Models of Vortices in Anisotropic Superconductors

Abstract

The line energy of a single magnetic vortex at an arbitrary direction in an anisotropic superconductor is evaluated in three different phenomenological models, the anisotropic London model, the anisotropic Ginzburg–Landau model, and the Lawrence–Doniach model. The vortex line energy in the anisotropic London model can be solved by Fourier transformation, but the core cutoff is not present in the model. Using the Klemm–Clem transformations of the free energy and vortex coordinates in the anisotropic Ginzburg–Landau model, a theorem relating the shape of the effective core cross-section in the anisotropic London model to that in the anisotropic Ginzburg–Landau model is presented. This theorem allows the anisotropic London model to be used in the entire regime $H_{c1} \leq H \ll H_{c2} $, regardless of the anisotropy and field direction. The resulting lower critical field $H_{c1} ( \theta _H )$ can exhibit a discontinuity, a kink, or be monotonic in its angular dependence. In addition, the transformations are applied to the case of a core cross-section of a single vortex in the Lawrence–Doniach model, and the angular dependence of the magnetic torque is calculated. The torque exhibits the expected dimensional crossover, having an angular dependence which reduces to that for an anisotropic Ginzburg–Landau superconductor very near to $T_c $, and to that for a lattice of coupled, parallel Josephson junctions well below the dimensional crossover temperature $T^ * $. In the vicinity of $T^ * $, however, dramatic oscillations in the angular dependence of the torque are predicted.