**9. Complementary analysis: Fractal dimension and power spectral density**

Two powerful techniques for further analysis in the study of surfaces are the fractal dimension and Power Spectral Density. These analysis are based on surface roughness. This

Measurement of the Nanoscale Roughness by

its shape under a scale transformation (Assis et al., 2008).

**9.1.1 Self-similar fractals** 

triangle ABC (Assis et al, 2008).

surfaces are classified as self-affine fractals.

a semi-full scale, and this fact characterizes a fractal.

**9.1.3 The concept of dimension** 

**9.1.2 Self-affine fractals** 

Atomic Force Microscopy: Basic Principles and Applications 167

Self-similar fractals are figures that are completely invariant under scale transformations. An example of self-similar fractal is the Sierpinski Triangle (Figure 22) that does not change

Fig. 22. The Sierpinski Triangle. The Triangle ADE is an exact copy in miniature of triangle ABC, depicting a self-similar fractal object. Triangles CDF and BEF are similarly related to

Self-affine fractals are a generalization of self-similar fractals. Self-affine fractal objects are composed of mini-copies of the original figure, but as the scale varies, the proportions are not maintained. Fractals are self-affine fractal invariant under anisotropic transformations. The surfaces of ultrathin films are often treated as self-affine fractal, since during the growth of the films there are two preferred directions of growth (Vicsek, 1989). Likewise most of the

The Euclidean dimension (D), popularly used, is a parameter that defines the geometry of an object. The Euclidean dimension is a parameter in the set of natural numbers in the interval [0, 3]. An object with D = 1, is associated with only one dimension, for example, a line. The dimension D = 2 describes plans objects and dimension D = 3 defines three-dimensional

Not all objects are treated in the field of Euclidean geometry. For many of them can be given

objects. Intuitively, D = 0 describes zero-dimensional objects, as a point, for example.

type of analysis requires more sophisticated equipment. Most SPMs have image analysis options in their software for additional analysis such as fractal dimension and power spectral density. In this section the analysis will be discussed.

#### **9.1 Fractal dimension**

The fractal dimension is a sophisticated parameter used to define the morphology of a surface, considering the roughness present. The surface morphology can be characterized qualitatively by its roughness and its fractal dimension (Guisbiers et al., 2007; Raoufi, 2010; Torkhova & Novikov, 2009; Yadav et al., 2011). The idea of using the concepts of fractal geometry in the study of geometric figures and irregular forms was popularized by Benoit B. Mandelbrot (Mandelbrot, 1982). Since then, such concepts have been used in various fields such as physics, chemistry, biology, materials science, among others.

A fractal is defined by the property of self-similarity or self-affinity, that is, they have the same characteristics for different variations in scale. The thumbnail is like the fractal as a whole and can be classified into self-similar (in the case parts of the fractal is identical to the original fractal) or self-affine (when parts of the fractal is statistically similar to the original). Often fractals are found in nature, for example, when we see the outline of a cloud, the forms produced by lightning, snowflakes, the shape of a cauliflower and especially the morphology of surfaces appear as fractal objects (Figure 21).

Fig. 21. Fractals in nature. (a) Outline of a cloud, (b) lightning, (c) snowflake, (d) surface of a cauliflower, (e) surface of a thin film of nickel oxide obtained AFM (Popsci, 2011; Saint-Marty Marty, 2011; Chaos Theory Dance, 2011).
