Highly efficient algorithm for percolative transport studies in two dimensions

Physical properties of a generalized model of multilayer adsorption of dimers

We investigate the transport properties of a complex porous structure with branched fractal architectures formed due to the gradual deposition of dimers in a model of multilayer adsorption. We thoroughly study the interplay between the orientational anisotropy parameter p_{0} of deposited dimers and the formation of porous structures, as well as its impact on the conductivity of the system, through extensive numerical simulations. By systematically varying the value of p_{0}, several critical and off-critical scaling relations characterizing the behavior of the system are examined. The results demonstrate that the degree of orientational anisotropy of dimers plays a significant role in determining the structural and physical characteristics of the system. We find that the Einstein relation relating to the size scaling of the electrical conductance holds true only in the limiting case of p_{0}→1. Monitoring the fractal dimension of the interface of the multilayer formation for various p_{0} values, we reveal that in a wide range of p_{0}>0.2 interface shows the characteristic of a self-avoiding random walk, compared to the limiting case of p_{0}→0 where it is characterized by the fractal dimension of the backbone of ordinary percolation cluster at criticality. Our results thus can provide useful information about the fundamental mechanisms underlying the formation and behavior of wide varieties of amorphous and disordered systems that are of paramount importance both in science and technology as well as in environmental studies.

Transport properties in multilayer adsorption of dimers

In this paper, we study the transport properties (percolation and conductivity) of a two-dimensional structure created by depositing dimers on a one-dimensional substrate where multilayer deposition is allowed. Specifically, we are interested in studying how the mentioned properties vary as a function of the height of the multilayer. The critical parameters of the percolation transition are calculated using finite-size scaling analysis, obtaining the scaling laws for the probability of percolation and the conductivity of the system. To calculate the electrical conductivity of the multilayer, we use the Frank-Lobb algorithm.

Emergent Insulator-Metal Transition with Tunable Optical and Electrical Gap in Thin Films of a Molecular Conducting Composite

Composites exhibit unique synergistic properties emerging when components with different properties are combined. The tuning of the energy bandgap in the electronic structure of the material allows designing tailor-made systems with desirable mechanical, electrical, optical, and/or thermal properties. Here, we study an emergent insulator-metal transition at room temperature in bilayered (BL) thin-films comprised of polycarbonate/molecular-metal composites. Temperature-dependent resistance measurements allow monitoring of the electrical bandgap, which is in agreement with the optical bandgap extracted by optical absorption spectroscopy. The semiconductor-like properties of BL films, made with bis(ethylenedithio)-tetrathiafulvalene (BEDT-TTF or ET) α-ET₂I₃ (nano)microcrystals as two-dimensional molecular conductor on one side and insulator polycarbonate as a second ingredient, are attributed to an emergent phenomenon equivalent to the transition from an insulator to a metal. This made it possible to obtain semiconducting BL films with tunable electrical/optical bandgaps ranging from 0 to 2.9 eV. A remarkable aspect is the similarity close to room temperature of the thermal and mechanical properties of both composite components, making these materials ideal candidates to fabricate flexible and soft sensors for stress, pressure, and temperature aiming at applications in wearable human health care and bioelectronics.

Observation and Manipulation of a Phase Separated State in a Charge Density Wave Material

The 1T polytype of TaS₂ has been studied extensively as a strongly correlated system. As 1T-TaS₂ is thinned toward the 2D limit, its phase diagram shows significant deviations from that of the bulk material. Optoelectronic maps of ultrathin 1T-TaS₂ have indicated the presence of nonequilibrium charge density wave phases within the hysteresis region of the nearly commensurate (NC) to commensurate (C) transition. We perform scanning tunneling microscopy on exfoliated ultrathin flakes of 1T-TaS₂ within the NC-C hysteresis window, finding evidence that the observed nonequilibrium phases consist of intertwined, irregularly shaped NC-like and C-like domains. After applying lateral electrical signals to the sample, we image changes in the geometric arrangement of the different regions. We use a phase separation model to explore the relationship between electronic inhomogeneity present in ultrathin 1T-TaS₂ and its bulk resistivity. These results demonstrate the role of phase competition morphologies in determining the properties of 2D materials.

Concurrence Percolation in Quantum Networks

Establishing long-distance quantum entanglement, i.e., entanglement transmission, in quantum networks (QN) is a key and timely challenge for developing efficient quantum communication. Traditional comprehension based on classical percolation assumes a necessary condition for successful entanglement transmission between any two infinitely distant nodes: they must be connected by at least a path of perfectly entangled states (singlets). Here, we relax this condition by explicitly showing that one can focus not on optimally converting singlets but on establishing concurrence-a key measure of bipartite entanglement. We thereby introduce a new statistical theory, concurrence percolation theory (ConPT), remotely analogous to classical percolation but fundamentally different, built by generalizing bond percolation in terms of "sponge-crossing" paths instead of clusters. Inspired by resistance network analysis, we determine the path connectivity by series and parallel rules and approximate higher-order rules via star-mesh transforms. Interestingly, we find that the entanglement transmission threshold predicted by ConPT is lower than the known classical-percolation-based results and is readily achievable on any series-parallel networks such as the Bethe lattice. ConPT promotes our understanding of how well quantum communication can be further systematically improved versus classical statistical predictions under the limitation of QN locality-a "quantum advantage" that is more general and efficient than expected. ConPT also shows a percolationlike universal critical behavior derived by finite-size analysis on the Bethe lattice and regular two-dimensional lattices, offering new perspectives for a theory of criticality in entanglement statistics.

Connectedness percolation in the random sequential adsorption packings of elongated particles

Connectedness percolation phenomena in the two-dimensional packing of elongated particles (discorectangles) were studied numerically. The packings were produced using random sequential adsorption off-lattice models with preferential orientations of the particles along a given direction. The partial ordering was characterized by the order parameter S, with S=0 for completely disordered films (random orientation of particles) and S=1 for completely aligned particles along the horizontal direction x. The aspect ratio (length-to-width ratio) of the particles was varied within the range ɛ∈[1;100]. Analysis of connectivity was performed assuming a core-shell structure of the particles. The value of S affected the structure of the packings, the formation of long-range connectivity, and the behavior of the electrical conductivity. The effects can be explained by taking accounting of the competition between the particles' orientational degrees of freedom and excluded volume effects. For aligned deposition, anisotropy in the electrical conductivity was observed with the values along the alignment direction σ_{x} being larger than the values in the perpendicular direction σ_{y}. Anisotropy in the localization of the percolation threshold was also observed in finite-sized packings, but it disappeared in the limit of infinitely large systems.

Sedimentation of a suspension of rods: Monte Carlo simulation of a continuous two-dimensional problem

The sedimentation of a two-dimensional suspension containing rods was studied by means of Monte Carlo (MC) simulations. An off-lattice model with continuous positional and orientational degrees of freedom was considered. The initial state before sedimentation was produced using a model of random sequential adsorption. During such sedimentation, the rods undergo translational and rotational Brownian motions. The MC simulations were run at different initial number densities (the numbers of rods per unit area), ρ_{i}, and sedimentation rates, u. For sediment films, the spatial distributions of the rods, the order parameters, and the electrical conductivities were examined. Different types of sedimentation-driven self-assembly and anisotropy of the electrical conductivity were revealed inside the sediment films. This anisotropy can be finely regulated by changes in the values of ρ_{i} and u.

Anisotropy in electrical conductivity of films of aligned intersecting conducting rods

Numerical simulations by means of the Monte Carlo method have been performed to study the electrical properties of a two-dimensional composite filled with rodlike particles. The main goal was to study the effect of the alignment of such rods on the anisotropy of its electrical conductivity. A continuous model was used. In this model, the rods have zero-width (i.e., infinite aspect ratio) and they may intersect each other. To involve both the low conductive host matrix and highly conductive fillers (rods) in the consideration, a discretization algorithm based on the use of a supporting mesh was applied. The discretization is equivalent to the substitution of rods with the polyominoes. Once discretized, the Frank-Lobb algorithm was applied to evaluate the electrical conductivity. Our main findings are (i) the alignment of the rods essentially affects the electrical conductivity and its anisotropy, (ii) the discrete nature of computer simulations is crucial. For slightly disordered system, high electrical anisotropy was observed at small filler content, suggesting a method to enable the production of optically transparent and highly anisotropic conducting films.

Anisotropy in electrical conductivity of two-dimensional films containing aligned nonintersecting rodlike particles: Continuous and lattice models

The electrical conductivity of two-dimensional films filled with rodlike particles (rods) was simulated by the Monte Carlo method. The main attention has been paid to the investigation of the effect of the rod alignment on the electrical properties of the films. Both continuous and lattice approaches were used. Intersections of particles were forbidden. Our main findings are (i) both models demonstrate similar behaviors, (ii) at low concentration of rods, both approaches lead to the same dependencies of the electrical conductivity on the concentration of the rods, (iii) the alignment of the rods essentially affects the electrical conductivity, (iv) at some concentrations of partially aligned rods, the films may be conducting only in one direction, and (v) the films may simultaneously be both highly transparent and electrically anisotropic.

Modeling the electrical properties of three-dimensional printed meshes with the theory of resistor lattices

The electrical properties of conducting meshes are investigated numerically by solving the related Kirchhoff equations with the Lanczos algorithm. The method is directly inspired by the recursion technique widely used to study the electronic and vibrational spectra of solids. The method is demonstrated to be very efficient and fast when applied to resistor networks. It is used to calculate equivalent resistances between arbitrary pairs of nodes in simple resistive lattices. When the resistance fluctuates statistically from bond to bond, the method makes it possible to evaluate the fluctuations of the electrical properties of the network. It is also employed to assign an effective bulk resistivity to a discrete conducting three-dimensional mesh.

Vertical drying of a suspension of sticks: Monte Carlo simulation for continuous two-dimensional problem

The vertical drying of a two-dimensional colloidal film containing zero-thickness sticks (lines) was studied by means of kinetic Monte Carlo (MC) simulations. The continuous two-dimensional problem for both the positions and orientations was considered. The initial state before drying was produced using a model of random sequential adsorption with isotropic orientations of the sticks. During the evaporation, an upper interface falls with a linear velocity in the vertical direction, and the sticks undergo translational and rotational Brownian motions. The MC simulations were run at different initial number concentrations (the numbers of sticks per unit area), p_{i}, and solvent evaporation rates, u. For completely dried films, the spatial distributions of the sticks, the order parameters, and the electrical conductivities of the films in both the horizontal, x, and vertical, y, directions were examined. Significant evaporation-driven self-assembly and stratification of the sticks in the vertical direction was observed. The extent of stratification increased with increasing values of u. The anisotropy of the electrical conductivity of the film can be finely regulated by changes in the values of p_{i} and u.

Dielectric and diamagnetic susceptibilities near percolative superconductor-insulator transitions

Coarse-grained superconductor-insulator composites exhibit a superconductor-insulator transition governed by classical percolation, which should be describable by networks of inductors and capacitors. We study several classes of random inductor-capacitor networks on square lattices. We present a unifying framework for defining electric and magnetic response functions, and we extend the Frank-Lobb bond-propagation algorithm to compute these quantities by network reduction. We confirm that the superfluid stiffness scales approximately as [Formula: see text] as the superconducting bond fraction p approaches the percolation threshold p _c . We find that the diamagnetic susceptibility scales as [Formula: see text] below percolation, and as [Formula: see text] above percolation. For models lacking self-capacitances, the electric susceptibility scales as [Formula: see text]. Including a self-capacitance on each node changes the critical behavior to approximately [Formula: see text].

Diffusion-driven self-assembly of rodlike particles: Monte Carlo simulation on a square lattice

The diffusion-driven self-assembly of rodlike particles was studied by means of Monte Carlo simulation. The rods were represented as linear k-mers (i.e., particles occupying k adjacent sites). In the initial state, they were deposited onto a two-dimensional square lattice of size L×L up to the jamming concentration using a random sequential adsorption algorithm. The size of the lattice, L, was varied from 128 to 2048, and periodic boundary conditions were applied along both x and y axes, while the length of the k-mers (determining the aspect ratio) was varied from 2 to 12. The k-mers oriented along the x and y directions (k_{x}-mers and k_{y}-mers, respectively) were deposited equiprobably. In the course of the simulation, the numbers of intraspecific and interspecific contacts between the same sort and between different sorts of k-mers, respectively, were calculated. Both the shift ratio of the actual number of shifts along the longitudinal or transverse axes of the k-mers and the electrical conductivity of the system were also examined. For the initial random configuration, quite different self-organization behavior was observed for short and long k-mers. For long k-mers (k≥6), three main stages of diffusion-driven spatial segregation (self-assembly) were identified: the initial stage, reflecting destruction of the jamming state; the intermediate stage, reflecting continuous cluster coarsening and labyrinth pattern formation; and the final stage, reflecting the formation of diagonal stripe domains. Additional examination of two artificially constructed initial configurations showed that this pattern of diagonal stripe domains is an attractor, i.e., any spatial distribution of k-mers tends to transform into diagonal stripes. Nevertheless, the time for relaxation to the steady state essentially increases as the lattice size growth.

Universality of the emergent scaling in finite random binary percolation networks

In this paper we apply lattice models of finite binary percolation networks to examine the effects of network configuration on macroscopic network responses. We consider both square and rectangular lattice structures in which bonds between nodes are randomly assigned to be either resistors or capacitors. Results show that for given network geometries, the overall normalised frequency-dependent electrical conductivities for different capacitor proportions are found to converge at a characteristic frequency. Networks with sufficiently large size tend to share the same convergence point uninfluenced by the boundary and electrode conditions, can be then regarded as homogeneous media. For these networks, the span of the emergent scaling region is found to be primarily determined by the smaller network dimension (width or length). This study identifies the applicability of power-law scaling in random two phase systems of different topological configurations. This understanding has implications in the design and testing of disordered systems in diverse applications.

Monte Carlo simulation of evaporation-driven self-assembly in suspensions of colloidal rods

The vertical drying of a colloidal film containing rodlike particles was studied by means of kinetic Monte Carlo (MC) simulation. The problem was approached using a two-dimensional square lattice, and the rods were represented as linear k-mers (i.e., particles occupying k adjacent sites). The initial state before drying was produced using a model of random sequential adsorption (RSA) with isotropic orientations of the k-mers (orientation of the k-mers along horizontal x and vertical y directions are equiprobable). In the RSA model, overlapping of the k-mers is forbidden. During the evaporation, an upper interface falls with a linear velocity of u in the vertical direction and the k-mers undergo translation Brownian motion. The MC simulations were run at different initial concentrations, p_{i}, (p_{i}∈[0,p_{j}], where p_{j} is the jamming concentration), lengths of k-mers (k∈[1,12]), and solvent evaporation rates, u. For completely dried films, the spatial distributions of k-mers and their electrical conductivities in both x and y directions were examined. Significant evaporation-driven self-assembly and orientation stratification of the k-mers oriented along the x and y directions were observed. The extent of stratification increased with increasing value of k. The anisotropy of the electrical conductivity of the film can be finely regulated by changes in the values of p_{i}, k, and u.

Electrical conductivity of a monolayer produced by random sequential adsorption of linear k-mers onto a square lattice

The electrical conductivity of a monolayer produced by the random sequential adsorption (RSA) of linear k-mers (particles occupying k adjacent adsorption sites) onto a square lattice was studied by means of computer simulation. Overlapping with predeposited k-mers and detachment from the surface were forbidden. The RSA process continued until the saturation jamming limit, p_{j}. The isotropic (equiprobable orientations of k-mers along x and y axes) and anisotropic (all k-mers aligned along the y axis) depositions for two different models-of an insulating substrate and conducting k-mers (C model) and of a conducting substrate and insulating k-mers (I model)-were examined. The Frank-Lobb algorithm was applied to calculate the electrical conductivity in both the x and y directions for different lengths (k=1 - 128) and concentrations (p=0 - p_{j}) of the k-mers. The "intrinsic electrical conductivity" and concentration dependence of the relative electrical conductivity Σ(p) (Σ=σ/σ_{m} for the C model and Σ=σ_{m}/σ for the I model, where σ_{m} is the electrical conductivity of substrate) in different directions were analyzed. At large values of k the Σ(p) curves became very similar and they almost coincided at k=128. Moreover, for both models the greater the length of the k-mers the smoother the functions Σ_{xy}(p),Σ_{x}(p) and Σ_{y}(p). For the more practically important C model, the other interesting findings are (i) for large values of k (k=64,128), the values of Σ_{xy} and Σ_{y} increase rapidly with the initial increase of p from 0 to 0.1; (ii) for k≥16, all the Σ_{xy}(p) and Σ_{x}(p) curves intersect with each other at the same isoconductivity points; (iii) for anisotropic deposition, the percolation concentrations are the same in the x and y directions, whereas, at the percolation point the greater the length of the k-mers the larger the anisotropy of the electrical conductivity, i.e., the ratio σ_{y}/σ_{x} (>1).

Divergence of dynamical conductivity at certain percolative superconductor-insulator transitions

Random inductor–capacitor (LC) networks can exhibit percolative superconductor-insulator transitions (SITs). We use a simple and efficient algorithm to compute the dynamicalconductivity σ(ω,p) of one type of LC network on large (4000 × 4000) square lattices, where δ = p − p(c) is the tuning parameter for the SIT. We confirm that the conductivity obeys a scaling form, so that the characteristic frequency scales as∝|δ|(νz) with νz ≈ 1.91, the superfluid stiffness scales as ϒ∝|δ|(t) with t ≈ 1.3, and the electric susceptibility scales as χE∝|δ|(−s) with s = 2νz − t ≈ 2.52. In the insulating state, the low-frequency dissipative conductivity is exponentially small, whereas in the superconductor, it is linear in frequency.The sign of Im σ(ω) at small ω changes across the SIT. Most importantly, we find that right at the SIT Re σ(ω) ∝ ω(t/νz−1)∝ω(−0.32), so that the conductivity diverges in the DC limit, in contrast with most other classical and quantum models of SITs.

Optoelectronic properties of graphene thin films deposited by a Langmuir-Blodgett assembly

Large area thin films of few-layered unfunctionalized graphene platelets are developed with fine control over the thickness. The thin films are obtained by a Langmuir-Blodgett assembly at the interface of graphene solution in N-methyl-2-pyrrolidone (NMP) and water, and their optoelectronic properties and conduction mechanism are investigated in relation to lateral flake size and thin film thickness. The electrical conductivity and carrier mobility are affected by the flake size (200 nm to 1 μm) and by the packing of the nanostructure platelet network. General effective medium theory is used to explain the thickness dependent conductivity and to determine the percolation threshold film thickness which was found to be about 10 nm (at a volume fraction of ~39%) for a Langmuir-Blodgett film of an average platelet lateral size of 170 ± 40 nm. The electronic behaviour of the material shows more similarities with polycrystalline turbostratic graphite than thin films of reduced graphene oxide, carbon nanotubes, or disordered conducting polymers. While in these systems the conduction mechanism is often dominated by the presence of an energy barrier between conductive and non-conductive regions in the network, in the exfoliated graphene networks the conduction mechanism can be explained by the simple two-band model which is characteristic of polycrystalline graphite.

Percolation with long-range correlated disorder

Long-range power-law correlated percolation is investigated using Monte Carlo simulations. We obtain several static and dynamic critical exponents as functions of the Hurst exponent H, which characterizes the degree of spatial correlation among the occupation of sites. In particular, we study the fractal dimension of the largest cluster and the scaling behavior of the second moment of the cluster size distribution, as well as the complete and accessible perimeters of the largest cluster. Concerning the inner structure and transport properties of the largest cluster, we analyze its shortest path, backbone, red sites, and conductivity. Finally, bridge site growth is also considered. We propose expressions for the functional dependence of the critical exponents on H.

Information theoretic aspects of the two-dimensional Ising model

We present numerical results for various information theoretic properties of the square lattice Ising model. First, using a bond propagation algorithm, we find the difference 2H(L)(w)-H(2L)(w) between entropies on cylinders of finite lengths L and 2L with open end cap boundaries, in the limit L→∞. This essentially quantifies how the finite length correction for the entropy scales with the cylinder circumference w. Secondly, using the transfer matrix, we obtain precise estimates for the information needed to specify the spin state on a ring encircling an infinitely long cylinder. Combining both results, we obtain the mutual information between the two halves of a cylinder (the "excess entropy" for the cylinder), where we confirm with higher precision but for smaller systems the results recently obtained by Wilms et al., and we show that the mutual information between the two halves of the ring diverges at the critical point logarithmically with w. Finally, we use the second result together with Monte Carlo simulations to show that also the excess entropy of a straight line of n spins in an infinite lattice diverges at criticality logarithmically with n. We conjecture that such logarithmic divergence happens generically for any one-dimensional subset of sites at any two-dimensional second-order phase transition. Comparing straight lines on square and triangular lattices with square loops and with lines of thickness 2, we discuss questions of universality.

Conductivity of Coniglio-Klein clusters

We performed numerical simulations of the q-state Potts model to compute the reduced conductivity exponent t/ν for the critical Coniglio-Klein clusters in two dimensions, for values of q in the range [1,4]. At criticality, at least for q<4, the conductivity scales as C(L) ~ L(-t/ν), where t and ν are, respectively, the conductivity and correlation length exponents. For q=1, 2, 3, and 4, we followed two independent procedures to estimate t/ν. First, we computed directly the conductivity at criticality and obtained t/ν from the size dependence. Second, using the relation between conductivity and transport properties, we obtained t/ν from the diffusion of a random walk on the backbone of the cluster. From both methods, we estimated t/ν to be 0.986 ± 0.012, 0.877 ± 0.014, 0.785 ± 0.015, and 0.658 ± 0.030, for q=1, 2, 3, and 4, respectively. We also evaluated t/ν for noninteger values of q and propose the conjecture 40 gt/ν = 72 + 20 g - 3g(2) for the dependence of the reduced conductivity exponent on q, in the range 0 ≤ q ≤ 4, where g is the Coulomb gas coupling.

Electronic durability of flexible transparent films from type-specific single-wall carbon nanotubes

The coupling between mechanical flexibility and electronic performance is evaluated for thin films of metallic and semiconducting single-wall carbon nanotubes (SWCNTs) deposited on compliant supports. Percolated networks of type-purified SWCNTs are assembled as thin conducting coatings on elastic polymer substrates, and the sheet resistance is measured as a function of compression and cyclic strain through impedance spectroscopy. The wrinkling topography, microstructure and transparency of the films are independently characterized using optical microscopy, electron microscopy, and optical absorption spectroscopy. Thin films made from metallic SWCNTs show better durability as flexible transparent conductive coatings, which we attribute to a combination of superior mechanical performance and higher interfacial conductivity.

Prospects for nanowire-doped polycrystalline graphene films for ultratransparent, highly conductive electrodes

Traditional transparent conducting materials such as ITO are expensive, brittle, and inflexible. Although alternatives like networks of carbon nanotubes, polycrystalline graphene, and metallic nanowires have been proposed, the transparency-conductivity trade-off of these materials makes them inappropriate for broad range of applications. In this paper, we show that the conductivity of polycrystalline graphene is limited by high resistance grain boundaries. We demonstrate that a composite based on polycrystalline graphene and a subpercolating network of metallic nanowires offers a simple and effective route to reduced resistance while maintaining high transmittance. This new approach of "percolation-doping by nanowires" has the potential to beat the transparency-conductivity constraints of existing materials and may be suitable for broad applications in photovoltaics, flexible electronics, and displays.

Fully transparent pixel circuits driven by random network carbon nanotube transistor circuitry

Optically transparent and mechanically flexible thin-film transistors have recently attracted attention for next generation transparent display technologies. Driving and switching transistors for transparent displays have challenging requirements such as high optical transparency, large-scale integration, suitable drive current (I(on)) in the microampere range, high on/off current ratio (I(on)/I(off)), high field-effect mobility, and uniform threshold voltage (V(th)). In this study, we demonstrate fully transparent high-performance and high-yield thin-film transistors based on random growth of a single-walled carbon nanotube (SWNT) network that are easy to fabricate. High-performance SWNT-TFTs exhibit optical transmission of 80% in visible wavelength, I(on)/I(off) higher than 10(3), and a high yield with reproducible electrical characteristics.

The elastic modulus, percolation, and disaggregation of strongly interacting, intersecting antiplane cracks

We study the modulus of a medium containing a varying density of nonintersecting and intersecting antiplane cracks. The modulus of nonintersecting, strongly interacting, 2D antiplane cracks obeys a mean-field theory for which the mean field on a crack inserted in a random ensemble is the applied stress. The result of a self-consistent calculation in the nonintersecting case predicts zero modulus at finite packing, which is physically impossible. Differential self-consistent theories avoid the zero modulus problem, but give results that are more compliant than those of both mean-field theory and computer simulations. For problems in which antiplane cracks are allowed to intersect and form crack clusters or larger effective cracks, percolation at finite packing is expected when the shear modulus vanishes. At low packing factor, the modulus follows the dilute, mean-field curve, but with increased packing, mutual interactions cause the modulus to be less than the mean-field result and to vanish at the percolation threshold. The "nodes-links-blobs" model predicts a power-law approach to the percolation threshold at a critical packing factor of p(c) = 4.426. We conclude that a power-law variation of modulus with packing, with exponent 1.3 drawn tangentially to the mean-field nonintersecting relation and passing through the percolation threshold, can be expected to be a good approximation. The approximation is shown to be consistent with simulations of intersecting rectangular cracks at all packing densities through to the percolation value for this geometry, p(c) = 0.4072.

Dispersive dielectric and conductive effects in 2D resistor-capacitor networks

How to predict and better understand the effective properties of disordered material mixtures has been a long-standing problem in different research fields, especially in condensed matter physics. In order to address this subject and achieve a better understanding of the frequency-dependent properties of these systems, a large 2D L × L square structure of resistors and capacitors was used to calculate the immittance response of a network formed by random filling of binary conductor/insulator phases with 1000 Ω resistors and 10 nF capacitors. The effects of percolating clusters on the immittance response were studied statistically through the generation of 10 000 different random network samples at the percolation threshold. The scattering of the imaginary part of the immittance near the dc limit shows a clear separation between the responses of percolating and non-percolating samples, with the gap between their distributions dependent on both network size and applied frequency. These results could be used to monitor connectivity in composite materials. The effects of the content and structure of the percolating path on the nature of the observed dispersion were investigated, with special attention paid to the geometrical fractal concept of the backbone and its influence on the behavior of relaxation-time distributions. For three different resistor-capacitor proportions, the appropriateness of many fitting models was investigated for modeling and analyzing individual resistor-capacitor network dispersed frequency responses using complex-nonlinear-least-squares fitting. Several remarkable new features were identified, including a useful duality relationship and the need for composite fitting models rather than either a simple power law or a single Davidson-Cole one. Good fits of data for fully percolating random networks required two dispersive fitting models in parallel or series, with a cutoff at short times of the distribution of relaxation times of one of them. In addition, such fits surprisingly led to cutoff parameters, including a primitive relaxation or crossover time, with estimated values comparable to those found for real dispersive materials.

Influence of nanotube length on the optical and conductivity properties of thin single-wall carbon nanotube networks

We study the optical and electrical properties of transparent conducting films made from length-sorted single-wall carbon nanotubes (SWCNT). Thin films of length-sorted SWCNTs, formed through filtration from a dispersing solvent onto a filter substrate ("buckypaper"), exhibit sharp changes in their optical properties and conductivity (sigma) with increasing SWCNT surface concentration. At a given surface concentration, tubes longer than 200 nm are found to form networks that are more transparent and conducting. We show that changes of sigma with SWCNT concentration can be quantitatively described by the generalized effective medium (GEM) theory. The scaling universal exponents describing the "percolation" transition from an insulating to a conducting state with increasing concentration are consistent with the two-dimensional (2D) percolation model. Shorter tubes and mixed length tubes form 3D networks. Furthermore, we demonstrate that the conductivity percolation threshold (x(c)) varies with the aspect ratio L as, x(c) approximately 1/L, a result that is also in accordance with the percolation theory. These findings provide a framework for engineering the optical and electrical properties of SWCNT networks for technological applications where flexibility, transparency, and conductivity are required.

Universality of liquid-gas Mott transitions at finite temperatures

We explain, in a consistent manner, the set of seemingly conflicting experiments on the finite temperature Mott critical point, and demonstrate that the Mott transition is in the Ising universality class. We show that, even though the thermodynamic behavior of the system near such critical point is described by an Ising order parameter, the global conductivity can depend on other singular observables and, in particular, on the energy density. Finally, we show that in the presence of weak disorder the dimensionality of the system has crucial effects on the size of the critical region that is probed experimentally.

Efficient algorithm for random-bond ising models in 2D

We present an efficient algorithm for calculating the properties of Ising models in two dimensions, directly in the spin basis, without the need for mapping to fermion or dimer models. The algorithm computes the partition function and correlation functions at a single temperature on any planar network of N Ising spins in O(N;{3/2}) time or less. The method can handle continuous or discrete bond disorder and is especially efficient in the case of bond or site dilution, where it executes in O(NlnN) time near the percolation threshold. We demonstrate its feasibility on the ferromagnetic Ising model and the +/-J random-bond Ising model and discuss the regime of applicability in cases of full frustration such as the Ising antiferromagnet on a triangular lattice.

Hysteresis and noise from electronic nematicity in high-temperature superconductors

An electron nematic is a translationally invariant state which spontaneously breaks the discrete rotational symmetry of a host crystal. In a clean square lattice, the electron nematic has two preferred orientations, while dopant disorder favors one or the other orientations locally. In this way, the electron nematic in a host crystal maps to the random field Ising model. Since the electron nematic has anisotropic conductivity, we associate each Ising configuration with a resistor network and use what is known about the random field Ising model to predict new ways to test for local electronic nematic order (nematicity) using noise and hysteresis. In particular, we have uncovered a remarkably robust linear relation between the orientational order and the resistance anisotropy which holds over a wide range of circumstances.

Computer simulation of electrical conductivity of colloidal dispersions during aggregation

The computation approach to the simulation of electrical conductivity of colloidal dispersions during aggregation is considered. We use the two-dimensional diffusion-limited aggregation model with multiple-seed growth. The particles execute a random walk, but lose their mobility after contact with the growing clusters or seeds. The two parameters that control the aggregation are the initial concentration of free particles in the system p and the concentration of seeds psi. The case of psi=1, when all the particles are the immobile seeds, corresponds with the usual random percolation problem. The other limiting case of psi=0, when all the particles walk randomly, corresponds to the dynamical percolation problem. The calculation of electrical conductivity and cluster analysis were done with the help of the algorithms of Frank-Lobb and Hoshen-Kopelman. It is shown that the percolation concentration phi c decreases from 0.5927 at psi=1 to 0 at psi --> 0. Scaling analysis was applied to study exponents of correlation length v and of conductivity t. For all psi>0 this model shows universal behavior of classical 2d random percolation with v approximately t approximately 4/3. The electrical conductivity sigma of the system increases during aggregation reaching up to a maximum at the final stage. The concentration dependence of conductivity sigma(phi) obeys the general effective medium equation with apparent exponent ta(psi) that exceeds t. The kinetics of electrical conductivity changes during the aggregation is discussed. In the range of concentration Pc(phi)<p<0.9527 the time of percolation cluster formation pi c decreases with increasing phi.

Percolating conduction in finite nanotube networks

The percolating conductance of a new class of nanocomposite thin-film transistors, with channels composed of isotropic ensembles of nanotubes or nanowires, is analyzed as a function of wire/tube density and channel length. The conductance exponents are validated against analytical results for short channel transistors, and against available experimental data for longer channel devices. Our plots of conductance exponents as a function of tube-to-tube coupling strength provide a unified framework to interpret future experiments and should help design better nanocomposite transistors.

Mean-field conductivity in a certain class of networks

We consider resistor networks, which are lattices with bonds represented by conductors and some of the bonds removed. It is known that effective medium theories predict that the effective conductivity of such networks is a linear function of the number of bonds present above the percolation threshold, but exact results for completely random networks deviate from linearity. We show that if instead we take a randomly chosen tree spanning the lattice and then start adding bonds to it at random, the conductivity changes linearly with the number of added bonds and coincides with the effective medium result for a given bond concentration. We also make comparisons with some related models.

Plasmon localization and local field distribution in metal-dielectric films

An exact and very efficient numerical method for calculating the effective conductivity and local-field distributions in random R-L-C networks is developed. Using this method, the local-field properties of random metal-dielectric films are investigated in a wide spectral range and for a variety of metal concentrations p. It is shown that for metal concentrations close to the percolation threshold (p=p(c)) and frequencies close to the resonance, the local-field intensity is characterized by a non-Gaussian, exponentially broad distribution. For low and high metal concentrations a scaling region is formed that is due to the increasing number of noninteracting dipoles. The local electric fields are studied in terms of characteristic length parameters. The roles of both localized and extended eigenmodes in Kirchhoff's Hamiltonian are investigated.

Colossal effects in transition metal oxides caused by intrinsic inhomogeneities

The influence of quenched disorder on the competition between ordered states separated by a first-order transition is investigated. A phase diagram with features resembling quantum-critical behavior is observed, even using classical models. The low-temperature paramagnetic regime consists of coexisting ordered clusters, with randomly oriented order parameters. Extended to manganites, this state is argued to have a colossal magnetoresistance effect. A scale T(*) for cluster formation is discussed. This is the analog of the Griffiths temperature, but for the case of two competing orders, producing a strong susceptibility to external fields. Cuprates may have similar features, compatible with the large proximity effect of the very underdoped regime.

Scaling law of resistance fluctuations in stationary random resistor networks

In a random resistor network we consider the simultaneous evolution of two competing random processes consisting in breaking and recovering the elementary resistors with probabilities W(D) and W(R). The condition W(R)>W(D)/(1+W(D)) leads to a stationary state, while in the opposite case, the broken resistor fraction reaches the percolation threshold p(c). We study the resistance noise of this system under stationary conditions by Monte Carlo simulations. The variance of resistance fluctuations <deltaR2> is found to follow a scaling law |p-p(c)|(-kappa(0)) with kappa(0) = 5.5. The proposed model relates quantitatively the defectiveness of a disordered media with its electrical and excess-noise characteristics.

Pore network as a model of porous media: comparison between nonhierarchical and hierarchical organizations of pores

A pore network is used as a model of a porous medium with hierarchical and single level organization of the pores. A microscopic diffusion or adsorption process is introduced that permits the study of both temporal evolution and effective properties of transport. The relationships between the model parameters and the measurable quantities are derived. The transport in media with a hierarchical organization of pores is shown to be qualitatively different from that in media lacking this organization. The question of experimental distinction between the two types of pore organization is also studied. We show that mercury intrusion porosimetry (MIP) measurements furnish valuable information about the pore organization in a sample, but that the exact deconvolution of the real pore size distribution from the experimental data is not straightforward. Our work provides indications for a correct interpretation of MIP results. Qualitative comparison with experiments show the pertinence of hierarchical models.

Transport on an annealed disordered lattice

We study the diffusion on an annealed disordered lattice with a local dynamical reorganization of bonds. We show that the typical rearrangement time depends on the renewal rate like tr approximately tau(alpha) with alpha not equal 1. This implies that the crossover time to normal diffusion in a slow rearrangement regime shows a critical behavior at the percolation threshold. Additional scaling relations for the dependence of the diffusion coefficient on the renewal rate are obtained. The derivation of scaling exponents confirms the crucial role of singly connected bonds in transport properties. These results are checked by numerical simulations in two and three dimensions.