A homogenization technique for the Boltzmann equation for low pressure chemical vapor deposition

SIAM Journal on Applied Mathematics

Volumn 64, Issue 1, 2003, Pages 196-215

  Gobbert, Matthias K ^a   Ringhofer, Christian ^b  

^a UNIVERSITY OF MARYLAND BALTIMORE COUNTY   (United States)

^b ARIZONA STATE UNIVERSITY   (United States)

Author keywords

Boltzmann equation; Boundary homogenization; Chemical vapor deposition; Microstructured surface; Rarefied gas dynamics

Indexed keywords

BOUNDARY CONDITIONS; FUNCTIONS; GEOMETRY; INTEGRAL EQUATIONS; LAPLACE TRANSFORMS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; MATRIX ALGEBRA; POLYNOMIALS; PROBABILITY DENSITY FUNCTION; PROBLEM SOLVING; THEOREM PROVING;

BOLTZMANN EQUATION; BOUNDARY HOMOGENIZATION; LOW PRESSURE CHEMICAL VAPOR DEPOSITION; RAREFIED GAS DYNAMICS;

LINEAR EQUATIONS;

EID: 1842509222     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139902393476     Document Type: Article

References (23)

1. H. Babovsky, Derivation of stochastic reflection laws from specular reflection, Transport Theory Statist. Phys., 16 (1987), pp. 113-126.
2. A. G. Belyaev, On Singular Perturbations of Boundary Problems, Ph.D. thesis, Moscow State University, Moscow, 1990 (in Russian).
3. T. S. Cale, T. H. Gandy, and G. B. Raupp, A fundamental feature scale model for low pressure deposition processes, J. Vac. Sci. Technol. A, 9 (1991), pp. 524-529.
4. C. Cercignani, The Boltzmann Equation and Its Applications, Appl. Math. Sci. 67, Springer-Verlag, New York, 1988.
5. A. Friedman and B. Hu, A non-stationary multi-scale oscillating free boundary for the Laplace and heat equations, J. Differential Equations, 137 (1997), pp. 119-165.
6. A. Friedman, B. Hu, and Y. Liu, A boundary value problem for the Poisson equation with multi-scale oscillating boundary, J. Differential Equations, 137 (1997), pp. 54-93.
7. M. K. Gobbert, T. S. Cale, and C. A. Ringhofer, One approach to combining equipment scale and feature scale models, in Process Control, Diagnostics, and Modeling in Semiconductor Manufacturing, M. Meyyappan, D. J. Economou, and S. W. Butler, eds., The Electrochemical Society Proceedings Series, vol. 95-4, 1995, pp. 553-563.
8. M. K. Gobbert, T. S. Cale, and C. A. Ringhofer, The combination of equipment scale and feature scale models for chemical vapor deposition via a homogenization technique, VLSI Des., 6 (1998), pp. 399-403.
9. M. K. Gobbert, T. P. Merchant, L. J. Borucki, and T. S. Cale, A multiscale simulator for low pressure chemical vapor deposition, J. Electrochem. Soc., 144 (1997), pp. 3945-3951.
10. M. K. Gobbert and C. Ringhofer, Mesoscopic scale modeling for chemical vapor deposition in semiconductor manufacturing, in Dispersive Transport Equations and Multiscale Models, N. B. Abdallah, A. Arnold, P. Degond, I. Gamba, R. Glassey, C. D. Levermore, and C. Ringhofer, eds., IMA Volumes in Mathematics and its Applications 136, Springer-Verlag, New York, 2003, pp. 133-150.
11. M. K. Gobbert and C. A. Ringhofer, An asymptotic analysis for a model of chemical vapor deposition on a microstructured surface, SIAM J. Appl. Math., 58 (1998), pp. 737-752.
12. M. K. Gobbert, C. A. Ringhofer, and T. S. Cale, Mesoscopic scale modeling of microloading during low pressure chemical vapor deposition, J. Electrochem. Soc., 143 (1996), pp. 2624-2631.
13. M. K. Gobbert, S. G. Webster, J.-F. Remacle, and T. S. Cale, A Spectral Galerkin Ansatz for the Deterministic Solution of the Boltzmann Equation on Irregular Domains, Technical report, University of Maryland, Baltimore County, Baltimore, MD, 2002.
14. H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), pp. 331-407.
15. A. Kersch and W. J. Morokoff, Transport Simulation in Microelectronics, Progress in Numerical Simulation for Microelectronics, 3, Birkhäuser Verlag, Basel, 1995.
16. A. Klar, A numerical method for kinetic semiconductor equations in the drift-diffusion limit, SIAM J. Sci. Comput., 20 (1999), pp. 1696-1712.
17. C. R. Kleijn and C. Werner, Modeling of Chemical Vapor Deposition of Tungsten Films, Progress in Numerical Simulation for Microelectronics, 2, Birkhäuser Verlag, Basel, 1993.
18. T. P. Merchant, M. K. Gobbert, T. S. Cale, and L. J. Borucki, Multiple scale integrated modeling of deposition processes, Thin Solid Films, 365 (2000), pp. 368-375.
19. C. Ringhofer, Computational methods for semiclassical and quantum transport in semiconductor devices, Acta Numer., 6 (1997), pp. 485-521.
20. C. Ringhofer, Space-time discretization of series expansion methods for the Boltzmann transport equation, SIAM J. Numer. Anal., 38 (2000), pp. 442-465.
21. C. Ringhofer, C. Schmeiser, and A. Zwirchmayr, Moment methods for the semiconductor Boltzmann equation on bounded position domains, SIAM J. Numer. Anal., 39 (2001), pp. 1078-1095.
22. C. Schmeiser and A. Zwirchmayr, Convergence of moment methods for linear kinetic equations, SIAM J. Numer. Anal., 36 (1998), pp. 74-88.
23. S. G. Webster, M. K. Gobbert, J.-F. Remacle, and T. S. Cale, Parallel numerical solution of the Boltzmann equation for atomic layer deposition, in Euro-Par 2002 Parallel Processing, B. Monien and R. Feldmann, eds., Lecture Notes in Comput. Sci. 2400, 2002, pp. 452-456.