Microscopic theory of the lower critical field in superconducting thin-film strips

Abstract

The lower critical field \(B_{c1}\) of a narrow superconducting thin-film strip sets the thermodynamic scale for vortex-free operation in a perpendicular magnetic field. The standard Pearl--London estimate requires a phenomenological vortex-core cutoff, because the London theory does not resolve the core. We formulate a microscopic theory for a dirty strip by solving the two-dimensional Usadel equations in the film plane, with the applied field included directly in the gauge-invariant momentum. Self-consistent vortex and Meissner solutions are computed at fixed field, and \(B_{c1}\) is obtained from their Gibbs-energy difference. The calculation resolves the vortex core and its finite-width deformation without introducing a cutoff. The resulting vortex self-energy is larger than the naive Pearl--London estimate and cannot, in general, be represented by a London logarithm with a single width-independent cutoff. The formulation applies at any \(T<T_c\) and provides a microscopic basis for predicting \(B_{c1}\) in superconducting nanostrips and related thin-film devices.