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”?

Of course one might look at many different criteria. We compared both schemes to the (computationally inefficient but more accurate) generalized Scharfetter-Gummel scheme [2]. The logarithmic absolute error for a range of different (electrostatic) potential differences at neighboring nodes (presuming Blakemore statistics) looks like this:

The picture on the left shows the error for the diffusion averaged flux and the picture on the right the error for the inverse activity based flux. The bold black lines highlight the same contour levels in both plots. The dashed lines show where generalized and modified schemes agree exactly: They agree when there is no diffusion (pure drift current, the horizontal dashed line). The diagonal dashed line is due to the consistency with the thermodynamic equilibrium. So judging from both pictures it seems that the diffusion average flux yields a considerably lower flux error. In both pictures, we fixed the arithmetic average between the electrochemical potentials. What if we raise the diffusion enhancement by increasing the average from 1.5 to 5?

In this case, the superiority of the diffusion-enhanced scheme is even more obvious! So is there possibly a general law behind this? Yes, there is! In [2], we managed to show that the diffusion averaged scheme depends more favorably on the diffusion enhancement than any inverse activity based scheme.

Of course there are other valid criteria which one might examine. In [2], for example, we also studied a full van Roosbroeck simulation of a pin diode. The absolute error in the IV curves also speaks in favor of the diffusion-enhanced scheme.

More details will be presented at the NUSOD 2017 conference (talk ThD2).

### Citations

- D. Scharfetter, H. Gummel,
*Large-signal analysis of a silicon read diode oscillator*, IEEE Transactions on Electron Devices **16**, 64-77 (1969) DOI: 10.1109/T-ED.1969.16566
- P. Farrell, T. Koprucki, J. Fuhrmann.
*Computational and Analytical Comparison of Flux Discretizations for the Semiconductor Device Equations beyond Boltzmann Statistics*, Journal of Computational Physics, **346**, 497-513 (2017), DOI: 10.1016/j.jcp.2017.06.023
- M. Bessemoulin-Chatard,
* A finite volume scheme for convection-diffusion equations with nonlinear diffusion derived from the Scharfetter-Gummel scheme,* Numerische Mathematik **121**, 637-670 (2012) DOI 10.1007/s00211-012-0448-x
- T. Koprucki, N. Rotundo, P. Farrell, D. H. Doan, J. Fuhrmann,
*On thermodynamic consistency of a Scharfetter-Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement,* Optical and Quantum Electronics **47**, 1327-1332 (2015) DOI: 10.1007/s11082-014-0050-9
- J. Fuhrmann,
*Comparison and numerical treatment of generalised Nernst-Planck models*, Computer Physics Communications **196**, 166-178 (2015) DOI 10.1016/j.cpc.2015.06.004