PHY 752 Spring 2015 -- Lecture 9 12/04/2015

PHY 752 Solid State Physics11-11:50 AM MWF Olin 107

Plan for Lecture 9:Reading: Chapter 8 in MPM; Electronic Structure

1. Linear combination of atomic orbital (LCAO) method

2. Slater and Koster analysis3. Wannier representation

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Linear combinations of atomic orbitals (LCAO) methods for analyzing electronic structure

Ra

Bloch wave:

in ne u k rk kr r

periodic function

Let a a R τ T

basis vectorlattice translation

Bloch wave identity:

in ne k Tk kr T r

LCAO basis functions with Bloch symmetry:

)(a

a nlm i a anlme

k

k τ T

Tr r τ T

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LCAO methods -- continued

LCAO basis functions with Bloch symmetry:

)( i aa n m alnlme k T

Tk r r τ T

k=0

k=p/2a

k=p/a

100( ) ( )Example for rnlm Ce r r

PHY 752 Spring 2015 -- Lecture 9 62/04/2015

LCAO methods -- continued – angular variationhttp://winter.group.shef.ac.uk/orbitron/

l=0 l=1

l=2

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LCAO methods -- continued – angular variation

While, for atoms the “z” axis is an arbitrary direction, for diatomic molecules and for describing bonds, it is convenient to take the “z” axis as the bond direction.

Atom symboll=0 m=0 s l=1 m=0 p m= ±1 p l=2 m=0 d m= ± 1 d m= ± 2 d

Bond symboll=0 l=0 s l=1 l=0 s l=1 p l=2 l=0 s l=1 p l= 2 d

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LCAO methods -- continued – bond types

ssspps

ppp ddp

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)

Approximate

LCAO basis funct

Bloch wavefunction

ions with Bloch symmetry:

(

:

)( ()

aa nlm

a nlm

i a anlm

a nlm

a nlm

e

X

k τ T

T

k kk

k r r τ T

r r

LCAO methods -- continued – Slater-Koster analysis

In this basis, we can estimate the electron energy by variationallycomputing the expectation value of the Hamiltonian:

HE

k k

kk k

' ' '

Terms in this expansion have the form:bai a a a

n l m nlb

mHe

k τ τ T

Tr τ r τ T

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LCAO methods – Slater-Koster analysis -- continued

' ' '

bai a b a an l m nlme H

k τ τ T

Tr τ r τ T

Notation in Slater-Koster paper

ba l m n k Tτ τ

' ' ' :a a an l m n

blmH r τ r τ T

PHY 752 Spring 2015 -- Lecture 9 122/04/2015

LCAO methods – Slater-Koster analysis -- continuedSimple cubic lattice

2 2 2 2 2 2 2 2 2

ˆ ˆ ˆ

ba ap aq ra

p q

r r

rl m np q rp q p q

Tτ τ x y z

, ,

,

( / ) (000) 2 (100) cos cos cos

+4 (110) cos cos cos cos cos cos ...

s s s s

s s

s s E E

E

on site NN

NNN

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LCAO methods – summary

)

Approximate

LCAO basis funct

Bloch wavefunction

ions with Bloch symmetry:

(

:

)( ()

aa nlm

a nlm

i a anlm

a nlm

a nlm

e

X

k τ T

T

k kk

k r r τ T

r r

In this basis, we can estimate the electron energy by variationallycomputing the expectation value of the Hamiltonian:

HE

k k

kk k

' ' '

' ' 'and also

Terms in this expansion have the form:a

ba

bi a a an l m nlm

i a a an l m nlm

b

b

He

e

k τ τ T

T

k τ τ T

T

r τ r τ T

r τ r τ T

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LCAO methods – summary

LCAO basis functions with Bloch symmetry:

)(a

a nlm i a anlme

k

k τ T

Tr r τ T

Is there a “best choice” for atom-centered functions?

Introduction to the Wannier representation

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Wannier representation of electronic states

Note: This formulation is based on the relationship between the Bloch and Wannier representations and does not necessarily imply an independent computational method.

Bloch wave:

in ne u k rk kr r

Bloch wave identity:i

n ne k Tk kr T r

3' '

Orthonorma ty:

'

li

n n nnd d k kr r k k

3

3

Wannier function in lattice cell , associated with band is given by:

W ( )( ) 2

in n

V d k e

n

p k T

kr T

T

r

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Wannier representation of electronic states -- continued

3

3

Wannier function in lattice cell , associated with band is given by:

W ( )( ) 2

in n

Vd k e

n

p k T

kr T

T

r

' '

Note that

( ( ')

:

)n n nnW W d d TTr T r T

Comment: Wannier functions are not unique since the the Bloch function may be multiplied by a k-dependent phase, which may generate a different function Wn(r-T).

PHY 752 Spring 2015 -- Lecture 9 172/04/2015

Example from RMP 84, 1419 (2012) by Mazari, Mostofi, Yates, Souza, and Vanderbilt