A spin–orbit scattering–enhanced high upper critical field at the LaAlO3/KTaO3(111) superconducting interface

Spin–orbit interaction is essential to enhance the in-plane upper critical field of two-dimensional superconductors. Here, we report the LaAlO3/KTaO3 (111) superconducting interface (T c,0 ≈ 0.475 K) with a high in-plane upper critical field (∼1.6 T), which is approximately 1.8 times the Pauli paramagnetic limit. The H − T superconducting phase diagram is well-fitted by the Klemm–Luther–Beasley (KLB) theory, and the relevant spin–orbit scattering (SOS) length is approximately 32 nm. Furthermore, normal-state magnetotransport measurements show signatures of weak antilocalization caused by strong spin–orbit coupling in LaAlO3/KTaO3 (111). The spin diffusion length derived from magnetotransport measurements was 40 nm at 2 K, which is comparable with the SOS length. The conformity of the phase diagram with the KLB theory and the consistency of normal state spin diffusion length and superconducting SOS length indicate that the high in-plane upper critical field at the LaAlO3/KTaO3 (111) superconducting interface is enhanced by SOS.

Superconductors with high upper critical fields attract a lot of attention, especially when such fields exceed the Clogston-Chandrasekhar limit [26,27] or the Pauli paramagnetic limit. Furthermore, the realization of superconducting materials that are resilient to strong external magnetic fields remains an important pursuit for fundamental research [28][29][30][31][32][33]. There are four possible factors responsible for the enhancement of the upper critical field [34]: spin-triplet superconducting pairing state, the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, spin-orbit scattering (SOS), and the Ising SOC. Each of them is an inspiring branch since they are related to unconventional superconductivity, especially topological superconductivity or Majorana zero modes, which have triggered enormous interest in recent years [35][36][37]. More details are provided in the next section.
In this paper, we report an enhanced high upper critical field at the amorphous LaAlO 3 /KTaO 3 (111) interface. Low-temperature measurements show that the in-plane upper critical field exceeds the Pauli paramagnetic limit. The H−T superconducting phase diagram can be well described by Klemm-Luther-Beasley (KLB) theory [38]. Furthermore, normal state magnetotransport measurements show signatures of weak antilocalization (WAL), and the effective SOC field deduced from the magnetoconductance is dominant compared with the effective dephasing field. The spin diffusion length derived from normal-state magnetotransport measurements is comparable to the spin-orbit scattering length derived from KLB fitting in the superconducting state. Our results reveal that the high in-plane upper critical field at the LaAlO 3 /KTaO 3 (111) superconducting interface is enhanced by SOC.

Method
Amorphous LaAlO 3 films (see figure A5 in Appendix A) were deposited on (111)-oriented KTaO 3 substrates by pulsed laser deposition technique (KrF, λ = 248 nm). Amorphous films were probably formed due to the large lattice mismatch between LaAlO 3 and KTaO 3 [39]. The lattice constant was 3.792 Å for LaAlO 3 and 3.989 Å for KTaO 3 , with a large lattice mismatch of 5.2%. The laser fluence was about 1 Jcm -2 , and the laser repetition rate was 1 Hz. During deposition, the substrate temperature was 400 • C, the oxygen pressure was 2 × 10 -5 mbar, and water vapor was not used [4].
The sub-Kelvin measurements were performed using a standard four-probe resistance configuration with a lock-in amplifier (Stanford Research System, SR830) in the alternate current mode in a dilution refrigerator (Oxford Instruments, Triton-400). All the sub-Kelvin measurements were performed with electric current along KTaO 3 [112] crystal axis. The excitation current was 100 nA, and the signal frequency was 7.9 Hz. The normal-state electrical transport and magnetotransport measurements were performed using the Van der Pauw configuration in a Physical Properties Measurement System (PPMS, Quantum Design). The direct current for normal-state measurements was 10 µA. Electrical contacts to the samples were bonded by ultrasonic bonding (25 µm diameter aluminum wires). Figure 1(a) shows the temperature-dependent resistance R(T) for the amorphous LaAlO 3 /KTaO 3 (111) interface. Superconducting transition was observed, with transition temperatures T c onset ≈ 0.67 K, T c ≈ 0.48 K, and T c zero ≈ 0.33 K. T c onset is the temperature at which R(T) first deviates from its linear dependence at a high temperature, T c is the temperature at which the resistance drops to half of its value at T c onset , and T c zero is the temperature at which the resistance drops indistinguishable from the noise floor. The transition width, defined between 20% and 80% of the normal state resistance (R n , measured at T = 0.67 K), is ∆T c ≈ 0.08 K (figure A 6 in Appendix B). The relative transition width ∆T c /T c was approximately 16.7%, which is comparable with that recorded in previous reports [4,5,7]. With an increasing magnetic field, the superconductivity is suppressed and the system gradually becomes weakly insulating. The isomagnetic R(T) curves show that resistance changes slightly as temperature decreases in a characteristic field (∼0.25 T, indicated by a short red arrow) that destroys the superconductivity, as shown in figure 1(b). As a signature of H-driven superconductor-insulator transition, the isotherm R(H ⊥ ) curves measured at different temperatures cross each other around 0.22 T, as shown in figure 1(c).

Superconducting transition and phase diagram
The temperature dependence of out-of-plane upper critical field H c2,⊥ , i.e. the H−T superconducting phase diagram, is shown in figure 1(d). For the definition of the upper critical field, H c2 is obtained from the position where the resistance drops to half of the normal state resistance (R n , measured at the cross-point of isotherm curves with a resistance of approximately 157.4 Ω). (For more details, see figure A 7 in Appendix B).
The temperature dependence of H c2,⊥ around T c,0 is well described by the Ginzburg-Landau (GL) theory, yielding a linearized equation [31,40] as follows: where Φ 0 is the flux quantum, T c,0 is the zero-field superconducting critical temperature, and ξ ab is the in-plane GL coherence length at T = 0 K. From the slope of the fitting curve, we can estimate the coherence length, ξ ab ≈ 74 nm. The H c2,⊥ data deviate from the 2D GL model at lower temperatures, suggesting an orbital effect in a perpendicular magnetic field (more details are discussed in section 3.2).
To further study the superconducting mechanism, we measured the isomagnetic R(T) and isotherm R(H // ) curves in a magnetic field applied parallel to the interface. As shown in figure 2(a), with an increasing magnetic field, the superconductivity is suppressed, and the system gradually becomes weakly insulating. These isomagnetic R(T) curves are similar to those measured at the out-of-plane magnetic field. However, the in-plane characteristic field (∼4.0 T, indicated by a short red arrow in figure 2(a)) destroying the superconducting state is much larger than the out-of-plane characteristic field (∼0.25 T). Furthermore, as shown in figure 2(b), the isotherm R(H // ) curves measured at different temperatures cross each other around 4.2 T, indicating a superconductor-insulator transition similar to the out-of-plane case but occurring at a much larger field. This suggests the strong anisotropy character of the superconductor.
The superconducting phase diagram in the in-plane magnetic field is shown in figure 2(c). The temperature dependence of H c2,// near T c,0 is well fitted by the 2D-GL model, yielding a square-root equation [31,40] as follows: where ξ c is the out-of-plane coherence length. The ξ c estimated from the slope of the 2D-GL fitting curve is approximately 2.4 nm, which is much smaller than ξ ab . The GL anisotropy estimated by ξ ab /ξ c is approximately 30.8, indicating the strong anisotropy character of the superconductor.

High in-plane upper critical field
Generally, superconductivity in type-II superconductors is destroyed by an external magnetic field in two ways: the orbital and paramagnetic pair-breaking effects [49]. The orbital effect leads to the formation of Abrikosov vortices. The orbital limiting field is given as H orb c2 at which vortex cores begin to overlap. Experimentally, H orb c2 at T = 0 is commonly derived from the slope of the H−T phase boundary at T c,0 as [50]: From the slope of the fitting curve in figure 1(d), we can estimate the H orb c2 at T = 0 is about 0.042 T, which is comparable with the measured out-of-plane upper critical field (≈0.042 T) at the lowest temperature (50 mK). This estimation means that the small out-of-plane upper critical field is dominated by the orbital effect.
When the layer thickness is smaller than the coherence length in layered materials in a parallel magnetic field, the orbital effect becomes negligible [49]. The superconductivity will be first destroyed by the paramagnetic pair-breaking effect. The paramagnetic pair-breaking effect originates from the Zeeman splitting of single electron energy levels. When a magnetic field is applied in the normal state, electrons become polarized due to the Zeeman effect (Pauli paramagnetism) [49]. In contrast, in the superconducting state, spin-singlet Cooper pairs (spin in the form of '↑↓ − ↓↑') with vanishing spin susceptibility due to the Meissner effect are not spin-polarized. Then, to polarize condensed electrons, the pairs must be broken. This destruction occurs when the Pauli paramagnetic energy E P = (1/2)χ n H 2 becomes comparable to the superconducting condensation energy is the spin susceptibility in the normal state [26,27,49], g is an electron g-factor, µ B is the Bohr magneton, N(0) is the density of states at the Fermi level, and ∆ 0 is the superconducting gap. The Pauli limiting upper critical field is then estimated as [26,27,49]: Generally, the superconducting gap function ∆ 0 is approximately 1.76k B T c,0 based on the Bardeen-Cooper-Schrieffer theory. Assuming g = 2 for a free electron, we obtained H P c2 ≈ 1.86T c,0 , as shown by the dashed line in figure 2(c).
In figure 2(c), H c2,// at lower temperature deviates from the 2D-GL fitting curve and monotonously increases to 1.6 T at 50 mK, exceeding H P c2 . As mentioned above, four possible factors are responsible for H c2 enhancement: spin-triplet superconducting pairing state, FFLO state, SOS, and Ising SOC. Each of the enhancement is discussed as follows: (a). Generally, a spin-triplet superconductor could have the parallel-aligned spin configuration (spin in the form of '↑↑' or '↓↓') in Cooper pairs, which is hardly affected by the Pauli paramagnetism, so the upper critical field can easily exceed the Pauli paramagnetic limit [51][52][53]. A Recent work show that the strong SOC may induce the coexistence of s-wave and p-wave pairing in KTaO 3 interface superconductor [54]. However, we could not confirm the possibility of spin-triplet superconducting pairing according to our experimental data. More experiments are needed to prove this, which is beyond the scope of this paper. (b). The FFLO state [55,56] originates from the paramagnetism of conduction electrons. Forming an FFLO state with inhomogeneous pairing densities favors the presence of a magnetic field, even above Pauli paramagnetic limit [49]. This system should be in clean limit ξ ab ≪ l [49,[55][56][57][58][59], where l is the mean free path. However, recent studies [4,5] did not find the mean free path l in KTaO 3 -based heterostructures to be higher than ξ ab, and neither did this work (l(2 K) ≈ 24 nm, ξ ab ≈ 74 nm, see the next paragraph), indicating that this system is not in the clean limit. (c). In dirty-limit layered superconductors with strong spin-orbit interaction (ξ ab ≫ l and τ ≪ τ SO , where τ and τ SO denote the total scattering time and characteristic SOS time, respectively), electron spins can be randomized by SOS, weakening the effect of spin paramagnetism [38,[60][61][62][63][64][65]. Therefore, H c2 can exceed H P c2 . This mechanism can be described by the microscopic KLB theory, which satisfies the following equation [38]: where ψ(x) is the digamma function. As shown in figure 2(c), the H c2,// is fitted much better by the KLB theory than by the 2D-GL model. The τ SO solved numerically following equation (5) is approximately 1.7 ps, comparable to the spin diffusion time found in the gate-induced oxide interfaces of LaAlO 3 /SrTiO 3 and LaAlO 3 /KTaO 3 [39,66]. These can preliminarily confirm that H c2 enhancement is caused by SOS. The next section shows more evidence about normal state magnetotransport measurements. (d). In some highly crystalline transition-metal disulfide (TMD) thin layers (MoS 2 , NbSe 2 , etc.), H c2 can easily reach four or five times of H P c2 . In the TMD system, the τ SO value calculated by the KLB theory is smaller than that of τ estimated by the transport [31]. This unphysical result indicates that H c2 enhancement in the TMD system cannot be explained by the KLB theory. Hence, a new interpretation termed 'Ising superconductivity' is proposed. In fact, the enhancement of H c2 in Ising superconductors is a consequence of the noncentrosymmetric structure with strong SOC and the breaking of in-plane inversion symmetry [67,68]. Taking a monolayer of MoS 2 as an example [69], the spins are locked to the out-of-plane direction due to the rotational symmetry. An in-plane magnetic field, therefore, has to compete with the Zeeman-like spin splitting built into the bands; thus, the superconducting state becomes more resilient to the paramagnetic effect. However, in the amorphous LaAlO 3 /KTaO 3 (111) interface, the value of τ SO calculated by the KLB theory is larger than that of τ estimated by the transport (see the next paragraph); this reasonable result indicates that the system conforms to the KLB theory rather than Ising superconductivity.

Weak antilocalization
To further confirm that SOC causes H c2 enhancement, we measured the magnetotransport in the normal state of LaAlO 3 /KTaO 3 . Hall measurements in figure 3(a) show a linear trend of R xy up to a magnetic field of 9 T, suggesting the absence of multiple carriers. Negative Hall slope coefficient R H indicates that the charge carriers are electrons, where R H = dR xy /dH. As plotted in figure 3(b), the sheet Hall carrier density n S and the carrier mobility µ can be calculated as n S = −1/(e·R H ) and µ = 1/(e·n S ·R S ), respectively, where e is the elementary charge and R S is the sheet resistance. At 2 K, the n S and µ are 2.4 × 10 14 cm -2 and 94.8 cm 2 V -1 s -1 , respectively, which are comparable to the values of a recent report [5].
We estimated the mean free path l of the conduction electrons using a single-band model, where l satisfies the following equation [31,70]: where h and k F are the Planck's constant and Fermi wave vector k F = (2πn S ) 1/2 , respectively. As shown in figure 3(c), the value of l at 2 K is approximately 24 nm and saturated at the lowest temperatures. This value is much smaller than the coherence length ξ ab ≈ 74 nm, excluding the possibility that the enhanced H c2 was caused by the FFLO state. The total scattering time τ can be estimated by the carrier mobility µ, where τ = m * µ/e, while m * is the effective mass of carriers, m * ≈ 0.36m 0 [71], and m 0 is the free electron mass. As shown in figure 3(c), the value of τ at 2 K is approximately 19 fs, which is much smaller than that of the SOS time τ SO (≈1.7 ps), indicating conformity to the KLB theory (τ ≪ τ SO ) in this system. This result proves that the high in-plane upper critical field at the LaAlO 3 /KTaO 3 (111) superconducting interface is enhanced by SOS rather than Ising SOC. MR in figure 3(d) show a sharp dip at a low field below 10 K, and the sharpness of the dip increases with decreasing temperature. This result manifests the WAL feature, caused by SOC [72]. To assess the influence of the spin-orbit interaction, we performed a quantitative analysis of the [∆σ(H)/σ 0 ]−H relation based on the Hikami-Larkin-Nagaoka (HLN) theory [73], which satisfies the following equation [73][74][75][76][77]: where ∆σ = σ(H) − σ(H = 0), σ(H) is the sheet conductance in an applied field H, and σ 0 = e 2 /πh is the quantum conductance. The 'fields' H n are defined by where H el = ℏ/(4eDτ el ), H so = ℏ/(4eDτ so ), H s = ℏ/(4eDτ s ), and H in = ℏ/(4eDτ in ) are the effective fields related to elastic, spin-orbit, magnetic, and inelastic scattering times, respectively. D = πℏ 2 n S µ/(m * e) is the diffusion coefficient. A K and C is the Kohler term, which accounts for orbital MR. Note that the dephasing rate τ −1 φ = 2τ −1 s + τ −1 in , and τ −1 s is the magnetic scattering rate mainly caused by magnetic impurities [76,77]. For details about the analysis of dedephasing rate, please see figure A8 in Appendix D.
The [∆σ(H)/σ 0 ] − H relation is shown in figure 4(a). The solid curves show the HLN fitting (for the fitting parameters, please see table A1 in Appendix C) based on equation (7), which can accurately reproduce the experimental results by adopting the parameters shown in figure 4(b). As shown in figure 4(b), the value of H so is approximately 0.1 T at 2 K, comparable to that of the amorphous LaAlO 3 /KTaO 3 (001) interface [39]. Furthermore, although all effective fields decrease with decreasing temperature, H so is always the biggest, indicating that SOC is dominant in this system. The calculated dephasing time (τ φ ) and spin relaxation time (τ so ) are shown in figure 4(c). Although the τ so (≈108 fs) at 2 K is an order of magnitude smaller than the τ SO (≈1.7 ps) derived from the superconducting phase diagram, the normal state spin diffusion length l so is comparable with the superconducting SOS length l SO . The reason is that the definitions of carrier diffusion coefficients between the superconducting state (D SC ) and normal state (D) are quite different [see equations (10) and (8), respectively]. The normal state spin diffusion length l so can be calculated by The obtained l so is approximately 40 nm at 2 K. D SC of a weak coupling superconductor in the dirty limit satisfies the following equation [78]: We obtained l SO = (D SC ·τ SO ) 1/2 ≈ 32 nm for the superconducting SOS length, which is well consistent with the normal state spin diffusion length l so , as shown in figure 4(d). This length consistency indicates a strong SOC in LaAlO 3 /KTaO 3 , whether in a superconducting or normal state. The repeatable results are shown in figures A9 and A10 in Appendix E for Sample #2. The high in-plane upper critical field at the LaAlO 3 /KTaO 3 (111) superconducting interface is indeed enhanced by SOS.

Conclusion
In conclusion, we have demonstrated that SOS enhanced the high in-plane upper critical field at the amorphous LaAlO 3 /KTaO 3 (111) interface. The in-plane H−T superconducting phase diagram can be well fitted by the KLB theory. Furthermore, the quantitative analysis of the HLN fitting revealed that the effective SOS field is dominant compared with the effective dephasing field. The spin diffusion length in the normal state is comparable to the SOS length derived from KLB fitting in the superconducting state. Results indicate that the high in-plane upper critical field at the LaAlO 3 /KTaO 3 (111) superconducting interface is enhanced by SOS. Our findings shed new light on the underlying physics of KTaO 3 -based heterostructures. It would improve understanding of the newly discovered superconductivity in the KTaO 3 (111) system.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.

Appendix D. Analysis of the deduced dephasing rate
The temperature dependence of the inelastic scattering rate τ −1 in and magnetic scattering rate τ −1 s deduced from equation (7) are shown in figure A 8(a). As mentioned before, the dephasing rate τ −1 φ = 2τ −1 s + τ −1 in . In figure A 8(b), the circle points show the temperature dependence of τ −1 φ . As shown in figure A 8(b), the τ −1 φ data does not follow the linear T dependence, which is caused by electron-electron interaction. This indicates that in this system, multiple scattering mechanisms could be involved in dephasing the electron's phase. Therefore, we fit the temperature dependence of τ −1 φ using the simple equation [75,79] as follows: where τ −1 φ (0) represents the zero-temperature dephasing rate, and A ee T and A ep T p represent the contributions from electron-electron and electron-phonon interactions, respectively. The value of p for electron-phonon interaction in 2D materials can vary between two and three depending on the effective dimensionality and disorder in the film [75,[80][81][82]. As shown by the red solid curve in figure 7(b), we obtained an excellent fit of the temperature-dependent τ −1 φ data with p ≈ 2.75. The deduced A ee and A ep are approximately 2.7 × 10 10 K −1 s −1 and 8 × 10 9 K −2.75 s −1 , respectively. The larger A ee than A ep means that the phase breaking is mainly caused by electron-electron interaction.
Theoretically, Altshuler et al [83] and other studies [84][85][86] provided 2D quantitative calculations for the electron-electron scattering rate: Considering τ ≈ 19 fs at T = 2 K for this sample, T < ℏ kBτ is satisfied, we choose equation (12) to estimate the electron-electron scattering rate τ −1 ee ≈ 10 10 s −1 , comparable to the term A ee T. The deduced τ −1 φ (0) ≈ 6.5 × 10 10 s −1 indicates a finite zero-temperature dephasing rate. Many different processes might cause a zero-temperature dephasing rate [75,84,86]. We suppose that this rate in this sample is due to magnetic impurities [74,75,77,84,86]. As shown in figure A 8(a), τ −1 s is comparable to τ −1 in and becomes saturated as temperature decreases, and the term τ −1 s is mainly caused by magnetic impurities.   (7). (f) Effective dephasing (Hφ) and effective spin-orbit fields (Hso) deduced from the data fitting of (e). (g) Dephasing (τφ) and spin relaxation times (τ so) deduced from the effective field shown in (f). (h) Dephasing (lφ) and spin diffusion length (lso) deduced from (g). The obtained spin diffusion length lso is approximately 18.5 nm at 2 K, calling a well consistence with the normal state spin diffusion length lso ≈ 18.3 nm, as obtained in figure A 9. This length consistency indicates a strong SOC in LaAlO3/KTaO3, whether in a superconducting or normal state. All data show the repeatable results of the sample reported in the main text.