Discretisation and Numerical Tests of a Diffuse-Interface Model with Ehrlich-Schwoebel Barrier

  • Felix Otto
  • Patrick Penzler
  • Tobias Rump
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 149)


We consider a step flow model for epitaxial growth, as proposed by Burton, Cabrera and Frank [3]. This type of model is discrete in the growth direction but continuous in the lateral directions. The effect of the Ehrlich-Schwoebel barrier, which limits the attachment rate of adatoms to a step from an upper terrace, is included. Mathematically, this model is a dynamic free boundary problem for the steps. In [6], we proposed a diffuse-interface approximation which reproduces an arbitrary Ehrlich-Schwoebel barrier. It is a version of the Cahn-Hilliard equation with variable mobility.

In this paper, we propose a discretisation for this diffuse-interface approximation. Our approach is guided by the fact that the diffuse-interface approximation has a conserved quantity and a Liapunov functional. We are lead to an implicit finite volume discretisation of symmetric structure.

We test the discretisation by comparison with the matched asymptotic analysis carried out in [6]. We also test the diffuse-interface approximation itself by comparison with theoretically known features of the original free boundary problem. More precisely, we investigate quantitatively the phenomena of step bunching and the Bales-Zangwill instability.


epitaxial growth Ehrlich-Schwoebel barrier phase-field model 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. Bales and A. Zangwill. Morphological instability of a terrace edge during step-flow growth. Phys. Rev. B 41(9) (1990), 5500–5508.CrossRefGoogle Scholar
  2. [2]
    M.O. Bristeau, R. Glowinski, and J. Périaux. Numerical methods for the Navier-Stokes equations. Applications to the simulation of compressible and incompressible viscous flows. In Finite elements in physics (Lausanne, 1986), pages 73–187. North-Holland, Amsterdam, 1987.Google Scholar
  3. [3]
    W.K. Burton, N. Cabrera, and F.C. Frank. The growth of crystals and the equilibrium structure of their surfaces. Philos. Trans. Roy. Soc. London. Ser. A. 243 (1951), 299–358.Google Scholar
  4. [4]
    G. Ehrlich and F.G. Hudda. Atomic view of surface diffusion: tungsten on tungsten. J. Chem. Phys. 44 (1966), 1036–1099.Google Scholar
  5. [5]
    G. Grün and M. Rumpf. Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87(1) (2000), 113–152.CrossRefGoogle Scholar
  6. [6]
    F. Otto, P. Penzler, A. Rätz, T. Rump, and A. Voigt. A diffuse-interface approximation for step flow in epitaxial growth. Nonlin. 17(2) (2004), 477–491CrossRefGoogle Scholar
  7. [7]
    F. Otto, P. Penzler, T. Rump. Discretisation and numerical test of a diffuse-interface model with Ehrlich-Schwoebel barrier. SFB preprint, 2004.Google Scholar
  8. [8]
    P.-A. Raviart and J.M. Thomas. A mixed finite element method for 2nd order elliptic problems. In Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), pages 292–315. Lecture Notes in Math., Vol. 606. Springer, Berlin, 1977.Google Scholar
  9. [9]
    R.L. Schwoebel and E.J. Shipsey. Step motion on crystal surfaces. J. Appl. Phys. 37 (1966), 3682–3686.CrossRefGoogle Scholar
  10. [10]
    U. Weikard. Numerische Lösungen der Cahn-Hilliard-Gleichung und der Cahn-Larché-Gleichung. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Oct. 2002.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Felix Otto
    • Patrick Penzler
      • Tobias Rump
        • 1
      1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany

      Personalised recommendations