Magnetic anisotropy and electronic structure in surface-supported single rare-earth atom magnets: a topical review

Surface-supported single rare-earth atom magnets represent an ultimate limit of magnetic miniaturisation, where information storage is reduced to the scale of an individual atom. At this limit, magnetism is intrinsically quantum mechanical and governed by the interplay of strong electron correlations, crystal-field effects, and spin-orbit coupling within the localized 4f shell. In this review, we summarise and analyse recent theoretical advances in the description of rare-earth adatoms, with particular emphasis on approaches that go beyond conventional static mean-field DFT+U , which may exhibit multiple metastable solutions and treat magnetic anisotropy in a semiclassical manner. We discuss a predictive framework combining relativistic density functional theory with an Anderson impurity model treatment of the multiconfigurational 4f shell (DFT+U (HIA)), enabling a consistent description of strong correlations, multiplet structure, and quantum tunnelling effects induced by transverse crystal-field terms.As a representative case, we review the electronic structure and magnetic anisotropy of Dy adatoms on insulating MgO and spin-polarised graphene/Ni substrates. For Dy@MgO, an apparent perpendicular anisotropy is strongly reduced by quantum tunnelling driven by transverse crystal-field terms, effectively shifting the easy axis towards the surface plane. In contrast, Dy@Gr/Ni(111) realises a robust perpendicular magnetic configuration at the single-atom level. Here, the large positive magnetic anisotropy energy arises from the interplay of crystal-field splitting and strong spin-orbit coupling within the Dy 4f shell, further reinforced by the exchange field generated by the ferromagnetic Ni substrate.We argue that Dy@Gr/Ni(111) may be viewed as a limiting atomic-scale analogue of perpendicular synthetic heterostructures used in magnetic memory technologies, where exchange coupling and strong anisotropy are engineered to stabilise nanoscale bits. The insights gained from these studies establish a microscopic design principle for achieving thermally robust magnetic anisotropy at the ultimate scaling limit and highlight the broader potential of strongly correlated rare-earth adatoms for atomic-scale spintronic applications.