Modeling multiscale effects on transients during chemical vapor deposition

Abstract

Several important manufacturing processes for integrated circuits (ICs) involve the flow of gaseous reactants over the wafer(s) on which the ICs are being made. We discuss a model in which reactive components of the gas phase do not collide; either because there is a dominant carrier species, or the pressure is low enough. The kinetic transport and reaction model consists of a system of transient linear Boltzmann equations for the reactive species in the flow. This model applies to a wide range of transport regimes, characterized by a wide range of Knudsen numbers as a function of pressure and length scale of interest. A numerical simulator based on a spectral Galerkin method in velocity space approximates each linear Boltzmann equation by a system of transient conservation laws in space and time with diagonal coefficient matrices, which are solved using the discontinuous Galerkin method. This deterministic solver gives direct access to the kinetic density that is the solution to the Boltzmann equation, as a function of position, velocity, and time. The availability of the kinetic density as a function of velocity is useful to analyze the underlying kinetic causes of macroscopic observables. Using chemical vapor deposition as an important application example, the influence of process parameters is studied in transient two-dimensional and three-dimensional features that represent structures seen during integrated circuit fabrication for a wide range of Knudsen numbers. The results highlight the capabilities of the KTRM and its implementation, and indicate that kinetic solvers may be needed for models characterized by Knudsen numbers on the order of unity. This regime applies both to feature scale simulators at certain operating conditions and to intermediate scale models used as part of multiscale simulators.