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 ${M}_{L}=0, \ifmmode\pm\else\textpm\fi{}1$. 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 $b>{b}_{c}$, 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 $\mathrm{Ni}{\mathrm{Fe}}_{t}{\mathrm{Cr}}_{2\ensuremath{-}t}{\mathrm{O}}_{4}$. Whereas superexchange energies maintain a Jahn-Teller stabilization below ${T}_{c}$ in Cu${\mathrm{Cr}}_{2}$${\mathrm{O}}_{4}$, despite collinear ${\mathrm{Cu}}^{2+}$-ion spins, magnetostrictive distortions below ${T}_{N}$ occur in FeO and CoO. Elastic restoring forces favor trigonal ($\ensuremath{\alpha}>60\ifmmode^\circ\else\textdegree\fi{}$) symmetry for octahedral-site ${\mathrm{Fe}}^{2+}$, but tetragonal ($\frac{c}{a}<1$) symmetry for ${\mathrm{Co}}^{2+}$ and ${\mathrm{V}}^{2+}$. In trigonal FeO, superexchange interactions also help stabilize the trigonal distortion, whereas in tetragonal CoO they do not. The compound LaV${\mathrm{O}}_{3}$ also has a spin-orbit coupling stabilization that is enhanced by a magnetostrictive distortion to tetragonal ($\frac{c}{a}<1$) symmetry below ${T}_{N}$. However, the isoelectronic compound PbCr${\mathrm{O}}_{3}$ shows no such distortion, presumably because it illustrates band antiferromagnetism together with spin-orbit-coupling stabilization. The low-spin ions ${\mathrm{Fe}}^{4+}$ and ${\mathrm{Co}}^{4+}$ also form collective $d$ orbitals in oxides with perovskite structure; electric, magnetic, and crystallographic data for SrFe${\mathrm{O}}_{3}$ and LaSr${\mathrm{Co}}_{2}$${\mathrm{O}}_{6}$ indicate collective $d$ electrons having transfer integrals in the narrow range ${b}_{c}