Generalized Einstein relation for nonparabolic multiple energy-band degenerate semiconductors

A generalized Einstein relation for electron gases of degenerate semiconductors with a system of typically two nonparabolic conduction band structures is derived and formulated in near-equilibrium condition based on using the semiclassical drift-diffusion equation. The result of this derivation shows that the Einstein relation for carrier gases in multiple energy bands generally depends on their mobilities unless the energy band structures are parabolic. The formulated results (a full formula and an approximate single-carrier-gas formula) are typically applied to GaAs and GaSb to calculate the values of the ratio of the diffusion coefficient to mobility for carriers in the lowest and upper conduction bands, and an in-depth investigation is made for the Einstein relation for conduction electrons in these semiconductors. It is shown, in particular, that highly degenerate GaAs exhibits the uncommon, peculiar property of the ratio of the diffusion coefficient to mobility for conduction electrons saturating in two different electron concentration, or Fermi energy, conditions, with increasing electron concentration or Fermi energy. It is also shown that for GaSb, the effect of the upper conduction band is so large that the use of any formula obtained for carriers in a single conduction band is generally unsuitable for describing the Einstein relation for conduction electrons in this semiconductor. In addition, a more generalized Einstein relation extended to a system of three nonparabolic energy band structures is also formulated and proposed in the paper, so as to be used to more accurately describe the Einstein relation for hole gases in degenerate semiconductors.