Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
Lecture 34 : Intrinsic Semiconductors
  The integral $ \int_0^\infty e^{-x}x^{1/2}dx$ is a gamma function $ \Gamma(3/2)$ whose value is $ \sqrt{\pi}/2$. Substituting this value, we get for the density of electrons in the conduction band
$\displaystyle \boxed{n=\frac{1}{4}\left(\frac{2m_ekT}{\pi\hbar^2}\right)^{3/2} e^{(E_F-E_c)/kT}= N_c e^{(E_F-E_c)/kT}}\eqno (A)$
  where
 
$\displaystyle N_c = \frac{1}{4}\left(\frac{2m_ekT}{\pi\hbar^2}\right)^{3/2}$
  One can in a similar fashion one can calculate the number density of holes, $ p$, by evaluating the expression
 
$\displaystyle p = \int_{E_v}^{-\infty} n_h(E)f_h(E)dE$
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