Surface barrier in mesoscopic type-I and type-II superconductors

We study the surface barrier for magnetic-field penetration in mesoscopic samples of both type-I and type-II superconductors. Our results are obtained from numerical simulations of the time-dependent Ginzburg-Landau equations. We calculate the dependence of the first field for flux penetration ${(H}_p)$ with the Ginzburg-Landau parameter $\left(\kappa \right)$ observing an increase of $H_p$ with decreasing κ for a superconductor-insulator boundary condition $\left[\right(\mathbf{\nabla }-iA)\Psi {|}_n=0\mathrm{}]$ while for a superconductor-normal boundary condition (approximated by the limiting case of $\Psi {|}_S=0)$ $H_p$ has a smaller value independent of κ and proportional to $H_c.$ We study the magnetization curves and penetration fields at different sample sizes and for square and thin-film geometries. For small mesoscopic samples we study the peaks and discontinuous jumps found in the magnetization as a function of magnetic field. To interpret these jumps we consider that vortices located inside the sample induce a reinforcement of the surface barrier at fields greater than the first penetration field $H_{p1\mathrm{}}.$ This leads to multiple penetration fields $H_{\mathrm{pi}}{=H}_{p1\mathrm{}}{,H}_{p2\mathrm{}}{,H}_{p3\mathrm{}},\dots$ for vortex entrance in mesoscopic samples. We study the dependence on sample size of the penetration fields $H_{\mathrm{pi}}.$ We explain these multiple penetration fields, extending the usual Bean-Livingston analysis by considering the effect of vortices inside the superconductor and the finite size of the sample.