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.
Highlights
The behavior of the gap function near a boundary of a BCS superconductor [1] is one of the most basic questions in superconductivity
The conclusion was reached that the gap at the boundary should vanish at the same temperature as the gap in the bulk, implying that the presence of a boundary in a large system does not alter the conclusions of BCS theory regarding the critical temperature associated with the disappearance of resistivity
The consequence of the gap behavior near the boundary that we find is that the superconducting state gets multiple critical temperatures: above the temperature at which the bulk of the system becomes normal, the boundary retains superconductivity
Summary
The behavior of the gap function near a boundary of a BCS superconductor [1] is one of the most basic questions in superconductivity. Possible superconductivity of boundaries in systems with normal bulk was previously invoked in connection with the increase of critical temperatures observed in granular elemental superconductors [9,10,11] The basis of this interpretation is that the. Let us compute the integral: drH (r) = dr (r) − drdr K0(r, r ) (r ) = This means that it is not possible to obtain the derivative of the order parameter near the boundary, contrary to the conclusions of Ref. Let us demonstrate that there is a second critical temperature Tc2 in a semi-infinite superconductor This effect appears due to the existence of energetically preferred inhomogeneous solutions of the linear gap equation near the boundary. The order parameter clearly has nonzero derivative on the boundary in contradiction to the CdGM condition (1)
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