The infrared conductivity of Na$_x$CoO$_2$: evidence of gapped states*

We present infrared ab-plane conductivity data for the layered cobaltate Na$_x$CoO$_2$ for $x=0.25, 0.30, 0.50$, and $0.75$. The Drude weight increases monotonically with hole doping, $1-x$. At the lowest hole doping level 0.75 the system resembles the normal state of underdoped cuprate superconductors with a scattering rate that varies linearly with frequency and temperature and there is an onset of scattering by a bosonic mode at 600 cm$^{-1}$. The hole doping level of $x=$0.30 sample shows a metallic behavior and a bosonic mode with a similar onset frequency as $x=$0.75. Doping to both $x=0.50$ and 0.25 causes a gap to develop in the optical conductivity at low temperatures and the material becomes an insulator. The spectral weight lost in the gap region is shifted to a prominent peak. We propose that the two gapped states of $x$=0.50 and 0.25 are pinned charge ordered states.