Optical signatures of the superconducting Goldstone mode in granular aluminum: Experiments and theory

Recent advances in the experimental growth and control of disordered thin films, heterostructures, and interfaces provide fertile ground for the observation and characterization of the collective superconducting excitations emerging below $T_c$ after breaking the $U\left(1\right)$ gauge symmetry. Here we combine THz experiments in a nanostructured granular Al thin film and theoretical calculations to demonstrate the existence of optically active phase modes, which represent the Goldstone excitations of the broken gauge symmetry. By measuring the complex transmission through the sample we identify a sizable and temperature-dependent optical subgap absorption, which cannot be ascribed to quasiparticle excitations. A quantitative modeling of this material as a disordered Josephson array of nanograins allows us to determine, with no free parameters, the structure of the spatial inhomogeneities induced by shell effects. Besides being responsible for the enhancement of the critical temperature with respect to bulk Al, already observed in the past, this spatial inhomogeneity provides a mechanism for the optical visibility of the Goldstone mode. By computing explicitly the optical spectrum of the superconducting phase fluctuations we obtain a good quantitative description of the experimental data. Our results demonstrate that nanograin arrays are a promising setting to study and control the collective superconducting excitations via optical means.

Figure 1

Real part (full circles) of the experimentally determined (normalized) dynamical conductivity ${\sigma }_1^{\exp }$ versus frequency $\nu$ at various temperatures below $T_c=2.74$ K. The solid lines are fits to the Mattis-Bardeen BCS prediction restricted to frequencies $\nu \ge 2\mathrm{\Delta }/\left(hc\right)$. The bars document the excessive conductivity ${\sigma }^{\mathrm{exc}}$, as defined in Eq. (1). The inset of the top panel shows the temperature dependence of the SC gap $\mathrm{\Delta }\left(T\right)$ (stars) extracted from the Mattis-Bardeen analysis of the optical conductivity, along with a BCS fit (solid line).

Figure 2

Sketch of the optical response of a Josephson junction array, modeled as in Eq. (2). Here the arrows represent the local spins on the array sites, connected by springs representing the local stiffnesses $J_{ij}$. To visualize its inhomogeneity, we set the arrow length proportional to the strength of the local stiffness. In the clean case, panel (a), the phase modes are decoupled from the transverse electromagnetic field, so the spins preserve their orientation (i.e., the phase of the SC order parameter is unchanged) and the radiation is not absorbed. On the other hand in the disordered case, panel (b), the spins respond to the incoming radiation with a local change of their relative direction that is larger when the system has lower phase rigidity (i.e., lower local $J_{ij})$. This leads to an inelastic response which absorbs part of the incoming radiation.

Figure 3

Distribution of the Josephson couplings $P(J_{ij}/J_0)$ for $R_N/R_Q=0.01$, as appropriate for the sample shown in Fig. 1. Following the experimental results of Ref. [38, 39, 40], we assume that the distribution of grain sizes is log-normal with a typical radius of 1 nm. The couplings $J_{ij}$ are computed exactly combining the exact solution of the BCS gap equation taking into account the coupling to other grains and the fluctuating spectral density (see Appendix pp1 for more details).

Figure 4

Optical conductivity (solid lines) obtained by using the $P(J/J_0)$ extracted from the shell model; see Fig. 3. The data points correspond to the normalized excess conductivity shown as bars in Fig. 1.

Figure 5

Optical conductivity (solid lines) obtained from the diluted $XY$ model, using a dilution level $p=0.205$ corresponding to a bimodal approximation for the microscopic distribution of Fig. 3. The data points correspond to the normalized excess conductivity shown as bars in Fig. 1.