Continuum Models for Surface Growth

  • Martin Rost
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 149)


As an introductory lecture to the workshop an overview is given over continuum models for homoepitaxial surface growth using partial differential equations (PDEs). Their heuristic derivation makes use of inherent symmetries in the physical process (mass conservation, crystal symmetry, ... ) which determines their structure. Two examples of applications are given, one for large scale properties, one including crystal lattice discreteness. These are: (i) a simplified model for mound coarsening and (ii) for the transition from layer-by-layer to rough growth. Virtues and shortcomings of this approach is discussed in a concluding section.


Surface evolution mound coarsening roughening transition 


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  1. [1]
    R.V. Kohn and F. Otto, Upper bounds on coarsening rates, Comm. Math. Phys. 229 (2002) 375–395.CrossRefGoogle Scholar
  2. [2]
    R.V. Kohn, X.D. Yan, Upper bound on the coarsening rate for an epitaxial growth model, Comm. Pure Appl. Math. 56 (2003) 1549–1564.CrossRefGoogle Scholar
  3. [3]
    B. Derrida, E. Domany, and D. Mukamel, An exact solution of the one dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69 (1992) 667–687; G. Schütz and E. Domany, Phase transitions in an exactly soluble one-dimensional asymmetric exclusion model, J. Stat. Phys. 72 (1993) 277–296.CrossRefGoogle Scholar
  4. [4]
    M. Biehl, this volume.Google Scholar
  5. [5]
    J. Krug, this volume.Google Scholar
  6. [6]
    P. Politi and J. Villain, Ehrlich-Schwoebel instability in molecular-beam epitaxy: A minimal model, Phys. Rev. B 54 (1996) 5114–5129.CrossRefGoogle Scholar
  7. [7]
    G. Ehrlich and F.G. Hudda, Atomic view of surface diffusion: Tungsten on Tungsten, J. Chem. Phys. 44 (1966) 1036–1099.Google Scholar
  8. [8]
    L. Schwoebel and E.J. Shipsey, Step motion on crystal surfaces, J. Appl. Phys. 37 (1966) 3682–3686.CrossRefGoogle Scholar
  9. [9]
    M. Siegert, Ordering dynamics of surfaces in molecular beam epitaxy, Physica A 239 (1997) 420–427.Google Scholar
  10. [10]
    W.W. Mullins, Flattening of a Nearly Plane Solid Surface Due to Capillarity, J. Appl. Phys. 30 (1959) 77–83.CrossRefGoogle Scholar
  11. [11]
    J. Villain, Continuum models of crystal growth form atomistic beams with and without desorption, J. de Physique I 1 (1991) 19–42.CrossRefGoogle Scholar
  12. [12]
    O. Pierre-Louis, M.R. D’Orsogna, and T.L. Einstein, Edge diffusion during growth: The kink Ehrlich-Schwoebel effect and resulting instabilities, Phys. Rev. Lett. 82 (1999) 3661–3664.CrossRefGoogle Scholar
  13. [13]
    J. Kallunki, J. Krug, M. Kotrla, Competing mechanisms for step meandering in unstable growth, Phys. Rev. B 65 (2002) 205411.CrossRefGoogle Scholar
  14. [14]
    P. Politi and J. Krug, Crystal symmetry, step-edge diffusion, and unstable growth, Surface Science 446 (2000) 89–97.CrossRefGoogle Scholar
  15. [15]
    P. Politi and J. Villain, Kinetic coefficients in a system far from equilibrium, in Surface Diffusion: atomistic and collective processes, Ed. M.C. Tringides, Plenum Press, New York (1997) 177–189.Google Scholar
  16. [16]
    S. van Dijken, L.C. Jorritsma, and B. Poelsema, Steering-Enhanced Roughening during Metal Deposition at Grazing Incidence, Phys. Rev. Lett. 82 (1999) 4038–4041.CrossRefGoogle Scholar
  17. [17]
    J. Yu, J.G. Amar, and A. Bogicevic, First-principles calculations of steering forces in epitaxial growth, Phys. Rev. B 69 (2004) 113406.CrossRefGoogle Scholar
  18. [18]
    W.K. Burton, N. Cabrera, F.C. Frank, The growth of crystals and the equilibrium of their surfaces, Phil. Trans. Roy. Soc. London A 243 (1951) 299–358.Google Scholar
  19. [19]
    P. Šmilauer, M. Rost, and J. Krug, Fast coarsening in unstable epitaxy with desorption, Phys. Rev. E 59 (1999) R6263–R6266.CrossRefGoogle Scholar
  20. [20]
    M. Rost (2004), unpublished.Google Scholar
  21. [21]
    D.E. Wolf, Computer simulation of molecular-beam epitaxy, in Scale Invariance, Interfaces and Non-Equilibrium Dynamics, Eds. A.J. McKane, M. Droz, J. Vannimenus, and D.E. Wolf, Plenum Press, New York (1995).Google Scholar
  22. [22]
    E. Somfai, D.E. Wolf, and J. Kertész, Correlated island nucleation in layer-by-layer growth, J. de Physique I 6 (1996) 393–401.CrossRefGoogle Scholar
  23. [23]
    M. Siegert, Coarsening dynamics of crystalline thin films, Phys. Rev. Lett. 81 (1998) 5481–5484.CrossRefGoogle Scholar
  24. [24]
    D. Moldovan and L. Golubovic, Interfacial coarsening dynamics in epitaxial growth with slope selection, Phys. Rev. E 61 (2000) 6190–6214.CrossRefGoogle Scholar
  25. [25]
    P. Politi, G. Grenet, A. Marty, A. Ponchet, and J. Villain, Instabilities in crystal growth by atomic or molecular beams, Physics Reports 324 (2000) 271–404.CrossRefGoogle Scholar
  26. [26]
    M. Rost and J. Krug, Coarsening of surface structures in unstable epitaxial growth, Phys. Rev. E 55 (1997) 3952–3957.CrossRefGoogle Scholar
  27. [27]
    P. Politi, Kink dynamics in a one-dimensional growing surface, Phys. Rev. E 58 (1998) 281–294CrossRefGoogle Scholar
  28. [28]
    L.H. Tang, P. Šmilauer, and D.D. Vvedensky, Noise-assisted mound coarsening in epitaxial growth, Eur. J. Phys#x1E02;2 (1998) 409–412.Google Scholar
  29. [29]
    H. Kallabis, L. Brendel, J. Krug, J., and D.E. Wolf, Damping of oscillations in layer-by-layer growth, Int. J. Mod. Phys. B 31 (1997) 3621–3634.Google Scholar
  30. [30]
    M. Rost and J. Krug, Damping of growth oscillations in molecular beam epitaxy: A renormalization group approach, J. de Physique I 7 (1997) 1627–1638.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Martin Rost
    • 1
  1. 1.Bereich Theoretische Biologie, Insitut für Zelluläre und Molekulare BotanikUniversität BonnBonnGermany

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