Dissipation due to bulk localized low-energy modes in strongly disordered superconductors

TL;DR

This paper develops a theory describing how ac dissipation in strongly disordered superconductors depends on temperature and frequency, highlighting the role of bulk localized low-energy modes in the dissipation process.

Contribution

The paper introduces a new theoretical framework for understanding ac dissipation in disordered superconductors, emphasizing the impact of localized collective modes on the quality factor.

Findings

Low-frequency dissipation dominated by bulk localized modes

Sharp decrease of Q with increasing frequency

Temperature dependence of Q resembles two-level systems

Abstract

We develop a theory of the temperature $T$ and frequency $\omega$ dependence of ac dissipation in strongly disordered superconductors featuring a pseudogap $\Delta_{P}$ in the single-particle spectrum. Our theory applies to the regime $T,\,\hbar\omega\ll\Delta_{\text{typ}}\ll\Delta_{P}$, where $\Delta_{\text{typ}}$ is the typical superconducting gap. The dissipation is expressed in terms of the quality factor $Q(T,\omega)$ of microwave resonators made of these materials. We show that low-$\omega$ dissipation is dominated by a new type of bulk localized collective modes. Due to the strongly nonuniform spectral density of these modes, $Q$ decreases sharply with frequency, while its temperature dependence exhibits a two-level-system-like growth as a function of $T$ for $T\ll T_{c}$. Our theory is applicable to InO$_x$, TiN, NbN, and similar strongly disordered materials. We further argue that the experimentally observed behavior of disordered films of granular Aluminum is explained by similar physics, although this case requires a separate theoretical analysis.