= 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 $e^{t-\eta}$ gives
$$
F_j(\eta)\simeq e^\eta,
$$
because 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 $F_{1/2}[(\mu-E_c)/(k_BT)]$ 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.
Back to article page
