Two-dimensional free-electron density of states

For nonrelativistic electrons in area $A$, with spin degeneracy $g_s=2$ and energy $E={\hslash }^2{k}^2/(2m)$, the density of one-particle states is independent of energy:

For a constant total density of states $D$ above a zero-energy band bottom, integrating the Fermi-Dirac occupation gives $N=D{k}_{B}T\ln (1+e^{\mu /(k_{B}T)})$. With ${ϵ}_F=N/D$, inversion gives the displayed chemical potential. It approaches ${ϵ}_F$ from below with an exponentially small correction at low temperature, rather than the usual three-dimensional quadratic correction. This formula assumes an ideal parabolic band extending to arbitrarily high energy.

Each allowed wavevector occupies area $(2\pi {)}^2/A$ under periodic boundary conditions, for each independent spin state. For parabolic dispersion, the number below energy $ϵ$ is $g_{s}Amϵ/(2\pi {\hslash }^2)$. Thus the often-quoted coefficient $Am/(2\pi {\hslash }^2)$ is per spin. Unpolarized electrons have $g_s=2$; the same convention must be used in particle counting and the Fermi energy. A linear Dirac dispersion has a different energy dependence.

A quadratic-dispersion Fermi gas in two dimensions has constant density of states $g$. At zero temperature, $N=g{E}_F$ and $U=g{E}_F^2/2$. Hence the mean energy is half the Fermi energy, independent of a common spin-degeneracy factor.

When the density of states is constant and the chemical potential is fixed at $E_F$, the thermal gain above $E_F$ cancels the thermal depletion below it after the energy range is extended to the whole real line. The finite lower edge leaves only an exponentially small correction of order $e^{-\beta E_F}$.

For constant density of states, weighting the difference between the Fermi-Dirac distribution and the zero-temperature step by $E-E_F$ produces a convergent integral that scales as ${\beta }^{-2}$. Hence the leading low-temperature energy correction is proportional to $T^2$.

For a two-dimensional Fermi gas of $N$ spin-one-half particles with Fermi energy $E_F$, the constant density of states makes the low-temperature internal energy correction and heat capacity at constant areaThus its low-temperature heat capacity obeys a power law with exponent one.