Problem $2(24$ points$)$

A P$-$N junction is created by doping the left side of a piece of silicon with ${10}^{15}$ atoms $\frac{?}{c{m}^3}$ of phosphorus and the right side with ${10}^{17}$ atoms$/c{m}^3$ of boron. Assume that the dopants are fully ionized, and assume the junction is at $x=0$ with the positive $x$ axis pointed to the right. Assume a temperature of $300$ K $,$ and assume the following values for silicon at $300$ K :

$E_g=1\mathrm{.}12eV,n_i={10}^{10}c{m}^{-3},N_C=3{10}^{19}c{m}^{-3},N_v=1{10}^{19}c{m}^{-3}.$

a$)$ Which side of the junction $($left or right$)$ is the P $-$side? Which side $($left or right$)$ is the N $-$side? $(2$ pts$)$

b$)$ Calculate $E_F-E_{Fi}$ on the N side, and $E_{F-}-E=$ on the P side, and write your answers in units of eV$.(4$ pts$)$

c$)$ Sketch $($by hand, roughly to scale$)$ an energy diagram for this P$-$N junction. Label the P$-$and Nsides, label $E_c,E_V,E_F,E_F,$ and label the values of $E_F-E_{Fi}$ on the $N$ side, and $E_F-E_F$ on the $P$ side from part $($b$)$ above. $(6$ pts$)$

d$)$ Calculate the built$-$in voltage $V_{bi}$ for this p$-$n junction, and write your answer in units of volts. $(2$ pts$)$

e$)$ Calculate the depletion widths on the N and P sides, respectively, $x_n$ and $x_p.$ Write your answers in units of centimeters $($cm$),$ meters $($m$),$ micrometers $($um$)$ and nanometers $($ nm $).(6$ pts $)$

f$)$ Calculate the magnitude of the maximum electric field $($in units of $\frac{V}{c}m),$ and also describe its direction. $(4$ pts$)$

g$)$ Calculate the value of $x$ at which $n=n_i($i$.$e$.$ at which pts$)$

h$)$ Calculate the value of $n$ at $x=0(4$ pts$)$