Thin superconductors in a perpendicular magnetic ac field. II. Circular disk

The electromagnetic response of a thin type-II superconductor disk of constant thickness d to an applied perpendicular magnetic field is calculated and compared with the response of a thin strip and of bars and cylinders in a longitudinal field. The strip and disk are characterized completely by their sheet resistivity, which may be either nonlinear and static or linear, complex, and frequency dependent. The equations of motion for the sheet current in the strip and disk are derived and discussed. Iterative and numerical-solution methods are presented that account for the infinities of the integral kernel and of the ideal screening current. The linear Ohmic response, realized during flux flow or thermally activated flux flow, is calculated for a jump in the applied field and for an ac field. Simple approximate expressions for the complex ac susceptibilities of an Ohmic strip and disk are given and compared with longitudinal geometry. The penetration of perpendicular flux into an Ohmic strip or disk is not a usual diffusion since a logarithmic infinity of the prpendicular-field component occurs at the edges at all times t and the sheet current always flows over the entire surface. Near the edges, the sheet current has a universal profile, with a maximum which initially penetrates with constant velocity v=0.77D/d and decreases as 1/ √t (D=ρ/${\mathrm{\mu }}_0$, where ρ is the resistivity). At large times t≫${\mathrm{\tau }}_0$ the current and magnetic moment decrease as exp(-t/${\mathrm{\tau }}_0$), where ${\mathrm{\tau }}_0$=0.2492ad/D for a strip with half width a and ${\mathrm{\tau }}_0$=0.1815ad/D for a disk with radius a.