Microwave loss and intermodulation in ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_y$ thin films

A superconducting microstrip hairpin resonator is used as a tool to study the linear and nonlinear microwave response of TBCCO thin films. We extract the surface resistance $R_s\mathrm{}\left(T\right),$ penetration depth $\lambda \mathrm{}\left(T\right),$ and intermodulation critical current $J_{\mathrm{IMD}}\mathrm{}\left(T\right)\mathrm{}$ for such a resonator at 4.2 GHz. The surface impedance results compare well with cavity perturbation measurements on portions of the same wafer. $J_{\mathrm{IMD}}\mathrm{}\left(T\right)\mathrm{}$ is compared against theoretical models for the intrinsic nonlinear response of $s$- and $d$-wave superconductors. We demonstrate that the dc critical current $J_c$ and $J_{\mathrm{IMD}}$ are distinct material parameters, describing different physical regimes.