The object of our investigation are layered and two-dimensional (2D) materials, such as cuprate and pnictide high-temperature superconductors, superconducting oxide interfaces, like LaAlO3/SrTiO3, disordered superconducting thin films, like InO, TiN, NbN, and transition metal dichalcogenides (TMDs), and graphene. Upon variation of the carrier density and/or disorder, these systems exhibit various novel phenomena, including a superconductor-to-insulator transition (SIT). Moreover, as a result of the competition between superconductivity and other phases (magnetic, charge-ordered, ...), many of these systems are characterized by a rich variety of intriguing behaviors, and their responses to different weak perturbations can be unusually sizable, opening the way to their exploitation in device engineering. In many cases, a major issue is the occurrence of an intrinsically inhomogeneous state: In cuprates, this state is characterized by dynamical or static charge ordering. The mechanism for the occurrence of CDWs and their role in determining the high critical temperature, the possible occurrence of filamentary superconductivity, and the anomalous properties of the metallic phase are central issues of this research field and will be investigated within the present project. The two-dimensional electron gas (2DEG) formed at oxide interfaces exhibits an inhomogeneous, likely rather filamentary, superconducting state, possibly realizing the long-sought quantum Griffiths phase, with rare regions affecting the properties of the system near the SIT. In oxide interfaces, TMDs, and disordered thin films, the inhomogeneity also affects the superfluid response, the current-voltage characteristics, and the properties of the magnetic-field-induced vortex lattice, calling for a systematic investigation of these effects. Finally, the investigation of the novel phenomena uncovered in field-effect-doped graphene adds a new chapter to the physics of 2D electron systems .
