# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 4 Evaluation Results and Discussions
### Specific Cases
For InAs, which has a narrow bandgap of 0.35 eV [5,16,25], the effect is substantially large, increasing by 15% in the nondegenerate condition, and decreasing by approximately 18% around \(n\) = 10\({}^{18}\) cm\({}^{-3}\). The nonparabolicity effect is much more significant with InSb, which has a very narrow bandgap of about 0.2 eV [16,25,26]. For this semiconductor, it is worth noting that a maximum error caused by the parabolic band approximation occurs in the nondegenerate condition for the possible practical range of doping densities, and amounts to 28%. Figure 4: Calculated values of the \(D/\mu\) ratios for conduction electrons as a function of the reduced FE for common direct-bandgap compound semiconductors at \(T_{L}\) = 300 K. The plot (**a**) is for \(\eta_{c}\) \(<\) 3; and the plot (**b**) for \(\eta_{c}\) \(>\) 3. Figures 4-6 have been obtained by considering only the lowest CB. However, the compound semiconductors used for these figures have another upper CB, in which it is possible for some electrons to occupy at very large Fermi energies, i.e., in heavy-doping conditions. In GaAs, the upper CB is separated only by 0.3 eV from the lowest CB, while it is separated by 0.4 eV or more than this energy in the other semiconductors (\(\Delta_{\Gamma-\mathrm{L}}\) = 0.3 eV for GaAs, \(\Delta_{\Gamma-\mathrm{X}}\) = 0.4 eV for InSb, \(\Delta_{\Gamma-\mathrm{L}}\) = 0.59 eV for InP, \(\Delta_{\Gamma-\mathrm{L}}\) = 0.72 eV for InAs, and \(\Delta_{\Gamma-\mathrm{X}}\) = 1.22 eV for GaN) [26]. It has been found at room temperature that for the semiconductors, except GaAs, the electron concentration in the upper CB is negligibly small at \(\eta_{c}\) = 10; this is not so for GaAs when \(\eta_{c}\) > 7 or \(n\) > 7.5 \(\times\) 10\({}^{18}\) cm\({}^{-3}\), as indicated by the extension of the \(D/\mu\) ratio curve for GaAs with dashes for values above \(D/\mu\) = 4.8 (\(\times\)26 mV) in Figure 5a. In GaAs, the upper CB consists of four equivalent L-valleys. The electron concentration in this CB, \(n_{\mathrm{L}}\), has been additionally calculated in thermal equilibrium for \(T_{L}\) = 300 K, and the ratio of \(n_{\mathrm{L}}\) to \(n_{\mathrm{T}}\) is plotted as a function of \(\eta_{c}\) in Figure 7 (\(n_{\Gamma}\) denotes the electron concentration in the lowest CB (\(\Gamma\)-valley). For the calculation of \(n_{\mathrm{L}}\), \(\alpha\) = 0.46 for the nonparabolicity [19] and 0.56\(n_{e}\), where \(m_{e}\) is the free electron rest mass, for the electron DOS effective mass [26] were used for the L-valleys, based on the isotropic band approximation. Figure 7 shows that at \(\eta_{c}\) = 10, the concentration ratio, \(n_{\mathrm{L}}/n_{\Gamma}\), amounts to 0.16, and at \(\eta_{c}\) = 12, it reaches up to 0.63, due to a very large value of the effective DOS, \(N_{c}\), of the upper CB, which is 24 times larger than that of the lowest CB. Therefore, for GaAs the electron population in the upper CB cannot be neglected and will affect the \(D/\mu\) ratio for conduction electrons  
Figure 5: The same as in Figure 4, but as a function of the electron concentration. The plot **(b)** is an enlarged detailed view of the plot **(a)** for the values in the nondegenerate condition. at very large Fermi energies or electron concentrations at room temperature. In addition, from Figure 7, the following very important fact can be deduced. If we assume that the energy separation (\(\Delta_{\Gamma-\mathrm{L}}\)) of GaAs is 0.4 eV, rather than 0.3 eV, then the concentration ratio at \(\eta_{c}=10\) will decrease to less than 1%, which is the value corresponding to the concentration ratio when the position of \(\eta_{c}\) has changed from 10 to 6.1 in Figure 7. This suggests that in semiconductors, the electronic property of the band structure above 0.4 eV from the lowest CB edge does not affect the \(D/\mu\) ratio for conduction electrons generally at room temperature, unless they are doped excessively so that \(\eta_{c}\) exceeds 10, even though they have multiple CB bands.