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Magnetic impurities in a superconductor induce Yu-Shiba-Rusinov (YSR) statesinside the superconducting gap, whose energy depends on the strength of thecoupling to the impurity and on the density of states (DOS) at the Fermi level.We consider DOS exhibiting a logarithmic or a power-law divergence at the Fermilevel due to Van Hove singularities (VHS) and high-order Van Hove singularities(HOVHS), respectively. We find that the energy of the YSR states has the samefunctional form as in the constant DOS scenario, with the effect of thesingularity being an enhancement of the effective coupling constants. Inparticular, the critical magnetic coupling strength at which the Shibatransition occurs is always lowered by a factor$1/\rho(\Delta/E_{\mathrm{c}}$), where $\Delta$ is the superconducting gap,$E_{\mathrm{c}}$ is the bandwidth, and $\rho(E)$ is the factor in DOS whichdiverges at $E=0$ for a VHS or HOVHS. Further, since the critical magneticcoupling is significantly reduced, a new regime becomes accessible where thetransition point is controlled by the non-magnetic coupling constant.Interestingly, the slope of the Shiba energy curve at the Shiba transition isindependent of impurity parameters and purely reflects the band structure.Additionally, we find that our main conclusions remain valid even when theFermi level is not precisely tuned to the Van Hove singularity, but insteadlies on an energy range of order the superconducting gap. Our results show thattuning a superconducting material towards a VHS or HOVHS enhances thepossibilities for engineering YSR states, and for characterizing thesuperconductor itself.

Effect of Van Hove singularities on Shiba states in two-dimensional $s$-wave superconductors