Introduction

Ever since the advent of superconductors, investigations of the mechanism of superconductivity have been at the forefront of condensed matter physics1,2,3,4,5,6,7,8. Recent years have seen the discovery of superconductivity in a plethora of engineered material systems, such as9,10 LaAlO3/SrTiO3, FeSe/SrTiO3 and twisted bilayer11 and trilayer12,13 graphene. These interfacial superconductors are of great interest due to the unconventional nature of their superconductivity, offering new routes towards pairing of electrons, as well as the tunability of their properties using electric field-effect gating14. For example, the presence of spin-orbit coupling and broken inversion symmetry at interfaces could enable new channels for Cooper pairing15, and for realizing a pairing state15,16 that can host zero-modes with non-Abelian statistics17,18. Insights into the pairing mechanisms in these material systems are thus of great interest, and an area of intense experimental and theoretical research.

In this work, we investigate the origins of superconductivity discovered at oxide-insulator/KTaO3 interfaces19 through chemical doping and electric field gating. We establish the doping dependence of Tc for KTaO3 (KTO) interfaces of different crystallographic orientations. A remarkable proportionality where Tc ∝ n2D over nearly an order of magnitude is observed at the EuO/KTO (111) interface. In contrast, no superconductivity is observed at the (001) interface of KTO down to 25 mK for all n2D, while the KTO (110) interface superconducts at a Tc as high as 1 K, intermediate between the (111) and (001) cases. Using electric field-effect gating, we tune both n2D, as well as the confinement of electrons to the interface. We find that Tc increases for both. We rule out the possibility that Tc is controlled by the phase stiffness temperature20. We then discuss possible scenarios for pairing and find that inter-orbital interactions mediated by the soft transverse optical phonon reproduces key findings in our measurements, particularly the extreme sensitivity of Tc to the crystallographic orientation of the interface, and the consistency with the linear dependence of Tc with n2D.

Results

Distinct n 2D dependences of T c at KTO (111), (001), and (110) interfaces

Two-dimensional electron gases (2DEGs) at KTO interfaces are obtained by growing an oxide overlayer EuO on (111), (001), and (110) KTO surfaces using molecular beam epitaxy (MBE). The charge carriers at the EuO/KTO interface originate from chemical doping of oxygen vacancies and/or Eu substitution for K near the interface with KTO19. The carrier density n2D in these samples is determined using Hall measurements at 10 K. n2D of the as-grown samples can be tuned by nearly a factor of 10 by varying the growth conditions (see Methods section). We find that the donor state of charge carriers in these samples evolves gradually from shallow to deeper energy as n2D increases, the details of which are presented in the Methods section and Supplementary Fig. 1.

The EuO/KTO(111) samples are labeled as KTO_#, with # being in the order of increasing n2D. Figure 1a shows the temperature dependence of the ratio of the sheet resistance Rs/RN (RN is the normal-state resistance at 4.5 K) for KTO (111) samples with different n2D. The superconducting transition defined by Rs/RN going to zero occurs at progressively higher temperatures with increasing n2D. We note that the variation in the sheet resistance in the normal state in these samples produces differences in the superconducting transition width. This behavior is expected for a 2D system when the conductance is enhanced by superconducting fluctuations. Figure. 1b shows the measurement of Rs(T) on EuO/KTO (001) and (110) samples. As reported before19, no superconducting transition can be measured for the (001) samples down to the lowest temperatures of ~25 mK. On EuO/KTO(110) samples, we observe that superconductivity emerges below about 1 K, which is similar to that reported for the LaAlO3/KTO (110) interface21. The transition temperature Tc for the KTO (110) samples also increases with n2D, but gradually saturates.

Fig. 1: Tc tuned by n2D through chemical doping.
figure 1

a, b The ratio of the resistance Rs/RN as a function of temperature measured for KTO (111) samples (a) and KTO (001), (110) samples (b), respectively, with varying n2D. The direction of the arrow indicates the increase of n2D. c, d n2D dependence of Tc for KTO (111) samples shown in (a) and KTO (001), (110) samples shown in (b), respectively. The red circle data point in (c) is taken from Ref. 59.

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When Tc (determined at 20% of RN) of all the KTO (111) samples are plotted against n2D, as shown in Fig. 1c, we observe a largely linear dependence of Tc on n2D. The solid line is a fit using Tc = a· n2D + b, where the coefficient a ≈ 0.2 × 10−13 K cm2, and the y-axis intercept b ≈ −0.1 K. We note that the fitting line extrapolates to near the origin of the axes, which implies that the undoped KTO (111) interface is near a doping induced critical point separating a band insulator and a superconducting phase. The linearity shown here is insensitive to the choice of Tc as shown in Supplementary Fig. 1.

Within the entire doping range, Tc versus n2D does not show a dome, suggesting that Tc may be further enhanced by increasing the doping level. In Fig. 1d we plot Tc versus n2D for KTO (001) and (110) interfaces, which show entirely different behavior than that of the KTO (111) interface: Tc cannot be measured, down to 25 mK, for all n2D for the KTO (001) interface; while Tc for the KTO (110) interface increases with n2D but saturates at a value about half that of the (111) case. We have also measured the Ginzburg-Landau coherence length (ξGL) in the KTO (111) samples with different n2D, and found that ξGL satisfies a scaling relation between Tc and Hall mobility. The details are presented in the Methods section and Supplementary Fig. 2.

Electrostatic tuning of superconductivity in a low n 2D sample

To gain further insight into the mechanism of the superconductivity, we have performed electrostatic gating measurements on the EuO/KTO (111) samples, in both the low and high n2D limits. It is expected that electric fields can tune a relatively large fraction of charge density in the low n2D sample in a back-gate geometry, given the large dielectric constant (~4500) of KTO at low temperatures22,23,24. Figure 2a shows a schematic of our measurement. A 100-nm thick Pt film deposited on the bottom side of KTO is used as the positive electrode, while the 2DEG at the EuO/KTO interface is grounded.

Fig. 2: Electrostatic gating on low n2D sample.
figure 2

a Schematic of the back gate measurement geometry. b Rs versus T measurements on KTO_1 for different VG. c Tc versus n2D controlled by electrostatic gating. Inset shows n2D as a function of VG. d V–I measurement for different VG at T = 0.1 K.

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Figure 2b shows the measurements of Rs(T) for different gate voltages VG on KTO_1, which has the lowest carrier density n2D = 1.25 × 1013 cm−2. For this sample, RN increases by over three times when VG is varied from 30 V to –20 V. Correspondingly, Tc becomes lower. We have measured n2D, which increases with VG monotonically, as shown in the inset of Fig. 2c. By using a parallel-plate capacitor approximation and the thickness of KTO substrate of 0.5 mm, the slope of n2D vs. VG at VG = 0 V yields a dielectric constant for KTO of ~5700, which is close to the known value for bulk KTO. When Tc is plotted against n2D as tuned by VG (Fig. 2c), we observe a linear dependence of Tc on n2D, i.e., Tc = 0.26 × 10−13 (K cm2)·n2D – 0.1 K. Crucially, the coefficient for n2D here is close to that found in the n2D dependence of Tc obtained through chemical doping shown in Fig. 1c. The consistency between the electrostatic gating and chemical doping measurements highlights the predominant role of the carrier density, regardless of its origin, in tuning Tc at the KTO interface. Figure 2d shows that the critical current Ic also increases monotonically with VG. We note that in recent ionic liquid gating measurements25 on KTO, Tc also changes monotonically with VG, though the inferred n2D does not.

Enhanced T BCS and a dome in T BKT by electrostatic gating

Next, we consider the effects of field-effect gating at higher carrier densities. Figure 3a shows Rs(T) measured on KTO_9 with n2D = 1.04 × 1014 cm−2, for different VG. As VG varies from 200 V to – 200 V, RN increases by about a factor of 2. At the same time, the superconducting transition occurs at higher temperatures, which is different than that seen on the low n2D sample. Utilizing the response of superconductivity to VG, we show in Fig. 3b that the superconducting state with zero resistance can be switched on or off reversibly when VG is varied between – 75 V and 200 V at T = 2.01 K. We note that a superconducting state is maintained at low temperature over the entire range of VG.

Fig. 3: Electrostatic gating for a high n2D sample.
figure 3

a Temperature dependence of Rs measured with different VG. The increment in VG shown in the plot is 50 V. b Reversible switching of the superconductivity by VG at T = 2.01 K. c Halperin-Nelson fit (solid line) to the Rs(T) data. d Enhanced conductivity Δσ/σn from superconducting fluctuations as a function of both VG (x-axis) and temperature T (y-axis). e Mean-field (blue squares) and BKT (red dots) transition temperatures TBCS and TBKT as a function of VG. f Log-Log plot of the V–I measurement; the red dashed line is V ∝ I3, indicating TBKT ~ 2.01 K. g Comparison of the ratio TBKT/TBCS with theory. h Hall mobility (left axis) and carrier density (right axis) as a function of VG.

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We found that the tunability of superconductivity here is due to an enhancement of the mean-field transition temperature TBCS with negative VG. The transition from superconducting to normal state shown in Fig. 3a can be precisely interpreted by the Halperin-Nelson (HN) formula26, which describes the increase in resistance with rising temperature due to the Berezinskii-Kosterlitz-Thouless (BKT) transition arising from superconducting phase fluctuations in 2D, as well as amplitude fluctuations of Cooper pairs at still higher temperatures as proposed by Aslamasov and Larkin (AL). The solid curve in Fig. 3c (and all those in Fig. 3a) is a fit to the data using Rs (T) = RN(1 + 4 A−2sinh2(b(T – TBKT)−1/2TBKT1/2))−1, where A and b are fitting parameters that depend on the energy of the vortex core and the phase stiffness27. We find that the vortex core energy extracted from the fits is close to the expected value for an XY model, indicating that inhomogeneity does not need to be considered to get physical values for the fitting parameters (similar findings to ours, though with a lower value of the vortex core energy, have been recently reported28). In the HN expression, Δσ/σN = 4A−2sinh2(b(T–TBKT)−1/2TBKT1/2) is a measure of the enhanced conductivity by superconducting fluctuations. Figure 3d shows a contour plot for Δσ/σN as a function of VG and temperature T, which is obtained from the HN analysis for measurements over the full range of VG. As VG goes negative, Δσ/σN shows monotonic enhancement for T ≥ 2.15 K, while at lower temperatures a local maximum in Δσ/σN is seen.

From the HN analysis, we obtain the TBKT, which is found to be close to Tc0, where Rs becomes zero. The mean-field TBCS is obtained from the inflection point29 of the HN fit (maximum of its first derivative, near 20% of RN), and has been confirmed by AL fits (Supplementary Fig. 3). Figure 3e shows how TBCS and TBKT evolve as a function of VG. TBCS is monotonically enhanced as VG goes to negative values, while TBKT shows a local maximum at VG ~ –75 V. We also measured the voltage-current (V–I) characteristic near Tc0 to determine TBKT independently. As shown in Fig. 3f, TBKT obtained through V-I measurements at VG = 0 V is ~2.01 K, very close to the results obtained by the HN fit. See also Supplementary Fig. 4 for V–I measurements at different VG.

We find that the non-monotonic dependence of TBKT on VG is caused by the enhancement of TBCS together with the increase of RN as VG decreases. The ratio of TBKT/TBCS from our data, as shown in Fig. 3g, follows closely the prediction for a 2D superconductor30, i.e., TBKT/TBCS = (1 + 0.173RN/Rc)−1, where Rc = ħ/e2 with ħ being the reduced Planck constant. The increase of RN by negative VG is mainly due to charge carriers being pushed closer to the EuO/KTO interface, where more disorder is present31,32. As shown in Fig. 3h, the Hall mobility decreases from about 37 cm2 V−1 m−1 to 17 cm2 V−1 m−1 as VG changes from 200 V to –200 V, while the variation in carrier density from Hall measurements is only ~5%. These observations show that, in addition to n2D as found earlier, the proximity of the carriers to the interface also increases Tc (TBCS), suggesting the interface itself is the origin of pairing. We also observe that the critical field is enhanced at negative VG, which is found to be mainly due to the decrease in mobility (Supplementary Figure 5).

Pairing mechanism at the KTO interfaces

We first discuss the marked proportionality of Tc with n2D observed for the KTO (111) interface. One possible explanation for this phenomenon is that Tc is limited by the phase stiffness of the superconducting state, where Tc ∝ EJ (TBKT) with EJ = ħ2ns(T)/4 m*, ns the 2D superfluid density, m* the effective mass of the electrons, and ns(0 K) ∝ n2D. This would give a linear relation between Tc and n2D. However, our analysis in Fig. 3c–h shows that TBKT is close to TBCS, and in fact this is the case for samples over the full doping range studied. The closeness of TBKT and TBCS is expected for a conventional 2D superconductor, where the temperature range over which BKT physics is dominant is small, while the initial resistance drop is due to amplitude fluctuations of Cooper pairs that control the transport over most of the temperature range associated with the superconducting transition. Therefore, in these KTO samples, the transition is mainly governed by TBCS.

The above discussion leads to the second possibility for the observed proportionality between Tc and n2D, which is that the pairing interaction is non-BCS like with a superconducting gap Δ ∝ EF (Fermi energy) instead of ħωD (Debye energy). Such a situation would occur in the anti-adiabatic limit33 with \(\hslash {\omega }_{D}\) > EF. Since in 2D, n2D scales with EF, one would obtain Tc ∝ Δ ∝ n2D. The estimated EF in our samples lies in the range of 10−80 meV, so pairing via the highest-energy longitudinal optical mode (LO4) with energy ~100 meV would be required. There are several potential issues with such a scenario. First, whether ħωD can by replaced by EF in the BCS expression for Tc is controversial5. Second, it would require that the BCS coupling constant be independent of n2D.

This leads us to consider pairing via the soft transverse optical (TO1) phonon mode34, the phonon responsible for the nearly ferroelectric behavior of KTO. We note that the standard gradient coupling of the electrons to TO modes vanishes as the momentum transfer q goes to 0, meaning that only pairs of phonons can couple to the electrons35. However, as pointed out in recent work36,37, inversion breaking from the TO1 mode leads to linear coupling to electrons for inter-orbital interactions among the three t2g orbitals: dxy, dyz, and dzx, which is illustrated in Fig. 4a. The displacements of Ta and O sites in the vertical direction, indicated by the black arrows, break inversion symmetry with respect to the horizontal plane joining the center of the d orbitals. As a result, the inter-orbital hopping amplitude of electrons along a Ta-O-Ta bond is no longer zero.

Fig. 4: Pairing via transverse optical phonons.
figure 4

a Illustration of inter-orbital hopping of electrons between adjacent Ta sites via the O py orbital that becomes allowed by inversion symmetry breaking displacements (shown by the black arrows) of O and Ta atoms due to a TO1 phonon. Here, 3d orbital profiles are shown for illustration purposes. The 5d orbitals for Ta have the same angular symmetries but possess extra nodes in the radial direction. b Calculated Tc versus n2D from TO1 phonon pairing for KTO (111) and (001) interfaces, the latter assuming λ001 = λ111/2. c–e Lattice structures formed by the first two layers of Ta and O atoms along the [111], [001], and [110] axes of KTO, respectively. Ta atoms in the upper and lower layers are shown in red and blue, respectively. The dark and light color of Ta-O-Ta bonds indicates the presence or absence of electron-TO1 phonon coupling, respectively. The degeneracy of the dxy, dyz, and dzx orbitals in energy for each of these KTO surfaces is indicated below each panel by their vertical positions, the degeneracy at the lowest energy (3, 1, 2) correlating with Tc. f Rashba-like splitting of bands for a KTO (111) bilayer due to displacements of Ta and O atoms perpendicular to the surface. g A similar displacement of atoms for the (001) surface does not produce a noticeable Rashba-like splitting.

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We show in detail in the Methods section that: (i) The TO1 mode energy (ωΤΟ1) increases as a function of n2D due to the electric field screening provided by the charge carriers. (ii) The BCS coupling constant λ (arising from the Rashba-like splitting of the energy bands with wavevector k) also increases with n2D. We take Tc = 1.14 ωΤΟ1(q = 2kF, n2D) exp[–(1 + λ)/λ]. Adjusting λ = c n2D < 1/ωΤΟ12 > by a free parameter c to scale the value of Tc to equal the experimental value of ~2 K at n2D = 1 × 1014/cm2, we obtain a nearly linear dependence of Tc on n2D for the KTO (111) interface, which is shown in Fig. 4b. Here, <> is a Fermi surface average (see Methods) and we assume that ωΤΟ12(q = 0) = c1 + c2 n2D as observed in SrTiO3, with c1 and c2 being constants.

This scenario also predicts the absence of superconductivity for the KTO (001) interface, i.e., the superconductivity is highly sensitive to the crystallographic orientation. This is due to the nature of the inter-orbital electron-TO1 phonon coupling and the effects of quantum confinement for electrons at the KTO-oxide interface. The degeneracy of the t2g orbitals for the 2DEG is lifted by quantum confinement depending on the crystalline surface normal, which is illustrated in Fig. 4c–e for the three KTO surface orientations. This lifting of degeneracy reduces the number of d orbitals participating in the inter-orbital hopping. Consequently, electrons and TO1 phonons are maximumly coupled for the KTO (111) surface where the three t2g orbitals are degenerate, while they are largely decoupled for the KTO (001) surface given the splitting between the dxy and dxz/yz orbitals. Shown in Fig. 4f are the calculated energy bands for the KTO (111) surface exhibiting large ‘Rashba’-like splitting due to inter-orbital hopping, while the energy bands for the KTO (001) surface shown in Fig. 4g do not exhibit a noticeable splitting. Further, we find that the electron-TO1 phonon coupling strength for the KTO (110) surface is intermediate because the inversion breaking does not occur along the [001] crystal axis (Fig. 4e) in the (110) plane, reflecting the reduced orbital degeneracy of (110) relative to (111). We note that the apparent saturation of Tc for the KTO (110) interface may be due to its unusual electronic structure, with the predicted Fermi surface intersecting the zone boundary for n2D ~ 7 × 1013 cm−2. This and other aspects of the (110) data will be presented in a future paper.

A more detailed analysis of pairing in KTO is highly non-trivial. It would also need to account for splitting of the TO1 mode due to both symmetry breaking and the electric field at the interface. As the inversion breaking is largest at the interface, this would be consistent with our observed increase of TBCS when the charge carriers are pushed closer to the interface by a negative VG. A detailed analysis also needs to consider (1) the influence of both LO and TO modes on Tc, (2) the carrier distribution n3D(z) along the interface normal (z) together with the strong interface orientation dependent electronic structure38,39,40, and (3) the influence of inter-orbital terms between the t2g and eg orbitals. We note that while the dependence of Tc on crystal orientation is not observed9,41,42 in SrTiO3 based 2DEGs (barring a recent exception43), this may be due to a weaker lifting44,45,46 of the degeneracy of the 3d orbitals (the confinement potential being weaker in STO than KTO). We also note that while the t2g orbitals in bulk KTaO3 are degenerate, it is not known to superconduct47. We believe this may be due to the relatively low density of states in doped bulk KTO samples. In 3D, the density of states, N, is proportional to kF whereas in 2D it is a constant. For the range of carrier densities we explore, N3D is significantly smaller than N2D, leading via λ to an exponential suppression of Tc in the 3D case (see Methods section and Supplementary Fig. 6).

Methods

Material growth

The growth steps include annealing the substrate at a temperature ~600 °C, and then exposing it to a flux of Eu atoms in the temperature range 375−550 °C, both carried out for pressures in the 10−10 Torr range. The annealing process promotes O vacancy formation near the surface, and subsequent exposure to Eu causes the formation of a thin layer of EuOx, as the Eu scavenges O from the substrate19. Following this step, the EuO overlayer is grown under a O2 partial pressure between 0.5–10 × 10−9 Torr with a final EuO thickness of ~25 nm. Our microscopy results19 indicated the presence of O vacancies and Eu substitution on the K sites in KTO, which would both promote the formation of an interfacial electron gas. Upon varying the temperature at which the KTO substrate is exposed to Eu in vacuum, we were able to vary the doping levels in the 2DEG. We note that the EuO/KTO interface can show transport anisotropy or a ‘stripe’-like phase in the (111) plane19. However, this does not influence the analysis of Tc, at which global superconductivity sets in as that is found to be the same for ‘stripe’ like samples at a given value of n2D. We also find that in samples with ‘stripe’ like behavior the anisotropy is strongly reduced when patterned into Hall bars vs those that are measured using a van der Pauw geometry, details of which are currently being studied. The samples presented here show weak or no anisotropy.

Evolution of donor states as a function of doping

n2D in these samples is determined using Hall measurements at 10 K. Supplementary Figure 1c shows the Hall resistance RH measured as a function of magnetic field for different KTO (111) samples. Notably, the field dependences of RH are all linear, with negative slopes, consistent with electron-like charge carriers. We found that the donor state of charge carriers evolves gradually from shallow to deeper energy as n2D increases. Supplementary Figure 1d shows the temperature dependence of n2D in the range 300 K–2 K. For samples with n2D(10 K) ≲ 5 × 1013 cm−2, the measured n2D(T) increases upon cooling. This suggests that charge carriers in these samples mainly come from shallow donors whose energies merge with the conduction band due to the nearly 20-fold increase in the dielectric constant in KTO upon cooling. For samples with n2D(10 K) ≳ 5 × 1013 cm−2, n2D(T) decreases upon cooling. This suggests that at higher doping levels charge carriers originate from deeper donor states48, to which some of them freeze out upon cooling. Supplementary Figure 1d also shows that regardless of the doping level in the as-grown samples, n2D remains nearly constant for temperatures below 10 K for all samples.

Scaling relation of coherence length

We have measured the Ginzburg-Landau coherence length (ξGL) in the KTO (111) samples with different n2D, by measuring the out-of-plane upper critical field Bc2 as a function of temperature. Bc2 shows a linear temperature dependence near Tc, from which we calculate ξGL using Bc2(T) = Φ0(1–T/Tc)/[2π(ξGL)2], where Φ0 is the magnetic flux quantum. Supplementary Figure 2a shows the data of Bc2 versus T for five samples. The Bc2 varies greatly among these samples with different n2D and Tc. Supplementary Figure 2b summarize the ξGL for all the samples measured, which lies in the range from 10 to 57 nm. We found that ξGL scales with (μ/Tc)1/2. As shown in Supplementary Fig. 2b, the ratio of ξGL to (μ/Tc)1/2 remains constant for samples over the entire n2D range. We found that this relation also holds when the superconductivity is tuned by VG (see Supplementary Fig. 5).

For a BCS superconductor in the dirty limit, it is expected that ξGL ∝ (lξBCS)1/2 with ξBCS being the BCS coherence length49. Using l ∝ vFμ and ξBCS ∝ vF/Tc with vF being the Fermi velocity, one gets ξGL ∝ vF(μ/Tc)1/2. For the KTO (111) interface, we observe that ξGL ∝ (μ/Tc) 1/2 instead. This would imply that vF is a constant, independent of n2D. One way this could happen is if there is a linear dispersion of energy versus momentum like in graphene, though this conflicts with the known electronic structure of the KTO conduction bands that have a parabolic dispersion around the Γ point.

Calculation of T c versus n 2D from TO1 mode pairing

Evaluation of TO1 mode energy

The BCS formula for pairing from a TO1 mode is taken to be Tc = 1.14 ωΤΟ1(q = 2kF, n2D) exp[–(1 + λ)/λ] where λ is the BCS coupling constant (any Coulomb pseudopotential, μ*, has been ignored). We first discuss the assumptions behind this expression, and then how to evaluate the various quantities. The first assumption is that the relevant pairing scale is controlled by the momentum transfer q of the TO1 mode for scattering around the Fermi surface, hence we set the cut-off energy to the maximum mode energy which is at q = 2kF. Therefore, the n2D dependence of Tc will come from any hardening of the TO1 mode along with the increase of kF with n2D since the TO1 mode has a substantial dispersion. The TO1 mode dispersion is taken from the low energy neutron scattering study of Farhi et al.50. There, the phonon dispersion is fit with an expression due to Vaks, and in the approximation where the anisotropy of the dispersion is ignored involves solving a 2 by 2 matrix that couples the transverse acoustic mode to the transverse optic one. We do this using the parameters of Ref. 50. evaluated at their lowest temperature of measurement (10 K). To proceed further, we need to know the n2D dependence of the mode energy at q = 0, as well as that of kF. For the former, we will assume that ωΤΟ12(q = 0, n2D) = c1 + c2 n2D; this relation is found in STO where doping of the bulk with free carriers has been observed51 (similar studies do not exist for bulk KTO). The resulting mode hardening is consistent with the electric field dependence of the Raman data of Fleury and Worlock52. c1 is set by the bulk value of ωΤΟ1 of 2.5 meV. We set c2 by the estimated value of ωΤΟ1 at n2D = 1014 cm−2 of 5.6 meV. The latter has been obtained from Ueno et al.53 that relates the field dependent dielectric function to n2D, with the relative dielectric function expressed as ε(E) = 4500/(1 + b E) with b = 8 × 10−7 (E in V m−1). The relevant “average” field F is then determined as in ref. 54. by assuming a triangular confining potential along z (the normal to the interface), and integrating the dielectric constant with respect to field up to F. That is, e·n2D = 2 0∫Fε0·ε(E)dE, where e is elementary charge and ε0 is 8.85 pF m−1. The mode energy is then given by the Lyddane-Sachs-Teller relation ωΤΟ12(q = 0, n2D) = ωΤΟ12(q = 0, n2D = 0) ε(0)/ε(F) that was found to be obeyed in the field dependent data of ref. 51. The reason we did not use this formalism over the entire range of n2D is that the resulting n2D dependence of ωΤΟ12(q = 0) deviates from the above assumed linear behavior (Supplementary Fig. 6e), which in turn has a detrimental impact on the functional dependence of Tc on n2D (Supplementary Fig. 6f, where a dome is found instead). In the future, this could be looked at by measuring the actual TO1 mode energy as a function of n2D in KTO (realizing that the TO1 mode polarized normal to the interface should harden more with n2D than the TO1 mode orthogonal to this).

To obtain the variation of kF with n2D, we assume the simple tight binding model of ref. 55, where the electronic structure is that of a (111) bilayer, with the parameters determined in order to reproduce the ARPES data from the (111) KTO surface of ref. 40. This results in a near neighbor hopping energy t of 1 eV, with the spin-orbit coupling ξ = 0.265 eV (in order to reproduce the quartet-doublet spin-orbit splitting of 0.4 eV at Γ). For kF, we take that of the larger of the two Fermi surfaces (the quartet splits into two doublets upon dispersing away from Γ), and set kF to its value along Γ-K of the surface Brillouin zone. n2D is gotten by determining the occupied area of the two Fermi surfaces at EF. The resulting dependence of EF and the TO1 mode energies with n2D is shown in Supplementary Fig. 6a. One can see the approximate linear dependence of EF with n2D (expected for a parabolic dispersion in 2D), with the deviation at larger n2D due to the approach to a van Hove singularity of the lower band of the quartet that occurs at the M point of the surface zone. As one can also see, the dependence of the mode energy at q = 2kF is not linear in n2D. This is to be expected, since in the approximation where the mode energy at q = 0 is small compared to that at q = 2kF, the mode energy should scale with q and thus kF, the latter going approximately as the square root of n2D. Also note that over most of the n2D range, the mode energy at q = 2kF is much smaller than EF, justifying the BCS (adiabatic) approximation for TO1 mode pairing. We note that although our experimental results, such as the orientational and gate-field dependences of superconductivity, suggest an underlying 2D electronic structure, we cannot rule out the existence of ‘3D’ electronic states at KTO (111) interfaces.

BCS coupling constant λ

For TO1 mode pairing, λ is given by:37 λ = c n2D < 1/ωΤΟ12(q, n2D)> where <>  represents an average over the Fermi surface. As the mode dispersion is quadratic in q, the Fermi surface average is trivial, resulting in λ = c·n2D/[ωΤΟ1(q = 2kF, n2D)·ωΤΟ1(q = 0, n2D)]. Here, n2D in the numerator comes from kF2 (kF2 = 2π·n2D). This factor arises from the electron-phonon vertex that is proportional to kF (that is, the effect of inversion breaking from the polar TO1 mode leads to a linear splitting of the energy bands with k). More will be described about this inversion breaking effect in the next section. The constant c subsumes the proportionality coefficient of the vertex along with the density of states at the Fermi energy, NF, that we assume is constant (equivalent to linearity of EF with n2D). This is in contrast to the bulk56 where NF goes as n3D1/3. The value of c would require a detailed microscopic theory. In lieu of that, we set c in order to give Tc of ~2 K at n2D = 1014 cm−2 (for this n2D, λ ~ 0.26 and so c = 2.85 when n2D is in units of 1013 cm−2 and ωTO1 in meV). The resulting λ and Tc versus n2D is plotted in Supplementary Fig 6b, c, respectively. Tc is then replotted in Supplementary Fig 6d assuming that λ is reduced by a factor of 2.

One can see a remarkable linear variation of Tc with n2D in Supplementary Fig. 6c, despite the non-linearity with n2D of both the BCS cut-off and the BCS coupling constant. We note that the variation of Tc with n2D is a result of the interplay between the dependences of ωTO1 and the electron-phonon coupling λ on n2D. Obtaining a linear behavior of Tc is sensitive to details of both dependences. However, the rise in Tc with n2D is a general finding, in part due to the increase of ωTO1 with n2D along with the strong dispersion of the mode. Therefore, we expect this rise will cease once the phonon wavevectors spanning the Fermi surface approach the zone boundary.

In Supplementary Fig. 6d, one sees a strong suppression of Tc, with values similar to those claimed by Ueno et al. for the (001) interface53 (a further reduction in λ would completely suppress Tc as we observe). Since λ is proportional to the square of the electron-phonon vertex, a mere reduction of the vertex by 21/2 is sufficient to cause an enormous reduction in Tc. As such, we expect Tc to be extremely sensitive to the interface orientation, which we address next. We note that in a more recent work25 using ionic liquid gating of the KTO (001) surface, superconductivity was not found.

Dependence of T c on interface orientation from TO1 mode pairing

Electron-phonon vertex

To obtain a significant electron-phonon vertex for a TO1 mode, one requires inter-orbital terms (intra-orbital ones instead lead to a coupling quadratic in the ion displacement35). Therefore, in a TO1 mode picture, one requires orbital degeneracy in order to get a substantial Tc. This qualitatively explains the observed dependence of Tc with interface orientation. For the (111) case (ignoring any trigonal distortion), the three t2g states are degenerate at Γ (being split into a lower quartet and an upper doublet by spin-orbit). We can contrast this with the (001) case, where confinement along [001] leads to the dxy state being pulled down substantially relative to the dxz/yz ones. This splitting acts to suppress the effect of inter-orbital terms, consistent with our lack of observation of superconductivity on (001) interfaces (in ref. 53, a Tc of <50 mK was observed). For (110), we expect something intermediate, because the effect of confinement in that case is to instead pull down the dxz/yz states relative to the dxy one (so, partial degeneracy).

To go beyond these qualitative observations, we need to consider the variation of the electronic structure with interface orientation, and we also need an estimate of the coupling of these electrons to the TO1 mode. To address the latter, we follow ref. 37, where the TO1 mode is considered to be a Slater mode (known to be a good approximation for the TO1 eigenvector). As they show, there are two contributions. One is the standard gradient one which only involves the metal ion motion and disappears as q goes to 0. The second is the inter-orbital terms that involves primarily the oxygen ion motion and does not vanish as q goes to 0.

The inversion splitting of the bands near Γ due to the TO1 mode gives rise to an electron-phonon vertex37 gTO1 = 2tʹcʹdkFa where tʹ is the derivative of the near-neighbor hopping with respect to the ion displacement, cʹ = (η−1/2 + η1/2) where η = 3MO/MTa, d is the zero-point displacement of the ions from excitation of the Slater mode given by d = (ℏ/2MSωTO1)1/2, MS is the mass of the Slater mode (3MO + MTa), and a is the lattice constant (3.9884 Å for KTO). The existence of tʹ allows coupling between one t2g orbital on a given Ta site with a different t2g orbital on a neighboring Ta site (which is not allowed if the Ta-O-Ta bond is linear). tʹ was obtained in ref. 37. from a frozen phonon calculation. Instead, we follow Ref. 36. which calculates the inter-orbital coupling due to displacement of the oxygen ion transverse to the Ta-O-Ta bond. This leads to a value of tʹ = 2tpd2/(Δpda) where tpd is the pdπ hopping and Δpd the energy difference between the O 2p and Ta 5d orbitals, noting that the oxygen ion is at a distance a/2 relative to the Ta ion. tpd and Δpd can be obtained from Mattheiss’ Slater-Koster fits57 for KTO. Doing this we obtain the values listed in Supplementary Table 1. tR is the inversion breaking hopping parameter that enters the electronic structure Hamiltonian55, and the TO1 mode energy listed is that of the bulk insulator at q = 0. The reason we included STO is to compare our crude approximation of tʹ to the more sophisticated one of Ref. 37. Our value for STO is about 50% larger than theirs (our value for KTO being similar to theirs for STO). The reason the STO value is larger is because of the softer TO1 mode coupled with the larger d (zero-point motion) due to the lighter Ti ion.

On the other hand, there are other inversion breaking terms that have been ignored here, and those couple the t2g electrons to the eg ones58. As those terms are proportional to the inversion breaking times the spin-orbit coupling divided by the t2g-eg splitting, they are presumably much larger in KTO than STO given its 20 times larger spin-orbit coupling ξ. These terms also exhibit orbital differentiation. For instance, for displacements along [001], the dxy states do not couple to the eg states, and therefore similar considerations to those above apply for these t2g-eg terms as well. These terms in turn could explain why Tc in KTO is much higher than in STO. But their inclusion results in a much more involved theory than that presented here, so we leave this to future work. In this context, note that our dispersion plots are based on displacements along the interface normal. In the dynamic case relevant for the calculation of Tc, one or both TO1 modes (or neither) come into play depending on the specific orbitals and interface orientation involved.

Regardless, the purpose of this exercise was to give a rough estimate of the size of tR due to dynamic coupling of the electrons to the TO1 mode. Its value above (~32.5 meV) is extremely large, as commented on in ref. 37, meaning that TO1 mode pairing is quite viable. In Fig. 4f, g and Supplementary Fig. 7a, we show the electronic structure in the bilayer approximation for the three interface orientations assuming this value of tR.

Splitting in the energy dispersion and interface orientation variation of T c

For KTO (111) (Fig. 4f), note the pronounced Rashba-like effect near Γ (displaced parabola) for the lowest band of the spin-orbit quartet. For (001) (Fig. 4g), note the small effect of inversion breaking. For (110) (Supplementary Figure 7a), note the small splitting along Γ-Z (similar to (001)) and the much larger splitting along Γ-M, along [1-10]. This (110) case leads to a large Fermi surface anisotropy, while the (001) case is isotropic and the (111) case only mildly anisotropic, since the Ta-O-Ta bonds are parallel to [001] but are not parallel to [1-10] in the (110) plane. These electron-TO1 mode couplings for different KTO surfaces are schematically illustrated in Fig. 4c–e of the main text using dark and light colors for the Ta-O-Ta bonds.

We now address the interface orientation variation of Tc from the variation of the inversion breaking effect with orientation. Note that the kF2 factor in the numerator of the BCS coupling constant λ comes from the square of the splitting of the bands due to inversion breaking, the latter going like αRkF, with the notation αR denoting that many (but not all) of these inter-orbital terms have a Rashba-like form. So, λ scales as αR2. αR is equal to a constant times tR. We denote this constant by δ (which has units of Angstroms). So, λ scales as δ2. In Supplementary Table 2 are values of δ gotten from the slope of the lowest energy band versus tR divided by |k | (k is taken to be near Γ), which we extract from that of the lowest band. An example is shown in Supplementary Fig. 7b.

The first thing to note from Supplementary Table 2 is that (001) will have a negligible λ. In fact, we can get an analytic estimate of what the δ ratio is from Ref. 58. The inversion breaking term for (111) is proportional to 31/2 (ignoring coupling to eg states). The inversion breaking for (001) is proportional instead to 2ξ/ε where ε is the splitting of the dxy state from dxz/yz. Therefore, the ratio of δ (111 to 001) is 31/2ε/2ξ. In our simple model, ε is equal to t. For our values (t = 1 eV, ξ = 0.265 eV), we get a ratio of 3.27, which approximately matches the numerical value from Supplementary Table 2 (3.78). Since λ goes as the square of this ratio, one sees that Tc on the (001) interface should be negligible.

For (110), it is difficult to give an estimate given the large anisotropy of the inversion breaking in the surface Brillouin zone coupled with the large anisotropy of the Fermi surface (the Fermi surface is predicted to be strongly elongated along [1-10]). Still the (110) δ values are much closer to the (111) value than the (001) value, so a Tc for (110) half that for (111) is reasonable. The upshot is that the Tc variation follows the breaking of orbital degeneracy, as shown by the last two columns of Supplementary Table 2. Here, “orbitals” refer to the t2g orbitals present in the low energy manifold of states.