The Einstein Relation

If we consider the non-uniformly doped semiconductor represented by the energy band diagram shown in Figure 1.1 and assume there are no electrical connections that the semiconductor is in thermal equilibrium, then the individual electron as hole currents must be zero. We can write

Figure 1.1 Energy-band diagram for a semiconductor in thermal equilibrium with a non-uniform donor impurity concentration.

(1)

If we assume quasi-neutrality so that n ≈ N_d (x). then we can rewrite Equation (1) as

(2)

Substituting the expression for the electric field into Equation (2), we obtain

(3)

Equation (3) is valid for the condition

(4a)

The hole current must also be zero in the semiconductor. From this condition we can show that

(4b)

Combining Equations (4a) and (4b) gives

(5)