Hall Transport in Granular Metals and Effects of Coulomb Interactions

We present a theory of Hall effect in granular systems at large tunneling conductance $g_{T}\gg 1$. Hall transport is essentially determined by the intragrain electron dynamics, which, as we find using the Kubo formula and diagrammatic technique, can be described by nonzero diffusion modes inside the grains. We show that in the absence of Coulomb interaction the Hall resistivity $\rho_{xy}$ depends neither on the tunneling conductance nor on the intragrain disorder and is given by the classical formula $\rho_{xy}=H/(n^* e c)$, where $n^*$ differs from the carrier density $n$ inside the grains by a numerical coefficient determined by the shape of the grains and type of granular lattice. Further, we study the effects of Coulomb interactions by calculating first-order in $1/g_T$ corrections and find that (i) in a wide range of temperatures $T \gtrsim \Ga$ exceeding the tunneling escape rate $\Ga$, the Hall resistivity $\rho_{xy}$ and conductivity $\sig_{xy}$ acquire logarithmic in $T$ corrections, which are of local origin and absent in homogeneously disordered metals; (ii) large-scale ``Altshuler-Aronov'' correction to $\sig_{xy}$, relevant at $T\ll\Ga$, vanishes in agreement with the theory of homogeneously disordered metals.