Introduction

The coexistence of two-dimensional (2D) superconductivity and spin-orbit-coupling (SOC) is key for the development of novel device concepts for quantum technologies that exploit unconventional electron pairing in topological phases1,2. Boundaries and defects in topological superconductors can host zero-energy Majorana modes, offering applications in quantum computation3,4,5,6,7,8. Although seminal proposals focused on hybrid superconductor-semiconductor structures9,10, the simultaneous presence of superconductivity and large tunable SOC in SrTiO3 quasi-two dimensional electron gases (q-2DEGs) offers a particularly appealing platform to explore unconventional and topological superconductive phases11,12. Particularly, q-2DEGs at the LaAlO3/SrTiO3 (LAO/STO) interface have been investigated intensively, where electrostatic gating allows to control the electron band filling within the t2g manifold13,14,15, enabling the electric field modulation of the superconducting phase diagram16,17,18,19,20,21,22 and Rashba SOC23,24,25. Interestingly, the multiband character of the LAO/STO q-2DEG together with large Rashba SOC promotes the appearance of singlet-triplet mixed pairings, which, in the presence of magnetic fields, could lead to nontrivial superconductive states26. However, while Anderson’s theorem holds for singlet pairing27, disorder is expected to reduce triplet pairing, potentially hindering the emergence of topological phases. It is therefore indispensable to approach a clean limit for superconductivity at the LAO/STO q-2DEGs to enhance the chances of detecting nontrivial topological features in these systems28.

In this work, we report on LAO/STO q-2DEGs with exceptionally large mobilities, which allow for achieving mean free paths (lmfp) comparable to the superconductive coherence length (ξ), approaching the clean limit even in the depletion regime. In LAO/STO interfaces the electronic conductivity is ensured by t2g electrons in Ti (dxy, dxz, dyz) orbitals. The quantum confinement of the q-2DEG splits the energy levels of the 3d orbitals, whose splitting is dictated by the out-of-plane effective mass m*, which depends on the degree of orbital wavefunctions overlapping along the confinement direction29. In conventional (001) oriented interfaces, the degenerated dxz/yz orbitals lie at higher energies, and their wave function delocalizes deeper into STO with a dielectric constant similar to that of the bulk. Electrons occupying these subbands encounter less scattering, resulting in higher electronic mobilities. In contrast, electrons in dxy orbitals lie at lower energies and are confined near the interface where the dielectric constant is reduced by the interfacial electric field and shows smaller electronic mobility due to the larger scattering caused by imperfections (Fig. 1b, c)30,31,32. The filling of these subbands can be precisely controlled with a gate voltage VG by continuous and reversible doping of electrons at the quantum well and monitored by Hall effect32. The critical temperature Tc shows a dome-shaped phase diagram, first increasing at low doping in the depletion region and then going through a maximum \({{{{{{{{\rm{T}}}}}}}}}_{c}^{max}\) to subsequently reduce for further increasing VG towards the overdoped region16,17,21. Attempts have been made to correlate this dome-shaped phase diagram with the filling of the subbands. It has been suggested that the superconducting order parameter emerges when electrons start filling dxz/yz subbands, whereas it is strongly suppressed or even absent below the Lifshitz transition15,16,21,23,32,33,34. In line with this argument, only one superconducting gap has been reported so far in (001)-oriented interfaces, implying the absence of dxy superconductivity for this orientation35. This has precluded the investigation of the superconductivity of the (001) LAO/STO quantum well in the dilute regime, where one single band is occupied. We stress that in this region of phase diagram, the average electronic mobility for dxy electrons usually reduces to \({\mu }_{{d}_{xy}}\) < 50 cm2/V.s, corresponding to a mean free path lmfp < 1nm, which leads these systems to the extreme dirty limit15,21,23,32,33,34. As we discuss below, we reach mobilities around two orders of magnitude larger for dxy electrons, which is crucial to observe superconductivity mediated by dxy bands.

Fig. 1: Schematic of the band structure and estimation of La/Al ratio using XPS.
figure 1

a Schematic illustration of LaAlO3/SrTiO3 (LAO/STO) heterostructure with two-dimensional electron gas (2DEG) at the interface. The zoomed-in view highlights the interfacial region of the crystal structure where the 2DEG exists. b, c Sketch of the quantum well in real and momentum space, respectively. Three non-degenerate t2g orbitals split due to the confinement potential: the dxz/yz orbitals (blue) are shifted higher in energy compared to the dxy orbitals (red). The conduction electrons, corresponding to dxy orbitals, are closer to the interface, exhibiting lower electron mobility compared to the mobility for dxz/yz orbitals, which extend deeper into the STO. d The reflection high-energy electron diffraction (RHEED) intensity as a function of time, depicting a layer-by-layer growth, from which thickness can be inferred for the LaAlO3 layer. e RHEED patterns obtained during the pulsed laser deposition (PLD) growth show a smooth film deposition. f Atomic force microscopy (AFM) topography images of a film. The well-aligned terraces indicate the surface smoothness. Scale bar: 500 nm. g The XPS spectra for all films. h The modulation of sheet resistance at room temperature as a function of the ratio of La/Al extracted from XPS spectra. i La/Al ratio in different samples extracted from scanning transmission electron microscopy coupled with electron energy-loss spectroscopy (STEM-EELS) and X-ray photoelectron spectroscopy (XPS) analysis.

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On the other hand, the origin of the 2DEG in LAO/STO interfaces has been discussed in terms of either oxygen vacancies36,37,38 or polar discontinuity39. The former mechanism depends on external doping, while the latter mechanism is supported in view of the observation of conducting interfaces only beyond a critical thickness for LAO40. A dependence of the interfacial conductivity on cation stoichiometry of the LAO layer was observed by Warusawithana et al.41, revealing that conductivity only appears in Al-rich LAO layers below a critical ratio La/Al < 0.97. However, how the variation of the La/Al ratio affects fundamental parameters such as carrier density, mobility or the superconducting transition remains an open issue. In this work, we have investigated the transport properties of LAO/STO interfaces that show a consistent variation of La/Al ratio and large mobilities μ ranging between 5 × 102–3 × 103 cm2/Vs, at low electrostatic doping (i.e., in the depletion region of the quantum well) and between 5 × 103–2 × 104 cm2/Vs at high doping (in the accumulation regime). These values are more than an order of magnitude larger than previously reported on superconducting interfaces15,21,23,32,33,34. We demonstrate that a small variation of the La/Al ratio in Al-rich LaAlO3, as identified in X-ray photoelectron spectroscopy (XPS) and scanning transmission electron microscopy coupled with electron energy-loss spectroscopy (STEM-EELS) measurements, provides an excellent degree of freedom, in addition to electrostatic doping, to consistently alter carrier densities, mobilities and, remarkably, the onset of superconductivity. In contrast to previous reports, where superconductivity is either absent or suppressed below the Lifshitz transition15,16,21,23,32,33,34, we observe a significant enhancement of Tc for electrostatic doping below the Lifshitz transition. We attribute this result to the large mobility, which allows the mean free path (at low doping) to be comparable to the superconducting coherence length (i.e. lmfp ~ ξ), which is key to achieving dxy-superconductivity.

Results

Film growth and characterization

Thin LAO films were grown by pulsed laser deposition (PLD) on top of (001) oriented TiO2-terminated STO substrates (methods for growth process details). The epitaxial layer-by-layer growth of the films was confirmed by intensity oscillations of reflection high-energy electron diffraction (RHEED), as shown in Fig. 1d, e. Films were investigated by Atomic Force Microscopy (Fig. 1f), which showed a smooth surface with single-unit cell steps. The film composition of La/Al ratio was investigated using X-ray photoemission spectroscopy (XPS). Here, we present results for two 5-unit cell (uc) (S1 and S2), one 10 uc thick (S3) and one 11 uc (S4) thick film. Figure 1g shows XPS spectra of Sr 3d, La 4d, Al 2s, and Al 2p peaks for all samples. As expected, thicker films have much weaker intensity of substrate peaks (Ti and Sr). From the areas of La 4d and Al 2p peaks, we estimated the ratio between La and Al, assuming relative sensitivity factors of 6.52 and 0.54, correspondingly and with an error of about ~± 1% (methods for more details). Due to charging effects, the analysis required an alignment of the core levels to the position of the Sr 3d peak. The results show that all films are La deficient (Fig. 1i), in agreement with previously published results on films grown by PLD and MBE (Fig. 2k)41. The variation in the LAO film stoichiometry is likely due to the laser energy density during the film growth (Supplementary Fig. 1). More details about the influence of growth parameters on the film composition are discussed in supplementary note 1.

Fig. 2: Superconducting phase diagram.
figure 2

a–d Resistance vs temperature curves measured between 0.4K - 0.03K at different gate voltage (VG) for samples with rLa/Al = 0.91, 0.90, 0.88, 0.84. e–h The corresponding color plot of normalized resistance as a function of temperature and VG. The color scale is the same for all panels. The dashed line in color plot represents VG for Lifshitz transition. i The modulation of critical temperature Tc as a function of total carrier density nT. Here the carrier density corresponding to the dashed line is extracted from polynomial extrapolation of the data in Fig. 4i–l. j The variation of maximum of superconducting \({{{{{{{{\rm{T}}}}}}}}}_{C}^{max}\) with La/Al ratio. k The dashed line, adapted from ref. 41, indicates insulating-to-metallic transition as a function of La/Al ratio. The data points represented by circles correspond to the values of the La/Al ratio in our samples, as estimated from XPS analysis. The phase diagram is divided into three different regions: superconducting (SC), normal metal (NM), and insulating (INS) based on the outcome of our result.

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Figure 3 shows the high-angle annular dark field (HAADF) atomic resolution images (Fig. 3a, d, g, and j) of LAO/STO heterostructures along with the La M elemental maps (Fig. 3b, e, h, and k) and the resulting averaged La/Al ratios for each atomic plane (Fig. 3c, f, i, and l) obtained using EEL spectroscopy. These data indicate an inhomogeneous distribution of chemical composition, with a higher concentration of La near the interface in all samples, and at the uppermost plane of the 5 u.c thick LAO films of S1 and S2. The inhomogeneous distribution of La and Al near the interface can be caused by interdiffusion, which is evident from atomic-resolution elemental maps of all elements, including oxygen, see Supplementary Fig. 2. In particular, the Ti L-edge elemental maps show that Ti is substituting Al within the first LAO layer in all samples. The colored regions of the averaged La/Al ratio in Fig. 3c, f, i, and l show the atomic planes having a moderate deviation of the La/Al ratio. Figure 1i provides a depiction of the average La/Al ratio derived from the EEL spectrum images and compares it with XPS spectra. While the values obtained through EEL differ somewhat from those obtained via XPS, they exhibit a consistent overall trend. This disparity could be attributed to the fact that all the samples exhibit a heightened concentration of La at the interface, a phenomenon that XPS may not be sensitive enough to detect.

Fig. 3: Estimation of La/Al ratio using STEM-EELS.
figure 3

The high-angle annular dark field (HAADF) atomic resolution image of LAO/STO interface, simultaneous ADF signal together with La elemental map and lateral averaged La/Al ratio of sample S1 (a, b and c), S2 (d, e and f), S3 (g, h and i) and S4 (j, k and l), respectively. The shaded region in c, f, i, l represents an area for which the average value of La/Al ratio is estimated. The error bars are estimated from the standard deviation with respect to the quantification fitting window parameters obtained from reference bulk LAO. The inset in b, e, h, and k shows the LaAlO3 structure viewed along the [110] crystallographic direction, with La in green, Al in blue, and O in red.

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Electrical transport in normal state

The sheet resistance at T = 300 K decreases for smaller La/Al ratios (Fig. 1h), indicating a correlation between electrical transport and stoichiometry. Since it is well-established that the electrical resistivity of the properly annealed crystalline LAO/STO interface does not depend on the thickness of the LAO overlayer42,43,44, thus the variation in electrical resistivity in our samples can be attributed to the difference in the La/Al ratio. We emphasize that all samples show a metallic behavior in all the analyzed range of temperatures (2K-300K), with little sign of an increase in resistance at low temperatures (see Supplementary Fig. 3), indicating the absence of weak localization effects45. As discussed below, this agrees with the remarkably large values of the electronic mobility observed in all samples. In addition, carrier density is almost temperature independent in the interval 2K-300 K (Supplementary Fig. 4) in contrast to the majority of reports where a large reduction of carrier density was observed at low temperatures attributed to charge freeze out on charged crystal defects such as oxygen vacancies46. This again proves the high quality of interfaces studied in this work.

We investigated the effect of the La/Al ratio (rLa/Al) on the evolution of carrier density at 2K as a function of back gate voltages VG. In the depleted regime, corresponding to negative or low positive VG, we found a linear response of the transverse Hall resistance RH as a function of the applied out-of-plane magnetic field (B⊥) in all samples (Fig. 4a–d). In contrast, in the overdoped regime, corresponding to large positive VG, we observed non-linear RH(B⊥), indicating that a second band is filled (Fig. 4e–h) as the VG is increased. The transition from the linear to non-linear RH(B⊥) is usually ascribed to a Lifshitz transition from dxy single-band to dxy + dxz/yz two-band transport regimes when the dxz/yz subband is occupied at higher VG (see Fig. 1b, c for sketch of band structure). The transition can be better observed by obtaining the Hall carrier density nH = B⊥/eRH through linear fits of RH(B⊥) in the limits B⊥ < 0.5T and B⊥ > 4T, respectively (Fig. 4i–l). The nH overlap at low VG in the linear regime (comparatively depleted region), in agreement with the single-band transport. In contrast, above the Lifshitz transition at larger VG, the two values of the nH deviate from each other.

Fig. 4: Gate voltage-dependent Hall carrier density.
figure 4

The Hall resistance as a function out-of-plane magnetic field, RH(B⊥), for LAO/STO samples S4, S3, S2, and S1 with rLa/Al = 0.91 (a, e), 0.90 (b, f), 0.88 (c, g), and 0.84 (d, h) respectively. This is within the low gate voltage (VG) region of the phase diagram with linear dependence in RH(B⊥) (a–d), and at 50V where RH(B⊥) display nonlinear dependence (e–h). The black-dashed line indicates a linear fit of RH(B⊥). The black solid line is a fit of the two-band model (eq 1 presented in methods). i-l The evolution of carrier density as a function of VG extracted from the linear fit of RH(B⊥) for B⊥ < 0.5T (open circle) and B⊥ > 4T (solid circle). The shaded area indicates the region of VG where RH(B⊥) is nonlinear i.e. two bands regime. In the shaded region, n(B⊥ < 0.5T) vs. VG curve is not representing the true value of carrier density due to nonlinear RH(B⊥), whereas, n(B⊥ > 4T) is nearly equal to total carrier density \({n}_{T}={n}_{{d}_{xy}}+{n}_{{d}_{xz/yz}}\). Two arrows are the selection of gate voltage for which RH(B⊥) curves are shown. All measurements were performed at T = 2K.

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In our samples, the nonlinearity is observed at rather low magnetic fields B⊥ ≤ 2T, as can be seen in Fig. 4e–h. This is in contrast to previous reports that showed nonlinear behavior in RH(B⊥) extending for magnetic fields B⊥ > 5T21,23,32,33. We can attribute this to the distinct values of the electronic mobility in our samples, which are one order of magnitude larger than in previous reports. This is supported by the quantitative analysis of carrier densities and mobilities for each 3d-subbands using two-band model fit (see methods for details).

Figure 5a displays the VG dependence of the total carrier density nT and the carrier densities \({n}_{{d}_{xy}}\) and \({n}_{{d}_{xz/yz}}\) for electrons occupying dxy and dxz/yz subbands, respectively. Similarly, Fig. 5b shows the data for the electronic mobilities \({\mu }_{{d}_{xy}}\) and \({\mu }_{{d}_{xz/yz}}\) as a function of VG. In agreement with previous reports, we find that the majority of carriers are populating the lower energy dxy orbitals (\({n}_{{d}_{xy}}\) ≫ \({n}_{{d}_{xz/yz}}\))21,23,32,33. However, \({n}_{{d}_{xy}}\)(\({\mu }_{{d}_{xy}}\)) and \({n}_{{d}_{xz/yz}}\)(\({\mu }_{{d}_{xz/yz}}\)) display a systematic increase (decrease) with the reduction of the rLa/Al ratio (Fig. 5a, b). The values of \({\mu }_{{d}_{xy}}\) and \({\mu }_{{d}_{xz/yz}}\) obtained in our samples are about 15 times larger than in previous reports21,23,32,33. This can be attributed to the cleaner interfaces obtained in the present case. In addition, the values of the carrier density in the dxz/yz band are significantly larger than those reported previously. For instance, in our samples with rLa/Al = 0.88 and 0.84, the fraction of carriers in dxz/yz bands exceeds 20%, as compared to less than 2% observed in previous works23,32,33. This implies that a small variation of the rLa/Al ratio strongly affects carrier densities and mobilities in both bands, providing an additional degree of freedom to electrostatic gating, where the range of doping is limited by dielectric properties of SrTiO3.

Fig. 5: Estimation of carrier density and mobility in each t2g band.
figure 5

a The evolution of total carrier density nT (solid circle) and carrier density for electrons occupied in lower energy dxy band, nd(xy), (open square) and upper dxz/yz band, nd(xz/yz) (solid triangle), as a function of VG, extracted from two-band model fit. b The extracted values of corresponding electron mobility μd(xy) and μd(xz/yz) as a function of VG. The shaded area represents two-band region in the phase diagram. c, d The estimated value of mean free path for dxy electrons \({l}_{mfp}^{{d}_{xy}}\) and dxz/yz electrons \({l}_{mfp}^{{d}_{xz/yz}}\) respectively. The shaded gray area is the estimated value of coherence length adapted from ref. 15.

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Superconducting properties

To investigate superconductivity, we measured the temperature dependence of the sheet resistance RS(T) between 0.03 and 0.45 K for different VG. Figure 2a–d show RS(T) curves at different VG and Fig. 2e–h display the corresponding color plot of normalized resistance as a function of T and VG. It is evident that the rLa/Al has also a strong effect on the formation of superconducting ground state. The maximum of superconducting critical temperature \({{{{{{{{\rm{T}}}}}}}}}_{C}^{max}\), extracted from TC-VG phase diagram, increases with reducing rLa/Al, as shown in the Fig. 2j. For rLa/Al = 0.91, the interface does not display any evidence of superconducting transition for any VG (Fig. 2a, e) despite being metallic down to the lowest temperature of 30 mK. A weak superconducting transition is observed for rLa/Al = 0.90 with a \({{{{{{{{\rm{T}}}}}}}}}_{C}^{max}\,\) ≈ 0.08 K at VG ≈ −60 V (Fig. 2b, f). In this sample, RS saturates to a residual resistance at low temperature, suggesting the presence of non-percolating superconducting regions in the sample, as shown in Supplementary Fig. 5 where superconducting regions embedded in a metallic matrix are sketched. On the other hand, a clear zero-resistance state with \({{{{{{{{\rm{T}}}}}}}}}_{C}^{max}\,\) ≈ 0.25K at VG < −50 V is achieved in the rLa/Al = 0.88 sample (Fig. 2c, g), where RS(T) displays a broad superconducting transition suggesting a competition between superconducting and metallic regions (Supplementary Fig. 5). Interestingly, confirming a systematic trend, a sharp superconducting transition is observed in the samples with the lowest rLa/Al = 0.84, with \({{{{{{{{\rm{T}}}}}}}}}_{C}^{max}\,\) ≈ ≈ 0.35K at gate voltages VG ≈ −50 V (Fig. 2d, h).

Discussion

Our results provide important insights into the nature of superconductivity in the oxide q-2DEGs. According to the conventional Bardeen-Cooper-Schrieffer (BCS) scenario, pairing through dxz/yz rather than dxy bands is more favorable for superconductivity due to the larger density of states of the former47. In line with this, both experimental and theoretical studies show either the absence or strong suppression of superconductivity in single dxy-band 2DEGs, in a region of the phase diagram where the electronic mobility lies below \({\mu }_{{d}_{xy}}\) < 50 cm2/V.s15,21,23,32,33,34. In striking contrast, the critical temperature Tc of the samples analyzed in this work is not suppressed below the Lifshitz transition. As shown in the phase diagram in Fig. 2g and h, for samples with rLa/Al = 0.88 and 0.84, the Tc increases substantially already below the Lifshitz transition, rising to >15% at maximum Tc, and further suppressed for higher doping. We attribute this remarkable observation to the large mobility of electrons in the dxy band, with values \({\mu }_{{d}_{xy}}\) > 600 cm2/V (Fig. 5b), i.e., more than an order of magnitude larger than usually reported elsewhere. Strikingly, as inferred from data displayed in Fig. 2i, the Lifshitz transition does not appear at a universal carrier density as suggested in previous observations21, indicating that the formation of the quantum well may evolve differently for different samples. These observations demonstrate that superconductivity at the LaAlO3/SrTiO3 interface can be achieved in the clean limit, opposite to previous reports where it was only reported in the dirty limit22,33, characterized by a reduced mean free path to lmfp < 1 nm in the depleted region. Our data (Fig. 5c, d) show that the lmfp in our samples is comparable to the ξ = 40-60 nm. This observation is important for the search of theoretically predicted topological superconductivity in these systems, as it should minimize drastically the detrimental effect of disorder on non-conventional electron pairing12,20,26. We also note that our results may allow studying pairing mechanism in single dxy-band superconductivity in (001)-oriented LaAlO3/SrTiO3 interfaces in the dilute limit mediated by ferroelectric fluctuations through soft transverse optical phonons48,49,50,51,52,53,54. This is in line with recent observations of enhanced transition temperatures near the ferroelectric quantum critical point of SrTiO355,56,57,58,59, and the possible co-existence of ferroelectricity and superconductivity59,60. Similar mechanisms have been proposed for the recently found superconductive KTaO3 interfaces, in which pairing occurs in 5d instead of 3d orbitals61. In particular, a soft optical mode induced by the inversion-breaking at the interface has been suggested for inter-orbital paring, a mechanism that could be general for quantum paraelectric interfaces62. Finally, we also show that both electrical transport and superconductivity in the LaAlO3/SrTiO3 q-2DEG are systematically correlated with cation stoichiometry of the LAO film. While the exact mechanism of this effect cannot be elucidated from the existing data, it may be related to the intrinsic charge compensation by defects41. Remarkably, our sample with the highest La/Al ratio (rLa/Al = 0.91, S4) has striking similarity with previously reported SrCuO2-capped q-2DEG showing temperature-independent carrier concentration and very high electron mobilities of up to 4 × 104 cm2/Vs, but no superconducting transition46. This suggests that the electrical transport in the LAO/STO interface is determined both by oxygen exchange kinetics and cation stoichiometry of the LAO film.

Methods

Thin films growth

Thin LAO films were grown on 5 × 5 mm2 large TiO2-terminated STO substrates by pulsed laser deposition (PLD). The laser spot area on the target was set to 2 mm2. The laser fluence was varied by changing the energy, which was measured after the focusing lens. The laser beam optics was adjusted to provide a true image of the laser aperture on the target. All films were deposited at an oxygen pressure of 10−4 mbar and a heater temperature of 800 oC. The target-to-substrate distance was set to 50 mm. The laser repetition rate was 1 Hz, corresponding to the growth rate of 40–50 pulses per unit cell layer. After deposition, all samples were annealed to 600 oC at the oxygen pressure of 300 mbar for one hour to eliminate oxygen vacancies that could be produced in the STO substrate during the deposition process.

X-ray photoemission spectroscopy (XPS)

XPS measurements have been performed by using Al Kα X-ray source (E = 1486.6 eV) and Argus CU hemispherical electron spectrometer from Scienta Omicron AB, P.O. Box 15120, SE-750 15 Uppsala Sweden. Due to unavoidable charging effects, core level peaks in spectra for all samples have been aligned to the position of Sr 3d peak, see Fig. 1g. La/Al ratios of all samples were estimated from analysis of La 4d and Al 2p peak areas using Scofield database for relative sensitivity factors (RSF) for La and Al of 6.52 and 0.537, correspondingly [J. H. Scofield, Lawrence Livermore Lab. Rept 1973, UCRL-51326]. Each value of La/Al ratio is an average of 3 consecutive measurements. The corresponding error was estimated to be ± 1%.

Scanning transmission electron microscopy - electron energy-loss spectroscopy (STEM-EELS)

STEM-EELS was performed along the [110] pseudocubic zone axis in an aberration-corrected Nion U-Hermes 100, operated at 60 kV, and equipped with Dectris ELA direct electron detector, which offers improved detective quantum efficiencies, narrower point spread functions, and superior signal-to-noise ratios. The convergence semi-angle was 32 mrad, the probe current 20 pA, and the collection semi-angle 75 mrad. The elemental maps were generated with Digital Micrograph’s EELS quantification tool. We used a reference EEL spectrum acquired from a control sample, a commercial LAO substrate. The background in the spectra was removed using a power law fit, followed by integrating the La M4, 5 (830 eV) and Al K (1560 eV) edges intensity by using integration windows with energy range of 900–1000 eV and 1575–1690 eV, respectively, excluding the near-edge fine structure in both cases. The fitting parameters were adjusted to obtain a La/Al elemental quantification of the control sample. The La/Al ratio of LAO thin films was calculated using the same procedure, and to neglect the influence of thickness and plural scattering in the quantification of all spectra in this work (including the reference) were acquired in regions with the same sample thickness (20 nm).

Hall carrier density and two-band model

The gate voltage-dependent Hall effect measurements were performed in Quantum Design PPMS system in Van der Pauw geometry. The Hall resistance (RH = VH/I was measured as a function of out-of-plane magnetic field, swept between ± 7T. The carrier density nH = B/eRH, reported in Fig. 4 of the main text, is extracted from RH in the magnetic field limit of B < 1T and B > 4T.

In the two-band regime analysis, reported in Fig. 5 of main text, the Hall resistance can be defined as

$${R}_{H}=\frac{B}{e}\frac{\frac{{n}_{{d}_{xy}}{\mu }_{{d}_{xy}}^{2}}{1\,+\,{\mu }_{{d}_{xy}}^{2}{B}^{2}}+\frac{{n}_{{d}_{xz/yz}}{\mu }_{{d}_{xz/yz}}^{2}}{1\,+\,{\mu }_{{d}_{xz/yz}}^{2}{B}^{2}}}{\left[\frac{{n}_{{d}_{xy}}{\mu }_{{d}_{xy}}}{1\,+\,{\mu }_{{d}_{xy}}^{2}{B}^{2}}+\frac{{n}_{{d}_{xz/yz}}{\mu }_{{d}_{xz/yz}}}{1\,+\,{\mu }_{{d}_{xz/yz}}^{2}{B}^{2}}\right]+{\left[\frac{{n}_{{d}_{xy}}{\mu }_{{d}_{xy}}^{2}B}{1\,+\,{\mu }_{{d}_{xy}}^{2}{B}^{2}}+\frac{{n}_{{d}_{xz/yz}}{\mu }_{{d}_{xz/yz}}^{2}B}{1\,+\,{\mu }_{{d}_{xz/yz}}^{2}{B}^{2}}\right]}^{2}}$$
(1)

Where, \({n}_{{d}_{xy}}\), \({n}_{{d}_{xz/yz}}\) and \({\mu }_{{d}_{xy}}\), \({\mu }_{{d}_{xz/yz}}\) are carrier densities and mobilities related to dxy and dxz/yz bands which are obtained by fitting RH(B⊥) using constraints for total carrier density \({n}_{T}={n}_{{d}_{xy}}+{n}_{{d}_{xz/yz}}\) and longitudinal resistance \(\frac{1}{{R}_{S}}=e({n}_{{d}_{xy}}{\mu }_{{d}_{xy}}+{n}_{{d}_{xz/yz}}{\mu }_{{d}_{xz/yz}})\).