Fermi Dirac Statistics (Semiconductors)

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)
\begin{align} w_{i} = \Pi_{k = 1}^{N_{i}-1} (g_{k} - k) = \frac{g_{i}!}{(g_{i}-N_{i})!} \end{align}

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)
\begin{align} W_{i} = \frac{g_{i}!}{N_{i}!(g_{i}-N_{i})!} \end{align}

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

(3)
\begin{align} W = \Pi_{i = 1}^{n} W_{i} = \Pi_{i = 1}^{n}\frac{g_{i}!}{(g_{i}-N_{i})!} \end{align}

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

(4)
\begin{align} N = \sum_{i = 1}^{n} N_{i} \end{align}

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)
\begin{align} f_{F}(E) = \frac{N_{i}(E)}{g(E)} = \frac{1}{1 + exp\left(\frac{E-E_{F}}{kT}\right)} \end{align}

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

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