Modified London Model for Type-II Superconductors

Abstract

A model is presented which modifies the London theory of superconductivity by incorporating solutions of the theory which are usually omitted owing to an infinity at the coordinate axis. These solutions are retained by limiting the current density to the finite critical current density previously calculated by London. The resulting radius at which this value is attained is assumed to be a constant, independent of field, and is related to the coherence distance of the Ginzburg-Landau theory. A circular boundary condition is then employed such that on the boundary the current density vanishes, and the interior of the circle contains one flux quantum. The radius of this circle is field-dependent, leading to the condition that the entire flux quantum is contained as magnetic flux in the normal core when the bounding radius equals that of the fluxoid core. At this field the current density vanishes, defining the upper critical field $H_{c2\mathrm{}}$. The Gibbs energy is then constructed and used to calculate the applied field and magnetization. From this calculation, a simple but rather accurate expression for the lower critical field $H_{c1\mathrm{}}$ is obtained. Considerations of the condensation energy lost in the normal core and in an assumed microscopic hole in the superconductor show that the energy is lower when a fluxoid is located over the hole, so that the fluxoid is pinned; one can also make an approximate calculation of the transport current density.