Our goal is to simulate charge transport in semiconductors, especially for general charge carrier statistics, which describe non-degenerate semiconductors. We consider Voronoi finite volume schemes for the discretization of the corresponding charge transport model and pay particular attention to the choices of flux approximations.
In the Boltzmann regime, which corresponds to a model for the charge carrier transport in degenerate semiconductor devices, the classical Scharfetter-Gummel flux discretization scheme [3] achieves numerical stability as well as preserves important physical properties such as thermodynamic consistency. A generalization of this approach to more general statistics while keeping these properties is desired and was introduced in [2]. However, this generalized Scharfetter-Gummel scheme becomes computational expensive. Thus, we need alternative flux discretization schemes.
But how to choose these discrete current densities?
In the literature, state-of-the-art modified Scharfetter-Gummel schemes are known to lower the computational costs [4, 5]. One of these schemes appears to be customary in parts of the device simulation community. The earliest reference we could find for this discrete current density is the source code of the SEDAN III simulator [5], therefore we call this scheme the Sedan scheme. Unfortunately, there seem to be no direct comparisons for the Sedan scheme. For this reason we direcly compare it to other state-of-the-art schemes.
The methods of comparison are based on [1]. Hence, this work can be seen as an extension of this paper. Noteworthy, is that for the simulation and the resulting computations depicted in Figure 1 a Julia-based solver VoronoiFVM.jl was used. This makes it possible to work with ForwardDiff.jl, a package based on automatic differentiation, which provides the ability to compute Jacobians automatically without needing to compute any derivatives.
Further details will be presented at the NUSOD-20 conference.
Figure 1: Left: The I-V curves computed with the different schemes for fixed mesh refinement and the simulation of a non-degenerate GaAs p-i-n diode. The reference solution was computed using the generalized Scharfetter-Gummel scheme on refinement level 10. Right: Convergence studies for the absolute errors of the total currents.
[1] P. Farrell, T. Koprucki, and J. Fuhrmann, “Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics”, J. Comput. Phys., vol. 346, pp. 497–513, 2017. [2] R. Eymard, J. Fuhrmann, and K. Gärtner, “A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems”, Numer. Math., vol. 102, no. 3, pp. 463–495, 2006. [3] D. Scharfetter and H. Gummel, “Large-signal analysis of a silicon Read diode oscillator”, IEEE Trans. Electr. Dev., vol. 16, pp. 64–77, 1969. [4] M. Bessemoulin-Chatard, “A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme”, Numerische Mathematik, vol. 121, pp. 637–670, 2012. [5] Z. Yu, and R. Dutton, “SEDAN III – A one-dimensional device simulator”, http://www-tcad.stanford.edu/tcad/programs/sedan3.html, 1988.
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