NaxCoO2 phase stability and hierarchical orderings in the O3/P3 structure family

The operation of Na-ion batteries is closely linked to the occurrence of phase transitions and Na ordering within the electroactive materials of their electrodes. We performed a first-principles study of ${\mathrm{Na}}_{x}{\mathrm{CoO}}_{2}$ ($0\ensuremath{\le}x\ensuremath{\le}1$) to establish phase stability among O3, P3, O1, and staged hybrid O1-O3 host structures and to determine low-energy Na orderings within each host. We predict several staircases of ground-state orderings, many of which are found to remain ordered at room temperature. At high Na content, the ground states belong to a family of vacancy row orderings in O3 with distorted ${\mathrm{CoO}}_{2}$ layers. In P3, where the available Na sites form a honeycomb network, we discover three families of hierarchical ground-state orderings at intermediate compositions that essentially consist of domains of the $x=1/2$ ground state periodically separated by antiphase boundaries. Our results agree with the experimentally reported voltage profile and the observed changes in stacking sequence with Na concentration.