= Solution
<Semiconductor doping> introduces controlled impurities to alter equilibrium carrier populations and <electrical conductivity>. In an <intrinsic semiconductor>, thermally excited <Electrons> in the <conduction band> and <electron holes> in the <valence band> have equal densities $n=p=n_i$. A shallow <donor impurity>, such as phosphorus in silicon, supplies an <Electron> from an impurity level just below the conduction-band edge. Once ionized it is positively charged, and the material is n type. An <acceptor impurity>, such as boron in silicon, accepts a valence <Electron> into a level just above the valence-band edge, leaving a mobile hole and a negatively charged ionized impurity; the material is p type.
With dopant densities $N_D,N_A$, charge neutrality requires
$$
p+N_D^+=n+N_A^-.
$$
For fully ionized, nondegenerate dopants at equilibrium, the <semiconductor mass-action law> gives $np=n_i^2$. In a substantially n-type sample, $N_D-N_A\gg n_i$, so
$$
\boxed{n\simeq N_D-N_A,\qquad p\simeq\frac{n_i^2}{N_D-N_A}.}
$$
The corresponding p-type result is $p\simeq N_A-N_D$ and $n\simeq n_i^2/(N_A-N_D)$. The majority population grows while the minority population decreases; adding donors does not simply add equal numbers of free <Electrons> and holes.
The <Fermi energy> shifts toward the conduction band for n-type material and toward the valence band for p-type material. In the nondegenerate approximation,
$$
n=N_Ce^{-(E_C-E_F)/(k_BT)},\qquad p=N_Ve^{-(E_F-E_V)/(k_BT)},
$$
where $N_C,N_V$ are effective band-edge state densities. At very low <temperature>, dopants may not be ionized and carrier freeze-out occurs. At intermediate temperatures, ionized dopants dominate the carrier density; at sufficiently high <temperature>, intrinsic excitation can dominate. Heavy doping can produce degenerate carrier statistics, invalidating these Boltzmann expressions.
The <electrical conductivity> is $\boxed{\sigma=e(n\mu_e+p\mu_h)}$. Doping usually raises the majority carrier density, but ionized impurity scattering can lower mobility, so conductivity need not scale indefinitely with dopant concentration. Joining p-type and n-type regions creates a <p-n junction>, with a <depletion region> and built-in <electric field> established by carrier diffusion and compensating drift. The fixed ionized dopants, not an absence of all charge, account for the depleted region's net <charge density>.
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