On the Fourfold Symmetry of H_c2 and Magnetization of d_x2-y2-Wave Superconductors for the in-Plane Magnetic Fields

For the d_x2-y2 superconductors, we investigate the fourfold symmetry of the upper critical fields and magnetization related to the torque under the parallel magnetic field to the ab plane of the crystal. We take into account the s-wave order parameter induced from the spatial variation of the d_x2-y2-wave order parameter. In addition, we investigate the effects of the shape of the Fermi surface. If the pairing interaction for the s-wave channel is attractive, the s-wave component can enhance the above fourfold symmetry. Depending on the structure of the Fermi surface, the fourfold symmetric anisotropy in the ab plane enhances or suppresses the fourfold symmetric behavior. There is also the case where the fourfold symmetric anisotropy of in the ab plane changes the sign of the fourfold symmetric behavior from that of the isotropic case.