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On thermodynamic consistency of a Scharfetter–Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement

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Abstract

Driven by applications like organic semiconductors there is an increased interest in numerical simulations based on drift-diffusion models with arbitrary statistical distribution functions. This requires numerical schemes that preserve qualitative properties of the solutions, such as positivity of densities, dissipativity and consistency with thermodynamic equilibrium. An extension of the Scharfetter–Gummel scheme guaranteeing consistency with thermodynamic equilibrium is studied. It is derived by replacing the thermal voltage with an averaged diffusion enhancement for which we provide a new explicit formula. This approach avoids solving the costly local nonlinear equations defining the current for generalized Scharfetter–Gummel schemes.

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Acknowledgments

The work has been supported by ERC-2010-AdG no. 267802  Analysis of Multiscale Systems Driven by Functionals (N.R.) and by Deutsche Forschungsgemeinschaft (DFG) within the collaborative research center 787 Semiconductor Nanophotonics (T.K.).

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Correspondence to Thomas Koprucki.

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Koprucki, T., Rotundo, N., Farrell, P. et al. On thermodynamic consistency of a Scharfetter–Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement. Opt Quant Electron 47, 1327–1332 (2015). https://doi.org/10.1007/s11082-014-0050-9

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Keywords

  • Scharfetter–Gummel scheme
  • Thermodynamic consistency