Spin-Orbit-Coupling Effects in Transition-Metal Compounds

Crystal-field splittings in a high-symmetry phase may leave an orbitally degenerate ground state. Three types of degeneracies are considered: (1) a twofold degeneracy that carries no orbital angular momentum, (2) a twofold degeneracy that carries an orbital angular momentum, and (3) a threefold degeneracy that carries an azimuthal angular momentum ML=0, ±1. In the first type, there is a competition between ferromagnetic superexchange coupling that stabilizes dynamic Jahn-Teller vibrational modes and a static Jahn-Teller distortion that introduces anisotropic superexchange interactions. In the second type, spin-orbit coupling removes the degeneracy, and the usual empirical rules for the sign of the superexchange coupling are applicable provided that the transfer integrals with near-neighbor ions take account of the geometrical modification of the orbitals by spin-orbit coupling. In the third type, there is a competition between (a) a magnetostrictive static distortion that enhances the spin-orbit-coupling stabilization below a magnetic-ordering temperature, and (b) a pure Jahn-Teller static distortion. However, from a knowledge of the structure the orbital configurations and their transfer integrals are known, and the usual empirical rules for superexchange coupling can be applied. Further, if the transfer integrals are b>bc, where bc is sharply defined, it is necessary to use a collective-electron band model. For narrow bands, spin-orbit-coupling energies may be large enough to split degenerate bands of collective-electron orbitals. This latter splitting appears to be illustrated by NbS₂ and WS₂, where the cationic occupation of trigonal-bipyramidal interstices optimizes spin-orbit-coupling stabilization. Ferromagnetic superexchange via dynamic Jahn-Teller correlations is illustrated by high-temperature LaMnO₃. The competition between spin-orbit-coupling and Jahn-Teller stabilizations is dramatically illustrated by the system NiFeₜCr_2-tO₄. Whereas superexchange energies maintain a Jahn-Teller stabilization below Tc in CuCr₂{O}₄$, despite collinear ${Cu}²⁺$-ion spins, magnetostrictive distortions below ${T}N$ occur in FeO and CoO. Elastic restoring forces favor trigonal (${α}>60^∘{}$) symmetry for octahedral-site ${Fe}²⁺$, but tetragonal ($c/a<1$) symmetry for ${Co}²⁺$ and ${V}²⁺$. In trigonal FeO, superexchange interactions also help stabilize the trigonal distortion, whereas in tetragonal CoO they do not. The compound LaV${O}₃$ also has a spin-orbit coupling stabilization that is enhanced by a magnetostrictive distortion to tetragonal ($c/a<1$) symmetry below ${T}N$. However, the isoelectronic compound PbCr${O}₃$ shows no such distortion, presumably because it illustrates band antiferromagnetism together with spin-orbit-coupling stabilization. The low-spin ions ${Fe}⁴⁺$ and ${Co}⁴⁺$ also form collective $d$ orbitals in oxides with perovskite structure; electric, magnetic, and crystallographic data for SrFe${O}₃$ and LaSr${Co}₂O₆ indicate collective d electrons having transfer integrals in the narrow range bc<b<bₘ, where bₘ is the maximum transfer integral for spontaneous band magnetism.