Static and dynamical anisotropy effects in the mixed state of d-wave superconductors

We describe effects of anisotropy caused by the crystal lattice in d-wave superconductors with s-wave mixing using the effective free energy approach. Only the d-wave order parameter field $d$ is introduced, while the effect of the s wave mixing as well as other effects breaking the rotational symmetry down to the fourfold symmetry of the crystal are represented by a single four (covariant) derivative term: $\eta d^{*}(\Pi _{y}^{2}-\Pi_{x}^{2})^{2}d $. The single vortex solution in a phenomenologically interesting range of parameters is almost identical to the two order parameters approach. We analytically consider the most general oblique lattice and orientation, but find that only rectangular body centered lattices are realized. A critical value $\eta _{c} $ at which a phase transition from the rectangular lattice to the square lattice takes place. The influence on the phase transition line is discussed. The formalism is extended to the time dependent anisotropic Ginzburg-Landau equations in order to calculate the effect of the anisotropy on the flux flow. The moving vortex structure is established and the magnetization as function of the current is calculated. Although the linear conductivity tensor is rotationally invariant due to the fourfold symmetry, the nonlinear one shows anisotropy. We calculate dependence of both direct and Hall I-V curves on the angle between the current and the crystal lattice orientation.