This page provides a brief description of the Fermi Dirac statistics which are used in the theoretical analysis/modeling of semiconductor devices. This statistics applies to fermions such as electrons, with half integral spin. A derivation based on more general laws of statistical mechanics, involving the grand canonical ensemble is available here.

Suppose a quantum system has N energy levels and the degeneracy of the i-th level is $g_{i}$, i.e. the $i^{th}$ energy level has $g_{i}$ quantum states. By the Pauli Exclusion principle, at most one particle is allowed in each quantum state.

Consider the $i^{th}$ level. A simple combinatorial argument shows that the total number of ways of arranging $N_{i}$ particles in the $i^{th}$ level is given by

(1)But fermions are indistinguishable, so some arrangements is in fact equivalent. Since there are $N_{i}!$ such arrangements, we divide by this number to get the independent number of ways of distributing $N_{i}$ particles in the $i^{th}$ level as:

(2)The total number of ways of arranging $(N_{1},N_{2},\ldots,N_{n})$ particles is thus given by

(3)W is the total number of ways in which N electrons can be arranged in the system, where

(4)We intend to maximize W by finding the optimal n-tuple $N_{i}$. The maximum W corresponds to the most probable configuration. For this, we use the method of Lagrange Multipliers. (To be updated).

The result is

(5)Here, $E_{F}$ is the Fermi Energy.