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Connecting Theory and Practice in Optoelectronics
In semiconductor device simulations, one often uses the so-called Boltzmann approximation (an exponential) to link Fermi energy levels to the densities of charge carriers. This corresponds to the classical Einstein relation between diffusion and mobility. However, in organic semiconductors, highly doped regions or at cryogenic temperatures the electron-hole plasma becomes degenerate and Fermi-Dirac statistics can no longer be neglected. The implications are two-fold: First, the distribution function F(η) may no longer be approximated by an exponential (Boltzmann approximation). Second, a generalized Einstein relation now governs the diffusive currents via a nonlinear density-dependent diffusion factor g (the diffusion enhancement). This factor reduces to one in the nondegenerate Boltzmann case and measures the degree of the degeneracy.
The Boltzmann approximation appears in both figures as a straight blue line. For low carrier densities it approximates the Fermi-Dirac integral as well as the Gauss-Fermi integral for organic semiconductors.
The classical Scharfetter-Gummel scheme  in combination with a Voronoï finite volume method provides a numerical solution to the drift-diffusion equations in nondegenerate semiconductors. However, how to approximate the numerical fluxes for degenerate semiconductors (leading to nonlinear diffusion), is still an active area of research. Recently, in  we compared two state-of-the-art numerical flux discretizations:
But which of two is the “best”? Read more of this post