Solid State Theory
INTRODUCTION TO BAND THEORY
Anton Grigoriev
www.uu.se !
Schrödinger equation
(1926)
ˆ
H" = E"
Nobel Prize (1933)
Erwin Schrödinger
!
Schrödinger equation (1926)
ˆ
H" = E"
ansatz for periodic potentials
and ‘bands’ of eigenvalues (1928)
Felix Bloch
!
Nobel Prize (1952)
[a1a2 a3 ] ,
reciprocal cell k in [b1b2b3 ]
cell r=(x,y,z) in
bi = 2"
!
!
a j # ak
V
Something to remember from Quantum Mechanics:
Sommerfeld theory
2
1 ikr
"=
e
V
hk
"=
2m
periodic boundary conditions:
e
ikx L
=e
iky L
or
=e
ikz L
2"n i
!
ki =
L
!
!
!
2
=1
4" 3
R V0 = N
3
3
kF
=
n
2
3"
Fermi distribution
1
f (") =
e
" #µ
kT
+1
Building a bulk:
Building a bulk (II):
"
potential
Bloch theorem
ikr
nk
nk
" =u e
" nk = $ c nk#K e
i( k#K ) r
K
!
$ h2
'
2
& (k " K ) " #)c k"K + *U K "K'c k"K ' = 0
%2m
(
K'
2a
2a
a
1/a
1/2
a
bcc empty lattice
[100] direction
bcc empty lattice
[100] direction
Energy, eV
Length, Å
n=2