Under equilibrium conditions, the number of elec­tronic defects of energy E is given by, ,

n(E) = N(E) • F(E)

where N(E) is the volume density of electronic levels that have energy E (known as the density of states) and F(E) is the probability that a given level is occu­pied, called the Fermi-Dirac distribution function.

N(E) is a function of energy. It is the maximum density of electrons of energy E allowed (per unit volume of crystal) by the Pauli exclusion principle. For a semiconductor, this has an approximately para­bolic behavior close to the band edges (i. e., N(E) « E1/2, refer to Figure 11).

where m* is the effective mass of an electron in the conduction band. Similarly, the effective valence band density of states is given by2,3

2ят*kT 3=2

‘{-t-

where m* is the effective mass of a hole in the valence band. Note that m* and mh are between two and ten times greater than the mass of a free electron. Also, per volume, these densities are approximately four orders of magnitude less than the typical atom den­sity in a solid.

The Fermi-Dirac distribution function is given

by2,3

F (E)

At 0 K, this implies that all energy levels are occupied up to Ef, the Fermi energy. This is a step function. The Fermi-Dirac function with respect to the energy is represented in Figure 12. At the Fermi energy, F(E) is 1=2. Above 0 K, some energy levels above Ef are occupied. This implies that some levels below Ef are empty.

Intrinsic

semiconductor

Figure 13 Characteristic electron energy band levels for a metal, an intrinsic semiconductor and an insulator, where Ec is the bottom of the conduction band, Ev is the top of the valence band, Eg is the band gap, and Ef is the Fermi level.