Superconductivity and localization

We present a review of theoretical and experimental works on the problem of mutual interplay of Anderson localization and superconductivity in strongly disordered systems. Superconductivity occurs close to the metal-insulator transition in some disordered systems such as amorphous metals, superconducting compounds disordered by fast neutron irradiation, etc. High-temperature superconductors are especially interesting from this point of view. Only bulk systems are considered in this review. The superconductor-insulator transition in purely two-dimensional disordered systems is not discussed. We start with a brief discussion of the modern aspects of localization theory including the basic concept of scaling, self-consistent theory and interaction effects. After that we analyze disorder effects on Cooper pairing and superconducting transition temperature as well as the Ginzburg-Landau equations for superconductors which are close to those for the Anderson transition. A necessary generalization of the usual theory of ''dirty'' superconductors is formulated which allows to analyze anomalies of the main superconducting properties close to the disorder-induced metal-insulator transition. Under very rigid conditions superconductivity may persist even in the localized phase (Anderson insulator). Strong disordering leads to considerable reduction of superconducting transition temperature T-c and to important anomalies in the behavior of the upper critical field H-c2. Fluctuation effects are also discussed. In the vicinity of the Anderson transition, inhomogeneous superconductivity appears due to statistical fluctuations of the local density of states. We briefly discuss a number of experiments demonstrating superconductivity close to the Anderson transition both in traditional and high-T-c superconductors. In traditional systems superconductivity is in most cases destroyed before the metal-insulator transition. In the case of high-T-c superconductors a number of anomalies show that superconductivity is apparently conserved in the localized phase before it is suppressed by a strong enough disorder.