Majorana bound states in magnetic impurity chains: effects of $d$-wave pairing

We consider an atomic chain of magnetic impurities on the surface of a spin-orbit coupled superconductor with a dominating $d$-wave and sub-dominating $s$-wave order parameters. In particular, we investigate the properties of the Majorana bound states (MBSs) emerging at the chain end points in the topological phase and how MBSs are affected by the $d$-wave order parameter. We provide a comprehensive picture by both studying time-reversal invariant and breaking superconducting substrates as well as chains oriented in different directions relative to the $d$-wave rotation. We show that increasing the $d$-wave order parameter significantly enhances the localization of MBSs and their protective mini-gap, as long as the direction along which the impurity chain is oriented does not cross any nodal lines of the gap function. Moreover, we find an extra gap-closing for a specific condensate and chain orientation within the topological phase, which we are able to attribute to simple geometrical effects in the corresponding two-dimensional limit. These results show how high-temperature $d$-wave superconductors can be used to significantly enhance the properties and stability of MBSs.