Abstract
We compare three thermodynamically consistent Scharfetter–Gummel schemes for different distribution functions for the carrier densities, including the Fermi–Dirac integral of order 1/2 and the Gauss–Fermi integral. The most accurate (but unfortunately also most costly) generalized Scharfetter–Gummel scheme requires the solution of an integral equation. Since one cannot solve this integral equation analytically, several modified Scharfetter–Gummel schemes have been proposed, yielding explicit flux approximations to the implicit generalized flux. The two state-of-the-art modified fluxes used in device simulation software are the diffusion-enhanced flux and the inverse activity coefficient averaging flux. We would like to study which of these two modified schemes approximates the implicit flux better. To achieve this, we propose a new method to solve the integral equation numerically based on Gauss quadrature and Newton’s method. This numerical procedure provides a highly accurate reference flux, enabling us to compare the quality of the two modified Scharfetter–Gummel schemes. We extend previous results (Farrell in J Comput Phys 346:497–513, 2017a) showing that the diffusion-enhanced ansatz leads to considerably lower flux errors for the Blakemore approximation to the physically more relevant Fermi–Dirac and Gauss–Fermi statistics.
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This work received funding via the Research Center Matheon supported by ECMath in Project D-CH11 and the DFG CRC 787 “Semiconductor Nanophotonics”.
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This article is part of the Topical Collection on Numerical Simulation of Optoelectronic Devices, NUSOD’ 17.
Guest edited by Matthias Auf der Maur, Weida Hu, Slawomir Sujecki, Yuh-Renn Wu, Niels Gregersen, Paolo Bardella.
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Farrell, P., Patriarca, M., Fuhrmann, J. et al. Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi–Dirac and Gauss–Fermi statistics. Opt Quant Electron 50, 101 (2018). https://doi.org/10.1007/s11082-018-1349-8
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DOI: https://doi.org/10.1007/s11082-018-1349-8