$(8$pts$)$ In a simple cubic system, please draw $($or sketch$)$ the following planes: $($a$)(100),($b$)(111),($c$)(221),($d$)(322).$ Use x$-$y$-$z system in your plots.

An abrupt silicon pn junction at zero bias has dopant concentrations of $N_a={10}^{17}c{m}^{-3},$ and $N_d=5{10}^{15}c{m}^{-3}.T=300K($a$)(5$ pts $)$ Calculate Fermi level in each side of the junction with respect to the intrinsic Fermi level. $($b$)(5$pts$)$ Determine the built$-$in potential of this junction. $($c$)(5$pts$)$Determine $x_n,x_p,$ and the peak electric field of this junction.