Quantum Percolation Model Research Articles

Electron weak localization in disordered films

The logarithmic temperature dependence of resistivity, commonly observed in disordered films, has generally been interpreted as evidence for electron weak localization, with its slope indicative of the inelastic scattering mechanism. In this work, we show that the 2D quantum percolation (QP) model, pertaining to disordered metallic films, predicts a sample-size dependent ln L conductance correction that is three times larger than that for the Anderson model. Moreover, when the film has a finite thickness, the coefficient of ln L decreases to about 2 3 of its 2D value for both the QP and the Anderson models. Since for disordered metallic films the QP model is more realistic than the Anderson model, which pertains to doped metallic films, it follows that many prior experimental results on metallic films have to be re-interpreted in regard to their implications about the inelastic scattering mechanism(s). These results have direct implications for the interpretation of experimental data.

Electron weak localization in disordered films.

The logarithmic temperature dependence of resistivity, commonly observed in disordered films, has generally been interpreted as evidence for electron weak localization, with its slope indicative of the inelastic scattering mechanism. In this work, we show that the two-dimensional (2D) quantum percolation (QP) model, pertaining to disordered metallic films, predicts a sample-size-dependent $\mathrm{ln}L$ conductance correction that is three times larger than that for the Anderson model. Moreover, when the film has a finite thickness, the coefficient of $\mathrm{ln}L$ decreases to about $\frac{2}{3}$ of its 2D value for both the QP and the Anderson models. These results have direct implications for the interpretation of experimental data.

Numerical studies of the Anderson transition in three-dimensional quantum site percolation

Numerical studies of the dimensionless conductance g in 3D metal - insulator systems are reported. A site quantum percolation model is defined. It consists of two semi-infinite ideal metal electrodes and a disordered sample of size located between them. The disorder of the sample is controlled by the metal fraction p of conducting particles randomly occupying the sites of the cubic lattice with probability p. A tight-binding one-electron Hamiltonian with diagonal disorder and a probability density of site energies of the form is considered. Magnetic field is also introduced into the model. The conductance g is calculated using the Landauer - Büttiker formula and the Green's function technique. It is found that above the classical percolation threshold - that is, for - a second critical point exists denoted as . In the region , , where is the localization length, the system is localized, while in the range where the conductance tends to indicate metallic-type behaviour. By fitting the estimated data on versus ln g to the approximate relation for the scaling function valid in the vicinity of the critical point, the critical conductance is estimated to be and the correlation length critical exponent is estimated to be . Using a finite-size scaling technique and are also found. Both estimates of are expressed in units of and are in good agreement with one another. It is found that in the region where the system indicates positive magnetoconductance typical for a disorder-induced localized states phase, while in the region the magnetoconductance is negative as expected for an extended states phase.

QUANTUM PERCOLATION

In this review we describe and analyze various numerical attempts at understanding transport in the Quantum Percolation Model. We conclude that in two-dimensions all states are localized, though not necessarily exponentially localized, and that transport at low temperatures is dominated by probabilistically exceptional necklace-like resonant states.

High- Tc superconductivity in a correlated disorder-induced localized system

A quantum-percolation model is used to discuss experimental transport properties of the YBa 2Cu 3− x Li x O 6.5+ y and Bi 2Sr 2CaCu 2 2− x Li x O 8+ y superconducting oxides. A cross-over from variable-range hopping conduction, in the insulating and semimetallic states, to Coulomb gap conduction in the “metallic” state is observed. This provides evidence of the correlated disorder-induced localized character of the unusual “metallic” dependence of high- T c superconductors.

Localization length exponent in quantum percolation.

Connecting perfect one-dimensional leads to sites $i$ and $j$ on the quantum percolation (QP) model, we calculate the transmission coefficient ${T}_{\mathrm{ij}}(E)$ at an energy $E$ near the band center and the averages of $\ensuremath{\Sigma}{\mathrm{ij}}^{}{T}_{\mathrm{ij}}$, $\ensuremath{\Sigma}{\mathrm{ij}}^{}{r}_{\mathrm{ij}}^{2}{T}_{\mathrm{ij}}$, and $\ensuremath{\Sigma}{\mathrm{ij}}^{}{r}_{\mathrm{ij}}^{4}{T}_{\mathrm{ij}}$ to tenth order in the concentration $p$. In three dimensions, all three series diverge at ${p}_{q}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.36}_{\ensuremath{-}0.02}^{+0.01}$, with exponents $\ensuremath{\gamma}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.82}_{\ensuremath{-}0.15}^{+0.10}$, $\ensuremath{\gamma}+2\ensuremath{\nu}$, and $\ensuremath{\gamma}+4\ensuremath{\nu}$. We find $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.38\ifmmode\pm\else\textpm\fi{}0.07$, differing from ``usual'' Anderson localization and violating the bound $\ensuremath{\nu}\ensuremath{\ge}2/d$ of Chayes et al. [Phys. Rev. Lett. 57, 2999 (1986)]. Thus, QP belongs to a new universality class.

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.

Magnetic penetration depth in the Chevrel-phase superconductors SnMo6S8-xSex and PbMo6S8-xSex.

A full account of positive-muon spin-rotation (${\mathrm{\ensuremath{\mu}}}^{+}$SR) studies on the systematics of the magnetic penetration depth \ensuremath{\lambda} in the isotropic Chevrel-phase superconductors ${\mathrm{SnMo}}_{6}$${\mathrm{S}}_{8\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Se}}_{\mathit{x}}$ (x=0,1,4,7) and ${\mathrm{PbMo}}_{6}$${\mathrm{S}}_{8\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Se}}_{\mathit{x}}$ (x=0,4) is presented. The absolute determination of \ensuremath{\lambda} from the \ensuremath{\mu}SR data necessitates a careful analysis of the \ensuremath{\mu}SR line shape and a comparison with simulated data, taking into account distortions of the flux-line lattice and a distribution of demagnetization factors in the sintered granular polycrystalline probes investigated. A correlation between ${\mathit{T}}_{\mathit{c}}$ and 1/${\ensuremath{\lambda}}^{2}$ is found which supports a recently developed quantum percolation model.

Anomalous composition dependence of Tc near Tcmax and microstructure of YBa 2Cu 3O 7−δ

Accurate measurements of T c ( δ) and ΔC p(δ) T c(δ) in the titled compound have shown sharp breaks in slope at or very near δ = δ 0 = 0.125. This behavior is explained by a supercell model which is similar to one previously proposed to explain the formation of the low-temperature tetragonal phase centered on x = x 0 = 0.125 in La 2− x (Ba,Sr) x CuO 4 alloys. Supercell ordering of resonant pinning centers is essential to produce coherent electron-phonon interactions mediated by such centers. Thus these data provide evidence for the crucial element in the quantum percolation model of high-temperature superconductivity.

Wave localization in disordered and fractal systems

Numerical study of wave localization in systems with disorder and/or fractal characteristics is considered. Computational techniques employed entail solving the time-dependent Schrödinger equation for particular initial wave packets. Efficient and accurate numerical algorithms based on Trotter-Suzuki product formulas are presented. Propagation of light through a semi-infinite medium with disorder in the transverse directions is studied. It is shown that propagation of wave packets in one direction coexists with localization in the other two dimensions: We refer to this effect as transverse localization. Localization properties of quantum particles on fractal networks are investigated for two very well-known tight-binding models: The quantum percolation model and the fracton model. Spatial behavior of eigenstates is considered in connection with theoretical predictions of superlocalization. In disagreement with these predictions regular exponential behavior is found.

Analysis of stochastic resonances in a two-dimensional quantum percolation model.

We locate stochastic resonances in a two-dimensional quantum percolation model by studying the variation of transmittance versus energy. We study the nature of the wave functions at the resonant energies using a multifractal analysis and compare the behavior of the multifractal spectrum of the resonant states with that for the localized states in the same model. The resonant states found by us appear to be necklace states described by Pendry.

Quantum percolation of high Tc materials: Tc dependence on carrier density

The characteristic dome-shaped dependence of T c on the carrier concentration has been analysed within a quantum percolation model. Our results indicate a universal behaviour of the critical exponent, γ, of the percolating islands, around 0.5–0.6. The same behaviour also appears for the percolation threshold, around 0.3, in all materials with high T c and possibly in superconductors with low T c that show a similar T c trend.

Quantum percolation and breakdown. Absence of the delocalisation transition in two dimensions

We estimate the transmittance of the quantum percolation model of Eggarter and Kirkpatrick on square lattices of various sizes using the vector recursion technique. We note from arguments of finite size scaling theory that there is no delocalisation transition for any degree of disorder in two dimensions. We also discuss the behaviour of the critical electric field for dielectric (Zener) breakdown in Anderson insulators.

IS THERE A DELOCALISATION TRANSITION IN A TWO-DIMENSIONAL MODEL FOR QUANTUM PERCOLATION?

We estimate the transmittance of the quantum percolation model of Eggarter and Kirkpatrick (1972) on the square lattice of various sizes using the vector recursion method. We note from finite size scaling that there is no delocalisation transition for any degree of disorder in two dimensions.

Feature article: Electronic transport in granular metal films†

Abstract A brief review is given of the theoretical framework for the description and calculation of electronic transport characteristics in granular metals, which are composite materials consisting of a random mixture of nanometer-sized metal and insulator grains. In the metal-rich regime, electrical conduction is by electron percolation through connected metallic networks. The formulation of an effective-medium theory is described for the calculation of this percolative aspect of the electrical transport. Even in the metallic regime, however, metallic conductivity behaviour is violated at low temperatures due to the electron localization effects caused by random scattering. A quantum percolation model is used to take into account the quantum-wave interference effects and to simulate both the temperature and magnetic-field dependence of the low-temperature electrical conductivity in thin granular metal films. The same theoretical model is also used to explore the mesoscopic transport behaviour of small g...

Quantum-percolation model of electronic transport in two-dimensional granular metal films.

We study the macroscopic and mesoscopic transport properties of granular metal films by using a quantum-percolation model. A granular metal film with metal fraction p is simulated by conducting grains randomly occupying the sites of a square lattice with probability p. Electrons are allowed to hop between nearest-neighbor grains. This model incorporates the granular property of a film and enables us to study the magnetoconductance in both the weak- and strong-scattering regimes. The calculated magnetoconductance exhibits a sign change as the temperature varies, as well as oscillations as a function of the applied magnetic field. Both are consistent with some recent experimental data. The universal conductance fluctuations and resonant-tunneling characteristics are also studied systematically in both the extended and the localized regimes of mesoscopic samples. At the percolation threshold, the localization behavior is studied by using both the finite-size-scaling method and the resonant-tunneling method. Our results show no evidence for superlocalization. The effect of a magnetic field on localized wave functions at the percolation threshold is also examined.

Mesoscopic transport properties of 2D disordered metallic films: a proposal for the observation of Anderson localization

The authors use the 2D quantum percolation model to simulate the mesoscopic transport properties of disordered metallic films. The conductance of a mesoscopic sample is calculated by the multichannel Landauer formula, with the transmission and reflection coefficients evaluated numerically with a recursive Green-function approach. The distribution of the conductance is found to exhibit three distinct types: normal, log-normal and between normal and log-normal, depending on the sample size L and metal fraction p. The variation of the mean with L yields the localization length and the root-mean-square conductance fluctuation Delta g is shown to have different regimes, including the universal-conductance fluctuation regime where Delta g is constant as a function of L and of the order of e2/h. Based on these results, they propose the use of mesoscopic samples of disordered metallic films for the direct experimental observation of Anderson localization for electrons. Two possible sample systems are discussed.

A Quantum Percolation Model for Magnetoconductance of Granular Metal Films

ABSTRACTA quantum percolation model is introduced to study the magnetoconductance of granular metal films. This model incorporates the granular property of the films and enables us to study the magnetoconductance in the strong scattering regime. Our calculations show a sign change in magnetoconductance as the temperature varies. There also exist oscillations in the magnetoconductance as the magnetic field is increased. Both observations are consistent with some recent experimental data.

Localization of waves in fractals: Spatial behavior.

Localization of a quantum particle on two-dimensional percolating networks is investigated numerically. Solving the time-dependent Schr\"odinger equation for particular initial wave packets we study the spatial behavior of eigenstates for two tight-binding models: the quantum percolation model and the fracton model. At the lower band edge of the fracton model localization lengths agree with theoretical scaling predictions. We address the question concerning the existence of superlocalization. Our results show no indication for superlocalized behavior of eigenstates.

Localization phase diagram for the energetically and substitutionally disordered Anderson/quantum percolation model

To study the localization of Frenkel excitons in binary systems, we consider a model that has features both of the Anderson model (diagonal disorder characterized by a probability distribution of width w) and of the quantum percolation model (substitutional disorder characterized by an occupational probability p for one of the components). With a finite-size scaling (phenomenological renormalization group) technique, and the concept of quantum connectivity, we calculate the position of the phase boundary separating localized from extended states in the w–p disorder plane. At the two endpoints of the boundary, we find that for the Anderson model the critical disorder is wc=15.95±0.25, and for the quantum percolation model the localization threshold is pq=0.477±0.011.