Normal-state bubbles and lamellae in type-I superconductors

Abstract

We report an extensive study of the formation of normal-state domains in type-I superconductors. Domain patterns are first considered theoretically. The magnetic interaction between domains is described in the framework of the ``current-loop'' model: the intermediate state is modeled by a set of loops of screening current encircling the domains and interacting as in the free space. This system is shown to be formally equivalent to a set of uniformly magnetized domains. An extension of the current-loop model is proposed to take into account the constraint of the magnetic shielding by the superconducting regions. We determine the free energy of a hexagonal array of cylindrical domains (bubbles) and of a lattice of infinitely long and parallel stripes. The equilibrium values of both the volume fraction of the normal phase and the domain size are calculated as functions of the magnetic field. A bubble-to-stripe transition is predicted to occur for a volume fraction of the normal phase about 0.3. Experimentally, normal-state domains are studied with the high-resolution magneto-optical imaging technique. The observed patterns consist in coexisting bubbles and disordered labyrinthine lamellae structures. We show evidence of the contribution of pinning on the position of domain interfaces. The average width of the lamellae is then analyzed as a function of the applied magnetic field and found to increase in good agreement with the predictions. In contrast, the average diameter of bubbles remains constant: it is almost independent of the magnetic interaction between domains. A very good agreement, over three decades of the magnetic Bond number, is found with the equilibrium diameter of an isolated bubble. The proposed constrained current-loop model is shown to provide significantly more accurate predictions than the current-loop model, in particular for small magnetic Bond numbers. Additionally, increasing the volume fraction of the normal phase results in a bubble-to-lamella transition, as predicted theoretically.