$14.$ INSTRINSIC FERMI ENERGY $(\backslash (\backslash$mathbf$\{10\}\backslash )$ points$)$ Temperature dependence of the intrinsic Fermi energy.

a$)$ For GaAs at room temperature, the conduction band effective density of states is $4\mathrm{.}7$ x $\backslash (10^\{17\}/\backslash$mathrm$\{$cm$\}^\{3\}\backslash )$ and valence band effective density of states is $\backslash (7\mathrm{.}0\backslash$times $10^\{18\}/\backslash$mathrm$\{$cm$\}^\{3\}\backslash ).$ How far is the intrinsic energy above$/$below midgap at $300$ K $?$

b$)$ At $100$ K $?$

c$)$ At zero degrees Kelvin?

d$)$ At $500$ K $?$

e$)$ Plot by hand an energy$-$versus$-$temperature plot. Let the conduction band energy and valence band energy be drawn as temperature$-$independent solid horizontal lines, and the midgap energy as a dashed line. Let the horizontal axis represent absolute temperature. Draw the intrinsic energy relative to mid$-$gap at all $4$ of the above temperatures, roughly to scale, assuming that GaAs has an approximately temperature$-$independent band gap of $1\mathrm{.}42$ eV $.$