Mathematical Modeling of Czochralski Type Growth Processes for Semiconductor Bulk Single Crystals

被引：0
作者：
W. Dreyer
P. -É. Druet
O. Klein
J. Sprekels
机构：
[1] Weierstrass Institute for Applied Analysis and Stochastics,
来源：
Milan Journal of Mathematics | 2012年 / 80卷
关键词：
35Q30; 35Q61; 45G05; 76W05; 80A20; 80A22; Czochralski method; crystal growth; traveling magnetic fields; radiative heat transfer; nonlinear PDE systems; Navier-Stokes equations; MHD equations; Maxwell’s equations; well-posedness; optimal control; first-order necessary optimality conditions; numerical simulation;
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摘要：
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., silicon (Si) or gallium arsenide (GaAs) 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.
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页码：311 / 332
页数：21
相关论文
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  • [1] Druet P.-É.(2008)Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right-hand side in Math. Meth. Appl. Sci. 32 135-166
  • [2] Druet P.-É.(2009)Existence of weak solutions to the time-dependent MHD equations coupled to the heat equation with nonlocal radiation boundary conditions Nonlin. Anal. RWA 10 2914-2936
  • [3] Druet P.-É.(2009)Existence for the stationary MHD-equations coupled to heat transfer with nonlocal radiation effects Cz. Math. J. 59 791-825
  • [4] Druet P.-É.(2010)Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary p-summable right-hand side Appl. of Math. 55 111-149
  • [5] Druet P.-É.(2011)Optimal control of three-dimensional state-constrained induction heating problems with nonlocal radiation effects SIAM J. Control Optim. 49 1707-1736
  • [6] Klein O.(2011)Noncompactness of integral operators modeling diffuse-gray radiation in polyhedral and transient settings, Integr Equ. Oper. Theory 69 101-111
  • [7] Sprekels J.(2002)Correct voltage distribution for axisymmetric sinusoidal modeling of induction heating with prescribed current, voltage, or power IEEE Transactions on Magnetics 38 1519-1523
  • [8] Tröltzsch F.(2005)Transient conductive-radiative heat transfer: Discrete existence and uniqueness for a finite volume scheme, Math Models Methods Appl. Sci. 15 227-258
  • [9] Yousept I.(2007)Development of a software for the numerical simulation of VCz growth under the influence of a traveling magnetic field J. Crystal Growth 303 161-164
  • [10] Druet P.-É.(2009)Regularization of state-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions SIAM J. Control Optim. 48 734-755
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