Question $6[10$ Marks$]$: Find the thickness of the undoped spacer layer $d_0$ so that the two$-$dimensional

electron gas concentration of an AlGaAs$/$GaAs heterojunction is $1\mathrm{.}25{10}^{12}c{m}^{-2}$ at zero gate bias.

Assume that the $n-$AlGaAs is doped to $1{10}^{18}c{m}^{-3}$ and has a thickness $d_1$ of $50nm,$ the Schottky

barrier height is $0\mathrm{.}89V$ and $\frac{E_C}{q}$ is $0\mathrm{.}23eV.$ Assume the permittivity of the AlGaAs is $12\mathrm{.}3.$