2.2.1.2 Distribution of Fermions

For a system of particles described by antisymmetric wave functions the Pauli exclusion principle implies that only one particle can be placed at each quantum state. To derive the distribution function the Gibbs distribution is applied to the subsystem of particles which are in the same quantum state. This is possible even in the presence of the exchange interaction because it only takes place inside the subsystem. Taking into account that the energy is equal to $n_{k}\epsilon_{k}$ the thermodynamic potential is given as:

$$\Omega_{k}=-k_{B}T\ln\sum_{n_{k}}\exp\biggl(\frac{\mu-\epsilon_{k}}{k_{B}T}\biggr)^{n_{k}},$$ (2.27)

where $k$ stands for all quantum numbers characterizing the particle state, $n_{k}$ is the number of particles in state $k$ and $\mu$ is the chemical potential of the system^2.6. According to the Pauli exclusion principle occupation numbers $n_{k}$ for fermions can only take values 0 or $1$. Thus the thermodynamic potential has the form:

$$\Omega_{k}=-k_{B}T\ln\biggl[1+\exp\biggl(\frac{\mu-\epsilon_{k}}{k_{B}T}\biggr)\biggr].$$ (2.28)

Since the average number of particles is defined through the derivative of the thermodynamic potential (2.28), with respect to the chemical potential $\mu$ by the expression:

$$\langle n_{k} \rangle = -\frac{\partial\Omega_{k}}{\partial\mu},$$ (2.29)

the expression for the average fermion number in state $k$ is:

$$\langle n_{k} \rangle = \frac{1}{\exp\bigl(\frac{\epsilon_{k}-\mu}{k_{B}T}\bigr)+1}.$$ (2.30)

It can be seen from (2.30) that all $n_{k}\leq 1$ and when $\exp\{(\mu-\epsilon_{k})/k_{B}T\}\ll 1$ the Boltzmann distribution function^2.7 is obtained. The normalization is obtained from the condition that the sum over all $\langle n_{k} \rangle$ is equal to the total number of particles $N$ in the system:

$$\sum_{k}\frac{1}{\exp\bigl(\frac{\epsilon_{k}-\mu}{k_{B}T}\bigr)+1}=N.$$ (2.31)

The normalization condition gives the chemical potential $\mu$ as an implicit function of the temperature $T$ and the total number of particles $N$ in the system.

For example, for the equilibrium electron gas in solids the number of electrons in the phase space element $d\vec{r}d\vec{k}$ can now be written as^2.8:

$$dN_{el}=\frac{2}{\exp\biggl(\frac{\epsilon(\vec{k})-\epsilon_{f}}{k_{B}T}\biggr)+1}\frac{d^{3}kd^{3}r}{(2\pi)^{3}}.$$ (2.32)

The physical meaning of the Fermi energy is that it is the largest electron energy at zero temperature. It is a boundary between occupied and free states. At non-zero temperatures electrons can also occupy states above the Fermi energy as shown in Fig. 2.4.

Figure 2.4: The Fermi-Dirac distribution at zero and finite temperatures.

S. Smirnov: