Nonlinear current flow in superconductors with restricted geometries

We calculate two-dimensional steady-state distributions of transport electric field E(x,y) and current density J(x,y) in superconductors with restricted geometries, such as films with macroscopic planar defects, faceted grain boundaries, current leads, flux transformers, and microbridges. We develop a hodograph method, which enables us to solve analytically Maxwell’s equations for E(x,y) and J(x,y), taking account of the highly nonlinear E-J characteristics of superconductors E=E_c(J/J_c)ⁿ,n≫1. Based on this approach, a very effective numerical method of solving the nonlinear Maxwell equations was also developed. We show that nonlinear current flows in restricted geometries exhibit orientational current-flow domains separated by domain walls of varying width, which remain different from the discontinuity lines of the Bean model, even in the critical state limit n⃗∞. The nonlinearity of E(J) gives rise to new length scales for E(x,y) and J(x,y) distributions, strong local enhancement of E(x,y) and long-range electric-field disturbances around planar defects on the scale L_⊥∼an much greater than the defect size a. For instance, a planar defect of length a>d/n in a film of thickness d produces a narrow (∼d/√n) magnetic-flux jet (domain of high electric field), which spans the entire current-carrying cross section. As a result, even small defects (a∼d/n), which occupy only a small fraction of the geometrical cross section, give rise to significant peaks of voltage and dissipation. This nonlinear current blockage by planar defects (high-angle grain boundaries, microcracks, etc.) essentially affects the global E-J characteristics and critical currents in superconductors.