= Solution
<Arsenic> has five valence electrons, whereas <silicon> has four. A substitutional arsenic atom can therefore supply an extra electron after participating in the silicon bonding network. It acts as a shallow <donor impurity>, introducing a localized level $E_D$ just below the <conduction band>. Ionizing that donor costs much less energy than excitation across the full <band gap>.
At the lowest temperatures, donor electrons remain bound and the <valence band> is filled. Mobile-carrier populations are negligible: this is <dopant freeze-out>. Initial heating releases donor electrons into the <conduction band>, producing a rapid rise in conductivity governed by the shallow-donor energy scale. The positive ionized donor charge compensates the released electronic charge; the bulk need not acquire net charge.
At intermediate temperatures, nearly all donors are ionized but intrinsic electron–hole generation is still small. In an uncompensated sample,
$$
n\simeq N_D,\qquad p\ll n,\qquad \sigma\simeq eN_Du_e,
$$
where $N_D$ is donor number density. The supply of donor electrons is finite, so further heating cannot keep creating them. This is the \b[extrinsic plateau] under the approximation of weakly varying mobility, and its <carrier density> is much larger than that of the undoped sample at the same temperature. In real material, scattering changes mobility, so donor exhaustion alone predicts a carrier-density plateau rather than guaranteeing perfectly flat conductivity.
At higher temperatures, intrinsic generation becomes comparable to and then exceeds the donor contribution. Once $n_i\gg N_D$, the populations approach $n\simeq p\simeq n_i$, and activation across the main gap causes the steeper rise. The relevant regimes are therefore \b[freeze-out, donor exhaustion, and intrinsic excitation], not three successive types of dopant.
\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2023/paper-2-semiconductor-regimes.png]
{title=Donor occupation, Fermi-level position and three conductivity regimes}
{description=Original illustrative model, not measurements of arsenic-doped silicon. The right panel solves charge neutrality with partial donor ionization and intrinsic carriers, assuming fixed equal electron and hole mobilities. Chosen parameters are $E_g=1.10\,\mathrm{eV}$, donor binding $0.050\,\mathrm{eV}$ and $N_c=N_v=100N_D(T/300\,\mathrm K)^{3/2}$. The left panel illustrates the resulting room-temperature energy ordering, including a Fermi level below the donor level.}
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