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:

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.