Weak localization in metallic granular media

We investigate the interference correction to the conductivity of a medium consisting of metallic grains connected by tunnel junctions. Tunneling conductance between the grains, $e^2{g}_{\mathrm{T}}∕\pi \hslash$, is assumed to be large, $g_{\mathrm{T}}⪢1$. We demonstrate that the weak localization correction to conductivity exhibits a crossover at temperature $T\sim g_{\mathrm{T}}^2\delta$, where $\delta$ is the mean level spacing in a single grain. At the crossover, the phase relaxation time determined by the electron-electron interaction becomes of the order of the dwell time of an electron in a grain. Below the crossover temperature, the granular array behaves as a continuous medium, while above the crossover the weak localization effect is largely a single-junction phenomenon. We elucidate the signatures of the granular structure in the temperature and magnetic field dependence of the weak localization correction.

Figure 1

(Color online) A typical diffusive trajectory in a granular array. The directed area $S_{\mathrm{eff}}$ consists of two components. The first one is the combined contribution of separate grains, see the first term in Eq. (5). The second component is the area under the coarse-grained trajectory, which is the counterpart of the directed area under an electron trajectory in a homogeneous disordered sample.

Figure 2

Second-order correction to the conductivity. Black circles represent the tunnel amplitudes, and dashed lines denote impurity scattering inside the grains.

Figure 3

Magnetic field dependence (31) of the conductivity at low temperatures, $T⪡T_{\mathrm{cr}}{g}_{\mathrm{T}}∕g_{\mathrm{cr}}$. The horizontal axis here is proportional to the logarithm of the applied magnetic field, $y=\ln (B{d}^2∕{\Phi }_0)+\ln (g_{\mathrm{gr}}∕g_{\mathrm{T}})$; the vertical axis is the magnetoresistance in dimensionless units, $\delta {\sigma }_{\mathrm{MR}}=(e^2∕2\hslash )f\left(y\right)$ with $f\left(y\right)=y+\ln (1+e^y)$, see Eq. (31). The crossover between $\ln B$ and $2\ln B$ dependences is clearly seen.

Figure 4

Sketch of the temperature dependence of the conductance in various magnetic fields: $B=0$ (1); $B⪡B_{\varphi }^{\mathrm{sg}}$ (2). The temperatures at which curve 2 departs considerably from curve 1 depend on the applied field; these temperatures are of the order of $T_{\mathrm{cr}}(B{d}^2∕{\Phi }_0)$ and $T_{\mathrm{cr}}{(B{d}^2∕{\Phi }_0)}^2(g_{\mathrm{gr}}∕g_{\mathrm{T}})$ for $B⪡({\Phi }_0∕d^2)(g_{\mathrm{gr}}∕g_{\mathrm{T}})$ and $B⪢({\Phi }_0∕d^2)(g_{\mathrm{gr}}∕g_{\mathrm{T}})$, respectively.