**9.1.1 Self-similar fractals**

166 Atomic Force Microscopy – Imaging, Measuring and Manipulating Surfaces at the Atomic Scale

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

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

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

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-

spectral density. In this section the analysis will be discussed.

morphology of surfaces appear as fractal objects (Figure 21).

Marty Marty, 2011; Chaos Theory Dance, 2011).

fields such as physics, chemistry, biology, materials science, among others.

**9.1 Fractal dimension** 

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

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 triangle ABC (Assis et al, 2008).

#### **9.1.2 Self-affine fractals**

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 surfaces are classified as self-affine fractals.

#### **9.1.3 The concept of dimension**

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

Not all objects are treated in the field of Euclidean geometry. For many of them can be given a semi-full scale, and this fact characterizes a fractal.

Measurement of the Nanoscale Roughness by

function will be:

squares.

Considering Δx = 1/*Ns* thus:

Atomic Force Microscopy: Basic Principles and Applications 169

Δ*hx x* ( ) ~ Δ

Besides the roughness exponent, we can determine other fractal characteristics such as the fractal dimension of self-related functions through methods such as the Box Counting method (Barabási & Stanley, 1995).The profile function h(x) shown in Figure 23 can be embedded within a dimension D = 2. To obtain the fractal dimension through Box Counting method, we can fill this space with objects of equal size to the Euclidian space. Thus, our measuring elements are elements of area with side Δx and area Δx². To find the amount of information necessary to cover the function, divide the domain of the function in *Ns* segments, each of length Δx = 1/*Ns*. Making use of equation (15) will require *Δh/Δx~*Δxα-1 to cover the function. Finally, the total number of measurement objects to cover the entire

From the definition of Fractal Dimension we have (Cruz et al., 2002):

The equation (18) for an object of dimension D becomes:

**9.1.5 Fractal dimensions using SPMs techniques** 

Δ 0

*D D <sup>f</sup>*

Equation (19) shows that the surface roughness is closely linked with the fractal dimension. Thus, by knowing the surface roughness exponent one can obtain the fractal dimension.

The techniques of scanning probe microscope allow us to find the roughness of surfaces. Figure 24 represents particles striking a one-dimensional surface. Each square represents one particle. The size of the surface in the horizontal direction is defined by an L number of

Using a mathematical tool and the Root Mean Square (RMS roughness) we can find the surface roughness (Rq) using the concepts of Laws Scale (Cruz et al., 2002; Lagally, 1990). In this case, the roughness follows the discrete growth model, shown in Figure 24 as a function

<sup>2</sup> <sup>2</sup>

<sup>1</sup> , , ( ,)

Where h(x, t) is the height of the column at x at time t. The term <hL(x, t)> is the average height of a given observation window L, whereas the brackets <…>x encompass the spatial

*q L x*

1

*x R Lt h xt h xt L*

*L*

of the time and the RMS function is rewritten as (Barabási & Stanley, 1995):

ln (<sup>Δ</sup> ) lim ln(1 /Δ ) *<sup>f</sup> <sup>x</sup> N x <sup>D</sup> <sup>x</sup>* 

(15)

<sup>α</sup> <sup>1</sup> *Nx Nx* <sup>Δ</sup> ~ .*<sup>s</sup>* <sup>Δ</sup> (16)

<sup>α</sup> <sup>2</sup> *Nx x* Δ ~ Δ (17)

(20)

(18)

(19)

#### **9.1.4 Fractal dimension to self-affine surfaces**

Figure 23 represents a two-dimensional self-affine function. This function is defined in the interval [0, 1] and may represent, for example, the profile of a mountain.

Fig. 23. Hypothetical self-affine function, h (x) represents the height h at position x, with x defined on the interval [0, 1]

Considering the object of study in this chapter, the surface has two preferred directions, one perpendicular to the surface where the roughness characterized by the height parameters, and the other along it. Since the scale is different between the two preferred directions, characterizing the object is an anisotropic transformation (Jurecka et al., 2010). To illustrate the anisotropic transformation, let's imagine that a self-affine surface, described by the function h(x) and expanded by a factor b in the surface dimension (x bx) then the dimension in the direction of growth must then be magnified by a factor *b*ߙ) leading *hbαh*). This way one can identify the similarities between the original surface and enlarged. Thus preserving the characteristics of an invariant transformation we have (Barabási & Stanley, 1995):

$$h(\mathbf{x}) \sim b^{-\alpha} h(b\mathbf{x}) \tag{14}$$

Where α, the roughness exponent, provides a quantitative measure of the imperfections of the surface morphology.

Equation (14) provides us with the information that an auto-order of this type is resized horizontally by a factor b, the vertical is just a different factor of *b*ߙ. In self-affine fractal objects, the difference between the two points vertically obeys the relation (14), generating (Barabási & Stanley, 1995):

$$
\Delta \mathbf{h}(\mathbf{x}) \sim \Delta \mathbf{x}^{\alpha} \tag{15}
$$

Besides the roughness exponent, we can determine other fractal characteristics such as the fractal dimension of self-related functions through methods such as the Box Counting method (Barabási & Stanley, 1995).The profile function h(x) shown in Figure 23 can be embedded within a dimension D = 2. To obtain the fractal dimension through Box Counting method, we can fill this space with objects of equal size to the Euclidian space. Thus, our measuring elements are elements of area with side Δx and area Δx². To find the amount of information necessary to cover the function, divide the domain of the function in *Ns* segments, each of length Δx = 1/*Ns*. Making use of equation (15) will require *Δh/Δx~*Δxα-1 to cover the function. Finally, the total number of measurement objects to cover the entire function will be:

$$N(\Delta \mathbf{x}) \sim N\_s \Delta \mathbf{x}^{a-1} \tag{16}$$

Considering Δx = 1/*Ns* thus:

168 Atomic Force Microscopy – Imaging, Measuring and Manipulating Surfaces at the Atomic Scale

Figure 23 represents a two-dimensional self-affine function. This function is defined in the

Fig. 23. Hypothetical self-affine function, h (x) represents the height h at position x, with x

Considering the object of study in this chapter, the surface has two preferred directions, one perpendicular to the surface where the roughness characterized by the height parameters, and the other along it. Since the scale is different between the two preferred directions, characterizing the object is an anisotropic transformation (Jurecka et al., 2010). To illustrate the anisotropic transformation, let's imagine that a self-affine surface, described by the function h(x) and expanded by a factor b in the surface dimension (x bx) then the dimension in the

identify the similarities between the original surface and enlarged. Thus preserving the

Where α, the roughness exponent, provides a quantitative measure of the imperfections of

Equation (14) provides us with the information that an auto-order of this type is resized

objects, the difference between the two points vertically obeys the relation (14), generating

characteristics of an invariant transformation we have (Barabási & Stanley, 1995):

horizontally by a factor b, the vertical is just a different factor of *b*

ߙ

) leading *hbαh*). This way one can

(14)

. In self-affine fractal

ߙ

interval [0, 1] and may represent, for example, the profile of a mountain.

**9.1.4 Fractal dimension to self-affine surfaces** 

defined on the interval [0, 1]

the surface morphology.

(Barabási & Stanley, 1995):

direction of growth must then be magnified by a factor *b*

() *h x b h bx* ~ ( )

$$N(\Delta \mathbf{x}) \sim \Delta \mathbf{x}^{\alpha - 2} \tag{17}$$

From the definition of Fractal Dimension we have (Cruz et al., 2002):

$$D\_f = \lim\_{\Delta x \to 0} \left( \frac{\ln N(\Delta x)}{\ln(1/\Delta x)} \right) \tag{18}$$

The equation (18) for an object of dimension D becomes:

$$D\_f = D - a \tag{19}$$

Equation (19) shows that the surface roughness is closely linked with the fractal dimension. Thus, by knowing the surface roughness exponent one can obtain the fractal dimension.

#### **9.1.5 Fractal dimensions using SPMs techniques**

The techniques of scanning probe microscope allow us to find the roughness of surfaces. Figure 24 represents particles striking a one-dimensional surface. Each square represents one particle. The size of the surface in the horizontal direction is defined by an L number of squares.

Using a mathematical tool and the Root Mean Square (RMS roughness) we can find the surface roughness (Rq) using the concepts of Laws Scale (Cruz et al., 2002; Lagally, 1990). In this case, the roughness follows the discrete growth model, shown in Figure 24 as a function of the time and the RMS function is rewritten as (Barabási & Stanley, 1995):

$$R\_q^2\left(L, t\right) \equiv \frac{1}{L} \sum\_{x=1}^{L} \langle \left[ h\left(\mathbf{x}, t\right) - \langle h\_L(\mathbf{x}, t)\rangle \right]^2 \rangle\_x \tag{20}$$

Where h(x, t) is the height of the column at x at time t. The term <hL(x, t)> is the average height of a given observation window L, whereas the brackets <…>x encompass the spatial

Fig. 24. Diagram illustrating the formation of a surface. Each square represents a particle arriving at this surface (Cruz T.G.S., 2002).

average in x, for all windows of the same size, and averaged over all windows chosen. Using the concepts of fractals, the self-affine function h(x) in this case can be compared to roughness function R(L). Thus, analogously to equation (15) we have:

$$R(L) \sim L^{\alpha} \tag{21}$$

Measurement of the Nanoscale Roughness by

roughness exponent (Cruz T.G.S., 2002).

contributes to the original surface.

Atomic Force Microscopy: Basic Principles and Applications 171

Fig. 25. For self-affine surfaces, the slope of the unsaturated region of roughness is the

Where *pk* is the kth frequency in direction x; *ql* is the lth frequency in direction y, z(xc, yd) is the height coordinate located at xc, yd, M is the number of points in the profile, N is the number of profiles, Δx, Δy, are sampling distances, i is the imaginary unit (Czifra & Horvath, 2011).

The PSD is able to separate the surface profile at various wavelengths, that is, the sum of the wavelengths generated by the PSD, make up the original form of the surface profile by a Fourier analysis (Figure 26). This information is relevant in discerning how each component

Fig. 26. Example of a wave decomposed by PSD. The sum of the four waves results in the

composed wave (adapted from Freitas, A. C. P., 2010).

Using equation 20 and plotting a graph of the logarithm of the values obtained from microscopic observations as a function of L we can obtain the roughness exponent α.

Figure 25 is an example of a graph of Rq versus L. For self-affine surfaces, the slope of the roughness in the region of non-saturated is the roughness exponent.

Thus, the roughness exponent can be found and provide the fractal dimension by equation (17).

#### **9.2 Power spectral density (PSD)**

The power spectral density (PSD) is a complementary analysis of surface roughness which gives information related to parameters of the roughness height and spacing. The PSD is a parameter used in micrographs which relates the Fourier Transform (FT) with the root mean square roughness (RMS). The relationship between the PSD, FT and RMS is described by (Park, 2011):

$$PSD = FT^2 = RMS^2 \tag{22}$$

A Discrete Fourier Transform (DFT) used for a 3D topography can be described by (Czifra & Horvath, 2011):

$$F(p\_k, q\_l) = \Delta \mathbf{y}.\\ \Delta \mathbf{x} \sum\_{\mathbf{d}=1}^{N} \sum\_{\mathbf{c}=1}^{M} \mathbf{z}(\mathbf{x}\_{\mathbf{c}}, \mathbf{y}\_{\mathbf{d}}) \mathbf{e}^{-i2\mathbf{n}(\mathbf{x}\_{\mathbf{c}} p\_k + \mathbf{y}\_{\mathbf{d}} q\_l)}\tag{23}$$

Fig. 24. Diagram illustrating the formation of a surface. Each square represents a particle

roughness function R(L). Thus, analogously to equation (15) we have:

roughness in the region of non-saturated is the roughness exponent.

average in x, for all windows of the same size, and averaged over all windows chosen. Using the concepts of fractals, the self-affine function h(x) in this case can be compared to

Using equation 20 and plotting a graph of the logarithm of the values obtained from

Figure 25 is an example of a graph of Rq versus L. For self-affine surfaces, the slope of the

Thus, the roughness exponent can be found and provide the fractal dimension by equation

The power spectral density (PSD) is a complementary analysis of surface roughness which gives information related to parameters of the roughness height and spacing. The PSD is a parameter used in micrographs which relates the Fourier Transform (FT) with the root mean square roughness (RMS). The relationship between the PSD, FT and RMS is described by

 ²² *PSD FT RMS* (22) A Discrete Fourier Transform (DFT) used for a 3D topography can be described by (Czifra &

> c d N M

d 1c 1 , <sup>Δ</sup>y.Δx z x ,y e *k l p q Fp q k l*

c d

i2π(x y )

(23)

microscopic observations as a function of L we can obtain the roughness exponent α.

(21)

arriving at this surface (Cruz T.G.S., 2002).

**9.2 Power spectral density (PSD)** 

(17).

(Park, 2011):

Horvath, 2011):

() *RL L*~

Fig. 25. For self-affine surfaces, the slope of the unsaturated region of roughness is the roughness exponent (Cruz T.G.S., 2002).

Where *pk* is the kth frequency in direction x; *ql* is the lth frequency in direction y, z(xc, yd) is the height coordinate located at xc, yd, M is the number of points in the profile, N is the number of profiles, Δx, Δy, are sampling distances, i is the imaginary unit (Czifra & Horvath, 2011).

The PSD is able to separate the surface profile at various wavelengths, that is, the sum of the wavelengths generated by the PSD, make up the original form of the surface profile by a Fourier analysis (Figure 26). This information is relevant in discerning how each component contributes to the original surface.

Fig. 26. Example of a wave decomposed by PSD. The sum of the four waves results in the composed wave (adapted from Freitas, A. C. P., 2010).

Measurement of the Nanoscale Roughness by

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In the example given in Figure 26, the composed wave equation is the sum of each of the four waves. The surface can thus be divided into height parameters and parameters of wavelength. The figure shows that the first wave contributes to the larger waves while the fourth wave contributes to reduced ripples
