Infrared Hall effect measurement of correlated metal Na$_{0.75}$CoO$_2$

Na$_{x}$CoO$_2$ has a layered Co-O plane where Co ions form a triangular bonding block in contrast with the square Cu moments of HTSC Cu-O plane. With varying x, the compound exhibits rich phases like superconductivity (x=0.3, H$_2$O intercalted), charge-ordering (x=0.5) and Curie-Weiss metal (x$\sim$0.7). Na$_{x}$CoO$_2$ thin film (x=0.75) was grown on SrTiO$_3$ substrate using PLD method. Resistivity and dc-Hall effect show same temperature dependence as those of single crystal. We studied ac Hall effect by measuring the complex Faraday rotation $\theta_{F}$ at infrared frequency ( 1100 cm$^{-1}$) for 30K$>$T$<$300K and in magnetic fields up to 8 Tesla. As T decreases, real part of $\theta_{F}$ shows a slop change where dRe($\theta_{F}$)/dT from negative to positive sign at T=100K. At the same T, Im($\theta_{F}$)shows a dispersive structure. This anomaly comes from $\sigma_xy$ while $\sigma_xx$ is monotonic with T. Possible origin of the unusual infrared Hall conductivity sigma$_xy$ is considered in terms of spin density wave, Na-ordering, and electronic Kagome lattice.