Comparison of numerical methods for transient CVD simulations

The numerical modeling of laminar reacting gas flows in thermal Chemical Vapor Deposition (CVD) processes commonly involves the solution of convection–diffusion-reaction equations for a large number of reactants and intermediate species. These equations are stiffly coupled through the reaction terms, which typically include dozens of finite rate elementary reaction steps with largely varying rate constants. The solution of such stiff sets of equations is difficult, especially when time-accurate transient solutions are required. The latter is important for the study of start-up and shut-down cycli, but also for the study of inherently transient CVD processes, such as Rapid Thermal CVD (RTCVD) and Atomic Layer Deposition (ALD). In this study various numerical schemes for multidimensional transient simulations of laminar reacting gas flows with homogeneous and heterogeneous chemical reactions are compared in terms of efficiency, accuracy and robustness. As a test case, we study the CVD process of silicon from silane, modeled according to the classical 16 species, 27 reactions chemistry model for this process as published by Coltrin and coworkers [M.E. Coltrin, R.J. Kee, G.H. Evans, J. Electrochem. Soc. 136 (1989) 819]. We validate our results by comparison to steady state solutions in the benchmark paper of Kleijn [ C.R. Kleijn, Thin Solid Films, 365 (2000) 294]. It is concluded that, for time-accurate transient simulations of the stiff chemistry problems in CVD modeling the conservation of the non-negativity of the species concentrations is much more important, and much more restrictive towards the time step size, than stability. For the time integration methods studied in this paper, we give suggestions for the associated optimal nonlinear and linear solvers, such that the total computational costs are reduced as much as possible.