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\sect{Ideal Fermi gas in semiconductor}

Given electron gas in a metal. The state uniparticle density
function is
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\[g\left(E\right)= g_{v}\left(E\right) + g_{c}\left(E\right)\]
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Assume that near the gap exists area  the proximity
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\[g_{c}\left(E\right)\simeq L\frac{V}{\left(2\pi\right)}\cdot
\left(2m_{c}\right)^{\frac{1}{2}}\left(E-E_{c}\right)^{\frac{1}{2}}\]
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\[g_{v}\left(E\right)\simeq L\frac{V}{\left(2\pi\right)}\cdot
\left(2m_{v}\right)^{\frac{1}{2}}\left(E_{v}-E\right)^{\frac{1}{2}}\]
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We define an occupation function of the holes in the valence band
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\[P^{k}\left(E-\mu\right)\equiv 1-f\left(E-\mu\right)\]
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Assume that the metal is attached to the reservoir 
with temperature ${\beta}$ and chemical potential ${\mu}$.
Assume that that ${E_{v}+T<<\mu<<E_{c}-T}$ and justify the
boltzman's approximation in the conductance band 
${f\left(E-\mu\right)\simeq e^{-\beta\left(E-\mu\right)}}$, 
and in the valence band 
${f^{k}\left(E-\mu\right)\simeq e^{\beta\left(E-\mu\right)}}$.
Now, find in the usual way the functions 
${N_{c}\left(\beta,\mu\right),N_{c}^{h}\left(\beta,\mu\right)}$.
The result is
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\[N_{c}\left(\beta,\mu\right)=2\cdot
V\left(\frac{m_{c}}{2\pi}\right)^{\frac{3}{2}}T^{\frac{3}{2}}exp\left(\frac{\mu-E_{c}}{T}\right)\]
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\[N_{v}\left(\beta,\mu\right)=2\cdot
V\left(\frac{m_{v}}{2\pi}\right)^{\frac{3}{2}}T^{\frac{3}{2}}exp\left(-\frac{\mu-E_{v}}{T}\right)\]
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Now, assume you begin with closed system in temperature zero. It's
given that then, the valence band is full and the conductive band is
empty. now you raise the temperature (The system is empty!). 
Assume ${m_{c}=m_{v}}$ and show that
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\[\mu\left(T\right)=\frac{E_{c}-E_{v}}{2}\]
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\[
N_{c}=N_{v}^{h}\propto
T\propto^{\frac{3}{2}}exp\left(-\frac{E_{g}/2}{T}\right)
\]




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