Anomalous gap-edge dissipation in disordered superconductors on the brink of localization

Superconductivity in disordered systems close to an incipient localization transition has been an area of investigation for many years, but many fundamentally important aspects are still not understood. It has been noted that in such highly disordered superconductors, anomalous spectral weight develops in their conductivity near and below the superconducting gap energy. In this work we investigate the low frequency conductivity in disordered superconducting NbN thin films close to the localization transition with time-domain terahertz spectroscopy. In the normal state, strong deviations from the Drude form due to incipient localization are found. In the superconducting state we find substantial spectral weight at frequencies well below the superconducting gap scale derived from tunneling. We analyze this spectral weight in the context of a model of disorder induced broadening of the quasiparticle density of states. We find that aspects of the optical and tunneling data can be consistently modeled in terms of this effect of mesoscopic disorder, showing that in this disorder and frequency range, quasiparticle effects and not collective modes are the source of low energy absorption. Interestingly, we also find that as a function of frequency the optical conductivity recovers to the normal state value much faster than any model predicts. This points to the nontrivial interplay of superconductivity and disorder close to localization.

Figure 1

The left axis is $T_c$ vs the dimensionless conductance parameter $k_{F}l$ for the samples used in this study. The thickness (unit is nm) of each sample is shown next to the data points. $T_c$ was defined by the temperature where the resistance is indistinguishable from zero. The green dashed line is $k_{F}l=1$. On the right axis is the depairing parameter $\eta$ extracted from optics and tunneling as discussed in the text.

Figure 2

(a) Real part of the optical conductivity at $1.1T_c$. (b) Real part and (c) imaginary parts of the optical conductivity at 1.5 K. (d) Real parts of optical conductivity at 1.5 K normalized by the normal state conductivity at $1.1T_c$ given in (a).

Figure 3

The red hollow squares represent the real parts of the optical conductivity at 1.5 K for four representative samples. The red dashed vertical curves show the optical energy gaps directly extracted from optics. The green dashed lines indicate the superconducting gaps extracted from tunneling. The green curves are created via a numerical solution to the MB formalism with superconducting gaps extracted from tunneling. The blue curves are simulations using the model of Larkin and Ovchinnikov.