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

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

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= − −

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

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

( )2 cos( ) cos( ) cos( )x y zE t k a k a k aε= − + +

Tight binding, fcc

mik

mE t e ρε ⋅= − ∑

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

Tight binding, fcc

Christian Gruber, 2008

Tight binding, fcc

http://www.phys.ufl.edu/fermisurface/

http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/tbtable.html

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

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

2000 Ig Nobel Prize for levitating a frog with a magnet

Andre Geim

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

( )( ) ( ) ( )( )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

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

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

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⋅

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