$($a$)$ Consider an n polysilicon$$silicon dioxide$$n$-$type silicon MOS structure. Let Nd $41015$ cm$3.$ Calculate the ideal fl at$-$band voltage for tox $20$ nm $200.($b$)$ Considering the results of part $($a$),$ determine the shift in fl at$-$band voltage for $($i$)$ Q ss $41010$ cm$2$ and $($ii$)$ Q ss $1011$ cm$2.($c$)$ Repeat parts $($a$)$ and $($b$)$ for an oxide thickness of tox $12$ nm $120.$