p-type minority carrier suppression and conductivity

A p-type semiconductor is an intrinsic semiconductor (Si, Ge) doped with a small amount of a trivalent (group III) impurity such as boron, aluminium, gallium or indium. Each impurity atom forms only three covalent bonds, so one bond is incomplete; it easily accepts an electron from the valence band, creating a hole. Such impurities are called acceptors ($N_a$). Holes are the majority carriers and electrons the minority carriers.

 Ec ------------------------
 EFi - - - - - - - - - - - -
 EFp .......................
 Ea  - o - o - o - (acceptors, ~0.05 eV)
 Ev ------------------------

Derivation of minority carrier suppression

1. Mass action law. In equilibrium, for any non-degenerate semiconductor,

$$np = n_i^2$$

2. Charge neutrality. Positive charges = negative charges:

$$p = n + N_a^-$$

At room temperature all acceptors are ionised, $N_a^- = N_a$.

3. Solve. Put $n = n_i^2/p$:

$$p^2 - N_ap - n_i^2 = 0 \;\Rightarrow\; p = \frac{N_a}{2} + \sqrt{\frac{N_a^2}{4}+n_i^2}$$

For normal doping $N_a \gg n_i$, so

$$p \approx N_a,\qquad n = \frac{n_i^2}{N_a}$$

4. Suppression. Compare with the intrinsic value:

$$\frac{n}{n_i} = \frac{n_i}{N_a} \ll 1$$

Adding acceptors raises $p$ above $n_i$ by the factor $N_a/n_i$ and lowers $n$ below $n_i$ by the same factor. The extra holes recombine with the thermally generated electrons. This reduction of the minority electron concentration is called minority carrier suppression.

Example: Si with $N_a = 10^{16}$: $p = 10^{16}$, $n = (1.45\times10^{10})^2/10^{16} = 2.1\times10^{4}\ \text{cm}^{-3}$.

Conductivity is due to holes

$$\sigma = e(n\mu_e + p\mu_h) = e\left(\frac{n_i^2}{N_a}\mu_e + N_a\mu_h\right)$$

Ratio of the two terms:

$$\frac{n\mu_e}{p\mu_h} = \frac{n_i^2}{N_a^2}\cdot\frac{\mu_e}{\mu_h}$$

For the example: $(1.45\times10^{10}/10^{16})^2\times3 \approx 6\times10^{-12}$. Even though $\mu_e \approx 3\mu_h$, the electron term is negligible, so

$$\sigma \approx eN_a\mu_h$$

Hence the conductivity of a p-type semiconductor is almost entirely due to holes and is fixed by the acceptor concentration.