# Thermal Energy Diffusion Incorporating Generalized Einstein Relation for Degenerate Semiconductors
## 1 Introduction  
The diffusion coefficient (\(D\)) and mobility (\(\mu\)) of carriers are critically important fundamental transport parameters for describing the properties of carrier transport in semiconductor devices and materials, and are interconnected to each other through a factor by the [PERSON] relation. Due to the theoretically sound interconnection between them, the diffusion coefficients are commonly determined from the mobility using the [PERSON] relation, which therefore plays a key role in semiconductor analysis and device design. However, the values of the factor, i.e., the \(D/\mu\) ratio, are generally given differently depending on the carrier concentration in an energy band and the electronic structure of the energy band, as well as the carrier temperature. The [PERSON] relation was initially derived for particles of Brownian motion based on the ideal gas law and [PERSON]'s law of diffusion [1]. Therefore, this classical [PERSON] relation is valid only for particles that obey Maxwell-Boltzmann statistics and have the same mass. Only in such a case, the \(D/\mu\) ratio is given by a constant, and is particularly equal to the value of the thermal voltage for charged particles developing an electric field. The conventional [PERSON] relation derived for nondegenerate semiconductors based on the use of Maxwell-Boltzmann statistics with the assumption that the carrier effective mass is constant, i.e., the energy band structure is parabolic, is given by:  
\[\frac{D}{\mu}=\frac{k_{\mathrm{B}}T_{L}}{q}, \tag{1}\]where \(k_{B}\) is the Boltzmann constant; \(T_{L}\) denotes the temperature of the semiconductor system in thermal equilibrium, i.e., the lattice temperature (or the carrier temperature) in K; and \(q\) is the positive electronic charge. Even though Equation (1) can be derived separately for carriers in semiconductors by applying the principle of detailed balance to the current equation, or in other ways, its theoretical basis is essentially the same as the classical [PERSON] relation stated above (the inclusion of \(q\) in Equation (1) is due to the charge of a carrier). On the bases of Fermi-Dirac statistics and the parabolic energy band assumption, a general [PERSON] relation that is valid for degenerate, as well as nondegenerate, semiconductors was derived earlier using the concept of the average kinetic energy of electrons in thermal equilibrium, which is given by [2]  
\[\frac{D}{\mu}=\left(\frac{k_{B}T_{L}}{q}\right)\frac{F_{1/2}(\eta_{c})}{dF_{1 /2}(\eta_{c})/d\eta_{c}},\] (2a) and it was shown that the \[D/\mu\] ratio for electrons is approximately equal to \[(2/3)\,\left(\epsilon_{f}-\epsilon_{c}\right)/q\], where \[\epsilon_{f}\] is the Fermi energy (FE), and \[\epsilon_{c}\] refers to the conduction band (CB) bottom energy, for metals and strongly degenerate semiconductors. The [PERSON] relation for carriers in semiconductors was also investigated using an expression \[(D/\mu=(1/q)d\zeta/d(\ln\ n))\] derived in terms of the chemical potential, \[\zeta\], and it was shown that in general, the \[D/\mu\] ratio depends on carrier concentration, rather than being constant, and is proportional to \[n^{2/3}\] for highly degenerate semiconductors [3]. By employing the principle of detailed balance, an expression equivalent to Equation (2a), but simpler, was also derived, such as [4; 5]: \[\frac{D}{\mu}=\left(\frac{k_{B}T_{L}}{q}\right)\frac{F_{1/2}(\eta_{c})}{dF_{1 /2}(\eta_{c})/d\eta_{c}},\]0  
In these equations, \(F_{1/2}(\eta_{c})\) and \(F_{-1/2}(\eta_{c})\) refer, respectively, to Fermi-Dirac integrals of order \(1/2\) and \(-1/2\), which is given as a function of the reduced Fermi energy defined by:  
\[\frac{D}{\mu}=\left(\frac{k_{B}T_{L}}{q}\right)\frac{F_{1/2}(\eta_{c})}{dF_{1 /2}(\eta_{c})/d\eta_{c}},\]1  
Note that Equation (2b) can be obtained from Equation (2a) if the recurrence relation of Fermi-Dirac integrals is utilized.