Orbital polarization effects on the magnetic anisotropy and orbital magnetism of clusters, films, and surfaces: A comparative study within tight-binding theory

The effects of orbital polarizations on the magnetic properties of transition-metal nanostructures are investigated in the framework of a self-consistent tight-binding theory. Three different approximations to the intra-atomic two-center Coulomb interactions are considered: (i) full orbital dependence of the direct and exchange Coulomb interactions U_mm^' and J_mm^' as given by atomic symmetry, (ii) orbital independent interactions U=U_mm^' and J=J_mm^', and (iii) orbital polarization (OP) approximation of the form HOP=-(B∕2)∑ᵢLᵢ², where Lᵢ refers to the orbital momentum operator at atom i and B to the Racah coefficient. Results are given for the local orbital magnetic moments ⟨L_iδ⟩ along high-symmetry magnetization directions δ and for the corresponding magnetic anisotropy energies ΔE_δγ of surfaces, films, and clusters of Fe, Co, and Ni. The quantitative differences between the approximations allow us to quantify the effects of orbital polarizations on ⟨L_iδ⟩ and ΔE_δγ. One observes that, with an appropriate choice of B, the OP ansatz yields a very good agreement with the rigorous orbital dependent calculations. The simplest orbital independent approach underestimates ⟨L_iδ⟩ and ΔE_δγ systematically. However, it provides a good qualitative description of the main general trends as a function of dimensionality, local environment, and d-band filling. Advantages and limitations of the various approaches are discussed.