Stochastic Modeling and Simultaneous Regulation of Surface Roughness and Porosity in Thin Film Deposition

Abstract

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 root-mean-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 trade-off is obtained in the closed-loop system between the two control objectives of surface roughness and porosity regulation.