First principles study of topological phase in chains of $3d$ transition metals

TL;DR

This study uses first principles calculations to explore the topological phases of 3d transition metal nanochains, revealing substrate effects and magnetic order influence on Majorana states, with potential for topological phases in deposited chains.

Contribution

It provides exact tight binding models and topological invariants for 3d transition metal chains, highlighting substrate effects and magnetic order stability on topological phases.

Findings

Topological phase exists in Mn and Co chains.

Substrate coupling can induce topological phases in all studied chains.

Non-collinear magnetic order is unstable in Fe and Co chains.

Abstract

Recent experiments have shown the signatures of Majorana bound states at the ends of magnetic chains deposited on a superconducting substrate. Here, we employ first principles calculations to directly investigate the topological properties of $3d$ transition metal nanochains (i.e., Mn, Cr, Fe and Co). In contrast to the previous studies [Nadj-Perge et al. Science 346, 602 (2014) and Ruby et al. Nano Lett. 17, 4473 (2017)], we found the exact tight binding models in the Wannier orbital basis for the isolated chains as well as for the surface--deposited wires. Based on these models, we calculate topological invariant of $\mathbb{Z}_2$ phase for all systems. Our results for the isolated chains demonstrate the existence of the topological phase only in the Mn and Co systems. We considered also a non-collinear magnetic order as a source of the non--trivial topological phase and found that this type of magnetic order is not a stable ground state in the Fe and Co isolated chains. Further studies showed that a coupling between the chain and substrate leads to strong modification of the band structure. Moreover, the analysis of the topological invariant indicates a possibility of emergence of the topological phase in all studied nanochains deposited on the Pb surface. Therefore, our results demonstrate an important role of the coupling between deposited atoms and a substrate for topological properties of nanosystems.