Jacques I. Pankove - Optical Processes in Semiconductors

In a simple parabolic band characterized by an effective mass m* containing a concentration of n carriers/cm3 at any temperature, the position of the Fermi level with respect to the band edge is related to the temperature T by the expression

(III-1)

where E is the carriers energy, E and both being measured with respect to the bottom of the parabolic band; k and h are Boltzmanns and Plancks constants, respectively. Equation (III-1) was solved by computer for various values of n, m*, and T. For this purpose, Eq. (III-1) was expressed in the intermediate form

where

The nomograph() permits a graphical determination of any parameter n, m*, , or T when any three of them are known. For example. if n and m* are known, they are joined by a straight line. From the intercept of this straight line with the X - axis one draws a straight line to the desired T tc find a value on the left-hand - scale. Transferring this value to the right-hand - scale, one strikes another straight line to T to find . If is negative, lies outside the parabolic band (i.e., inside the energy gap).

By way of example, the diagram illustrates how one finds the position of the Fermi level at T = 80K in n- type GaAs having a carrier concentration n = 1 1017 cm-3. The three steps labeled 1, 2, 3, lead to 3 meV.

Of course, this nomograph is valid only for the case of a temperature-independent free-carrier density and, therefore, it does not apply in the temperature range where carrier freeze-out may occur. Furthermore, a gross approximation is made by assuming that the band is parabolic over the entire range of energies over which the distribution function is appreciable. The neglect of band tailing and of nonpara-bolicity makes the value determined by this nomograph a maximum value.