Origin of a 1 g and e g ′ orderings in Na x CoO 2

It has often been suggested that correlation effects suppress the small ${e}_{g}^{\ensuremath{'}}$ Fermi-surface pockets of ${\text{Na}}_{x}{\text{CoO}}_{2}$ that are predicted by LDA, but absent in ARPES measurements. It appears that within the dynamical mean-field theory (DMFT) the ARPES results can be reproduced only if the on-site energy of the ${e}_{g}^{\ensuremath{'}}$ complex is lower than that of the ${a}_{1g}$ complex at the one-electron level, prior to the addition of local correlation effects. Current estimates regarding the order of the two orbital complexes range from $\ensuremath{-}200$ to 315 meV in terms of the energy difference. In this work, we perform density-functional theory calculations of this one-electron splitting $\ensuremath{\Delta}={ϵ}_{{a}_{1g}}\ensuremath{-}{ϵ}_{{e}_{g}^{\ensuremath{'}}}$ for the full two-layer compound, accounting for the effects of Na ordering, interplanar interactions and octahedral distortion. We find that ${ϵ}_{{a}_{1g}}\ensuremath{-}{ϵ}_{{e}_{g}^{\ensuremath{'}}}$ is negative for all Na fillings and that this is primarily due to the strongly positive Coulomb field created by ${\text{Na}}^{+}$ ions in the intercalant plane that disproportionately affects the extended ${a}_{1g}$ orbital. We discuss also the effects of octahedral compression and multiorbital filling on the value of $\ensuremath{\Delta}$ as a function of Na content. Our results indicate that if the ${e}_{g}^{\ensuremath{'}}$ pockets are indeed suppressed, that can only be due to nonlocal correlation effects beyond the standard DMFT.