Abstract
After almost twenty years of intense work on the celebrated LaAlO_{3}/SrTiO_{3}system, the recent discovery of a superconducting twodimensional electron gas (2DEG) in (111)oriented KTaO_{3}based heterostructures injects new momentum to the field of oxides interface. However, while both interfaces share common properties, experiments also suggest important differences between the two systems. Here, we report gate tunable superconductivity in 2DEGs generated at the surface of a (111)oriented KTaO_{3} crystal by the simple sputtering of a thin Al layer. We extract the superfluid stiffness of the 2DEGs and show that its temperature dependence is consistent with a nodeless superconducting order parameter having a gap value larger than expected within a simple BCS weakcoupling limit model. The superconducting transition follows the BerezinskiiKosterlitzThouless scenario, which was not reported on SrTiO_{3}based interfaces. Our finding offers innovative perspectives for fundamental science but also for device applications in a variety of fields such as spinorbitronics and topological electronics.
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Introduction
Potassium tantalate KTaO_{3} is a band insulator with a 3.6 eV gap that retains a cubic perovskite structure down to the lowest temperature^{1}. Like strontium titanate (SrTiO_{3}), it is a quantum paraelectric material on the verge of a ferroelectric instability that is characterized by a large permittivity at low temperature (ϵ_{r} ≃ 5000)^{1,2}. Both materials can be turned into metal by electron doping, through oxygen vacancies, for example. Because of their common properties, it was suggested that superconductivity should also occur in doped KTaO_{3}. However, while superconductivity was discovered more than half a century ago in bulk SrTiO_{3}^{3}, all the attempts to induce bulk superconductivity in KTaO_{3}have failed so far^{4}. Using ionic gating, Ueno et al. could generate a superconducting 2DEG at the surface of (001)KTaO_{3}, albeit at a very low temperature (T_{c} ≃ 40 mK)^{5}. Later explorations of KTaO_{3} 2DEGs did not evidence any superconductivity until the beginning of the year 2021, when two articles reported the discovery of superconducting 2DEG formed at the interface between (111)KTaO_{3} and insulating overlayers of LaAlO_{3}or EuO^{6,7}. An empiric increase of T_{c} with electron density was proposed with a maximum value of 2.2 K for doping of ≈ 1.04 × 10^{14}e^{−} × cm^{−2} ^{6}, which is almost one order of magnitude higher than in the LaAlO_{3}/SrTiO_{3} interface^{8}. An electric field effect control of the T_{c} was also demonstrated in a Hall bar device^{7} and a domeshaped superconducting phase diagram similar to that of SrTiO_{3}based interfaces was derived^{9,10}. Following this discovery, the (110)oriented KTaO_{3} interface was also found to be superconducting with T_{c} ≃ 1 K^{11}. It was recently proposed that the soft transverse optical mode, involved in the quantum paraelectricity, could be responsible for electron pairing in KTaO_{3} interfaces. The coupling amplitude between this phonon mode and electrons is expected to be maximum in the (111) orientation and minimum in the (001) one, which would explain the hierarchy in T_{c} observed in these superconducting 2DEGs^{12}.
In conventional superconductors, well described by the Bardeen–Cooper–Schrieffer (BCS) theory, the superconducting transition is controlled by the breaking of Cooper pairs as the temperature exceeds the energy scale set by the superconducting gap. However, in twodimensional superconductors, the superfluid stiffness, i.e., the energy associated with the phase rigidity of the superconducting condensate, can be comparable to the pairing energy, allowing for a T_{c} suppression driven by the loss of phase coherence. In this case, the transition is expected to belong to the Berezinskii–Kosterlitz–Thouless (BKT) universality class, where the transition is controlled by the unbinding of topological vortexantivortex pairs^{13,14,15}. Critical magnetic field measurements in (111)KTaO_{3} 2DEGs, both in the perpendicular and in the parallel geometry, set an upper bound, d ≈ 5 nm, on the extension of the 2DEG in the substrate^{6}. This is lower than the superconducting coherence length, ξ ≈ 10–15 nm^{6}, which confirms that the superconducting 2DEG is within the 2D limit. In addition, the presence of disorder, which has been identified in this system^{6,7}, is also expected to lower the superfluid rigidity and reinforce the role of phase fluctuations. Even though the measurements of the currentvoltage characteristics in ref. 6 could be compatible with indirect signatures of a BKT transition, a direct measurement of the superfluid stiffness is required to properly address this issue^{16}.
Here, we show that a 2DEG can be generated at the surface of a (111)oriented KTaO_{3} crystal simply by sputtering a very thin Al layer. The deposition of Al leads to the reduction of Ta ions as evidenced by Xray photoelectron spectroscopy (XPS) and leads to the formation of an interfacial gatetunable superconducting 2DEG. We use resonant microwave transport to measure the complex conductivity of the 2DEG and extract the temperaturedependent superfluid stiffness J_{s}(T). Our results are consistent with a nodeless superconducting order parameter in a rather strongcoupling regime (\({{\Delta }}(0)/{k}_{{{{{{{{\rm{B}}}}}}}}}{T}_{{{{{{{{\rm{c}}}}}}}}}^{0}\) = 2.3). Taking into account the presence of disorder and finitefrequency effect, we show that the superconducting transition follows the Berezinskii–Kosterlitz–Thouless model, which was not observed on SrTiO_{3}based interfaces.
Results
2DEGs were generated by dc sputtering of a very thin Al layer on (111)oriented KTaO_{3}`substrates at a temperature between 550 and 600 °C. The preparation process is detailed in the Methods section. Prior to deposition, we measured the insitu Xray photoelectron spectra (XPS) of the Ta 4f valence state (Fig. 1a top) of the KTaO_{3} substrate. The spectra show the sole presence of Ta^{5+} states (4f_{5/2} and 4f_{3/2}), indicating the expected stoichiometry of the substrate. The Ta 4f core levels were then measured after growing 1.8–1.9 nm of Al and transferring the sample in vacuum to the XPS setup. The bottom graph in Fig. 1a shows the Ta 4f core level spectra with additional peaks corresponding to reduced states of Ta i.e., Ta^{4+} and Ta^{2+}. The deeper and lighter shades of samecolored peaks correspond to 4f_{5/2} and 4f_{3/2} split peaks. The reduction of Ta^{5+} to Ta^{4+} upon Al deposition indicates the formation of oxygen vacancies at the surface of KTaO_{3}, which in turn suggests the formation of a 2DEG. The Ta^{2+} signal may be due to the presence of a small amount of Ta in the AlO_{x} layer (akin to the situation in AlOx/STO^{17}) or reflects the presence of small clusters of oxygen vacancies around some Ta ions, reducing their apparent valence state. We monitored the Al oxidation state by measuring the Al 2p core levels after exposure of the sample to the atmosphere, which evidenced full oxidation of the Al layer into AlO_{x}. Thus, as in the AlO_{x}/SrTiO_{3} system, the 2DEG is formed through a redox process by which oxygens are transferred from the KTaO_{3} substrate to the Al overlayer^{17,18,19}.
The structure of the AlO_{x}/KTaO_{3} (111) interface has been imaged by scanning transmission electron microscopy (STEM). Figure 1b depicts the highangle annular dark field (HAADF)  STEM image in crosssection. The electron energy loss spectroscopy (EELS) indicates that a small amount of K and Ta diffuse into the AlO_{x} layer. In contrast, the Al signal decays very rapidly in KTaO_{3}, indicating no Al diffusion into KTaO_{3}. Our fabrication method based on the sputtering of a thin Al film has already been successfully implemented to generate 2DEGs on (001)oriented KTaO_{3} substrates showing a fivefold enhancement of the Rashba spin–orbit coupling as compared to SrTiO_{3}^{20}. In the present work, four samples, labeled A, B, C, and D, have been investigated by transport measurement at low temperature in a dilution refrigerator (see Methods section for fabrication parameters).
Figure 2a shows the resistance vs temperature curve of sample A on a wide temperature range revealing a superconducting transition at T_{c} ≃ 0.9 K. In Fig. 2b, we plot the T_{c} as a function of the 2D carrier density, n_{2D} for the different samples studied and compare their values with those extracted from ref. 6. Our results confirm the trend observed in the literature (T_{c} increases with the carrier density) and demonstrate that our growth method, while being much easier to implement than the molecular beam epitaxy of a rareearth element such as Eu or the pulsed laser deposition of a complex oxide, is able to produce good quality samples with similar T_{c}. The resistance vs temperature curves of sample B measured for different values of a magnetic field applied perpendicularly to the sample plane are shown in Fig. 2c. The temperature dependence of the critical magnetic field is consistent with a LandauGinsburg model near T_{c}, \({\mu }_{0}{H}_{c}(T)=\frac{{{{\Phi }}}_{0}}{2\pi {\xi }_{\parallel }^{2}(T)}\), taking into account an inplane superconducting coherence length \({\xi }_{\parallel }={\xi }_{\parallel }(0){(1\frac{T}{{T}_{{{{{{{{\rm{c}}}}}}}}}})}^{\frac{1}{2}}\). We found ξ_{∥}(T = 0) ≈ 27 nm, which is comparable with the value reported in ref. 6.
Although KTaO_{3} is a quantum paraelectric material like SrTiO_{3}, its permittivity is reduced by a factor of five as compared to SrTiO_{3}, making the electric field effect less efficient in a backgating configuration^{1,2}. To overcome this difficulty, we prepared an AlO_{x}/KTaO_{3} sample using a thinner substrate (150 μm). After cooling the sample, the backgate voltage was first swept to its maximum value V_{G} = 200 V while keeping the 2DEG at the electrical ground. This forming procedure is commonly applied on SrTiO_{3}based interfaces to ensure the reversibility of the gate sweeps in further gating sequences^{21}. Figure 3 shows the sheet resistance of sample C as a function of temperature for different values of the gate voltage between −40 and 200 V. Electrostatic gating induces both a modulation of the normalstate resistance and a variation of the superconducting critical temperature. For negative gate voltages corresponding to a depleted quantum well, R vs T curves exhibit a quasireentrant behavior: the resistance first decreases and then upturns upon further cooling^{22,23}. This is characteristic of disordered superconducting thin films in which superconductivity only exists locally, forming a network of isolated islands surrounded by an insulating medium that precludes percolation. While the decrease of resistance marks the emergence of superconductivity inside the islands, the upturn of resistance at lowtemperature results from the opening of a gap in the excitation spectrum, which prevents the flow of quasiparticles across islands. Hence, the resistance does not reach zero, indicating that the superconducting order does not extend at a long range. As carriers are added upon increasing the gate voltage, the resistance curves flatten at low temperature, and the 2DEG eventually reaches a true zero resistive state (V_{G} > −25 V). Longrange superconducting order is established through Josephson coupling between the islands. Further doping makes the network of islands denser and increases the coupling between islands resulting in a “homogeneouslike” superconducting 2DEG at high doping. The resulting superconducting phase diagram is shown in Fig. 3b, where the resistance is plotted in color scale as a function of temperature and electron density extracted by combining the Hall effect and gate capacitance measurements^{10,24}. In this experiment, the carrier density was tuned from n_{2D} ≃ 0.95 × 10^{13}e^{−} × cm^{−2} to n_{2D} ≃ 2.2 × 10^{13}e^{−} × cm^{−2}, which is not sufficient to explain the modulation of the normal resistance by more than one order of magnitude. This indicates that the gate voltage not only controls the carrier density but also modifies deeply the electronic properties of the 2DEG, in particular the electronic mobility, in agreement with the previous reports^{7}.
We further investigated the superconducting KTaO_{3} 2DEG by measuring its superfluid stiffness J_{s}, which is the energy scale associated with the phase rigidity of the superconducting condensate. J_{s} is related to the imaginary part of the complex conductivity σ(ω) = σ_{1}(ω) − iσ_{2}(ω) of the superconductor that accounts for the transport of Cooper pairs at finitefrequency ω. This is a direct probe of the superconducting order parameter that provides important information on the nature of the superconducting state. In the lowfrequency limit, ℏω ≪ Δ, a superconductor displays an inductive response to an ac electrical current and σ_{2}(ω) = \(\frac{1}{{L}_{{{{{{{{\rm{k}}}}}}}}}\omega }\), where L_{k} is the kinetic inductance of the superconductor that diverges at T_{c}^{25}. The superfluid stiffness is then directly related to L_{k}
where ℏ is the reduced Planck constant and e is the electron charge.
We used resonant microwave transport to extract L_{k} below T_{c} and determine the superfluid stiffness of sample D as a function of temperature. The method, which was successfully applied to superconducting SrTiO_{3}based interfaces, is illustrated in Fig. 4a and described in detail in refs. 26,27. In short, the KTaO_{3} sample is embedded into a parallel RLC resonant electrical circuit made with surface mount microwave devices (SMD). The capacitance of the circuit is dominated by the KTaO_{3} substrate contribution (C_{KTO}) due to its large intrinsic permittivity. The total inductance of the circuit, \({L}_{{{{{{{{\rm{tot}}}}}}}}}(T)=\frac{{L}_{1}{L}_{{{{{{{{\rm{k}}}}}}}}}(T)}{{L}_{1}+{L}_{{{{{{{{\rm{k}}}}}}}}}(T)}\), includes the contribution of an SMD inductor (L_{1}) and the contribution of the kinetic inductance L_{k} of the superconducting 2DEG below T_{c}. Finally, an SMD resistor R_{1} imposes that the dissipative part of the circuit impedance remains close to 50 Ω in the entire temperature range, ensuring a good impedance matching with the microwave circuitry. The circuit resonates at the frequency \({\omega }_{0}=\frac{1}{\sqrt{{L}_{{{{{{{{\rm{tot}}}}}}}}}{C}_{{{{{{{{\rm{KTO}}}}}}}}}}}\), which is accessed by measuring the reflection coefficient of the sample circuit \({{\Gamma }}(\omega )=\frac{{A}^{{{{{{{{\rm{in}}}}}}}}}}{{A}^{{{{{{{{\rm{out}}}}}}}}}}=\frac{Z(\omega ){Z}_{0}}{Z(\omega )+{Z}_{0}}\). The resonance manifests itself as a peak in the real part of the circuit impedance, Z(ω), accompanied by a π phase shift^{26}. The height and the width of the peak are controlled by the dissipative part of the circuit impedance. In the superconducting state, the 2DEG conductance acquires a kinetic inductance L_{k} that generates a shift of ω_{0} towards high frequencies with respect to the normal state (Fig. 4b). The temperaturedependent superfluid stiffness \({J}^{\exp }\), extracted from the resonance shift and Eq. (1) is presented in Fig. 4c (blue circles).
Discussion
The flattening of the \({J}^{\exp }\) curve below 1 K supports a fully gapped behavior, i.e., an absence of nodes in the order parameter. The purpled dashed line (J^{BCS}) shows an attempt to fit the experimental curve with a standard BCS expression \({J}_{{{{{{{{\rm{s}}}}}}}}}^{{{{{{{{\rm{BCS}}}}}}}}}(T)/{J}_{{{{{{{{\rm{s}}}}}}}}}(0)=({{\Delta }}(T)/{{\Delta }}(0))\tanh ({{\Delta }}(T)/{k}_{{{{{{{{\rm{B}}}}}}}}}T)\)^{25}, where Δ(T) is the superconducting gap obtained numerically by a selfconsistent solution of the BCS equation, so that it vanishes at the meanfield temperature \({T}_{{{{{{{{\rm{c}}}}}}}}}^{0}\) (i.e., the temperature at which Cooper pairs form). Since J_{s}(0) is fixed by the experimental value at the lowest temperature, the only free parameter is then the ratio \({{\Delta }}(T=0)/{k}_{{{{{{{{\rm{B}}}}}}}}}{T}_{{{{{{{{\rm{c}}}}}}}}}^{0}\), that determines the curvature of the J_{s}(T) curve. As one can see, even using a relatively strongcoupling value \({{\Delta }}(0)/{k}_{{{{{{{{\rm{B}}}}}}}}}{T}_{{{{{{{{\rm{c}}}}}}}}}^{0}\) = 2.3, from the fit of the lowtemperature curve, one obtains \({T}_{{{{{{{{\rm{c}}}}}}}}}^{0}\) ≃ 2.2 K, that is larger than the experimental T_{c}. To fit the data in the whole temperature range with the BCS expression only, one would then need an unreasonably large value (\({{\Delta }}(0)/{k}_{{{{{{{{\rm{B}}}}}}}}}{T}_{{{{{{{{\rm{c}}}}}}}}}^{0}\) ≃ 6), a result that we checked to hold irrespectively of the exact functional BCS form used to fit the stiffness. Here we follow a different approach and interpret the rapid drop of J_{s}(T) below the BCS fit as a BKT signature, as we will discuss below. This interpretation is supported by a second striking observation that holds regardless of any specific consideration about its temperature dependence: the T = 0 value of the stiffness J_{s}(T = 0) ≃ 7.3 K is of the same order as T_{c} ≃ 2.2 K. It is worth noting that in conventional superconductors, where the superfluid density n_{s}(T = 0) is close to the carrier density n_{2D}, the stiffness at zero temperature is of the order of the Fermi energy, and then several orders of magnitude larger than \({T}_{{{{{{{{\rm{c}}}}}}}}}^{0}\). A strong reduction of J_{s}(0) is instead observed in 2Dsuperconductors, where disorder strongly reduces n_{s} with respect to n_{2D} already at T = 0^{28,29,30,31,32,33,34,35,36,37,38}. In the dirty limit, in which the elastic scattering rate 1/τ is much larger than the superconducting gap, only a fraction of carriers, n_{s}/n_{2D} ≃ 2Δ(0)/(ℏ/τ), forms the superconducting condensate. In a singleband picture, an estimate of the superfluid stiffness is obtained from Δ(0) and the normal resistance R_{N}, \({J}_{{{{{{{{\rm{s}}}}}}}}}\simeq \frac{\pi \hslash {{\Delta }}(0)}{4{e}^{2}{R}_{{{{{{{{\rm{N}}}}}}}}}}\). Using the previously estimated value of Δ(0) ≃ 5 K and R_{N} ≃ 1300 Ω, we obtain J_{s} ≃ 11.8 K, close to the measured value (\({J}_{{{{{{{{\rm{s}}}}}}}}}^{\exp }(T=0)\) ≃ 7.3 K), which is consistent with the dirty limit.
The superfluid density of the 2DEG can be directly deduced from the stiffness through the formula \({n}_{{{{{{{{\rm{s}}}}}}}}}=\frac{4m}{{\hslash }^{2}}{J}_{{{{{{{{\rm{s}}}}}}}}}\), where m is the effective mass of superconducting electrons. In the case of (111)KTaO_{3} 2DEGs, the conduction band is derived from the bulk J = 3/2 states with a Fermi Surface formed by a hexagonal contour inside a sixfold symmetric starshaped contour^{39,40}. Considering an average effective mass m ≃ 0.5m_{0}, the corresponding superfluid density n_{s} extracted from \({J}_{{{{{{{{\rm{s}}}}}}}}}^{\exp }\) is n_{s} ≃ 1.8 × 10^{12} e^{−} cm^{−2}, which is about 2.5% of the total carrier density (n_{2D} = 7.5 × 10^{13} e^{−} cm^{−2} for sample D). This very low ratio is comparable with previous findings in LaAlO_{3}/SrTiO_{3} interfaces^{26,41,42}. Although such reduced superfluid density is consistent with the dirty limit, KTaO_{3}(111) 2DEG is a multiband system^{40}, in which superconductivity may involve only specific bands, as also suggested in SrTiO_{3}^{26}.
The reduced dimensionality and the suppression of the energy scale associated with the stiffness represent the prerequisites to observe BKT^{13,14,15} physics, since it makes the BKT temperature scale T_{BKT} associated with the unbinding of vortexantivortex pairs far enough from \({T}_{{{{{{{{\rm{c}}}}}}}}}^{0}\)^{43}. The most famous hallmark of the BKT transitions is the discontinuous jump to zero of J_{s} at T_{BKT} < T_{c} with a universal ratio J_{s}(T_{BKT})/T_{BKT} = 2/π^{44}. Such a prediction, theoretically based on the study of the 2D XY model^{13,14,15}, has been successfully confirmed in superfluid He films^{45}. In practice, the experimental observation of the BKT transition in real superconductors is more subtle. Indeed, in thin films, the suppression of n_{s} (and then J_{s}) with disorder comes along with an increasing inhomogeneity of the SC background, that is predicted to smear out the discontinuous superfluiddensity jump^{35,46,47,48} into a rapid downturn, as observed experimentally via the direct measurement of the inverse penetration depth^{31,32,33,34,35,36,37,38} or indirectly via the measurement of the exponent of the nonlinear IV characteristics near T_{c}^{16,28,29,30}. In the case of SrTiO_{3}based interfaces, the direct measurement of J_{s} is rather challenging, and the few experimental reports available so far do not evidence a BKT jump^{26,41,42}.
Within the BKT approach, the effect of vortexlike topological excitations provides additional suppression of J_{s} with respect to the BCS dependence discussed above, driven only by quasiparticle excitations. To provide a fit of \({J}_{{{{{{{{\rm{s}}}}}}}}}^{\exp }\), we then solved numerically the renormalizationgroup (RG) equations of the BKT theory for the superfluid stiffness and vortex fugacity. As input parameters of the RG equations, we used the BCS temperature dependence of the stiffness. As mentioned above, the lowtemperature part is fully captured by the BCS approximation, and for the estimated \({{\Delta }}(0)/{k}_{{{{{{{{\rm{B}}}}}}}}}{T}_{{{{{{{{\rm{c}}}}}}}}}^{0}\) ratio, the dirtylimit and the cleanlimit expressions of \({J}_{{{{{{{{\rm{s}}}}}}}}}^{{{{{{{{\rm{BCS}}}}}}}}}\) provide the same result. We also included the finitefrequency effects in our calculation^{43,49,50}. Indeed, even though the resonance frequency (about 0.5 GHz) is still small as compared to the optical gap (2Δ ~ 10 K ~ 200 GHz) it can nonetheless lead to nonnegligible effects, in particular a rounding of the jump and suppression of the stiffness at a temperature slightly larger than the one where the dc resistivity vanishes^{32,33,38}, as indeed observed in our case. The resistivity itself is consistently fitted with the interpolating Halperin–Nelson formula^{50}, which accounts for BKTlike fluctuations between T_{BKT} and \({T}_{{{{{{{{\rm{c}}}}}}}}}^{0}\), and for standard Gaussian fluctuations above \({T}_{{{{{{{{\rm{c}}}}}}}}}^{0}\). Finally, to account for spatial inhomogeneities, we introduce a gaussian distribution of local T_{c} and J_{s} with variance σ_{G} centered around \({T}_{{{{{{{{\rm{c}}}}}}}}}^{0}\) and \({J}_{{{{{{{{\rm{s}}}}}}}}}^{\exp }(0)\). As seen in Fig. 4c, the result of the fitting procedure (dashed red line) is in very good agreement with experimental data both for the superfluid stiffness and the resistance, considering a very small inhomogeneity, σ_{G} = 0.02. Details on the fitting procedure are given in the Method Section.
Although KTaO_{3} and SrTiO_{3} have many common properties, the superconducting phases of their interfacial 2DEG exhibit noticeable differences. Whereas a pure BCS weakcoupling limit with Δ(0)/k_{B}T_{c} ≃ 1.76 provides a very good description of superconductivity in SrTiO_{3}based interfaces^{26,27}, we found a stronger value of the coupling for KTaO_{3} (Δ(0)/k_{B}T_{c} ≃ 2.3). Such an important difference, which must be traced back to the pairing mechanism, is a strong constraint on the possible origin of superconductivity in these two materials. In addition, BKT physics was not observable in SrTiO_{3} for which a simple BCS model without phase fluctuations was sufficient to fit the J_{s}(T) curves with a very good accuracy^{27}. This may suggest more bosoniclike superconductivity in KTaO_{3}based interfaces (in the highly doped regime), as evidenced by the large separation between the pairing scale, set by Δ, and the phasecoherence scale, set by the small value of the superfluid stiffness. Recent measurements of the inplane critical field in KTaO_{3}based interfaces suggested that the order parameter could be a mixture of swave and pwave pairing components induced by strong spinorbit coupling^{51}. While we can not rule out this possibility, the saturation of the J_{s}(T) curve below T_{c}/2 seen in Fig. 4b suggests a dominance of the fully gapped swave component. In addition, despite the complex band structure of the KTaO_{3}(111) interfaces, we have not observed any signatures suggesting multigap superconductivity in our data. Further experiments, including tunneling spectroscopy, are therefore necessary to understand the nature of superconductivity in KTaO_{3}based interfaces.
Methods
Sample fabrication
Prior to deposition, KTaO_{3} (111) substrates from MTI corporation were annealed at 600 ^{∘}C for 1 h in vacuum. Then, the thin Al layer was deposited in a dc magnetron sputtering system (PLASSYS MP450S) under a base pressure of the vacuum chamber lower than 5 × 10^{−8} mbar. During Al deposition, the Ar partial pressure and the dc power were kept fixed at 5 × 10^{−4} mbar and 10 W, respectively. The deposition rate for Al was 0.66 Å/s. Table 1 below summarizes the deposition parameters for the different samples.
XPS analysis
Xray photoelectron spectroscopy (XPS) was measured using a nonmonochromatized Mg K_{α} source (hν = 1253.6 eV) in an Omicron NanoTechnology GmbH system with a base pressure of 5 × 10^{−10} mbar. The operating current and voltage of the source was 20 mA and 15 kV, respectively. Spectral analysis to determine different valence states of Ta were carried out using the CasaXPS software. Adventitious carbon was used as a charge reference to obtain the Ta 4f_{5/2} peak position for the fitting. The energy difference and the ratio of the area between 4f_{5/2} and 4f_{3/2} peaks for all the Ta valence states were constrained according to the previously reported values.
STEM characterization
STEMHAADF and STEMEELS measurements have been done at 100 keV using a Cs corrected Nikon STEM microscope and a Gatan modified EELS spectrometer equipped with a MerlinEM detector.
Theoretical analysis of J _{s}(T)
In order to account for vortex excitations, we solved the BKT RG equations^{15,43,44} for the vortex fugacity \(g=2\pi {e}^{\mu /({k}_{B}T)}\), with μ the vortexcore energy, and the rescaled stiffness K ≡ πJ_{s}/k_{B}T:
where \(\ell=\ln (a/{\xi }_{0})\) is the RGscaled lattice spacing with respect to the coherence length ξ_{0}, that controls the vortex sizes and appears as a shortscale cutoff for the theory. The initial values at ℓ = 0 are set by the BCS fitting J^{BCS}(T) of \({J}_{{{{{{{{\rm{s}}}}}}}}}^{\exp }\), and the renormalized stiffness is given by the largescale behavior, J_{s} = (k_{B}T/π)K(ℓ → ∞). The ratio μ/J_{s} = 0.87, similar to the one found in other conventional superconductors^{34,35,37,38}, is used as a free (temperatureindependent) parameter, which controls the strength of stiffness renormalization due to bound vortices below T_{BKT}^{43}. To account for finitefrequency effects, we further include a dynamical screening of vortices^{49,50} via an effective frequencydependence dielectric function ε(ω) which enters in the complex conductivity of the film as \(\sigma (\omega )=\frac{4{J}^{{{{{{{{\rm{BCS}}}}}}}}}{e}^{2}}{i\omega {\hslash }^{2}\varepsilon (\omega )}\). At zero frequency ε(ω) is real and ε_{1}(0) = K(0)/K(ℓ → ∞) = J^{BCS}/J_{s} so one recovers the usual static result. At finitefrequency ε(ω) develops an imaginary part due to the vortex motion, that can be expressed in the first approximation^{49} as \({\varepsilon }_{2}\simeq {({r}_{\omega }/\xi )}^{2}\), where ξ is the vortex correlation length and r_{ω} is a finite length scale set in by the finite frequency of the probe, i.e. \({r}_{\omega }=\sqrt{\frac{14{D}_{v}}{\omega }}\), with D_{v} the vortex diffusion constant of the vortices. The main effect of ε_{2} is to induce a small tail above T_{BKT} for the finitefrequency stiffness, as given by J_{s} = ℏ^{2}ωσ_{2}(ω)/(4e^{2}), as we indeed observe in the experiments. Here we follow the same procedure outlined in ref. 38 to compute ε(ω), and in full analogy, with this previous work, we find a very small vortex diffusion constant D_{v} ~ 10^{10} nm^{2}/s. The correlation length ξ(T) also enters the temperature dependence of the resistivity above T_{BKT}, that follows the usual scaling R/R_{N} = 1/ξ^{2}(T). To interpolate between the BKT and Gaussian regime of fluctuations, we use the wellknown Halperin–Nelson expression^{43,46,50} \({\xi }_{HN}(T)=\frac{2}{A}\sinh \left(\frac{b}{\sqrt{t}}\right)\) where t = (T − T_{BKT})/T_{BKT}, and we set A = 2.5 and b = 0.27, consistent with the theoretical estimate of b ≃ 0.2 that we obtain from the value of μ^{35,38,43}. Finally, to account for the possible inhomogeneity of the sample, we consider the extension of the previous method to the case where the overall complex conductivity of the sample is computed in the selfconsistent effectivemedium approximation^{52} as the solution of the following equation:
Here σ_{i}(ω) denotes the complex conductivity of a local superconducting puddle with stiffness J_{i} and local \({T}_{{{{{{{{\rm{c}}}}}}}}}^{i}\), that are taken with a Gaussian distribution P_{i} with variance σ_{G} centered around the BCS fit of \({J}_{{{{{{{{\rm{s}}}}}}}}}^{\exp }\). For each realization J_{i} we then compute the J_{s,i} from the solution of the BKT equations (2) (3), we determine the corresponding complex conductivity σ_{i}(ω) and we finally solve Eq. (4) to get the average \({J}_{{{{{{{{\rm{s}}}}}}}}}^{{{{{{{{\rm{BKT}}}}}}}}}=({\hslash }^{2}/4{e}^{2})\omega {\sigma }_{2}(\omega )\) below T_{c} and the average σ_{1}(ω = 0) ≡ 1/R^{HN} above T_{c}, i.e., the dashed lines reported in Fig. 4c. Further details about the implementation of the effectivemedium approximation can be found in refs. 16,38. The main effect of inhomogeneity is to contribute slightly to the suppression of J_{s} with respect to J^{BCS} before T_{BKT}. In our case, we checked that inhomogeneity, if present, is very small, and a σ_{G} = 0.02 is enough to account for the measured temperature dependences.
Data availability
The authors declare that the data that support the findings of this study are available within the article. All other relevant data are available from the corresponding authors upon request.
References
Fujii, Y. & Sakudo, T. Dielectric and optical properties of KTaO_{3}. J. Phys. Soc. Jpn. 41, 888–893 (1976).
Fleury, P. A. & Worlock, J. M. Electricfieldinduced Raman scattering SrTiO_{3} and KTaO_{3}. Phys. Rev. 174, 613 (1968).
Schooley, J. F., Hosler, W. R. & Cohen, M. L. Superconductivity in semiconducting SrTiO_{3}. Phys. Rev. Lett. 12, 474–475 (1964).
Thompson, J. R., Boatner, L. A. & Thomson, J. O. Very lowtemperature search for superconductivity in semiconducting KTaO_{3}. J. Low Temp. Phys. 47, 467 (1982).
Ueno, K. et al. Discovery of superconductivity in KTaO_{3} by electrostatic carrier doping. Nature Nano. 6, 408–412 (2011).
Liu, C. et al. Twodimensional superconductivity and anisotropic transport at KTaO_{3} (111) interfaces. Science 371, 716–721 (2021).
Chen, Z. et al. Electric field control of superconductivity at the LaAlO_{3}/KTaO_{3} (111) interface. Science 372, 721–724 (2021).
Reyren, N. et al. Superconducting interfaces between insulating oxides. Science 317, 1196–1199 (2007).
Caviglia, A. D. et al. Electric field control of the LaAlO_{3}/SrTiO_{3} interface ground state. Nature 456, 624–627 (2008).
Biscaras, J. et al. Twodimensional superconducting phase in LaTiO_{3}/SrTiO_{3} heterostructures induced by highmobility carrier doping. Phys. Rev. Lett. 108, 247004 (2012).
Chen, Z. et al. Twodimensional superconductivity at the LaAlO_{3}/KTaO_{3} (110) heterointerface. Phys. Rev. Lett. 126, 026802 (2021).
Liu, C. et al. Tunable superconductivity at the oxideinsulator/KTaO_{3} interface and its origin. Preprint at arXiv:2203.05867 (2022).
Berezinskii, V. L. Destruction of longrange order in onedimensional and twodimensional systems possessing a continuous symmetry group. II. Quantum systems. Sov. Phys. JETP 34, 610 (1972).
Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in twodimensional systems. J. Phys. C 6, 1181 (1973).
Kosterlitz, J. M. The critical properties of the twodimensional xy model. J. Phys. C 7, 1046 (1974).
Venditti, G. et al. Nonlinear IV characteristics of twodimensional superconductors: BerezinskiiKosterlitzThouless physics versus inhomogeneity. Phys. Rev. B 100, 064506 (2019).
Vaz, D. C. et al. Mapping spincharge conversion to the band structure in a topological oxide twodimensional electron gas. Nature Mat. 18, 1187–1193 (2019).
VicenteArche, L. M. et al. Metal/SrTiO_{3} twodimensional electron gases for spintocharge conversion. Phys. Rev. Mat. 5, 064005 (2021).
Rödel, T. C. et al. Universal fabrication of 2D electron systems in functional oxides. Adv. Mater. 28, 1976–1980 (2016).
VicenteArche, L. M. et al. Spincharge interconversion in KTaO_{3} 2D electron gases. Adv. Mater. 33, 2102102 (2021).
Biscaras, J. et al. Limit of the electrostatic doping in twodimensional electron gases of LaXO_{3} (X = Al, Ti)/SrTiO_{3}. Sci. Rep. 4, 6788 (2014).
Jaeger, H. M., Haviland, D. B., Orr, B. G. & Goldman, A. M. Onset of superconductivity in ultrathin granular metal films. Phys. Rev.B 40, 182–196 (1989).
Orr, B. G., Jaeger, H. M. & Goldman, A. M. Local superconductivity in ultrathin Sn films. Phys. Rev.B 32, 7586(R) (1985).
Singh, G. et al. Effect of disorder on superconductivity and Rashba spinorbit coupling in LaAlO_{3}/SrTiO_{3} interfaces. Phys. Rev. B 96, 024509 (2017).
Tinkham, M. Introduction to Superconductivity 2nd edn (Dover Publications, Inc., 2004).
Singh, G. et al. Competition between electron pairing and phase coherence in superconducting interfaces. Nat. Commun. 9, 407 (2018).
Singh, G. et al. Gap suppression at a Lifshitz transition in a multicondensate superconductor. Nature Mat. 18, 948–954 (2019).
Epstein, K., Goldman, A. M. & Kadin, A. M. Vortexantivortex pair dissociation in twodimensional superconductors. Phys. Rev. Lett. 47, 534 (1981).
Epstein, K., Goldman, A. M. & Kadin, A. M. Renormalization and the KosterlitzThouless transition in a twodimensional superconductor. Phys. Rev. B 27, 6691 (1983).
Fiory, A. T., Hebard, A. F. & Glaberson, W. I. Superconducting phase transitions in indium/indiumoxide thinfilm composites. Phys. Rev. B 28, 5075 (1983).
Turneaure, S. J., Lemberger, T. R. & Graybeal, J. M. Effect of thermal phase fluctuations on the superfluid density of twodimensional superconducting films. Phys. Rev. Lett. 84, 987 (2000).
Crane, R. W. et al. Fluctuations, dissipation, and nonuniversal superfluid jumps in twodimensional superconductors. Phys. Rev. B 75, 094506 (2007).
Liu, W., Kim, M., Sambandamurthy, G. & Armitage, N. P. Dynamical study of phase fluctuations and their critical slowing down in amorphous superconducting films. Phys. Rev. B 84, 024511 (2011).
Kamlapure, A. et al. Measurement of magnetic penetration depth and superconducting energy gap in very thin epitaxial NbN films. Appl. Phys. Lett. 96, 072509 (2010).
Mondal, M. et al. Role of the vortexcore energy on the BerezinskiiKosterlitzThouless transition in thin films of NbN. Phys. Rev. Lett. 107, 217003 (2011).
Misra, S., Urban, L., Kim, M., Sambandamurthy, G. & Yazdani, A. Measurements of the magneticfieldtuned conductivity of disordered twodimensional Mo_{43}Ge_{57} and InOx. Superconducting films: evidence for a universal minimum superfluid response. Phys. Rev. Lett. 110, 037002 (2013).
Yong, J., Lemberger, T., Benfatto, L., Ilin, K. & Siegel, M. Robustness of the BerezinskiiKosterlitzThouless transition in ultrathin NbN films near the superconductorinsulator transition. Phys. Rev. B 75, 184505 (2013).
Ganguly, R., Chaudhuri, D., Raychaudhuri, P. & Benfatto, L. Slowing down of vortex motion at the BerezinskiiKosterlitzThouless transition in ultrathin NbN films. Phys. Rev. B 91, 054514 (2015).
Bareille, C. et al. Twodimensional electron gas with sixfold symmetry at the (111) surface of KTaO_{3}. Sci. Rep. 4, 3586 (2014).
Bruno, F. Y. et al. Band structure and spinorbital texture of the (111)KTaO_{3}2D electron gas. Adv. Electron. Mater. 5, 1800860 (2019).
Manca, N. et al. Bimodal phase diagram of the superfluid density in LaAlO_{3}/SrTiO_{3} revealed by an interfacial waveguide resonato. Phys. Rev. Lett. 122, 036801 (2019).
Bert, J. A. et al. Gatetuned superfluid density at the superconducting LaAlO_{3}/SrTiO_{3} interface. Phys. Rev. B 86, 060503(R) (2012).
Benfatto, L., Castellani, C. & Giamarchi, T. in 40 Years of BerezinskiiKosterlitzThouless Theory (ed. Josè, J. V.) Ch. 5 (World Scientific,2013).
Nelson, D. R. & Kosterlitz, J. M. Universal jump in the superfluid density of twodimensional superfluids. Phys. Rev. Lett. 39, 1201–1205 (1977).
McQueeney, D., Agnolet, G. & Reppy, J. D. Surface superfluidity in dilute ^{4}He^{3}He mixtures. Phys. Rev. Lett. 52, 1325 (1984).
Benfatto, L., Castellani, C. & Giamarchi, T. Broadening of the BerezinskiiKosterlitzThouless superconducting transition by inhomogeneity and finitesize effects. Phys. Rev. B 80, 214506 (2009).
Mondal, M. et al. Phase fluctuations in a strongly disordered swave NbN superconductor close to the metalinsulator transition. Phys. Rev. Lett. 106, 047001 (2011).
Maccari, I., Benfatto, L. & Castellani, C. Broadening of the BerezinskiiKosterlitzThouless transition by correlated disorder. Phys. Rev. B 96, 060508 (R) (2017).
Ambegaokar, V., Halperin, B. I., Nelson, D. R. & Siggia, E. D. Dynamics of superfluid films. Phys. Rev. B 21, 1806 (1979).
Halperin, B. I. & Nelson, D. R. Resistive transition in superconducting films. J. Low. Temp. Phys. 36, 599 (1979).
Zhang, G. et al. Spontaneous rotational symmetry breaking in KTaO_{3} interface superconductor. Preprint at arXiv:2111.05650v2 (2021).
Kirkpatrick, S. Percolation and conduction. Rev. Mod. Phys. 45, 574 (1973).
Acknowledgements
This work was supported by the ANR QUANTOP ProjectANR19CE470006 grant, by the QuantERA ERANET Cofund in Quantum Technologies (Grant Agreement N. 731473) implemented within the European Union’s Horizon 2020 Program (QUANTOX), by EU under project MORETEM ERCSYN (grant agreement No 951215) and by Sapienza University of Rome, through the projects Ateneo 2019 (Grant No. RM11916B56802AFE) and Ateneo 2020 (Grant No. RM120172A8CC7CC7), and by the Italian MIUR through the Project No. PRIN 2017Z8TS5B.
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N.B. and M.B. proposed and supervised the study. S.M. and H. W. prepared the samples and performed XPS experiments and their analysis. A.G. performed the STEM and EELS analysis. G.C.M., G.S., and S.M. performed the dc and microwave transport experiments and analysed the data with input from M.B., J.L., and N.B. L.B. conducted the BKT analysis of microwave data. N.B., M.B., and L.B. wrote the manuscript with input from all authors. All authors discussed the results and contributed to their interpretation.
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Mallik, S., Ménard, G.C., Saïz, G. et al. Superfluid stiffness of a KTaO_{3}based twodimensional electron gas. Nat Commun 13, 4625 (2022). https://doi.org/10.1038/s4146702232242y
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DOI: https://doi.org/10.1038/s4146702232242y
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