Hall Resistivity of Granular Metals

Abstract

We calculate the Hall conductivity σxy and resistivity ρxy of a granular system at large tunneling conductance gT ≫ 1. We show that in the absence of Coulomb interaction the Hall resistivity depends neither on the tunneling conductance nor on the intragrain disorder and is given by the classical formula ρxy = H/(n * ec), where n * differs from the carrier density n inside the grains by a numerical coefficient determined by the shape of the grains. The Coulomb interaction gives rise to logarithmic in temperature T correction to ρxy in the range Γ T min(gT Ec, E Th ), where Γ is the tunneling escape rate, Ec is the charging energy and E Th is the Thouless energy of the grain. does not depend on the mean free path and allows one to experimentally determine the carrier concentration n. Recently, much attention from both experimental and theoretical sides has been paid to granular systems (see a review [1] and references therein). Although various physical quantities have been calculated in different regimes, the Hall transport in the granular matter has not been addressed theoretically yet. In this work we calculate the Hall conductivity(HC) of a granular system and Coulomb interaction corrections to it in the metallic regime, when the intergrain tunneling conductance G T = (2e 2 / )g T is large, g T ≫ 1 (further we set = 1).Technically, calculating HC σ xy appears to be more complicated than calculating the longitudinal conductivity(LC) σ xx . The granularity of the system is ensured by the condition that the conductance G 0 = 2e 2 g 0 of the grain is much larger than the tunneling conductance G T , g 0 /g T ≫ 1. In this limit the main contribution to σ xx comes from the tunnel barriers between the grains rather than from scattering on impurities inside the grains. In the absence of Coulomb interaction LC equalswhere a is the size of the grains and d is the dimensionality of the system. Therefore when studying longitudinal transport one can neglect electron dynamics inside the grains, which simplifies calculations significantly. On the contrary, for Hall transport one is forced to take the intragrain electron dynamics into account, since the Hall current originates from the transversal drift in crossed magnetic and electric fields inside the grains. As we find in this work the intragrain electron dynamics can be included within the diagrammatic approach by considering higher diffusion modes inside the grain. This procedures accounts for the finiteness of the ratio g T /g 0 , and allows one, in principle, to study both LC and HC of the granular system for arbitrary ratio g T /g 0 . The obtained results reproduce the solution of the classical electrodynamics problem for a granular medium (e.g. the formulacan be obtained as a series in g T /g 0 ). Quantum effects (e.g. Coulomb interaction and weak localization) may be incorporated into this scheme afterwards.We perform calculations for magnetic fields H such that ω H τ ≪ 1, where ω H = eH/mc is the Larmor frequency and τ is the electron scattering time inside the grai...

Electronic conduction properties of indium tin oxide: single-particle and many-body transport

“…The presence of marked UCFs suggest that there must exist a large amount of point defects in artificially synthesized ITO nanostructures, In this section, we discuss the electrical-transport properties of inhomogeneous ITO ultrathin films (average thickness ≈ 5-15 nm) which reveal new many-body physical phenomena that are absent in homogeneous disordered systems. These new physical properties, including logarithmic temperature dependences of both longitudinal electrical conductivity and Hall transport in a wide range of temperature, have recently been theoretically predicted [52][53][54][55][56][57], but not yet experimentally tested in detail.…”

“…Hence, an observation of the FITC processes occurring in porous ITO films implies that the constituent ITO nanoparticles are metallic. Indeed, in section 4, we will discuss that the metallic feature of ITO nanoparticles has provided a powerful platform to experimentally test the recent theories of granular metals [52][53][54][55][56][57]109].…”

“…They found that there also exists a correction to the Hall resistivity in the wide temperature range g Tδ k B T min(g T E c , E Th ), where E Th = D 0 /a 2 is the Thouless energy of a grain of radius a, D 0 is the electron diffusion constant in the grain, and min(g T E c , E Th ) denotes the minimum value of the set (g T E c , E Th ). The resulting Hall resistivity is given by [56,57] ρ xy (T ) = ρ xy,0 + δρ xy…”

“…Apart from the longitudinal electrical conductivity, Kharitonov and Efetov [56,57] have investigated the influence of Coulomb interaction on the Hall resistivity, ρ xy , by taking the electron dynamics inside individual grains into account. They found that there also exists a correction to the Hall resistivity in the wide temperature range g Tδ k B T min(g T E c , E Th ), where E Th = D 0 /a 2 is the Thouless energy of a grain of radius a, D 0 is the electron diffusion constant in the grain, and min(g T E c , E Th ) denotes the minimum value of the set (g T E c , E Th ).…”

“…The responsible electron dephasing (electron-electron scattering, electron-phonon scattering, and spin-orbit scattering) processes are discussed. Furthermore, we show that this class of material provides a very useful platform for experimentally testing the recent theories of granular metals [52][53][54][55][56][57]. In the last case, ultrathin ITO films can be intentionally made to be slightly inhomogeneous or granular, while the coupling between neighboring grains remains sufficiently strong so that the system retains global metalliclike conduction (section 4).…”