Chapter 6 Electronic Structure and Periodic Properties of ...
Electronic Structure Theory for Periodic Systems: The ...
Transcript of Electronic Structure Theory for Periodic Systems: The ...
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Electronic Structure Theory for Periodic Systems: The Concepts
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Christian Ratsch
Institute for Pure and Applied Mathematics and Department of Mathematics, UCLA
Motivation
• There are 1020 atoms in 1 mm3 of a solid.
• It is impossible to track all of them.
• But for most solids, atoms are arranged periodically
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Outline
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Part 1
• Periodicity in real space
• Primitive cell and Wigner Seitz cell
• 14 Bravais lattices
• Periodicity in reciprocal space
• Brillouin zone, and irreducible Brillouin zone
• Bandstructures
• Bloch theorem
• K-points
Part 2
• The supercell approach
• Application: Adatom on a surface (to calculate for example a diffusion barrier)
• Surfaces and Surface Energies
• Surface energies for systems with multiple species, and variations of stoichiometry
• Ab-Initio thermodynamics
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Primitive Cell and Wigner Seitz Cell
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
primitive cell
Wigner-Seitz cell
𝑹 = 𝑁1𝒂1+𝑁2𝒂2
Non-primitive cell
a1
a2
a1
a2
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Primitive Cell and Wigner Seitz Cell
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Face-center cubic (fcc) in 3D:
𝑹 = 𝑁1𝒂1+𝑁2𝒂2+𝑁3𝒂3
Primitive cell
Wigner-Seitz cell
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How Do Atoms arrange in Solid
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Atoms with no valence electrons: (noble elements) • all electronic shells are filled • weak interactions • arrangement is closest packing
of hard shells, i.e., fcc or hcp structure
Atoms with s valence electrons: (alkalis) • s orbitals are spherically
symmetric • no particular preference for
orientation • electrons form a sea of negative
charge, cores are ionic • Resulting structure optimizes
electrostatic repulsion (bcc, fcc, hcp)
Atoms with s and p valence electrons: (group IV elements) • Linear combination between s and p
orbitals form directional orbitals, which lead to directional, covalent bonds.
• Structure is more open than close-packed structures (diamond, which is really a combination of 2 fcc lattices)
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14 Bravais Lattices
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
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Reciprocal Lattice
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
• The reciprocal lattice is a mathematically convenient construct; it is the space of the Fourier transform of the (real space) wave function.
• It is also known as k-space or momentum space
𝒃1 = 2𝜋𝒂2 × 𝒂3
𝒂1 ∙ 𝒂2 × 𝒂3
𝒃2 = 2𝜋𝒂3 × 𝒂1
𝒂2 ∙ 𝒂1 × 𝒂3
𝒃3 = 2𝜋𝒂1 × 𝒂2
𝒂3 ∙ 𝒂1 × 𝒂2
Example: Honeycomb lattice
Real space
Reciprocal space
The reciprocal lattice is defined by
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Brillouin Zone
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Body-centered cubic
Cubic structure
3D The Brillouin Zone is the Wigner-Seitz cell of the reciprocal space
Square lattice (2D) square lattice
Hexagonal lattice (2D) hexagonal lattice
The Irreducible Brillouin Zone is the Brillouin Zone reduced by all symmetries
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Outline
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Part 1
• Periodicity in real space
• Primitive cell and Wigner Seitz cell
• 14 Bravais lattices
• Periodicity in reciprocal space
• Brillouin zone, and irreducible Brillouin zone
• Bandstructures
• Bloch theorem
• K-points
Part 2
• The supercell approach
• Application: Adatom on a surface (to calculate for example a diffusion barrier)
• Surfaces and Surface Energies
• Surface energies for systems with multiple species, and variations of stoichiometry
• Ab-Initio thermodynamics
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From Atoms to Molecules to Solids: The Origin of Electronic Bands
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Silicon
Discrete states Localized discrete states
Many localized discrete states, effectively a continuous band
EC
EV EV
EC
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Bloch Theorem
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Suppose we have a periodic potential where is the lattice vector
𝑹 = 𝑁1𝒂1+𝑁2𝒂2+𝑁3𝒂3
𝑹
Bloch theorem for wave function (Felix Bloch, 1928):
ѱ𝒌 𝒓 = 𝑒𝑖𝒌∙𝒓𝑢𝒌 𝒓
𝑢𝒌 𝒓 + 𝑹 = 𝑢𝒌 𝒓 phase factor
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The meaning of k
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Free electron wavefunction
𝑘 = 0
𝑘 ≠ 0
𝑘 = 𝜋/𝑎
ѱ𝒌 𝒓 = 𝑒𝑖𝒌∙𝒓𝑢𝒌 𝒓
𝑘 = 2𝜋/𝑎
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Band structure in 1D
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
periodic potential
𝜀
𝜋𝑎
0
Repeated zone scheme
𝜋𝑎 3𝜋
𝑎 0
𝜀
𝜋𝑎
0
𝜀
weak interactions Only first parabola
𝜋𝑎
0
𝜀
Extended zone scheme of similar split at 𝑘 = 𝜋
𝑎, 𝑘 = 2𝜋
𝑎
3𝜋𝑎
𝜋𝑎 0
𝜀 Reduced zone scheme
𝜋𝑎 0
𝜀
Energy of free electron in 1D
𝜀 =ℏ2𝑘2
2𝑚
𝜀
• We have 2 quantum numbers k and n
• Eigenvalues are En(k) Energy bands
2𝜋𝑎 2𝜋
𝑎
2𝜋𝑎
2𝜋𝑎
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Band Structure in 3D
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
• In principle, we need a 3+1 dimensional plot.
• In practise, instead of plotting “all” k-points, one typically plots band structure along high symmetry lines, the vertices of the Irreducible Brillouin Zone.
Example: Silicon
• The band gap is the difference between highest occupied band and lowest unoccupied band.
• Distinguish direct and indirect bandgap.
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Insulator, Semiconductors, and Metals
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
The Fermi Surface EF separates the highest occupied states from the lowest un-occuppied states
gap is order kT Large gap No gap
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Fermi Surface
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
In a metal, some (at least one) energy bands are only partially occupied by electrons. The Fermi energy εF defines the highest occupied state(s). Plotting the relation 𝜖𝑗 𝒌 = 𝜖𝐹 in reciprocal space yields the Fermi surface(s).
Example: Copper
Fermi Surface
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How to Treat Metals
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Fermi distribution function fF enters Brillouin zone integral:
• A dense k-point mesh is necessary.
• Numerical limitations can lead to errors and bad convergence.
Occ
upat
ion
energy 0
2
EF
• Possible solutions:
• Smearing of the Fermi function: artificial increase kBTel ~ 0.2 eV; then extrapolate to Tel = 0.
𝑓 𝜖𝑗 = 1
1 + 𝑒(𝜖𝑗−𝜖𝐹)/𝑘𝐵𝑇𝑒𝑒
• Tetrahedron method [P. E. Blöchl et al., PRB 49, 16223 (1994)]
• Methfessel-Paxton distribution [ PRB 40, 3616 (1989) ]
• Marzari-Vanderbilt ensemble-DFT method [ PRL 79, 1337 (1997) ]
𝑛 𝒓 = � � 𝑓(𝜖𝑗) 𝜓𝑗,𝒌(𝒓) 2 𝑑3𝒌Ω𝐵𝐵
Ω𝐵𝐵𝑗
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k-Point Sampling
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Charge densities (and other quantities) are represented by Brillouin-zone integrals replace integral by discrete sum:
• For smooth, periodic function, use (few) special k-points (justified by mean value theorem).
• Most widely used scheme proposed by Monkhorst and Pack.
• Monkhorst and Pack with an even number omits high symmetry points, which is more efficient (especially for cubic systems)
Brillouin zone
Irreducible Brillouin zone
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Outline
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Part 1
• Periodicity in real space
• Primitive cell and Wigner Seitz cell
• 14 Bravais lattices
• Periodicity in reciprocal space
• Brillouin zone, and irreducible Brillouin zone
• Bandstructures
• Bloch theorem
• K-points
Part 2
• The supercell approach
• Application: Adatom on a surface (to calculate for example a diffusion barrier)
• Surfaces and Surface Energies
• Surface energies for systems with multiple species, and variations of stoichiometry
• Ab-Initio thermodynamics
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Supercell Approach to Describe Non-Periodic Systems
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
A infinite lattice A defect removes periodicity
Define supercell to get back a periodic system
Size of supercell has to be tested: does the defect “see itself”?
the unit cell the supercell
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Adatom Diffusion Using the Supercell Approach
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Transition state theory (Vineyard, 1957):
)/exp(0 kTED d∆−Γ=
∏∏
−
=
==Γ 13
1*
3
10 N
j j
N
j j
ν
νAttempt frequency
Diffusion on (100) Surface
Adsorption Transition site
adtransd EEE −=∆
slab thickness
Vacuum separation
Need to test
Lateral supercell size
Supercell
Lateral supercell size
Vacuum separation
slab thickness
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Surface Energies
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Surface energy is the energy cost to create a surface (compared to a bulk configuration)
areaNE DFT
Surface)(
21 )( µγ −
=
)(DFTENµ
: Energy of DFT slab calculation
: Number of atoms in slab
: Chemical potential of material
Example: Si(111)
0 5 10 151.34
1.35
1.36
1.37
1.38
Surfa
ce e
nerg
y pe
r ato
m (e
V)
Number of layers
Surface
Bulk system
∞
∞
∞
∞
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Ab-Initio Thermodynamics to Calculate Surface Energies
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Example: Possible surface structures for InAs(001)
α(2x4) α2(2x4) α3(2x4)
β(2x4) β2(2x4) β3(2x4)
For systems that have multiple species, things are more complicated, because of varying stoichiometries
In atom
As atom
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Surface Energy for InAs(100)
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Equilibrium condition:
with
AsAsInInDFT
slabInAsSurfaceInAs NNE µµγ −−= −−)(
)(bulkii µµ ≤
)(bulkInAsInAsAsIn µµµµ ==+
α2 (2x4)
This can be re-written as:
AsInAsInAsInDFT
slabInAsSurfaceInAs NNNE µµγ ))( ( −−−= −−
with )()()( bulk
AsAsbulk
Inbulk
InAs µµµµ ≤≤−
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Conclusions
Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014
Thank you for your attention!!