Effective interaction between the interpenetrating Kagomé lattices in ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$

A multiorbital model for a $\mathrm{Co}{\mathrm{O}}_2$ layer in ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$ is derived. In this model, the kinetic energy for the degenerate $t_{2g}$ orbitals is given by indirect hopping over oxygen, leading naturally to the concept of four interpenetrating Kagomé lattices. Local Coulomb interaction couples the four lattices and an effective Hamiltonian for the interaction in the top band can be written in terms of fermionic operators with four different flavors. Focusing on charge- and spin-density instabilities, a big variety of possible metallic states with spontaneously broken symmetry are found. These states lead to different charge, orbital, spin, and angular momentum ordering patterns. The strong superstructure formation at $x=0.5$ is also discussed within this model.

Figure 1

(Color online) Schematic figure of a $\mathrm{Co}{\mathrm{O}}_2$ plane drawn with cubic unit cells. The edge sharing of the oxygen octahedra around the Co ions is visualized. The edges of the cubes are oriented along the coordinate system $(x,y,z)$. The triangular lattice of the cobalt is spanned by the vectors ${\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3$ $({\mathbf{a}}_1+{\mathbf{a}}_2=-{\mathbf{a}}_3)$. $a=\mid {\mathbf{a}}_i\mid$ is the lattice spacing. An oxygen $2p$ orbital and the cobalt $t_{2g}$ orbitals hybridizing with it by $\pi$ hybridization are shown.

Figure 2

The original Brillouin zone (BZ) of the triangular lattice consists of four reduced BZs around the $\Gamma$ point (0) and the three $M$ points (1,2,3). The symmetry points of the reduced BZs—$M^{\prime }$, $K^{\prime }$, and ${\Gamma }^{\prime }$—are symmetry points for the tight-binding model in Eq. (1) due to the higher periodicity of the bands. It is therefore sufficient to draw the bands along the lines ${\Gamma }^{\prime }\text{−}{K}^{\prime }\text{−}{M}^{\prime }\text{−}{\Gamma }^{\prime }$. The Fermi surface (FS) for $x=0.5$ lies at $E_{\mathbf{k}}^1\approx 3.16t$. The density of states per spin and per reduced Brillouin zone $D$ is given in units of $1∕t$. It has a logarithmic singularity at $E=2t$.

Figure 3

(Color online) Charge-ordering instability with finite expectation value $⟨{\tilde{n}}_0^1⟩$ leading to a charge enhancement or reduction on one Kagomé lattice. The folding of the BZ and the splitting of the pockets is shown (the double dotted line in the BZ indicates a triply degenerate pocket). On the right, a quasiparticle state that is in this case just a Kagomé lattice state is drawn.

Figure 4

(Color online) The spin-density-wave pattern corresponding to a finite expectation value $⟨S_z^1⟩$ is shown on the left. This pattern is stabilized if ${\gamma }_2>0$ in Eq. (38). The pattern on the right corresponds to a finite order parameter $⟨S_z^1+S_z^2+S_z^3⟩$, which is stabilized for ${\gamma }_2<0$.

Figure 5

(Color online) Ordering instabilities, described by real off-diagonal SU(4) generators $Q^{4–9}$. (a) $Q^4$ breaks neither translational nor rotational symmetry. (b) shows the ordering corresponding to $Q^6$. The ordering shown in (c) is described in real space and in reciprocal space by the same matrix ${\tilde{Q}}^7={\tilde{K}}^7$ and breaks translational symmetry. The corresponding BZ is shown in Fig. 3.

Figure 6

(Color online) Transitions to time-reversal symmetry-breaking states, where the expectation value of the orbital angular momentum $⟨\vec{L}⟩$ on the Co sites is finite. (a) States where the angular momentum does not lie in the plane. ${\tilde{K}}^{10}={\tilde{Q}}^{10}$. (b) States with angular momentum in the plane. ${\tilde{K}}^{13}=-{\tilde{Q}}^{13}$. (c) The folding of the BZ and the hybridization of the bands for (a) are shown. Dotted lines indicate doubly degenerate bands.

Figure 7

(Color online) The dimensionless coupling constants ${\tilde{\Lambda }}_r^{\mathrm{c}∕\mathrm{s}}=9{\Lambda }^{\mathrm{c}∕\mathrm{s}}∕\left(2U\right)$ as functions of $\alpha =U^{\prime }∕U$. The relations $U=U^{\prime }+2J_{\mathrm{H}}$ and $J^{\prime }=J_{\mathrm{H}}$ are assumed to hold. The solid (dashed) lines denote the charge (spin) coupling constants.

Figure 8

(Color online) Two different Na superstructures in ${\mathrm{Na}}_{1∕2}\mathrm{Co}{\mathrm{O}}_2$. The left one does not break rotational symmetry and would drive a charge-ordering as shown in Fig. 5c. The right one is in fact realized in ${\mathrm{Na}}_{1∕2}\mathrm{Co}{\mathrm{O}}_2$; it is obtained from the right one by shifting the Na chains along the arrows. This shift is due to the Coulomb repulsion of the Na ions.

Figure 9

(Color online) The charge-ordering pattern corresponding to the matrix $K^1-K^7$ consists of alternating rows of $d_x$ and $d_y-d_z$ orbitals. On the left-hand side, the original BZ, the orthorhombic BZ due to the charge ordering, and the experimentally observed reduced orthorhombic BZ (dark) are shown.