Extrinsic Semiconductors

N-type

N-type materials are where a material is doped with donor impurities (a negative charge).

The concentration of donors is far greater than the intrinsic carrier concentration:

\[N_D >> n_i\]

Electrons are the majority carriers while holes are in the minority:

\[n_n\approx N_D\text{ (majority)}\] \[p_n\approx \frac{n_i^2}{N_D}\text{ (minority)}\]

Notice that we use a subscript $n$ to indicate that it is n-type.

P-type

P-type materials are where a material is doped with acceptor impurities (a positive charge).

The concentration of acceptors is far greater than the intrinsic carrier concentration:

\[N_A >> n_i\]

Holes are the majority carriers while electrons are in the minority:

\[p_p\approx N_A\text{ (majority)}\] \[n_p\approx \frac{n_i^2}{N_A}\text{ (minority)}\]

Approximations

Notice that many approximations can be used when the material is sufficiently doped. Be aware that at high temperatures, making approximations becomes less accurate.

True Concentrations

More correct equations for concentrations are as follows:

\[n = \frac{N_D-N_A}{2} +\sqrt{\left(\frac{N_D-N_A}{2}\right)^2+n_i^2}\] \[p = \frac{N_A-N_D}{2} +\sqrt{\left(\frac{N_A-N_D}{2}\right)^2+n_i^2}\]

Doping Both Donors and Acceptors

If $N_D = N_A$, it will act as an intrinsic semiconductor where $n=p=n_i$.

If $N_D > N_A > > n_i$, ignore $n_i$.

\[n_n = N_D-N_A\] \[p_n = \frac{n_i^2}{n_n}\]

If $N_A > N_D > > n_i$, ignore $n_i$.

\[p_p = N_A-N_D\] \[n_p = \frac{n_i^2}{p_p}\]

Bear in mind that $np = n_i^2$ will remain true for lightly to moderately doped materials.