Introduction

Oxide heterojunctions are ideal platforms for exploring a remarkable variety of emergent phenomena1. In particular, the superconducting electron gases residing at the interface between two insulators have attracted considerable attention2,3,4,5,6,7,8,9. Previous studies have revealed that the conductive interfaces in the SrTiO3 (STO)-based heterojunctions exhibit a surprisingly enriched cascade of unusual properties including the coexistence of superconductivity and ferromagnetism10,11,12, and pseudogap-like behavior13. The more recently discovered KTaO3 (KTO)-based heterojunctions also host two-dimensional (2D) superconductivity8,9 and anomalous Hall effect14. Most interestingly, application of electric fields can effectively modulate the strength of spin-orbit coupling (SOC)15,16,17, the carrier density18,19, and the disorder level20, consequently it can control the Tc and even lead to a superconducting-to-insulator quantum phase transition18,19,20. Despite that KTO and STO are isostructural compounds with similar band structures21, the two systems are distinct from each other in many aspects. The conductive electrons in STO and KTO are contributed by the Ti 3d and Ta 5d bands, respectively22, thus the KTO-based superconducting interface (SI) should have a stronger SOC. Furthermore, whilst superconductivity has been realized at the (001)-, (110)- and (111)-oriented STO-based interfaces with comparable Tc2,3,7, the KTO (001)-based interface is non-superconducting14,23, whereas the KTO (110) and KTO (111)-based interfaces show superconductivity with Tc around 1 and 2 K, respectively8,9.

Several recent experimental observations, e.g., peculiar stripe phase in the normal state (although needs further confirmation)8,24 and unusual doping dependence of the upper critical field (Hc2)25, imply possible unconventional superconductivity at the KTO-based interface. Further evidence for the novelty of the superconductivity can stem from its responses to the external electric fields. It has been argued that the back gate voltage applied to the LaAlO3/KTO(111) (LAO/KTO) interface predominantly controls the effective disorder (and consequently the electron scattering rate) rather than the carrier density20. At the EuO/KTO(111) interface, both the Tc and Hc2 are reported to be sensitive to the carrier density in the gating process25. On the other hand, investigations of the electric-field-control SOC, which may be suggestive of unconventional superconducting pairing mechanisms17, are still lacking for the KTO-based devices.

Under certain circumstances, the SOC effect can play an essential role in determining the physics in oxide heterostructures (most notably at LAO/STO interface close to the Lifshitz point)26, due to the inversion symmetry breaking at the interface15,16,17. For these interfaces, the behaviors of both the normal and the superconducting states are to be understood only with the SOC taken into account. For instance, the magnetotransport properties of the normal state usually reflect the influence of SOC in terms of the weak localization or weak antilocalization effects27,28, whereas the unusually high Hc2 and potential unconventional pairing in the superconducting state can be triggered by the complex contributions of SOC17,29. Moreover, it has been shown that the SOC at the STO-based SIs can be effectively manipulated by the application of electric fields, leading to intricate evolution of physical properties as revealed experimentally15,16,17. In this sense, it is important to explore the role of SOC at the KTO-based SIs and its influence on 2D superconductivity8,25.

This work reports the growth of high-quality EuO (111) thin films on KTO (110) substrates. The interface between them is proved to host 2D superconductivity. Most intriguingly, we find a large tunability of Tc and SOC at the EuO/KTO(110) interface by applying an electric field across KTO substrates. Based on these observations, we propose that in addition to carrier density and effective disorder, the SOC strength has a significant impact on Tc. This may be linked to the unconventional nature of the superconductivity at the KTO-based interface.

Results and discussion

Characterization of the EuO/KTO(110) heterostructures

High-quality EuO (111) thin films were grown on (110)-oriented KTO substrates using a molecular beam epitaxy (MBE) system (see Methods for details). Bulk EuO crystallizes in a cubic structure with lattice constant a = 5.145 Å. Stoichiometric EuO is an insulator with a band-gap of 1.12 eV at room temperature30. Fig. 1a shows a schematic illustration for our EuO/KTO(110) heterostructure. To confirm the quality of films, we performed the scanning transmission electron microscopy (STEM) measurements. Figure 1b shows a cross-section with KTO [001] orientation in the plane (another direction along KTO [1\(\bar 1\)0] is shown in Supplementary Fig. 1a). Due to a certain lattice mismatch (~5%) between EuO (111) and KTO (110) surfaces, the epitaxial EuO at the interface is distorted within a thickness of approximately 2 atomic layers (red square in Supplementary Fig. 2); single crystallinity is recovered beyond this region. Atomic-scale energy-dispersive x-ray spectroscopy (EDS) shows a relatively clear interface in which the diffusion of Eu exists in the superficial layer of KTO (the white dotted region in Supplementary Fig. 2). Electron energy loss spectroscopy (EELS) peaks of Eu also suggest that the Eu doping in KTO persists up to approximately 3 atomic layers crossing the interface (Supplementary Fig. 3). θ-2θ X-ray diffraction (XRD) pattern confirms that our samples have good single crystallinity (Fig. 1c). A fit using the angle of the Laue oscillation peaks yields the film thickness of about 7 nm (Supplementary Fig. 1b). The films also exhibit good surface flatness with a root-mean-squared roughness around 0.343 nm (Supplementary Fig. 4).

Fig. 1: Structural characterization and transport properties of EuO/KTO(110).
figure 1

a A schematic diagram for the epitaxial growth of EuO (111) on KTO (110). b STEM image of the EuO/KTO(110) interface along [001] direction. The red square indicates the interface region. c θ-2θ X-ray diffraction (XRD) pattern specifying the well-orientated EuO (111) film on KTO (110). d Sheet resistance as a function of temperature of samples #1 and #2 measured in a wide temperature range. Inset: a sketch of the van der Pauw method for the measurements of sheet resistance and Hall resistance. e Sheet resistance as a function of temperature shows superconducting transitions at low temperatures. Inset: Hall resistance as a function of the magnetic field measured at 2 K. f Carrier density ns and Hall mobility μ (inset) plotted against T.

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The transport properties were measured using the Van der Pauw method (inset of Fig. 1d). The samples are metallic in the whole temperature (T) range as shown in Fig. 1d, indicating the formation of electron gases at the interfaces. Both samples undergo a superconducting transition at low temperatures. For sample #1, \(T_{{{\mathrm{c}}}}^{{{{\mathrm{onset}}}}}\) is 1.35 K, and the zero resistance is observed at \(T_{{{\mathrm{c}}}}^{{{{\mathrm{zero}}}}}\)= 1.06 K (Fig. 1e). The magnetic-field (H)-dependent Hall resistance RHall measured at T = 2 K confirms that the charge carriers are electron-type for both samples (inset of Fig. 1e). In Fig. 1f we plot the 2D Hall carrier density ns and the Hall mobility μ (extracted from the Hall and sheet resistance data) versus temperature. For sample #1 (#2), ns is 8.6 (9.0) × 1013 cm−2 and μ is 86 (128) cm2 V−1 s−1 at 2 K. Compared to LAO/KTO(110) interface9, both ns and Tc in our samples are higher, consistent with the results for the (111)-oriented devices8. During the growth process, Eu atoms have a strong capability to uptake oxygen from the surface layer of the KTO; this effect may cause the higher ns (and consequently the enhanced Tc) in the EuO/KTO heterostructures.

2D superconductivity

We measured the T-dependent 2D sheet resistance Rsheet under magnetic fields applied perpendicular and parallel to the interface to investigate the nature of this interfacial superconductivity. As shown in Fig. 2a and b, the superconductivity is remarkably suppressed by a magnetic field of ∼0.4 T and ∼6 T applied perpendicular and parallel to the interface, respectively. Such strong anisotropy indicates the 2D nature of superconductivity. To further verify this, we fit our data to the Ginzburg-Landau theory for a 2D superconductor31:

$$\begin{array}{l}\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{c}}}}}\left( T \right) = \left[ {{{\Phi }}_0/2\pi \xi _{{{{\mathrm{GL}}}}}^2\left( 0 \right)} \right]\left[ {1 - \left( {T/T_{{{\mathrm{c}}}}} \right)} \right],\\ \mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{ab}}}}}\left( T \right) = \left[ {{{\Phi }}_0\sqrt {12} /2\pi \xi _{{{{\mathrm{GL}}}}}\left( 0 \right)d_{{{{\mathrm{sc}}}}}} \right]\left[ {1 - \left( {T/T_{{{\mathrm{c}}}}} \right)} \right]^{1/2},\end{array}$$
(1)

where ξGL is the Ginzburg–Landau coherence length, Ф0 is the flux quantum, and dsc is the superconducting layer thickness. The T/Tc dependence of upper critical fields for H//ab (\(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{ab}}}}}{{{\mathrm{ }}}}\)) and H//c (\(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{c}}}}}{{{\mathrm{ }}}}\)) (Fig. 2c) are determined from the Rsheet-T curves shown in Fig. 2a, b, respectively. The fits to Eq. 1 yield the zero temperature limit values \(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{c}}}}}{{{\mathrm{ }}}}\)(0) = 0.45 T and \(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{ab}}}}}{{{\mathrm{ }}}}\)(0) = 6.65 T, corresponding to an anisotropic ratio of ~15; meanwhile, the fits also give ξGL(0) = 27.03 nm and dsc = 6.35 nm. A similar analysis for sample #2 was shown in Supplementary Fig. 5. Since the growth conditions of #1 and #2 are different, μ0H c2 and dsc of these two samples also differ slightly. Nonetheless, dsc is smaller than ξGL in both samples, confirming the 2D nature of the superconductivity at the EuO/KTO(110) interface. Besides, dsc is much larger than the diffusion depth (~0.5 nm) of Eu atoms (Supplementary Fig. 3). The mean free path of the conducting electrons can be estimated as lmfp = (h/e2)(1/kFRsheet) in a single-band model (\(k_{{{\mathrm{F}}}} = \sqrt {2{{{\mathrm{\pi }}}}n_{{{\mathrm{s}}}}}\) is the Fermi wave number, h is the Planck constant, e is the elementary charge)32. Using the measured Rsheet(2 K) and ns(2 K), we estimated the lmfp of sample #1 to be 59.3 nm, which is larger than the ξGL. We note that though our EuO/KTO(110) interfaces are much cleaner compared to the LAO/KTO(110) and EuO/KTO(111) interfaces wherein lmfp < ξGL8,9, these SIs are still in the dirty limit (Methods).

Fig. 2: Superconducting transition and the upper critical field of EuO/KTO(110).
figure 2

Temperature-dependent sheet resistance under different magnetic fields a out-of-plane (H//c) and b in-plane (H//ab). c T/Tc-dependent upper critical field μ0Hc2, extracted from the 50% normal-state resistance out-of-plane and in-plane. The estimated Pauli paramagnetic limit (μ0HP) is marked with a blue dashed line in c.

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With solid evidence for 2D superconductivity, we further examine the expected behaviors of the Berezinskii-Kosterlitz-Thouless (BKT) transition in our devices33. The BKT transition, a transition from unpaired vortexes and anti-vortexes to bound vortex-antivortex pairs, can result in a V∝Iα power-law dependence and can be characterized by a transition temperature TBKT where α(TBKT) = 334. To reveal such characteristics, we measure the current-dependent voltage (I-V curves). The data for sample #1 is displayed in Fig. 3a (see Supplementary Fig. 6a for sample #2). By fitting the I-V curve in the nonlinear range (Fig. 3b and c), we attain an exponent α approaching 3 at TBKT = 1.01 K. Apart from the I-V method, the TBKT can also be estimated from the formula Rsheet(T) = R0exp[-b(T/TBKT-1)−1/2], where R0 and b are material parameters. Application of such fit to the measured Rsheet(T) yields TBKT = 1.17 K (inset of Fig. 3c). TBKT obtained from these two approaches appears to be close to \(T_{{{\mathrm{c}}}}^{{{{\mathrm{zero}}}}}\), again pointing towards the 2D nature of the superconductivity.

Fig. 3: The Berezinskii-Kosterlitz-Thouless transition.
figure 3

a I-V curves measured at different temperatures. b I-V curves plotted in a logarithmic-logarithmic scale with the same color codes as in (a). The black solid line represents V ∝ I3, which is used to infer the BKT transition temperature TBKT. c T-dependence of the power-law exponent α (V ∝ Iα) obtained from the linear fits of the curves in (b), in the range of transition (where the I-V relation is no longer linear). Inset: [dln(Rsheet)/dT]−2/3 plotted against T [we use the zero-field data of Rsheet(T) shown in Fig. 2b]. A linear extrapolation from the high-T linear section (red dashed line) crosses the T-axis at TBKT = 1.17 K.

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For a 2D weak coupling BCS superconductor, the parallel critical field can be determined by the Chandrasekhar-Clogston limit (Pauli paramagnetic limit):35,36 μ0HP ≈ 1.76kBTc /\(\sqrt 2\)μB, where kB and μB are the Boltzmann’s constant and Bohr magneton, respectively. Taking Tc = TBKT = 1.01 K, we have μ0HP = 1.874 T, which reaches only 28% of \(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{ab}}}}}{{{\mathrm{ }}}}\)(0) (blue dashed line in Fig. 2c). Several factors can enhance the Pauli limit appreciably, such as strong-coupling superconductivity and many-body effects37,38. In our samples, the most likely reason for the large \(\mu _0H_{{{\mathrm{c}}}}^{//{{{\mathrm{ab}}}}}\) could be the strong SOC originating from the inversion symmetry breaking at the interface and the relatively heavy tantalum ions29,39, which can be verified by the electric-field manipulation that we will discuss in the following section.

Electric-field control of superconductivity

To explore the effect of electric fields, our samples were made into 20 × 100 μm2 Hall bar devices (inset of Fig. 4a). A schematic diagram of the device (Supplementary Fig. 7) and the manufacturing process are presented in Methods. The superconductivity can be successfully tuned by applying a gate voltage (VG) across KTO (Fig. 4a). As VG varies from −180 to 150 V, \(T_{{{\mathrm{c}}}}^{{{{\mathrm{onset}}}}}\) (\(T_{{{\mathrm{c}}}}^{{{{\mathrm{mid}}}}}\)) increases from 0.96 (0.89) to 1.69 (1.54) K, respectively, highlighting an enhancement exceeding 70% (Fig. 4b). Meanwhile, at T = 3.8 K, ns shows a moderate change, achieving the maximum value at VG = −80 V, whereas μ varies monotonically from 73 to 211 cm2 V−1 s−1 over the entire gating voltage range (Fig. 4c). As RHall(H) maintains linearity at all gate voltages (Supplementary Fig. 8b), the mean-free path lmfp and the effective disorder kFlmfp at different voltages can be estimated using the 3.8 K data of Rsheet and ns. Over the ramping range of VG from 150 to −180 V, kFlmfp evolves from 75 to 27 and lmfp varies from 32.2 to 11.4 nm (Supplementary Fig. 9). Such variation of kFlmfp and lmfp is much smaller than that of LAO/KTO(111) interface20. For the latter, it is shown that the tuning effect of electric fields highly depends on the mobility (disorder level) of the sample20. As the mobility is relatively high in our single-crystalline heterostructures with cleaner interfaces, the electric-field controlling of the effective disorder scattering is less efficient compared to that reported in ref. 20.

Fig. 4: Electric field tunability of the superconducting state.
figure 4

a Temperature-dependent sheet resistance at different gating voltage (VG). Inset: photograph of the device used in the gating process. It was fabricated into a six-probe Hall bar configuration. b VG-dependent onset and midpoint of Tc. c VG-dependent carrier density ns and Hall mobility μ measured at T = 3.8 K. d VG-dependent Pauli limit (μ0HP), in-plane upper critical field (\(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{ab}}}}}{{{\mathrm{ }}}}\)), and spin-orbit energy (εso). Inset: VG-dependent out-of-plane upper critical field (\(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{c}}}}}{{{\mathrm{ }}}}\)). e VG-dependent GL coherence length (ξGL) and superconducting layer thickness (dsc). f The spin-orbit energy-dependent midpoint of Tc (\(T_{{{\mathrm{c}}}}^{{{{\mathrm{mid}}}}}\)). The purple dashed line is a linear fit to the data at VG < −80 V.

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The limited ability of the electric fields in controlling the carrier density and disorder suggests that there should be an alternative origin for the continuous increase of Tc, especially for VG < −80 V where ns and Tc show anticorrelated behavior. In LAO/STO systems, the spin-orbit coupling, whose strength directly affects \(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{ab}}}}}{{{\mathrm{ }}}}\), is believed to contribute to stabilizing the 2D superconductivity29,39,40. To examine the validity of such scenario in our devices, we trace the evolution of \(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{c}}}}}{{{\mathrm{ }}}}\), \(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{ab}}}}}{{{\mathrm{ }}}}\), and spin-orbit energy (εso) during the gating process. As shown in Fig. 4d, We plot the comparison between \(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{c}}}}}{{{\mathrm{ }}}}\) (inset of Fig. 4d), μ0HP, and \(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{ab}}}}}{{{\mathrm{ }}}}\)(Fig. 4d) (at T = 0.1 K, calculated using the methods shown in Supplementary Figs. 10 and 11) upon varying VG. The much higher \(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{ab}}}}}{{{\mathrm{ }}}}\) relative to μ0HP (Fig. 4d) is likely to be caused by a strong spin-orbit interaction and/or VG-dependent electron wave function widths at the interface (which can be understood as the superconducting layer thickness dsc): in an effective model for 2D superconductor with strong SOC, we have17,41:

$$\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{ab}}}}}\left( {T = 0} \right) = \sqrt {\frac{{1.76\hbar k_{{{\mathrm{B}}}}T_{{{\mathrm{c}}}}}}{{\left[ {3\mu _{{{\mathrm{B}}}}^2\tau _{{{{\mathrm{so}}}}} + D\left( {d_{{{{\mathrm{sc}}}}}{{{\mathrm{e}}}}} \right)^2/3} \right]}}}$$
(2)

where τso is the spin-orbit relaxation time (\(\varepsilon _{{{{\mathrm{so}}}}} = \hbar /\tau _{{{{\mathrm{so}}}}}\)), D is the diffusion constant obtained from the slope of the out-of-plane upper critical field: [-d(\(\mu _0H_{{{{\mathrm{c}}}}2}^{//{{{\mathrm{c}}}}}{{{\mathrm{ }}}}\))/dT]T=Tc = 4kB/πDe (we use the data in Supplementary Fig. 5c to fit D because sample #2 has similar μ0Hc2 with the gating sample)41. The fits of μ0Hc2 using Eq. 1 allow us to determine the evolution of ξGL and dsc upon the continuous changing of VG from 150 to −180 V (Fig. 4e). As the VG varies from 150 to −180 V, ξGL (dsc) decreases from 32 (14) to 22 (2.7) nm (an illustration of VG-dependent dsc is shown in Supplementary Fig. 12), ξGL is much larger than dsc at all VG, especially for VG < 0, confirming the 2D nature of the superconductivity at EuO/KTO (110) interface in the gating process. Taking Tc = \(T_{{{\mathrm{c}}}}^{{{{\mathrm{mid}}}}}\), we also obtain the VG-dependent εso as plotted in Fig. 4d. With VG ramping from 150 to −180 V, εso increases from 1.2 to 21.1 meV with a large ratio of ~17.6, revealing a strong tunability of the SOC at the EuO/KTO(110) interface. The plot of Tc versus εso (Fig. 4f) indicates that Tc is predominantly controlled by different factors in two ranges of VG separated by VG = −80 V where Tc (εso) exhibits a clear kink. From VG = 150 to −80 V, Tc rises rapidly with decreasing VG. Considering the increasing ns with decreasing VG in this range, the increase of Tc is predominantly driven by the variation of carrier density. By contrast, as VG is swept from −80 towards −180 V, Tc increases linearly with εso (Fig. 4f), whereas ns gradually decreases. Therefore, we conclude that the increase of Tc is directly related to the enhancement of spin-orbit scattering in the range of VG from −80 to −180 V (Fig. 4b).

The present work is the first report on the electric-field-controlled SOC at the KTO-based SI. Here we briefly compare it with that in the STO-based SI. Distinct from the barely tunable SOC at the LAO/STO(110) interface42, the tunability of SOC in our samples is considerably large. Nonetheless, a relationship between Tc and εso similar to that displayed in Fig. 4d has been observed in LAO/STO(100) SI (see Supplementary Fig. 13a–c for our analysis based on the published data)15,16. We also noticed that, for the LAO/STO(111) SI, a dome-shaped VG-dependence of both Tc and εso has been established17, with a roughly linear relationship between these two (Supplementary Fig. 13d); such a close link between Tc and εso is proposed to hint at potential unconventional superconductivity pairing mechanism17,43,44. In our case, the unconventional linear relationship between the electric-field-control SOC and superconductivity implies the possibility of unconventional pairings at the KTO-based SIs, which definitely deserves further studies.

SOC relaxation time in magnetotransport

The strength of SOC can also be evaluated from the normal-state perpendicular magnetoresistance. Here, we measured the magnetoresistance at T = 3.8 K under different VG (Supplementary Fig. 14). In the diffusive regime, the field-dependent quantum correction to conductivity Δσ(H) can be described by the Maekawa-Fukuyama (MF) model17,27:

$$\begin{array}{l}\frac{{\Delta \sigma (H)}}{{\sigma _0}} = {{\Psi }}\left( {\frac{H}{{H_{{{\mathrm{i}}}} + H_{{{{\mathrm{so}}}}}}}} \right) + \frac{1}{{2\sqrt {1 - \gamma ^2} }}{{\Psi }}\left( {\frac{H}{{H_{{{\mathrm{i}}}} + H_{{{{\mathrm{so}}}}}\left( {1 + \sqrt {1 - \gamma ^2} } \right)}}} \right)\\ \quad \quad \quad \quad - \frac{1}{{2\sqrt {1 - \gamma ^2} }}{{\Psi }}\left( {\frac{H}{{H_{{{\mathrm{i}}}} + H_{{{{\mathrm{so}}}}}\left( {1 - \sqrt {1 - \gamma ^2} } \right)}}} \right) - \frac{{AH^2}}{{1 + CH^2}}\end{array}$$
(3)

here σ0 = 2e2/h is the quantum conductance, \(\Psi \left( x \right) = \ln \left( x \right) + \Psi \left[ {1/2 + \left( {1/x} \right)} \right]\) [where \(\Psi \left( x \right)\) is a digamma function and \(\gamma = g\mu _{{{\mathrm{B}}}}H/4eD_{{{{\mathrm{tr}}}}}H_{{{{\mathrm{so}}}}}\)], Hi and Hso are the inelastic and spin-orbit effective fields, respectively. The last term including parameters A and C is a Kohler term that originated from the classical orbital magnetoresistance. Combining the 2D nature of the superconductivity in the gating process, the diffusion coefficient Dtr can be expressed as: \(D_{{{{\mathrm{tr}}}}} = v_{{{\mathrm{F}}}}^2\tau /2\) (\(v_{{{\mathrm{F}}}} = \hbar \sqrt {2{{{\mathrm{\pi }}}}n_{{{\mathrm{s}}}}} /m^ \ast\) is the Fermi velocity, where m* is the effective electron mass). The relaxation time τ for elastic scattering can be extracted from Rsheet based on the Drude model: \(\tau = m^ \ast /{{{\mathrm{e}}}}^2n_{{{\mathrm{s}}}}R_{{{{\mathrm{sheet}}}}}\). By applying the fits to the MF model (Eq. 3) to the magnetoconductance (Fig. 5a), we obtained the parameters Hi,so (Fig. 5b) and A,C (Supplementary Fig. 15) at different voltages by assuming \(g = 2\) and m* = me45 (the variation of m* within the reasonable range (~0.5–1.0 me) does not change the qualitative conclusions, see Methods). Hso increases with VG decreasing, in agreement with the VG dependence of εso shown in Fig. 4d and verifies the enhancement of SOC at large negative VG.

Fig. 5: Spin-orbit scattering effect in magnetotransport.
figure 5

a The normalized transverse magnetoconductance [Δσ(H) = 1/Rsheet(H)-1/Rsheet(0)] measured at T = 3.8 K under different VG. H is applied perpendicular to interface. We ignore the Hall term due to its small amplitude (Supplementary Fig. 8). The curves are shifted vertically for clearance. The black solid lines are fits to the Maekawa-Fukuyama model (see text). b The VG-dependent effective fields Hi and Hso (see text) extracted from the fitting in (a). c The evolution of the relaxation times for inelastic scattering (τi), spin-orbit scattering (τso), and elastic scattering (τ) upon varying VG. d τ-dependent τso−1 determined from the magnetoconductance (MC) and τso determined from the upper critical field. The dashed lines are linear fits to guide the eyes.

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The evolution of spin-orbit relaxation time τso and the inelastic relaxation time τi can be further derived from the effective fields: \(H_{{{{\mathrm{so}}}},{{{\mathrm{i}}}}} = \hbar /4eD_{{{{\mathrm{tr}}}}}\tau _{{{{\mathrm{so}}}},{{{\mathrm{i}}}}}\). In Fig. 5c we plot all three relaxation times τi, τso and τ against VG. τso is the smallest among them, which means that the spin-orbit scattering in the KTO-based SI is strong and dominates the decoherence process. More intriguingly, as shown in Fig. 5d, we have τso−1 ∝ τ; this is consistent with the expectation for the D’yakonov-Perel’ (DP) mechanism of spin relaxation46. The DP scenario describes the spin precession around the spin-orbit field between scatterings that leads to the spin dephasing; such mechanism is consistent with Rashba-type SOC at the interface46. However, we mention that if we plot the τso extracted from εso (Fig. 4d) determined from Hc2 against τ (Fig. 5d), it shows τso ∝ τ, i.e., τso obtained from Hc2 and the magnetoconductance exhibit distinct behaviors. The relationship of τso ∝ τ corresponds to the Elliott-Yafet (EY) mechanism47,48 describing spin-flip scatterings. Hence, the spin-orbit scattering that affects the pair-breaking effect of the Zeeman field and that contributes to the quantum correction of charge transport in the normal state are assigned to the EY and DP mechanisms, respectively. A possible explanation for this discrepancy is that Cooper pair formation and the normal-state electrical transport are dominated by electrons occupying different conduction channels or subbands at the interface. Similar behavior has also been observed for the LAO-STO system41. More exotic probabilities, such as SOC-enhanced spin susceptibility in the superconducting state (which naturally enhanced the Pauli limit critical field) or unconventional superconducting pairings49,50,51, are to be verified by future investigations.

To conclude, high-quality single-crystalline EuO (111) thin films have been grown on KTO (110) substrates. The large anisotropy of Hc2 and the characteristics of a BKT transition show that the interface between them is a 2D superconductor. The remarkable response of Tc to the applied VG is proved to be predominantly linked to the high tunability of SOC strength under external electric fields. τso obtained from Hc2 and the magnetoconductance manifests the typical behaviors expected for the EY and DP spin-relaxation mechanisms, respectively, implying the complexity of the SOC effects at the EuO-KTO SIs. Our results demonstrate that the SOC should be considered as an important factor controlling the 2D superconductivity and might lead to unconventional superconductivity at the KTO-based interface. Further theoretical investigations are needed to elucidate such unusual interplay between the electric-field-control SOC and superconductivity.

Methods

Growth of EuO/KTO(110) heterostructures and device fabrication

EuO (111) thin films were grown on (110)-orientated KTO single crystal using a molecular beam epitaxy system with a base pressure of 4 × 10−10 mbar. The samples size are 5 × 5 mm2. Before growth, the KTO substrates were pre-annealed at 600 °C for 1 h and then cooled down to growth temperature. The deposition rate of Eu was 0.2 Å/s, calibrated by a quartz-crystal monitor. The depositions were performed at 400 °C. The oxygen pressures during the growth of the two samples are 1.9 × 10−9 mbar for #1, 2.0 × 10−9 mbar for #2. After growth, the samples were cooled down to room temperature with no oxygen supply. A 3–4-nm-thick germanium was prepared to protect the sample from further oxidation when exposed to air.

The hall bar devices were prepared using standard optical lithography and Argon etching techniques. The etching thickness is 40 nm which is much larger than the thickness of EuO films.

Scanning transmission electron microscopy (STEM) and transport measurements

The slices for STEM were prepared from selected areas using Carl Zeiss Crossbeam 550 L and the high angle annular dark field scanning transmission electron microscopy (HADDF STEM) was obtained using a probe Cs-corrected JEOL-ARM200F NEOARM.

The transport measurements were carried out in a 3He cryostat (HelioxVT, Oxford Instruments) and a commercial Quantum Design PPMS with a dilution refrigerator insert.

Verification of the dirty-limit scenario

For a weak coupling BCS superconductor, the BCS coherence length \(\xi _{{{{\mathrm{BCS}}}}} = \left( {\hbar v_{{{\mathrm{F}}}}} \right)/\left( {{{{\mathrm{\pi }}}}1.76k_{{{\mathrm{B}}}}T_{{{\mathrm{c}}}}} \right)\). In the negative VG regime for our sample, \(l_{mfp}/\xi _{{{{\mathrm{BCS}}}}} = \left( {{{{\mathrm{\pi }}}}1.76k_{{{\mathrm{B}}}}T_{{{\mathrm{c}}}}\tau } \right)/\hbar\) ranges from 0.05 to 0.07. Thereby, the condition \(l_{{{{\mathrm{mfp}}}}} \ll \xi _{{{{\mathrm{BCS}}}}}\) for the dirty-limit superconductors is still valid at our SIs. This result validates the application of Eq. 2 in the main text which describes the upper critical field of a 2D superconductor in the dirty limit.

Effective mass of electrons

Due to the relatively low mobility of our samples, our available experimental probes fail to resolve quantum oscillations in magnetoresistance down to 0.1 K (Supplementary Fig. 16). In ref. 45, the effective mass of electrons m* ≈ 0.62me under high magnetic fields. To verify the influence of the effective mass of electrons on our experimental conclusions, we assumed m* = 0.5me and reanalyzed our data, as shown in Supplementary Fig. 17. All parameters (Hi, Hso, A, C) show the same magnitude and trend as the results obtained by assuming m* = me; the consistency is particularly good for Vg < 0. Therefore, we propose that our fits do not strongly depend on the value of m*, and the variation of m* within the reasonable range (~0.5–1.0 me) does not change the qualitative conclusions.