# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 5 Conclusions  
The currently used generalized [PERSON] 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 [PERSON] relation (a macroscopic equation and a formula) for degenerate semiconductors with isotropic nonparabolic energy bands was derived from the semiclassical momentum balance equation based on a drift-diffusion approximation by assuming a near-equilibrium transport condition. For the derivation, the novel concept of the effective temperature of a carrier gas has been introduced for the generalization of the classical kinetic theory for nonideal gases of carriers in semiconductors. The new generalized [PERSON] relation takes into account the carrier thermal energy diffusion effect completely, so that it can correctly reflect the effect of band nonparabolicity on the \(D/\mu\) ratio for carriers in degenerate semiconductors with isotropic nonparabolic energy bands. A theoretical analysis of the validity limit of the new Einstein relation for applied electric fields was also performed. The result indicates that it is generally valid for applied electric fields of the full linear regime for conduction electrons in semiconductors. Many important nonparabolicity effects on the \(D/\mu\) ratio for carriers in semiconductors were newly observed, and are summarized in the following. For any given FE, band nonparabolicity causes the \(D/\mu\) ratio to always increase, and the nonparabolicity effect is largest in the nondegenerate condition. However, the effect is very different for a given electron concentration. In other words, band nonparabolicity causes the \(D/\mu\) ratio to always increase in the nondegenerate condition, but it causes the \(D/\mu\) ratio to always decrease in the high degenerate condition. Therefore, quite an interesting feature appears, such that for semiconductors with isotropic nonparabolic energy bands, when the electron concentrations are equal to the value of \(2.33N_{\mathrm{c}}\), the \(D/\mu\) ratios are always nearly equal to the value of \(1.77k_{\mathrm{B}}T_{L}/q\), irrespective of the degree of nonparabolicity of their CBs. The results calculated for common direct-bandgap compound semiconductors have additionally shown the following specific aspects for the nonparabolicity effect on the \(D/\mu\) ratio for conduction electrons. For compound semiconductors that have a very wide direct bandgap, such as GaN, the nonparabolicity effect is negligibly small and can be disregarded. It is also negligibly small in the nondegenerate condition for direct-bandgap compound semiconductors, such as GaAs and InP, that have a relatively large bandgap energy. However, it can be significantly large in heavy-doping conditions in these semiconductors. This is another newly observed and critically important aspect. On the other hand, for compound semiconductors, such as InAs and InSb, that have a narrow direct bandgap, the nonparabolicity effect is substantially large in both the nondegenerate condition and the high degenerate condition. Therefore, the use of the parabolic band approximation for these semiconductors can give rise to considerable errors for the diffusion coefficients of electrons. The dependency of the \(D/\mu\) ratio and the FE on the electron concentration (or doping density in homogeneous materials) for a given semiconductor can be easily determined from the figures presented in the work. Therefore, they should be quite useful for semiconductor device design and material analysis. The generalized Einstein relation formulated in the present work can be utilized to more accurately analyze various depletion phenomena in semiconductor devices and materials in thermal equilibrium or near-equilibrium conditions, and also minority carrier diffusion properties in low-field or charge-neutral active regions of semiconductor devices, typically such as \(p\)-\(n\) diodes and bipolar transistors.