Carrier Concentration

Under thermodynamic equilibrium, $np=n_i^2(T)$ holds independent of the doping level — n-type doping increases $n$, but proportionally decreases $p$ (and vice versa for p-type), so the product stays fixed at a given temperature.

The donor atom needs a small but finite ionization energy ($E_d$, typically tens of millielectronvolts) to release its extra electron into the conduction band. At sufficiently low temperature $k_BT \ll E_d$, and most donors remain un-ionized — this is a particularly significant issue in wide-bandgap materials (GaN, SiC), where $E_d$ is typically larger.

This "window" — where $n\approx N_d$ stably — is exactly the temperature range where real devices are designed to operate: warm enough for full ionization, cool enough that intrinsic carriers don't "swamp" the doping-defined behavior.

$n_i\propto\exp(-E_g/2k_BT)$ — this exponential dependence means even a modest difference in $E_g$ (e.g. GaN's 3.4 eV vs silicon's 1.12 eV) reduces $n_i$ by many orders of magnitude at the same temperature — this directly explains why SiC/GaN devices retain doping-defined behavior even at 300°C, where silicon is already fully intrinsic.