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We consider an atomic chain of magnetic impurities on the surface of aspin-orbit coupled superconductor with a dominating $d$-wave and sub-dominating$s$-wave order parameters. In particular, we investigate the properties of theMajorana bound states (MBSs) emerging at the chain end points in thetopological phase and how MBSs are affected by the $d$-wave order parameter. Weprovide a comprehensive picture by both studying time-reversal invariant andbreaking superconducting substrates as well as chains oriented in differentdirections relative to the $d$-wave rotation. We show that increasing the$d$-wave order parameter significantly enhances the localization of MBSs andtheir protective mini-gap, as long as the direction along which the impuritychain 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 orientationwithin the topological phase, which we are able to attribute to simplegeometrical effects in the corresponding two-dimensional limit. These resultsshow how high-temperature $d$-wave superconductors can be used to significantlyenhance the properties and stability of MBSs.

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