$4\mathrm{.}2$ Consider a silicon PN step junction diode with $N_d={10}^{16}c{m}^{-3}$ and $N_a=5{10}^{15}c{m}^{-3}.$ Assume $T=300K.$

$($a$)$ Calculate the built$-$in potential ${}_{bi}.$

$($b$)$ Calculate the depletion$-$layer width $(W_{\text{d}\text{e}\text{p}})$ and its length on the N side $(x_n)$ and $P$ side $(x_p).$

$($c$)$ Calculate the maximum electric field.

$($d$)$ Sketch the energy band diagram, electric potential, electric field distribution, and the space$-$charge profile.

$($e$)$ Now let $N_a={10}^{18}c{m}^{-3}.$ Repeat $($a$),($b$),$ and $($c$).$ Compare these to the previous results. How have the depletion widths changed?