Semiconductors: Numerical Problems Set (1)

Question

In a pure germanium sample at 300 K, the intrinsic carrier concentration is $n_i = 2.5 \times 10^{19}$ m$^{-3}$. The sample is doped with donor atoms of density $N_D = 2.5 \times 10^{23}$ m$^{-3}$. Find (a) the electron concentration, (b) the hole concentration in the doped sample and (c) the ratio of majority to minority carriers.

Solution — Step by Step

Why This Works

The mass-action law $n_e n_h = n_i^2$ comes from the equilibrium between thermal generation and recombination of electron-hole pairs. Doping doesn’t change this product — it only redistributes between $n_e$ and $n_h$. Boost one, the other drops by the same factor.

The huge ratio ($10^8$) is why doped semiconductors behave so differently from intrinsic ones — the minority carrier population is essentially negligible for most conduction calculations.

Alternative Method

You can solve the quadratic from charge neutrality: $n_e - n_h = N_D$ along with $n_e n_h = n_i^2$. For $N_D \gg n_i$, the quadratic gives $n_e \approx N_D$ to high precision. Most exam problems use this approximation directly.

Common Mistake