Fermi dirac distribution

Presented by-
Md Ahsan Halimi
Scholar No: 19-3-04-105
Dept. of ECE, NIT Silchar
“Introduction of Fermi Dirac Distribution Function”
1

Contents
Some Basic Concept
Fermi-statistics and Bose Statistics
Postulates of Fermi particles
Fermi Dirac Distribution Function
Conclusion
References
2

Some basic concepts
 Fermi level :- Fermi level is the highest energy state occupied by
electrons in a material at absolute zero temperature.
 Fermi energy:-This is the maximum energy that an electron can
have at 0K. i.e. the energy of fastest moving electron at 0K. It is
given by,
𝐸 𝐹 =
1
2
𝑚𝑣 𝐹
2
 Fermi velocity (𝑣 𝐹):- It is the velocity of electron at Fermi level.
 The band theory of solids gives the picture that there is a sizable
gap between the Fermi level and the conduction band of the
semiconductor. At higher temperatures, a larger fraction of the
electrons can bridge this gap and participate in electrical
conduction.
3

Fermi-statistics and Bose Statistics
 The wave function of a system of identical particles must be either
symmetrical (Bose) or antisymmetrical (Fermi) in permutation of a
particle of the particle coordinates (including spin). It means that there
can be only the following two cases:
1. Fermi-Dirac Distribution
2. Bose-Einstein Distribution
 The differences between the two cases are determined by the nature of
particle. Particles which follow Fermi-statistics are called Fermi-
particles (Fermions) and those which follow Bose-statistics are called
Bose- particles (Bosones).
 Electrons, positrons, protons and neutrons are Fermi-particles, whereas
photons are Bosons. Fermion has a spin 1/2 and boson has integral spin.
Let us consider this two types of statistics consequently.
4

Different types of systems considered
 Distinguishable particles >(Fermions when spin is not
considered)
 Indistinguishable particles that obey Pauli exclusion principle
> (Fermions)
 Indistinguishable particles that doesn't obey Pauli exclusion principle
>(Bosons)
5

Postulates of Fermi Particle
 Particles are indistinguishable.
 Particles obey Pauli principle.
 Each quantum state can have only one particle.
 Each particle has one half spin.
 𝒈𝒊 be the quantum states associated with 𝒊 𝒕𝒉
energy
level.
 𝑵𝒊 is the no. of particles associated with 𝒊 𝒕𝒉 energy level.
 For a particular value of N, there is only one
distribution
6
N2 NnN1 ……………….

Fermi -Dirac distribution function
(Derivation)
Consider now the ith energy level with degeneracy gi. For this level,
the total no. of ways of arranging the particles is:
Consider all energy level, the permutation among themselves Now
the Ni particles can have Ni! Permutations
We now apply, the other two assumptions, namely conservation of
particles and energy.
7
)!(
!
)1)......(2)(1(
ii
i
iiiii
Ng
g
Ngggg


 

n
i iii
i
n
NgN
g
NNNNQSo
1
321
)!(!
!
),.......,,(,
constUEN
constNN
i
i
i
i
i





Contd…
 Stirling approximation (x>>1)
 Lagrangian multiplier method for lnQ
 Now we proceed in the standard fashion, by applying Stirling’s
approximation to lnQ, and then using the method of Lagrange
multipliers to maximize Q.
8
0ln
11









 
n
i
ii
j
n
i
i
jj
NE
N
N
N
Q
N

XXXX  ln!ln
𝑔𝑗
𝑁𝑗
= 1 + 𝑒−(𝛼+𝛽𝐸 𝑗)
𝑁𝑗
𝑔𝑗
=
1
1 + 𝑒−(𝛼+𝛽𝐸 𝑗)

Contd…
For i=j,
𝑁𝑖
𝑔𝑖
=
1
1 + 𝑒−(𝛼+𝛽𝐸 𝑖)
=
1
1 + 𝑒(𝐸 𝑖−𝐸 𝐹)/𝑘𝑇
;
𝛼 =
𝐸 𝐹
𝑘𝑇
, 𝛽 = −
1
𝑘𝑇
And because energy level is continuous,
𝑁 𝐸 𝑑𝐸 =
𝑔 𝐸 𝑑𝐸
1 + 𝑒(𝐸−𝐸 𝐹)/𝑘𝑇
 g(E)dEis the number of available states in the energy range E and E+dE
 Number of particles between E and E+dE is given by
N(E)dE=f(E)*g(E)dE
 f(E) is the probability that a state at energy E is occupied by a particle
𝑓 𝐸 =
𝑁 𝐸
𝑔(𝐸)
=
1
1 + 𝑒(𝐸−𝐸 𝐹) 𝑘𝑇
9

Contd…
 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. Whereas (1- f(E))
gives the probability that energy state E will be occupied by a
hole.
 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.
10

12
Fermi-Dirac distribution: Consider T  0 K
For E > EF :
For E < EF :
0
)(exp1
1
)( F 

 EEf
1
)(exp1
1
)( F 

 EEf
E
EF
0 1 f(E)

13
Classical limit
For sufficiently large  we will have (-)/kT>>1, and in this limit
kT/)(
e)f( 
 
 (5.47)
This is just the Boltzmann distribution. The high-energy tail of the Fermi-Dirac
distribution is similar to the Boltzmann distribution. The condition for the
approximate validity of the Boltzmann distribution for all energies  0 is that
1 kT/
e  (5.48)

Fermi -Dirac distribution function:-
14
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Fermi Dirac Distribution function
Energy (eV)
FermiDiracDistributionfunction
T1=50 K
T2=100 K
T3=300 K
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Fermi Dirac Distribution function of particle density with Energy
Energy (eV)
FermiDiracDistributionfunction
T1=50 K
T2=100 K
T3=300 K

References
1. Statistical Physics (2nd Edition), F. Mandl, Manchester
Physics, John Wiley & Sons, 2008,
ISBN 9780471915331.
2. H.J.W. Muller-Kirsten, Basics of Statistical Physics,
2nd ed., World Scientific, ISBN: 978-981-4449-53-3.
15

Fermi dirac distribution

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