Dome-shaped magnetic phase diagram of thermoelectric layered cobaltites

Using muon spin spectroscopy we have found that, for both Na$_x$CoO$_2$ (0.6 $\leq x \leq$ 0.9) and 3- and 4-layer cobaltites, a common low temperature magnetic state (which in some cases is manifest as an incommensurate spin density wave) forms in the CoO$_2$ planes. Here we summarize those results and report a dome-shaped relation between the transition temperature into the low-$T$ magnetic state and the composition $x$ for Na$_x$CoO$_2$ and/or the high-temperature asymptotic limit of thermopower in the more complex 3- and 4-layer cobaltites. This behavior is explained using the Hubbard model on two-dimensional triangular lattice in the CoO$_2$ plane.