Structural and Magnetic Properties of Small 4d Transition Metal Clusters: Role of Spin-Orbit Coupling

The spin and orbital magnetic moments, as well as the magnetic anisotropy energy (MAE), of small 4d transition metal (TM) clusters are systematically studied by using the spin-orbit coupling (SOC) implementation of the density-functional theory (DFT). The effects of spin-orbit interactions on geometrical structures and spin moments are too weak to alternate relative stabilities of different low-lying isomers. Remarkable orbital contributions to cluster magnetic moments are identified in Ru, Rh, and Pd clusters, in contrast to immediate quenching of the atomic orbital moment at the dimer size in other elemental clusters. More interestingly, there is always collinearity between total spin and orbital moments (antiferromagnetic or ferromagnetic coupling depends on the constituent atoms whose 4d subshell is less or more than half-filled). The clusters preserve the validity of Hund's rules for the sign of orbital moment. The calculations on MAEs reveal the complicated changes of the easy axes in different structures. The perturbation theory and the first-principles calculations are compared to emphasize how MAEs evolve with cluster size. Finally, large orbital moments combined with strong spin-orbit coupling are proposed to account for large MAEs in Ru, Rh, and Pd clusters.