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 da) when either the penetration depth obeys λ<d/2 or, if λ>d/2, the two-dimensional screening length obeys Λ=2λ²/da for the following cases: (a) magnetic flux Φz(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 Hₐ, and (c) focusing of magnetic flux into the hole when a magnetic field Hₐ 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.