Vortex states at low temperature in disordered thin and thick films of a-Mo$_x$Si$_{1-x}$

We have measured the ac complex resistivity in the linear regime, as well as dc resistivity, for thick ($100$, $300$ nm) amorphous (a-)Mo$_x$Si$_{1-x}$ films at low temperatures ($T > 0.04$ K) in constant fields $B$. The critical behavior associated with the second-order transition has been observed for both dc and ac resistivities, which is similar to that observed for granular indium films. This is the first convincing evidence for the vortex glass transition (VGT) in the homogeneously disordered low-$T_C$ superconductors containing microscopic pinning centers. We have found that the VGT persists down to $T\sim0.1T_{C0}$ up to $B\sim0.9B_{C2}(0)$, where $T_{C0}$ and $B_{C2}(0)$ are the mean-field transition temperature and upper critical field at $T=0$, respectively. At $T\to0$ the VGT line $B_g(T)$ extrapolates to a field below $B_{C2}(0)$, indicative of the presence of a $T=0$ quantum-vortex-liquid phase in the region $B_g(0)<B<B_{C2}(0)$.
For thin ($4$ nm) films the ($T=0$) field-driven superconductor-insulator transition takes place at $B_C$. We have not obtained evidence of the metallic quantum liquid phase below $B_C$, while in $B>B_C$ an anomalous negative magnetoresistance (MR) suggesting the presence of the localized Cooper pairs has been observed. The negative MR is commonly observed for thin films; however, for thick films the MR is always positive. This means that the two-dimensionality plays an important role in the appearance of the negative MR (or localized Cooper pairs). The negative MR is no longer visible as the field is applied parallel to the film surface, consistent with the view that mobile vortices, as well as localized Cooper pairs, are present in $B>B_C$.