Geometrical Phase Transition on WO 3 Surface
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p draft Geometrical Phase Transition on WO Surface
Abbas Ali Saberi a) School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5531, Tehran,Iran (Dated: 24 May 2018)
A topographical study on an ensemble of height profiles obtained from atomic force microscopy techniques on variousindependently grown samples of tungsten oxide WO is presented by using ideas from percolation theory. We findthat a continuous ’geometrical’ phase transition occurs at a certain critical level-height δ c below which an infiniteisland appears. By using the finite-size scaling analysis of three independent percolation observables i.e., percolationprobability, percolation strength and the mean island-size, we compute some critical exponents which characterize thetransition. Our results are compatible with those of long-range correlated percolation. This method can be generalizedto a topographical classification of rough surface models.PACS numbers: 05.40.-a, 64.60.ah, 68.35.Ct, 89.75.DaThe growth of rough surfaces and interfaces with many oftheir scaling and universality properties has attracted the at-tention of statistical physicists over the last three decades .One of the less studied subjects is the one concerned with thetopographical properties of the self-affine surfaces. In this let-ter, we present a systematic investigation of percolation tran-sition on an ensemble of experimentally grown WO surfaces.This gives some ’geometrical’ characteristic exponents whichwell describe a geometrical phase transition at a certain level-height below which an infinite cluster-height (or island) ap-pears. The results of our work are closely related to the statis-tical properties of the size distribution of the islands and theircoastlines together with their fractal properties.The tungsten oxide WO surface is one of the most interest-ing metal oxides. It has been investigated extensively becausefor its distinctive applications such as electrochromic ,photochromic , gas sensor , photo-catalyst , and photo-luminescence properties . Many properties of the WO thinfilms are related to their surface structure as well as to theirsurface topography and statistics such as grain size and heightdistribution. These properties can also be affected during thegrowth process by either deposition method or imposing ex-ternal parameters on the grown surfaces such as annealingtemperature .In order to have an ensemble of WO height profiles, 34samples were independently, and under the same conditions,deposited on glass microscope slides with an area 2.5 cm × ∼ × − Pa. The thick-ness of the deposited films was chosen to be about 200 nm,and measured by stylus and optical techniques. Using atomicforce microscopy (AFM) techniques, we have obtained 300height profiles from the grown rough surfaces with resolutionof 1/256 µ m in the scale of 1 µ m × µ m (an example is shownin Fig. 1). The AFM scans were performed in various non-overlapping domains (10 images from each sample) from thecentric region of the deposited samples in order for the profilesto be statistically independent. a) Electronic mail: a [email protected] & [email protected]
The percolation problem is an example of the simplestpure geometrical phase transitions with nontrivial critical be-havior, and it is closely related to the surface topography. Letus suppose a sample of height profile { h ( r ) } is a topograph-ical landscape. Now, let us imagine flooding this landscapeand coloring the parts above the water level white and the restblack. If the water level is high, there will be small discon-nected islands, while if it is low, there will be disconnectedlakes. There is however a critical value of the sea level forwhich there is one large supercontinent and one large ocean.As long as the original height profile has a gaussian distribu-tion with only short-range correlations, it is believed that thelarge-scale properties of the coastlines correspond to standardpercolation cluster boundaries and thus should be describedby the theory of Schramm-Loewner evolution (SLE) (see for a review of SLE). However, the height profiles of WO surface are quite different due to the relevant contribution oflong-range correlations . Hence, the corresponding coast-lines of the height profiles belong statistically to a differentuniversality class .To define islands on a WO surface, a cut is made at acertain height h δ = h h ( r ) i + δ p h [ h ( r ) − h h ( r ) i ] i := 0 ,where the symbol h··i denotes spatial averaging. Each island(cluster-height) is then defined as a set of nearest-neighborconnected sites of positive height. We show that there is acritical level height denoted by the dimensionless parameter FIG. 1. (Color online) AFM image of WO thin film in scale 1 µ m × µ m with resolution of 1/256 µ m. ( δ - δ c ) L ν P s -5 0 5 1000.20.40.60.81 L = 32L = 64L = 96L = 128L = 192L = 256 ν = 1.90(30) δ P s : = p e r c o l a ti onp r ob a b ilit y -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.800.20.40.60.81 δ = δ c FIG. 2. (Color online) Main: probability P s for the presence of aspanning island as a function of δ measured for different lattice sizes L . The curves cross at a critical level height δ c = − . . Inset:data collapse for the P s curves of different L with ν = 1 . and δ c = − . . δ = δ c , at which a continuous percolation transition occurs.To study the finite-size scaling (FSS) properties and mea-sure percolation observables which characterize the criticalbehavior, we consider a box of dynamic size L × L from thecentric region of each original AFM sample. For each latticesize ≤ L ≤ , each percolation observable was aver-aged over all number of AFM samples.The first quantity we measure is the probability P s that at eachlevel height δ , an infinite island spans two opposite boundariesof the box in just a specific direction, say y -direction. Ideally,the curves obtained for different lattice sizes cross at a singlepoint, marking the critical level height δ c . As shown in Fig. 2,the measured curves cross at δ = δ c , implying that the scalingdimension of the percolation probability P s is zero.According to scaling theory , one expects that all the mea-sured curves should obey the scaling form P s ( δ ) = P s [( δ − δ c ) L /ν ] , (1)where the exponent ν characterizes the divergence of the cor-relation length ξ (proportional to the spatial extent of the is-lands) near the percolation threshold, ξ ∼ | δ − δ c | − ν .We measure the values of the exponent ν and the crossingpoint of the curves δ c by utilizing the data collapse. The qual-ity of the collapse of the curves is measured by defining afunction S ( ν, δ ) of the chosen values ν and δ (the smaller S is indicative of a better quality of the collapse − see andthe appendix of ). We find its minimum S min ∼ . for ν = 1 . and δ c = − . . Inset of Fig. 2 illustratesthe collapse of all the P s curves, within the achieved accuracy,onto a universal function by using the estimated values for ν and δ c .Percolation strength P ∞ , which measures the probabilitythat a point on a level height δ belongs to the largest island(or equivalently, the fraction of sites in the largest cluster), is another quantity defined as order parameter in the percolation.As shown in Fig. 3, we have computed P ∞ as a function of δ .We find that the data follows the scaling form P ∞ ( δ ) = L − ˜ β P ∞ [( δ − δ c ) L /ν ] . (2)Our best estimate for the exponent ˜ β ( ˜ β = β/ν in percolationtheory) is ˜ β = 0 . which was obtained by optimizing thequality of the collapsed data. The data collapse is shown inthe inset of Fig. 3.Another independent observable is the mean island size χ = P ′ s s n s P ′ s sn s , (3)where n s denotes the average number density of islands ofsize s , and the prime on the sums indicates the exclusion ofthe largest island in each measurement.As presented in Fig. 4, the obtained curves χ ( δ ) for differ-ent lattice sizes have their maximum around the critical level δ c . We find that χ ( δ c ) ∼ L ˜ γ with ˜ γ = 1 . ( ˜ γ = γ/ν in percolation theory) − see inset of Fig. 4(a). By using theexponents ˜ γ and ν , it is possible to achieve a data collapseaccording to the scaling form χ ( δ ) = L ˜ γ χ [( δ − δ c ) L /ν ] , (4)which is shown in Fig. 4(b).A point which remains is determination of the universal-ity class which the observed percolation transition belongs to.The values we find for the exponents are obviously not thoseof short-range correlated percolation. In order to see whetherthey fit with long-range correlated percolation, we calculatethe two-point correlation function G ( x , x ′ ) which is definedas the probability that two points of distance r = | x − x ′ | ,at the critical level height δ c , belong to the same island. Thebest fit to our data shows that G is a decreasing function ofthe distance with a power-law behavior G ( r ) ∼ r − η and η = 0 . . δ P ∞ : = p e r c o l a ti on s t r e ng t h -2 -1 0 1 200.20.40.60.81 L = 32L = 64L = 96L = 128L = 192L = 256 ( δ - δ c ) L ν L . P ∞ -10 -5 0 5 1000.511.5 FIG. 3. (Color online) The percolation strength P ∞ of the islands asa function of δ for different lattice sizes L . Inset: collapse of the datain the suitably rescaled scales. L
128 256 ~ L χ ( δ c ) ( δ - δ c ) L ν L - . χ -10 0 10 2000.050.10.150.20.25 L = 32L = 64L = 96L = 128L = 192L = 256 ( b ) δ χ : = m ea n i s l a nd s i ze -1 0 1050010001500200025003000 δ c ( a ) FIG. 4. (Color online) (a) mean island size χ as a function of δ fordifferent lattice sizes L . Inset: χ at the critical level cut δ c ≃ − . as a function of L . (b) data collapse for χ for different lattice sizes,according to Eq. (4). In the past, several papers have dealt with percolation withlong-range correlations in which correlations decrease with increasing r . A completely different percolation withlong-range correlations has also been proposed by Sahimi in which correlations increase as r does . From the abovereferences, our results are in agreement with in which thevalue of ν is consistently lower than the prediction ν = 2 /η given by , for η ≤ in two dimensions.In order to examine whether our exponents satisfy well-knownscaling and hyperscaling relations, we have also computed thefractal dimension of the islands at the critical level δ c , andfound to be d f = 1 . . It is thus straightforward to see thatour obtained exponents satisfy, within the statistical errors, thefollowing scaling relations for d = 2 , d f = d − ˜ β = 12 ( d + ˜ γ ) = 12 ( d + 2 − η ) . (5)In conclusion, analysis of independent percolation observ-ables on an ensemble of height profiles obtained by AFMimages of WO surfaces, revealed a continuous geometricalphase transition at a certain critical level height δ c . We com-puted some critical exponents which can be regarded as topo-graphical characteristics of the WO surfaces. This methodmay lead to a topographical classification of self-affine roughsurfaces by computing the exponents ν , ˜ β , ˜ γ and η for differ-ent growth models. I would like to thank J. Cardy for his useful comments andM. Vincon for critical reading of the manuscript. I also ac-knowledge financial support from INSF grant. A. L. Barabsi and H. E. Stanley,
Fractal Concepts in Surface Growth (Cam-bridge University Press, Cambridge, 1995). C. G. Granqvist, Handbook of Electrochromic Materials (Elsevier, Amster-dam, 1995). P. R. Bueno, F. M. Pontes, E. R. Leite, L. O. S. Bulhoes, P. S. Pizani, P. N.Lisboa-Filho and W. H. Schreiner, J. Appl. Phys. , 2102 (2004). R. Azimirad, O. Akhavan and A. R. Moshfegh, J. Electrochem. Soc. ,E11 (2006). S. L. Kuai, G. Bader and P. V. Ashrit, App. Phys. Lett., , 221110 (2005). Y. Takeda, N. Kato, T. Fukano, A. Takeichi and T. Motohiro, J. Appl. Phys., , 2417 (2004). C. O. Avellaneda and L. O. S. Bulhes, Solid State Ionics, , 117 (2003). Y. S. Kim, S-C. Ha, K. Kim, H. Yang, S-Y. Choi, Y. T. Kim, J. T. Park, C.H. Lee, J. Choi, J. Paek and K. Lee, Appl. Phys. Lett., , 213105 (2005). E. Gyrgy, G. Socol, I. N. Mihailescu, C. Ducu and S. Ciuca, J. Appl. Phys., , 093527 (2005). H. Kawasaki, T. Ueda, Y. Suda and T. Ohshima, Sens. Actuators B, ,266 (2004). M. A. Gondal, A. Hameed, Z. H. Yamani and A. Suwaiyan, Chem. Phys.Lett., , 111 (2004). M. Feng, A. L. Pan, H. R. Zhang, Z. A. Li, F. Liu, H. W. Liu, D. X. Shi, B.S. Zou and H. J. Gao, Appl. Phys. Lett., , 141901 (2005). G. R. Jafari, A. A. Saberi, R. Azimirad, A. Z. Moshfegh and S. Rouhani, J.Stat. Mech., P09017, (2006). M. Sahimi,
Applications of Percolation Theory (Taylor & Francis, London,1994). D. Stauffer and A. Aharony,
Introduction to Percolation Theory , 2nd ed.(Taylor & Francis, London, 1994). O. Schramm, Isr. J. Math. , 221 (2000). J. Cardy, Ann. Physics , 81 (2005). A. Weinrib and B. I. Halperin, Phys. Rev. B, , 413 (1983);ibid.A. Wein-rib, , 387 (1984). A.A. Saberi, M. A. Rajabpour and S. Rouhani, Phys. Rev. Lett. , 044504(2008). K. Binder and D.W. Heermann,
Monte Carlo Simulation in StatisticalPhysics , Springer, Berlin, (1997). N. Kawashima and N. Ito, J. Phys. Soc. Jpn. , 435 (1993). S. M. Bhattacharjee and F. Seno, J. Phys. A , 6375 (2001). J. Houdayer and A.K. Hartmann, Phys. Rev. B , 014418 (2004). A. Coniglio, C. Nappi, L. Russo, and F. Peruggi, J. Phys. A , 205 (1977). M. B. Isichenko and J. Kalda, J. Nonlinear Sci. , 255 (1991). S. Prakash, S. Havlin, M. Schwartz, and H.E. Stanley, Phys. Rev. A ,R1724 (1992). M. Sahimi, Phys. Rep. , 213 (1998). M. Sahimi, J. Phys. I , 1263 (1994);AIChE. J. , 229 (1995). M. Sahimi and S. Mukhopadhyay, Phys. Rev. E , 3870 (1996); M. A.Knackstedt, M. Sahimi, and A. P. Sheppard, Phys. Rev. E61