Magneto-structural properties and magnetic anisotropy of small transition-metal clusters: a first-principles study

Ab initio density-functional calculations including spin-orbit coupling (SOC) have been performed for Ni and Pd clusters with three to six atoms and for 13-atom clusters of Ni, Pd, and Pt, extending earlier calculations for Pt clusters with up to six atoms (2011 J. Chem. Phys. 134 034107). The geometric and magnetic structures have been optimized for different orientations of the magnetization with respect to the crystallographic axes of the cluster. The magnetic anisotropy energies (MAE) and the anisotropies of spin and orbital moments have been determined. Particular attention has been paid to the correlation between the geometric and magnetic structures. The magnetic point group symmetry of the clusters varies with the direction of the magnetization. Even for a 3d metal such as Ni, the change in the magnetic symmetry leads to small geometric distortions of the cluster structure, which are even more pronounced for the 4d metal Pd. For a 5d metal the SOC is strong enough to change the energetic ordering of the structural isomers. SOC leads to a mixing of the spin states corresponding to the low-energy spin isomers identified in the scalar-relativistic calculations. Spin moments are isotropic only for Ni clusters, but anisotropic for Pd and Pt clusters, orbital moments are anisotropic for the clusters of all three elements. The magnetic anisotropy energies have been calculated. The comparison between MAE and orbital anisotropy invalidates a perturbation analysis of magnetic anisotropy for these small clusters.