Control configuration selection and distributed nonlinear control of surface roughness in a sputtering process

Abstract

This work focuses on control configuration selection and nonlinear control of the roughness of a onedimensional surface in a sputtering process including two surface micro-processes, diffusion and erosion. The fluctuation of surface height of such a sputtering process can be approximately described by the stochastic Kuramoto-Sivashinsky equation (KSE), a fourth-order nonlinear stochastic partial differential equation (PDE). First, both a spatially distributed control configuration and a spatially invariant control configuration are investigated and the spatially distributed control configuration is found necessary to achieve the desired control objectives. The control problem is then formulated as the one of controlling the surface roughness by manipulating the spatially distributed gas composition across the surface. To perform the nonlinear controller design, we initially formulate the stochastic KSE into a system of infinite nonlinear stochastic ordinary differential equations (ODEs) by using Galerkin’s method. A finite-dimensional approximation of the stochastic KSE is then derived that captures the dominant mode contribution to the surface roughness. A nonlinear feedback controller is then designed based on the finite-dimensional approximation. Subsequently, the necessity of regulating the surface roughness through the spatially distributed control configuration and the effectiveness of the proposed nonlinear controller are demonstrated by applying the nonlinear feedback controller to the kinetic Monte Carlo (kMC) model of the sputtering process in numerical simulations.