= Solution
There are three useful <temperature> regimes.
At low <temperature>, <dopant freeze-out> keeps most donor <electrons> bound. Warming ionizes donors and increases conductivity through the rising mobile-electron concentration. For an uncompensated dilute donor system, negligible holes and <spin degeneracy> two, <charge neutrality with partial donor ionization> gives
$$
\frac{n^2}{N_D-n}=\frac{N_c}{2}e^{-E_b/k_{\rm B}T}.
$$
Deep in freeze-out, $n\ll N_D$, so
$$
n\simeq\sqrt{\frac{N_DN_c}{2}}e^{-E_b/(2k_{\rm B}T)}.
$$
The activation scale is set by the shallow donor binding, rather than the whole <band gap>.
In the extrinsic regime, donors are largely ionized and $n\simeq N_D$ is nearly constant. The <temperature> dependence of conductivity then mainly follows the <electron mobility>. Increasing <phonon> scattering can make conductivity fall as <temperature> rises, even though donors remain ionized; impurity scattering can give a different mobility trend at lower <temperature>.
At sufficiently high <temperature>, thermally generated <electron>–hole pairs overwhelm the dopant population. \b[The intrinsic, gap-activated increase in conductivity returns.] The boundaries depend on donor concentration, compensation and scattering, so no one exponential describes the entire doped-sample curve.
Back to article page
