EEE 315 - Electrical
Properties of Materials
Lecture 7
Carrier distribution functions
Of particular interest is the probability
density function of electrons, called the
Fermi function.
Maxwell-Boltzmann distribution is also
provided.
Fermi-Dirac distribution
function
The Fermi-Dirac distribution function,
also called Fermi function, provides the
probability of occupancy of energy
levels by Fermions.
Fermions are half-integer spin particles,
which obey the Pauli exclusion principle.
The Pauli exclusion principle postulates
that only one Fermion can occupy a
single quantum state.
Fermi level
Density of states tells us how many states exist at a given energy E.
The Fermi function f(E) specifies how many of the existing states at
the energy E will be filled with electrons. The function f(E)
specifies, under equilibrium conditions, the probability that an
available state at an energy E will be occupied by an electron. It is a
probability distribution function.
EF = Fermi energy or Fermi level
k = Boltzmann constant = 1.38 1023 J/K
= 8.6  105 eV/K
T = absolute temperature in K
Fermi-Dirac distribution: Consider T  0 K
For E > EF :
f ( E  EF ) 
1
 0
1  exp ()
For E < EF :
f ( E  EF ) 
1
1
1  exp ()
E
EF
0
1
f(E)
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Fermi-Dirac distribution: Consider T > 0 K
If E = EF then f(EF) = ½
E  EF  3kT
If
then
 E  EF 
exp 
  1
 kT 
Thus the following approximation is valid:
  ( E  EF ) 
f ( E )  exp 

kT


i.e., most states at energies 3kT above EF are empty.
If
E  EF  3kT
then
 E  EF 
exp 
  1
 kT 
 E  EF 
f ( E )  1  exp 

 kT 
So, 1f(E) = Probability that a state is empty, decays to zero.
Thus the following approximation is valid:
So, most states will be filled.
kT (at 300 K) = 0.025eV, Eg(Si) = 1.1eV, so 3kT is very small
in comparison.
6
Temperature dependence of Fermi-Dirac distribution
7
1.0
0K
50 K
100 K
150 K
200 K
250 K
300 K
350 K
400 K
0.8
F(E)
0.6
0.4
0.2
0.0
-0.20
-0.15
-0.10
-0.05
0.00
E-EF
0.05
0.10
0.15
0.20
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The Maxwell Boltzmann
distribution
The Maxwell Boltzmann applies to noninteracting particles, which can be distinguished
from each other. This distribution function is
also called the classical distribution function
since it provides the probability of occupancy for
non-interacting particles at low densities. Atoms
in an ideal gas form a typical example of such
particles. The Maxwell-Boltzmann distribution
function is given by:
A plot of the three distribution functions,
the Fermi-Dirac distribution, the MaxwellBoltzmann distribution and the BoseEinstein distribution is shown in Figure
below.
Probability of occupancy versus energy of the Fermi-Dirac, the
Bose-Einstein and the Maxwell-Boltzmann distribution.
Fermi energy and average energy of
electrons
 The free fermions that occupy the
lowest energy states form a sphere in
momentum space. The surface of this
sphere is the Fermi surface.
 The total energy of a Fermi sphere of
fermions is given by
Where,
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