Fermi–Dirac statistics

Fermi-Dirac statistics (after the Italian physicist Enrico Fermi (1901-1954) and the British physicist Paul Dirac (1902-1984)) is a concept in physical quantum statistics. It describes the macroscopic behaviour of a system consisting of many identical particles of the fermion type and applies, for example, to the electrons that provide electrical conductivity in metals and semiconductors.

The starting points of the Fermi-Dirac statistics are:

None of the states of the individual particles can be occupied by more than one particle (Pauli principle).
If you exchange two particles with each other, you do not get a new state (which would have to be counted extra in the statistical consideration), but the same as before (principle of indistinguishability of the same particles).

The Fermi distribution gives the probability in an ideal Fermi gas at a given absolute temperature a state of energy is occupied by one of the particles. In statistical physics, the Fermi distribution is derived from the Fermi-Dirac statistics for similar fermions for the important special case of interaction freedom.

For a complete description of Fermi-Dirac statistics, see quantum statistics. For a simplified derivation see ideal Fermigas.

Description

General formula

In a system of temperature $T\!\,$ the Fermi distribution is W(E) , which measures the occupation probability:

$W(E)={\frac {1}{\exp {\left({\frac {E-\mu }{k_{{\mathrm {B}}}T}}\right)}+1}}$

with

of the energy for the state of a particle,
the chemical potential μ $\mu$ (For $T=0\,\mathrm {K}$ holds μ $\mu =E_{F}$ , where $E_{\rm {F}}$ called the Fermi level),
of the thermal energy $k_{\mathrm {B} }T$ , where is $k_{\mathrm {B} }=8{,}617\;3303\;(50)\cdot 10^{-5}\,\mathrm {eV} /\mathrm {K}$ the Boltzmann constant.

If the energy $E\,$ calculated from the lowest possible single-particle state, $E_{\rm {F}}\,$ also called Fermi energy. The occupation probability for a state with Fermi level energy $E=E_{\rm {F}}\!\,$ is at all temperatures:

$W(E=E_{\rm {F}})={\frac {1}{e^{0}+1}}={\frac {1}{2}}\ .$

To calculate the particle density ⟨ $\langle n(E)\rangle$ $E\,$ prevailing at energy e.g. for electrons in a metal, the Fermi distribution must still be $D(E)\!\,$ multiplied by the density of states

$\langle n(E)\rangle =W(E)\cdot D(E)\ .$

At absolute temperature zero

At the absolute temperature zero point $T=0\,{\mathrm {K}}$ the Fermi gas as a whole is in its energetically lowest possible state, i.e. in the ground state of the many-particle system. Since (given a sufficiently large number of particles) not all particles can occupy the one-particle ground state according to the Pauli principle, there must be particles in excited one-particle states even at the absolute temperature zero $T=0\,{\mathrm {K}}$ particles must be in excited one-particle states. This can be vividly described with the notion of a Fermi lake: each added fermion occupies the lowest possible energy state, which is not yet occupied by another fermion. The "filling level" is determined by the density of the occupiable states and the number of particles to be accommodated.

Accordingly, the Fermi distribution for temperature $T=0\,\mathrm {K}$ a sharp jump at the Fermi energy $E_{{\mathrm {F}}}=\mu \!\,$ , which is therefore also called the Fermi edge or Fermi limit (see figure).

All states with $E<E_{\rm {F}}$ are occupied, since here $W(E)=1$ , i.e., the probability of encountering one of the fermions in such a state is one.
None of the states with $E>E_{\rm {F}}$ is occupied, since here holds: $W(E)=0$ , i.e. the probability of encountering one of the fermions in such a state is zero.

The Fermi level at $T=0\,{\mathrm {K}}$ is therefore determined by the number and energetic distribution of states and the number of fermions to be accommodated in those states. Only one energy difference appears in the formula. If one gives the magnitude of the Fermi energy alone, it is the energy difference of the highest occupied to the lowest possible one-particle state. For illustration or quick estimation of temperature-dependent effects, this quantity is often expressed as a temperature value - the Fermi temperature:

$T_{{\mathrm {F}}}={\frac {E_{{\mathrm {F}}}}{k_{{\mathrm {B}}}}}\!\,$ .

At the Fermi temperature, the thermal energy $k_{{\mathrm {B}}}T\!\,$ equal to the Fermi energy. This term has nothing to do with the real temperature of the fermions, it is only used to characterize energy ratios.

At finite temperatures

The Fermi distribution gives the occupation probability in the equilibrium state at temperature } $T>0\,\mathrm {K}$ Starting from $T=0\,{\mathrm {K}}$ , states above the Fermi energy $E_{\mathrm {F} }(T=0\,\mathrm {K} )$ occupied by fermions. In exchange, an equal number of states below the Fermi energy remain empty and are called holes.

The sharp Fermi edge is symmetrically located around $E_{\mathrm {F} }$ located in an interval of total width $\sim 4k_{\mathrm {B} }T$ rounded off ("softened", see Fig.). States at lower energies are still nearly full ( $W\lessapprox 1$ ), while those at higher energies are very weak ( $0<W\ll 1$ ).

Since the same number of particles is still to be distributed among the possible states with density of states $D(E)\!\,$ , the Fermi energy can shift with temperature: If the density of states in the excited particle region is smaller than in the holes, the Fermi energy increases, in the opposite case it decreases.

In the temperature range $T\ll T_{\mathrm {F} }\,$ the system is called a degenerate Fermi gas, because the occupation of the states is largely determined by the Pauli principle (exclusion principle). This leads to the fact that all states with $E<E_{\mathrm {F} }$ have the same probability (of nearly one) of being occupied; this involves a large energy range compared to the softening interval.

For energies of at least some $k_{\mathrm {B} }T$ above $E_{\mathrm {F} }$ , i.e., for $E-E_{\mathrm {F} }\gg k_{\mathrm {B} }T$ , the Fermi distribution can be approximated by the classical Boltzmann distribution:

$W(E)\propto \exp {\left(-{\frac {E-E_{\mathrm {F} }}{k_{\mathrm {B} }T}}\right)}$ .

At very high temperatures

"Very high temperatures" are those well above the Fermi temperature, i.e., $T\gg T_{\mathrm {F} }\Leftrightarrow k_{\mathrm {B} }T\gg E_{\mathrm {F} }$ . Because this makes the softening interval very large, so that even for energies well above the Fermi energy the occupation probability is noticeably different from zero, particle number conservation leads to the Fermi energy being below the lowest occupiable level. The Fermi gas then behaves like a classical gas, it is not degenerate.

Fermi distribution for different temperatures, increasing rounding with increasing temperature (red line: T = 0 K)

Fermi distribution for metals

For the conduction electrons in a metal, the Fermi energy $E_{\rm {F}}\!\,$ is a few electron volts, corresponding to a Fermi temperature $T_\mathrm{F} \!\,$ of a few 10,000 K. As a consequence, the thermal energy $k_{\mathrm {B} }T$ is much smaller than the typical width of the conduction band. This is a degenerate electron gas. Therefore, the contribution of electrons to the heat capacity is negligible even at room temperature and can be accounted for in perturbation theory. The temperature dependence of the Fermi energy is very small (meV range) and is often neglected.

Questions and Answers

Q: What is Fermi-Dirac statistics?

A: Fermi-Dirac statistics is a branch of quantum statistics that is used to describe the macroscopic state of a system made of many similar particles.

Q: Who is Fermi-Dirac statistics named after?

A: Fermi-Dirac statistics is named after Enrico Fermi and Paul Dirac.

Q: What is an example of a system that can be described using Fermi-Dirac statistics?

A: One example of a system that can be described using Fermi-Dirac statistics is the state of electrons in metals and semimetals, in order to describe electrical conductivity.

Q: What assumptions are made in Fermi-Dirac statistics?

A: Fermi-Dirac statistics makes two assumptions: 1) none of the states of the particles can hold more than one particle (known as Pauli exclusion principle), and 2) exchanging a particle for another similar particle will not lead to a new state, but will give the same state (known as identical particles).

Q: What does the Fermi distribution tell us?

A: The Fermi distribution tells us with what probability a Fermi gas, at a given temperature and energy level, will have a particle in the given state.

Q: What is another name for the Pauli exclusion principle?

A: The Pauli exclusion principle is also known as the exclusion principle.

Q: What is a Fermi gas?

A: A Fermi gas is a group of fermions that are at a low enough temperature to exhibit quantum effects.