Introduction

General scaling arguments indicate that metallic states are absent at zero temperature in weakly interacting and weakly disordered two-dimensional (2D) electron systems1. Nevertheless, apparent metallic ground states in proximity to 2D superconductivity have been observed in various experiments2,3,4,5,6,7,8,9. Thus the existence and nature of the 2D quantum (zero-temperature) metal has been a matter of long-standing debate10,11,12,13,14,15. At the electrostatically gated LaAlO3/SrTiO3 interface 2D superconductor16,17, a superconducting dome is formed with the appearance of a pseudogap on the negative gating side18,19,20. Interestingly, metallic behaviour emerges upon quenching of the 2D superconductor, an aspect that has generally not been emphasized in previous reports on LaAlO3/SrTiO3. The coexistence of metallic and pseudogap behaviour suggests that further examination of the connection between the superconducting and metallic ground states would be insightful. This calls for experiments that can independently manipulate the key parameters of disorder and carrier density. Previous experiments, using a single gate from the back of the undoped SrTiO3 substrate, simultaneously influences both the carrier density and interface scattering from the tail of the electron distribution envelope19,21,22, making a disentanglement of singly gated behaviour difficult.

Here we implement dual electrostatic gates, providing systematic control over the effective interfacial disorder, carrier distribution, and density, enabling a wide-range mapping of the low-temperature phase diagram.

Results

Operation of the dual-gate device

First, we describe the fundamental operation of the dual electrostatic gates21 (structure shown in Fig. 1a), which are used to manipulate the electrostatic boundary conditions of the asymmetric quantum well within SrTiO3 (Fig. 1b, c). The confinement potential is bounded sharply at the interface by the wide-gap insulator LaAlO3 but extends gradually into the quantum paraelectric SrTiO3 following an electric field-dependent dielectric function. Carriers close to the interface experience higher scattering, compared to carriers extending further into the clean SrTiO3 substrate. Voltages applied to the top-gate (VTG) predominantly control the density of carriers confined close to the interface, but the electronic thickness remains nearly constant (Fig. 1b). When VTG is decreased, the overall scattering is reduced due to the decrease of carriers distributed close to the interface, giving rise to an increase of mobility for VTG > 0.6 V in Fig. 1d (Below 0.6 V in Fig. 1d, localization behaviour appears at low carrier density, along with reduced mobility.). In contrast, the back-gate primarily controls the tail of the quantum well extending into the SrTiO3 substrate. With decreasing back-gate voltage (VBG), the thickness of the conduction layer decreases dramatically such that the narrowly confined carriers are strongly scattered by the interfacial disorder (Fig. 1c) and the mobility decreases (Fig. 1e). Importantly, the large and strongly electric field-dependent dielectric response of SrTiO3 greatly enhances the dual-gate tunability compared to conventional semiconductor heterostructures23. A complete presentation of dual-gating normal-state transport and Poisson–Schrödinger simulations in the presence of the nonlinear dielectric response, can be found in ref. 21.

Fig. 1
figure 1

Operation of the dual-gate device. a The dual-gate device is formed by applying a Au electrode on top of the eight unit-cell epitaxial LaAlO3 layer and a Ag electrode from the back of the SrTiO3 substrate with 0.5 mm thickness. The channel width is 400 μm with Hall bars patterned by predeposited AlOx hard mask. b, c Schematic diagram of the dual-gate operation for top- and back-gates, respectively. The thin lines represent the electron envelope wavefunction perpendicular to the interface at different gate voltages colour-matched to the thick solid lines, which represent the corresponding confinement potential. d, e Top- and back-gate modulation of carrier density (circles) and Hall mobility (squares) at T = 600 mK. The straight solid black lines are linear fits to the circles. The red and blue curves are estimates of mobility based on the measured sheet resistivity and fitted extrapolation of Hall density. The ranges of the solid lines correspond to the experimentally accessible tuning ranges

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Top-gate modulation of the interface ground state

Figure 2a plots the sheet resistivity versus temperature (R–T) curves with varying VTG and fixed VBG = 0 V. In the low VTG regime, the resistivity upturns slightly as temperature decreases, which may indicate an initial tendency toward localization. With increasing VTG, the resistivity at T = 400 mK (denoted as RN) decreases and crosses h/e2 ~ 26 kΩ. In this regime, the R–T curves are nearly temperature independent. We note that this behaviour is rather different from metal–insulator transitions in conventional semiconductor systems14,15. When RN falls below ~2 kΩ, the resistivity exhibits a drop with decreasing temperature, followed by a saturating finite resistance approaching zero temperature (also Fig. 2a inset). The saturating resistivity at the lowest measured temperature can be as much as two orders of magnitude lower than RN. Upon further increasing VTG, the resistivity drops below the measurement noise limit, indicating macroscopic superconductivity. We take here a functional definition of the superconducting transition temperature (TC) as the temperature at which the sheet resistivity drops below 1% of RN. Importantly, we find that the R–T curves exhibit a two-step feature upon transitioning into the superconducting state. These sequential resistive drops with decreasing temperature can be identified by peaks in the second derivative of the R–T curves (denoted as TP and TF in Fig. 2b). The top-gate tuned phase diagram with four distinct regimes is thus obtained by taking TP, TF, and TC as boundaries as shown in Fig. 2c. We will discuss the assignment of these states in detail below, but note here that the boundaries of the phase diagram extrapolate to zero temperature, implying different ground states.

Fig. 2
figure 2

Top-gate modulation of the interface ground state. a Sheet resistivity versus temperature (R–T) curves as a function of VTG from −0.36 to 1.80 V with fixed VBG = 0 V. Inset: magnification of the data plotted against inverse temperature. b The second-order derivative calculated using a spline fit of the R–T curves shown in a for VTG from 0.34 to 1.80 V. Red circles and blue squares indicate the local maxima of the peaks, defining TP and TF. Curve colours are matched between a and b. Dashed lines are guides to the eye. c Phase diagram extracted from a and b. Red circles, blue squares, and blue triangles represent TP, TF, and TC, respectively. Normal state here indicates a non-superconducting state. Error bars are defined by the full width of the second derivative peaks for TP and TF. TC is defined as the temperature at which the sheet resistivity drops to 1% of the normal-state resistivity RN (400 mK). The lowest temperature measured for our R–T curves is 40 mK, so data points estimated to be <40 mK are plotted with arrows. Background colours are guides to the eye indicating the different regimes

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Back-gate modulation of the interface ground state

The counterpart R–T curves and phase diagram under varying VBG at fixed VTG = 1.8 V are shown in Fig. 3. Following the analysis of the second derivative of the R–T curves (Fig. 3a, b), a phase diagram is obtained as a function of VBG (Fig. 3c). Note that back-gate modulation produces a rather different phase diagram from the top-gate case (shown previously in Fig. 2): TP monotonically decreases with VBG; both TF and TC are non-monotonic, and TC exhibits a complete dome; on both sides of the dome, the resistivity saturates at finite values approaching zero temperature19. A key finding here is the deviation between TP and TC on the negative back-gate side of the dome. The behaviour of TP and TC is qualitatively consistent across the dome with the pseudogap observations from tunneling spectroscopy20. This implies a correspondence between TP and gap opening.

Fig. 3
figure 3

Back-gate modulation of the interface ground state. a Sheet resistivity versus temperature (R–T) curves as a function of VBG from −46 to 90 V with fixed VTG = 1.8 V. b The second-order derivative calculated using a spline fit of the R–T curves shown in a. Red circles and blue squares indicate the local maxima of the peaks, defining TP and TF. Curve colours are matched between a and b. c Phase diagram for back-gating with fixed VTG = 1.8 V. Red circles, blue squares, and blue triangles represent TP, TF, and TC, respectively. Normal state here indicates a non-superconducting state. Error bars are defined by the full width of the second derivative peaks for TP and TF. TC is defined as the temperature at which the sheet resistivity drops to 1% of the normal state RN (400 mK). The lowest temperature measured for our R–T curves is 40 mK, so data points estimated to be <40 mK are plotted with arrows. Background colours are guides to the eye indicating the different regimes

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Discussion

These results demonstrate that a variety of emergent ground states are achievable solely by electrostatic gates. To understand their nature, we first discuss the different length scales for disorder, inhomogeneity, and superconductivity at the interface. Owing to finite interdiffusion, the interface where the 2D superconductor resides has disorder on a range comparable to the lattice constant20,24. This disorder length scale (~1 nm) is much smaller than either the low-temperature electron mean free path or the Ginzburg–Landau superconducting coherence length of the system (both ~100 nm). Thus the interdiffusion-induced disorder can be considered homogeneous on the relevant electronic lengths. Despite this, a highly inhomogeneous, patchy superfluid density with a typical length scale of micrometers has been observed in scanning probe measurements25. We associate this patchy superfluid density to the formation of superconducting puddles at the interface. The only other feature of the system with a similar large length scale is the domain structure in SrTiO3 that forms below the cubic to tetragonal phase transition at 105 K. However, the spatial distribution of the superfluid density appears decoupled from the tetragonal domain boundaries, which follows high symmetry directions25,26.

Therefore, we conclude that the formation of superconducting puddles here is a fundamental and emergent consequence of disorder. This is consistent with the results of computational studies indicating that microscopic disorder in 2D naturally gives rise to phase separation and thus emergent mesoscopic superconducting puddles27,28. Hence, we expect that inter-puddle phase coupling should play an essential role. In addition, we note that the features in the R–T data are remarkably similar to experiments of isolated superconducting islands on thin non-superconducting metallic films, where global phase coherence is mediated by inter-island (puddle) Josephson coupling3,5,7,9. Based on the combined analysis of magnetoresistance and current–voltage characteristics shown in Supplementary Figure 2, we interpret the first drop in resistance as the onset of pairing and formation of local emergent superconducting puddles (TP: pairing temperature), consistent with the opening of a pseudogap20. With decreasing temperature, a second drop in resistivity occurs with the onset of long-range phase coupling limited by fluctuations (TF: onset of macroscopic phase fluctuations). Further lowering temperature, the formation of global phase coherence leads to macroscopic 2D superconductivity (TC: defined here as 1% RN).

Extrapolating the characteristic temperatures (i.e. TP, TF, and TC) to zero, we can now construct the full gate-dependent phase diagram as shown in Fig. 4a, representing gate-tuned ground states. From left to right in Fig. 4a, we see that the phase diagram crosses a rich set of distinct states: a normal (non-superconducting) state without apparent pairing; a local superconducting puddle state without inter-puddle phase coupling; a phase fluctuating state in which macroscopic coherence is impaired by quantum fluctuations; and a global phase coherent 2D superconducting state. In particular, the system goes through a superconductor–metal transition when phase fluctuations destroy global coherence (across the dotted line). The dual electrostatic gates systematically control not only the interface carrier density but also the carrier distribution and the associated strength of interface disorder. VTG monotonically drives the proportional rise of TP, TF, and TC (Fig. 2c). This indicates that the increase of interfacial carrier density by VTG turns on superconducting pairing in local puddles and the interface progressively transitions into a macroscopic superconducting phase only after the puddles are fully phase coupled. On the other hand, the back-gate predominantly controls the sampling of interfacial disorder. This is highlighted by the diverging relationship between TP and TC on the negative side of the back-gate dome (Fig. 3c). With decreasing VBG, the associated increase of the effective disorder induces stronger phase fluctuations among the puddles, leading to the collapse of TC, although TP is increasing and remains finite. While the pairing strength (TP) could be related to various factors including a gate-tuned spin–orbit coupling29, the 2D superconductor–metal transition below TP is governed by macroscopic phase fluctuations.

Fig. 4
figure 4

Ground-state phase diagram by dual gates. a Sheet resistivity mapping as a function of VTG and VBG measured at T = 40 mK. The colour scale is based on interpolated resistivity. Circles and squares are extracted from the peaks of the second-order derivative calculated using a spline fit of the resistivity-versus-VTG curves. Error bars are defined by the full width of the second derivative peaks. Triangles are the gate voltages at which the resistivity drops to 1% of normal-state resistance. The circles, squares, and triangles indicate the voltages where TP, TF, and TC extrapolate to zero temperature, respectively. The dotted line represents a superconductor–metal phase transition line. Normal state here indicates a non-superconducting state. b Schematics of the four different ground states in the phase diagram. The red arrows represent the superconducting phase of individual puddles. N normal state, S superconducting state

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We emphasize here the distinction between the observed metallic state and a finite temperature crossover. In ordinary 2D metal thin films30 that conform to the scaling theory of localization1, the residual resistivity is much less than a quantum of resistance and has very weak temperature dependence. By contrast, the experiment here has a tunable resistivity at the lowest accessible temperature that can range from far below to far above a quantum of resistance (for example, in Fig. 2a), implying the presence of strong interactions. These interactions can create pronounced superconducting phase fluctuations, stabilizing metallic ground states over a wide range of parameters accessed here by dual electrostatic gating.

Finally, we note that many aspects of the phase diagram of Fig. 4 should be quite general for 2D superconductivity in the presence of disorder. The visibility and separation of the fluctuation regimes here are large due to several favourable characteristics. The superconducting gap is small (~40 μeV) such that disorder that is small relative to normal state scattering is still significant with respect to superconductivity. Furthermore, the carrier density is small, such that fluctuations on the scale of the superconducting coherence length are highly relevant. We note that very recent reports of gate-induced 2D superconductivity in boron nitride encapsulated van der Waals materials, such as monolayer WTe2, are in a similar regime of low TC and low carrier density and show a similar two-stage resistive transition for a range of Local superconducting puddles8,31,32. For different materials systems, the relative scale of disorder and density will determine the width and relevance of these intermediate fluctuation regimes, but they should be a generic feature of disordered 2D superconductivity.

In conclusion, we have presented the phase diagrams of a superconducting interface by independent control of carrier density and disorder. We find the onset of global superconductivity of the system is dominated by the long-range phase coherence of preformed superconducting puddles. The destruction of macroscopic phase coherence induces a 2D superconductor–metal transition in all measurable directions approaching zero temperature.

Methods

Sample

Ultraviolet photolithography was used to pattern an AlOx Hall bar hard mask (Fig. 1a; channel width 400 μm) on the SrTiO3 substrate. The eight unit-cell epitaxial LaAlO3 was grown with pulsed laser deposition at 800 °C and in 1 × 10–5 Torr O2, after pre-annealing at 950 °C in 5 × 10–6 Torr O2. After growth, the sample was post-annealed in situ at 600 °C for 3 h at 0.2 bar O2; this extended high-temperature post-annealing improved the quality of the LaAlO3 epitaxial layer with respect to stability at high top-gate voltage. The gold top-gate electrode was deposited by electron-beam evaporation with a shadow mask. The silver back-gate electrode was applied by using silver conductive paste. Hall bar contacts were made with ultrasonic Al wire-bonding. In operation, direct current (DC) voltages are applied to the top- and back-gates with respect to the channel.

Magnetotransport measurements

Resistivity measurements were performed using typically a 100 nA switched DC current. Strong nonlinear Hall effects are observed at T = 600 mK. To estimate the total mobile carriers from the Hall effect, we used the high field limit of the slope (between 13 and 14 T) for the Hall resistivity versus magnetic field curves.