Keywords

1 Introduction

During the deposition process of thin films, there are different morphologies of structures formed depending on the deposition type, process parameters, films, and substrate types [1]. Scanning probe microscopy (SPM) techniques such as atomic force microscope (AFM) are used to study the surface morphology of various thin films and coatings [2,3,4,5]. The micrographs obtained from the SPM techniques are used to undertake roughness analyses such as statistical [6, 7] and fractal measurements [8,9,10,11,12]. Fractal methods offer a detailed description of lateral roughness [13] and the nature of the surface morphology can be captured [14]. Although fractal characterization is widely reported in the literature [6, 12, 15,16,17], very little is reported on the relationship between the fractal measurements and the structure type/morphologies of the films. Therefore, the purpose of this work is to generate fractal profiles (using Minkowski functionals, autocorrelation, height-height correlation, and power spectral density functions) based on theoretical/synthetic surfaces of different morphologies.

2 Methods

Various synthetic morphologies of thin films were produced using scanning probe microscopy (SPM) software Gwydion (Fig. 1). These films depict different structural types that are obtained through various deposition processes such as sputtering and thermal spray. These structures are columnar, ballistic, fibrous, and pile-up structures (Fig. 1) and represent some of the most common morphologies observed in thin films. The process of creating synthetic (simulated) surfaces in Gwydion software are described elsewhere [18, 19]. All the images were single-layer, with a maximum height of 1000 nm and a scan area of 3 × 3 μm2. The fractal analyses of the simulated AFM images were undertaken according to the flowchart in Fig. 2. To validate the simulated fractal analyses, fractal values of a typical columnar AFM of CdTe thin films sputtered on glass substrates (Fig. 3) were computed and compared to the simulations. This process was iterative until comparable results were obtained (e.g., Fig. 4). Subsequently, all computations were conducted for the other simulated structure and results presented in Table 1.

Fig. 1
figure 1

Illustrating simulated surfaces of thin films consisting of various structural morphologies a columnar b ballistic c fibrous, and d pile-up particles. Corresponding 3D images are shown as insets on each image

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Fig. 2
figure 2

Flowchart illustrating the image and fractal analyses procedures

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Fig. 3
figure 3

Obtained from Camacho-Espinosa et al. [22] under open access creative commons

a SEM micrograph along the cross-section of CdTe thin films deposited by RF magnetron sputtering. The white arrows show columnar structures of the films perpendicular to the substrate. b Showing the AFM image (scan area of 0.5 × 0.5 μm2) at the top surface of the films. Recalibrated 3D AFM image of the CdTe films.

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Fig. 4
figure 4

Bi-logarithmic plots for power spectral density (PSDF) against the spatial frequency (k) of (a) typical columnar CdTe films deposited on glass substrates and b the corresponding simulated profile plot. The shapes of the profiles are comparable and are characterized by withers at the transition region between the flat and the linear areas of the PSDF profile

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Table 1 Illustrating various fractal analyses results from different simulated (synthetic) structures of thin films
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3 Results and Discussions

The results of the fractal analyses of the simulated AFM surfaces of thin films are presented in Table 1. A short description of the results in Table 1 is as follows:

  • Minkowski connectivity (X): Negative values dominate the X for columnar, ballistic, and fibrous structures whereas positive dominates for pile-up particles. The profiles vary with the type of structures.

  • Minkowski boundary: There are significant differences; while columnar and ballistic tend to nearly Gaussian profiles, the maximum values of boundary lengths for fibrous, and pile-up are skewed right and left, respectively.

  • Minkowski volume: The profiles for columnar, ballistic and pile-up particles are symmetrical about V = 0.5, and exhibit S-shape [11, 20, 23]. The fibrous structures are asymmetrical and exhibit quarter-circle shaped Minkowski volume.

  • Power spectral density: For columnar surface structures, the profile has a flat region at low frequencies and linearly decreasing PSD at high frequency with withers at the transition point [21, 24, 25]. For ballistic surfaces, the 1-d PSD profile consists of flat region and nonlinearly decreasing PSD. The flat region is not clear in fibrous surfaces whereas the pile-up surfaces have distinct flat and linear regions at low and high spatial frequencies respectively.

  • Areal autocorrelation (ACF): For columnar surfaces, the profile exhibit oscillatory behavior with decreasing and increasing values at low and high shifts respectively. The ACF decreases sharply to nearly r = 1.0 and then nearly remains constant for ballistic and fibrous. For pile-up surfaces, the ACF profile exhibit U-shape.

  • Height-height correlation (HCF): The HCF increases with r for all surfaces up to certain values. At very large r (mounded surface characteristics) oscillatory behavior of the profile was observed for columnar and ballistic surface structures [12, 23, 25]. The flat region (at large r) is not distinct for ballistic surfaces. The HCF decreases at nearly constant r at the end of the flat region for columnar and pile-up surfaces.

4 Conclusion

The profile plots of the most common fractal analyses of thin film surfaces of different synthetic morphologies have been presented. The surfaces were generated using Gwydion software and a typical validation of the columnar structure showed that the software provides a good approximation of deposited films. Profiles of Minkowski functionals, autocorrelation, height-height correlation and power spectral density functions of the synthetic morphologies (columnar, ballistic, fibrous and pile-up particles) presented in Table 1 will be a useful reference in relating the fractal results to the films’ deposition techniques and conditions.