Density Functional Theory for Electrons in Materials Richard M. Martin

1
Density Functional Theoryfor Electrons in
MaterialsRichard M. Martin
Prediction of Phase Diagram of Carbon at High P,T
Bands in GaAs
2
Outline

• Pseudopotentials
• Ab Initio -- Empirical
• Bloch theorem and bands in crystals
• Definition of the crystal structure and Brillouin
  zone in programs used in the lab (Friday)
• Plane wave calculations
• Iterative methods
• Krylov subspaces
• Solution by energy minimization Conjugate
  gradient methods
• Solution by residual minimization (connnection
  toVASP code that will be used by Tuttle)
• Car-Parrinello ab initio simulations
• Examples

3
Bloch Theorem and Bands

• Crystal Structure Bravais Lattice Basis

Atoms
Points or translation vectors
Crystal
Space group translation group point group
Translation symmetry - leads to
Reciprocal Lattice Brillouin Zone Bloch
Theorem ..
4
a2
b2
b2
a2
a1
b1
a1
Wigner-Seitz Cell
b1
Brillouin Zone
Real and Reciprocal Lattices in Two Dimensions
5
a3
a2
a1
Simple Cubic Lattice Cube is also Wigner-Seitz
Cell
6
Body Centered Cubic Lattice
7
Face Centered Cubic Lattice
8
z
y
X
ZnS Structure with Face Centered Cubic Bravais
Lattice
NaCl Structure with Face Centered Cubic Bravais
Lattice
9
z
z
X
X
R
L
y
L
L
K
G
S
y
G
X
D
X
W
S
W
M
U
D
X
K
X
U
X
X
z
A
H
z
D
L
H
L
K
T
P
G
D
y
G
H
S
S
H
N
M
K
X
y
x
Brillouin Zones for Several Lattices
10
Example of Bands - GaAs
T.-C. Chiang, et al PRB 1980

• GaAs - Occupied Bands - Photoemission Experiment
  - Empirical pseudopotential
• Ab initio LDA or GGA bands almost as good for
  occupied bands -- BUT gap to empty bands much too
  small

11
Transition metal series
L. Mattheisss, PRB 1964

• Calculated using spherical atomic-like potentials
  around each atom
• Filling of the d bands very well described in
  early days - and now - magnetism, etc.
• Failures occur in the transition metal oxides
  where correlation becomes very important

12
Standard method - Diagonalization

• Kohn- Sham self Consistent Loop
• Innner loop solving equation for wavefunctions
  with a given Veff
• Outer loop iterating density to self-consistency
• Non-linear equations
• Can be linearized near solution
• Numerical methods - DIIS, Broyden, etc. (D.
  Johnson) See later - iterative methods

13
Empirical pseudopotentials

• Illustrate the computational intensive part of
  the problem
• Innner loop solving equation for wavefunctions
  with a given Veff
• Greatly simplified program by avoiding the
  self-consistency
• Useful for many problems
• Description in technical notes and lab notes

14
Iterative methods

• Have made possible an entire new generation of
  simulations
• Innner loop This is where the main computation
  occurs
• Many ideas - all with both numerical and a
  physical basis
• Energy minimization - Conjugate gradients
• Residual minimization - Davidson, DIIS, ...
• See lectures of E. de Sturler

Used in Lab
15
Car-Parrinello Simulations

• Elegant solution where the optimization of the
  electron wavefunctions and the ion motion are all
  combined in one unified algorithm

16
Example

• Prediction of Phase Diagram of Carbon

M. Grumbach, et al, PRB 1996

• Above 5 Mbar C prdicted to behave like Si -
  Tmelt decreases with P

17
Conclusions

• The ground state properties are predicted with
  remarkable success by the simple LDA and GGAs.
• Accuracy for simple cases gives assuarnce in
  complex cases
• Iterative methods make possible simulations far
  beyond anything done before
• Car-Parrinello ab initio simulations
• Greatest problem at present Excitations
• The Band Gap Problem