$$

$(40$ pts$)$ A p $p^{+}-n-p$ Si BJT $($base width, $W_b=1m$ and Area $=2{10}^{-4}c{m}^2)$ has the following parameters at $T=300K.$

$[$ Emitter $][$ Base$][Collector]$

$Na={10}^{18}c{m}^{-3},Nd={10}^{16}c{m}^{-3},Na={10}^{15}c{m}^{-3}$

${}_n=0\mathrm{.}1s,{}_p=2500ps,{}_n=2s$

${}_n=150c\frac{m^2}{V}s,{}_n=1500,{}_n=1500$

${}_p=100,{}_p=400,{}_p=450$

a$)(5$ pts$)$ Sketch the energy band diagram at equilibrium condition from Emitter to Collector. Find contact potentials in E$-$B and B$-$C junctions.

b$)(5$ pts$)$ Calculate the equilibrium depletion width and the maximum electric field in B$-$C junction.

c$)(10$ pts $)$ Calculate the reverse saturation current in B$-$C junction

d$)(5$ pts$)$ Sketch the energy band diagram at the active mode. Clearly show the quasi$-$Fermi levels.

e$)(15$ pts$)$ Calculate $,B,$ and $.$