Quantitative topographic analysis of fractal surfaces by scanning tunneling microscopy
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Philadelphia,
(Received 8 January 1990; accepted 5 July 1990)
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 A, 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 l//-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. I. INTRODUCTION
Quantification of the topographic structure of surfaces is critical to the study of a number of phenomena including adsorption on catalytic surfaces, fracture, aggregation of colloids, and vapor deposition.1 The study of the morphologies of surfaces involved in, or resulting from, these processes can lead to a better understanding of the processes themselves, perhaps even at atomistic levels. Regardless of the characterization technique, topographic analysis often involves concepts of fractal geometry which are increasingly utilized to describe the morphologies of surfaces, both in mathematical modeling of the processes and in experimental characterization of surface structures.2 Fractal descriptions characterize a deviation of the surface from Euclidian geometry by relating parameters, mass, and length, for example, by a noninteger exponent called the fractal dimension. Different dimensions are calculated from different measurement procedures. Typical dimensions are the box dimension, the similarity dimension, the Hausdorff dimension, and the spectral density.3 In some cases these dimensions are equal or can be related through simple expressions. Most surfaces are not strictly fractal in that they exhibit upper
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