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Chap 6 Electron motion in the crystal
Objectives
At the end of this Chapter, you should:
1.Be able to identify a wave packet in a crystal.
2. Be able to calculate the effective mass at different points in
the band.
3. Understand the Electron quasi-momentum k .
4. Understand the difference of electron motion in k space and
real space.
5. Understand the concept of the hole, and understand its
motion behavior.
When I started to think about it, I felt that the main problem
was to explain how the electrons could sneak by all the ions in
a metal. To my delight，by straight Fourier analysis, I found
that the wave differed from the plane wave of free electrons
only by a periodic modulation
F. BLOCH
When an external field (electric and magnetic fields, etc.) is applied
to the crystal, the electrons in the crystal does not just feel the role
of the external field, but also feel the role of the crystal in a
periodic potential. Normally, the external field is much weaker than
the crystal periodic potential. Because the periodic field intensity is
generally 108 V/cm, very difficult for the external electric field to
achieve such magnitude. Therefore, the movement of electrons in
the external field in crystal must be discussed based on the
eigenstate basis of the periodic field.
6.1 The quasi-classical description of Bloch electrons：
There are two methods used:
Solving the single-electron wave equation in the presence of the
external field.
Under certain conditions, solving the electronic movement with
external field as a quasi-classical particle.
Wave equation with external field
 2 2

  2m   U  r   V   E


Typically, we can only approximately solve the wave equation
in the presence of the external field. Very hard.
Another method could be considered if：
The field is weak and constant.
Neglect the electronic transitions between different energy
bands.
Neglect the electron diffraction, interference and collision.
This method can be simply used and show a clear physical image.
We are willing to adopt it.
Classical particle have defined energy and momentum, however, for
microscopic particles governed by quantum mechanics, it is valid.
Wave packet is often used for the description of quantum state
classically, which means that the particles is spatially distributed at
r0 with uncertainty of △r. The momentum is k0 with uncertainty
of k . r0 is treated as the centre of the particle, rk results from
the uncertainty princible.
Wave packet and electron velocity:
In crystal, the quasi-classical movement of e- can be described
by wave packet composed of Bloch functions. There are
eigenstates with different energy in the wave packet. So, time
factor should be considered for the Bloch functions.
Firstly, consider the 1D case, assume the wave packet is
composed of wave functions in the range of k centered at k0.
k is small and we can think that:
uk  x   uk0  x 
independent of k .
For defined k , the Bloch function with time factor is,
 k  x, t   eikxt uk  x 
 k   E k  /
the neighboring k’ states will be mixed with quantum states k0:
Wavelet packet
  x, t   
k0  2k
k0  2k
e
i  kx t 
uk  x  dk
k0  2k
i  kx t 
 uk0  x  
k0  2k
set
dk
 d 
  k   0  
 
 dk k0
k  k0  
  x, t   uk0  x  e
 uk0  x  e
e
uk  x   uk0  x 
i  k0 x 0t 
i  k0 x 0t 

   d   

 2k exp i  x   dk k t    d
0 
 

k
2

2sin


 x   d  t 
dk k0 

x   ddk k t
k
2
0
For the analysis of wave packet movement, only need to analyse
2 (that is, the probability distribution).
2
  x, t 
2
 sin k  x   d  t  
2 
2 
dk k0  
2
 uk0  x  
k 


 2k  x   ddk k0 t  
 d 
set w  x  
 t
 dk k0
sin 2k w
k
2 w
2
The wave function centered
2
in the range of
,
k
The center of wave packet
is : w＝0.

2
k
0
2
k
w
1  dE 
 d 
There is x  
 t 
 t
 dk k0
 dk k0
E k    k 
If the wave packet is a quasi-particle, the particle velocity is
dx 1  dE 
v  k0  
 

dt
d
k

 k0
The width of Brillouin zone: 2/a ，assume that k is small,
generally require
2
2
that is
 a
k 
k
a
For 3D case, the velocity of electron is:
v
1
k E
For the dispersion of Bloch wave, a stable wave packet requires
that the wave vector range △k should be very small. k  2
a
Considering the uncertainty relation
px  x  k x  x 
x  a
2
suggesting that if the wave packet size is much larger than the cell
size, the movement of electrons can be described in terms of wave
packet. For transportation, only when the electron mean free path is
much larger than the cell size, the electrons in the crystal can be
solved as a quasi-classical particle. The motion speed of wave
packet (group velocity) is equal to the average velocity of the
particles in the center of wave packet .
important fact 1:
The average velocity of electrons are fully determined by energy
and wave vector, which is independent of time and space.
Therefore, average velocity will keep unchanged without
attenuation. This means: the electrons can not be scattered by
static atoms. The resistance of a strictly periodic crystal is zero.
This point is entirely different from the concept of free electron
theory: ion (scattering centers) will affect the average velocity
(drift) of the electrons.
In other words: If the electron is in a certain state  k , as long as
the lattice periodicity is kept unchanged, the electrons will be in
constant motion throughout the crystal with the same velocity.
Because the lattice influence on the propagation velocity, have
been included in the definition of energy.
important fact 2:
the direction of electron velocity in k space is the direction of
the energy gradient, ie. perpendicular to the energy surface.
Therefore, the direction of motion of electrons is determined by
the shape of the energy surface, under normal circumstances, in
k space, the surface is not spherical. therefore, generally the
direction of v is not k direction. The following figure is a more
accurate reflection of the Bloch electrons.
when the energy surface is a sphere, or in some special direction, v
and k are in the same direction. For the 1D case,
At the bottom and top of the band, E(k) reaches its extremity:
dE
0
So, v＝0.
dk
at a certain point of the band,
The electron velocity is
maximum, which
completely different from
the fact that the velocity of
free electrons is always
monotonically increase
with the energy.
d2E
0
2
dk
E(k)
v(k)
NEF model explanation:
at the FBZcenter, the electron can be approximately regarded as the
plane wave, v is proportional to the k. With the increase of k, it is
around the border of the FBZ, and the influence of lattice scattering
increase. At the Brillouin zone boundary, a strong Bragg reflection
make the scattered wave and incident wave equal, so the wave
velocity is zero. This result can be applied to all the radiated wave
propagation in a periodic crystal.
6.2. Electron quasi-momentum k ：
Set the external field force on electron is F. In dt time, the work of
external field on the electrons is Fv dt
According to the principle of work and power
F  vdt  dE  k E  dk
dk 

1
F 
v  0
v  k E
dt 

In parallel direction of v, dk/dt and the component of F are
equal; when F is perpendicular to velocity v, the principle of
work and power can not be used to discuss the changes in the
electron energy state, but we can still prove that in the direction
perpendicular to the speed, dk/dt and the component of F is
also equal.
dk
F 
dt
k is the electron quasi-momentum, not the real momentum of
Bloch electrons in a strict way. The strict meaning of the rate of
momentum change is the force exerted on electrons, while the rate
of change of quasi-momentum is only determined by the external
field force, here, the potential field of lattice force is not included,
which has been embodied in the quasi-momentum.
For free electrons, k=p/
is the electron momentum.
For Bloch electrons
 

  ikr
 (r )  e  k  (r )
i
i




ik  r
 nk  (e unk (r ))
i
i



ik  r 
 k  nk  e
unk (r )
i
Bloch wave is not the eigen function of momentum. In crystal
periodic field, k is the expanded concept of momentum, quasimomentum or electron-lattice momentum.
6.3 The accelerated velocity and effective mass
the external field is dealt in a way of the combination of classic
way and the band theory.The basic relationship of quasi-classical
motion is:
dr
1
  n ( k )   k En ( k )
dt
dk
 F  e  E ( r , t )   n (k )  B ( r , t ) 
Newton law
dt
In addition, assume that the band index n is a constant, that is,
ignoring the electron transitions between bands.
Based on the above electron motion equation, we can with ease
get the electron acceleration.
a）1D case
dv d  1 dE  1 dk d 2 E
a
  
 2 

dt dt  dk 
dt dk
the effective mass
m 
2
d2 E
dk 2
F
 
d2 E
dk 2
2
F  m
dv
dt
Due to the periodic field, the acceleration form is obviously not consistent
with the electron inertial mass
In periodic field, the function between E(k) and k is not a parabola,
therefore, m* is not constant, dependent of E and k.
At the band bottom, E(k), minimum
At the band top
E(k), maximam
d2E
0
2
dk
d2E
0
2
dk
m*>0；
m*<0 .
b) 3D case
dv d  1
 1 dk
a
  k E  
 kk E
dt dt 
dt

in component form, written as
dv d  1 E
a 
 
dt dt  k
 1 3 dk 

 
k
 1 dt

2E
 2  F 
k k 
 1
1
3
＝1, 2, 3
 E 



k
 
The matrix form
 2E

2

k
x

 vx 
 2E
1
 

v

y
2
 
 k y k x
v 
 z
 2
  E
 k z k x

2E
k x k y
2E
k y2
2E
k z k y
Compared with Newton's laws
2E 

k x k z 
 Fx 
2

 E  
 Fy
k y k z   
  Fz 
2
 E 
k z2 
1
v F
m
1
Here, we substitute
with a second-order tensor
m
 2E

2

k
x

 2
 1  1   E
 m   2  k k
y
x
 2
  E
 k z k x

2E
k x k y
2E
k y2
2E
k z k y
2E 

k x k z 
2E 

k y k z  effective mass tensor

2
 E 
k z2 
Which is a symmetric tensor. At the same time, the point group
symmetry of the crystal can reduce the tensor. For isotropic
crystal, it degenerates into a scalar.
The role of the effective mass is to embody the periodic field, so
that we can determine the acceleration of electrons only by the
external field force.
taking kx、ky、kz as the major axis of the tensor, the matrix
can be diagonalized as:
 2E
 2
 k x
 1  1 
 m   2  0


 0

where
0
2E
k y2
0
  1
0   
  mx
 
0  0
 

2E  
 0
2
k z  
dv y
1
dv x
1
  Fy ,
  Fx ,
dt m y
dt
mx
0
1
my
0

0 


0 

1 


mz 
dv z
1
  Fz
dt m z
The direction of the acceleration is not necessarily consistent with
the direction of the external field force, which is determined by the
inverted effective mass tensor. The electron effective mass commonly
can be experimentally obtained by the electronic specific heat:
 exp m*

0
m
0 is the free electronic specific heat coefficient, exp is experimental
value.
Example: seeking electron effective mass of s states for
simple cubic crystal .
E  k    s  J 0  2 J1  cos k x a  cos k y a  cos k z a 
2a 2 J1 cos k a

2E

, 1, 2, 3
k k  

0
kx , ky, kz is the direction of principle axis of tensor, so,

x
m 

y
m 
mz 
2
2 E
k x2
2
2E
k 2y
2
2 E
kz2



2
2
2a J1
2
2
2a J 1
2
2a 2 J1
cos kx a 
 cos k a 
1
1
y
 cos kz a 
1
The principle component of effective mass are inversely proportional
to J1. the bigger interatomic distance, the smaller the J1, thus, the
bigger the effective mass.
2
At the bottom point of energy 
mx  my  mz  m  2  0
band : k = (0, 0, 0)，
2a J 1
The effective mass tensor degraded to be a scalar.
mx 0
0
2




m    0 mx 0   2  0
2a J 1

 0

0 mx 
 

,
At the band top R: k   ,

a
a
a


mx  my  mz  m  
2
2
2a J 1
0
At the bottom and the top of the band, the electron effective
mass is isotropic, and is a scalar, due to the cubic symmetry.
At X point：


k   , 0, 0 
a

mx  
2
2
2a J 1
 0,
my  mz 
2
2
2a J 1
0
Normally, effective mass is a tensor. In exceptional circumstances,
it can degrade to be a scalar (positive or negative)
Near the bottom of the band, effective mass is always positive. Near
the top of the band, effective mass is always negative. At the band
bottom and top, E (k) take the maximum and minimum values​​,
respectively, with positive and negative divalent derivative.
Free electrons
Quantum
number
k
(unlimited quantized
value）
2
energy
Velocity
k2
E
2m
k
v
m
Bloch electrons
n , k (value in the first Brillouin zone)
Generally there is no simple forms
En  k 
 1

 n (k )   k En (k )

Bloch wave
Wave
function
1 ik r
k 
e
V
Plane wave function
 n,k  eikrun,k  r 
un ,k  r  uncertain simple form，
un,k  r   un,k  r  Rn 