Feedback control of surface roughness in sputtering processes using the stochastic Kuramoto–Sivashinsky equation

Abstract

This work focuses on control of surface roughness in sputtering processes including two surface micro-processes, diffusion and erosion. The fluctuation of surface height of such sputtering processes can be described by the stochastic Kuramoto–Sivashinsky equation (KSE), a fourth-order stochastic partial differential equation (PDE). Specifically, we consider sputtering processes, including surface diffusion and erosion, on a one-dimensional lattice and design feedback controllers based on stochastic PDEs to regulate the surface roughness at desired levels. We initially reformulate the stochastic KSE into a system of infinite stochastic ordinary differential equations (ODEs) by using modal decomposition. A finite-dimensional approximation of the stochastic KSE is then derived that captures the dominant mode contribution to the surface roughness. A state feedback controller is designed based on the finite-dimensional approximation to control the surface roughness. Feedback control of surface roughness in three different sputtering processes with different sputtering yield functions and different ratios of erosion and diffusion rates is subsequently studied. Kinetic Monte-Carlo simulations are first performed to simulate the evolution of the surface height fluctuation in the three sputtering processes. Then, a systematic identification approach is used to identify the parameters of the stochastic KSE models describing the sputtering processes by using the data from kinetic Monte-Carlo simulations. Specifically, the evolution of state covariance of the stochastic KSE models is directly obtained from multiple kinetic Monte-Carlo simulation runs. The correlations between model parameters and the state covariance of the stochastic KSE models are established and the parameters of the stochastic KSE models are subsequently computed by using least-mean-square fitting so that the evolution of the surface roughness computed from the stochastic KSE models is consistent with that computed from kinetic Monte-Carlo simulations. Feedback controllers are designed and applied to kinetic Monte-Carlo models of the sputtering processes to control the surface roughness to desired levels.