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

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.

Microscopic derivation of superconductor-insulator boundary conditions for Ginzburg-Landau theory revisited. Enhanced superconductivity at boundaries with and without magnetic field

“…We call this enhancement 2 boundary state. It was found in one, two, and three-dimensional superconductors in tight binding BCS model and for one dimensional continuous BCS model [19]. An earlier study of the standard three dimensional continuous BCS model concluded that boundary state is absent: the gap near the boundaries is neither suppressed nor enhanced [17] or weakly suppressed [18].…”

Boundary Superconductivity in the BCS Model

“…This result can be viewed as a rigorous justification of the numerical observations in [15]. Numerics shows that the ratio T R c pvq{T R c pvq can be as large as 1.06, see [15,Fig. 2].…”

“…This seems to implicitly assume that the effect of the boundary on the critical temperature is negligible. Recent numerical computations [3,15], however, indicate that the Cooper-pair wave function can localize near the boundary, leading to an increase in the critical temperature compared to its bulk value. In this paper, we shall give a rigorous proof of the occurrence of this phenomenon in the simplest setting of one dimension, with δ-interactions among the particles.…”

“…To prove (3.1), we use the variational principle with a trial function mimicking the ground state found in [15]. We choose ψ λ ǫ pxq " e ´ǫ|x| `λgpxq, where λ P R and the Fourier transform p g is real, continuous and centered at 2 ?…”

“…In [15] it was observed by numerical computations that the effect of boundary superconductivity disappears in the weak coupling limit, in the sense that…”