Quantum Percolation Threshold Research Articles

Finite-size effect on quantum percolation in topological insulators

We study the finite-size effect on quantum percolation in two-dimensional topological insulators. We demonstrate that the percolation threshold in topological insulators strongly depends on the localization length of the edge states in small clusters due to the finite-size effect. Also, we explain why the percolation threshold in the corresponding classical model determines the lower bound of the quantum percolation threshold in topological insulators. In addition, we extend the percolation model to a more general scenario, where the system is composed of both topological and trivial clusters. We find that the quantum percolation threshold can be less than the classical percolation threshold due to quantum tunneling of the edge states.

Direct observation of quantum percolation dynamics

Abstract Percolation, describing critical behaviors of phase transition in a geometrical context, prompts wide investigations in natural and social networks as a fundamental model. The introduction of quantum coherence and superposition brings percolation into quantum regime with more fascinating phenomena and unique features, which, however, has not been experimentally explored yet. Here we successfully map these large-scale porous structures into a photonic chip using femtosecond laser direct writing techniques and present an experimental demonstration of quantum transport in hexagonal percolation lattices, probed by coherent light. A quantum percolation threshold of 80% is observed in the prototyped laser-written lattices with up to 1,600 waveguides, which is significantly larger than the classical counterpart of 63%. We also investigate the spatial confinement by localization parameters and exhibit the transition from ballistic to diffusive propagation with the decrease of the occupation probability. Direct observation of quantum percolation may deepen the understanding of the relation among materials, quantum transport, geometric quenching, disorder and localization, and inspire applications for quantum technologies.

Electron-electron interaction effect on longitudinal and Hall transport in thin and thickAgx(SnO2)1−xgranular metals

We study the temperature behaviors of the Hall coefficient, ${R}_{H}$, and longitudinal conductivity, $\ensuremath{\sigma}$, of two series of ${\mathrm{Ag}}_{x}({\mathrm{SnO}}_{2}){}_{1\ensuremath{-}x}$ $(x$ being the Ag volume fraction) granular films lying in the metallic regime. The first (second) series of films are $\ensuremath{\sim}500$ $(\ensuremath{\sim}9)$ nm thick and constitute a three- (two-) dimensional granular array. In the high-$x$ regime with ${g}_{T}\phantom{\rule{0.16em}{0ex}}\ensuremath{\gg}\phantom{\rule{0.16em}{0ex}}{g}_{T}^{c}$ $({g}_{T}$ being the dimensionless intergrain tunneling conductance, and ${g}_{T}^{c}$ being the critical tunneling conductance at the percolation threshold), we observe a ${R}_{H}\ensuremath{\propto}lnT$ law from $\ensuremath{\sim}6$ to 300 K. This $lnT$ behavior is independent of array dimensionality. We also observe a $\ensuremath{\sigma}\ensuremath{\propto}lnT$ law in the temperature range $\ensuremath{\sim}20$--100 K. Below $\ensuremath{\sim}10$ K, the temperature behavior of $\ensuremath{\sigma}$ changes to a $\sqrt{T}$ dependence in thick films, while it changes to a different $lnT$ dependence in thin films. The overall ${R}_{H}$ and $\ensuremath{\sigma}$ characteristics can be explained by the electron-electron interaction (EEI) effects in the presence of granularity. As $x$ is reduced and approaches the percolation threshold, we found that the $lnT$ dependences of ${R}_{H}$ and $\ensuremath{\sigma}$ still hold for a wide temperature range. We propose an explanation for the long-standing puzzle of the $\ensuremath{\sigma}\ensuremath{\propto}lnT$ dependence, which has previously been frequently observed in composite systems near the quantum percolation threshold, as arising from the same EEI effect considered in the recent theory of granular metals.

Transport in tight-binding bond percolation models.

Most of the investigations to date on tight-binding, quantum percolation models focused on the quantum percolation threshold, i.e., the analog to the Anderson transition. It appears to occur if roughly 30% of the hopping terms are actually present. Thus, models in the delocalized regime may still be substantially disordered, hence analyzing their transport properties is a nontrivial task which we pursue in the paper at hand. Using a method based on quantum typicality to numerically perform linear response theory we find that conductivity and mean free paths are in good accord with results from very simple heuristic considerations. Furthermore we find that depending on the percentage of actually present hopping terms, the transport properties may or may not be described by a Drude model. An investigation of the Einstein relation is also presented.

Giant Hall effect of (Ni90Co10) x (SiO2)1− x nanogranular films at percolation threshold

The series of (Ni90Co10) x (SiO2)1− x nanogranular films were fabricated using the magnetron sputtering technique. The Hall effect of these films increased as the fraction of permalloy metal composition decreases near percolation. The maximum saturated Hall resistivity was found to be 4.32 µΩ cm in (Ni90Co10)55(SiO2)45 film at room temperature, about two orders of magnitude larger than that of pure ferromagnetic metallic films. The atomic fraction at the quantum percolation threshold was accurately proved with the Rutherford backscattering (RBS) for the first time.

Room Temperature Giant Hall Effect in (Ni61Fe39) x (Al2O3)1−x Percolating Nanogranular Films

A series of (Ni61Fe39) x (Al2O3)1−x films were prepared using the magnetron sputtering technique. The maximum of saturated Hall resistivity 4.06 μΩ⋅cm was observed in (Ni61Fe39)60(Al2O3)40 nanogranular film at room temperature, which was about two orders larger than that of pure ferromagnetic metallic films. For the first time, the composition in the atomic fraction at the quantum percolation threshold was accurately characterized by the Rutherford backscattering (RBS) channelling technique in order to substantiate the newly metal-insulator fraction of percolation threshold in the Giant Hall Effect. It also was identified from the TEM image of an as-deposited sample that this large enhancement of the Hall resistivity coefficient was interrelated with the percolation threshold. With a different annealing temperature (T A ) up to 300 °C , the saturated Hall resistivity of (Ni61Fe39)60(Al2O3)40 granular film decreased only a little, which showed its good thermal stability for potential application.

Localization-delocalization transitions in a two-dimensional quantum percolation model: von Neumann entropy studies

In two-dimensional quantum site-percolation square lattice models, the von Neumann entropy is extensively studied numerically. At a certain eigenenergy, the localization-delocalization transition is reflected by the derivative of von Neumann entropy which is maximal at the quantum percolation threshold ${p}_{q}$. The phase diagram of localization-delocalization transitions is deduced in the extrapolation to infinite system sizes. The nonmonotonic eigenenergies dependence of ${p}_{q}$ and the lowest value ${p}_{q}\ensuremath{\simeq}0.665$ are found. At localized-delocalized transition points, the finite scaling analysis for the von Neumann entropy is performed and it is found the critical exponents $\ensuremath{\nu}$ not to be universal. These studies provide an evidence that the existence of a quantum percolation threshold ${p}_{q}<1$ in the two-dimensional quantum percolation problem.

Dynamical aspects of two-dimensional quantum percolation

The existence of a quantum-percolation threshold ${p}_{q}<1$ in the two-dimensional (2D) quantum site-percolation problem has been a controversial issue for a long time. By means of a highly efficient Chebyshev expansion technique we investigate numerically the time evolution of particle states on finite disordered square lattices with system sizes not reachable up to now. After a careful finite-size scaling, our results for the particle's recurrence probability and the distribution function of the local particle density give evidence that indeed extended states exist in the 2D percolation model for $p<1$.

Local distribution approach to disordered binary alloys

We study the electronic structure of the binary alloy and (quantum) percolation model. Our study is based on a self-consistent scheme for the distribution of local Green functions. We obtain detailed results for the density of states, from which the phase diagram of the binary alloy model is constructed, and discuss the existence of a quantum percolation threshold.

CPA theory for the giant Hall effect in disordered nanoscale systems

We present a theory based on the Green's-function formalism to study the so-called giant Hall effect in disordered nanoscale systems in which the metallic sites build up percolating networks. The disordered Green's functions are solved in the coherent potential approximation (CPA). This method is an extension of the Blackman-Esterling-Beck approach. The crucial point is that we are able to predict the quantum percolation threshold near which the Hall coefficient is enhanced by several orders of magnitude.

Localization effects in quantum percolation

We present a detailed study of the quantum site percolation problem on simple cubic lattices, thereby focusing on the statistics of the local density of states and the spatial structure of the single particle wave functions. Using the kernel polynomial method we refine previous studies of the metal-insulator transition and demonstrate the nonmonotonic energy dependence of the quantum percolation threshold. Remarkably, the data indicates a ``fragmentation'' of the spectrum into extended and localized states. In addition, the observation of a chequerboardlike structure of the wave functions at the band center can be interpreted as anomalous localization.

Giant Hall effect in metal/insulator composite films

Metal/insulator composite materials have been fabricated by using co-sputtering at room temperature. Hall effect has been found to be greatly enhanced, when the metal volume fraction decreases from 1 to ∼0.5, in both magnetic and non-magnetic composite films. In non-magnetic Cu x –(SiO 2) 1− x composite films, nearly three orders of magnitude enhancement in the Hall coefficient is observed. This large enhancement of the Hall coefficient is not only significantly larger than the prediction of the classical percolation theory, but also occurs at a metal concentration identified to be the quantum percolation threshold, which has been accounted for in the framework of the local quantum interference effect. The greatly enhanced extraordinary Hall effect has been observed in different magnetic composites; however, it decreases with increasing temperature. Recently, we have fabricated soft magnetic composite films by using sputtering, which show a nearly temperature-independent giant Hall effect from 5 to 300 K.

Quantum Percolation and the Anderson Transition

Percolation has been studied extensively and a number of applications have been proposed due to its simplicity. One of its most interesting applications in solid state physics is the metal-insulator transition. Since the quantum phase transition is defined at absolute zero of temperature, the classical percolation picture is insufficient and the quantum effects should be taken into account. Thus localization due to the quantum interference effect becomes very important. In the classical percolation, we consider conducting and insulating elements. We define the portion of the conducting part to be p. With the increase of the conducting part, the current begins to flow once the conducting part percolates throughout the system, and the system becomes metallic. This metal-insulator transition occurs at the percolation threshold p = pc. However, in the quantum percolation (QP), even if there is already an infinite cluster, the wave function may be localized and the system remains to be in the insulating regime. With further increase of p, the localized wave function becomes delocalized at the quantum percolation threshold, p = pq(≥ pc). The MITs of non-interacting electron systems with randomness are often described as the Anderson transition (AT). If the QP belongs to the universality class of AT, our understandings of the QP are deepened, and it has been discussed whether the QP belongs to the universality classes of the AT or not. The most direct way is to estimate precisely the critical exponent of the QP and compare it with that for the AT. However, the exponent for the QP is at variance. For example, the critical exponent ν for the divergence of correlation and localization lengths ranges from 0.38 to 2.1. Using the finite size scaling of the energy level statistics, 16) we have recently studied the three dimensional (3D) bond percolation problems and estimated ν and pq. We have studied the effect of breaking the time reversal symmetry, 18) which is very important for the AT. The decrease of pq by breaking the time reversal symmetry and the values of ν are consistent with the conjecture that the QP is the AT. In this paper, we give further support of this conjecture by showing that the localization-delocalization transition occurs in 2D systems in the presence of time reversal but in the absence of the spin-rotational symmetry. This is consistent with the 2D Anderson transition that

Quantum interference and the giant Hall effect in percolating systems

We show that in metallic percolating networks the wave nature of the charge carriers can significantly modify the classical picture of the Hall effect, especially in the metal concentration range $x\ensuremath{\approx}{x}_{q},$ where x denotes the metal volume, and ${x}_{q}$ the quantum-percolation threshold. Calculations based on the model of local quantum interference effect are shown to give a consistent and quantitative account of the recent experimental findings on the (ordinary) giant Hall effect.

Quantum percolation in power-law diluted chains

We investigate the quantum percolation problem in a diluted chain with long-range hopping amplitudes. Each bond is activated with probability p( r)= p 1/ r α , where r is the distance between two sites and α characterizes the range of the interactions. The average participation ratio of all eigenstates is used as a measure of the wave-functions localization length. We found that, above a quantum percolation threshold p 1 ( q) , true extended states appear for α<1.5. In the regime of 1.5< α<2.0 there is no truly extended states even in the presence of a spanning cluster. Instead, a phase of critical wave-functions sets up.

Giant Hall effect in nonmagnetic granular metal films.

Nearly 3 orders of magnitude enhancement in the Hall coefficient is observed in Cu(x)-(SiO(2))(1--x) granular films. This large enhancement of the Hall coefficient not only is significantly larger than the prediction of the classical percolation theory, but also occurs at a metal concentration identified to be the quantum percolation threshold. Measurements of the electron dephasing length and magnetoresistance, plus the TEM characterization of microstructures, yield a physical picture consistent with the mechanism of the local quantum interference effect.

Three-Dimensional Quantum Percolation Studied by Level Statistics

Three-dimensional quantum percolation problems are studied by analyzing energy level statistics of electrons on maximally connected percolating clusters. The quantum percolation threshold p q , which is larger than the classical percolation threshold p c , becomes smaller when magnetic fields are applied, i.e., p q ( B =0)> p q ( B ≠0)> p c . The critical exponents are found to be consistent with the recently obtained values of the Anderson model, supporting the conjecture that the quantum percolation is classified onto the same universality classes of the Anderson transition. Novel critical level statistics at the percolation threshold is also reported.

Classical percolation threshold and resistance versus temperature behaviour of RuO 2-glass films

Two transition concentrations of the conducting component in RuO 2-glass films are defined. The first is identified with the classical percolation threshold ν c and its value has been found as lying usually in the range 0.02–;0.04. At the second, denoted ν q, the temperature coefficient of resistance, measured at room temperature, changes its sign and its is found that percolation ν q is most often between 0.12 and 0.16. The relatively small value of ν c is interpreted within the framework of a model including segregation. A segregation coefficient is defined as particle size ratio χ = D/ d of the glass and RuO 2 particle mean diameters D and d, respectively, with χ ranging from 10 to 150. Our computer simulation results ν c( χ = 1)≅0.16 and ν c( χ = ∞)≅0.02 agree with the literature data for uncorrelated site-percolation and random-void models, respectively. The transition observed at ν q is interpreted as quantum percolation threshold, separating a system with localized states for ν < ν q from a system with extended states for ν > q. This is illustrated by results of numerical studies of the dimensionless conductance g in a quantum percolation model; the calculations are made using Landauer-Büttiker formula and a Green's function method. Studies of conductance versus temperature characteristics from 4.2 to 300 K show G = a + bT y low-temperature behaviour with y≅0.33 for compositions below and above ν q but close to it. An interpretation of this behaviour is given within a localization/delocalization picture assuming that inelastic scattering processes play a dominant role in the transport properties. Other approaches like those based on electron-electron interaction effects and hopping conduction are also discussed.

Spectral statistics near the quantum percolation threshold.

The statistical properties of spectra of a three-dimensional quantum bond percolation system are studied in the vicinity of the metal-insulator transition. In order to avoid the influence of small clusters, only regions of the spectra in which the density of states is rather smooth are analyzed. Using the finite-size scaling hypothesis, the critical quantum probability for bond occupation is found to be ${p}_{q}=0.33\ifmmode\pm\else\textpm\fi{}0.01$ while the critical exponent for the divergence of the localization length is estimated as $\ensuremath{\nu}=1.35\ifmmode\pm\else\textpm\fi{}0.10$. This later figure is consistent with the one found within the universality class of the standard Anderson model.

Simulation of the mesoscopic transport properties of disordered metallic films

Abstract An extension of Sheng and Zhang's 2D quantum percolation model of disordered metal films to 3D systems has been described. Our simulation studies, carried out on sc lattice for the ratio of Fermi energy E to the hopping matrix element U , E / U = 0.01, have shown that for the metal concentration p = 0.34 being close to the classical site-percolation threshold p c ≅ 0.312, i.e. in the strong scattering regime, 3D dimensionless conductance g = G /( e 2 / h ) decreases exponentially with the linear size L of the sample. The localization length 0.36 has been found. On the other side, for the metal concentration p = 0.6, g tends to increase linearly with L . These results are consistent with the ones obtained for 3D quantum site-percolation by Soukoulis et al. that close to the center of the band the value of the quantum percolation threshold p q ≅ 0.54.