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Application of Mie theory and fractal models to determine the optical and surface roughness of Ag–Cu thin films Article in Optical and Quantum Electronics · June 2017 DOI: 10.1007/s11082-017-1079-3





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Opt Quant Electron (2017)49:256 DOI 10.1007/s11082-017-1079-3

Application of Mie theory and fractal models to determine the optical and surface roughness of Ag–Cu thin films Ștefan Țălu1 · Ram Pratap Yadav2 · Ashok Kumar Mittal3 · Amine Achour4 · Carlos Luna5 · Mohsen Mardani6 · Shahram Solaymani7 · Ali Arman6 · Fatemeh Hafezi6 · Azin Ahmadpourian8 · Sirvan Naderi9 · Khalil Saghi6 · Alia Méndez10 · Gabriel Trejo11

Received: 22 March 2017 / Accepted: 12 June 2017 © Springer Science+Business Media, LLC 2017

Abstract Thin films of Ag/Cu were deposited by reactive DC magnetron sputtering on (001)-oriented Si and glass substrates for various deposition times (4–24 min). These films were characterized by atomic force microscopy (AFM), and a power law scaling was performed on the obtained micrographs to investigate the self-affine nature of the sample morphology, which is indicative of a fractal structure. We applied the Higuchi’s algorithm to the AFM data to determine the fractal dimension of each sample, and the Hurst exponents were computed. The deposition time dependences of these parameters and the grain size distributions estimated from the UV–visible spectra using the Mie theory, & Ali Arman [email protected] 1

Discipline of Descriptive Geometry and Engineering Graphics, Department of AET, Faculty of Mechanical Engineering, Technical University of Cluj-Napoca, 103-105 B-dul Muncii St., 400641 Cluj-Napoca, Romania


Department of Physics, Motilal Nehru National Institute of Technology, Allahabad, UP 211004, India


Department of Physics, University of Allahabad, Allahabad, UP 211002, India


Institut National de la Recherche Scientifique (INRS), 1650 Boulevard Lionel-Boulet, Varennes, QC J3X 1P7, Canada


Facultad de Ciencias Fı´sico Matema´ticas, Universidad Auto´noma de Nuevo Leo´n, Av. Universidad s/n, San Nicola´s de los Garza, Nuevo Leo´n 66455, Mexico


Vacuum Technology Group, ACECR, Sharif Branch, Tehran, Iran


Department of Physics, Science and Research Branch, Islamic Azad University, Tehran, Iran


Young Researchers and Elite Club, Arak Branch, Islamic Azad University, Arak, Iran


Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran


Centro de Quı´mica-ICUAP Beneme´rita Universidad Auto´noma de Puebla, Ciudad Universitaria Puebla, Puebla 72530, Mexico


Center of Research and Technological Development in Electrochemistry (CIDETEQ), Parque Tecnolo´gico Sanfandila, A.P. 064, Pedro Escobedo C.P. 76703, Quere´taro, Mexico


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allowed us to describe a particle formation mechanism during the deposition process, in which the length of continuous paths of conductive particles increases as the deposition time is increased. In agreement with this explanation, the electrical resistance decreased with the increment of the deposition time. Keywords Ag–Cu thin films · AFM · Fractal analysis · Surface topography

1 Introduction The remarkable progress in the research fields related to low dimensional materials have allowed the exploration of new physical phenomena and have opened up promising avenues to develop astonishing and innovative technologies (Ferrari et al. 2015; Lhuillier et al. 2015; Jariwala et al. 2014); giant magnetoresistance (Yuasa et al. 2004; Ghodselahi and Arman 2015), emission of spin waves (Berger 1996), new sources of magnetic anisotropy (Skumryev et al. 2003), superparamagnetism and its combination with luminescent properties (Evans et al. 2010), engineered photocatalytic activities (Oveisi et al. 2010; Ouldhamadouche et al. 2017) and the surface enhanced Raman scattering effect (Bader and Parkin 2010) are only few examples of the astonishing physicochemical behaviors uniquely related to these materials. Such phenomena display a great potential in a variety of technological applications such as spintronics (Bader and Parkin 2010), sensor technologies (Stewart et al. 2008), energy storage (Achour et al. 2017), environmental remediation (Mohamed and Salam 2014) and nanomedicine (Luna et al. 2015; Nafiujjaman et al. 2015). Confinement and surface effects arising from the nanometer-scale dimensions govern the unique properties of low dimensional materials (Luo et al. 2013; T¸a˘lu 2015; Luna et al. 2016); consequently, descriptive parameterizations are required for the accurately quantification of the material dimensions and surface topography. Specifically, the comprehensive description of the surface roughness of these materials is an intriguing and crucial challenge necessary for the understanding of their properties and their potential technological exploitation (T¸a˘lu et al. 2014). Atomic force microscopy (AFM) represents an outstanding imaging tool for the examination and analysis of nanostructures and surfaces (T¸a˘lu et al. 2014, 2015, 2016), especially in thin films deposited by sputtering techniques (Kalb et al. 2004; Gelali et al. 2012; Stach et al. 2015; Arman et al. 2015). The three-dimensional (3D) surface topography of these materials can be engineered controlling the deposition variables, such as the deposition time (Ahmadpourian et al. 2016), DC power (Ghobadi et al. 2016a), deposition pressure (Assuncao et al. 2003; Molamohammadi et al. 2015) and substrate temperature (Ghobadi et al. 2016b), among others, and its study by AFM have generated crucial information about the nature of surfaces (Reyes-Vidal et al. 2015; Stach et al. 2017; Elenkova et al. 2015). In this regards, it has been pointed out that surfaces are not generally random or stationary processes due to surface parameters depend on the scan length and on the measuring instrument (Molamohammadi et al. 2015; Ramazanov et al. 2015; Arman et al. 2015). In consequence, new models for surface roughness measure have been proposed to determine its properties at all scales using scale-independent parameters. In this regards,


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several recent studies have shown that natural and fabricated surfaces display self-affine features, therefore their topography can be described using scale-independent parameters by fractal (T¸a˘lu et al. 2014, 2015a, b, 2016a, b; Stach et al. 2015; Arman et al. 2015; Me´ndez et al. 2015; Singh et al. 2016; Yadav et al. 2012, 2014a, b, 2015a, b) or multifractal (T¸a˘lu et al. 2014, 2015; Yadav et al. 2012, 2014, 2015) analyses of the AFM images. Fractal analysis provides surface information different from traditional analysis, and it helps to understand the nature of the surface geometry (Yadav et al. 2015). The physical properties of the system are affected by it; hence, fractal approaches are appropriate tools to determine the complexity and irregularity of thin films (Yadav et al. 2012, 2014a, b, 2015a, b; Singh et al. 2016) and understanding surface morphology(Singh et al. 2016; Yadav et al. 2012, 2014). In the present study, the 3D surface morphology of Ag/Cu thin films synthesized by direct-current magnetron sputtering varying the deposition time from 4 to 24 min have been investigated by AFM imaging, and a fractal approach has been implemented for the analysis of the obtained 3D AFM images. In addition, the UV–visible absorption and the electrical resistance of these samples have been studied as functions of the deposition time and the complexity/irregularity of the samples.

2 Experimental procedure 2.1 Cleaning of the thin film supports Prior to thin film deposition and in order to avoid any contamination of surfaces, glass and silicon substrates (1 cm 9 1 cm in dimensions) were carefully washed with soap and water, followed by ultra-sonication during 5 min in acetone without heating. After washing the silicon and glass substrates, they were dried using a nitrogen gun.

2.2 Preparation of sample deposition Thin films of Ag/Cu were deposited by reactive DC magnetron sputtering on (001)oriented Si and glass substrates using a silver/copper target with 99.99% purity. The silver/copper atomic ratio of the target was equal to 0.639, as it was confirmed by the energy-dispersive X-ray spectroscopy analysis shown in reference Ahmadpourian et al. (2016). The deposition chamber was equipped by two (rotary) mechanical pumps that achieved a pressure of 10−3 torr, and another turbo (diffusion) pump that achieved a base pressure of 10−5 torr. The distance between the target and substrate holder was fixed at 5 cm. The thin films were deposited at constant power of 20 W without intentional heating. The deposition time was varied from 4 to 24 min with a time step of 4 min. Table 1 summarizes the experimental parameters used during film deposition.

2.3 Sample characterization The nanostructured surface morphology of the films was investigated by tapping mode (TM) atomic force microscopy (AFM). The average grain size and the surface roughness of each sample were determined from the corresponding AFM data using the WSxM software (version 5.0). The optical properties of layers were studied using UV–visible spectroscopy


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Table 1 Interface width (w), grain size, Hurst exponent (H), Fractal dimension (Df) and electrical resistance of sputtered Ag–Cu thin films deposited at different times Deposition time (min)

Interface width (w) nm

Grain size [nm]

Fractal dimension (Df)

Hurst exponent (H)

Resistance (Ω)




1.44 ± 0.05

0.56 ± 0.05





1.28 ± 0.05

0.72 ± 0.05





1.50 ± 0.05

0.50 ± 0.05





1.51 ± 0.04

0.49 ± 0.04





1.48 ± 0.05

0.52 ± 0.05





1.26 ± 0.05

0.75 ± 0.05


and the particle size distribution was calculated from the obtained spectra using the Mie theory and compared with AFM results (Bohren and Huffman 1998; Naderi et al. 2012; Dalouji et al. 2016; Balan et al. 2007; Christian and O’Reilly 1986).

2.4 Mie theory The absorption cross section obtained from Mie theory using the dipole Plasmon approximation for spherical particles is given by the following expression (Bohren and Huffman 1998; Naderi et al. 2012): ( ) 18pVe3=2 e00 m ð1Þ Qabs ðkÞ ¼ k ðe0 þ 2em Þ2 þ e002 where V is the volume of the spherical particles, λ is the radiation wavelength, em is the medium dielectric constant, and e0 is the real part of the dielectric function. When the denominator reaches its minimum, i.e. e0 ¼ 2em , resonant conditions are satisfied and the absorption reaches its maximum value. The imaginary part of the dielectric function e00 is: (Naderi et al. 2012; Dalouji et al. 2016; Balan et al. 2007) e00 ðx; ri Þ ¼ e00bulk þ

3x2p vF 4x3 ri


where νF is the Fermi velocity and ωp is the Plasmon frequency. The Plasmon frequency of metallic nanoparticles is: xp;b xp ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2em


where xp;b is the volume plasmon frequency (Balan et al. 2007; Christian and O’Reilly 1986). The particle size distribution was calculated from the experimental absorption spectrum by fitting the absorption cross section obtained from Mie theory with the experimental absorption spectrum of Ag/Cu samples (Ahmadpourian et al. 2016).


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3 Methods 3.1 Surface roughness A random rough surface can be described mathematically as h(i, j), where h(i, j) is the rough surface height distribution at point (i, j). For the rough surfaces, we assume that the height fluctuation is a random field with respect to the position. Root mean square (RMS) roughness is one of the most important parameters needed to characterize such surface. Analytically, it can be estimated as: (Yadav et al. 2015) 2 L L 312 Z Z 1 ð4Þ wL ¼ 4 fhði; jÞ  hhði; jÞigdxdy5 L 0


where L is the length of the measure surface and hhði; jÞi is the average surface height that it can be analytically expressed as: (Yadav et al. 2015) 1 hhði; jÞi ¼ 2 L

ZL Z L hði; jÞdxdy 0



These parameters are real and impose certain restrictions on the mathematical description of the random rough surface. An accurate estimation is important to understand the surface roughness because it provides the basic information about the surface texture. Our knowledge of surface roughness depends on the range of measurement and the instrumental resolution. On the other hand, the concept of fractal is very useful for describing rough surfaces. The idea of the fractal geometry is closely associated with the properties of invariance under a change of scale. The simplest fractal object is a selfsimilar object that is invariant under similarity transformations.

3.2 Higuchi’s Fractal dimension Higuchi’s algorithm (Higuchi 1988) computes the fractal dimension of a vertical section of the surface from the discrete height data that is obtained by AFM, for example. The height data of the vertical section can be denoted by X (1), X (2)…X (N). The Higuchi’s algorithm constructs sub-sequences: (Higuchi 1988) Xkm : XðmÞ; Xðm þ kÞ; Xðm þ 2kÞ. . .Xðm þ pkÞ   ðN  MÞ for m ¼ 1; 2; 3. . .k; where p ¼ int k The ‘length’ Lm(k) of each segment Xmk is then calculated as: ( ) ! p X N1 =k Lm ðkÞ ¼ jXðm þ ikÞ  Xðm þ ði  1ÞkÞj pk i¼1

ð6Þ ð7Þ


where N is the total number of sample points and N1 pk denotes the normalization constant. Lm(k) is not a ‘length’ in Euclidean sense. It represents the normalize sum of absolute values of difference in pair of points at distance k with initial point m.


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Therefore the length of curve is given by: Pk Lm ðkÞ LðkÞ ¼ m¼1 k


Higuchi’s algorithm for determining the fractal dimension is based on how the length L (k) of the curve, which represents the series of k samples as a unit, scales with k as: L ðkÞ  K Df


where Df is the fractal dimension and characterizes the complexity/irregularity of the surface. Df is calculated by the least square linear best fit procedure in the linear regression of the log–log representation using Eq. (10): y ¼ ax þ b with a = Df according to following relation: P P P n ðxk yk Þ  xk yk Df ¼ P P n ðx2k Þ  n ðxk Þ2



where yk ¼ ln L ðkÞ; xk ¼ ln k, k = k1,k2,………kmax and n denotes the number of k values for which the linear regression is calculated. The standard deviation of Df is estimated as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P P ffi xk yk  b yk n y2k  Df  P P 2  SDf ¼ ð13Þ xk ðn  2Þ n x2k  P P here b ¼ 1n ð yk  Df xk Þ with standard deviation rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 1 S Df x2k Sb ¼ ð14Þ n Finally, we computed the Hurst exponent from the fractal dimension using the relation (Yadav et al. 2015) H ¼ 2  Df


4 Results and discussion Figure 1 depicts the UV–visible absorption spectra of the six studied samples and the best fitted curves using Eq. (1). The particle size distributions obtained from these fittings are shown in Fig. 2. From AFM images (Fig. 3), the average particle diameters obtained for the samples 1–6 are 19.29, 44.37, 47.23, 54.26, 36.85 and 47.43 nm, respectively (Table 1). We observe that the particle size distributions obtained from the fitting of the UV– visible absorption are in good agreement with the AFM results for samples 1–5. However, for sample 6 the average diameter calculated from the optical properties is about 5 times smaller than the value determined by AFM. This difference is probably due to the limitations of the AFM lateral resolution to study nanoparticles of few nanometers in size.


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Fig. 1 UV–visible spectra of the six studied samples (dots) and the corresponding fitting curves (solid lines) obtained using the dipole Plasmon approximation model

Figure 3 shows AFM images of Ag/Cu thin films deposited by reactive DC magnetron sputtering on (001)-oriented Si and glass substrates at different deposition time (4, 8, 12, 16, 20 and 24 min respectively). These images indicate that the surface structures and surface roughness of samples are significantly modified when the deposition time is varied. We notice that granules of various sizes, irregular shapes and separations are present in the films. The values of interface width (w) of thin films were measured using the method described in Sect. 3.1. The obtained values are summarized in Table 1. From this table, it can be seen that the surface roughness of the films does not vary monotonically with increasing deposition


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Fig. 2 Particle size distributions estimated from the fits of Eq. (1) to the UV–visible spectra

time: it increases for deposition times from 4 to 12 min and then, it decreases for larger deposition times, being the film obtained after a deposition time equal to 12 min the film with greater roughness. This tendency can be explained as follows. During the first minutes of deposition, grains and inter-grain valleys appear. Then, at larger deposition times, the valleys are filled and the surface roughness is reduced. Afterwards, the progressive agglomeration of particles and the formation of new islands yield to the increment of the surface roughness (Yadav et al. 2012).


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Fig. 3 AFM images of Ag/Cu thin films deposited by reactive DC magnetron sputtering on (001)-oriented Si and glass substrates at different deposition time: 4, 8, 12, 16, 20 and 24 min, respectively

In addition, it is important to highlight the evolution of the grown crystalline phases during the deposition process. X-ray diffraction (XRD) studies of these Ag/Cu thin films, carried out in a previous work (Kalb et al. 2004), reveal a complex evolution of the crystallized phases as the deposition time is increased. XRD results showed that the Ag and Cu tend to grow as separated phases with nanometric sizes due to the immiscible character of these elements, and the Cu appeared partially oxidized. However, samples prepared with low deposition times (specifically, 4 and 8 min) presented additional face centered cubic nanocrystals with a preferable crystallization orientation along the [111] direction. Interface width (w) only characterizes the large-scale properties of the surface height fluctuations and it is only sensitive to peak values of surface profile (Singh et al. 2016; Yadav et al. 2012, 2014) . Also, it provides only vertical information and misses out important characteristics of the surface profile. Thin film evolution under non-equilibrium conditions is expected to produce a selfaffine surface following a power law scaling (Lhuillier et al. 2015; Yadav et al. 2012, 2014a, b, 2015; Singh et al. 2016) .These characteristics are investigated using fractal analysis that gives the information about irregularity/complexity of a surface (Singh et al. 2016; Yadav et al. 2012, 2014a, b) . For this reason fractal analysis is highly needed for thin films surface assessment. In order to distinguish the microstructure of Ag/Cu thin films surfaces for various deposition times, we estimated the fractal dimension and Hurst exponents from the AFM images. For this purpose the method described in Sect. 3.2 is applied to the height data of 256 linear traces in fast scan.


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Fig. 4 log L(k) as function of log k The red solid line is the best-fit curve, whose slope gives the value of fractal dimension. (Color figure online)

A graph is plotted between the average value of curve length L(k) and k at double logarithmic scale. Figure 4 depicts the plot between L(k) and k at double logarithmic scale for the film deposited during 4 min. The curves for the rest of the films are not shown here. The solid red line corresponds to the least square fit to the data, whose slope gives the value of fractal dimension (Df). The average value of Df for each film gives an idea about the change of surface morphology (Table 1). A larger value of Df corresponds to a more jagged morphological structure, while smaller one indicates a smoother structure (Yadav et al. 2015). The measure values of Df versus pixels for each thin film deposited for various time is shown in Fig. 5. The Hurst exponent (H) of each surface is computed by Eq. (12). The plot between H versus pixels for each surface is depicted in Fig. 6, and the average value of H for each thin film is listed in Table 1. Values of H varying between 0 \ H \ 1 illustrate the fractal characteristics of the system (Yadav et al. 2015). From this data, we observe that H lies between 0.5 and 1 for all samples. The greater fractal dimension and lower Hurst exponent value is obtained for the thin film deposition time of 16 min. Values H [ 0.5 indicate that a long-term memory effect occurs during deposition and a persistence behavior exists. Average fractal dimension, average Hurst exponent and electrical resistance are plotted as a function of deposition time in Fig. 7. The average fractal dimension and the average Hurst exponent exhibit a non-monotonic dependence on deposition time probably due to the complex above-discussed growth mechanism of the films. Also, from this figure it is observed that the electrical resistance decreases as the deposition time increases. The thin film with deposition time of 4 min has the highest electrical resistance, which is around 4000 Ω, and it gradually decreases with the increment of the deposition time. This may be due to the fact that size of particles increases with increasing deposition time; thus, the contact between particles is more frequent for larger deposition times. It yields to the formation of continuous paths of conductive particles, and in consequence, the electrical resistance is reduced.


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Fig. 5 Fractal dimensions as a function of the analyzed linear traces (256) for each film prepared at different deposition time

Fig. 6 Hurst exponents for different horizontal sections of each thin film surface profile

5 Conclusions A fractal approach has been performed to investigate the complexity and irregularity of Ag/Cu thin films deposited by reactive DC magnetron sputtering at different deposition time (from 4 to 24 min). The found describing parameters were estimated and correlated with the deposition time. Also, optical and electrical properties were measured.


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Fig. 7 Fractal dimension, Df (upper panel), Hurst exponent, H (middle panel) and electrical resistance, Ω (lower panel) graph for sputtered Ag–Cu thin films deposited at different times

It was found that the interface width exhibited a maximum for a deposition time of 12 min, and the fractal dimension computed using the Higuchi´s algorithm and Hurst exponent exhibited complex non-monotonic dependences on the deposition time. These behaviors have been explained with the initial occurrence of inter-grain valleys at the first minutes of deposition, and their subsequent filling and particle agglomeration produced at larger deposition times. In agreement with this hypothesis, the electrical resistance decreased as the deposition time increased, suggesting the increment of the contacts between particles as the deposition time increases. On the other hand, the UV–visible spectra were measured and analyzed using the Mie theory with the dipole Plasmon approximation, obtaining particle size distributions in agreement with the AFM observations with the exception of the sample obtained at largest deposition time, probably due to limitations of the AFM lateral resolution to study nanoparticles of few nanometers in size. Funding Neither author has a financial or proprietary interest in any material or method mentioned. Compliance with ethical standards Conflict of interest The authors declare that they have no competing interests.

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Application of Mie theory and Fractal models to determine…

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