We investigate theoretically and experimentally the statistical properties of the inhomogeneous order-parameter distribution (OPD) at the verge of the superconductor-insulator transition (SIT). We find within two prototype fermionic and bosonic models for disordered superconductors that one can identify a universal rescaling of the OPD. By performing scanning-tunneling microscopy experiments in three samples of NbN with increasing disorder we show that such a rescaling also describes the experimental data with excellent accuracy. These results can provide a breakthrough in our understanding of the SIT

Oxide heterostructures are of great interest both for fundamental and applicative reasons. In particular the two-dimensional electron gas at the LaAlO3/SrTiO3 or LaTiO3/SrTiO3 interfaces displays many different physical properties and functionalities. However there are clear indications that the interface electronic state is strongly inhomogeneous and therefore it is crucially relevant to investigate possible intrinsic electronic mechanisms underlying this inhomogeneity. Here the electrostatic potential confining the electron gas at the interface is calculated self-consistently, finding that the electron confinement at the interface may induce phase separation, to avoid a thermodynamically unstable state with a negative compressibility. This provides a generic robust and intrinsic mechanism for the experimentally observed inhomogeneous character of these interfaces.PACS numbers: 73.43.Nq,73.21.Fg, The two-dimensional electron gas (2DEG) that forms at the interface of two insulating oxides, like LaAlO 3 /SrTiO 3 and LaTiO 3 /SrTiO 3 (hereafter generically referred to as LXO/STO) [1][2][3][4], exhibits a rich phenomenology, such as a gate-tunable metal-tosuperconductor transition [5][6][7][8], a magnetic-field-tuned quantum criticality [9], and inhomogeneous magnetic responses [10][11][12][13][14][15]. Tunneling [16,17] and SQUID magnetometry [18] provide clear evidence of an inhomogeneous interface on both micro-and nanoscopic scales. Transport measurements report further signs of inhomogeneity and a percolative metal-to-superconductor transition with a sizable fraction of the 2DEG never becoming superconducting down to the lowest accessible temperatures [19][20][21][22]. For both fundamental reasons and applicative purposes, like device design, it is crucial to identify possible intrinsic mechanisms that may render the 2DEG so strongly inhomogeneous via a phase separation (PS). This is precisely the focus of the present work.Here, we identify a very effective electron-driven mechanism leading to PS, based on the confinement of the 2DEG at the interface. From customary self-consistent calculations of the confining potential well in semiconductors, it is well known [23] that a finite lateral extension usually renders the 2DEG more compressible than its strictly 2D counterpart. This effect is much stronger in LXO/STO than in ordinary semiconductor interfaces, due to the huge dielectric constant of STO, allowing for much larger electron densities, with a strong amplification of the self-consistent adjustments of the confining potential. As a consequence, a non-rigid band structure arises, which varies with the local electron density: an increased electron density is accompanied by a corresponding increase of the positive countercharges (due to oxygen vacancies and/or polarity catastrophe [24][25][26][27][28][29]), from which the interfacial electrons are introduced and restoring the overall charge neutrality. For small-to-moderate increases of electron and countercharge densities the potential well deepens and the e...

The two-dimensional electron gas at the LaTiO3/SrTiO3 or LaAlO3/SrTiO3 oxide interfaces becomes superconducting when the carrier density is tuned by gating. The measured resistance and superfluid density reveal an inhomogeneous superconductivity resulting from percolation of filamentary structures of superconducting "puddles" with randomly distributed critical temperatures, embedded in a nonsuperconducting matrix. Following the evidence that superconductivity is related to the appearance of high-mobility carriers, we model intrapuddle superconductivity by a multiband system within a weak coupling BCS scheme. The microscopic parameters, extracted by fitting the transport data with a percolative model, yield a consistent description of the dependence of the average intrapuddle critical temperature and superfluid density on the carrier density

According to the Goldstone theorem the breaking of a continuous U(1) symmetry comes along with the existence of low-energy collective modes. In the context of superconductivity these excitations are related to the phase of the superconducting (SC) order parameter and for clean systems are optically inactive. Here we show that for strongly disordered superconductors phase modes acquire a dipole moment and appear as a subgap spectral feature in the optical conductivity. This finding is obtained with both a gauge-invariant random-phase approximation scheme based on a fermionic Bogoliubov-de Gennes state as well as with a prototypical bosonic model for disordered superconductors. In the strongly disordered regime, where the system displays an effective granularity of the SC properties, the optically active dipoles are linked to the isolated SC islands, offering a new perspective for realizing microwave optical devices.

We propose a model for the two-dimensional electron gas formed at the interface of oxide heterostructures that includes a Rashba spin-orbit coupling proportional to an electric field oriented perpendicularly to the interface. Taking into account the electron density dependence of this electric field confining the electron gas at the interface, we report the occurrence of a phase separation instability (signaled by a negative compressibility) for realistic values of the spin-orbit coupling and of the electronic band-structure parameters at zero temperature. We extend the analysis to finite temperatures and in the presence of an in-plane magnetic field, thereby obtaining two phase diagrams which exhibit a phase separation dome. By varying the gating potential the phase separation dome may shrink and vanish at zero temperature into a quantum critical point where the charge fluctuates dynamically. Similarly the phase separation may be spoiled by a planar magnetic field even at zero temperature leading to a line of quantum critical points.

Motivated by recent experimental data on thin film superconductors and oxide interfaces, we propose a random-resistor network apt to describe the occurrence of a metal-superconductor transition in a two-dimensional electron system with disorder on the mesoscopic scale. We consider low-dimensional (e.g. filamentary) structures of a superconducting cluster embedded in the twodimensional network and we explore the separate effects and the interplay of the superconducting structure and of the statistical distribution of local critical temperatures. The thermal evolution of the resistivity is determined by a numerical calculation of the random-resistor network and, for comparison, a mean-field approach called effective medium theory (EMT). Our calculations reveal the relevance of the distribution of critical temperatures for clusters with low connectivity. In addition, we show that the presence of spatial correlations requires a modification of standard EMT to give qualitative agreement with the numerical results. Applying the present approach to an LaTiO 3 /SrTiO 3 oxide interface, we find that the measured resistivity curves are compatible with a network of spatially dense but loosely connected superconducting islands.

Several experiments reveal the inhomogeneous character of the superconducting state that occurs when the carrier density of the two-dimensional electron gas formed at the LaXO3/SrTiO3 (X=Al or Ti) interface is tuned above a threshold value by means of gating. Re-analyzing previous measurements, that highlight the presence of two kinds of carriers, with low and high mobility, we shall provide a description of multi-carrier magneto-transport in an inhomogeneous two-dimensional electron gas, gaining insight into the properties of the physics of the systems under investigation. We shall then show that the measured resistance, superfluid density, and tunneling spectra result from the percolative connection of superconducting "puddles" with randomly distributed critical temperatures, embedded in a weakly localizing metallic matrix. We shall also show that this scenario is consistent with the characteristics of the superconductor-to-metal transition driven by a magnetic field. A multi-carrier description of the superconducting state, within a weak-coupling BCS-like model, will be finally discussed.

We present a theory for the pseudo-gap state recently observed at the LaAlO3/SrTiO3 interface, based on superconducting islands embedded in a metallic background. Superconductivity within each island is BCS-like, and the local critical temperatures are randomly distributed, some of them necessarily exceeding the critical temperature for global percolation to the zero resistance state. Consequently, tunneling spectra display a suppression of the density of states and coherence peaks already well above the percolative transition. This model of inhomogeneous superconductivity accounts well for the experimental tunneling spectra. The temperature dependence of the spectra suggests that a sizable fraction of the metallic background becomes superconducting by proximity effect when the temperature is lowered.PACS numbers: 74.78.Fk, 74.55.+v, 74.20.De The observation of a two-dimensional (2D) metallic state at the interface of two insulating oxides [1][2][3][4], and the subsequent demonstration of its gate-tunable metal-to-superconductor transition [5][6][7][8], have attracted much attention in the last decade. Numerous experiments indicate that the 2D electron gas (EG) is inhomogeneous: Transport measurements reveal a large width of the superconducting (SC) transition, suggesting charge inhomogeneity [9][10][11][12], and, at lower carrier density, a saturation to a plateau with finite resistance, which is a clear signature of the percolating character of the metal-to-superconductor transition. Magnetometry [13][14][15][16][17][18], tunneling [19], and piezo-force spectroscopy [20] experiments report submicrometric inhomogeneities. It seems likely that inhomogeneities at nanometric scales [21] coexist with structural inhomogeneities at micrometric scales [22]. Into this picture arguably enter the recent superconductor-insulator-metal tunnel spectroscopy measurements [23], that detect a state with finite resistance, but SC-like density of states (DOS). The experiments are carried out in LaAlO 3 /SrTiO 3 (LAO/STO) interfaces, by depositing a metallic Au electrode on the insulating LAO layer, and then driving a tunnel current I (by means of a bias voltage V ) between the electrode and the 2DEG. The size of the electrode measures several hundreds µm (orders of magnitude larger than the inhomogeneities [21]). The carrier density of the 2DEG is tuned by means of a back-gating voltage V G across the STO slab. At very low temperature, T = 30 mK, the measurements reveal a gap in the DOS at the Fermi energy (E F ) over the whole range V G ∈ [−300, 300] V, accompanied by more or less pronounced coherence peaks above the gap, signaling SC coherence and pairing as the origin of DOS suppression. In the carrier depleted regime (V G < 0), the suppression occurs even when global superconductivity is absent down to the lowest accessible temperatures, thereby highlighting the inhomogeneous character of the state formed by regions with SC pairing embedded in the non-SC matrix; the DOS at E F is diminished and the coherence peaks are bro...