Fermi-Dirac Statistics

FERMI-DIRAC STATISTICS

Fermi - Dirac statistics deals with the particles having half integral spin like electrons. They are known as Fermi particles or Fermions.

Definition

Fermi distribution function gives the distribution of electrons among the various energy levels as a function of temperature.

It is a probability function F(E) of an electron occupancy for a given energy level at absolute temperature.

It is given by

where

E - Energy of the level whose occupancy is being considered

E_F - Fermi energy level

k - Boltzmann's constant

T - Absolute temperature

The probability value F(E) lies between 0 and 1.

i. If F (E) = 1, the energy level is occupied by an electron.

ii. If F (E) = 0, the energy level is vacant ie., it is not occupied by the electron.

iii. If F (E) = 0.5 or 1/2 then there is a 50% chance for the electron occupying in that energy level.

ANNA UNIVERSITY SOLVED PROBLEM

Problem 2.3

Use Fermi distribution function to obtain the value of F(E) for E - E_F = 0.01 eV at 200 K. (A.U. May 2015)

Given data

E – E_F = 0.01 eV = 0.01 × 1.6 × 10^-19 J = 1.6 × 10^-21 J

Temperature T = 200 K

Boltzmann's constant k = 1.38 × 10^-23 JK^-1

Solution

We know that 

Substituting the given values, we have

Effect of Temperature on Fermi Function

The dependence of Fermi distribution function on temperature and its effect on the occupancy of energy level is shown in fig. 2.10 (a) and (b).

Case (i) Probability of occupation for E < E_F at T = 0 K

When T = 0 K and E < E_F then applying the values in the expression, we have

Thus at T = 0 K, there is 100% chance for the electrons to occupy the energy levels below Fermi energy level ie., all the energy levels are occupied by the electrons.

Case (ii) Probability of occupation for E > E_F at T = 0 K

When T = 0 K and E > E_F, then applying the values in the expression 

Thus, there is 0% chance for the electrons to occupy the energy level above Fermi energy level ie., all the energy levels above Fermi energy level are not occupied by the electrons (empty).

From the above two cases, at T = 0 K the variation of F (E) for different energy values become a step function as shown in fig 2.10(a).

Case (iii) Probability of occupation at ordinary temperature

At ordinary temperature, the value of the probability function starts reducing from 1 for energy values E slightly less than E_F

With the increase of temperature, i.e., T > 0 K, Fermi function F (E) varies with E as shown in fig. 2.10(b).

At any temperature other than 0 K and E = E_F

Hence, there is 50% chance for the electrons to occupy Fermi energy level ie., the value of F (E) becomes 1/2 at E = E_F

This result is used to define Fermi energy level.

Fermi energy level

i. It is the energy level at any finite temperature above 0 K at which the probability of electron occupation is 1/2 or 50%.

ii. It is also the energy level of maximum energy of wore the filled states at 0 K.

Further for E > E_F, the probability value falls off rapidly to zero (Fig 2.10(b)).

Case (iv) At high temperature

When kT >> E_F or T → ∞, the electrons lose their quantum mechanical character. Now, Fermi distribution function reduces to classical Boltzmann distribution.

Uses of Fermi distribution function

i. It gives the probability of an electron occupancy for a given energy level at a given temperature.

ii. It is very useful to find the number of free electrons per unit volume at a given temperature.