$4\mathrm{.}19$ Consider a Schottky diode with the doping profile shown in Fig. $4-53.$ Assume that the built$-$in potential ${}_{bi}$ is $0\mathrm{.}8$ V $.$

$\backslash$table$[[$Metal$,$N type Si$]]$

FIGURE $4-53$

$($a$)$ Sketch $\frac{1}{C^2}$ vs$.V($the reverse bias voltage$)$ qualitatively. Do not find numerical values for $C.$

$($b$)$ Sketch the electric field profile for the bias condition when $W_{\text{d}\text{e}\text{p}}=2m.$ Again, do not find numerical values for the electric field.

$($c$)$ What is the potential drop across the junction in part $($b$)?$

$($d$)$ Derive an expression of $C$ as a function of $V$ for $W_{\text{d}\text{e}\text{p}}>1m.$