The Fed-Dirac Integral

We assumed that the Boltzmann approximation was valid. If the Boltzmann approximation does not hold, as

(1)

If we again make a change of variable and let

(2a)

and also define

(2b)

then we can rewrite Equation (1) as

(3)

The integral is defined as

(4)

Figure 1.1 The Fermi-Dirac integral F_1/2 as a function of the Fermi energy.

This function, called the Fermi-Dirac integral, is a tabulated function of the variable η_F. Figure 1.1 is a plot of the Fermi-Dirac integral. Note that if  η_F  > 0, then E_F > E_c; thus the Fermi energy is actually in the conduction hand.

We may use the same general method to calculate the thermal equilibrium con centration of holes, We obtain

(5)

where

(6a)

and

(6b)

The integral in Equation (5) is the same Fermi-Dirac integral defined by Equation (4) although the variables have slightly different definitions. We may note that if η'_F > 0, then the Fermi level is in the valence hand.