[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content

Advertisement

Springer Nature Link
Log in

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

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bird, G.A. (1976). Molecular Gas Dynamics, Oxford University Press.

  2. Cale T.S., Bloomfield M.O., Richards D.F., Jansen K.E., Gobbert M.K. (2002). Integrated multiscale process simulation. Comp. Mater. Sci. 23:3–14

    Google Scholar 

  3. Cale T.S., Merchant T.P., Borucki L.J., and Labun A.H. (2000). Topography simulation for the virtual wafer fab. Thin Solid Films 365(2):152–175

    Article  Google Scholar 

  4. Carrillo J.A., Gamba I.M., Majorana A., and Shu C.-W. (2003). A WENO-solver for the transients of Boltzmann–Poisson system for semiconductor devices: Performance and comparisons with Monte Carlo methods. J. Comput. Phys. 184:498–525

    Article  MATH  MathSciNet  Google Scholar 

  5. Cercignani, C. (1988). The Boltzmann Equation and Its Applications, Vol.67 of Applied Mathematical Sciences. Springer-Verlag.

  6. Cercignani, C. (2000). Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations, Cambridge Texts in Applied Mathematics. Cambridge University Press.

  7. Chorin A.J. (1972). Numerical solution of Boltzmann’s equation. Comm. Pure Appl. Math. 25:171–186

    MATH  MathSciNet  Google Scholar 

  8. Cockburn, B., Karniadakis, G.E., and Shu, C.-W. (eds.) (2000). Discontinuous Galerkin Methods: Theory, Computation and Applications, Vol.11 of Lecture Notes in Computational Science and Engineering. Springer-Verlag.

  9. Fatemi E., and Odeh F. (1993). Upwind finite difference solution of Boltzmann equation applied to electron transport in semiconductor devices. J. Comput. Phys. 108:209–217

    Article  MATH  MathSciNet  Google Scholar 

  10. Gobbert, M.K., and Cale, T.S. (2001). A feature scale transport and reaction model for atomic layer deposition. In Swihart, M.T., Allendorf, M.D., and Meyyappan, M. (eds.), Fundamental Gas-Phase and Surface Chemistry of Vapor-Phase Deposition II, volume 2001–13, The Electrochemical Society Proceedings Series pp. 316–323

  11. Gobbert, M.K., and Cale, T.S. (2006). A kinetic transport and reaction model and simulator for rarefied gas flow in the transition regime. J. Comput. Phys. (in press).

  12. Gobbert M.K., Prasad V., and Cale T.S. (2002). Modeling and simulation of atomic layer deposition at the feature scale. J. Vac. Sci. Technol. B 20(3):1031–1043

    Article  Google Scholar 

  13. Gobbert M.K., Prasad V., and Cale T.S. (2002). Predictive modeling of atomic layer deposition on the feature scale. Thin Solid Films 410:129–141

    Article  Google Scholar 

  14. Gobbert M.K., Webster S.G., and Cale T.S. (2002). Transient adsorption and desorption in micrometer scale features. J. Electrochem. Soc. 149(8):G461–G473

    Article  Google Scholar 

  15. Grad H. (1949). On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2:331–407

    MATH  MathSciNet  Google Scholar 

  16. Kersch, A., and Morokoff, W.J. (1995). Transport Simulation in Microelectronics, Vol.3 of Progress in Numerical Simulation for Microelectronics. Birkhäuser Verlag, Basel

  17. Markovich, P.A., Ringhofer, C.A., and Schmeiser, C. (1990). Semiconductor Equations, Springer-Verlag.

  18. Poupaud F. (1991). Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers. Asymptotic Anal. 4:293–317

    MATH  MathSciNet  Google Scholar 

  19. Reed W.H., and Hill T.R. (1973). Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM

    Google Scholar 

  20. Remacle J.-F., Flaherty J.E., and Shephard M.S. (2003). An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems. SIAM Rev. 45(1):53–72

    Article  MATH  MathSciNet  Google Scholar 

  21. Ringhofer C. (2000). Space–time discretization of series expansion methods for the Boltzmann transport equation. SIAM J. Numer. Anal. 38(2):442–465

    Article  MATH  MathSciNet  Google Scholar 

  22. Ringhofer C., Schmeiser C., and Zwirchmayr A. (2001). Moment methods for the semiconductor Boltzmann equation on bounded position domains. SIAM J. Numer. Anal. 39(3):1078–1095

    Article  MATH  MathSciNet  Google Scholar 

  23. Schmeiser C., and Zwirchmayr A. (1998). Convergence of moment methods for linear kinetic equations. SIAM J. Numer. Anal. 36(1):74–88

    Article  MATH  MathSciNet  Google Scholar 

  24. Webster S.G. (2004). Stability and convergence of a spectral Galerkin method for the linear Boltzmann equation. Ph.D.thesis, University of Maryland, Baltimore County

    Google Scholar 

  25. Webster, S.G., Gobbert, M.K., and Cale, T.S. (2002). Transient 3-D/3-D transport and reactant–wafer interactions: Adsorption and desorption. In Timans, P., Gusev, E., Roozeboom, F., Ozturk, M., and Kwong, D.L. (eds.), Rapid Thermal and Other Short-Time Processing TechnologiesIII, volume 2002–11, The Electrochemical Society Proceedings Series, pp. 81–88

  26. Webster, S.G., Gobbert, M.K., Remacle, J.-F., and Cale, T.S. (2002). Parallel numerical solution of the Boltzmann equation for atomic layer deposition. In Monien, B., and Feldmann, R. (eds.), Euro-Par 2002 Parallel Processing, Vol. 2400 of Lecture Notes in Computer Science, Springer-Verlag, pp. 452–456

Download references

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Samuel G. Webster
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Timothy S. Cale
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthias K. Gobbert.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords

Search

Navigation