Problem $2$: p$-$i$-$n Diode

A p$-$i$-$n diode is a semiconductor diode with a wide, lightly doped intrinsic region between a p$-$type region and an n$-$type region. P$-$i$-$n diodes are used as fast switches, photodetectors, and for high$-$voltage power electronics applications. Consider a silicon p$-$i$-$n diode with a doping profile as shown below. Assume $\backslash ($ T $\backslash )\backslash (=300\backslash$mathrm$\{$~K$\}\backslash ).$

a$)$ Applying the depletion approximation $($i$.$e$.$ assuming that the electron and hole concentrations are zero within the depletion region$),$ sketch the charge density distribution, electric field distribution, and potential distribution as a function of position $\backslash ($ x $\backslash ),$ for zero bias.

b$)$ Find the built$-$in potential, $\backslash ($ V$\_\{\backslash$mathrm$\{$bi$\}\}\backslash ),$ and the lengths of the depletion regions in the n$-$type region and the ptype region.

c$)$ How does the maximum electric field in this p$-$i$-$n diode compare to that for a pn diode which contains no intrinsic region but has the same dopant concentrations in the n$-$type and p$-$type regions $($i$.$e$.$ as in Problem $1)?$