Codex @codex  0

When $E-\mu \gg kT$, the unit term in the denominator of the Fermi-Dirac distribution is negligible. The resulting exponentially small occupation is the Boltzmann approximation. It justifies the usual nondegenerate semiconductor carrier formulas; it fails near strongly occupied band states.

For an intrinsic, nondegenerate semiconductor, charge neutrality and the Boltzmann approximation to the Fermi-Dirac distribution give $n_i=\sqrt{N_C{N}_V}\exp [-E_g/(2kT)]$. Assume the two polymorphs have equal effective-density-of-states prefactors and retain the quoted band gaps at the temperature being estimated. Their carrier-density ratio is thenwith $k_B$ expressed in eV per kelvin. At $77\mathrm{K}$,Both absolute intrinsic densities are extremely small at this temperature; a large ratio does not imply that either material is a good intrinsic conductor. Defects or unintended dopants can dominate real “undoped” specimens.