= Solution
\b[Doping changes the mobile carrier population by introducing controlled impurity levels.] In an <intrinsic semiconductor>, the filled <valence band> is separated from the nearly empty <conduction band> by a <band gap> $E_g$. Thermal excitation produces an electron-hole pair, so the <Electron> and hole <number densities> obey $n=p=n_i$. A <electron hole> represents a missing valence-band <Electron> and behaves as a mobile positive <charge carrier>. For nondegenerate bands in thermal <equilibrium>,
$$
n=N_Ce^{-(E_C-E_F)/(k_BT)},\qquad
p=N_Ve^{-(E_F-E_V)/(k_BT)},\qquad
np=N_CN_Ve^{-E_g/(k_BT)}=n_i^2.
$$
Here $N_C,N_V$ are the <effective density of states>, $E_C,E_V$ the band edges, and $E_F$ denotes the <chemical potential>, conventionally called the <Fermi level>. The <semiconductor mass-action law> holds in this nondegenerate equilibrium regime.
In a tetrahedrally bonded elemental <semiconductor> such as <silicon>, a substitutional group-V atom has an extra outer <Electron> relative to the host. It is a <donor impurity>: a shallow level just below $E_C$ can ionise, releasing an <Electron> to the <conduction band> and leaving a fixed positively charged donor. The resulting <n-type semiconductor> has <Electrons> as majority carriers and holes as <minority carriers>. Conversely a group-III substitution supplies one fewer bonding <Electron>. An <acceptor impurity> can take an <Electron> from the <valence band>, leaving a mobile <electron hole> and a fixed negative acceptor; the resulting <p-type semiconductor> has holes as majority carriers.
The impurity state is shallow because the <Electron> or hole responds to a screened Coulomb potential and an <effective mass>, rather than the bare atomic potential. A simple hydrogenic donor estimate is
$$
E_C-E_D\simeq\frac{m^*}{m_e}\frac{13.6\,\mathrm{eV}}{\epsilon_r^2},
$$
where $\epsilon_r$ is the relative <permittivity>. This is a useful scaling model, with real band anisotropy and central-cell effects omitted. Dopants are fixed in the lattice; their ionisation changes mobile carrier numbers without making a uniformly doped bulk sample carry a large net <electric charge>.
The <charge neutrality> condition is
$$
n+N_A^-=p+N_D^+.
$$
When shallow dopants are almost fully ionised and $N_D-N_A\gg n_i$, an n-type region has $n\simeq N_D-N_A$ and $p\simeq n_i^2/n$. A p-type region similarly has $p\simeq N_A-N_D$ and $n\simeq n_i^2/p$. Thus compensation by the opposite impurity type reduces the majority-carrier density; the total dopant concentration is not generally the mobile <carrier density>. At sufficiently high impurity density the <degenerate semiconductor> regime requires the full <Fermi-Dirac distribution> and these exponential approximations fail.
There are three useful <temperature> regimes. At low <temperature>, <dopant freeze-out> leaves many shallow levels un-ionised. In the extrinsic regime, the dopants are largely ionised and set the majority-carrier density. At high enough <temperature>, intrinsic pair creation dominates and $n,p$ approach $n_i$. The <Fermi level> moves towards the <conduction band> with donor doping and towards the <valence band> with acceptor doping. The <electrical conductivity> is
$$
\boxed{\sigma=e(n\mu_e+p\mu_h)}.
$$
Doping therefore controls conductivity and its carrier type, but ionised impurities also scatter carriers and can reduce <charge carrier mobility>. <Carrier density> and mobility must both be considered.
\b[A p-n junction forms when p-type and n-type regions are joined within a semiconductor.] Initially the concentration gradients drive <diffusion current>: <Electrons> diffuse from the n region to the p region and holes in the opposite direction. Recombination removes mobile majority carriers near the join, exposing fixed positive donors on the n side and fixed negative acceptors on the p side. This forms a <depletion region> and an <electric field> directed from n to p. Its <drift current> opposes diffusion. At equilibrium the net current vanishes and the <Fermi level> is uniform, although the band edges vary spatially through <band bending>.
For an abrupt nondegenerate junction with fully ionised dopants, its <built-in potential> is
$$
\boxed{V_{\rm bi}=\frac{k_BT}{e}\ln\frac{N_AN_D}{n_i^2}}.
$$
Under the <depletion approximation>, the space-charge densities are $-eN_A$ on a p-side width $w_p$ and $+eN_D$ on an n-side width $w_n$. Integrating <Poisson equation> and imposing zero field outside gives $N_Aw_p=N_Dw_n$ and
$$
W=w_p+w_n=\sqrt{\frac{2\epsilon_sV_{\rm bi}}e\left(\frac1{N_A}+\frac1{N_D}\right)}.
$$
Most of the depletion width therefore extends into the more lightly doped side. The schematic shows the impurity levels before contact and the resulting space charge and potential after contact; there is no externally extractable equilibrium battery voltage from the built-in field alone.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2018/paper-2-d12-doping-junction.png]
{title=Shallow donor and acceptor levels, and charge, field and potential in an equilibrium abrupt p-n junction}
A forward bias, with p positive relative to n, reduces the barrier from $V_{\rm bi}$ towards $V_{\rm bi}-V$ and injects <minority carriers> across the junction. In the ideal low-injection, diffusion-controlled regime this produces the <Shockley diode equation>, $I=I_s[e^{eV/(k_BT)}-1]$. Reverse bias increases the barrier and the <depletion region>; only a small thermally generated minority-carrier current flows before breakdown. Reverse-biased depletion regions also give a voltage-dependent <junction capacitance>. Real devices have recombination, leakage, series resistance and breakdown mechanisms beyond the ideal law. \b[Controlled doping creates majority and minority carriers, and their junction produces a depletion barrier that makes rectification possible.]
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