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Optimal Control of Semiconductor Melts by Traveling Magnetic Fields

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Abstract

In this paper, the optimal control of traveling magnetic fields in a process of crystal growth from the melt of semiconductor materials is considered. As controls, the phase shifts of the voltage in the coils of a heater-magnet module are employed to generate Lorentz forces for stirring the crystal melt in an optimal way. By the use of a new industrial heater-magnet module, the Lorentz forces have a stronger impact on the melt than in earlier technologies. It is known from experiments that during the growth process, temperature oscillations with respect to time occur in the neighborhood of the solid-liquid interface. These oscillations may strongly influence the quality of the growing single crystal. As it seems to be impossible to suppress them completely, the main goal of optimization has to be less ambitious, namely one tries to achieve oscillations that have a small amplitude and a frequency which is sufficiently high such that the solid-liquid interface does not have enough time to react to the oscillations. In our approach, we control the oscillations at a finite number of selected points in the neighborhood of the solidification front. The system dynamics is modeled by a coupled system of partial differential equations that account for instationary heat conduction, turbulent melt flow, and magnetic field. We report on numerical methods for solving this system and for the optimization of the whole process. Different objective functionals are tested to reach the goal of optimization.

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Funding

This work was supported by DFG Research Center MATHEON, “Mathematics for Key Technologies” in Berlin, project C9.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623, Berlin, Germany

    Peter Nestler, Nico Schlömer & Fredi Tröltzsch

  2. Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS), Mohrenstr. 39, 10117, Berlin, Germany

    Olaf Klein & Jürgen Sprekels

  3. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Jürgen Sprekels

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  1. Peter Nestler
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  2. Nico Schlömer
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Correspondence to Fredi Tröltzsch.

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Nestler, P., Schlömer, N., Klein, O. et al. Optimal Control of Semiconductor Melts by Traveling Magnetic Fields. Vietnam J. Math. 47, 793–812 (2019). https://doi.org/10.1007/s10013-019-00355-5

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