Fractal superconductivity near localization threshold

We develop a semi-quantitative theory of electron pairing and resulting superconductivity in bulk poor conductors in which Fermi energy $E_F$ is located in the region of localized states not so far from the Anderson mobility edge $E_c$. We review the existing theories and experimental data and argue that a large class of disordered films is described by this model. Our theoretical analysis is based on the analytical treatment of pairing correlations, described in the basis of the exact single-particle eigenstates of the 3D Anderson model, which we combine with numerical data on eigenfunction correlations. Fractal nature of critical wavefunctions correlations is shown to be crucial for the physics of these systems. We identify three distinct phases: critical superconductive state formed at $E_F=E_c$, superconducting state with a strong pseudogap, realized due to pairing of weakly localized electrons and insulating state realized at $E_F$ still deeper inside localized band. The critical superconducting phase is characterized by the enhancement of the transition temperature with respect to BCS result, by the inhomogeneous spatial distribution of superconductive order parameter and local density of states. The major new feature of the pseudo-gaped state is the presence of two independent energy scales: superconducting gap $Delta$, that is due to many-body correlations and a new pseudogap energy scale $Delta_P$ which characterizes typical binding energy of localized electron pairs and leads to the insulating behavior of the resistivity as a function of temperature above superconductive $T_c$. Two gap nature of the pseudo-gaped superconductor is shown to lead to a number of unusual physical properties.