$10.($a$)$ Find the threshold voltages of a conventional and a delta$-$doped heterostructure

AIGaAs$-$GaAs FETs.

$($b$)$ Evaluate the variations of these threshold voltages for two$-$monolayer fluctuations in

AlGaAs layer thickness.

Assuming that one monolayer $=38,$ in AIGaAs, the Schottky bamer height is $0\mathrm{.}9$ V$,$ the

conduction$-$band discontinuity is $0\mathrm{.}3$ eV$,$ the uniform doping in the conventional HEFT is

l$0$l$8$ ~m$-$~ with a thickness of $40$ nm$,$ the delta doping is located $40$ nm from the metal$-$

semiconductor interface, with a sheet charge density of $1\mathrm{.}5$~ $10$l$2$ cm$-2,$ and the dielectric

permittivity for AlGaAs is assumed to be $10-$l$2$ F$/$cm$.$