Multifractals and noise in metal-insulator mixtures

Tremblay, A.–M. S.; Fourcade, B.; Breton, P.

doi:10.1016/0378-4371(89)90282-3

Cited by 36 publications

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“…3 (closed symbol). It is seen that w J ഠ 0.13 which is much less than that found near p c [5,13] (hence the deliberate use of the suffix J in w J to emphasize the fact that the regime of concern corresponds to p ¿ p c ). Upon comparison of (3) with (1), it is seen that the scaling function g͑z͒, to a first approximation, is given by g͑z͒ Ӎ 1 1 z 2 and I o ϳ ͑R o S͒ 21͞2 .…”

Section: Predictable Electrical Breakdown In Compositesmentioning

confidence: 95%

“…The authors also derived theoretical bounds for x, 0.5 $ x $ 0.5͓1 2 1͞t͑2 1 w J ͔͒ so that breakdown currents were expected to lie between two bounds. Here, t is the electrical conductivity exponent and w J k͞t, k being the noise exponent [5].In this Letter, we present systematic measurements of electrical resistance, particularly in the Joule regime of a composite system of carbon high-density polyethylene (HDPE) up to breakdown. The breakdown in a sample has the nature of a first-order transition: as soon as the current from a constant current source exceeds a certain value I b , the sample resistance R starts increasing uncontrollably and becomes unsteady.…”

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Predictable Electrical Breakdown in Composites

Mukherjee

Bardhan

Heaney³

1999

Phys. Rev. Lett.

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The Joule regime at large electric fields in composites is presented in the context of a conduction phase diagram in the field-concentration plane. A sample suffers breakdown when the field is too large. The resistance up to breakdown is described by a universal curve as a function of field. It is found that the ratio of the breakdown resistance to the zero-field resistance assumes a fixed value Y at breakdown. Y is found to be 1.37 in carbon high-density polyethylene composites and is independent of carbon fraction and external conditions but depends on the nature of the conductor. A quantity which is independent of conducting material is defined. Results are compared with previous data.PACS numbers: 72.80.Ng, 05.70.Jk, 72.20.Ht Application of finite force (F, mechanical or electrical) in disordered systems usually results in a nonlinear response leading to some sort of catastrophic behavior in the extreme limit (e.g., fracture in mechanical systems and dielectric breakdown or burning in electrical systems). In recent years there has been a renewed interest in the problem of catastrophic phenomena [1] although the problem of non-Ohmic electrical conductivity in the precatastrophic regime in various disordered systems has been studied for a long time [2,3]. However, there have been very few attempts so far to describe the behavior of a system over the full range of the applied force. Such a study holds the promise of unraveling many important aspects such as precursor effects, predictability, and the effect of disorder on the nature of breakdown. Yagil et al.[4] carried out somewhat limited measurements of I-V curves in thin semicontinuous metallic films of Ag and Au. Focusing on breakdown events, it was concluded that breakdown currents I b in the films scale as I b ϳ B 2x , where B is the normalized third harmonic component (see below) generated as a result of Joule heating. Breakdown was assumed to occur when the sample resistance R exhibited the first irreversible discontinuity as a function of applied current I. The exponent x was measured to be 0.48 and 0.41 in films of Ag and Au, respectively. The authors also derived theoretical bounds for x, 0.5 $ x $ 0.5͓1 2 1͞t͑2 1 w J ͔͒ so that breakdown currents were expected to lie between two bounds. Here, t is the electrical conductivity exponent and w J k͞t, k being the noise exponent [5].In this Letter, we present systematic measurements of electrical resistance, particularly in the Joule regime of a composite system of carbon high-density polyethylene (HDPE) up to breakdown. The breakdown in a sample has the nature of a first-order transition: as soon as the current from a constant current source exceeds a certain value I b , the sample resistance R starts increasing uncontrollably and becomes unsteady. Let R o R͑0͒ be the linear or zero-field resistance, R b R͑I b ͒ be the breakdown resistance and Y R b ͞R o . It is found that for p . p J , where p is the (volume) fraction of conducting component (carbon) and p J is a fraction characteristic of the system ...

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Section: Predictable Electrical Breakdown In Compositesmentioning

confidence: 95%

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confidence: 99%

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Predictable Electrical Breakdown in Composites

Mukherjee

Bardhan

Heaney³

1999

Phys. Rev. Lett.

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“…The right-hand side therefore gives a rigorous lower boundary for χ e (β), although not a directly computable bound since g e is not known exactly. Equation ( 11) also corresponds in a sense to the assumption of 'gap scaling' which is precisely what does not occur for multifractal exponents [27][28][29][30].…”

Section: Effective-medium Approximationmentioning

confidence: 99%

Critical behaviour of non-linear susceptibility in random non-linear resistor networks

Zhang¹

1996

J. Phys.: Condens. Matter

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The critical behaviour of non-linear susceptibility of a two-component composite is studied in this paper. The first component of fraction p is non-linear and obeys a current-field (J -E) characteristic of the form J = g 1 v + χ 1 v β while the second component of fraction q is linear with J = g 2 v. Near the percolation threshold p c or q c , we examine the conductorinsulator (C-I) limit (g 2 = 0) and superconductor-conductor (S-C) limit (g 2 = +∞). For the C-I limit and p > p c , the effective linear and non-linear response functions behave as g e ≈ (p − p c ) t and χ e (β) ≈ (p − p c ) t 2 (β) , respectively. For the S-C limit and q < q c , g e and χ e (β) are found to diverge as g e ≈ (q c − q) −s and χ e (β) ≈ (q c − q) s 2 (β) . Within the effectivemedium approximation, the exponents are found to be s = t = 1 and s 2 (β) = t 2 (β) = (β +1)/2, p c = 1/d and q c = (d − 1)/d. By using a connection between the non-linear response of the random non-linear composite problem and the resistance or conductance fluctuations of the corresponding random linear composite problem, the exponents t 2 (β) and s 2 (β) are found to berespectively, where t (s) is the conductivity exponent in a C-I(S-C) composite, d is the dimension of the composite and ν is the correlation-length exponent in d dimensions, κ((β + 1)/2) and κ ((β + 1)/2) are given by ψ R ((β + 1)/2) + [(β + 1)/2](dν − ζ R ) = κ((β + 1)/2) + dν, ψ G ((β + 1)/2) + [(β + 1)/2](dν − ζ G ) = κ ((β + 1)/2) + dν, where ψ R ((β + 1)/2)(ψ G ((β + 1)/2)) characterizes the scaling of the [(β + 1)/2]th cumulant of the global resistance (conductance) distribution due to local resistance (conductance) fluctuations in the corresponding linear C-I(S-C) composites, ζ R = t − (d − 2)ν and ζ G = s + (d − 2)ν.We prove that t 2 (β) is a monotonically increasing function of β while s 2 (β) is a monotonically decreasing function of β, which have the following special values:The critical behaviour of the non-linear susceptibility in a C-I composite is very different from that of the non-linear susceptibility in a S-C composite and some unexpected results of the critical behaviour of non-linear susceptibility in a C-I network are reported in this paper.

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“…These values agree well with the experimental ones in (1). Noise have been studied extensively in many percolating systems like composites both experimentally and theoretically but mostly in ohmic regimes [9]. Recently, the bias-dependent behaviour of noise near the percolation threshold has been measured [10].…”

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Finite coherence length of thermal noise in percolating systems

Bardhan

Mukherjee

2002

Phys. Rev. B

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Noise has been measured in two types of coductor-insulator mixtures as a function of bias and composition. It was marked by a huge increase in magnitude as the resistance increased only slightly due to Joule heating. The noise (resistance) current scale Is(Ir) for nonlinearity were found to scale with the linear resistance Ro as Is(Ir) ∼ Ro −xs(xr ) where the exponent xs is equal to 0.80 and 0.68 in carbon-wax and carbon-polyethylene respectively and xr ≈ 0.5. It is shown that the large increase of noise in nonohmic regime as well as the differences between the noise and resistance exponents are due to the finite-sized inequilibrium thermal fluctuations whose coherence length is same as the correlation length of the underlying percolating systems. A expression for xs is derived. PACS numbers: 72.70.+m, 72.20.Ht, 72.60.+g The low frequency resistance noise, also known as 1/fnoise, is a very common phenomenon[1] in normal conductors and is increasingly being used as a tool for studying, particularly, disordered systems [2]. While its origin remains controversial, some of its features are wellestablished. In particular, the correlation length of resistance fluctuations (ξ s ) is commonly assumed to be of the order of microscopic lengths [1]. This implies that the noise power should be inversely proportional to the number of fluctuators or the system volume L d , d being the system dimensionality. But there are cases where relatively large increases in the noise levels under certain experimental conditions have been explained in terms of coarsening of the coherent volume with correlation lengths increasing to macroscopic scale. This, other things remaining same, amounts to an increase in theFor example, the large increase in the noise at the onset of charge density wave motion in certain conductors upon application of an electric field has been ascribed to the finite coherence length of a charge density wave domain [3]. Analogous phenomena are the well-know critical opalescence in various systems [4] where the static correlation length diverges as a critical point is approached. Interestingly, in systems in nonequilibrium states but not necessarily near any critical point, the correlation length of fluctuations at equal time is also of macroscopic order [5].Earlier [6], the resistance behaviour of various composite samples (random mixtures of conductors and insulators) exhibiting substantial Joule heating was invesigated as a function of biasing currents. At low currents, the resistance R = V /I had a bias-independent value R o but increased at high currents. It was found that the current I r at which the sample resistance starts increasing scaled with R o as I r ∼ R −xr o where x r ≈ 0.5 is the onset exponent for resistance. In this letter, we report on systematic measurements of low frequency resistance noise as a function of currents in two composites systems of carbonwax (C-W) and carbon-polymer (C-HDPE). Samples under fixed currents were in nonequilibrium steady states (below breakdown) with temperature grad...

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Multifractals and noise in metal-insulator mixtures

Cited by 36 publications

References 60 publications

Predictable Electrical Breakdown in Composites

Predictable Electrical Breakdown in Composites

Critical behaviour of non-linear susceptibility in random non-linear resistor networks

Finite coherence length of thermal noise in percolating systems

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