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.

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.

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 $N_c$.