Past exam of the chemistry A course of the University of Cambridge / 2025 / Paper 2 / A10 / a

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.

Using the same intrinsic-carrier approximation at $300\mathrm{K}$ givesSimilar mass density and coordination do not prove equal electron and hole effective masses; this result isolates the exponential band-gap effect.

At $500\mathrm{K}$, the same approximation givesAs temperature rises, the relative advantage from a fixed gap difference becomes smaller, even while the intrinsic carrier densities of both phases increase.