Advertisement

Edge Diffusion in Phase-Field Models for Epitaxial Growth

  • Andreas Rätz
  • Axel Voigt
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 149)

Abstract

A phase-field model is proposed to describe step-flow in epitaxial growth. In this model the motion of steps or island boundaries of discrete atomic layers on an epitaxial growing film is determined by the time evolution of an introduced phase-field variable. We use formally matched asymptotic expansion to determine the asymptotic limit of vanishing interfacial thickness and show the reduction to classical sharp interface models of Burton-Cabrera-Frank type with edge diffusion.

Keywords

step-flow model phase-field approximation edge diffusion 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. Bänsch, F. Haußer and A. Voigt, Finite element method for epitaxial growth with thermodynamic boundary conditions. SIAM J. Sci. Comput. (2005), to appear.Google Scholar
  2. [2]
    F. Haußer and A. Voigt, (this volume).Google Scholar
  3. [3]
    W.K. Burton, N. Cabrera and F.C. Frank, The growth of crystals and the equilibrium structure of their surfaces. Phil. Trans. Roy. Soc. London Ser. A 243 (1951), 299–358.Google Scholar
  4. [4]
    J.W. Cahn, C.M. Elliott and A. Novick-Cohen, The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. Euro. J. Appl. Math. 7 (1996), 287–301.Google Scholar
  5. [5]
    C.M. Elliott, H. Garcke, Existence results for diffusive surface motion laws. Adv. Math. Sci. Appl. 7 (1997), 465–488.Google Scholar
  6. [6]
    P.C. Fife and O. Penrose, Interfacial dynamics for thermodynamically consistent phase-field models with nonconserved order parameter. Electron. J. Differential Equations 16 (1995), 1–49.Google Scholar
  7. [7]
    A. Karma and M. Plapp, Spiral surface growth without desorption. Phys. Rev. Lett. 81 (1998), 4444–4447CrossRefGoogle Scholar
  8. [8]
    J. Krug, (this volume).Google Scholar
  9. [9]
    J. Krug, Four lectures on the physics of crystal growth. Physica A 318 (2002), 47–82.Google Scholar
  10. [10]
    F. Liu and H. Metiu, Stability and kinetics of step motion on crystal surfaces. Phys. Rev. E 49 (1994), 2601–2616.CrossRefGoogle Scholar
  11. [11]
    O. Pierre-Louis, Phase field models for step flow. Phys. Rev. E 68 (2003), 021604.CrossRefGoogle Scholar
  12. [12]
    A. Räatz and A. Voigt Phase-field model for island dynamics in epitaxial growth. Appl. Anal. 83 (2004), 1015–1025.CrossRefGoogle Scholar
  13. [13]
    C. Ratsch, M.F. Guyre, R.E. Caflisch, et al., Level-set method for island dynamics in epitaxial growth. Phys. Rev. B 65 (2002), 195403.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Andreas Rätz
    • Axel Voigt
      • 1
    1. 1.Crystal Growth groupresearch center caesarBonnGermany

    Personalised recommendations