Ac magnetic response of mesoscopic type-II superconductors

Abstract

The response of mesoscopic superconductors to an ac magnetic field is numerically investigated on the basis of the time-dependent Ginzburg-Landau equations. We study the dependence with frequency $\ensuremath{\omega}$ and dc magnetic field ${H}_{\mathrm{dc}}$ of the linear ac susceptibility $\ensuremath{\chi}{(H}_{\mathrm{dc}},\ensuremath{\omega})$ in square samples with dimensions of the order of the London penetration depth. At ${H}_{\mathrm{dc}}=0$ the behavior of $\ensuremath{\chi}$ as a function of $\ensuremath{\omega}$ agrees very well with the two-fluid model, and the imaginary part of the ac susceptibility, ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega}),$ shows a dissipative maximum at the frequency ${\ensuremath{\nu}}_{o}{=c}^{2}/(4\ensuremath{\pi}\ensuremath{\sigma}{\ensuremath{\lambda}}^{2}).$ In the presence of a magnetic field a second dissipation maximum appears at a frequency ${\ensuremath{\omega}}_{p}\ensuremath{\ll}{\ensuremath{\nu}}_{0}.$ The most interesting behavior of mesoscopic superconductors can be observed in the $\ensuremath{\chi}{(H}_{\mathrm{dc}})$ curves obtained at a fixed frequency. At a fixed number of vortices, ${\ensuremath{\chi}}^{\ensuremath{''}}{(H}_{\mathrm{dc}})$ continuously increases with increasing ${H}_{\mathrm{dc}}.$ We observe that the dissipation reaches a maximum for magnetic fields right below the vortex penetration fields. Then, after each vortex penetration event, there is a sudden suppression of the ac losses, showing discontinuities in ${\ensuremath{\chi}}^{\ensuremath{''}}{(H}_{\mathrm{dc}})$ at several values of ${H}_{\mathrm{dc}}.$ We show that these discontinuities are typical of the mesoscopic scale and disappear in macroscopic samples, which have a continuous behavior of $\ensuremath{\chi}{(H}_{\mathrm{dc}}).$ We argue that these discontinuities in $\ensuremath{\chi}{(H}_{\mathrm{dc}})$ are due to the effect of nascent vortices which cause a large variation of the amplitude of the order parameter near the surface before the entrance of vortices.