Filamentarity and inhomogeneities in low dimensional superconductors

In this Thesis, I investigate the problem of the interplay between disorder and low dimensionality in superconductors. From the microscopic point of view, I show that the presence of impurities in the superconducting condensate can produce a pairbreaking effect at the Lifshitz transition in multibands superconductors, avoiding or at least circumventing Anderson’s Theorem. This is consistent with the observed suppression of the superconducting critical temperature Tc in SrTiO3 -based heterostructures as a function of the gate potential Vg . This study allows us to disentangle microscopic from mesoscopic disorder in SrTiO3 -based interfaces: microscopic impurities are in fact necessary to explain the suppression of Tc observed when multiband superconductivity is involved; the global behavior is instead well captured only if the strongly inhomogeneous nature of such compounds is considered. Indeed, disorder can also appear on a mesoscopic length scale. The electronic condensate can in fact segregate into regions large enough to define a local phase but small compared with the sample. The reasons behind this inhomogeneity of the superconducting order parameter can be several, depending on the system in exam: in oxides heterostructures it may be connected to the thermodynamic instability caused by the electrostatic potential confining the electron gas at the interface, in transition metal dichalcogenides the instability may be introduced by the gating ionic liquid, whereas sometimes the phase separation can be caused by the presence of microscopic impurities in addition with the competition of superconductivity with another order parameter, e.g., the charge ordering in the case of cuprates. Such a phase separation may appear as a filamentary structure. Concerning mesoscopic inhomogeneities, I will focus on two different materials. In SrTiO3 -based eterostructures, where filamentary superconductivity is embedded in a metallic matrix, I study the consequences of filamentarity in transport properties, assuming a priori a fractal-like organization of the electronic condensate and calculating the complex conductance with a Random Impedance Network model. The geometry of the filamentary structure plays a crucial role, especially in the behavior of the superfluid stiffness as a function of the temperature. Although the motivation of this work is connected to resonant microwave experiments performed on a LaAlO3 /SrTiO3 interface, the results are quite general and can in principle be applied to other materials displaying filamentary superconductivity. In cuprates, I focus on the phase competition as the most likely reason of filamentarity. The charge ordering-superconducting (CO-SC) competition is studied by means of Monte Carlo simulations within an anisotropic Heisenberg model accounting for the basic physical symmetries involved, the out-of-plane pseudospin component mapping two possible charge density waves (CDW) variants while the in-plane component standing for the SC order parameter. The anisotropy term α is taken as the control parameter, tuning the transition from Berezinskii-Kosterlitz-Thouless (BKT) to the charge ordered state. The phase diagram T c vs α is studied both in the clean case and in a random field, the presence of microscopic impurities being necessary to stabilize the clustering of charge ordered domains and the appearance of filamentary superconductivity.