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\sect{Ideal Fermi gas in semiconductor}

Consider a gas of electrons in a semiconductor, 
the temperature is $T$, and the chemical potential is ${\mu}$. 
The single particle density of states ${g(E) = g_{v}(E)+g_{c}(E)}$
consists of valence and conduction bands, separated 
by a gap ${E_g=E_c-E_v}$. In the vicinity of the energy gap, 
one can use the following approximation:

\[
g_{c}\left(E\right)\approx 2\frac{V}{\left(2\pi\right)^2}\cdot\left(2m_{c}\right)^{\frac{3}{2}}\left(E-E_{c}\right)^{\frac{1}{2}}
\]

\[
g_{v}\left(E\right)\approx 2\frac{V}{\left(2\pi\right)^2}\cdot\left(2m_{v}\right)^{\frac{3}{2}}\left(E_{v}-E\right)^{\frac{1}{2}}
\]

The electron has Fermi occupation ${f(E-\mu)}$, 
optionally it is customary to define an occupations 
function  ${\tilde{f}(E-\mu)=1-f}$ for the holes. 

\Dn

(a) What are the occupation functions of the electrons
in the conduction band, and of the holes in the valance band, 
in the Boltzmann approximation.

\Dn

(b) What is the condition for the validity of this approximation?
Assume that this condition is satisfies in the following items.

\Dn

(c) Derive expressions for the number of electrons $N_c(\beta,\mu)$
and for the number of holes $N_v(\beta,\mu)$. 
in the conductance and valence band respectively.
Explain how the product $N_cN_v$ could be optionally deduced 
from the law of mass action.  

\Dn

(d) Consider a {\em closed} system, such that at $T=0$ the valence band 
is fully occupied, while the conductance band is empty. 
The temperature is raised to~$T$. Find the chemical potential 
and evaluate $N_c(T)$ and $N_v(T)$.

\Dn


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