Critical state in disk-shaped superconductors

Abstract

We have calculated the magnetic fields and currents occurring in a disk-shaped superconductor (radius ≫thickness) in the critical state in a self-consistent way using finite-element analysis. We find that the field shielded (or trapped) in the center of the disk is roughly equal to $J_c$d, where d is the thickness of the disk. The shielding currents also create radial fields which are or order $J_c$d/2 on the disk surface. For low applied fields $H_{\mathrm{appl}<J_c\mathrm{d}}$ these self-field effects dominate, leading to a deviation of the local field direction from the applied field, which can exceed 90 deg towards the outside perimeter of the disk. If $J_c$d is large, as is the case for ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ single crystals at 4.2 K, self-field effects persist up to several telsa applied field. The field dependence of the calculated magnetic moment in the self-field dominated regime is independent of whether $J_c$ is weakly or strongly (∝1/H) dependent on field. The calculations were validated by comparison to both magnetic and resistive measurements on a disk-shaped section in ${\mathrm{Nb}}_3$Sn tape.