often a long way from the experimental information that was coming in from Fermi surface and optical studies.

This recognition of the sensitivity of & (k) to the form of u (r) generated a new philosophy: The determination of electronic band structure came to be regarded as an art, where one tried to build up a model of the energy surfaces, represented analytically by a simple function (e.g. a pseudo-

H. Jones, The Theory of Brillouin Zones and Electronic States in Crystals. North- Holland Publ., Amsterdam, 1960; J. Callaway, Energy Band Theory. Academic Press, New York, 1964; A. Nussbaum, Solid State Phys. 18, 165-272 (1966).

6 R. E. Peierls, Quantum Theory of Solids. Oxford Univ. Press (Clarendon), London and New York, 1955; J. M. Ziman, Principl-es of the Theory of Solids, p. 39. Cam- bridge Univ. Press, London and New York, 1964; C. Kittel, Introduction to Solid State Physics, 3rd. ed. Wiley, New York, 1966.

4 J. M. ZIMAN

potential) with only a few adjustable parameters but consistent both with experiment and with some rough algebraic approximation to a solution to the basic Schrodinger equation. As I have indicated, the success of this procedure has revolutionized the whole theory of metals in recent years.

To make further progress, however, it was essential to combine technique and art into a science. We have now, I believe, entered a neotechnic phase, where the heavy engineering of machine computation has been greatly refined, and brought under more delicate control. We have un- covered the mathematical connections between the various methods, and can understand the circumstances under which they should converge. We have a better rationale of empirical models, and are able to transform the experimental fitting of calculated band structures into adjustments of the starting potentials. In other words, we have learned to treat this problem as one where algebraic analysis, numerical computation, and physical intuition all have their part.

The present account may be considered an extension of an earlier course of lectures16 and draws heavily on the admirable reviews by Treusch and Heine2 as well as on the various more specialized articles in the volume on Energy Band Structure in the series edited by Alder et a1.8 But I have tried to bring up to date (Spring 1970) the literature on the theory of the various methods-and even if it contains little that is truly original, an individual point of view is perhaps of value in a survey of a large field.

The whole article really falls into two parts. In Sections 2-7, I have endeavored to derive, by general physical intuition (based mathematically on tight-binding, cellular, and pseudopotential concepts) , the main characteristics of Bloch functions in various materials, drawing attention to those features which depend essentially on the nature of the constituent atoms and those which are governed by their arrangement in a lattice. These sections could stand on their own, as a hand-waving account of the subject, but are meant as a background to the more solemn discussion in Sections 8-15 of the mathematical techniques, which, in their turn, exemplify or justify these intuitive arguments.

2. THE LATTICE POTENTIAL

The potehial U(r) is not just any function satisfying the periodicity condition (1.2) ; real solids are made up of atoms, which come in a limited

J. M. Ziman, in Theory of Condensed Matter (F. Bassani, G. Cagliotti, and J. M. Ziman, eds.), pp. 3-23. Internat. Atomic Energy Agency, Vienna, 1968. J. Treusch, Feskiurper Problem 7, 18-72 (1967). B. Alder, S. Fernbach, and M. Rotenberg, eds., Methods Cornputat. Phys. 8 (Energy Bands in Solids) (1968).


range of shapes and sizes. Hypothetically, let us set up a low density array of independent atoms, and squeeze it uniformly until we get to the correct distance between the nuclei. This, of course, does violence to the wave functions of the outer electrons on the atoms, which are forced into contact. Because of the electron-electron interaction, there can never be precise equality between the one-electron potentials for the free atoms and those to be used in the Bloch problem.

But the consequences of this interaction depend upon the material. In an inert noble gas, the outer closed shell of electrons may be slightly com- pressed, but there is no fundamental change. The existing electrons remain bound, while any extra electron would simply see a succession of nonoverlapping, spherically symmetric potentials, as if for the free neutral atoms (Fig. la). Similarly, in an ionic crystal, once the valence electrons have been transferred from the metallic atoms, we find an assembly of closed-shell ions, each nearly spherically symmetrical and capable of binding all its electrons. An itinerant electron can see the Coulomb forces exerted by neighboring ions, but since these are always arranged so as to cancel one anothers charges as nearly as possible, the effect is simply to distort the potential function in the interstitial region without significantly lowering the barriers between the ionic spheres (Fig. lb) .

In a metal or semiconductor, however, the eigenstates of the valence electrons are profoundly modified, and become the extended Bloch wave functions that we propose to study. In the long run, we cannot avoid a complete self-consistent calculation, putting the electron density of these new states back into the one-electron Hartree potential from which they are to be computed. This is now the main barrier to progress in the theory of band structure, but will not, as I have said, be dealt with in detail here. Nevertheless, the general principle of local charge neutrality is very powerful. The total number of valence electrons in a unit cell must be exactly equal to the charge of the ions in that cell. This can be ensured automatically by the heuristic device of assuming that each ion continues to carry the charge cloud of electrons that it had in the neutral atom, even if these clouds now interpenetrate one another. In other words, we calculate by Poissons equation the electrostatic potential v , (r) produced by the nucleus and all the electrons in a neutral atom-a spherically symmetrical function falling off rapidly to zero beyond a certain atomic radius. But this radius is somewhat larger than that available for the atom in the unit cell of the crystal, so that when we superpose potentials to form the periodic function

u(r) = - 0, I

we cannot ignore the effects of overlap between neighbors. This is, of course,

Solid Ne

No+

-T NoCl

ct-

+T

FIG. 1. Making a solid from neutral atoms: (a) rare gas, (b) ionic crystal, (c) metal.


the mechanism by which the valence electrons are set free for metallic conduction (Fig. lc) .

This picture, although nearly consistent with the electron densities inferred from X-ray diffraction by metal^,^ is certainly not exact, and does not seem to be formally deducible from first principles except as a grossly linearized approximation to the Thomas-Fermi equations. But other methods, which treat the ions as swimming in a sea of free electrons, have to grapple with all the problems of screening, correlation, and exchange, and yield, eventually, much the same type of result.* The real question is not whether the potential in a metal or semiconductor can be represented fairly accurately as a superposition of spherically symmetrical functions centered on the lattice sites, as in (2.1), but exactly what form of neutral atom or neutral pseudoatom potential is correct in a particular material. Although sometimes treated as if any reasonable approximation were good enough, the nature of this potential in the outer regions, where it overlaps its neighbors, can have a significant effect on the overall electronic structure of the crystallo; but this is a topic into which we have agreed not to stray.

The strategy of the attack on the Bloch problem is enormously simplified by the spherical symmetry of the potential va(r) . The numerical integration of a general three-dimensional partial differential equation is a formidable problem, unless we can set in motion the powerful machinery of the angular momentum representation. This can achieve far greater economies of effort than are obtained at a later stage by using the finite point group symmetries of the lattice.

In algebraic language, we try to make use of solutions of equations of the type

for specially chosen values of the energy parameter K ~ . These are always linear combinations of the set of functions

- V2x (r) + va (r)x (r) = K ~ X (r) (2.2)

where Ylrn(r) is a spherical harmonic in the direction cosines of the vector r. The radial function satisfies the one-dimensional differential equation

which can easily be integrated numerically from a nonsingdar value at T = 0.

P. J. Brown and W. H. Taylor, in The Physics of Metals (J. M. Ziman, ed.), pp. 317-39. Cambridge Univ. Press, London and New York, 1969.

0 J. M. Ziman, Proc. Phys. Soc. 91, 701-723 (1967).

8 J. M. ZIMAN

Several of the band-structure methods do indeed depend upon this scheme-e.g., in the LCAO method the attempt is made to construct the Bloch states entirely out of atomic orbitals which are, of course, functions of the type XI,,, for which the radial functions also satisfy the boundary condition of converging as T -+ 00. The same functions also provide the core states of the OPW and pseudopotential methods. But there is a serious difficulty in any general scheme for representing the wave function #k solely in terms of eigenfunctions of each of the atomic potentials that overlap to make u (r) . This is well known to students of the subject, but the reason does not seem to have been made clear. It appears to be as follows.

Below the energy zero of u.(r) [i.e. for & < u.( a)], the eigenfunctions of (2.2) represent bound states, whose mutual interaction decreases as we go to lower energies: a computational scheme based on these functions would converge like a power series in the overlap integral between orbitals centered on neighboring atoms, which behaves like exp[ - a! ( -&) 12]. On the other hand, for positive energies, the functions xl,,,(r) belong to a continuum of propagating states, each modified by the scattering from a single atomic center: these would be combined into a perturbation series in powers of a parameter such as ( I t I/&), where I t I measures the t matrix (i.e. scattering amplitude) from each atom. For both these series, therefore, & = 0 is a branch point, which can only be bridged by a connection formula valid in both regions-as derived e.g. by the APW and KKR methods.

But when there is substantial overlap of potentials (Fig. 2 ) , a no-mans land is created in the range of energy from & = 0 down to the top of the actual potential barrier between neighboring cells of the lattice. In this region the bound states of v,(r) have been destroyed. This effect is fund* mental, for it turns a discrete spectrum into a continuum, which cannot be described by an ordinary unitary transformation in an invariant Hilbert space. On the other hand, any attempt to continue the t-matrix formalism below & = 0 is frustrated because we have no way of defining and manipu- lating this operator at points where two potential functions overlap in space. In this important range of energies, therefore, neither scheme of

Free propagation and scattering h = n

FIQ. 2. The language of atomic orbitals and atomic scattering cannot be used where the potentials overlap.


FIQ. 3. The muffin-tin approximation for the lattice potential.

approximation can be convergent, and we must seek an alternative frame- work of basis states for the representation of Bloch functions. This argument indicates that the construction of a mufin-tin potential

(Fig. 3) is more than just a convenient approximation. We do this by drawing about each site in the lattice an atomic sphere (of radius R, say) which does not intersect its neighbors. Within each such sphere we define a spherically symmetric function vyT(r) which follows as closely as possible the total potential 0 (r) inside that cell. This leaves a region of complicated shape between the spheres: in the simplest approximation, the interstitial potential is taken to be a constant, which we shall call the mufin-tin zero, i.e.

W I s ( r ) = E M T Z . (2.5)

It is clear that this approximation greatly simplifies our problem, and by-passes all the mathematical difficulties mentioned above. The zero of the energy scale is moved to EMTz, and u M T ( T ) replaces v.(r) in Eqs. (2.3)- (2.6). Since W(r) is now defined as a superposition of nonoverlapping potentials, there is no objection to our describing the wave function within each such sphere entirely in terms of eigenfunctions of local angular

(a) (b)

FIG. 4. MufEn-tin potential (b) for clowpacked lattice (a).

10 .J. M. ZIMAN

n

(a) (b)

FIG. 5. (a) Lattice potential for diamond structure with section along line AA. (b) Muffin-tin approximation showing large jump at sphere boundaries to give correct average interstitial potential.

momentum,) above and below the energy at which such states become bound.

In most materials, there is no difficulty in finding atomic spheres of reasonable size (i.e. nearly inscribed in the unit cells of the lattice) Fithin which it is justifiable to assume that vYT is spherically symmetric. The dangerous approximation is (2.5) -the assumption that the potential is flat throughout the interstitial region. In a close-packed metal (Fig. 4) , the error is thought to be negligible, although careful bookkeeping is necessary in defining &YTz by an average value of UIs(r),ll because the whole position of the valence band relative to the core states can depend on this parameter.1 In ionic crystals the required correction begins to be significant,I2 whilst in a typieal diamond lattice semiconductor the interstitial potential may vary over a range of energy comparable with the whole width of the valence band, from the lattice minimum along the bonds between neighboring atoms to a maximum at the center of the most capacious interstitial void of the crystal (Fig. 5). l1 T. L. Loucks, Augmented Plane Wave Method. Benjamin, New York, 1967:

l* P. D. DeCicco, Phys. Rev. 163, 931 (1967). D. A. Liberman, Phys. Rev. 168, 704-705 (1967).


(b)

FIG. 6. Separation of (a) the total lattice potential, 'U into (b) a muffin-tin potential VMT and an interstitial potential 'UIS.

In such a case, if we use some average value of 'UIs(r) for the energy in the interstitial region, we are causing uMT to jump at the boundary of the atomic sphere, and may tend to bind some states at an energy where they would otherwise be free to propagate along the ('valleys" of the chemical bonds. To improve the calculation, we must divide the total lattice potential into two separate functions of position-a genuine muffin-tin potential 'UMT(r) which is continuous on the surfaces of the spheres and flat between them, together with an interstitial potential 'UIs(r) which varies as required in the interstitial region but is defined to be flat across the potential wells of the atoms (Fig. 6). We may write

'U(r) = ' U d r ) + * U I S ( ~ ) (2.6)

by choosing E M T Z as the energy of an equipotential surface of 2) (r) approxi-

12 J. M. ZIMAN

I I

FIO. 7. (a) Superposing atomic potentials, u., we get (b) the lattice potential U. This may be decomposed (c) into a muffin-tin potential UMT and an interstitial potential VIE, but may also be represented (d) as the superposition of muffin-tin well potentials, UMT and external atomic potentials ~ X T .

mating to a set of nonintersecting atomic spheres, and measuring both .UHT(r) and UIs(r) from this level as zero.

As we shall see (Sections 11-15) the APW and KICR methods use angular momentum representations (2.3) for the wave functions inside the muffin-tin wells, and plane wave representations in the interstitial region. Since UIS is well defined and periodic throughout the crystal, its Fourier components are only required at reciprocal lattice vectors. In the usual case where the total potential (2.1) is nearly exactly a linear superposition of spherically symmetric functions, these components depend only on the form factor from each atomic contribution. In practice, therefore, the scheme (2.6) is approximately equivalent (Fig. 7) to dividing each atomic potential v,(T) , into an inner part which supplies the potential

v , (T ) - v.(R) for T I R

I0 for T > R (2.7) ~ Y T ( T ) =

for each muffin-tin well, and an outer part, which contributes

for 1 I R i v.(r) - v.(R) for T > R (2.8) m X T ( T ) = to the potential in the interstitial region. But by choosing va(R) for the muffin-tin zero, we have ignored any effects due to the potential from one cell overlapping into the atomic spheres of its neighbors. Here, as else- where, if v.(T) ranges too far outside an atomic sphere, great care must be exercised in defining an absolute scale of energy for the muflk-tin zero, core states, etc., in the crystal potential.


3. RELATIVISTIC HAMILTONIANS

In heavy elements, relativistic effects in the atomic energy levels-in particular, spin-orbit interactions-are known to be large enough to be significant in the energy band structure, especially in the h e details of the F e d surface. The theory of such effects is now well established, although there has been a tendency to expound the algebra within the context of one or other of the special methods of band-structure calculation. The technique is, in fact, quite general, and will be discussed at this point because the corrections arise from a change in the form of the initial one- electron Hamiltonian of the problem. In other words, we are called upon to solve a partial differential equation of somewhat greater complexity than the Schrodinger equation (1.1) , taking explicit account of the electron spin states I =I=$); in practice, however, this problem is always simplified, as in the nonrelativistic case (2.2) , by the spherical symmetry of the atomic potential or muEn-tin potential in which basis functions are being constructed. There are two competing techniques, which are almost exactly equivalent.

The formally exact procedurela is to set up the Dirac equation, whose Hamiltonian,

contains the Dirac matrices a and 8, and acts upon a bi-spinor wave function with four components, two for each spin. But of these the positive energy components are much the larger, and dominate the calculation^.^^ In the interstitial regions of the crystal, where the forces acting on an electron are very weak, the difference between this component of the wave function and the ordinary nonrelativistic solution corresponds to the relativistic correction to the kinetic energy of the particle, which is negligible at such low velocities relative to c, the velocity of light.

Within the atomic core, however, the spatial variation of the potential becomes large, and must be taken into account. But the transformation properties of the solutions of the Dirac equation under the rotation group are thoroughly understood. In a spherically symmetrical potential the states are classified by the quantum numbers X and p , related to the ordinary orbital quantum numbers 1 and m of (2.3) by the rules

1 = x; j = l - + ( A > 0 ) 1 = - A - 1; j = l + + ( A < O )

XD = m.p + m&3 + V) (3.1)

(3.2)

I* P. Soven, Phys. Rev. 187, A1706-1717, A1717-1725 (6965); T. L. Loucks, ibid. 189, A1333 (1965); l&, 506 (1966); S. Takada, Progr. Theoret. Phys. S8, 224 (1966); Y. Onodera and M. Okazaki, J . Phys. Soc. Japan 21, 1273-1281 (1966). J. M. Ziman, Elements of Advanced Quantum Theory, p. 191. Csmbridge Univ. Press, London and New York, 1969.

L

14 J. M. ZIMAN

with p going from j to - j for a given value of A. The positive energy components of the wave function may then be written

XA,, = ~ A ( T ) C C(Z3j; cc - ma, m.) Yz#-%(r) I m.) (3.3) %-zkllf

where Clebsch-Gordan coefficients define the mixture of spin states to be used with the corresponding spherical harmonics in the (orbital variables. The orbital quantum number, X runs through both positive and negative integers (excluding zero), to allow for the doubling of the total number of states by the spin.

( r ) . Although the Dirac equation seems only to contain first derivatives of the various components of the wave function, these are coupled together. Instead of the radial Schrodinger equation (2.6)) we have to solve the pair of differential equations

The real labor comes in computing the radial function

(3.4)

where F-A is the radial part of the negative energy component of the complete bi-spinor wave function to which (3.3) belongs. Equations (3.3) and (3.4) provide the basis functions for any expansion of the relativistic Bloch functions within the atomic sphere.

But the velocity of light, in these (atomic Units, is twice the inverse of the fine structure constant, so that the factor l/c (2 X 137)+ is very small. It is easy to eliminate the radial function FA from the two equations, thereby constructing a second order differential equation for RA alone. To order 1/8, this may be written

+ ( K 2 - U a ) 2 6 i ~ + ___ ( l + A) ..) = 0, (3.5) - r dr whose solution is quite sufficiently accurate for our present purposes.

It will be seen at once that (3.5) differs from the nonrelativistic radial equation (2.4) by terms of order 1/8. Of these, the first two are independent of X, and hence correspond to corrections to the detailed behavior of the


wave function within the atomic core-the Darwin and mass-velocity terms, which are insensitive to spin and may be treated as small modific* tions to the atomic potential. But the final term depends upon the magnitude and sign of X, and hence varies with the magnitude and orientation of the orbital moment of the electron relative to its spin. This term, therefore, corresponds to the spin-orbit interaction.

As pointed out by Treusch,15 this formula can be obtained much more directly by starting from the Pauli equatiqn16J7 which is derived from the Dirac equation by a canonical transformation in which the "negative energy" components are reduced to zero. To order 1/$, this has the effect of merely adding to the Hamiltonian of the nonrelativistic Schrodinger equation ( 1.1) the operators

XI, + XMV + XSO = (-1/8) V4 + (1/23) V'U - ( i / c " ) ~ * (VV,+, V) (3.6)

where u is the usual electron spin operator. The first two terms, being spin- independent, may be ignored as before. But when we look for solutions to this equation in a spherically symmetrical potential we arrive precisely at the basis functions (3.3), with the radial function (RA(r) satisfying exactly the equation (3.5), and we verify that XSo gives rise to the final term in (1 + A). To this order of accuracy, therefore, there is nothing to be gained, except mathematical snob value, by introducing the Dirac formalism. We discover in practice, moreover, that it is difficult to calculate the magnitude of the spin-orbit correction from first principles, and that empirical values deduced from atomic spectroscopic levels often give better results.16J8

II. Mathematical Insights and Physical Concepts

4. THE LCAO REPRESENTATION Our physical intuition that a crystal is merely an assembly of atoms,

brought close together and allowed to interact, is expressed mathematically by trying to represent the Bloch functions as linear combinations of atomic orbitals (LCAO's) . It is convenient to replace the pair of angular momentum quantum numbers ( I , m) in (2.2)-(2.4) by a single symbol L, together

16 J. Treusch, Phys. Status Solidi 19, 603 (1967). 16 L. Liu, Phys. Rev. 126, 1317-1328 (1962). 11 P. C. Chow and L. Liu, Phys. Rev. 140, A1817-1826 (1965); J. B. Conklin, L. E.

I* F. Herman, C. D. Kuglin, K. F. Cuff, and R. L. Kortum, Phys. Rev. Lett. 11, 541-545 Johnson, and G. W. Pratt, ibid. 157, A1282-1294 (1965).

(1963).

16 J. M. ZIMAN

with a principal quantum number n, and to choose these in such a way that when K~ = &.L the radial function &(r) + 0 as r + 00. We thus assume that our basis functions (2.3) are restricted to bound states of the free atom potential v.(r) , symbolized by

xlm(r) = bnL(r). (4.1)

The most general Bloch function of wave-vector k that may be constructed out of such functions is of the form

(r) = Cexp(ik-2) C a , , ~ b , ~ ( r - 2). (4.2) q k L C A O 1 nL

The problem is to find the best set of coefficients an^. The obvious procedure is to use these as variational parameters, with (4.2) as a trial function, in minimizing the expectation value of the Hamiltonian of the Schrodinger equation (1.1). By standard elementary arguments, we find that we must solve the set of linear equations

C { ( E ~ L - G)DnL,neLr(k) + V n L , n ~ L ~ ( k ) ) ~ n ~ ~ ~ = 0 (4.3) nrLt

for all values of n and L included in the sum (4.2).

symmetry they are easily expressed as lattice Fourier transforms, i.e. The coefficients in (4.3) are a bit elaborate, but because of translational

vnL,,#L,(k) = C exp(ik.2) Vn~.n#~p(2)

Dn~,nt~#(k) = C exp(zk.2)DnL,n~L'(2), I

(4.4) 1

whose direct lattice components are independent of & and k:

V n ~ , n ~ ~ r ( f ) = C 1 bnt(r)2ra(r + It)bnrL'(r + 2) df

D , , L , ~ ~ L * ( ~ ) = 1 bnL(r)bntLr(r + 2) d31..

(4.5) It*

and

(4.6)

These integrals, involving the overlap of functions centered on two or three neighbohg sites, are obviously rather nasty, and can only be evaluated with great labor. Nevertheless, the above formulation seems quite explicit, and was for long regarded as capable of giving a solution to the band- structure problem to any required degree of accuracy.

This is the algebraic justscation for one of the standard elementary principles of solid state physicethat as the atoms come together, each of their bound states becomes broadened into a band of Bloch functions of


&

I

FIG. 8. Bands formed by combining atomic orbitals as atoms are brought together.

appropriate symmetry (Fig. 8). The broadening occurs, quite sharply, because bound-state functions decay exponentially outside the atomic sphere, so the overlap integrals are very small until the atoms come close together. Then, as the bands arising from distinct levels begin to overlap one another in energy, they hybridize into the linear combinations that would be obtained by solving the above equations for a, ,~. We thus refer to the s-p band, the 3d band, etc. identifying the bands by the atomic levels from which they are ultimately derived.

As has already bem remarked, however, this confidence in the LCAO or tight-binding method is misplaced. The overlap of wave functions eventually leads, to an overlap of potentials, destroying the bound states (except for resonances, about which more in Section 7) and liberating the electrons into nearly free bands in which the quantum numbers of the atomic levels are almost irrelevant (Fig. 9). The principle advertised in Fig. 8 is only of limited applicability, and is without significance for bands above the mutEn-tin zero.

Even when there is no overlap of neighboring potentials, the representation (4.2) is functionally incomplete: the set of bound states (4.1) of the atomic potential v.(r) does not contain the continuum of scattered free-electron eigenfunctions of the Schrodinger equation at positive energies.l9 It is impossible, therefore, to describe states in a metal or semiconductor in and above the valence and conduction bands, for we know from other calculations that these resemble free-electron waves in the

If o,(r) were of infinite range-e.g. a Coulomb potential-it might perhaps supply enough basis functions out of its infinite set of bnund states. But then the super- posed potentials would push the mdn-tin zero down to an infinite negative energy, where the errors produced by overlaps (Fig. 3) are incorrigible.

18 J. M. ZIMAN

s-p bonds

( b)

FIG 9. (a) Conventional LCAO description of formation of metallic conduction band. (b) Nearly-freeelectron description.

interstitial space. We note, e.g., that the coefficients (4.4)-(4.6) contain no algebraic mechanism to ensure that the width of any band above & ~ T z should approximate to that of a nearly free electron band in the appropriate extended zone of k space. To put it crudely: the LCAO method absolutely fails the empty lattice test, for it reduces to a nullity when applied to the hypothetical model crystal with 'u (r) E 0.

On the other hand, this is a legitimate description of the bound bands, below &MTZ. As is well known, the deeper bound state functions bnL(r) decay rapidly to zero outside the atomic core, so that the overlap integrals (4.5) and (4.6) become very small as we go down in energy, and the solutions of (4.3) are very narrow bands centered about each atomic level &nL. The computation of such bands therefore provides a natural field of exercise for the method. Unfortunatelyz0 it may be quite misleading to use a conventional determinant of Bloch functions to describe the many- electron states in such circumstances. There is evidence, for example20* that the effects of electron correlation may best be accounted for by treating each electron as if it occupied a nearly independent localized orbital on a p,articular atom. The construction of a band of one-electron tight-bound Bloch functions in such a case may then be without physical significance.

Having abandoned the notion of calculating band structures from first principles by the tight-binding method, we can still learn something from the general form of Eqs. (4.3). These are made rather clumsy by the non-

ao N. F. Mott, Phil. Mag. 6, 287 (1961). *Or K. Thornber, Sci. Progr. Ozford 67, 149-168 (1969).


orthogonality of the basis functions (4.2)) as a consequence of the non- orthogonality, in integrals such as (4.6), of atomic orbitals centered on different sites. But this is not an essential complication, for the simplifying assump tion

may be justified in principle by supposing that the atomic orbitals (4.1) have already been recombined by a suitable unitary transformation so as to make (4.6) vanish for I # 0. This is .a standard property of Wannier function^,^ but can be achieved in many other ways.

Looking at the coefficients (4.5)) we can apply group theory to obtain powerful selection rules governing the choice of L, L', and 1 in various types of l a t t i ~ e . ~ It is also reasonable to assume that these coefficients become negligibly small when they refer to lattice sites a t any great distance apart. Thus, there need only be a few distinct coefficients of importance: we have arrived at one of the standard interpolation schemes or parametric representations of band structure theory.21 The energy &(k) is supposed to be a root of the secular equation

& . n s ~ p (k) = & L . ~ ~ L * (4.7)

det l l [ E n ~ - E(k) 1 & L , ~ ~ L ' + V,L,,tL.(k) II = 0 (4.8) where V,L.,,'L~ (k) is given by a Fourier series such as (4.4). The numbers V,L,,!L~(Z) are then treated as parameters to be adjusted empirically to bring &(k) into agreement with experiment. This approach has been used, e.g., by Dresselhaus and DresselhausZ2 to give exact analytical formulas for the valence and conduction bands in silicon and germanium, and is also the basis of many schemes for d bands in transition metals (see Sections 7 and 15).

It must be emphasized, however, that the actual values of the parameters arrived at in this way cannot be interpreted physically as overlap integrals of atomic orbitals. To a large extent, this type of representation is imposed upon the band structure by the symmetry of the crystal.

To illustrate this, suppose, as in the case of many monovalent metals, that there is only a single nearly free electron conduction band to be fitted to the shape of the Fermi surface. The function &(k) must necessarily be periodic in the reciprocal lattice; it must have a Fourier representation in the direct lattice :

&(k) = &O + C exp(zk.I)&(I). (4.9) I*'

But the Brillouin zone has the point group symmetries of the crystal, so that many of the coefficients &(I) must be equal to one another, or zero.

J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498-1524 (1954). I' G. Dresselhaus and M. S . Dresselhaus, Phys. Rev. 160, 649-679 (1967).

20 J. M. ZIMAN

It is obvious, e.g., that & ( I ) must have the same value for all twelve nearest neighbors in the fcc lattice, so that the leading terms of (4.9) would take the form

E (k) = Eo + 4E (100) X { cos +ak, cos +ak, + cos iak, cos +ak, + cos auk, cos Baku). (4.10)

If now we give to & (100) the value - P/4ma2, this function will behave like a simple free-electron energy-momentum relation over a wide range of energy. Indeed, by the addition of further terms in &(211), etc., the dis- torted and multiply connected Fermi surfaces of the noble metals can be fitted very exactly in this way.23 But (4.10) is just the formula at which we should have arrived from (4.4) and (4.8)) assuming that a single s-like atomic orbital dominated the band structure. It is obviously incorrect to interpret this special value of &( 100) as an overlap integral Vlo.lo( 100) between bound s states on the neighboring atoms. The best we can do is a circular argument in which the functions bla(r) are discovered to be the Wannier functions of the free-electron Bloch functions we are trying to construct.

More generally, &(k) is a many-valued function of k in the reduced zone, as well as being periodic in the reciprocal lattice, and cannot, therefore, be represented by a single Fourier series such as (4.9). But roots of the secular equation (4.8) can be provided for any number of bands by introducing enough different values of angular momentum L and principal quantum number n. Again group theory will be our guide in choosing these quantum numbers and in reducing the secular determinant to manageable dimensions.

Such representations can always be justified formally by an appeal to the theory of Wannier functions; but empirical success in any particular case does not alone warrant a claim that the Bloch functions are of essentially tight-binding character. Thus, despite the excellent fit to the observed band structures of Si and Ge obtained by Dresselhaus and Dresselhaus= using only s- and p-type parameters, we know that, starting from actual atomic potential, the LCAO method cannot match the accuracy of OPW or KKR calculations on these materials. It would not be quantitatively cofrect, therefore, to say that the Bloch states of diamond lattice semiconductors are constructed simply out of the bonding and antibonding combinations of the hybridized spa orbitals of the free atoms.

This is really very unfortunate, for it demonstrates a grave weakness in

*a F. Garcia-Moliner, Proc. Phys. Soe. 72, 996 (1958); D. J. Roaf, Phil. Trans. Roy. SOC. London Ser. A 266, 135 (1962).


the whole quantum theory of the condensed state. In the realms allotted by custom to theoretical chemistry, the LCAO representation reigns supreme, for it has provided excellent qualitative understanding of phenomena such as bonding, resonance, electron transfer, etc. This point of view has proved its worth not only in organic chemistry but in such fields M the invention of new compound semiconductors, where the principle of saturated covalent bonding almost determines the chemistry of the system. To give an extreme example: the appearance of semiconducting properties and optical band gaps in many amorphous materials is easily explained chemically on this basis. Yet according to conventional solid-state theory, such glassy materials lack any of the long-range order required to give rise to band gaps by coherent electron diffraction.

In other words, there is an intellectual gulf between the qualitative principles of chemical theory and the mathematical techniques that have proved their accuracy in the calculation of electron states in regular crystals. The LCAO representation, by providing algebraic continuity between the Bloch functions of the condensed material and the atomic or molecular orbitals of the separate constituents, has immense explanatory power, yet seems to fail the ultimate test of mathematical convergence and precision. A determined effort is urgently needed to bridge this gulf (see Section 15).

Finally, it may be remarked that the tighbbinding method provides a valuable model or [ % ~ y ~ ~ ~ ~ in the investigation of hypothetical properties of ordered and disordered systems. It is easy enough to show, e.g., that a single band such as (4.9) is the solution of an infinite set of equations in the direct lattice, such as

{ & l - &)a1 + c V( t , 2 ) U l f = 0. (4.11) In the electron case, at is the amplitude of the wave function on the Zth site, but these equations have obvious analogies for phonons, magnons, excitons, etc. In the case of a regular lattice, the substitution

11

ar = aexp(zk.2) (4.12)

leads automatically to (4.3) ; but the effects of disorder can be simulated by randomly varying the overlap integral V(2, 2) between neighbors, or by changing the bound-state energy &I from site to site in the lattice, or by setting up lattices with topological di~order.2~ In such cases, physical verisimilitude is less important than a full understanding of the consequences of abandoning the shelter of Blochs theorem.

f( J. M. Zman, Nature 296, 1187-1192 (1965). J. M. Zman, J . Phys. C. 2, 1230-1247 (1969).

22 J. M. ZIMAN

5. THE CELLULAR METHOD

The weakness of the LCAO method may be ascribed to the use of basis functions satisfying atomic boundary conditions which are of no significance in a crystal. Surely, then, it is better to use functions that satisfy the natural crystal boundary conditions in the first place?

Strictly speaking, the only boundary condition on a Bloch function is that it should not become infinite at a large distance. But the Bloch theorem,

#k(r + 2) = eXP(zk.2)#k(r)t (5.1) together with the condition of continuity, can be read as a condition on #k at the face of a single cell of the lattice: the value of the function at any point r = Rb of the boundary is related by this phase factor to its value at another point R b + 2, typically on the opposite polyhedral face-to which it might be carried by an appropriate lattice translation. We also ensure that the outward gradient of #k ( R b ) normal to the face of the cell is properly matched to ensure continuity of slope, i.e.

vn#k(% + 2) = - eXP(zk.2) v n # k ( R b ) . (5.2) The solution of the Schrodinger equation (1.1) within the cell, and satisfying these conditions for all points R b , must be a Bloch function of energy & and wave vector k. This is the theorem (whose proof need not be attempted) at the heart of the cellular method of band-structure calculation.

This argument is attractive in principle and played an important part in the early history of band-structure calculations: yet our experience in the last decade suggests that it has outlived its practical usefulness. Without wishing to take too strong a stand on a controversial topic, I would argue that the algebra of this method, although capable of yielding accurate

FIG. 10. Spherical potential inside fcc unit cell.


FIG. 11. Spherically symmetric potentials in a diamond lattice, showing ridges of potential along cell hoiindrtries (cf. Fig. 6a).

numerical results,26 is not well adapted to the physical features of the system, and hence lacks the power of a general canonical technique.

In the first place, we are not free in practice to exploit the hypothetical possibility of solving the Schrodinger equation in just any potential with the overall symmetry of the unit cell. We are forced to assume that U(r) is spherically symmetric right out to the boundaries of the cell (cf. Fig. 10) so that we may URC the familiar angular momentum representation (2.3). Thus, we write

vW) = C C L ~ I ( ~ , &) YL(r) , (5.3) L

where &(r, &) is n solution of the radial equation (2.4) at energy & in the potential U(r). Then we adjust the coefficients CL, and vary the energy & until (5.1) and (5.2) are satisfied as well as possible for a sufficient number of points Rb on the cell boundary. This restriction to spherically symmetrical potentials is serious, for we must either assume (unphysically) that the potential has sharp ridges over parts of the cell boundary (Fig. ll), or else that we are dealing with a muffin-tin potential (Section 2) , for which other standard methods (APW or IiIiR) are more naturally adapted.

The technical efficiency of the cellular method depends upon the fullest use of group theory, by which functions of various angular momentum are gathered into linear combinations-lattice harmonics-that already transform as irreducible representations of the symmetry group of the wave vector k.2i This means that the method is mainly useful for the calculation of energy cigenvalues in symmetry directions of the Brillouin zone, and

* S. L. Altmnnn and C. J. Bradley, Proc. Phys. Soe. 86, 915-931 (1965); S. L. Altmann, S. L. Altmann, B. Davies, and A. R. Harford, J . Phys. C. 1, 1633-1636 (1968).

ibid. A244, 141-152, 153-165 (1968).

24 J. M. ZIMAN

would not be applicable a t all to problems where the ions have moved slightly from their strict lattice sites-as in calculating the electron-phonon interaction-or where the structure is topologically disordered as in a liquid metal. Again, the randomness of the set of points on the boundary where fitting is to take place, although essential for computational accuracy, introduces an element of arbitrariness into the algebra, blocking any abstract analytical transformation of the theory into other, physically visualizable formulae. That is why we have discovered no means of drawing from the cellular method the new intuitive concepts of band theory-pseudopotentials, resonance bands, etc.

The fundamental conceptual weakness of the method is that the polyhedral Wigner-Seitz cell of the direct lattice is an artificial mathematical construct whose boundaries do not delineate any significant physical characteristic of the crystal potential.% They are drawn through the smoothest and flattest regions of the potential, where the Schrodinger equation is easily solved anyway. It is perverse to cut across such a feature- less region with an imaginary plane, solve the equation on both sides of the cut, and then have to go to a lot of trouble to rejoin the pieces correctly. A measure of this extra labor is the computation required to verify the empty lattice test: going up to spherical harmonics of order 12, and ensuring continuity at 256 boundary points, we arrive extremely close to the trivial result & = k2. Mathematical techniques in theoretical physics are not to be judged simply on the accuracy of the answers they give when correctly programmed into a very powerful computer: it is important that we should understand clearly what is going on, and have the means for quick approximate calculations in simplified cases.

The cellular method does, however, yield one very important simple formula. Let us look for solutions at the center of the zone, at k = 0; the Bloch conditions (5.1) tells us that the wave function now has the periodicity of the lattice. We further assume that the unit cell has become a sphere of radius r,--this does little more violence to reality than the previous assumption that the potential was spherically symmetrical within the whole polyhedral cell. The condition of continuity of slope (5.2) is then automatically satisfied if our solution has zero gradient on the boundary of the sphere. When the radial part of the s-wave solution of the radial Schrodinger,equation (2.4) has this property, i.e.

P

Cmo(r, c)/arlr=% = 0, (5.4)

m Paradoxically, the reciprocal construction in an imagined space-the Brillouin zone-seems much more real. This is because the lines and planes of atoms ip crystals are very definite geometrical features, giving rise to strong diffraction effecta in these regions of reciprocal space.


then this single term in (2.3) is sufficient. By adjusting the parameter E until (5.4) is satisfied, we determine the energy Eo of a Bloch state of the crystal.

In the literature on the cellular method, this is merely the starting point for further calculations-e.g. at other points in the zone-which we would prefer nowadays to express in APW or OPW language. But (5.4) is valuable in its own right because it tells us directly one of the key parameters in any band structure-the energy of the bottom of the free band. In many respects, &,, plays the role of the zero of kinetic energy in the crystal; Bloch states above Eo are best described as free-electron waves modified by diffraction from the lattice; below Go they are essentially atomic wave functions overlapping to form narrow tightrbound bands. The whole pattern of the band structure as observable in many properties such as cohesion, X-ray emission, optical spectra, etc., depends upon a correct determination of G ~ . This, in its turn, can be subtly influenced by the assumptions made in setting up the one-electron potential 0 (r) . But one can see at once that most of these assumptions, together with (5.4) itself, do not depend on any details of the crystal structure, but are functions only of the atomic volume. To within the sphericizing approximation, therefore, these are Lnonstructural propertiesO; this is the simple explana- tion for such familiar phenomena as the smallness of the change of many electronic properties of metals when they melt.

The Wigner-Seitz condition (5.4) is very explicit, and well suited for direct computation. But for close-packed structures, where the muf6n-tin approximation (2.7) is nearly correct, one can rewrite this in another very instructive form, as follows.10*28

The radius R of the muffin-tin well must be less than r.. In the region r. < r < R we are effectively in free space, with EMTZ as the zero of energy. Write

the solutions of the radial Schrodmger equation for 1 = 0 in this region are the spherical Bessel functions of zero order, jo( KT) and no ( KT) . A combination of these has to be found to join on to the computed solution within U M T ( T ) . But this is a well-known problem in scattering theory, whose solution is given in any standard textbook of quantum mechanics:

%(r, go) = j O ( K T ) - trqn ~ I O ( K ) ~ K T ) (5.6)

&o - &YTZ = K2; (5.5)

where ~ I ~ ( K ) is the phase shift for s-wave scattering from the potential U M T ( T ) . The condition (5.4) now reads

tan I f o ( K ) = jO(Krs)/%(Krs)- (5.7) * P. Lloyd, &roc. ~ O C . 90, 207-216 (1967).

26 J. M. ZIMAN

It is obvious, however, that Eo cannot be very distant from the muffin-tin zero. For small values of the (momentum K, it is usual to approximate to the phase shift by introducing a scattering length, a, such that

tanqO(K) M - K U (5.8)

(expressed, of course, in atomic units). This formula is valid if I a I < r.-a reasonable property of even a very

deep potential well; indeed the scattering length of a potential is usually smaller than its geometrical radius, except when there is a bound state or resonance near the energy zero. For an attractive potential, where a would be negative, Eo < E M T z . This makes K pure imaginary; but the algebra is still correct by analytic continuation. The bottom of the band lies a little below the muffin-tin zero, because the electrons can still tunnel coherently through the low barriers between the atomic spheres (Fig. 12). Generally speaking, however, (5.8) is a small quantity, so that the position of EO on an absolute energy scale is mainly determined by E Y T z , which depends, in its turn, on conditions in the interstitial regions between the atomic spheres, and upon the approximations made in overlapping atomic potentials, screening electrons from one another, etc.

It is even possible for a to be positive, as it might be in a rare gas solid; the bottom of the conduction band would then lie somewhat above the muffin-tin zero, showing that the extra electron is gently repelled by the atoms of the solid.l0

The spherical approximation for the cell would obviously fail in a lattice of low coordination number, such as a diamond-type semiconductor. The interstitial potential (2.6) would then not be negligible, but could appear in (5.9), say, only as a rough average correction to the muffin-tin zero. We may then ask whether the bottom of the band lies below the lowest (pass dong the bond valleys between neighboring atoms, or whether it takes on some higher value because of the mountains in the interstitial regions (see Fig. Ga) . To answer this question, one might determine Eo more exactly by solving the general equations of the cellular method (5.1)) (5.2)) at k = 0 in the proper polyhedral cell. Perhaps this method has its uses after all!

FIG. 12. Bottom of free band lies just below muffin-tin zero.

T H E CALCULATION OF BLOCH FUNCTIONS 27

6. THE NFE REPRESENTATION

We now adopt an entirely different point of view toward the Schrodinger equation (1.1). The obvious way to bolve this is to exploit its lattice periodicity by a Fourier transformation. The reciprocal lattice representation of a Bloch function,

#k = c ak-8 exp[i(k - g) .r], (6.1) g

automatically satisfies the Bloch theorem (5.1). Substituting in ( l . l ) , we get a set of linear equations for the coefficients ak-g, i.e.

{ I k - g 1 - E(k))ak-g -k c ugt-gak-gr = 0. (6.2) g

The potential (1.2) now appears only through its Fourier coefficients, which depend only on the differences of the reciprocal lattice vectors g and g. In the typical case of a superposition of potentials, m in (2.1), this becomes

= U(r) exp[i(g - g) or] d f 1

In a Bravais lattice, S(g - g) is unity for all reciprocal lattice vectors g and g; we are left simply with the corresponding Fourier transform of a single atomic, cellular, or muffin-tin potential, normalized to the volume of the unit cell. The introduction of the actual structure factor S(g - g) in more complicated cases is obviously trivial in principle, if tiresome in practice.

The infinite set of linear equations (6.2) can only be solved by successive approximation. We might assume, e.g., that the coefficient for g = 0 in (6.1) is much larger than the others, and that the Fourier components u.(g - g) are all quite small. After some labor, we might arrive at an expression in the form of an infinite series, e.g.

&(k) - k2 = ~ ~ ( 0 ) + C l2 + c ( ) + (6.4) g+ok2 - I k + g I2 g.g

This formula is conventionally interpreted as a perturbation expansion; the Bloch state is supposed to look like a free-electron wave, of energy k2, perturbed by the crystal potential.

28 J. M. ZIMAN

Unfortunately, this approximation procedure is not convergent. We can easily see this by considering the analogous problem of calculating the effect due to a single potential well, va( r ) put in the path of our beam of free electrons. We might write down the corresponding perturbation expansion for a scattering amplitude or t matrix, i.e.

where sums over reciprocal lattice vectors are replaced by integrals over the wave vectors of all intermediate states. The successive terms in (6.4) thus correspond to successive orders of Born approximation in an estimate of t.(O) . But remember that k2 is of the order of the Fermi energy-a few electron volts-whereas va(r) has bound states down to the inner X-ray levels. In these circumstances, a Born series, (6.4) or (6.5)) cannot be convergent : the higher Fourier components of the potential, and the higher terms in the series, become dominant, and the assumptions made in arriving at this approximation are not valid.

This method is evidently useless as a computational technique. What are we to do? Before embarking upon the theory of pseudopotentds, in which these holes are patched up, let us look at this heuristically as a problem capable of a physically intuitive solution.

The idea is to treat the t matrix of a single center ~ t s a fair approximation to the right-hand side of (6.4). In writing down (6.5)) t a ( K ) has been given the same dimensions as v a ( K ) by multiplying the usual scattering amplitude f(0) by 4r and dividing by the volume of a unit cell. The very first term in (6.4)) the average potential v.(O) , can also be interpreted as the first Born approximation to the scattering amplitude. By replacing this by the true forward scattering amplitude, and writing

(6.6) &(k) - k2 M t. (O) , we effectively sum the whole series (6.4).

The point is, of course, that we can calculate t , ( K ) by much better methods than the perturbation series (6.5). For a spherically symmetrical potential, in particular, we need only evaluate the partial wave phase shifts 711 toobtain the expression

t .(e) = 4rivj(e)

= - (4TN/K) C (21 + 1) sin 711 exp(ir]Z)Pt(cos e) (6.7) for scattering through the angle e on the energy surface & = K ~ . Thus, from a knowledge of these phase shifts for scattering by a single atomic, ionic, or muffin-tin potential well, we can estimate the change in the energy of a

1


free electron propagating in a lattice of such atoms. It is gratifying to note that (6.6) and (6.7) agree with the Wignedeitz formulas (5.8) and (5.9) in the special case where the s-wave phase shift is small, and the others zero.

What we are really doing is trying to represent the Bloch functions inside the muffin-tin potential by an actual combination of spherical harmonics and radial functions, as in (2.3) , that satisfies the true radial Schrodinger equation (2.4) at the energy &. This is, of course, a much more realiitic procedure than trying to expand this function in plane waves- an expansion which simply does not converge. As we shall see in due course, this is the secret of the success of the APW and KKR methods.

But even for very weak potentials, the perturbation series (6.4) is only valid near the bottom of a band. What happens when the wave vector k is large enough for the Fermi surface to intersect a zone boundary? We know by elementary arguments that we must use degenerate perturbation theory. All the coeflicients (Yk-G in (6.1) that refer to waves capable of being mixed by such zone boundaries are collected together, and we solve explicitly the equations

for this finite set. In conventional perturbation theory, we then treat each eigenvector of this matrix as an unperturbed state, and calculate the effects of all the other terms in the complete set of equations (6.2) by successive approximations.

The same result is achieved by folding the infinite secular determinant of the original equations (6.2) on to a matrix of the form (6.8). This con- sists of the following elementary transformation.20 Suppose that our secular equation may be written

with the reciprocal lattice vectors partitioned into the set G and the set h # G. The matrix identity

29 N. W. Ashcroft, Phil. Mug. 8, 2055 (1963); V. Heine,,in The Physics of Metals (J. M. Ziman, ed.), pp. 1-61. Cambridge Univ. Press, London and New York, 1969.

30 J. M. ZIMAN

shows that (6.9) can be reduced to the vanishing of the determinant of I C - B*A-B I which has only the dimensions G, GI. This is exact: but provided A-I is not singular we may evaluate the coefficients of B*A-lB by a perturbation expansion, so that our equation takes the form

{ I k - G l2 - &}ak-~ + rGG*(Ytk-GJ = 0, (6.11) GI

in which

(6.12)

This is an important general procedure, which can hasten a numerical calculation and also provide formal justification for a simple parametrization scheme for a band structure.

Here again, in the simple free-electron case, such a Born series will usually diverge. But by analogy with (6.5) we are induced to try the approximation

r G G S(G - G)t,(G - G) (6.13)

where, as in (6.3), we might include a structure factor for non-Bravais lattices, but where the Fourier component of a single atomic potential is replaced by an element of the corresponding t matrix. We assert that the diffraction of a conduction electron by the Bragg planes of the crystal is determined by the amplitude of the scattering actually capable of being produced by each atom.

This argument, although not exact as it stands, is at the heart of the familiar nearly free electron (NFE) approximation, whose striking success in the interpretation of Fermi surface experiments is well known. A strong atomic potential, with many bound states, may have very large Fourier components at very short wavelengths, yet these will not give rise to large band gaps if the phase shifts in (6.7) are small. The main features of the Fermi surface (if not of the whole band structure) can be discovered by solving a set of linear equations like (6.11) for the small number of zone boundaries actually intersected by this surface. It is assumed that the geometrical and topological properties of such surfaces are familiar to the reader.4

But the assumptions made in (6.6) and (6.10), although physically plausible, cannot be justified mathematically. The series (6.4) is not the same as (6.5), so that there is no reason for supposing them to have the same sum even if they did converge. In principle, the energy of a Bloch state may be expressed as a power series in the t matrix of a single center,


with terms corresponding to various orders of multiple scattering by different atoms in the lattice,a but this has not turned out to be a very profitable procedure except for the general theory of disordered systems.

The trouble is that we have no simple way of evaluating the t matrix off the energy shell.al The partial wave formula (6.7) is only valid for scattering between two states of the same energy: thus, when I G - G I exceeds, say, I 2k 1, the quantity tn(G - G) is not well defined.

It is true that we could attempt to solve one or other of the integral equations from which the series (6.5) is derived, and hence construct t,(K) as a general function. But this is a laborious procedure, whose final product is arbitrary and artificial. More profitably, we may argue as follows.

The effect of a given atom (or muffin-tin well) on a Bloch state of energy & seems not to depend upon the nature of the wave function inside the atomic sphere, but is determined solely by the scattering amplitude at that energy-e.g. by the phase shifts 711 produced in the various partial waves. We may therefore treat the atom as a black box, and replace u.(r) by any other weak potential, w,(r), which gives rise to exactly the same scattering amplitude at this energy. We can use its Fourier components, w,(K), in place of u,(K) in the general set of equations (6.2). If wa(r) has no bound states, however, the perturbation expansions for the folded coefficients rGG8 in (6.9) may be assumed to converge, so that the NFE formulas can be evaluated exactly. Indeed, an approximation analogous to (6.13),

IGGg S(G - G)w.(G - G), (6.14) may be good enough for most practical purposes.

This is the essence of the argument for the introduction of a model potential, or pseudopotential in place of the true atomic potential-a technique which has revolutionized our thinking about the electronic structure of metals and semiconductors in the past decade. The above justification, if somewhat vague, is obviously very flexible; it is still valid when the lattice is perturbed by phonons, or even thoroughly disordered as in a liquid. Combined with a theory of linear screening, it yields the method of neutral pseudoatoms-a zeroth order approximation for almost all the properties of simple metals.2 From a knowledge of w,(r) [or its Fourier transform, wa(K)] for a particular metal, we. can deduce its band structure,

0 J. L. Beeby and S. F. Edwards, Proc. Roy. SOC. A274, 39.5411 (1963); J. L. Beeby,

a1 J. M. Ziman, Elements of Advanced Quantum Theory, p. 127. Cambridge Univ. ibid. A279, 82-96 (1964); I(. H. Bennemann, Phys. Rev. la, A1045 (1964).

Press, London and New York, 1969.

32 J. M. ZIMAN

FIG. 13. Different pseudopotentiah, w1, wa with different wave functions $1, $9 inside the atom may represent the true potential v with true wave function $.

electrical resistivity, phonon spectrum alloying behavior, phase stability, etc., by quite elementary arguments.a2

It must be emphasized, however, that w,(r) is not fully determined by the above considerations, nor is it truly a simple function of r. There are many different weak potentials with the same phase shifts as v.(r) at a given energy (Fig. 13)-but there does not seem to be a useful algorithm for a local model potential, without bound states, capable of reproducing the scattering amplitude over a wide range of energies. As we shall see at length in Sections 9, 10, and 14, there are many different interpretations of the pseudopotential concept, and many alternative derivations of similar

Finally, it should be remarked that actual band structures are sensitive only to a few special features of the pseudopotential. This we learn by using the NFE equations (6.9) , as an empirical scheme, whose parameters row we adjust to bring the Fermi surface into agreement with experiment. The

form@as.

* M. L. Cohen and V. Heine, Solid State Phys. 24,38 (1970); V. Heine and D. Weah, ibid. 24, 249 (1970).


Fro. 14. Fourier transform of pseudopotential (atomic form factor) for Al. Fitting the Fermi surface fixes the points X. A1 data from V. Heine, in The Physics of Metals (J. M. Ziman, ed.), pp. 1-61. Cambridge Univ. Press, London and New York, 1969. Fermi surface data from N. W. Ashcroft, Phil. Mag. 8, 2055 (1963).

surprising discovery is the close fit that can be obtained by a simple NFE model-in Al, e.g., a 4 X 4 matrix with only two adjustable parameters llll and rzm is needed.2e Taking (6.11) at its face value, this means that just two Fourier components of the pseudopotential fix the band structure; w,(K) is determined accurately at only two points (Fig. 14). This is not an obvious consequence of the perturbation-type theories upon which such schemes are supposedly based, yet it cannot be just an accident!

7. d BANDS AND RESONANCES

We have been looking so far at a rather oveisimplified picture of electronic structure-narrow tight-bound bands below the muffin-tin zero, with nearly free electron bands above. In the transition metals. these c a t e gories become confused: a narrow band arising from the d levels of the atoms lies within a broad band of s electrons. Even without the additional complications of magnetic polarization, this situation requires careful analysis.

The simplest description is to treat the d electrons and s electrons as distinct entities, whose occasional interaction or interchange (((s-d scattering) gives rise to special phenomena such as electrical resistivity. The s electrons are then assumed to be quite free, with energy k2, while the five degenerate d states, b,,,(r), of each atom are combined into a typical tight-bound band of the type discussed in Section 4. There is indeed a considerable literature in which this description of the electronic structure of metals such aa Fe is taken for granted.

In practice, we cannot calculate the width and okher properties of this d band with any accuracy, so we fall back upon the empirical LCAO representation (4.8). The d states of the atom have the same energy, Ed,

34 J. M. ZIMAN

but different magnetic quantum numbers m = 1, . . ., 5, so that we have to find the eigenvalues of a 5 X 5 matrix

= Ed am,* + vmmf (k) . (7.1) The elements Vm,t(k) are functions of k, defined by (4.4) but depending on overlap integrals such as (4.5). In the spirit of the empirical scheme, suitably symmetrbed versions of these integrals are then treated as ad- j ustable parameters.

Can we now incorporate the s electrons in the same description? In principle, we could extend our set of atomic orbitals to include the s and p states of the free atom, and adding a further four rows and columns to the matrix (7.1) we should presumably have enough parameters to give a fair account of the band structure.

But, as we have already remarked, the LCAO representation is quite unsatisfactory for free bands, where we should naturally employ the NFE representation (6.9). This tells us to find eigenvalues of the matrix

XGG~ = I k - G 1 ~ G G + FGG, (7.2) whose rows and columns are labeled by the reciprocal lattice vectors occurring in a plane wave expansion (6.1)-or, more precisely, in an expansion in orthogonalized or augmented plane waves. Here again, the pseudopotential matrix elements r G G t are often treated as empirically adjustable parameters, whose values may turn out to be quite small. For the number of electrons conventionally assigned to the s band in the transition metals, (7.2) would describe a nearly spherical Fermi surface of the usual free-electron mass.

To gain the advantages of both representations, let us simply combine them. In other words, let us supposethat linear combination of d atomic orbitals and

*k(r) = Ce*l.l C *h(r -

where we assume, for the moment, that properties as exp (i(k - G) -r } . We should eigenvalues of a matrix

6

I nrpl

our true Bloch functions are a plane waves, i.e.

l ) + C ak-G4k-G (7.3) G

4k-G has essentially the same then be called upon to find the

of which (7.1) and (7.2) are diagonal submatrices.


G L

k

FIG. 15. Band structure of Cu in the (111) direction, showing splitting and hybridization of NFE s-p band and d band of type A where they cross [after V. Heine, in The Physics of Metals (J. M. Ziman, ed.), pp. 1-61. Cambridge Univ. Press, London and New York, 19697.

This expression is called the model HamiEtonian.ss If the hybridization coefficients Y.,,G. happened to be zero, we should be back at the simple model of noninteracting s and d bands; otherwise, these have the effect of distorting the simple bands a little, and splitting them apart at points in k space where they cross. This splitting is, of course, a standard feature of the band structure of transition metals, as calculated by more direct methods (Fig. 15).

At this stage, for completeness, we should write down explicit functional expressions for Y,,,G#, obtained by treating (7.3) in much the same way as (4.2). We should arrive at a complicated function of k, depending upon various overlap integrals similar to (4.5) but with the atomic orbital bntu replaced by a plane wave part &--0. Using some standard approximations, these integrals can be evaluated-or at least expressed-in terms of arbitrary parameters. As with the LCAO components, group theoretical sym- metrization reduces the number of such parameters to two or three, which may be adjusted to describe the behavior of these coefficients in practice. Indeed, with so many parameters at our disposal, there is little difficulty in using the model Hamiltonian as an accurate interpolation scheme for the band structure of transition metals. As explained in detail by Ehrenreich and H o d g e ~ , ~ ~ this is a valuable working tool in the investigation of such

F. M. Mueller, Phys. Rev. 163, 659-669 (1967). See also L. Hodges, H. Ehrenreich, and N. D. Lang, ibid. 162, 50.55ri26 (1966) ; J. Friedel, in The Physics of Metals: I. Electrons (J. hl. Ziman, ed.), pp. 340-408; Cambridge Univ. Press, London and New York, 1969. H. Ehrenreich and L. Hodges, dlethods Computat.Phys. 8, 149-192 (1968).

36 J. M. ZIMAN

complex phenomena as magnetism, Fermi surface experiments, and optical properties.

But this whole interpretation seems to be forbidden by our previous arguments. In Sections 4-6 it was maintained that all atomic bound states would disappear above the muffin-tin zero, leaving only nearly free Bloch states. What did we overlook in that discussion?

The answer is that atomic levels of high angular momentum (Fig. 16) are not completely destroyed by the overlap of potentials, but become virtual or resonance levels. This phenomenon, which is quite familiar in atomic and nuclear physics, arises as follows (Fig. 17). The radial Schro- dinger equation contains the term 1(1+ 1) /?, which behaves like the potential of a centrifugal force repelling the electron from the region of the nucleus. In a bound state of high angular momentum (e.g., 11 2 ) the electron is confined to the annular space between this barrier and the ordinary external Coulomb potential of the atom. When atoms are brought together, this outer barrier may not be completely lost, but may still inter- pose a hill through which the electron in the original atomic level must tunnel if it is to escape. Thus, although we may now be above the muffin-tin zero, we find a strong tendency for the wave function to concentrate within the atom as we pass through the energy Ed. This effect cannot be perfectly sharp (as it would be for a true bound state) but must be spread over a width W , which would depend in detail on the characteristics of the centrifugal barrier and of the self-consistent potential of the atom.

I

FIQ. 16. 3d and 4s radial functions [after J. Friedel, in The Physics of Metals: I Electrons (J. M. Ziman, ed.), pp. 340408. Cambridge Univ. Press, London and New York, 19691.


FIO. 17. The addition of the centrifugal energy Z(1 + 1)/P to an atomic potential u. gives rise to an effective potential with a bound state at &b. Overlapping to produce muffin-tin potentids UMT turns this into a resonance at g, above the miiffin-tin zero.

A wave function of the form (7.3)-something like an atomic bound state with a free-electron part outside-is therefore a reasonable trial function for the Bloch function. But the algebraic and computational disadvantages of such a hybrid representation can be avoided by an extension of the general NFE procedure. In Section 6, we argued in favor of treating each muffin-tin well as a black box, whose only known properties were its partial wave phase shifts T , J ~ ( & ) .

For any given potential, these phase shifts may be computed directly by solving the radial Schrodinger equation (2.4) and matching the logarithmic derivative of &(r) to the corresponding free wave components on the surface of the atomic sphere, i.e.

(7.5)

In fact, we do not need to carry out such an analysis in detail; in m y standard treatise on quantum theory (see also Section 15) it is shown that the phase shift near a resonance should behave like

(7.6)

showing that it passes rapidly but continuously through the value 7r/2 at E = E, (Fig. 18).

v ( & ) M tan- [*w/(&r - &)I,

38 J. M. ZIMAN

5,

FIG. 18. Variation of phase shift with energy at a resonance.

This strong dependence of the phase shift on energy obviously has significant effects on the band structure. Anticipating a formal proof in Section 13, let us improve the NFE formula (6.10), by replacing the t matrix of the muffin-tin well by another abstract operator, the reaction matriz (or K matrix) to which it is analytically related.a1 A plausible justification of this step is that the t matrix refers to genuine, time-ordered scattering events, whereas the reaction matrix describes noncausal stationary states of interaction between a localized potential well and a particle, such as one would expect in a true Bloch function. Anyway, the effect is to replace the factor sin 1 1 exp(iv1) by tan v 1 in the usual scattering amplitude formula (6.7), i.e.

K.(e) = - ( ~ T N / K ) C (21 + 1) tan vr(&) Pl(cos e). (7.7) 1

If now, in place of (6.10), we write

r G G t X S ( G - G)K,(G - G), (7.8) and put this into the NFE equations (6.9), we shall obviously get a serious perturbation of the band structure at any singularity of the form of (7.6). It is easy to show, in simple cases,a5 that the picture of a d band crossing and hybridizing with an s band can be reproduced in this way.

From this standpoint we can comprehend some important general features of the band structure of transition and noble metals. It is clear, e.g., that width of the d band, and its position within the s band, depend mainly upon the width of the corresponding muffin-tin rescmance, and its energy relative to the muffin-tin zero. These, in their turn, are very sensitive

16 J. M. Ziman, Proc. Phys. Soc. 86, 337-353 (1965).


to assumptions made about the screening or overlap of superposable potentials in the interstitial region (Section 2) . The wide range of variation in computed estimates of these parameters is thus quite expIi~able.~OJ6

One can also argue convincingly that these features should not be very much affected by melting. If there is little change of atomic volume, the width and position of the Lpseudoatom resonance will stay nearly the same: in the liquid as in the crystalline solid, the electron energy spectrum must surely be governed by the ability of each atom to swallow up a large density of electrons over a narrow range of energy in these quasi-stationary internal states. But of course it is quite a different matter to give a formal proof of this principle in a disordered assembly.

With this section we complete what may be described as the hand waving phase of the argument. Without any pretence of rigor or algebraic accuracy, I have been trying to show what the band structure problem is really about and what characteristics of the atomic potentials and their spatial arrangement really determine the electronic spectrum. As in most branches of theoretical physics, these general principles, once thoroughly grasped, are a faithful guide through the labyrinth of algebraic symbolism that has been thrown together by a generation of earnest devotees of this mysterious cult. We shall now venture further within the shrine, in an attempt to record and analyze the actual liturgical exercises and theological doctrines. In prosaic language, we shall now study in detail the mathematical techniques that are in practical use for exact ab init io computations of electronic band structure in crystalline solids.

111. Pseudopotentials

8. ORTHOGONALIZED PLANE WAVES

We have it in mind now to make a determined quantitative attack on the band structure of some actual system. In the first instance we should choose some relatively simple material, such as an alkali metal, where the superposition of atomic potential

Frederick Seitz - Solid State Physics, Vol.26

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