Charge disproportionation and spin ordering tendencies inNaxCoO2

The strength and effect of Coulomb correlations in the (superconducting when hydrated) $x\ensuremath{\approx}1∕3$ and ``enhanced'' $x\ensuremath{\approx}2∕3$ regimes of ${\mathrm{Na}}_{x}{\mathrm{CoO}}_{2}$ are evaluated using the correlated band theory $\mathrm{LDA}+\mathrm{U}$ (local density approximation of Hubbard U) method. Our results, neglecting quantum fluctuations, are (1) allowing only ferromagnetic order, there is a critical ${U}_{c}=3\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, above which charge disproportionation occurs for both $x=1∕3$ and $x=2∕3$, (2) allowing antiferromagnetic order at $x=1∕3$, ${U}_{c}$ drops to $1\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ for disproportionation, (3) disproportionation and gap opening occur simultaneously, and (4) in a ${\mathrm{Co}}^{3+}\ensuremath{-}{\mathrm{Co}}^{4+}$ ordered state, antiferromagnetic coupling is favored over ferromagnetic, while below ${U}_{c}$ ferromagnetism is favored. A comparison of the calculated Fermi level density of states compared to reported linear specific heat coefficients indicates enhancement of the order of five for $x\ensuremath{\sim}0.7$, but negligible enhancement for $x\ensuremath{\sim}0.3$. This trend is consistent with strong magnetic behavior and local moments (Curie-Weiss susceptibility) for $xg0.5$ while there is no magnetic behavior or local moments reported for $xl0.5$. We suggest that the phase diagram is characterized by a crossover from effective single-band character with $U⪢W$ for $xg0.5$ into a three-band regime for $xl0.5$, where $U\ensuremath{\rightarrow}{U}_{\mathrm{eff}}\ensuremath{\leqslant}U∕\sqrt{3}\ensuremath{\sim}W$ and correlation effects are substantially reduced.