Scale Invariance in Epitaxial Growth

  • D. D. Vvedensky
  • A. Zangwill
  • C. N. Luse
  • C. Ratsch
  • P. Šmilauer
  • M. R. Wilby
Part of the Institute for Nonlinear Science book series (INLS)


We examine three manifestations of scale invariance during epitaxial growth from the standpoint of a single atomistic model. The first two manifestations are in the submonolayer regime of growth, where clusters are formed but have not yet begun to coalesce. Depending on the material deposited and the substrate, the clusters can exhibit either an effective fractal dimension or a Euclidean dimension. The cluster size distribution function in this regime also exhibits scaling as a function of time. Interestingly, this distribution function and its scaling properties can be accurately described in terms of rate equations that omit entirely the influence of any fluctuations. An altogether different picture emerges in the regime of long deposition times. To analyze the scaling properties of the growing surface in this limit, we use exact discrete Langevin equations that are obtained directly from the rules of our simulation model. A regularization procedure is then used to convert these discrete equations into (infinite-order) continuum partial differential equations. The truncation of these continuum equations is achieved by performing a renormalization-group analysis, which leads asymptotically to a fourth-order nonlinear stochastic differential equation that is identical to that proposed independently by Wolf and Villain and by Lai and Das Sarma. Large-scale Monte Carlo simulations in d = 1, 2, and 3 substrate dimensions show that for all spatial dimensions the observed exponents correspond to those obtained from our continuum equations. This shows that vapor deposition is an example of a driven, nonequilibrium process that evolves to a nontrivial scale-invariant structure under the influence of input noise.


Epitaxial Growth Scale Invariance Shot Noise Langevin Equation Scanning Tunneling Microscopy Image 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A.Y. Cho, Thin Solid Films 100, 291 (1983).ADSCrossRefGoogle Scholar
  2. [2]
    B.A. Joyce, Rep. Prog. Phys. 48, 1595 (1985).CrossRefGoogle Scholar
  3. [3]
    J.D. Weeks and G.H. Gilmer, Adv. Chem. Phys. 40, 157 (1979).CrossRefGoogle Scholar
  4. [4]
    A. Madhukar and S.V. Ghaisas, Crit. Rev. Solid State Mater. Sci. 13, 1434 (1987).Google Scholar
  5. [5]
    T. Shitara, D.D. Vvedensky, M.R. Wilby, J. Zhang, J.H. Neave, and B.A. Joyce, Phys. Rev. B 46, 6815–6825 (1992).ADSCrossRefGoogle Scholar
  6. [6]
    C. Godrèche, ed., Solids Far From Equilibrium (Cambridge University Press, Cambridge, 1991).Google Scholar
  7. [7]
    F. Family and T. Vicsek, Dynamics of Fractal Surfaces (World Scientific, Singapore, 1991).zbMATHGoogle Scholar
  8. [8]
    E. Kopatzki, S. Günther, W. Nichtl-Pecher, and R.J. Behm, Surf. Sci. 284, 154 (1993).ADSCrossRefGoogle Scholar
  9. [9]
    R.Q. Hwang, J. Schroder, C. Gunther, and R.J. Behm, Phys. Rev. Lett. 67, 3279 (1991).ADSCrossRefGoogle Scholar
  10. [10]
    T.A. Witten and L.M. Sander, Phys. Rev. B 27, 5686 (1983).MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    A. Zangwill, Physics at Surfaces (Cambridge University Press, Cambridge, 1988).Google Scholar
  12. [12]
    V. Bortolani, N.H. March, and M.R Tosi, Interaction of Atoms and Molecules With Solid Surfaces (Plenum, New York, 1990).Google Scholar
  13. [13]
    V.P. Zhdanov, Elementary Physicochemical Processes on Solid Surfaces (Plenum, New York, 1991).Google Scholar
  14. [14]
    Z. Zhang, Y.-T. Lu, and H. Metiu, Surf. Sci. 255, L543 (1991).Google Scholar
  15. [15]
    C.-L. Liu and J.B. Adams, Surf. Sci. 265, 262 (1992).ADSCrossRefGoogle Scholar
  16. [16]
    Z. Zhang and H. Metiu, J. Chem. Phys. 93, 2087 (1990).ADSCrossRefGoogle Scholar
  17. [17]
    T. Ala-Nissila and S.C. Ying, Phys. Rev. B 42, 10264 (1990).ADSCrossRefGoogle Scholar
  18. [18]
    H.C. Kang and W.H. Weinberg, J. Chem. Phys. 90, 2824 (1989).ADSCrossRefGoogle Scholar
  19. [19]
    K. Binder, in Monte Carlo Methods in Statistical Physics, edited by K. Binder (Springer-Verlag, Berlin, 1979), pp. 1–45.Google Scholar
  20. [20]
    S. Clarke and D.D. Vvedensky, Phys. Rev. Lett. 58, 2235 (1987).ADSCrossRefGoogle Scholar
  21. [21]
    H. Yan, Phys. Rev. Lett. 68, 3048 (1992).ADSCrossRefGoogle Scholar
  22. [22]
    D.A. Kessler, H. Levine, and L.M. Sander, Phys. Rev. Lett. 69, 100 (1992).ADSCrossRefGoogle Scholar
  23. [23]
    P. Šmilauer and D.D. Vvedensky, Phys. Rev. B 48, 17603 (1993).ADSCrossRefGoogle Scholar
  24. [24]
    J.H. Neave, RJ. Dobson, B.A. Joyce, and J. Zhang, Appl. Phys. Lett. 47, 100 (1985).ADSCrossRefGoogle Scholar
  25. [25]
    J. Sudijono, M.D. Johnson, C.W. Snyder, M.B. Elowitz, and B.G. Orr, Phys. Rev. Lett. 69, 2811 (1992).ADSCrossRefGoogle Scholar
  26. [26]
    G. Zinmeister, Thin Solid Films 2, 497 (1968); 4, 363 (1969); 7, 51 (1971).ADSCrossRefGoogle Scholar
  27. [27]
    B. Lewis and D.S. Campbell, J. Vac. Sci. Technol. 4, 209 (1967).ADSCrossRefGoogle Scholar
  28. [28]
    J.A. Blackman and A. Wilding, Europhys. Lett. 16, 115 (1991).ADSCrossRefGoogle Scholar
  29. [29]
    R.L. Drake, in Topics in Current Aerosol Research, vol. 3, part 2, edited by G.M. Hidy and J.R. Brock (Pergamon, Oxford, 1972), pp. 201–376.Google Scholar
  30. [30]
    T. Vicsek and F. Family, Phys. Rev. Lett. 52, 1669 (1984).ADSCrossRefGoogle Scholar
  31. [31]
    H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York, 1971).Google Scholar
  32. [32]
    C. Ratsch, A. Zangwill, P. Šmilauer, and D.D. Vvedensky, Phys. Rev. Lett. 72, 3194 (1994).ADSCrossRefGoogle Scholar
  33. [33]
    P. Meakin, T. Viscek, and F. Family, Phys. Rev. B 31, 564 (1985).ADSCrossRefGoogle Scholar
  34. [34]
    F. Family, Physica A 168, 561 (1990).ADSCrossRefGoogle Scholar
  35. [35]
    E. Medina, T. Hwa, M. Kardar, and Y.-C. Zhang, Phys. Rev. A 39, 3053 (1989).MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    J. Villain, J. Phys. I 1, 19 (1991).CrossRefGoogle Scholar
  37. [37]
    A. Zangwill, C.N. Luse, D.D. Vvedensky, and M.R. Wilby, in Interface Dynamics and Growth, edited by K.S. Liang, M.P. Anderson, R.F. Bruinsma, and G. Scoles (Materials Research Society, Pittsburgh, 1992), pp. 189–198Google Scholar
  38. [37a]
    A. Zangwill, C.N. Luse, D.D. Vvedensky, and M.R. Wilby, Surf. Sci. 274, L529 (1992).CrossRefGoogle Scholar
  39. [38]
    D.D. Vvedensky, A. Zangwill, C.N. Luse, and M.R. Wilby, Phys. Rev. E 48, 852 (1993).ADSCrossRefGoogle Scholar
  40. [39]
    N.G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).zbMATHGoogle Scholar
  41. [40]
    R.F. Fox and J. Keizer, Phys. Rev. A 43, 1709 (1991).MathSciNetADSCrossRefGoogle Scholar
  42. [41]
    T.G. Kurz, Math. Prog. Stud. 5, 67 (1976)Google Scholar
  43. [41a]
    T.G. Kurz, Stoch. Proc. Appl. 6, 223 (1978).CrossRefGoogle Scholar
  44. [42]
    P. Nozières and F. Gallet, J. Phys. (Paris) 45, 353 (1987).Google Scholar
  45. [43]
    A. Dìaz-Guilera, Phys. Rev. A 45, 8551 (1992).ADSCrossRefGoogle Scholar
  46. [44]
    D. Wolf and J. Villain, Europhys. Lett. 13, 389 (1990).ADSCrossRefGoogle Scholar
  47. [45]
    Z.-W. Lai and S. Das Sarma, Phys. Rev. Lett. 66, 2348 (1991).ADSCrossRefGoogle Scholar
  48. [46]
    C.N. Luse and A. Zangwill, Phys. Rev. B 48, 1970 (1993).ADSCrossRefGoogle Scholar
  49. [47]
    M.R. Wilby, D.D. Vvedensky, and A. Zangwill, Phys. Rev. B 46, 12896 (1992); (errata) Phys. Rev. B 47, 16068 (1993).ADSCrossRefGoogle Scholar
  50. [48]
    P.A. Maksym, Semicon. Sci. Technol. 3, 594 (1988).ADSCrossRefGoogle Scholar
  51. [49]
    D.D. Vvedensky and S. Clarke, Surf. Sci. 225, 373 (1990).ADSCrossRefGoogle Scholar
  52. [50]
    S. Das Sarma, J. Vac. Sci. Technol. B 10, 1695 (1992).Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • D. D. Vvedensky
  • A. Zangwill
  • C. N. Luse
  • C. Ratsch
  • P. Šmilauer
  • M. R. Wilby

There are no affiliations available

Personalised recommendations