A PN$+$ silicon diode has NA $=10^16$ cm$^-3$ in P and ND $=10^18$ cm$^-3$ in N$,$ Tp$,=$ tn $=1$ us and A $=1\mathrm{.}2$x$10^-4$ cm$^2.$ Determine: $($a$)$ the conductivities of the P and N regions; $($b$)$ the diode current when the applied voltage in the reverse direction is $0\mathrm{.}01$ V; $($c$)$ under Va $=0\mathrm{.}3$ V$,$ the excess minority carrier concentration at x$\text{'}=1$ um in N; $($d$)$ under Va $=0\mathrm{.}5$ V$,$ the electron current at x$\text{'}=1$ um in the N region.