Theory of Sodium Ordering in NaxCoO2
  The ordering of Na ions in Na$_x$CoO$_2$ is investigated systematically by combining detailed density functional theory (DFT) studies with model calculations. Various ground state ordering patterns are identified, and they are in excellent agreement with avaliable experimental results. Our results suggest that the primary driving force for the Na ordering is the screened Coulomb interaction among Na ions. Possible effects of the Na ordering on the electronic structure of the CoO$_2$ layer are discussed. We propose that the nonexistence of a charge ordered insulating state at $x = 2/3$ is due to the lack of a commensurate Na ordering pattern, whereas an extremely stable Na ordering at $x = 0.5$ enhances the charge ordering tendency, resulting in an insulating state as observed experimentally. 
Peihong Zhang, Rodrigo B. Capaz, Marvin L. Cohen, and Steven G. Louie