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PROBLEM $2(10$ POINTS$)$

A silicon step junction maintained at room temperature $(300$K$)$ is doped such that: $E_F=E_V+\frac{E_G}{4}$ on the $p-$side and $E_F=E_C-\frac{E_G}{3}$ on the $n-$side. The bandgap of Silicon is $1\mathrm{.}1$ eV $.$

$($a$)$ Draw the equilibrium energy band diagram for this junction. Draw neatly and carefully.

$($b$)$ From the diagram, determine the built$-$in potential $V_{bi}.$