7 research outputs found
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Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals
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 ..
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Optimal control of semiconductor melts by traveling magnetic fields
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 condution, 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
Optimal control of semiconductor melts by traveling magnetic fields
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 condution, 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
Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals
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
Optimal Control of Three-Dimensional State-Constrained Induction Heating Problems with Nonlocal Radiation Effects
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 existence and uniqueness of the state. The regularity analysis associated with the time-harmonic Maxwell equations is also studied. In the second part of the paper, existence and uniqueness of the solution of the corresponding linearized equation are shown. With this result at hand, the differentiability of the control-to-state operator is derived. Finally, based on the theoretical results, first order necessary optimality conditions for an associated optimal control problem are established