Density-functional theory of the magnetic anisotropy of nanostructures: an assessment ofdifferent approximations

Abstract

We discuss the multiple technical choices that have to be made in ab initio density-functionalcalculations of the magnetic anisotropy of supported nanostructures: (i) choice of theexchange–correlation functional, (ii) degree of optimization of the geometry of theadsorbate/substrate complex, (iii) magnetic anisotropy energy calculated self-consistently orvia the ‘force theorem’, (iv) calculations based on slab models of the substrate or using aGreen’s function describing a semi-infinite substrate, (v) full potential approach oratomic-sphere approximation. Using isolated Fe and Co atoms on Pt(111) as an examplewe demonstrate that by using a judicious combination of relatively crude approximations(complete neglect of structural relaxation, local exchange–correlation functional,...) seemingly good agreement with experimental anisotropy energies can beachieved, while the calculated orbital moments remain small. At a higher levelof theory (relaxed adsorbate/substrate complex, gradient-corrected functionals,...) providing a realistic geometry of the adsorbate/substrate complex and hence a correctdescription of the interaction between the magnetic adatom and its ligands, anisotropyenergies are also in semi-quantitative agreement with experiment, while the orbitalmoments of the adatoms are much too small. We suggest that the anisotropy energiesprovided by both approaches should be considered as lower limits of the real anisotropies.Without relaxation the ligand effect coupling the orbital moments of the adatom to theheavy atoms of the substrate is underestimated, while in a relaxed adsorbate/substratecomplex the lack of orbital dependence of the exchange potential combined with a stronghybridization of adatom and substrate states leads to a strong underestimation of theorbital moment. We have briefly explored the influence of post-density-functionalcorrections. Adding a modest on-site Coulomb repulsion to the d states of the adatom (in aDFT+U approach) leads to a modest increase of spin and orbital moments of the adatomaccompanied by a slow decrease of the induced moments, leaving the anisotropy energyalmost unchanged.