Download Free Electron theory :Quantum Mechanical Treatment

Free Electron theory :Quantum
Mechanical Treatment
Basic Assumption : A metal crystal consists of positive ions whose valence electrons are
free to move between the ions as if they constituted an electron gas
Density of Available states D(E) and Density of filled States N(E)
D(E) : total number of available electron states per unit energy range at E
N(E):
Number of
range at E
states that are filled with electrons per unit energy
Density of Available states D(E) and Density of filled States N(E)
Parameter of free electron gas at absolute zero
Parameter of free electron gas at absolute zero
Parameter of free electron gas at absolute zero
AVERAGE VELOCITY AND VELOCITY OF ELECTRON AT FERMI LEVEL
Average Velocity
Velocity at Fermi Level
3
1
E0  EF  mv 02
5
2
 2  3 2 N 


EF 
2m  V 
 2 E0 
So v 0  

m


1
2
1 2
 mv F
2
1

2
v F  (3 n) 3
m
2/3
N

n  V 


Fermi Energy at temperature T is given by
  2 kT 2 
EF  EF 0 1 
(
) 
 12 EFo 
So fermi energy is not constant but decreases as temperature rises.
Above expression is applicable only when kT<<EF .
Electrical Conductivity
On the basis of free electron model, electrons at 0 K fill a sphere of radius kFo
in the wave number space kFo is known as Fermi wave vector. Fermi surface is
the surface of maximum energy.
On application of field E, each electron acquires a certain additional velocity
dv. This is equivalent to the displacement of the Fermi Sphere by dk.
Accordinglly, F  eE   dk  m d
dt
dt
Ee
d  
dt
m
This additional velocity is acquired
in the characteristic time 
 e
dv 
E
m
Current density J  nedv
ne 2

E
m
ne 2
But J  E 
E
m
Hence electrical conductivi ty
ne 2

m
Mean free path Fo  vFo , where vFo is the
velocity at fermi surface,

vFo   (3 2 n)1/ 3
m
Electrical Conductivity
Hence electrical conductivi ty
ne 2

m
Mean free path
Fo  vFo ,
where vFo is the
velocity at fermi surface,
ne 2 Fo

mvFo
Since only quantity on the RHS which depends upon temperature is F
F 
1
T
Thus

1
T
Electrical Conductivity
The kinetic energy of the electron is
1
3
mc 2  kT
2
2
3kT
and c 
m
At room temperature , the drift velocity imparted to the electrons by
the applied electric field is very much smaller than the average
thermal velocity. The average distance travelled by an electron
between two successive collisions in the presence of applied field is
known as free mean path . The time taken by an electron between
two successive collisions is known as mean collision time of the
electron c.
Hence the time taken by the electrons in traversing the distance  will
be decided mainly by rms velocity.
Now  c 

c

m
3kT
Mobility
• Mobility of the electron μ is defined as the steady state drift
velocity<vd> per unit electric field.
d
e


E
m
ne 2
e

 ne.
m
m
  ne
1
m
m
 
 2
 ne ne 
Where  (resistivit y )
ne ne 2


m
m
•The electrical conductivity σ depends on two factors ,the charge
density n and their mobility. These two quantities depend on
temperature.
•In metals n is constant and μ decreases slightly with
temperature and hence with increase of temperature ,the
conductivity decreases.
•In semiconductors the exponential increase of n with
temperature is responsible for increase of conductivity with
temperature.
• In insulator n remains constant and above certain temperature
μ increase exponentially resulting in dielectric breakdown.
For the thermal conductivity of a metal in terms of temperature.
Relaxation Time
Relaxation Time can be defined as the time taken for the drift
velocity to decay to 1/e of its initial value
Let assume that the applied field is cut off after the drift velocity of
the electron has reached its steady value. Drift velocity after this
instant is governed by
d d

 m
dt

d d
dt

m
d

 d (t )   d (0) exp( t /  )
at t  
 (0)
 d (t )  d
e
Differenti ating equation

d
 d (t )   d
dt

Vd(0)is the steady state drift velocity
Collision time
vd  0
the change in average velocity on collision is opposite. Hence the rate of change of
average velocity is given by
Q.1/Tut 9
The relaxation time and root mean square velocity of the electron
at room temperature are 2.5x10-14 s and 1x105 m/s. Calculate the
value of mean free path of the electron.
Q.2/Tut 9
The resistivity of a metal at temperature 20°C is 1.69 x 10-8 ohm m
and concentration of the free electrons in metal, ne = 8.5x1028/m3.
Calculate root mean square velocity (c), relaxation time (τ), mean
free path (λ), mobility of electrons (μ) and value of electrical
conductivity (σ) on the basis of classical free electron theory.
Semiconductor
For intrinsic semiconductors like silicon and germanium, the Fermi level
is essentially halfway between the valence and conduction bands.
Although no conduction occurs at 0 K, at higher temperatures a finite
number of electrons can reach the conduction band and provide some
current.