Vortex nucleation in superconductors within time-dependent Ginzburg-Landau theory in two and three dimensions: Role of surface defects and material inhomogeneities

We use time-dependent Ginzburg-Landau theory to study the nucleation of vortices in type-II superconductors in the presence of both geometric and material inhomogeneities. The superconducting Meissner state is metastable up to a critical magnetic field, known as the superheating field. For a uniform surface and homogeneous material, the superheating transition is driven by a nonlocal critical mode in which an array of vortices simultaneously penetrate the surface. In contrast, we show that even a small amount of disorder localizes the critical mode and can have a significant reduction in the effective superheating field for a particular sample. Vortices can be nucleated by either surface roughness or local variations in material parameters, such as T-c. Our approach uses a finite-element method to simulate a cylindrical geometry in two dimensions and a film geometry in two and three dimensions. We combine saddle-node bifurcation analysis along with a fitting procedure to evaluate the superheating field and identify the unstable mode. We demonstrate agreement with previous results for homogeneous geometries and surface roughness and extend the analysis to include variations in material properties. Finally, we show that in three dimensions, surface divots not aligned with the applied field can increase the superheating field. We discuss implications for fabrication and performance of superconducting resonant frequency cavities in particle accelerators.