Aluminum $($work function $4\mathrm{.}1$ eV $)$ is in contact with silicon $($electron affinity $4\mathrm{.}05$ eV $,$ dielectric coefficient ${}_r=12\mathrm{.}0,$ effective density of states in the conduction band $N_C=2\mathrm{.}86{10}^{19}c{m}^{-3},$ effective density of states in the valence band $N_V=2\mathrm{.}66{10}^{19}c{m}^{-3},$ and bandgap $1\mathrm{.}12$ eV at room temperature$)$ doped with ${10}^{17}c{m}^{-3}$ of arsenic at room temperature.

$($a$)$ Calculate the work function of the silicon