WIAS Preprint No. 1738, (2012)

Discretization scheme for drift-diffusion equations with a generalized Einstein relation


  • Koprucki, Thomas
    ORCID: 0000-0001-6235-9412
  • Gärtner, Klaus

2010 Mathematics Subject Classification

  • 65N08 35K55


  • generalized Einstein relation, generalized Scharfetter-Gummel scheme, drift-diffusion equations, non-Boltzmann statistic distributions, diffusion enhancement


Inspired by organic semiconductor models based on hopping transport introducing Gauss-Fermi integrals a nonlinear generalization of the classical Scharfetter-Gummel scheme is derived for the distribution function F(η)=1/(exp(-η)+γ). This function provides an approximation of the Fermi-Dirac integrals of different order and restricted argument ranges. The scheme requires the solution of a nonlinear equation per edge and continuity equation to calculate the edge currents. In the current formula the density-dependent diffusion enhancement factor, resulting from the generalized Einstein relation, shows up as a weighting factor. Additionally the current modifies the argument of the Bernoulli functions.

Appeared in

  • Opt. Quantum Electron., 45 (2013) pp. 791--796.

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