Quasiparticle focusing of bound states in two-dimensional $s$-wave superconductors

TL;DR

This paper develops an analytical theory linking the energy dispersion of 2D electrons to the spatial structure of Yu-Shiba-Rusinov bound states induced by magnetic impurities in superconductors, validated by tight-binding calculations.

Contribution

It introduces a simple analytical expression connecting Fermi surface features to YSR state shapes, enabling predictions for experimental observations.

Findings

Flatter Fermi surface segments enhance local density of states around impurities.

Analytical approximation matches tight-binding calculations across various lattices.

Theory successfully predicts YSR state shapes in NbSe2.

Abstract

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. By means of a saddle-point approximation we study the scattering processes giving rise to YSR states in gapped, two-dimensional superconductors. Further, we develop a theory which relates through a simple analytical expression an arbitrary energy dispersion of normal electrons in a two-dimensional host to the spatial features of the YSR states. Namely, we find that flatter segments of the Fermi surface with large Fermi velocity enhance the local density of states (LDOS) around the impurity. Our analytical approximation is quantitatively accurate against tight-binding calculations on various lattices with different Fermi surfaces, and it allows to predict the shape and orientation of YSR states observed in scanning tunneling spectroscopy experiments. We illustrate our results with a model of $\mathrm{NbSe}_2$.