Charge and Spin Order on the Triangular Lattice: Na_xCoO₂ at x=0.5

The nature of electronic states due to strong correlation and geometric frustration on the triangular lattice is investigated in connection to the unconventional insulating state of Na_xCoO₂ at x=0.5. We study an extended Hubbard model using a spatially unrestricted Gutzwiller approximation. We find a new class of charge and spin ordered states at x=1/3 and x=0.5 where antiferromagnetic (AFM) frustration is alleviated via weak charge inhomogeneity. At x=0.5, we show that the √3a×2a off-plane Na dopant order induces weak √3a×1a charge order in the Co layer. The symmetry breaking enables successive √3a×1a AFM and 2a×2a charge- or spin-ordering transitions at low temperatures. The Fermi surface is truncated by the 2a×2a hexagonal zone boundary into small electron and hole pockets. We study the phase structure and compare to recent experiments.