Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 237

since a hole isdefined as the absence of anelectron. We maythenwrite

where is the energy of a hole and we have used the nondegeneracy con-

ditionfor holes The number of holes is

The energy of ahole(from the bottom of theconduction band) is

Therefore,similar to(a):

The product of the concentrations of electrons and holes does not de-

pend on thechemical potential as we see by multiplying(S.4.104.3)

and (S.4.104.8):

We did not use the fact thatthere are noimpurities. The onlyimportant

assumption isthat whichimpliesthat thechemical potential

is not too close toeither the conduction or valencebands.

c) Since, in the case of anintrinsic semiconductor (everyelectron
in the conduction bandleaves behind ahole in thevalenceband), we can
write, using (S.4.104.9),