Analytic approximations for the Fermi energy of an ideal Fermi gas

Abstract

An important function in semiconductor‐device analysis and transport theory is the widely tabulated Fermi‐Dirac integral, F (η) =2π^−1/2F^∞₀[exp(x−η)+1]⁻¹f dx, f=x^1/2, which relates, for example, the Fermi energy ηkT to the carrier density N=FN₀ in a parabolic semiconductor band (N₀=effective density of states). We show that the classical or Boltzmann approximation to this integral (η=lnF, η≲−2) is extended to cover the Fermi‐energy range of semiconductor lasers (η≲+2) by the expression η=lnF+2^−3/2F and by other simple differentiable approximations applicable to higher degeneracy (η≲7) or to nonparabolic bands (f≠x^1/2).