Normalized complete Fermi-Dirac integral ($F_j(\eta )$)

The convention $F_j(\eta )=\Gamma (j+1{)}^{-1}{\int }_0^{\infty }{x}^j/[1+e^{x-\eta }]dx$ is useful for parabolic-band carrier densities. In three dimensions, a spin-degenerate single valley has $n=2[m{k}_{B}T/(2\pi {\hslash }^2){]}^{3/2}{F}_{1/2}[(\mu -E_c)/(k_{B}T)]$. Other conventions omit the gamma normalization, so prefactors must be stated.

For equal-mass conduction and valence edges at $0,-E_g$ and fixed $\mu =0$, $n=N(T)F_{1/2}(0)$ and $p=N(T)F_{1/2}[-E_g/(k_{B}T)]$, where $N(T)=2[m{k}_{B}T/(2\pi {\hslash }^2){]}^{3/2}$. Other states pinning the chemical potential can compensate the unequal carrier charges; imposing intrinsic charge neutrality in addition would change the problem.

Expanding the Fermi denominator as an alternating exponential series gives $F_{1/2}(0)=(1-2^{-1/2})\zeta (3/2)\simeq 0.765147$. Thus a band with chemical potential pinned exactly at its lower edge has $n=F_{1/2}(0)N_c$, rather than the classical result $N_c$.