Past exam of the condensed matter physics course of the University of Oxford / 2012 / 3 / c / Solution

With the supplied fixed band gap and effective masses, the intrinsic carrier concentration isNumerically,For uncompensated, approximately ionized donor impurities, charge neutrality and the semiconductor mass-action law requireThere is no abrupt extrinsic-to-intrinsic transition. An order-of-magnitude lower crossover density is thereforeRobust extrinsic dominance requires $N_D\gg n_i$; at the nominal equality, $n=1.618N_D$ and $p=0.618N_D$, so intrinsic contributions are already appreciable. For a quantitative tolerance $p/n\le \eta$, the minimum ionized dopant density is $n_i(1-\eta )/\sqrt{\eta }$. For example, $\eta =0.1$ would require $N_D\ge 2.85n_i$ at the upper temperature.

Using the nominal crossover estimate and the complete-ionization approximation at room temperature, $n\simeq N_D$ andFor the twofold neutral-donor degeneracy, a consistency calculation gives about $91.3\%$ ionization at $300\mathrm{K}$, raising this minority density to $1.30\times 10^{15}{\mathrm{m}}^{-3}$, about $9.6\%$ above the complete-ionization estimate. Stronger extrinsic dominance uses more donors and correspondingly fewer electron holes. Actual germanium has temperature-dependent band parameters and multiple valleys; the numerical estimate consistently uses the fixed effective data supplied rather than adding an unsupplied multiplicity.