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Surfaces of $d$-wave superconductors may host a substantial density ofzero-energy Andreev states. The zero-energy flat band appears due to atopological constraint, but comes with a cost in free energy. We have recentlyfound that an adjustment of the surface states can drive a phase transitioninto a phase with finite superflow that breaks time-reversal symmetry andtranslational symmetry along the surface. The associated Doppler shifts ofAndreev states to finite energies lower the free energy. Direct experimentalverification of such a phase is still technically difficult and controversial,however. To aid further experimental efforts, we use the quasiclassical theoryof superconductivity to investigate how the realization and the observabilityof such a phase are influenced by sample geometry and surface ruggedness. Phasediagrams are produced for relevant geometric parameters. In particular,critical sizes and shapes are identified, providing quantitative guidelines forsample fabrication in the experimental hunt for symmetry-breaking phases.

Spontaneous symmetry-breaking at surfaces of $d$-wave superconductors: influence of geometry and surface ruggedness