DSpace at VNU: A continuum percolation model in an anisotropic medium: dimensional crossover?

Physica A 316 (2002) – 12 www.elsevier.com/locate/physa A continuum percolation model in an anisotropic medium: dimensional crossover? Nguyen Van Liena;∗ , Dang Dinh Toib , Nguyen Hoai Namc a Institute of Physics, P.O Box 429 Bo Ho, Hanoi 10000, Viet Nam Faculty, Hanoi State University, 90 Nguyen Trai, Hanoi, Viet Nam c Physics Faculty, Hanoi University of Education, Cau-giay, Hanoi, Viet Nam b Physics Received 18 December 2001 Abstract We propose a two-dimensional continuum percolation model in an anisotropic medium, which consists of parallel chains of sites coupled to each other weakly via rare “impurities” By checking simultaneously two percolations, parallel and perpendicular to the chain direction, we show that while there is a quasi-one-dimensional to two-dimensional crossover in the percolation radius of ÿnite systems as the “impurity” density s increases, in the limit of inÿnite systems two percolations are equivalent in the sense that their main characters are, respectively, coincident, regardless of s The proposed model is assumed to be used for describing hopping conduction networks in the compounds such as conjugated polymers or porous silicon c 2002 Elsevier Science B.V All rights reserved PACS: 64.60.Ak; 64.60.Cn; 05.40.+j Keywords: Two-dimensional percolation; Anisotropic medium; Dimensional crossover The model Recently, there has been a number of works reporting an observation of dimensional crossovers in the temperature dependence behaviour of the variable-range hopping (VRH) conductivity in various compounds, such as the conjugated polymers [1] or porous silicon [2] In di erence from that induced by the electric ÿeld [3], the crossovers reported in Refs [1,2] are believed to be associated with the speciÿc ∗ Corresponding author Tel.: +84-4-843-5917; fax: +84-4-8349050 E-mail address: nvlien@iop.ncst.ac.vn (N Van Lien) 0378-4371/02/$ - see front matter c 2002 Elsevier Science B.V All rights reserved PII: S - ( ) - N Van Lien et al / Physica A 316 (2002) – 12 microstructure of materials It was shown that in conducting conjugated polymers far from the insulator-to-metal transition the conducting network consists of parallel linear chains which are weakly coupled to each other via rare impurities (or small metallic particles) At low doping levels (impurity concentration) [1] and=or low temperatures [2] the typical interchain hopping rate E is much less than the typical intrachain rate I that gives rise to the quasi-one dimensional (Q1D) behaviour of conductivity With increasing doping level (or temperature) the interchain rate E increases rapidly and at some critical doping level (temperature) it becomes comparable to the intrachain rate I , i.e., two directions, parallel and perpendicular to the chain direction, become equal in percolation This may result in a Q1D to 2D (3D) crossover in temperature dependence behaviour of VRH conductivity It is widely accepted that the percolation method is one of the best approaches in describing the hopping conduction properties [4,5] To formulate a percolation model corresponding to the experimental systems of Refs [1,2] and to calculate its percolation characters in the 2D case are the aim of the present work Let us consider the (L × L)-square with sides parallel to the x- and y-axis of the Decartesian coordinates system Inside the square n linear chains of sites (atoms) are generated in the way that they are parallel with the x-direction and regularly separated from each other Along each chain the sites are randomly arranged by the Poisson distribution with an average separation being equal to (the length units in the simulation) Next, the impurities (small metallic islands) are modelled by sites added at random into the interchain spaces with a density s ¡ In the result, we have, in general, a quasi-anisotropic 2D system of sites with random coordinates as displayed schematically in Fig Due to a randomness of site coordinates this system may be thought of as a continuum percolation model in a structurally anisotropic medium [5 –7] The model is thus characterized by two physical parameters: the “chain density” = n=L 1 and the interchain “impurity” density s ¡ (the linear density of site in a chain is equal to 1) In the limit of n = and s = 0, we have a ÿnite 1D system of length L In the other limit of s → the system should tend to behave like a quasi-isotropic 2D system For a given , 1=L 1, and a given s ¡ 1, the system is generally anisotropic in the sense that in the direction longitudinal to the chains the percolation proceeds easier than that in the perpendicular direction In other words, for the same ÿnite sample with a low density s, the threshold radius Rc(t) of percolating from the upper to the lower edge of the square sample (see, Fig 1) in the direction perpendicular to the chains (transverse percolation—TP) should be greater than the corresponding radius Rc(l) of percolating from the left to the right edge along the chain direction (longitudinal percolation—LP): Rc(t) =Rc(l) ¿ Certainly, with increasing the density s, the ratio Rc(t) =Rc(l) decreases and it is natural to assume that there must exist a critical density s = sc such that for all s ¿ sc two percolation radii are practically coincident Such a change in the relation between two percolation radii could perhaps produce a dimensional crossover in the temperature dependence behaviour of the VRH conductivity since the latter is mainly determined by the typical hopping length, which is in turn associated with the corresponding percolation radius [5] Deÿnitely, the critical density sc depends on the chain density However, even for a given , it is still not N Van Lien et al / Physica A 316 (2002) – 12 Fig The simulation model: medium is anisotropic, but percolating process is isotropic (circle problem) clear (i) how the crossover behaviour depends on the sample size L (if there has a crossover in an inÿnite system) and (ii) if other percolation characters, namely, the percolation exponent [4,5] and the fractal dimension D [4], experience the same crossover As an attempt to ÿnd answers to these questions, in the present work both percolations, TP and LP, are simultaneously simulated in samples with di erent sizes L and di erent values of s for some typical values of the chain density The assumed dimensional crossover will be checked by comparing corresponding characters of LP and TP, taking into account the ÿnite size e ects Calculations and numerical results Given a sample of size L with and s ¡ 1, the percolations are examined over random sites from the left edge towards the right edge of the simulation square (i.e., along the chain direction (LP) in Fig 1) and from the upper edge towards the lower one (TP) As usual [4,5], regarding a particularity of the edge to edge percolation problem, the periodical boundary condition is only applied in the direction perpendicular to the percolating direction For each direction the threshold radius can be evaluated using the standard algorithm for the so-called r-percolation problem [4,5] Given a distance R, any two sites i and j with distance rij are assumed connected by a bond if rij R or disconnected if rij ¿ R The threshold percolation radius is then deÿned as the lowest R such that the bonds rij R form a percolating network connecting two edges Such a percolating network N Van Lien et al / Physica A 316 (2002) – 12 is often called the critical cluster (the largest cluster of connected sites at percolation) and the number of sites belonging to it is called the cluster mass (see later, Fig as an illustration of critical clusters) Thus, for each realization of site coordinates we can calculate the percolation radii Rc(l) (L) and Rc(t) (L), as well as, the critical cluster masses M (l) (L) and M (t) (L) for LP and TP, respectively In general, the quantities Rc(l); (t) (L) and M (l); (t) (L) uctuate from one random site realization to another Their averages over site realizations will deÿne the percolation radii and the critical masses for samples of given size L: Rc(l); (t) (L) = Rc(l); (t) (L) and M(l); (t) (L) = M (l); (t) (L) , respectively The fact that both quantities Rc(l); (t) (L) and M(l); (t) (L) depend on the sample size L is a natural consequence of the ÿnite size e ects of the investigated problem In order to study these e ects and to ÿnd the percolation radii, as well as other percolation characters, corresponding to an inÿnite system, for given values of and s the simulations have been performed for samples of di erent sizes: L = 100, 200, 400, 600, 800 and 1000 The number of random site realizations over which the averages are taken is k = 5000 for the case L = 100 and decreases as L increases in such a way that the number k × L2 , i.e., the total “area” used in checking percolation, are almost constant for all the cases under study In simulations, along with the average quantities Rc(l); (t) (L) and M(l); (t) (L) we always calculate the moments [4,5]: Rc(l); (t) (L) = (Rc(l); (t) (L) − Rc(l); (t) (L))2 1=2 M (l); (t) (L) = (M (l); (t) (L) − M(l); (t) (L))2 ; 1=2 (1) ; (2) and also the next-order moments, which are used to estimate the statistical errors of simulation results The quantities Rc(l); (t) , which measure the width of transition regions of the spanning clusters and depend on the sample size L, can be used to deÿne the percolation exponents (l); (t) , following the well-known ÿnite-size scaling relation [4,5,8] Rc(l); (t) (L) ˙ L−1= (l); (t) : (3) In Fig 2, the simulation data of −ln Rc(l); (t) (L) are plotted versus ln L for the case = 0:1 and some values of the impurity density s, s = 0:05, 0.1, 0.2, 0.3, and 0.4 (from bottom) For each value of s the simulation points are described by the solid circles and the triangles for the LP and TP, respectively Everywhere in this and following ÿgures the error bars not exceed the symbol sizes It is clear from Fig that for each s the simulation points for both LP and TP follow quite well the scaling relation of Eq (3) More importantly, the slopes of all ÿtting straight lines, solid lines for LP and dashed lines for TP, fall well into a narrow value range: 0:71±0:02, which corresponds to = 1:41 ± 0:04 We note that what observed for some values of s in Fig.2 is also maintained for unshown cases with other values of s, ¡ s ¡ Therefore, within the statistical error (less than 3%), our simulation data suggest that for the model under study with a given there exists only one critical exponent for both LP and TP and for any s of ¡ s ¡ N Van Lien et al / Physica A 316 (2002) – 12 5 −lnδℜc (l),(t) (L) 4 lnL (l) • (t) Fig −ln Rc (L)( ) and −ln Rc (L)( ) are plotted versus ln L for the cases of s (from bottom): s = 0:05, 0.1, 0.2, 0.3, 0.4 = 0:1 with di erent Next, for each studied case the percolation radii rc(l); (t) (s) ≡ Rc(l); (t) (L → ∞), corresponding to an inÿnite system, can be determined using the scaling relation [4,5,8]: |rc(l); (t) − Rc(l); (t) (L)| ˙ L−1= (l); (t) : (4) In Fig 3, the quantities Rc(l); (t) (L) are plotted against L−1= (l); (t) for the case of = 0:1 and di erent values of s: s = 0:01, 0.05, 0.1, 0.2, 0.3, and 0.4 (from top), using (l); (t) determined directly from Fig It is clear that for any ÿnite sample size L, for all values of s under study the TP radius Rc(t) (L) (triangles) is always higher than the LP one Rc(l) (L) (solid circles) However, with increasing L two percolation radii seem to change by di erent directions: Rc(l) (L) increases, while Rc(t) (L) decreases Such a di erence on direction of the ÿnite size e ects associated with LP and TP is similar to that discussed in Ref [6] The ÿnite size e ect in the LP, as thoroughly discussed in Ref [9], is mostly associated with the sites located close to the boundaries For the TP, the e ect can be understood by the fact that if the distance between two adjacent parallel chains is ÿxed (i.e., given ), then the longer the chain length L is, the shorter the optimum percolation path between them becomes The fact that the ÿnite size e ects for the LP and TP are in opposite directions is a remarkable characteristic of the studied percolation model On the other side, importantly, as is evident in Fig 3, for any density s under study and for both the LP and TP the simulation points obey well the scaling relation of Eq (4) And therefore, the asymptotics of the ÿtting straight lines (solid or dashed for LP or TP, respectively) give the corresponding percolation radii (rc(l) or rc(t) ) of inÿnite systems Very impressively, although for any ÿnite L the radius Rc(t) (L) is always considerably larger than Rc(l) (L) (the smaller s, the larger discrepancy), due to a di erence on the direction of ÿnite size e ects as mentioned above for each s under study in the limit of L → ∞ two radii tend to the practically coincided limits: rc(l) (s) = rc(t) (s) ≡ rc (s) 6 N Van Lien et al / Physica A 316 (2002) – 12 c ℜ (l),(t) (L) 0.02 0.04 0.06 −1/ν(l),(t) L (l); (t) • Fig The ÿnite size e ect of percolation radii Rc (L) are plotted against L−1= (l); (t) ( and for LP (l) and TP (t), respectively) for the cases of = 0:1 with di erent s (from top): s = 0:01, 0.05, 0.1, 0.2, 0.3, (l); (t) 0.4 The extrapolations of ÿtting straight lines yield the values of rc , corresponding to inÿnite systems The radius rc (s) decreases with increasing s and with a relative error less than 3% Fig gives rc (s) ≈ 4:987, 3.534, 2.898, 2.271, 1.943, and 1.726 for s = 0:01, 0.05, 0.1, 0.2, 0.3, and 0.4, respectively Thus, our simulation results suggest that while there is a di erence between percolation radii of the LP and TP in ÿnite systems, for any impurity density ¡ s ¡ 1, in the limit of inÿnite systems two correspondingly limiting radii are coincident One more important character of the percolation problem under study as mentioned above is the scaling behaviour of the critical cluster mass, i.e., the feature of the average total number of sites M(L) ≡ M (L) belonging to the largest percolative cluster at threshold in dependence on the sample size L as L → ∞ It was originally suggested by Mandelbrot [10] and is now widely believed that the critical cluster exhibits a fractal structure and its mass scales with L as M(L)L→∞ ˙ LD ; where D is called the fractal dimension (5) N Van Lien et al / Physica A 316 (2002) – 12 12 lnM(L) 10 4 lnL • Fig ln M(l) ( ) and ln M(t) ( ) are plotted against ln L for the cases of = 0:1 with di erent s (from top): s = 0:05, 0.1, 0.2, 0.3, 0.4 The slopes of ÿtting straight lines give the fractal dimension D In the present simulation for given L, , and s the mass M was counted for both LP (M(l) ) and TP (M(t) ) In Fig 4, we plot ln M(l) (L) (solid circles) and ln M(t) (L) (triangles) against ln L for the case = 0:1 and some values of the impurity density: s = 0:05, 0.1, 0.2, 0.3, and 0.4 (from top) The straight lines (solid or dashed for LP or TP, respectively) are the best ÿts of simulation points to the relation of Eq (5) It is clear that for all the cases under study the simulation data follow well the Mandelbrot’s scaling relation of Eq (5) And very interestingly, the fractal dimensions D as measured by the slopes of the ÿtting lines are practically the same for both LP and TP and for all s under study From Fig we obtain D = 1:94 ± 0:04 We like here to note that for any sample of ÿnite L, as can be seen in Fig 4, the mass M(t) (L) of TP is always larger than the mass M(l) (L) of LP This is even more evident when we compare the cluster in Fig 5a (for LP) with that in Fig 5b (for TP) or the cluster in Fig 5c (for LP) with that in Fig 5d (for TP) for s = 0:1 or 0.3, respectively However, importantly, for all cases under study both these masses scale with the sample size L with the same value of the power D in Eq (5) 8 N Van Lien et al / Physica A 316 (2002) – 12 Fig Examples of critical clusters for sample of L = 400 and (b) and (d): TP for s = 0:1 and 0.3, respectively = 0:1 (a) and (c): LP for s = 0:1 and 0.3; Thus, with respect to all three characters (percolation exponent, percolation radius, and fractal dimension) our simulation results in Figs 2– for the case = 0:1 show that for inÿnite samples two percolation, LP and TP, are equivalent, i.e., the system could be considered isotropically 2D percolative, regardless of s The similar calculations have also been performed for some other values of the chain density in the range 0:01 6 0:2 (in real systems the distance between adjacent chains is always much larger than the average distance between nearest neighbouring sites in a chain [11]) The obtained results are qualitatively similar to those for the case of = 0:1, presented in Figs 2– They all suggest an isotropic 2D behaviour of percolation in inÿnite systems with a single as well as D for both LP and TP, regardless of s And, very surprisingly, within statistical errors the values of these characters seem to be also insensitive to a change of in the range of values under study although the limiting percolation radii rc(l); (t) (L → ∞) and the ÿnite-size critical density sc (L) clearly depend on (see the next section, Fig 6) Thus, while the N Van Lien et al / Physica A 316 (2002) – 12 0.8 sc(L) 0.6 0.4 0.2 0 0.004 0.008 0.012 −1 L Fig The size-dependent “critical” densities sc (L) for ÿnite systems are plotted versus L−1 for some values of (from top): 0.05, 0.1, 0.2 The curves are freely drawn as a guide percolation radii depend on both and s, the percolation exponent ≈ 1:41 ± 0:04 and the fractal dimension D ≈ 1:94 ± 0:04 may be considered universal at least for the ranges of parameters under study Discussion Our simulation results (Figs 2– and unshown for other values of and s) suggest that in the investigated model (1) there have a single critical exponent and a single fractal dimension which are independent of percolation directions (LP or TP), of the impurity density s with ¡ s ¡ 1, and of in the range of most practical interest 0:01 6 0:2 and (2) though for ÿnite systems the threshold radius of the transverse 10 N Van Lien et al / Physica A 316 (2002) – 12 percolation Rc(t) (L) is always larger than that of the longitudinal percolation Rc(l) (L), for inÿnite systems two radii are coincident and the single radius rc ≡ Rc(t) (L → ∞) = Rc(l) (L → ∞) decreases with increasing and s Thus, in inÿnite systems for given and s in the ranges of values under study two percolations, LP and TP, are equivalent in the sense that all their characters, , D and threshold radius, are, respectively, coincident In other words, the system becomes then isotropically 2D percolative, regardless of an anisotropicity in its structure In order to understand the obtained results we assume that, generally, in inÿnite systems an anisotropicity in percolation is not associated with the medium topological structure, but rather with the rule of generating percolation clusters (an anisotropicity of medium should be averaged out as the system size extends to inÿnity) In a 2D isotropic and homogeneous medium, for example, two di erent rules, namely, the circle (isotropic) rule and the forward (directed) one, as well known, lead to di erent classes of percolation, characterized by di erent values of and D: for isotropic percolation = = 4=3 and D = D0 = 91=48 (exact values [4]); for directed percolation ≈ 1:73, ≈ 1:1 [12] In the present model though the system is anisotropic, the generating ⊥ rules are isotropic (circle) and identical for both LP and TP, therefore two percolations (LP and TP) should belong to the same class of the same and the same D The nature (e.g., the topological structure) of the system should be manifested in the values of these quantities: ≈ 1:41 and D ≈ 1:94 We have no idea to explain quantitatively the di erences between the values of and D obtained in the present model and those of and D0 mentioned above, but only assuming that they may be related to a di erence on topological structure of two corresponding systems Besides, we like to note that the obtained values of and D are not in contrast to the well-known percolation relation D = d − ÿ= , where d = and ÿ = 5=36 [13] Concerning the percolation radius rc (s) of inÿnite systems we have some guesses for the limiting cases When s is close to 1, the system could be seen quasi-isotropic, and therefore, rc (s) should be close to the value 1.19, determined from the percolation relation rc2 = Bc with Bc = 4:5 for the circle problem [5] In the opposite limit, when s → 0, the radius rc will be bounded by the distance between adjacent chains, which is equal to 1= (i.e., 10 for data in Fig 3) Thus, we have 1:19 rc (s) −1 , which was really observed in simulation data (particularly, in Fig for = 0:1) Now, while there is no crossover on percolation radius for inÿnite systems, one can ask if there is any quasi-crossover in ÿnite systems To ÿnd an answer to this question, with a given for each sample size L of 100 L 1000 we have searched if there is a crossover on the percolation radius as the density s varies Note again that the hopping conductivity is mainly determined by the percolation radius For each L our simulations show that there really exists a size-dependent “critical” density sc (L) such that Rc(t) (L) ¿ Rc(l) (L) for all s ¡ sc (L), while for all s ¿ sc (L) two radii are coincident Certainly, the density sc (L) also depends on For a given , as can be seen in Fig 6, sc (L) strongly decreases with increasing L The fact that sc (L) monotonously decreases as L increases unambiguously implies that in the limit of inÿnite L two This reminds us of the generating rule in constructing fractals: each rule leads to a class of fractals with a well-deÿned D (see Refs [10,13]) N Van Lien et al / Physica A 316 (2002) – 12 11 percolation radii are coincident with whatever small s In other words, in consistence with Fig 3, in inÿnite systems the percolation becomes isotropically 2D In ÿnite systems an anisotropicity in medium structure, in sample shape, or in boundary conditions [14] may unequally a ect the percolation processes along di erent directions, and therefore, lead to a di erence between corresponding percolation radii This discussion is valid for not only 2D systems and we assume that the main conclusions of this work should be qualitatively applied for the 3D systems, where there are even more choices for percolating processes in any direction Experimentally, the dimensional crossover in percolation in ÿnite systems can be manifested, for example, in the temperature dependence behaviour of the Mott VRH conductivity: (T ) ˙ exp[ − (T0 =T )x ] with x depending on the dimensionality [1,2] Reedijk et al [1] observed a clear transition from the Q1D regime of x = 1=2 [15] to the 3D regime of x = 1=4 in the (T )-behaviour of conjugated polymer PMBTh doped with FeCl3 as the dopping level c increases from c ¡ c0 = 0:12 to c ¿ c0 It is believed that such a transition is induced by an increase of the interchain connectivity Theoretically, the temperature dependence of the hopping conduction of anisotropic Q1D compounds have been extensively discussed for decades [15,16] More recently, Zvyagin [11] has analysed the hopping conduction in systems, corresponding to the limiting case of our model with s = Samukhin et al [17] suggested a fractal network model for describing the nonmetallic conduction of conducting polymers As for our model displayed in Fig 1, which is assumed relevant to describe such material as that in [1], the Monte-Carlo simulation of the hopping conductivity as a function of temperature for di erent values of s and has just been done in [18] Conclusion The percolation is simulated in a 2D continuum model, which consists of regularly arranged parallel chains of sites coupled to each other weakly via rare “impurities” By comparing two percolations, along and perpendicular to the chain direction (LP and TP) we are able to show that (1) in inÿnite systems two percolations are equivalent, i.e., the system is isotropically 2D percolative, regardless of the chain density , 1, as well as the impurity density s, s ¡ and (2) in ÿnite systems there is a crossover on the percolation radius in the sense that there exists a size-dependent critical density sc such that for s ¡ sc the percolation radius of TP is larger than that of LP, while for s ¿ sc two radii are coincident To explain the simulation data we assume that in inÿnite systems an anisotropicity in percolation is not associated with the medium topological structure, but rather with the rule of generating percolation clusters The observed crossover on percolation radii in ÿnite systems may be useful for describing the dimensional crossover in temperature dependence behaviour of hopping conductivity observed in various compound materials Note that for a single inÿnite 1D system x = 12 N Van Lien et al / Physica A 316 (2002) – 12 Acknowledgements One of the authors (V.L.N.) thanks for ÿnancial supports from National Center for Theoretical Sciences, Hsinchu, Taiwan, where this work was ÿnally completed We thank the Referee for very valuable comments which give a signiÿcal improvement in the last version of paper The Institute of Nuclear Science and Technique (Hanoi) is acknowledged for generous computer facilities This work was in part supported by the collaboration fund from Solid State Group, Lund University (Sweden) and by the Natural Science Council of Vietnam References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] J.A Reedijk, H.C.F Martens, H.B Brom, M.A.J Michels, Phys Rev Lett 83 (1999) 3904 A.I Yakimov, et al., J Phys.: Condens Matter (1997) 889 D Shahar, Z Ovadyahu, Phys Rev Lett 64 (1990) 2293 D Stau er, A.A Ahrony, An Introduction to Percolation Theory, 2nd Edition, Taylor & Francis, London, 1994 B.I Shklovskii, A.L Efros, Electronic Properties of Doped Semiconductors, Springer, Berlin, 1984 (Chapter 5) V Lien Nguyen, A Rubio, Solid State Commun 95 (1995) 833 V Lien Nguyen, E Canessa, Phys Rev E 57 (1998) 2467; V Lien Nguyen, E Canessa, 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each chain the sites are randomly arranged... temperature) the interchain rate E increases rapidly and at some critical doping level (temperature) it becomes comparable to the intrachain rate I , i.e., two directions, parallel and perpendicular... an average separation being equal to (the length units in the simulation) Next, the impurities (small metallic islands) are modelled by sites added at random into the interchain spaces with a