Is equilibrium carrier concentration dependent on doping density? If it is, then how to know how much the doping density should be for a particular semiconductor device?
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Carrier concentration in thermal equilibrium is dependent on doping concentration.
Carrier concentration in thermal equilibrium is governed by:
$$n = N_C\cdot e^{-\frac{E_C - E_F}{kT}}$$ $$p = N_V\cdot e^{-\frac{E_F - E_V}{kT}}$$
With \$E_C\$ and \$E_V\$ the conduction and valence band energies, and \$E_F\$ the Fermi level. These laws are a consequence of statistics of where electrons/holes might be on average. It should be noted that \$n\cdot p = E_CE_V\cdot e^{-\frac{E_C-E_V}{kT}}\$ is only dependent on the material and is therefore usually defined as
$$n\cdot p = n_i^2$$
When semiconductors are doped deliberately, a dopant is used to disturb the equality between \$n\$ and \$p\$.
A donor is a dopant that has a very easy time of "donating" one of its electrons to the conduction band. It introduces an electron with an energy state just smaller than \$E_C\$ making the extra electron act like it was part of the conduction band. In first order approximation, adding a concentration of \$N_D\$ donors will add just as many electrons, so you would find:
$$\begin{cases} n = p + N_D\\ n\cdot p = n_i^2 \end{cases} \Rightarrow \begin{cases} p = \frac{\sqrt{N_D^2 + 4n_i^2} - N_D}{2}\\ n = \frac{\sqrt{N_D^2 + 4n_i^2} + N_D}{2} \end{cases}$$
An acceptor is a dopant that has a very easy time of "accepting" electrons from the valence band. It introduces an energy state just larger than \$E_V\$, causing electrons to easily get trapped by the state, leaving behind an additional hole in the valence band. In first order approximation, adding a concentration of \$N_A\$ acceptors will add just as many holes, so you would find:
$$\begin{cases} p = n + N_A\\ n\cdot p = n_i^2 \end{cases} \Rightarrow \begin{cases} p = \frac{\sqrt{N_A^2 + 4n_i^2} + N_A}{2}\\ n = \frac{\sqrt{N_A^2 + 4n_i^2} - N_A}{2} \end{cases} $$
IIRC the approximation breaks down for degenerate semiconductors (extremely high doped regions). Also remember that this approximation only works for dopants with energy levels near the conduction/valence band.
