  Magnetic impurities in a superconductor induce Yu-Shiba-Rusinov (YSR) states
inside the superconducting gap, whose energy depends on the strength of the
coupling 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 Fermi
level 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 same
functional form as in the constant DOS scenario, with the effect of the
singularity being an enhancement of the effective coupling constants. In
particular, the critical magnetic coupling strength at which the Shiba
transition 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 which
diverges at $E=0$ for a VHS or HOVHS. Further, since the critical magnetic
coupling is significantly reduced, a new regime becomes accessible where the
transition point is controlled by the non-magnetic coupling constant.
Interestingly, the slope of the Shiba energy curve at the Shiba transition is
independent of impurity parameters and purely reflects the band structure. Our
results show that tuning a superconducting material towards a VHS or HOVHS
enhances the possibilities for engineering YSR states, and for characterizing
the superconductor itself.
