WIAS Preprint No. 2331, (2016)

Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics


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

2010 Mathematics Subject Classification

  • 35Q99 82D37 65M08, 65N08, 74S10


  • finite volume method, flux discretization, Scharfetter--Gummel scheme, Fermi--Dirac statistics, degenerate semiconductors, van Roosbroeck system, semi-conductor device simulation, nonlinear diffusion, diffusion enhancement




For a Voronoï finite volume discretization of the van Roosbroeck system with general charge carrier statistics we compare three thermodynamically consistent numerical fluxes known in the literature. We discuss an extension of the Scharfetter--Gummel scheme to non-Boltzmann (e.g. Fermi--Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.

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