Complete electrodynamics of a BCS superconductor with $\mu$eV energy scales: microwave spectroscopy on titanium at mK temperatures

Abstract

We performed resonant microwave measurements on superconducting titanium (Ti) down to temperatures of 40~mK, well below its critical temperature $T_\mathrm{c} \approx 0.5$~K. Our wide frequency range 3.3-40~GHz contains the zero-temperature energy gap $2\Delta_0$ and allows us to probe the full electrodynamics of the superconducting state, including excitations across the gap and the low-frequency responses of superfluid condensate and thermal quasiparticles. The observed behavior follows the predictions of the BCS-based Mattis-Bardeen formalism, which implies that superconducting Ti is in the dirty limit, in agreement with our determination of the scattering rate. We directly determine the temperature dependence of the energy gap, which is in accordance with BCS predictions, and $2\Delta_0/k_\mathrm{B}T_\mathrm{c}\approx3.5$ with $\Delta_0 \approx$ 75~$\mu$eV. We also evaluate the penetration depth, and we characterize the behavior of superconducting Ti in external magnetic field.