This topic is currently not treated at the WIAS

Semiconductor crystals are used in computers, mobile phones, laser devices and solar cells. During their production, electromagnetic fields are often used to produce heat by induction. Moreover, Lorentz forces, generated by time dependent electro magnetic fields, can improve the melt motion during crystal growth processes. Their modeling leads to a system of coupled partial differential equations (PDEs) consisting of the following components:
  1. Time--harmonic Maxwell's equations to describe the electromagnetic fields, with a temperature-dependent electrical conductivity. These equations are solved on domains consisting of many subdomains with different materials, for illustration see the following figure:
    Figure 1: A liquid encapsulated Czochralski (LEC) crystal growth configuration with a heater-magnet modules (see below)
  2. Nonlinear heat equation with nonlocal radiation boundary conditions.
  3. Navier-Stokes equations in the Boussinesq approximation.
The Research Group Thermodynamic Modeling and Analysis of Phase Transitions studies the coupled system of PDEs within the following projects:
  1. In the framework of the Matheon project C9 existence and uniqueness of solutions to the above system of PDEs is proved. Moreover, optimal control problems are studied.
  2. Within the interdisciplinary project Kristallzüchtung im Magnetfeld - KristMAG® (Crystal Growth in Magnetic Fields) the use of traveling magnetic fields generated by special heater-magnet modules (HMM) are investigated. Those HMM consists of coils that are placed above each other (see figure below). They substitute inside the pressure chamber of the growth device the usual meander formed resistance heater units.
    Figure 2: Zoom into the liquid encapsulated Czochralski (LEC) crystal growth configuration with HMM.
    LEC crystal growth processes as well as processes according to the vapour-pressure-controlled Czochralski - VCz strategy are investigated. The simulations reveal the stabilizing effect on the melt due to the Lorentz force. This fact improves the quality of the grown semiconductor single crystals.
  3. Within the 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) it is studied how time dependent magnetic magnetic fields can be used to improve the yield during the production of multi-crystalline solar silicon. To this end the evolution of impurities within the melt is studied. Moreover, effects near the triple line are considered where the interface between melt and crystal meets the crucible wall. Finally, global simulations of temperature and the magnetic field inside the overall growth apparatus are performed.

Publications

  Articles in Refereed Journals

  • P.-É. Druet, Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations, Discrete and Continuous Dynamical Systems, 8 (2015), pp. 479--496.
    Abstract
    We show that Lp vector fields over a Lipschitz domain are integrable to higher exponents if their generalized divergence and rotation can be identified with bounded linear operators acting on standard Sobolev spaces. A Div-Curl Lemma-type argument provides compact embedding results for such vector fields. We investigate the regularity of the solution fields for the low-frequency approximation of the Maxwell equations in time-harmonic regime. We focus on the weak formulation 'in H' of the problem, in a reference geometrical setting allowing for material heterogeneities.

  • 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.

  • W. Dreyer, P.-É. Druet, O. Klein, J. Sprekels, Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals, Milan Journal of Mathematics, 80 (2012), pp. 311--332.
    Abstract
    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.

  • P.-É. Druet, O. Klein, J. Sprekels, F. Tröltzsch, I. Yousept, Optimal control of three-dimensional state-constrained induction heating problems with nonlocal radiation effects, SIAM Journal on Control and Optimization, 49 (2011), pp. 1707--1736.
    Abstract
    The paper is concerned with a class of optimal heating problems in semiconductor single crystal growth processes. To model the heating process, time-harmonic Maxwell equations are considered in the system of the state. Due to the high temperatures characterizing crystal growth, it is necessary to include nonlocal radiation boundary conditions and a temperature-dependent heat conductivity in the description of the heat transfer process. The first goal of this paper is to prove the existence and uniqueness of the solution to the state equation. The regularity analysis associated with the time harmonic Maxwell equations is also studied. In the second part of the paper, the existence and uniqueness of the solution to the corresponding linearized equation is shown. With this result at hand, the differentiability of the control-to-state mapping operator associated with the state equation is derived. Finally, based on the theoretical results, first oder necessary optimality conditions for an associated optimal control problem are established.

  • P.-É. Druet, Existence for the stationary MHD equations coupled to heat transfer with nonlocal radiation effects, Czechoslovak Mathematical Journal, 59 (2009), pp. 791--825.

  • P.-É. Druet, Existence of weak solution to time-dependent MHD equations coupled to heat transfer with nonlocal radiation boundary conditions, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 10 (2009), pp. 2914--2936.

  • P. Rudolph, M. Czupalla, N. Dropka, Ch. Frank-Rotsch, F.-M. Kiessling, O. Klein, B. Lux, W. Miller, U. Rehse, O. Root, Crystal growth from melt in combined heater-magnet modules, Journal of the Korean Crystal Growth and Crystal Technology, 19 (2009), pp. 215-222.

  • O. Klein, Ch. Lechner, P.-É. Druet, P. Philip, J. Sprekels, Ch. Frank-Rotsch, F.M. Kiessling, W. Miller, U. Rehse, P. Rudolph, Numerical simulations of the influence of a traveling magnetic field, generated by an internal heater-magnet module, on liquid encapsulated Czochralski crystal growth, Magnetohydrodynamics. Consultants Bureau, New York (US). Consultants Bureau, New York. Translation from: Magnitnaya Gidrodinamika., 45 (2009), pp. 557-567.

  • J. Geiser, O. Klein, P. Philip, A finite volume scheme designed for anisotropic heat transfer in general polyhedral domains, Advances in Mathematical Sciences and Applications, 18 (2008), pp. 43--67.

  • O. Klein, Ch. Lechner, P.-É. Druet, P. Philip, J. Sprekels, Ch. Frank-Rotsch, F.M. Kiessling, W. Miller, U. Rehse, P. Rudolph, Numerical simulation of Czochralski crystal growth under the influence of a traveling magnetic field generated by an internal heater-magnet module (HMM), Journal of Crystal Growth, 310 (2008), pp. 1523-1532.

  • CH. Lechner, O. Klein, P.-É. Druet, Development of a software for the numerical simulation of VCz growth under the influence of a traveling magnetic field, Journal of Crystal Growth, 303 (2007), pp. 161-164.

  • J. Geiser, O. Klein, P. Philip, Numerical simulation of temperature fields during the sublimation growth of SiC single crystals, using WIAS-HiTNIHS, Journal of Crystal Growth, 303 (2007), pp. 352-356.

  • J. Geiser, O. Klein, P. Philip, Influence of anisotropic thermal conductivity in the apparatus insulation for sublimation growth of SiC: Numerical investigation of heat transfer, Crystal Growth & Design, 6 (2006), pp. 2021--2028.

  • J. Geiser, O. Klein, P. Philip, Transient numerical study of temperature gradients during sublimation growth of SiC: Dependence on apparatus design, Journal of Crystal Growth, 297 (2006), pp. 20-32.

  • P. Philip, O. Klein, Transient conductive-radiative heat transfer: Discrete existence and uniqueness for a finite volume scheme, Mathematical Models & Methods in Applied Sciences, 15 (2005), pp. 227--258.

  • O. Klein, P. Philip, J. Sprekels, Modeling and simulation of sublimation growth of SiC bulk single crystals, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 6 (2004), pp. 295--314.

  • O. Klein, P. Philip, Transient numerical investigation of induction heating during sublimation growth of silicon carbide single crystals, Journal of Crystal Growth, 247 (2003), pp. 219--235.

  • O. Klein, P. Philip, Transient temperature phenomena during sublimation growth of silicon carbide single crystals, Journal of Crystal Growth, 249 (2003), pp. 514--522.

  Contributions to Collected Editions

  • M. Hintermüller, D. Hömberg, O. Klein, J. Sprekels, F. Tröltzsch, C4 -- PDE-constrained optimization with industrial applications, in: MATHEON -- Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 207--222.

  • O. Klein, J. Sprekels, SHOWCASE 13 -- Growth of semiconductor bulk single crystals, in: MATHEON -- Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 224--225.

  • 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.

  • P.-É. Druet, Weak solutions to a model for crystal growth from the melt in changing magnetic fields, in: Optimal Control of Coupled Systems of Partial Differential Equations, K. Kunisch, G. Leugering, J. Sprekels, F. Tröltzsch, eds., 158 of Internat. Series Numer. Math., Birkhäuser, Basel et al., 2009, pp. 123--137.

  • P. Rudolph, Ch. Frank-Rotsch, F.-M. Kiessling, W. Miller, U. Rehse, O. Klein, Ch. Lechner, J. Sprekels, B. Nacke, H. Kasjanov, P. Lange, M. Ziem, B. Lux, M. Czupalla, O. Root, V. Trautmann, G. Bethin, Crystal growth in heater-magnet modules --- From concept to use, in: Proceedings of the International Scientific Colloquium Modelling for Electromagnetic Processing (MEP2008), Hannover, October 26--29, 2008, E. Braake, B. Nacke, eds., Leibniz University of Hannover, 2008, pp. 79-84.

  • P.-É. Druet, Weak solutions to a model for global heat transfer arising in crystal growth from the melt with applied magnetic fields, in: Optimal Control of Coupled Systems of PDE, Workshop, March 2--8, 2008, 5 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2008, pp. 597--599.

  • O. Klein, P.-É. Druet, J. Sprekels, Ch. Lechner, P. Philip, Ch. Frank-Rotsch, F.-M. Kiessling, W. Miller, U. Rehse, P. Rudolph, Numerical simulations of the influence of a traveling magnetic field, generated by an internal heater magnetic module, on Czochralski crystal growth, in: Proceedings of the International Scientific Colloquium Modelling for Electromagnetic Processing (MEP2008), Hannover, October 26--29, 2008, E. Braake, B. Nacke, eds., Leibniz University of Hannover, 2008, pp. 91-96.

  • J. Geiser, O. Klein, P. Philip, tt WIAS-HiTNIHS: Software tool for simulation in crystal growth for SiC single crystal: Application and methods, in: Proceedings of ``International Congress of Nanotechnology'', San Francisco, USA, November 7 -- 10, 2004, 2005.

  • J. Sprekels, O. Klein, P. Philip, K. Wilmanski, Optimal control of sublimation growth of SiC crystals, in: Mathematics --- Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.-J. Krebs, eds., Springer, Berlin [u.a.], 2003, pp. 334--343.

  Talks, Poster

  • P.-E. Druet, Optimization of crystal growth using magnetic fields: Some mathematical aspects, The 18th European Conference on Mathematics for Industry 2014 (ECMI 2014), Minisymposium 36: Recent Trends in Modeling, Analysis, and Simulation of Induction Heat Treatments, June 9 - 14, 2014, Taormina, Italy, June 13, 2014.

  • P.-E. Druet, Some results on distributional solvability vs. higher regularity of the fields in low-frequency electromagnetics, IFIP TC7.2 Workshop Electromagnetics -- Modelling, Simulation, Control and Industrial Applications, May 13 - 17, 2013, WIAS, Berlin, May 13, 2013.

  • 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.

  • J. Sprekels, Mathematical challenges in the industrial growth of semiconductor bulk single crystals, Kick-off Symposium Mathematics: Official Opening of the Felix Klein Building, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, January 13, 2012.

  • J. Sprekels, Optimal control problems arising in the industrial growth of bulk semiconductor single crystals, 21st International Symposium on Mathematical Programming (ISMP), Invited Session ``Optimization Applications in Industry I'', August 19 - 24, 2012, Technische Universität Berlin, August 21, 2012.

  • J. Sprekels, Optimal control problems arising in the industrial growth of bulk single semiconductor crystals, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica ``F. Casorati'', Italy, September 11, 2012.

  • P.-E. Druet, On existence and regularity results for the equations of magnetohydrodynamics in complex geometries, Seminar of International Research Training Group 1529 ``Mathematical Fluid Dynamics'', Technische Universität Darmstadt, June 14, 2011.

  • J. Sprekels, Mathematical challenges in the industrial growth of semiconductor bulk single crystals, Multiphase and Multiphysics Problems, September 25 - 30, 2011, Riemann International School of Mathematics, Verbania, Italy, September 26, 2011.

  • J. Sprekels, Real-life crystal growth: Turbulence, magnetic fields, heat transfer via radiation, and free boundaries, Interfaces and Discontinuities in Solids, Liquids and Crystals (INDI2011), Special Session: Between Mechanics and Mathematics: The ``Non-smooth'' View by Michel Frémond, June 20 - 23, 2011, Gargnano (Brescia), Italy, June 21, 2011.

  • J. Sprekels, Technological and mathematical problems in the industrial growth of semiconductor bulk single crystals, Workshop ``Nonlinear Diffusion: Algorithms, Analysis and Applications'', June 6 - 8, 2011, Warwick Mathematics Institute, UK, June 7, 2011.

  • J. Sprekels, Technological and mathematical problems in the industrial growth of semiconductor bulk single crystals, Academy of Sciences of the Czech Republic, Institute of Mathematics, Prague, March 15, 2011.

  • 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, 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.

  • J. Sprekels, Mathematical problems in industrial crystal growth: Radiation, magnetic fields, and free boundaries, Symposium zur Angewandten Mathematik (zum 100. Geburtstag von Lothar Collatz), October 7 - 8, 2010, Universität Hamburg, October 7, 2010.

  • J. Sprekels, Praktische und mathematische Probleme bei der Züchtung von Halbleiter-Einkristallen, Technische Universität Chemnitz, Forschungsseminar Numerik, January 5, 2010.

  • J. Sprekels, Technical and mathematical problems in the Czochralski growth of single crystals, Workshop ``New Directions in Simulation, Control and Analysis for Interfaces and Free Boundaries'', January 31 - February 6, 2010, Mathematisches Forschungsinstitut Oberwolfach, February 1, 2010.

  • P.-É. Druet, Higher integrability of the magnetic field for 3D induction heating and MHD problems, 14th Belgian-French-German Conference on Optimization, September 14 - 18, 2009, Leuven, Belgium, September 15, 2009.

  • O. Klein, Modulierung und Simulation der LEC GaAs Züchtung in Heizer-Magnetmodulen, 10. Kinetikseminar & 6. Workshop Angewandte Simulation in der Kristallzüchtung, April 1 - 3, 2009, Deutsche Gesellschaft für Kristallwachstum und Kristallzüchtung e.V., Griebnitzsee, April 2, 2009.

  • O. Klein, Numerical simulation of Czochralski crystal growth stabilized by a traveling magnetic field generated by an internal heater-magnet module (HMM), EUROTHERM 84 - ``Thermodynamics of Phase Changes'', May 25 - 27, 2009, Université Catholique de Louvain, Namur, Belgium, May 25, 2009.

  • J. Sprekels, Problems in the industrial growth of semiconductor crystals: Radiative transfer, convection, magnetic fields, and free boundaries, Seminar of the Institute of Applied Mathematics and Information Technology, Università di Pavia, Dipartimento di Matematica ``F. Casorati'', Italy, September 23, 2009.

  • J. Sprekels, Problems in the industrial growth of semiconductor crystals: Radiative transfer, convection, magnetic fields, and free boundaries, BMS Days 2009, Berlin Mathematical School, Urania Berlin, February 17, 2009.

  • P.-É. Druet, Weak solutions to a model for global heat transfer arising in crystal growth from the melt with applied magnetic fields, Workshop ``Optimal Control of Coupled Systems'', March 2 - 7, 2008, Mathematisches Forschungsinstitut Oberwolfach, March 5, 2008.

  • P.-É. Druet, On weak solutions to the system of magnetohydrodynamics coupled to heat transfer, including nonlocal radiation effects, Oberseminar Analysis, Technische Universität Dresden, Institut für Analysis, January 31, 2008.

  • O. Klein, Numerical simulations of the influenece of a traveling magnetic field, generated by an internal heater magnetic module, on Czochralski crystal growth, International Scientific Colloquium ``Modelling for Electromagnetic Processing'' (MEP 2008), October 26 - 29, 2008, Leibniz Universität Hannover, Hannover, October 27, 2008.

  • J. Sprekels, Problems in the industrial growth of semiconductor crystals: Radiative heat transfer, convection, magnetic fields and free boundaries, The Legacy of John Crank -- Developments in Time Dependent PDE's, Diffusion and Free Boundary Problems, July 9 - 11, 2008, Brunel University, Uxbridge, UK, July 10, 2008.

  • O. Klein, Ch. Lechner, P.-É. Druet, P. Philip, J. Sprekels, Ch. Frank-Rotsch, F.-M. Kiessling, W. Miller, U. Rehse, P. Rudolph, Numerical simulation of Czochralski crystal growth under the influence of a traveling magnetic field generated by an internal heater-magnet module (HMM), The 15th International Conference on Crystal Growth, Salt Lake City, USA, August 12 - 17, 2007.

  • O. Klein, P. Philip, J. Geiser, D. Siche, J. Wollweber, Numerical investigation of the influence of an anisotropic insulation felt on the temperature field in a PVT growth apparatus, The 15th International Conference on Crystal Growth, Salt Lake City, USA, August 12 - 17, 2007.

  • J. Geiser, O. Klein, P. Philip, Numerical simulation of temperature fields during the sublimation growth of SiC single crystals using ttfamily WIAS-HiTNIHS, 5th International Workshop on Modeling in Crystal Growth, Bamberg, September 10 - 13, 2006.

  • O. Klein, Ch. Lechner, P.-É. Druet, Numerical simulation of VCz growth with a traveling magnetic field, 5th International Workshop on Modeling in Crystal Growth, Bamberg, September 10 - 13, 2006.

  • O. Klein, Modeling Czochralski crystal growth under the influence of electromagnetic forces, Recent Advances in Free Boundary Problems and Related Topics (FBP2006), September 14 - 16, 2006, Levico, Italy, September 15, 2006.

  • J. Geiser, O. Klein, Numerical simulations of sublimation growth for SiC single crystals: Anisotropy materials, radiation and transient heat-transfer, Eighth U.S. National Congress on Computational Mechanics, Session: Complex Models in Solid and Fluid Mechanics: Methods and Applications in Multi-physical and Multi-scaling Problems (2), July 24 - 28, 2005, U.S. Association for Computational Mechanics, Austin, USA, July 27, 2005.

  • O. Klein, Neue Entwicklungen in der Simulation der Sublimationszüchtung mit WIAS-HiTNIHS: Anisotropie, Semitransparenz, Nutzeroberfläche, DGKK Arbeitskreis Angewandte Simulation in der Kristallzüchtung, November 2 - 4, 2005, Deutsche Gesellschaft für Kristallwachstum und Kristallzüchtung e.V., Heigenbrücken, November 3, 2005.

  • O. Klein, Numerical simulations of sublimation growth for SiC single crystal: Induction heating, anisotropy materials, radiation and Considerations for dealing with several coils within the project KristMag, Treffen der Numeriker-Gruppe des KristMag-Projektes, Universität Hannover, Institut für Elektrothermische Prozesse, September 7, 2005.

  • O. Klein, Optimierung des Temperaturfeldes bei der Sublimationszüchtung von SiC Einkristallen, DGKK Arbeitskreis Angewandte Simulation in der Kristallzüchtung, February 5 - 6, 2004, Deutsche Gesellschaft für Kristallwachstum und Kristallzüchtung e.V., Volkach, February 5, 2004.

  • C. Meyer, O. Klein, P. Philip, A. Rösch, J. Sprekels, F. Tröltzsch, Optimal"-steuerung bei der Herstellung von SiC-Einkristallen, MathInside---Überall ist Mathematik, event of the DFG Research Center ``Mathematics for Key Technologies'' on the occasion of the Open Day of Urania, Berlin, September 13, 2003.

  • O. Klein, P. Philip, J. Sprekels, K. Wilmanski, Numerical simulations and control of SiC bulk single crystal growth, Statusseminar zum BMBF-Fördergebiet "`Ausgewählte Gebiete der Mathematik"', Frankfurt am Main, December 11 - 12, 2000.