Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
Lecture 34 : Intrinsic Semiconductors
  giving
$\displaystyle \boxed{ E_F = \frac{E_c+E_v}{2}}\eqno (C)$
  i.e. the Fermi level lies in the middle of the forbidden gap . Note that there is no contradiction with the fact that no state exists in the gap as $ E_F$ is only an energy level and not a state.
  By substituting the above expression for Fermi energy in (A) or (B), we obtain an expression for the number density of electrons or holes ( $ n=p=n_i$)
 
$\displaystyle \boxed{n_i=\frac{1}{4}\left(\frac{2kT}{\pi\hbar^2}\right)^{3/2}(m_em_h)^{3/4}e^{-\Delta/2kT}}\eqno (D)$
  where $ \Delta$ is the width of the gap.
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