Problem $1$: $(\backslash (\backslash$mathbf$\{20\}\backslash )$ points$)$ Following questions, consider the below low frequency gate capacitance $($per unit area$)$ vs$.$ gate voltage characteristic for a metal gate p$-$channel MOSFET.

$($a$)$ Match approximate regions or points of above figure among the below list. $(6$ points$)$

$($b$)$ What is the oxide capacitance per unit area? $(2$ points$)$

$($c$)$ Draw band diagram for each points from $1$ to $6.($Hint : diagram should include $\backslash ($ E$\_\{$C$\},$ E$\_\{$V$\}\backslash ),$ and Fermi energy level for the both metal and semiconductor; work function of semiconductor is greater than work function of metal$)(12$ points$)$