Anisotropic superconductivity: (s+d)-wave model, (d+s)-wave model and the effect of normal impurities

We study in three and two dimensions (3D and 2D), some of the superconducting properties of anisotropic hybrid ($s+d$)- and ($d+s$)-wave superconductor models (depending whether the $s$ or $d$ component predominates). The anisotropy is contained in the fermion-fermion pairing interaction as reflecting structural anisotropies of the kind found in layered high-$T_c$ cuprates, in particular the orthorhombic ones. We show that even within the weak-coupling formalism and for an isotropic Fermi surface (so that the interaction has a lower symmetry than the Fermi surface), we obtain enhanced values of $\frac{{\Delta }_M}{T_c}$ (${\Delta }_M$ is the maximum value of the gap energy) compared to the BCS one, as well as enhanced $T_c$ values. Depending on the anisotropy parameters we can get a four-leaf clover kind of shape for the gap in κ space in 2D, with the size of the leaves in one direction being different from one of those in the perpendicular direction. We also find unusual shapes for the density of states; in particular we can recover the "gapless" case similar to the one of the purely $d$-wave model of the literature. We study two kinds of anisotropic pairings: the first one, in 3D and 2D yields an ($s+d$) pairing; the other one, in 2D, yields a ($d+s$) pairing and is more specifically adapted to orthorhombic high-$T_c$ cuprates, the $s$ component disappearing in the tetragonal case where the distortion is absent and for which the purely $d$ pairing is recovered. We then concentrate on this last ($d+s$) model for which we study the effects of normal impurities, with a striking result: compared to the $d$-wave case where the residual density of states on the Fermi surface at $T=0\mathrm{}$ K, increases with the impurity concentration continuously toward the normal state value, in the ($d+s$) case in contrast, the residual density of states abruptly drops to zero for an impurity concentration slightly below the critical one where $T_c$ vanishes; this is accompanied by a change in the symmetry of the order parameter for varying impurity concentration. We suggest that such features might be tested experimentally.