Feedback Control Of Surface Roughness Research Articles

Feedback control of surface roughness in a one-dimensional Kardar-Parisi-Zhang growth process.

Control of generically scale-invariant systems, i.e., targeting specific cooperative features in nonlinear stochastic interacting systems with many degrees of freedom subject to strong fluctuations and correlations that are characterized by power laws, remains an important open problem. We study the control of surface roughness during a growth process described by the Kardar-Parisi-Zhang (KPZ) equation in (1+1) dimensions. We achieve the saturation of the mean surface roughness to a prescribed value using nonlinear feedback control. Numerical integration is performed by means of the pseudospectral method, and the results are used to investigate the coupling effects of controlled (linear) and uncontrolled (nonlinear) KPZ dynamics during the control process. While the intermediate time kinetics is governed by KPZ scaling, at later times a linear regime prevails, namely the relaxation toward the desired surface roughness. The temporal crossover region between these two distinct regimes displays intriguing scaling behavior that is characterized by nontrivial exponents and involves the number of controlled Fourier modes. Due to the control, the height probability distribution becomes negatively skewed, which affects the value of the saturation width.

Model parameter estimation and feedback control of surface roughness in a sputtering process

This work focuses on model parameter estimation and model-based output feedback control of surface roughness in a sputtering process which involves two surface micro-processes: atom erosion and surface diffusion. This sputtering process is simulated using a kinetic Monte Carlo (kMC) simulation method and its surface height evolution can be adequately described by the stochastic Kuramoto–Sivashinsky equation (KSE), a fourth-order nonlinear stochastic partial differential equation (PDE). First, we estimate the four parameters of the stochastic KSE so that the expected surface roughness profile predicted by the stochastic KSE is close (in a least-square sense) to the profile of the kMC simulation of the same process. To perform this model parameter estimation task, we initially formulate the nonlinear stochastic KSE into a system of infinite nonlinear stochastic ordinary differential equations (ODEs). A finite-dimensional approximation of the stochastic KSE is then constructed that captures the dominant mode contribution to the state and the evolution of the state covariance of the stochastic ODE system is derived. Then, a kMC simulator is used to generate representative surface snapshots during process evolution to obtain values of the state vector of the stochastic ODE system. Subsequently, the state covariance of the stochastic ODE system that corresponds to the sputtering process is computed based on the kMC simulation results. Finally, the model parameters of the nonlinear stochastic KSE are obtained by using least-squares fitting so that the state covariance computed from the stochastic KSE process model matches that computed from kMC simulations. Subsequently, we use appropriate finite-dimensional approximations of the identified stochastic KSE model to design state and output feedback controllers, which are applied to the kMC model of the sputtering process. Extensive closed-loop system simulations demonstrate that the controllers reduce the expected surface roughness by 55 % compared to the corresponding values under open-loop operation.

Nonlinear Feedback Control of Surface Roughness Using a Stochastic PDE: Design and Application to a Sputtering Process

In this work, we develop a method for nonlinear feedback control of the roughness of a one-dimensional surface whose evolution is described by the stochastic Kuramoto-Sivashinsky equation (KSE), a fourthorder nonlinear stochastic partial differential equation. We initially formulate the stochastic KSE into a system of infinite nonlinear stochastic ordinary differential equations 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 to control the surface roughness. An analysis of the closed-loop nonlinear infinite-dimensional system is performed to characterize the closed-loop performance enforced by the nonlinear feedback controller in the closed-loop infinite-dimensional system. The effectiveness of the proposed nonlinear controller and the advantages of the nonlinear controller over a linear controller resulting from the linearization of the nonlinear controller around the zero solution are demonstrated through numerical simulations. Finally, a successful application of a stochastic KSE-based nonlinear feedback controller to the kinetic Monte Carlo model of a sputtering process is also demonstrated.

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

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.

Feedback control of surface roughness of GaAs (0 0 1) thin films using kinetic Monte Carlo models

In this work, we follow the methodology presented in Lou and Christofides [Estimation and control of surface roughness in thin-film growth using kinetic Monte-Carlo models, Chem. Eng. Sci. 58 (2003a) 3115–3129] to study estimation and control of surface roughness of gallium arsenide (GaAs) (0 0 1) thin films during deposition in a horizontal-flow quartz reactor using triisobutylgallium (TIBGa) and tertiarybutylarsine (TBAs) as precursors with H2 as the carrier gas. The adsorption of TIBGa onto the surface and the migration of Ga atoms on the surface are considered as the two rate-limiting steps in the film growth and are explicitly modeled within a kinetic Monte Carlo simulation framework. The energy barrier and the pre-exponential factor of the migration rate of Ga atoms on the surface used in the simulations are initially determined by fitting the simulation results to experimental data reported in Law et al. [Analysis of the growth modes for gallium arsenide metalorganic vapor-phase epitaxy, J. Appl. Phys. 88 (2000) 508–512]. Then, a roughness estimator is constructed that allows computing estimates of the surface roughness of the GaAs thin films at a time-scale comparable to the real-time evolution of the process using discrete on-line roughness measurements. The estimator involves a kinetic MC simulator based on multiple small-lattice models, an adaptive filter used to reduce roughness stochastic fluctuations and an error compensator used to reduce the error between the roughness estimates and the roughness measurements. The roughness estimates are fed to a proportional–integral (PI) feedback controller which is used to control the surface roughness to a desired level by manipulating the substrate temperature. Application of the proposed estimator/controller structure to the process model based on a large-lattice kinetic Monte Carlo simulator demonstrates successful regulation of the surface roughness to the desired level. The proposed approach is shown to be superior to PI control with direct use of the discrete roughness measurements. The reason being that the available measurement techniques do not provide measurements at a frequency that is comparable to the time-scale of the dominant film growth dynamics.

Feedback Control of Surface Roughness in a Deposition Process Using a Stochastic PDE

This paper focuses on feedback control of surface roughness in a deposition process, based on a stochastic partial differential equation (PDE) which describes the fluctuation of surface height in the spatial domain. Specifically, we focus on control of surface roughness in a deposition process on a 1-dimensional lattice, whose fluctuation of surface height can be described by the Edwards-Wilkinson equation, a secondorder stochastic PDE. We initially refonnulate the stochastic PDE into a system of infinite stochastic ordinary differential equations by using modal decomposition. A finite-dimensional approximation of the Edwards-Wilkinson equation 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. Analysis of the closed-loop system shows that the controller can drive the surface roughness governed by the infinite-dimensional system to desired levels. The effectiveness of the proposed method is demonstrated by numerical simulations.

Feedback control of growth rate and surface roughness in thin film growth

AbstractA method for multivariable feedback control of surface roughness and growth rate in thin film growth using kinetic Monte‐Carlo (MC) models is proposed. The method is applied to the process of thin film growth in a stagnation flow geometry including atom adsorption, desorption, and surface migration as the three processes that shape film microstructure and determine film growth rate. A multiscale model involving coupled partial differential equations to model the gas phase and a kinetic MC simulator, based on a large lattice, for the modeling of film growth, is used to simulate the process. A roughness and growth rate estimator constructed allowed computing estimates of the surface roughness and growth rate at a time‐scale comparable to the real‐time evolution of the process. The estimator involves kinetic MC simulators based on small lattice models, adaptive filters used to reduce stochastic fluctuations of the outputs of the small lattice MC simulator outputs, and measurement error compensators used to reduce the errors between the estimates and measurements. The interactions between the inputs (substrate temperature and inlet precursor mole fraction) and outputs (growth rate and surface roughness) in the closed‐loop system are studied and found to be significant. A multivariable feedback controller, which uses the state estimator and explicitly compensates for the effect of input/output interactions, is designed to simultaneously regulate the growth rate and surface roughness by manipulating substrate temperature and inlet precursor mole fraction. Application of the estimator/controller structure to the multiscale process model demonstrates successful regulation of the surface roughness and growth rate to the desired set point values. This approach is shown to be superior to control of the growth rate and surface roughness using two independent feedback control loops.