Matthias K. Gobbert

Simulation Techniques for the
Boltzmann Equation in Gas Dynamics

This page can be reached via my homepage at http://www.math.umbc.edu/~gobbert.

Historical Note

Ludwig Eduard Boltzmann was born on February 20, 1844 in Vienna (Austria) and died on September 05, 1906 in Duino near Trieste (Italy). While on his first professor position in Graz (Austria) in 1872, he published the ground-braking paper that introduced the equation bearing his name. Pictures of the young Boltzmann, the old Boltzmann, his tombstone, and a detail of his tombstone.

Overview

Selected processes in the manufacture of integrated circuits are unified by the property that mixtures of reactive gases flow through the processing equipment at low pressures. These processes include thermally activated processes such as low pressure chemical vapor deposition and atomic layer deposition, as well as plasma enhanced processes such as sputter deposition and plasma enhanced chemical vapor deposition. Modeling these processes involves combinations of complex transport and reaction models at the equipment scale (reactor scale, typical length scale 10 cm), at the scale of device features on the wafer (feature scale, typical length scale less than 1 µm), as well as at various intermediate scales of interest (mesoscopic scale, typical length scales ranging from 0.1 mm to 1.0 cm). To model transport, it is often valid to use continuum fluid dynamics on the reactor scale, but at the typically very low pressures used in industry, kinetic flow descriptions have to be used on the smaller length scales.

Together with Timothy S. Cale (Department of Chemical Engineering, Rensselaer Polytechnic Institute), we have developed a kinetic transport and reaction model based on a system of Boltzmann transport equations for gas dynamics that is valid on both the feature scale and the mesoscopic scale. The Boltzmann transport equation is a integro partial differential equation; the left-hand side describes the convective transport of the molecules, and the right-hand side models collisions between the molecules. For the applications under consideration, it is possible to reduce the problem to a system of linear Boltzmann equations for the reactive species.

A Galerkin approach is used to convert each Boltzmann equation to a system of linear hyperbolic equations. In collaboration with Christian Ringhofer (Department of Mathematics, Arizona State University), basis functions were chosen that make the coefficient matrices in this system diagonal. The discontinous Galerkin method (DGM) is then used to solve the systems of hyperbolic equations. We use the implementation of the method in the code DG, developed at the Scientific Computation Research Center, Rensselaer Polytechnic Institute.

The fundamental aim of this work is to develop an efficient, predictive multiscale simulator for the processes under consideration. The numerical methods and models have been validated on each scale individually already. Mathematical analysis is needed to develop rational linkages between models on vastly different length scales. To obtain an efficient overall simulator, we will use parallel computing both for each individual model as well as overall by running several feature scale models in parallel.

People Involved

Christian Ringhofer, Professor of Mathematics, Department of Mathematics, Arizona State University
Timothy S. Cale, Professor of Chemical Engineering, Isermann Department of Chemical and Biological Engineering and Focus Center - New York, Rensselaer: Interconnections for Hyperintegration, Rensselaer Polytechnic Institute

People Formerly Involved

Mark L. Breitenbach, part-time graduate student, M.S. (non-thesis) December 2004; continued working for the Department of Defense
Samuel G. Webster, graduate student, Department of Mathematics and Statistics, University of Maryland, Baltimore County Ph.D. May 2004; continued as tenure-track Assistant Professor at the Department of Mathematics and Computer Science, Hillsdale College, Hillsdale, MI
Vinay Prasad, post-doctoral associate, Focus Center - New York, Rensselaer: Interconnections for Hyperintegration, Rensselaer Polytechnic Institute; continued on to private industry
Steven C. Foster, undergraduate student, Department of Mathematics and Statistics, University of Maryland, Baltimore County, B.S. August 2003; continued to the graduate program at the Department of Mathematics, University of North Carolina at Chapel Hill

Chemical Vapor Deposition

Movie of gas concentration filling a two-dimensional cross-section of an infinite trench; both third coordinate dimension and color indicate value of of the gas concentration.
Movie of slice plots of gas concentration filling a three-dimensional corner of semi-infinite trenches; color indicates value of the gas concentration: blue = 0, red = 1. The gas enters the domain at the gas-phase interface at x₃ = 0.25. It reaches the mean wafer surface at x₃ = 0 rapidly, before it more slowly fills the interior of the trench.
The feature fills with gas more rapidly than typical processing times. These results were computed using 8 processors on a Beowulf cluster. Results by Samuel G. Webster.

Atomic Layer Deposition

Number density in feature with aspect ratio 4 at times 1 ns, 5 ns, 10 ns, 20 ns, 40 ns, 50 ns. The solution was computed on a cluster of Linux PCs using four processors. The height shows the value of the solution; the four colors indicate the subdomain of each processor. The feature fills with gas more rapidly than typical processing times. Click here for a complete movie of the solution. Results by Samuel G. Webster.

Numerical Studies for the Discontinuous Galerkin Method

Scalar transport solved with DG, refinement level 2, at times 0, 20, 40, 60, 80, 100, The solution was computed on a cluster of Linux PCs using four processors. The height shows the value of the solution; the four colors indicate the subdomain of each processor. The effect of the local mesh refinement and coarsening can be observed. Click here for a complete movie of the solution. Results by Steven C. Foster.
Test of Boltzmann solver using DG on square movie of component k = 2, jpeg file of one frame.
Finite difference solver on square short mpeg-movie long mpeg-movie

Acknowledgements

This research uses significant hardware supplied by the National Science Foundation equipment grant DMS-0215373, "Scientific Computing Research Environments for the Mathematical Sciences (SCREMS)," (with Jonathan Bell, Madhu Nayakkankuppam, and Florian Potra), 2002-2005, $150,000.
Samuel G. Webster was supported in part by an UMBC DRIF Research Assistantship Support, "Parallel Multi Scale Simulation of Atomic Layer Deposition," AY 2002-03, $7,000.
Steven C. Foster was partially supported by an UMBC Undergraduate Research Award, "Numerical Simulation of Atomic Layer Epitaxy using MPI," AY 2001-02, $1,500.
This research was partially supported by the National Science Foundation grant DMS-9805547, "Computational Methods for the Simulation of Chemical Vapor Deposition on Rough Surfaces," 08/01/98-07/31/01, $68,000.

Publications Resulting From This Research

In reverse chronological order

The following list is a partial list with a focus on student involvement in the work; see my own publication list for additional papers as well as for links to reprints and preprints.

Matthias K. Gobbert, Samuel G. Webster, and Timothy S. Cale. A Galerkin Method for the Simulation of the Transient 2-D/2-D and 3-D/3-D Linear Boltzmann Equation. Submitted.
Matthias K. Gobbert and Timothy S. Cale. A Kinetic Transport and Reaction Model and Simulator for Rarefied Gas Flow in the Transition Regime. Journal of Computational Physics, accepted (2005).
Matthias K. Gobbert, Mark L. Breitenbach, and Timothy S. Cale. Cluster Computing for Transient Simulations of the Linear Boltzmann Equation on Irregular Three-Dimensional Domains. In: Vaidy S. Sunderam, Geert Dick van Albada, Peter M. A. Sloot, and Jack J. Dongarra, editors, Computational Science - ICCS 2005, Lecture Notes in Computer Science, vol. 3516, pp. 41-48, Springer-Verlag, 2005.
Samuel G. Webster. Stability and Convergence of a Spectral Galerkin Method for the Linear Boltzmann Equation. Ph.D. thesis, University of Maryland, Baltimore County, May 2004.
Matthias K. Gobbert and Christian Ringhofer. Mesoscopic Scale Modeling for Chemical Vapor Deposition in Semiconductor Manufacturing. In: Naoufel Ben Abdallah, Anton Arnold, Pierre Degond, Irene M. Gamba, Robert T. Glassey, C. David Levermore, and Christian Ringhofer, editors, Dispersive Transport Equations and Multiscale Models, The IMA Volumes in Mathematics and its Applications, vol. 136, pp. 133-149, Springer-Verlag, 2004.
Matthias K. Gobbert and Christian Ringhofer. A Homogenization Technique for the Boltzmann Equation for Low Pressure Chemical Vapor Deposition. SIAM Journal on Applied Mathematics, vol. 64, no. 1, pp. 196-215, 2003.
Steven C. Foster. Application of the Boltzmann Equation to the Modeling of Atomic Layer Deposition with Performance Studies. Senior Thesis, University of Maryland, Baltimore County, August 2003.
Steven C. Foster. Performance Studies for the Discontinuous Galerkin Method Applied to the Scalar Transport Equation. UMBC Review: Journal of Undergraduate Research and Creative Works, vol. 4, pp. 36-47, 2003.
Vinay Prasad, Matthias K. Gobbert, Max Bloomfield, and Timothy S. Cale. Improving Pulse Protocols in Atomic Layer Deposition. In: B. M. Melnick, T. S. Cale, S. Zaima, and T. Ohta, editors, Advanced Metallization Conference 2002, pp. 709-715, Materials Research Society, 2003.
Samuel G. Webster, Matthias K. Gobbert, and Timothy S. Cale. Transient 3-D/3-D Transport and Reactant-Wafer Interactions: Adsorption and Desorption. In: P. Timans, E. Gusev, F. Roozeboom, M. Ozturk, and D. L. Kwong, editors, Rapid Thermal and Other Short-Time Processing Technologies III, The Electrochemical Society Proceedings Series, vol. 2002-11, pp. 81-88, 2002.
Matthias K. Gobbert, Samuel G. Webster, Jean-François Remacle, and Timothy S. Cale. A Spectral Galerkin Ansatz for the Deterministic Solution of the Boltzmann Equation on Irregular Domains. Technical Report, University of Maryland, Baltimore County, 2002.
Samuel G. Webster, Matthias K. Gobbert, Jean-François Remacle, and Timothy S. Cale. Parallel Numerical Solution of the Boltzmann Equation for Atomic Layer Deposition. In: Burkhard Monien and Rainer Feldmann, editors, Euro-Par 2002 Parallel Processing, Lecture Notes in Computer Science, vol. 2400, pp. 452-456, Springer-Verlag, 2002.
Steven C. Foster, Matthias K. Gobbert, and Jean-François Remacle. Performance Studies on the Discontinuous Galerkin Method for Solving the Scalar Transport Equation. Technical Report, University of Maryland, Baltimore County, 2002.
Matthias K. Gobbert, Vinay Prasad, and Timothy S. Cale. Predictive Modeling of Atomic Layer Deposition on the Feature Scale. Thin Solid Films, vol. 410, pp. 129-141, 2002.
Matthias K. Gobbert, Vinay Prasad, and Timothy S. Cale. Modeling and Simulation of Atomic Layer Deposition at the Feature Scale. Journal of Vacuum Science & Technology B, vol. 20, no. 3, pp. 1031-1043, 2002.
Matthias K. Gobbert, Samuel G. Webster, and Timothy S. Cale. Transient Adsorption and Desorption in Micrometer Scale Features. Journal of The Electrochemical Society, vol. 149, no. 8, pp. G461-G473, 2002.
Matthias K. Gobbert, Vinay Prasad, and Timothy S. Cale. A Feature Scale Model for Atomic Layer Deposition. In: T. Wade, editor, Proceedings of the Eighteenth International VLSI Multilevel Interconnection Conference, pp. 413-417, IMIC, 2001.
Matthias K. Gobbert and Timothy S. Cale. A Feature Scale Transport and Reaction Model for Atomic Layer Deposition. In: M. T. Swihart, M. D. Allendorf, and M. Meyyappan, editors, Fundamental Gas-Phase and Surface Chemistry of Vapor-Phase Deposition II, The Electrochemical Society Proceedings Series, vol. 2001-13, pp. 316-323, 2001.

Background References

In alphabetical order

Engelbert Broda, Ludwig Boltzmann: Man, Physicist, Philosopher, Ox Bow Press: Woodbridge, Connecticut, 1983.
Carlo Cercignani, editor, Kinetic Theories and the Boltzmann Equation, Lecture Notes in Mathematics, no. 1048, Springer-Verlag, 1984.
Carlo Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, 1988.
Carlo Cercignani, Ludwig Boltzmann: The Man Who Trusted Atoms, Oxford University Press, 1998.
Carlo Cercignani, Rarefied Gas Dynamics: From Basic Concepts to Actual Computations, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2000.
E. G. D. Cohen and W. Thirring, editors, The Boltzmann Equation: Theory and Applications, Proceedings of the International Symposium "100 Years Boltzmann Equation" in Vienna, 4th-8th September 1972, Acta Physica Austriaca, Supplementum X, Springer-Verlag, 1973.
Alfred Kersch and William J. Morokoff, Transport Simulation in Microelectronics, Birkhauser series Progress in Numerical Simulation for Microelectronics, no. 3, 1995.
Mikhail N. Kogan, Rarefied Gas Dynamics, Plenum Press, 1969.
G. N. Patterson, Molecular Flow of Gases, Wiley, 1956.

Simulation Techniques for the Boltzmann Equation in Gas Dynamics