Interface-based tuning of Rashba spin-orbit interaction in asymmetric oxide heterostructures with 3d electrons

The Rashba effect plays important roles in emerging quantum materials physics and potential spintronic applications, entailing both the spin orbit interaction (SOI) and broken inversion symmetry. In this work, we devise asymmetric oxide heterostructures of LaAlO3//SrTiO3/LaAlO3 (LAO//STO/LAO) to study the Rashba effect in STO with an initial centrosymmetric structure, and broken inversion symmetry is created by the inequivalent bottom and top interfaces due to their opposite polar discontinuities. Furthermore, we report the observation of a transition from the cubic Rashba effect to the coexistence of linear and cubic Rashba effects in the oxide heterostructures, which is controlled by the filling of Ti orbitals. Such asymmetric oxide heterostructures with initially centrosymmetric materials provide a general strategy for tuning the Rashba SOI in artificial quantum materials.

In this manuscript, the authors report on the observation of a transition between two different regimes of Rashba spin-orbit interaction in an asymmetric SrTiO3-based oxide heterostructures. The overall context of the study is quite timely and I find the experimental signatures reported here interesting for the community. However, at this stage I am not convinced that the measurements and the analysis are sufficient to support the conclusions of the paper. My main criticism is that the claims entirely rely on the fitting of magneto-resistance curves through a complex formula (ILP model) that, in my opinion, includes too many fitting parameters to be relevant. In addition, several critical assumptions in their approach are not justified. I cannot give my recommendation to publish this article in Nature Communications, unless the authors find a way to strengthen their analysis, and provide additional experimental signatures of the cubic to linear Rashba transition.
-In the context of LAO/STO interfaces, magneto-conductance curves are often fitted by Maekawa-Fukuyama formula which is also derived within a weak-localisation framework (for instance in ref 25 and 51). How does that compare with the ILP model used by the authors ? In ref 25, the authors also include a Zeeman splitting term, which is absent in the ILP model.
-In ref 51, the authors introduce also a "Kohler" term that account for classical magnetoresistance. This should correspond to the alpha term in front of the B2 term in the formula given line 238. However, the authors have used an approximation and not the full magnetic field dependence that includes a saturation of classical magneto-resistance at high magnetic field. The MR curves of thin STO layers (8 uc, 10 uc) could be very well fitted by adding a full "Kohler" term and a single BSO3 term making the discussion on the cubic to linear Rashba transition irrelevant. This term could be negligible but this must be justified.
-Related to the previous comments, I recommend the authors to clarify the meaning of the gamma, delta and alpha coefficients in the expression of the resistance (line 238). I also would like to see the value of these fitting parameters for the different thicknesses of STO.
-One way to disentangle the BSO terms from the Bphi one and give consistency to the analysis is to look at the temperature dependence of the magneto-resistance in the temperature range of validity of WL theory (<10K). The Bphi term should display a power law type of variation with temperature with a characteristic exponent that depends on the scattering mechanism that limits phase coherence (e-e, e-phonon). On the other hand, BSO terms, which only depend on the band structure, should not be affected by the temperature.
-Hall effect in LAO/STO interfaces is known to be non-linear at high doping level because of multiband transport. According to fig 1, the cubic to linear Rashba transition takes place between two bands (red and blue) of different masses. I would therefore expect very different mobilities in the two bands that should result in a transition between a linear Hall effect and a non-linear one at doping around 0.01 e/Ti.
-In their calculation, the authors assume a perfect crystal structure, which is in contradiction with the structural analysis. This aspect is really critical as the main claim (transition between two regime of Rashba SO) relies on an apparent agreement between theory and experiment. I can hardly imagine that the transition at 0.01e/Ti is sufficiently robust to any kind of imperfection in the structure.
-The authors assume that electrons are homogeneously distributed in the STO. I would expect that for the highest thicknesses of STO, one should recover the same situation than in LAO/STO heterostructures, with electrons confined at the interfaces. In this case, the low doping regime corresponding to the transition at 0.01e/Ti could be a pure artefact.
-The resistance vs temperature curves should be shown for the different samples. Do they exhibit the characteristic upturn in the resistance expected for WL ? -The authors should clearly indicate what is the exact formula used in the fit of magnetoresistance.
Reviewer #3 (Remarks to the Author): • Key results: The paper of Lin et al. reports the study of the transport properties of LaAlO3/SrTiO3/LaAlO3 heterostructures formed by two non-equivalent interfaces. They propose this system as a good candidate to study the spin-orbit Rashba effect, which is a really interesting property for the spintronic community. One of the main results given by this work is that, by varying the SrTiO3 layer thickness, the authors achieved to tune the carrier density and observed experimentally a transition from a cubic to linear (more precisely cubic+linear, if I understood correctly) Rashba effect, the linear Rashba effect being only predicted theoretically until now.
• Validity: The manuscript is globally well written. Some precisions seem however necessary to judge about the complete validity of the paper.
• Originality and significance: The observation of the linear Rashba effect will undoubtedly motivate further study to get a better understanding of this system and to get a better control of this interaction. It would thus be interesting to submit it for discussion. It is however not certain that this paper will get an audience out of the spintronic and oxide community.
• Data & methodology: The magneto-transport measurements are associated with some analysis of the atomic structure performed by STEM and some support from first-principles calculations.
For the DFT-based tight binding calculations, I would suggest to add more information about the parameters which were used and the methodology. In particular, it will be easier for the reader to have everything in hand than to collect all the information from previous references. What is really from "first principles" and which parameters (energy splitting, atomic spin-orbit interaction, asymmetry strength coefficient) are chosen to fit with the experience? In Ref. 40, some parameters are obtained from DFT calculations, but performed on systems with different geometries and in-plane lattice parameters. Was it modified for the current paper? L. 142-143, the authors say that the results are supported by DFT calculations and refer to the supplementary information. I did not see any detail about the DFT calculations (which code was used, which Exc functional…? (Wien2k + GGA?)). How was the value given in Fig. S4d (purple square) obtained?
With a slab geometry to simulate the effect of the internal electric field (with a STO thickness of 3-4 u.c.?), and an in-plane lattice parameter equal to the experimental or calculated one? How are defined the values represented by the dashed lines in the same figures (also calculations?).
About the chemical analysis, it is said that the two interfaces show clear cation intermixing, but that they remain non-equivalent. Was such analysis performed for different STO layer thicknesses? Or only for the 20 u.c.? From what is said, the out-of-plane STO layer lattice parameter is lower than expected mainly due to electrostrictive effect. Such effect should be linked with the asymmetry of the interfaces, and thus it should depend on the atomic structure at these interfaces. Did the authors notice any difference for the low and high layer thicknesses (that is, below and above 20 u.c., the thickness at which the transition occurs)?
The "rough" approximation which consists in averaging the number of electrons per Ti should be maybe discussed a little. The localization of the electrons near the interface is important as it depends on the internal electric field, which will itself have some impact on the Rashba splitting. Moreover, such approximation is quite contradictory with the fact that all the study relies on having two non-equivalent interfaces.
Is it possible to have some information on the method used to fit the parameters for Fig. S6? Above 20 u.c. the red and black points are superimposed, right? Why is the error increasing (of a factor 3) above 20 u.c.? This increase of the error is higher than the difference of errors between the fits with and without the linear contribution.
In the supplementary paragraph S4, how are the effective mass determined? Also, I have two questions for my understanding of the results: 1) If the carrier in STO are mainly here due to the polarity of LAO, can the authors explain a little more why the carrier density varies as a function of the STO thickness?
2) Why do the authors finally show some evidence of the linear Rashba effects, while other did not? Is it linked with the high carrier density? The special geometry of the system? From the bandstructure, I can understand why the cubic Rashba is seen first, and the linear appears only after, but if we forget the band energy ordering, the linear Rashba should be also present for the standard LAO/STO interface (?). I did not get the hypothesis that the dxy orbital being localized at the interface, the corresponding electrons would not participate to the charge transport.
• Conclusions: Regarding my previous comments, most of the conclusion are certainly reliable, but some of them would greatly benefit of more details (in particular about the methods).
• References: ok • Clarity and context: In the abstract, I would suggest to change "transition between cubic and linear" by "transition between cubic and linear+cubic".
• Particular part of the manuscript that you feel is outside the scope of your expertise: I am not expert in transport measurement and the full meaning of the ILP equations and their use is quite new for me. Sorry if some of my questions were "naive" in this sense.

Response to the referees' comments
(The referee's comments are quoted and our point-to-point responses in blue. Changes made in the revised manuscript are marked in blue.) Reviewer #1 (Remarks to the Author): It is a very interesting article dealing about magneto-transport -Rashba SOI in STO/LAO related system. They clearly evidence that they can tune / modulate the Rashba SOI using a different geometry, Id est, LAO/STO/LAO. For this system, the STO orbital ordering is inverted (yz/xz lower that xy) as compared to the traditional LAO/STO, resulting in different linear vs cubic rashba SOI possibility. Furthermore they clearly show that the effect can be tube by the STO thickness and band filling.
Modeling, microscopy and magneto-transport are convincing and well explained.
Response: We thank the positive evaluation of our work from the reviewer.
Reviewer #2 (Remarks to the Author): In this manuscript, the authors report on the observation of a transition between two different regimes of Rashba spin-orbit interaction in an asymmetric SrTiO 3 -based oxide heterostructures. The overall context of the study is quite timely and I find the experimental signatures reported here interesting for the community.
Response: We thank the reviewer for his/her recognition of the value of our study.
However, at this stage I am not convinced that the measurements and the analysis are sufficient to support the conclusions of the paper. My main criticism is that the claims entirely rely on the fitting of magnetoresistance curves through a complex formula (ILP model) that, in my opinion, includes too many fitting parameters to be relevant. In addition, several critical assumptions in their approach are not justified. I cannot give my recommendation to publish this article in Nature Communications, unless the authors find a way to strengthen their analysis, and provide additional experimental signatures of the cubic to linear Rashba transition.
Response: We thank the reviewer for these critical comments. We agree with the reviewer that the fitting to the magnetotransport data is quite complex, but it is still a reliable method and has been utilized to investigate the SOC in various systems, such as 2DEG in semiconductor quantum well, two dimensional  (2002)]. Nevertheless, to address the concern of the referee, we improved the manuscript on the following three aspects: First, we have provided more details about the magnetoresistance fitting and the RT data to further validate the method used in this manuscript.
Second, we performed X-ray linear dichroism (XLD) experiments on the LAO//STO/LAO structures with 8 uc and 30 uc STO, as shown in Fig.R1. The XLD data confirm that the d xz/yz orbitals are the first available states in the LAO//STO/LAO structure as predicted, while the d xy is the lowest state in the conventional LAO/STO structures. And as the STO thickness decreases, more electrons occupy the d xz/yz orbitals, which support the modulation effect of the STO thickness. The details of the XLD experiment are as follow: Linearly polarized X-ray absorption spectroscopy (XAS) experiments were carried out by adjusting the incident angle of the X-ray beam. As shown in Fig. R1a, the in-plane (IP) component is obtained by the normal incident X-ray, while the out-of-plane (OP) one by the grazing incident X-ray. All spectra were acquired by recording the total electron yield (TEY) at Ti L 2,3 -edge. These linearly polarized X-rays will excite the electrons from the Ti 2p core level of 2p1/2 and 2p3/2 states to the unoccupied d orbital, and thus the intensity of the XAS (I IP and I OP ) reflects population of the empty states. Moreover, the absorption of the X-ray shows strong dependence on the photon polarization with respect to the lattice direction, i.e. IP and OP components will excite more electrons to the d xy (dx 2 -y 2 ) and d xz/yz (dz 2 ), respectively, for the electrons from the Ti 2p core level. Therefore, the sign of XLD data, (I IP -I  are available [D. Pesquera et al, Phys. Rev. Lett. 113, 156802(2014)]. This means that the d xz/yz orbital is occupied first, which is consistent with the calculation. Such a feature with different electrons populations in different orbitals is called orbital polarization, whose strength can be characterized as 2(I IP -I OP ) /(I IP + I OP ). The strengths for the structures with 30 uc STO and 8 uc STO at the t 2g main peak of L 3 edge are 3.50% and 4.94%, respectively, meaning more electrons occupy the d xz/yz orbitals at the structure with 8 uc STO. This is consistent with our Hall measurement data that the Fermi level increases as the STO thickness decreases.
Third, as presented in the manuscript, the density-functional-theory (DFT)-based tight-binding (TB) calculation supports the interpretation of the magnetotransport measurements. In the revised manuscript, we provide more details on the calculation, as suggested by the Reviewer 3.

Fig. R1 | X-ray linear dichroism (XLD) measurements of LAO//STO/LAO at Ti L 2,3 -edge. (a)
Schematic experimental set-up for the linearly polarized X-ray absorption spectroscopy (XAS) at Ti L 2,3edge with the total electron yield (TEY) detection mode at room temperature. In this measurement configuration, the polarization direction of the linearly polarized X-rays was achieved by modulating the X-ray incidence angle. Here, the X-ray with normal and grazing (30°) incidence correspond to the inplane (E∥IP) and majority out-of-plane (E∥OP) polarized components, respectively.  To specify the reason we chose the ILP model, we add a sentence "which considers the k-dependent SOI" in second paragraph of page 11.
-In ref 51, the authors introduce also a "Kohler" term that account for classical magneto-resistance. This should correspond to the alpha term in front of the B2 term in the formula given line 238. However, the authors have used an approximation and not the full magnetic field dependence that includes a saturation of classical magneto-resistance at high magnetic field. The MR curves of thin STO layers (8 uc, 10 uc) could be very well fitted by adding a full "Kohler" term and a single BSO3 term making the discussion on the cubic to linear Rashba transition irrelevant. This term could be negligible but this must be justified.
Response: Thanks for the comment. We agree with the reviewer that if the magnetic field is strong enough, the B 2 term might be not applicable, which may influence our fitting model for the magnetoconductance data. However, to disqualify the B 2 term, the magnetic field should satisfy that μH >1, wherer μ is mobility. Therefore, for the maximum magnetic field (9 T) applied in MR measurement, this will require μ > 1.1*10 3 cm 2 V -1 s -1 , which is far larger than the mobility of the carriers in the LAO//STO/LAO structures. Second, we aware that Kohler's rule is the relationship that governs how the applied transverse magnetic field modifies the resistance of a materials, and can be summarized as Therefore, we do not think that it is reasonable to include the full magnetic field dependent Kohler term in the fitting model, which will also complicate the fitting process as more fitting parameters are involved (the model in ref 51).
-Related to the previous comments, I recommend the authors to clarify the meaning of the gamma, delta and alpha coefficients in the expression of the resistance (line 238). I also would like to see the value of these fitting parameters for the different thicknesses of STO.
Response: Thanks very much for the suggestion. Among these three parameters, γ is the reciprocal Van der Pauw constant due to the method we used here, which is a constant, ln(2)/π, for all the fitting procedures. α is the coefficient for the magnetoresistance due to the Lorentz force and δ is the field independent component of conductivity of the samples. We have updated the meanings for the all these parameters in the revised manuscript, page 12. The fitted parameters of δ and α are listed in the table below.  Figure R2 shows the temperature dependent fitting parameters B so1 , B so3 and B φ for the structures with 8 uc and 30 uc STO, respectively. For both structures, the B φ increases as the temperature increases, which is consistent with the behavior of the phase coherence property of the conducting carriers.
The  the redistribution of carriers from the two d xz/yz orbitals is not that significant when the Rashba transition occurs. Therefore, the mobilities of the carriers from these two orbitals are expected to exhibit only modest change due to their different effective masses. As a result, the non-linear Hall effect was not observed in our structures when the Rasha transition occurs.
-In their calculation, the authors assume a perfect crystal structure, which is in contradiction with the structural analysis. This aspect is really critical as the main claim (transition between two regime of Rashba SO) relies on an apparent agreement between theory and experiment. I can hardly imagine that the transition at 0.01e/Ti is sufficiently robust to any kind of imperfection in the structure. To clarify this, we added one sentence in the revised manuscript "We should note that this critical carrier concentration is an intrinsic property of STO and determined by its electronic band structure." in page 6.

8
-The authors assume that electrons are homogeneously distributed in the STO. I would expect that for the highest thicknesses of STO, one should recover the same situation than in LAO/STO heterostructures, with electrons confined at the interfaces. In this case, the low doping regime corresponding to the transition at 0.01e/Ti could be a pure artefact.
Response: Thanks very much for the comment. We agree with the reviewer that if the STO is thick enough, the scenario may recover to the one like that of the conventional LAO/STO heterostructure, due to the relaxation of the STO lattice and/or imperfect layer-by-layer growth of the STO layer. However, in the STO thickness range studied in this report, the low doping regime is still at the state where the strain from LAO holds, as supported by XRD measurement (Fig. S4) and the STEM strain mapping data ( Fig. 2 and S5). We have added the temperature dependent resistance in the revised supplementary materials as Fig. S8. To indicate procedures clearly, we added the sentence "which are obtained after removing the classical Lorentz components" in the first paragraph of page 13.

Reviewer #3 (Remarks to the Author):
• Key results: The paper of Lin et al. reports the study of the transport properties of LaAlO3/SrTiO3/LaAlO3 heterostructures formed by two non-equivalent interfaces. They propose this system as a good candidate to study the spin-orbit Rashba effect, which is a really interesting property for the spintronic community.
One of the main results given by this work is that, by varying the SrTiO3 layer thickness, the authors achieved to tune the carrier density and observed experimentally a transition from a cubic to linear (more precisely cubic+linear, if I understood correctly) Rashba effect, the linear Rashba effect being only predicted theoretically until now.
Response: Thanks very much for the reviewer's positive evaluation of our work.
• Validity: The manuscript is globally well written. Some precisions seem however necessary to judge about the complete validity of the paper.
Response: Thanks very much for the comment. In the revised manuscript, we have revised and improved the manuscript accordingly.
• Originality and significance: The observation of the linear Rashba effect will undoubtedly motivate further study to get a better understanding of this system and to get a better control of this interaction. It would thus be interesting to submit it for discussion. It is however not certain that this paper will get an audience out of the spintronic and oxide community.
Response: We agree with the reviewer that the observation reported in the manuscript is interesting for the spintronic and oxide community. Besides, we also expect that it will attract audience beyond this community because the spin-orbit interaction is a fundamental property of materials and it has been found to play important roles in various exotic materials ranging from layered graphene-like materials to cold • Data & methodology: The magneto-transport measurements are associated with some analysis of the atomic structure performed by STEM and some support from first-principles calculations.
For the DFT-based tight binding calculations, I would suggest to add more information about the parameters which were used and the methodology. In particular, it will be easier for the reader to have everything in hand than to collect all the information from previous references.
What is really from "first principles" and which parameters (energy splitting, atomic spin-orbit interaction, directions of the xy orbital, respectively. The term includes atomic spin-orbit interaction, whose strength is 19.2 meV, as estimated from the DFT calculated orbital splitting at Γ point. The last term =< | | / > is an antisymmetric hopping between xy and yz/xz orbitals along the x/y direction.
It describes the asymmetry due to the built-in electric field, which results in the Rashba spin splitting. In the bulk STO, vanishes due to inversion symmetry, while at the LAO/STO interface is 20 meV." We also added a description of how the STO lattice constant is calculated in the revised supplementary material (supplementary information 3).
About the chemical analysis, it is said that the two interfaces show clear cation intermixing, but that they remain non-equivalent. Was such analysis performed for different STO layer thicknesses? Or only for the 20 u.c.? From what is said, the out-of-plane STO layer lattice parameter is lower than expected mainly due to electrostrictive effect. Such effect should be linked with the asymmetry of the interfaces, and thus it should depend on the atomic structure at these interfaces. Did the authors notice any difference for the low and high layer thicknesses (that is, below and above 20 u.c., the thickness at which the transition occurs)?
Response: Thanks very much for the reviewer's comment. At this moment, we only did the STEM characterization for the structure with 20 uc STO. In this work, we did not focus on the electrostrictive effect because the transition of SOC occurs because of the modulation of the carriers introduced to the Ti site, which is an intrinsic property of the 3d orbitals of Ti in STO layer. In general, the imperfect interfaces provide a complementary way to achieve an energy gain to compensate the effect of the polar field from the LAO, which will influence the build-in field experienced by the STO layer. We agree with the reviewer that more systematic TEM investigations on the interfaces of the structure with different STO thickness may give us more information about the development of the built-in field in the STO layer, but this will require a large amount of experimental resources and time and we plan to devote a future project to this specific task.
The "rough" approximation which consists in averaging the number of electrons per Ti should be maybe discussed a little. The localization of the electrons near the interface is important as it depends on the internal electric field, which will itself have some impact on the Rashba splitting. Moreover, such approximation is quite contradictory with the fact that all the study relies on having two non-equivalent interfaces.
Response: Thanks very much for the comment. We agree with the reviewer that the calculated number of electrons per Ti is a "rough" approximation to estimate the doping level of the STO layer. As expected from the designed structure with two non-equivalent interfaces, the electrons distribution across the whole STO layer will show a gradual increase from the p-type interface to the n-type interface. The estimated electron number per Ti site gives only an average value, which does not reflect the local deviations along the STO layer. To give more discussions about this point, we have added the sentences "In the rough estimation, the carriers are assumed to be homogeneously doped in the entire STO layer, which does not reflect the local deviation of the electron filling across the STO layer. As expected from the designed LAO/STO/LAO structure with two non-equivalent interfaces, the electrons distribution across the whole STO layer should show a gradual increase from the p-type interface to the n-type interface. Furthermore, the localized electrons near the interfaces may have an important influence on the internal electric field and impact on the Rashba splitting. Nevertheless, the carrier density estimated from the Hall effect measurement exhibits a consistent trend of dependence on the STO layer thickness." in the first paragraph of page 9.
Is it possible to have some information on the method used to fit the parameters for Fig. S6? Above 20 u.c.
the red and black points are superimposed, right? Why is the error increasing (of a factor 3) above 20 u.c.?
This increase of the error is higher than the difference of errors between the fits with and without the linear contribution.
Response: Yes, the red and black points in Fig. S6 are superimposed for structures with above the 20 uc STO, indicating that both models give identical fitting results at this regime. The error bars are obtained as the total absolute values of the differences between the calculated and the measured conductivity. The larger errors for the structures with above 20 uc STO might be due to the larger component of the field independent conductivity, as seen from the fitting parameters in Table R1. However, as both models give almost the same errors and fitting parameters (Bso1 from the model including linear and cubic term is trivial for structures with above 20 uc STO), it will not influence the validity of our fitting results.
To give more information and discussion about the error analyses, we have added the sentences "To make the comparison quantitatively, we calculated the errors from both fitting models, which are obtained as the total absolute values of the differences between the calculated and the measured conductivity. As shown in Fig. S6c, the errors from the fitting with the linear Rashba term are smaller than that without the linear term, for the samples with STO thickness less than 20 uc. And for the samples with more than 20 uc STO, the fittings with and without the linear Rashba term result in almost similar error values." in the end of the last paragraph of the page 11 in the revised supplementary materials. 2) Why do the authors finally show some evidence of the linear Rashba effects, while other did not? Is it linked with the high carrier density? The special geometry of the system? From the bandstructure, I can understand why the cubic Rashba is seen first, and the linear appears only after, but if we forget the band energy ordering, the linear Rashba should be also present for the standard LAO/STO interface (?). I did not get the hypothesis that the dxy orbital being localized at the interface, the corresponding electrons would not participate to the charge transport.
Response: Thanks very much for the comment. Yes, we agree with the reviewer that the linear Rashba term should be present for the standard LAO/STO interface, however, which has not been reported so far.
As shown in Fig