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

6 pages, 4 figures + 6 pages (supplemental material), to be published in Phys. Rev. B Rapid communication

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• Spontaneous generation of fractional vortex-antivortex pairs at single edges of high-Tc superconductors
• Enhanced chiral edge currents and orbital magnetic moment in chiral $d$-wave superconductors from mesoscopic finite-size effects
• Impurity strength-temperature phase diagram with phase crystals and competing time-reversal symmetry breaking states in nodal $d$-wave superconductors
• Time-reversal symmetry breaking phase and gapped surface states in d-wave nanoscale superconductors