Vortices in d-wave superconductors

Vortex state solutions are studied in mean field theory for the simplest microscopic model of unconventional superconductivity, the nearest neighbor tight-binding model on a square lattice with on-site and nearest neighbor interactions, which has d- and extended s-wave mean field ground states for suitable values of the parameters. Vortex states of this model are studied by solving the Bogoliubov-de Gennes equations self-consistently in position space for finite lattices with various boundary conditions. The results of such calculations are interpreted in terms of the appropriate Ginzburg-Landau (GL) free energy, and the GL equations themselves are integrated to test our understanding of the microscopic calculations. The order parameter fields for this problem may be described either as complex functions on horizontal and vertical nearest neighbor bonds or as local linear combinations of these with s- and d-wave symmetry. The d-wave component of a single vortex resembles the vortex order parameter field of a conventional superconductor. However the s-wave component, which is nonzero in a ring around the vortex core and which dies off at long distances like 1r2, has a nontrivial internal structure involving domains and extra point nodes. The effects of a small orthorhombic distortion are discussed as are the implications of our results for scanning tunneling experiments.