Spanning lengths of percolation clusters

The spanning length of a percolation cluster is defined as the difference between the maximum and minimum coordinates of the cluster with respect to some chosen direction. It is statistically related to the number size of the cluster by an exponent that differs from the iriverse dimension that would characterize a compact cluster. This exponent for large percolation clusters in simple cubic lattice sites was studied by the Monte Carlo technique, and results are presented. Previous theoretical treatments of this exponent and its relationship with other critical exponents are discussed.In the present paper we shall refer exclusively to the site percolation problem, and all our definitions will be within that context.     

Percolative conduction in the half-metallic-ferromagnetic and ferroelectric mixture of (La, Lu, Sr)MnO3

The immiscibility between rhombohedral La(5/8)Sr(3/8)MnO3 and hexagonal LuMnO3 leads to a microm-scale heterogeneous mixture of half-metallic-ferromagnetic and insulating-ferroelectric phases. Electronic conduction of the mixture exhibits nearly ideal percolation behavior in the paramagnetic state with a threshold of 0.224(5) metal volume fraction and a resistivity scaling exponent t=2.1+/-0.1, consistent with the predicted universal behavior of classical percolation. However, far below T(C), t increases to 2.4+/-0.1, probably resulting from intergrain tunneling. Therefore, this system represents a unique example of the temperature-induced crossover from universal to nonuniversal behavior of t.     

Temperature- and voltage-dependent trap generation model in high-k metal gate MOS device with percolation simulation

High-k metal gate stacks are being used to suppress the gate leakage due to tunneling for sub-45 nm technology nodes.The reliability of thin dielectric films becomes a limitation to device manufacturing,especially to the breakdown characteristic.In this work,a breakdown simulator based on a percolation model and the kinetic Monte Carlo method is set up,and the intrinsic relation between time to breakdown and trap generation rate R is studied by TDDB simulation.It is found that all degradation factors,such as trap generation rate time exponent m,Weibull slope β and percolation factor s,each could be expressed as a function of trap density time exponent α.Based on the percolation relation and power law lifetime projection,a temperature related trap generation model is proposed.The validity of this model is confirmed by comparing with experiment results.For other device and material conditions,the percolation relation provides a new way to study the relationship between trap generation and lifetime projection.     

Precise calculation of the dynamical exponent of two-dimensional percolation

We report numerical data obtained on the special-purpose computer PERCOLA for the exponent of the electrical conductivity of 2D percolation. The extrapolation yields and a correction to the scaling exponent=1.2±0.2.     

Two-dimensional metal-insulator transition as a percolation transition in a high-mobility electron system

By carefully analyzing the low temperature density dependence of 2D conductivity in undoped high-mobility n-GaAs heterostructures, we conclude that the 2D metal-insulator transition in this 2D electron system is a density inhomogeneity driven percolation transition due to the breakdown of screening in the random charged impurity disorder background. In particular, our measured conductivity exponent of approximately 1.4 approaches the 2D percolation exponent value of 4/3 at low temperatures and our experimental data are inconsistent with there being a zero-temperature quantum critical point in our system.     

Morphological differences between Bi, Ag and Sb nano-particles and how they affect the percolation of current through nano-particle networks

Nano-particles of Bi, Ag and Sb have been produced in an inert gas aggregation source and deposited between lithographically defined electrical contacts on SiN. The morphology of these films have been examined by atomic force microscopy and scanning electron microscopy. The Bi nano-particles stick well to the SiN substrate and take on a flattened dome shape. The Ag nano-particles also stick well to the SiN surface; however they retain a more spherical shape. Whereas, many of the Sb nano-particles bounce off the SiN surface with only a small fraction of the Sb nano-particles aggregating at defects resulting in a non-random distribution of the clusters. These nano-scale differences in the film morphology influence the viability of applying percolation theory to in situ macroscopic measurements of the film conductivity, during the deposition process. For Bi and Ag nano-particles the increase in conductivity follows a power law. The power law exponent, t, was found to be 1.27 ±0.13 and 1.40 ±0.14, for Bi and Ag respectively, in agreement with theoretical predictions of t ≈1.3 for 2D random continuum percolation networks. Sb cluster networks do not follow this model and due to the majority of the Sb clusters bouncing off the surface. Differences in the current onset times and final conductance values of the films are also discussed.     

Superconductivity fluctuations in Bi(Pb) based granular ceramics superconductors: evidence for fractal behavior

We report precise measurements of the electrical resistivity in three different Bi(Pb) based granular ceramics superconductors. We show that a single critical exponent (5/6) describes the superconductivity fluctuations. Such a critical exponent indicates a fractal behavior of the superconducting path. Our results thus indicate a strict two dimensional fluctuation percolation set (below the superconductivity onset temperature 195 K down toT _c), and provide some proof for Tarascon et al. shell conductivity path hypothesis. We estimate the shell thickness to be of the order of 10 Å.     

Scaling behavior of permeability and conductivity anisotropy near the percolation threshold

We use the finite-size scaling method to estimate the critical exponent that characterizes the scaling behavior of conductivity and permeability anisotropy near the percolation thresholdp _c . Here is defined by the scaling lawk _l /k _t –1(p–p _c ), wherek _t andk _t are the conductivity or permeability of the system in the direction of the macroscopic potential gradient and perpendicular to this direction, respectively. The results are (d=2)0.819±0.011 and (d=3)0.518±0.001. We interpret these results in terms of the structure of percolation clusters and their chemical distance. We also compare our results with the predictions of a scaling theory for due to Straley, and propose that (d=2)=t- _B , wheret is the critical exponent of the conductivity or permeability of the system, and _B is the critical exponent of the backbone of percolation clusters.     

Scaling of particle trajectories on a lattice

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. On both lattices, for most concentrations of the scatterers the trajectories close exponentially fast. For special critical concentrations infinitely extended trajectories can occur which exhibit a scaling behavior similar to that of the perimeters of percolation clusters.At criticality, in addition to the two critical exponents =15/7 andd _f=7/4 found before, the critical exponent =3/7 appears. This exponent determines structural scaling properties of closed trajectories of finite size when they approach infinity. New scaling behavior was found for the square lattice partially occupied by rotators, indicating a different universality class than that of percolation clusters.Near criticality, in the critical region, two scaling functions were determined numerically:f(x), related to the trajectory length (S) distributionn _s, andh(x), related to the trajectory sizeR _s (gyration radius) distribution, respectively. The scaling functionf(x) is in most cases found to be a symmetric double Gaussian with the same characteristic size exponent =0.433/7 as at criticality, leading to a stretched exponential dependence ofn _S onS, n_Sexp(–S ^6/7). However, for the rotator model on the partially occupied square lattice an alternative scaling function is found, leading to a new exponent =1.6±0.3 and a superexponential dependence ofn _S onS.h(x) is essentially a constant, which depends on the type of lattice and the concentration of the scatterers. The appearance of the same exponent =3/7 at and near a critical point is discussed.     

First-Passage Percolation, Semi-Directed Bernoulli Percolation, and Failure in Brittle Materials

We present a two-dimensional, quasistatic model of fracture in disordered brittle materials that contains elements of first-passage percolation, i.e., we use a minimum-energy-consumption criterion for the fracture path. The first-passage model is employed in conjunction with a semi-directed Bernoulli percolation model, for which we calculate critical properties such as the correlation length exponent v ^sdir and the percolation threshold p _c ^sdir . Among other results, our numerics suggest that v ^sdir is exactly 3/2, which lies between the corresponding known values in the literature for usual and directed Bernoulli percolation. We also find that the well-known scaling relation between the wandering and energy fluctuation exponents breaks down in the vicinity of the threshold for semi-directed percolation. For a restricted class of materials, we study the dependence of the fracture energy (toughness) on the width of the distribution of the specific fracture energy and find that it is quadratic in the width for small widths for two different random fields, suggesting that this dependence may be universal.     

Bulk transport properties and exponent inequalities for random resistor and flow networks

The properties of random resistor and flow networks are studied as a function of the density,p, of bonds which permit transport. It is shown that percolation is sufficient for bulk transport, in the sense that the conductivity and flow capacity are bounded away from zero wheneverp exceeds an appropriately defined percolation threshold. Relations between the transport coefficients and quantities in ordinary percolation are also derived. Assuming critical scaling, these relations imply upper and lower bounds on the conductivity and flow exponents in terms of percolation exponents. The conductivity exponent upper bound so derived saturates in mean field theory.Research supported by the NSF under Grant No. DMR-8314625Research supported by the DOE under Grant No. DE-AC02-83ER13044     

Sample-to-sample fluctuations in the conductivity of a disordered medium

We investigate the sample-to-sample fluctuations in the conductivity of a random resistor network—equivalently, in the diffusivity of a disordered medium with symmetric hopping rates. We argue that whenever the effective conductivity ^* is strictly positive, then the fluctuations are normal, i.e., proportional to (volume)^–1/2. If the local conductivities are allowed to be zero, then ^* vanishes when approaching the percolation thresholdp _c. Close top _c the fluctuations are anomalous. From the renormalization group on hierarchical lattices we find that atp _c fluctuations and mean scale in the same fashion, i.e., there is no independent scaling exponent for the fluctuations.     

Density inhomogeneity driven percolation metal-insulator transition and dimensional crossover in graphene nanoribbons

Transport in graphene nanoribbons with an energy gap in the spectrum is considered in the presence of random charged impurity centers. At low carrier density, we predict and establish that the system exhibits a density inhomogeneity driven two dimensional metal-insulator transition that is in the percolation universality class. For very narrow graphene nanoribbons (with widths smaller than the disorder induced length scale), we predict that there should be a dimensional crossover to the 1D percolation universality class with observable signatures in the transport gap. In addition, there should be a crossover to the Boltzmann transport regime at high carrier densities. The measured conductivity exponent and the critical density are consistent with this percolation transition scenario.     

Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation

For independent translation-invariant irreducible percolation models, it is proved that the infinite cluster, when it exists, must be unique. The proof is based on the convexity (or almost convexity) and differentiability of the mean number of clusters per site, which is the percolation analogue of the free energy. The analysis applies to both site and bond models in arbitrary dimension, including long range bond percolation. In particular, uniqueness is valid at the critical point of one-dimensional 1/x–y² models in spite of the discontinuity of the percolation density there. Corollaries of uniqueness and its proof are continuity of the connectivity functions and (except possibly at the critical point) of the percolation density. Related to differentiability of the free energy are inequalities which bound the specific heat critical exponent in terms of the mean cluster size exponent and the critical cluster size distribution exponent ; e.g., 1+ (/2–1)/(–1).Research supported in part by NSF Grant PHY-8605164Research supported in part by the NSF through a grant to Cornell UniversityResearch supported in part by NSF Grant DMS-8514834     

Bond percolation with parallelism restriction on square lattices

As a simple approximation for the ±J spin glass we studied bond percolation on square lattices. However, two neighboring chains of ferromagnetic bonds are required for spins to be regarded as connected. We determine the percolation thresholdp _c =0.8282±0.0002 and the critical exponent =0.75 _–0.05 ^+0.02 for this specific percolation by means of Monte-Carlo simulation on square lattices (up to 150×150).     

Critical Behavior of Nonlinear Properties in Percolating Superconductor/Nonlinear-Normal Conductor Networks

The critical behavior of nonlinear response in random networks of superconductor/nonlinear-normal conductors below the percolation threshold is investigated. Two cases are examined: (i) The nonlinear normal conductor has weakly nonlinear current (i)-voltage (ν) response of the form ν = ri + bi^α (bi^α-1《t and α > 1). Both the crossover current density and the crossover electric field are introduced to mark the transition between the linear and nonlinear responses of the network and are found to have power-law dependencies ～(f_c - f)^H and ～(f_c - f)^M as the percolation threshold f_c of the superconductor is approached from below, where H = ν_d - s_d > 0, M = ν_d > 0, ν_d and s_d are the correlation length exponent and the critical exponent of linear conductivity in percolating S/N system respectively; (ii) The nonlinear-normal conductor has strongly nonlinear ν-i response, i.e., i = X^να The effective nonlinear response X_e, behaves as X_e ～(f_c - f)^-W(α), where W ( α ) is the critical exponent of the nonlinear response x_e(α) and is α-dependent in general. The results are compared with recently published data, reasonable agreement is found.     

Influence of inter cell resonant tunneling on the out-of-plane electronic transport behavior in layered high T<Subscript>c</Subscript> cuprates

The influence of inter unit cell resonant tunneling between the copper-oxygen planes on the c-axis electronic conductivity (σ_c) in normal state of optimal doped bilayer high T_c cuprates like Bi₂Sr₂CaCu₂O_8+x is investigated using extended Hubbard Hamiltonian including resonant tunneling term (T₁₂) between the planes in two adjoining cells. The expression for the out-of-plane (c-axis) conductivity is calculated within Kubo formalism and single particle Green's function by employing Green's function equations of motion technique within meanfield approximation. On the basis of numerical computation, it is pointed out that the renormalized c-axis conductivity increases exponentially with the increment in inter cell resonant tunneling. The effect of T₁₂ on renormalized c-axis conductivity is found to be prominent at low temperatures as compared to temperatures above room temperature (~300 °K). The Coulomb correlation suppresses the variation of renormalized c-axis conductivity with temperature, while renormalized c-axis conductivity increases on increasing carrier concentration. These theoretical results are viewed in terms of existing c-axis transport measurements.     

The influence of impurities on the quasiparticle tunneling current between weakly coupled quasi-one-dimensional charge density waves

The dc and ac conductivity of a tunneling junction between two impure quasi-one-dimensional charge density wave (CDW) systems is calculated. The non-magnetic impurities are considered in the self-consistent Born approximation (SCBA). Impurities modify the density of states (DOS) of the pure CDW system for quasiparticles inside the energy region of the gap 2(T). As in the pure case, the theory predicts in addition to a tunneling current which is proportional to the product of the DOS a term proportional to the cosine of the order parameter phase difference. In the case of a normal state/CDW junction, analytical expressions are obtained forT=0 showing deviations from the pure case. The linear ac conductivity is obtained by the scaling relation between the dc and the ac response.