2/04/2015phy 752 spring 2015 -- lecture 91 phy 752 solid state physics 11-11:50 am mwf olin 107 plan...
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2/04/2015PHY 752 Spring Lecture 93TRANSCRIPT
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
PHY 752 Spring 2015 -- Lecture 9 22/04/2015
PHY 752 Spring 2015 -- Lecture 9 32/04/2015
PHY 752 Spring 2015 -- Lecture 9 42/04/2015
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
PHY 752 Spring 2015 -- Lecture 9 52/04/2015
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
PHY 752 Spring 2015 -- Lecture 9 72/04/2015
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
PHY 752 Spring 2015 -- Lecture 9 82/04/2015
LCAO methods -- continued – bond types
ssspps
ppp ddp
PHY 752 Spring 2015 -- Lecture 9 92/04/2015
PHY 752 Spring 2015 -- Lecture 9 102/04/2015
)
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
PHY 752 Spring 2015 -- Lecture 9 112/04/2015
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
PHY 752 Spring 2015 -- Lecture 9 132/04/2015
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
PHY 752 Spring 2015 -- Lecture 9 142/04/2015
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
PHY 752 Spring 2015 -- Lecture 9 152/04/2015
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
PHY 752 Spring 2015 -- Lecture 9 162/04/2015
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