The magnetocrystalline anisotropy energy $E_{ anis}$ for free-standing chains (quantum wires) and rings (quantum corrals) of Fe-adatoms $N=$(2...48) is determined using an electronic tight-binding theory. Treating spin-orbit coupling non-perturbatively, we analyze the relationship between the electronic structure of the Fe $d$-electrons and $E_{ anis}(n_{ d})$, for both the chain and ring conformations. We find that $E_{ anis}(N)$ is larger for wires than for rings or infinite monolayers. Generally $E_{ anis}(n_{ d})$ decreases in chains upon increasing $N$, while for rings $E_{ anis}(n_{ d})$ is essentially independent of $N$. For increasing $N$, $E_{ anis}(n_{ d})$ in corrals approaches the results for freestanding monolayers. Small rings exhibit clear odd-even oscillations of $E_{ anis}(N)$. Within our theoretical framework we are able to explain the experimentally observed oscillations of $E_{ anis}(n_{ d})$ 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.
Categories: cond-mat