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Linear Parameter-varying Approach for Modeling and Control of Rapid Thermal Processes

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  • Control Theory and Applications
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Abstract

In the present paper, a new approach is presented to model and control single wafer rapid thermal processing (RTP) systems. In the past decade, RTP has achieved acceptance as the mainstream technology for semiconductor manufacturing. Thermal processing is one of the most efficient ways to control the phase-structure properties. Moreover, the time duration of RTP systems reduces the so-called thermal budget significantly compared to the traditional methods. RTP implementation is based on the use of light from heating lamps to provide a heat flux. This process is highly nonlinear due to the radiative heat transfer and material properties. By invoking the first principles-based models, we develop in this paper a linear parameter-varying (LPV) model to directly account for the nonlinearities within the system. The model is then discretized into a high-order LPV model; thereafter, principal component analysis (PCA) method is utilized to reduce the number of LPV model’s scheduling variables, followed by the use of proper orthogonal decomposition (POD) for model order reduction. Using the reduced order model, we then design a gain-scheduled controller to satisfy an induced L2 gain performance for tracking of a temperature profile and show improvement over other controller design methods suggested in the literature.

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References

  1. J. Ebert, D. De Roover, L. Porter, V. Lisiewicz, S. Ghosal, R. Kosut, and A. Emami-Naeini, “Model-based control of rapid thermal processing for semiconductor wafers,” in Proc. of American Control Conference, pp. 3910–3921, 2004.

    Google Scholar 

  2. Y. J. Lee, B. Khuri-Yakub, and K. C. Saraswat, “Temperature measurement in rapid thermal processing using acoustic techniques,” Review of scientific instruments, vol. 65, no. 4, pp. 974–976, 1994.

    Article  Google Scholar 

  3. J. Liu and Y.-S. Chen, “Simulation of rapid thermal processing in a distributed computing environment,” Numerical Heat Transfer: Part A: Applications, vol. 38, no. 2, pp. 129–152, 2000. [click]

    Article  Google Scholar 

  4. F. Fasching, S. Halama, and S. Selberherr, Technology CAD systems. Springer Science & Business Media, 2012.

    Google Scholar 

  5. E. Dassau, B. Grosman, and D. R. Lewin, “Modeling and temperature control of rapid thermal processing,” Computers & chemical engineering, vol. 30, no. 4, pp. 686–697, 2006. [click]

    Article  Google Scholar 

  6. W. S. Yoo, T. Fukada, I. Yokoyama, K. Kang, and N. Takahashi, “Thermal behavior of large-diameter silicon wafers during high-temperature rapid thermal processing in single wafer furnace,” Japanese journal of applied physics, vol. 41, no. 7R, p. 4442, 2002.

    Article  Google Scholar 

  7. H. Aling, S. Banerjee, A. K. Bangia, V. Cole, J. Ebert, A. Emami-Naeini, K. F. Jensen, I. G. Kevrekidis, and S. Shvartsman, “Nonlinear model reduction for simulation and control of rapid thermal processing,” in Proc. of American Control Conference, 1997, pp. 2233–2238.

    Google Scholar 

  8. A. Theodoropoulou, R. A. Adomaitis, and E. Zafiriou, “Model reduction for optimization of rapid thermal chemical vapor deposition systems,” IEEE Transactions on Semiconductor Manufacturing, vol. 11, no. 1, pp. 85–98, 1998.

    Article  Google Scholar 

  9. Y. M. Cho and P. Gyugyi, “Control of rapid thermal processing: A system theoretic approach,” IEEE Trans. Control Systems Technology, vol. 5, no. 6, pp. 644–653, 1997. [click]

    Article  Google Scholar 

  10. D. De Roover, A. Emami-Naeini, J. L. Ebert, and R. L. Kosut, “Trade-offs in temperature control of fast-ramp RTO and RTA systems,” in 7th Intl Conf. on Advanced Thermal Processing of Semiconds, RTP, vol. 99. Citeseer, 1999.

    Google Scholar 

  11. E. Zafiriou, R. Adomaitis, and G. Gattu, “An approach to run-to-run control for rapid thermal processing,” in Proc. of American Control Conference, 1995, pp. 1286–1288.

    Google Scholar 

  12. C. D. Schaper, M. M. Moslehi, K. C. Saraswat, and T. Kailath, “Modeling, identification, and control of rapid thermal processing systems,” Journal of the Electrochemical Society, vol. 141, no. 11, pp. 3200–3209, 1994.

    Article  Google Scholar 

  13. C. D. Schaper, “Real-time control of rapid thermal processing semiconductor manufacturing equipment,” in Proc. of American Control Conference, 1993, pp. 2985–2990.

    Google Scholar 

  14. R. S. Gyurcsik and T. J. Riley, “A model for rapid thermal processing: Achieving uniformity through lamp control,” IEEE Trans. Semiconductor Manufacturing, vol. 4, no. 1, pp. 9–13, 1991. [click]

    Article  Google Scholar 

  15. P. P. Apte and K. C. Saraswat, “Rapid thermal processing uniformity using multivariable control of a circularly symmetric 3 zone lamp,” IEEE Trans. Semiconductor Manuf., vol. 5, no. 3, pp. 180–188, 1992. [click]

    Article  Google Scholar 

  16. T. F. Edgar, S. W. Butler, W. J. Campbell, C. Pfeiffer, C. Bode, S. B. Hwang, K. Balakrishnan, and J. Hahn, “Automatic control in microelectronics manufacturing: Practices, challenges, and possibilities,” Automatica, vol. 36, no. 11, pp. 1567–1603, 2000. [click]

    Article  MATH  Google Scholar 

  17. M. Trudgen, S. Z. Rizvi, and J. Mohammadpour, “Linear parameter-varying approach for modeling rapid thermal processes,” Proc. of American Control Conference (ACC), IEEE, pp. 3243–3248, 2016. [click]

    Google Scholar 

  18. J. Mohammadpour and C. W. Scherer, Eds., Control of linear parameter varying systems with applications. Springer Science & Business Media, 2012.

    Google Scholar 

  19. H. Lord, “Thermal and stress analysis of semiconductor wafers in a rapid thermal processing oven,” IEEE Trans. Semiconductor Manufacturing, vol. 1, no. 3, pp. 105–114, 1988.

    Article  Google Scholar 

  20. V. Borisenko and P. J. Hesketh, Rapid Thermal Processing of Semiconductors, Springer Science & Business Media, 2013.

    Google Scholar 

  21. A. Virzi, “Computer modelling of heat transfer in czochralski silicon crystal growth,” Journal of crystal growth, vol. 112, no. 4, pp. 699–722, 1991.

    Article  Google Scholar 

  22. I. Jolliffe, Principal Component Analysis, 2nd ed., Springer, NY.

  23. A. Kwiatkowski and H. Werner, “PCA-based parameter set mappings for LPV models with fewer parameters and less overbounding,” IEEE Transactions on Control Systems Technology, vol. 16, no. 4, pp. 781–788, 2008. [click]

    Article  Google Scholar 

  24. S. Rizvi, J. Mohammadpour, R. Tòth, and N. Meskin, “A kernel-based PCA approach to model reduction of linear parameter-varying systems,” IEEE Transactions on Control Systems Technology, vol. 24, no. 5, pp. 1883–1891, 2016. [click]

    Article  Google Scholar 

  25. P. Holmes, Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, 2012.

    Book  MATH  Google Scholar 

  26. S. M. Hashemi and H. Werner, “LPV modelling and control of Burgers equation,” in Proc. 18th IFAC World Congress, 2011.

    Google Scholar 

  27. L. Sirovich, “Turbulence and the dynamics of coherent structures,” Quarterly of applied mathematics, vol. 45, pp. 561–571, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Apkarian, P. Gahinet, and G. Becker, “Self-scheduled H ∞ control of linear parameter-varying systems: a design example,” Automatica, vol. 31, no. 9, pp. 1251–1261, 1995.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Electrical & Computer Engineering, The University of Georgia, Athens, GA, 30602, USA

    Mark Trudgen & Javad Mohammadpour Velni

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  1. Mark Trudgen
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  2. Javad Mohammadpour Velni
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Correspondence to Javad Mohammadpour Velni.

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Recommended by Associate Editor Yang Tang under the direction of Editor Myo Taeg Lim.

Mark Trudgen received his B.S. and M.S. degrees in mechanical engineering and his Ph.D. degree in engineering. He joined the University of Georgia as a research assistant in the Complex Systems Control Laboratory under the direction of Dr. Velni in August 2013. In January 2017 he joined the University of Georgia as a lecturer in the school of electrical and computer engineering. His current research is in low-order LPV model development and robust control design for automotive and manufacturing applications.

Javad Mohammadpour Velni received his B.S. and M.S. degrees in electrical engineering and his Ph.D. degree in mechanical engineering. He joined University of Georgia as an assistant professor of electrical engineering in Aug. 2012. He has published over 100 articles in international journals and conferences, served in the editorial boards of ASME and IEEE conferences on control systems and edited two books on large-scale systems (2010) and LPV systems modeling and control (2012). His current research is on secure control of cyber physical systems, coverage control of multi-agent systems, and data-driven model learning and control of complex distributed systems.

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Trudgen, M., Velni, J.M. Linear Parameter-varying Approach for Modeling and Control of Rapid Thermal Processes. Int. J. Control Autom. Syst. 16, 207–216 (2018). https://doi.org/10.1007/s12555-016-0788-x

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  • DOI: https://doi.org/10.1007/s12555-016-0788-x

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