Hostname: page-component-f554764f5-wjqwx Total loading time: 0 Render date: 2025-04-11T17:34:18.562Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantitative topographic analysis of fractal surfaces by scanning tunneling microscopy

Published online by Cambridge University Press:  31 January 2011

Morgan W. Mitchell  and
Dawn A. Bonnell
Morgan W. Mitchell
Affiliation:
The University of Pennsylvania, Department of Materials Science and Engineering, Philadelphia, Pennsylvania 19104
Dawn A. Bonnell
Affiliation:
The University of Pennsylvania, Department of Materials Science and Engineering, Philadelphia, Pennsylvania 19104
Get access

Abstract

The applicability of models based on fractal geometry to length scales of nanometers is confirmed by Fourier analysis of scanning tunneling microscopy images of a sputter deposited gold film, a copper fatigue fracture surface, and a single crystal silicon fracture surface. Surfaces are characterized in terms of fractal geometry with a Fourier profile analysis, the calculations yielding fractal dimensions with high precision. Fractal models are shown to apply at length scales to 12 Å, at which point the STM tip geometry influences the information. Directionality and spatial variation of the topographic structures are measured. For the directions investigated, the gold and silicon appeared isotropic, while the copper fracture surface exhibited large differences in structure. The influences of noise in the images and of intrinsic mathematical scatter in the calculations are tested with profiles generated from fractal Brownian motion and the Weierstrass-Mandelbrot function. Accurate estimates of the fractal dimension of surfaces from STM data result only when images consist of at least 1000–2000 points per line and 1/f-type noise has amplitudes two orders of magnitude lower than the image signal. Analysis of computer generated ideal profiles from the Weierstrass-Mandelbrot function and fractional Brownian motion also illustrates that the Fourier analysis is useful only in determining the local fractal dimension. This requirement of high spatial resolution (vertical information density) is met by STM data. The fact that fractal models can be used at lengths as small as nanometers implies that continued topographic structural analyses may be used to study atomistic processes such as those occurring during fracture of elastic solids.

Type
Articles
Information
Journal of Materials Research , Volume 5 , Issue 10 , October 1990 , pp. 2244 - 2254
Copyright
Copyright © Materials Research Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Pfeifer, P., Appl. of Surf. Sci. 18, 146–164 (1984).CrossRefGoogle Scholar
2Mandelbrot, B., Fractal Geometry of Nature (W. H. reeman & Co., San Francisco, CA, 1982).Google Scholar
3Feder, J., Fractals (Plenum Press, New York, 1988).CrossRefGoogle Scholar
4Mecholsky, J. J., Mackin, T., and Passoja, D. E., Adv. in Cer. Vol. 22: Fractography of Glasses and Ceramics (American Ceramic Society, Inc., 1988), pp. 127–134.Google Scholar
5Passoja, D. E., Adv. in Cer. Vol. 22: Fractography of Glasses and Ceramics (American Ceramic Society, Inc., 1988), pp. 127–134.Google Scholar
6Passoja, D. E. and Amborski, D. J., “Fracture Profile Analysis by Fourier Transform Methods”.Google Scholar
7Mandelbrot, B., Passoja, D., and Paullay, A., Nature 308, 721–722 (1984).CrossRefGoogle Scholar
8Underwood, E. and Banerji, K., Mat. Sci. and Eng. 80, 1–14 (1986).CrossRefGoogle Scholar
9Binnig, G. and Rohrer, H., Helv. Phys. Acta 55, 726–729 (1982).Google Scholar
10 See, for example, a number of papers in Proc. Int. Conf. on STM 1987, edited by R. Feenstra, J. Vac. Sci. and Technol., 257–556 (1988).Google Scholar
11Bonnell, D. A. and Clarke, D. R., J. Am. Ceram. Soc. 71 629–637 (1988).CrossRefGoogle Scholar
12Bonnell, D. A., Mat. Sci. and Eng. A105/106, 55–63 (1988).CrossRefGoogle Scholar
13Bonnell, D. A. and Angelopoulos, M., Synth. Met. 33, 301–310 (1989).CrossRefGoogle Scholar
14Voss, R., in Fundamental Algorithms for Computer Graphics, edited by Earnshaw, R. A. (Springer-Verlag, New York, 1985), pp. 805–835.CrossRefGoogle Scholar
15Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vettering, W. T., Numerical Recipes (Cambridge University Press, Cambridge, U. K., 1986), pp. 381–453.Google Scholar
16Berry, M. V. and Lewis, Z. V., Proc. R. Soc. London 370, 459–484 (1980).Google Scholar
17Coster, M. and Chermat, J. L., Int. Met. Rev. 28, 228–250 (1983).CrossRefGoogle Scholar
18Dubuc, B., Quiniou, J. F., Rouques-Carmes, C., Tricot, C., and Zucker, S. W., Phys. Rev. A 39, 1500–1512 (1989).CrossRefGoogle Scholar
19Stoll, E. and Marti, O., Surf. Sci. 181, 222–229 (1987).CrossRefGoogle Scholar
20Meakin, P., Ramanlal, R., Sander, L. M., and Ball, R. C., Phys. Rev. A 34, 5091–5103 (1986).CrossRefGoogle Scholar
21Family, F. and Vicsek, T., J. Phys. A 18, L75 (1984).CrossRefGoogle Scholar
22Kardar, M., Parisi, G., and Zhang, Y. C., Phys. Rev. Lett. 56, 889 (1986).CrossRefGoogle Scholar
23Meakin, P. and Jullien, R., Europhys. Lett. 9, 71–76 (1989).CrossRefGoogle Scholar
24Williford, R. E., Scripta Metall. 22, 149–154 (1988).Google Scholar