Chapter 41, Problem 34P

To determine

To show:

(a) that when the number of electrons in the conduction band equals the number of holes in the valence band, it implies

$\frac{N_c}{\exp \frac{{∆E}_c}{kT}+1 }= \frac{N_v}{\exp \frac{{∆E}_v}{kT}+1 }$

in which ∆E_c =  E_c - E_F and ∆E_v = - (E_v - E_F).

(b) that under the conditions when the Fermi level is in the gap between the two bands and its distance from each band is large relative to kT

$E_{f }= \frac{E_c+ E_v}{2}+\frac{kT ln \left(\frac{N_v}{N_c}\right)}{2}$