Download FREE ELECTRON THEORY


Equation of motion of an electron with an applied
electric and magnetic field.
dv
me
 eE  ev  B
dt


This is just Newton’s law for particles of mass me
and charge (-e).
The use of the classical equation of motion of a
particle to describe the behaviour of electrons in
plane wave states, which extend throughout the
crystal. A particle-like entity can be obtained by
superposing the plane wave states to form a
wavepacket.

The velocity of the wavepacket is the group
velocity of the waves. Thus
2
k2
E 
2me
d 1 dE
k
p
v



dk
d k me me

p k
So one can use equation of mdv/dt
 dv v 
me 
   eE  ev  B
 dt  

(*)
= mean free time between collisions. An electron
loses all its energy in time


In the absence of a magnetic field, the applied E
results a constant acceleration but this will not
cause a continuous increase in current. Since
electrons suffer collisions with
 phonons
 electrons
v
 The additional term me   cause the velocity v to
 
decay exponentially with a time constant
the applied E is removed.

when
The Electrical Conductivty

In the presence of DC field only, eq.(*) has the
steady state solution
e
v
E
me
e
e 
me
Mobility for
electron
a constant of
proportionality
(mobility)

Mobility determines how fast the charge carriers
move with an E.

Electrical current density, J
J  n (  e )v

e
v
E
me
N
n
V
Where n is the electron density and v is drift
velocity. Hence
ne 
J
E
me
2
Ohm’s law
J  E
ne 

me
2
Electrical conductivity
Electrical Resistivity and Resistance

1

R
L
A
Collisions



In a perfect crystal; the collisions of electrons are
with
thermally
excited
lattice
vibrations
(scattering of an electron by a phonon).
This
electron-phonon
scattering
gives
a
temperature dependent  ph (T ) collision time
which tends to infinity as T 0.
In real metal, the electrons also collide with
impurity
atoms,
vacancies
and
other
imperfections, this result in a finite scattering
time  0 even at T=0.

The total scattering rate for a slightly imperfect
crystal at finite temperature;
1
1
1


  ph (T )  0
Due to phonon

Due to imperfections
So the total resistivity ρ,
me
me
me
 


  I (T )   0
2
2
2
ne 
ne  ph (T )
ne  0
Ideal resistivity
Residual resistivity
This is known as Mattheisen’s rule and illustrated in
following figure for sodium specimen of different
purity.
Residual resistance ratio
Residual resistance ratio = room temp. resistivity/ residual resistivity
and it can be as high as
106
for highly purified single crystals.
impure
pure
Temperature

Collision time
 ( RT )sodium  2.0 x107 (  m)1


n  2.7 x10 m
28
pureNa
 5.3x1010 (  m) 1
can be found by taking
me  m
Taking
 residual
3
vF  1.1x106 m / s ;
and
m
ne2
2.6 x1014 s
at RT
7.0 x1011 s
at T=0
l  vF
l ( RT )  29nm
l (T  0)  77 m
These mean free paths are much longer than the interatomic
distances, confirming that the free electrons do not collide with the
atoms themselves.
Thermal conductivity, K
Due to the heat tranport by the conduction electrons
Kmetals >> Knon metals
Electrons coming from a hotter region of the metal carry
more thermal energy than those from a cooler region, resulting in a
net flow of heat. The thermal conductivity
1
K  CV vF l
3
where
CV
is the specific heat per unit volume
vF is the mean speed of electrons responsible for thermal conductivity
since only electron states within about
of  F change their
kBT
occupation as the temperature varies.
l is the mean free path; l  vF and Fermi energy  F  12 mevF2
2
2
2
1
1

N
T
2

nk
BT
K  CV vF2 
kB ( )
 F 
3
3 2 V
TF me
3me
where Cv 
2
T 
Nk B 

2
 TF 
Wiedemann-Franz law
ne 
 
me
2
K
 2 nk B2T
3me
The ratio of the electrical and thermal conductivities is independent of the
electron gas parameters;
K
 2  kB 
8
2


2.45
x
10
W

K
 
T
3  e 
2
Lorentz
number
K
L
 2.23x108W K 2
T
For copper at 0 C