$10$pts $(2$pts for each$)$

For a long$-$channel nMOSFET with $t_{ox}=2\mathrm{.}4nm$ and $N_{\text{s}\text{u}\text{b}}=2{10}^{17}c{m}^{-3},$

$($a$)$ Estimated subthreshold slop S in $m\frac{V}{?}$ decade not very far away from strong inversion. Notice that S may degrade in deep subthreshold region, which does not need to be considered here.

$($b$)$ To obtain Ion$/$Ioff $={10}^7,$ what is the minimal voltage swing?

$($c$)$ If S need to be less than $90m\frac{V}{?}$ decade under the constraint of fixed $t_{ox}=2\mathrm{.}4nm,$ what is the constraint on $N_{\text{s}\text{u}\text{b}}.$

$($d$)$ Derive the expression of $g_m$ and $\frac{g_m}{I_D}$ in the subthreshold region when $V_{DS}$ is enough to put the nMOSFET into saturation.

$($e$)$ For the gate stack of $Hf\frac{O_2}{S}i{O}_2(3n\frac{m}{5}nm)$ with the same $N_{\text{s}\text{u}\text{b}}=2{10}^{17}c{m}^{-3},$ what is the S now?