# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 4 Evaluation Results and Discussions
### General Cases
The \(D/\mu\) ratios for the other different values of \(\beta\) are not shown in the figure, but their values are obtained between the curve for \(\beta\) = 0 and the curve 1. Therefore, no crossover point, such as \(P_{D}\), is obtained with the existing formula. It is shown in Figure 3 that, when the electron concentration is larger than 2.33\(N_{c}\), the \(D/\mu\) ratios increase considerably with increasing electron concentration, whereas when it is smaller than 2.33\(N_{c}\), they do not change considerably. Especially when \(n<0.1N_{c}\), the \(D/\mu\) ratios are constant and do not vary with the electron concentration (as in the nondegenerate condition of \(\eta_{c}<-3\)). It is quite interesting to note for a given material that for \(n<2.33N_{c}\), band nonparabolicity causes the \(D/\mu\) ratio  
Figure 3: The same as in Figure 2, but as a function of the electron concentration normalized to the effective density of states (DOS), \(N_{c}\). The dashed curve 1 displays the \(D/\mu\) ratio calculated with the existing formula, Equation (6), for \(\beta\) = 0.18.  
to increase (property 1), but for \(n>2.33N_{c}\), it causes the \(D/\mu\) ratio to decrease (property 2). In Section 1, it was explained that a larger value of \(\beta\) always leads to a larger value of the \(D/\mu\) ratio for a given FE. In the nondegenerate condition (\(\eta_{c}<-3\) or \(n<0.1N_{c}\)), the \(D/\mu\) ratio is always constant, not depending on the FE and therefore not depending on the electron concentration. Therefore, we observe property 1 for \(n<2.33N_{c}\). The reason for property 2 is complicated, but can be explained as in the following. The DOS decreases as the value of \(\beta\) decreases. Therefore, if the CB is strongly degenerated by a very large concentration of electrons, the FE is required to increase considerably inside of the band in order for the very large concentration of electrons to be accommodated within the band if the value of \(\beta\) is small. As a result, the band will be populated by electrons up to a high energy state for a small value of \(\beta\) due to the increased FE. This will eventually give rise to a large average kinetic energy of electrons, so that the diffusivity of electrons will be increased substantially, if constant mobility is assumed. Therefore, as noted in Figure 3, a substantial increase in the \(D/\mu\) ratio is possible, especially when \(\beta=0\), for a very large electron concentration. It is very important to note that the nonparabolicity effect on the \(D/\mu\) ratio can be quite substantial even for a small value of \(\beta\) at a very large carrier concentration. In other words, if a semiconductor material is heavily doped, the parabolic band approximation can lead to a very large error in the \(D/\mu\) ratio. The electron concentration dependencies of the diffusion coefficient and the FE for a given semiconductor material can be easily determined from Figures 2 and 3. In other words, we can find the value of the \(D/\mu\) ratio and thus the value of the diffusion coefficient from Figure 3 with known values of \(n\) (or \(N_{D}\) for homogeneous materials with dopants fully ionized), \(N_{c}\), \(\mu\), and \(\beta\), and thereafter the FE value also from Figure 2.