Temperature-concentration phase diagram of P 2 -Na x CoO 2 from first-principles calculations

Temperature-concentration phase diagrams for ${\text{Na}}_{x}{\text{CoO}}_{2}$ $(0.5\ensuremath{\le}x\ensuremath{\le}1)$ are obtained by combining density functional theory (DFT) in the generalized gradient approximation (GGA) and in the GGA with Hubbard $U$ correction $(\text{GGA}+U)$ with the cluster expansion and Monte Carlo simulation technique. In the GGA, holes are delocalized over the Co layer, while in $\text{GGA}+U$ the charges on the Co layer completely localize, forming distinct ${\text{Co}}^{3+}$ and ${\text{Co}}^{4+}$ cations. The leading interactions in GGA are long-range in-plane electrostatics and relaxation effects, whereas in $\text{GGA}+U$ Co-Co interactions dominate. Comparison of ground states, $c$-lattice parameter, and Na1/Na2 ratio with experimental results consistently suggests that GGA is a better approximation for $0.5\ensuremath{\le}x\ensuremath{\le}0.8$.