$3.[10$ points each for $($a$)-($b$),5$ points for $($c$)$; total $25$ point$]$ Consider a silicon bar that is doped a non$-$uniform distribution of phosphorus atoms given by $()=0/0$ where $0=1016$ cm$3$ and $0=0\mathrm{.}1$ cm$.$ The semiconductor is subjected to an electric field $=5$ V$/$cm along the x$-$direction. Assume $=1000$ cm$2/$V $$ s$,=500$ cm$2/$V$$s$,$ and the temperature $$ is $300$ K$.($a$)$ Derive an expression for the electron current density $()$ as a function of position x$,$ accounting for both drift and diffusion components. $($b$)$ Determine the total average current I through the semiconductor if the semiconductor has a cross sectional area of $=0\mathrm{.}01$ cm$2$ and the length of the semiconductor L is $0\mathrm{.}2$ cm$.($c$)$ Discuss qualitatively how the non$-$uniform doping profile affects the electric field distribution within the semiconductor and how this might affect the accuracy of the drift$-$diffusion model.