(528e) Multivariable Model Predictive Control of Surface Roughness and Slope in a Thin Film Growth Process
Thin film growth of semiconductor materials has attracted significant research attention due to its importance in many applications. For example, surface morphology of the thin films strongly influence the electrical and photovoltaic properties in the manufacturing of microelectronic and photovoltaic devices. Recent studies have demonstrated that the light trapping efficiency of thin-film solar cells is determined by surface roughness and mean surface slope of the thin film [1, 2].
Significant research efforts have been made over the last ten years towards a rational approach of modeling and control of thin film growth and manufacturing of desired thin films with certain surface morphology; see, for example, the book  for an overview of key results and reference. Two general modeling approaches can be used in the modeling of the thin film growth process: kinetic Monte Carlo (kMC) methods and stochastic partial differential equation (PDE) models. KMC methods were initially introduced to simulate thin film microscopic processes based on the microscopic rules and the thermodynamic and kinetic parameters obtained from experiments and molecular dynamics simulations. Recently, we have initiated an effort towards modeling and control of surface mean slope which strongly influences the light reflectance and transmittance properties of thin films. In this direction, we have studied dynamics and lattice size dependence of surface mean slope  and predictive control of both surface roughness and slope using stochastic PDEs in one spatial dimension . Recent research work has also focused on the computationally efficient multiobjective optimization and predictive control of microscopic and multiscale systems using in situ adaptive tabulation . However, model predictive control of both surface roughness and slope using stochastic PDEs in two spatial dimensions, an important problem from a practical standpoint, to optimize the light trapping efficiency during thin film manufacturing processes has not been studied yet.
Motivated by these considerations, this work focuses on the development of a multivariable model predictive controller that simultaneously regulates thin film surface roughness and mean slope to optimize light reflectance and transmittance during thin film manufacturing. Both substrate temperature and deposition rate are used as the manipulated variables. Surface roughness and surface slope are defined as the root-mean-squares of the surface height profile and the surface slope profile, respectively. The dynamics of the evolution of the thin film surface height profile are assumed to be described by an Edwards-Wilkinson-type equation, a second-order stochastic partial differential equation, in two spatial dimensions. Analytical solutions of the expected surface roughness and surface slope are obtained on the basis of the Edwards-Wilkinson equation and are used in the controller design. The dependence of model parameters of the Edwards-Wilkinson equation on the manipulated variables is used in the formulation of the predictive controller to predict the influence of the control action on the surface roughness and slope at the end of the growth process. The model predictive controller involves constraints on the magnitude and rate of change of the control action and optimizes a cost that involves penalty on both surface roughness and mean slope from the set-point values. The controller is applied to the two-dimensional Edwards-Wilkinson equation and the two-dimensional kMC models and is shown to successfully regulate surface roughness and mean slope to set-point values at the end of the deposition that yield desired film reflectance and transmittance.
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