Time-reversal symmetry breaking and gapped surface states due to spontaneous emergence of new order in $d$-wave nanoislands

TL;DR

This paper demonstrates that in nanoscale $d$-wave systems, spontaneous time-reversal symmetry breaking induces a complex extended $s$-wave order parameter, gapping surface states and leading to vortex phenomena near the surfaces.

Contribution

It reveals the spontaneous emergence of a complex extended $s$-wave order parameter in $d$-wave nanoislands, breaking TRS and gapping surface states, which was not previously understood.

Findings

TRS breaking occurs at low temperatures in $d$-wave nanoislands.

Surface Andreev states are gapped by a new order parameter.

Vortex-antivortex pairs form near surfaces in large nanoislands.

Abstract

We solve the Bogoliubov-de Gennes equations self-consistently for the $d$-wave order parameter in nanoscale $d$-wave systems with [110] surfaces and show that spontaneous time-reversal symmetry (TRS) breaking occurs at low temperatures due to a spontaneously induced complex order parameter of extended $s$-wave symmetry. The Andreev surface bound states, which are protected by a one-dimensional (1D) topological invariant in the presence of TRS, are gapped by the emergence of this new order parameter. The extended $s$-wave order parameter is localized within a narrow region near the surfaces, which is consistent with the fact that topological protection of the gapless Andreev surface states is characterized by the 1D topological invariant. In this TRS-breaking phase, not only is the complex order parameter induced, but also the $d$-wave order parameter itself becomes complex. Furthermore, the disappearance of topological protection brings about novel vortex phenomena near the surfaces. We show that vortex-antivortex pairs are formed in the extended $s$-wave order parameter along the surfaces if the side length of a nanoisland or the width of an infinitely long nanoribbon is relatively large.