Gate dependence of upper critical field in superconducting (110) LaAlO3/SrTiO3 interface

The fundamental parameters of the superconducting state such as coherence length and pairing strength are essential for understanding the nature of superconductivity. These parameters can be estimated by measuring critical parameters such as upper critical field, Hc2. In this work, Hc2 of a superconducting (110) LaAlO3/SrTiO3 interface is determined through magnetoresistive measurements as a function of the gate voltage, VG. When VG increases, the critical temperature has a dome-like shape, while Hc2 monotonically decreases. This relationship of independence between the variation of Tc and of Hc2 suggests that the Cooper pairing potential is stronger in the underdoped region and the coherence length increases with the increase of VG. The result is as for high temperature superconducting cuprates and it is different than for conventional low temperature superconductors.

different gate voltages (− 5 V, 0 V and 25 V) under zero magnetic field. Temperature for which R s drops below the detection limit of the measuring equipment defines the critical temperature, T C Zero (Fig. 1). The curve of T C Zero as a function of V G (in zero magnetic field) has a dome-like shape with a maximum at 25 V (Fig. 1 inset). The domelike shape of the T C Zero (V G ) curves was previously observed in LAO/STO and (001) LaTiO 3 /SrTiO 3 (LTO/STO) interfaces 7,20,23 . A dome-like shape with a maximum at V G = 25 V is also obtained for the variation of the critical current I c (V G ). The critical current, I c , was determined at 50 mK from I-V curves measurements (see Supplementary Information, Fig. S1). The T C Zero (V G ) and I c (V G ) similar dome-like dependencies indicate a strong influence of the carrier density on T C Zero and I c . From the measurements of R s with magnetic field (up to 1.6 T) applied perpendicular to the surface of the interface at a fixed temperature and gate voltage one can determine the normal state resistance, R n . An example of R s (B) curves at 50 mK and for V G from − 25 to + 200 V is presented in Fig. 2a. Inset to Fig. 2a shows for selected (T, V G ) values how R n is determined. Namely, R n is the resistance of the cross point between the fitting lines of the steepest part of the R s (B) experimental curve and of the region where R s almost saturates. For a fixed temperature (50 mK), one R n (V G ) curve is plotted in Fig. 1 inset. Enhancement of V G decreases R n . Curve is non-linear and the decrease rate is smaller for higher V G .
It is remarkable that V G can tune superconducting and normal state characteristics such as T C Zero and R n , respectively. Results suggest that V G influences 2DEG superconductivity features and the carrier density. Our results are consistent with literature 20 .
Upper critical field as a function of gate voltage. For a fixed temperature and V G , the upper critical field H c2 is determined as the field where resistance is 10%, 50% of R n in the R s (B) curves (Fig. 2a inset, right corner). This methodology was used to determine H c2 for different superconducting systems [24][25][26] . We note that magnetic field B is applied in this work only perpendicular to the interface surface. Reported articles 19,20 indicate that  H c2 shows large anisotropy when B is applied in-plane and out-of-plane. At 50 mK, the H c2 10% (V G ) or H c2 50% (V G ) ( Fig. 2 inset, left corner) curves are unexpectedly without a dome-like shape that is specific for the T C Zero (V G ) and I c (V G ) (Figs 1 and S1) curves. Namely, for a lower V G , H c2 monotonically increases ( Fig. 2 inset, left corner). At the same time, a decrease of V G below V G, max = 25 V counter-intuitively produces the decrease of both T C Zero and I c (Figs 1 and S1).
The differential curves dR s /dB as a function of B, obtained from R s (B) data ( Fig. 2a) for different V G and at 50 mK, are presented in Fig. 2b. Each curve displays a peak (marked with an arrow in Fig. 2b). The peak shifts to higher B and its intensity increases when V G decreases. The magnetic field of the peak, denoted H c2 peak , as a function of V G (at 50 mK) is shown in Fig. 2b, inset. A dome-like shape is not obtained and, as expected, this curve has a similar behavior as H c2 10% (V G ) or H c2 50% . (V G ) curves. Furthermore, the values of H c2 peak and of H c2 50% are close to each other, but we shall keep both parameters in the discussion because of their different background: H c2 50% is arbitrary taken, while H c2 peak defines the inflexion point of the R s (B) cur. The inflexion point may have a physical meaning and, hence, H c2 peak can be more sensitive to external factors than H c2 50% . Figure 3 shows curves of H c2 50% (T) for different V G from − 5 V up to 200 V. For all gate voltages, H c2 50% increases with temperature decrease. Curves of H c2 50% . (T) shift into the region of higher values of H c2 50% -T for decreasing V G . This enhancement occurs even for negative V G , where a lower V G induces a lower T C Zero . Moreover, the slope (dH c2 50% /dT | Tc ) of the H c2 50% (T) curve approaching H c2 50% (T c ) = 0 systematically increases when V G decreases. The overall observed tendency does not change, if we use in the analysis H c2 10% or H c2 peak instead of H c2 50% (Fig. S2 a,b). According to Ginzburg-Landau (GL) theory, near T c the upper critical field is a linear function of (T c − T) and smoothly saturates when lowering the temperature. Our results suggest an anomalous unconventional behavior of the upper critical field vs. temperature, and it is no longer appropriate to describe this dependence using the GL theory. The H c2 10%, 50%, peak (T = 0 K) values obtained by linear extrapolation of e data at low temperature are increasing with decreasing V G . The determination of H c2 at T = 0 K lacks precision. However, if we estimate the relative H c2 -increase (δ H c2 ) at a finite temperature (e.g. 60 mK) between the curves for V G = − 5 V and V G = 0 V, a criterion closer to R s = 0 shows a lower value (i.e. δ H c2 10% = 1.3%, while δ H c2 50% = 5.5%). This finds its understanding in the following: The emergence of R s = 0 is due to the global superconductivity. With increasing magnetic field, a finite resistance occurs, and, hence, global superconductivity disappears. However, local superconductivity still exists before the system recovers to normal state. Thus, it is reasonable that we observe a larger H c2 for a more negative V G . This indicates that Cooper pair can persist up to a higher magnetic field than the upper critical field corresponding to global superconductivity.
Superconductor-insulator transition. The magnetic-field-induced superconductor-insulator transition(SIT) 27 shows a characteristic fan-shaped pattern of R s (B) isotherms crossing at one point. For example, in Fig. 4a the SIT cross point for V G = − 5 V is at 810 mT. The R s (T) curves extracted from R s (B) show a plateau for 810 mT (Fig. 4b). The plateau separates two regimes. Therefore, the magnetic field drives a continuous quantum phase transition from a superconducting 2DEG to a weakly insulating state. Further finite-size scaling analysis shows that the data can be collapsed onto a bi-value curve (Fig. 4a, inset). It results that the crossing point is a quantum critical point (QCP), at which the phase transition occurs. Literature often show magnetic-field-induced SIT for different 2D superconductors 27,28 , including for interfaces such as LAO/STO and (001) LTO/STO 29,30 .
As already noted, H c2 (T) for a fixed V G depends on the criterion adopted for the H c2 determination in the case of a broad transition 26,31 and the H c2 (T = 0 K) cannot be determined in a reliable manner. This situation questions the intrinsic nature of the H c2 -enhancement (for a lower V G ) in the underdoped region. Due to this, here we use another method to directly determine the zero -temperature upper critical field,  being similar to that of H c2 50% (T = 0 K) (Fig. 5). One reason for a non-precise determination of = ⁎ H (T 0 K) c2 vs. V G is: when V G ≠ − 5 V, e.g., a V G = 75 V is used for the construction of the magnetic-field-induced SIT pattern, the crossing point of the R s (B) isotherms transforms into a field domain centered at 0.55 T and extending over ± 0.05 T (Fig. S3a). In this case, the conventional power-law scaling behavior fails to describe the quantum criticality (Fig. S3b). Multiple quantum criticality was also found and reported for the (001) LTO/STO interface 29 . The phenomenon of multiple critical exponents suggests an unconventional critical behavior of SIT in the (110) LAO/ STO interface, and it will be discussed elsewhere.
Phase diagram. The superconducting phase diagram of the (110) LAO/STO interface is presented in Fig. 5.
As already addressed in the previous sections, the upper critical field H c2 independently of the criterion for its determination monotonically increases when V G decreases, while T C Zero displays a dome-like curve with a maximum. In the inset to Fig. 5 is shown the GL coherence length ξ 0 determined from H c2 as a function of V G . The most striking anomalous result is enhancement of H c2 accompanied by the decrease of ξ 0 for decreasing V G , i.e. for the carrier depletion reflected by the dome-like curve in the underdoped region. Results indicate that the Cooper pairing potential is stronger in the underdoped region.  A systematic increase in H c2 with decreasing doping has been reported both in high-T c cuprates and iron-based superconductors 31,32 . This is usually considered as an evidence for the existence of a so-called 'pseudogap' state, in which the bosonic pairs form above T c but cannot condensate into superconducting state due to dilution of pairs 33 . Recently, the planar tunneling spectroscopy study in 2DEG at a (001) LAO/STO interface has shown that the energy gap ∆ increases with charge carrier depletion in both underdoped and overdoped regions 9 . And, the coherence-peak-broadening parameter Γ derived from the Dynes fit, that is related to the strength of the superconducting pairing interaction, increases steeply with decreasing V G . GuangLei Cheng et al. 10 recently reported that the Cooper pairs form at temperatures well above the superconducting transition temperature of the (001)LAO/STO superconducting system. In the experiments they used a superconducting single-electron transistor and they observed that pairs condensate at low magnetic fields and temperatures. The physical understanding of the processes at (110) LAO/STO interface is analogous to that of hole-doped cuprates 31 , namely, the pairing potential is stronger and the Ginzburg-Landau coherence length ξ 0 decreases in the underdoped region as V G decreases. The trend differs from that of superfluid density and superconducting transition temperature 7,14 . Namely, the superfluid density decreases with decreasing V G providing that the phase fluctuation is important 14,34,35 . The observation of ∆ /Γ scaled with T c in ref. 9 implies that the limited quasiparticle lifetime controls T c effectively 9 , and the reduction in T c versus ∆ was attributed to a competing order parameter or to a weak phase coherence. In addition, in LAO/STO system the spin-orbit coupling is non negligible and strongly depends on V G 36,37 . Both conventional and unconventional pairing mechanisms have been considered to describe superconductivity in interfaces 5,6,16,38 . For example, the spin-orbit coupling 36,37 and the coexistence of superconductivity and ferromagnetism 11-13 may indicate formation of possible exotic superconducting states such as finite momentum Cooper pairing 39,40 . One has also to consider a different orbital reconstruction between (001) and (110) systems 20,41 . Superconducting properties and Rashba spin-orbit coupling can be largely tuned by controlling selective orbital occupancy in different crystal orientations 20 . For the underdoped region, it has been reported that a Lishiftz transition is observed at (001) interface 42,43 . The (110) system is expected to be characterized only by a 3d xz /d yz filled electronic state; thus the superconducting properties of the (110) interface system can be substantially different from that of the (001) system.

Conclusions
We systematically investigated the upper critical field as a function of gate voltage by ultralow temperature magnetoresistance measurements in superconducting 2DEG of a (110) LAO/STO interface. We found that upper critical field increases as the gate voltage decreases. Two independent methods to determine the upper critical field give a similar trend. This implies that the pairing potential is stronger in the underdoped region. This observation is similar to recent reports that consider a pseudogap-like behavior at the (001) LAO/STO interface. Our results for an interface with a different orientation contribute to understanding of the pairing mechanism of superconductivity at LAO/STO interface.

Method
A five-unit-cell LaAlO 3 thin film was grown on the (110) SrTiO 3 substrate (500 μ m thickness) by pulsed laser deposition. Details were described in ref. 19. A metallic back gate was evaporated and attached to the rear of the substrate. Leakage current was low (below the maximum value of 5 nA at V G = 200 V). Standard four-terminal resistance measurements were made using wedge-bonding contacts. The sample was cooled in a dilution refrigerator with a base temperature of 10 mK. The measurement current is sufficiently low (~50 nA) to avoid sample heating at ultralow temperatures. To ensure the reversible behavior of the superconductivity, the gate voltage was ramped up to 200 V after cooling down. Perpendicular magnetic field B was applied to the sample (interface) surface and the field direction is the same for all measurements.