Numerical Simulation of Thermal Oxidation Process in Semiconductor Devices Using Mixed—Hybrid Finite Elements

  • Riccardo Sacco
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 509)


In this lecture, we present the basic mathematical and numerical tools for the simulation of the thermal oxidation process in semiconductor device technology. The mathematical model is first reviewed, emphasizing that it requires at each time level the solution of a diffusion-reaction problem and a fluid-structure interaction problem. Then, mixed-hybrid finite elements are introduced for the numerical approximation of each differential subproblem, with a detailed discussion of the static condensation procedure to eliminate the mixed variables in favor of the hybrid Lagrange multipliers. Several numerical examples are included to validate the accuracy and stability of the proposed computational procedure.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. Arnold, F. Brezzi, Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates, Math. Modeling and Numer. Anal. 19-1 (1985) 7–32.MathSciNetGoogle Scholar
  2. [2]
    F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Verlag, New York, 1991.zbMATHGoogle Scholar
  3. [3]
    P. Causin, Mixed-Hybrid Galerkin and Petrov-Galerkin Finite Element Formulations in Fluid Mechanics, Ph. D. thesis, Università degli Studi di Milano, web-site: (2002).Google Scholar
  4. [4]
    C. Carstensen, P. Causin, R. Sacco, A Posteriori Dual-Mixed (Hybrid) Adaptive Finite Element Error Control for Lamè and Stokes Equations, Numer. Math. (101), 309–332 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    P. Causin, R. Sacco, A dual-mixed hybrid formulation for fluid mechanics problems: mathematical analysis and application to semiconductor process technology, Comp. Meth. Appl. Mech. Engrng. (192), 593–612 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    P. Causin, M. Restelli, R. Sacco, A simulation system based on mixed-hybrid finite elements for thermal oxidation in semiconductor technology, Comput. Methods Appl. Mech. Engrg. (193), 3687–3710 (2004).zbMATHCrossRefGoogle Scholar
  7. [7]
    B. Cockburn, J. Gopalakhrisnan, A characterization of hybridized mixed methods for second order elliptic problems, SIAM J. Numer. Anal. (42), 283–301 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    B. Cockburn, J Gopalakhrisnan, Error analysis of variable degree mixed methods for elliptic problems via hybridization, Math. Comp. (74), 1653–1677 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    M. Farhloul, M. Fortin, Dual hybrid methods for the elasticity and the Stokes problems: a unified approach, Numer. Math. 76-4 (1997) 419–440.CrossRefMathSciNetGoogle Scholar
  10. [10]
    S.E. Hansen, M.D. Deal, SUPREM-IV.GS User’s Reference Manual, Stanford University, 1994.Google Scholar
  11. [11]
    L. Herrmann, Elasticity equations for incompressible and nearly—incompressible materials by a variational theorem, AIAA Jnl. 3–10 (1965) 1896–1900.Google Scholar
  12. [12]
    L. Herrmann, R. Taylor, K. Pister, On a variational theorem for incompressible and nearly-incompressible orthotropic elasticity, Int. J. Solids Structures 4 (1968) 875–883.zbMATHCrossRefGoogle Scholar
  13. [13]
    T.J.R. Hughes, The Finite Element Method—Linear static and dynamic finite element analysis (Cap. 4), Prentice-Hall (1987).Google Scholar
  14. [14]
    J.P. Peng, D. Chidambarrao, G.r. Srinavasan, NOVEL. A nonlinear viscoelastic model forar thermal oxidation of silicon, COMPEL, 10(4) (1991) 341–353.Google Scholar
  15. [15]
    C. Rafferty, Stress effects in silicon oxidation—simulation and experiment, Ph.D. thesis, Stanford University (1990).Google Scholar
  16. [16]
    V. Rao, T.J.R. Hughes, K. Garikipati, On modeling thermal oxidation in silicon. Part i-ii, Int. J. Num. Meth. Engr. 47 (2000) 341–377.zbMATHCrossRefGoogle Scholar
  17. [17]
    P. Raviart, J. Thomas, Primal hybrid finite element methods for 2nd order elliptic equations, Math. Comp. 31–138 (1977) 391–413.CrossRefMathSciNetGoogle Scholar
  18. [18]
    P. Riviart, J. Thomas, A mixed finite element method for second order elliptic problems, in: I. Galligani, E. Magenes (Eds.), Mathematical Aspects of Finite Element Methods, I, Springer-Verlag Berlin, 1977.Google Scholar
  19. [19]
    J. Roberts, J. Thomas, Mixed and hybrid methods, in: P. Ciarlet, J. Lions (Eds.), Finite Element Methods, Part I, North-Holland Amsterdam, 1991, vol. 2.Google Scholar

Copyright information

© CISM, Udine 2009

Authors and Affiliations

  • Riccardo Sacco
    • 1
  1. 1.Dipartimento di Matematica “F. Brioschi”Politecnico di MilanoItaly

Personalised recommendations