Electron correlation in the two-dimensional triangle lattice of Na$_x$CoO$_2$

Magnetism of layered cobaltites Na$_x$CoO$_2$ with $x$ = 0.6 and 0.9 has been investigated by a positive muon spin rotation and relaxation ($\mu^+$SR) spectroscopy together with magnetic susceptibility and specific heat measurements, using single crystal samples in the temperature range between 250 and 1.8 K. Zero-field (ZF-) $\mu^+$SR measurements on Na$_{0.9}$CoO$_2$ indicates a transition from a paramagnetic to an incommensurate spin density wave state at 19 K(=$T_{\sf SDW}$). The anisotropic ZF-$\mu^+$SR spectra suggest that the oscillating moments of the {\sf IC-SDW} directs along the c-axis. Since Na$_{0.6}$CoO$_2$ is paramagnetic down to 1.8 K, the magnitude of $T_{\sf SDW}$ is found to strongly depend on $x$.This behavior is well explained using the Hubbard model within a mean field approximation on two-dimensional triangle lattice in the CoO$_2$ plane. Also, both the appearance of the {\sf IC-SDW} state by the change in $x$ and the magnitude of the electronic specific heat parameter of Na$_{0.6}$CoO$_2$ indicate that Na$_x$CoO$_2$ is unlikely to be a typical strongly correlated electron system.