PHYSICAL REVIEW B 75, 104204 共2007兲

Piezoresistivity and tunneling-percolation transport in apparently nonuniversal systems N. Johner,* P. Ryser, and C. Grimaldi LPM, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

I. Balberg The Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel 共Received 6 December 2006; published 21 March 2007兲 We formulate a generalized tunneling-percolation model that describes the electrical properties of conductorinsulator cellular disordered systems. This model predicts and explains the experimentally observed universal and nonuniversal behaviors of such properties and the crossover between them. Studying both the conductance and the corresponding piezoresistive response coefficient we show, by using effective-medium theories and Monte Carlo calculations, that the crossover of the conductivity-percolation exponent is reflected by a corresponding crossover in the behavior of the above coefficient, thus providing a more practical means to characterize the critical region of the system. DOI: 10.1103/PhysRevB.75.104204

PACS number共s兲: 72.20.Fr, 64.60.Fr, 72.60.⫹g

I. INTRODUCTION

The critical behavior of dc transport in percolating conductor-insulator composites is well known to be characterized by the power-law behavior of the electrical conductance of the system: G ⯝ G0共p − pc兲t ,

共1兲

where G0 is a prefactor, p is the fractional content of the conducting phase and pc is its critical value below which G vanishes. While the standard theory of percolation predicts a universal conductivity exponent t that depends only on the dimensionality D of the system,1 a property common to many composites is the nonuniversality of dc transport. This behavior is manifested by conductivity exponents t whose values depend on material characteristics. The generally accepted explanation for the deviation from the predictions of standard percolation theory has been attributed to the diverging distribution h共g兲 of the local conductances g, as g → 0, in random resistor networks.2 For example, in the random void 共RV兲 model,3 insulating spheres are embedded randomly in a continuous conducting medium and the diverging behavior of h共g兲 stems from the distribution of the conducting “neck” widths between neighboring spheres. The other known example is the tunneling-percolation 共TP兲 model4 in which the interparticle conductance mechanism is tunneling. The corresponding distribution of the local conductances, h共g兲, diverges then, for g → 0, when the tunneling distances between neighboring particles are broadly distributed. Despite their success in providing a well-defined theoretical background for the onset of nonuniversality, both the RV and the TP models, in their original formulations, predict values of the dc conductance exponent t which do not account quantitatively for the observed ones. In particular, the corresponding experimentally found t values are generally larger than the prediction of the RV model and smaller than expected from the TP theory.5,6 Attempts were made, however, for these basic models to account for the abovementioned discrepancies with the experimental results. This was done for the RV model by using a generalized distribu1098-0121/2007/75共10兲/104204共9兲

tion function that may have a stronger divergence that can lead to values of t in much better accord with the relevant experimental data.7 Concerning the TP model, some advances were made by modifying the distribution function of the tunneling distances by taking into account the actual intersurface separations between adjacent conducting particles.5 However, although giving rather satisfying results for some systems, others were found to still have too low values of t in comparison with the theoretical predictions, eluding therefore a systematic description in terms of the TP mechanism. Recently, a solution to the long-standing discrepancy between the TP scenario and the experimental data5 has been proposed by considering the relation between more realistic distributions of the tunneling distances and the proximity p − pc to the percolation threshold.8 The main feature of these distributions is that, for two- or three-dimensional homogeneous random systems, the distribution function h共g兲 of the tunneling conductances g between nearest-neighbor particles is not divergent for g → 0, but, rather, it has a peak at relatively low g values, say at gm, which are much smaller than gmax, the maximum possible tunneling conductance value in the system.8 Hence by approaching the percolation threshold from above, the macroscopic conductance G is determined initially by a rapidly increasing h共g兲 and then by a decreasing h共g兲 as g → 0. Following this dependence the t value obtained from Eq. 共1兲 will be larger than the universal one for the larger p − pc values, while sufficiently close to the percolation threshold 共for which the relevant g’s are equal or lower than gm, and for which the distribution function of g does not diverge for g → 0兲 G will be characterized by the universal value of the exponent t. An immediate consequence of the above scenario is that the conductance exponent values may depend quite strongly on the distance p − pc from the percolation threshold pc. Consequently, the t values extracted from computational or experimental data are p − pc dependent. Therefore attempts to extract experimentally the value of t by assuming, as it is usually the case, that t is independent of p − pc can lead to results that may be considered to indicate a nonuniversal TP behavior while the critical behavior is universal.

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A numerical analysis based on a TP model has recently confirmed this scenario,9 pointing out also the difficulty of a possible experimental verification of a p − pc dependent critical exponent. In principle, this difficulty follows the relatively poor resolution of the p values that can be obtained in available experimental systems. In particular, one would need a large number of different p values in the very narrow critical range of small enough p − pc values. Hence while the preparation of a large set of samples may lead to the most direct verification of the theory, it is nevertheless quite a difficult route for extracting the variations of t in general and in the critical regime in particular. In this paper we suggest an alternative way that could enable us to follow the basic feature of theoretical predictions in a less demanding way. For that matter we consider here a quantity which is more sensitive to the local conductance distribution function than G. Within the TP mechanism of transport, the natural candidate for such quantity is the piezoresistivity ⌫, i.e., the relative change of the resistivity of the system upon the application of an external strain. Indeed, as shown in detail in various studies,6,10 ⌫, that is defined as d ln共G兲 / d⑀, displays a logarithmic divergence of the form ln共p − pc兲 as p → pc when the conductance exponent is nonuniversal, while ⌫ is almost p independent when t is universal. This behavior, which follows the sensitivity of the individual conductances to the interparticle distance, is then a direct consequence of the TP mechanism of nonuniversality.4 In particular, the value of t depends explicitly on the mean interparticle tunneling distance a,4,11 and thus can be varied by applying an external strain to the system. Correspondingly, for strongly varying 共but not diverging兲 distribution functions h共g兲 such as those considered in Refs. 8 and 9, ⌫ is expected to display a p − pc dependence 关roughly of the form ln共p − pc兲兴 as long as p is much larger than pc, followed by a constant 共p independent兲 behavior for small p − pc values for which the system has entered the “true⬙ critical universal regime. From the experimental verification point of view, this will still demand samples in the very close vicinity of pc but there will be no need for many of them and/or the exact value of the corresponding p values as needed from the extraction of t by fits to the behavior depicted in Eq. 共1兲. In this paper, the issue of an “apparent” TP nonuniversality is investigated then by considering the model that we introduced in Ref. 9, which is a variant of the segregated TP model already studied in Ref. 11. We show below that within the effective-medium approximation, the dc transport exponent t crosses over from a nonuniversal value to the asymptotic universal limit as p − pc decreases and, at the same time, the piezoresistive response ⌫ changes from a logarithmic behavior ln共p − pc兲 to a constant asymptotic regime. Such p − pc dependence of ⌫ is confirmed by numerical Monte Carlo calculations which were in the range where the expected p − pc dependence of t is not observed. The structure of the paper is as follows. In Sec. II we introduce our tunneling-percolation model and obtain an approximated formula for the tunneling conductance distribution function, which is used in Sec. III where an effectivemedium approximation for the conductance and the

piezoresistive response are derived. In Sec. IV we report our Monte Carlo calculations on two-dimensional resistor networks and in Sec. V we turn to our conclusions. II. MODEL

In the tunneling-percolation theory of transport, it is assumed that conducting particles are dispersed in an insulating medium, and that current can flow through electron tunneling between the conducting particles. The dominant tunneling process is between nearest-neighbor particles because of the exponential decay of the tunneling probability with the interparticle distance which essentially excludes farther particles to participate in the process.5 In this situation, transport properties are governed by the distribution function of the nearest-neighbor interparticle distances which, in general, depends upon the particle density, the dimensionality, and the interparticle interactions. In the present study, we consider the scenario in which the conducting particles are distributed in a particular particles network within the insulating phase, giving rise to an interparticle distribution function very different from those characterizing homogeneous particle dispersions. Our model corresponds then to cellular materials, where large regions of insulating phase forbid an homogeneous distribution of the conducting particles.11,12 A typical example of this situation is provided by the microstructure of ruthenium-based thickfilm resistors 共TFRs兲, in which large glassy grains, typically of linear size of few micrometers, segregate the much smaller conducting particles, of 10– 100 nm diameter, in the spaces left between neighboring large glassy grains.12 After the firing process, the resulting microstructure is characterized by quasi-one-dimensional chains of conducting particles embedded in a glassy matrix. During firing the glass melts and sinters in the gaps left between the conducting particles of a chain which can be separated then by a glassy layer. Correspondingly, the conduction mechanism between the particles in a chain is tunneling. The above described particular structure leads to the relatively simple model of segregated tunneling-percolation systems that is illustrated in Fig. 1 for a two-dimensional lattice.11 In Fig. 1共a兲 the full circles represent impenetrable conducting spherical particles of diameter ␾ that are constrained to occupy channels of length L and width ␾. These channels 共or links兲 form a square lattice and can be either occupied by a chain of N conducting particles, placed at random, or empty. In order to assure electrical connectivity between two occupied channels, each intersection 共or node兲 between channels is occupied by a particle, while the conductance of an empty link is assumed to be 0. We consider tunneling between adjacent particles as the only conduction mechanism. Hence if the distance between the centers of two particles is r 关see Fig. 1共c兲兴, the interparticle tunneling conductance ␴ is given by

冉

␴共r兲 = ␴0 exp −

冊

2共r − ␾兲 , ␰

共2兲

where ␰ is the tunneling decay constant and the prefactor ␴0 can be set in our model as equal to unity.

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ticles are impenetrable, PN can be easily obtained by considering a system of N point particles 共␾ = 0兲 placed randomly within the effective length L − N␾. The equivalence with the original model is assured by assigning to the particle i the coordinate xi = ri − i␾ with 0 艋 xi 艋 L − N␾. Now, PN can be constructed by noticing that there are N! ways to order N particles each having a probability dxi / 共L − N␾兲 of being comprised between xi and xi + dxi. Hence PN = CN␪共x1 − x0兲¯␪共xN − xN−1兲dx1dx2 ¯ dxN ,

共6兲

where CN = N! / 共L − N␾兲N and ␪共x兲 = 0 for x ⬍ 0 and ␪共x兲 = 1 for x 艌 0. From the above Eq. 共6兲, and by using xi = ri − i␾, the distribution function PL共r1 , r2 , . . . , rN兲 reduces to N+1

PL共r1,r2, . . . ,rN兲 = CN 兿 ␪共ri − ri−1 − ␾兲.

共7兲

i=1

FIG. 1. 共a兲 Square lattice bond tunneling-percolation model used in the present work. The filled circles denote conducting impenetrable particles of diameter ␾. One particle is assigned to each intersection between the channels 共nodes兲 and each bond 共channel兲 has a probability p of being occupied by N particles. 共b兲 A channel containing N = 5 impenetrable particles randomly placed within a length L. 共c兲 Tunneling conductance between two adjacent particles whose centers are separated by a distance r.

Let us consider now a single channel as that shown in Fig. 1共b兲. We denote by ri the distance between the center of the ith particle and the origin which we place at the center of one of the nodes. In this way, as shown in Fig. 1共b兲, r0 = 0 and rN+1 = L + ␾ stand for the two fixed particles at the nodes while the remaining N particles have coordinates ␾ ⬍ r1 ⬍ r2 ⬍ ¯ ⬍ rN ⬍ L. The resulting tunneling conductance of a link is therefore given by

冉兺

N+1

g共r1,r2, . . . ,rN兲 =

i=1

1 ␴共ri − ri−1兲

冊

Equations 共5兲 and 共7兲 permit us to find an approximate analytical formula for hL共g兲. In fact, close to the percolation threshold, the network resistance is dominated by the bonds that have the smallest conductances, so that, in this limit, hL共g兲 can be calculated analytically by noticing that, for small g, the dominant contribution comes from the lowest interparticle conductance among the N + 1 conductances of the link. In this way we obtain from Eqs. 共5兲 and 共7兲 hL共g兲 ⬇ 共N + 1兲

= 共N + 1兲

hL共g兲 =

冕

L

0

dr1 ¯

冕

L

drN PL共r1,r2, . . . ,rN兲

0

L

dr1 PL共r1兲␦关g − e−2共r1−␾兲/␰兴,

PL共r1兲 =

冕

L

dr2 ¯

=

冉

冕

L

drN PL共r1,r2, . . . ,rN兲

0

r1 − ␾ ␳ 1− 1 − ␳␾ L共1 − ␳␾兲

冊

␳L−1

⫻␪共r1 − ␾兲␪关L − 共N − 1兲␾ − r1兴,

共9兲

where ␳ = N / L is the number density of particles. From the latter result one can obtain the mean interparticle distance aL: aL =

冕

L

drrPL共r兲 =

0

␾+L , ␳L + 1

and, finally, find that hL共g兲 reduces to hL共g兲 ⬇ ␳L

冋

1 − ␣L 共1 − ␣L兲ln共g兲 1+ g ␳L + 1

⫻␪共g − gmin兲␪共gmax − g兲,

drN PL共r1,r2, . . . ,rN兲

共8兲

where PL共r1兲 is the probability of having the first particle at distance r1 from the origin:

L

共10兲

册

␳L−1

共11兲

where gmax = 1, gmin = exp关−2共L − N␾兲 / ␰兴, and

0

⫻␦关g − g共r1,r2, . . . ,rN兲兴.

冕

0

and the subscript L stands for the finite length L of the links. The function hL共g兲 is the distribution function of the conductance for occupied channels which we evaluate as follows. Let us first define PN = PL共r1 , r2 , . . . , rN兲dr1dr2 ¯ drN as the probability to find the first particle in 关r1 , r1 + dr1兴, the second in 关r2 , r2 + dr2兴, and so on. The function hL共g兲 can then be obtained from PN as

冕

0

共3兲

共4兲

dr1 ¯

⫻␦关g − e−2共r1−␾兲/␰兴

If a given channel has a probability p of being occupied by N particles and probability 1 − p of being empty, we will have that the probability that a channel has conductance between g and g + dg is given by ␳L共g兲dg, where

␳L共g兲 = 共1 − p兲␦共g兲 + phL共g兲,

L

0

−1

.

冕

共5兲

Since this system is one dimensional and the conducting par104204-3

␣L = 1 −

␰/2 . aL − ␾

共12兲

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JOHNER et al.

finite, hL共g兲 cannot grow indefinitely due to the nonzero lower cutoff gmin = exp关−2共L − N␾兲 / ␰兴. This leads to the nonmonotonic behavior of hL共g兲, displayed in Fig. 2, and characterized by a maximum of hL共g兲 which, by setting dhL共g兲 / dg = 0 in Eq. 共11兲, is at g = gm = exp共N − 1兲gmin. In passing we note then that the present quasi-one-dimensional system, with infinitely long channels, is a prototype of tunneling-percolation systems that demonstrate a genuine nonuniversal critical behavior.11 III. EFFECTIVE-MEDIUM APPROXIMATION

FIG. 2. The distribution function of the channel conductances for channels of length L / ␾ = 10 occupied by N = 5 impenetrable particles. Open circles are the numerical calculations, while the solid lines are the approximate solution of Eq. 共11兲. The dashed lines are the L / ␾ → ⬁ limit of hL共g兲, Eq. 共13兲, for the same value of the particle density ␳.

In Fig. 2 we compare the approximate solution 共11兲 that is valid for small g 共solid lines兲 with a numerical calculation of hL共g兲 共empty circles兲 obtained by recording the conductance of 106 realizations of one channel of length L / ␾ = 10 occupied by N = 5 particles. As expected, the asymptotic formula 共11兲 agrees with the numerical results over a broader range of g values when ␰ / ␾ is sufficiently small. Furthermore, the results of Fig. 2 confirm that the general trend of hL共g兲 is characterized by a strong peak whose intensity increases as ␰ / ␾ is reduced and that this behavior is correctly reproduced by Eq. 共11兲. The origin of such behavior is easily understood by taking the L / ␾ → ⬁ limit of Eq. 共11兲 which, by keeping ␳␾ constant, leads to the same power-law behavior of the Kogut-Straley distribution function:2 h⬁共g兲 ⬇ N共1 − ␣⬁兲g−␣⬁␪共gmax − g兲,

共13兲

where

␣⬁ = 1 −

␰/2 , a⬁ − ␾

共14兲

and, from Eq. 共12兲, a⬁ = ␳−1 is the mean interparticle distance for an infinitely long channel. Hence by lowering g, the initial increase of hL共g兲 in Fig. 2 follows the behavior of the Kogut-Straley distribution 共dashed lines兲 but, since L / ␾ is

The fact that in the L / ␾ → ⬁ limit hL共g兲 tends to the Kogut-Straley distribution function has interesting consequences in relation to the transport properties. It is in fact well known that Eq. 共13兲 is a necessary condition to have a nonuniversal exponent of the conductivity.2 For example, for two-dimensional networks, the exponent t is equal to 1 / 共1 − ␣⬁兲 for ␣⬁ ⲏ 0.23, while for lower values of ␣⬁ the dc transport becomes universal with t ⯝ 1.3.13–16 The fact that for finite L / ␾ values the distribution function hL共g兲 displays the behavior plotted in Fig. 2 suggests that the network conductance G follows Eq. 共1兲, with the universal exponent t ⯝ 1.3, only very close to the percolation threshold pc, since only for the very low g values, hL共g兲 deviates substantially from the Kogut-Straley power-law behavior. As shown below, this behavior can be well described by employing an effective-medium approximation 共EMA兲 to the conductance network. Such analysis will be shown to be useful for the understanding of the numerical Monte Carlo results discussed in the next section. Within the EMA, the average network conductance G satisfies the integral equation:1

冕

⬁

dg

0

␳L共g兲共G − g兲 = 0. g + 共z/2 − 1兲G

共15兲

Here, z is the number of bonds per site and ␳L共g兲 is the bond conductance distribution function defined in Eq. 共4兲. In the two-dimensional square lattice z = 4, and by using Eqs. 共4兲 and 共15兲, we can present the EMA equation for our problem in terms of hLg by

冕

⬁

0

dg

hL共g兲 共p − pc兲 = g+G pG

共16兲

with pc = 2 / z = 1 / 2 which is the EMA percolation threshold for this square lattice. The numerical solution of Eq. 共16兲, with hL共g兲 as given by its asymptotic form Eq. 共11兲, is plotted in Fig. 3共a兲 as a function of p − pc, for different values of ␰ / ␾. All curves have been calculated for N = 5 particles in a channel with a length of L = 10␾. For ␰ / ␾ = 1 共dotted line兲 the EMA conductance G follows apparently a straight line when plotted in the log-log scale, suggesting that Eq. 共1兲 applies to the whole range of the considered p − pc values. A closer look reveals that G actually deviates from a simple power-law behavior and shows, as a function of p − pc, an initially slightly faster decay followed, for sufficiently small values of p − pc, by a power-law behavior with the exponent

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FIG. 3. 共a兲 EMA conductance as a function of the proximity to the percolation threshold for the segregated tunneling-percolation model for different values of ␰ / ␾. Here, N = 5 and L = 10␾. 共b兲 Corresponding p-dependent transport exponent as given by Eq. 共17兲.

t which corresponds to the EMA universal value of ¯t0 = 1. This feature is much more evident for ␰ / ␾ = 0.6 共dashed lines兲 and ␰ / ␾ = 0.4 共solid lines兲 where the initial decay of G is steeper as ␰ / ␾ is reduced while the universal exponent ¯t0 = 1 is recovered only for the relatively small values of p − pc
10−3. To quantify the behavior of G as p − pc → 0 it is useful to introduce the “p-dependent” conductivity exponent that we define “locally” as t共p兲 =

d ln共G兲 , d ln共p − pc兲

共17兲

and shown in Fig. 3共b兲. By reducing the value of ␰ / ␾, it is clearly seen that t共p兲 acquires a stronger p − pc dependence which would result in an apparent nonuniversality with t共p兲 ⬎¯t0 = 1. This is apparent if one tries to fit G values according with Eq. 共1兲 for the interval of the larger p − pc values. As p → pc, the above local exponent asymptotically tends to the universal EMA value of ¯t0 = 1, independently of the value of the tunneling factor ␰ / ␾. The results plotted in Fig. 3 stem from the strong dependence of hL共g兲 which, as shown in Fig. 2, is nonmonotonic with a characteristic maximum at a given value of g. Upon lowering ␰ / ␾, the maximum of hL共g兲 increases in amplitude and shifts to lower values of g. This feature, combined with the factor 1 / 共g + G兲 in the integrand of Eq. 共16兲, which favors the small g region of integration for small G values 共i.e., small p − pc values兲, leads to a change in the local exponent as the percolation transition is approached from above. A similar situation has been recently described in Ref. 8 for the case of the three-dimensional homogeneous distribution of conducting spherical particles, which displays indeed a microscopic conductances behavior qualitatively similar to the ones presented in Fig. 2. As already stressed in the Introduction, the local conductivity exponent t共p兲, as defined in Eq. 共17兲, is a useful quantity to characterize the crossover between nonuniversal and

universal behaviors, but it is nevertheless of limited practical use when applied to real systems since it would require the measurement of G for a large number of different p values in the close vicinity of pc, where the experimental capability of sample preparation with a fine resolution of p values is limited.8 This problem, however, can be circumvented, at least in principle, by looking at the p dependence of the piezoresistive response given by the change of conductance when an external strain ⑀ is applied to the system. That the total conductance G is rather sensitive to an external applied strain is easily appreciated by the exponential dependence of the interparticle conductance ␴, Eq. 共2兲, on the tunneling distance r, so that the change r → r共1 + ⑀兲 would also affect the link 共or channel兲 conductance g. To quantify the piezoresistive response within EMA, let us first rewrite Eq. 共16兲 as follows: p − pc 1 1 = , 兺⬘ N⬘ i gi + G pG

共18兲

where the primed summation is restricted to the occupied channels 共gi ⫽ 0兲 which are a fraction p = N⬘ / N of the total channel number N forming the square lattice. For simplicity, let us assume that the conducting and insulating phases have identical elastic constants so that the system is elastically homogeneous. In this way, the applied strain ⑀ is uniform and gi and G reduce to gi → gi共⑀兲 ⯝ gi关1 − ␥共gi兲⑀兴,

共19兲

G → G共⑀兲 ⯝ G关1 − ⌫⑀兴,

共20兲

where ⑀ has been assumed infinitesimal, ␥共gi兲 = −d ln gi兩共⑀兲 / d⑀兩⑀=0 is the piezoresistive coefficient for a given channel, and ⌫=−

冏

d ln共G兲 d⑀

冏

共21兲

⑀=0

is the piezoresistive response coefficient of the whole system. Once the above expressions are substituted in Eq. 共18兲, the terms linear in ⑀ enable us to find the following EMA equation for ⌫:

⌫=

兺i ⬘关gi␥共gi兲/共gi + G兲2兴 兺i ⬘关gi/共gi + G兲2兴

=

冕

⬁

dg关ghL共g兲/共g + G兲2兴␥共g兲

0

冕

⬁

, dg关ghL共g兲/共g + G兲 兴 2

0

共22兲 where in the second equality we have restored the continuum representation by using hL共g兲 = N1⬘ 兺i⬘␦共g − gi兲. Within the approximation scheme derived in the previous section, for small values of g the function ␥共g兲 in Eq. 共22兲 is approximately given by ln共1 / g兲. The p − pc dependence of the resulting ⌫ is plotted then in Fig. 4共a兲 for the same parameter values that were used for Fig. 3. Two different behaviors are clearly discernible. In the not too close vicinity of pc, ⌫ increases approximately as ln关1 / 共p − pc兲兴, and the increase is stronger as ␰ / ␾ is smaller. Upon the approach to pc the pi-

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FIG. 4. 共a兲 Piezoresistive response coefficient as a function of the proximity to the threshold. The results were obtained from the EMA equation for the same parameter values of Fig. 3. 共b兲 The same coefficient plotted as a function of the sample conductance.

ezoresistivity response coefficient crosses over to a region where ⌫ is independent of p − pc. The crossover position depends on ␰ / ␾ and moves towards the percolation threshold as ␰ / ␾ is lowered. When ⌫ is plotted as a function of the EMA conductance G, as in Fig. 4共b兲, a logarithmic behavior of the form

⌫ ⬀ ln共1/G兲

共23兲

is obtained for the interval of the larger G values. This interval is broadened towards the smaller G values as the value of ␰ / ␾ is lowered. We can define then a crossover G value, that we denote G*, that characterizes the transition in the G dependence of ⌫. In Fig. 5共a兲 we illustrate how the value of G* can be extracted from the fits of ⌫ to the large, G ⬎ G*, and low, G ⬍ G*, intervals of the conductance range. The corresponding ␰ / ⌽ dependence of the crossover conductance is shown in Fig. 5共b兲 for several values of ␰ 共filled squares兲. The behavior of ⌫ can be understood by assuming that the crossover G* is given actually by the bond conductance value for which hL共g兲 approaches zero. As shown in Fig. 2共c兲, for small values of ␰ / ␾, hL共g兲 is sharply peaked at g = gm = exp共N − 1兲gmin and then falls rapidly to zero for lower values of g. In this case the estimate G* ⬇ gm can be used. The logarithmic dependence Eq. 共23兲 for G ⲏ G* ⬇ gm can then be derived from Eq. 共22兲 by noticing that the function g / 共g + G兲2 appearing in the integrals is strongly peaked at g = G, especially for small G values, so that the main contribution to ⌫ comes from g ⯝ G, and thus Eq. 共23兲 follows by the approximation ␥共g兲 ⯝ ln共1 / g兲 for small values of g. On the other hand, for G ⱗ G* ⬇ gm, ⌫ is independent of G and can be approximated by the G → 0 limit of Eq. 共22兲 which reads

FIG. 5. 共a兲 Dependence of the piezoresistive response coefficient on the sample conductance, as obtained by the EMA calculation, for L = 10␾, N = 5, and ␰ / ␾ = 1. The solid lines are linear fits to the low and large conductances and the crossover value is determined by their intersection. 共b兲 Dependence of the crossover conductance value on ␰ / ⌽ for the same L and N. The analytic bounds to the observed behavior are determined by the channel conductance where the corresponding distribution function Eq. 共11兲 vanishes, gmin = exp关−2共L − N⌽兲 / ␰兴, or is maximized, gm = exp共N − 1兲gmin.

lim ⌫ ⬅ ⌫0 =

G→0

冕

⬁

dg关hL共g兲/g兴␥共g兲

0

冕

⬁

.

共24兲

dg关hL共g兲/g兴

0

The same reasoning can be followed also when ␰ / ⌽ is large, so that the corresponding hL共g兲 has a much weaker dependence as shown in Fig. 2共a兲, provided that G* is approximated now by the g value for which hL共g兲 vanishes, i.e., G* ⬇ gmin. Hence we obtained analytic bounds for the G* dependence on the value of ␰ / ␾: G* ⬇ gm for small ␰ / ␾ values and G* ⬇ gmin for the larger ␰ / ␾ values. These two bounds of G* are displayed in Fig. 5共b兲. The comparison with the G* values extracted from the EMA results 共solid squares兲 indicates that the above estimates can be considered as reasonable bounds to the G* vs ␰ / ⌽ behavior.

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FIG. 6. 共a兲 Our Monte Carlo results of the system conductance as a function of the proximity to the threshold that is determined by the size of the sample used兲, for L = 10␾, N = 5, and ␰ / ␾ = 1. The dashed line is a fit to Eq. 共25兲. 共b兲 Our Monte Carlo results of the piezoresistance response coefficient for the same parameter values of 共a兲. IV. MONTE CARLO RESULTS

Let us present now our Monte Carlo results on the conductance and the piezoresistance for the model introduced in Sec. II. We construct a n ⫻ n square lattice, with n = 256, where each link of length L has a probability p of being occupied by N impenetrable particles and a probability 1 − p of being empty. The calculated distribution function of the occupied link conductance has already been discussed in Sec. II and plotted in Fig. 2. Once the lattice is generated, we calculate the conductance using a Fourier-accelerated conjugate-gradient algorithm as described in Ref. 17. For each p value, we average the results over 20 different realizations of the lattice, and the different values of p have been chosen to range from p − pc = 0.5 down to p − pc = 0.003, where pc = 1 / 2 is the percolation threshold for the square lattice. Instead of fitting the so-obtained conductance G with Eq. 共1兲, we extract the conductivity exponent t from G = G0关p − pc共n兲兴t ,

共25兲

where pc共n兲 is the percolation threshold for our finite-size systems with n ⫻ n nodes. The use of Eq. 共25兲 rather than its infinite-size limit Eq. 共1兲 is meant to simulate the situation encountered in experiments, where the percolation threshold, in our case pc共n兲, is not known a priori and must be obtained by a direct fit to Eq. 共25兲. We have checked that, by changing the size of the network,18 pc − pc共n兲 scales as n−1/␯, where ␯ = 4 / 3 is the correlation length exponent in two dimensions. The piezoresistive response is obtained by calculating, for a given lattice realization, the conductance G when a homogeneous strain ⑀ = 0.01 is applied to the lattice. The difference

FIG. 7. Our Monte Carlo results of the piezoresistance response coefficient as a function of the system conductance for different values of ␰ / ␾. This is for L = 10 and N = 5.

⌬G with respect to the case without strain is then used to evaluate ⌫ via −⌬G / ⑀G. In Fig. 6 we present then typical results for G and ⌫ as functions of p − pc共n兲 as obtained from our Monte Carlo calculations for N = 5 particles and ␰ / ␾ = 1. The critical exponent extracted from the fit of the conductance shown in Fig. 6共a兲 is t = 1.42± 0.08 which is slightly above the universal value t0 ⯝ 1.3 for two-dimensional networks. From our discussion of the EMA results in the last section, such relatively large values of ␰ / ␾ are not expected to give rise to a strong p dependence of the transport exponent t as observed in Fig. 6共a兲 and our Monte Carlo data for G can be reasonably well fitted by an almost universal power law with constant exponent in a wide range of p values. In Fig. 6共b兲 we show our calculated piezoresistive response coefficient as a function of p − pc共n兲 for the same parameter values of Fig. 6共a兲. For bond concentration values larger than p − pc共n兲 ⯝ 0.03, ⌫ follows approximately a logarithmic behavior while, for lower concentrations, it saturates at ⌫ ⯝ 4.75. This trend is very similar to that reported in the last section for our EMA calculations of the piezoresistive response 关Fig. 4共a兲兴 and suggests a crossover between quasinonuniversality at high p values and universality at lower p’s, despite the apparently accurate power-law behavior displayed by G in Fig. 6共a兲. The tunneling-percolation nature of the crossover is verified in Fig. 7 where ⌫ is plotted against G for several values of the tunneling factor ␰ / ␾ and for the fixed L / ␾ = 10 and

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Eq. 共24兲 共crosses兲 which is identical to the piezoresistive response of a linear chain of conductances distributed according to hL共g兲. This suggests that, very close to the percolation threshold, ⌫ is insensitive to the topology of the percolating cluster, as it is also verified by computing the averaged piezoresistive response obtained from a single channel occupied by N impenetrable spheres, that is also shown in Fig. 8. V. CONCLUSION

FIG. 8. Comparison of the xi / ⌽ dependencies of the asymptotic value of the piezoresistive response coefficient that were obtained from EMA, from Monte Carlo calculations, and from the averaged piezoresistive response of a single bond with L = 10 and N = 5.

N = 5 values. As ␰ / ␾ is reduced, the point below which ⌫ saturates moves to lower values of G, in accord with the EMA results shown in Fig. 4共b兲. By following the interpolation procedure described in Fig. 5共a兲, the values of crossover conductance G* 关gray triangles in Fig. 5共b兲兴 match very closely those obtained from our EMA calculations. The Monte Carlo calculations reported above show clearly that, as we suggested, the piezoresistive response is a much more sensitive quantity for the investigation of the TP crossover between nonuniversality and universality than the conductance itself, whose crossover is masked by the limited number of p values and the increasing fluctuations of G as the percolation threshold is approached from above. From the random-resistor-network point of view, such sensitivity of ⌫ can also be understood by noticing that, away from the percolation threshold, the current flows through the less resistive links in the network so that the bond conductance distribution function hL共g兲 is probed, basically, only for the larger g values. On the contrary, close to the percolation threshold, the current is forced to flow through bonds with much lower g values, so that the conductivity of the system is basically dominated by the lowest such conductance and therefore the behavior of hL共g兲 as g → 0 is directly probed. Since hL共g兲 for our TP model does not diverge as g → 0, the resulting piezoresistance for p close to pc is expected to be independent of p, as for any universal TP system. This situation is illustrated in Fig. 8 where we have plotted the asymptotic limit ⌫0 = limp→pc⌫ extracted from our Monte Carlo results 共filled circles兲 together with the EMA formula

*Email address: [email protected] Kirkpatrick, Rev. Mod. Phys. 45, 574 共1973兲. 2 P. M. Kogut and J. P. Straley, J. Phys. C 12, 2151 共1979兲. 3 B. I. Halperin, S. Feng, and P. N. Sen, Phys. Rev. Lett. 54, 2391 共1985兲; S. Feng, B. I. Halperin, and P. N. Sen, Phys. Rev. B 35, 1 S.

In this paper we have addressed the problem of dc transport properties in a tunneling-percolation model of segregated 共or cellular兲 systems. We have shown that the distribution function hL共g兲 of the channel conductances g has a nonmonotonic behavior characterized by a strong peak at some low g value, in qualitative agreement with other tunneling-percolation models such as that characterizing a homogeneous distribution of conducting spheres distributed randomly in the three-dimensional space. The resulting macroscopic conductance G does not follow, within an EMA approximation, a simple power-law behavior, but rather displays a p-dependent transport exponent, whose value approaches the universal limit very close to the percolation threshold pc. We have shown that such crossover from a nonuniversal-like to a universal-like behavior can be studied very efficiently by the behavior of the piezoresistive response coefficient ⌫, which displays a logarithmic increase for relatively large p − pc values followed by a p-independent ⌫ as the system crosses over the universal regime. We have carried out also numerical Monte Carlo calculations of G and ⌫, finding that G can be fitted by a simple power law with a single p-independent exponent, whose value is nevertheless larger than its universal limit. We have interpreted this result as due to a nonuniversal-universal crossover, as obtained in our EMA analysis, which contrary to the latter is masked by the limited number of p values and the increasing error bars of G as p → pc. By performing Monte Carlo calculation of the piezoresistive response, we have nevertheless been able to clearly detect this crossover. We interpreted those observations as due to the fact that ⌫ is much more sensitive to the tunneling-percolation features than the values of G. We conclude then that, at least in principle, this result could be used in tunneling-percolation materials to investigate the origin of their nonuniversal exponents, by recording ⌫ as a function of p − pc or G. ACKNOWLEDGMENTS

This work was supported in part by the Swiss National Science Foundation 共Grant No. 200021-103486兲 and in part by the Israel Science Foundation.

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