Chemical potential of a two-dimensional ideal Fermi gas ($\mu ={ϵ}_F+k_{B}T\ln (1-e^{-{ϵ}_F/(k_{B}T)})$)

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.