Derivation of the Fermi-Dirac distribution function


We start from a series of possible energies, labeled Ei. At each energy we can have gi possible states and the number of states that are occupied equals gifi, where fi is the probability of occupying a state at energy Ei.

The numer of possible ways - called configurations - to fit gi fi electrons in gi states, given the restriction that only one electron can occupy each state, equals:

This equation is obtained by numbering the individual states and exchanging the states rather than the electrons. This yields a total number of gi! possible configurations. However since the empty states are all identical, we need to divide by the number of permutations between the empty states, as all permutations can not be distinghuished and can therefore only be counted once. In addition, all the filled states are indistinguishable from each other, so we need to divide also by all permutations between the filled states, namely gifi!.

The number of possible ways to fit the electrons in the number of avalable states is called the multiplicity function.

The multiplicity function for the whole system is the product of the multiplicity functions for each energy Ei

Using Stirling’s approximation, one can eliminate the factorial signs, yielding:

The total number of electrons in the system equals N and the total energy of those N electrons equals E. These system parameters are related to the number of states at each energy, gi, and the probability of occupancy of each state, fi, by:

and

According to the basic assumption of statistical thermodynamics, all possible configurations are equally probable. The multiplicity function provides the number of configurations for a specific set of occupancy probabilities, fi. The multiplicity function sharply peaks at the thermal equilibrium distribution. The occupancy probability in thermal equilibrium is therefore obtained by finding the maximum of the multiplicity function, W, while keeping the total energy and the number of electrons constant.

For convenience, we maximize the logarithm of the multiplicity function instead of the multiplicity function itself. According to the Lagrange method of undetermined multipliers, we must maximize the following function:

where a and b need to be determined. The maximum multiplicity function is obtained from:

which can be solved, yielding:

or

which can be written in the following form

with b = 1/b and EF = -a/b. The symbol EF was chosen since this constant has units of energy and will be the constant associated with this probability distribution.

Taking the derivative of the total energy, one obtains:

Using the Lagrange equation, this can be rewritten as:

Any variation of the energies, Ei, can only be caused by a change in volume, so that the middle term can be linked to a volume variation dV.

Comparing this to the thermodynamic identity:

one finds that b = kT and S = k lnW . The energy, EF, equals the energy associated with the particles, m.

The comparison also identifies the entropy, S, as being the logarithm of the multiplicity function, W, multiplied with Boltzmann’s constant.

The Fermi-Dirac distribution function then becomes:


©. B. Van Zeghbroeck, 1998