Minority suppression and donor-defined conduction

Minority carrier suppression is the reduction of the minority carrier concentration below the intrinsic value $n_i$ when a semiconductor is doped. In n-type material the holes are suppressed; in p-type the electrons are suppressed. It follows from the mass action law $np = n_i^2$: if doping raises one carrier type, the other must fall in the same ratio.

Proof for n-type semiconductor

Consider Si doped with $N_d$ donors per cm³ (P, As, Sb). Each donor level lies about 0.05 eV below $E_c$, so at room temperature practically all donors are ionised: $N_d^+ = N_d$.

 Ec -------------------------
 Ed  - * - * - * - (donors, ~0.05 eV)
 EFn ........................
 EFi - - - - - - - - - - - - -
 Ev -------------------------

1. Charge neutrality:

$$n = p + N_d$$

2. Mass action law (thermal equilibrium):

$$np = n_i^2,\quad n_i^2 = N_cN_ve^{-E_g/kT}$$

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

$$n^2 - N_dn - n_i^2 = 0 \;\Rightarrow\; n = \frac{1}{2}\left[N_d + \sqrt{N_d^2+4n_i^2}\right]$$

4. Normal doping, $N_d \gg n_i$:

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

So the electron concentration is fixed by the donor concentration only, not by temperature or band gap, and the holes are suppressed by the factor $n_i/N_d$.

Conduction is defined by the donors

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

The ratio of the hole term to the electron term is $\dfrac{n_i^2\mu_h}{N_d^2\mu_e}$, which is tiny. Hence

$$\sigma \approx eN_d\mu_e$$

Example: Si with $N_d = 10^{16}$: $n = 10^{16}$, $p = 2.1\times10^{4}$ cm⁻³; hole/electron current ratio $\approx 7\times10^{-13}$. $\sigma = 1.6\times10^{-19}\times10^{16}\times1350 = 2.16$ S cm⁻¹.

Conclusions

• Electron concentration $\approx N_d$ (donor defined).
• Conductivity $\approx eN_d\mu_e$ (donor defined), and it can be set accurately by controlling the doping.
• This holds over the extrinsic range of temperature; at very high T, $n_i$ becomes comparable to $N_d$ and the material turns intrinsic.