Microwave Response of Superconductors with Paramagnetic Impurities

TL;DR

This work develops a microscopic, quasiclassical framework to predict how dilute paramagnetic impurities modify the microwave response of conventional spin-singlet superconductors. By solving Eilenberger equations with configurational-averaged T-matrices for non-magnetic and paramagnetic impurities, the authors connect the sub-gap impurity-band spectrum to observable quantities such as $n_s$, $\sigma(\omega)$, $\lambda(\omega)$, $\delta f$, and $Q$ in GHz resonators. The key findings include a non-monotonic, low-temperature anomaly in the superfluid density driven by sub-gap states, the emergence of a finite sub-gap DOS with width $\gamma$ that produces residual surface resistance, and a pronounced sensitivity of cavity frequency shifts and quality factors to both non-magnetic and paramagnetic disorder. These results provide a rigorous mechanism for residual losses in superconducting resonators and offer quantitative diagnostics for impurity content in materials used for SRF cavities and related devices.

Abstract

We develop theoretical methods to predict the effects of paramagnetic impurities on the microwave response of conventional spin-singlet superconductors. Our focus is on superconducting devices and resonators with low concentrations of impurities and exchange interactions with conduction electrons. We connect the sub-gap quasiparticle spectrum generated by pair-breaking to the frequency and temperature dependence of the conductivity for superconductors operating at microwave frequencies. We report theoretical results for superconducting device performance -- dissipation, quality factor and frequency shift anomalies -- based on self-consistent calculations of the current response and penetraion of the electromagnetic field at the vacuum-superconducting interface. Key results include the prediction of a non-monotonic anomaly in the low-frequency superfluid fraction and penetration depth at very low temperatures related to the sub-gap quasiparticle spectrum. Dissipation of microwave power is predicted from intra- and inter- impurity band transitions at GHz frequencies at low temperatures, including a physical mechanism responsible for residual resistance. We predict anomalies in the resonant frequency, $f(T)$, and quality factor, $Q(T)$, of high-Q SRF cavities operating in the GHz range at low-temperatures that are sensitive to non-magnetic and paramagnetic impurity disorder.

Figures (16)

• Figure 1: Leading self-energy diagrams in the quasiclassical theory of superconductivity. (a) electron-phonon self energy, (b) mean-field electron-electron self energy, (c) electron-impurity self energy in the T-matrix approximation, and (d) the coupling of charge currents to an electromagnetic field with $q=({\bf q},\omega_m)$.
• Figure 2: T-matrix for multiple scattering of quasiparticles and pairs by an impurity with transition matrix $\widehat{U}({\bf p}',{\bf p})$. Intermediate states are described by the quasiclassical propagator, $\widehat{\mathfrak{G}}({\bf p},\varepsilon_n)$.
• Figure 3: Configurational averaged T-matrix for scattering of quasiparticles and pairs from an impurity selected from an ensemble of paramagnetic impurities. The leading order term is defined by the vertex $u_s^2\,\widehat{g}'$ and intermediate states are described by the configurational averaged quasiclassical propagator, $\widehat{g}=N_f\,\widehat{\mathfrak{G}}$.
• Figure 4: Suppression of the transition temperature, $T_c$, as a function of $\Gamma_{\text{S}}$ (impurity concentration) for a range of $\bar{u}_{\text{S}}$ spanning the Born limit, $\bar{u}_{\text{S}}\ll 1$, to maximal pair-breaking for $\bar{u}_{\text{S}}=1$.
• Figure 5: Pair-breaking suppression of the superconducting gap $\Delta$ by paramagnetic impurities for $\Gamma_{\text{S}}/\pi T_{c_0}=0.2$ and the same range of $\bar{u}_{\text{S}}$ as in Fig. \ref{['fig-Tc_vs_GammaS']}.