Critical-field anisotropy and fluctuation conductivity in granular aluminum films

We have measured the upper critical field $H_{c2\mathrm{}}\left(\theta \right)$ for extreme type-II granular aluminum films much thicker than the coherence length and have found them to display a strong temperature-dependent anisotropy ($\frac{H_{\parallel }}{H_{\perp }}\gg 1$). The temperature dependence of the parallel critical field, $H_{\parallel } \mathrm{}\left(T\right)\mathrm{}$, shows an infinite slope near $T_c$, which we interpret as an indication that these films have a layered structure. The perpendicular critical field, $H_{\perp }\mathrm{}\left(T\right)\mathrm{}$, has an upward curvature, reminiscent of the behavior observed in ${\mathrm{}\left(\mathrm{SN}\right)\mathrm{}}_x$ and some layered compounds. As a result, the anisotropy ratio decreases strongly as the temperature is lowered. We interpret this behavior as a transition towards zero dimensionality (decoupled grains). We have also measured the fluctuation conductivity ${\sigma }_s$ above $T_c$. We find that, for films with high values of normal-state resistivity, ${\sigma }_s$ follows a power law characteristic of zero dimensionality far above $T_c$, and characteristic of two dimensionality closer to $T_c$, in agreement with the proposed interpretation of the critical-field data.