Magnetic anisotropy of fcc transition-metal clusters: Role of surface relaxation

The magnetic-anisotropy energy (MAE) of fcc transition-metal (TM) clusters $(19\mathrm{}<~N<~79)$ is determined using two different semiempirical schemes. First within a tight-binding calculation for the d band, the equilibrium geometries of the clusters, which are built by adding successive atomic shells around a central atom, are obtained by means of the fictitious Lagrangian method introduced by Car and Parrinello [Phys. Rev. Lett. 55, 2471 (1985); 60, 204 (1988)]. In this approach, both atomic and electronic structures are treated simultaneously in the minimization algorithm, a procedure that reveals the existence of a highly nonuniform relaxation profile of the intershell spacings in the clusters, which may contract as well as expand, in agreement with the results found at surfaces of TM’s. In a second step, treating the spin-orbit coupling nonperturbatively and also within the framework of a d-band Hamiltonian, we analyze the role of this complex interlayer-spacing distribution on the magnetoanisotropic behavior of the clusters. In all cases, we perform single-point energy calculations, for two different directions of the magnetization $\delta ,$ on the previously optimized geometries using the Car-Parrinello method. The MAE shows a complicated behavior as a function of cluster size, bond length, and d-band filling. Moreover, by investigating different relaxations, it is shown that the existence of this nonuniform pattern of interatomic distances causes appreciable changes in the magnitude of the MAE and can be also at the origin of reorientations of the magnetization in the particles. We conclude that this kind of structural transformations are essential for quantitative predictions of the MAE in cluster systems.