Quasiparticle focusing of Yu-Shiba-Rusinov states in two-dimensional singlet superconductors

A magnetic impurity on a superconducting substrate induces in-gap Yu-Shiba-Rusinov (YSR) bound states, whose intricate spatial structure crucially influences the possibilities of engineering collective impurity states (e.g., Shiba chains, lattices...) and topological superconductivity. The spatial structure is complex as it blends together the information about substrate band structure, pairing function and impurity coupling. We use a generalized saddle-point approximation to deal with a point-like magnetic impurity in a gapped two-dimensional superconductor and we unveil a simple analytical relationship between, on the one hand, the real-space anisotropy of decay and oscillations of YSR states, and on the other hand, the momentum-space anisotropy of the Fermi surface, Fermi velocity, and pairing function. We reveal explicitly a non-trivial quasiparticle focusing effect in real-space that appears both in a power-law prefactor, as for impurity states in normal metals, but also in the exponential decay factor due to the superconducting gap. Our analytical approximation is quantitatively accurate against tight-binding calculations in lattice models. We study models relevant for transition metal dichalcogenides and also discuss the six-fold anisotropy of YSR state observed in scanning tunneling spectroscopy experiments of magnetic impurities on NbSe₂.