Effect of radiation-induced defects on the superfluid density and optical conductivity of overdoped La$_{2-x}$Sr$_x$CuO$_4

Using a combination of time-domain THz spectroscopy (TDTS) and mutual inductance measurements, we have investigated the low-energy electrodynamic response of overdoped La$_{2-x}$Sr$_x$CuO$_4$ films that have been exposed to ion irradiation. Films went through three rounds of irradiation (2, 4, and 6 $\times 10^{13}$ ions/cm$^2$) and mutual inductance and TDTS experiments were performed between each step. Together with the as-grown film, this gives four different levels of disorder. The transport scattering rate that is measured directly in the THz experiments is an approximately linear function of the radiation dose at all temperatures. This is consistent with a proportionate increase in elastic scattering. In the superconducting state we find that the relation between $T_c$, the superfluid density, and the scattering rates are quantitatively at odds with the predictions based on the extant theory of Abrikosov-Gorkov-like pair breaking in a dirty $d$-wave superconductor. Increasing disorder causes only a small change in the superconducting transition temperature for the overdoped films, but the changes to the $\omega \sim 0$ superfluid density are much larger.