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 [1] 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 [2] we compared two state-of-the-art numerical flux discretizations:

- Diffusion averaged flux approximation [3, 4]
- Inverse activity based flux approximations [5]

But which of two is the “best”? Read more of this post