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A Galerkin Method for the Simulation of the Transient 2-D/2-D and 3-D/3-D Linear Boltzmann Equation

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Abstract

Many production steps used in the manufacturing of integrated circuits involve the deposition of material from the gas phase onto wafers. Models for these processes should account for gaseous transport in a range of flow regimes, from continuum flow to free molecular or Knudsen flow, and for chemical reactions at the wafer surface. We develop a kinetic transport and reaction model whose mathematical representation is a system of transient linear Boltzmann equations. In addition to time, a deterministic numerical solution of this system of kinetic equations requires the discretization of both position and velocity spaces, each two-dimensional for 2-D/2-D or each three-dimensional for 3-D/3-D simulations. Discretizing the velocity space by a spectral Galerkin method approximates each Boltzmann equation by a system of transient linear hyperbolic conservation laws. The classical choice of basis functions based on Hermite polynomials leads to dense coefficient matrices in this system. We use a collocation basis instead that directly yields diagonal coefficient matrices, allowing for more convenient simulations in higher dimensions. The systems of conservation laws are solved using the discontinuous Galerkin finite element method. First, we simulate chemical vapor deposition in both two and three dimensions in typical micron scale features as application example. Second, stability and convergence of the numerical method are demonstrated numerically in two and three dimensions. Third, we present parallel performance results which indicate that the implementation of the method possesses very good scalability on a distributed-memory cluster with a high-performance Myrinet interconnect.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD, 21250, USA

    Matthias K. Gobbert & Samuel G. Webster

  2. Focus Center – New York, Rensselaer: Interconnections for Hyperintegration, Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, CII 6015, 110 8th Street, Troy, NY, 12180-3590, USA

    Timothy S. Cale

  3. Department of Mathematics and Computer Science, Hillsdale College, 33E. College Street, Hillsdale, MI, 49242, USA

    Samuel G. Webster

Authors
  1. Matthias K. Gobbert
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  2. Samuel G. Webster
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  3. Timothy S. Cale
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Correspondence to Matthias K. Gobbert.

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Gobbert, M.K., Webster, S.G. & Cale, T.S. A Galerkin Method for the Simulation of the Transient 2-D/2-D and 3-D/3-D Linear Boltzmann Equation. J Sci Comput 30, 237–273 (2007). https://doi.org/10.1007/s10915-005-9069-1

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  • DOI: https://doi.org/10.1007/s10915-005-9069-1

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