Spin-orbital interaction for face-sharing octahedra: Realization of a highly symmetric SU(4) model

Specific features of orbital and spin structure of transition-metal compounds in the case of the face-sharing ${\text{MO}}_6$ octahedra are analyzed. In this geometry, we consider the form of the spin-orbital Hamiltonian for transition-metal ions with double $\left(e_g^{\sigma }\right)$ or triple $\left(t_{2g}\right)$ orbital degeneracy. Trigonal distortions typical of the structures with face-sharing octahedra lead to splitting of $t_{2g}$ orbitals into an $a_{1g}$ singlet and $e_g^{\pi }$ doublet. For both doublets ($e_g^{\sigma }$ and $e_g^{\pi }$), in the case of one electron or hole per site, we arrive at a symmetric model with the orbital and spin interaction of the Heisenberg type and the Hamiltonian of unexpectedly high symmetry: SU(4). Thus, many real materials with this geometry can serve as a testing ground for checking the prediction of this interesting theoretical model. We also compare general trends in the spin-orbital (“Kugel-Khomskii”) exchange interaction for three typical situations: those of ${\text{MO}}_6$ octahedra with common corner, common edge, and the present case of common face, which has not been considered yet.

Figure 1

A chain of face-sharing octahedra. Large (red) and small (blue) circles denote metal and ligand ions, respectively.

Figure 2

(a) Magnetic atom (M) surrounded by trigonally distorted oxygen (O) octahedron in transition-metal compounds with face-sharing octahedra. The global trigonal coordinate is shown. Trigonal distortion is determined by the angle $\theta$; the value $cos{\theta }_0=1/\sqrt{3}$ corresponds to undistorted octahedron. Magnetic atoms form a quasi-one-dimensional chain directed along the $z$ axis. (b) Crystal field splitting of $d$ orbitals of the magnetic atom. The splitting of $t_{2g}$ levels $\left({\Delta }_1\right)$ is due to both the trigonal distortions of oxygen octahedra and contribution from neighboring M atoms to the crystal field. The sign of ${\Delta }_1$ can be different depending on the type of distortions.

Figure 3

Angle $\alpha$ versus M1-O-M2 angle $\beta =\pi -2\theta$; $k=0.1,r_0=2\text{Å}$.

Figure 4

Hopping integral for $e_g^{\sigma }$ orbitals versus M1-O-M2 angle $\beta =\pi -2\theta$; $k=0.1,r_0=2\text{Å}$.

Figure 5

Hopping integral for $e_g^{\pi }$ orbitals versus M1-O-M2 angle $\beta =\pi -2\theta$; $k=0.1,r_0=2\text{Å}$.