Chapter 4 Electron theory of solid
In a theory which has given results like these, there
must certainly be a great deal of truth.
H.A. Lorentz
The history

Classical theory of free electron by P. Drude and H.A.
Lorentz
the valence electrons in metal are treated as classical
particles such as gas molecules, or “ideal gas”

Quantum theory of free electron by Sommerfeld
Electron should obey FD distribution. The motion of
electron in crystal should be treated in the framework of
quantum mechanics, that is, in terms of Schrödinger
equation. Motion of electron is in a uniform potential.

L.Brillouin and Bloch’s theory
Motion of electron is in the periodic potential of
lattice: Nearly free electron model
1
§4.1 Energy state of electron gas
1. Sommerfeld model
Quantum free electron gas, no-interaction, obeys FD
statistics.
2. Free electron gas in a 3D box
(1) The potential box (infinite potential well)
V ( x, y, z )  0, 0  x, y, z  L
V ( x, y, z )  , x, y, z  L
(2) Schrödinger equation
2m
   2 E  0 (inside the box)

2
(3) Solution
Separation of variables:
 ( x, y, z )  1 ( x ) 2 ( y ) 3 ( z )
Periodic boundary conditions:
 ( x  L, y, z )   ( x, y, z )
 ( x, y  L, z )   ( x, y, z )
 ( x, y , z  L)   ( x, y , z )
2
2 2

k
1

E

 (r ) 
e ,
2m
V
2
2
2
kx 
nx , k y 
ny , kz 
nz
L
L
L

ik r

3. State in k space





(1) k space k  k x i  k y j  k z k

(2) The density of representative points in k space:
V
(2 ) 3
4. Density of State and equi-energy surface
(1) The DOS g E  
dZ
dE
(2) The equi-energy surface within free electron
approximation
A sphere with radius
g ( E )  4Vc (
k
2mE

2m 3 / 2 1 / 2
) E
2

3
§4.2 Fermi energy of electron gas
1. Fermi distribution and Fermi level
F 

 N V
Fermi energy E F  
1
f E    E  E  k T
F
B
e
1
T 0K:
f ( E )  1, E  EF 0
f ( E )  0, E  EF 0
f ( E )  1, E  E F  several k BT
T 0K:
f ( E )  1 / 2, E  E F
f ( E )  0, E  E F  several k BT
EF is not fixed, it depends on temperature T .
Fermi temperature TF  E F / k B
Fermi wave vector k F  2mE F / 
2. Fermi energy at T  0 K
(1) the idea N   f ( Ei )
i
(2) the derivation
N 
E F0
0
f ( E ) g ( E )dE →
(3) the average energy E  3EF / 5
0
4
E F0
h 2 3n 2 / 3

( )
2m 8
3. Fermi energy at T  0 K
N 

0
EF 
E F0
2  3 / 2 f
f ( E ) g ( E )dE  C  E (  )dE
3 0
E
  2 k BT 2 
( 0 ) 
1 
12
EF 

5
→
§4.3 Heat capacity of the electron gas in
metal
1. Introduction
Why the heat capacity of electron is so small?
— Only those near the Fermi surface are counted
2. The derivation of CVe
2
2

3 0
5 2  k BT  
5 2  k BT  
Ee  EF 1    0    E0 1    0  
5  12  EF  
 12  EF  



Cve
 2 k BT
 Ee 

kB 0
 

T
2
EF

V
6
§4.4 Bloch wave
1. Introduction
The electrons in a crystal is not absolutely free, they
are confined in a periodic
potential.
To
solve
the
many-body
problem:
(1) adiabatic approximation
(2) single electron approximation
(3) periodic field approximation
2. General features of electron motion in a periodic
potential

using Bloch wave to describe

has energy band structure
3. Bloch wave
(1) Bloch theorem


 
 e uk ( r ) , uk ( r )  uk ( r  Rl )
 
 
ik  Rl  

 k ( r  Rl )  e  k ( r )

 k ( r )
or

ik r
7
(2) proof of Bloch theorem
Using the translational symmetry of crystals:
[ Hˆ , TˆRl ]  0
(3) The physical meaning of Bloch functions
Free motion of electrons as well as confinement by
periodic potential.
An illustration of Bloch wave
4. The Brillouin zone
the reduced Brillouin zone (1st Brillouine zone): The
region enclosed by the perpendicular bisector of lines
(planes) connecting the nearest reciprocal lattice points.




k   1b1   2 b2   3b3
j 
lj
Nj
，
Nj
2
 lj 
Nj
2
8
two typical examples showing electron motions in 1D
periodic field：
 Kronig-Penney model
 Nearly free electron model
9
§4.5 Kronig-Penney model
1. Introduction
a simplified model proposed by Kronig and Penney
in 1930 which can exhibit energy band structures of
electrons in periodic potential.
the problem can be strictly solved
V(x)
V0
-a -b 0
c a
x
a=b+c
2. Single-electron Schrödinger equation
d 2 2m

( E  V )  0 ，V ( x)  V ( x  na)
dx2 2
 ( x )  eikxu( x )
3. Solutions in different regions
 in the potential well: 0  x  c V  0
10
let  2 
2mE
2
 in
the
2 
potential
barrier:
 b  x  0 V  V0
let
2m(V0  E )
2
4. The constants A0 , B0 , C0 , D0
at x  0, c ， u and
du
should be continuous in both
dx
regions
 2  2
sinh b  sin c  cosh b  cos c  cos ka
2
5. Simplifications and discussions
Suppose V0   and b  0 , but V0 b is limited.
 2ab
 p , then we have
2
sin a
P
 cos a  cos ka
a
Let lim
P sin a
 cos a
a
-4
-
-3
-2
+1

0
-1
P=3/2
11
2
3
4 a

Energy band structure: E ~ k
Some general features:
（1）energy bands and forbidden bands
（2） E ~ k is even function
（3）location of forbidden bands
（4）the width of energy bands
（5）periodicity
（6）number of electrons
6. Scheme of energy band structures （ Page 146 of
textbook）
 extended zone scheme: different bands are drawn
in different zone
12
 periodic zone scheme: every band is drawn in
every zone
 reduced zone scheme: all bands are drawn in the
1st zone
13
§4.6
Nearly
free
electron
non-degenerate perturbation
model:
1. The idea of perturbation theory
The band structure of a crystal can often be explained
by the nearly free electron model for which the band
electrons are treated as perturbed only weakly by the
periodic potential of the ion cores. This model answers
almost all the qualitative questions about the behavior of
electrons in metals.
2. Single-electron Schrödinger equation
 2 2




V
(
x
)

  E ，V ( x)  V ( x  na)
2
m


V ( x )  V0   Vn e
'
i
2
nx
a
n
2
i

2
'
ˆ
ˆ
ˆ
H  H0  H '  (
  V0 )   Vn e
2m
n
2
nx
a
Using non-degenerate perturbation theory:
2
Ek 
Ek0

Ek( 2 )
2m Vn
 2k 2

'
2 2
2m
n  2k 2   2 (k 
n)
a
14
 k   k0  k(1)
2


i nx
*
a


2mVn e
1 ikx

e 1   '
2 2 
L
2 2
2
n
 k   (k 
n) 


a

3. Failure of non-degenerate perturbation theory
non-degenerate perturbation theory ： zero-order
energy is non-degenerate Ek0  Ek0'
（1）at the zone boundary
k
Kn  n
2
n

n
，k'  k 
， Ek0  Ek0'
2
a
a
a
Ek( 2 )   ,  k(1)  
（2）near the zone boundary
k
Kn
K
(1   ) ， k '   n (1   )
2
2
E k0  E k0'
Failure of non-degenerate perturbation theory!
Should consider degenerate perturbation theory.
15
§4.7 Nearly free electron model: degenerate
perturbation
1. Energy states at or near the zone boundary
k
Kn
K
(1   ) ， k '   n (1   )
2
2
(1) zero-order wavefunction
 0  A k0  B k0'  A
1 ikx
1 ik ' x
e B
e
L
L
(2) the energy
2
E  Tn (1  2 )  Vn  4Tn2 2
K
 2 K n 2  2 n 2
Tn 
( ) 
( ) （kinetic energy for k  n ）
2
2m 2
2m a
(3) discussions
  0 （at zone boundary）
k
K
n
K n n

，k'   n  
2
a
2
a
E  Tn  Vn ， E  Tn  Vn ，
forbidden bands at zone boundary： E g  2Vn
  0 （near zone boundary）
 Ek0  Ek0'  Vn
 Ek0  Ek0'  Vn
2. General features of dispersion relations ( E ~ k )
16
§4.8 Velocity and acceleration of electron
motion in crystal
1. The motivation
The response of electron in crystal to external
electromagnetic fields can be regarded as a classical free
electron with effective mass m *
2. Average velocity of electron motion

  1
v ( k )  k E ( k )

3.
The variation of state k in external electromagnetic
fields

dk
  

 e[  v  B]
dt

quasi-momentum: k
4. acceleration and effective mass
(1) 1D case
dv 1  2 E
 2 2 F ， F  m*a
dt  k
1
1 2E

m*  2 k 2
17
effective mass m * : includes both the inertial mass and
the effect of periodic potential
effective mass of electrons at the bottom and top of
energy band：
k
Kn
K
(1   )  k0   k ， k '   n (1   )  k0   k
2
2
（ k0 
Kn
K
，  k  n   k0  ）
2
2
2
2
E  E min  * （ k）
2mbottom
*
at the bottom： mbottom

m
0
 2 k02
1
m Vn
2
2
E  Emax  * （ k）
2mtop
*
at the top： mtop

m
0
 2 k02
1
m Vn
(2) 3D case
2
 1   E  
v 2
F
  k k  
1
1
reciprocal effective mass tensor：    2
 m * 
18
 2E 



k

k




it can be diagonalized if properly choosing axis:
1
 m*
m*
(3) summary
 m * is related to energy band structure
 m * is related to symmetries in crystal
 m * can be positive or negative

dv  1 
dk 

  F ， 
F
dt  m * 
dt
19
§4.9
Conductor,
insulator
semiconductor
and
1. Introduction
Although there are many electrons in solids, some of
them have good electric conductivity, some are less and
others do not have any conductivity.
2. The classification of energy bands
 filled bands: all the ２ N states are occupied by
electrons
 empty bands: no state in the ２ N states
 conduction band ： the ２ N states are partially
occupied
3. The conducting behaviors of energy bands
(1) when there is no external field (   0 )
all the three kinds of energy bands remains neutral,
no current
(2) when an external field is applied (   0 )
electrons in filled bands: no current
electrons in unfilled bands: induce current
20
4. The band structures of conductor, semiconductor and
insulator
In addition to a series of occupied bands in a
conductor, there are partially occupied energy bands
which can induce current flow. These partially occupied
bands are the so-called conduction bands.
In a non-conductor, a series of lower energy bands
are exactly occupied by electrons and the higher energy
bands are all empty. Since no current can be caused by
filled bands, it’s not conducting although there are many
electrons.
Both
semiconductor
and
insulator
belong
to
non-conductor. The difference is that a semiconductor
has small band gap (< 2eV) while an insulator has a large
gap.
5. Some examples
 alkali metals
 alkaline earth metals
 elements in column IV
21
6. Nearly filled bands and holes
(1) The idea

 
j  ev (k )

when a k state is empty, the total current of energy
bands is as if caused by a positive charge e with

velocity the same as that of electron in the k state.
Such empty state is called hole.
(2) Hole motion in electromagnetic fields

dv
e   
  * (  v  B )
dt
me
let mh*  me*  0

dv
e   
 * (  v  B )
dt mh
The hole can be regarded as a particle with positive
effective mass m h* and positive charge e .
(3) the momentum and energy of holes
It is generally recognized that holes and electrons
 1

obey the same classical laws by v (k )  k E (k ) and

1 dk 
F
 dt



the quasi-momentum: kh  ke


the energy: E (kh )   E (ke )
22
Homework
P136 1，2
P164 4，7
Addition：在低温下，金属钾的摩尔热容量的实验结果
可以写成
c  (2.08 T  2.57 T 3 ) 毫焦/摩尔·开
若１摩尔钾有 N  61023 个电子，试求钾的费米温度和
德拜温度。
23