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Matheon Project C9: Simulation and Optimization of Semiconductor Crystal Growth from the Melt Controlled by Traveling Magnetic Fields

(Project was called "Numerical simulation and control of sublimation growth of semiconductor bulk single crystals" until 05/2010, the beginning of the present funding period of the DFG Research Center Matheon.)

Quick-Links

• Research Team, Collaboration, Funding
• Crystal growth techniques
• Modeling and optimization of sublimation growth
• Modeling and optimization of growth from the melt
• Modeling
• Mathematical Analysis
• The Simulation Software
• Current and Future Tasks
• Publications
• Presentations

Research Team, Collaboration, Funding

Head Persons of Research Team

Prof. Dr. Jürgen Sprekels, Dr. Olaf Klein
(Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany),
Prof. Dr. Fredi Tröltzsch
(Department of Mathematics, Technical University (TU), Berlin, Germany).

Current Research Team and Topical Classification

Dr. Pierre-Etienne Druet, Dr. Olaf Klein, Prof. Dr. Jürgen Sprekels, Prof. Dr. Fredi Tröltzsch, Dr. Irwin Yousept, Vili Dhamo (associated member).
Pierre-Etienne Druet, Olaf Klein, and Jürgen Sprekels are members of the research group Thermodynamic Modeling and Analysis of Phase Transition at WIAS;
Fredi Tröltzsch, Irwin Yousept and Vili Dhamo are members of the research group Modelling - Numerics - Differential Equations, Section: Optimization in PDEs at the Department of Mathematics of TU Berlin.
At WIAS, this project belongs to the application Crystal Growth, Especially under the Influence of Magnetic Fields in the Main Application Area Phase Transitions and Multi-functional Materials.

Related Projects:

• WIAS-HiTNIHS.

In Collaboration With

Leibniz Institute of Crystal Growth (IKZ), Berlin, Germany

Funding

This project is currently supported by the Deutsche Forschungsgemeinschaft (DFG) within the DFG Research Center Matheon "Mathematics for Key Technologies".

Crystal growth techniques

Semiconductor crystals like silicon (Si), silicon carbide (SiC), gallium arsenide (GaAs), and aluminum nitride (AlN) crystals have important applications in key technologies.

Industrial applications of these semiconductor crystals require the availability of large-diameter, low-defect crystals, and a high growth rate during the production process is desirable to reduce production time and costs.

• The most important industrial technique for growth of Si and GaAs bulk single crystals is the growth from the melt, especially by the Czochralski method and their variants, i.e. Liquid Encapsulated Czochralski (LEC) and Vapor pressure controlled Czochralski (VCz).

  In this kind of processes, a rotating crystal seed is dipped into the melt that is contained in a rotating crucible. The seed is then slowly pulled out of the melt, and a single crystal solidifies. Since the melt flow is turbulent, impurities may find their way into the growing crystal, and temperature oscillations with small frequencies may occur below the crystal. This should be avoided, since this would lower the quality of the grown crystal.

  To achieve this aim, one applies time dependent electromagnetic fields to the melt, which is electrically conducting so that a Lorentz force is induced into it in their presence. This Lorentz force can be used to influence the melt motion.

• The state-of-the-art procedure for the production of SiC and AlN bulk single crystals is the Physical Vapor Transport (PVT) (also called Modified Lely Method), which involves sublimation of the source material. Even though substantial progress has been made in recent years, the technological problems remain challenging, and only partial solutions exist.

This project is carried out in close cooperation with the Leibniz Institute of Crystal Growth (IKZ) in Adlershof.

Modeling and optimization of sublimation growth

From 2002 to 2007, C9 dealt with the sublimation growth of semiconductor bulk single crystals. In this technological process, polycrystalline powder is heated via magnetic induction to temperatures up to 3000 K. The powder evaporates and is transported through a gas cavity to a cooled seed where it recrystallizes as a single crystal.

Mathematically, one has to solve an optimal control problem with pointwise state and control constraints on a complicated, nonsmooth cylindrical domain for a PDE system featuring the following components:

1. Maxwell's equations for the electromagnetic field
2. the (nonlinear) heat equation, coupled to nonlocal radiation boundary conditions, where
3. the physical coefficients are strongly temperature-dependent

For this PDE system, necessary and sufficient optimality conditions, as well as numerical methods, were established for new classes of state-constrained optimal control problems with nonlocal radiation boundary conditions.

These investigations were finished successfully both from the mathematical and the industrial viewpoint; the software WIAS-HiTNIHS developed in C9 was licensed to IKZ.

Further informations about activties on modeling and optimal control of sublimation growth can be found here.

Modeling and optimization of growth from the melt

In 2007, C9 shifted its efforts to model variants of the Czochralski growth of bulk semiconductor crystals from the melt. The reason for this was that the use of traveling magnetic fields (TMF), a special kind of time dependent magnetic fields to control the turbulent melt flow, has drawn increasing attention from crystal growing industry worldwide.

The interest in these problems was triggered by the project KRISTMAG of the Technologiestiftung Berlin (TSB). In this project, a WIAS group headed by O. Klein and J. Sprekels joined forces with IKZ and two industrial partners. A technological breakthrough was achieved by demonstrating that traveling magnetic fields can be successfully applied for this task. A special heater magnet module (HMM) was developed, and a number of patent applications were filed. The groundbreaking nature of these developments was recognized by the Innovationspreis (innovation prize) Berlin-Brandenburg 2008. The software WIAS-HiTNIHS played an important role in these investigations. Presently, many companies worldwide are implementing the new technology into their growth apparatus, trying to find the optimal configurations.

From the mathematical viewpoint, crystal growth from the melt is even more difficult to treat than sublimation growth, since, in addition to the features a.)--c.), the fluid flow has to be accounted for, modeled by the Boussinesq approximation of the Navier-Stokes equations for thermally and electrically conducting fluids.

The mathematical problem concerns the optimal control and simulation the equations of temperature-dependent magnetohydrodynamics coupled to the energy balance, on a complicated nonsmooth domain. In addition, almost all physical coefficients are strongly temperature-dependent, the coefficient functions are discontinuous, nonlinear and nonlocal radiation boundary conditions occur. A mathematical breakthrough was achieved in the thesis by P.-E. Druet, who showed existence for this system.

In accordance with the needs of our partners, we will study optimal control problems arising from the use of traveling magnetic fields to influence melt flow during crystal growth from the melt. Issues of special interest are:

• Eddies in the melt flow should have a suitable form to keep impurities apart from the growing crystal.
• The amplitudes of temperature oscillations below the crystals should be minimized, while their frequencies must be kept sufficiently large.

Mathematically, one has to deal with new classes of control and state constrained optimization problems for the very difficult PDE system described above. The results in the thesis by P.-E. Druet have made it possible to tackle control problems.

Modeling

For the investigation of the very complex phenomena happening in crystal growth from the melt, no fundamentally new model is developed in the present funding period of the project: It is going to rely on the modeling experience gained in the former funding periods of the project, as well as in the project KRISTMAG.

A reliable model for global heat transfer in high-temperatures systems was already developed and applied in our project to deal with sublimation growth. The most important point is to include in the model the radiative heat transfer which is the dominant effect in the huge cavities inside of the crystal growth furnace. The gas convection is neglected in comparison to the radiation. More detailed descriptions of the model for global heat transfer are to find on the subpage sublimation growth.

Two important issues related to the melt flow in the crucible are the modeling of the thermal convection, that is, the buoyancy, and of the coupling to the traveling magnetic field. The project relies on an approach that has proved its viability in the project KRISTMAG, and is generally well accepted in crystal growth.
The buoyancy is described by the Boussinesq approximation of compressible fluids. The density fluctuations in the fluid are regarded as due to temperature variations only, and are considered significant only as a source for buoyancy.
The coupling to the magnetic field is described by the approximation of low magnetic Reynolds numbers: The motion induced current is neglected with respect to the conduction current. Therefore, the magnetic field is coupled to the temperature only.

Finally, the definition of appropriate goals for the optimization is also part of our work.

Mathematical Analysis

To rigorously deal with optimal control problems related to the heating of the growth apparatus and the achievment of certain flow properties, we have to develop an existence theory for the complex coupled system of partial differential equations consisting of the Maxwell system, the Navier-Stokes system, and the heat equation with nonlocal and nonlinear terms.

The next challenge consists in a step by step proof that the (as epxected) very weak existence theory allows to mathematically derive optimality conditions, the foundation of efficient optimization algorithm.
Due to the complexity of the phenomena occuring in crystal growth from the melt, it is expected that the optimization will be confronted with some nonstandard objective functionals.

The Simulation Software

The following software packages were licensed to the Leibniz Institute of Crystal Growth (IKZ).

WIAS-HiTNIHS (P. Philip and O. Klein) WIAS-HiTNIHS is a simulation tool for stationary and transient temperature distributions in axisymmetric technical systems that are subject to intense heating. The simulator accounts for heat transfer by radiation through cavities and allows for temperature-dependent material parameters and anisotropic thermal conductivity. It can be used to compute traveling magnetic fields and the resulting Lorentz forces acting on conducting liquids. WIAS-HiTNIHS was one of the foundations of the project KRISTMAG. See description of WIAS-HiTNIHS for more informations. WIAS-MatConE (O. Klein) WIAS-MatConE is a tool to provide prototypical GUIs for creating and editing files that are used as inputs for simulation software. Typical examples are material data files and control files. WIAS-MatConE provides a fast and flexible way to generate GUIs for prototypical software without having to deal with the details of GUI development.             Left-hand: A temperature computation for GaAs Czochralski crystal growth using WIAS-HiTNIHS.

Further results and descriptions on modeling and analysis, obtained in the period 2002-2007 on sublimation growth, can be found on the subpage sublimation growth of the project.

Current and Future Tasks

In accordance with the needs of our partners, we will study optimal control problems arising from the use of traveling magnetic fields to influence melt flow during crystal growth from the melt. Issues of special interest are:

• Eddies in the melt flow should have a suitable form to keep impurities apart from the growing crystal.
• The amplitudes of temperature oscillations below the crystals should be minimized, while their frequencies must be kept sufficiently large.

Mathematically, one has to deal with new classes of control and state constrained optimization problems for the very difficult PDE system described above. The results in the thesis by P.-E. Druet have made it possible to tackle control problems.

Publications

The following list restricts to the publications on crystal growth from the melt. The publications concerning the former topic of Project C9, sublimation growth, are listed on the subpage sublimation growth of the project.

Theses

1. P-E. Druet: Analysis of a coupled system of partial differential equations modeling the interaction between melt flow, global heat transfer, and applied magnetic fields in crystal growth. Doctoral thesis, successfully defended at the Department of Mathematics, Humboldt University of Berlin, Germany, Febuary, 2009.

   Using the following link, you can download the thesis: thesis by P.-E. Druet

Submitted Articles

3. O. Klein, Ch.Lechner, P-E.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, submitted June, 9th, 2009.

2. P-E. Druet, O.Klein, J.Sprekels, F.Tröltzsch , I.Yousept:Optimal control of 3D state-constrainded induction heating problems with nonlocal radiation effects , SIAM Journal Contr. Optim., submitted May 29th, 2009.

   The preprint of this article is available at Matheon Preprint 579 .

1. V. Dhamo and F. Tröltzsch:Some aspects of reachability for parabolic boundary control problems with control constraints , Computational Optimization and Applications, submitted Oct. 17th, 2008.

   The preprint of this article is available at Matheon Preprint 551 .

Articles in Refereed Scientific Journals

16. P-E Druet: Existence for the stationary MHD-equations coupled to heat transfer with nonlocal radiation effects , Cz. Math. J., accepted for publication.

15. P-E Druet: Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary p-summable right-hand side , Applications of Mathematics, accepted for publication.

14. E. Casas, F. Tröltzsch:First- and second-order optimality conditions for a class of optimal control problems with quasilinear elliptic equations, SIAM J. Control and Optimization, accepted for publication.

    The preprint of this article is available at Matheon Preprint 474 .

13. E. Casas, J-C de los Reyes, F. Tröltzsch:Sufficient second order optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Control and Optimization, accepted for publication.

    The preprint of this article is available at Matheon Preprint 473 .

12. J-C. de los Reyes, P. Merino, J. Rehberg, F. Tröltzsch:Optimality conditions for state-constrained PDE control problems with time-dependent controls, accepted for publication.

    The preprint of this article is available at Matheon Preprint 472 .

11. M. Hintermüller, I. Yousept: A sensitivity-based extrapolation technique for the numerical solution of state-constrained optimal control problems, ESAIM COCV, online published first .

    The preprint of this article is available at Matheon Preprint 421 .

10. C. Meyer, I. Yousept: Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions, Comput. Optim. Appl., online published first .

9. C. Meyer, I. Yousept: Source representation strategy for optimal boundary control problems with state constraints , SIAM J. on Control and Optimization, 48 (2009), 734-755.

8. F. Tröltzsch, I. Yousept: Source representation strategy for optimal boundary control problems with state constraints , J. Analysis and its Applications, 28 (2009), 189-203.

7. F. Tröltzsch, I. Yousept: A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints, Comput. Optim. Appl., 42 (2009), 43-63.

6. P-E Druet, J. Naumann: On the existence of weak solutions to a stationary one-equation RANS model with unbounded eddy viscosities, Ann. Univ. Ferrara, 55 (2009), 67-87.

5. P-E Druet: Existence of weak solutions to the time-dependent MHD-equations coupled to heat transfer with nonlocal radiation boundary conditions, Nonlinear Analysis. Real World Applications, 10 (2009), 2914-2936.

4. P-E Druet: Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right-hand side in L-p (p>=1), Math. Meth. Appl. Sci., 32 (2009), 135-166.

3. J-C. de los Reyes, I. Yousept: Regularized state-constrained boundary optimal control of the Navier-Stokes equations, J. Math. Anal. Appl., 356 (2009), 257-279.

2. M. Hintermüller, I. Yousept: Mesh independence of semismooth Newton methods for Lavrentiev-regularized state constrained optimal control problems, Numerische Mathematik, 108 (2008), 571-603.

1. O. Klein, F. Luterotti, R. Rossi: Existence and asymptotic analysis of a phase field model for supercooling, Quarterly of Applied mathematics, 64 (2006), 291-319.

Articles in Conference Issues of Refereed Scientific Journals

2. O. Klein, Ch. Lechner, P-E. 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), J. Crystal growth, 310 (2008), 1523-1532. (Proceedings of the 15th International Conference on Crystal Growth (ICCG-15), August 12-17, 2007, Salt Lake City, USA.)

1. O. Klein, Ch. Lechner, P-E. Druet: Development of a software for the numerical simulation of VCz growth under the influence of a traveling magnetic field, J. Crystal growth, 303 (2007), 161-164. (Proceedings of 5th International Workshop on "Modeling in Crystal Growth" (IWMCG-5), September 10 - 13, 2006, Bamberg, Germany.)

Articles in Conference Proceedings and Collections

4. P-E Druet: Weak solutions to a model for crystal growth from the melt in changing magnetic fields, In Optimal Control of Coupled Systems of PDE, K. Kunisch, G. Leugering, J. Sprekels, F. Tröltzsch editors, Birkhäuser, Basel, to appear.

3. P. Rudolph, Ch. Frank-Rotsch, F-M. Kießling, W. Miller, U. Rehse, O. Klein, Ch. Lechner, J. Sprekels, B. Nacke, H. Kasjanow, P. Lange, M. Ziem, B. Lux, M. Czupalla, O. Root, V. Trautmann, G. Bethin: Crystal growth in heatermagnet modules - from concept to use, In Procedings of the International Scientific Colloquium Modelling for Electromagnetic Processing (MEP2008), E. Baacke and B. Nacke editors, Leibniz University of Hannover, 2008, 79-84.

   The text of this article is available at http://www.modlab.lv/publications/MEP2008/pdfs/91-96.pdf.

2. O. Klein, Ch. Lechner, P-E. 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 Czochralski crystal growth, In Procedings of the International Scientific Colloquium Modelling for Electromagnetic Processing (MEP2008), E. Baacke and B. Nacke editors, Leibniz University of Hannover, 2008, 91-96.

   The text of this article is available at http://www.modlab.lv/publications/MEP2008/pdfs/103-108.pdf.

1. J-C. de los Reyes and I. Yousept: Boundary optimal flow control with state constraints, Proc. Appl. Math. Mech., 2007. Published online

   The preprint of this article is available at Matheon Preprint 422 .

Miscellaneous Articles

2. Ch. Frank-Rotsch, P. Rudolph, O. Klein, R-P. Lange, B. Nacke: Vorrichtung und Verfahren zur Herstellung von Kristallen aus elektrisch leitenden Schmelzen (Device and Method for Producing Crystals from Electroconductive Melt) , Patentschrift(patent specification) of granted patent 10 2007 028 548 German Patent and Trade Mark Office, July, 17th, 2009.

1. P-E Druet: Higher integrability of the Lorentz force for weak solutions to Maxwell's equations in complex geometries, Preprint No.\ 1270, Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany.

   The preprint is available at Wias Preprint 1270 .

Presentations

The following list restricts to the presentations on crystal growth from the melt. The presentations about sublimation growth, the former topic of Project C9, are listed on the subpage sublimation growth of the project.

International Conference Plenary Lectures

1. J. Sprekels: Problems in the industrial growth of semiconductor crystals: Radiative heat transfer, convection, magnetic fields and free boundaries, Conference The legacy of John Crank - Developments in Time Dependent PDE's, Diffusion and Free Boundary Problems, Brunel University, Uxbridge, 07/10/2008.

International Conference Talks

8. O. Klein (joined work with P.-E. Druet, J. Sprekels, Ch. Lechner (Vienna), P. Philip (LMU Munich), Ch. Frank-Rotsch, F.-M. Kießling, W. Miller, U. Rehse, P. Rudolph (IKZ, Berlin): 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'', Namur, Belgium, 25/05/2009.

7. F. Tröltzsch: On the verification of bang-bang properties for parabolic boundary control problems, Oberwolfach, Germany, 27/01/2009.

6. O. Klein (et al.): Numerical simulations of the influence of a traveling magnetic field, generated by an internal heater-magnet module, on Czochralski crystal growth , International scientific colloquium Modeling for electromagnetic processing (MEP 2008), Hannover, Germany, 26/10/2008.

5. F. Tröltzsch: On the optimal control of some heating processes, 60th birthday of J. Sprekels, Berlin, Germany, 22/10/2008.

4. I. Yousept: On a state-constrained optimal control problem involving nonlocal radiation interface conditions, DMV, Erlangen, Germany 16/09/2008.

3. I. Yousept: On a state-constrained optimal control problem involving nonlocal radiation interface conditions, SIAM conference on Optimization, Boston, USA, 11/05/2008.

2. P-E. Druet: Weak solutions to a model for crystal growth from the melt in changing magnetic fields, Conference Optimal Control of Coupled Systems of PDE, Oberwolfach, Germany, 04/03/2008.

1. O. Klein: Modeling Czochralski crystal growth under the influence of electromagnetic forces, Workshop Recent advances in free boundary problems and related topics (FBP 2006), Levico, Italy, 15/09/2006.

Lecture to the General Public

Talks at Colloquiums, Workshops, and Seminars

2. O. Klein (joined work with P.-E. Druet, J. Sprekels, Ch. Lechner (Vienna), P. Philip (LMU Munich), Ch. Frank-Rotsch, F.-M. Kießling, W. Miller, U. Rehse, P. Rudolph (IKZ, Berlin): Modellierung und Simulation der LEC GaAs Züchtung in Heizer-Magnetmodulen, 10. DGKK Kinetik-Seminar joinded with 6. DGKK Workshop "Angewandte Simulation in der Kristallzüchtung", Postdam, Germany, 02/04/2009.

1. P-E. Druet: On weak solutions to the system of MHD coupled to heat transfer including nonlocal radiation boundary conditions, Oberseminar analysis, TU Dresden, Germany, 31/01/2008.

Poster Presentations

2. O. Klein (joined work with Ch. Lechner, P-E. Druet, P. Philip, J. Sprekels, Ch. Frank-Rotsch, F. M Kießling, W. Miller, U. Rehse and P. Rudolph): Numerical simulation of Czochralski crystal growth under the influence of a traveling magnetic field generated by an internal heater-magnet module (HMM), 15th International Conference on Crystal Growth (ICCG15), Salt Lake City, USA, 12-17/08/2007.

1. O. Klein (joined work with W.Dreyer, P-E. Druet, Ch. Lechner and J. Sprekels): Numerical simulation of VCz growth with a traveling magnetic field, 5th International Workshop on Modeling in Crystal Growth (IWMCG-5), Bamberg, Germany, 10-13/09/2006.

Further informations about the activties in the period 2002-2007 on modeling and optimal control of sublimation growth can be found on the subpage sublimation growth

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Last update: Pierre-Étienne Druet. on June 03, 2010.