In magnetic chains on superconductors, so-called Shiba bands are formed within the superconducting gap of the host. Inside the Shiba bands, a minigap can be induced around zero energy by forming a magnetic spin spiral state or by a large spin-orbit coupling in the system. In the spirit of the bulk-edge correspondence principle, if the band structure is topological, zero energy bound states (Majorana states) can be found at both ends of finite chains. To have a quantitative and realistic description of these systems, we solve the Kohn—Sham—Dirac Bogoliubov-de Gennes equations within the Korringa—Kohn—Rostoker multiple scattering theory [1,2]. With examples, we show that physical quantities that are antisymmetric with respect to the Fermi energy, e.g., the singlet order parameter, can be used to prove band inversion of the system. Moreover, through the manipulation of the magnitude of the magnetic moments, we explore the conditions for the formation of topological phases in the chains and compare this to the change of Shiba state energies in single adatoms. Finally, by adding a non-magnetic overlayer between the superconductor and the chain, we explore the topological properties of a large variety of systems in terms of changing the crystallographic direction of the chain and the magnetic configuration.

 

[1] B. Nyári, A. Lászlóffy, G. Csire, L. Szunyogh, B. Ujfalussy, Topological superconductivity from first principles. I. Shiba band structure and topological edge states of artificial spin chains, Physical Review B 108, 134512 (2023)

[2] A. Lászlóffy, B. Nyári, G. Csire, L. Szunyogh, B. Ujfalussy, Topological superconductivity from first principles. II. Effects from manipulation of spin spirals: Topological fragmentation, braiding, and quasi-Majorana bound states, Physical Review B 108, 134513 (2023)