Influence of surface electron scattering on subgap states in helical atom chains on a superconductor

Within the Bogoliubov--de Gennes equations formalism, we study subgap quasiparticle states in a helical magnetic atom chain placed on top of a three-dimensional superconductor. The surface of the superconductor is modeled as an impenetrable barrier for electrons. As a result, for a surface impurity, spherically symmetric scattering is suppressed, and we consider scattering with angular momentum $l=1$. For a single impurity, we find that the density of states near the surface corresponding to a Yu-Shiba-Rusinov state falls off proportionally to the distance from the impurity to the minus fourth power, opposed to the inverse-square falloff law characteristic for an impurity in the bulk, which is typically used for interpretation of scanning tunneling microscopy experiments. For a helical atom chain, we calculate the band spectrum of hybridized Yu-Shiba-Rusinov states and the Majorana number and analyze the structure of Majorana edge modes. In the limit of large coherence lengths, the Majorana wave function in the chain falls off proportionally to the inverse square of the distance from the edge of chain, which is faster than for a magnetic chain in the bulk of a superconductor. The energy of the corresponding edge mode is proportional to the inverse square of the length of the chain. When the the atomic spins form a planar helix, the system belongs to the Altland-Zirnbauer symmetry class BDI. By calculating the corresponding topological invariant we demonstrate that in this case one edge of the chain can host up to three Majorana modes.