$2.(15$ points$)$ Beginning with band alignment before the

contact, draw $($by hand$)$ bandstructure $($E vs$.$ x$)$ of the

following structures and indicate EC$,$ EV$,$ EF$,$BI and other

parameters: $$EC $,$EV where applicable:

$($a$)$ p$-$n

$+$

homo$-$junction in GaAs ;

$($b$)$ p

$+$

$-$N heterojunction of p$+$

$-$GaAs and n$-$Al$0\mathrm{.}3$Ga$0\mathrm{.}7$As with conduction band offset $$EC $/$EG $=$

$0\mathrm{.}7$;

$($c$)$ p

$+$

$-$P heterojunction of p

$+$

$-$GaAs and p$-$Al$0\mathrm{.}3$Ga$0\mathrm{.}7$As with conduction band offset $$EC $/$EG $=$

$0\mathrm{.}7$

$($d$)$ Schottky junction of Au$/$p$-$GaAs with Schottky barrier $0\mathrm{.}8$ eV$.$

Hint: begin with band alignment with respect to vacuum; consider depletion width vs$.$ doping.