Orbital degeneracy as a source of frustration in ${\mathrm{LiNiO}}_2$

Motivated by the absence of cooperative Jahn-Teller effect and of magnetic ordering in ${\mathrm{LiNiO}}_2$, a layered oxide with triangular planes, we study a general spin-orbital model on the triangular lattice. A mean-field approach reveals the presence of several singlet phases between the $\mathrm{SU}\left(4\right)$ symmetric point and a ferromagnetic phase, a conclusion supported by exact diagonalizations of finite clusters. We argue that one of the phases, characterized by a large number of low-lying singlets associated to dimer coverings of the triangular lattice, could explain the properties of ${\mathrm{LiNiO}}_2$, while a ferro-orbital phase that lies nearby in parameter space leads to a new prediction for the magnetic properties of ${\mathrm{NaNiO}}_2$.

Figure 1

${\mathrm{ANiO}}_2$ structure. $\mathrm{Ni}$ ions are located in the middle of the $\mathrm{O}$ octahedra.

Figure 2

(a) The ${\mathcal{C}}_3$ axis and one of the ${\mathcal{C}}_2$ axes of the point group of the $\mathrm{Ni}$ site in the ${\mathrm{ANiO}}_2$ structure (the other two $C_2$ axes are obtained by applying ${\mathcal{C}}_3$). (b) Orbital states of the seventh $d$-electron of ${\mathrm{Ni}}^{3+}$ on the backgound of the network of oxygen octahedra.

Figure 3

The six neighbors of site 1.

Figure 4

Zero temperature phase diagram of the model on two sites as a function of $t^{\prime }∕t$ and $J_H∕\tilde{U}$. We have indicated the spin and orbital parts. Along the thick line at $t^{\prime }∕t=1$ and $J_H<0$ the ground state is triply degenerate: the orbital part becomes $\mathrm{SU}\left(2\right)$ symmetric, and the orbital triplet $\mid aa⟩$, $\mid ab⟩+\mid ba⟩$ and $\mid bb⟩$ with the spin singlet forms the ground-state wave function. Along the $t^{\prime }∕t=-1$ and $J_H<0$ line, the orbital triplet consists of $\mid aa⟩$, $\mid ab⟩-\mid ba⟩$, and $\mid bb⟩$. At the $\mathrm{SU}\left(4\right)$ points the ground state is 6-fold degenerate.

Figure 5

Energy spectrum of the spin singlet sector in the tetrahedron as a function of $t^{\prime }∕t$ for the $J_H=J_p=0$ case. For $t^{\prime }∕t$ between 0.2 and 1 the ground state is well separated from the rest of the states, and it is the adiabatic continuation of the $\mathrm{SU}\left(4\right)$ singlet state at the $t^{\prime }∕t=1$ point (full line). The $\mathrm{SU}\left(4\right)$ singlet nature of the ground state is lost at around $t^{\prime }∕t=0.2$ (denoted by an arrow), and in the region $-0.7<t^{\prime }∕t<0.2$ three levels [a non-degenerate level (solid line) and a two-fold degenerate level (dashed line)] go together. Finally, at $t^{\prime }∕t=-0.7$ the symmetry of the ground state changes, indicating the appearance of the third phase (dashed-dotted line).

Figure 6

Phase diagram of the spin-orbital model with $\mathit{J}={\mathit{J}}_{\mathit{H}}=2{\mathit{J}}_p$ and $\mathit{U}=\tilde{\mathit{U}}-\mathit{J}$ on a tetrahedron, based on exact diagonalization (see also Fig. 5). Phase boundaries in bold lines belong to level crossings in the ground state energy. A further singlet-to-singlet transition is identified in the vicinity of an antilevel crossing (dashed line). The degeneracy (apart from the trivial spin degeneracy) of each state is also indicated.

Figure 7

The phase diagram of the effective model on a tetrahedron based on the spin-orbital decoupling scheme (30).

Figure 8

Mean-field phase diagram on a 16-site cluster as a function of hopping integral versus Hund’s coupling. The grey phase is the ferromagnetic phase, with the classical phase boundaries shown (see Sec. 5B).

Figure 9

Spin and orbital structure in the singlet phases of the mean-field phase diagram. Solid line indicates $\mathrm{AF}$, dashed line $\mathrm{FM}$ spin correlations.

Figure 10

Schematic representation of the orbital orderings in the spin ferromagnetic case.

Figure 11

Energy level scheme of a 12 site diamond-like cluster with periodic boundary conditions and compatible with the 3-sublattice LRO. Shown are the levels which can be associated with the ferro-orbital $T^y$ order (denoted by $\Gamma 1$ and $\Gamma 2$), collinear phase ($\Gamma 1$, $M1$, and $\Gamma 3$) and the lowest states constituting the Anderson tower of the ${120}^{°}$ antiferro-orbital phase ($\Gamma 1$, $K1$, and $\Gamma 4$). The first letter refers to the momentum of the state. We have encircled the level crossings that we used to determine the phase boundary ($t^{\prime }∕t=-0.20$ and $t^{\prime }∕t=0.35$).

Figure 12

The 12-site cluster with periodic boundary conditions, and the associated Brillouin zone.

Figure 13

Exact diagonalizations: phase diagram for the 12-site cluster.

Figure 14

Low-lying singlets for $t^{\prime }∕t=0$ and $J∕U=0.008$.

Figure 15

Two stable states for a 12-site cluster: the dashed line represents the cluster. $t^{\prime }∕t=0$ and $J∕U=0.008$.

Figure 16

Two stable states for another 12-site cluster: the dashed line represents the cluster. $t^{\prime }∕t=0$ and $J∕U=0.008$.

Figure 17

${\theta }_{CW}$ as a function of $J_F$ for C and D phases (see Fig. 8): $J∕U=0.064$ and $t^{\prime }∕t=-0.1$ for the solid line (D phase), $t^{\prime }∕t=0.1$ for the dashed line (C phase).