This topic is currently not treated at the WIAS

Currently, solar cells are mainly produced by using multi-crystalline silicon. A part of the overall production costs of a usable solar module is due to crystal growth from the silicon melt. The aim of modern crystal growth processes is the reduction of these costs while also improving the quality of the grown crystals.
The joint research project "Anlagen- und Verfahrensentwicklung sowie Absatz einer neuen Technologie zur Kristallisation von Solar-Silizium (AVANTSOLAR, Equipment and process development as well as sale of a new technology for the crystallization of solar silicon)" investigated until June 2011, how to improve the yield during the production of multi-crystalline solar silicon by means of time dependent electro magnetic fields. The research group Thermodynamic Modeling and Analysis of Phase Transitions contributed to this project by modelling and simulations to the topics:
  1. Calculation of the distribution of wanted as well as unwanted impurities within the electric conducting melt.
  2. Effects in the vicinity of the triple-line, where crystal, melt and crucible wall meet each other.
  3. Temperature distribution and electromagnetic fields in the overall growth apparatus
In this context, the following system of partial differential equations (PDEs) is investigated:
  1. Maxwell's equations to describe the electromagnetic fields.
  2. Nonlinear heat equation with coupling to nonlocal radiation boundary conditions.
  3. Navier-Stokes equations in the Boussinesq approximation with coupling to reaction-diffusion equations.
  4. Free boundary problems of Stefan-Gipps-Thomson-type; these are two phase Stefan problems with surface curvature and under-cooling.
Moreover, the existence and uniqueness of solutions to the above coupled system of PDEs and corresponding problems of optimal control were investigated in the project C9 of the DFG research center Matheon until May 2014.


  Articles in Refereed Journals

  • P.-É. Druet, Some mathematical problems related to the second order optimal shape of a crystallization interface, Discrete and Continuous Dynamical Systems, 35 (2015) pp. 2443--2463.
    We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient.

  • CH. Kudla, A.T. Blumenau, F. Büllesfeld, N. Dropka, Ch. Frank-Rotsch, F. Kiessling, O. Klein, P. Lange, W. Miller, U. Rehse, U. Sahr, M. Schellhorn, G. Weidemann, M. Ziem, G. Bethin, R. Fornari, M. Müller, J. Sprekels, V. Trautmann, P. Rudolph, Crystallization of 640 kg mc-silicon ingots under traveling magnetic field by using a heater-magnet module, Journal of Crystal Growth, 365 (2013) pp. 54--58.

  Contributions to Collected Editions

  • N. Dropka, W. Miller, U. Rehse, P. Rudolph, F. Bullesfeld, U. Sahr, O. Klein, D. Reinhardt, Numerical study on improved mixing in silicon melts by double-frequency TMF, in: Proceedings of the 16th International Conference on Crystal Growth (ICCG-16), J. Crystal Growth, 318, 2011, pp. 275--279.

  Talks, Poster

  • O. Klein, Modeling and numerical simulation of the application of traveling magneticfields to stabilize crystal growth from the melt, 12th International Conference on Free Boundary Problems: Theory and Applications, June 11 - 15, 2012, Frauenchiemsee, June 11, 2012.

  • W. Dreyer, P.-É. Druet, F. Duderstadt, Ch. Grützmacher, O. Klein, D. Reinhardt, J. Sprekels, Ch. Frank-Rotsch, F.-M. Kiessling, W. Miller, U. Rehse, P. Rudolph, Ch. Lechner, Crystal growth under the influence of traveling magnetic fields, Status Seminar AVANTem SOLAR, Griebnitzsee, September 9 - 10, 2010.

  • O. Klein, K. Afanasiev, F. Duderstadt, W. Dreyer, S. Eichler, J. Sprekels, On effects in the vicinity of the solid melt interface during crystal growth from the melt, The 16th International Conference on Crystal Growth (ICCG-16) in conjunction with The 14th International Conference on Vapor Growth and Epitaxy (ICVGE-14), Beijing, China, August 8 - 13, 2010.

  • O. Klein, Modeling and simulations for directional solidification of solar-grade silicon under the influence of traveling magnetic fields, The 16th International Conference on Crystal Growth (ICCG-16) in conjunction with The 14th International Conference on Vapor Growth and Epitaxy( ICVGE-14), August 8 - 13, 2010, Beijing, China, August 10, 2010.

  • O. Klein, Numerical simulation of the use of traveling magnetic fields to stabilize crystal growth from the melt, Warsaw Seminar on Industrial Mathematics (WSIM'10), March 18 - 19, 2010, Warsaw University of Technology, Poland, March 18, 2010.