# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 4 Evaluation Results and Discussions
### Specific Cases  
The \(D/\mu\) ratios for conduction electrons at room temperature (\(T_{L}=300\) K) in common direct-bandgap compound semiconductors have been calculated specifically using Equation (31), and the results are presented as a function of the reduced FE in Figure 4, and also as a function of the electron concentration in Figure 5 (replotted from Figure 4). In the figures, the \(D/\mu\) ratios for InSb and InAs are displayed only up to their values corresponding to \(n=6N_{c}\) and \(n=12N_{c}\), respectively, since these semiconductors have a small bandgap energy as mentioned below. For the calculations, the values of the nonparabolicity parameter \(\alpha\) for the semiconductors were taken from the material parameter data in Reference [16] or [24], and mean values taken from those data in Reference [16], or [24], and Reference [25] for the effective electron mass \(m_{c}\), shown in Table 1.  
The dependencies of the \(D/\mu\) ratios on the electron concentration (or doping density in homogeneous materials), and the FE for the semiconductors, can be readily determined from Figures 4 and 5. For this purpose, an enlarged detailed plot is presented additionally in Figure 5b for accurate values of the \(D/\mu\) ratios in the nondegenerate condition. These figures can be used conveniently for semiconductor device design and material analysis, as well as Figures 2 and 3, which are particularly useful for any arbitrary value of \(\beta\). \begin{table}
\begin{tabular}{c c c c c c} \hline  & **InSb** & **InAs** & **InP** & **GaAs** & **GaN** \\ \hline \(\alpha\) & 5.72 eV\({}^{-1}\) & 2.73 eV\({}^{-1}\) & 0.67 eV\({}^{-1}\) & 0.64 eV\({}^{-1}\) & 0.19 eV\({}^{-1}\) \\ \(\beta\) & 0.1479 & 0.0706 & 0.0173 & 0.0165 & 0.0049 \\ \(m_{c}\) & 0.0135\(m_{e}\) & 0.025\(m_{e}\) & 0.077\(m_{e}\) & 0.067\(m_{e}\) & 0.19\(m_{e}\) \\ \(N_{c}\) & \(3.936\times 10^{16}\) cm\({}^{-3}\) & \(9.919\times 10^{16}\) cm\({}^{-3}\) & \(5.31\times 10^{17}\) cm\({}^{-3}\) & \(4.352\times 10^{17}\) cm\({}^{-3}\) & \(2.078\times 10^{18}\) cm\({}^{-3}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Material parameter values used for the calculation of the \(D/\mu\) ratios for conduction electrons at \(T_{L}=300\) K.  
Figure 6 shows the difference in values of the \(D/\mu\) ratios due to band nonparabolicity for the conduction electrons of the common compound semiconductors, as plotted as a function of the electron concentration. In the figure, the set of solid curves represents the \(D/\mu\) ratios displayed in Figure 5, and the set of dashed curves represents the \(D/\mu\) ratios calculated with the parabolic band approximation (\(\beta\) = 0) for the same temperature (\(T_{L}\) = 300 K). For accurate comparison in the nondegenerate condition, an enlarged detailed plot is presented additionally in Figure 6b, where a horizontal dotted line is drawn to mark the value of the \(D/\mu\) ratio at the crossover points. At \(T_{L}\) = 300 K, it is \(D/\mu\) = 45.7 mV. From Figures 5 and 6, the following is observed as compared to the values obtained by the parabolic band approximation. For GaN, which has a very small value of \(\alpha\) (the present work used \(\alpha\) = 0.19), due to the very wide bandgap of 3.44 eV [5,25], the nonparabolicity effect is very small and can be neglected (approximately 4% around \(n\) = 5 \(\times\) 10\({}^{19}\) cm\({}^{-3}\)). For GaAs and InP, which have relatively large bandgap energies of 1.42 eV and 1.35 eV, respectively [5,25], the effect is also small and can be neglected, causing an increase of only 4% in the nondegenerate condition (the case for GaAs is almost the same as that for InP). However, it is significantly large in heavy-doping conditions, and causes a decrease of approximately 10% around \(n\) = 10\({}^{19}\) cm\({}^{-3}\) for InP (for GaAs the effect is discussed separately later in this case).