Anomalous frequency and temperature dependent scattering in the dilute metallic phase in lightly doped-SrTiO$_3$

The mechanism of superconductivity in materials with aborted ferroelectricity and the emergence of a dilute metallic phase in systems like doped-SrTiO$_3$ are outstanding issues in condensed matter physics. This dilute metal has features both similar and different to those found in the normal state of other unconventional superconductors. We have investigated the optical properties of the dilute metallic phase in doped-SrTiO$_3$ using THz time-domain spectroscopy. At low frequencies the THz response exhibits a Drude-like form as expected for typical metal-like conductivity. We observed the frequency and temperature dependencies to the low energy scattering rate $\Gamma(\omega, T) \propto (\hbar\omega)^2 + (p \pi k_BT)^2 $ expected in a conventional Fermi liquid, despite the fact that densities are too small to allow current decay through electron-electron scattering. However we find the lowest known $p$ values of 0.39-0.72. As $p$ is 2 in a canonical Fermi liquid and existing models based on energy dependent elastic scattering bound $p$ from below to 1, our observation lies outside current explanation. Our data also gives insight into the high temperature regime and shows that the temperature dependence of the resistivity derives in part from strong T dependent mass renormalizations.