Fermi Level in Intrinsic semiconductors:

• For intrinsic semiconductors, it can be shown that fermi energy level EF lies midway between valence band and conduction.
• At any temperature T >$0 ^\circ \ K$, where$n_e : Number \ of \ electrons \ in \ conduction \ band, \ n_v= $Number of holes in valence band.

• We have,$n_e=N_C \ e^{-(E_C-E_F)/KT} $
  where$N_C=Efficiency $ density of states in conduction band
  and$n_v = N_V \ e^{-(E_F-E_V)/KT} $
  where$N_V=$ effective density of states in valance band

• For best approximation:$N_C = N_V \ $
  For intrinsic semiconductor$n_c= n_v$
  $N_C .\ e^{-(E_C-E_F)/KT} = N_V. \ e^{-(E_F-E_V)/KT} $

  Hence,$\dfrac {e^{-(E_C-E_F)/KT}}{\ e^{-(E_F-E_V)/KT}}=\dfrac {N_V}{N_C}$

$Hence ,\ e^{-(E_C-E_F-E_F+E_V)/KT} = \dfrac {N_V}{N_C}$

$Hence ,\ e^{-(E_C+E_V-2E_F)/KT} = \dfrac {N_V}{N_C}$

as$N_V=N_C=1$

$ e^{-(E_C+E_V-2E_F)/KT} = 1$ * Taking log on both sides we get,$ -(E_C+E_V-2E_F)/KT=0$
Hence,$ (E_C+E_V)=2E_F $ * $\dfrac {(E_C+E_V)}{2}=E_F $, Thus an intrinsic semiconductor lies at the center of the forbidden energy gap in the Fermi level.