Fermi energy

The Fermi energy $E_{F}$ is the energy of the highest occupied state of a system of fermions at zero temperature. It corresponds to the zero-temperature chemical potential of the system, so that $\mu=E_{F}$ when $T=0$. At nonzero temperatures, thermal effects add energy to the fermions, so the Fermi energy is no longer well-defined. However, for temperatures far below the Fermi temperature (see below), it remains a good approximation, $\mu\simeq E_{F}$.

Since energy determines a surface in phase space, the Fermi energy surface divides occupied states (inside) from unoccupied ones (outside). It is known as the Fermi surface.

Explanation#

The Pauli exclusion principle prevents multiple fermions from occupying the same state. This causes the fermions to be "pushed apart" in different states, despite there not being any repulsive potential. This inherent repulsive behavior is characteristic of fermions. Because of this repulsion, even at zero temperature, most fermions will be in excited states, as there can only be one in the ground state. This gives fermionic systems a certain irreducible amount of energy that cannot be removed even at zero temperature. The energy of the highest occupied state at zero temperature is known as the Fermi energy. This quantity can be derived from purely mathematical considerations.

Related quantities#

Several quantities can be derived from the Fermi energy. The Fermi momentum is $p_{F}=\sqrt{ 2mE_{F} }$, the Fermi velocity is $v_{F}=p_{F}/m$, the Fermi wavevector is $k_{F}=p_{F}/\hbar$. These are the momentum, velocity and wavevector of the highest energy particle in the Fermi gas (i.e. the one whose kinetic energy is the Fermi energy). The wave vector can also be expressed from the particle density $n=N/V$ directly as $k_{F}=(3\pi ^{2}n)^{1/3}$.

The Fermi temperature is defined as $T_{F}=E_{F}/k_{B}$, where $k_{B}$ is the Boltzmann constant. It can be thought of as the temperature at which thermal effects become significant compared to quantum effects (specifically the Pauli exclusion principle). It is a useful metric to determine how "sharp" the Fermi-Dirac distribution is. At $T\ll T_{F}$, it is almost a Heaviside step function across $E_{F}$, but when $T\gg T_{F}$, it broadens to the point that there is no clear boundary and becomes indistinguishable from the Maxwell-Boltzmann statistic.

Since the Pauli exclusion principles causes zero- and low-temperature fermions to be subject to degenerate pressure, $T_{F}$ is also sometimes known as the degeneracy temperature, which is particularly important in later stages of stellar evolution when stars collapse into white dwarfs (degenerate electron gas) or neutron stars (degenerate neutron gas). This temperature is not necessarily very low: for instance, for electrons in metals, it is typically in the range of $\sim 10^{4}$ kelvins. In neutron stars, it is $>10^{7}$ kelvins.