Consider a MOS capacitor with an n$-$type silicon substrate. A metal$$semiconductor work function difference of $=0\mathrm{.}3$ is required. Determine the silicon doping concentration required to meet this specification when the gate is $($a$)$ n$+$ polysilicon, $($b$)$ p$+$ polysilicon, and $($c$)$ aluminum. Modified work function for Al$,=3\mathrm{.}2$ and modified electron affinity for Si$,=3\mathrm{.}25.$