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Tight binding
Tight binding does not include electron-electron interactions
22
02 4A
MOAe A
Z eHm r rπε−
= ∇ −−∑
( )( ) ( )1 2 3 1 2 3, ,
expk a al m n a
i lk a mk a nk a c r la ma naψ ψ= ⋅ + ⋅ + ⋅ − − −∑ ∑
Assume a solution of the form.
atomic orbitals: choose the relevant valence orbitals
http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tightbinding/tightbinding.php
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Tight binding
a MO k k a kH Eψ ψ ψ ψ=
MO k k kH Eψ ψ=
( )( ) ( )1 2 3 1 2 3, ,
expk a al m n a
i lk a mk a nk a c r la ma naψ ψ= ⋅ + ⋅ + ⋅ − − −∑ ∑
1 2 3nearest neighbors
exp( ( )) small terms
small terms
a a MO a m a MO mm
k a a a
c H c H i hk a jk a lk a
E c
ψ ψ ψ ψ
ψ ψ
+ ⋅ + ⋅ + ⋅ +
= +
∑
There is one equation for each atomic orbital
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Tight binding, one atomic orbital
For only one atomic orbital in the sum over valence orbitals
1 2 3nearest neighbors
exp( ( )) small terms
small terms
a a MO a m a MO mm
k a a a
c H c H i hk a jk a lk a
E c
ψ ψ ψ ψ
ψ ψ
+ ⋅ + ⋅ + ⋅ +
= +
∑
1 2 3nearest neighbors
exp( ( ))k a a a a a MO a a a MO mm
E c c H c H i hk a jk a lk aψ ψ ψ ψ ψ ψ= + ⋅ + ⋅ + ⋅∑
mikk
mE t e ρε ⋅= − ∑
( ) ( )a MO ar H rε ψ ψ= ( ) ( )a MO a mt r H rψ ψ ρ= − −
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Tight binding, simple cubic
mik
mE t e ρε ⋅= − ∑
2 2*
2 2
22
md E tadk
= =Effective mass
( )( )2 cos( ) cos( ) cos( )
y yx x z zik a ik aik a ik a ik a ik a
x y z
E t e e e e e e
t k a k a k a
ε
ε
−− −= − + + + + +
= − + +
Narrow bands → high effective mass
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Density of states (simple cubic)
Calculate the energy for every allowed k in the Brillouin zone
( )2 cos( ) cos( ) cos( )x y zE t k a k a k aε= − + +
http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/tbtable.html
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( )2 cos( ) cos( ) cos( )x y zE t k a k a k aε= − + +
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Tight binding, fcc
mik
mE t e ρε ⋅= − ∑
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Density of states (fcc)
Calculate the energy for every allowed k in the Brillouin zone
http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/tbtable.html
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Tight binding, fcc
Christian Gruber, 2008
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Tight binding, fcc
http://www.phys.ufl.edu/fermisurface/
Page 11
http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/tbtable.html
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Student Projects
Plot the dispersion relation for hexagonal crystals
Calculate the density of states for, CNTs, or BN
Draw the missing Fermi surfaces
Calculate the thermodynamic properties based on a calculated DOS
Make a similar table for the plane wave method
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1
2
3 1ˆ ˆ2 23 1ˆ ˆ
2 2
a ax ay
a ax ay
= +
= −
Graphene
Two atoms per unit cell
Graphene has an unusual dispersion relation in the vicinity of the Fermi energy.
1a
2ac1
C1
c1
c2
C2
C2
c1
C1
C1
C2
C2
c2
C1
C1
C1
C2
C2
C2
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2000 Ig Nobel Prize for levitating a frog with a magnet
Andre Geim
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ψa
ψb
( )( ) ( ) ( )( )1 2 1 2 1 2,
expz zk a p a b p b
l m
i lk a mk a c r la ma c r la maψ ψ ψ= ⋅ + ⋅ − − + − −∑
k kH Eψ ψ=
2 carbon atoms / unit cell
The standard guess for the wave function in the tight binding model is
For graphene, the valence orbitals are pz orbitals
Substitute this wave function into the Schrödinger equation
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( )( ) ( ) ( )( )1 2 1 2 1 2,
expz zk a p a b p b
l m
i lk a mk a c r la ma c r la maψ ψ ψ= ⋅ + ⋅ − − + − −∑
k kH Eψ ψ=
small terms
small terms
m
m
ika a a b a b
m
ika a a b a b
m
c H c H e
E c c e
ρ
ρ
ψ ψ ψ ψ
ψ ψ ψ ψ
⋅
⋅
+ +
⎛ ⎞= + +⎜ ⎟
⎝ ⎠
∑
∑
m sums over the nearest neighbors
Multiply by and integrate( )*zp a rψ
2 carbon atoms / unit cell
the orbital for the atom at l = 0, m = 0.
0
1
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small terms
+ small terms
m
m
ika b a b b b
m
ika b a b b b
m
c H e c H
E c e c
ρ
ρ
ψ ψ ψ ψ
ψ ψ ψ ψ
− ⋅
⋅
+ +
⎛ ⎞= +⎜ ⎟
⎝ ⎠
∑
∑
Multiply by and integrate
To get a second equation for ca and cb
( )*zp b rψ
the orbital for the atom at l = 0, m = 0.
2 carbon atoms / unit cell
k kH Eψ ψ=
Write as a matrix equation 0 1
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0
m
m
ika a a b
m a
ikbb a b b
m
H E H eccH e H E
ρ
ρ
ψ ψ ψ ψ
ψ ψ ψ ψ
⋅
− ⋅
⎡ ⎤−⎡ ⎤⎢ ⎥
=⎢ ⎥⎢ ⎥− ⎣ ⎦⎢ ⎥
⎣ ⎦
∑
∑
m sums over the nearest neighbors.
There will be two eigen energies for every k.
N orbitals / unit cell results in N bands
Tight binding graphene
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There will be two eigen energies for every k.
Tight binding graphene
3 31 exp exp2 2 2 2
m y yik x x
m
k a k ak a k ae i iρ⋅⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟= + + + −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
∑
1a
2a
1
2
3 1ˆ ˆ2 23 1ˆ ˆ
2 2
a ax ay
a ax ay
= +
= −
0
m
m
ik
m
ik
m
E t e
t e E
ρ
ρ
ε
ε
⋅
− ⋅
− −=
− −
∑
∑
unit cell
1k a⋅ 2k a⋅
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3 31 exp exp2 2 2 2
03 31 exp exp2 2 2 2
y yx x
y yx x
k a k ak a k aE t i i
k a k ak a k at i i E
ε
ε
⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− − + − + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ =
⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− + − − + − + −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
( )2 2
3 31 exp exp2 2 2 2
3 3 3exp 1 exp2 2 2 2 2 2
3 3exp exp2 2 2
y yx x
y y yx x x
yx x
k a k ak a k ai i
k a k a k ak a k a k aE t i i i
k ak a k ai i
ε
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞+ − + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞− − + − − + + − − + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ ⎞⎛ ⎞+ − + + − +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
0
3 12 2 2y yxk a k ak ai
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
=⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟+ − +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠
Solve for the dispersion relation