The number of electrons in the conduction band of an insulator or intrinsic semiconductor is governed chiefly by the Fermi factor. Since the valence band in these materials is nearly filled and the conduction band is nearly empty, the Fermi energy EF is generally midway between the top of the valence band and bottom of the conduction band, i.e., at Eg/2, where Eg is the band gap between the two bands and the energy is measured from the top of the valence band.

(a) In silicon, Eg ≈ 1.0 eV. Show that in this case the Fermi factor for electrons at the bottom of the conduction band is given by exp(–E/2kT) and evaluate this factor. Discuss the significance of this result if there are 1022 valence electrons per cubic centimeter and the probability of finding an electron in the conduction band is given by the Fermi factor.

(b) Repeat the calculation in (a) for an insulator with a band gap of 6.0 eV.