# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 3 Derivation of the New Generalized Einstein Relation
In addition, the momentum density is given by \(\mathbf{P}=n\mathbf{<}\hbar\mathbf{k>}=n\hbar\mathbf{k}_{o}\), where the angle brackets mean the average over all of the electrons, so that \(\mathbf{k}_{o}\) denotes the weighted average wave vector in \(\mathbf{k}\)-space. The temperature tensor is obtained as (see Appendix A):  
\[\mathbf{T}=\frac{1}{nk_{B}}\int\frac{1}{m_{k}}(\hbar\mathbf{k}-\hbar\mathbf{k }_{o})(\hbar\mathbf{k}-\hbar\mathbf{k}_{o})d^{3}k, \tag{25}\]  
where \(m_{k}=m_{c}(1+2\alpha\varepsilon)\), which is the electron mass associated with the group velocity of an electron wave packet at the state of \(\mathbf{k}\)[13; 16; 19]. The magnitude of \(m^{*}(w)\) is not equal to that of \(<\!\!m_{k}\!\!>\!=m_{c}(1+2\alpha w)\), which is defined by \(m_{o}^{*}(w)\), for nonparabolic energy bands, and lies between the magnitudes of \(m_{o}^{*}(w)\) and \(m_{c}(1+2\alpha w)^{3}\), depending on the electric field strength [20]. At low applied electric fields, \(m^{*}(w)\) and \(m_{o}^{*}(w)\) are approximately equal in magnitude in stationary transport conditions. In order to derive the new generalized Einstein relation strictly, a near-equilibrium transport (NET) condition is assumed, such that at each local point of a semiconductor region, the average drift kinetic energy of carriers is negligibly very small so that their average kinetic energy, \(w\), does not change perceivably by an applied electric field from the value of their average thermal kinetic energy obtained in near-equilibrium, i.e., the average thermal kinetic energy that is given for carriers of the same concentration in thermal equilibrium. In the following, this thermal energy will be referred to as the equilibrium average thermal kinetic energy, and the corresponding nondisplaced distribution function of carriers as the equilibrium distribution function. It should be mentioned that in semiconductor devices, the carrier concentration in near-equilibrium generally differs and can deviate significantly from its value in thermal equilibrium when an electric field has been applied. A typical example is the case of carriers being injected from the junction into an opposite-type semiconductor region and transported diffusively as minority carriers in a \(p\)-\(n\) junction diode. The NET condition assumed then means collaterally that: (1) the carriers practically retain thermal equilibrium with the lattice during their transport by releasing their energy gained from the electric field to the lattice rapidly through scattering, so that the transport of carriers is stationary (i.e., under a local-transport condition); (2) since the carriers are little heated and hence their distribution function does not expand perceivably by a local electric field, it is only displaced in \(\mathbf{k}\)-space in proportion to the strength of the field (generally, a local driving force, electric or diffusive), so that the transport characteristic of carriers is linear (i.e., carrier velocity response to a local driving force is linear); and (3) since the average drift kinetic energy of carriers is so small, the displacement of the distribution function is not large, so that the carrier concentration can be approximated properly by using the equilibrium distribution function of carriers centered at \(\mathbf{k}=0\) for isotropic nonparabolic energy bands where the DOS is distributed nonuniformly in \(\mathbf{k}\)-space. The reason is associated with the spherical symmetricity existing in both the structure of the energy band and the distribution function of the carriers, which causes the average of the antisymmetric part of the carrier distribution in \(\mathbf{k}\)-space to be approximately zero in the NET condition. Note that for parabolic energy bands, the carrier concentration is not changed by any mere displacement of the distribution function in \(\mathbf{k}\)-space due to a uniformly distributed DOS in \(\mathbf{k}\)-space.