Proximity Effects in Single- and Two-Band Superconducting Heterostructures: A Time-Dependent Ginzburg-Landau Approach

In this work, we study the proximity effects in a single- and two-band superconducting three-dimensional heterostructure, described by two condensates (condensate 1 and condensate 2) in the presence of an external magnetic field perpendicular to the heterostructure. The distance between the interfaces of both condensates is given by the parameter \(\lambda '\). We solve the time-dependent Ginzburg-Landau equations considering a Josephson-like coupling to explore properties such as magnetization, Gibbs free energy, and the Abrikosov vortex state. We propose three cases: case 1, both condensates are composed of a single-band; case 2, the condensates are composed of two bands; and case 3, condensate 1 has a single-band and condensate 2 has two bands. As a result, we highlight the variation of the first critical field and the novel vortex configurations induced by the proximity effect between the superconducting condensates. This phenomenon substantially influences the arrangement of vortices in each of the superconducting bands.