Enhanced Cooper pairing versus suppressed phase coherence shaping the superconducting dome in coupled aluminum nanograins

The development of the fundamental superconducting (SC) energy scales—the SC energy gap $\mathrm{\Delta }$ and the superfluid stiffness $J$—of granular aluminum, i.e., thin films composed of coupled nanograins, is studied by means of optical THz spectroscopy. Starting from well-coupled grains, $\mathrm{\Delta }$ grows as the grains are progressively decoupled, causing the unconventional increase of $T_c$ with sample resistivity. When the grain coupling is suppressed further, $\mathrm{\Delta }$ saturates while the critical temperature $T_c$ decreases, concomitantly with a sharp decline of $J$, delimiting a SC dome in the phase diagram. This crossover to a phase-driven SC transition is accompanied by an optical gap surviving into the normal state above $T_c$. We demonstrate that granular aluminum is an ideal testbed to understand the interplay between quantum confinement and global SC phase coherence due to nanoinhomogeneity.

Figure 1

Superconducting dome and dynamical conductivity. (a) Critical temperature $T_c$ as a function of the normal-state resistivity (measured at 5 K) of granular Al films studied in this work. Yellow symbols refer to the samples displayed in panels below. $T_c$ encloses a domelike superconducting phase with low-, optimal- and high-resistivity regimes. (b)–(d) (Normalized) spectra of ${\sigma }_1\left(\nu \right)$ and ${\sigma }_2\left(\nu \right)$ of samples located on the left (sample 1), the right (sample 3), and at the maximum (sample 2) of the SC dome. The solid lines are fits to the Mattis-Bardeen theory. Note that the fit on ${\sigma }_1$ disregards the low-frequency range due to excessive conductivity beyond Mattis-Bardeen theory.

Figure 2

Superconducting energy scales. $T_c,\mathrm{\Delta }\left(0\right),J\left(0\right),J_{\mathrm{\Delta }}\left(0\right)$ (expressed in units of temperature), and $\mathrm{\Delta }\left(0\right)/k_B{T}_c$ as a function of normal-state resistivity (measured at 5 K) of granular Al films. $T_c$ (green diamonds) encloses a superconducting dome with a maximum around $700 \mu \mathrm{\Omega }$ cm where $T_c$ is enhanced by nearly a factor of 3 compared to the bulk value. $\mathrm{\Delta }\left(0\right)$ (olive stars) follows the increase of $T_c$ on the left side of the dome for LR samples while it saturates in the HR regime. This is reflected in the ratio $\mathrm{\Delta }\left(0\right)/k_B{T}_c$ which increases from the weak-coupling value 1.78 (dotted line) to 2.25 when crossing from the left to the right side of the dome. The calculation of superfluid stiffness from ${\sigma }_2\left(\nu \right)$ and from $\mathrm{\Delta }\left(0\right)$, i.e., $J\left(0\right)$ and $J_{\mathrm{\Delta }}\left(0\right)$, is subject to an uncertainty reflected by the shaded area. $J\left(0\right)$ follows approximately a $1/{\rho }_{\mathrm{dc}}$ behavior, as expected from Eq. (2) for a constant value of $\mathrm{\Delta }\left(0\right)$, and becomes comparable to $\mathrm{\Delta }\left(0\right)$ in the HR regime. Dashed lines are guides to the eye.

Figure 3

Temperature evolution of spectral gap. (a) and (b) Temperature dependence of normalized ${\sigma }_1\left(\nu \right)$ of a granular Al sample in the high- and optimal-resistivity regimes [samples 2 and 3 in Fig. 1]. In case of the HR sample, the suppression of ${\sigma }_1\left(\nu \right)$ below $T_c=2.55$ K (dashed lines) persists up to $T=2.8$ K (solid lines), whereas the spectral gap closes right at $T_c$ in the LR regime. (c) Temperature dependence of the spectral gap for samples from the optimal (stars, sample 2) and high resistivity regimes (diamonds, sample 3). The blue data traces ${\rho }_{\text{dc}}\left(T\right)$ of the HR sample. For the HR sample, deviations from the BCS prediction for $\mathrm{\Delta }\left(T\right)/\mathrm{\Delta }\left(0\right)$ (black solid line) appear already at $T/T_c\lesssim 1$, where $\mathrm{\Delta }$ is anomalously large. The persistence of a gap across $T_c$ (empty diamonds) is in striking resemblance with strongly disordered or correlated superconductors.