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Magnetic impurities on a superconductor induce sub-gap Yu-Shiba-Rusinov (YSR)bound states, localized at the impurity site and fading away from it fordistances up to several nanometers. In this article, we present a theoreticalmethod to calculate the spatial distribution of the YSR spectrum of atwo-dimensional superconductor with arbitrary Fermi contours (FCs) in thepresence of magnetic impurities. Based on the Green's Function (GF) formalism,we obtain a general analytical expression by approximating an arbitrary contourshape to a regular polygon. This method allows us to show the connectionbetween the spatial decay (and, hence, the extension) of YSR states and theshape of the FC of the host superconductor. We further apply this formalism tocompute the evolution of YSR states in the presence of a nearby impurity atom,and compare the results with Scanning Tunneling Microscopy (STM) measurementson interacting manganese dimers on the $\beta$-Bi2Pd superconductor. The methodcan be easily extended to any arbitrary number of magnetically coupledimpurities, thus, providing a useful tool for simulating the spectralproperties of interacting YSR states in artificial atomic nanostructures.

Yu-Shiba-Rusinov states in 2D superconductors with arbitrary Fermi contours