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In this study, indium tin oxide (ITO) thin films were prepared by electron beam evaporation method on float glass substrates at room temperature (RT). The surface morphology and dynamic scaling behavior of the films were studied by atomic force microscopy (AFM). It was found that average surface roughness values decreased as the film thickness increased from 100 nm to 350 nm. Fractal geometry and statistical physics techniques have been used to study a variety of irregular films within a common framework of the variance thickness. The Hurst exponent H and growth exponent ? for ITO thin films were determined to be 0.73 ? 0.01 and 0.078, respectively. Based on these results, we suggest that the growth of ITO thin films can be described by the combination of the Edwards-Wilkinson equation and Mullins diffusion equation.

Indium tin oxide (ITO) is an n-type wide band-gap (3.3 - 4.3 eV) semiconductor which by doping with tin, the density of the carrier can be increased up to the Mott critical ((1020/cm^{3}), and the highly degenerated semiconductor is formed. This semiconductor shows high transmission in the visible and near-IR regions of the spectrum [1,2]. It is widely used as a transparent electrode in various optoelectronic devices such as solar cells, liquid crystal displays (LCDs), organic light-emitting diodes (OLEDs) and other flat panel displays (FPDs) [3,4]. Numerous deposition techniques have been utilized to prepare ITO thin films such as DC/RF magnetron sputtering [

ITO thin films, with different thicknesses (100 to 350 nm), were prepared by electron beam evaporation method on polished float (soda-lime) glass substrates at room temperature. The target material used in this study was an ITO pellet (Merck Co.) with nominal 99.9% purity In_{2}O_{3}:SnO_{2} (90 wt% and 10 wt%, respectively). Before loading the glass substrates into the chamber, they were ultrasonically cleaned in acetone, ethanol and deionizer (DI) water for 10 min. Finally, they were dried with nitrogen gas. The optimum conditions of films depositing were achieved in the presence of oxygen with an initial vacuum (base pressure) of 1 × 10^{–}^{6} mbar, an accelerating voltage of 1 - 10 kV, electron beam current 10 - 12 mA. For control the deposition rate, the oxygen (99.99%) is introduced into the deposition system from a steel tube through a calibrated leak valve. By using the data from a quartz crystal thickness measurement system, the electron gun current is automatically adjusted to have a constant deposition rate. The deposition rate was 0.1 nms^{–}^{1} and the thickness of the ITO thin films varied in the range of 100 - 350 nm. After prepare films, each sample was taken out for immediately ex situ measurements to study the surface morphology of the films. The surface morphology of the thin films was characterized with an Atomic Force Microscope (AFM-Park Scientific Instruments) under ambient conditions. The scan sizes were 0.25 × 0.25 μm^{2} and 1 × 1 μm^{2}. All the surface images were obtained in the contact mode using silicon nitride tips with approximate tip radius of 10 nm, and the height of the surface relief was recorded at a resolution of 256 pixels × 256 pixels. A variety of scans were acquired at random locations on the film surface. To analyses the AFM images, the topographic image data were converted into ASCII data.

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

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

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 [

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

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:

The interesting results (RMS and R_{a}) have been plotted in _{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 [_{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 [

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.

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:

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:

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:

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

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

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 [

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

where E + 1 is the dimension of the embedded space (E = 1 for a profile; E = 2 for a plan) [

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

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 [

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

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, [

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.

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.