Zero energy modes in a superconductor with ferromagnetic adatom chains and quantum phase transitions

TL;DR

This paper investigates Majorana zero energy modes in a superconductor with ferromagnetic adatom chains, analyzing their properties, decay lengths, and phase transitions with implications for experimental detection.

Contribution

It provides a comparative analysis of MZEM behavior under self-consistent and non self-consistent superconducting order, revealing critical phenomena and decay length dependencies.

Findings

Superconducting gap drops discontinuously at the topological transition.

Decay length of MZEM is linearly related to exchange coupling, chemical potential, and superconducting order.

Wider chains host MZEM at smaller exchange couplings, with decay length slope increasing with chain width.

Abstract

We study Majorana zero energy modes (MZEM) that occur in a s-wave superconducting surface, at the ends of a ferromagnetic (FM) chain of adatoms, in the presence of Rashba spin-orbit interaction (SOI) considering both non self-consistent and self-consistent superconducting order. We find that in the self-consistent solution the average superconducting gap function over the adatom sites has a discontinuous drop with increasing exchange interaction at the same critical value where the topological phase transition occurs. We also study the MZEM for both treatments of superconducting order and find that the decay length is a linear function of the exchange coupling strength, chemical potential and superconducting order. For wider FM chains the MZEM occur at smaller exchange couplings and the slope of the decay length as a function of exchange coupling grows with chain width. Thus we suggest experimental detection of different delocalization of MZEM in chains of varying widths. We discuss similarities and differences between the MZEM for the two treatments of the superconducting order.