In a simplified model of an un-doped semiconductor, the actual distribution of energy states may be replaced by one in which there are  N_v, states in the valence band, all these states having the same energy E_v, and N_c states in the conduction band, all these states having the same energy E_c,. The number of electrons in the conduction band equals the number of holes in the valence band.  

(a) Show that this last condition implies that in which 

(b) If the Fermi level is in the gap between the two bands and its distance from each band is large relative to kT, then the exponentials dominate in the denominators. Under these conditions, show that and that, if N_v ≈ N_c, the Fermi level for the un-doped semiconductor is close to the gap's center.

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Part (a) N, exp(AE,IkT) + 1 exp(AE,/kT) + 1 N. AE, = E, - E and AE, = -(E, – Ep). Part (b) (E, + E,) Er = kT In(N,IN) 2 2