Continuous model of critical field-cooled state in a wide superconducting strip

We present a continuous model describing the magnetic-field and sheet-current profiles in a thin and wide superconducting strip cooled through the transition temperature (T_c) in a uniform magnetic field. The metastable field-cooled (FC) state results from complex interplay between the flux expulsion dynamics at the edges and bulk vortex pinning. We assume that the vortices continue to escape the strip within some temperature range near T_c enabling local equilibration near the edges. Away from edges, however, bulk pinning prevents global equilibration of the flux dome inside. These two processes generate a FC state which is composed of five regions: a frozen region near the center, two intermediate pinning regions where the critical current flows, and two flux-free (Meissner) regions near the edges. A characteristic feature of this state is that the current changes sign inside the flux-flow region. Using a complex-analysis formulation, we derive analytic integral representations for the field and current profiles and obtain closed-form expressions in the limit of narrow flux-free regions. The model provides a basis for quantitative interpretation of magneto‑optical or scanning‑probe measurements on field‑cooled strips for better understanding of flux-escape processes governing flux-dome equilibration.