Hall Resistivity of Granular Metals

We calculate the Hall conductivity ${\sigma }_{xy}$ and resistivity ${\rho }_{xy}$ of a granular system at large tunneling conductance $g_T\gg 1$. 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 ${\rho }_{xy}=H/(n^{*}ec)$, 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 ${\rho }_{xy}$ in the range $\Gamma \lesssim T\lesssim \min (g_T{E}_c,E_{\mathrm{Th}})$, where $\Gamma$ is the tunneling escape rate, $E_c$ is the charging energy, and $E_{\mathrm{Th}}$ is the Thouless energy of the grain.

Figure 1

Left: Diagrams for the Hall conductivity ${\sigma }_{xy}^{(0)}$ of the granular system [Eq. (14)]. The diffuson ${\overline{D}}_{\nearrow }$ connecting contact ${\mathbf{s}}_1$ to ${\mathbf{s}}_0$ is shown. External tunneling vertices (wavy lines) must be attached in 4 possible ways. Three more diffusons ${\overline{D}}_{\searrow },{\overline{D}}_{\swarrow },{\overline{D}}_{\nwarrow }$ connecting contacts ${\mathbf{s}}_2,{\mathbf{s}}_3,{\mathbf{s}}_4$ to ${\mathbf{s}}_0$, respectively, must also be taken into account. Right: Classical picture of Hall conductivity of a granular system. The current $I_y=G_T{V}_y$ running through the grain in the $y$ direction causes the Hall voltage drop $V_H=R_H{I}_y$ between its opposite banks in the $x$ direction, which is also applied to the contacts (the total voltage drop per lattice period in the $x$ direction is 0) giving the Hall current $I_x=G_T{V}_H=G_T^2{R}_H{V}_y$ [Eq. (3)].