Hopping conductance and macroscopic quantum tunneling effect in three dimensional Pbx(SiO2)1−x nanogranular films

We have studied the low-temperature electrical transport properties of ${\mathrm{Pb}}_x{({\mathrm{SiO}}₂)}_{1\ensuremath{-}x}$ ($x$ being the Pb volume fraction) nanogranular films with thicknesses of $\ensuremath{\sim}1000$ 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 $x\ensuremath{\lesssim}0.50$, the resistivities $\ensuremath{\rho}$ as functions of temperature $T$ obey a $\ensuremath{\rho}\ensuremath{\propto}exp(\mathrm{\ensuremath{\Delta}}/{k}_BT)$ relation ($\mathrm{\ensuremath{\Delta}}$ being the local superconducting gap and ${k}_B$ the Boltzmann constant) below the superconducting transition temperature ${T}_c$ ($\ensuremath{\sim}7$ 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. This effect is a consequence of the competition between resistance decrease due to the occurrence of superconductivity on isolated Pb grains and the enhancement of excitation resistance due to the opening of the energy gap on the grains. For the $0.60\ensuremath{\lesssim}x\ensuremath{\lesssim}0.72$ films, the resistivities sharply decrease with decreasing temperature just below ${T}_c$, and then show a 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.