# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 3 Derivation of the New Generalized Einstein Relation
The corresponding generalized macroscopic equation and formula for valence holes in an isotropic nonparabolic band can be written directly from Equations (29)-(31). Equation (31) contains several Fermi-Dirac integrals, but they can be evaluated comfortably using a computer software package for mathematics. In addition, analytical expressions for approximating all of these integrals in a convenient form with reasonable accuracy is obtained in Reference [21]. Also note that in the nondegenerate condition, the Fermi-Dirac integrals included in both the numerator and denominator in Equation (31) are almost canceled, resulting in the following simple expression of the \(D/\mu\) ratio that does not depend on the FE:  
\[\mathbf{Q}=nk_{B}\mathbf{T}+\mathbf{v}\mathbf{P}+\mathbf{P}(\mathbf{v}- \mathbf{v}_{o})\bigg{(}\mathbf{v}_{o}\equiv\frac{1}{m_{o}^{*}(w)n}\mathbf{P} \bigg{)}, \tag{23}\]2  
In other words, in the nondegenerate condition, the population of carriers over energy states in an energy band is not dense, so that Maxwell-Boltzmann statistics can be used as a very good approximation. The validity limit of the new generalized macroscopic equation and formula for applied electric fields is discussed. Monte Carlo simulations performed for bulk materials at \(T_{L}=300\) K show that the average electron energy (or the electron temperature) remains almost constant up to an electric field strength \(E=2\) kV/cm for GaAs [22], and \(E=3\) kV/cm for InP [23]. The doping densities of the materials were \(N_{D}=2\)\(\times\)\(10^{17}\) cm\({}^{-3}\) and \(N_{D}=10^{17}\) cm\({}^{-3}\), respectively. The drift velocity for \(E=2\) kV/cm is \(v=9\)\(\times\)\(10^{6}\) cm/s for the former case, and \(v=9.6\)\(\times\)\(10^{6}\) cm/s for the latter, when calculated by \(v=\mu E\) (with \(\mu=4500\) cm\({}^{2}\)/V-s for GaAs and \(\mu=3200\) cm\({}^{2}\)/V-s for InP). It has been shown by calculations that the ratio of the drift kinetic energy, \(m^{\star}(w)v^{2}/2\), to the average thermal kinetic energy in thermal equilibrium, which is equal to \(3k_{B}T_{E}/2\), is negligibly small and less than 5% in both cases. This fact is in agreement with the assumption adopted in the derivation. Such a theoretical analysis indicates that the new macroscopic equation and formula can be valid for applied electric fields up to the upper velocity and field bounds of the linear regime (in the velocity-field relation) for conduction electrons of direct-bandgap semiconductors (\(v<10^{7}\) cm/s due to possible velocity overshoot). In this type of semiconductor, which usually features high electron mobility, the mean thermal velocity (\(v_{th}\)) of electrons is commonly large due to their small effective mass (typically, for GaAs \(v_{th}\approx 4.5\)\(\times\)\(10^{7}\) cm/s in the nondegenerate condition at room temperature), so that the upper velocity bound of the linear regime is generally high for direct-bandgap semiconductors. The generalized Einstein relation expressed by the macroscopic equation and the formula is dependent only on the spatial variation of the FE, i.e., the carrier concentration, and independent of the electric field. Therefore, the validity limit discussed above for direct-bandgap semiconductorsgenerally holds for other types of semiconductors as well (of course, in the validity limits of isotropic nonparabolic band approximations). Since it has been derived on the basis of the BTE, its applications should be limited within the validity limits of the BTE. Note that this equation is valid only when the spatial variation of the potential in a semiconductor device is slow on the scale of the De Broglie wave length, which is typically 10 to 20 nm [16]. Therefore, for a small scaled device where the carrier concentration changes rapidly with position, the BTE itself is invalid.