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Disorder and magnetic field induced Bose-metal state in two-dimensional Ta$_x$(SiO$_2$)$_{1-x}$ granular films

The origin of the intermediate anomalous metallic state in two-dimensional superconductor materials remains enigmatic. In the present paper, we observe such a state in a series of $\sim$9.0 nm thick Ta$_x$(SiO$_2$)$_{1-x}$ ($x$ being the volume fraction of Ta) nanogranular films. At zero field, the $x$ $\gtrsim$ 0.75 films undergo a Berezinskii-Kosterlitz-Thouless transition as transform from normal to superconducing states upon cooling. For the $x$ $\lesssim$ 0.71 films, the resistance increases with decreasing temperature from 2 K down to 40 mK. A normal state to anomalous metallic state transition is observed in the $x$ $\simeq$ 0.73 film, i.e., near the transition temperature, the resistance of the film decreases sharply upon cooling as if the system would cross over to superconducting state, but then saturates to a value far less than that in normal state. When a small magnetic field perpendicular to the film plane is applied, the anomalous metallic state occurs in the $x$ $\gtrsim$ 0.75 films. It is found that both disorder and magnetic field can induce the transition from superconductor to anomalous metal and their influences on the transition are similar. For the the magnetic field induced case, we find the sheet resistance $R_{\square}(T,H)$ ($T$ and $H$ being the temperature and the magnitude of magnetic field) data near the crossover from the anomalous metal to superconductor and in the vicinity of the anomalous metal to insulator transition, respectively, obey unique scaling laws deduced from the Bose-metal model. Our results strongly suggest that the anomalous metallic state in the Ta$_x$(SiO$_2$)$_{1-x}$ granular films is bosonic and dynamical gauge field fluctuation resulting from superconducting quantum fluctuations plays a key role in its formation.