= Solution
{c}
At $300\,\mathrm K$, the two-dimensional <effective density of states> values are
$$
N_c\simeq2.81\times10^{11}\,\mathrm{cm}^{-2},\qquad N_v\simeq4.43\times10^{12}\,\mathrm{cm}^{-2}.
$$
Using $E_g=0.36\,\mathrm{eV}$ and $k_BT\simeq0.02585\,\mathrm{eV}$ gives
$$
\boxed{n=p=n_i=\sqrt{N_cN_v}\,e^{-0.36/(2\cdot0.02585)}\simeq1.06\times10^9\,\mathrm{cm}^{-2}.}
$$
For the doped sample, estimate the intrinsic-to-extrinsic crossover by $n_i(T_*)\simeq n_d=10^9\,\mathrm{cm}^{-2}$. With the masses and gap held fixed, $n_i(T)=A T e^{-B/T}$, where
$$
A\simeq3.72\times10^9\,\mathrm{cm}^{-2}\mathrm K^{-1},\qquad B=\frac{E_g}{2k_B}\simeq2089\,\mathrm K.
$$
Solving $AT_*e^{-B/T_*}=10^9\,\mathrm{cm}^{-2}$ gives
$$
\boxed{T_*\simeq298\,\mathrm K\approx300\,\mathrm K.}
$$
It is a smooth carrier crossover, not a thermodynamic phase transition. Donors are essentially ionized near this temperature because their binding energy, calculated next, is much smaller than $k_BT_*$. At substantially higher temperatures intrinsic pairs dominate; below the crossover and above appreciable donor freeze-out, the majority density approaches $n_d$.
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