An N$-$channel MOSFET with N$+-$poly gate is fabricated on a $15$ cm P$-$type Si wafer

with oxide fixed charge density $=$ q $81010$cm$2,$ W $=50$m$,$ L $=2$m$,$ Tox $=5$ nm$.$

$($a$)$ Determine the flat$-$band voltage, Vfb$.$

$($b$)$ What is the threshold voltage, Vt$?$

$($c$)$ A circuit designer requested N$-$MOSFET with Vt $=0\mathrm{.}5$ V from a device engineer.

It was not allowed to change the gate oxide thickness. If you are the device

engineer, what can you do$?$ Give specific answers including what type of

equipment to use.