WIAS Preprint No. 2575, (2019)

Mathematical modeling of semiconductors: From quantum mechanics to devices


  • Kantner, Markus
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Mittnenzweig, Markus
    ORCID: 0000-0002-8502-1702
  • Rotundo, Nella

2010 Mathematics Subject Classification

  • 35K57 80M30 81S22, 82D37


  • Semiconductor modeling, drift-diffusion system, open quantum system,, Lindblad operator, reaction-diffusion systems, detailed balance condition, gradient structure, thermodynamically consistent coupling




We discuss recent progress in the mathematical modeling of semiconductor devices. The central result of this paper is a combined quantum-classical model that self-consistently couples van Roosbroeck's drift-diffusion system for classical charge transport with a Lindblad-type quantum master equation. The coupling is shown to obey fundamental principles of non-equilibrium thermodynamics. The appealing thermodynamic properties are shown to arise from the underlying mathematical structure of a damped Hamitlonian system, which is an isothermal version of so-called GENERIC systems. The evolution is governed by a Hamiltonian part and a gradient part involving a Poisson operator and an Onsager operator as geoemtric structures, respectively. Both parts are driven by the conjugate forces given in terms of the derivatives of a suitable free energy.

Appeared in

  • Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 269--293, DOI 10.1007/978-3-030-33116-0 .

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