Magnetic-field and current-density distributions in thin-film superconducting rings and disks

We show how to calculate the magnetic-field and sheet-current distributions for a thin-film superconducting annular ring (inner radius a, outer radius b, and thickness d≪a) when either the penetration depth obeys λ<d/2 or, if λ>d/2, the two-dimensional screening length obeys Λ=2λ²/d≪a for the following cases: (a) magnetic flux Φ_z(a) trapped in the hole in the absence of an applied magnetic field, (b) zero magnetic flux in the hole when the ring is subjected to an applied magnetic field H_a, and (c) focusing of magnetic flux into the hole when a magnetic field H_a is applied but no net current flows around the ring. We use a similar method to calculate the magnetic-field and sheet-current distributions and magnetization loops for a thin, bulk-pinning-free superconducting disk (radius b) containing a dome of magnetic flux of radius a when flux entry is impeded by a geometrical barrier.