# (659c) Gray-Box Modeling of 300mm Czochralski Single-Crystal Si Production Process

Authors:
Kyoto University
Kyoto University
Kyoto University
Kyoto University
SUMCO Corporation
SUMCO Corporation
The Czochralski (CZ) process is a dominant method for manufacturing large cylindrical single-crystal ingots for the electronics industry. To understand the phenomena of CZ process and efficiently operate it, many models and control methods have been proposed, but their industrial application results were not reported [Abdollahi et al. (2014), Gevelber and Stephanopoulos (1987), Irizarry-Rivera and Seider (1997), Ng and Dubljevic (2012), and Winkler et al. (2010)]. To the authorsâ?? best knowledge, only two papers reported modeling of industrial CZ processes [Lee et al. (2005) and Zheng et al. (2015)]. Lee et al. (2005) developed linear step response models which predict the crystal radius from the crystal pulling rate and the heater temperature. The models were used for determining the operation condition of a CZ process that produces bulk crystal silicon ingots with the diameters of 200 and 300 mm. However, the accuracy of the developed models might not be sufficient since they cannot describe the process nonlinearity caused by various factors such as the radiative heat transfer. Zheng et al. (2015) constructed a nonlinear first-principle model which calculates the crystal radius and the crystal growth rate from the heater input, the crystal pulling rate, and the crucible rise rate. They showed an application example in a CZ process that produces 300 mm single-crystal silicon ingots. In the first-principle model, the Eulerâ??Laplace equation formula is used to estimate the meniscus height, which is necessary to calculate the crystal radius. However, the Euler-Laplace formula might not be appropriate to calculate the dynamic change of meniscus height since it is derived under the steady-state assumption and the meniscus height changes with time. Thus, there is a room for the improvement of the model.

In this research, a statistical model is embedded into the first-principle model proposed by Zheng et al. (2015) to further improve the model accuracy, since a first-principle model of .the dynamics of the meniscus height is difficult to build. The statistical model is used for estimating a key parameter in the nonlinear first-principle model, which significantly affects the meniscus height. Moving window partial least squares (MWPLS) is used to develop the statistical model since it can cope with the time-varying characteristics of CZ process, which originate from changes in the crystal length and the crucible position. The developed gray-box model was applied to a 300 mm CZ single-crystal silicon production process. It was confirmed that the accuracy of the developed model was much better than the model proposed by Zheng et al. (2015).

References

J. Abdollahi, M. Izadi, and S. Dubljevic. Model predictive temperature tracking in crystal growth processes. Comput. Chem. Eng., 71:323â??330, 2014.

M. A. Gevelber and G. Stephanopoulos. Dynamics and control of the Czochralski process I. Modelling and dynamic characterization. J. Cryst. Growth, 84:647â??668, 1987.

R. Irizarry-Rivera and W. D. Seider. Model-predictive control of the Czochralski crystallization process part I. Conduction-dominated melt. J. Cryst. Growth, 178:593â??611, 1997.

K. Lee, D. Lee, J. Park, and M. Lee. MPC based feedforward trajectory for pulling speed tracking control in the commercial Czochralski crystallization process. Int. J. Cont. Autom., 3:252â??257, 2005

J. Ng and S. Dubljevic. Optimal control of convection-diffusion process with time-varying spatial domain: Czochralski crystal growth. J. Proc. Cont., 21:1361â??1369, 2011.

J. Winkler, M. Neubert, and J. Rudolph. Nonlinear model-based control of the Czochralski process I: Motivation, modeling and feedback controller design. J. Cryst. Growth, 312:1005â??1018, 2010.

Z. Zheng, T. Seto, S. Kim, M. Kano, T. Fujiwara, M. Mizuta, and S. Hasebe. Development of a first principle model of Czochralski process and its verification with real industrial data. APCChE 2015, Melbourne, Australia, Sep. 2015.

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