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The electrical transport properties of Pb$_x$(TiO$_2$)$_{1-x}$ ($x$ being the Pb volume fraction and ranging from $\sim$0.45 to $\sim$0.69) granular films are investigated experimentally. The charging energy of the Pb granules is reduced to less than the superconducting gap of Pb granules for the low temperature insulating films by using high-$k$ dielectric TiO$_2$ as the insulating matrix. For the insulating films in the vicinity of the superconductor-insulator transition, Cooper-pair hopping governs the low-temperature hopping transport. For these films, the low-temperature magnetoresistance is positive at low field and the resistivity vs temperature for Cooper-pair hopping obeys an Efros-Shklovsii-type variable-range-hopping law. A crossover from Cooper-pair-dominated hopping to single-electron-dominated hopping is observed with decreasing $x$. The emergence of single-electron-dominated hopping in the more insulating films is due to the causation that the intergrain Josephson coupling becomes too weak for Cooper pairs to hop between adjacent superconducting Pb granules., Comment: 8 pages and 6 figures

We systematically investigated the low-temperature transport properties of a series of NbN epitaxial films with thickness $t$ ranging from $\sim$2.0 to $\sim$4.0 nm. The films undergo a superconductor-insulator transition (SIT) with decreasing film thickness, and the critical sheet resistance for the SIT is close to the quantum resistance of Cooper pairs $h/4e^2$ (6.45 k$\Omega$). Besides the Berezinski-Koterlitz-Thouless transition, a magnetic-field-driven SIT is observed in those two-dimensional (2D) superconducting films (2.6 nm $\lesssim t \lesssim 4.0$ nm). Interestingly, it is found that the low-temperature magnetoresistance isotherms do not cross at a single fixed point but at a well-distinguished region for these superconducting films. The dynamical critical exponent obtained by analyzing these magnetoresistance isotherms is divergent as the quantum critical point is being approached. The behavior of the dynamical critical exponent, originating from quenched disorder at ultralow temperatures, provides direct evidence for the occurrence of quantum Griffiths singularity in the quantum phase transition process of the films. The field-driven anomalous metal (quantum metal) state does not appear in these films. Our results suggest that the quantum Griffiths singularity not only occurs in the highly crystalline 2D superconductors with superconductor-metal transition but also in those with SIT., Comment: 8 pages and 7 figures

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., Comment: The manuscript contains 8 pages, and 8 figures and one Supplemental Material file

The microstructure and electrical transport of Cr$_x$(SiO$_2$)$_{1-x}$ nanogranular films with Cr volume faction $x$$\simeq$0.67 and 0.72 are systematically investigated. The transmission electron microscopy images and elemental mappings indicate that the films are quite inhomogeneous: some Cr granules directly connect to others while some Cr granules with size $\sim$1 to $\sim$3\,nm disperse in the SiO$_2$ dielectric matrix. For each film, the Hall coefficient $R_H$ varies linearly with the natural logarithm of temperature, i.e., $\Delta R_H$$\propto$$\ln T$, above $\sim$60\,K, saturates at $\sim$60\,K, and retains the saturating value below $\sim$60\,K. The temperature dependence of Hall coefficient can be explained by the recent theory in granular metals and originates from \emph{virtual diffusion} of electrons through the metallic granules. For the conductivity $\sigma$, a robust $\Delta \sigma$$\propto$$\sqrt{T}$ law is observed from $\sim$50 down to 2\,K. The behavior of the conductivity stems from the ``Altshuler-Aronov" correction, whose influence on the Hall coefficient is not present in the films., Comment: 5 pages, and 3 figures

We have studied the low-temperature electrical transport properties of Pb$_x$(SiO$_2$)$_{1-x}$ ($x$ being the Pb volume fraction) nanogranular films with thicknesses of $\sim$1000 nm and $x$ spanning the dielectric, transitional, and metallic regions. It is found that the percolation threshold $x_c$ lies between 0.57 and 0.60. For films with $x$$\lesssim$0.50, the resistivities $\rho$ as functions of temperature $T$ obey $\rho\propto\exp(\Delta/k_BT)$ relation ($\Delta$ being the local superconducting gap and the $k_B$ Boltzmann constant) below the superconducting transition temperature $T_c$ ($\sim$7 K) of Pb granules. The value of the gap obtained via this expression is almost identical to that by single electron tunneling spectra measurement. The magnetoresistance is negative below $T_c$ and its absolute value is far larger than that above $T_c$ at a certain field. These observations indicate that single electron hopping (or tunneling), rather than Cooper pair hopping (or tunneling) governs the transport processes below $T_c$. The temperature dependence of resistivities shows reentrant behavior for the 0.50$<$$x$$<$0.57 films. It is found that single electron hopping (or tunneling) also dominates the low-temperature transport process for these films. The reduction of the single electron concentration leads to an enhancement of the resisivity at sufficiently low temperature. For the 0.60$\lesssim$$x$$\lesssim$0.72 films, the resistivities sharply decrease with decreasing temperature just below $T_c$, and then show dissipation effect with further decreasing temperature. Treating the conducting paths composed of Pb particles as nanowires, we have found that the $R(T)$ data below $T_c$ can be well explained by a model that includes both thermally activated phase slips and quantum phase slips., Comment: 8 pages, 6 figures