# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 2 The Effective Temperature of Electrons in Thermal Equilibrium  
[PERSON]'s dispersion relation for isotropic nonparabolic CBs is given by:  
\[\epsilon-\epsilon_{c}=\frac{\hbar^{2}k^{2}}{2m_{c}[1+\alpha(\epsilon-\epsilon_{c} )]}, \tag{10}\]  
where \(\hbar\) is the reduced Plank's constant; \(k\) is the magnitude of the wave vector \(\mathbf{k}\); and \(m_{c}\) denotes the electron effective mass at \(\epsilon_{c}\), as defined previously. The electron concentration and the thermal kinetic energy density of electrons can be evaluated simultaneously from the following density integral moment equation by introducing a moment parameter \(\lambda\):  
\[\mathcal{D}=\frac{1}{4\pi^{3}}\int(\epsilon-\epsilon_{c})^{\lambda}f(\mathbf{ k})d^{3}k, \tag{11}\]  
where \(f(\mathbf{k})\) is the probability function for an electron near \(\mathbf{k}\), and the integration is over all \(\mathbf{k}\)-space within the band (in the first Brillouin zone). If \(\lambda=0\), Equation (11) yields the electron concentration, and if \(\lambda=1\), the thermal kinetic energy density. In thermal equilibrium, the distribution function \(f(\mathbf{k})\) is given by the isotropic Fermi-Dirac distribution function. Therefore, by substitution, we have:  
\[\mathcal{D}=\frac{1}{\pi^{2}}\int_{0}^{\infty}\frac{\left(\epsilon-\epsilon_{c }\right)^{\lambda}k^{2}dk}{1+\exp\left(\epsilon-\epsilon_{f}\right)/\left(k_{ B}T_{L}\right)}. \tag{12}\]  
Differentiating Equation (10) with respect to \(k\) gives:  
\[de=\frac{\hbar^{2}k}{m_{c}[1+2\alpha(\epsilon-\epsilon_{c})]}dk. \tag{13}\]  
Substitution of Equations (10) and (13) into Equation (12) yields:  
\[ \begin{array}{c}\mathcal{D}=4\pi\left(\frac{2m_{c}}{\hbar^{2}}\right)^{3/2} \int_{\epsilon_{c}}^{\infty}\frac{\left(\epsilon-\epsilon_{c}\right)^{\lambda +1/2}\sqrt{1+\alpha(\epsilon-\epsilon_{c})}[1+2\alpha(\epsilon-\epsilon_{c})]} {1+\exp\left[\left(\epsilon-\epsilon_{f}\right)/\left(k_{B}T_{L}\right)\right] }d\epsilon\\ =\frac{2}{\sqrt{\pi}}N_{c}(k_{B}T_{L})^{\lambda}\int_{0}^{\infty}\frac{\chi^{ \lambda+1/2}\sqrt{1+\beta\lambda}(1+2\beta\lambda)}{1+\exp\left(\chi-\eta_{c} \right)}d\chi,\end{array}  \tag{14}\]  
where \(h\) is [PERSON]'s constant and a dimensionless parameter \(\beta=ak_{B}T_{L}\) has been introduced. The integration variable is defined by \(x=(\epsilon-\epsilon_{c})/(k_{B}T_{L})\), and \(N_{c}\), the effective DOS, is given by:  
\[N_{c}=2\left(\frac{2\pi k_{B}T_{L}m_{c}}{\hbar^{2}}\right)^{3/2}. \tag{15}\]  
In the present work, the square-root part involved in the integral of Equation (14) is approximated by choosing a suitable fitting number for the coefficient of the linear term empirically based on the following expansion property:  
\[\sqrt{1+\beta\chi}=1+\frac{1}{2}\beta\chi-\frac{1}{8}(\beta\chi)^{2}+\cdots \approx 1+\frac{2}{5}\beta\chi. \tag{16}\]  
The accuracy of this empirical approximation for Equation (14) has been numerically calculated in the case of \(\lambda=1\) (larger errors are resultant for \(\lambda=1\) than \(\lambda=0\)), and the results are depicted in Figure 1. Note that the approximation is quite accurate with a maximum error of less than 2% over a very wide range of \(\eta_{c}\) for the extensive variation of \(\beta\) values. In the same figure, calculated errors induced by the binomial approximation are also depicted in dashed curves typically for \(\beta=0.12\) and \(0.18\) for comparison. As shown, the errors of this approximation increase largely with the FE for \(\eta_{c}>3\), and in particular, the error is significantly larger for the large value of \(\beta=0.18\) at Fermi energies of \(\eta_{c}>5\).