# Boundary states with elevated critical temperatures in Bardeen-Cooper-Schrieffer superconductors

## TL;DR

This paper reveals that in standard BCS superconductors, the superconducting gap persists at the boundaries at higher temperatures than in the bulk, indicating multiple critical points and boundary-specific phase transitions.

## Contribution

The study demonstrates, both analytically and numerically, that boundary regions in BCS superconductors can remain superconducting at higher temperatures than the bulk, challenging the traditional single critical point paradigm.

## Key findings

- Boundary superconductivity persists at higher temperatures than bulk.
- Multiple critical points exist for boundary and bulk transitions.
- Analytical and numerical solutions confirm boundary states in BCS models.

## Abstract

Bardeen-Cooper-Schrieffer (BCS) theory describes a superconducting transition as a single critical point where the gap function or, equivalently, the order parameter vanishes uniformly in the entire system. We demonstrate that in superconductors described by standard BCS models, the superconducting gap survives near the sample boundaries at higher temperatures than superconductivity in the bulk. Therefore, conventional superconductors have multiple critical points associated with separate phase transitions at the boundary and in the bulk. We show this by revising the Caroli-De Gennes-Matricon theory of a superconductor-vacuum boundary and finding inhomogeneous solutions of the BCS gap equation near the boundary, which asymptotically decay in the bulk. This is demonstrated for a BCS model of almost free fermions and for lattice fermions in a tight-binding approximation. The analytical results are confirmed by numerical solutions of the microscopic model. The existence of these boundary states can manifest itself as discrepancies between the critical temperatures observed in calorimetry and transport probes.