  We investigate long-range chiral magnetic interactions among adatoms mediated
by surface states spin-splitted by spin-orbit coupling. Using the Rashba model,
the tensor of exchange interactions is extracted wherein a pseudo-dipolar
interaction is found besides the usual isotropic exchange interaction and the
Dzyaloshinskii-Moriya interaction. We find that, despite the latter
interaction, collinear magnetic states can still be stabilized by the
pseudo-dipolar interaction. The inter-adatom distance controls the strength of
these terms, which we exploit to design chiral magnetism in Fe nanostructures
deposited on Au(111) surface. We demonstrate that these magnetic interactions
are related to superpositions of the out-of-plane and in-plane components of
the skyrmionic magnetic waves induced by the adatoms in the surrounding
electron gas. We show that, even if the inter-atomic distance is large, the
size and shape of the nanostructures dramatically impacts on the strength of
the magnetic interactions, thereby affecting the magnetic ground state. We also
derive an appealing connection between the isotropic exchange interaction and
the Dzyaloshinskii-Moriya interaction, which relates the latter to the first
order change of the former with respect to the spin-orbit coupling. This
implies that the chirality defined by the direction of the
Dzyaloshinskii-Moriya vector is driven by the variation of the isotropic
exchange interaction due to the spin-orbit interaction.

Mesoscale and Nanoscale Physics