Conduction In Granular Metals Research Articles

Drude conductivity of a granular metal

We present a complete derivation of the analog to Drude conductivity in granular metals using diagrammatic methods. The derivation of Drude conductivity in a homogeneous metal using Kubo linear response theory is somewhat involved, with the evaluation of a singular double integral being required to cancel the diamagnetic response. We first provide a history and careful derivation of this standard result, before analyzing the equivalent situation in the model of a granular metal as a lattice of metallic grains connected by tunneling. A singular triple integral occurs here, and its evaluation gives the result expected from the Einstein relation and Fermi’s golden rule.

Superconducting fluctuation corrections to the thermal conductivity in granular metals

The first-order superconducting fluctuation corrections to the thermal conductivity of a granular metal are calculated. A suppression of thermal conductivity proportional to ${T}_{c}∕(T\ensuremath{-}{T}_{c})$ is observed in a region not too close to the critical temperature ${T}_{c}$. As $T\ensuremath{\simeq}{T}_{c}$, a saturation of the correction is found, and its sign depends on the ratio between the barrier transparency and the critical temperature. In both regimes, the Wiedemann-Franz law is violated.

Electrical properties of bismuth nanocrystal–glass composite

Composites of bismuth nanocrystals and glass matrix have been obtained from Bi 0.33Ge 0.67O 1.84 and Bi 0.57Si 0.43O 1.72 glasses as a result of thermal treatment in hydrogen atmosphere. We have shown that the temperature dependence of the conductivity of bismuth germanate glass containing Bi nanocrystals, reduced in hydrogen exhibits crossover from T −1/4 to T −1/2 dependence in the temperature range between 170 and 190 K. Characteristic parameters describing conductivity were determined and their values were discussed within the models describing conductivity in granular metals. Conductivity in Bi–GeO 2 system may be described within the model of variable-range hopping between localized centers including Coulomb gap. On the other hand, in Bi–SiO 2 system the ES model is not a good explanation of conductivity. We suggest that Zvyagin’s model of conduction by hopping via virtual states (no Coulomb gap effects) may quite well describe conductivity in this system with high concentration of grains.

Conduction in Granular Metals by Virtual Tunneling on the Fractal Percolation Cluster

We discuss the calculation of the dc conductivity of granular metals on the insulating side of the metal-insulator transition. The granular structure is represented by a fractal percolation cluster and the problem of virtual tunneling-assisted conductivity is shown to be related to estimating the number of minimal paths for the problem of chemical distance metric on the fractal. The resulting conductivity temperature dependence has the form ln σ const - (T 0 /T) x , where x = ζ/(D B + ζ), D B is the fractal dimensionality of the backbone cluster and ζ is the superlocalization exponent. This gives the value of x close to 0.4 both in two and three dimensions that agrees fairly well with the experimental value x 1/2 for many granular conductors.

Conduction in granular metals by hopping via virtual states

Conduction in granular metals by hopping via virtual states

Conduction in granular metals by hopping via virtual states

Abstract It is shown that the conductivity of granular metals on the insulating side of the metal-insulator transition can be essentially determined by intergrain tunnelling via virtual localized states of intermediate grains. This mechanism gives rise lo the temperature dependence of the conductivity in the form In σ ≈ N constant – (T o/T)x, where x ≈ 0.4, and to a large enhancement of the conductivity.

Hopping conductivity in granular metals near the insulator–metal transition

Percolation theory of activated hopping conductivity on metal grains embedded in a dielectric matrix has been developed. The Coulomb gap, reduced by screening, has been shown to close at the percolation threshold. The Hall constant has been estimated assuming greater mobility of holes compared to that of electrons due to smaller variation in the ionization potential of grains than in the electron affinity. The electric conductivity has been calculated as a function of the volume fraction of metal and the temperature. A few factors including localization of electrons in the Coulomb potential wells and multiple ionization of grains have been found responsible of the temperature dependence usually referred to as the variable range hopping.

Hopping conduction in granular metals

We modify some previous transport theories which consider only the nearest-neighbor hopping in granular metals, and compare the modified theory with the experiment of Abeles et al. In particular, we carry out a critical path analysis, and show that, with practical material parameters, the modified theory gives a value of T 0( x) in better agreement with the experiment, for the temperature-dependent conductivity σ∝ exp [−(T 0(x)/T) n] with x=metal volume fraction and n= 1 2 . We have also done a resistor network simulation, and shown that the temperature range over which the exponent n= 1 2 is rather wide and compare reasonably with the experiment of Abeles et al.

On hopping conductivity in granular metals

Theory of hopping conductivity on spherical metal grains below the percolation threshold in a dielectric matrix has been developed. Screening of the potential beyond the percolation radius of grains reduces the energy of creating the electron–hole pairs to zero at the percolation threshold. The electric conductivity has been calculated as a function of the metal volume fraction and temperature. A few factors including partial localization of electrons and multiple ionization of grains have been found responsible for the temperature dependence usually referred to as variable range hopping.

Modelling of conduction in granular metals

The Charging energy model of conduction in granular metals has been investigated. In this approach electrical conduction is modelled by a resistor lattice built from conductances, G ij = G 0exp(2χs ij+E ij/kT). χ is the decay rate of the electron wave function in an insulator, k is Boltzmann constant and s ij and E ij are intergrain separation and intergrain charging energy respectively. The critical path method used to find the conductivity, σ, of the model gives the temperature dependence of σ ∼ exp[-(T 1/T) 1 2 ] in agreement with widely observed experiments. The temperature, T 1, was expressed in terms of the model's parameters, T 1 = 6AχB c/zϱk. Here A is the width of the distribution of tunnelling distances, B c is the constant of order 1.5 in three dimension, z is the coordination number of the lattice and p is the density of grain charging energies. Numerical simulations of the model has been performed. Data have been carefully analysed. The influence of finite size of the model and pre-exponential terms appearing in the formula for σ has been considered. An excellent agreement between the critical path analysis and numerical data has been found in the limit Aχ/ϱ⪢1.

Conduction in granular metals-the effect of the grain separation distribution

A charging energy model of conduction in granular metals has been considered. It has been shown that the temperature dependence of the DC conductivity, sigma approximately exp(-(T1/T)12/), interpreted as a crossover between high- and low-temperature behaviour can in fact be explained as resulting from flat parts of both the density of states and the distribution of tunnelling distances. It has also been shown that such distributions can appear in real metal-insulator composites. Models of hopping conduction in regular lattices and in continuous systems of hard and soft grains have been proposed. A comparison with experiments has been given as well.

Conduction in granular metals

Abstract The conductivity in cermets and in granular films is known to vary as exp [-(To/T)1/2], but the existing theories for such a behaviour have been subjected to criticism. A few new possible models for the observed behaviour are considered here. Most of them cannot explain the observations. The best of those that we examined is based on correlated hopping in a Coulomb gap.

Numerical Simulation of Hopping Conductivity in Granular Metals

ABSTRACTElectrical conduction in granular metals is simulated by mapping the hopping conductance between pairs of metal grains onto a simple cubic lattice with bonds between neighbors. By considering a log-normal distribution of grain sizes and the effect of disorder potential, the numerically calculated network conductance exhibit clear deviation from simple activation. Plotting -log a vs. T-½, where σ denotes conductivity and T the temperature, gives good straight line behavior with slopes comparable to those measured experimentally. Our results are noted to differ from those of Adkins et al.

Field Effect Experiments on Discontinuous Metal Films

ABSTRACTWe describe experiments in which the energetics of electrical conduction in discontinuous metal films are probed by inducing carriers electrostatically into the films. Both conductivity and thermopower show field effects that are generally of the expected form but they are very much smaller than the classic model for conduction in these systems would predict. Furthermore, in some cases at least, they can be shown to result from dielectric relaxation processes rather than from direct modification of basic transport energetics.We discuss possible explanations, including that large potential disorder is present and that transport is controlled by energy structure that moves with the chemical potential, as would be the case i correlation energies determined carrier mobility. Experiments that measure the charging energies of the metal particles directly suggest that correlation is important, and we argue that progress in understanding conduction in granular metals depends on development of a model in which correlation energies play a critical role in controlling transport.

Conduction in granular metals-variable-range hopping in a Coulomb gap?

The author sets out the model for conduction in disordered systems by variable-range tunnelling in a Coulomb gap. Detailed analysis of experimental data for three different types of granular metal shows that the model cannot apply to these systems. He concludes that the problem of the mechanism of conduction in granular metals remains unsolved.

Conduction in granular metals with potential disorder

The author presents a detailed analysis of conduction in discontinuous metals based on the traditional model but taking into account both the statistical aspects of relevant structural features and the presence of potential disorder. The analysis predicts conductivity with a temperature dependence close to simple activation at all temperatures. It is argued that an essential feature of the physics of transport in these systems has yet to be identified and incorporated into a satisfactory detailed model.

Concentration dependence of hopping conductivity in granular metals.

An analysis of conductivity in a model of identical metallic spheres randomly distributed in an insulator is presented. Using an approach based on ideas of percolation theory, the equation is derived which defines the limiting value for the separations between neighboring particles along the ``optimal path.'' The volume and structure of the current-carrying backbone are discussed and its contribution to the bulk conductivity is obtained.

Hopping processes of conduction in granular disordered systems

A detailed analysis of the hopping conduction in granular metals is presented. The shape of the density of states and the process of conduction are resulting from the distribution of the sizes of the metallic grains. It is shown that many processes for the hopping conduction can be simultaneously observed. Under some conditions related to the range of temperature and to the width of the grain size distribution only one of these processes will be dominant. A brief application to real systems gives results which are in good agreement with the experimental data.

Transport phenomenon in granular metals

An improved calculation model in the percolation theory is used to determine the d.c. conductivity in granular metals. The model is shown to be adequate for explaining the simultaneous contribution of the laws ln σ∼ T −γ with γ=1/4, 1/3, 1/2 in the same range of temperature: this results from a constant density of states. One of these laws is only dominant for a certain distribution of the grain sizes. The crossover from γ=1/4 to γ=1/2 with increasing temperature is only possible for a density of states of the form р(∈)=р 0 + a ∈ 2 . The results are shown to be dependent on the shape of the density of states.

Quantum size effect for hopping conduction in granular metals

The electrical conductivity in granular metals is usually considered as a thermally activated tunneling process. The authors present here an improved calculation model, which shows that the contribution of each grain to the conduction process is not obvious. This model allows the determination of the conductivity in such systems, as a limit of the general behaviour when the dimension of the grains is very much reduced and the temperature very low. It seems that the quantum size effect is increased when the particles are embedded in solid matrix.