Abstract
What causes Cooper pairs to form in unconventional superconductors is often elusive because experimental signatures that connect to a specific pairing mechanism are rare. Here, we observe distinct dependences of the superconducting transition temperature T_{c} on carrier density n_{2D} for electron gases formed at KTaO_{3} (111), (001) and (110) interfaces. For the (111) interface, a remarkable linear dependence of T_{c} on n_{2D} is observed over a range of nearly one order of magnitude. Further, our study of the dependence of superconductivity on gate electric fields reveals the role of the interface in mediating superconductivity. We find that the extreme sensitivity of superconductivity to crystallographic orientation can be explained by pairing via interorbital interactions induced by an inversionbreaking transverse optical phonon and quantum confinement. This mechanism is also consistent with the dependence of T_{c} on n_{2D}. Our study may shed light on the pairing mechanism in other superconducting quantum paraelectrics.
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Introduction
Ever since the advent of superconductors, investigations of the mechanism of superconductivity have been at the forefront of condensed matter physics^{1,2,3,4,5,6,7,8}. Recent years have seen the discovery of superconductivity in a plethora of engineered material systems, such as^{9,10} LaAlO_{3}/SrTiO_{3}, FeSe/SrTiO_{3} and twisted bilayer^{11} and trilayer^{12,13} 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 fieldeffect gating^{14}. For example, the presence of spinorbit coupling and broken inversion symmetry at interfaces could enable new channels for Cooper pairing^{15}, and for realizing a pairing state^{15,16} that can host zeromodes with nonAbelian statistics^{17,18}. 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 oxideinsulator/KTaO_{3} interfaces^{19} through chemical doping and electric field gating. We establish the doping dependence of T_{c} for KTaO_{3} (KTO) interfaces of different crystallographic orientations. A remarkable proportionality where T_{c} ∝ n_{2D} 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 n_{2D}, while the KTO (110) interface superconducts at a T_{c} as high as 1 K, intermediate between the (111) and (001) cases. Using electric fieldeffect gating, we tune both n_{2D}, as well as the confinement of electrons to the interface. We find that T_{c} increases for both. We rule out the possibility that T_{c} is controlled by the phase stiffness temperature^{20}. We then discuss possible scenarios for pairing and find that interorbital interactions mediated by the soft transverse optical phonon reproduces key findings in our measurements, particularly the extreme sensitivity of T_{c} to the crystallographic orientation of the interface, and the consistency with the linear dependence of T_{c} with n_{2D}.
Results
Distinct n _{2D} dependences of T _{c} at KTO (111), (001), and (110) interfaces
Twodimensional 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 KTO^{19}. The carrier density n_{2D} in these samples is determined using Hall measurements at 10 K. n_{2D} of the asgrown 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 n_{2D} 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 n_{2D}. Figure 1a shows the temperature dependence of the ratio of the sheet resistance R_{s}/R_{N} (R_{N} is the normalstate resistance at 4.5 K) for KTO (111) samples with different n_{2D}. The superconducting transition defined by R_{s}/R_{N} going to zero occurs at progressively higher temperatures with increasing n_{2D}. 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 R_{s}(T) on EuO/KTO (001) and (110) samples. As reported before^{19}, 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 LaAlO_{3}/KTO (110) interface^{21}. The transition temperature T_{c} for the KTO (110) samples also increases with n_{2D}, but gradually saturates.
When T_{c} (determined at 20% of R_{N}) of all the KTO (111) samples are plotted against n_{2D}, as shown in Fig. 1c, we observe a largely linear dependence of T_{c} on n_{2D}. The solid line is a fit using T_{c} = a· n_{2D} + b, where the coefficient a ≈ 0.2 × 10^{−13} K cm^{2}, and the yaxis 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 T_{c} as shown in Supplementary Fig. 1.
Within the entire doping range, T_{c} versus n_{2D} does not show a dome, suggesting that T_{c} may be further enhanced by increasing the doping level. In Fig. 1d we plot T_{c} versus n_{2D} for KTO (001) and (110) interfaces, which show entirely different behavior than that of the KTO (111) interface: T_{c} cannot be measured, down to 25 mK, for all n_{2D} for the KTO (001) interface; while T_{c} for the KTO (110) interface increases with n_{2D} but saturates at a value about half that of the (111) case. We have also measured the GinzburgLandau coherence length (ξ_{GL}) in the KTO (111) samples with different n_{2D}, and found that ξ_{GL} satisfies a scaling relation between T_{c} 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 n_{2D} limits. It is expected that electric fields can tune a relatively large fraction of charge density in the low n_{2D} sample in a backgate geometry, given the large dielectric constant (~4500) of KTO at low temperatures^{22,23,24}. Figure 2a shows a schematic of our measurement. A 100nm 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.
Figure 2b shows the measurements of R_{s}(T) for different gate voltages V_{G} on KTO_1, which has the lowest carrier density n_{2D} = 1.25 × 10^{13} cm^{−2}. For this sample, R_{N} increases by over three times when V_{G} is varied from 30 V to –20 V. Correspondingly, T_{c} becomes lower. We have measured n_{2D}, which increases with V_{G} monotonically, as shown in the inset of Fig. 2c. By using a parallelplate capacitor approximation and the thickness of KTO substrate of 0.5 mm, the slope of n_{2D} vs. V_{G} at V_{G} = 0 V yields a dielectric constant for KTO of ~5700, which is close to the known value for bulk KTO. When T_{c} is plotted against n_{2D} as tuned by V_{G} (Fig. 2c), we observe a linear dependence of T_{c} on n_{2D}, i.e., T_{c} = 0.26 × 10^{−13} (K cm^{2})·n_{2D} – 0.1 K. Crucially, the coefficient for n_{2D} here is close to that found in the n_{2D} dependence of T_{c} 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 T_{c} at the KTO interface. Figure 2d shows that the critical current I_{c} also increases monotonically with V_{G}. We note that in recent ionic liquid gating measurements^{25} on KTO, T_{c} also changes monotonically with V_{G}, though the inferred n_{2D} does not.
Enhanced T _{BCS} and a dome in T _{BKT} by electrostatic gating
Next, we consider the effects of fieldeffect gating at higher carrier densities. Figure 3a shows R_{s}(T) measured on KTO_9 with n_{2D} = 1.04 × 10^{14} cm^{−2}, for different V_{G}. As V_{G} varies from 200 V to – 200 V, R_{N} 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 n_{2D} sample. Utilizing the response of superconductivity to V_{G}, we show in Fig. 3b that the superconducting state with zero resistance can be switched on or off reversibly when V_{G} 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 V_{G}.
We found that the tunability of superconductivity here is due to an enhancement of the meanfield transition temperature T_{BCS} with negative V_{G}. The transition from superconducting to normal state shown in Fig. 3a can be precisely interpreted by the HalperinNelson (HN) formula^{26}, which describes the increase in resistance with rising temperature due to the BerezinskiiKosterlitzThouless (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 R_{s} (T) = R_{N}(1 + 4 A^{−2}sinh^{2}(b(T – T_{BKT})^{−1/2}T_{BKT}^{1/2}))^{−1}, where A and b are fitting parameters that depend on the energy of the vortex core and the phase stiffness^{27}. 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 reported^{28}). In the HN expression, Δσ/σ_{N} = 4A^{−2}sinh^{2}(b(T–T_{BKT})^{−1/2}T_{BKT}^{1/2}) is a measure of the enhanced conductivity by superconducting fluctuations. Figure 3d shows a contour plot for Δσ/σ_{N} as a function of V_{G} and temperature T, which is obtained from the HN analysis for measurements over the full range of V_{G}. As V_{G} goes negative, Δσ/σ_{N} shows monotonic enhancement for T ≥ 2.15 K, while at lower temperatures a local maximum in Δσ/σ_{N} is seen.
From the HN analysis, we obtain the T_{BKT}, which is found to be close to T_{c0}, where R_{s} becomes zero. The meanfield T_{BCS} is obtained from the inflection point^{29} of the HN fit (maximum of its first derivative, near 20% of R_{N}), and has been confirmed by AL fits (Supplementary Fig. 3). Figure 3e shows how T_{BCS} and T_{BKT} evolve as a function of V_{G}. T_{BCS} is monotonically enhanced as V_{G} goes to negative values, while T_{BKT} shows a local maximum at V_{G} ~ –75 V. We also measured the voltagecurrent (V–I) characteristic near T_{c0} to determine T_{BKT} independently. As shown in Fig. 3f, T_{BKT} obtained through VI measurements at V_{G} = 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 V_{G}.
We find that the nonmonotonic dependence of T_{BKT} on V_{G} is caused by the enhancement of T_{BCS} together with the increase of R_{N} as V_{G} decreases. The ratio of T_{BKT}/T_{BCS} from our data, as shown in Fig. 3g, follows closely the prediction for a 2D superconductor^{30}, i.e., T_{BKT}/T_{BCS} = (1 + 0.173R_{N}/R_{c})^{−1}, where R_{c} = ħ/e^{2} with ħ being the reduced Planck constant. The increase of R_{N} by negative V_{G} is mainly due to charge carriers being pushed closer to the EuO/KTO interface, where more disorder is present^{31,32}. As shown in Fig. 3h, the Hall mobility decreases from about 37 cm^{2} V^{−1} m^{−1} to 17 cm^{2} V^{−1} m^{−1} as V_{G} 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 n_{2D} as found earlier, the proximity of the carriers to the interface also increases T_{c} (T_{BCS}), suggesting the interface itself is the origin of pairing. We also observe that the critical field is enhanced at negative V_{G}, 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 T_{c} with n_{2D} observed for the KTO (111) interface. One possible explanation for this phenomenon is that T_{c} is limited by the phase stiffness of the superconducting state, where T_{c} ∝ E_{J} (T_{BKT}) with E_{J} = ħ^{2}n_{s}(T)/4 m^{*}, n_{s} the 2D superfluid density, m* the effective mass of the electrons, and n_{s}(0 K) ∝ n_{2D}. This would give a linear relation between T_{c} and n_{2D}. However, our analysis in Fig. 3c–h shows that T_{BKT} is close to T_{BCS}, and in fact this is the case for samples over the full doping range studied. The closeness of T_{BKT} and T_{BCS} 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 T_{BCS}.
The above discussion leads to the second possibility for the observed proportionality between T_{c} and n_{2D}, which is that the pairing interaction is nonBCS like with a superconducting gap Δ ∝ E_{F} (Fermi energy) instead of ħω_{D} (Debye energy). Such a situation would occur in the antiadiabatic limit^{33} with \(\hslash {\omega }_{D}\) > E_{F}. Since in 2D, n_{2D} scales with E_{F}, one would obtain T_{c} ∝ Δ ∝ n_{2D}. The estimated E_{F} in our samples lies in the range of 10−80 meV, so pairing via the highestenergy longitudinal optical mode (LO4) with energy ~100 meV would be required. There are several potential issues with such a scenario. First, whether ħω_{D} can by replaced by E_{F} in the BCS expression for T_{c} is controversial^{5}. Second, it would require that the BCS coupling constant be independent of n_{2D}.
This leads us to consider pairing via the soft transverse optical (TO1) phonon mode^{34}, 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 electrons^{35}. However, as pointed out in recent work^{36,37}, inversion breaking from the TO1 mode leads to linear coupling to electrons for interorbital interactions among the three t_{2g} orbitals: d_{xy}, d_{yz}, and d_{zx}, 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 interorbital hopping amplitude of electrons along a TaOTa bond is no longer zero.
We show in detail in the Methods section that: (i) The TO1 mode energy (ω_{ΤΟ1}) increases as a function of n_{2D} due to the electric field screening provided by the charge carriers. (ii) The BCS coupling constant λ (arising from the Rashbalike splitting of the energy bands with wavevector k) also increases with n_{2D}. We take T_{c} = 1.14 ω_{ΤΟ1}(q = 2k_{F}, n_{2D}) exp[–(1 + λ)/λ]. Adjusting λ = c n_{2D} < 1/ω_{ΤΟ1}^{2} > by a free parameter c to scale the value of T_{c} to equal the experimental value of ~2 K at n_{2D} = 1 × 10^{14}/cm^{2}, we obtain a nearly linear dependence of T_{c} on n_{2D} for the KTO (111) interface, which is shown in Fig. 4b. Here, <> is a Fermi surface average (see Methods) and we assume that ω_{ΤΟ1}^{2}(q = 0) = c_{1} + c_{2} n_{2D} as observed in SrTiO_{3}, with c_{1} and c_{2} 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 interorbital electronTO1 phonon coupling and the effects of quantum confinement for electrons at the KTOoxide interface. The degeneracy of the t_{2g} 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 interorbital hopping. Consequently, electrons and TO1 phonons are maximumly coupled for the KTO (111) surface where the three t_{2g} orbitals are degenerate, while they are largely decoupled for the KTO (001) surface given the splitting between the d_{xy} and d_{xz/yz} orbitals. Shown in Fig. 4f are the calculated energy bands for the KTO (111) surface exhibiting large ‘Rashba’like splitting due to interorbital 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 electronTO1 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 T_{c} for the KTO (110) interface may be due to its unusual electronic structure, with the predicted Fermi surface intersecting the zone boundary for n_{2D} ~ 7 × 10^{13} 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 nontrivial. 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 T_{BCS} when the charge carriers are pushed closer to the interface by a negative V_{G}. A detailed analysis also needs to consider (1) the influence of both LO and TO modes on T_{c}, (2) the carrier distribution n_{3D}(z) along the interface normal (z) together with the strong interface orientation dependent electronic structure^{38,39,40}, and (3) the influence of interorbital terms between the t_{2g} and e_{g} orbitals. We note that while the dependence of T_{c} on crystal orientation is not observed^{9,41,42} in SrTiO_{3} based 2DEGs (barring a recent exception^{43}), this may be due to a weaker lifting^{44,45,46} of the degeneracy of the 3d orbitals (the confinement potential being weaker in STO than KTO). We also note that while the t_{2g} orbitals in bulk KTaO_{3} are degenerate, it is not known to superconduct^{47}. 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 k_{F} whereas in 2D it is a constant. For the range of carrier densities we explore, N_{3D} is significantly smaller than N_{2D}, leading via λ to an exponential suppression of T_{c} 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 EuO_{x}, as the Eu scavenges O from the substrate^{19}. Following this step, the EuO overlayer is grown under a O_{2} partial pressure between 0.5–10 × 10^{−}^{9 }Torr with a final EuO thickness of ~25 nm. Our microscopy results^{19} 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) plane^{19}. However, this does not influence the analysis of T_{c}, at which global superconductivity sets in as that is found to be the same for ‘stripe’ like samples at a given value of n_{2D}. 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
n_{2D} in these samples is determined using Hall measurements at 10 K. Supplementary Figure 1c shows the Hall resistance R_{H} measured as a function of magnetic field for different KTO (111) samples. Notably, the field dependences of R_{H} are all linear, with negative slopes, consistent with electronlike charge carriers. We found that the donor state of charge carriers evolves gradually from shallow to deeper energy as n_{2D} increases. Supplementary Figure 1d shows the temperature dependence of n_{2D} in the range 300 K–2 K. For samples with n_{2D}(10 K) ≲ 5 × 10^{13 }cm^{−2}, the measured n_{2D}(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 20fold increase in the dielectric constant in KTO upon cooling. For samples with n_{2D}(10 K) ≳ 5 × 10^{13 }cm^{−2}, n_{2D}(T) decreases upon cooling. This suggests that at higher doping levels charge carriers originate from deeper donor states^{48}, to which some of them freeze out upon cooling. Supplementary Figure 1d also shows that regardless of the doping level in the asgrown samples, n_{2D} remains nearly constant for temperatures below 10 K for all samples.
Scaling relation of coherence length
We have measured the GinzburgLandau coherence length (ξ_{GL}) in the KTO (111) samples with different n_{2D}, by measuring the outofplane upper critical field B_{c2} as a function of temperature. B_{c2} shows a linear temperature dependence near T_{c}, from which we calculate ξ_{GL} using B_{c2}(T) = Φ_{0}(1–T/T_{c})/[2π(ξ_{GL})^{2}], where Φ_{0} is the magnetic flux quantum. Supplementary Figure 2a shows the data of B_{c2} versus T for five samples. The B_{c2} varies greatly among these samples with different n_{2D} and T_{c}. Supplementary Figure 2b summarize the ξ_{GL} for all the samples measured, which lies in the range from 10 to 57 nm. We found that ξ_{GL} scales with (μ/T_{c})^{1/2}. As shown in Supplementary Fig. 2b, the ratio of ξ_{GL} to (μ/T_{c})^{1/2} remains constant for samples over the entire n_{2D} range. We found that this relation also holds when the superconductivity is tuned by V_{G} (see Supplementary Fig. 5).
For a BCS superconductor in the dirty limit, it is expected that ξ_{GL} ∝ (lξ_{BCS})^{1/2} with ξ_{BCS} being the BCS coherence length^{49}. Using l ∝ v_{F}μ and ξ_{BCS} ∝ v_{F}/T_{c} with v_{F} being the Fermi velocity, one gets ξ_{GL} ∝ v_{F}(μ/T_{c})^{1/2}. For the KTO (111) interface, we observe that ξ_{GL} ∝ (μ/T_{c}) ^{1/2} instead. This would imply that v_{F} is a constant, independent of n_{2D}. 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 T_{c} = 1.14 ω_{ΤΟ1}(q = 2k_{F}, n_{2D}) 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 cutoff energy to the maximum mode energy which is at q = 2k_{F}. Therefore, the n_{2D} dependence of T_{c} will come from any hardening of the TO1 mode along with the increase of k_{F} with n_{2D} 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 n_{2D} dependence of the mode energy at q = 0, as well as that of k_{F}. For the former, we will assume that ω_{ΤΟ1}^{2}(q = 0, n_{2D}) = c_{1} + c_{2} n_{2D}; this relation is found in STO where doping of the bulk with free carriers has been observed^{51} (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 Worlock^{52}. c_{1} is set by the bulk value of ω_{ΤΟ1} of 2.5 meV. We set c_{2} by the estimated value of ω_{ΤΟ1} at n_{2D} = 10^{14 }cm^{−2} of 5.6 meV. The latter has been obtained from Ueno et al.^{53} that relates the field dependent dielectric function to n_{2D}, 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·n_{2D} = 2 _{0}∫^{F}ε_{0·}ε(E)dE, where e is elementary charge and ε_{0} is 8.85 pF m^{−1}. The mode energy is then given by the LyddaneSachsTeller relation ω_{ΤΟ1}^{2}(q = 0, n_{2D}) = ω_{ΤΟ1}^{2}(q = 0, n_{2D} = 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 n_{2D} is that the resulting n_{2D} dependence of ω_{ΤΟ1}^{2}(q = 0) deviates from the above assumed linear behavior (Supplementary Fig. 6e), which in turn has a detrimental impact on the functional dependence of T_{c} on n_{2D} (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 n_{2D} in KTO (realizing that the TO1 mode polarized normal to the interface should harden more with n_{2D} than the TO1 mode orthogonal to this).
To obtain the variation of k_{F} with n_{2D}, 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 spinorbit coupling ξ = 0.265 eV (in order to reproduce the quartetdoublet spinorbit splitting of 0.4 eV at Γ). For k_{F}, we take that of the larger of the two Fermi surfaces (the quartet splits into two doublets upon dispersing away from Γ), and set k_{F} to its value along ΓK of the surface Brillouin zone. n_{2D} is gotten by determining the occupied area of the two Fermi surfaces at E_{F}. The resulting dependence of E_{F} and the TO1 mode energies with n_{2D} is shown in Supplementary Fig. 6a. One can see the approximate linear dependence of E_{F} with n_{2D} (expected for a parabolic dispersion in 2D), with the deviation at larger n_{2D} 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 = 2k_{F} is not linear in n_{2D}. This is to be expected, since in the approximation where the mode energy at q = 0 is small compared to that at q = 2k_{F}, the mode energy should scale with q and thus k_{F}, the latter going approximately as the square root of n_{2D}. Also note that over most of the n_{2D} range, the mode energy at q = 2k_{F} is much smaller than E_{F}, justifying the BCS (adiabatic) approximation for TO1 mode pairing. We note that although our experimental results, such as the orientational and gatefield 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 n_{2D} < 1/ω_{ΤΟ1}^{2}(q, n_{2D})> 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·n_{2D}/[ω_{ΤΟ1}(q = 2k_{F}, n_{2D})·ω_{ΤΟ1}(q = 0, n_{2D})]. Here, n_{2D} in the numerator comes from k_{F}^{2} (k_{F}^{2} = 2π·n_{2D}). This factor arises from the electronphonon vertex that is proportional to k_{F} (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, N_{F}, that we assume is constant (equivalent to linearity of E_{F} with n_{2D}). This is in contrast to the bulk^{56} where N_{F} goes as n_{3D}^{1/3}. The value of c would require a detailed microscopic theory. In lieu of that, we set c in order to give T_{c} of ~2 K at n_{2D} = 10^{14} cm^{−2} (for this n_{2D}, λ ~ 0.26 and so c = 2.85 when n_{2D} is in units of 10^{13} cm^{−2} and ω_{TO1} in meV). The resulting λ and T_{c} versus n_{2D} is plotted in Supplementary Fig 6b, c, respectively. T_{c} 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 T_{c} with n_{2D} in Supplementary Fig. 6c, despite the nonlinearity with n_{2D} of both the BCS cutoff and the BCS coupling constant. We note that the variation of T_{c} with n_{2D} is a result of the interplay between the dependences of ω_{TO1} and the electronphonon coupling λ on n_{2D}. Obtaining a linear behavior of T_{c} is sensitive to details of both dependences. However, the rise in T_{c} with n_{2D} is a general finding, in part due to the increase of ω_{TO1} with n_{2D} 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 T_{c}, with values similar to those claimed by Ueno et al. for the (001) interface^{53} (a further reduction in λ would completely suppress T_{c} as we observe). Since λ is proportional to the square of the electronphonon vertex, a mere reduction of the vertex by 2^{1/2} is sufficient to cause an enormous reduction in T_{c}. As such, we expect T_{c} to be extremely sensitive to the interface orientation, which we address next. We note that in a more recent work^{25} using ionic liquid gating of the KTO (001) surface, superconductivity was not found.
Dependence of T _{c} on interface orientation from TO1 mode pairing
Electronphonon vertex
To obtain a significant electronphonon vertex for a TO1 mode, one requires interorbital terms (intraorbital ones instead lead to a coupling quadratic in the ion displacement^{35}). Therefore, in a TO1 mode picture, one requires orbital degeneracy in order to get a substantial T_{c}. This qualitatively explains the observed dependence of T_{c} with interface orientation. For the (111) case (ignoring any trigonal distortion), the three t_{2g} states are degenerate at Γ (being split into a lower quartet and an upper doublet by spinorbit). We can contrast this with the (001) case, where confinement along [001] leads to the d_{xy} state being pulled down substantially relative to the d_{xz/yz} ones. This splitting acts to suppress the effect of interorbital terms, consistent with our lack of observation of superconductivity on (001) interfaces (in ref. ^{53}, a T_{c} 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 d_{xz/yz} states relative to the d_{xy} 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 interorbital 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 electronphonon vertex^{37} g_{TO1} = 2tʹcʹdk_{F}a where tʹ is the derivative of the nearneighbor hopping with respect to the ion displacement, cʹ = (η^{−1/2} + η^{1/2}) where η = 3M_{O}/M_{Ta}, d is the zeropoint displacement of the ions from excitation of the Slater mode given by d = (ℏ/2M_{S}ω_{TO1})^{1/2}, M_{S} is the mass of the Slater mode (3M_{O} + M_{Ta}), and a is the lattice constant (3.9884 Å for KTO). The existence of tʹ allows coupling between one t_{2g} orbital on a given Ta site with a different t_{2g} orbital on a neighboring Ta site (which is not allowed if the TaOTa bond is linear). tʹ was obtained in ref. ^{37}. from a frozen phonon calculation. Instead, we follow Ref. ^{36}. which calculates the interorbital coupling due to displacement of the oxygen ion transverse to the TaOTa bond. This leads to a value of tʹ = 2t_{pd}^{2}/(Δ_{pd}a) where t_{pd} is the pdπ hopping and Δ_{pd} 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. t_{pd} and Δ_{pd} can be obtained from Mattheiss’ SlaterKoster fits^{57} for KTO. Doing this we obtain the values listed in Supplementary Table 1. t_{R} is the inversion breaking hopping parameter that enters the electronic structure Hamiltonian^{55}, 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 (zeropoint 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 t_{2g} electrons to the e_{g} ones^{58}. As those terms are proportional to the inversion breaking times the spinorbit coupling divided by the t_{2g}e_{g} splitting, they are presumably much larger in KTO than STO given its 20 times larger spinorbit coupling ξ. These terms also exhibit orbital differentiation. For instance, for displacements along [001], the d_{xy} states do not couple to the e_{g} states, and therefore similar considerations to those above apply for these t_{2g}e_{g} terms as well. These terms in turn could explain why T_{c} 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 T_{c}, 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 t_{R} 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 t_{R}.
Splitting in the energy dispersion and interface orientation variation of T _{c}
For KTO (111) (Fig. 4f), note the pronounced Rashbalike effect near Γ (displaced parabola) for the lowest band of the spinorbit quartet. For (001) (Fig. 4g), note the small effect of inversion breaking. For (110) (Supplementary Figure 7a), note the small splitting along ΓZ (similar to (001)) and the much larger splitting along ΓM, along [110]. 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 TaOTa bonds are parallel to [001] but are not parallel to [110] in the (110) plane. These electronTO1 mode couplings for different KTO surfaces are schematically illustrated in Fig. 4c–e of the main text using dark and light colors for the TaOTa bonds.
We now address the interface orientation variation of T_{c} from the variation of the inversion breaking effect with orientation. Note that the k_{F}^{2} 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 α_{R}k_{F}, with the notation α_{R} denoting that many (but not all) of these interorbital terms have a Rashbalike form. So, λ scales as α_{R}^{2}. α_{R} is equal to a constant times t_{R}. We denote this constant by δ (which has units of Angstroms). So, λ scales as δ^{2}. In Supplementary Table 2 are values of δ gotten from the slope of the lowest energy band versus t_{R} 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 3^{1/2} (ignoring coupling to e_{g} states). The inversion breaking for (001) is proportional instead to 2ξ/ε where ε is the splitting of the d_{xy} state from d_{xz/yz}. Therefore, the ratio of δ (111 to 001) is 3^{1/2}ε/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 T_{c} 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 [110]). Still the (110) δ values are much closer to the (111) value than the (001) value, so a T_{c} for (110) half that for (111) is reasonable. The upshot is that the T_{c} variation follows the breaking of orbital degeneracy, as shown by the last two columns of Supplementary Table 2. Here, “orbitals” refer to the t_{2g} orbitals present in the low energy manifold of states.
Data availability
All data presented here in the paper are available via the Harvard Dataverse at https://doi.org/10.7910/DVN/TKTVK1.
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Acknowledgements
We thank Ivar Martin, Peter Littlewood, Alex Edelman, Arun Paramekanti and James Rondinelli for discussions. All research presented here is supported by the Materials Science and Engineering Division, Office of Basic Energy Sciences, U.S. Department of Energy. The use of facilities at the Center for Nanoscale Materials was supported by the U.S. DOE, BES under Contract No. DEAC0206CH11357. C.L. acknowledges partial financial support from the College of Arts and Sciences, University at Buffalo, SUNY.
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Synthesis of samples was carried out by C.L. with assistance from D.H., J.P., H.Z., and A.B. Measurements in PPMS and with the adiabatic demagnetization option were carried out by C.L. with assistance from B.F., J.S.J., and J.P. Measurements in the dilution fridge at the Center for Nanoscale Materials were carried out by C.L. with assistance from X.Z. and D.J. Analysis of the data was carried out by C.L. with assistance from M.N. and A.B. All theoretical work presented here was carried out by M.N.; A.B. supervised all experimental work and analysis. The paper was written by C.L., M.N., and A.B. All authors contributed to discussions regarding the paper.
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Liu, C., Zhou, X., Hong, D. et al. Tunable superconductivity and its origin at KTaO_{3} interfaces. Nat Commun 14, 951 (2023). https://doi.org/10.1038/s41467023363092
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DOI: https://doi.org/10.1038/s41467023363092
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