Calculate the thermal-equilibrium electron and hole concentrations in germanium for a given doping concentration.
Consider a germanium sample at T=300 \mathrm{~K} in which N_{d}=2 \times 10^{14} \mathrm{~cm}^{-3} and N_{a}=0. Assume that n_{i}=2.4 \times 10^{13} \mathrm{~cm}^{-3}.
Again, from Equation (4.60), the majority carrier electron concentration is
\begin{array}{c}n_{0}=\frac{\left(N_{d}-N_{a}\right)}{2}+\sqrt{\left(\frac{N_{d}-N_{a}}{2}\right)^{2}+n_{i}^{2}} \\ \end{array} (4.60)
n_{0}=\frac{2 \times 10^{14}}{2}+\sqrt{\left(\frac{2 \times 10^{14}}{2}\right)^{2}+\left(2.4 \times 10^{13}\right)^{2}} \cong 2.028 \times 10^{14} \mathrm{~cm}^{-3}
The minority carrier hole concentration is
p_{0}=\frac{n_{i}^{2}}{n_{0}}=\frac{\left(2.4 \times 10^{13}\right)^{2}}{2.028 \times 10^{14}}=2.84 \times 10^{12} \mathrm{~cm}^{-3}
Comment
If the donor impurity concentration is not too different in magnitude from the intrinsic carrier concentration, then the thermal-equilibrium majority carrier electron concentration is influenced by the intrinsic concentration.