Skip to main content

Percolation and fractal properties of thin gold films

Conference Lectures

Part of the Lecture Notes in Mathematics book series (LNM,volume 1035)

Abstract

Transmission electron micrographs of thin evaporated gold films with thickness varying from 6 to 10 nm were analyzed by computer. The films cover the range from electrically insulating to conducting and thus span the 2D percolation threshold. The computer analysis allows the direct comparison of actual geometric cluster statistics with both the scaling theory of percolation and Mandelbrot's fractal geometry. We find that Au-Au and Au-substrate interactions set a natural correlation length of order 10nm. Small clusters are dominated by these effects and have simple almost-circular shapes. At larger scales, however, the irregular connected clusters are ramified with a perimeter linearly proportional to area. Near the percolation threshold the large scale power-law correlations and area distributions are consistent with the scaling theory of 2nd order phase transitions. In the fractal interpretation, we demonstrate that the boundary of all clusters is a fractal of dimension D=2 while the largest cluster boundary has a fractal dimension Dc≈1.9. Moreover, many of the usual analytic scaling relations between universal exponents are shown to have fractal geometric basis.

Keywords

  • Fractal Dimension
  • Large Cluster
  • Pair Correlation Function
  • Percolation Cluster
  • Cluster Boundary

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. See for example B. A. Abeles in Applied Solid State Science, edited by R. Wolfe (Academic, New York, 1976) Vol. 6, p. 1; B. A. Abeles, H. L. Pinsh, and J. I. Gittleman, Phys. Rev. Lett. 35, 247 (1976); or C. J. Lobb, M. Tinkham, and W. J. Skocpol, Solid State Comm. 27, 1253 (1978).

    Google Scholar 

  2. See the excellent review by D. Stauffer, Phys. Reports 54, 1, (1979) and references therein.

    CrossRef  ADS  Google Scholar 

  3. L. P. Kadanoff et al., Rev. Mod. Phys. 39, 395 (1967).

    CrossRef  ADS  Google Scholar 

  4. S. Kirkpatrick, A.I.P. Conf. Proc. 50, 99 (1977) and A.I.P Conf. Proc. 58, 79 (1979).

    Google Scholar 

  5. H. E. Stanley J. Phys. A 10, L211 (1977).

    Google Scholar 

  6. P. L. Leath, Phys. Rev. B14, 5046 (1976).

    CrossRef  ADS  Google Scholar 

  7. R. J. Harrison, G. H. Bishop, and G. D. Quinn, J. Stat. Phys. 19, 53 (1978).

    CrossRef  ADS  Google Scholar 

  8. J. W. Halley and T. Mai, Phys. Rev. Lett. 43, 740 (1979).

    CrossRef  ADS  Google Scholar 

  9. R. B. Laibowitz, E. I. Allessandrini, and G. Deutscher, Phys. Rev. B25, 2965 (1982).

    CrossRef  ADS  Google Scholar 

  10. R. F. Voss, R. B. Laibowitz, and E. I. Alessandrini, Phys. Rev. Lett 49, 1141 (1982).

    CrossRef  ADS  Google Scholar 

  11. A. Kapiltunik and G. Deutscher, Phys. Rev. Lett. 49, 1444 (1982).

    CrossRef  ADS  Google Scholar 

  12. R. B. Laibowitz and A. N. Broers, in Treatise on Materials Science and Technology, (Academic Press, New York, 1982), Vol. 24 p. 237.

    Google Scholar 

  13. The same metal-insulator assymetry is seen in the Pb films on Ge substrates [11].

    Google Scholar 

  14. For a general discussion of fractals see B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco 1982) and references therein. Chapter 13 deals specifically with percolation.

    MATH  Google Scholar 

  15. Y. Gefen, A. Aharony, B. B. Mandelbrot, and S. Kirkpatrick, Phys. Rev. Lett. 47, 1771 (1981).

    CrossRef  ADS  MathSciNet  Google Scholar 

  16. S. Lovejoy, Science 216, 185 (1982).

    CrossRef  ADS  Google Scholar 

  17. The multiple possibilities for "defining" a dimension are extensively discussed in both refs. 2 and 14.

    Google Scholar 

  18. H. Kunz and B. Souillard, J. Stat. Phys. 19, 77 (1978) and A. Coniglio and L. Russo, J. Phys. A 12, 545 (1979).

    CrossRef  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1983 Springer-Verlag

About this paper

Cite this paper

Voss, R.F., Laibowitz, R.B., Alessandrini, E.I. (1983). Percolation and fractal properties of thin gold films. In: Hughes, B.D., Ninham, B.W. (eds) The Mathematics and Physics of Disordered Media: Percolation, Random Walk, Modeling, and Simulation. Lecture Notes in Mathematics, vol 1035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073258

Download citation

  • DOI: https://doi.org/10.1007/BFb0073258

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12707-9

  • Online ISBN: 978-3-540-38693-3

  • eBook Packages: Springer Book Archive