Fermi Dirac Statistics - Civil Engineering (CE) PDF Download

Fermi-Dirac Statistics

Fermi energy:
According to quantum free electron theory, free electron energies are quantised. Thus if there are N free electrons there exists N closely spaced energy levels. Since electrons are distributed among these energy levels according to Paulis exclusion principle. At absolute zero temperature, two electrons with least energy having opposite spins occupy lowest available energy level. The next two electrons with opposite spins occupy next energy level. And so on. In this way N/2 energy levels are occupied, and remaining N/2 energy levels are vacant. The top most energy level occupied at absolute zero temperature is called Fermi Energy level. The energy corresponding to that energy level is called Fermi Energy. Fermi Energy can be defined as energy of that energy level below which all energy levels are completely occupied and above which all energy levels completely empty. It is denoted by E_F Thus Fermi energy represents maximum energy that electrons can have at absolute zero temperature.

Fermi factor:
We know that at T=0 K, all energy levels below E_F are completely filled and all energy

levels above E_F are completely empty. As the temperature is increased, some of the electrons absorb energy and start migrating to energy level above E_F . As a result there exists some vacant energy levels below E_F and filled energy levels above E_F . Now it becomes difficult to say whether a particular energy level is occupied or vacant. The probability of occupation of any energy level is given by a mathematical function

f(E)is called Fermi factor. It is defined as follows.
Fermi factor is a function, which gives the probability of occupation of an energy level E at the given temperature T for a material in thermal equilibrium. 

3.10 The dependence of Fermi factor on Energy and Temperature
The variation of Fermi factor with Energy for various temperatures can be discussed as follows
(i) At absolute zero temperature (T=0):
(a) When T=0 and E < E_F

∴ since f(E)=1, at T=0, all energy levels below Fermi level are occupied.

(b) When T=0 and E > E_F

∴ At T=0 all the energy levels above Fermi energy level are vacant.
The graph of f(E) against E at T=0 is as shown in the fig
(ii) For temperature T > 0;
(a) When energy E < E_F

∴ f(E) is less than 1 but greater than 1/2
(b) When energy E > E_F

∴ f(E) = 1/2

The curve obtained is as shown in the fig.
It is clear from the above discussion that irrespective of the temperature the probability of occupation of the Fermi energy is always 1/2.
Therefore Fermi energy is considered as the most probable energy that electron possess in the solid.

Density of States
The density of states is defined as the number of energy levels available per unit volume per unit energy centered at E. It is denoted by g(E). The product g(E)dE gives the number of states per unit volume between the energy levels E and E+dE. Then the number of electrons per unit volume having energies between E and E+dE is given by
N(E)dE = g(E)dE × f(E)
Where f(E) is Fermi factor.
Expression for density of states:
The expression for the density of state is given by

The above equation is called density of energy states.The graph of g(E) against E is as shown in the fig

Expression for Fermi Energy at T=0K: Number of electrons per unit volume having energies between E and E+dE is given by
N(E)dE = g(E)dE × f(E)

At T=0 only energy levels up to E_F are filled.Therefore f(E)=1 for E < E_F

Therefore the number of electrons per unit volume is given by

Fermi energy at T > 0K :
The Fermi energy at any temperature in general is given by

where E_F0 represents Fermi energy at T=0 K
Except at very high temperature it can be shown that
E_F = E_F0

Fermi Temperature,(TF ):
The temperature at which the average thermal energy of the free electron in a solid becomes equal to the Fermi energy at T=0K is called Fermi temperature. Since thermal energy of an electron is given by kT.
When T = TF

E_F = kT_F 
∴ T_F = E_F/k

Fermi Velocity,(VF ):
The velocity of the electrons whose energy is equal to Fermi energy is called Fermi velocity.

The Fermi-Dirac Distribution
The Fermi-Dirac Distribution gives the distribution of electrons among the permitted energies E and E+dE. It is given by
N(E)dE = g(E)dE × f(E)

where g(E)dE is called density of states given by

and f(E) is called Fermi factor given by

The graph of N(E)dE against E gives the distribution of electrons among the available energy levels in the solid at the given temperature. The distribution is known as Fermi-Dirac Distribution. The graph representing the distribution is shown below.
It is clear from the graph that N(E) increase with the temperature becomes maximum for

E = E_F and then decreases to zero for the energies E > E_F , at the temperature T=0 K. At higher temperatures some of the electrons occupying energy levels below Fermi energy absorb energy and move to higher energy levels above E_F . It is clear from the graph that even at very high temperatures only few electrons occupying energy levels closer to Fermi energy absorb energy and move to higher energy levels above Fermi energy. Electrons at lower energy levels are not at all disturbed.