# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 3 Derivation of the New Generalized Einstein Relation
Thus, it should be noted that the generalized Einstein relation to be derived in this section is invalid if any of these detailed three subconditions of the NET condition is not satisfied. Therefore, it follows from the assumption that: (1) all of the drift kinetic energy tensor components of \(v_{x}P_{\beta}\), as well as \(P_{x}(v_{\beta}-v_{\beta\beta})\), can be neglected; and (2) all of the non-diagonal components of \(\mathbf{T}\), i.e., \(T_{a\beta}\) with \(\alpha\
eq\beta\), completely vanish, due to the existence of the isotropic symmetricity in both the band structure and the distribution function in the NET condition. In addition, all of the diagonal components of \(\mathbf{T}\), i.e., \(T_{a\beta}\) with \(\alpha=\beta\), are almost equal in magnitude (not exactly equal for nonparabolic energy bands due to a displacement of the distribution function). As a result, a scalar electron temperature \(T\) can be defined justly so that \(\mathbf{T}=T\delta\), where \(\delta\) is the unit dyad. From these results of (1) and (2), Equation (24) reduces to the following drift-diffusion equation:  
\[\mathbf{J}=q\mu n\mathbf{E}+\mu\
abla Q, \tag{26}\]  
where  
\[Q=nk_{B}T. \tag{27}\]  
In the NET condition, the average thermal kinetic energy remains the same as the equilibrium average thermal kinetic energy, so that the electron temperature \(T\) becomes dependent only on the local FE, which in turn uniquely determines the electron concentration, or vice versa. Therefore, \(Q=Q(n,T)\) becomes a function of only one independent variable, \(\eta_{c}\). In other words, \(Q=Q(\eta_{c}(n(x,y,z)))\). Consequently, the application of the chain rule for differentiation to Equation (26) results in:  
\[\mathbf{J}=q\mu n\mathbf{E}+qD\
abla n, \tag{28}\]  
where  
\[D=\frac{1}{q}\frac{dQ}{d\eta_{c}}\left(\frac{dn}{d\eta_{c}}\right)^{-1}\mu \tag{29}\]  
and  
\[Q=\ nk_{B}T_{E}\left(=\frac{2}{3}W\right). \tag{30}\]  
Note that, in Equation (30), the electron temperature \(T\) has been reduced to the effective temperature of electrons in thermal equilibrium, \(T_{E}\), due to the fact that \(T\approx T_{E}\) under the NET condition, and \(W\) is the total thermal kinetic energy density in thermal equilibrium, as defined in Section 2. Equation (29), including Equation (30), is the new, much more generalized, macroscopic equation of the [PERSON] relation which can take into account the effect of thermal energy diffusion of carriers completely for degenerate semiconductors with isotropic nonparabolic energy bands. It is worth noting that this new generalized macroscopic equation reduces to Equation (1), the conventional [PERSON] relation for nondegenerate semiconductors with parabolic energy bands, for \(T_{E}=T_{L}\), and also to Equation (4), or (2b), the generalized [PERSON] relation for degenerate semiconductors with parabolic energy bands, for \(T_{E}=F_{3/2}(\eta_{c})/F_{1/2}(\eta_{c})T_{L}\) and \(n=N_{c}F_{1/2}(\eta_{c})\), since \(dQ/d\eta_{c}=k_{B}T_{L}n\) then. In the NET condition, the concentration \(n\) is approximately equal to the concentration value given by the equilibrium distribution function of carriers for an arbitrary value of the FE (generally quasi-FE), as described in subcondition (3). Therefore, substituting Equations (18) and (21), or Equation (19), into Equations (29) and (30) yields the following new generalized formulation of the [PERSON] relation for electrons in isotropic nonparabolic energy bands:  
\[\mathbf{Q}=nk_{B}\mathbf{T}+\mathbf{v}\mathbf{P}+\mathbf{P}(\mathbf{v}- \mathbf{v}_{o})\bigg{(}\mathbf{v}_{o}\equiv\frac{1}{m_{o}^{*}(w)n}\mathbf{P} \bigg{)}, \tag{23}\]0  
where the recurrence property of Fermi-Dirac integrals has been used such that:  
\[\mathbf{Q}=nk_{B}\mathbf{T}+\mathbf{v}\mathbf{P}+\mathbf{P}(\mathbf{v}- \mathbf{v}_{o})\bigg{(}\mathbf{v}_{o}\equiv\frac{1}{m_{o}^{*}(w)n}\mathbf{P} \bigg{)}, \tag{23}\]1  
For \(\beta=0\), the new formula reduces to Equation (2b), as well as Equation (1) in the nondegenerate condition.