Introduction

Ever since the discovery of a conducting channel in LaAlO3/SrTiO31 and the subsequent observations of magnetism2 and superconductivity3, the vast majority of oxide interface research has focussed on synthesising intrinsically-doped heterostructures featuring narrow conducting channels () with two-dimensional carrier densities n2D in the 1012–1014 cm−2 range4,5,6,7,8,9. At such interfaces, it has been shown4,10,11,12,13,14 that symmetry-lowering and quantum confinement lift the Ti t2g degeneracy, so that the dxy orbital lies at lower energy than the dxz,yz orbitals. X-ray absorption spectroscopy4 reveals a band splitting of ~50 meV for n2D ~ 1013 cm−2 and theoretical approaches indicate that this increases with n2D, reaching ~0.25 eV at 3 × 1014 cm−212. Regardless of the total n2D, the splitting should gradually vanish below the interface, until the electronic structure resembles that of bulk SrTiO3 with degenerate dxy,xz,yz orbitals creating a Fermi surface at the centre of the Brillouin zone15. The lengthscale over which this degeneracy is regained - i.e. the total distance over which the interface induces electronic reconstruction - remains unknown, despite being a vital prerequisite for building layered 3D oxide devices.

Probing this lengthscale requires the synthesis of LaAlO3/SrTiO3 heterostructures with significantly more carriers (and correspondingly deeper conducting channels) than the norm. Previously, high n2D heterostructures have only been grown in reducing environments16,17, creating bulk-like conducting layers hundreds of microns thick (n2D ≥ 1016 cm−2) in which the broken symmetry of the interface plays no role. However, interfaces with have until now remained unexplored: at these intermediate n2D, electrons “spill over” from the interface and begin to occupy states lying deeper within the SrTiO3. The principal focus of our work is therefore to track the evolution of the electronic structure and its crossover from 2D interfacial to 3D bulk-like behaviour within this range of carrier densities. For n2D ≥ 6.9 × 1014 cm−2, we report the first instance of Shubnikov-de Haas (SdH) oscillations from an ultra-high mobility electron gas (μH ~ 104 cm2V−1s−1) for small magnetic fields parallel to the interface. The absence of such oscillations from the low-field perpendicular magnetoresistance indicates that these carriers originate from an anisotropic 3D Fermi surface (FS); our first-principles calculations of the sub-interfacial electronic structure reveal dominant dxz,yz orbital occupancy, which is consistent with our experimental data. Superconductivity remains confined within 20 nm of the interface, while the 3D FS characteristic of bulk doped SrTiO3 gradually emerges with increasing n2D. Together, our results imply the existence of a region below the interface whose electronic structure differs from that of the bulk, with a minimum thickness of 100 nm imposed by the cyclotron radius. This discovery has important implications for oxide devices seeking to functionalise interfacial electronic reconstructions.

Results

During sample growth, three mechanisms exist for carrier-doping the LaAlO3/SrTiO3 interface: (a) intrinsic self-doping via the polar catastrophe18 (leading to a maximum n2D = 3.3 × 1014 cm−2), (b) oxygen vacancy doping19 (contributing 2e− per O2− vacancy) and (c) cation intermixing9 (an unbalanced switching of La3+ for Sr2+ and Al3+ for Ti4+). Since our principal aim is to explore the evolution of the electronic structure for n2D > 5 × 1014 cm−2 (far beyond the upper limit imposed by the polar catastrophe) and cation intermixing is difficult to control in a pulsed laser deposition (PLD) chamber, we use O2− vacancy doping to achieve the high n2D values necessary for this project. To this end, we synthesise LaAlO3/SrTiO3 heterostructures at an intermediate O2 pressure (10−3 mbar), without any post-annealing procedure (further growth and characterisation details may be found in the Methods and Supplementary Material). The lack of annealing guarantees a high O2− vacancy concentration and hence a large n2D, while the intermediate growth pressure ensures that these vacancies do not penetrate far into the SrTiO3 substrate. Low pressure growth (10−6 mbar) without annealing16,17 has previously been shown to result in macroscopic substrate conduction, with n2D ≥ 5 × 1015 cm−2; in contrast, our method of synthesis consistently yields heterostructures with as-grown Hall carrier densities in the 1014–1015 cm−2 range, which we will refer to as “series B”. For comparative purposes, we have also annealed certain heterostructures (“series A”) at high O2 pressures, yielding n2D ~ 1013 cm−2. A-type interfaces are comparable to the majority of those previously studied in the literature3,5,6, in which carrier injection is dominated by the polar catastrophe. Both series exhibit coexistent superconductivity (SC) and ferromagnetism (FM), a comprehensive analysis of which may be found in ref. 20. For quantitative continuity in the present work, we focus on two specific samples A and B, with as-grown n2D = 2.3 × 1013 cm−2, 6.9 × 1014 cm−2 at T = 0.1 K and SC channel thicknesses d = 18 ± 1 nm, 9 ± 1 nm respectively. Sample B has a back gate beneath the SrTiO3: n2D increases to 2.4 × 1015 cm−2 (d = 19 ± 2 nm) at gate voltage Vg = 350 V. The heterostructure withstands Vg = 500 V with no discernible leakage current and the substrate capacitance ~ 1 nF is comparable to values measured in annealed LaAlO3/SrTiO3 heterostructures with lower n2D5,6 (see Supplementary section 1). Such conditions can only be achieved if the bulk of the SrTiO3 substrate is insulating: this confirms that O2− vacancies have not penetrated deep into the SrTiO3 and are restricted to the neighbourhood of the interface.

We probe the electronic structure and FS geometry using SdH oscillations in the magnetoresistance (MR) Rxx(H) (Fig. 1a). Two magnetic field orientations are principally considered: H ⊥ (001) (H⊥) and H//[110] (H//), where the [100] directions correspond to the crystallographic axes of the SrTiO3 substrate and [001] points out-of-plane. Sample A does not display any SdH effect for either orientation. In contrast, sample B exhibits strong oscillations for H// as low as 2.5 T, with faint oscillations also emerging for H⊥ > 6 T. However, data acquired with an in-plane field H//[010] do not show any oscillations up to 4 T. Symmetry dictates that the plane of a 2D FS in LaAlO3/SrTiO3 must lie parallel to the interface; any such FS will therefore lack states with out-of-plane momenta and cannot exhibit any SdH effect for in-plane fields. It is therefore immediately clear that the oscillations which we observe with H//[110] must originate from an anisotropic 3D FS.

Figure 1
figure 1

Shubnikov-de Haas oscillations and effective mass of the high mobility electron gas below the interface.

(a), Oscillatory components of the MR Rxx(H⊥,//)/Rbg in samples A and B. The background Rbg was determined using a polynomial fit to the raw Rxx(H) data for 1.5 T < H⊥ < 9 T and 2 T < H// < 4 T. Data for sample B were acquired at n2D = 2.4 × 1015 cm−2, i.e. Vg = 350 V. The vertical scale for these graphs is identical. (b), Temperature variation of the in-plane SdH oscillation amplitude in sample B (Vg = 350 V). (c), Amplitude suppression with increasing temperature for four in-plane SdH MR oscillation peaks/troughs Rpk(T) (indicated by arrows in (b)). Symbols represent oscillation amplitude data-points, while the four colour-coordinated lines are least-squares fits using equation 1 to calculate m*. Due to the low resistance of sample B (~1 Ω), the SdH effect for H ⊥ (001) creates voltage fluctuations of ~1 nV, which is the noise threshold in our measurements. Consequently, oscillations were only visible below T = 0.1 K and m* could not be determined for these carriers.

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For H//[110], the SdH oscillations in sample B are sufficiently pronounced for us to extract the effective mass m* and the Dingle temperature TD (a measure of the scattering) from their temperature-dependent amplitude (Fig. 1b). The magnitude of the oscillatory resistance is given by:

where Rbg is the background resistance. Fitting this equation to the oscillation amplitude (Fig. 1c) yields m* = 1.24 ± 0.1 me and TD = 1.4 ± 0.4 K. m* is similar to values previously reported for the LaAlO3/SrTiO3 2DEG7,8, although our TD is lower which implies a higher carrier mobility in our heterostructures. To estimate this mobility, we initially calculate the Hall mobility μH = 1/n2DeRxx, where Rxx(Vg = 0) = 0.28 Ω/□ and we assume single-band transport. This yields an exceptionally high Hall mobility μH = 32000 cm2V−1s−1, setting a new record for pure LaAlO3/SrTiO3 and rivalling the best epitaxial SrTiO3 films21.

In order to justify such a high mobility, we evaluate the Drude scattering time τdr ≡ m*μH/e = 23 ps, which is more than an order of magnitude greater than the Dingle scattering time ps. An alternative estimate of the scattering time in sample B may be extracted from the field at which a SdH effect first appears, using the quantum oscillation emergence condition ωcτSDH ~ 1 (where ωc ≡ Be/m* is the cyclotron frequency and B the magnetic field strength). For H//[110], oscillations are visible above 2.5 T: this corresponds to τSDH = 2.8 ps, which is also shorter than τdr suggested by our high μH. It is likely that four factors contribute to this disparity: firstly, all scattering events suppress quantum oscillations and contribute to τD, while only back-scattering influences τdr and the Drude conductivity. Similar variance between τD and τdr can be seen in other LaAlO3/SrTiO3 heterostructures7. Secondly, the finite thickness of the conducting channel in our heterostructures may postpone the emergence of any SdH effect, until the applied field is sufficiently large for the diameter of the cyclotron orbits to fall below this thickness. Thirdly, superconducting fluctuations at fields below ~2.5 T effectively “short-circuit” our heterostructures, reducing our ability to probe transport from carriers deeper below the interface. Finally, our single-band estimate for μH is an over-simplification, since multiband transport is expected for carrier densities above the Lifshitz transition in LaAlO3/SrTiO312,22. A three-band approximation to the field-dependent Hall coefficient (see Supplementary section 2) suggests a minority contribution from a high-mobility band with μH ≈ 8000 cm2V−1s−1. The total number of conduction bands in our heterostructures and their field-dependent mobilities remain unknown, so we cannot obtain a more precise value for the mobility of these quantum-oscillating carriers. However, it is clear that our SdH effect, resistivity and Hall data all indicate the presence of a high-mobility band with an anisotropic FS and μH ~ 104 cm2V−1s−1.

The fact that our measured TD is lower than than those reported for the LaAlO3/SrTiO3 2DEG7,8 also suggests that the band whose FS generates the in-plane oscillations lies within an extremely clean region of our heterostructures, far from the cation defects and magnetic scattering expected at oxygen-deficient PLD-grown LaAlO3/SrTiO3 interfaces. To determine the location of these high-mobility carriers more precisely, we examine the evolution of the SdH oscillation frequencies with field-effect doping, obtained from the peaks in fast Fourier transforms (FFTs) of Rxx(H//,⊥) (Fig. 2a,b). The Onsager relation links the peak frequency F with the extremal area S of the FS normal to the applied field via : since the size of the FS should be proportional to the carrier density, it is useful to compare F(Vg) with our experimentally-determined total n2D as well as the superconducting critical temperature Tc (which varies strongly with the local three-dimensional carrier density n3D23,24). Once the interfacial carrier density exceeds n3D ~ 1020 cm−3, we expect a gradual suppression of SC leading to a dome in Tc(Vg)5,6; this is indeed observed (Fig. 2c). However, the in-plane oscillation frequency F// is independent of Vg, implying that the FS area S ⊥ [110] responsible for these oscillations remains roughly constant upon field-effect doping. Furthermore, F//(Vg) displays no correlation with Tc(Vg) or n2D(Vg): the FS (and hence the density of states) of the SC band(s) is being influenced by field-effect doping, but the FS of the high-mobility band is not. Field-effect doping should have a similar effect on all occupied bands within the same spatial region. Therefore, the only possible explanation for this decoupling between Tc(Vg) and F//(Vg) is that the SdH-oscillating electron gas must be spatially separated from superconductivity, i.e. the high-mobility carriers lie below the SC channel.

Figure 2
figure 2

Evolution of the Shubnikov-de Haas oscillations and carrier density with increasing gate voltage.

(a), Variation of the oscillating component of Rxx(H//) for sample B (left panel) at Vg ≥ 0, with FFTs of the raw data (right panel). For Vg < 0 the noise level rises and it is not possible to identify oscillations: this is a well-known phenomenon and has been attributed to emergent inhomogeneity6. (b), Oscillating components of Rxx(H⊥) for Vg ≥ −100 V in sample B (left panel) with associated FFTs (right panel). The two peaks in the FFTs are indicated by grey and red arrows; for Vg = −100 V, the peaks merge. (c), Vg dependence of various properties of sample B, including Tc (above), SdH frequencies F//,2⊥ and n2D (below). Tc is measured from Rxx(T) data (see Supplementary Fig. S2) and the errors in F//,2⊥ correspond to the FFT peak widths at 80% of their maximum height (from (a),(b)). n2D(Vg) obtained from the Hall coefficient follows the values expected from the sample capacitance C(Vg) (see Supplementary Figs. S1b,S3b for raw capacitance and Hall data). We attribute the fall in n2D above Vg = 350 V to charge-trapping deep within the SrTiO3. Tc(Vg) forms a dome: since d = 19 nm at Vg = 350 V and SrTiO3 is SC for 5.5 × 1017 cm−3 ≤ n3D ≤ 5 × 1020 cm−323,24, we estimate a maximum conducting channel thickness W ~ 20 μm due to the combination of carrier injection and electron gas decompression6. In practice, we anticipate due to the extremely high n2D at the interface which will locally suppress SC: even at Vg = 350 V, every carrier in sample B could be accommodated in merely 3 nm of SrTiO3 doped at 0.5e−/unit cell.

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The gate evolution of Rxx(H⊥) is very different from Rxx(H//), with two Vg-dependent peaks appearing in the FFTs (Fig. 2b). One of these (F1⊥, grey arrows) lies below 20 T and is suppressed for large Vg: although this frequency seems too low to originate from the dxy interfacial bands (which form a larger FS at much lower n2D7,8), spin-orbit splitting may create a series of small FS for high n2D at the interface. The other peak (F2⊥, red arrows) mirrors n2D(Vg) as Vg increases, saturating and broadening at ~40 T for large Vg. This implies that F2⊥ also cannot arise from a dxy 2DEG at the interface, since for back-gate doping at the interfacial dxy occupancy should not change significantly: instead, carriers move deeper into the SrTiO3. It is therefore tempting to link this peak with the 40 T mode from de Haas-van Alphen experiments25 on δ-doped bulk SrTiO3; however the light 3D band whose spherical FS was shown to be responsible for the 40 T oscillation15 is only occupied for n3D > 6.7 × 1017 cm−3, by when SrTiO3 already shows SC24. Since d ≤ 20 nm for sample B20, n3D ≤ 5.5 × 1017 cm−3 below the SC channel, ruling out any occupancy of this light band. We therefore attribute F2⊥ to the gradual population of the 3D FS from the first occupied band in bulk doped SrTiO3, which is formed by degenerate Ti 3dxy,xz,yz orbitals and remains approximately isotropic for such low .

It is clear that the in-plane SdH effect F// in our data is unrelated not only to F2⊥, but also to any previously reported 2D7,8 or 3D16 quantum oscillations in LaAlO3/SrTiO3. Instead, our oscillations originate from a highly anisotropic FS (since there is no SdH effect for H⊥ < 6 T), which occupies a clean intermediate region between the interface and the bulk. We estimate the minimum thickness of this region using the cyclotron radius : since a depth of at least 2rg is necessary to establish SdH oscillations for H//(001), we use (where Φ0 is the magnetic flux quantum and we assume a spherical FS for simplicity), obtaining 2rg ~ 140 nm at 2.5 T.

To understand the origin of these in-plane SdH oscillations, we calculate the evolution of the sub-interfacial orbital occupancy (which determines the FS symmetry) with increasing n2D. The majority of electronic structure calculations for LaAlO3/SrTiO3 to date have only considered the first few layers below the interface for n2D ≤ 1014 cm−2 and are of limited use in our heterostructures. We have therefore performed first-principles calculations of the depth-dependent band structure in LaAlO3/SrTiO3 for n2D = 3 × 1013, 3 × 1014 and 8 × 1014 cm−2, specifically chosen to approach our experimental n2D in samples A, B (Vg ~ 0) and B (Vg > 0) respectively. Our calculated orbital occupancies are plotted in Fig. 3a and can also be seen in Fig. 4a–c: although computational power limits us to considering the first 10 unit cells below the interface, this is already sufficient to reveal the FS anisotropy responsible for our in-plane SdH effect.

Figure 3
figure 3

Orbital occupancy and Fermiology calculations at high carrier densities in LaAlO3/SrTiO3.

(a), Layer-dependent orbital populations for n2D = 3 × 1013, 3 × 1014 and 8 × 1014 cm−2. Data are plotted on a logarithmic scale with a lower cut-off of 10−3 electrons per Ti atom. (b),(c), Fermi surfaces of the interfacial dxy (red) and dxz,yz bands (green) projected onto the (001) and (110) planes for n2D = 8 × 1014 cm−2 (thin lines) and 3 × 1014 cm−2 (thick lines). The FS are calculated using a tight-binding model in which the parameters are fitted to bands calculated from first principles (see Supplementary Material for the band structure). For simplicity, our diagram ignores the hybridization which lifts the degeneracy at the band crossing points; taking this into account would split the doubly-degenerate FS into two. (d), Effective mass variation for dxz electrons within the (001) and (110) planes: thin and thick lines denote results at n2D = 8 × 1014 and 3 × 1014 cm−2. The planar angles are measured from the [100] and [001] axes respectively. for dyz electrons in the (001) plane is equivalent to for the dxz FS rotated by 90° around [001]. A sketch of the mass variation expected in the (110) plane for a bulk degenerate dxy,xz,yz FS is also shown (dashed brown line). (e),(f), Schematics illustrating the extremal FS orbits normal to magnetic fields along [001], [010] and [110] for a degenerate dxy,xz,yz FS (e) and a dxz,yz-dominated FS (f). Only H//[110] in the dxz,yz-dominated case probes the small, light FS cross-section whose presence we infer from our in-plane SdH oscillations.

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Figure 4
figure 4

Electronic structure, orbital character and depth-dependent phase emergence at the LaAlO3/SrTiO3 interface.

(a–c), Schematics illustrating the phase and carrier distributions for sample A (a), B at low Vg (b) and B at large Vg (c). The approximate local carrier density within the SrTiO3 is indicated by the colour shading: pale yellow denotes undoped bulk SrTiO3, while higher carrier density regions are either red (dxy), green (dxz,yz) or brown (degenerate dxy,xz,yz) depending on the dominant orbital character. dxy ferromagnetism is present regardless of the carrier density, but remains tightly confined to the interface20, i.e. at the top of the red zone. For comparison, we also plot the calculated orbital occupancies for the first ten TiO2 layers below the interface at n2D = 3 × 1013 (a), 3 × 1014 (b) and 8 × 1014 cm−2 (c). The index “1” refers to the TiO2 layer closest to the interface. Red and green bars correspond to dxy and dxz,yz orbitals respectively. (d), Qualitative illustration of the depth-dependent influence of field-effect doping on the local carrier density, which we deduce from our transport data. The line colour indicates the variation in the dominant orbital character with depth. (e),(f), Cartoon Fermi surfaces of the sub-interfacial high-mobility dxz,yz electron gas (e) and the bulk doped SrTiO3 (f) which develops progressively for large n2D.

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The central result from these calculations is a crossover from dxy to dxz,yz occupancy as we move away from the interface. Close to the interface and for small n2D, dxy states dominate due to quantum confinement, as expected10,12. The absence of a clear SdH signal from the 2D dxy interfacial FS in sample A is due to scattering from local moments10 and the large Rashba spin-orbit coupling; we note that there are no reports of a 2D SdH effect in FM LaAlO3/SrTiO3 in the literature. The important new result from our calculations is the creation of a conducting “tail” deeper below the interface for large n2D, with a disproportionate occupation of dxz,yz orbitals. For example, the dxz,yz:dxy ratio in layer 9 for n2D = 8 × 1014 cm−2 is 2.8:1, significantly greater than the 2:1 expected in bulk SrTiO3. A recent study of top-gated SrTiO3 also hints at a low density “tail” of carriers persisting over at least 50 TiO2 layers, independently of the total n2D13. While the majority of carriers occupy tightly-bound bands close to the interface, the back-gate field in our sample B should reduce the quantum confinement and expand the “tail” still further into the SrTiO3: this competition between confinement and decompression is responsible for the weak variation of F//(Vg) (Fig. 2c). We therefore identify a dxz,yz-dominated FS as the source of our in-plane SdH effect.

The strong asymmetry in our observed SdH effect (i.e. the absence of oscillations for small H⊥) may be explained by considering the FS geometry. In Fig. 3b,c, we plot the calculated (001) and (110) extremal cross-sections of the interfacial FS at n2D = 3 × 1014 and 8 × 1014 cm−2. The elliptical cross-section of the dxz,yz FS implies that our previously-calculated rg will be scaled by kF[001]/kF[110] = 0.73, reducing the minimum thickness over which the electronic structure deviates from that of bulk SrTiO3 to ~100 nm. Furthermore, the variation in |kF| across the FS drives a corresponding modulation in the effective band mass , shown for the (001) and (110) planes in Fig. 3d. Electrons in the (001) plane are significantly heavier and hence more easily scattered: therefore, SdH oscillations will only emerge for . Our measured F// ~ 25 T is clearly too small to originate from the large interfacial FS projections in Fig. 3c: instead, our in-plane oscillations are generated by a similarly-shaped smaller FS deeper below the interface, where n3D is lower. The overall symmetry of the dxz,yz FS does not vary significantly with depth and hence our effective mass argument justifying the suppression of oscillations for H⊥ remains valid. In the (110) plane, the average band mass of the carriers is , which only allows for a small electron-phonon coupling λ ~ 0.8 when compared with our measured m* = 1.24 me (since . However, we note that SdH experiments on both LaAlO3/SrTiO3 and n-type SrTiO3 heterostructures have persistently yielded small effective masses7,8,26.

Identifying the role of in determining the emergence of SdH oscillations allows us to make a profound statement regarding the shape of the in-plane oscillating FS. In Fig. 3d, we sketch the approximate dependence in the (110) plane expected for a degenerate (bulk-like) dxy,xz,yz FS. Here, the variation is similar to that in the (001) plane, though with a 180° rather than 90° period. We attribute the absence of oscillations for small H⊥ to the presence of heavy carriers in the (001) plane: therefore, the emergence of oscillations at small H// implies that cannot rise significantly at 0°. Consequentially, the FS within this sub-interfacial region must be flattened along the [001] direction in comparison with the bulk, i.e. the dxy,xz,yz degeneracy is lifted and the dxz,yz orbitals are shifted to lower energy. To illustrate this point further, in Fig. 3e,f we sketch dxy,xz,yz and dxz,yz-dominated Fermi surfaces, comparing the shapes of their extremal orbits perpendicular to [001], [010] and [110]. The low-frequency SdH oscillations which we observe with H//[110] must originate from a FS whose extremal orbits are composed exclusively of light carriers (i.e. the FS cross-sectional area must be small): it is clear that this condition is only satisfied for the dxz,yz-dominated FS.

Discussion

What is the physical origin of this change in the FS? We note that the shape of our dxz,yz FS is similar to that calculated by Mattheiss27 using a crystal-field parameter D which was subsequently shown to be too large15. Since D is related to the tetragonal structure of SrTiO3, our renormalised electronic structure may result from strain effects at the interface - such as the compression from the LaAlO3 layer - which are known to influence the 2DEG28. Studies of the 2D–3D crossover in δ-doped SrTiO3 films26 (in which strain should be absent) have not revealed the dxz,yz-dominated intermediate FS which we observe; nevertheless it remains unclear whether a long-range interface-induced change in D or the spin-orbit coupling is responsible for our results. Finally, our determination of the FS orbital character assumes the SrTiO3 tetragonal c-axis lies parallel to [001]: since orthogonal tetragonal domains are expected for T < 105 K, this may not initially seem plausible. However, an offset surface potential exists between domains with c//[001] and c//[100] in LaAlO3/SrTiO3, requiring substantial charge transfer to equalise the chemical potential29. This increases the carrier density in domains with c//[001], so transport predominantly occurs within these regions. Previous transport studies of La-doped SrTiO3 have also indicated a prevalence of [001]-oriented domains30.

We summarise the evolution of the LaAlO3/SrTiO3 interface with n2D in Fig. 4, where we schematically represent the spatial distribution of SC together with the approximate n3D variation and our calculated depth-dependent dxy and dxz,yz orbital occupancies (Fig. 4a–c). At low carrier densities (Fig. 4a), dxy orbitals dominate and the charge is concentrated within a few unit cells of the interface. Electrons in the top TiO2 layer tend to localise10,31, creating an inhomogeneous patchwork of FM zones above a narrow (≤20 nm) SC channel20.

As n2D increases (Fig. 4b), FM and SC both remain present at the interface. However, a high-mobility dxz,yz “tail” of minimum thickness 100 nm develops below the interface, generating an anisotropic 3D FS which exhibits SdH oscillations for small in-plane fields. Together, the appearance of this SdH effect, its independence from n2D(Vg) and Tc(Vg) and its absence in small perpendicular fields indicate that dxz,yz orbital occupancy is favoured over dxy to a depth of at least 120 nm below the interface. Unfortunately, it is not possible to accurately determine the maximum depth reached by this “tail”, since the carrier density very close to the interface (where we expect the majority of the carriers to reside) is unknown. However, our data do enable us to comment on the O2− vacancy penetration depth, which we already believe to be small since the capacitance of our B-type samples is comparable to values seen in annealed heterostructures. The high electron mobility within the “tail” region is primarily a consequence of the low carrier density (which leads to a small FS and low effective mass), but a lack of crystal defects (e.g. O2− vacancies) below the interface may also play an important role. Recently, ultra-high mobility carriers (μH ~ 50,000 cm2V−1s−1) have been observed in SrCuO2-capped LaAlO3/SrTiO3 heterostructures, in which O2− vacancy formation is suppressed32. This suggests that although the carriers in our B-type heterostructures originate from O2− vacancies, these vacancies may be confined close to the interface (or in the LaAlO3 layer) while the electrons which they donate are redistributed deeper within the SrTiO3. This concept is supported by the absence of any parasitic SrTiO3 surface conduction in our heterostructures (whose presence would be expected in the case of deep O2− vacancy penetration), as well as theoretical work which indicates that O2− vacancies preferentially inhabit the LaAlO3 surface rather than the interface33. Ideally, future theoretical work should examine the evolution of the electronic structure in the “tail” as a function of O2− vacancy density and location. It also remains to be determined whether the absence of superconductivity from the “tail” region is merely due to a sub-critical carrier density, or if the dxz,yz orbital character also plays some role.

At the maximum n2D which we are able to simulate (Fig. 4c), only the top TiO2 monolayer at the interface still has a dxy character, with dxz,yz states dominating below. We illustrate the effects of a back-gate electric field in Fig. 4d: as Vg increases, the carrier density in the superconducting channel rises and a shift to the overdoped side of the superconducting dome occurs (as seen in Fig. 2c). In parallel, electrons in the “tail” decompress away from the interface due to band-bending from the electric field, migrating hundreds of nanometres into the bulk. This migration creates the 3D FS responsible for the SdH oscillations which we observe with H⊥(001), whose frequency scales with the total carrier density. Between the interface and the bulk, the carrier density of the dxz,yz-dominated region remains roughly constant: electrons which it “loses” to deeper-lying bulk states are replaced by electrons from the interface. The presence of a large carrier population below the interface results in a screening of the electric field, thus explaining the relatively small increase of d to 19 nm at Vg = 350 V compared to d > 40 nm reported at much smaller back-gate fields in the literature34. Finally, Figs. 4e,f display exaggerated sketches illustrating the evolution of the FS as we move deeper into the SrTiO3, from dxz,yz domination (Fig. 4e) to a gradual recovery of dxy,xz,yz degeneracy (Fig. 4f) over a lengthscale ≥ 120 nm. While the microscopic origins of this long-distance evolution are still unclear, our work shows that functional oxide devices can reliably hope to profit from a renormalised electronic structure tens of nanometres away from a symmetry-breaking interface.

Methods

Two series of LaAlO3/SrTiO3 heterostructures, “A” and “B”, were grown using a standard pulsed laser deposition system manufactured by Twente Solid State Technology B.V., equipped with a reflection high-energy electron diffraction (RHEED) facility. We use 0.5 mm thick commercial 5 × 5 mm SrTiO3 (001) “STEP” substrates from Shinkosha: these are HF-treated for TiO2 termination and cleaned by the manufacturer, then vacuum-packed for shipping. We do not perform any additional surface cleaning or annealing prior to deposition: the substrates are loaded directly into our PLD chamber, which is subsequently evacuated to base pressure prior to back-filling with 10−3 mbar O2. The substrate is then heated to growth temperature (800°C). Series A and B both feature 10 unit cells of LaAlO3, deposited using a total incident laser energy of 9 mJ focussed onto a 6 mm2 rectangular spot. The O2 pressure and substrate temperature were maintained at 10−3 mbar and 800°C respectively for both sample series throughout the deposition process. Subsequently, A-type samples underwent an annealing stage: after cooling to 500°C at 10−3 mbar, the O2 pressure was increased to 0.1 bar. The temperature was held at 500°C for 30 minutes before natural cooling to 20°C in 0.1 bar O2. In contrast, B-type samples were cooled naturally to 20°C in 10−3 mbar O2.

To fabricate Hall bars on these LaAlO3/SrTiO3 films, we first defined contact pad areas using photolithography with AZ5214 photoresist. 2 nm Ti followed by 8 nm Au were evaporated directly onto the LaAlO3 surface; the remaining photoresist was removed by soaking in acetone for 30 minutes, then rinsed in IPA. Sample B also had an Au-Ti back gate deposited across the entire base of the SrTiO3 substrate prior to fabrication. The Hall bars were defined using a similar photolithography process and the Hall bar mesas etched using a dry Ar ion technique (at a slow rate of 1 Å s−1 to avoid any substrate heating). The Hall bar width was 80 μm and the voltage contact separation 660 μm. Multiple Hall bars were fabricated on each 5 × 5 mm substrate: tests showed that the Hall bars were electrically isolated from each other (thus ruling out any parasitic conduction from the SrTiO3 surface) and displayed similar transport properties (indicating that our heterostructures are homogeneous). Prior to measurement, the Hall bars were mounted in thermally-conductive chip-carriers, with electrical contacts made using 10 μm Au wires ball-bonded to the Au-Ti contact pads.

Transport data were acquired in a cryogen-free dilution refrigerator, using an AC technique with two digital lock-in amplifiers and a current source outputting 500 nA at 19 Hz. This value was chosen to maximise the signal-to-noise ratio whilst minimising sample heating below 0.1 K. Our noise threshold is approximately 1 nV. The substrate capacitance was measured with femtoFarad sensitivity for gate voltages up to 500 V using a General Radio 1621 manual capacitance bridge. All results presented in this work were qualitatively reproducible over a 6-month period comprising numerous cool-downs of both samples. A total of 6 “A-type” and 4 “B-type” heterostructures were fabricated in our laboratory using identical “recipes” to those detailed above: all samples displayed similar behaviour to those discussed in the present work.