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Effects of interface width scaling and spatial correlations on Ising systems with rough boundaries
Physica A 291 (2001) 375–386
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E!ects of interface width scaling and spatial correlations on Ising systems with rough boundaries F.D.A. Aar˜ao Reis Instituto de F sica, Universidade Federal Fluminense, Avenida Litorˆanea s/n, 24210-340 Niter oi RJ, Brazil Received 15 June 2000
Abstract We studied the ferromagnetic Ising model on two-dimensional systems with rough boundaries and several thickness distributions. First, we considered very long strips with discretized Gaussian distributions of widths with mean 36L612. Systems with 5xed interface width W and with increasing roughness (W ∼ L , with 0661) were analysed. We also considered strips with columns’ heights correlations, where the di!erence in the heights of neighbouring columns was 0 or 1, with 06 . 0:5. Free energies per spin, fL , and initial susceptibilities, L , were calculated using transfer matrix techniques at the critical temperature of the two-dimensional Ising model. The scaling of fL and of the 5nite-size estimates of the ratio of exponents (= )L was analysed. In systems with small and slowly increasing W ( ¡ 0:2), correlated or not, extrapolations of fL and (= )L to L → ∞ give accurate estimates of the critical parameters f∞ and = . In systems with rapidly increasing roughness, the deviations of those estimates from the exact values indicate the presence of large corrections in 5nite-size scaling relations. The comparison of systems with nearly the same shows that spatial correlations have no systematic e!ect on corrections to scaling. It suggests the reliability of uncorrelated models or related approximate systems for the description of scaling properties of thin magnetic 5lms and other low-dimensional structures. c 2001 Elsevier Science B.V. All rights reserved. PACS: 05.50.+q; 75.40.−s; 75.70.−i Keywords: Ising model; Surface roughness; Critical behaviour; Two-dimensional scaling; Low-dimensional systems
E-mail address: [email protected]!.br (F.D.A. Aar˜ao Reis). c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 5 1 5 - X
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1. Introduction Magnetic systems with 5nite dimensions have attracted much interest in last years due to developments in thin 5lms technology and their increasing number of applications. Experiments on several nanostructured systems have shown the importance of the geometry to their physical properties [1– 6]. The inHuence of surface roughness on critical behaviour and 5nite-size scaling relations is one of the interesting problems in this 5eld, and has been considered in experimental [1,3,4] and theoretical [7–12] works. Recently, we investigated that problem in two- and three-dimensional Ising systems with some uncorrelated roughness patterns [9 –12]. These works considered the ferromagnetic Ising model in strips and in thin 5lms with Gaussian distributions of thicknesses [9,10] or with a single partially 5lled layer [11,12]. Some interesting conclusions were obtained, for instance those concerning the scaling of critical temperatures with the average thickness [10,12]. Remarkable deviations from the expected 5nite-size scaling relations were observed in systems with a single partially 5lled layer and in systems with rapidly increasing interface width. The limitations of those studies were to consider only systems with uncorrelated surface roughness and to consider a small variety of surface patterns. The absence of correlations may be a good approximation for systems with small Huctuations in the average thickness, but it is not expected to be realistic when the interface width (the rms deviation of the local thickness distribution) is large. The aim of the present work is to consider the e!ects of interface width scaling and spatial correlations in the 5nite-size scaling relations of magnetic systems with rough boundaries. For that purpose, we will study two-dimensional systems (in5nitely long strips) with several roughness patterns. The possibility of heights correlations will be considered, as well as di!erent relations between the interface width W and the average thickness L, which determine a slow or rapid increase of roughness as L increases. In a real deposition process, the average thickness L is typically proportional to the deposition time, and W is related to L [13] as W ∼ L :
(1)
Several values of the exponent will be considered here ( = 0 represents the cases of 5xed roughness). We will study strips with average thicknesses up to L = 12 and local thicknesses varying from 1 to 17. We will use transfer matrix techniques [14,15] to calculate the free-energy per spin (fL ) of very long strips, and will obtain magnetic susceptibilities from numerical derivatives of fL . The calculations will be done at the critical temperature Tc of the two-dimensional Ising ferromagnet (kB Tc =J = 2:269 : : :) [17], where the critical parameters are exactly known. Models of thin 5lms would certainly be more suitable for the description of real systems, while strictly one-dimensional models (strips) have a small number of applications [5]. However, in previous works we observed that some relations between the
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roughness pattern and the scaling properties in two-dimensional models (strips) parallel some results in the corresponding three-dimensional systems (5lms). For instance, when W increases rapidly with L ( = 1), the scaling of pseudocritical temperatures of strips [9] and the scaling of Tc of 5lms [10] have remarkable corrections when compared to the corresponding uniform systems. Moreover, in systems with a single partially 5lled layer, oscillations in the thermodynamic quantities of strips [11], at 5xed temperature, parallel the oscillations of Tc in 5lms [12]. We consider that some conclusions of the present work will also justify the study of two-dimensional systems and will suggest interpretations of some experimental results for thin 5lms. This work is organized as follows. In Section 2, we will present details of the strips geometry and the methods of calculation. In Section 3, we will analyse the scaling of free energies and in Section 4, we will analyse the scaling of critical exponents obtained from magnetic susceptibilities. In Section 5 we summarize our results and present our conclusions.
2. Models and methods of calculation We will study strips of length N = 105 sites and one rough surface, as illustrated in Fig. 1. This length is suLcient to provide accurate estimates of physical quantities for an in5nitely long system. Free boundary conditions will be considered. The coupling J ¿ 0 is constant for all nearest-neighbour pairs. For each average thickness L, the values of physical quantities presented below will be averages over four estimates, each obtained from a di!erent realization of the given disorder distribution. First, we will consider six models of strips with uncorrelated distributions of local thicknesses, labelled 1A–1F. In these cases, the width of each column is chosen from discretized Gaussian distributions with mean L and variance ML2 =2, with no correlation in the heights of neighbouring columns. Then, the interface width W (Eq. (1)) is proportional to ML. The values of ML considered here are (1A) ML = 1; (1B) ML = L1=5 ; (1C) ML = L1=3 ; (1D) ML = L1=2 ; (1E) ML = L=5; (1F) ML = 3. Local heights in the range 16H 617 are allowed. In cases 1A and 1F, the strip has constant roughness for
Fig. 1. Long strips with a distribution of local thicknesses of mean L and rms Huctuation (interface width) W .
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Table 1 Labels of strips with uncorrelated (1A–1F) and correlated (2A–2F) roughness, exponents (exact or estimated), range of interface widths W of strips with 36L612 and asymptotic estimates of f∞ and (= ) obtained from the 5ts in Figs. 1– 4. Errors in the last digit, shown in parentheses, are standard deviations exact = 0:929695 : : : and (= )exact = 1:75 given by the 5tting procedure. Exact values are f∞ Strips
Wmin − Wmax
f∞
1A 1B 1C 1D 1E 1F 2A 2B 2C 2D 2E
0 0:2 0:333 : : : 0:5 1 0 0 0:11 0:18 0:25 0:54
0:71 − 0:71 0:87 − 1:16 1:00 − 1:60 1:16 − 2:26 0:33 − 1:67 2:12 − 2:12 0:61 − 0:61 0:70 − 0:82 0:75 − 0:98 0:83 − 1:19 0:32 − 1:00
0:92971(1) 0:9280(1) 0:9239(3) 0:893(1) 0:911(1) 0:9319(4) 0:92969(1) 0:9294(2) 0:9280(3) 0:9059(8) 0:9268(9)
1:7506(7) 1:73(4) 1:77(2) 1:80(2) 1:62(7) 1:719(3) 1:749(1) 1:74(3) 1:77(3) 1:82(4) 1:78(6)
any average thickness L, thus = 0; they are models of small and large roughness, respectively. In cases 1B–1E, W increases with L as in Eq. (1) ( ¿ 0). Case 1B represents a very slow increase of surface roughness, while case 1E is the extreme of a rapidly increasing roughness. The exponents of all systems are shown in Table 1. It is relevant to recall that only the limiting cases 1A and 1E were analysed in previous works [9,11]. We will also consider 5ve models of strips with correlated distributions of local thicknesses, labelled 2A–2E. In order to generate these strips, we will use the corresponding Gaussian distributions 1A–1E, but now we will introduce the condition that the di!erence of the heights of neighbouring columns cannot exceed 1. Consequently, the surfaces will be locally smooth. During the construction of the strip (column after column), the height of a new column (Hnew ) depends on the height of the previous column, Hpr , as follows. First, we choose a height Hc from the given Gaussian distribution (1A–1E). If Hc ¡ Hpr , then Hnew = Hpr − 1, i.e., the height decreases just one unit; if Hc ¿ Hpr , then Hnew = Hpr + 1, i.e., the height increases just one unit; if Hc = Hpr , then Hnew = Hpr . It is important to notice that the relations between W and L in cases 2A–2E are very di!erent from the corresponding relations in 1A–1E due to the correlations. In the correlated cases, W and L were measured in simulations of very long strips, and they assumed the scaling form in Eq. (1). The exponents were then estimated and are shown in Table 1. We also show in Table 1 the range of the interface width W measured in each case, where the minimum corresponds to L = 3 and the maximum to L = 12. The exponents for cases 1B and 2C are nearly the same, although di!erent distributions were used to generate the strips (notice that we are comparing an uncorrelated case with a correlated one). The same occurs with cases 1D and 2E. These properties will be useful to analyse the e!ect of correlations on 5nite-size scaling relations.
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The above method of choosing the columns’ heights generates strips with a limited range of local thicknesses. The existence of a maximum allowed local thickness (H = 17) is essential for the application of transfer matrix techniques for very long strips. Alternatively, we might have used deposition models with long times of di!usion and suitable energy barriers. However, it would not be helpful because our aim is to gain some insight into the general e!ects of correlations instead of understanding the behaviour of realistic models. The calculation of thermodynamic quantities proceeds as follows. We consider the ferromagnetic Ising model with nearest neighbour interactions of strength J in a uniform longitudinal 5eld h. The largest Lyapunov exponent 0L for a strip of average width L and length N =105 is calculated in the usual way [16]. Starting from an arbitrary initial vector C0 , one generates the transfer matrices Ti that connect columns i and i + 1, and applies them successively, to obtain N i=1 Ti C0 1 0 : (2) L = ln N C0 The average free energy per site is then fLave (T; h)=(1=L) 0L , in units of −J=kB T (thus, the minus sign that appears in the numerical values of the free energies, in J=kB T units, will be omitted). The initial susceptibility of a strip, L (T ), is given by @2 fLave (T; h) L (T ) = : (3) @h2 T; h=0 It is obtained numerically from the free energies calculated at (T; h = 0) and (T; h = 10−4 ), with h also in units of J=kB T . Our calculations will be done at the critical temperature Tc of the two-dimensional Ising model (kB Tc =J = 2:2691853 : : :) [17]. At that point, the asymptotic value of the (exact) free energy is exactly known: f∞ = 0:929695398 : : : [18], in units of −J=kB T . fL is expected to converge to f∞ with corrections in powers of 1=L when free edges are considered. The initial susceptibility at Tc scales as [19,20]
L (Tc ) ∼ L ;
(4)
with = = 1:75. The calculation of 5nite-size estimates (= )L (Section 4) is a useful method to test the convergence of the L data to the expected scaling form (Eq. (4)). In the following sections, standard extrapolations of the 5nite-size estimates fL and (= )L will be performed. Since accurate estimates of f∞ and = are obtained in the extrapolations of data of uniform strips (with periodic or free boundaries), the accuracy of the estimates in a system with rough surfaces is a test for the presence of roughness-induced corrections to scaling. In other words, remarkable e!ects of corrections to scaling are related to the deviations of the estimated f∞ and = from their exact values.
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3. Free energy scaling In Figs. 2a–d, we show fL versus 1=L for the uncorrelated strips. The data for strips with 5xed roughness (cases 1A and 1F) and for uniform strips (with free boundaries) are shown in Fig. 2a. Figs. 2b–d show results for strips with increasing roughness ( ¿ 0). The curves shown in Figs. 2a–d are 5ts of each set of data to second-degree polynomials in 1=L, using L = 5 to L = 12 (in case 1F, only data with 76L612 were considered because W is very large). The estimates of f∞ obtained from those 5ts (1=L → 0) are shown in Table 1. Fits to polynomials of other degrees do not improve the estimates of f∞ . In the uniform strips and in case 1A, the convergence of fL is very good; the relative di!erence between the exact and the estimated f∞ do not exceed 0:002%. Indeed it was already shown in previous works [9,11] that a small and 5xed roughness had weak e!ects on the scaling of thermodynamic quantities in strip geometry. Then very accurate estimates of f∞ are obtained from the 5ts of fL including up to second-order corrections in 1=L. In case 1F, where the interface width is very large but constant, an accurate estimate of f∞ is also obtained. The same occurs in case 1B, where the roughness slowly increases with L. In both cases, the di!erence from the exact value is nearly 0:2%. In case 1C, the estimate of f∞ deviates 0:6% from the exact value. In cases 1D and 1E
Fig. 2. Free energy per spin of strips with uncorrelated surface roughness and average thickness L: (a) uniform strips (5lled squares), cases 1A (open squares) and 1F (crosses); (b) case 1B (up triangles); (c) case 1C (down triangles); (d) cases 1D (diamonds) and 1E (+). Curves are second-degree polynomials that best 5t each set of data.
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Fig. 3. Free energy per spin of strips with correlated surface roughness and average thickness L: (a) case 2A (open squares); (b) case 2B (stars); (c) case 2C (up triangles); (d) cases 2D (down triangles) and 2E (diamonds). Curves are second-degree polynomials that best 5t each set of data.
the deviations are much larger, indicating a very slow convergence of the free energy scaling. In Figs. 3a–d we show fL versus 1=L for cases 2A–2E (strips with correlated roughness) and the 5ts of each set of data to second-degree polynomials in 1=L. In Table 1, we also show the estimates of f∞ obtained from those 5ts. The convergence of fL to f∞ in case 2A is as good as in case 1A or in the uniform strips. The same is observed in case 2B, where the roughness increases very slowly ( ≈ 0:1); the deviation of f∞ from the exact value is 0:03%. In case 2C, the deviation of f∞ is nearly the same of case 1B (see Figs. 2b and 3c). Those systems have approximately the same exponent , but 2C is correlated and 1B is not. On the other hand, case 2D have exponent intermediate between cases 1C and 1D, but shows much slower convergence of fL than those uncorrelated systems. Finally, cases 2E and 1D have nearly the same , but the convergence of fL in the correlated system is much better (Figs. 2d and 3d). This analysis proves that the presence of correlations do not lead (systematically) to more accurate estimates of f∞ when systems with the same roughness scaling (same ) are compared. It means that spatial correlations do not imply weaker corrections to scaling relations. The range of the interface width W (Table 1) also cannot be used to explain the deviations of the estimates of f∞ . These estimates could certainly be improved in all cases if suitable extrapolation variables were used. For instance, an extrapolation variable 1=L! , with ! = 1, could be
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used instead of 1=L. However, the above discussion indicates that no general relation between these variables and spatial correlations or the range of W are expected. On the other hand, our results show that small deviations in the free energy scaling of uniform strips are found in cases of 5xed roughness or slowly increasing roughness ( ¡ 0:2), although for larger no general trend is found. Finally, we can extend the previous calculation of the conformal anomaly [11] to systems with increasing roughness. In an in5nitely long strip of width L, it is expected that the free energy per spin at criticality has the scaling form [21,22] fs M f L = f∞ + (5) + 2 + ::: ; L L where 12 fs is the surface free energy (which depends on the boundary conditions), and M is a quantity that characterizes the universality class of the system. For free boundaries, M=c=24 (with f in units of −J=kB T ), where c is the conformal anomaly or the value of the central charge of the Virasoro algebra [21,23]. For the two-dimensional Ising model, c = 1=2. For strips with small and constant roughness (case 1A) we have previously obtained an accurate estimate of c, in agreement with the exact value [11]. In cases 2B and 2C (slowly increasing roughness), we obtain M = 0:070 ± 0:011 and M=0:061±0:014, respectively, from the 5ts in Figs. 3b and c. They give c = 0:53±0:09 and c = 0:47 ± 0:11, respectively, both values consistent with the two-dimensional Ising universality class. In the other cases with ¿ 0, much less accurate estimates are obtained. 4. Susceptibility scaling The standard method to obtain two-dimensional critical exponents from calculations in strips is to obtain their 5nite-size estimates (e!ective exponents) and extrapolate them to L → ∞. In the case of the ratio of exponents = (Eq. (3)), the 5nite-size estimates are ln[L =L−1 ] : (6) ≡ L ln[L=(L − 1)] In Figs. 4a–d, we show (= )L versus 1=L for uncorrelated strips (1A–1F). In Figs. 5a–d, we show (= )L versus 1=L for the strips with heights’ correlations (2A– 2E). The 5tting curves are the second or third-degree polynomials in 1=L that give the estimates of (1=L → 0) closest to the exact value. Those estimates are shown in Table 1. In the strips with small and 5xed roughness (1A and 2A), the estimates of = di!er less than 0:1% from the exact value, similarly to the case of uniform strips. It con5rms the convergence of the scaling relation (4) with the typical (rapidly decreasing) corrections in 1=L. In case 1F, where the interface width is constant but large, the 5t shown in Fig. 4a do not give an accurate estimate of = . Fits with higher order polynomials do not improve those estimates. This behaviour di!ers from the free energy scaling, which do not present remarkable corrections in case 1F (see Section 3).
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Fig. 4. (a)–(d) Finite-size estimates of the ratio of exponents (= )L of strips with uncorrelated surface roughness and average thickness L. Symbols are the same of Fig. 2. Curves are second (cases 1C, 1D and 1F) or third (uniform strips, cases 1A, 1B and 1E) degree polynomials that best 5t each set of data.
Fig. 5. (a)–(d) Finite-size estimates of the ratio of exponents (= )L of strips with correlated surface roughness and average thickness L. Symbols are the same of Fig. 3. Curves are third-degree polynomials that best 5t each set of data.
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In case 2B, where the interface width slowly increases, we obtain a very accurate estimate of = . In cases 1B and 2C, which have nearly the same , the estimates of = di!er nearly 1% from the exact value. Increasingly larger deviations are found in cases 1C, 2D, 1D, 2E and 1E, respectively. Then, also in the case of susceptibility scaling, the presence of correlations do not lead to a better convergence of 5nite-size estimates to their asymptotic values, when systems with nearly the same are compared. That convergence is rapid for slowly increasing roughness (typically ¡ 0:2) if the interface widths are small. Larger exponents or large interface widths (W ¿ 2 for L ∼ 10, as in case 1F) typically lead to larger di!erences between the estimated and the exact = , indicating stronger corrections to 5nite-size scaling relations due to surface roughness.
5. Summary and conclusion We studied the ferromagnetic Ising model in strips with one rough surface considering several roughness patterns, in order to analyse the dependence of 5nite-size scaling relations on interface width scaling and on spatial correlations. First, we considered strips with discretized Gaussian distributions of widths, thus with no correlation in the columns’ heights: two cases of 5xed roughness (1A and 1F) and four cases where the interface width increased with the average thickness (1B– 1E). Subsequently, we introduced correlations in the columns’ heights by restricting the di!erence between neighbouring heights to a maximum 1 (cases 2A–2E). Using transfer matrix techniques, we calculated the free energy per spin and the initial susceptibility at the critical temperature of the two-dimensional Ising model. The importance of corrections to 5nite-size scaling relations was investigated by studying the convergence of the free energy fL and the e!ective exponents (= )L as L → ∞. The convergence of fL and (= )L to their asymptotic values in systems with constant and small roughness has no remarkable di!erence when compared to uniform systems. It parallels the weak corrections to the scaling of critical temperatures of thin 5lms with constant surface roughness [10]. This result cannot be extended to systems with large interface widths, as shown in case 1F, where the convergence of (= )L was slower. If the roughness increases slowly with L (typically ¡ 0:2), then the estimates of f∞ and = are also very near the exact values. It proves that scaling relations have weak corrections in these conditions when compared to uniform systems, independently of the presence of spatial correlations. As the exponent increases, those estimates typically deviate from the exact values, which indicates the presence of larger corrections to scaling. However, spatial correlations have no systematic inHuence on those deviations when systems with nearly the same are compared. Consequently, it is not possible to make general quantitative predictions for systems with rapidly increasing roughness. It is interesting to discuss the consequences of extending these qualitative ideas to three-dimensional systems, i.e., thin 5lms with rough surfaces. First, the negligible corrections to scaling in strips with small suggest that no remarkable di!erence in the
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shift exponents # (of Tc scaling) is expected in good deposition conditions, i.e., when the interface width is small and slowly increasing. This suggestion agrees with the proposal of Farle et al. [2] that the scaling of Tc may be used to test the quality of the deposition process. That proposal was based on experimental results on Gd 5lms, and experiments of other groups also gave # exponents in agreement with uniform 5lms’ models [1] (an exception appears only in the limit of a perfect layer-by-layer growth, which leads to non-smooth Tc (L) curves if continuous L are considered [12]). Moreover, our results concerning the e!ects of spatial correlations suggest that unrealistic growth models (e.g. with uncorrelated surface roughness) may give reliable qualitative information on scaling properties of real systems, although they do not represent accurately their growth and morphology. It gives additional support to our previous comparisons of experimental results on thin 5lms and theoretical results obtained from models with uncorrelated surface roughness [10,12]. However, it is important to stress that these suggestions have to be con5rmed by studies of three-dimensional systems. Finally, we also consider that this work is a new motivation for studying the magnetism of models that incorporate only the basic aspects of the growth conditions, but that may describe the scaling of thermodynamic quantities in real systems. Acknowledgements This work was partially supported by CNPq and FAPERJ (Brazilian agencies). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
F. Huang, M.T. Kief, G.J. Mankey, R.F. Willis, Phys. Rev. B 49 (1994) 3962. M. Farle, B. Baberschke, U. Stetter, A. Aspelmeier, F. Gerhardter, Phys. Rev. B 47 (1993) 11 571. G. Garreau, M. Farle, E. Beaurepaire, B. Baberschke, Phys. Rev. B 55 (1997) 330. Q. Jiang, H.-N. Yang, G.-C. Wang, Surf. Sci. 373 (1997) 181. J. Shen, R. Skomski, M. Klaua, H. Jenniches, S. Sundar Manoharan, J. Kirschner, Phys. Rev. B 56 (1997) 2340. H.J. Elmers, J. Hauschild, H. HUoche, U. Gradmann, H. Bethge, D. Heuer, U. KUohler, Phys. Rev. Lett. 73 (1994) 898. H.W. Diehl, A. NUusser, Z. Phys. B 79 (1990) 69. M. Pleimling, W. Selke, Eur. Phys. J. B 1 (1998) 385. F.D.A. Aar˜ao Reis, Phys. Rev. B 55 (1997) 11 084. F.D.A. Aar˜ao Reis, Phys. Rev. B 58 (1998) 394. F.D.A. Aar˜ao Reis, J. Phys. A 31 (1998) 9105. F.D.A. Aar˜ao Reis, Phys. Rev. B 62 (2000) 6565. A.-L. BarabWasi, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, New York, 1995. M.P. Nightingale, in: V. Privman (Ed.), Finite Size Scaling and Numerical Simulations of Statistical Systems, World Scienti5c, Singapore, 1990. H.W.J. BlUote, M.P. Nightingale, Physica A 112 (1982) 405. A. Crisanti, G. Paladin, A. Vulpiani, in: H.K. Lotsch (Ed.), Products of Random Matrices in Statistical Physics, Springer Series in Solid State Sciences, Vol. 104, Springer, Berlin, 1993. L. Onsager, Phys. Rev. 65 (1944) 117. R.J. Baxter, J. Phys. C 6 (1973) L445.
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[19] M.E. Fisher, Critical phenomena, in: M.S. Green (Ed.), Proceedings of the International School of Physics Enrico Fermi, Course LI, Varenna, 1970, Academic, New York, 1971. [20] M.N. Barber, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, Vol. 8, Academic, New York, 1983. [21] H.W. BlUote, J.L. Cardy, M.P. Nightingale, Phys. Rev. Lett. 56 (1986) 742. [22] I. AXeck, Phys. Rev. Lett. 56 (1986) 746. [23] M.A. Virasoro, Phys. Rev. D 1 (1970) 2933.
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E!ects of interface width scaling and spatial correlations on Ising systems with rough boundaries F.D.A. Aar˜ao Reis Instituto de F sica, Universidade Federal Fluminense, Avenida Litorˆanea s/n, 24210-340 Niter oi RJ, Brazil Received 15 June 2000
Abstract We studied the ferromagnetic Ising model on two-dimensional systems with rough boundaries and several thickness distributions. First, we considered very long strips with discretized Gaussian distributions of widths with mean 36L612. Systems with 5xed interface width W and with increasing roughness (W ∼ L , with 0661) were analysed. We also considered strips with columns’ heights correlations, where the di!erence in the heights of neighbouring columns was 0 or 1, with 06 . 0:5. Free energies per spin, fL , and initial susceptibilities, L , were calculated using transfer matrix techniques at the critical temperature of the two-dimensional Ising model. The scaling of fL and of the 5nite-size estimates of the ratio of exponents (= )L was analysed. In systems with small and slowly increasing W ( ¡ 0:2), correlated or not, extrapolations of fL and (= )L to L → ∞ give accurate estimates of the critical parameters f∞ and = . In systems with rapidly increasing roughness, the deviations of those estimates from the exact values indicate the presence of large corrections in 5nite-size scaling relations. The comparison of systems with nearly the same shows that spatial correlations have no systematic e!ect on corrections to scaling. It suggests the reliability of uncorrelated models or related approximate systems for the description of scaling properties of thin magnetic 5lms and other low-dimensional structures. c 2001 Elsevier Science B.V. All rights reserved. PACS: 05.50.+q; 75.40.−s; 75.70.−i Keywords: Ising model; Surface roughness; Critical behaviour; Two-dimensional scaling; Low-dimensional systems
E-mail address: [email protected]!.br (F.D.A. Aar˜ao Reis). c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 5 1 5 - X
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1. Introduction Magnetic systems with 5nite dimensions have attracted much interest in last years due to developments in thin 5lms technology and their increasing number of applications. Experiments on several nanostructured systems have shown the importance of the geometry to their physical properties [1– 6]. The inHuence of surface roughness on critical behaviour and 5nite-size scaling relations is one of the interesting problems in this 5eld, and has been considered in experimental [1,3,4] and theoretical [7–12] works. Recently, we investigated that problem in two- and three-dimensional Ising systems with some uncorrelated roughness patterns [9 –12]. These works considered the ferromagnetic Ising model in strips and in thin 5lms with Gaussian distributions of thicknesses [9,10] or with a single partially 5lled layer [11,12]. Some interesting conclusions were obtained, for instance those concerning the scaling of critical temperatures with the average thickness [10,12]. Remarkable deviations from the expected 5nite-size scaling relations were observed in systems with a single partially 5lled layer and in systems with rapidly increasing interface width. The limitations of those studies were to consider only systems with uncorrelated surface roughness and to consider a small variety of surface patterns. The absence of correlations may be a good approximation for systems with small Huctuations in the average thickness, but it is not expected to be realistic when the interface width (the rms deviation of the local thickness distribution) is large. The aim of the present work is to consider the e!ects of interface width scaling and spatial correlations in the 5nite-size scaling relations of magnetic systems with rough boundaries. For that purpose, we will study two-dimensional systems (in5nitely long strips) with several roughness patterns. The possibility of heights correlations will be considered, as well as di!erent relations between the interface width W and the average thickness L, which determine a slow or rapid increase of roughness as L increases. In a real deposition process, the average thickness L is typically proportional to the deposition time, and W is related to L [13] as W ∼ L :
(1)
Several values of the exponent will be considered here ( = 0 represents the cases of 5xed roughness). We will study strips with average thicknesses up to L = 12 and local thicknesses varying from 1 to 17. We will use transfer matrix techniques [14,15] to calculate the free-energy per spin (fL ) of very long strips, and will obtain magnetic susceptibilities from numerical derivatives of fL . The calculations will be done at the critical temperature Tc of the two-dimensional Ising ferromagnet (kB Tc =J = 2:269 : : :) [17], where the critical parameters are exactly known. Models of thin 5lms would certainly be more suitable for the description of real systems, while strictly one-dimensional models (strips) have a small number of applications [5]. However, in previous works we observed that some relations between the
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roughness pattern and the scaling properties in two-dimensional models (strips) parallel some results in the corresponding three-dimensional systems (5lms). For instance, when W increases rapidly with L ( = 1), the scaling of pseudocritical temperatures of strips [9] and the scaling of Tc of 5lms [10] have remarkable corrections when compared to the corresponding uniform systems. Moreover, in systems with a single partially 5lled layer, oscillations in the thermodynamic quantities of strips [11], at 5xed temperature, parallel the oscillations of Tc in 5lms [12]. We consider that some conclusions of the present work will also justify the study of two-dimensional systems and will suggest interpretations of some experimental results for thin 5lms. This work is organized as follows. In Section 2, we will present details of the strips geometry and the methods of calculation. In Section 3, we will analyse the scaling of free energies and in Section 4, we will analyse the scaling of critical exponents obtained from magnetic susceptibilities. In Section 5 we summarize our results and present our conclusions.
2. Models and methods of calculation We will study strips of length N = 105 sites and one rough surface, as illustrated in Fig. 1. This length is suLcient to provide accurate estimates of physical quantities for an in5nitely long system. Free boundary conditions will be considered. The coupling J ¿ 0 is constant for all nearest-neighbour pairs. For each average thickness L, the values of physical quantities presented below will be averages over four estimates, each obtained from a di!erent realization of the given disorder distribution. First, we will consider six models of strips with uncorrelated distributions of local thicknesses, labelled 1A–1F. In these cases, the width of each column is chosen from discretized Gaussian distributions with mean L and variance ML2 =2, with no correlation in the heights of neighbouring columns. Then, the interface width W (Eq. (1)) is proportional to ML. The values of ML considered here are (1A) ML = 1; (1B) ML = L1=5 ; (1C) ML = L1=3 ; (1D) ML = L1=2 ; (1E) ML = L=5; (1F) ML = 3. Local heights in the range 16H 617 are allowed. In cases 1A and 1F, the strip has constant roughness for
Fig. 1. Long strips with a distribution of local thicknesses of mean L and rms Huctuation (interface width) W .
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Table 1 Labels of strips with uncorrelated (1A–1F) and correlated (2A–2F) roughness, exponents (exact or estimated), range of interface widths W of strips with 36L612 and asymptotic estimates of f∞ and (= ) obtained from the 5ts in Figs. 1– 4. Errors in the last digit, shown in parentheses, are standard deviations exact = 0:929695 : : : and (= )exact = 1:75 given by the 5tting procedure. Exact values are f∞ Strips
Wmin − Wmax
f∞
1A 1B 1C 1D 1E 1F 2A 2B 2C 2D 2E
0 0:2 0:333 : : : 0:5 1 0 0 0:11 0:18 0:25 0:54
0:71 − 0:71 0:87 − 1:16 1:00 − 1:60 1:16 − 2:26 0:33 − 1:67 2:12 − 2:12 0:61 − 0:61 0:70 − 0:82 0:75 − 0:98 0:83 − 1:19 0:32 − 1:00
0:92971(1) 0:9280(1) 0:9239(3) 0:893(1) 0:911(1) 0:9319(4) 0:92969(1) 0:9294(2) 0:9280(3) 0:9059(8) 0:9268(9)
1:7506(7) 1:73(4) 1:77(2) 1:80(2) 1:62(7) 1:719(3) 1:749(1) 1:74(3) 1:77(3) 1:82(4) 1:78(6)
any average thickness L, thus = 0; they are models of small and large roughness, respectively. In cases 1B–1E, W increases with L as in Eq. (1) ( ¿ 0). Case 1B represents a very slow increase of surface roughness, while case 1E is the extreme of a rapidly increasing roughness. The exponents of all systems are shown in Table 1. It is relevant to recall that only the limiting cases 1A and 1E were analysed in previous works [9,11]. We will also consider 5ve models of strips with correlated distributions of local thicknesses, labelled 2A–2E. In order to generate these strips, we will use the corresponding Gaussian distributions 1A–1E, but now we will introduce the condition that the di!erence of the heights of neighbouring columns cannot exceed 1. Consequently, the surfaces will be locally smooth. During the construction of the strip (column after column), the height of a new column (Hnew ) depends on the height of the previous column, Hpr , as follows. First, we choose a height Hc from the given Gaussian distribution (1A–1E). If Hc ¡ Hpr , then Hnew = Hpr − 1, i.e., the height decreases just one unit; if Hc ¿ Hpr , then Hnew = Hpr + 1, i.e., the height increases just one unit; if Hc = Hpr , then Hnew = Hpr . It is important to notice that the relations between W and L in cases 2A–2E are very di!erent from the corresponding relations in 1A–1E due to the correlations. In the correlated cases, W and L were measured in simulations of very long strips, and they assumed the scaling form in Eq. (1). The exponents were then estimated and are shown in Table 1. We also show in Table 1 the range of the interface width W measured in each case, where the minimum corresponds to L = 3 and the maximum to L = 12. The exponents for cases 1B and 2C are nearly the same, although di!erent distributions were used to generate the strips (notice that we are comparing an uncorrelated case with a correlated one). The same occurs with cases 1D and 2E. These properties will be useful to analyse the e!ect of correlations on 5nite-size scaling relations.
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The above method of choosing the columns’ heights generates strips with a limited range of local thicknesses. The existence of a maximum allowed local thickness (H = 17) is essential for the application of transfer matrix techniques for very long strips. Alternatively, we might have used deposition models with long times of di!usion and suitable energy barriers. However, it would not be helpful because our aim is to gain some insight into the general e!ects of correlations instead of understanding the behaviour of realistic models. The calculation of thermodynamic quantities proceeds as follows. We consider the ferromagnetic Ising model with nearest neighbour interactions of strength J in a uniform longitudinal 5eld h. The largest Lyapunov exponent 0L for a strip of average width L and length N =105 is calculated in the usual way [16]. Starting from an arbitrary initial vector C0 , one generates the transfer matrices Ti that connect columns i and i + 1, and applies them successively, to obtain N i=1 Ti C0 1 0 : (2) L = ln N C0 The average free energy per site is then fLave (T; h)=(1=L) 0L , in units of −J=kB T (thus, the minus sign that appears in the numerical values of the free energies, in J=kB T units, will be omitted). The initial susceptibility of a strip, L (T ), is given by @2 fLave (T; h) L (T ) = : (3) @h2 T; h=0 It is obtained numerically from the free energies calculated at (T; h = 0) and (T; h = 10−4 ), with h also in units of J=kB T . Our calculations will be done at the critical temperature Tc of the two-dimensional Ising model (kB Tc =J = 2:2691853 : : :) [17]. At that point, the asymptotic value of the (exact) free energy is exactly known: f∞ = 0:929695398 : : : [18], in units of −J=kB T . fL is expected to converge to f∞ with corrections in powers of 1=L when free edges are considered. The initial susceptibility at Tc scales as [19,20]
L (Tc ) ∼ L ;
(4)
with = = 1:75. The calculation of 5nite-size estimates (= )L (Section 4) is a useful method to test the convergence of the L data to the expected scaling form (Eq. (4)). In the following sections, standard extrapolations of the 5nite-size estimates fL and (= )L will be performed. Since accurate estimates of f∞ and = are obtained in the extrapolations of data of uniform strips (with periodic or free boundaries), the accuracy of the estimates in a system with rough surfaces is a test for the presence of roughness-induced corrections to scaling. In other words, remarkable e!ects of corrections to scaling are related to the deviations of the estimated f∞ and = from their exact values.
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3. Free energy scaling In Figs. 2a–d, we show fL versus 1=L for the uncorrelated strips. The data for strips with 5xed roughness (cases 1A and 1F) and for uniform strips (with free boundaries) are shown in Fig. 2a. Figs. 2b–d show results for strips with increasing roughness ( ¿ 0). The curves shown in Figs. 2a–d are 5ts of each set of data to second-degree polynomials in 1=L, using L = 5 to L = 12 (in case 1F, only data with 76L612 were considered because W is very large). The estimates of f∞ obtained from those 5ts (1=L → 0) are shown in Table 1. Fits to polynomials of other degrees do not improve the estimates of f∞ . In the uniform strips and in case 1A, the convergence of fL is very good; the relative di!erence between the exact and the estimated f∞ do not exceed 0:002%. Indeed it was already shown in previous works [9,11] that a small and 5xed roughness had weak e!ects on the scaling of thermodynamic quantities in strip geometry. Then very accurate estimates of f∞ are obtained from the 5ts of fL including up to second-order corrections in 1=L. In case 1F, where the interface width is very large but constant, an accurate estimate of f∞ is also obtained. The same occurs in case 1B, where the roughness slowly increases with L. In both cases, the di!erence from the exact value is nearly 0:2%. In case 1C, the estimate of f∞ deviates 0:6% from the exact value. In cases 1D and 1E
Fig. 2. Free energy per spin of strips with uncorrelated surface roughness and average thickness L: (a) uniform strips (5lled squares), cases 1A (open squares) and 1F (crosses); (b) case 1B (up triangles); (c) case 1C (down triangles); (d) cases 1D (diamonds) and 1E (+). Curves are second-degree polynomials that best 5t each set of data.
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Fig. 3. Free energy per spin of strips with correlated surface roughness and average thickness L: (a) case 2A (open squares); (b) case 2B (stars); (c) case 2C (up triangles); (d) cases 2D (down triangles) and 2E (diamonds). Curves are second-degree polynomials that best 5t each set of data.
the deviations are much larger, indicating a very slow convergence of the free energy scaling. In Figs. 3a–d we show fL versus 1=L for cases 2A–2E (strips with correlated roughness) and the 5ts of each set of data to second-degree polynomials in 1=L. In Table 1, we also show the estimates of f∞ obtained from those 5ts. The convergence of fL to f∞ in case 2A is as good as in case 1A or in the uniform strips. The same is observed in case 2B, where the roughness increases very slowly ( ≈ 0:1); the deviation of f∞ from the exact value is 0:03%. In case 2C, the deviation of f∞ is nearly the same of case 1B (see Figs. 2b and 3c). Those systems have approximately the same exponent , but 2C is correlated and 1B is not. On the other hand, case 2D have exponent intermediate between cases 1C and 1D, but shows much slower convergence of fL than those uncorrelated systems. Finally, cases 2E and 1D have nearly the same , but the convergence of fL in the correlated system is much better (Figs. 2d and 3d). This analysis proves that the presence of correlations do not lead (systematically) to more accurate estimates of f∞ when systems with the same roughness scaling (same ) are compared. It means that spatial correlations do not imply weaker corrections to scaling relations. The range of the interface width W (Table 1) also cannot be used to explain the deviations of the estimates of f∞ . These estimates could certainly be improved in all cases if suitable extrapolation variables were used. For instance, an extrapolation variable 1=L! , with ! = 1, could be
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used instead of 1=L. However, the above discussion indicates that no general relation between these variables and spatial correlations or the range of W are expected. On the other hand, our results show that small deviations in the free energy scaling of uniform strips are found in cases of 5xed roughness or slowly increasing roughness ( ¡ 0:2), although for larger no general trend is found. Finally, we can extend the previous calculation of the conformal anomaly [11] to systems with increasing roughness. In an in5nitely long strip of width L, it is expected that the free energy per spin at criticality has the scaling form [21,22] fs M f L = f∞ + (5) + 2 + ::: ; L L where 12 fs is the surface free energy (which depends on the boundary conditions), and M is a quantity that characterizes the universality class of the system. For free boundaries, M=c=24 (with f in units of −J=kB T ), where c is the conformal anomaly or the value of the central charge of the Virasoro algebra [21,23]. For the two-dimensional Ising model, c = 1=2. For strips with small and constant roughness (case 1A) we have previously obtained an accurate estimate of c, in agreement with the exact value [11]. In cases 2B and 2C (slowly increasing roughness), we obtain M = 0:070 ± 0:011 and M=0:061±0:014, respectively, from the 5ts in Figs. 3b and c. They give c = 0:53±0:09 and c = 0:47 ± 0:11, respectively, both values consistent with the two-dimensional Ising universality class. In the other cases with ¿ 0, much less accurate estimates are obtained. 4. Susceptibility scaling The standard method to obtain two-dimensional critical exponents from calculations in strips is to obtain their 5nite-size estimates (e!ective exponents) and extrapolate them to L → ∞. In the case of the ratio of exponents = (Eq. (3)), the 5nite-size estimates are ln[L =L−1 ] : (6) ≡ L ln[L=(L − 1)] In Figs. 4a–d, we show (= )L versus 1=L for uncorrelated strips (1A–1F). In Figs. 5a–d, we show (= )L versus 1=L for the strips with heights’ correlations (2A– 2E). The 5tting curves are the second or third-degree polynomials in 1=L that give the estimates of (1=L → 0) closest to the exact value. Those estimates are shown in Table 1. In the strips with small and 5xed roughness (1A and 2A), the estimates of = di!er less than 0:1% from the exact value, similarly to the case of uniform strips. It con5rms the convergence of the scaling relation (4) with the typical (rapidly decreasing) corrections in 1=L. In case 1F, where the interface width is constant but large, the 5t shown in Fig. 4a do not give an accurate estimate of = . Fits with higher order polynomials do not improve those estimates. This behaviour di!ers from the free energy scaling, which do not present remarkable corrections in case 1F (see Section 3).
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Fig. 4. (a)–(d) Finite-size estimates of the ratio of exponents (= )L of strips with uncorrelated surface roughness and average thickness L. Symbols are the same of Fig. 2. Curves are second (cases 1C, 1D and 1F) or third (uniform strips, cases 1A, 1B and 1E) degree polynomials that best 5t each set of data.
Fig. 5. (a)–(d) Finite-size estimates of the ratio of exponents (= )L of strips with correlated surface roughness and average thickness L. Symbols are the same of Fig. 3. Curves are third-degree polynomials that best 5t each set of data.
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In case 2B, where the interface width slowly increases, we obtain a very accurate estimate of = . In cases 1B and 2C, which have nearly the same , the estimates of = di!er nearly 1% from the exact value. Increasingly larger deviations are found in cases 1C, 2D, 1D, 2E and 1E, respectively. Then, also in the case of susceptibility scaling, the presence of correlations do not lead to a better convergence of 5nite-size estimates to their asymptotic values, when systems with nearly the same are compared. That convergence is rapid for slowly increasing roughness (typically ¡ 0:2) if the interface widths are small. Larger exponents or large interface widths (W ¿ 2 for L ∼ 10, as in case 1F) typically lead to larger di!erences between the estimated and the exact = , indicating stronger corrections to 5nite-size scaling relations due to surface roughness.
5. Summary and conclusion We studied the ferromagnetic Ising model in strips with one rough surface considering several roughness patterns, in order to analyse the dependence of 5nite-size scaling relations on interface width scaling and on spatial correlations. First, we considered strips with discretized Gaussian distributions of widths, thus with no correlation in the columns’ heights: two cases of 5xed roughness (1A and 1F) and four cases where the interface width increased with the average thickness (1B– 1E). Subsequently, we introduced correlations in the columns’ heights by restricting the di!erence between neighbouring heights to a maximum 1 (cases 2A–2E). Using transfer matrix techniques, we calculated the free energy per spin and the initial susceptibility at the critical temperature of the two-dimensional Ising model. The importance of corrections to 5nite-size scaling relations was investigated by studying the convergence of the free energy fL and the e!ective exponents (= )L as L → ∞. The convergence of fL and (= )L to their asymptotic values in systems with constant and small roughness has no remarkable di!erence when compared to uniform systems. It parallels the weak corrections to the scaling of critical temperatures of thin 5lms with constant surface roughness [10]. This result cannot be extended to systems with large interface widths, as shown in case 1F, where the convergence of (= )L was slower. If the roughness increases slowly with L (typically ¡ 0:2), then the estimates of f∞ and = are also very near the exact values. It proves that scaling relations have weak corrections in these conditions when compared to uniform systems, independently of the presence of spatial correlations. As the exponent increases, those estimates typically deviate from the exact values, which indicates the presence of larger corrections to scaling. However, spatial correlations have no systematic inHuence on those deviations when systems with nearly the same are compared. Consequently, it is not possible to make general quantitative predictions for systems with rapidly increasing roughness. It is interesting to discuss the consequences of extending these qualitative ideas to three-dimensional systems, i.e., thin 5lms with rough surfaces. First, the negligible corrections to scaling in strips with small suggest that no remarkable di!erence in the
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shift exponents # (of Tc scaling) is expected in good deposition conditions, i.e., when the interface width is small and slowly increasing. This suggestion agrees with the proposal of Farle et al. [2] that the scaling of Tc may be used to test the quality of the deposition process. That proposal was based on experimental results on Gd 5lms, and experiments of other groups also gave # exponents in agreement with uniform 5lms’ models [1] (an exception appears only in the limit of a perfect layer-by-layer growth, which leads to non-smooth Tc (L) curves if continuous L are considered [12]). Moreover, our results concerning the e!ects of spatial correlations suggest that unrealistic growth models (e.g. with uncorrelated surface roughness) may give reliable qualitative information on scaling properties of real systems, although they do not represent accurately their growth and morphology. It gives additional support to our previous comparisons of experimental results on thin 5lms and theoretical results obtained from models with uncorrelated surface roughness [10,12]. However, it is important to stress that these suggestions have to be con5rmed by studies of three-dimensional systems. Finally, we also consider that this work is a new motivation for studying the magnetism of models that incorporate only the basic aspects of the growth conditions, but that may describe the scaling of thermodynamic quantities in real systems. Acknowledgements This work was partially supported by CNPq and FAPERJ (Brazilian agencies). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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