# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 4 Evaluation Results and Discussions
### Specific Cases
Since essentially the \(D/\mu\) ratio, or the diffusion coefficient, for electrons represents the average property of a single electron for diffusion, the average \(D/\mu\) ratio for the total conduction electrons in GaAs can be obtained by taking a weighted average using the electron concentrations of the lowest and upper CBs as follows:  
Figure 6: The values of the \(D/\mu\) ratios (solid curves), shown in Figure 5, are compared with those of the \(D/\mu\) ratios (dashed curves) calculated with the parabolic band approximation (\(\beta=0\)) for the same temperature. The plot (**b**) is an enlarged detailed view of the plot (**a**) for the values in the nondegenerate condition. \[\left(\frac{D}{\mu}\right)_{\text{average}}=\frac{n_{\Gamma}(D/\mu)_{\Gamma}+n_{ \text{L}}(D/\mu)_{\text{L}}}{n_{\Gamma}+n_{\text{L}}}. \tag{34}\]  
Thus, it has been calculated and presented in Figure 8 as a function of the reduced FE in the plot (a) and also as a function of the total electron concentration, i.e., \(n=n_{\Gamma}+n_{\text{L}}\), in the plot (b). In this figure, the average \(D/\mu\) ratio calculated in the case of \(\beta=0\) (for both the lowest and upper CBs) are additionally shown by a dashed curve for comparison. These two curves are also compared in the figure with the other additional curve obtained by considering the lowest CB only, i.e., completely neglecting the existence of the upper CB (\(n=n_{\Gamma}\) in this case). All of the results are for \(T_{L}=300\) K.  
Figure 8 shows that the error arising from completely neglecting the electron population of the upper CB increases notably for \(\eta_{c}>7\) or \(n>7.4\times 10^{18}\) cm\({}^{-3}\) (the concentration value corresponding to the average \(D/\mu\) ratio when \(\eta_{c}=7\)), and becomes dramatically large for \(\eta_{c}>10\) or \(n>1.5\times 10^{19}\) cm\({}^{-3}\) (the concentration value corresponding to the average \(D/\mu\) ratio when \(\eta_{c}=10\)). The most interesting and important feature observed from this figure is that the average \(D/\mu\) ratio becomes saturated, with a maximum value \(D/\mu=6.1\) (\(\times 26\) mV), and then decreases. The saturation occurs at \(\eta_{c}=10.2\) or \(n=1.6\times 10^{19}\) cm\({}^{-3}\) (when \(n_{\text{L}}/n_{\Gamma}=0.17\) from Figure 7). It should be noted that since the FE is at least 0.036 eV less than the bottom energy of the upper CB for \(\eta_{c}<10.2\) (the saturation point value) at \(T_{L}=300\) K, the \(D/\mu\) ratio for \(n_{\text{L}}\) is approximately equal to its value given in the nondegenerate condition, i.e., 26 mV (\(=\)\(k_{\text{B}}T_{L}/\mu\)), for this range of \(\eta_{c}\) (refer to the case of \(\eta_{c}=-1.4\) and \(\beta=0.012\), the value of \(\beta\) for the L-valleys at \(T_{L}=300\) K, in Figure 2). This value of the \(D/\mu\) ratio is in particular approximately seven times smaller than that of the \(D/\mu\) ratio for \(n_{\Gamma}\) noted at \(\eta_{c}=10.2\), which amounts to 6.9 (\(\times 26\) mV) in Figure 8a or Figure 8b. As a result, the increasing rate of the average per electron of the total kinetic pressure, or kinetic energy density, of electron gases in the CBs should be reduced continuously with increasing \(n_{\text{L}}\), resulting in the appearance of the saturation property. Note in Figure 8 that a similar saturation property also appears for the average \(D/\mu\) ratio (\(\beta=0\)) for the same reason, but the effect of \(n_{\text{L}}\) on the saturation property is more severe in this case since the saturation is produced at a reduced maximum value of \(D/\mu=5.9\) (\(\times 26\) mV). The reason is that the concentration ratio \(n_{\text{L}}/n_{\Gamma}\) increases more rapidly with increasing \(\eta_{c}\) when \(\beta=0\), which can be verified by observing that the saturation is caused by a smaller value of \(\eta_{c}\) in the plot (a) and also a smaller value of \(n\) in the plot (b). Figure 7: Calculated electron concentration ratio of \(n_{\text{L}}\) to \(n_{\Gamma}\) as a function of the reduced FE for GaAs in thermal equilibrium (\(T_{L}=300\) K).