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We solve the Bogoliubov-de Gennes equations self-consistently for d-wavesuperconductors with [110] surfaces. We find spontaneous breaking oftime-reversal symmetry (TRS) caused by the spontaneous occurrence of a complexorder parameter with an extended s-wave symmetry along the [110] surfaces. Inthe TRS-breaking phase, the d-wave order parameter itself becomes complex. Wealso show that vortex-antivortex pairs can appear along the [110] surfaces ofnanoribbons. These pairs are detectable with surface sensitive probes.

Time-reversal symmetry breaking phase and gapped surface states in d-wave nanoscale superconductors