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We introduce a strongly correlated mean-field theory of the ionic Hubbardmodel on the triangular lattice with alternating stripes of site energy usingBarnes-Coleman slave bosons. We study the paramagnetic phases of this theory atthree quarters filling, where it is a model of Na_0.5CoO_2, Rb_0.5CoO_2, andK_0.5CoO_2. This theory has two bands of fermionic quasi-particles: one ofwhich is filled or nearly filled and hence weakly correlated; the other ishalf-filled or nearly half-filled and hence strongly correlated. Furtherresults depend strongly on the sign of the hopping integral, t. The light bandis always filled for t>0, but only becomes filled for |Delta/t|>=1.5 for t<0,where Delta is the difference in the site energies of the two sublattices. Ametal--charge transfer insulator transition occurs at |Delta/t|=5.0 for t>0 and|Delta/t|=8.0 for t<0. In the charge transfer insulator complete chargedisproportionation occurs: one sublattice is filled and the other in halffilled. We compare our results with exact diagonalisation calculations andexperiments on Na_0.5CoO_2, and discuss the relevance of our results toRb_0.5CoO_2 and K_0.5CoO_2. In particular we propose a resolution of seeminglycontradictory experimental results on Na_0.5CoO_2. Many experiments suggestthat there is a charge gap, yet quantum oscillations are observed suggestingthe existence of quasiparticle states at arbitrarily low excitation energies.We argue that the heavy band is gapped while the light band, which containsless than 1 charge carrier per 100 unit cells, remains ungapped.

Ionic Hubbard model on a triangular lattice for Na_0.5CoO_2, Rb_0.5CoO_2 and K_0.5CoO_2: Mean-field slave boson theory