(463e) Model Predictive Control of Czochralski Crystal Growth Process with Time-Varying Spatial Domain

Czochralski crystal growth is an important industrial process utilized for the production of semiconductor materials whereby large single ingots are grown by drawing a crystal seed from a melt by mechanical actuator. The control objectives are to obtain a high-purity crystal with constant radius by controlling the trajectory of the mechanical pulling arm actuator and the temperature of the furnace where the crystal is being grown, see [1],[2],[9]. A parabolic partial differential equation (PDE) model for the temperature distribution of the crystal is developed from first-principles continuum mechanics to preserve the time-varying spatial domain dynamical features, see [4],[12]. The PDE is cast in cylindrical coordinates with algebraic expression for crystal radial variance due to both the temperature and the rate at which the crystal is being drawn from the melt. The evolution of the temperature distribution and radius of the crystal are coupled to the pulling actuator subsystem with dynamics modelled as a second order ordinary differential equation (ODE) for rigid body mechanics, see [14]. Control of the temperature distribution and crystal radius is realized by the heat input at the boundary and by the force applied to the mechanical subsystem. This work considers the implementation of a model-predictive control strategy to maximize the control objectives based on the PDE model, an algebraic equation and ODE system describing the dynamics of the process, see [3],[9],[10]. Finally, we discuss the process dynamics of the closed-loop predictive control implementation in conjunction with numerical simulation results, see [5].

References:

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[7] Derby, J. and Brown, R. (1986b). Thermal-capillary analysis of czochralski and liquid encapsulated czochralski crystal growth. Part II. Processing strategies. J. Cryst. Growth, no.75, p.227-240.

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[9] Irizarry-Rivera, R., and Seider, W. D. (1997). Model-predictive control of Czochralski crystallization process. Part I. Conduction-dominated melt. Journal of Crystal Growth, no. 178, p.593-611.

[10] Irizarry-Rivera, R., and Seider, W. D. (1997). Model-predictive control of Czochralski crystallization process. Part II. Reduced-order convection model. Journal of Crystal Growth, no. 178, p.612-633.

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[13] Ray, W. (1981). Advanced Process Control. McGraw-Hill, New York, New York.

[14] Wang,P.K.C.(1990).Stabilization and control of distributed systems with time-dependent spatial domains. J. Optim. Theor. & Appl., no.65, 331-362.

[15] Wang, P.K.C. (1995). Feedback control of a heat diffusion system with time-dependent spatial domains. Optim. Contr. Appl. & Meth., no.16, p.305-320.