Microwave Response of Superconducting Devices with Paramagnetic Impurities

We report new theoretical results on the role and impact of dilute concentrations of paramagnetic impurities on the microwave response of superconducting circuits and resonators. We examine the sub-gap quasiparticle spectrum as a function of the impurity potential - both non-magnetic and exchange interactions - and concentration of paramagnetic impurities. We connect the spectra to the frequency and temperature dependence of the a.c. conductivity for Nb and Al superconducting elements operating at microwave frequencies. We specifically highlight the effects of small concentrations of impurities with strong exchange interactions in the current carrying regions of superconducting elements under microwave excitation. In this limit we show that there is substantial power absorption at low temperatures at GHz frequencies. We report calculations of the dissipation from both intra-band transitions at low temperatures, ħω ∈ ε_imp ≪ ∆, , where ε_imp is the impurity band, as well as inter-band transitions between the nearly occupied negative energy band, -ε_imp, and the nearly unoccupied positive energy band, +ε_imp. We compare our results for the conductivity, penetration depth, frequency anomalies and quality factors at microwave frequencies with recent experiments on Nb and Ta superconducting devices and resonators [1,2,3].

[1] A. Romanenko et al, Phys. Rev. App. 13, 034032 (2020).

[2] K. Crowley et al, Phys. Rev. X 13, 041005 (2023).

[3] A. Datta et al, Superconductor Science and Technology 37 095006 (2024).