Charge disproportionation and spin ordering tendencies inNaxCoO2

The strength and effect of Coulomb correlations in the (superconducting when hydrated) $x\approx 1∕3$ and “enhanced” $x\approx 2∕3$ 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 $U_c=3\mathrm{eV}$, 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 $1\mathrm{eV}$ for disproportionation, (3) disproportionation and gap opening occur simultaneously, and (4) in a ${\mathrm{Co}}^{3+}-{\mathrm{Co}}^{4+}$ 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 $x\sim 0.7$, but negligible enhancement for $x\sim 0.3$. This trend is consistent with strong magnetic behavior and local moments (Curie-Weiss susceptibility) for $x>0.5$ while there is no magnetic behavior or local moments reported for $x<0.5$. We suggest that the phase diagram is characterized by a crossover from effective single-band character with $U⪢W$ for $x>0.5$ into a three-band regime for $x<0.5$, where $U\to U_{\mathrm{eff}}⩽U∕\sqrt{3}\sim W$ and correlation effects are substantially reduced.