Gate-tunable giant nonreciprocal charge transport in noncentrosymmetric oxide interfaces

A polar conductor, where inversion symmetry is broken, may exhibit directional propagation of itinerant electrons, i.e., the rightward and leftward currents differ from each other, when time-reversal symmetry is also broken. This potential rectification effect was shown to be very weak due to the fact that the kinetic energy is much higher than the energies associated with symmetry breaking, producing weak perturbations. Here we demonstrate the appearance of giant nonreciprocal charge transport in the conductive oxide interface, LaAlO3/SrTiO3, where the electrons are confined to two-dimensions with low Fermi energy. In addition, the Rashba spin–orbit interaction correlated with the sub-band hierarchy of this system enables a strongly tunable nonreciprocal response by applying a gate voltage. The observed behavior of directional response in LaAlO3/SrTiO3 is associated with comparable energy scales among kinetic energy, spin–orbit interaction, and magnetic field, which inspires a promising route to enhance nonreciprocal response and its functionalities in spin orbitronics.

In this manuscript, authors studied the nonreciprocal charge transport in noncentrosymmetric oxide interfaces. They successfully observed nonreciprocal charge transport in the conductive oxide interface LaAlO3/SrTiO3, which shows the characteristic behavior for polar systems and found that it can be tuned and enhanced by gating. The results are impressive, clearly showing the potential of oxide interfaces. I believe that results are worth publishing in Nature Communications. However, I think the authors should further consider and clarify the following point before accepting the manuscript.
1. LaAlO3/SrTiO3 show the in-plane negative magnetoresistance as shown in Figs. 1 d and 2 a. What is the microscopic origin of it? Is it related with the out-of-plane magnetoresistance (weak (anti-)localization) discussed in Supplementary Fig. 5? What is the temperature dependence of the magnetoresistance? I also want to know the relation between the negative magnetoresistance and the sign of nonreciprocal charge transport. Fig. 2 c, which show the decrease below 10K. They attributed it to the decrease of conductivity in LaAlO3/SrTiO3. I think it is better to show the temperature dependence of ∆R_xx (raw data) in supplementary information.

Authors show the temperature dependence of ∆R_xx/R_xx in
3. In the inset of Fig.2 b, authors draw the schematic of electronic band structure of LaAlO3/SrTiO3. I want to know the typical quantitative value of the energy and carrier density of Lifshitz transition. Is it close to the present case (n~10^14 cm^3 according to the Supplementary  Fig. 4)?
4. I agree that deviation from the B-linear behavior of R_2ω might come from the higher order terms. Why does it show increasing (not decreasing) behavior? Again, is it related with the negative magnetoresistance? 5. I am interested in the detailed field-angle dependence of nonreciprocal resistance. It seems that field-angle dependence for xy plane and that for zy plane are different. This means that signals of nonreciprocal charge transport deviate from the simple formula ∆R∝I•(P×B). What is the potential reason? Is it related with the fact that LaAlO3/SrTiO3 interface is fourfold symmetrical?
6. In addition to Ref. 29 (Sci. Adv. 3, e1602390 (2017)), I recommend authors to cite two more related works which studied the enhanced nonlinear superconducting transport in noncentrosymmtric systems. The work reports the appearance of non reciprocal charge transport, tunable by electric field effect, in the 2DEG at the LaAlO3/SrTiO3 interface. The authors perform DC and AC measurements of the resistance as a function of the direction of the injected current and report the behavior of the non reciprocal response as a function of gate voltage, magnetic field and angle between the current direction axis and magnetic field orientation. The magnitude of the non reciprocal response found by the authors is larger than what has been found in many polar materials. They stress that this is due not only to the large Rashba spin-orbit interaction energy, but especially to the relative strength of the Rashba spin-splitting energy when compared to the Fermi energy. The authors highlight the interesting point that the LAO/STO system shows many properties and, more interestingly, these are characterized by a special reciprocal energy scale.
The work is well written in the first part while the second, dealing with the AC measurements, is difficult to follow. There are also some important issues which should be cleared by the authors.
• The authors write that "The strong asymmetric Vg dependence of the nonreciprocal response is a consequence of the Vg dependent Rashba spin-orbit interaction in combination with the n-3 dependence". On the other hand, the authors observe a change in the carrier density from 1.65 to 1.9 x1013 cm-2 (for Vg=0 and 200V respectively). This variation is weak, as also remarked by the authors, if compared with typical values found for LAO/STO 2DEG (see for instance A. Joshua et al., Nature Communications 3, 1129Communications 3, (2012). This seems to indicate that, although the carrier density does not change much, the Rashba spin-orbit interaction raises considerably with increasing gate voltage. The authors should provide more details on this issue. For instance, they could analyze the magnetoconductance curves (supplementary figure 5) to estimate the Rashba scattering parameters as a function of the gate voltage.
• The curves shown in Figure 1d and 2a remind the magnetoresistance hysteresis shown by Ayno et al. in Physical Review Materials 2, 031401(R) (2018) and attributed to a magnetothermal effect. The authors should provide evidence that this effect is not at play in their case. They should specify the field sweeping rate used. Moreover, high currents for both DC (30μA) and AC (200μA) measurements were used. Why did they chose such values? What happens if the bias current is reduced?
Our responses: We greatly appreciate reviewer's a number of important comments and advices as well as positive remarks. Below is the detailed discussions and corrections we made in response to reviewer's comments.

#1.
Reviewer's comments: LaAlO 3 /SrTiO 3 show the in-plane negative magnetoresistance as shown in Figs. 1 d and 2 a. What is the microscopic origin of it? Is it related with the out-of-plane magnetoresistance (weak (anti-)localization) discussed in Supplementary Fig. 5? What is the temperature dependence of the magnetoresistance? I also want to know the relation between the negative magnetoresistance and the sign of nonreciprocal charge transport.
Our responses: We appreciate reviewer's comments for important issue, which we should have paid attention to. In response to reviewer's comment, we added following sentence in the revised manuscript. "This negative in-plane MR in LAO/STO was attributed to the anisotropic deformation of the Fermi surface upon increasing Zeeman energy, which results in suppressed interband scattering and reduced sheet resistance 40 ." -When the out-of-plane magnetic field was applied, quantum interference effect to the diffusive transport becomes significant, especially at low temperature. Thus, the mechanism of in-plane MR and out-of-plane MR are different, although both MR highly rely on the spin-orbit interaction.
In LAO/STO, a negative MR in response to the out-of-plane magnetic field can be observed for a large negative gate voltage (−80 V g ), indicating the dominance of weak localization due to weak spin-orbit interaction ( Supplementary Fig. 5). As the applied gate voltage increased, a negative MR gradually turned into a positive MR, indicating that the charge transport relies on the weak anti-localization ( Supplementary Fig. 5). On the other hand, when the in-plane field was applied, the negative MR get stronger as the applied gate voltage increased. Thus, the gate-dependent evolution of MR is opposite to each other between in-plane and out-of-plane MR. However, we cannot argue that both MRs (in-plane and out-of-plane) are independent each other, because both mechanisms highly rely on the strength of spin-orbit interaction.
-The in-plane negative MR shows maximum magnitude at around 8 K. It is progressively suppressed as the temperature is raised but still clearly visible at 20K, in agreement with previous report [Phys. Rev. Lett. 115, 016803 (2015)]. In response to reviewer's comments, we added the temperature dependent in-plane MR curves in Supplementary Fig. S10.
-As we discussed, the sign of nonreciprocal charge transport depends on the direction of polarity and magnetic field as follows, ∆ = ( ) − (− ) ∝ • ( × ). If the direction of magnetic field or polarity is reversed, ΔR changes sign. According to the explanation of Diez et al. (Phys. Rev. Lett. 115, 016803 (2015)), the negative in-plane MR arises from the suppressed interband scattering due to anisotropic deformation of the Fermi surface. Thus, the sign of negative inplane MR does not change for the reversal of B y or P z . As the reviewer perceived, it seems that both mechanisms are highly correlated, because both effects depend on the strength of spin-orbit coupling.
In response to reviewer's comment, we added following sentence in the revised manuscript. "Interestingly, the negative in-plane MR also increases significantly with applying positive V g and can be collapsed into a single curve by a rescaling of the magnetic field B  B/B* (B* is a density dependent value) 40 (see Supplementary Fig. 9). Mechanisms of nonreciprocal charge transport and negative in-plane MR could be highly correlated because both effects depend on the anisotropic deformation of the Fermi surface" #2. Reviewer's comments: Authors show the temperature dependence of ∆R xx /R xx in Fig. 2 c, which show the decrease below 10K. They attributed it to the decrease of conductivity in LaAlO 3 /SrTiO 3 . I think it is better to show the temperature dependence of ∆R xx (raw data) in supplementary information.
Our responses: Following reviewer's suggestion, we have added the temperature dependence of ∆R xx in Fig. 2c and in-plane MR curves (raw data) in supplementary Fig. 10. Below 10 K, nonreciprocal responses ∆R xx /R xx decreases with decreasing temperature. We believe that it would be associated with quantum interference effect, which get stronger below 10 K. Other possible explanation is the slight decrease of carrier concentration when the temperature is lowered below 10 K (see the inset of Supplementary Fig. 4b). The decrease of carrier concentration away from the Lifshitz transition reduces spin-orbit coupling, so does the nonreciprocal response.
In response to reviewer's valuable comment, we modified Fig. 2c and Supplementary Fig. 4b and we added Supplementary Fig. 10 and 11, and following sentences in the revised manuscript. "Another possible explanation is the slight decrease of a carrier concentration when the temperature is lowered below 10 K ( Supplementary Fig. 4b). The decrease of a carrier concentration away from the Lifshitz transition reduces the spin-orbit coupling, so does the nonreciprocal response."

#3. Reviewer's comments:
In the inset of Fig.2 b, authors draw the schematic of electronic band structure of LaAlO 3 /SrTiO 3 . I want to know the typical quantitative value of the energy and carrier density of Lifshitz transition. Is it close to the present case (n~10 14 cm -2 according to the Supplementary Fig. 4)?
Our responses: Thanks for pointing out important issue. According to literatures, typical quantitative values of the carrier density for Lifshitz transition were 1.68±0.18×10 13 cm -2 by Joshua [Nat. Commun. 3, 1129 (2012)]. If we estimate Fermi energy based on 2D free electron model, this value of n corresponds to ε F ~ 20.1 meV. In our study, the obtained value of n s with zero gate voltage is ~ 1.61×10 13 cm -2 at 8 K and 1.56×10 13 cm -2 at 2 K (for device B, Supplementary Fig. 4b). This value is slightly less than the Lifshitz transition. Thus, we observed strongly enhanced nonreciprocal response with increasing V g across zero voltage. Note that there is slight sample to sample variation in n, so does gate voltage required to induce the Lifshitz transition.
In response to the reviewer's comments, we added following sentences in the revised manuscript. "According to the literature by Joshua et al. 10 , typical quantitative values of the carrier density for Lifshitz transition were 1.68±0.18×10 13 cm -2 . This value corresponds to the Fermi energy of ~ 20.1 meV within the 2D free electron model. Not that the obtained value of n s with V g = 0 is ~ 1.61×10 13 cm -2 at 8 K (for device B, Supplementary Fig. 4). This value is slightly less than the Lifshitz transition. Thus, Lifshitz transition could occur with increasing V g across zero voltage." #4. Reviewer's comments: I agree that deviation from the B-linear behavior of R 2ω might come from the higher order terms. Why does it show increasing (not decreasing) behavior? Again, is it related with the negative magnetoresistance?
Our responses: We thank for reviewer's positive remarks and constructive questions. The high order terms depend on the equation, as follow nd = ( + + ( )). The appearance of higher order term as we discussed depends on the relative strength between magnetic field energy and Rashba spin-orbit splitting energy. Basically, nonreciprocal charge transport occurs due to the imbalance of Fermi momentum between leftward and rightward carriers. At B y << λ, this imbalance linearly increases with increasing field. When B y ~ λ, it strongly enhanced with increasing field. Thus, the high order dependences are reinforcing components to the nonreciprocal response rather than detrimental components.
-As we mentioned in our previous response, large negative in-plane MR is associated with the spin-orbit interaction along with Zeeman energy, which drives a highly anisotropic Fermi surface, leading to suppressed interband scattering and reduced sheet resistance. The anisotropic deformation of Fermi surface gets stronger with increasing magnetic field at high field regime, so does the negative in-plane MR. As reviewer perceived, it seems that both mechanisms are highly correlated, because both effects are contingent on the strength of spin-orbit coupling.
In response to reviewer's valuable comments, we added following sentences in the revised manuscript. "Interestingly, the negative in-plane MR also increases significantly with applying positive V g and can be collapsed into a single curve by a rescaling of the magnetic field B  B/B* (B* is a density dependent value) 40 (see Supplementary Fig. 9). Mechanisms of nonreciprocal charge transport and negative in-plane MR could be highly correlated because both effects depend on the anisotropic deformation of Fermi surface" #5. Reviewer's comments: I am interested in the detailed field-angle dependence of nonreciprocal resistance. It seems that field-angle dependence for xy plane and that for zy plane are different. This means that signals of nonreciprocal charge transport deviate from the simple formula ∆R∝I·(P×B). What is the potential reason? Is it related with the fact that LaAlO 3 /SrTiO 3 interface is fourfold symmetrical?
Our responses: We thank reviewer for bringing attention to the angle-dependent behavior of nonreciprocal charge transport. Similar to the angle dependent in the xy plane, the R 2ω is largest at B‖y axis (θ = 90°, 270°) for the zy plane. It is noticeable, however, that the R 2ω does not simply scale with B y for the rotation in the zy plane. This different angle-dependent R 2ω for the rotation in the zy-plane is due to the higher-order dependence on magnetic field at high field regime. When the field is rotated in the xy-plane, the direction of field is always orthogonal to the direction of polarization (see figure below). Thus, the orthogonal component of B with respect to P is constant. Because ∆ ∝ • ( × ), component of I parallel to ( × ), which varies as • cos during the field rotation in the xy plane, is subject to ΔR. Thus, we observe sinusoidal R 2ω for field rotation in the xy-plane as shown in Fig. 3b. On the other hand, when the field is rotated in the zy-plane, the component of field, which is orthogonal to the direction of polarization, is not constant and varies as • sin (see figure below). Because the ΔR has additional higher order dependence on B at high magnetic field regime, the variation of ΔR become more significant at high field regime, making sharp increase of ΔR near 90 degree and 270 degree in the zy-plane. At relatively low magnetic field, where the ΔR is linear to the B, R 2ω displays sinusoidal behavior for the rotation of the magnetic field in the zy-plane (see Supplementary Fig. 18).
In response to reviewer's valuable comment, we added following sentences in the revised manuscript. "The higher order dependence on the applied magnetic field also reflects on the different behavior of R 2ω in between xy-and zy-plane rotations (shown in Fig. 3c). When the magnetic field is rotated in the xy-plane, the direction of the field is always orthogonal to the direction of the polarization. Because ∆ ∝ • ( × ), R 2ω displays sinusoidal behavior for the rotation of the magnetic field in the xy-plane. On the other hand, when the field is rotated in the zy-plane, the orthogonal component of the field to the direction of the polarization is not constant and varies as • sin . Because the ΔR has additional higher order dependences on B at a high magnetic field regime, the variation of R 2ω becomes more significant at high fields, making sharp increase of R 2ω near 90° and 270° in the zy-plane (Fig. 3c). At a relatively low magnetic field, where ΔR is linear to B, R 2ω displays sinusoidal behavior for the rotation of the magnetic field in the zy-plane (see Supplementary Fig. 18) Our responses: We appreciate that the reviewer provided and reminded us for other related reports. We agree that those pioneering works should be discussed in our introduction. In response to reviewer's comments, we added these references in the revised manuscript.
In short, the major changes we made in response to the reviewer's comments are as follows.
The work is well written in the first part while the second, dealing with the AC measurements, is difficult to follow. There are also some important issues which should be cleared by the authors.
Our responses: We appreciate reviewer's a number of important comments and advices as well as positive remarks. In response to reviewer's comments, we modified part of AC measurements to improve legibility for general audiences. Thanks to reviewer's comments, our revised manuscript has been improved significantly. Below is the detailed discussions and corrections we made in response to reviewer's comments.

#1. Reviewer's comments:
The authors write that "The strong asymmetric V g dependence of the nonreciprocal response is a consequence of the V g dependent Rashba spin-orbit interaction in combination with the n -3 dependence". On the other hand, the authors observe a change in the carrier density from 1.65 to 1.9 x10 13 cm -2 (for V g =0 and 200 V respectively  Fig. 7). The V g dependent variation of spin splitting ∆ was more significant than that of the carrier density. Therefore, as reviewer emphasized, gate-tuned Rashba interaction mainly accounts for the observed V g dependence of the nonreciprocal response in this system.
In response to reviewer's comments, we added Supplementary Fig. 6 and 7 and Note 1 and following sentence in the revised manuscript. "Further analyses of out-of-plane MR curves within a Maekawa-Fukuyama theory were discussed in Supplementary Note 1 and Fig. 6 and 7. Result showed that the strong enhancement of the Rashba spin-orbit interaction across the Lifshitz point ( Supplementary Fig. 7a)." "The carrier concentration of the studied LAO/STO system exhibits gradual increase with increasing gate voltage but its variation is very weak (Supplementary Fig. 7c and 8). In contrast, the estimated Rashba spin splitting energy is significantly enhanced with increasing V g (Supplementary Fig. 7a). Therefore, gate-tuned Rashba interaction mainly accounts for the observed V g dependence of the nonreciprocal response in this system."

#2. Reviewer's comments:
The curves shown in Figure 1d and  Our responses: We agree with reviewer's important concern and appreciate important comments. The paper from Ayino [Phys. Rev. Mater. 2, 031401(R) (2018)] showed the magnetoresistance (MR) hysteresis induced by a magnetothermal effect. The magnetoresistance hysteresis is the resistance difference between + to -field sweep and -to + field sweep measurements. The nonreciprocal charge transport in our report is the resistance difference between measurements with +I and -I currents. Thus, we are dealing with different phenomena.
-In the report of Ayno et al, the MR hysteresis appears in the vicinity of the small magnetic field and disappear by increasing the magnetic field over 1T. This behavior occurs when magnetic anisotropy energy overcome the thermal effect. Thus, it occurs when the temperature was lowered below ~ 800 mK. In our study, the nonreciprocal responses persist over several tenth Kelvin. And it is negligible in the vicinity of the small magnetic field, and increases linearly with increasing field, and finally it diverges with higher order dependence at high magnetic field. Therefore, the overall behavior of the studied nonreciprocal response is completely different and is not related with magnetothermal effect. The MR measurement was done with magnetic field sweeping rate of 10 mT/s.
-Our measurement was typically done with the current of DC (30 μA) and AC (200 μA) to clearly observe the nonreciprocal response. As can be seen in Figure 4d, the nonreciprocal response linearly proportional to the electric current as ∆ ∝ • ( × ). As the reviewer commented, if the bias current is reduced, the nonreciprocal response linearly decreases. Applying high current may introduce deviation from linear relationship due to the heating effect. Thus, we chose moderate high current of DC (30 μA) and AC (200 μA) to obtain clear enough signal to noise ratio.
In response to reviewer's comments, we added following sentences. In the revised manuscript, "Measurement were done with magnetic field sweeping rate of 10 mT/s." In Supplementary Fig. 11, "We note that the observed ΔR xx is not associated with magnetothermal effect, which appears in the vicinity of the small magnetic field at very low temperature (< ~ 800 mK) [Phys. Rev. Mater. 2, 031401(R) (2018)]."

#3.
Reviewer's comments: The authors write that "The nonreciprocal response is nearly negligible for V g < 0 V, while it stiffly increases upon applying positive V g ( fig. 2b)". On the other hand, the crossover between weak localization and weak anti-localization (indicating the increase in Rashba spin-orbit coupling) takes place at much lower gate voltages, between V g =-40 V and V g =-80 V (supplementary fig. 5). The authors should comment on this discrepancy.
Our responses: We appreciate the reviewer's careful proofreading and valuable comments. The crossover between weak localization and weak anti-localization are associated with relative scale between phase coherence length and spin diffusion length. Thus, the crossover between weak localization and weak anti-localization may occur before the Lifshitz transition, where the spin-orbit coupling starts to increase more strongly. As reviewer suggested, we estimated spin-orbit splitting by fitting with Maekawa-Fukuyama (MF) theory to the out-of-plane MR. Results show that even before the Lifshitz transition, spin-orbit coupling slightly increases with increasing V g (Supplementary Fig. 7a). Thus, the change of relative scale between phase coherence length and spin diffusion length may occurs before the Lifshitz transition ( Supplementary Fig. 7c). In contrast, the nonreciprocal responses would directly rely on the strength of spin-orbit coupling as follows ∆ ∝ • ( × ), where P is the polarization. Here, the polarization P is directly associated with the Rashba spin-orbit coupling, which is proportional to the potential gradient. We also note that there is slight sample to sample variation in n s , so does gate voltage required to induce Lifshitz transition and transition between weak localization and weak anti-localization.
In response to the reviewer's comments, we added further discussions and analysis on the weaklocalization and weak-antilocalization and discrepancy with the Lifshitz transition in the Supplementary Note 1 and Fig. 6 and 7.
Our responses: Thanks for careful proofreading. According to the Supplementary Fig. 4, the estimated value of n s was ~ 1.61×10 13 cm -2 at 8 K and 1.56×10 13 cm -2 at 2 K, which was estimated from device B.
The results in the main text were obtained from device A. The estimated sheet carrier density (n s ) were nearly identical between device A and device B. For device A, the estimated n s was ~ 1.56×10 13 cm -2 at 2 K. Supplementary Fig. 8 shows the estimated n s upon varying V g measured for device D. In this case, the estimated n s was ~ 1.65×10 13 cm -2 at 8 K and V g = 0. We also note that there is slight sample to sample variation in n s , so does gate voltage required to induce Lifshitz transition.
In response to the reviewer's comments, we added inset in Supplementary Fig. 4b, which clearly displays values of carrier density at low temperatures.

#5. Reviewer's comments:
In Figure 1 the panel showing the SEM picture of the device is missing.
Our responses: Fig. 1c in the revised manuscript displays the SEM image of the studied device A and we improved contrast of this figure.
#6. Reviewer's comments: The saturation of the nonreciprocal response at high V g cannot be seen in Figure 2b.
Our responses: We thank for the reviewer's careful proofreading. As can be seen in Fig. 2b, the nonreciprocal response significantly increases with increasing V g across 0 V. But its enhancement become less effective when V g increases further. As reviewer pointed out, we didn't observe the saturation.
But overall behavior appears to show the reduced enhancement with increasing V g further.
In response to reviewer's comments, we added following sentence in the revised manuscript. "Therefore, gate-controlled Rashba interaction mainly accounts for the observed V g dependence of the nonreciprocal response in this system. As shown in Fig. 2b, the observed nonreciprocal response is nearly negligible for V g < 0 V, while it stiffly increases upon applying positive V g , in consistent with the Lifshitz transition across zero gate voltage." In short, the major changes we made in response to the reviewer's comments are as follows. Authors answered all the questions appropriately and manuscript is now suitably revised. Although I believe that it can be ready for acceptance, I have another minor question after reading the response to comment 3 (typical quantitative value of the energy and carrier density of Lifshitz transition). According to the authors, carrier density of their sample is slightly less than the Lifshitz transition and Lifshitz transition could occur with increasing Vg across zero voltage. Is there any signature (or anomaly) reflecting the Lifshitz transition in Rω or R2ω? I think it may be an important future issue.
Our responses: We greatly appreciate reviewer's positive remarks and important comments. In addition to the direct measurement of band structure, the signature of Lifshitz transition can also be evidenced through various analysis of transport properties. For example, the SdH oscillation allow us to investigate the Fermi surface. In this specific system of LAO/STO, Lifshitz transition closely associated with the strength of Rashba spin-orbit interaction. Thus, the analysis of out-of-plane magnetoresistance curve based on MF theory could exhibit abrupt increase of spin-orbit interaction at the Lifshitz point, as shown in supplementary Figure 7. These analysis requires multiple fitting procedure with many parameters. For ac measurement, the linear components of R upon applying out-of-plane field could be used for such time-consuming analysis to find out Lifshitz point. On the other hand, the nonreciprocal R2 is a physical property that is directly associated with the strength of spin-orbit interaction and can be utilized for the estimation of the size of Rashba constant. Thus, sudden increase of R2 can be regarded as the signature of the Lifshitz point in LAO/STO system.
Another signature of Lifshitz transition in LAO/STO is the change of symmetry in angular dependence of magnetoresistance, as done in PNAS 110, 9633-9638 (2013). We agree that it is an interesting issue for future work. At this point, we observed that the symmetry of angular dependent R2do not vary across the Lifshitz transition and only the magnitude of R2increases steeply across the transition, as shown in Fig. 4a.
In response to the reviewer's comments, we added following sentence in the revised manuscript. "As the nonreciprocal response R2 is directly associated with the strength of the Rashba spin-orbit interaction, the sudden increase of R2 could be regarded as the signature of the Lifshitz transition in this LAO/STO system." In this revised version, the authors included the analysis of the magnetoconductance (MC) curves as a function of the gate voltage. This analysis is important to reply to some questions both the First Reviewer and I asked during the first review stage. However, the data added and their analysis raise many doubts, in my mind.
Our responses: We greatly appreciate reviewer's careful proofreading and a number of valuable comments. We agree that we should have paid more attention on the analysis of magnetoconductance (MC) to produce high-precision analysis and to improve the completeness of our manuscript. In order to produce more reliable analysis, we performed additional experiment with newly fabricated device with further caution to get more reliable results, which allowed us to get more precise fitting analysis on MC. Thanks to reviewer's important comments, our analyses on MC have been significantly improved and the obtained results became more reliable and consistent with previous reports on MC.

#1. Reviewer's comments:
In the Maekawa and Fukuyama formula, the minimum of the differential MC curve gives an estimation of the B_so field (W. Knapp et al., Phys. Rev B 53, 3912 (1996)). Thus, for B_so fields in the order of 1T (as is typically found in LAO/STO devices) the MC data should be taken up to several Tesla, in order to have a clear picture of the evolution of the curve minimum. In Supplementary Fig. 6, on the other hand, the authors show MC data up to 1.2T. This is too low to obtain a reliable fit of the curves.
Our responses: We appreciate reviewer for reminding us of valuable literature on WL/WAL. We have performed additional experiment for MC measurement with newly fabricated device in order to get more robust MC results with reduced signal to noise. We have extended the range of fitting up to 4 T to cover the minimum of the differential MC curves ( Supplementary Fig. 6 in revised manuscript). We also noticed that the fitting to MC results start to deviate at high field as the Vg was increased far above from the Lifshitz point due to orbital mangnetoresistance. This behavior is also reported in previous reports [Nat. Comm. 6, 6028 (2015)]. Thus, our analysis is primary focused on the range of Vg around Lifshtiz point, where we could obtain excellent fitting, as shown in Supplementary Fig. 6.
We also followed the analysis done in ref [15] and added results in Supplementary Fig. 8. Our results show that the charge transport evolving from being dominated ( * ~0 .62 ) to being , dominated ( * ~2 .3 ) across the Lifshitz transition, in consistent with previous report.
As the reviewer mentioned, the change of effective mass modifies the values of the diffusion constant D, the scattering times, and the Rashba spin-orbit interaction constant α. If we solve for Rashba spin-orbit interaction constant = Therefore, if we assume fixed effective mass, we could obtain sudden increase of Rashba spinorbit interaction across the Lifshitz transition. If we assume linear variation of Rashba spin-orbit interaction, we could obtain sudden increase of effective mass across the Lifshitz transition.
In short, we performed the analysis on MC results obtained from newly fabricated device based on MF theory. Further analysis on the other parameters, such as so and was obtained with fixed electron mass following the ref.
[13] (Supplementary Fig. 7). In response to reviewer's comments, we also performed estimation of effective mass based on the assumption of linear variation of Rashba spin-orbit interaction following the ref.