Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
Lecture 34 : Intrinsic Semiconductors
  Using this the density of electrons in the conduction band ( $ n$) may be written as follows.
$\displaystyle n$ $\displaystyle =$ $\displaystyle \int_{E_c}^\infty e^{(E_F-E)/kT} \frac{1}{2\pi^2}\left(\frac{2m_e}{\hbar^2}\right)^{3/2}(E-E_c)^{1/2}dE$
$\displaystyle =$ $\displaystyle \frac{1}{2\pi^2}\left(\frac{2m_e}{\hbar^2}\right)^{3/2} \int_{E_c}^\infty e^{(E_F-E)/kT}(E-E_c)^{1/2}dE$
$\displaystyle =$ $\displaystyle \frac{1}{2\pi^2}\left(\frac{2m_e}{\hbar^2}\right)^{3/2} \int_0^\infty e^{(E_F-xkT-E_c)/kT}(kT)^{3/2}x^{1/2}dx$
$\displaystyle =$ $\displaystyle \frac{1}{2\pi^2}\left(\frac{2m_ekT}{\hbar^2}\right)^{3/2}e^{(E_F-E_c)/kT} \int_0^\infty e^{-x}x^{1/2}dx$
  where we have substituted
 
$\displaystyle x = \frac{E-E_c}{kT}$
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