Nonlinear transport current flow in superconductors with planar obstacles

Abstract

We present a detailed description of a hodograph method, which enables us to calculate analytically the two-dimensional distributions of the electric field $\mathbf{E}(\mathbf{r})$ and transport current density $\mathbf{J}(\mathbf{r})$ in superconductors, taking into account their highly nonlinear $E\ensuremath{-}J$ characteristics. The method gives a unique solution $\mathbf{E}(\mathbf{r})$ of nonlinear steady-state Maxwell's equations for given boundary conditions, showing applicability limits of the critical state model and pointing out where it breaks down. The nonlinear problem of calculation of $\mathbf{J}(\mathbf{r})$ by the hodograph method reduces to solving a linear equation for the electric potential $\ensuremath{\varphi}(\mathbf{E}),$ or the current stream function $\ensuremath{\psi}(\mathbf{E})$ as a function of $\mathbf{E}.$ For the power-law characteristics ${E=E}_{c}{(J/J}_{c}{)}^{n},$ calculation of $\mathbf{E}(\mathbf{r})$ and $\mathbf{J}(\mathbf{r})$ can be mapped onto solutions of the London equation with the inverse screening length $\ensuremath{\beta}=(n\ensuremath{-}1)/2\sqrt{n}$ in the hodograph space ${(E}_{x}{E}_{y}).$ We give general methods of solving the hodograph equations and obtain closed-form analytical solutions for particular current flows. The method is applied to calculate distributions of $\mathbf{E}(\mathbf{r})$ and dissipation in superconductors with macroscopic planar defects, such as high-angle grain boundaries, microcracks, etc. Current patterns around planar obstacles are shown to break up into domains of different orientations of $\mathbf{J},$ separated by current domain walls. We calculate the structure of the current domain walls, whose width depends both on the geometry of current flow and the exponent n. These domain walls differ from the current discontinuity lines of the Bean model even in the limit $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}.$ We obtained a solution for current flow past a planar defect of length $2a$ in an infinite superconductor and showed that the defect causes strong local electric-field enhancement and long-range disturbances of $\mathbf{E}(\mathbf{r})$ on length scales ${L}_{\ensuremath{\perp}}\ensuremath{\sim}\mathrm{an}\ensuremath{\gg}a$ and ${L}_{\ensuremath{\Vert}}\ensuremath{\sim}a\sqrt{n}\ensuremath{\gg}a$ perpendicular and parallel to the mean current flow, respectively. This solution also exhibits large stagnation regions of magnetic flux near planar defects, universal distributions of $\mathbf{J}(\mathbf{r})$ in the critical state limit, $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty},$ and local flux flow regions near the edges. We calculate Joule heating for nonlinear current flow and show that planar defects cause significant excess dissipation, which affects ac losses and local thermal instabilities in superconductors.