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Approximate self-affine characterization of two-point rough surface statistics simulated by Eden algorithm
Vacuum 58 (2000) 158}165
Approximate self-a$ne characterization of two-point rough surface statistics simulated by Eden algorithm夽 Liliya A. Vulkova , Ivailo S. Atanasov , O.I. Yordanov * Institute of Electronics, Bulgarian Academy of Sciences, boul. **Tzarigradsko Choussee& ++ 72, Soxa 1784, Bulgaria American University in Bulgaria, Blagoevgrad 2700, Bulgaria
Abstract We study the self-a$ne properties of growing surfaces generated by using version A of the Eden model (R. Jullien and R. Botet, J Phys A:Math Gen 1985;18:2279). The analysis is based on two-point surface statistical quantities. A three-segmented spectral model is suggested and shown to be in agreement with the data related to the sample autocovariance function at all stages of the growth. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Semi-empirical models and model calculations; Surface structure; Morphology; Roughness; Topography
1. Introduction Technologically important thin "lms are typically grown by using deposition and aggregation processes. As a result the "lms are at some resolution inevitably rough with convoluted surface morphology. An important, though formidable, physical problem is to model the growth processes and subsequently to adequately characterize the resulting morphology. The di$culties arise from the very nature of the growth phenomena; namely, being non-equilibrium processes, they are not amenable to treatments based on the methods of the classical statistical physics. Because of this one has to rely on computer models. There are two possible approaches in the computer modeling of
夽
Paper presented at the 11th International School on Vaccum, Electron and Ion Technologies, 20}25 September 1999, Varna, Bulgaria. * Correspondence address: Institute of Electronics, Bulgarian Academy of Sciences, boul. `Tzarigradsko ChousseeH a 72, So"a 1784, Bulgaria. Tel.: #359-73-88543; fax: #359-73-80-828. E-mail address: [email protected] (O.I. Yordanov). 0042-207X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 0 0 ) 0 0 1 6 4 - 0
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rough morphology growth. Within the "rst, one attempts to develop a realistic model for the speci"c technological conditions of interest, incorporating as many as possible features as necessary to reproduce speci"c phenomena. In this study we employ the second approach, within which some, in principle signi"cant physical details, are deliberately neglected, provided however, the simpli"ed model renders some important characteristic(s) of the surface. Such models are more easily implemented, run faster and allow to conduct extensive studies of the characteristics in focus. Perhaps, the single most exciting "nding in this area for the last two decades is that the majority of the natural and man-made surfaces exhibit self-a$ne symmetry; see [1] and the references therein. In a widely used terminology, these surfaces are called fractal surfaces [2]. The self-a$nity is the main property which we study and characterize in this paper. Because of their complexity, the morphologies are described in statistical terms only. In the majority of recent papers devoted to fractal surfaces, the authors focus on the behavior of the global `widtha (variance) m(¸, t) of the height pro"le h(x, t), where x is the position over a substrate of total lateral length ¸ and t denotes the time since the onset of the process. Most of the theoretical studies [3}5] also concentrate on deriving expressions for m(¸, t). The attractiveness of the surface width as a quantity for investigations comes from that discovered by Family and Vicsek dynamic scaling [6]. This scaling, established "rst in computer simulations, provides in fact a simple model of m(¸, t) for a variety of rough morphologies and allows to characterize the structure in terms of just two independent exponents, s, called roughness exponent and z the dynamic exponent. Considerably more detailed information about the surface morphologies, however, is compressed into the behavior of the autocovariance function (ACF); note that m is a single-point statistical quantity and equals just the value of the ACF at zero. Our "rst objective in this work is to continue the systematic study of fractal surface morphologies with the aim to build and ascertain simple phenomenological ACF models for them. Preceding developments and results along this line can be found in Refs. [7}12]. The surfaces we consider here are generated using the version A of the Eden model [13]. Details of our implementation of this model are presented in the next section, which also contains aspects of the sample ACF estimation from the simulated pro"les. The basic facets of the approximate self-a$ne model, which we propose here to describe the behavior of the sample ACF at all stages of growth, are given in Section 3. In particular, we stress on the importance of reckoning with various biases of the sample ACF. The outcome of the non-linear least-squares "t to the data and retrieving of the model's parameters are discussed in Section 4. The main results of the study are summarized in the conclusions.
2. Version A of the Eden model: self-a7nity of the Eden surfaces We implement a version of the Eden algorithm [14] for cluster aggregation in two dimensions. In the nomenclature of [13] the algorithm is referred as version A. Accordingly, the surface grows on a lattice (substrate) with unit constant and length ¸ by adding particles one at a time. The latter allows to count the time by the number of particles t already aggregated. All unoccupied sites adjacent to the surface accept the incoming particle with equal probability. Periodic boundary conditions are imposed at the edges of the lattice. At the outset, the surface is assumed #at. The interface pro"le function h(x, t ) is de"ned by the height of the maximum occupied side along each column.
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At 10 selected times t "t /¸ , the growth is halted and both the power spectrum S(k , t ) and H H J H the sample ACF associated with h(x, t ) are evaluated. In order to avoid the e!ects in#icted by the H imposed arti"cial periodic boundary conditions, we take the pro"les de"ned over the middle half-part of the total substrate only, with a length of ¸"¸ /2. The spectra are calculated on the basis of the periodogram estimate using FFT [15], whereas the sample ACF according to the so-called biased estimate 1 *\J (1) AQ(x , t )" hK (x , t )hK (x #x , t ), G H G J H J H ¸ G where x "l*x, l"0, 1,2¸!1, *x is the substrate spacing and hK (x, t),h(x, t)!1h (t)2. The J * mean height 1h (t)2 is evaluated on the basis of its sample mean h "1/¸ * h(x , t). * * G G Although the discussion below focuses on the lattice sizes of ¸"512, simulations were carried out for lengths of ¸"1024 as well. In the former case 700 realizations are done, whereas in the latter } 500. For every realization both the spectra and the sample ACF were evaluated and the results for every t were averaged. These averaged quantities, along with their sample variances, are H used in the analysis below. As noted before, the fractal nature of the Eden surfaces can be revealed by looking at the dependence of the surface width m(¸, t) as a function of ¸ [6,13]. Alternative and statistically more reliable approach [11] is to consider the graphs of the sample structure function (SF) DQ(x , t ) at J H "xed times t . This function is related to the sample ACF through H DQ(x , t )"2(AQ(0, t )!AQ(x , t )); J H H J H we remark that DQ is non-negative for every x . In Fig. 1, we show graphs of the SF in a log}log J scale for three di!erent times, see the legends. The initial linear branches are indicative of the power-law behavior of DQ and hence the existence of fractal, self-a$ne hierarchy for small to intermediate spatial scales. Note that both the slopes and the extent of the linear branches increase with increasing t.
Fig. 1. Graphs of the sample structure function in a log}log scale for three snapshots during the surface growth. The linear form of the graphs for small and intermediate lags indicates the self-a$nity of the roughness morphology. The values of the slopes are given in the legends.
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3. Approximate self-a7ne model of Eden surfaces The methodology for building phenomenological models of fractal surfaces which we employ consists of constructing appropriate multisegmented power-law functions [7}12]. That this approach is applicable to Eden surfaces is suggested in Fig. 2, where pro"le spectra, corresponding to 10 unevenly spaced moments, are plotted in a log}log scale. The time extent used in Fig. 2 covers the entire growth and evolution of the simulated aggregate. It is seen that the spectra evolution undergoes a steady build up of a right (large wavenumber) power-law spectral branch, supplemented by one or perhaps two white-noise-like components at intermediate and small wave number regions. Physically, this corresponds to the generation and enlargement through a coalescence of clusters until a single large cluster "lls up the entire lattice. The "nal cluster contains a self-a$ne hierarchy of subclusters forming a fractal-like morphology. Accordingly, we shall attempt to approximate the spectra at all times by the following three segment piece-wise continuous function:
Ak\?(k /k )\?@ ("k"/k )?? , k )"k")k , @ ? @ ? $ ? k )"k")k , Ak\?("k"/k )?@ , ? @ @ @ S(k)" Ak\?, k )"k")k , @ , 0, otherwise,
(2)
where k "p/¸ , and k "p/*x, are the fundamental and the Nyquist wavenumbers, respective$ , ly. The relatively #at, noise-like, form of the left and the middle regions of the spectra shown in Fig. 2, suggests that both a and a should be small compared to the main branch spectral exponent ? @ a, especially at early times. The time dependence of the surface statistics is accounted for by assuming that the spectral parameters vary with time. In addition, the presence of an overall white-noise component can be identi"ed in most of the studied pro"les. This component manifests itself in the graphs of the sample ACF as a peak at zero,
Fig. 2. Spectra of the roughness pro"les for ten selected times during the evolution: t"4, 8, 16, 32, 64, 128, 256, 512, 1024, and 2048, from the bottom to the top.
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see below. The noise component is accounted for by augmenting S(k) with
A S (k)" U L U L 0
k )"k")k , $ , otherwise,
(3)
where A is a constant. This brings the total number of free parameters of the model to seven: a , U L ? a , a, A, k , k , and A . The contributions of the various spectral components to the ACF of the @ ? @ U L model, A(x, t), is additive and can be calculated exactly in terms of type (1, 2) hypergeometric functions, F , [7}12]. The leading asymptotic term of A(x, t) for intermediate distance lags, A(x, t)&A(0, t)!q\?x?\, constitutes the self-a$nity of the model. Here, the parameter q, called topothesy, depends on the spectral constant A and a only. The topothesy is a measure of the intensity of the small, fractality scales of the structure. The leading term also shows that the slopes measured from the graphs of the SF, Fig. 1, are given by a!1. On the other hand, the expression for A(0, t) relates the roughness exponent s to a through s"(a!1)/2, see [11]. The higher asymptotic orders account for the "nite-size e!ects and show that the self-a$nity of the model is only approximate. They allow to introduce an additional parameter * large scale correlation length * which for the model (2) is given by l "(a!1)k\. A detailed study of the approximate @ self-a$nity can be found in [16,17].
4. Illustration of the model applicability and discussions We validate the model not by "tting (2) to the periodogram estimate of the pro"le spectra, but rather by carrying out a non-linear least-squares "t to the sample ACF [7}12]. This approach avoids the di$culties associated with the practical implementation of the classical spectral analysis [15]. As a trade-o!, one has to cope with the speci"c biases of the estimate (1). There are two sources of bias for the sample ACF. The "rst is brought about by using the factor (1/¸) (rather than 1/(¸!l)) in (1) and has a magnitude of (x /¸)A(x, t), where A(x, t) is the &true' ACF of the interface J [12]. The second bias is caused by the use in (1) of the sample, rather that the true mean height. The derivation of an expression for the second bias is somewhat lengthy and will be presented elsewhere. It is equivalent and in fact more convenient to reckon with the biases by "tting AQ with its theoretical mean (regarding the sample ACF itself as random "eld). The latter is expressed in terms of the true ACF, for which we substitute the model's ACF. A typical result of the outlined procedure is exempli"ed in Fig. 3, where we show the sample ACF (open circles) for a morphology generated at relatively early stage of the growth: t"32. The error bars inside each circle indicate the sample variances of AQ(x, t), related to every di!erent x. The theoretical mean of the sample ACF, yielding the best "t, is depicted by the solid line. The values of the model function's parameters that render the "t are: a "0.178$0.73;10\; ? a "0.957$0.29;10\; a"1.661$0.06; A"0.605$0.71;10\; k "0.075$0.86;10\; @ ? k "0.113$0.27;10\; and A "0.040$0.07. Using these values we compute q"3.27 and @ U L l "9.96. The model function itself is drawn by the dashed line. The inset gives some details of the initial branch of the ACF. It is seen that an exceptionally accurate "t has been obtained for all x. The average per point relative discrepancy is 19%. In the considered case the noise component is negligible, and accumulated within the large white-noise-like left branch of the model. The
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Fig. 3. Sample ACF (circles), model's ACF (dashed line) and the "tted mean of the sample ACF (solid line) corresponding to a pro"le recorded at t"32. The inset provides a focused view of the initial branch of the ACFs. For the values of the model's parameters rendering the best "t see the text.
Fig. 4. The same, except for the inset, as Fig. 4, however, for a pro"le generated at time t"2048.
incorporated in the ACF biases, although small in magnitude, have an important contribution to the good agreement, compare with the model's graph. Another example * corresponding to a late, saturated, regime * is shown in Fig. 4. More speci"cally, the surface pro"le is recorded at t"2048. The nomenclature of the symbols is the same as in Fig. 3. At such late stage of the growth, the middle segment of the spectrum seems to collapse and the "t has been carried out with k "k thereby eliminating the parameter a . The values of the @ ? @ remaining parameters yielding the best "t are: a "!0.5$2.587; a"2.023$0.014; ? A"0.322$0.018; k "(0.825$0.015);10\; and A "0.382$0.242. Again a very good "t ? U L is obtained, the average discrepancy per point now is just 1.5%. It is also evident that even the left branch of the spectrum practically disappears * compare the values of k and k * which ? $ explains the estimated large error bounds for a . At this regime, the surface may be viewed as ?
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having a simple fractal morphology characterized by a single exponent a. The value of a de"nes a roughness exponent of s"0.511$0.007, which indicates that the surface belongs to the Kardar}Parisi}Zhang universality class. The latter was suggested in a number of numerical simulations, but our result for s seems closer than that obtained before to the value of 0.5 speci"c for this class (within two standard deviations). It is interesting to note the large discrepancy between the model's ACF and the mean of the sample ACF, which emphasizes the importance of the biases for values of a close to 2. Note also that the corresponding values of topothesy, q"20.95 and the large-scale correlation length, l "124.3 pertain to the shape of A(x, t), rather than the sample ACF. The model was tested on all the pro"les recorded at the selected times and found to be accurate to the same degree as in the cases discussed above.
5. Conclusions In summary, we simulate a number of rough surface morphologies using version A of the Eden aggregation model. We con"rm the earlier established fact that this model leads to fractal-like surfaces by employing an approach based on the behavior of the sample structure function rather than the properties of the surface width. Using this, we constructed a three-segment spectral model of the two-point statistics of the pro"les, which incorporates the self-a$ne symmetry of the structure. The model was veri"ed by a non-linear least-squares "t to the sample autocovariance function. A key point of the "tting procedure is the explicit account for the biases of the sample ACF. The e!ect of the biases is especially pronounced for large values of the main spectral branch exponent a. On the bases of the retrieved parameters, we conclude that the exponent of the main spectral branch increases with time until it reaches saturation at about a"2.023$0.014. This value puts the Eden surface roughness exponent within two standard deviations from roughness exponent which characterizes the Kardar}Parisi}Zhang universality class.
Acknowledgements The authors thank A. Oliver and E. Dimitrova for their assistance in the programming and preparation of the manuscript.
References [1] [2] [3] [4] [5] [6] [7] [8]
BarabaH si A-L, Stanley HE. Fractal concepts in surface growth. Cambridge: Cambridge University Press, 1995. Mandelbrot BB. The fractal geometry of nature. San Francisco: Freeman, 1982. Kardar M, Parisi G, Zhang YC. Phys Rev Lett 1986;56:889. Krug. J. Phys Rev A 1987;36:5456. Bouchaud JP. Cates ME. Phys Rev E 1993;47:R1455. Family F, Vicsek T. J Phys A 1985;18:L75. Yordanov OI, Ivanova K. Surf Sci 1995;331:1043. Holzwarth M, Wi{ing M, Simeonova DS, Tzanev S, Snowdon KJ, Yordanov OI. Surf Sci 1995;331:1093.
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[9] Yordanov OI, Ivanova K, Michalev MA. In: Groll H, Nedkov I, editors. Proceedings of NATO Advanced Research Workshop Microwave Physics and Technique, October 1996, Sozopol, Bulgaria. [10] Yordanov OI, Ivanova K, Michalev MA. High Technology, NATO ASI Series 3, Vol. 33, 1997. p. 267. [11] Yordanov OI, Guissard A. Physica A 1997;238:49. [12] Atanasov IS, Yordanov OI. In: Ausloos M, Nedkov I, editors. Proceedings of NATO Advanced Reasearch Workshop on `Nano-crystalline and Thin Film Magnetic Oxidesa. Sozopol, Bulgaria, Dordrecht: Kluwer Academic Publishers, 1998. p. 291. [13] Jullien R, Botet R. J Phys A 1985;18:2279. [14] Eden M. In: Proceedings of 4th Berkely Symposium on Mathematical Statistics and Probability, Vol. 4. Berkeley: University of California Press, 1961. p. 223. [15] Press WH, Flaunery BP, Teukolsky SA, Vetterling WT. Numerical recipes. Cambridge: Cambridge University Press, 1986. [16] Yordanov OI, Nickolaev NI. Phys Rev E, Rapid Commun 1994;49:R2517. [17] Yordanov OI, Nickolaev NI. Physica D 1997;101:116.
Approximate self-a$ne characterization of two-point rough surface statistics simulated by Eden algorithm夽 Liliya A. Vulkova , Ivailo S. Atanasov , O.I. Yordanov * Institute of Electronics, Bulgarian Academy of Sciences, boul. **Tzarigradsko Choussee& ++ 72, Soxa 1784, Bulgaria American University in Bulgaria, Blagoevgrad 2700, Bulgaria
Abstract We study the self-a$ne properties of growing surfaces generated by using version A of the Eden model (R. Jullien and R. Botet, J Phys A:Math Gen 1985;18:2279). The analysis is based on two-point surface statistical quantities. A three-segmented spectral model is suggested and shown to be in agreement with the data related to the sample autocovariance function at all stages of the growth. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Semi-empirical models and model calculations; Surface structure; Morphology; Roughness; Topography
1. Introduction Technologically important thin "lms are typically grown by using deposition and aggregation processes. As a result the "lms are at some resolution inevitably rough with convoluted surface morphology. An important, though formidable, physical problem is to model the growth processes and subsequently to adequately characterize the resulting morphology. The di$culties arise from the very nature of the growth phenomena; namely, being non-equilibrium processes, they are not amenable to treatments based on the methods of the classical statistical physics. Because of this one has to rely on computer models. There are two possible approaches in the computer modeling of
夽
Paper presented at the 11th International School on Vaccum, Electron and Ion Technologies, 20}25 September 1999, Varna, Bulgaria. * Correspondence address: Institute of Electronics, Bulgarian Academy of Sciences, boul. `Tzarigradsko ChousseeH a 72, So"a 1784, Bulgaria. Tel.: #359-73-88543; fax: #359-73-80-828. E-mail address: [email protected] (O.I. Yordanov). 0042-207X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 0 0 ) 0 0 1 6 4 - 0
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rough morphology growth. Within the "rst, one attempts to develop a realistic model for the speci"c technological conditions of interest, incorporating as many as possible features as necessary to reproduce speci"c phenomena. In this study we employ the second approach, within which some, in principle signi"cant physical details, are deliberately neglected, provided however, the simpli"ed model renders some important characteristic(s) of the surface. Such models are more easily implemented, run faster and allow to conduct extensive studies of the characteristics in focus. Perhaps, the single most exciting "nding in this area for the last two decades is that the majority of the natural and man-made surfaces exhibit self-a$ne symmetry; see [1] and the references therein. In a widely used terminology, these surfaces are called fractal surfaces [2]. The self-a$nity is the main property which we study and characterize in this paper. Because of their complexity, the morphologies are described in statistical terms only. In the majority of recent papers devoted to fractal surfaces, the authors focus on the behavior of the global `widtha (variance) m(¸, t) of the height pro"le h(x, t), where x is the position over a substrate of total lateral length ¸ and t denotes the time since the onset of the process. Most of the theoretical studies [3}5] also concentrate on deriving expressions for m(¸, t). The attractiveness of the surface width as a quantity for investigations comes from that discovered by Family and Vicsek dynamic scaling [6]. This scaling, established "rst in computer simulations, provides in fact a simple model of m(¸, t) for a variety of rough morphologies and allows to characterize the structure in terms of just two independent exponents, s, called roughness exponent and z the dynamic exponent. Considerably more detailed information about the surface morphologies, however, is compressed into the behavior of the autocovariance function (ACF); note that m is a single-point statistical quantity and equals just the value of the ACF at zero. Our "rst objective in this work is to continue the systematic study of fractal surface morphologies with the aim to build and ascertain simple phenomenological ACF models for them. Preceding developments and results along this line can be found in Refs. [7}12]. The surfaces we consider here are generated using the version A of the Eden model [13]. Details of our implementation of this model are presented in the next section, which also contains aspects of the sample ACF estimation from the simulated pro"les. The basic facets of the approximate self-a$ne model, which we propose here to describe the behavior of the sample ACF at all stages of growth, are given in Section 3. In particular, we stress on the importance of reckoning with various biases of the sample ACF. The outcome of the non-linear least-squares "t to the data and retrieving of the model's parameters are discussed in Section 4. The main results of the study are summarized in the conclusions.
2. Version A of the Eden model: self-a7nity of the Eden surfaces We implement a version of the Eden algorithm [14] for cluster aggregation in two dimensions. In the nomenclature of [13] the algorithm is referred as version A. Accordingly, the surface grows on a lattice (substrate) with unit constant and length ¸ by adding particles one at a time. The latter allows to count the time by the number of particles t already aggregated. All unoccupied sites adjacent to the surface accept the incoming particle with equal probability. Periodic boundary conditions are imposed at the edges of the lattice. At the outset, the surface is assumed #at. The interface pro"le function h(x, t ) is de"ned by the height of the maximum occupied side along each column.
160
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At 10 selected times t "t /¸ , the growth is halted and both the power spectrum S(k , t ) and H H J H the sample ACF associated with h(x, t ) are evaluated. In order to avoid the e!ects in#icted by the H imposed arti"cial periodic boundary conditions, we take the pro"les de"ned over the middle half-part of the total substrate only, with a length of ¸"¸ /2. The spectra are calculated on the basis of the periodogram estimate using FFT [15], whereas the sample ACF according to the so-called biased estimate 1 *\J (1) AQ(x , t )" hK (x , t )hK (x #x , t ), G H G J H J H ¸ G where x "l*x, l"0, 1,2¸!1, *x is the substrate spacing and hK (x, t),h(x, t)!1h (t)2. The J * mean height 1h (t)2 is evaluated on the basis of its sample mean h "1/¸ * h(x , t). * * G G Although the discussion below focuses on the lattice sizes of ¸"512, simulations were carried out for lengths of ¸"1024 as well. In the former case 700 realizations are done, whereas in the latter } 500. For every realization both the spectra and the sample ACF were evaluated and the results for every t were averaged. These averaged quantities, along with their sample variances, are H used in the analysis below. As noted before, the fractal nature of the Eden surfaces can be revealed by looking at the dependence of the surface width m(¸, t) as a function of ¸ [6,13]. Alternative and statistically more reliable approach [11] is to consider the graphs of the sample structure function (SF) DQ(x , t ) at J H "xed times t . This function is related to the sample ACF through H DQ(x , t )"2(AQ(0, t )!AQ(x , t )); J H H J H we remark that DQ is non-negative for every x . In Fig. 1, we show graphs of the SF in a log}log J scale for three di!erent times, see the legends. The initial linear branches are indicative of the power-law behavior of DQ and hence the existence of fractal, self-a$ne hierarchy for small to intermediate spatial scales. Note that both the slopes and the extent of the linear branches increase with increasing t.
Fig. 1. Graphs of the sample structure function in a log}log scale for three snapshots during the surface growth. The linear form of the graphs for small and intermediate lags indicates the self-a$nity of the roughness morphology. The values of the slopes are given in the legends.
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3. Approximate self-a7ne model of Eden surfaces The methodology for building phenomenological models of fractal surfaces which we employ consists of constructing appropriate multisegmented power-law functions [7}12]. That this approach is applicable to Eden surfaces is suggested in Fig. 2, where pro"le spectra, corresponding to 10 unevenly spaced moments, are plotted in a log}log scale. The time extent used in Fig. 2 covers the entire growth and evolution of the simulated aggregate. It is seen that the spectra evolution undergoes a steady build up of a right (large wavenumber) power-law spectral branch, supplemented by one or perhaps two white-noise-like components at intermediate and small wave number regions. Physically, this corresponds to the generation and enlargement through a coalescence of clusters until a single large cluster "lls up the entire lattice. The "nal cluster contains a self-a$ne hierarchy of subclusters forming a fractal-like morphology. Accordingly, we shall attempt to approximate the spectra at all times by the following three segment piece-wise continuous function:
Ak\?(k /k )\?@ ("k"/k )?? , k )"k")k , @ ? @ ? $ ? k )"k")k , Ak\?("k"/k )?@ , ? @ @ @ S(k)" Ak\?, k )"k")k , @ , 0, otherwise,
(2)
where k "p/¸ , and k "p/*x, are the fundamental and the Nyquist wavenumbers, respective$ , ly. The relatively #at, noise-like, form of the left and the middle regions of the spectra shown in Fig. 2, suggests that both a and a should be small compared to the main branch spectral exponent ? @ a, especially at early times. The time dependence of the surface statistics is accounted for by assuming that the spectral parameters vary with time. In addition, the presence of an overall white-noise component can be identi"ed in most of the studied pro"les. This component manifests itself in the graphs of the sample ACF as a peak at zero,
Fig. 2. Spectra of the roughness pro"les for ten selected times during the evolution: t"4, 8, 16, 32, 64, 128, 256, 512, 1024, and 2048, from the bottom to the top.
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see below. The noise component is accounted for by augmenting S(k) with
A S (k)" U L U L 0
k )"k")k , $ , otherwise,
(3)
where A is a constant. This brings the total number of free parameters of the model to seven: a , U L ? a , a, A, k , k , and A . The contributions of the various spectral components to the ACF of the @ ? @ U L model, A(x, t), is additive and can be calculated exactly in terms of type (1, 2) hypergeometric functions, F , [7}12]. The leading asymptotic term of A(x, t) for intermediate distance lags, A(x, t)&A(0, t)!q\?x?\, constitutes the self-a$nity of the model. Here, the parameter q, called topothesy, depends on the spectral constant A and a only. The topothesy is a measure of the intensity of the small, fractality scales of the structure. The leading term also shows that the slopes measured from the graphs of the SF, Fig. 1, are given by a!1. On the other hand, the expression for A(0, t) relates the roughness exponent s to a through s"(a!1)/2, see [11]. The higher asymptotic orders account for the "nite-size e!ects and show that the self-a$nity of the model is only approximate. They allow to introduce an additional parameter * large scale correlation length * which for the model (2) is given by l "(a!1)k\. A detailed study of the approximate @ self-a$nity can be found in [16,17].
4. Illustration of the model applicability and discussions We validate the model not by "tting (2) to the periodogram estimate of the pro"le spectra, but rather by carrying out a non-linear least-squares "t to the sample ACF [7}12]. This approach avoids the di$culties associated with the practical implementation of the classical spectral analysis [15]. As a trade-o!, one has to cope with the speci"c biases of the estimate (1). There are two sources of bias for the sample ACF. The "rst is brought about by using the factor (1/¸) (rather than 1/(¸!l)) in (1) and has a magnitude of (x /¸)A(x, t), where A(x, t) is the &true' ACF of the interface J [12]. The second bias is caused by the use in (1) of the sample, rather that the true mean height. The derivation of an expression for the second bias is somewhat lengthy and will be presented elsewhere. It is equivalent and in fact more convenient to reckon with the biases by "tting AQ with its theoretical mean (regarding the sample ACF itself as random "eld). The latter is expressed in terms of the true ACF, for which we substitute the model's ACF. A typical result of the outlined procedure is exempli"ed in Fig. 3, where we show the sample ACF (open circles) for a morphology generated at relatively early stage of the growth: t"32. The error bars inside each circle indicate the sample variances of AQ(x, t), related to every di!erent x. The theoretical mean of the sample ACF, yielding the best "t, is depicted by the solid line. The values of the model function's parameters that render the "t are: a "0.178$0.73;10\; ? a "0.957$0.29;10\; a"1.661$0.06; A"0.605$0.71;10\; k "0.075$0.86;10\; @ ? k "0.113$0.27;10\; and A "0.040$0.07. Using these values we compute q"3.27 and @ U L l "9.96. The model function itself is drawn by the dashed line. The inset gives some details of the initial branch of the ACF. It is seen that an exceptionally accurate "t has been obtained for all x. The average per point relative discrepancy is 19%. In the considered case the noise component is negligible, and accumulated within the large white-noise-like left branch of the model. The
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Fig. 3. Sample ACF (circles), model's ACF (dashed line) and the "tted mean of the sample ACF (solid line) corresponding to a pro"le recorded at t"32. The inset provides a focused view of the initial branch of the ACFs. For the values of the model's parameters rendering the best "t see the text.
Fig. 4. The same, except for the inset, as Fig. 4, however, for a pro"le generated at time t"2048.
incorporated in the ACF biases, although small in magnitude, have an important contribution to the good agreement, compare with the model's graph. Another example * corresponding to a late, saturated, regime * is shown in Fig. 4. More speci"cally, the surface pro"le is recorded at t"2048. The nomenclature of the symbols is the same as in Fig. 3. At such late stage of the growth, the middle segment of the spectrum seems to collapse and the "t has been carried out with k "k thereby eliminating the parameter a . The values of the @ ? @ remaining parameters yielding the best "t are: a "!0.5$2.587; a"2.023$0.014; ? A"0.322$0.018; k "(0.825$0.015);10\; and A "0.382$0.242. Again a very good "t ? U L is obtained, the average discrepancy per point now is just 1.5%. It is also evident that even the left branch of the spectrum practically disappears * compare the values of k and k * which ? $ explains the estimated large error bounds for a . At this regime, the surface may be viewed as ?
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having a simple fractal morphology characterized by a single exponent a. The value of a de"nes a roughness exponent of s"0.511$0.007, which indicates that the surface belongs to the Kardar}Parisi}Zhang universality class. The latter was suggested in a number of numerical simulations, but our result for s seems closer than that obtained before to the value of 0.5 speci"c for this class (within two standard deviations). It is interesting to note the large discrepancy between the model's ACF and the mean of the sample ACF, which emphasizes the importance of the biases for values of a close to 2. Note also that the corresponding values of topothesy, q"20.95 and the large-scale correlation length, l "124.3 pertain to the shape of A(x, t), rather than the sample ACF. The model was tested on all the pro"les recorded at the selected times and found to be accurate to the same degree as in the cases discussed above.
5. Conclusions In summary, we simulate a number of rough surface morphologies using version A of the Eden aggregation model. We con"rm the earlier established fact that this model leads to fractal-like surfaces by employing an approach based on the behavior of the sample structure function rather than the properties of the surface width. Using this, we constructed a three-segment spectral model of the two-point statistics of the pro"les, which incorporates the self-a$ne symmetry of the structure. The model was veri"ed by a non-linear least-squares "t to the sample autocovariance function. A key point of the "tting procedure is the explicit account for the biases of the sample ACF. The e!ect of the biases is especially pronounced for large values of the main spectral branch exponent a. On the bases of the retrieved parameters, we conclude that the exponent of the main spectral branch increases with time until it reaches saturation at about a"2.023$0.014. This value puts the Eden surface roughness exponent within two standard deviations from roughness exponent which characterizes the Kardar}Parisi}Zhang universality class.
Acknowledgements The authors thank A. Oliver and E. Dimitrova for their assistance in the programming and preparation of the manuscript.
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