The magnetic anisotropy energy (MAE) of $3d$ transition-metal clusters was determined by calculating self-consistently the effects of the spin-orbit coupling on the spin-polarized charge distribution and on the electronic spectrum for different orientations of the magnetization. The MAE shows a complicated, nonperturbative behavior as a function of cluster size, structure, bond length, and $d$-band filling. In agreement with experiment, the MAE of small clusters is found to be considerably larger than in the corresponding crystals, often even larger than in thin films. Remarkably, the in-plane anisotropy can be of the same order of magnitude as the off-plane anisotropy.

• Received 13 March 1995

DOI:https://doi.org/10.1103/PhysRevLett.75.326