Localized anisotropic superconductors

We address the question of whether an anisotropic gap of dx2−y2 symmetry is compatible with localized states in the normal phase. The issue is important for high-Tc superconductors for which a superconductor-to-insulator transition is observed, together with a number of experiments that support d-wave pairing. We prove that d-wave superconductivity is compatible with a localized normal state. When the coherence length is of the order of the lattice constant, the effects of localization are important. We find a re-entrant behaviour of superconductivity in the strongly disordered phase. There is a growing body of experimental evidence for high-Tc superconductors that indicates that the pairing state is of dx2−y2 symmetry [1]. For superconductors with an anisotropic order parameter, both magnetic and non-magnetic impurities are pair breaking. For d-wave symmetry, the effect of non-magnetic impurities is equivalent to that of magnetic impurities in s-wave superconductors [2]. Perturbation theory for the impurity scattering introduces a mean free path ` for the extended states, and the standard treatment indicates that anisotropic superconductivity is destroyed when ξ0/` = 1/π [3], with ξ0 the coherence length. On the other hand, the charge dynamics for oxide superconductors is basically two dimensional, and it is known from the scaling theory of localization that in two dimensions all one-particle states are localized [4]. This conclusion is unchanged by the presence of electron–electron interaction [4, 5]. In fact, the experimental evidence from resistivity measurements for low levels of doping is consistent with a divergent resistivity as T → 0 that is cut off by the superconducting transition at T = Tc [6]. The resistivity shows an upturn at a characteristic temperature Tmin that is apparent when Tc < Tmin. Qualitatively, Tmin corresponds to the temperature scale for which the inelastic scattering length is comparable to the localization length. Conversely, if the elastic mean free path is much bigger than the coherence length, the localization effects are not important, and the variation of Tc with disorder will be given by the usual pair-breaking expressions. For the regime with Tc < Tmin, it is clear that superconductivity becomes established at a temperature low enough for the effects of localization to dominate the normal-state transport properties. The purpose of the present work is to present a treatment of anisotropic superconductivity that incorporates the fact that the states from which the superconducting state is built up are localized, and reconcile two seemingly conflicting properties: the observed insulator–superconductor transition, and anisotropic pairing. We show that, if ξ0/a 1, with a being the lattice constant, superconductivity is destroyed for small values of the disorder, and the localization effects are not important. In this case the critical value of the disorder is such that ξ0 = `/π λ, with λ the localization length. If ξ0/a 0953-8984/98/347587+09$19.50 c © 1998 IOP Publishing Ltd 7587 7588 A G Rojo and C A Balseiro is of order unity—as is the case for the oxide superconductors—when disorder increases, localization effects play a role before superconductivity is destroyed by conventional pairbreaking scattering. In this case, the dependence of the critical temperature on the disorder deviates from the celebrated Abrikosov–Gor’kov–Maki (AGM) theory [3]. We discuss the cases of p-wave and d-wave superconductivity. For concreteness we consider fermions on a lattice described by the following Hamiltonian: H = H0 − U ∑