Friedel oscillations and Majorana zero modes in inhomogeneous superconductors

TL;DR

This paper investigates how inhomogeneities affect superconducting order parameters, Friedel oscillations, and Majorana zero modes, revealing their dependence on symmetry and implications for experimental detection.

Contribution

It demonstrates the link between order parameter symmetry, Friedel oscillations, and Majorana modes, and proposes Josephson STM as a tool for probing these phenomena.

Findings

Friedel oscillations depend on superconducting symmetry.

Competing order parameters can create trivial regions near topological boundaries.

Majorana states' positions are affected by inhomogeneities and can be probed via Josephson STM.

Abstract

We study the modulations of the superconducting order parameter in the vicinity of edges, magnetic and non-magnetic impurities by self-consistently solving the gap equations of a system with competing interactions in the Cooper channel. It is shown that the presence or absence of Friedel-like oscillations of the superconducting order parameter crucially depends on its symmetry and can, hence, be used to obtain information about the symmetry properties of the condensate. Furthermore, the appearance of competing order parameters at inhomogeneities is discussed. We show that this can lead to the presence of a topologically trivial region close to the boundary of a system that is topologically nontrivial in its bulk. The resulting shift in position of the Majorana bound states is demonstrated to significantly affect its signatures in Josephson-junction experiments. We discuss Josephson scanning tunneling microscopy as a probe to resolve the Friedel-like oscillations as well as the spatial texture of competing s-wave superconductivity and Majorana bound states in the vicinity of the edge of the system.