# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 1 Introduction
Using an approximate [PERSON]'s dispersion relation [10; 11], a more generalized [PERSON] relation for degenerate semiconductors with nonparabolic energy bands was also derived earlier [12], and the result has often been utilized [13; 14; 15]. This existing generalized [PERSON] relation for electrons in nonparabolic CBs is given by:  
\[\frac{D}{\mu}=\left(\frac{k_{B}T_{L}}{q}\right)\frac{F_{1/2}(\eta_{c})}{dF_{1 /2}(\eta_{c})/d\eta_{c}},\]4  
where \(\alpha\) is the nonparabolicity parameter, usually expressed in the unit of eV\({}^{-1}\), for the isotropic CB; and \(F_{3/2}(\eta_{c})\) is the Fermi-Dirac integral of order \(3/2\). Note that Equation (6) was obtained by assuming that \(\alpha(\epsilon-\epsilon_{c})<<1\), where \(\epsilon\) is the electron energy, in order to simplify the density integral (refer to Equation (14)) in a closed form by the binomial approximation. Concerning the formula of Equation (6), however, two theoretically serious defects are pointed out here: (1) due to the adoption of the binomial approximation for the derivation, it produces large errors for large values of \(\alpha\) or \(T_{L}\) (see Figure 1); and (2) in the nondegenerate condition, it reduces to the conventional [PERSON] relation, Equation (1), derived for parabolic energy bands, since all of the Fermi-Dirac integrals are almost equal in value if \(\eta_{c}<-3\). However, this aspect is physically improper. The reason is explained in the following. Let us consider two different cases of CBs, in which both bands have the same value of \(m_{\rm c}\) (the electron effective mass at the band bottom), but one is parabolic (\(\alpha=0\)) and the other is nonparabolic (\(\alpha>0\)). For the case of \(\alpha=0\), the DOS in three-dimensional \({\bf k}\)-space (momentum space) is uniformly equal throughout the space. However, for the case of \(\alpha>0\), the DOS increases continuously as the magnitude of \({\bf k}\) increases from zero (the energy minimum point) [16], since the energy-dependent effective mass for electrons increases continuously as the energy increases. Therefore, for a given FE, the average kinetic energy of electrons is always given more largely in the case of \(\alpha>0\), resulting in a larger diffusivity of electrons for a fixed mobility, as already explained above. In other words, if nonparabolicity exists in an energy band, we always have a larger \(D/\mu\) ratio for any values of \(\eta_{c}\), i.e., even for the nondegenerate condition, than that expected by the parabolic band approximation. If the binomial approximation is applied for the density integral, the electron concentration can be approximated by:  
\[\frac{D}{\mu}=\left(\frac{k_{B}T_{L}}{q}\right)\frac{F_{1/2}(\eta_{c})}{dF_{1 /2}(\eta_{c})/d\eta_{c}},\]5  
Substituting this equation into Equation (4), we simply obtain Equation (6). Therefore, the existing generalized [PERSON] relation for nonparabolic energy bands is a simple modification of the carrier concentration to the generalized [PERSON] relation for parabolic energy bands, so that it disregards the carrier thermal energy diffusion effect arising from the thermal kinetic motion of carriers. The consequence is that it produces a physically improper \(D/\mu\) ratio, as can be verified in Section 4. The formulas of the [PERSON] relation (1), (2b), (4), and (6) were derived in thermal equilibrium or near-equilibrium conditions by various approaches, typically based on the principle of detailed balance or the Boltzmann transport equation (BTE), respectively. In the present paper, a new, much more generalized [PERSON] 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 in Section 3, and formulated to be utilized for semiconductors, particularly with accuracy for direct-bandgap semiconductors where their lowest CB structures can be approximated by [PERSON]'s dispersion relation.