Strong enhancement of bulk superconductivity by engineered nanogranularity

It is now well established, both theoretically and experimentally, that very small changes in the size of isolated nanograins lead to substantial nonmonotonic variations, and sometimes enhancement, of the mean-field spectroscopic gap of conventional superconductors. A natural question to ask, of broad relevance for the theory and applications of superconductivity, is whether these size effects can also enhance the critical temperature of a bulk granular material composed of such nanograins. Here we answer this question affirmatively. We combine mean-field, semiclassical, and percolation techniques to show that engineered nanoscale granularity in conventional superconductors can enhance the critical temperature by up to a few times compared to the nongranular bulk limit. This prediction is valid for three-dimensional and also quasi-two-dimensional samples, provided the thickness is much larger than the grain size. Our model takes into account an experimentally realistic distribution of grain sizes in the array, charging effects, tunneling by quasiparticles, and limitations related to the proliferation of thermal fluctuations for sufficiently small grains.

Figure 1

Sketch of the Josephson array of nanograins at three different temperatures: (a) $T\ll T_c$, (b) $T\lesssim T_c$, and (c) $T\gtrsim T_c$, where $T_c$ is the critical temperature of the grain in the bulk limit. The distribution of grain sizes is Gaussian with typical average $\sim 5$ nm and variance 1 nm. Due to size effects the array is highly inhomogeneous as each grain has a different critical temperature. Red (gray) grains are (no longer) superconducting at that temperature. The phase of the superconducting gap is indicated by an arrow. The transition in the array occurs at a temperature for which either phase fluctuations induce the loss of phase coherence or there are not enough superconducting grains to form a cluster permeating the whole of the material. We have found a substantial enhancement of the array critical temperature for different average grain size and variance, packings, electron-phonon coupling, and normal state resistance. For an optimal grain packing, see Fig. 4, the critical temperature of the array can be up to a few times higher than for a bulk nongranular material.

Figure 2

The critical temperature of the array in units of the bulk critical temperature of the material against $R_N$ for a cubic array with ${\epsilon }_F=10.2$ eV, ${\epsilon }_D=9.5$ meV, $\overline{R}=5$ nm, $\sigma =1.0$ nm, and $\lambda =0.3$. The blue line shows the critical temperature due to percolation given by finding the temperature at which $p=p_c$. The red line shows the critical temperature due to phase fluctuations found by solving Eq. (A28). Close to 2.5 $\mathrm{k}\Omega$ these two lines cross, meaning the transition that breaks global-phase coherence goes from being percolation to phase-fluctuation driven. The shaded region shows the range of parameters for which the array will be globally superconducting. This crossover corresponds to the sharp tail seen at large resistance in the following figures.

Figure 3

$T_c^{\text{Array}}$ in units of the bulk material critical temperature, against $R_N$ for a cubic array with ${\epsilon }_F=10.2$ eV, ${\epsilon }_D=9.5$ meV, and $\lambda =0.25$. Top: Gaussian distribution of sizes with mean $\overline{R}=5$ nm, and variance $\sigma =0.1$ nm (blue, solid line), 0.6 nm (red), 1.0 nm (yellow), and 1.4 nm (green). Results are weakly dependent on $\sigma$ (for $\sigma >1$ Å) as the typical scale for which shell effects in the size distribution are not randomized is much smaller. Bottom: $\sigma =1.0$ nm and $\overline{R}=5$ nm (blue), 7 nm (red), 11 nm (yellow), and 17 nm (green). $T_c^{\text{Array}}$ becomes independent of $R_N$, due to the decreasing importance of quasiparticle tunneling, and then gradually decreases for increasing $\overline{R}$ due to the weakening of shell effects.

Figure 4

$T_c^{\text{Array}}$ in units of the bulk material critical temperature against $R_N$ with ${\epsilon }_F=10.2$ eV, ${\epsilon }_D=9.5$ eV, $\overline{R}=5$ nm, and $\sigma =1.0$ nm. Top: A cubic array with $\lambda =0.2$ (blue), $0.25$ (red), $0.3$ (yellow), $0.35$ (green), and $0.40$ (black). Increasing $\lambda$ suppress size effects resulting in a behavior which is closer to the bulk. Bottom: $\lambda =0.25$ for a cubic (blue), bcc (red), and fcc (yellow) array. A smaller $p_c$ substantially enhances $T_c^{\text{Array}}$ by allowing the removal of more grains from the superconducting cluster so that the remaining ones have a higher $T_c$.

Figure 5

The superconducting electronic specific heat of the array in units of the normal electronic specific heat as a function of the temperature for $\lambda =0.2$, $z=6$, $\overline{R}=5$ nm, $\sigma =0.5$ nm, and different values of the tunneling resistance $R_N=50\Omega$ (blue), 100 $\Omega$ (red), 500 $\Omega$ (yellow), and 1000 $\Omega$ (green). As the resistance increases the peak of the specific heat becomes broader. This is a signature of a percolation-driven transition.