Itinerant and localized magnetism on the triangular lattice: sodium rich phases of Na${}_x$CoO${}_2$

We study the interplay between correlation, itinerant ferromagnetism and local moment formation on the electron doped triangular lattice of sodium cobaltates Na${}_x$CoO${}_2$. We find that strong correlation renormalizes the Stoner criterion and stabilizes the paramagnetic state for $x<x_c\simeq 0.67$. For $x>x_c$, ferromagnetic (FM) order emerges. The enhanced Na dopant potential fluctuations play a crucial role in the sodium rich phases and lead to an inhomogeneous FM state, exhibiting nonmagnetic Co${}^{3+}$ patches, antiferromagnetic (AF) correlated regions, and FM clusters with AF domains. Hole doping the band insulator at x=1 leads to the formation of local moments near the Na vacancies and AF correlated magnetic clusters. We explain recent observations by neutron, $\mu$SR, and NMR experiments on the evolution of the magnetic properties in the sodium rich phases.