Percolation Scenario Research Articles

On the dilemma between percolation processes and fluctuating pairs as the origin of the enhanced conductivity above the superconducting transition in cuprates

The confrontation between percolation processes and superconducting fluctuations to account for the observed enhanced in-plane electrical conductivity above but near T c in cuprates is revisited. This dilemma is currently an open and debated question, whose solution would contribute to the phenomenological understanding of the emergence of superconductivity in these compounds. The cuprates studied here, La1.85Sr0.15CuO4, Bi2Sr2CaCu2O, and Tl2Ba2Ca2Cu3O10, have a different number of superconducting CuO2 (ab)-layers per unit-cell length and different Josephson coupling between them, and are optimally-doped to minimize T c -inhomogeneities. The excellent chemical and structural quality of these optimally-doped samples also contribute to minimize the effect of extrinsic -inhomogeneities, a crucial aspect when analyzing the possible presence of intrinsic percolative processes. Our analyses also cover the so-called high reduced-temperature region, up to the resistivity rounding onset ɛ onset. By using the simplest form of the effective-medium theory, we show that possible emergent percolation processes alone cannot account for the measured enhanced conductivity. In contrast, these measurements can be quantitatively explained using the Gaussian–Ginzburg–Landau (GGL) approach for the effect of superconducting fluctuations in layered superconductors, extended to ɛ onset by including a total energy cutoff, which takes into account the limits imposed by the Heisenberg uncertainty principle to the shrinkage of the superconducting wavefunction. Our present analysis confirms the adequacy of this cutoff, which was introduced heuristically, and that the effective periodicity length is controlled by the relative Josephson coupling between superconducting layers, two long standing debated aspects of the GGL approaches for multilayered superconductors. These conclusions are reinforced by analyzing, as an example, one of the recent works that allegedly discards the superconducting fluctuations scenario while supporting a percolative scenario for the enhanced conductivity above T c in cuprates.

Phase boundary near a magnetic percolation transition

Motivated by recent experimental observations [Rowley et al. in Phys Rev 96:020407, 2017] on hexagonal ferrites, we revisit the phase diagrams of diluted magnets close to the lattice percolation threshold. We perform large-scale Monte Carlo simulations of XY and Heisenberg models on both simple cubic lattices and lattices representing the crystal structure of the hexagonal ferrites. Close to the percolation threshold $$p_\mathrm{c}$$ , we find that the magnetic ordering temperature $$T_\mathrm{c}$$ depends on the dilution p via the power law $$T_\mathrm{c} \sim |p-p_\mathrm{c}|^\phi $$ with exponent $$\phi =1.09$$ , in agreement with classical percolation theory. However, this asymptotic critical region is very narrow, $$|p-p_\mathrm{c}| \lesssim 0.04$$ . Outside of it, the shape of the phase boundary is well described, over a wide range of dilutions, by a nonuniversal power law with an exponent somewhat below unity. Nonetheless, the percolation scenario does not reproduce the experimentally observed relation $$T_\mathrm{c} \sim (x_\mathrm{c} -x)^{2/3}$$ in PbFe $$_{12-x}$$ Ga $$_x$$ O $$_{19}$$ . We discuss the generality of our findings as well as implications for the physics of diluted hexagonal ferrites.

A polymerization scenario of the liquid–liquid transition of GeI4

SnI4 and GeI4 have been shown to exhibit similar polyamorphic nature. We examine the microscopic nature of the liquid–liquid transition in GeI4 by conducting an isothermal–isobaric molecular dynamics simulation for the system composed of rigid tetrahedral molecules. The model allows us to semiquantitatively discuss the structural properties of liquid GeI4 below 1 GPa. We define a physical bond between the nearest intermolecular iodine sites satisfying the conditions of forming the metallic I2 bond. We then focus on the formation of molecular clusters in dynamic networks of the bonds. The clusters are mainly formed by molecules whose nearest pairs are in edge-to-edge, face-to-edge, and vertex-to-edge orientations. The clusters grow as pressure increases, and the onset of percolation is observed below 1 GPa. The finite-size scaling analysis for the percolation probability identifies that the threshold pressure is GPa for the present model, which is near the extension of the boundary between the two liquid phases. We thus speculate that the liquid–liquid transition of GeI4 is attained by polymerization, i.e. percolation of molecular networks. The same percolation scenario with a slight modification such as introducing a bootstrap mechanism is expected to apply to the transition in liquid SnI4.

Electronic structure change of NiS2−xSex in the metal-insulator transition probed by X-ray absorption spectroscopy

The electronic structure change of NiS2−xSe x as a function of Se concentration x has been studied by Ni L-edge X-ray absorption spectroscopy (XAS). The XAS spectra show distinct features in Ni L3 edge, indicating whether the system is insulating or metallic. These features can be semi-quantitatively explained within the framework of the configurational interaction cluster model (CICM). In the S-rich region, relatively large charge-transfer energy (Δ ~ 5 eV) from ligand p to Ni 3d states and a little small p-d hybridization strength (V pdσ ~ 1.1 eV) can reproduce the experimental spectra in the CICM calculation, and vice versa in the Se-rich region. Our analysis result is consistent with the Zaanen-Sawatzky-Allen scheme that the systems in S-rich side (x ≤ 0.5) are a charge transfer insulator. However, it also requires that the Δ value must change abruptly in spite of the small change of x near x = 0.5. As a possible microscopic origin, we propose a percolation scenario where a long range connection of Ni[(S,Se)2]6 octahedra with Se-Se dimers plays a key role to gap closure.

A statistical mechanical phase transition to turbulence in a model shear flow

It is becoming increasingly clear that the strong spatial and temporal fluctuations observed in a narrow Reynolds number regime around the laminar–turbulent transition in shear flows can best be understood using the concepts and techniques from a seemingly unrelated discipline – statistical mechanics. During the last few years, a consensus has begun to emerge that these phenomena reflect an underlying non-equilibrium phase transition exhibited by a model of interacting particles on a crystalline lattice, directed percolation, that seems very far from fluid mechanics. Now, Chantry et al. (J. Fluid Mech., vol. 824, 2017, R1) have developed a truncated-mode computation of a model shear flow, capable of simulating systems far larger and longer than any previous study and have for the first time generated enough statistical data that a high-precision test of theory is feasible. The results broadly confirm the theory, extending the class of flows for which the directed percolation scenario holds and removing any remaining doubts that non-equilibrium statistical mechanical critical phenomena can be exhibited by the Navier–Stokes equations.

Possible crossover to percolation scenario near superfluid-Bose-glass transition

We discuss magnetically ordered (“superfluid”) phase near quantum transition to Bose-glass phase in a simple modeling system, Heisenberg antiferromagnet in spatial dimension d>2 in external magnetic field with disorder in exchange coupling constants. Our analytical consideration is based on hydrodynamic description of long-wavelength excitations and it is valid in the entire critical region near the quantum critical point (QCP). We demonstrate that the system behaves in full agreement with predictions by Fisher et al. (Phys. Rev. B 40, 546 (1989)) in close vicinity of QCP. On the other hand, we show that many recent experimental and numerical results obtained in various 3D systems can be described by our formulas using percolation critical exponents. Then, it is a possibility that a percolation critical regime arises in the ordered phase in some 3D systems not very close to QCP.

Detection of the superconducting transition and magnetic flux trapping in a niobium micro-ring by using micro-Hall sensors

An InAs heterostructure-based micro-Hall sensor was used to study the magnetic properties of a superconducting Nb micro-ring, enabling observation of magnetic phenomena such as diamagnetism onset and magnetic flux trapping in the 20-μm-diameter sample. The superconducting diamagnetism of the micro-ring was observed to develop slowly from T = 7.5 K down to 5 K and showed a notably sharp and substantial drop at 7.0 K, the zero-resistivity temperature obtained from transport measurements on a strip-patterned sample. The observed superconducting transition is discussed in terms of a percolation scenario. In magnetic-field-cooling measurements, the Hall signal from the magnetic flux trapped in the Nb ring at 4.5 K was detected at a sufficiently high level for quantitative comparison with the estimate.

Nonpercolative nature of the metal-insulator transition and persistence of local Jahn-Teller distortions in the rhombohedral regime ofLa1−xCaxMnO3

Evolution of the average and local crystal structure of Ca-doped LaMnO$_{3}$ has been studied across the metal to insulator (MI) and the orthorhombic to rhombohedral (OR) structural phase transitions over a broad temperature range for two Ca concentrations ($x = 0.18; 0.22$). Combined Rietveld and high real space resolution atomic pair distribution function (PDF) analysis of neutron total scattering data was carried out with aims of exploring the possibility of nanoscale phase separation (PS) in relation to MI transition, and charting the evolution of local Jahn-Teller (JT) distortion of MnO$_{6}$ octahedra across the OR transition at T$_{S}$ $\sim$ 720 K. The study utilized explicit two-phase PDF structural modeling, revealing that away from TMI there is no evidence for nanoscale phase coexistence. The local JT-distortions disappear abruptly upon crossing into the metallic regime both with doping and temperature, with only small temperature-independent signature of quenched disorder being observable at low temperature as compared to CaMnO$_{3}$. The results hence do not support the percolative scenario for the MI transition in La$_{1-x}$Ca$_{x}$MnO$_{3}$ based on PS, and question its ubiquity in the manganites. In contrast to LaMnO3 that exhibits long range orbital correlations and sizeable octahedral distortions at low temperature, the doped samples with compositions straddling the MI boundary exhibit correlations (in the insulating regime) limited to only $\sim$ 1 nm with observably smaller distortions. In $x = 0.22$ sample local JT-distortions are found to persist across the OR transition and deep into the R-phase (up to $\sim$ 1050 K) where they are crystallographically prohibited. Their magnitude and subnanometer spatial extent remain unchanged.

Griffiths phase and temporal effects in phase separated manganites

Phenomenological description of relaxation phenomena in magnetic and transport properties of perovskite manganites has been presented. The approach is based on generalization of some hypotheses appropriate to the Preisach picture of magnetization process for half-metallic ferromagnets and on an assumption that in doped manganites the phase separated state exists near the magnetic ordering temperature. For systems with the percolation type of a ferromagnet–paramagnet transition, distinctive features in relaxation of magnetization and resistivity have been found. The relaxation is shown to be most pronounced near the transition temperature, and to be an approximately logarithmic function of time. The theoretical results replicate a broad spectrum of behavior observed experimentally on time dependence of magnetization and resistivity of CMR systems and allow a direct comparison with available experimental data. We propose an additional experimental test to distinguish between the percolation scenario of magnetic and transport transitions in doped manganites, and the ferromagnetic polaron picture. In particular, an anomalously slow relaxation to zero of the order parameter can be considered as a key feature of the Griffiths-like phase transition in doped manganites. It is also shown that a system with the Griffiths-like state will exhibit nonequilibrium aging and rejuvenation phenomena, which in many aspects resemble that of a spin glass. We hope that experimental observation of a set of time decay properties will provide a settlement of apparently conflicting results obtained for different characteristics of phase-separated manganites.

On the threshold concentration of sticks providing formation of a percolating cluster in mixed matrix membranes

At the present time, many publications on mixed matrix membranes (MMM) are concerned with the study of membranes containing nonspherical nanoparticles with high aspect ratios. Macroscopic characteristics of MMMs and their variations are known to be controlled by the parameters of a percolating cluster composed of incorporated particles in the host membrane. However, conditions providing the formation of percolating structures in films, the thickness of which is comparable to the dimensions of incorporated particles, can be appreciably different from the percolation scenario in isotropic infinite systems.In this work, the Monte Carlo method is used to define the conditions providing the formation of a percolating cluster composed of sticks at varying concentrations of sticks in finite membrane and at different ratios between membrane thickness and dimensions of sticks. In the above systems, percolation-induced changes show a threshold character and take place in a narrow concentration interval. Parameters of the as-formed MMMs are found to be unstable and sensitive to both film thickness and aspect ratio of sticks.

Forest-Fire Analogy to Explain thebValue of the Gutenberg-Richter Law for Earthquakes

The Drössel-Schwabl model of forest fires can be interpreted in a coarse-grained sense as a model for the stress distribution in a single planar fault. Fires in the model are then translated to earthquakes. I show that when a second class of trees that propagate fire only after some finite time is introduced in the model, secondary fires (analogous to aftershocks) are generated, and the statistics of events becomes quantitatively compatible with the Gutenberg-Richter law for earthquakes, with a realistic value of the b exponent. The change in exponent is analytically demonstrated in a simplified percolation scenario. Experimental consequences of the proposed mechanism are indicated.

Understanding the Bose glass phase via a percolation scenario

We carry out a detailed investigation of the Bose glass (BG) phase that interrupts a direct transition from a superfluid to a Mott insulating phase in a two dimensional disordered Bose Hubbard Model. The accurate distinction between various phases is done via computing a number of relevant physical quantities such as a ratio of the sites containing integer density of the bosons to that of the total number of sites, compressibility, or a close variant of it, that is the relative fluctuation of the boson density and the entropy. Quite intriguingly, assuming the appearance of a spanning cluster of sites with integer occupancy to signal the onset of the BG phase, we find that the percolation threshold is same as that of random site percolation model in two dimensions. This similarity between two completely different systems is thought to be accidental, but may open up more studies in the future to critically examine the issue. Further, with the data obtained from our analysis, we have been able to obtain a complete phase diagram defined by the local interparticle interaction parameter (U) and the chemical potential (μ) averaged over several random configurations at different disorder values.

Triple-gap superconductivity of MgB2-(La,Sr)MnO3composite

The interplay of superconductivity and magnetism was studied in a composite prepared from ferromagnetic half-metallic La${}_{0.67}$Sr${}_{0.33}$MnO${}_{3}$ (LSMO) nanoparticles and the $s$-wave superconductor MgB${}_{2}$. A few principal effects have been found. With the onset of MgB${}_{2}$ superconductivity, a spectacular drop of the sample resistance was detected and complete superconductivity was observed at temperatures up to 20 K. The basic nanocomposite characteristics (critical temperature, current-voltage dependence, percolation threshold, etc.) are strongly affected by the half-metallic LSMO and, most probably, cannot be quantitatively explained within the framework of a conventional percolation scenario. Point contact (PC) spectroscopy was used to measure directly the superconducting energy coupling. For small voltage, an excess current and doubling of the PC normal-state conductance were detected. Conductance peaks corresponding to three energy gaps are clearly observed. Two of these gaps we identified as enhanced ${\ensuremath{\Delta}}_{\ensuremath{\pi}}$ and ${\ensuremath{\Delta}}_{\ensuremath{\sigma}}$ gaps originating from the MgB${}_{2}$; the third gap ${\ensuremath{\Delta}}_{\mathrm{tr}}$ is more than three times larger than the largest MgB${}_{2}$ gap. The temperature behavior of ${\ensuremath{\Delta}}_{\mathrm{tr}}$ does not follow the BCS dependence. The experimental results have a natural and qualitative explanation within the phase-coherency scenario of proximity-induced superconductivity. Specifically, at low temperature, a $p$-wave spin-triplet condensate with pairing energy ${\ensuremath{\Delta}}_{\mathrm{tr}}$ is essentially sustained in LSMO but is incapable of displaying a long-range superconducting response because of a phase-disordered state. The proximity coupling to MgB${}_{2}$ restores the long-range phase coherency of the superconducting state, which, in turn, enhances the superconducting state of the MgB${}_{2}$.

Structural Phase Separation in K0.8Fe1.6+xSe2 Superconductors

Structural investigations on the K0.8Fe1.6+xSe2 superconducting materials have revealed microstripes arising from phase separation. Two coexisting structural phases can be characterized by modulations of q1 = 1/5[3a* + b*], the antiferromagnetic phase K0.8Fe1.6Se2, and q2 = 1/2[a* + b*], the superconducting phase KyFe2Se2, respectively. These stripe patterns also show visible correlation with the anisotropic assembly of superconducting particles along the [110] and [11̅0] directions. In addition to the notable stripe structures, a nanoscale phase separation also appears in the present superconducting system as clearly observed by high-resolution transmission electron microscopy. Certain nanoscale structural features in this heterogeneous system can be qualitatively understood by the percolation scenario.

The origins of dragon-kings and their occurrence in society

A society is a medium with a complex structure of one-to-one relations between people. Those could be relations between friends, wife–husband relationships, relations between business partners, and so on. At a certain level of analysis, a society can be regarded as a gigantic maze constituted of one-to-one relationships between people. From a physical standpoint it can be considered as a highly porous medium. Such media are widely known for their outstanding properties and effects like self-organized criticality, percolation, power-law distribution of network cluster sizes, etc. In these media supercritical events, referred to as dragon-kings, may occur in two cases: when increasing stress is applied to a system (self-organized criticality scenario) or when increasing conductivity of a system is observed (percolation scenario). In social applications the first scenario is typical for negative effects: crises, wars, revolutions, financial breakdowns, state collapses, etc. The second scenario is more typical for positive effects like emergence of cities, growth of firms, population blow-ups, economic miracles, technology diffusion, social network formation, etc. If both conditions (increasing stress and increasing conductivity) are observed together, then absolutely miraculous dragon-king effects can occur that involve most human society. Historical examples of this effect are the emergence of the Mongol Empire, world religions, World War II, and the explosive proliferation of global internet services. This article describes these two scenarios in detail beginning with an overview of historical dragon-king events and phenomena starting from the early human history till the last decades and concluding with an analysis of their possible near future consequences on our global society. Thus we demonstrate that in social systems dragon-king is not a random outlier unexplainable by power-law statistics, but a natural effect. It is a very large cluster in a porous percolation medium. It occurs as a result of changes in external conditions, such as supercritical load, increase in system elements’ sensitivity, or system connectivity growth.

Simulation of crack growth during fracture of heterogeneous materials

A cellular automaton model for describing the fracture of mechanically loaded heterogeneous materials has been constructed. Two extreme scenarios of the fracture process have been revealed, i.e., the dispersion (percolation) scenario, according to which defects accumulate uniformly throughout the volume of the material, and the correlated scenario (growth of predominantly a single source), which have been observed during the fracture of real materials. It has been shown that, in the case of the correlated fracture, a crack grows through the mechanism of ejection of double kinks of its front. In the intermediate case, the process occurs according to both scenarios: first, the slow accumulating (percolation) fracture and, then, the rapid correlated fracture; by the time the latter process begins, a self-organized critical state with a power-law size distribution of cracks typical of it has been formed.

Evolution of the conducting phase topology at the percolation threshold in colossal magnetoresistance manganites: A magnetic small-angle neutron scattering study

We have studied by small-angle neutron scattering the magnetic field evolution of the magnetic phase separation in a colossal magnetoresistive (CMR) ${\text{Pr}}_{0.7}{\text{Ca}}_{0.3}{\text{MnO}}_{3}$ crystal, in a wide range of scattering vectors $Q$, down to $2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\text{ }{\text{\AA{}}}^{\ensuremath{-}1}$ and up to ${10}^{\ensuremath{-}1}\text{ }{\text{\AA{}}}^{\ensuremath{-}1}$. Access to such small $Q$ values allows measuring the ${Q}^{\ensuremath{-}4}$ scattering from interfaces between ferromagnetic and antiferromagnetic phases appearing along the phase separation. We were able to monitor its evolution from the very beginning of the insulator-to-metal transition, characteristic of the CMR effect. Our results confirm the percolation scenario of a field-induced conducting phase of micrometric size growing within the initial antiferromagnetic insulating phase. Moreover, we show that the CMR transition occurs when the transformation mechanism starts to be driven by large-scale strain field minimization.

Multiple-scale dynamic leaching of a municipal solid waste incineration ash

Predicting the impact on the subsurface and groundwater of a pollutant source, such as municipal solid waste (MSW) incineration ash, requires a knowledge of the so-called "source term". The source term describes the manner in which concentrations in dissolved elements in water percolating through waste evolve over time, for a given percolation scenario (infiltration rate, waste source dimensions, etc.). If the source term is known, it can be coupled with a model that simulates the fate and transport of dissolved constituents in the environment of the waste (in particular in groundwater), in order to calculate potential exposures or impacts. The standardized laboratory upward-flow percolation test is generally considered a relevant test for helping to define the source term for granular waste. The LIMULE project (Multiple-Scale Leaching) examined to what extent this test, performed in very specific conditions, could help predict the behaviour of waste at other scales and for other conditions of percolation. Three distinct scales of percolation were tested: a laboratory upward-flow percolation column (30 cm), lysimeter cells (1-2 m) and a large column (5 m) instrumented at different depths. Comparison of concentration data collected from the different experiments suggests that for some non-reactive constituents (Cl, Na, K, etc.), the liquid versus solid ratio (L/S) provides a reasonable means of extrapolating from one scale to another; if concentration data are plotted versus this ratio, the curves coincide quite well. On the other hand, for reactive elements such as chromium and aluminium, which are linked by redox reactions, the L/S ratio does not provide a means of extrapolation, due in particular to kinetic control on reactions. Hence extrapolation with the help of coupled chemistry-transport modelling is proposed.

Forward–backward rapidity correlations in a two-step scenario

We argue that two-step models, like String and Glasma models, with creation first of sources extended in rapidity that after locally decay into particles, lead to long range forward–backward correlations due to fluctuations in the number or the colour of the sources, and to a similar expression for the F – B correlation parameter b. In the simplest String percolation scenario, b increases with centrality or the number of participating nucleons. However, asymptotically it decreases with energy. This is different from the Glasma model where b increases with both, centrality and energy.

Locally ordered regions in the phase transition in the systems with a finite-range correlated quenched disorder

The locally ordered regions (LOR) in the phase transition in disordered systems are studied. There are two parts in this paper. One part is to report our numerical results on the one-dimensional saddle point equation of the Ginzburg–Landau Hamiltonian with random temperature in the presence of an ordering field. The disordered system is modelled as a lattice, on which each cell has a local reduced temperature. The random part of the local reduced temperature is distributed in the Gaussian form. The one-dimensional saddle point equation is solved numerically. The average, the fluctuation and the correlation length of the solution are calculated. The scaling relations for these quantities with the temperature, the ordering field and the disorder strength are derived. The numerical data are fitted with the scaling relations well. Another part is to discuss qualitatively the phase diagram of the finite-range correlated disordered systems. There are two proposed classes for the phase transition in connection with the LOR. One class is described by the percolative scenario, in which the phase transition is inhomogeneous. In the percolative scenario the percolation of the LOR dominates the phase transition. In another class, the phase transition is homogeneous, and can be described by the renormalization group (RG) with replica symmetry breaking (RSB). In the RG with RSB, there is nothing to do with the percolation of LOR. We shall show that these two theories, which seem contradictory, may describe two parts of the whole phase diagram. Whether the phase transition is homogeneous or inhomogeneous depends on the interaction between the LOR. If the interaction between the LOR is strong enough, the phase transition is percolative and inhomogeneous. If the interaction between the LOR is weak, the phase transition is homogeneous. The interaction between the LOR is discussed with the numerical solution on the saddle point equation.