Feedback control of surface roughness using stochastic PDEs

The surface roughness of thin films deposited from gas phase precursors is an important variable to control, because it strongly affects the quality of such films. The modern integrated circuit technology depends strongly on the uniformity and microstructure of deposited thin films (Granneman, 1993). Due to the increasingly stringent requirements on the quality of such films, on-line estimation and control of thin film deposition becomes important. In a thin film growth process, the film is directly formed from microscopic random processes (for example, particle adsorption, desorption, migration and surface reaction). Therefore, the stochastic nature of thin film growth processes must be fully considered in the modeling and control of the surface roughness of thin films. The desire to understand and control the thin film micro-structure has motivated extensive research on fundamental mathematical models describing the deposition processes, which include (1) kinetic Monte-Carlo methods (for example, Gillespie, 1976; Fichthorn and Weinberg, 1991; Lam and Vlachos, 2001), and (2) stochastic partial-differential equations (PDEs) (for example, Edwards and Wilkinson, 1982; Villain, 1991; Vvedensky et al., 1993). The kinetic Monte-Carlo simulation method can be used to predict average properties of the thin film (which are of interest from a control point of view, for example, surface roughness), by explicitly accounting for the microprocesses that directly shape thin film microstructure. Recently, a methodology for feedback control of thin film growth using kinetic Monte-Carlo models has been developed in (Lou and Christofides, 2003a,b). The methodology leads to the design of (a) real-time roughness estimators by using multiple small lattice kinetic Monte-Carlo simulators, adaptive filters and measurement error compensators, and (b) feedback controllers based on the real-time roughness estimators. The method was successfully applied to control surface roughness in a GaAs deposition process using an experimentally determined kinetic Monte-Carlo process model (Lou and Christofides, 2004b). Other approaches have also been developed to: (a) identify linear models from outputs of kinetic Monte-Carlo simulators and perform controller design by using linear control theory (Siettos et al., 2003), and (b) construct reduced-order approximations of the master equation (Gallivan and Murray, 2003). However, the fact that kinetic Monte-Carlo models are not available in closed-form makes very difficult to perform model-based controller design directly on the basis of kinetic Monte-Carlo models. To achieve better closed-loop performance, it is desirable to design feedback controllers on the basis of deposition process models. This motivates research on feedback control of deposition processes based on stochastic PDE models of thin film growth. Stochastic PDE models have been developed to describe the evolution of the height profile for surfaces in a variety of physical and chemical processes. Examples include deposition processes including adsorption and surface relaxation (Edwards and Wilkinson, 1982; Vvedensky, 2003), crystal growth from atomic-beams with and without desorption (Villain, Correspondence concerning this article should be addressed to P. D. Christofides; e-mail: pdc@seas.ucla.edu.