Problem $3(20$ points$)$

A P$-$N junction is created by doping the left side of a piece of silicon with ${10}^{17}$ atoms $\frac{?}{c{m}^3}$ of phosphorus and the right side with ${10}^{17}$ atoms$/$cm ${?}^3$ of boron. Assume that the dopants are fully ionized, and assume the junction is at $x=0$ with $x+$ pointed to the right. Assume that the band gap of silicon is $E_E=1\mathrm{.}12eV,$ and that $E_g,E_C,E_V,$ and $E_{Fi}$ do not change with temperature.

Assume that the intrinsic carrier concentration in silicon $($Si$)$ varies according to the following plot:

TEMPERATUREINK

a$)$ Calculate values of $E_{F-}{E}_{Fi}$ on the $N$ side and $E_{Fi}{E}_F=0$ the $P$ side for this junction at $300$ K $.(4$ pts$)$

b$)$ Sketch $($by hand, roughly to scale$)$ an energy diagram for this P$-$N junction at $300$ K $.$ Label the P and $N-$sides, and label $E_C,E_v,E_{F,}{E}_F$ Also label the values of $E_F-E_{Fi}$ on the $N$ side and $E_F-E_F$ on the $P$ side that you calculated in a$)$ above. $(4$ pts$)$

c$)$ Calculate values of $E_F-E_{Fi}$ on the $N$ side and $E_{F_i}{E}_F$ on the $P$ side for this junction at $1000$ K $.(4$ pts$)$

d$)$ Sketch $($by hand, roughly to scale$)$ an energy diagram for this P$-$N junction at $1000$K$.$ Label the Pand $N-$sides, and label $E_{C,}{E}_V,E_{Fi},E_F.$ Also label the values of $E_F-E_{Fi}$ on the $N$ side and $E_{Fi}-E_F$ on the $P$ side that you calculated in c$)$ above. $(4$ pts$)$