Nonperturbative theory of magnetocrystalline anisotropy energyfor wires and rings of Fe adatoms

The magnetocrystalline anisotropy energy ${\mathrm{E}}_{\mathrm{anis}}$ for free-standing chains (quantum wires) and rings of Fe adatoms N=(2,ldots,48) is determined using an electronic tight-binding theory. Treating spin-orbit coupling nonperturbatively, we analyze the relationship between the electronic structure of the Fe d electrons and ${\mathrm{E}}_{\mathrm{anis}}$(${\mathrm{n}}_{\mathrm{d}}$), for both the chain and ring conformations. We find that ${\mathrm{E}}_{\mathrm{anis}}$(N) is larger for wires than for rings or infinite monolayers. Generally ${\mathrm{E}}_{\mathrm{anis}}$(${\mathrm{n}}_{\mathrm{d}}$) decreases in chains upon increasing N, while for rings ${\mathrm{E}}_{\mathrm{anis}}$(${\mathrm{n}}_{\mathrm{d}}$) is essentially independent of N. For increasing N, ${\mathrm{E}}_{\mathrm{anis}}$(${\mathrm{n}}_{\mathrm{d}}$) in rings approaches the results for free-standing monolayers. Small rings exhibit clear odd-even oscillations of ${\mathrm{E}}_{\mathrm{anis}}$(N). Within our theoretical framework we are able to explain the experimentally observed oscillations of ${\mathrm{E}}_{\mathrm{anis}}$(${\mathrm{n}}_{\mathrm{d}}$) during film growth with a period of one monolayer. Finally, a generalization of Hund's third rule on spin-orbit coupling to itinerant ferromagnets is proposed.