# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 1 Introduction
The former simpler Equation (2b) is commonly referred to as the generalized [PERSON] relation for electrons, but it is valid only for semiconductors having parabolic energy bands. This generalized [PERSON] relation is often formulated in a different form, such as [6; 7; 8]:  
\[\frac{D}{\mu}=\left(\frac{k_{B}T_{L}}{q}\right)\frac{F_{1/2}(\eta_{c})}{dF_{1 /2}(\eta_{c})/d\eta_{c}},\]2  
where \(n\) is the electron concentration. However, this form is equivalent to Equation (2b), due to the following identities:  
\[\frac{D}{\mu}=\left(\frac{k_{B}T_{L}}{q}\right)\frac{F_{1/2}(\eta_{c})}{dF_{1 /2}(\eta_{c})/d\eta_{c}},\]3  
where \(N_{c}\) denotes the effective density of states (DOS) for parabolic CB cases, and the recurrence property of Fermi-Dirac integrals has been used. The experimentally established [PERSON]'s law asserts that the diffusive flux of particles at a local spatial point is simply proportional to the spatial gradient of the particle concentration. In this macroscopic law, the proportional parameter (the diffusion coefficient) represents the average property of a particle for diffusion. However, we see in Equation (2b) that, due to the microscopic property of carriers' quantum mechanical behavior, the additional factor \(F_{1/2}(\eta_{c})/F_{-1/2}(\eta_{c})\) appears when Fermi-Dirac statistics are adopted. This causes the \(D/\mu\) ratio, representing an average property, to depend on a spatial position, even though all of the electrons have the same effective mass and are subject to the same macroscopic temperature. The direct reason for the appearance of the additional factor is, of course, due to the carrier concentration being given by a Fermi-Dirac integral in Fermi-Dirac statistics. However, the underlying physical reason for its appearance beyond such a somewhat mathematical reason has not yet been addressed, and is clarified in detail in the following. Let us consider a fixed number \(N\) of degenerate electrons distributed in a certain parabolic CB. We assume that they have been restricted to be redistributed over energy states according to Maxwell-Boltzmann statistics by some external means. Recall that Maxwell-Boltzmann statistics permits any number of electrons to occupy the same quantum (spin) state [9]. Now, if we remove the external restricting condition, it is expected that a large fraction of the electrons sharing the same quantum states with a higher probability of occupation at a lower energy will jump over to higher-energy empty states to accommodate [PERSON]'s exclusion principle, the key law in Fermi-Dirac statistics. This will give rise to an increase in the average kinetic energy of electrons from the average value that resulted from the Maxwell-Boltzmann statistics, essentially leading to a larger kinetic pressure of the electron gas. Therefore, for a given mobility, the diffusivity of the \(N\) electrons will be larger due to their greater energetic kinetic motion, on average. In fact, this effect is reflected by the additional factor. Note that the factor \(F_{1/2}(\eta_{c})/F_{-1/2}(\eta_{c})\) is always larger than unity for any values of \(\eta_{c}\). Thus, for a parabolic energy band, a larger value in the \(D/\mu\) ratio is always generated even for the nondegenerate condition (\(\eta_{c}<-3\)) when Fermi-Dirac statistics is applied. In semiconductors, nonparabolicity always exists in their energy band structures. The nonparabolicity effect on carrier transport is technically important, particularly for direct-bandgap compound semiconductors with high electron mobility, since they can have significant nonparabolicity in their central conduction valleys, which feature isotropic constant surfaces of energy. This necessitates a more generalized [PERSON] relation, beyond Equation (2b) or Equation (4), which can properly take into account the nonparabolicity effect.