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 $M_L=0, \pm 1$. 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 $b>\mathrm{}{b}_c$, where $b_c$ 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 Nb${\mathrm{S}}_2$ and W${\mathrm{S}}_2$, 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 LaMn${\mathrm{O}}_3$. The competition between spin-orbit-coupling and Jahn-Teller stabilizations is dramatically illustrated by the system $\mathrm{Ni}{\mathrm{Fe}}_t{\mathrm{Cr}}_{2-t}{\mathrm{O}}_4$. Whereas superexchange energies maintain a Jahn-Teller stabilization below $T_c$ in Cu${\mathrm{Cr}}_2$ ${\mathrm{O}}_4$, despite collinear ${\mathrm{Cu}}^{2+}$-ion spins, magnetostrictive distortions below $T_N$ occur in FeO and CoO. Elastic restoring forces favor trigonal ($\alpha >60\mathrm{}°$) symmetry for octahedral-site ${\mathrm{Fe}}^{2+}$, but tetragonal ($\frac{c}{a}<1\mathrm{}$) symmetry for ${\mathrm{Co}}^{2+}$ and ${\mathrm{V}}^{2+}$. In trigonal FeO, superexchange interactions also help stabilize the trigonal distortion, whereas in tetragonal CoO they do not. The compound LaV${\mathrm{O}}_3$ also has a spin-orbit coupling stabilization that is enhanced by a magnetostrictive distortion to tetragonal ($\frac{c}{a}<1\mathrm{}$) symmetry below $T_N$. However, the isoelectronic compound PbCr${\mathrm{O}}_3$ shows no such distortion, presumably because it illustrates band antiferromagnetism together with spin-orbit-coupling stabilization. The low-spin ions ${\mathrm{Fe}}^{4+}$ and ${\mathrm{Co}}^{4+}$ also form collective $d$ orbitals in oxides with perovskite structure; electric, magnetic, and crystallographic data for SrFe${\mathrm{O}}_3$ and LaSr${\mathrm{Co}}_2$ ${\mathrm{O}}_6$ indicate collective $d$ electrons having transfer integrals in the narrow range $b_c<b<\mathrm{}{b}_m$, where $b_m$ is the maximum transfer integral for spontaneous band magnetism.