1. Instrinsic charge carrier concentration

The concentration of thermally-generated electron-hole pairs varies (lots!) with temperature, and is:

\[n_i = B \cdot T^{3/2} \cdot \exp\left(\dfrac{-E_g}{2 k_B T}\right) \;\; \mathrm{\#/cm^3}\]

2. Doping

Adding impurities to the crystal from Group III or IV elements (replacing an Si atom with another like B or P) also adds extra electrons or holes to the structure.

Impurities which add extra free electrons are called donors, with concentration $N_D \;\; \mathrm{cm^{-3}}$.

Impurities which add extra (free) holes are acceptors: $N_A \;\; \mathrm{cm^{-3}}$.

Even when doped, a semiconductor crystal has no net charge

\[\sum \text{charges} = q \left( +N_D + p - N_A - n \right) = 0\]

This, combined with the fundamental relationship

\[n\cdot p = n_i^2\]

Allows us to calculate the concentration of free electrons and holes for any doping condition (always choose $+$ version):

\[n = \dfrac{(N_D - N_A) \pm \sqrt{(N_D - N_A)^2 + 4 n_i^2}}{2} \; \text{ and } \; p = \dfrac{n_i^2}{n}\]

or, solved the other way:

\[p = \dfrac{(N_A - N_D) + \sqrt{(N_A - N_D)^2 + 4 n_i^2}}{2} \; \text{ and } \; n = \dfrac{n_i^2}{p}\]

Q: Are the above two sets of solutions equivalent?

Q: Does it matter which set to use depending on which of $N_D$ or $N_A$ is larger?

When we dope in practice, the impurities $N_D$ and/or $N_A$ are in the range $10^{14} \text{ to } 10^{21} \; \mathrm{cm^{-3}}$, which is much greater than $n_i$ at normal temperatures. This results in us using the following approximations all the time:

\[\text{when $N_D$ is largest: } n \approx N_D \text{ and } p \approx n_i^2/N_D\]
\[\text{when $N_A$ is largest: } p \approx N_A \text{ and } n \approx n_i^2/N_A\]

$\rightarrow$ Notice how only the minority carrier concentrations are affected by temperature (holes for n-type doping, electrons for p-type doping).

3. Conclusions

  • $n_i$ varies lots with temperature

  • We can change the concentration of free electrons ($n$) and (free) holes ($p$) by doping.

  • Minority carrier concentrations are greatly affected by temperature. Majority carriers not so much.