A MOS structure is made with a p$-$type Si substrate with a doping concentration of $2\mathrm{.}7$x$1018/$cm$3,$ oxide thickness of $1\mathrm{.}1$ nm with a dielectric constant of $3\mathrm{.}9,$ and a metal work function of $4\mathrm{.}5$ eV$.$ You can consider room temperature conditions with an applied voltage of $1$ V at the metal gate. Determine the following quantities in the case of an ideal interface $($Qox $=0).$

a$)$ The flatband voltage, VFB $.$

b$)$ The surface potential, S

c$)$ The electric field in the oxide, EOX $.$

d$)$ The electric field in the silicon at the surface, ES

e$)$ The depletion region depth, WD$.$