Mesoscopic disorder and intrinsic charge instability in oxide heterostructures

After Ohtomo and Hwang detected a high-mobility two-dimensional electron gas (2DEG) at the interface between the two insulating perovskite oxides Strontium Titanate (SrTiO$_3$) and Lanthanum Aluminate (LaAlO$_3$) \cite{Ohtomo2004}, an increasingly intense theoretical and experimental investigation has been devoted to these systems. Oxide heterostructures are of great interest for both fundamental and applicative reasons. In particular, the two-dimensional electron gas at the LaAlO$_3$/SrTiO$_3$ (LAO/STO) interfaces displays many different properties and functionalities. It can be made superconducting when its carrier density is tuned by means of a gate voltage \cite{Reyren2007,Caviglia2008}, thus opening the way to voltage-driven superconducting devices. Also, it exhibits magnetic properties \cite{Ariando2011,Li2011,Bert2011,Dikin2011,Mehta2012,Bert2012}; displays a strong and tunable \cite{Caviglia2010,Caprara2012} Rashba spin-orbit coupling; it is extremely two-dimensional, having a lateral extension $\approx \SI{5}{\nano\meter}$, thereby enhancing the effect of disorder due to extrinsic and/or intrinsic sources. Similar results hold true also for other oxides such as LaTiO$_3$/SrTiO$_3$ \cite{Biscaras2010,Biscaras2012}, where the role of Lanthanum Aluminate is taken by Lanthanum Titanate LaTiO$_3$ (LTO). In these kind of systems, superconductivity is possibly related to the presence of high-mobility carriers (HMC) and low-mobility carriers (LMC), which presence is revealed by magneto-transport experiments, and seems to develop as soon as high-mobility carriers appear \cite{Biscaras2012}, when the carrier density is tuned above a threshold value by means of gate voltage, V$_g$. Along with these intriguing physical properties, there are clear experimental indications that the interface electronic state is strongly inhomogeneous, as revealed in various magnetic experiments \cite{Ariando2011,Li2011,Bert2011,Bert2012}, in tunneling spectra \cite{Ristic2011}, and possibly in piezoforce microscopy \cite{Feng2013}. It seems that inhomogeneities at nanometric scale coexist with larger (e.g., micrometric) scale inhomogeneities, revealed by the occurrence of striped textures in the current distribution \cite{Kalisky2013} and in the surface potential \cite{Ilani2013}. The inhomogeneous character of the 2DEG may be responsible of the anomalously large width of the superconducting transition, which is not compatible with any reasonable superconducting fluctuations (e.g., Aslamazov-Larkin contribution to the paraconductivity) \cite{Caprara2011}, but instead it is well accounted for assuming that the 2DEG consists of superconducting ``puddles '' embedded in weakly localizing metallic background. This scenario could also help to explain the measurements carried out at low carrier density, showing a saturation plateau with finite resistance, a clear signature of the percolating character of the metal-to-superconductor transition. \\ \\ The aim of the present thesis is to assemble all these experimental evidences into a coherent overall theoretical framework. The outline is as follows. \\ In chapter \ref{ch:overview} we review some of the experimental facts about LAO/STO and LTO/STO interface, stressing our attention on the peculiar transport properties, both in the normal and in the superconducting state. \\ In chapter \ref{ch:sust} we first provide new, compelling evidences of the inhomogeneous character of the 2DEG. We consider the phenomenological assumption that the electron gas consists of a metallic sea, where only LMC are present, hosting metallic islands, where LMC and HMC coexist and become superconducting below a randomly distributed critical temperature. Basing on this idea, we extend previous multicarrier analyses of magneto-transport measurements to deal with inhomogeneous systems and show how Hall resistivity, superfluid density and tunneling spectra measurements can be explained within this scenario. The connection between superconductivity and the appearing of the high-mobility carriers is also investigated. \\ Chapter \ref{ch:QW} is devoted to formulate a possible justification of the inhomogeneous scenario and the appearing of two type of carriers whose distribution is not uniform in space. The mechanism that we propose is based on the quantum confinement experienced by the 2DEG because of the strong electric field perpendicular to the interface. We will show that such an \emph{intrinsic} mechanism is due to the fact that the confining potential well depends on the electron density, therefore giving rise to a non-rigid band structure which evolves as a function of the electron density too. In a wide range of reasonable parameters, compatible with the experiments, the non-rigidity determines a negative compressibility in the uniform system, which it avoids by phase separation. The densities of the phase separated regions are thus found by Maxwell construction and depend on the doping level. The phase separation scenario also naturally explains the inhomogeneous distribution of low- and high-mobility carriers, and their behavior as a function of the doping level. In chapter \ref{ch:dds} we explore the fluctuations of the superconducting order parameter in the vicinity of the metal-to-superconductor phase transition, that is reached as the gate voltage is changed. In the case of superconductivity, the order parameter has an amplitude and a phase, which can both fluctuate according to well identified scenarios. We address the possibility that a new type of fluctuations occurs in the superconducting 2DEG at LAO/STO and LTO/STO interfaces, with an anomalous dynamics. In particular, we show that the superconducting-to-metal quantum phase transition displays anomalous scaling properties, which can be explained by density driven superconducting critical fluctuations. \\ Finally, the concluding remarks will be given in chapter \ref{ch:conclusion}. We will critically discuss the merits and limitations of our approach and provide a brief outlook on future work