Random Resistor Network Research Articles

Electrical properties of disordered films of van der Waals semiconductor WS2 on paper.

One of the primary objectives in contemporary electronics is to develop sensors that are not only scalable and cost-effective but also environmentally sustainable. To achieve this goal, numerous experiments have focused on incorporating nanomaterial-based films, which utilize nanoparticles or van der Waals materials, on paper substrates. In this article, we present a novel fabrication technique for producing dry-abraded van der Waals films on paper, demonstrating outstanding electrical characteristics. We assess the quality and uniformity of these films by conducting a spatial resistivity characterization on a 5 × 5 cm2 dry-abraded WS2 film with an average thickness of 25 μm. Employing transfer length measurements with varying channel length-to-width ratios, we extract critical parameters, including sheet resistance and contact resistance. Notably, our findings reveal a resistivity approximately one order of magnitude lower than previous reports. The film's inherent disorder manifests as an asymmetric distribution of resistance values for specific geometries. We explore how this behavior can be effectively modeled through a random resistance network (RRN), which can reproduce the experimentally observed resistance distribution. Finally, we investigate the response of these devices under applied uniaxial strain and apply the RRN model to gain a deeper understanding of this process.

Effective electrical conductivity of random resistor networks generated using a Poisson–Voronoi tessellation

We studied the effective electrical conductivity of dense random resistor networks (RRNs) produced using a Voronoi tessellation when its seeds are generated by means of a homogeneous Poisson point process in the two-dimensional Euclidean space. Such RRNs are isotropic and in average homogeneous; however, local fluctuations of the number of edges per unit area are inevitable. These RRNs may mimic, e.g., crack-template-based transparent conductive films. The RRNs were treated within a mean-field approach. We found an analytical dependency of the effective electrical conductivity on the number of conductive edges (resistors) per unit area, nE. The effective electrical conductivity is proportional to nE when nE≫1.

Green's functions for random resistor networks.

We analyze random resistor networks through a study of lattice Green's functions in arbitrary dimensions. We develop a systematic disorder perturbation expansion to describe the weak disorder regime of such a system. We use this formulation to compute ensemble-averaged nodal voltages and bond currents in a hierarchical fashion. We verify the validity of this expansion with direct numerical simulations of a square lattice with resistances at each bond exponentially distributed. Additionally, we construct a formalism to recursively obtain the exact Green's functions for finitely many disordered bonds. We provide explicit expressions for lattices with up to four disordered bonds, which can be used to predict nodal voltage distributions for arbitrarily large disorder strengths. Finally, we introduce a novel order parameter that measures the overlap between the bond current and the optimal path (the path of least resistance), for a given resistance configuration, which helps to characterize the weak and strong disorder regimes of the system.

Cooperation in bioluminescence: understanding the role of autoinducers by a stochastic random resistor model

Quorum sensing is a communication mechanism adopted by different bacterial strains for the regulation of gene transcription. It takes place through the exchange of molecules called autoinducers. Bioluminescence is an emergent threshold phenomenon shown by some bacteria strains. Its precise relationship to quorum sensing is a debated topic, particularly regarding the role of the different autoinducers used by bacteria. In this paper, assuming a direct relationship between bioluminescence and quorum sensing, we investigate the role of multiple autoinducers in the bioluminescence response of Vibrio harveyi, considered as a model bioluminescent strain, due to its quorum sensing circuitry involving an array of three different autoinducers. Experiments on mutants of this bacterium, obtained by suppression of one or more autoinducers, reveal their relative non-trivial relevance and cooperative interaction patterns. The proposed analysis is implemented on a regular lattice, whose nodes represent microbial entities equipped with charges, which represent the ability to up/down regulate the gene expression. Quorum sensing results from a Coulomb-type field, produced by the charges. In analogy with random resistor network models, the lattice is permeated by an effective current which accounts for the amount and distribution of the charges. We propose that the presence of different autoinducers correspond to a different up/down regulation of gene expression, i.e., to a different way to account for the charges. Then, by introducing a modulation of the charge dependence into the current flowing within the network, we show that it is able to describe the bioluminescence exhibited by V. harveyi mutants. Furthermore, modulation of the charge dependence allows the interactions between the different autoinducers to be taken into account, providing a prediction regarding the data obtainable under specific growth conditions.Graphical abstract

Modelling the minimum conductivity of graphene using random resistor networks

Modelling the minimum conductivity of graphene using random resistor networks

Continuous Mott Transition in Moiré Semiconductors: Role of Long-Wavelength Inhomogeneities.

Recent experiments in moiré transition metal dichalcogenide materials have reported the observation of a continuous bandwidth-tuned transition from a metal to a paramagnetic Mott insulator at a fixed filling of one electron per moiré unit cell. The electrical transport measurements reveal a number of puzzling features that are seemingly at odds with the theoretical expectations of an interaction-induced, but disorder-free, bandwidth-tuned metal-insulator transition. In this Letter, we include the effects of long-wavelength inhomogeneities, building on the results for a continuous metal-insulator transition at fixed filling in the clean limit. We examine the effects of mesoscale inhomogeneities near the critical point on transport using the framework of random resistor networks, highlighting the salient differences from a simple percolation-based picture. We place our results in the context of recent and ongoing experiments.

Percolation threshold and electrical conductivity of conductive polymer composites filled with curved fibers in two-dimensional space.

Quantifying the influence of fiber curvature on the percolation behavior of flexible conductive fiber and further on the electrical conductivity of conductive polymer composites (CPCs) is crucial for the design of CPCs. This study considers CPCs as a random packing of soft curved discorectangles (CDCRs) in a polymer matrix. The geometry of CDCR is developed, and an inter-CDCR contact detection algorithm is used to generate a random packing structure of CDCRs. The effects of aspect ratio α and bending central angles θ of CDCR on the percolation threshold ϕc of the overlapped CDCR system in a two-dimensional plane are then investigated using the finite-size scaling method. The result reveals that ϕc decreases monotonically as α grows and increases monotonically as θ rises. A shape-independent power law formula, denoted as ϕc = 2.2015 A-0.8172dex is developed to quantify the relationship between the Adex and ϕc. A comparison of our numerical simulations, published data, and predictions verifies the reliability and universality of the fitting model. Subsequently, a resistor network searching algorithm (RNSA) is proposed to construct the random resistor network model (RRNM). A power law model, denoted as is developed to evaluate the effects of the normalized reduced density (η - ηc)/ηc on the effective conductivity σeff of CPC. Comparing our predictions with data from the literature and our simulation verifies the reliability of our RNSA and the fitting model. This paper's methodology and findings may provide a theoretical hint for the CPC's design.

Thermo-electro-optical properties of seamless metallic nanowire networks for transparent conductor applications.

Rapid reaction time, high attainable temperatures, minimum operating voltage, excellent optical transmittance, and tunable sheet resistance are all desirable properties of transparent conductors, which are important thin-film components in numerous electronic devices. A seamless nanowire network (NWN) refers to a structure composed of nanowires that lack interwire contact junctions, resulting in a continuous and uninterrupted network arrangement. This seamless nature leads to unique properties, including high conductivity and surface area-to-volume ratios, which make it a promising candidate for a vast application range in nanotechnology. Here, we have conducted an in-depth computational investigation to study the thermo-electro-optical properties of seamless nanowire networks and understand their geometrical features using in-house computational implementations and a coupled electrothermal model built in COMSOL Multiphysics software. Sheet resistance calculations were performed using Ohm's law combined with Kirchhoff circuit laws for a random resistor network and compared with those obtained employing COMSOL. In this work, aluminium, gold, copper, and silver nanowires are the materials of choice for testing the transparent conduction performance of our systems. We have studied a wide range of tuning parameters, including the network area fraction, the width-to-depth aspect ratio, and the length of the nanowire segments. We obtained corresponding figures of merit (optical transmittance versus sheet resistance) and temperature distributions to provide a complete characterization of the performance of real-world transparent conductors idealized with seamless NWNs. Our analysis accounted for the thermo-electro-optical responses of the NWNs and the inspection of various controlling parameters depending on system design considerations to shed light on how the electrical transport, optical qualities, and thermal management of these systems can be optimized.

Comprehensive Model for the Thermoelectric Properties of Two-Dimensional Carbon Nanotube Networks

Networks of semiconducting single-walled carbon nanotubes (SWCNTs) are interesting thermoelectric materials due to the interplay between CNT and network properties. Here, we present a unified model to explain charge and energy transport in SWCNT networks. We use the steady-state master equation for the random resistor network containing both intra- and intertube resistances, as defined through their one-dimensional density of states that are modulated by static Gaussian disorder. The tube-resistance dependence on the carrier density and disorder is described through the Landauer formalism. Electrical and thermoelectric properties of the network are obtained by solving Kirchhoff's laws through a modified nodal analysis, where we use the Boltzmann-transport formalism to obtain the conductivity, Seebeck coefficient, and electronic contribution to the thermal conductivity. The model provides a consistent description of a wide range of previously published experimental data for temperature and charge-carrier-density-dependent conductivities and Seebeck coefficients, with energetic disorder being the main factor to explain the experimentally observed mobility upswing with carrier concentration. Moreover, we show that, for lower disorder energies, the Lorentz factor obtained from the simulation is in accordance with the Wiedemann-Franz law for degenerate-band semiconductors. At higher disorder, deviations from simple band behavior are found. Suppressed disorder energy and lattice thermal conductivity can be the key to higher thermoelectric figures of merit, zT, in SWCNT networks, possibly approaching or even exceeding zT = 1. The general understanding of transport phenomena will help the selection of chirality, composition, and charge-carrier density of SWCNT networks to improve their efficiency of thermoelectric energy conversion.

Random-resistor network modeling of resistance hysteresis of vanadium dioxide thin films

The resistance hysteresis of vanadium dioxide (VO2) is a key feature in revealing mechanisms of a phase transition as well as emerging applications. In this study, a dynamical model based on random-resistor networks is developed to simulate the transport properties of VO2 thin films. The reversible metal–insulator phase transition of each microscopic domain is captured by a modified Landau-type functional. The proposed model enables analysis of not only the formation of conducting filaments driven by an electric field, but also the thermal-driving reversal curves of resistance hysteresis. It is shown that the appearance of a hysteresis loop as well as the aggregation of metallic domains can be tuned via the interactions of each domain with its neighbors and the substrate. The interaction effects are vital for the persistence of metallic domains, which can re-trigger the insulating-to-metallic transition by a subthreshold voltage bias with the delay time much longer than the transition switching time. These results are in agreement with experimental observations and can be helpful in developing VO2-based key components ranging from infrared bolometers to the volatile resistive switches for neuromorphic computing.

Complex network growth model: Possible isomorphism between nonextensive statistical mechanics and random geometry.

In the realm of Boltzmann-Gibbs statistical mechanics, there are three well known isomorphic connections with random geometry, namely, (i) the Kasteleyn-Fortuin theorem, which connects the λ → 1 limit of the λ-state Potts ferromagnet with bond percolation, (ii) the isomorphism, which connects the λ → 0 limit of the λ-state Potts ferromagnet with random resistor networks, and (iii) the de Gennes isomorphism, which connects the n → 0 limit of the n-vector ferromagnet with self-avoiding random walk in linear polymers. We provide here strong numerical evidence that a similar isomorphism appears to emerge connecting the energy q-exponential distribution ∝ e (with q = 4 / 3 and β ω = 10 / 3) optimizing, under simple constraints, the nonadditive entropy S with a specific geographic growth random model based on preferential attachment through exponentially distributed weighted links, ω being the characteristic weight.

Electrical conductivity of nanorod-based transparent electrodes: Comparison of mean-field approaches.

We mimic nanorod-based transparent electrodes as random resistor networks (RRNs) produced by the homogeneous, isotropic, and random deposition of conductive zero-width sticks onto an insulating substrate. We suppose that the number density (the number of objects per unit area of the surface) of these sticks exceeds the percolation threshold, i.e., the system under consideration is a conductor. We computed the electrical conductivity of random resistor networks versus the number density of conductive fillers for the wire-resistance-dominated case, for the junction-resistance-dominated case, and for an intermediate case. We also offer a consistent continuous variant of the mean-field approach. The results of the RRN computations were compared with this mean-field approach. Our computations suggest that, for a qualitative description of the behavior of the electrical conductivity in relation to the number density of conductive wires, the mean-field approximation can be successfully applied when the number density of the fillers n>2n_{c}, where n_{c} is the percolation threshold. However, note the mean-field approach slightly overestimates the electrical conductivity. We demonstrate that this overestimate is caused by the junction potential distribution.

Ultra-Highly Sensitive DNA Detection with Conducting Polymer-Modified Electrodes: Mechanism, Manufacture and Prospects for Rapid e-PCR

At low copy number, sequence detection by polymerase chain reaction (PCR) requires up to 30 cycles (amplification 109) to produce a reliably detectable concentration of fluorescently-labelled amplicons. The cycle number and hence detection time is determined by the analytical sensitivity of the detector. Hybridisation of complementary DNA strands to oligonucleotide-modified conducting polymer electrodes yields an increase in the charge transfer resistance for the ferri-ferrocyanide redox couple. We demonstrate sensors using screen-printed carbon electrodes modified with a conducting polymer formed from a monomer pre-functionalised with complementary oligonucleotide, with pM sensitivity for short sequences and aM for bacterial lysate, with a response time-scale of 5 min. The response is due to the variation of electrical resistance within the polymer film. We develop a mechanism based on repulsion from the solution interface of dopant anions by the charge associated with surface-bound DNA. With results for >160 single-use sensors, we formulate a response model based on percolation within a random resistor network and highlight challenges for large-scale manufacture of such sensors. Such sensors used for label-free electrochemical detection for PCR (e-PCR) would decrease the required cycle number from 30 to less than 10 and would offer a much simplified instrument construction.

Third-order Electrical Conductivity of the Charge-ordered Organic Salt α-(BEDT-TTF)2I3

We performed third-order electrical conductivity measurements on the organic conductor $\alpha$-(BEDT-TTF)$_2$I$_3$ using an ac bridge technique sensitive to nonlinear signals. Third-order conductance $G_3$ is clearly observed even at low electric fields, and interestingly, $G_3$ is critically enhanced above the charge-order transition temperature $T_{\rm CO}=136$~K. The observed frequency dependence of $G_3$ is incompatible with a percolation model, in which a Joule heating in a random resistor network is relevant to the nonlinear conduction. We instead argue the nonlinearity of the relaxation time according to a phenomenological model on the mobility in materials with large dielectric constants, and find that the third-order conductance $G_3$ corresponds to the third-order electric susceptibility $\chi_3$. Since the nonlinear susceptibility is known as a probe for higher-order multipole ordering, the present observation of the divergent behavior of $G_3$ above $T_{\rm CO}$ reveals an underlying quadrupole instability at the charge-order transition of the organic system.

Electrical conductivity of random metallic nanowire networks: an analytical consideration along with computer simulation.

The current interest in the study of the 2D systems of randomly deposited metallic nanowires is inspired by a combination of their high electrical conductivity with excellent optical transparency. Metallic nanowire networks show great potential for use in numerous technological applications. Although there are models that describe the electrical conductivity of the random nanowire networks through wire resistance, junction resistance, and number density of nanowires, they are either not rigorously justified or contain fitting parameters. We have proposed a model for the electrical conductivity in random metallic nanowire networks. We have mimicked such random nanowire networks as random resistor networks (RRN) produced by the homogeneous, isotropic, and random deposition of conductive zero-width sticks onto an insulating substrate. We studied the electrical conductivity of these RRNs using a mean-field approximation. An analytical dependency of the electrical conductivity on the main physical parameters (the number density and electrical resistances of these wires and of the junctions between them) has been derived. Computer simulations have been performed to validate our theoretical predictions. We computed the electrical conductivity of the RRNs against the number density of the conductive fillers for the junction-resistance-dominated case and for the case where the wire resistance and the junction resistance were equal. The results of the computations were compared with this mean-field approximation. Our computations demonstrated that our analytical expression correctly predicts the electrical conductivity across a wide range of number densities.

A Hypothetical Approach to Cosmic Inflation Incorporating Binary Entropy

The current paper presents an alternative hypothesis for the termination of cosmic inflation based on Huang’s model of spacetime involving the movement of a superfluid through a random resistor network. Using this model, we previously derived a mathematical relationship between the velocity of a reference frame and the probability that a random bond is intact. As an extension of this finding, the permutations of open and closed bonds are now shown to represent potential microstates, thus providing a means of relating motion within the network to binary entropy. Applying this concept to cosmic inflation, termination of this process is an expected consequence of the changes in binary entropy associated with the increasing velocity of expansion.

Improved Electrical, Thermal, and Thermoelectric Properties Through Sample‐to‐Sample Fluctuations in Near‐Percolation Threshold Composite Materials

AbstractEffective medium theories (EMT) are powerful tools to calculate sample averaged thermoelectric material properties of composite materials. However, averaging over the heterogeneous spatial distribution of the phases can lead to incorrect estimates of the thermoelectric transport properties and the figure of merit ZT in compositions close to the percolation threshold. This is particularly true when the phases’ electronic properties are rather distinct leading to pronounced percolation effects. The authors propose an alternative model to calculate the thermoelectric properties of multi‐phased materials that are based on an expanded nodal analysis of random resistor networks (RRN). This method conserves the information about the morphology of the individual phases, allowing the study of the current paths through the phases and the influence of heterogeneous charge transport and cluster formation on the effective material properties of the composite. The authors show that in composites with strongly differing phases close to the percolation threshold the thermoelectric properties and the ZT value are always dominated exclusively by one phase or the other and never by an average of both. For these compositions, the individual samples display properties vastly different from EMT predictions and can be exploited for an increased thermoelectric performance.

Left–right crossings in the Miller–Abrahams random resistor network and in generalized Boolean models

We consider random graphs G built on a homogeneous Poisson point process on Rd, d≥2, with points x marked by i.i.d. random variables Ex. Fixed a symmetric function h(⋅,⋅), the vertexes of G are given by points of the Poisson point process, while the edges are given by pairs {x,y} with x≠y and |x−y|≤h(Ex,Ey). We call GPoissonh-generalized Boolean model, as one recovers the standard Poisson Boolean model by taking h(a,b)≔a+b and Ex≥0. Under general conditions, we show that in the supercritical phase the maximal number of vertex-disjoint left–right crossings in a box of size n is lower bounded by Cnd−1 apart from an event of exponentially small probability. As special applications, when the marks are non-negative, we consider the Poisson Boolean model and its generalization to h(a,b)=(a+b)γ with γ>0, the weight-dependent random connection models with max-kernel and with min-kernel and the graph obtained from the Miller–Abrahams random resistor network in which only filaments with conductivity lower bounded by a fixed positive constant are kept.

Nonlinear electrical transport in Fe3O4-decorated graphene nanoplatelets

This paper investigates the nonlinear properties of graphene nanoplatelets (GNPs), decorated with Fe3O4 magnetic nanoparticles (MaNPs). Nanocomposite MaNP/GNP samples were prepared by a solvothermal method with three different MaNP loading concentrations of 17 wt%, 28 wt% and 40 wt%, and deposited on a metallic interdigitated electrode (IDE). Three different models are proposed to assess measurements, with the objective to explain the electronic transport in the nanocomposites. At first, a thermionic transport model is proposed to fit the DC nonlinear current–voltage characteristics for the three concentrations. It is observed that the barrier height decreases to 0.312, 0.310 and 0.281 eV, following a decrease in the MaNP loading. A second model, dynamic random resistor network (DRRN) further shows that the impedance of IDE increases following the decreasing MaNP loading rate, 40 wt% > 28 wt% > 17 wt%, and that charge transport takes place through a resistor–capacitor (RC) rectifying percolating network. Finally, impedance spectroscopy performed at different applied DC biases shows that a constant phase element (CPE) is necessary in the equivalent circuit in order to fit the Cole–Cole plot AC response of the IDE, instead of the classical parallel RC circuit. The presence of the CPE confirms the hypothesis of random phenomena occurring in the transport according to the DRRN model. CPE is associated with a spatial distribution of different RC circuits, due to disorderness that arises from inhomogeneities in the Fe3O4–GNP samples.

Quantifying the influence of graphene film nanostructure on the macroscopic electrical conductivity

Graphene films have emerged as a promising nanostructured material class to exploit graphene's outstanding nanoscopic properties on the macroscale. Their potential applications include solar cells (Eda et al 2008 Appl. Phys. Lett. 92, 233305; Müllen et al 2008 Nano Lett. 8, 323–7), antennas (Zhang et al 2018 Electronics 7, 285; Song et al 2018 Carbon 130, 164–9), or electromagnetic interference shielding (Zhou et al 2017 Nanoscale 9, 18613–8; Wan et al 2017 Carbon 122, 74–81; Wang et al 2018 Small 14, 1704332), all of which require a high electrical conductivity. While an outstanding electrical conductivity is a key feature of pristine graphene monolayers, the transfer to the macroscale is challenging. Here, we combined theory and experiment to quantify the impact of specific structural graphene film properties. We synthesized graphene films with systematically varied flake sizes, studied their electrical conductivities, and found excellent agreement to simulations with a three-dimensional random resistor network model. In a further percolation-type study, we computed the critical share of non-conductive elements in a graphene film θ c = 10% where a substantial loss of electrical conductivity occurs. We prepared mixed films from graphene and graphene oxide to validate the threshold experimentally. In combination, experiments and simulations provide a coherent picture of how the graphene film microstructure is related to the macroscopic electrical conductivity (Rizzi et al 2018 ACS Appl. Mater. Interfaces 10 43088–94; Rizzi et al 2019 Comput. Mater. Sci. 161, 364–70). Our findings provide valuable insights for the production of highly conductive graphene-based macro-materials.