$4.$ Fermi Dirac Statistics $(16$ Points$)$

Ge is doped with phosphorous, an n$-$type impurity, that produces an donor energy level, $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$d$\}\}\backslash ),$ located $0\mathrm{.}012$ eV below the conduction band edge, $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$c$\}\}\backslash ).$ The bandgap energy, $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$g$\}\}\backslash ),$ for Ge is $0\mathrm{.}68$ eV and its Fermi level, $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$f$\}\}\backslash ),$ is located $0\mathrm{.}2$ eV above its intrinsic level for the doping level present. Assume the temperature is $300$ K $.$

a$.$ Sketch an energy band diagram for this Ge material. Label the conduction band edge, $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$C$\}\}\backslash ),$ the valence band edge, $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$V$\}\}\backslash ),$ the Fermi level, $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$F$\}\}\backslash ),$ the intrinsic Fermi level, $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$i$\}\}\backslash ),$ and the donor energy level, $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$d$\}\}\backslash ),$ showing numerical values for them as appropriate.

b$.$ If the difference between the Fermi level and the donor energy level is $0\mathrm{.}128$ eV $,$ use the Fermi Dirac statistics to calculate the probability of finding an electron at $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$d$\}\}\backslash ).$

c$.$ If the difference between the Fermi level and the conduction band edge is $0\mathrm{.}140$ eV $,$ use the Fermi Dirac statistics to calculate the probability of finding an electron at $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$c$\}\}\backslash ).$

d$.$ The probability calculated in part b above should be VERY small. Explain why the Fermi$-$Dirac calculation produces such a small probability of finding an electron at $\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$mathrm$\{$d$\}\}\backslash )$ and the physical significance of that small probability.