$17.$ UNBIASED p$-$n JUNCTION $(10$ points$)$

Consider a p$-$n junction with p$-$type side on the left and n$-$type side on the right.

a$)$ Assume the p$-$n junction is under zero bias. Let the donor doping be TWICE the acceptor doping. What is the relative width of the depleted p$-$side, compared to the total width?

What is the relative width of the depleted $\backslash ($ n $\backslash )-$side compared to the total width?

b$)$ Keeping in mind your answer to part a$),$ draw this p$-$n junction at ZERO BIAS, below, where $\backslash ($ x$=0\backslash )$ represents the metallurgical junction between the p $-($left$)$ and n $-($right$)$ sides. Indicate on the $\backslash ($ x $\backslash )-$axis the extent of the depletion region on the $\backslash ($ p $\backslash )-$side with an appropriate variable name, and the depletion on the $\backslash (\backslash$mathbf$\{$n$\}\backslash )-$side with an appropriate variable name. The magnitude of the energy gap Eg is indicated, left. Consider that the sign of the local curvature of the conduction and valence bands in the depletion region is determined by the local charge density. Your drawing should indicate the Fermi energy, conduction band energy, and valence band energy, with appropriate variable names. Indicate with a vertical arrow the total band$-$bending in the junction with the label $\backslash ($ q V$\_\{$b i$\}\backslash ).$