Scaling behavior of the longitudinal and Hall resistivities in indium films

We have made simultaneous measurements of the longitudinal ${\rho }_{\mathrm{xx}}$ and Hall ${\rho }_{\mathrm{xy}}$ resistivities for granular films of indium with different thicknesses ($\overline{t}=12,$ 20, and 60 nm) and the normal-state resistivities $({\rho }_{\mathrm{xxn}}=2\mathrm{}–5\times {10}^{-5}\Omega \mathrm{m})$ at temperatures $T$ down to 0.5 K and in fields $B$ up to 7 T. A striking scaling behavior expressed as ${\rho }_{\mathrm{xy}}\mathrm{}\left(T\right)=A{\rho }_{\mathrm{xx}}{\mathrm{}\left(T\right)\mathrm{}}^{\beta }$ has been observed for all the films studied irrespective of $\overline{t}$ and ${\rho }_{\mathrm{xxn}}$ at fixed fields which are not close to a critical field $B_C.$ In the mixed state ($B<\mathrm{}{B}_C$ and $T<\mathrm{}{T}_C,$ where $T_C$ is the transition temperature), the coefficient $A$ and exponent β $(=2\mathrm{}–3)$ are nearly independent of the field strength $B$ and the film thickness $\overline{t}.$ In the vicinity of the field-driven superconductor-insulator transition, we have found the unusual insulating region $B_{\mathrm{xxC}}<B<\mathrm{}{B}_{\mathrm{xyC}},$ where $B_{\mathrm{xxC}}$ and $B_{\mathrm{xyC}}$ represent critical fields determined by ${\rho }_{\mathrm{xx}}$ and ${\rho }_{\mathrm{xy}},$ respectively, which tends to grow with increasing ${\rho }_{\mathrm{xxn}}$ and/or decreasing $\overline{t}.$ This region is similar to one found previously by Paalanen, Hebard, and Ruel in amorphous ${\mathrm{InO}}_x$ thin films, and the present result is consistent with their notion that the insulating region corresponds to the Bose-glass insulator.