Effects of Nonmagnetic Impurities and Subgap States on the Kinetic Inductance, Complex Conductivity, Quality Factor, and Depairing Current Density
Takayuki Kubo
Abstract
We investigate how a combination of a nonmagnetic impurity scattering rate $\ensuremath{\gamma}$ and finite subgap states parametrized by Dynes $\mathrm{\ensuremath{\Gamma}}$ affects various physical quantities relevant to superconducting devices made from extreme type-II $s$-wave superconductors. All the calculations are based on the Eilenberger formalism of the BCS theory. It is well known that the optimum impurity concentration minimizes the surface resistance ${R}_{s}$. We find the optimum $\mathrm{\ensuremath{\Gamma}}$ can also reduce ${R}_{s}$ by one order of magnitude for a clean superconductor ($\ensuremath{\gamma}/{\mathrm{\ensuremath{\Delta}}}_{0}<1$) and a few tens of % for a dirty superconductor ($\ensuremath{\gamma}/{\mathrm{\ensuremath{\Delta}}}_{0}>1$). Here, ${\mathrm{\ensuremath{\Delta}}}_{0}$ is the pair potential for the idealized ($\mathrm{\ensuremath{\Gamma}}\ensuremath{\rightarrow}0$) BCS superconductor for $T\ensuremath{\rightarrow}0$. Also, we find a nearly ideal ($\mathrm{\ensuremath{\Gamma}}/{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremath{\ll}1$) clean-limit superconductor exhibits a frequency-independent ${R}_{s}$ for a broad range of frequency $\ensuremath{\omega}$, which can significantly improve $Q$ of a compact cavity with a few tens of GHz frequency. As $\mathrm{\ensuremath{\Gamma}}$ or $\ensuremath{\gamma}$ increases, ${R}_{s}$ obeys the ${\ensuremath{\omega}}^{2}$ dependence. The subgap-state-induced residual surface resistance ${R}_{\mathrm{res}}$ is also studied, which can be detected by a high-$Q$ three-dimensional resonator. We calculate the kinetic inductance ${L}_{k}(\ensuremath{\gamma},\mathrm{\ensuremath{\Gamma}},T)$ and the depairing current density ${J}_{d}(\ensuremath{\gamma},\mathrm{\ensuremath{\Gamma}},T)$, which are monotonic increasing and decreasing functions of $(\ensuremath{\gamma},\mathrm{\ensuremath{\Gamma}},T)$, respectively. Measurements of $(\ensuremath{\gamma},\mathrm{\ensuremath{\Gamma}})$ of device materials can give helpful information on improving $Q$, engineering ${L}_{k}$, and ameliorating ${J}_{d}$ via materials processing.