We present a fully relativistic first-principles-based theoretical approach for the
calculation of the spectral properties of magnetic impurities on the surface of a
superconducting substrate, providing a material specific framework for the investigation
of the Yu-Shiba-Rusinov (YSR) states. By using a suitable orbital decomposition of
the local densities of states we discuss in great detail the formation of the YSR
states for an Mn adatom and for two kinds of Mn dimers placed on the Nb(110) surface
and compare our results to recent experimental findings. In the case of the adatom
we find that the spin-orbit coupling slightly shifts some of the YSR peaks and also
the local spin polarization on the Nb atoms has marginal effects on their positions.
Moreover, by scaling the exchange field on the Mn site we could explain the lack of
the dx(2)-y(2)-like YSR state in the spectrum. While our results for a close packed
ferromagnetic dimer are in satisfactory agreement with the experimentally observed
splitting of the YSR states, in the case of an antiferromagnetic dimer we find that
the spin-orbit coupling is not sufficiently large to explain the splitting of the
YSR states seen in the experiment. Changing the relative orientation of the magnetic
moments in this dimer induces splitting of the YSR states and also shifts their energy,
leading even to the formation of a zero bias peak in the case of the deepest YSR state.
