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

## TL;DR

This paper investigates how the geometry and surface roughness of $d$-wave superconductors affect the emergence and detection of a symmetry-breaking phase driven by surface Andreev states, providing practical guidelines for experiments.

## Contribution

It offers a detailed analysis of how sample shape and surface features influence the phase transition into a symmetry-breaking state in $d$-wave superconductors, aiding experimental design.

## Key findings

- Critical sizes and shapes for phase emergence identified
- Surface ruggedness impacts the stability of the symmetry-breaking phase
- Guidelines for sample fabrication to observe the phase provided

## Abstract

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