Boltzmann limit of the normalized Fermi-Dirac integral

For $F_j(\eta )=\Gamma (j+1{)}^{-1}{\int }_0^{\infty }{t}^j/(e^{t-\eta }+1)dt$, the nondegenerate limit is $\eta \ll -1$. Replacing the denominator by $e^{t-\eta }$ givesbecause the remaining integral is $\Gamma (j+1)$. Expanding in powers of fugacity gives $F_j(\eta )=e^{\eta }-e^{2\eta }/2^{j+1}+\cdots$.

Thus a carrier density proportional to $F_{1/2}[(\mu -E_c)/(k_{B}T)]$ reduces to the familiar exponential occupation when the chemical potential lies several thermal energies below the band edge. At $\mu =E_c$, the expansion parameter is one and the classical replacement is not controlled. The normalization of the Normalized complete Fermi-Dirac integral must be stated before prefactors from different conventions are compared.