Disproportionation and Critical Interaction Strength in Na$_x$CoO$_2$:   Concentration Dependence
  We present results of studies of charge disproportionation (CD) and spin differentiation in Na$_x$CoO$_2$ using the correlated band theory approach (local-density approximation+Hubbard U:LDA+U). The simultaneous CD and gap opening can be followed through a first order charge disproportionation transition. By comparison with experiments, we propose a value of the Coulomb repulsion strength U($x$) that has significant dependence on the carrier concentration $x$, for which we obtain a phase diagram. The connection between the oxygen height and effects of on-site correlation is also reported. 
K.-W. Lee and W. E. Pickett