Question #43718

Consider a semiconductor at absolute temperature $T$ with a band gap $E_g$ such that $E_g \gg k_B T$, where $k_B$ is the Boltzmann constant. The density of states for the conduction and valence bands are given by $g_c(E) = A_c \sqrt{E - E_c}$ for $E \ge E_c$ and $g_v(E) = A_v \sqrt{E_v - E}$ for $E \le E_v$, respectively. The intrinsic carrier concentration, calculated assuming classical Maxwell-Boltzmann statistics for both bands, is $n_i$. The semiconductor is heavily doped n-type such that its Fermi level $E_F$ exactly coincides with the conduction band edge $E_c$. Let the actual product of electron and hole concentrations in this degenerate state be denoted by $(np)_{\text{actual}}$. Given the definite integral ratio $\frac{\int_0^\infty \frac{\sqrt{x}}{1+e^x} dx}{\int_0^\infty \sqrt{x} e^{-x} dx} = \alpha$, which of the following expressions correctly represents $(np)_{\text{actual}}$?