Dimensional crossover in a mesoscopic superconducting loop of finite width

Abstract

Superconducting structures with a size of the order of the superconducting coherence length $\ensuremath{\xi}(T)$ have a critical temperature ${T}_{c},$ oscillating as a function of the applied perpendicular magnetic field H (or flux $\ensuremath{\Phi}).$ For a thin-wire superconducting loop, the oscillations in ${T}_{c}$ are perfectly periodic with H (this is the well-known Little-Parks effect), while for a singly connected superconducting disk the oscillations are pseudoperiodic, i.e., the magnetic period decreases as H grows. In the present paper, we study the intermediate case: a loop made of thick wires. By increasing the size of the opening in the middle, the disklike behavior of ${T}_{c}(H)$ with a quasilinear background [characteristic of three-dimensional (3D) behavior] is shown to evolve into a parabolic ${T}_{c}(H)$ background (2D), superimposed with perfectly periodic oscillations. The calculations are performed using the linearized Ginzburg-Landau theory, with the proper normal/vacuum boundary conditions at both the internal and external interfaces. Above a certain crossover magnetic flux $\ensuremath{\Phi},$ ${T}_{c}(\ensuremath{\Phi})$ of the loops becomes quasilinear, and the flux period matches with the case of the filled disk. This dimensional transition is similar to the 2D-3D transition for thin films in a parallel magnetic field, where vortices enter the material as soon as the film thickness $t>1.8\ensuremath{\xi}(T).$ For the loops studied here, the crossover point appears for $w\ensuremath{\approx}1.8\ensuremath{\xi}(T)$ as well, with w the width of the wires forming the loop. In the 3D regime, a ``giant vortex state'' is established, where superconductivity is concentrated near the sample's outer interface. The vortex is then localized inside the loop's opening.