Temperature and frequency dependence of the mid-infrared Hall effect in Ca$_x$Sr$_{1-x}$RuO$_3$

Ca$_x$Sr$_{1-x}$RuO$_3$ compounds exhibit unusual properties, such as metamagnetism, quantum criticality, non-Fermi liquid behavior and an anomalous Hall effect that continue to challenge the condensed matter community. The mid-infrared (115-238 meV) complex Faraday, Kerr, and Hall angles are studied in Ca$_x$Sr$_{1-x}$RuO$_3$ films. The magneto-optical signals in transmission are up to an order of magnitude larger than those obtained in reflection for the same sample. The frequency dependence of the low-temperature magneto-optical signals in SrRuO$_3$ is in good qualitative and quantitative agreement with first-principles band calculations [Z. Fang et al., Science 2003]. Striking qualitative similarities and differences are observed in the temperature dependence of the Hall angle at three probe energies (dc, 120~meV and 224~meV) in CaRuO$_3$ and SrRuO$_3$.