3. Results and Discussions

3.1. Structural Properties

First, in order to study the microstructure of the ITO thin films, prepared at room temperature, the X-ray diffractometer (XRD) measurements were performed.

Figure 1 shows a typical XRD pattern of the ITO films. It is noted that no diffraction peak except a broad background (2θ = 15˚ - 35˚) is observed for the thin film. This feature indicates an amorphous and/or a nanocrystalline structure for the prepared ITO film [3].

3.2. Evolution Features of ITO Thin Films

Figures 2(a)-(d) show the AFM images of ITO thin films deposited on a glass substrate on the over scan area of 1 × 1 μm^{2}. The one-dimensional cross section scans of surface profiles are also plotted in Figures 3(a)-(d) of the ITO thin films with various thicknesses of 100 nm, 170 nm, 250 nm and 350 nm, respectively. Two typical morphological features are recognized readily by visual inspection of Figures 2 and 3. The first feature is that, the ITO thin films surfaces show continuous island-like structures, and with increasing the film thickness, these islands become smaller in both lateral and vertical directions. This evolution feature can be more easily observed, from the corresponding surface profiles of these films, in Figure 3. The second feature is that, the interface width w (or, root mean square roughness, RMS) as a function of film thickness where valleys, mountains and island clusters become smaller decreased as film thickness increased.

Figure 1. A typical XRD pattern of the ITO films prepared at room temperature.

(a)(b)(c)(d)

Figure 2. An AFM image series of ITO thin films: (a) 100 nm; (b) 170 nm; (c) 250 nm; (d) 350 nm.

(a)(b)

Figure 3. One-dimensional AFM surface profile scans of ITO thin films for: (a) 100 nm; (b) 170 nm; (c) 250 nm; (d) 350 nm.

The roughness calculation is the simple and the most used parameter for observation of changes in surface topography. In quantitative analyses on AFM images, it is known that the height roughness R_{a} and RMS have been used to describe the surface morphology. R_{a} is defined as the mean value of the surface height relative to the center plane, and RMS is the standard deviation of the surface height within the given area [20,21].

Denoting by h(i, j), the height of the surface measured by AFM at the point (i, j), N ´ N the total number of points at which the surface heights have been measured then the interface width w value of the surface is defined as [20]:

(1)

where represents the heights mean value of surface and defined as below.

(2)

Interface width is very attractive because compute simplicity and has ability to summarize the surface roughness by a single value. The average roughness is another simple statistical measure and it is defined as:

(3)

The interesting results (RMS and R_{a}) have been plotted in Figure 4" target="_self"> Figure 4. From this figure, it is noted that the average roughness R_{a} and w values decreased with increasing the film thickness. This behavior is due to the reflecting nucleation, coalescence and continuous film growth processes, i.e. Volmer-Weber type initial growth [22]. The above analyses indicate that average roughness R_{a} and w are strongly affected by the degree of aggregation and cluster size of the thin films. The different cluster size influences the surface roughness of the films [23].

However, these parameters are rather inadequate to provide a complete description of the irregularity of thin film surfaces [24,25]. These simple statistical measurements give only height information, and therefore cannot fully characterize the surface.

3.3. Scaling Behavior of ITO Thin Films

Figure 5 shows the rescaled range (R/S) of ITO thin films. The Hurst rescaled range analysis is a technique proposed by Henry Hurst in 1951 [26] to test presence of correlations in empirical time series. The main idea behind the R/S analysis is that one looks at the scaling behavior of the rescaled cumulative deviations from the mean, or the distance the system travels as a function of time. Consider a time series in prices of length P. This time series is then transformed into a time series of logarithmic returns of length N = P − 1 such that

Figure 4. w (RMS) and R_{a} for ITO thin films.

(4)

Time period is divided into m contiguous sub-periods of length n, such that. Each sub-period is labeled by I_{a}, with. Then, each element in I_{a} is labeled by N_{k}, such that. For each subperiod I_{a} of length n the average is calculated as:

(5)

Thus, M_{a} is the mean value of the contained in the sub-period I_{a} of length n. Then, we calculate the time series of accumulated departures from the mean (X_{k}_{,a}) for each sub-period I_{a}, defined as:

(6)

As can be seen from Equation (6), the series of accumulated departures from the mean always will end up with zero. Now, the range that the time series covers relative to the mean within each sub-period is defined as:

(7)

The next step is to calculate the standard deviation for each sub-period I_{a},

(8)

Then, the range for each sub-period () is rescaled by the corresponding standard deviation (). Recall that we had m contiguous sub-periods of length n. Thus, the average R/S value for length or box n is

(9)

Now, the calculations from Equations (4)-(9) must be repeated for different time horizons. This is achieved by successively increasing n and repeating the calculations until we have covered all integer ns. One can say that R/S analysis is a special form of box-counting for time series. However, the method was developed long before the concepts of fractals. After having calculated R/S values for a large range of different time horizons n, we plot versus By performing a leastsquares regression with as the dependent variable and as the independent one, we find the slope of the regression which is the estimate of the Hurst exponent H [27], that is

(10)

The Hurst exponent (H) and the fractal dimension D_{f }are related as [28]:

(11)

where E + 1 is the dimension of the embedded space (E = 1 for a profile; E = 2 for a plan) [3]. The Hurst exponent, who varies between 0 and 1, describes the fractal characteristics of time series. In theory, H = 0.5 means that the time series is independent, but as mentioned above the process need not be Gaussian e.g. the system obeys a random walk [29,30]. If characterizes the persistence of the time series (called the memory effect). It is also a main characteristic of non-linear dynamical systems that there is a sensitivity to initial conditions which implies that such a system in theory would have an infinite memory. The persistence implies that if the series has been up or down in the last period then the chances are that it will continue to be up or down, respectively, in the next period. This behavior is also independent of the time scale we are looking at. When, we have anti-persistence. This means that whenever the time series have been up in the last periodit is more likely that it will be down in the next period. Thus, an anti-persistent time series will be choppier than a pure random walk with H = 0.5 [31].

Figure 5 presents the rescaled range analysis of ITO thin films were prepared on glass slide substrates at room temperature by electron beam evaporation technique. From this graph, the H value was obtained by means of a statistical regression as the slope of the straight line. The average slope of the curves represent the Hurst exponent H = 0.73 ± 0.01, which characterizes the time series structure. The process is a persistent process, i.e., long memory. The experimental results show that Hurst coefficient did not more change with increase thickness, i.e., the Hurst exponent for all samples are very close, this suggests that the dynamics of roughness formation may be quite similar during the increase thickness of the ITO thin films.

The self-affine structure of the films is further confirmed by measurements of another important parameter β (growth exponent), which characterizes the time evolution of a self-affine surface in the following way: provided that the deposition time t is lower than a saturation value t_{x}, the interface width w is proportional to t^{β}: as in

(12)

Since the deposition time t and the film thickness d are proportional by the (constant) deposition rate r (d = rt), to evaluate β we can report w as function of d using samples grown at different deposition times [32]. Figure 6 describes the dependence of the interface width w on thickness d for different ITO thin films. The thin solid line is the best liner fit indicating the growth exponent β, and the result of the fitting gives β = 0.078. We compare the scaling exponents (H and β) determined for ITO thin films with exponents suggested by theoretical models. In fact, ITO growth behavior is between noise-driven Mullins diffusion model and noise-induced Edwards-Wilkinson model.

Figure 6. log-log plot of w versus thickness of ITO thin films. Thin solid line indicates the growth exponent β.

Combining the smoothing mechanism with random fluctuations, one can describe the growth process by a Langevin equation in the form:

(13)

where, is the result of the Gibbs-Thompson relation describing the thermal equilibrium interface between the vapour and solid. The term is the small slope expansion of the surface curvature, and the prefactor v is proportional to the surface tension coefficient. The second term is the result of surface diffusion due to the curvature induced chemical potential gradient. The prefactor is proportional to the surface diffusion coefficient. We can see that the general Langevin equation is a combination of the EW equation and Mullins diffusion equation. The scaling parameter cannot be applied to Equation (12) because there is crossover from EW and Mullins equation. Therefore the value is between the EW model, and the Mullins diffusion model, [33]. From the above analysis, we obtain the Hurst exponent (roughness exponent) H = 0.73 ± 0.01 and, respectively. From the measured roughness exponent and growth exponent, we suggest that our ITO thin films growth behaviour is the combination of the Mullins diffusion model and the Edwards-Wilkinson model.

4. Conclusion

In this work, we have analyzed the changes of surface morphology and dynamic scaling behavior of ITO thin films prepared by electron beam deposition method on glass substrate for different film thicknesses. AFM images of the ITO thin films reveal the formation of a porous granular surface, while the surface roughness values are decreasing from 24.8 nm to 4.13 nm with increasing of the film thickness from 100 nm to 350 nm. In order of to quantify the dynamic scaling and irregularity of the ITO thin films in more detail, we employ the fractal concept. The Hurst exponent (roughness exponent) was determined by applying the rescaled range (R/S) analysis method. From the measured roughness exponent H and growth exponent β, the dynamic scaling was observed clearly after numerical correlation analysis. The growth process can be described by the combination of Mullins diffusion Model and Edwards-Wilkinson model.

5. Acknowledgements

The authors would like to thank Dr. G. R. Jafari, Department of Physics, Shahid Beheshti University, Tehran, for many useful discussions and for his help in the R/S analysis of the films.

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