Electrons
Fermi Surfaces
Empty lattice approximation
ky
kx
e

ik r
e
   
ik r iG r
e
Empty lattice approximation
Empty lattice approximation
W. Harrison, Phys. Rev. 118 p. 1182 (1960)
aluminum
Fermi surface for free electrons
n
EF
 D( E )dE

2m 

n
3
2
2 2 3
EF

EdE

2
2
3
2
EF 
3 n 

2m
 2 k F2
EF 
2m
Constructing Fermi surface
4
3
2
No Fermi surface in the 1st Brillouin zone
2d square
2N electron states in a
Brillouin zone
The Fermi surface
strikes the Brillouin
zone boundary at 90o.
2d hexagonal
2d centered
rectangular
Student project
Replot the Fermi surfaces in 2-D plotting just the surface.
It would be best if we use a program to draw the Fermi
surfaces dynamically.
Fermi surface for fcc in the empty lattice approximation
Valence 4
The flat planes are edges of the Brillouin
zone boundary, not the Fermi surface.
Band structure calculations
Start with the full Hamiltonian.
Z Ae 2
Z A Z B e2
e2
2 2
2 2
H  
i  
A  


i 2me
A 2m A
i , A 4 0 riA
i  j 4 0 rij
A B 4 0 rAB
Everything you can know is contained in this Hamiltonian.
Usually this is too difficult to solve.
Electrons in a crystal
Fix the positions of the nuclei (Born Oppenheimer approximation) and
consider the many electron Hamiltonian.
H elec
Z Ae2
e2
2 2
 
i  

i 2me
i , A 4 0 riA
i  j 4 0 rij
This is still too difficult. Neglect the electron-electron interactions.
Separation of variables
H elec
Z Ae 2
e2
2 2
 
i  

i 2 me
i , A 4 0 riA
i  j 4 0 rij
The electronic Hamiltonian separates into the molecular orbital
Hamiltonians.
Helec(r1,r2,...rn) = HMO(r1) + HMO(r2) + ... HMO(rn)
elec(r1,r2,...rn) = |MO(r1)MO(r2) ... MO(rn)>
H MO
Z Ae 2
 2


 
2me
A 4 0 r  rA
Solving the molecular orbital Hamiltonian
H MO
Z Ae2
2


 
2me
A 4 0 r  rA
Band structure calculations:
Plane wave method
Tight binding (LCAO+)
Plane wave method
2 2


   U MO (r )  E
2m
Write U and  as Fourier series.
 

iG  r
e
U MO ( r )  
U
G

G
For a periodic lattice of Coulomb potentials:
  Ze
U MO (r ) 
4 0
2

j
1
 
r  rj

 Ze
V0
2
 
iG r
e

2

G G
volume of a unit cell
http://lampx.tugraz.at/~hadley/ss1/crystaldiffraction/fourier.php
Plane wave method
2 2


   U (r )  E
2m
 
 


iG  r
ik  r
U MO ( r )  
U G e
 (r )  
C k e


G


k
k
 
  
 2 k 2  ik r
i ( G  k  )r
ik  r
 E
Ck e  
U G C k e
C k e
 


2m
G
k
k
Must hold for each Fourier coefficient.
  
k  G  k

  
k  k  G
 2k 2

U G Ck G  0
 E  Ck  


G
 2m

Central equations (one for every k in the first Brillioun zone)
Plane wave method
The central equations can be written as a matrix equation.


MC  EC
Diagonal elements:
2  
M ii 
k  Gi
2m


2
Ze 2
Off-diagonal elements: M ij  
 
V  0 Gi  G j


2
For Z = 0, this results in the empty lattice approximation.
Plane wave method
Plane wave method
fcc Z= 0
empty lattice
Plane wave method bcc
Muffin tin potentials, pseudopotentials
Ze 2

U (r )  
4 0 r
inside a radius R and is constant outside
Si
Bachelor thesis Benedikt Tschofenig
http://www.quantum-espresso.org/
Tight binding
Tight binding does not include electron-electron interactions
H MO
Z Ae2
 2
2


  V (r ) 

 
r
2me
2me
4

 rA
A
0
Assume a solution of the form.
k 

l ,m ,n
 
 
 
exp i lk  a1  mk  a 2  nk  a3


 


c

r

la

ma

na

 aa
1
2
3
a
atomic orbitals:
choose the relevant
valence orbitals
http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tightbinding/tightbinding.php
Tight binding
k 

l ,m ,n
 
 
 
exp i lk  a1  mk  a 2  nk  a3


 


c

r

la

ma

na

 aa
1
2
3
a
H MO k  E k k
 a H MO  k  E k  a  k
ca  a H MO  a 

cm  a H MO  m
 
 
 
exp(i ( hk  a1  jk  a 2  lk  a3 ))  small terms
nearest neighbors m
 E k ca  a  a  small terms
There is one equation for each atomic orbital
Tight binding, one atomic orbital
ca  a H MO  a 

 
 
 
exp(i ( hk  a1  jk  a 2  lk  a3 ))  small terms
cm  a H MO  m
nearest neighbors m
 E k ca  a  a  small terms
For only one atomic orbital in the sum over valence orbitals

E k ca  a  a  ca  a H MO  a 
ca  a H MO  m
 
 
 
exp(i ( hk  a1  jk  a 2  lk  a3 ))
nearest neighbors m
one atomic orbital
Ek    t  e
 
ik   m
m


   a  r  H MO  a  r 

 
t    a  r  H MO  a  r   m 
Tight binding, simple cubic
E    t e

E  t e
ik x a
e
 ik x a
e
m
ik y a
 
ik   m
e
 ik y a
 e ik z a  e  ik z a

   2t  cos( k x a )  cos( k y a )  cos( k z a ) 
Effective mass
2
2
m  2 
d E 2ta 2
dk 2
*
Narrow bands  high effective mass
Density of states (simple cubic)
Calculate the energy for every allowed k in the Brillouin zone
E    2t  cos( k x a )  cos( k y a )  cos( k z a ) 
http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/tbtable.html
E    2t  cos( k x a )  cos( k y a )  cos( k z a ) 
Tight binding, fcc
E    t e
m
 
ik   m
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/
Graphene
C1 C2

a
C2
c1 C2
1 C1
C1
c1 C2
c2

a
C2
2
C1
C1
c1 C2
3
1
a xˆ  a yˆ
2
2
3
1

a2 
a xˆ  a yˆ
2
2

a1 
C1 C2
Two atoms per unit cell
Graphene has an unusual dispersion relation in the vicinity of the
Fermi energy.