Interplay of frustration, magnetism, charge ordering, and covalency in the ionic Hubbard model for ${\text{Na}}_{0.5}{\text{CoO}}_2$

We investigate the ionic Hubbard model on a triangular lattice at three-quarters filling. This model displays a subtle interplay between metallic and insulating phases and between charge and magnetic orders. We find crossovers among Mott, charge transfer and covalent insulators, and magnetic order with large moments that persist even when the charge transfer is weak. Our theoretical results are consistent with experiments on $A_{0.5}{\text{CoO}}_2$ $\left(A=\text{K},\text{Na}\right)$ and identify these materials as correlated covalent insulators.

Figure 1

(Color online) Variation in the charge transfer with $\Delta$ for different Coulomb repulsion energies. The Lanczos results on an 18-site cluster for: $U/\left|t\right|=0$ (stars), 15 (open symbols), and 100 (filled symbols) are shown. The $U=0$ results are in excellent agreement with the infinite lattice tight-binding (tb) result (dashed-dotted line) (Ref. 24). Strong-coupling results (dashed lines) are also shown for comparison.

Figure 2

(Color online) Variation in the charge gap, ${\Delta }_c$, with $\Delta$ for $U/\left|t\right|=15$ and 100 (the charge gap is zero for $U=0$). The tendency toward an insulating state is greater for $t<0$ than for $t>0$. Dashed lines show the strong-coupling limit: $U⪢\Delta ⪢t$. Note that although strong correlations are essential for creation of the charge gap they are not required for the charge transfer, cf. Fig. 1. Note that, for $U⪢\Delta$ with $U$ large, the charge gap is robust against the value of $U$.

Figure 3

(Color online) Evolution from a charge-transfer insulator to a covalent insulator. The energy dependence of the spectral density, $A\left(\omega \right)$, is shown for two different parameter regimes of the model with $t<0$ and $U=15\left|t\right|$. The spectral density in the upper panel $\left(\Delta =10\left|t\right|\right)$ is that characteristic of a charge-transfer insulator (Ref. 15) there is a single weakly correlated band largely associated with the $B$ sites and lying between lower Hubbard band (LHB) and upper Hubbard band (UHB) (separated by $\sim U$) that are largely associated with the $A$ sites and ${\Delta }_c\sim \Delta$. In contrast, the lower panel $\left(\Delta =\left|t\right|\right)$ shows a spectral density characteristic of a covalent insulator (Ref. 15): there are only small difference between $A$ and $B$ sites, the separation of the LHB and UHB is $>U$ and ${\Delta }_c\sim \left|t\right|$.

Figure 4

(Color online) The magnetic moment as a function of $\Delta$ for $t<0$ and various $U$. The magnetic moment on the $A$ sites, $m_A$, (main panel) is strongly enhanced when $U⪢\left|t\right|$. The inset shows the moment on the $B$ sublattice. $m_B$ is much smaller, reduced by $\Delta$, and only weakly dependent on $U$. These results demonstrate that, for large $U$ and $\Delta \gtrsim \left|t\right|$, the electrons on the $A$ sublattice are much more strongly correlated than those on the $B$ sublattice despite the small charge transfer (see Fig. 1).