Hopping conductance and macroscopic quantum tunneling effect in three dimensional ${\mathrm{Pb}}_x{\left({\mathrm{SiO}}_2\right)}_{1-x}$ nanogranular films

We have studied the low-temperature electrical transport properties of ${\mathrm{Pb}}_x{\left({\mathrm{SiO}}_2\right)}_{1-x}$ ($x$ 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 a $\rho \propto exp(\mathrm{\Delta }/k_{B}T)$ relation ($\mathrm{\Delta }$ being the local superconducting gap and $k_B$ the 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. 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 $0.60\lesssim x\lesssim 0.72$ 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 $R\left(T\right)$ data below $T_c$ can be well explained by a model that includes both thermally activated phase slips and quantum phase slips.

Figure 1

Bright-field TEM images for ${\mathrm{Pb}}_x{\left({\mathrm{SiO}}_2\right)}_{1-x}$ films with $x$ values of (a) 0.47, (b) 0.50, (c) 0.60, and (d) 0.65. The insets in (a) and (c) are the selected-area electron-diffraction patterns of corresponding films, and the insets in (b) and (d) show the corresponding grain-size distribution histograms.

Figure 2

Normalized resistivity as a function of temperature from 300 down to 10 K for the films with $x\simeq 0.47,0.53,0.57,0.60$, and 0.74.

Figure 3

(a) Logarithm of resistivity as a function of $T^{-1/2}$ from 80 down to 6 K for the $x\simeq 0.47,0.48$, and 0.49 films. The symbols are the experimental data and the straight solid lines are least-squares fits to Eq. (1). (b) Logarithm of resistivity as a function of the reciprocal of temperature from 10 down to 2 K for the same films in (a). The symbols are the experimental data and the straight solid lines are least-squares fits to Eq. (2). The inset represents the logarithm of resistivity as a function of $T^{-1/2}$ from 8 to 2 K for the $x\simeq 0.48$ film.

Figure 4

(a) Logarithm of resistivity as a function of the reciprocal of temperature from 10 down to 2 K under different fields for the $x\simeq 0.48$ film. (b) Current $I$ versus voltage $V$ at different temperatures for the $x\simeq 0.50$ film.

Figure 5

(a) Resistivity versus temperature under different field for the $x\simeq 0.52$ film. (b) Resistivity versus magnetic field at different temperatures. The ${\mu }_0$ in the title of the abscissa is the permeability of free space.

Figure 6

(a) Resistivity as a function of temperature for $x\simeq 0.60$ and 0.65 films. Inset: Resistance versus temperature below $\sim 6.5$ K for the two films. The solid curves are fits to Eq. (4). (b) Resistivity as a function of temperature under different fields for the $x\simeq 0.60$ film. Inset: Resistance versus temperature under 5 T below $\sim 4.5$ K for the film. The solid curve is the fit to Eq. (4).