Realistic indirect spin-interactions between magnetic impurities on a metallic Pb(110) surface

Motivated by recent experiments, here we study the indirect interactions between magnetic impurities deposited on top of a clean Pb(110) surface induced by the underlying conduction electrons. Our approach makes use of \textit{ab initio} calculations to characterize the clean Pb(110) surface and avoids self-consistency, a feature that greatly reduces the computational cost. In combination with 2$^{\text{nd}}$ order perturbation theory in the microscopic \textit{s-d} exchange parameter $J_{K}$ between a magnetic adatom and the conduction electrons, we are able to systematically derive the Ruderman-Kittel-Kasuya-Yosida (RKKY), the Dzyaloshinskii-Moriya (DM) and the anisotropic tensor interactions emerging at the Pb(110) surface between magnetic impurities. The only adjustable parameter is $J_{K}$, which is fitted to reproduce the experiments. Our results show important anisotropy effects arising both from the rectangular geometry of the (110) unit cell, and from the strong Rashba spin-orbit interaction due to the broken inversion symmetry at the Pb(110) surface. In addition to Pb(110), the characterization of the indirect spin interactions described here could be extended to other realistic metallic surfaces for weakly-coupled impurities, and would enable to fabricate atomic-size nanostructures with engineered interactions and on-demand magnetic properties, anticipating useful applications in nanotechnology.