Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors

The currently used generalized Einstein relation for degenerate semiconductors with isotropic nonparabolic energy bands produces physically improper results, as well as losing numerical accuracy for large values of nonparabolicity parameters at room temperature. Therefore, a new generalized Einstein relation (a macroscopic equation and a formula) is derived from the semiclassical momentum balance equation based on a drift-diffusion approximation, by introducing a new concept of the effective temperature of a carrier gas for generalization of the classical kinetic theory for nonideal gases of carriers in semiconductors. The proposed formula takes into account the carrier thermal energy diffusion effect completely, so that it can accurately reflect the effect of band nonparabolicity on the ratio of the diffusion coefficient to the mobility for carriers in degenerate semiconductors. From the results evaluated with the formula, new and critically important nonparabolicity effects are observed. It is shown that the new generalized Einstein relation is valid for applied electrical fields of the full linear regime. In addition, useful figures are also presented, from which the ratio of the diffusion coefficient to mobility, as well as the Fermi energy, can be easily determined from the electron concentration, or doping density, for a given semiconductor material.