Morphological, microstructural and optical properties supremacy of ...

N.K. Sahoo et al. / Applied Surface Science 253 (2007) 3455–3463 3457macroscopic dielectric response will also depend on theseparameters. The definition of the dielectric function as thedipole moment per unit volume suggests that the magnitude ofthe dielectric response of a polycrystalline or amorphous filmwith respect to a standard spectrum can provide a simple,convenient and contactless means of measuring the density ofthe film. The macroscopic dielectric response of a thin film istherefore connected in an intimate way with the, grainstructures, compositional and microstructural parameters thatdetermine its other physical properties. It is probably for thesereasons earlier researchers have predicted the microstructuraldensifications in the composite film through refractive indexanalyses [24–26]. There are also several other model basedtechniques such as Tauc–Lorentz (TL) parametric and Wempleand Di Domenico’s single-effective oscillator models that canbe used to probe the microstructural interaction process deeper[32–34].With the advent of atomic force microscopy the visualizationof microstructure and its derived functional correlationparameters in thin films have added better understanding aswell as interpreting physical properties and phenomena.Morphological measurements and other derived factors likeautocorrelation and height–height correlation functions havecontributed substantially to this understanding. In the presentwork we have made use of these parametric analyses to explainthe superior properties of the certain composite films.4. Analyses with autocorrelation and height–heightcorrelation functionsAlthough, at the first sight, thin film morphology appears tobe random, a close analysis reveals volumes of correlatedproperties. The best way the thin film morphology can beprobed is by analyzing its autocorrelation function (ACF) aswell as height–height correlation function (HHCF) [35]. Spatialautocorrelation is, conceptually as well as empirically, the twodimensionalequivalent of redundancy. It measures the extent,to which the occurrence of an event in a designated unitconstrains, or makes more probable, the occurrence of an eventin its neighboring unit. In other words, the autocorrelationfunction can be used to detect non-randomness in surface dataas well as to identify an appropriate space or time series modelif the data are not random. Within the AFM measurements weusually evaluate the one-dimensional autocorrelation function(A x ) determined only from the height profiles (z n + m,l and z n,l )in the fast scanning axis which can be evaluated from thediscrete (N M) AFM data values as [36],A x ðt x Þ¼1NðMmÞX NXMml¼1 n¼1z nþm;l z n;l (2)Here t x is the distance between the two points in the x-direction(fast scanning axis) in pixel units and m refers to the currentdata point. In the present studies several co-deposited thin filmsof gadolinia–silica systems have been analyzed for theirmorphologies as well as autocorrelation functions using thedata and information acquired through the Solver P47H atomicforce microscope (AFM) measurements. The autocorrelationfunctions of pure film and the composites have shown verydifferent trends in their evolution. However, in most of the codepositedfilms, such functions can be fitted well with theappropriate realistic Gaussian models. Height–height correlationfunction is also very intimately associated with the autocorrelationand very prominently represents the grain structuresin thin film morphology. Mathematically, it is given by [37],H x ðr x Þ¼1NðMmÞX NXMml¼1 n¼1ðz nþm;l z n;l Þ 2 (3)Here H x is the one-dimensional function and r x is the distancebetween the two points in the x-direction (fast scanning axis) inpixel units. In order to get insight into the dynamic behavior andthe growth processes related evolution of morphology in thepure and composite films, we calculated the height–heightcorrelation function H(r) based on the AFM data. Like mostthin film experiments, here also it is expected that for each film,log H(r) should increase linearly with log r for small distance r,depicting a power-law behavior represented by, H(r) / r 2a , andthen leading to saturation for large r. At the initial regime of thecorrelation function there exist variant slopes representingdifferent microstructural evolutions in composite and purefilms. With this concept, we have tried to fit the height–heightcorrelation function with the self-affine properties using therelationship [38], 2a rHðrÞ ¼2w1 2 exp(4)jwhere r is the relative distance between a pair of points on thesurface, w the interface width, a the roughness exponent and j isthe lateral correlation length. All these parameters have veryspecial significances with the respect to the qualitative as wellas quantitative behavior of the surface morphology. Forinstance, a, the roughness exponent, signifies the how ‘‘wiggly’’the local slope is. It is also an indicative of the ‘‘jaggedness’’of the topography as well as signifies the relativecontribution of high frequency fluctuations to the roughness.Its value is directly correlated with ‘‘Herst’’ exponent thatdefine the fractal dimension of the surfaces. On the very shortlength scale, this parameter a is connected with the fractaldimension as ‘‘D =3 a’’. It is worth mentioning that thefractal dimension is the measure of contribution of two- andthree-dimensional mechanism during the film growth. Twodimensionalmechanism of growth gives the value of fractaldimension close to two- and three-dimensional close to 3. Weconsider that fractal dimension is one of the most informativeparameter of surface geometry. Moreover, most experimentalstudies have indicated that fractal dimension and thermodynamicgrowth condition of film are very much interconnected.The lateral correlation length, j, describes the largest distancein which the height is still correlated. It is to be further notedthat, j provides a length scale which distinguishes the shortrangeand long-range behaviors of the rough surface. Theinterface width, w, is a measure of the surface height fluctuation,i.e., it signifies the RMS fluctuation about the mean height.