The dependence of film surface roughness and porosity on lattice size in a porous thin film deposition process is studied via kinetic Monte Carlo simulations on a triangular lattice. For sufficiently large lattice size the steady-state value of the expected film porosity has a weak dependence on the lattice size and the steady-state value of the expected surface roughness square varies linearly with lattice size. An analysis of the film morphology based on a stochastic partial differential equation description of the film surface morphology supports and explains the findings of the numerical simulations.
This work focuses on stochastic modeling and simultaneous regulation of surface roughness and porosity for a porous thin film deposition process modeled via kinetic Monte Carlo (kMC) 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. The definition of the surface height profile is first introduced for a porous thin film deposition taking place in a triangular lattice. The dynamics of surface height of the thin film are described by an Edwards-Wilkinson (EW) type equation, which is a second-order linear stochastic partial differential equation (PDE). The rootmean-square (RMS) surface roughness is chosen as one of the controlled variables. Subsequently, an appropriate definition of film site occupancy ratio (SOR) is introduced to represent the extent of porosity inside the film and is chosen as the second to-be-controlled variable. A deterministic ordinary differential equation (ODE) model is postulated to describe the time evolution of the film SOR. The coefficients of the EW equation of surface height and of the deterministic ODE model of the film SOR are estimated on the basis of data obtained from the kMC simulator of the deposition process using least-squares methods, and their dependence on substrate temperature is determined. The developed dynamic models are used as the basis for the design of a model predictive control algorithm that includes a penalty on the deviation of the surface roughness square and film SOR from their respective set-point values. Simulation results demonstrate the applicability and effectiveness of the proposed modeling and control approach in the context of the deposition process under consideration. When simultaneous control of surface roughness and porosity is carried out, a balanced tradeoff is obtained in the closed-loop system between the two control objectives of surface roughness and porosity regulation.
in Wiley InterScience (www.interscience.wiley.com).A method is developed 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. Initially a nonlinear stochastic PDE is formulated into a system of infinite nonlinear stochastic ordinary differential equations by using Galerkin's 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. To demonstrate the method, the model predictive controller is applied to the stochastic Kuramoto-Sivashinsky equation, and the kinetic Monte Carlo model of a sputtering process to regulate the surface roughness at a desired level.
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