WIAS Preprint No. 2263, (2016)

Numerical methods for drift-diffusion models


  • Farrell, Patricio
    ORCID: 0000-0001-9969-6615
  • Rotundo, Nella
  • Doan, Duy Hai
  • Kantner, Markus
    ORCID: 0000-0003-4576-3135
  • Fuhrmann, Jürgen
    ORCID: 0000-0003-4432-2434
  • Koprucki, Thomas
    ORCID: 0000-0001-6235-9412

2010 Mathematics Subject Classification

  • 25N08 35K55


  • Scharfetter-Gummel scheme, thermodynamic consistency, Drift-diffusion equations, non-Boltzmann statistic distributions, diffusion enhancement




The van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume discretization scheme and recent efforts to generalize this approach to general statistical distribution functions.

Appeared in

  • P. Farrell, N. Rotundo, D.H. Doan, M. Kantner, J. Fuhrmann, Th. Koprucki, Chapter 50 ``Drift-Diffusion Models'' in Volume 2 of Handbook of Optoelectronic Device Modeling and Simulation: Fundamentals, Materials, Nanostructures, LEDs, and Amplifiers , J. Piprek, ed., Series in Optics and Optoelectronics, CRC Press Taylor & Francis Group, 2017, pp. 733--771.

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