Flicker (1/f) noise in percolation networks: A new hierarchy of exponents

Noise and Electrical Characteristics of Composites Filled with Onion-like Carbon Nanoparticles

Polymer matrix composites filled with carbon nanoparticles are promising materials for many applications, but their properties strongly depend on the particle features, concentration and distribution within the matrix. Here we present a study of the electrical resistivity and the low-frequency voltage fluctuation of composites based on epoxy resin filled with onion-like carbon (OLC) of different sizes (40-250 nm) above the percolation threshold, which should clarify the electrical transport characteristics in these materials. Electrical measurements were performed in the temperature range of 78 to 380 K, and voltage noise analysis was carried out from 10 Hz to 20 kHz. At low temperatures (below 250 K), thermally activated tunneling, variable-range hopping and generation-recombination of charge carriers take place. Above 250 K, the rapid expansion of the matrix with the temperature increases the resistivity, but above ~330 K, the conductivity of the matrix becomes significant. Quasi one-dimensional electrical transport is observed in composites with the smallest particles (40 nm), while in composites with the largest particles (220-250 nm), the dimensionality of the electrical transport is higher. The temperature dependence of the electrical conductivity of composites with smaller particles is more sensitive to matrix expansion.

Low-Frequency Dynamics of Strongly Correlated Electrons in (BEDT-TTF)2X Studied by Fluctuation Spectroscopy

Fluctuation spectroscopy measurements of quasi-two-dimensional organic charge-transfer salts (BEDT-TTF) 2 X are reviewed. In the past decade, the method has served as a new approach for studying the low-frequency dynamics of strongly correlated charge carriers in these materials. We review some basic aspects of electronic fluctuations in solids, and give an overview of selected problems where the analysis of 1 / f -type fluctuations and the corresponding slow dynamics provide a better understanding of the underlying physics. These examples are related to (1) an inhomogeneous current distribution due to phase separation and/or a percolative transition; (2) slow dynamics due to a glassy freezing either of structural degrees of freedom coupling to the electronic properties or (3) of the electrons themselves, e.g., when residing on a highly-frustrated crystal lattice, where slow and heterogeneous dynamics are key experimental properties for the vitrification process of a supercooled charge-liquid. Another example is (4), the near divergence and critical slowing down of charge carrier fluctuations at the finite-temperature critical endpoint of the Mott metal-insulator transition. Here also indications for a glassy freezing and temporal and spatial correlated dynamics are found. Mapping out the region of ergodicity breaking and understanding the influence of disorder on the temporal and spatial correlated fluctuations will be an important realm of future studies, as well as the fluctuation properties deep in the Mott or charge-ordered insulating states providing a connection to relaxor or ordered ferroelectric states studied by dielectric spectroscopy.

Analysis of the interface characteristics of CVD-grown monolayer MoS2 by noise measurements

We investigated the current-voltage and noise characteristics of two-dimensional (2D) monolayer molybdenum disulfide (MoS₂) synthesized by chemical vapor deposition (CVD). A large number of trap states were produced during the CVD process of synthesizing MoS₂, resulting in a disordered monolayer MoS₂ system. The interface trap density between CVD-grown MoS₂ and silicon dioxide was extracted from the McWhorter surface noise model. Notably, generation-recombination noise which is attributed to charge trap states was observed at the low carrier density regime. The relation between the temperature and resistance following the power law of a 2D inverted-random void model supports the idea that disordered CVD-grown monolayer MoS₂ can be analyzed using a percolation theory. This study can offer a viewpoint to interpret synthesized low-dimensional materials as highly disordered systems.

Suppression of Low-Frequency Electronic Noise in Polymer Nanowire Field-Effect Transistors

The authors report on the reduction of low-frequency noise in semiconductor polymer nanowires with respect to thin-films made of the same organic material. Flicker noise is experimentally investigated in polymer nanowires in the range of 10-10(5) Hz by means of field-effect transistor architectures. The noise in the devices is well described by the Hooge empirical model and exhibits an average Hooge constant, which describes the current power spectral density of fluctuations, suppressed by 1-2 orders of magnitude compared to thin-film devices. To explain the Hooge constant reduction, a resistor network model is developed, in which the organic semiconducting nanostructures or films are depicted through a two-dimensional network of resistors with a square-lattice structure, accounting for the different anisotropy and degree of structural disorder of the active nanowires and films. Results from modeling agree well with experimental findings. These results support enhanced structural order through size-confinement in organic nanostructures as effective route to improve the noise performance in polymer electronic devices.

Linear polymers in disordered media: the shortest, the longest, and the mean self-avoiding walk on percolation clusters

Long linear polymers in strongly disordered media are well described by self-avoiding walks (SAWs) on percolation clusters and a lot can be learned about the statistics of these polymers by studying the length distribution of SAWs on percolation clusters. This distribution encompasses 2 distinct averages, viz., the average over the conformations of the underlying cluster and the SAW conformations. For the latter average, there are two basic options, one being static and one being kinetic. It is well known for static averaging that if the disorder of the underlying medium is weak, this disorder is redundant in the sense the renormalization group; i.e., differences to the ordered case appear merely in nonuniversal quantities. Using dynamical field theory, we show that the same holds true for kinetic averaging. Our main focus, however, lies on strong disorder, i.e., the medium being close to the percolation point, where disorder is relevant. Employing a field theory for the nonlinear random resistor network in conjunction with a real-world interpretation of the corresponding Feynman diagrams, we calculate the scaling exponents for the shortest, the longest, and the mean or average SAW to 2-loop order. In addition, we calculate to 2-loop order the entire family of multifractal exponents that governs the moments of the the statistical weights of the elementary constituents (bonds or sites of the underlying fractal cluster) contributing to the SAWs. Our RG analysis reveals that kinetic averaging leads to renormalizability whereas static averaging does not, and hence, we argue that the latter does not lead to a well-defined scaling limit. We discuss the possible implications of this finding for experiments and numerical simulations which have produced widespread results for the exponent of the average SAW. To corroborate our results, we also study the well-known Meir-Harris model for SAWs on percolation clusters. We demonstrate that the Meir-Harris model leads back up to 2-loop order to the renormalizable real-world formulation with kinetic averaging if the replica limit is consistently performed at the first possible instant in the course of the calculation.

Scanning noise microscopy on graphene devices

We developed a scanning noise microscopy (SNM) method and demonstrated the nanoscale noise analysis of a graphene strip-based device. Here, a Pt tip made a direct contact on the surface of a nanodevice to measure the current noise spectrum through it. Then, the measured noise spectrum was analyzed by an empirical model to extract the noise characteristics only from the device channel. As a proof of concept, we demonstrated the scaling behavior analysis of the noise in graphene strips. Furthermore, we performed the nanoscale noise mapping on a graphene channel, allowing us to study the effect of structural defects on the noise of the graphene channel. The SNM method is a powerful tool for nanoscale noise analysis and should play a significant role in basic research on nanoscale devices.

Non-Gaussian fluctuation in the charge transport of Si nanochains

The stability of the tunneling charge transport of a tangle of Si nanochains is investigated at high bias voltages using a micromanipulator in a scanning electron microscope. We confirm that the influence of electron injection due to the electron beam of a scanning electron microscope on the charge transport properties of nanochains is negligible when the electrode gap is small and the bias voltage is large. Under such conditions, current-time curves show large fluctuations. We find that the fluctuation is not a simple Brownian motion, but its distribution function can be fitted well by a Lévy distribution. Its origin is discussed in terms of percolation theory.

Percolation of diffusional creep: a new universality class

We study the percolation aspects of diffusional "Coble" creep on heterogeneous grain boundary networks, assuming free grain boundary sliding. A novel percolation threshold is obtained for the honeycomb lattice when two representative types of grain boundaries are randomly distributed, p(cc)=0.5416+/-0.0036. The creep viscosity diverges near the percolation threshold with power-law exponents t=1.69+/-0.09 and s=1.88+/-0.12, different from the standard conduction and rigidity percolation exponents. The moments of both the force and flux distributions all conform to finite-size scaling at p(cc), but with new exponents. These new scaling behaviors seen in the creeping system are proposed to arise from the unique coupling of both force and flux balances in the network.

Advective transport in percolation clusters

We simulate advective transport in bond percolation clusters at the critical point. We compute the histogram of flow speeds in each bond of the backbone and find the multifractal spectrum for two-dimensional lattices with linear dimension L2000 and in three dimensions for L250 . We demonstrate that in the limit of large systems all the negative moments of the velocity distribution become ill-defined. However, to model transport, the velocity histogram should be weighted by the flux to obtain a well-defined mean travel time. Finally, we use continuous time random walk theory to demonstrate that anomalous transport is observed whose characteristics can be related to the multifractal properties of the system.

Space-charge-limited current fluctuations in organic semiconductors

Low-frequency current fluctuations are investigated over a bias range covering Ohmic, trap-filling, and space-charge-limited current regimes in polycrystalline polyacenes. The relative current noise power spectral density S(f) is constant in the Ohmic region, steeply increases at the trap-filling transition region, and decreases in the space-charge-limited-current region. The noise peak at the trap-filling transition is accounted for within a continuum percolation model. As the quasi-Fermi level crosses the trap level, intricate insulating paths nucleate within the Ohmic matrix, determining the onset of nonequilibrium conditions at the interface between the insulating and conducting phase. The noise peak is written in terms of the free and trapped charge carrier densities.

Weight-driven growing networks

We study growing networks in which each link carries a certain weight (randomly assigned at birth and fixed thereafter). The weight of a node is defined as the sum of the weights of the links attached to the node, and the network grows via the simplest weight-driven rule: A newly added node is connected to an already existing node with the probability which is proportional to the weight of that node. We show that the node weight distribution n (w) has a universal tail, that is, it is independent of the link weight distribution: n (w) approximately w(-3) as w-->infinity . Results are particularly neat for the exponential link weight distribution when n (w) is algebraic over the entire weight range.

Multifractality in a broad class of disordered systems

We study multifractality in a broad class of disordered systems which includes, e.g., the diluted x-y model. Using renormalized field theory we analyze the scaling behavior of cumulant averaged dynamical variables (in case of the x-y model the angles specifying the directions of the spins) at the percolation threshold. Each of the cumulants has its own independent critical exponent, i.e., there are infinitely many critical exponents involved in the problem. Working out the connection to the random resistor network, we determine these multifractal exponents to two-loop order. Depending on the specifics of the Hamiltonian of each individual model, the amplitudes of the higher cumulants can vanish and in this case, effectively, only some of the multifractal exponents are required.

Resistance noise scaling in a dilute two-dimensional hole system in GaAs

We have measured the resistance noise of a two-dimensional (2D) hole system in a high mobility GaAs quantum well, around the 2D metal-insulator transition (MIT) at zero magnetic field. The normalized noise power S(R)/R(2) increases strongly when the hole density p(s) is decreased, increases slightly with temperature (T) at the largest densities, and decreases strongly with T at low p(s). The noise scales with the resistance, S(R)/R(2) approximately R2.4, as for a second order phase transition such as a percolation transition. The p(s) dependence of the conductivity is consistent with a critical behavior for such a transition, near a density p(*) which is lower than the observed MIT critical density p(c).

Maps of electric current density and hydrodynamic flow in porous media: NMR experiments and numerical simulations

The electric current density in percolation clusters was mapped with the aid of a NMR microscopy technique monitoring the spatial distribution of spin precession phase shifts caused by the currents. A test structure and a quasi-two-dimensional random-site percolation model object filled with an electrolyte solution were examined and compared with numerical simulations based on potential theory. The current density maps permit the evaluation of histograms and of volume-averaged current densities as a function of the probe volume radius as relationships characterizing transport in the clusters. The current density maps are juxtaposed to velocity maps acquired in flow NMR experiments in the same objects. It is demonstrated that electric current and hydrodynamic flow lead to transport patterns deviating in a characteristic way due to the different dependencies of the transport resistances on the pore channel width.

Resistance and resistance fluctuations in random resistor networks under biased percolation

We consider a two-dimensional random resistor network (RRN) in the presence of two competing biased processes consisting of the breaking and recovering of elementary resistors. These two processes are driven by the joint effects of an electrical bias and of the heat exchange with a thermal bath. The electrical bias is set up by applying a constant voltage or, alternatively, a constant current. Monte Carlo simulations are performed to analyze the network evolution in the full range of bias values. Depending on the bias strength, electrical failure or steady state are achieved. Here we investigate the steady state of the RRN focusing on the properties of the non-Ohmic regime. In constant-voltage conditions, a scaling relation is found between <R>/<R>(0) and V/V(0), where <R> is the average network resistance, <R>(0) the linear regime resistance, and V0 the threshold value for the onset of nonlinearity. A similar relation is found in constant-current conditions. The relative variance of resistance fluctuations also exhibits a strong nonlinearity whose properties are investigated. The power spectral density of resistance fluctuations presents a Lorentzian spectrum and the amplitude of fluctuations shows a significant non-Gaussian behavior in the prebreakdown region. These results compare well with electrical breakdown measurements in thin films of composites and of other conducting materials.

Multifractal structure of the harmonic measure of diffusion-limited aggregates

The method of iterated conformal maps allows one to study the harmonic measure of diffusion-limited aggregates with unprecedented accuracy. We employ this method to explore the multifractal properties of the measure, including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing. We resolve probabilities as small as 10(-35), and present an accurate determination of the generalized dimensions and the spectrum of singularities. We show that the generalized dimensions D(q) are infinite for q<q*, where q* is of the order of -0.2. In the language of f(alpha) this means that alpha(max) is finite. The f(alpha) curve loses analyticity (the phenomenon of "phase transition") at alpha(max) and a finite value of f(alpha(max)). We consider the geometric structure of the regions that support the lowest parts of the harmonic measure, and thus offer an explanation for the phase transition, rationalizing the value of q* and f(alpha(max)). We thus offer a satisfactory physical picture of the scaling properties of this multifractal measure.

Multifractal current distribution in random-diode networks

Recently it has been shown analytically that electric currents in a random-diode network are distributed in a multifractal manner [O. Stenull and H. K. Janssen, Europhys. Lett. 55, 691 (2001)]. In the present paper we investigate the multifractal properties of a random diode network at the critical point by numerical simulations. We analyze the currents running on a directed percolation cluster and confirm the field-theoretic predictions for the scaling behavior of moments of the current distribution. It is pointed out that a random diode network is a particularly good candidate for a possible experimental realization of directed percolation.

Multifractal properties of resistor diode percolation

Focusing on multifractal properties we investigate electric transport on random resistor diode networks at the phase transition between the nonpercolating and the directed percolating phase. Building on first principles such as symmetries and relevance we derive a field theoretic Hamiltonian. Based on this Hamiltonian we determine the multifractal moments of the current distribution that are governed by a family of critical exponents [psi(l)]. We calculate the family [psi(l)] to two-loop order in a diagrammatic perturbation calculation augmented by renormalization group methods.

Noisy random resistor networks: renormalized field theory for the multifractal moments of the current distribution

We study the multifractal moments of the current distribution in randomly diluted resistor networks near the percolation threshold. When an external current is applied between two terminals x and x(') of the network, the lth multifractal moment scales as M((l))(I)(x,x(')) approximately equal /x-x'/(psi(l)/nu), where nu is the correlation length exponent of the isotropic percolation universality class. By applying our concept of master operators [Europhys. Lett. 51, 539 (2000)] we calculate the family of multifractal exponents [psi(l)] for l>or=0 to two-loop order. We find that our result is in good agreement with numerical data for three dimensions.

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.

Scaling and universality in electrical failure of thin films

We describe the electrical failure of thin films as a percolation in two-dimensional random resistor networks. We show that the resistance evolution follows a scaling relation expressed as R approximately epsilon(-&mgr;) where epsilon = (1-t/tau), tau is the time of electrical failure of the film, and &mgr; is the same critical exponent appearing in the scaling relation between R and the defect concentration. For uniform degradation the value of &mgr; is universal. The validity of this scaling relation in the case of nonuniform degradation is proved by discussing the case in which the failure is due to a filamentary defect growth. The existence of this relation allows predictions of failure times from early time measurements of the resistance.

Multifractal properties of the random resistor network

We study the multifractal spectrum of the current in the two-dimensional random resistor network at the percolation threshold. We consider two ways of applying the voltage difference: (i) two parallel bars, and (ii) two points. Our numerical results suggest that in the infinite system limit, the probability distribution behaves for small i as P(i) approximately 1/i, where i is the current. As a consequence, the moments of i of order q</=q(c)=0 do not exist and all currents of value below the most probable one have the fractal dimension of the backbone. The backbone can thus be described in terms of only (i) blobs of fractal dimension d(B) and (ii) high current carrying bonds of fractal dimension going from 1/nu to d(B).

Geometry and dynamics of invasion percolation with correlated buoyancy

We study an invasion percolation model for drainage where the disorder comes partly from capillary thresholds and partly from height differences in a rough self-affine landscape. As a function of the buoyancy, the geometry of the invaded clusters changes dramatically. Long-range correlations from the fracture topography induce a double cluster structure with strings and compact blobs. A characteristic length is introduced comparing the width of the capillary threshold distribution and gravity effects at the pore scale. We study electrical properties of percolating clusters. Current distributions along percolating clusters are shown to be multifractal and sensitive to the buoyancy.

Scaling for the critical percolation backbone

We study the backbone connecting two given sites of a two-dimensional lattice separated by an arbitrary distance r in a system of size L at the percolation threshold. We find a scaling form for the average backbone mass: <M(B)> approximately L(dB)G(r/L), where G can be well approximated by a power law for 0<or=x<or=1: G(x) approximately x(psi) with psi=0.37+/-0.02. This result implies that <M(B)> approximately L(dB-psi)r(psi) for the entire range 0<r<L. We also propose a scaling form for the probability distribution P(M(B)) of backbone mass for a given r. For r approximately L, P(MB) is peaked around L(dB), whereas for r<<L, P(M(B)) decreases as a power law, M(-tauB)B, with tauB approximately 1.20+/-0.03. The exponents psi and tauB satisfy the relation psi=dB(tauB-1), and psi is the codimension of the backbone, psi=d-dB.