A theory of dissipative nonlinear conductivity, $sigma_1(omega,H)$, of s-wave superconductors under strong electromagnetic fields at low temperatures is proposed. Closed-form expressions for $sigma_1(H)$ and the surface resistance $R_s(omega,H)$ are
obtained in the nonequilibrium dirty limit for which $sigma_1(H)$ has a significant minimum as a function of a low-frequency $(hbaromegall k_BT)$ magnetic field $H$. The calculated microwave suppression of $R_s(H)$ is in good agreement with recent experiments on alloyed Nb resonator cavities. It is shown that superimposed dc and ac fields, $H=H_0+H_acosomega t$, can be used to reduce ac dissipation in thin film nanostructures by tuning $sigma_1(H_0)$ with the dc field.
We present detailed experimental and theoretical investigations of hotspots produced by trapped vortex bundles and their effect on the radio-frequency (rf) surface resistance $R_s$ of superconductors at low temperatures. Our measurements of $R_s$ com
bined with the temperature mapping and laser scanning of a 2.36 mm thick Nb plate incorporated into a 3.3 GHz Nb resonator cavity cooled by the superfluid He at 2 K, revealed spatial scales and temperature distributions of hotspots and showed that they can be moved or split by thermal gradients produced by the scanning laser beam. These results, along with the observed hysteretic field dependence of $R_s$ which can be tuned by the scanning laser beam, show that the hotspots in our Nb sample are due to trapped vortex bundles which contain $sim 10^6$ vortices spread over regions $sim 0.1-1$ cm. We calculated the frequency dependence of the rf power dissipated by oscillating vortex segments trapped between nanoscale pinning centers, taking into account all bending modes and the nonlocal line tension of the vortex driven by rf Meissner currents. We also calculated the temperature distributions caused by trapped vortex hotspots, and suggested a method of reconstructing the spatial distribution of vortex dissipation sources from the observed temperature maps. Vortex hotspots can dominate the residual surface resistance at low temperatures and give rise to a significant dependence of $R_s$ on the rf field amplitude $H_p$, which can have important implications for the rf resonating cavities used in particle accelerators and for thin film structures used in quantum computing and photon detectors.
In this Comment we show that the statements made in PRB85, 014505 (2012) regarding our work (PRL 100, 227007 (2008))) are incorrect because they result from model artifacts. We address the issues neglected in PRB85, 014505 (2012) and discuss their im
portance for a more consistent theory of thermally-activated hoping of vortices in thin films and the interpretation of experimental data.
