Crystal radius and temperature regulation in Czochralski crystallization process

In this paper we explore modelling and regulation techniques for temperature and crystal growth control in an industrially important process of Czochralski crystal growth. First, a geometrical model for both crystal growth and radius evolution dynamics is developed and a controller is designed based on an input-output linearization to regulate and track a desired reference crystal radius. The pulling based regulated crystal radius control provides a time-varying shape (geometry) evolution of the underlying time-varying partial differential equation describing the crystal temperature dynamics. A low dimensional time-varying parabolic PDE model obtained by Galerkin method is used for regulation of the triple point temperature (melt-solid-encapsulant intersection). The coupling among finite-dimensional crystal radius and triple point temperature regulation is given by the perturbation in the crystal growth parameters and it is induced by possible solid-melt interface temperature fluctuations. We provide a unified regulation framework for such a complex coupled system of crystal growth and temperature regulation in order to improve operational and economic features associated with the Czochralski crystal growth process.

[1]  Philippe Martin,et al.  Boundary control of a nonlinear Stefan problem , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[2]  Stevan Dubljevic,et al.  Optimal control of convection–diffusion process with time-varying spatial domain: Czochralski crystal growth , 2011 .

[3]  Talid Sinno,et al.  Modeling Microdefect Formation in Czochralski Silicon , 1999 .

[4]  Thierry Duffar,et al.  Crystal growth processes based on capillarity : czochralski, floating zone, shaping and crucible techniques , 2010 .

[5]  M. Krstić,et al.  Backstepping boundary control of Burgers' equation with actuator dynamics , 2000 .

[6]  P. K. C. Wang,et al.  Stabilization and control of distributed systems with time-dependent spatial domains , 1990 .

[7]  Stevan Dubljevic,et al.  Computation of empirical eigenfunctions of parabolic PDEs with time-varying domain , 2012, 2012 American Control Conference (ACC).

[8]  P. K. C. Wang.  FEEDBACK CONTROL OF A HEAT DIFFUSION SYSTEM WITH TIME‐DEPENDENT SPATIAL DOMAIN , 1995 .

[9]  Mark J. Balas,et al.  Finite-dimensional control of distributed parameter systems by Galerkin approximation of infinite dimensional controllers☆ , 1986 .

[10]  J. Derby,et al.  On the dynamics of Czochralski crystal growth , 1987 .

[11]  Robert A. Brown,et al.  Theory of transport processes in single crystal growth from the melt , 1988 .

[12]  Svein I. Sagatun,et al.  Exponential Stabilization of a Transversely Vibrating Beam by Boundary Control Via Lyapunov’s Direct Method , 2001 .

[13]  Jan Winkler,et al.  Nonlinear model-based control of the Czochralski process II: Reconstruction of crystal radius and growth rate from the weighing signal , 2010 .

[14]  P. Christofides,et al.  Crystal temperature control in the Czochralski crystal growth process , 2001 .

[15]  Kazumfumi Lto,et al.  Finite-dimensional compensators for infinite-dimensional systems via Galerkin-type approximation , 1990 .

[16]  Jan Winkler,et al.  Nonlinear model-based control of the Czochralski process I: Motivation, modeling and feedback controller design , 2010 .

[17]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[18]  Nam-Ho Kim.  Introduction to Nonlinear Finite Element Analysis , 2014 .

[19]  György Szabó,et al.  Thermal strain during Czochralski growth , 1985 .

[20]  Stevan Dubljevic,et al.  Optimal boundary control of a diffusion–convection-reaction PDE model with time-dependent spatial domain: Czochralski crystal growth process , 2012 .