= Solution
With a positive charge magnitude e, the relaxation-time equation for electrons is $m_e^*\dot{\mathbf v}=-e\mathbf E-m_e^*\mathbf v/\tau$. Steady drift is opposite the field, $\mathbf v_d=-e\tau\mathbf E/m_e^*$, so the positive <electron mobility> and <electrical conductivity> are
$$
\boxed{\mu_e=e\tau/m_e^*,\qquad\mathbf j=-ne\mathbf v_d=ne\mu_e\mathbf E,\qquad
\sigma=ne^2\tau/m_e^*.}
$$
This is the weak-field <Drude model>; the mobility is a magnitude, not the signed velocity-to-field ratio.
For an intrinsic nondegenerate semiconductor, $n_i=p_i=\sqrt{N_cN_v}e^{-E_g/(2k_BT)}$. Equal electron and hole masses give $N_c=N_v$ and intrinsic chemical potential midway between the band edges. At300K, the exponential is $e^{-19.34}=3.98\times10^{-9}$. Using the later supplied mass $0.015m_e$ gives
$$
N_c=2\left(\frac{m^*k_BT}{2\pi\hbar^2}\right)^{3/2}=4.61\times10^{22}\,\mathrm{m^{-3}},\qquad
\boxed{n_i\simeq1.84\times10^{14}\,\mathrm{m^{-3}}.}
$$
This is many orders below a typical metal's carrier density. Since $\sigma_i=en_i(\mu_e+\mu_h)$, ordinary finite carrier mobilities leave its intrinsic conduction negligible on a metallic scale; the gap alone does not specify an exact conductivity without the mobilities.
Under the explicitly stipulated full donor ionization, $n\simeq N_d=10^{24}\,\mathrm{m^{-3}}$ and the supplied mobility is $0.0100\,\mathrm{m^2\,V^{-1}\,s^{-1}}$. Therefore
$$
\boxed{\sigma=eN_d\mu_e=1.60\times10^3\,\mathrm{S\,m^{-1}}.}
$$
The band diagram places the intrinsic chemical potential at midgap and a shallow isolated donor level just below the conduction edge. The doped chemical potential moves upward. Quantitatively, the <degeneracy test for a fully ionized light-mass donor gas> gives $N_d/N_c=21.7$, so a nondegenerate logarithmic chemical-potential formula is inconsistent. The parabolic-band zero-temperature estimate is $E_{F,d}-E_c=\hbar^2(3\pi^2N_d)^{2/3}/(2m^*)=0.243\,\mathrm{eV}$; its300K value is about0.241eV. The diagram accordingly shows the doped chemical potential inside the conduction band, not between an isolated donor level and the edge.
The <hydrogenic donor state> scaling using the supplied hydrogen values is
$$
\boxed{E_B=\frac{m^*/m_e}{\epsilon_r^2}(13.6\,\mathrm{eV})=0.630\,\mathrm{meV},\qquad
a^*=\frac{\epsilon_r}{m^*/m_e}a_0=63.6\,\mathrm{nm}.}
$$
The binding is much smaller than $k_BT=25.9\,\mathrm{meV}$, indicating a shallow donor on the isolated-state estimate. However, the mean donor spacing is only $N_d^{-1/3}=10\,\mathrm{nm}$, well below the orbit radius. The <donor-state overlap estimate> invalidates an independent isolated-level ionization model at this concentration. Shallow binding supports abundant mobile carriers, but the <full ionization versus isolated donor occupation> distinction means full ionization cannot be justified solely by comparing binding with temperature while simultaneously retaining isolated levels and a degenerate chemical potential. The conductivity above follows the question's carrier-density assumption; an overlapping impurity/conduction-band description is needed for a self-consistent microscopic account. <University Physics Volume 3 by Samuel J. Ling and others>[§9.6, donor levels and semiconductor doping; isolated-level picture distinguished here from the high-density limit]{ref}
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2010/paper-1-pa2010-donor-bands.png]
{title=Donor band reference}
{description=The assumed density is degenerate; the isolated donor line is a reference, not an independent-level model.}
{source=paper-1-pa2010-donor-bands.py}
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