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Connecting Theory and Practice in Optoelectronics
Whenever you try to answer a research question by using numerical simulation, you start by developing a (or reusing an already developed) mathematical model. Thereafter, you are developing (or reusing already developed) software to perform your numerical simulation. This produces data that you are now analysing and visualising to interpret the discovered results. Finally, you might want to write it all down in an article and publish it on the arXiv and/or in a scientific journal. In this published form the results consist – apart from a couple of figures or tables – mainly of text. In most cases mathematical models, software, data, visualisations and so on are not or not fully shown. This makes it difficult for editors, reviewers and readers alike to fully grasp the research and its results, see e.g. How to get your simulation paper accepted. Moreover, it makes it difficult to validate and in many cases impossible to reproduce the results.
In a time when scholarly publication was limited to printed journals and books it was simply not feasible to provide long rows of numbers not to mention interactive 3D figures or a moving series of pictures. However, with the advent of the digital age and its easy accessible and easy to use infrastructures and tools there is no excuse for not publishing the full research story – and that does not only consist of plain text.
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