Exact solution to the linear Ginzburg–Landau equation for a single-band type-II superconductor confined in a thin-film structure

We solve exactly the linear Ginzburg–Landau equation for a single-band bulk superconductor confined to a thin-film structure. In order to achieve a confinement effect, we replace the constant mass with an effective mass varying with position. We break the isotropic behaviour of the bulk superconductor through the semi-confinement effect. We show that the obtained macroscopic order parameter of the type-II superconducting condensate is expressed via the Laguerre polynomials. Here, one observes first signatures of the anisotropy in y and z directions through the analytical expression of the quadratic Landau coefficient. Next, we restrict the superconductor from both sides to the symmetric thin film. The macroscopic order parameter is now expressed via the Jacobi polynomials. We also obtain a generalized symmetric expression of the quadratic Landau coefficient, which exhibits strong dependence on the kinetic energy y and z components. Finally, we break a symmetric confinement again and show that under such a generalized case, the quadratic Landau coefficient dependence on the kinetic energy y and z components drastically changes according to the sharply varying asymmetric confinement profile. We presented the relations that connect the surface and bulk critical fields.