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We use Time-Dependent Ginzburg-Landau theory to study the nucleation ofvortices in type II superconductors in the presence of both geometric andmaterial inhomogeneities. The superconducting Meissner state is meta-stable upto a critical magnetic field, known as the superheating field. For a uniformsurface and homogenous material, the superheating transition is driven by anon-local critical mode in which an array of vortices simultaneously penetratethe surface. In contrast, we show that even a small amount of disorderlocalizes the critical mode and can have a significant reduction in theeffective superheating field for a particular sample. Vortices can be nucleatedby either surface roughness or local variations in material parameters, such asTc. Our approach uses a finite element method to simulate a cylindricalgeometry in 2 dimensions and a film geometry in 2 and 3 dimensions. We combinesaddle node bifurcation analysis along with a novel fitting procedure toevaluate the superheating field and identify the unstable mode. We demonstrateagreement with previous results for homogenous geometries and surface roughnessand extend the analysis to include variations in material properties. Finally,we show that in three dimensions, suface divots not aligned with the appliedfield can increase the super heating field. We discuss implications forfabrication and performance of superconducting resonant frequency cavities inparticle accelerators.

physical review. b./physical review. b

Role of surface defects and material inhomogeneities for vortex  nucleation in superconductors within time-dependent Ginzburg-Landau theory in  2 and 3 dimensions