The Hall effect in the variable-range-hopping regime

Friedman, L.; Pollak, M.

doi:10.1080/01418638108222584

Cited by 75 publications

(28 citation statements)

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“…The hydraulic conductivity is closely related to the critical conductance (e.g. [4,[43][44][45][46] and analogue for electrical conductivity [47]), but the relationship is somewhat complex and the details are not required in the present context. The existence of an infinite path in an infinite system does not guarantee the existence of finite interconnected paths in finite systems, and there is a spread of values of p, for which 'percolation' occurs in finite systems.…”

Section: Application Of Cluster Statistics Of Percolation Theory To Tmentioning

confidence: 99%

Distribution of hydraulic conductivity in single scale anisotropy

Hunt

Blank

Skinner

2006

Philosophical Magazine

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The cluster statistics of percolation theory are used to find the distributions of hydraulic conductivity, K, of anisotropic (truncated) random fractal media. Rescaling of variables to transform anisotropic to isotropic media also produces deformations of, for example experimental volumes, and the resulting non-equidimensional shapes may generate interesting size effects on K. Previously, the most likely value of K was obtained by comparing the correlation length from percolation theory with system dimensions, a procedure analogous to those developed for hopping conduction in disordered systems to calculate the longitudinal conductivity of thin films. The result probably explains the frequent tendencies of measurements of K in anisotropic fracture networks and agricultural soils to increase with the scale of measurement, similarly to how the longitudinal conductivity of a thick film would be larger than the corresponding conductivity of a thin film (three-rather than two-dimensional conduction). However, the same procedure applied to the conductivity in the perpendicular direction (analogous to the transverse electrical conductivity of a thin film) shows a diminishing function of spatial scale. Collectively, these 'scale effects' disappear if the shape of the experimental volume is selected to maintain the relationships of conduction in the various directions as the scale of the experiment is increased analogously to equidimensional volumes in isotropic media. The increase in K is, thus, merely due to an increase in the dimensionality of conduction from one to three with increasing system size. The paper, thus, provides a solid argument against a common assumption in the porous media communities that the connectivity of highly conducting regions of a medium should increase with increasing scale of measurement.

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Section: Application Of Cluster Statistics Of Percolation Theory To Tmentioning

confidence: 99%

Distribution of hydraulic conductivity in single scale anisotropy

Hunt

Blank

Skinner

2006

Philosophical Magazine

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“…is satisfied for every bond (ij) in the triad [18] and S 123 = sgn[h · (r 12 × r 13 )] accounts for the orientation of the triangle with respect to h. Taking into account that the Hall current density is given by j H = σ xx E H and I = σ xx L 2 E, we obtain the following expression for the anomalous Hall conductivity [19,20]:…”

mentioning

confidence: 99%

“…Since the nodes are far away from each other, we may regard them as independent sources of the Hall emf. Thus the problem reduces to finding the Hall emf developed at a single isolated node of the backbone in response to the current flowing through the node [19,20,21]. Applying the above described two-step iterative procedure to a single node with an associated triad of sites shown in Fig.1, we obtain:…”

mentioning

confidence: 99%

“…Following Refs. [19,20,21], we now assert that the Hall conductivity in the system is associated exclusively with the current carrying backbone of the percolating cluster. The backbone can be visualized as a supernetwork of macrobonds and nodes, with typically three macrobonds intersecting at a node.…”

mentioning

confidence: 99%

“…We believe that the best approach to this problem is the one used in Refs. [19,20,21] to investigate the problem of ordinary Hall effect in hopping conduction regime. This approach starts from the assumption that the problem can be solved perturbatively in the second, Hall term in Eq.(6).…”

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Anomalous Hall Effect in Ferromagnetic Semiconductors in the Hopping Transport Regime

Burkov

Balents

2003

Phys. Rev. Lett.

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We present a theory of the Anomalous Hall Effect (AHE) in ferromagnetic (Ga,Mn)As in the regime when conduction is due to phonon-assisted hopping of holes between localized states in the impurity band. We show that the microscopic origin of the anomalous Hall conductivity in this system can be attributed to a phase that a hole gains when hopping around closed-loop paths in the presence of spin-orbit interactions and background magnetization of the localized Mn moments. Mapping the problem to a random resistor network, we derive an analytic expression for the macroscopic anomalous Hall conductivity σ AH xy . We show that σ AH xy is proportional to the first derivative of the density of states ̺(ǫ) and thus can be expected to change sign as a function of impurity band filling. We also show that σ AH xy depends on temperature as the longitudinal conductivity σxx within logarithmic accuracy.PACS numbers: 75.50. Pp,73.50.Jt,72.20.Ee Diluted magnetic semiconductors (DMS) are very interesting new materials, still not understood in full detail. Although it is well accepted that the origin of ferromagnetism in Ga 1−x Mn x As, the most thoroughly studied DMS, is a hole-mediated effective interaction between the localized S = 5/2 Mn spins, the appropriate description of this interaction, and a closely related question of the nature of hole carriers, are still controversial issues. Samples with the highest Curie temperatures show metallic behavior [1] and are well described by the valenceband hole-fluid model [2]. On the other hand, for low Mn concentrations, i.e. when x is less than about 0.03, (Ga,Mn)As is insulating and the hole transport likely occurs in the Mn impurity band [3]. The low-temperature conduction in this case is by phonon-assisted hopping.Anomalous Hall Effect (AHE) has been an important characterization tool for itinerant ferromagnets [4,5,6] and played an important role in the experimental study of DMS ferromagnetism as well [7]. Experimentally AHE is manifested as an additional term in the Hall resistivity of the sample, usually assumed to be proportional to the magnetization:Standard theories of AHE in metallic ferromagnets [4,6] attribute it to the modification of impurity scattering in the presence of spin-orbit interactions. Recently, however, it was realized that a purely geometrical mechanism of AHE is possible [8,9,10,11]. In particular, a k-space Berry phase theory of AHE [8] has been proposed, and applied specifically to (Ga,Mn)As in the metallic transport regime. This theory is very successful in describing AHE in metallic (Ga,Mn)As, and we believe that the Berry-phase mechanism of AHE should be relevant in general for all itinerant ferromagnets. In this Letter we propose an analogous geometricalphase theory of AHE in (Ga,Mn)As, but in the case when conduction is by hopping between strongly localized states in the impurity band, rather than through extended valence-band states. Our approach is based on a modified Holstein's theory of the Hall effect in hopping conduction [12] (for related w...

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Flow, Transport, and Reaction in Porous Media: Percolation Scaling, Critical‐Path Analysis, and Effective Medium Approximation

Hunt

Sahimi

2017

Reviews of Geophysics

125

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We describe the most important developments in the application of three theoretical tools to modeling of the morphology of porous media and flow and transport processes in them. One tool is percolation theory. Although it was over 40 years ago that the possibility of using percolation theory to describe flow and transport processes in porous media was first raised, new models and concepts, as well as new variants of the original percolation model are still being developed for various applications to flow phenomena in porous media. The other two approaches, closely related to percolation theory, are the critical‐path analysis, which is applicable when porous media are highly heterogeneous, and the effective medium approximation—poor man's percolation—that provide a simple and, under certain conditions, quantitatively correct description of transport in porous media in which percolation‐type disorder is relevant. Applications to topics in geosciences include predictions of the hydraulic conductivity and air permeability, solute and gas diffusion that are particularly important in ecohydrological applications and land‐surface interactions, and multiphase flow in porous media, as well as non‐Gaussian solute transport, and flow morphologies associated with imbibition into unsaturated fractures. We describe new applications of percolation theory of solute transport to chemical weathering and soil formation, geomorphology, and elemental cycling through the terrestrial Earth surface. Wherever quantitatively accurate predictions of such quantities are relevant, so are the techniques presented here. Whenever possible, the theoretical predictions are compared with the relevant experimental data. In practically all the cases, the agreement between the theoretical predictions and the data is excellent. Also discussed are possible future directions in the application of such concepts to many other phenomena in geosciences.

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The Hall effect in the variable-range-hopping regime

Abstract: The calculated Hall mobility p~ due to hopping transport in spatially random systems is extended to include site-energy disorder. In the low-temperature, variable-range-hopping limit, an upper limit to p~ is found and its temperature dependence is established.

Cited by 75 publications

References 12 publications

Distribution of hydraulic conductivity in single scale anisotropy

Distribution of hydraulic conductivity in single scale anisotropy

Anomalous Hall Effect in Ferromagnetic Semiconductors in the Hopping Transport Regime

Flow, Transport, and Reaction in Porous Media: Percolation Scaling, Critical‐Path Analysis, and Effective Medium Approximation

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