# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 3 Derivation of the New Generalized Einstein Relation  
The derivation uses the semiclassical momentum balance equation that was derived from the BTE by the method of moments for describing the macroscopic transport properties of electrons in a single conduction valley with a nonparabolic energy band structure (for Equation (23), refer to Appendix A) [13; 17]:  
\[\frac{\partial\mathbf{P}}{\partial t}+qn\mathbf{E}+\
abla\cdot\mathbf{Q}= \bigg{(}\frac{\partial\mathbf{P}}{\partial t}\bigg{)}_{c} \tag{22}\]  
and  
\[\mathbf{Q}=nk_{B}\mathbf{T}+\mathbf{v}\mathbf{P}+\mathbf{P}(\mathbf{v}- \mathbf{v}_{o})\bigg{(}\mathbf{v}_{o}\equiv\frac{1}{m_{o}^{*}(w)n}\mathbf{P} \bigg{)}, \tag{23}\]  
Figure 1: Approximation errors in percentage vs. the reduced Fermi energy (FE) \(\eta_{c}\) as calculated for various values of \(\beta\). The solid curves display the errors induced by the empirical approximation, and the dashed curves for \(\beta\) = 0.12 and 0.18 by the binomial approximation. where \(t\) is the time; \(\mathbf{P}\), \(\mathbf{v}\), and \(\mathbf{T}\) are the momentum density, the average velocity (i.e., the drift velocity), and the electron temperature tensor, respectively; \(\mathbf{E}\) is the electric field; and \(\mathbf{v}_{o}\) represents a drift velocity equal to the electron velocity at the wave vector \(\mathbf{k}_{o}\) in \(\mathbf{k}\)-space (\(\mathbf{k}_{o}\) and \(m_{o}^{*}(w)\) are defined following Equations (24) and (25), respectively). The variable \(\mathbf{Q}\) thus represents the total kinetic energy density tensor that has combined three tensor quantities: the first quantity corresponds to a kinetic pressure tensor of the electron gas due to the random thermal motion of electron particles deviated from their average velocity in velocity space [17], or their average momentum in \(\mathbf{k}\)-space (see Equation (25)); the second corresponds to a kinetic energy density tensor associated with the group drift motion in real space; and the third corresponds to a kinetic energy density tensor appearing due to the difference in magnitude between the two velocities of \(\mathbf{v}\) and \(\mathbf{v}_{o}\). For parabolic energy bands, \(\mathbf{v}\) equals \(\mathbf{v}_{o}\), so that this density tensor becomes zero. In particular, the divergence of the second tensor quantity represents the rate of local outflow of the momentum density, and is responsible for the appearance of a velocity convection term in the momentum balance equation when expressed in terms of drift velocity. This convection term is usually neglected in high-field transport device simulations for numerical convenience. If the total energy density tensor \(\mathbf{Q}\) is written in matrix form, then \(Q_{\alpha\beta}=nk_{B}T_{\alpha\beta}+v_{\alpha}P_{\beta}+P_{\alpha}(v_{ \beta}-v_{o\beta})\) for \(\alpha\) and \(\beta=x,y,z\) (space coordinates) with \(T_{\alpha\beta}\), \(v_{\alpha}\), \(v_{\beta}\), \(P_{\alpha}\), \(P_{\beta}\), and \(v_{o\beta}\) the corresponding components of \(\mathbf{T}\), \(\mathbf{v}\), \(\mathbf{P}\), and \(\mathbf{v}_{o}\), respectively. The collision term in Equation (22) accounts for the rate of change in momentum density due to carrier scattering. Under the usual relaxation-time approximation [18], this term becomes \(-\mathbf{P}/\tau_{m}(w)\), where \(\tau_{m}(w)\) is the average momentum relaxation time, so that Equation (22) in the steady-state condition is simplified to  
\[\mathbf{J}=q\mu n\mathbf{E}+\mu\
abla\cdot\mathbf{Q}, \tag{24}\]  
where \(\mu=q\tau_{m}(w)/m^{*}(w)\) is the mobility, with \(m^{*}(w)\) being the average conductivity effective mass (the mobility is thus dependent on \(w\), but a simple notation has been used). The energy-dependent parameters \(\tau_{m}(w)\) and \(m^{*}(w)\) are usually determined by Monte Carlo simulations in stationary transport conditions. The momentum density and the average velocity are related to each other through the average conductivity effective mass by \(\mathbf{P}=nm^{*}(w)\mathbf{v}\).