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.
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