Abstract
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.
Highlights
The diffusion coefficient (D) and mobility (μ) of carriers are critically important fundamental transport parameters for describing the properties of carrier transport in semiconductor devices and materials, and are interconnected to each other through a factor by the Einstein relation
A new, much more generalized Einstein relation for degenerate semiconductors with isotropic nonparabolic energy bands is derived rigorously based on the semiclassical momentum balance equation that was obtained from the Boltzmann transport equation (BTE), by assuming a near-equilibrium transport condition, as specified in Section 3, and formulated to be utilized for semiconductors, with accuracy for direct-bandgap semiconductors where their lowest conduction band (CB) structures can be approximated by Kane’s dispersion relation
Several important features noted are: (1) band nonparabolicity causes the D/μ)Γ + nL ( (D/μ) ratios to always increase for any given Fermi energy (FE); (2) for a given material, the D/μ ratio is constant and not dependent on the FE in the nondegenerate condition (η c < −3), whereas it is linearly dependent on the FE for η c > 3; (3) the relative amount of the nonparabolicity effect onAppl
Summary
The diffusion coefficient (D) and mobility (μ) of carriers are critically important fundamental transport parameters for describing the properties of carrier transport in semiconductor devices and materials, and are interconnected to each other through a factor by the Einstein relation. The conventional Einstein relation derived for nondegenerate semiconductors based on the use of Maxwell-Boltzmann statistics with the assumption that the carrier effective mass is constant, i.e., the energy band structure is parabolic, is given by: D k T. Einstein relation that is valid for degenerate, as well as nondegenerate, semiconductors was derived earlier using the concept of the average kinetic energy of electrons in thermal equilibrium, which is given by [2]. A new, much more generalized Einstein relation for degenerate semiconductors with isotropic nonparabolic energy bands is derived rigorously based on the semiclassical momentum balance equation that was obtained from the BTE, by assuming a near-equilibrium transport condition, as specified, and formulated to be utilized for semiconductors, with accuracy for direct-bandgap semiconductors where their lowest CB structures can be approximated by Kane’s dispersion relation. The effective temperature of electrons in nonparabolic energy bands is derived first, and the new generalized Einstein relation is derived and formulated in the subsequent section
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