Transport Properties of Granular Metals at Low Temperatures

We investigate transport in a granular metallic system at large tunneling conductance between the grains, $g_T\gg 1$. We show that at low temperatures, $T\le g_T\delta$, where $\delta$ is the mean energy level spacing in a single grain, the coherent electron motion at large distances dominates the physics, contrary to the high-temperature ($T>g_T\delta$) behavior where conductivity is controlled by the scales of the order of the grain size. In three dimensions we predict the metal-insulator transition at the bare tunneling conductance $g_T^C=(1/6\pi )\ln (E_C/\delta )$, where $E_C$ is the charging energy of a single grain. Corrections to the density of states of granular metals due to the electron-electron interaction are calculated. Our results compare favorably with the logarithmic dependence of resistivity in the high-$T_c$ cuprate superconductors indicating that these materials may have a granular structure.

Figure 1

Diagrams describing the conductivity of granular metals: diagram (a) corresponds to ${\sigma }_0$ in Eq. (2a), and it is the analog of Drude conductivity. Diagrams (b)–(e) describe the first order correction to the conductivity of granular metals due to electron-electron interaction. The solid lines denote the propagator of electrons, and the dashed lines describe effective screened electron-electron propagator. The tunneling vertices are described by the circles. The sum of diagrams (b) and (c) results in the conductivity correction $\delta {\sigma }_1$ in Eq. (2a). The other two diagrams, (d) and (e), result in the correction $\delta {\sigma }_2$.