Multiple odd-frequency superconducting states in buckled quantum spin Hall insulators with time-reversal symmetry

TL;DR

This paper explores multiple odd-frequency superconducting states in buckled quantum spin Hall insulators like silicene when coupled with conventional s-wave superconductors, revealing mechanisms for both spin-singlet and spin-triplet odd-$requency$ pairing.

Contribution

It identifies two distinct mechanisms generating odd-$requency$ pairing in QSHIs and analyzes their dependence on edge type and doping, advancing understanding of unconventional superconductivity.

Findings

Both spin-singlet and spin-triplet odd-$requency$ pairs exist in QSHIs.

Spin-singlet pairs are due to inter-orbital processes and appear at edges and in heavily doped bulk.

Spin-triplet pairs require a superconducting order gradient, naturally occurring at armchair edges.

Abstract

We consider a buckled quantum spin Hall insulator (QSHI), such as silicene, proximity-coupled to a conventional spin-singlet, s-wave superconductor. Even limiting the discussion to the disorder-robust s-wave pairing symmetry, we find both odd-frequency ($\omega$), spin-singlet and spin-triplet pair amplitudes and where both preserve time-reversal symmetry. Our results show that there are two unrelated mechanisms generating these different odd-$\omega$ pair amplitudes. The spin-singlet state is due to the strong inter-orbital processes present in the QSHI. It is exists generically at the edges of the QSHI, but also in the bulk in heavily doped regime if an electric field is applied. The spin-triplet state requires a finite gradient in the proximity-induced superconducting order along the edge, which we find is automatically generated at the atomic scale for armchair edges but not at zigzag edges. In combination these results make superconducting QSHIs a very exciting venue for investigating not only the existence of odd-$\omega$ superconductivity, but also the interplay between different odd-$\omega$ states.