WIAS Preprint No. 1689, (2012)

Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals


  • Dreyer, Wolfgang
  • Druet, Pierre-Étienne
    ORCID: 0000-0001-5303-0500
  • Klein, Olaf
    ORCID: 0000-0002-4142-3603
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604

2010 Mathematics Subject Classification

  • 35Q30 35Q61 45G05 76W05 80A20 80A22


  • Czochralski method, crystal growth, traveling magnetic fields, radiative heat transfer, nonlinear PDE systems, Navier--Stokes equations, MHD equations, Maxwell's equations, well-posedness, optimal control, first-order necessary optimality conditions, numerical simulation




This paper deals with the mathematical modeling and simulation of crystal growth processes by the so-called Czochralski method and related methods, which are important industrial processes to grow large bulk single crystals of semiconductor materials such as, e.,g., gallium arsenide (GaAs) or silicon (Si) from the melt. In particular, we investigate a recently developed technology in which traveling magnetic fields are applied in order to control the behavior of the turbulent melt flow. Since numerous different physical effects like electromagnetic fields, turbulent melt flows, high temperatures, heat transfer via radiation, etc., play an important role in the process, the corresponding mathematical model leads to an extremely difficult system of initial-boundary value problems for nonlinearly coupled partial differential equations. In this paper, we describe a mathematical model that is under use for the simulation of real-life growth scenarios, and we give an overview of mathematical results and numerical simulations that have been obtained for it in recent years.

Appeared in

  • Milan J. Math., 80 (2012) pp. 311--332 .

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