$(30$ points$)$ Consider an n$-$channel enhancement mode metal oxide semiconductor field effect transistor $($MOSFET$)$ made out of silicon with substrate doping Na $=2$ x $1017$ cm$3,$ an oxide thickness of $7$ nm$,$ a channel width to length ratio of $350$ and operated at $300$K$.$

a$)$ Describe the main principles of operation of this transistor. Why is it called n$-$channel given that the substrate is p$-$type?

b$)$ What polarity of gate voltage with respect to the substrate is required to obtain conduction between drain and source? What happens if the opposite voltage polarity is applied to the gate?

c$)$ Calculate the threshold voltage.

d$)$ Calculate the channel resistance at a gate voltage of $2\mathrm{.}2$ V and very low drain voltage. Hint: use the mobility on the equation sheet. If you did not get an answer to part $($c$)$ you may assume $1\mathrm{.}5$ V$.$

e$)$ When the gate voltage is a few volts larger than the threshold voltage, describe and explain pinchoff and how it affects the drain$-$source current.

f$)$ Calculate the saturation current at a gate voltage of $2\mathrm{.}2$ V$.$

g$)$ Calculate the transconductance at a gate voltage of $2\mathrm{.}2$ V$.$