# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 4 Evaluation Results and Discussions  
### General Cases  
The \(D/\mu\) ratios for various values of \(\beta\) have been calculated numerically for electrons using Equation (31), and the results are plotted as a function of the reduced FE in Figure 2. The range of \(\beta\) values was decided in due consideration of the range of variation of \(\alpha\) values of the major important semiconductors at room temperature. Note that if \(\beta>0\), the \(D/\mu\) ratio is always larger than \(k_{B}T_{L}/q\) even for the nondegenerate condition, as reasoned in connection with Equation (6) in Section 1. Several important features noted are: (1) band nonparabolicity causes the \(D/\mu\) ratios to always increase for any given FE; (2) for a given material, the \(D/\mu\) ratio is constant and not dependent on the FE in the nondegenerate condition (\(\eta_{c}<-3\)), whereas it is linearly dependent on the FE for \(\eta_{c}>3\) (to be defined as the high degenerate condition in the following); (3) the relative amount of the nonparabolicity effect on the \(D/\mu\) ratio is largest in the nondegenerate condition for a given material (however, it should be noted that this is the case only when the \(D/\mu\) ratio is plotted as a function of the FE). In particular, for the large value of \(\beta=0.18\), the \(D/\mu\) ratio is noted to increase significantly by 1.35 times in the nondegenerate condition; and (4) the \(D/\mu\) ratios, including the case of parabolic energy bands (\(\beta=0\)), increase a very large amount with increasing FE in the high degenerate condition. This indicates that the quantum mechanical effect of electron degeneracy on the \(D/\mu\) ratio is very large for a highly degenerate electron gas. Figure 2: Calculated values of the \(D/\mu\) ratios for electrons as a function of the reduced FE for selected various values of \(\beta\). The dashed curve 1 displays the \(D/\mu\) ratio calculated for electrons with the existing formula, Equation (6), for \(\beta=0.06\), and the dashed curve 2 for \(\beta=0.18\). In the same figure, the \(D/\mu\) ratios calculated for electrons using the existing formula Equation (6) are also plotted typically for \(\beta\) = 0.06 (dashed curve 1) and \(\beta\) = 0.18 (dashed curve 2). It is worth noting that there are substantial differences in value between the results of the new formula and the existing formula. The results of the existing formula show that the \(D/\mu\) ratios are not affected by the amounts of band nonparabolicity in the nondegenerate condition. An additional physically improper aspect is that band nonparabolicity causes the \(D/\mu\) ratios to decrease quite significantly below the values obtained for \(\beta\) = 0 (parabolic energy band case), even for the small value of \(\beta\) = 0.06, in the high degenerate condition. In Figure 3, the \(D/\mu\) ratios for various values of \(\beta\) are plotted as a function of the electron concentration that has been normalized to the effective DOS, \(N_{c}\). This figure shows a quite interesting feature, such that there exists a point, designated by \(P_{D}\), near which all of the curves of the \(D/\mu\) ratios come to cross, and thus all have nearly the same value. The crossover of these curves arises near the point of \(D/\mu\) = 1.77\(k_{B}T_{L}/q\) and \(n\) = 2.33\(N_{c}\). This means that, for semiconductors with isotropic nonparabolic CBs, when the electron concentrations are equal to the value of 2.33\(N_{c}\), the \(D/\mu\) ratios for conduction electrons are always nearly equal to the value of 1.77\(k_{B}T_{L}/q\), irrespective of the degree of nonparabolicity of their CBs. The corresponding Fermi energies to this value of \(D/\mu\) are noted to be approximately between 0.5\(k_{B}T_{L}/q\) [eV] (\(\eta_{c}\) = 0.5) and 1.5\(k_{B}T_{L}/q\) [eV] (\(\eta_{c}\) = 1.5) when measured from the band bottom energy (refer to Figure 2). In the figure, the \(D/\mu\) ratio calculated for electrons with the existing formula, typically for \(\beta\) = 0.18, is also plotted with the curve 1 (dashed).