Gangshi Hu

Model predictive control of nonlinear stochastic PDEs: Application to a sputtering process

In this work, we develop a method for model predictive control of nonlinear stochastic partial differential equations (PDEs) to regulate the state variance, which physically represents the roughness of a surface in a thin film growth process, to a desired level. We initially formulate a nonlinear stochastic PDE into a system of infinite nonlinear stochastic ordinary differential equations (ODEs) by using Galerkins method. A finite-dimensional approximation is then derived that captures the dominant mode contribution to the state variance. A model predictive control problem is formulated based on the finite-dimensional approximation so that the future state variance can be predicted in a computationally efficient way. The control action is computed by minimizing an objective function including penalty on the discrepancy between the predicted state variance and a reference trajectory, and a terminal penalty. An analysis of the closed-loop nonlinear infinite-dimensional system is performed to characterize the closed-loop performance enforced by the model predictive controller. The model predictive controller is initially applied to the stochastic Kuramoto-Sivashinsky equation (KSE), a fourth-order nonlinear stochastic PDE. Simulation results demonstrate that the proposed predictive controller can successfully drive the norm of the state variance of the stochastic KSE to a desired level in the presence of significant model parameter uncertainties. In addition, we consider the problem of surface roughness regulation in a one-dimensional ion-sputtering process. The predictive controller is applied to the kinetic Monte Carlo model of the sputtering process to successfully regulate the expected surface roughness to a desired level.

Lattice-size Dependence and Dynamics of Surface Mean Slope in a Thin Film Deposition Process

Abstract This work focuses on the study of the dynamic behavior and lattice size dependence of the surface root-mean-square slope in a porous thin film deposition process taking place on a triangular lattice. The simulation results indicate that the expected mean slope square reaches quickly a steady-state value and exhibits a very weak dependence with respect to lattice size variation. The simulation findings are corroborated by an analysis of appropriate finite-difference discretizations of surface height profiles computed by an Edwards-Wilkinson-type partial differential equation that can be used to describe the dynamics of surface height profile in the thin film deposition process under consideration.

Model predictive control of film porosity in thin film deposition

A model predictive controller is developed to regulate film porosity and minimize its fluctuation in thin film deposition. The deposition process is modeled via kinetic Monte Carlo (kMC) simulation on a triangular lattice. The microscopic events involve atom adsorption and migration and allow for vacancies and overhangs to develop. Appropriate definitions of film site occupancy ratio (SOR), i.e., fraction of film sites occupied by particles over total number of film sites, and its fluctuation are introduced to describe film porosity. Deterministic and stochastic ordinary differential equation (ODE) models are also constructed to describe the time evolution of film SOR and its fluctuation. The coefficients of the ODE models are estimated on the basis of data obtained from the kMC simulator of the deposition process using least-square methods. The developed ODE models are used as the basis for the design of model predictive control (MPC) algorithms that include penalty on the film SOR and its variance to regulate the expected value of film SOR at a desired level and reduce run-to-run fluctuations. Simulation results demonstrate the applicability and effectiveness of the proposed film porosity control method in the context of the deposition process under consideration.

Stochastic modeling of film porosity in thin film deposition

This work focuses on modeling of film porosity in thin film deposition using stochastic differential equations. A deposition process is modeled via kinetic Monte Carlo (kMC) simulation on a triangular lattice. The microscopic process events involve atom adsorption and migration and the film growth allows for vacancies and overhangs to develop inside the film. Appropriate definitions of film site occupancy ratio (SOR), i.e., fraction of film sites occupied by particles over total number of film sites, and its fluctuation are introduced to describe film porosity. Deterministic and stochastic ordinary differential equation (ODE) models are also derived to describe the time evolution of film SOR and its fluctuation. The coefficients of the ODE models are estimated on the basis of data obtained from the kMC simulator of the deposition process using least-square methods. Simulation results demonstrate the applicability and effectiveness of the proposed film porosity modeling methods in the context of the deposition process under consideration.

Simultaneous Regulation of Surface Roughness and Porosity in Thin Film Growth

Abstract Abstract This work focuses on simultaneous control of surface roughness and film porosity in a porous thin film deposition process modeled via kinetic Monte Carlo simulation on a triangular lattice. The microscopic model of the thin film growth process includes adsorption and migration processes. Vacancies and overhangs are allowed inside the film for the purpose of modeling thin film porosity. Appropriate closed-form dynamic models are first derived to describe the evolution of film surface roughness and porosity and used as the basis for the design of a model predictive control algorithm that includes penalty on the deviation of surface roughness and film porosity from their respective set-point values. Closed-loop simulations demonstrate that when simultaneous control of surface roughness and porosity is carried out, a balanced trade-off is obtained in the closed-loop system between the two control objectives of surface roughness and porosity regulation.

Dynamic output feedback covariance control of linear stochastic dissipative partial differential equations

In this work, we develop a method for dynamic output feedback covariance control of the state covariance of linear dissipative stochastic partial differential equations (PDEs) using spatially distributed control actuation and sensing with noise. Such stochastic PDEs arise naturally in the modeling of surface height profile evolution in thin film growth and sputtering processes. We begin with the formulation of the stochastic PDE into a system of infinite stochastic ordinary differential equations (ODEs) by using modal decomposition. A finite-dimensional approximation is then obtained to capture the dominant mode contribution to the surface roughness profile (i.e., the covariance of the surface height profile). Subsequently, a state feedback controller and a Kalman-Bucy filter are designed on the basis of the finite-dimensional approximation. The dynamic output feedback covariance controller is subsequently obtained by combining the state feedback controller and the state estimator. The steady-state expected surface covariance under the dynamic output feedback controller is then estimated on the basis of the closed-loop finite-dimensional system. An analysis is performed to obtain a theoretical estimate of the expected surface covariance of the closed-loop infinite- dimensional system. Applications of the linear dynamic output feedback controller to the linearized stochastic Kuramoto- Sivashinsky equation are presented.

Dependence of film surface roughness on surface migration and lattice size in thin film deposition

This work focuses on the study of the dependence of film surface roughness on surface migration and lattice size in thin film deposition processes. Two different models of thin film deposition processes, in both one-dimension and two dimensions, are considered: random deposition with surface relaxation model and deposition/migration model. Surface roughness is defined as the root-mean-squares of the surface height profile and is found to evolve (starting from a flat initial surface zero value) to steady-state values at large times. A linear and a logarithmic dependence of surface roughness square on lattice size are observed in the one-dimensional and two-dimensional lattice models, respectively, in both the random deposition with surface relaxation model and the deposition/migration model with zero activation energy contribution from each neighboring particle. Furthermore, a stronger lattice-size dependence is found in the deposition/migration model when the migration activation energy contribution from each neighboring particle becomes significant.

Simultaneous regulation of thin film surface mean slope and roughness for light trapping optimization using predictive control

This work focuses on the development of a model predictive control algorithm to simultaneously regulate the surface slope and roughness of a thin film growth process to optimize thin film light reflectance and transmittance. Specifically, a thin film deposition process modeled on a one-dimensional triangular lattice that involves two microscopic processes: an adsorption process and a migration process, is considered. Kinetic Monte Carlo (kMC) methods are used to simulate this thin film deposition process. To characterize the surface morphology and to evaluate the light trapping efficiency of the thin film, surface roughness and surface slope are introduced as the root mean squares of the surface height profile and surface slope profile. An Edwards-Wilkinson (EW)-type equation with appropriate computed parameters is used to describe the dynamics of the surface height profile and predict the evolution of the root-mean-square (rms) roughness and rms slope. A model predictive control algorithm is then developed on the basis of the EW equation model to regulate the rms slope and the rms roughness at desired levels by optimizing the substrate temperature at each sampling time. Closed-loop simulation results demonstrate the effectiveness of the proposed model predictive control algorithm in successfully regulating the rms slope and the rms roughness at desired levels that optimize thin film light reflectance and transmittance.

Simultaneous regulation of film thickness, surface roughness and porosity in a multiscale thin film growth process

This work focuses on simultaneous regulation of film thickness, surface roughness and porosity in a multiscale model of a thin film growth process using the inlet precursor concentration as the manipulated input. Specifically, a continuous macroscopic partial differential equation model is used to describe the dynamics of the gas phase. The thin film growth process is modeled via a microscopic kinetic Monte Carlo simulation model on a triangular lattice with vacancies and overhangs allowed to develop inside the film. Closed-form dynamic models of thin film surface profile and porosity are developed and used as the basis for the design of a model predictive control algorithm to simultaneously regulate film thickness, surface roughness and film porosity. Simulation results demonstrate the applicability and effectiveness of the proposed modeling and control approach by applying the proposed controller to the multiscale model.

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

This work focuses on control configuration selection and nonlinear control of the roughness of a one-dimensional 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 Galerkins 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.