Intrinsic Carrier Concentration
In a pure (intrinsic) semiconductor, the number of free electrons equals the number of holes, and both depend strongly on temperature and the material’s bandgap. This is approximated by ni = A × T3/2 × exp(−Eg / 2kT), where Eg is the bandgap energy, T is absolute temperature, k is Boltzmann’s constant in eV/K, and A is a material-dependent prefactor (about 5.2×1015 cm-3K-3/2 for silicon).
Because the exponential term dominates, intrinsic carrier concentration rises very rapidly with temperature and falls sharply with a larger bandgap — which is why wide-bandgap materials like silicon carbide stay more insulating at high temperatures than narrow-gap materials like germanium. This is a standard approximation used for general semiconductor-physics reference.