= Solution
{c}
With equal constant <electron mobility> and <hole mobility> $\mathcal M$, the sheet <electrical conductivity> is $\sigma=e\mathcal M(n+p)$. In the intrinsic regime, $n+p\simeq2n_i\propto T e^{-E_g/(2k_BT)}$, so
$$
\log\sigma=\mathrm{constant}+\log T-\frac{E_g}{2k_BT}.
$$
It decreases steeply with increasing $1/T$ on the high-temperature side. There is a small logarithmic curvature rather than a perfectly straight activation line.
In the fully ionized extrinsic regime, <charge neutrality> and the <semiconductor mass-action law> give $n-p=n_d$, $np=n_i^2$, hence
$$
n+p=\sqrt{n_d^2+4n_i^2},\qquad\sigma\simeq e\mathcal M n_d\quad(n_i\ll n_d).
$$
This gives a nearly horizontal region passing through $\sigma(200\,\mathrm K)=0.01\,\Omega^{-1}$. At lower temperatures <semiconductor carrier freeze-out> decreases the ionized donor density and bends the curve downward again.
To locate that bend realistically, include <donor charge-state degeneracy>. For a neutral donor with two spin states and an empty donor with one state, the ionized density is
$$
n_d^+=\frac{n_d}{1+2(n/N_c)e^{E_B/(k_BT)}}.
$$
Solve $n-n_i^2/n=n_d^+$ for the mobile density. In the asymptotic freeze-out regime, where $p$ is negligible and $n\ll n_d$,
$$
n\simeq\sqrt{\frac{n_dN_c}{2}}e^{-E_B/(2k_BT)},\qquad\log\sigma\simeq\mathrm{constant}+\frac12\log T-\frac{E_B}{2k_BT}.
$$
Its activation scale $E_B/2\simeq3.14\,\mathrm{meV}$ is much smaller than the intrinsic scale $E_g/2=180\,\mathrm{meV}$.
Within the specified interval the low-temperature downturn is modest: $N_c(30\,\mathrm K)\simeq2.81\times10^{10}\,\mathrm{cm}^{-2}$, and the partial-ionization equation gives $n(30\,\mathrm K)\simeq0.65n_d$. The sample is not yet deep in the asymptotic freeze-out limit at 30 K. Calibrating the mobility to the stated 200 K value gives representative values
$$
\boxed{\sigma(30\,\mathrm K)\simeq0.0066,\quad\sigma(200\,\mathrm K)=0.010,\quad\sigma(300\,\mathrm K)\simeq0.024,\quad\sigma(600\,\mathrm K)\simeq1.39\quad\text{in }\Omega^{-1}.}
$$
The donor statistical weights were not explicitly supplied, so they are a stated assumption for these illustrative curve details; a one-state donor approximation changes the low-temperature rounding but not the three physical regimes or their activation scales.
\Image[/past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2019-inas-conductivity-regimes.png]
{title=Two-dimensional InAs conductivity from intrinsic carriers to incipient donor freeze-out}
{description=Original ideal-model calculation using the supplied gap, masses and donor density, the derived 2D donor binding energy, twofold neutral-donor spin weight and equal temperature-independent carrier mobilities. The vertical scale is normalized to the stated 200 K conductivity. The plotted range is 30 to 600 K; donor freeze-out is only beginning at its cold end. The dashed line is the fully ionized extrinsic approximation.}
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