Ginzburg-Landau calculations for a superconducting cylinder in a magnetic field

Abstract

Self-consistent solutions of the Ginzburg-Landau system of nonlinear equations, which describe the behavior of the order parameter $\ensuremath{\psi}$ and the magnetic-field distribution B in a long superconducting cylinder of finite radius R in external magnetic field H, provided that, there are no vortices inside the superconductor, are studied by using numerical method. The lower and upper critical fields of the cylinder, ${H}_{c1}^{(0)}$ and ${H}_{c2}^{(0)},$ are found as functions of the radius R, temperature T, and parameter $\ensuremath{\kappa}$ of the Ginzburg-Landau theory. For type-I superconductors one has ${H}_{c1}^{(0)}{=H}_{c2}^{(0)};$ for type-II superconductors one has ${H}_{c1}^{(0)}<{H}_{c2}^{(0)}.$ In small fields $H<{H}_{c1}^{(0)}$ the superconductor is in stable Meissner phase (with $\ensuremath{\psi}\ensuremath{\sim}1$ and $B\ensuremath{\sim}0).$ It is found, that for type-II superconductors the state with $\ensuremath{\psi}\ensuremath{\sim}1$ is unstable in the fields $H>{H}_{c1}^{(0)},$ and the superconductor passes to a new stable state. In this state the external field begins to penetrate freely into a superconductor in a form of a finite width ring, which is situated near the surface of the cylinder, where the order parameter is strongly suppressed (the rim-suppressed state). The field ${H}_{c1}^{(0)}$ differs from the lower critical field ${H}_{c1},$ at which the field begins to penetrate into the bulk superconductor in a form of vortices. When the field H is increased further, this ring layer (or, the rim) widens, while the order parameter remaines finite $(\ensuremath{\psi}\ensuremath{\ne}0)$ only near the center of the cylinder. In the field ${H=H}_{c2}^{(0)}$ the order parameter finally vanishes everywhere and the metal passes into the normal state. For $R\ensuremath{\gg}\ensuremath{\lambda}$ the field ${H}_{c2}^{(0)}$ coincides with the upper critical field ${H}_{c2},$ at which the mixed vortex state terminates. The intervals of R, T, and $\ensuremath{\kappa},$ where the rim-suppressed state can exist, are found.