Keywords

1 Introduction

1.1 Background

The concept of fractals has been accepted to describe most natural systems and currently finding extensive application in the characterisation of surfaces [1, 2]. In most cases, random systems such as surface microscopy behave like fractals, i.e. their geometrical units resemble one another at all scales (at both low and high magnification similar features are identified) [3,4,5]. It, therefore, means that fractals are not sensitive to the scale of resolution and fractal analysis can be used to extract more information about a surface as compared to the conventional statistical methods [6]. The concept of fractals in thin film structures was clearly described, for the first time, in 1985 by Yehoda and Messier [7]. They argued that the presence of low-density regions (network of voids) of films microstructures was the reason for the fractal behaviour of thin films. In thin film surfaces, fractal dimension is used as an analytical index to measure how the morphological features vary on scaling [2]. The fractal analysis provides information on roughness exponent, correlation length, shift (or lattice size) and pseudo-topothesy besides the fractal dimension of thin films [8]. These parameters offer a detailed description of spatial patterning, segmentation, texture and lateral roughness of the surface morphology [9]. As such extensive literature exists on fractal analysis of surfaces of thin films [10,11,12,13,14,15,16,17,18,19] and several methods of fractal analysis, have been developed. The objective of this paper is to provide a very short overview of the most commonly used methods and summarize some of the published results of fractal analysis of thin films. The article may assist researchers expecting to use these methods in thin film imaging applications.

1.2 A Review of Common Fractal Analysis Methods

1.2.1 Autocorrelation Function

The autocorrelation function (ACF) shows the dependence of a signal on its own at different time shifts. In thin films, ACF describes the self-affine characteristics of surfaces and is used to derive fractal parameters such as roughness exponent, correlation length and fractal dimension (D) [12]. From various literatures [20, 21], the autocorrelation function (A(r)) along the direction of fast scan (x-direction) is expressed in terms of height function z(i, j) as follows.

$$ A\left( {r = ld} \right) = \frac{1}{{m\left( {m - l} \right)w^{2} }}\sum\limits_{j = 1}^{m} {\sum\limits_{i = 1}^{m - l} {z\left( {i + l,j} \right)z\left( {i,j} \right)} } $$
(1)

where d is the horizontal distance between two adjacent image features and l is the preceding feature of the point (m) of interest.

1.2.2 Height-Height Correlation Function

Various researchers have used height-height correlation function (H(r)) to illustrate the self-affine and mounded characteristics of surfaces of thin films [1, 6, 12, 13]. Mathematically, one-dimensional H(r) of the m × m area of surface micrograph in the direction of the fast scan of the AFM probe is given as follows [22].

$$ H\left( {r = ld} \right) = \frac{1}{{m\left( {m - l} \right)}}\sum\limits_{j = 1}^{m} {\sum\limits_{i = 1}^{m - l} {\left[ {z\left( {i + l,j} \right) - z\left( {i,j} \right)} \right]^{2} } } $$
(2)

A bi-logarithmic plot of H(r) versus r reveals two regimes as described by Yadav et al. [12] and it has been shown that the fractal dimension (D) is determined by fitting a power law within the linear region (small values of r) of the plot whereas roughness exponent, correlation lengths and Hurst exponents are determined by best-curve at the nonlinear region (large values of r). Detailed applications of H(r) are reported elsewhere [1, 12, 20].

1.2.3 Power Spectral Density Function

The power spectral density function uses a fast Fourier transform algorithm of the height functions (H_st) of the surface as shown in the logarithmic diagram in Fig. 1a. The fractal dimension is computed as a function of the average power (S) of the height spectra over the area under study and is determined as follows [23].

Fig. 1
figure 1

(adapted from Carpinteri et al. [23] with permission from Elsevier, copyright order number 501448072)

Illustrating the spectral determination of the fractal dimension through (a) fast Fourier transform of surface and (b) the double log plot of power versus spatial frequency

Full size image
$$ S = \frac{1}{{l^{2} N_{j} }}\sum\limits_{1}^{{N_{j} }} {\left| {H_{st} } \right|^{2} } $$
(3)

where Nj is the number of points within the digital area whose linear size is defined by l. Within the highly correlated region (self-affine surfaces), S obeys the power law in the form \( \varvec{S} = \varvec{k}_{\varvec{j}}^{{{\mathbf{ - }}\varvec{\beta}{\mathbf{ - 1}}}} \) where, kj is the radial spatial frequency and the slope of the curve in Fig. 1b is defined by β [24]. From this method, D can be determined as follows.

$$ \varvec{D} = \frac{{{\mathbf{7}} -\varvec{\beta}}}{{\mathbf{2}}} $$
(4)

1.2.4 Box-Counting Method

In this method, the fractal features are covered with a single box, which is subsequently divided into four quadrants. Each of the quadrants is further divided into four quadrants, and this is repeated in a loop until the minimum size of each box is equal to the resolution of the data [25, 26]. Then for each case, the number of boxes (N) covering the fractal features are counted, and its logarithm is plotted versus the size of boxes (h). The fractal dimension (D) is determined from the maximal slope coefficient of the double log plot defined as follows [25, 27, 28]. The method is illustrated in Fig. 2.

Fig. 2
figure 2

Illustrating the box-counting method for dragon curve fractals. In this method, N is the number of the boxes and h is the length scale (size of each box)

Full size image
$$ \varvec{D} = \mathop {{\mathbf{lim}}}\limits_{{\varvec{h}{\mathbf{ \to 0}}}} - \frac{{{\mathbf{log}}\,\varvec{N}\left( \varvec{h} \right)}}{{{\mathbf{log}}\,\varvec{h}}} $$
(5)

1.2.5 Triangulation Method

The computation of D in triangulation method (as known as prism counting) is based on approximating the area of the surface using successive pyramids and computing their lateral regions as illustrated in Fig. 2 [23, 29]. The area under study is covered by a square patch, and then one pyramid on the four angles of the square is created (Fig. 3a). The square is further subdivided into four quadrants, and then on each quadrant, a pyramid is created so that a total of 8 pyramids are generated (Fig. 3b).

Fig. 3
figure 3

Triangulation fractal dimension scheme of the dragon curve fractal. The a, b, c, d represents the repetitive steps followed in constructing the pyramids for each rectangular space occupied by the fractal features

Full size image

The procedure is repeated to generate 16 pyramids (Fig. 3c), 32 pyramids (Fig. 3d) and so forth until the base length of each pyramid (r) is equal to the resolution of the digital data of the image. The apparent area (A) of each prism is then computed for each r. The slope of the bi-logarithmic plot of r versus A is used to compute D as follows [23].

$$ D = 2 - \mathop {\lim }\limits_{r \to 0} - \frac{{{ \log }\,{\text{A}}\left( r \right)}}{{{ log }\,r}} $$
(6)

2 Overview of Published Literature

There is considerable published literature on the fractal analysis of thin films. The general observation from literature is that fractal methods provide a detailed description of the microstructure and evolution of the physical structures during thin film deposition. Through, fractal methods, the evolution of surface complexity of structure with the deposition parameters has been detailed [1, 13, 17, 24, 30]. The fractal dimension has been shown to increase/decrease with the substrate temperature, power and films’ thickness [1, 31, 32]. In a recent study, the effect of deposition time on the sputtering of Ti thin films on glass substrate was reported and shown that the fractal dimension increases with the deposition time [33]. Ţălu et al. [34] reported on the variation of fractal dimension of Ni–C prepared through the combination of radio frequency sputtering and plasma enhanced chemical vapor deposition (PEVD) techniques at varying times of 7, 10, 13 min on silicon and glass substrates. The highest fractal dimension was obtained at 10 min while the lowest obtained at 13 min of deposition. The fractal dimension has also been shown to decrease with increase in PEVD deposition pressure [35]. The power spectral density of 10 and 20 nm gold thin films has been reported [36]. The effect of annealing temperatures on the fractal properties of ITO thin films deposited by electron beam evaporation has been reported [37, 38]. Using height-height correlation method, Raoufi et al. [37] reported that fractal dimension of ITO films increases with the annealing temperature. Similar results were reported for the same films using power spectral density method [3]. In a similar study, Raoufi [38] reported that the lower annealing temperature, the slower the decrease of the fractal dimension. The fractal dimension of AlN epilayers sputtered on alumina was shown to increase with the substrate temperature [31]. The effect of deposition power and substrate temperature of Al thin films on steel substrates has been described by power spectral density function [24, 30]. The relationship between the statistical and fractal measurements of thin films has also been reported by various researchers [14, 24, 30, 39]. The general finding from these reports is that there is no direct relationship between root mean square and average roughness and fractal dimension.

Table 1 provides a summary of some of the fractal methods used in thin film analysis. The results of the correlation among the properties, deposition methods/parameters and fractal characteristics of the thin films are also included in Table 1.

Table 1 Selected published articles and key results on the fractal analysis of thin films
Full size table

3 Conclusions

Fractal techniques offer powerful tools for characterisation and segmentation of surface structures of thin films. The fractal dimension is the basic measure of irregularity and discontinuities of the surface properties. Various methods have been used for computation of fractal dimension of thin films some of which include, autocorrelation, height-height correlation functions, power spectrum and box counting methods. The advantage of these techniques over the statistical techniques of surface analysis of thin films is that they describe the lateral development of the surface features rather than just the vertical features. Through fractal analysis, researchers can deeply understand the growth and scaling of thin films during deposition at various process parameters and techniques. Despite the extensive use of fractal methods, there has been no much efforts to determine the best technique for analysing thin films. Future researches on fractal studies, therefore, should undertake comparative studies of the behaviour of different techniques on the morphological data to determine the suitable algorithm for specific films. There is also increasing application of multifractal analysis for films’ surfaces exhibiting multifractal behaviours.