The Einstein relation for degenerate semiconductors with nonuniform band structures

Modification of the Einstein equation for semiconductors with nonparabolic energy bands and doped nonuniformly with impurity atoms is suggested. The suggestion is based on a new approximation of the Fermi-Dirac integral of order 1/2, namely, F_1/2(η_n), where η_n is the reduced Fermi level for electrons. The relation reduces to that for semiconductors with parabolic energy bands and doped uniformly with impurity atoms under appropriate boundary conditions. A comparison of the calculated and exact results for F_1/2(η) is found to be very encouraging