Conduction current and generalized einstein relations for degenerate semiconductors and metals

Abstract

Abstract Much confusion exists for conduction in degenerate materials. The complete current expressions are derived from the Boltzmann transport equation. Spatial dependence of all effect, including the variation of band edge, quasi‐Fermi level, chemical potential, and temperature is taken into account. Two different forms of diffusion current D ∇ n and ∇( D ′ n ), used in recent work on semiconductors, are obtained and the relationship between them is discussed. Expressions for D, D ′, as well as other transport coefficients, are given in terms of band integrals for an arbitrary band structure. Several forms of a generalized Einstein relation are derived. Effects of a magnetic field are included when the effective mass approximation applies. Discrepancies in detail with the literature are frequently noted.