WIAS Preprint No. 2424, (2017)

Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi--Dirac and Gauss--Fermi statistics


  • Farrell, Patricio
    ORCID: 0000-0001-9969-6615
  • Patriarca, Matteo
  • Fuhrmann, Jürgen
    ORCID: 0000-0003-4432-2434
  • Koprucki, Thomas
    ORCID: 0000-0001-6235-9412

2010 Mathematics Subject Classification

  • 65N08 35K55


  • Scharfetter--Gummel schemes, (organic) semiconductors, nonlinear diffusion, ermodynamic consistency, finite volume scheme, Gauss--Fermi integral, Fermi--Dirac integral




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. We propose a new method to solve this integral equation numerically based on Gauss quadrature and Newton's method. We discuss the quality of this approximation and plot the resulting currents for Fermi--Dirac and Gauss--Fermi statistics. Finally, by comparing two modified (diffusion-enhanced and inverse activity based) Scharfetter--Gummel schemes with the more accurate generalized scheme, we show that the diffusion-enhanced ansatz leads to considerably lower flux errors, confirming previous results (J. Comp. Phys. 346:497-513, 2017).

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