Question

Compute the relative probability of finding an electron at the bottom of the conduction band relative to the probability of finding an electron at the top of the valence band in a silicon crystal at a temperature of (a) 3 K and (b) 300 K. Use Fermi-Dirac statistics. Compare your answer with the one given by the Boltzmann equation.

Answer 1

The probability of finding an electron with energy between E and E+dE is given by the fermi-dirac distribution "f(E)". The bottom of the conduction band has energy E_c while the top of the valence band has energy E_v. For Silicon, E_c-E_v=1.1eV.

Thus the probability of finding an electron at the bottom of conduction band is:

Similarly, the probability of finding electron at the top of valence band is:

For an intrinsic semiconductor at T=0K, the fermi energy E_f is exactly in between conduction and valence band i.e.

Thus the relative probability is given by:

(a) at T=3K

(b) at T=300K

from boltzmann distribution, the ratio would be

thus at T=3K, the ration is

and at T=300K,

P.S. : Here we do not need to consider the density of states "g(E)", since g(E)*f(E) is the density of electrons with energy E to E+dE and not the probability.