Crystal symmetry-dependent Orbital Rashba Edelstein effect in epitaxial CuO thin film

Domain: biology
Published: 2026-03-24
Editorial outcome: major_revision

Abstract
Orbital angular momentum has recently been demonstrated as a promising approach to manipulate magnetic order in spintronic devices with high efficiency. While the generation of orbital angular momentum is generally attributed to electron orbital hopping and hybridization, its experimental manifestations have often been discussed in an isotropic manner. In crystalline systems, however, orbital hybridization is inherently constrained by crystal symmetry, suggesting that symmetry can play a decisive role in shaping the anisotropy of orbital angular momentum -related responses. In this work, we report a crystal-symmetry-dependent orbital Rashba-Edelstein effect in a distinct epitaxially grown CuO thin film with four-fold crystal symmetry. A crystal-symmetry-dependent sign change of spin torque efficiency is observed in the CuO/ferromagnetic heterostructures, verified by second harmonic Hall measurement and current-induced perpendicular magnetization switching. The experimental results, together with the first-principles calculation, indicate the existence of a strong four-fold anisotropy of orbital Rashba-Edelstein effect in crystalline CuO, which is a direct manifestation of the crystal symmetry’s impact on the anisotropy of orbital angular momentum related response. These findings establish crystal symmetry as a key factor governing the anisotropy of orbital angular momentum -related phenomena and provide a symmetry-based framework for orbitronic functionalities in advanced spintronic devices.

Introduction
Leveraging angular momentum transport in solids is essential for spintronic application. For example, spin angular momentum (SAM) in the form of spin current, originating from spin Hall effect (SHE)1,2 or Rashba-Edelstein effect (REE)3 in system with strong spin-orbit coupling (SOC)3,4, has been an effective method to manipulate magnetic order 5-9. Recent studies revealed that the magnetic order can also be switched in magnetic systems with negligible SOC 10,11 by a wave packet of orbital angular momentum (OAM)12,13, i.e., the orbital current. As a counterpart of spin current, orbital current can be generated through orbital Hall effect (OHE)10,14 or orbital Rashba-Edelstein Effect (OREE)15,16, and can be exerted on magnetic moment after being converted into spin current by the intrinsic SOC in the magnetic layer17 or by an inserted high SOC layer 18. Besides the magnetization switching, this new approach to transfer angular momentum to the spin system also inspires novel magneto-electrical responses like orbital Rashba-Edelstein magneto-resistance 15, unidirectional orbital magneto-resistance 19, etc.

An important approach to manipulate angular momentum in solid is the symmetry design of the crystalline. Taking SAM as an example, it has been widely reported that manipulating the crystal symmetry—such as through rotation or distortion—can tune both the inversion symmetry breaking energy and the SOC energy 13. This enables effective control over the efficiency of SAM generation and facilitates the emergence of unconventional SASM 7,20-22. As for OAM, its transport behavior is fundamentally induced by the electron’s orbital hopping phenomenon, e.g., d-orbital hopping in transition metals 23,24. In principle, the change of crystal symmetry condition might change the orbital hopping paths or induce additional orbital hybridization, which could modify the OAM strength and even flip the sign. However , current works on OAM generation have been primarily focused on polycrystalline or amorphous thin films, such as Ti 10, Cr 25, Mn 26, Ru 27 and CuO 28, etc. Despite of the great potential to comprehensively reveal the OAM generation mechanism, the role of crystal symmetry in anisotropic OAM related response has remained unexplored experimentally.

In this work, we demonstrate that OREE is crystal-symmetry-dependent in an epitaxially grown CuO thin film. Unlike previous studies that used amorphous CuO formed by natural oxidation29-31, we have epitaxially grown a crystalline CuO <111> thin film exhibiting four-fold rotation symmetry. When attaching the CuO to a Ni layer, we proved that the observed spin torque efficiency strength strongly depends on the angle α between the injected current direction and the <01-1> direction of CuO. More importantly, the spin torque efficiency reverses its sign when α changes by 45°, exhibiting a four-fold symmetry corresponding to the crystal symmetry of CuO. Our first-principles calculation indicates that this crystal symmetry dependence is attributed to the interplay of anisotropic OREE and spin Rashba-Edelstein effect in CuO.

Results
Crystalline structure analysis of the epitaxial CuO thin film
As shown in Fig. 1a, CuO is of a monoclinic crystal structure, where the cyan-colored plane represents the (111) crystallographic plane. We epitaxially deposited a high-quality crystalline CuO onto an MgO (001) substrate by pulse laser deposition at a temperature of 500 ℃. Fig. 1b illustrates the epitaxial relationship between the CuO (111) thin film and the MgO (001) substrate. The CuO [01-1] direction aligns with the MgO [110] direction, which exhibits a relatively low lattice mismatch of approximately 1.2%. This small mismatch enables the CuO thin film to be epitaxially grown on the MgO substrate 32. Fig. 1c plots the θ-2θ scan curve from X-ray diffraction (XRD) measurement of a 20-nm-thick CuO thin film on MgO (001) single-crystal substrate. A clear peak at the position of 38.55° is observed, confirming the successful growth of CuO (111) crystal. Through the X-ray photoelectron spectroscopy (XPS) spectra, it is confirmed that the thin film sample contains only divalent copper (Cu²⁺), with no indication of lower oxidation states such as Cu⁺, confirming that CuO is the only phase in the sample.

Moreover, although the monoclinic CuO (111) plane lacks an intrinsic rotational symmetry, our XRD φ scan result for MgO (111) and CuO (11-1) crystal plane in Fig 1d indicates that our deposited CuO (111) film exhibits a clear four-fold rotation symmetry. This is because MgO’s [110] orientation possesses four equivalent directions, i.e., [110], [1-10], [-110] and [-1-10]. And CuO’s [01-1] orientation can accommodate each equivalent direction, giving rise to the four-fold rotation symmetry in CuO. To calculate the lattice parameters, the relative positions of two non-parallel CuO planes, i.e., (11-1) and (200), are obtained from the φ-scan (see Fig. 1e). These two in-plane vectors, along with the surface normal vector, constitute a set of three non-coplanar vectors. With three non-coplanar vectors, the structure of the CuO can be calculated via the reciprocal space vector (RSV)33. By calculation, we determined that the deposited CuO thin film has a standard monoclinic structure, with a =4.701±0.002 Å, b =3.429 ±0.002 Å, c =5.098±0.002 Å and β =99.4 °. The monoclinic structure of CuO is further verified by the transmission electron microscopy (TEM) in Fig. 1f together with the fast Fourier transform ( FFT) analysis in the inset.

Electrical transport properties measurements
To explore the OREE in our CuO with a four-fold rotation symmetry, we further deposit a 8-nm-thick Ni layer on CuO by magnetron sputtering. Ni is chosen because it has been reported to exhibit a higher orbital-to-spin conversion efficiency than other metallic ferromagnets like Co and CoFeB 17. For electrical measurement, the thin films are then fabricated into 5-μm-wide Hall bar devices by standard lithography and ion milling process. The experimental setup is illustrated in Fig. 2a. To implement the second-harmonic Hall measurement, an ac current I is applied by Keithley 6221, while two Stanford SR830 lock-in amplifiers are used to detect the first harmonic and second harmonic voltage signals. We denote the angle between the injected current direction in the Hall bar and the MgO [100] crystal orientation as α, and the angle between the injected current direction and the in-plane magnetic field direction as φ.

When an ac current I ∝ sin(ωt) passes through the CuO/Ni interface, an orbital accumulation is expected to be generated via OREE and be converted to spin current in Ni layer due to the strong SOC effect. The spin-current-induced magnetic oscillation in Ni layer then generates harmonic signals. Fig. 2b and 2c show the standard first harmonic transverse resistance R and second harmonic transverse resistance R as a function of φ in a device with α = 45°. The amplitude and frequency of applied I is 3.5 mA and 317.3 Hz, respectively. The applied magnetic field is 2500 Oe. All the electrical measurements are implemented at room temperature unless otherwise specified. Typically, the relationship between R and φ can be described by the following equation 34:

𝑅2𝜔(𝜑) = (𝑅 𝐵 DLT+𝑅2𝜔)cos⁡𝜑+2𝑅 𝐵 FLT +𝐵Oe(2cos3𝜑−cos𝜑)
𝑥𝑦 AHE ∇𝑇 PHE
𝐵
eff
𝐵ext
𝐵𝑧 𝐵𝑧
−2𝑅 DLTcos2𝜑+𝑅 FLT,
PHE AHE
𝐵ext 𝐵
eff

where𝑅 and 𝑅 denote the anomalous Hall resistance and planar Hall resistance, respectively; 𝐵 and 𝐵 represent the damping-like and field-like effective fields generated by the conventional in-plane spin torque; 𝐵𝑧 and 𝐵𝑧 are the damping-like and field-like effective fields from out-of-plane spin torque; 𝐵 is the oersted field. The effective magnetic field 𝐵 is defined as 𝐵 =⁡𝐵 +𝐵 −𝐵 , where 𝐵 and 𝐵 is the demagnetization field and the anisotropy field of Ni, respectively. 𝑅2𝜔 represent the contribution of the anomalous Nernst effect. Here, we mainly focus on the discussion of 𝐵 which is the main driving force for magnetization switching.

As shown in Equation (1), the contribution of 𝐵 in the second-harmonic measurement is in the shape of cos𝜑 term. Fig. 2d plots the obtained cos𝜑 term R [cos(φ)] , i.e., the value of 𝑅 𝐵 D 𝑦 LT +𝑅2𝜔, as a function of 1/𝐵 for two devices where α = 45° and 90°, respectively. In such a case, the slope of the fitting linear line can in principle represent the value of 𝐵 . Interestingly, as shown in Fig. 2d, the slopes of these two devices clearly show opposite signs, indicating that 𝐵 𝑦 at α = 45° and 90° has opposite direction. To further confirm the observed anisotropy and sign reversal phenomena, we systematically varied α from 0° to 360° with a step of 15°. Here, we use ξ, defined as 𝐵 /𝑗 with j being the applied current density, to quantify the efficiency of the damping-like torque generated per unit current density. As shown in Fig. 2d, starting from α = 0°, when α increases, ξ decreases in the negative direction, reverses its sign, and then increases in the positive direction, reaching its maximum positive value at 45°. Beyond this point, ξ decreases in the positive direction, changes the sign again, and reaches its maximum negative value at 90°. This pattern repeats four times, resulting in a clear four-fold symmetric trend in the 0°-360° range. This four-fold anisotropy of ξ is consistent with the four-fold crystal rotation symmetry in CuO that has been proved in Fig. 1d, indicating that crystal field can significantly impact OREE.

To validate this sign-reversal phenomenon in a more direct and visible way, we deposited a CuO/Pt (2 nm)/[Co/Ni] (4 nm) stack with a perpendicular magnetic anisotropy (PMA) and conducted current-induced perpendicular magnetization switching experiment in the corresponding Hall bar devices. The inserted Pt layer not only guarantees the PMA in [Co/Ni] multilayer, but also serves as a good orbital-to-spin conversion layer. Fig. 3b plots its transverse resistance R as a function of out-of-plane magnetic field H. A quasi-square loop with a coercive field of 250 Oe is observed, confirming the PMA property. In the current-induced magnetization switching experiment, a current pulse with a 30-μs-width is applied, and a small current is applied to detect the magnetic state via the change of R after 5 s cooling down.

Figure 3b and 3c plot the switching curve as a function of applied current pulse in the two devices where α = 0° (MgO [100] direction) and 45° (MgO [110] direction), respectively. For both switching, an external magnetic field of 400 Oe is applied along the current direction to break the in-plane symmetry and enable deterministic switching. Clearly, these two devices show opposite switching polarities, which is consistent with our harmonic measurement in Fig. 2e, i.e., ξ in the devices where α = 0° and 45° are opposite. We also note that the critical switching current for the device with α = 0° is around 15 mA, which is much larger than the critical switching current (~8 mA) for the device with α = 45°. The difference in critical switching current is consistent as well with the absolute value difference of ξ that is revealed in Fig. 2e. The switching ratio is relatively small in our device, which can possibly be attributed to the following two reason: 1) ξ is objectively small in the device, but the bulk PMA anisotropic energy of [Co/Ni] is high, therefore only a small part of [Co/Ni] can be switched; 2) the small ξ induces a large switching current density, resulting in severe Joule-heating induced demagnetization 35. However, the small switching ratio doesn’t affect the clarity of the switching polarity. Our switching experiments provide strong support for the harmonic measurement and further validate the observed anisotropic OREE in CuO.

First-principles calculation for Rashba effect on CuO (111) plane
To further shed light on the underlying physical mechanism of the observed anisotropic OREE, we conducted first-principles calculations on CuO with four-fold rotation symmetry36,37 . See calculation details in supplementary materials. Fig. 4a and 4b plot the OAM and SAM susceptibilities36,37 due to the OREE and the spin Rashba-Edelstein effect, respectively, as a function of the Fermi energy E with respect to the true Fermi energy (0 eV). The red (blue) line in Fig. 4a and 4b indicates the energy band for the case where α = 0° (45°). Clearly, the OAM in CuO is a negative value and can dramatically decrease by 2 times from 0° to 45°, while the spin Rashba effect is a negative value and can dramatically increase by 4 times from 0° to 45°. Our results reveal that both the OREE and the spin Rashba effect in the four-fold crystalline CuO exhibits a significant four-fold symmetry tendency with sign change.

The observed sign change can be attributed to the competition between OAM and SAM susceptibilities in CuO. The total spin current intensity achieved in the system can be described by 𝛿𝑆 +𝑐𝛿𝐿 , where 𝛿𝑆 and 𝛿𝐿 are the SAM and OAM intensities and c is the orbital-to-spin conversion coefficient. For any value of c (to best fit our results, here we adapt 0.07) (see the table in Fig. 4c), the total intensity of 𝛿𝑆 +𝑐𝛿𝐿 is is dominated by OAM and negative when =0 whereas 𝛿𝑆 +𝑐𝛿𝐿 is dominated by SAM and positive when =45. Figure 4d shows the total intensity for the particular value c=0.07. Note that the total intensities for =0 and 45 have opposite signs in a wide range of E. At the true Fermi energy E=0 eV, in particular, the intensity for 45 is two times larger than that for =0 in magnitude, in agreement with the experimental result. As shown in Fig. 4e, our experimentally achieved angular-dependence of ξ can be well fitted by the OAM and SAM competition model. Note that such an OAM and SAM competition model has already been used to explain the inversed OREE phenomena at a LaAlO3/SrTiO3 interface37. The major difference is that, in their study37, SAM has a crystal-symmetry-dependent anisotropy while OAM remain isotropic to crystal symmetry. This is in direct contrast to our findings where both OAM and SAM are anisotropic.

In summary, we have experimentally demonstrated that the crystal symmetry of crystalline CuO (111) can induce a crystal-symmetry-dependent OREE in a CuO/Ni heterostructure. The four-fold OREE is well validated by second-harmonic Hall measurement and the current-induced switching experiment. The first-principles calculations reveal that the observed four-fold symmetry of the OREE originates from the intrinsic crystalline symmetry of the four-fold crystalline CuO film. Moreover, the interplay between the anisotropic OREE and SRE explains the observed sign reversal of the damping-like torque. These findings underscore the critical role of crystal symmetry in governing the anisotropy of OAM-related orbital transport phenomena, providing new insights into crystal-field-controlled orbital transport, such as symmetry allowed OAM generation to achieve field-free magnetization switching and promoting the development of future orbitronic devices.

Methods
Sample preparation: The epitaxial CuO thin film samples were deposited by pulsed laser deposition (PLD) with oxygen pressure 100mTorr at 500℃ using a KrF (λ = 248 nm) excimer laser. The laser fluence was 1.2J cm-2 with a repetition rate of 2 Hz for CuO. After the deposition, the films cool down to the room temperature with the cooling rate of 10℃/min. Then we in-situ deposit the ferromagnetic layer through magnetron sputtering at room temperature.

Device fabrication and electrical measurement: The CuO/FM heterostructure was patterned into Hall cross devices in 5-μm-width using a combination of maskless laser writer (TuoTuo Technology) and ion beam etching. We fabricate the Hall bar devices which the longitudinal directions along various MgO orientations with 15° steps. The harmonic Hall voltage measurement is driven by Keithley 6221 and the first and second harmonic Hall voltage signal is read by two different lock-in amplifier SR830. The magnetization switching measurement is driven by 30-μs-wide electrical current pulses generated with Keithley 6221. Lock-in amplifier SR830 is used to read out the anomalous Hall voltage of Hall cross with a 100 A d.c. current by Keithley 6221.

First-principles calculations: The orbital and spin Rashba-Edelstein effects in a three-layer CuO (111) film were studied using first-principles calculations based on the FLAPW method implemented in the FLEUR code with GGA + U (U = 6.0 eV, J = 0.98 eV). Tight-binding parameters were extracted from 242 Wannier functions generated via WANNIER90 using 484 Kohn–Sham states. Linear response theory was applied on a 400 × 400 k-mesh to compute the Edelstein responses, which were symmetrized over fourfold in-plane rotations. Details of the calculation is provided in the Supplementary Section 1.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In spintronics involving spin angular momentum has a long history of symmetry considerations — for example, in the study of spin Hall effects, Rashba/Dresselhaus interactions, or anisotropic magnetoresistance, where symmetry governs allowed spin–orbit interactions and torques. Crystal symmetry has been, however, not deeply explored in orbitronics (OAM-based spintronics), which is astonishing, since the pioneering theoretical works of Go relate OAM-generation to orbital hybridization and, thus, to crystal symmetry.\nThis manuscript investigates the influence of crystal symmetry on orbital angular momentum (OAM) generation in spintronic devices. The authors present evidence for a crystal-symmetry-dependent orbital Rashba-Edelstein effect (OREE) in epitaxially grown CuO thin films with four-fold symmetry. Using electrical measurements and first-principles calculations, they report a symmetry-dependent sign change in spin torque efficiency, highlighting the importance of incorporating crystal symmetry into orbitronics research and its potential impact on future spintronic device design.\nThis work addresses a fundamental gap in orbitronics by directly linking crystal symmetry to orbital angular momentum (OAM) generation through combined experimental and theoretical analysis. By moving beyond the conventional isotropic treatment of OAM, it establishes a symmetry-based framework for understanding and controlling orbital Rashba–Edelstein effects. These insights could transform both the fundamental understanding of angular momentum transfer in solids and the design of next-generation spintronic devices, where symmetry-controlled OAM offers a route to enhanced or reversible spin–orbit torques. More broadly, the distinctions between spin angular momentum (SAM), an intrinsic electron property, and OAM, tied to the motion of electrons, remain elusive. In this light, the paper contributes to the ongoing exploration of these phenomena.\nAlthough the presented results are convincing, I strongly encourage the authors to address the following points, which could further strengthen the manuscript and provide additional support for their conclusions, and I look forward to receiving a revised submission.\n# Major Comments\n* Mechanism of OAM generation: The manuscript attributes OAM transport to electron orbital hopping (Go mechanism). Is this the only conceivable mechanism? The role of orbital hybridization (Ref. 13) should be discussed, specifying which orbitals contribute for $\\alpha = 0^\\circ$ and $45^\\circ$.\n* Crystal symmetry claims:\n* The abstract and text make strong statements about the absence of direct evidence linking OAM to crystal symmetry; either provide supporting references or soften these claims.\n* Why is the OREE anisotropic while the spin Rashba-Edelstein effect (SREE) is isotropic? Both are forms of angular momentum, and their cartesian components may depend on crystal symmetry.\n* Experimental evidence for OAM:\n* Spin torque may arise from both OAM and SAM. Can the OREE contribution be distinguished, e.g., via SARPES or another experimental method?\n* Clarify whether OAM generation by OREE produces an OAM current, and specify its direction and polarization relative to the Ni magnetization.\n* Current-induced switching:\n* Data gaps from $-10$ mA to $+10$ mA (Fig. 3) and points outside the hysteresis curve need explanation.\n* Ensure the CuO crystal structure remains unaffected by Pt; has this been verified by XRD or LEED?\n* Causality vs. consistency: Statements linking the four-fold anisotropy of $\\xi$ to CuO crystal symmetry are strong; consistency is not proof of causality and should be carefully justified.\n* Theoretical calculations:\n* Why is OAM much larger than SAM (in absolute value, Fig. 4)?\n* Explain the necessity of the averaging procedure in the Supplemental Material; could selecting $X$ and $v$ along $\\alpha = 0^\\circ$ and $45^\\circ$ be sufficient?\n* Reference energies in Figs. S1, S2, and Fig. 4 seem inconsistent; consider using a single reference, e.g., the VBM.\n* Did direct calculations $\\langle S\\rangle = \\langle \\psi | S | \\psi \\rangle$ and $\\langle L \\rangle = \\langle \\psi | r \\times p | \\psi \\rangle$ yield the same trends as the Berry-curvature-type approach?\n# Minor Comments\n* Terminology: Clarify what is meant by “unconventional four-fold symmetry” (page 2).\n* Sample preparation and characterization:\n* Substrate description is inconsistent: text states MgO(110) (page 3), Figure 1 caption states MgO(001).\n* Fig. 1b (top view of CuO): four-fold symmetry is not clearly visible. LEED imaging is recommended to confirm symmetry.\n* Measurement details:\n* How is the current angle $\\alpha$ varied in steps of $15^{\\circ}$ in the Hall bar?\n* Explain the connection between $B_\\mathrm{DLT}$ and OREE.", "reviewer_2": "In this work, the authors report the observation of a four-fold anisotropy in the Rashba Edelstein effect in single-crystalline CuO thin films. By combining with the first-principle calculations, the authors attribute the anisotropic Rashba Edelstein effect to the orbital transport. Research on orbital transport is pivotal, as it paves the way for the development of orbitronics. However, the distinction between spin and orbital contributions in this work relies exclusively on theoretical calculations. Experimental validation or corroborating evidence is needed to firmly establish this interpretation. Below I list several questions.\n1. The XRD θ-2θ scan of the CuO thin film exhibits a broad peak, indicating reduced crystalline quality that may be attributed to sample inhomogeneity and small grain size. It is essential to discuss whether these structural imperfections could influence the observed anisotropic Rashba-Edelstein effect, as they might affect spin/orbital transport or torque generation mechanisms.\n2. The authors report a four-fold anisotropy in the current-induced in-plane damping-like torque. Their first-principles calculations attribute this symmetry-dependent torque primarily to the anisotropic orbital Rashba-Edelstein effect, in contrast to the expected isotropy of the spin Rashba-Edelstein effect. However, the experimental methodology for disentangling the individual contributions from spin and orbital moments remains elusive. It would be valuable if the authors could address this point, perhaps by proposing an experimental approach to clearly distinguish the spin and orbital contributions.\n3. As shown in Fig. 2e, the current-induced in-plane damping-like torque exhibits an approximate, but not perfect, four-fold symmetry. Notably, the values at α = 45° and 315° are not equivalent. What could account for this deviation from the ideal symmetry?\n4. The current-induced magnetic switching is presented in Fig. 3 for two devices with α angles of 0° and 45°. Perpendicular switching was achieved in both cases through the application of an external magnetic field. It should be noted that the direction of the applied field differed between the two devices (along 0° and 45°, respectively). The effect of the magnetic field orientation, therefore, merits a systematic investigation.\n5. Unlike the perpendicular switching in Fig. 3, which requires external magnetic field assistance, the CuO thin film may be exploited to achieve field-free in-plane magnetic switching. This approach completely obviates the need for an external magnetic field, thus circumventing related effects.\n6. In first-principle calculations, a parameter *c* is introduced to represent the orbital-to-spin conversion coefficient. In my opinion, the value of *c* should depend on the material used as the orbital-to-spin conversion layer (e.g., Pt or Ni). Is this correct?\n7. Have the authors considered the potential influence of the substrate? The observed Rashba-Edelstein effect in this study may also originate from the interface between the CuO film and the MgO substrate, analogous to the interfacial effect reported in a previous work (e.g., Adv. Mater. 34, 2109449 (2022)). It would be important to include a discussion on this aspect to better clarify the origin of the observed phenomena.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "R. Xiao et al. presents a compelling study on the crystal-symmetry-dependent Orbital Rashba-Edelstein Effect (OREE) in epitaxial CuO thin films. The authors demonstrate, for the first time, that orbital angular momentum generation can be directly governed by the underlying crystal symmetry, leading to a four-fold anisotropy and sign reversal of the spin–orbit torque efficiency. The work is highly innovative, filling a key gap in current orbitronic research where OREE has been treated as isotropic. Overall, the results are novel and technically rigorous. Before recommending publication, I would suggest that the authors address a few minor comments for further improvement.\n1. The authors prepared sample as Ni over CuO layer. The properties of the Ni layer might change due to oxidization and induce angular dependence. Did the authors rule out this possibility?\n2. Discussing the efficiency of CuO OREE, the author demonstrates the efficiency of OREE by using an effective field per current density. It would be better if they provide or discuss a comparison between the CuO OREE efficiency and the spin–orbit torque efficiency of Pt, to better benchmark the effectiveness of CuO as an orbital torque source.\n3. The authors mentioned this finding can be applied to advanced spintronic devices, can the author elaborate on a specific picture or scenario of how the anisotropic OREE could be utilized in practical applications\nMinor comments:\n1. The manuscript refers to “Ref. 38” when discussing the LaAlO₃/SrTiO₃ interface OREE, but this reference number does not appear in the reference list. Please verify and correct the numbering or add the missing citation to ensure consistency.\n2. Throughout the manuscript, the term “first-principal calculation” is repeatedly used. The correct expression should be “first-principles calculation”.", "reviewer_5": "Rui Xiao et al. investigated the crystal symmetry dependent orbital Rashba Edelstein effect in four-fold crystalline CuO/Ni heterostructure. Moreover, the crystal-symmetry-dependent spin torque efficiency sign reversal was obtained in the heterostructure. Crystal symmetry dependent orbital Rashba Edelstein effect provides an interesting way to control orbital current and improve torque efficiency in orbitronic devices. Before recommending its acceptance, several issues need to be clarified as following:\n1) The generation of orbital current inducing magnetization switching in CuO-based heterostructures was reported through orbital Rashba Edelstein effect at CuO/Cu interface (such as PRL 2020, 125 177201; APL 2022, 121 072404; Adv. Mater. 2023, 35, 2300858). In this manuscript, the authors reported that the orbital Rashba Edelstein effect in a CuO/Ni film without CuO/Cu interface. How to explain the source of orbital Rashba Edelstein effect?\n2) In the case of the CuO/Ni film, the Ni film is single crystal, polycrystalline, or amorphous? The crystal-symmetry-dependent orbital Rashba Edelstein effect may be related to the crystal structure of Ni. How to rule out this possibility?\n3) In Fig. 3b,c, the Rxy data are quite sparse, could the authors have lost some of them?\n4) Breaking the mirror symmetry can realize the current-driven magnetization switching without magnetic field. This artcle fucus on the crystal symmetry dependent switching, thus the authors should provide field-free current-induced magnetization switching, which will improve the quality of the manuscript.\n5) The spin torque efficiency of CuO/Ni(10nm) sample (Fig. S5) is markedly improved compared to the CuO/Ni(8nm) (Fig. 2e). Please elaborate on this difference.\n6) Regarding the CuO/Pt[CoNi]n with 2 nm Pt, I suggest to add some discussions about thinner Pt (such as 1 nm) to reduce spin current from the spin Hall effect."}, "author_response": "Response to reviewers’ comments\nThe referee’s comments are quoted in blue and our responses in black. Revision\nrelated parts are highlighted.\nReviewer #1 (Remarks to the Author):\nIn spintronics involving spin angular momentum has a long history of symmetry\nconsiderations — for example, in the study of spin Hall effects, Rashba/Dresselhaus\ninteractions, or anisotropic magnetoresistance, where symmetry governs allowed\nspin–orbit interactions and torques. Crystal symmetry has been, however, not deeply\nexplored in orbitronics (OAM-based spintronics), which is astonishing, since the\npioneering theoretical works of Go relate OAM-generation to orbital hybridization\nand, thus, to crystal symmetry.\nThis manuscript investigates the influence of crystal symmetry on orbital angular\nmomentum (OAM) generation in spintronic devices. The authors present evidence for\na crystal-symmetry-dependent orbital Rashba-Edelstein effect (OREE) in epitaxially\ngrown CuO thin films with four-fold symmetry. Using electrical measurements and\nfirst-principles calculations, they report a symmetry-dependent sign change in spin\ntorque efficiency, highlighting the importance of incorporating crystal symmetry into\norbitronics research and its potential impact on future spintronic device design.\nThis work addresses a fundamental gap in orbitronics by directly linking crystal\nsymmetry to orbital angular momentum (OAM) generation through combined\nexperimental and theoretical analysis. By moving beyond the conventional isotropic\ntreatment of OAM, it establishes a symmetry-based framework for understanding and\ncontrolling orbital Rashba–Edelstein effects. These insights could transform both the\nfundamental understanding of angular momentum transfer in solids and the design of\nnext-generation spintronic devices, where symmetry-controlled OAM offers a route to\nenhanced or reversible spin–orbit torques. More broadly, the distinctions between\nspin angular momentum (SAM), an intrinsic electron property, and OAM, tied to the\nmotion of electrons, remain elusive. In this light, the paper contributes to the ongoing\nexploration of these phenomena.\nAlthough the presented results are convincing, I strongly encourage the authors to\naddress the following points, which could further strengthen the manuscript and\nprovide additional support for their conclusions, and I look forward to receiving a\nrevised submission.\nResponse: We thank the referee’s positive feedback for our work. We addressed the\nreferee’s concerns in the following.\n# Major Comments\n1. Mechanism of OAM generation: The manuscript attributes OAM transport to\nelectron orbital hopping (Go mechanism). Is this the only conceivable mechanism? The\nrole of orbital hybridization (Ref. 13) should be discussed, specifying which orbitals\ncontribute for and\n∘ ∘\n𝛼𝛼 = 0 45\nResponse: We thank the referee for the important question and comment. Our OREE\nmechanism is indeed based on the orbital hybridization between different orbitals. In\norder to unveil the contribution of the orbital types to the OAM generation, the orbitals\nwhich are dominant around the true Fermi energy of the CuO film are shown in the\nfollowing figure.\nFig. R1. The orbital-projected density of states (pDOS) of the CuO film for Cu d\norbitals with (dark), (red), (green), (blue), and (purple).\n𝟐𝟐 𝟐𝟐 𝟐𝟐\n𝒅𝒅𝒛𝒛 𝒅𝒅𝒛𝒛𝒛𝒛 𝒅𝒅𝒚𝒚𝒛𝒛 𝒅𝒅𝒚𝒚𝒛𝒛 𝒅𝒅𝐱𝐱 −𝒚𝒚\nFig. R1 shows the orbital-projected density of states (pDOS) and of the CuO film with\nrespect to the Fermi energy . Note that and orbitals of Cu atoms are\ndominant around the true Fermi energy ( ). These o2rbitals could generate OAM\n𝜀𝜀 𝑑𝑑𝑥𝑥𝑥𝑥 𝑑𝑑𝑧𝑧\nand when hybridized with or orbitals. For , and\n𝜀𝜀 = 𝜇𝜇\norbital hybridization generates OAM . Also, for ∘ ,\n𝐿𝐿𝑥𝑥 𝐿𝐿𝑥𝑥 𝑑𝑑𝑧𝑧𝑥𝑥 𝑑𝑑𝑥𝑥𝑧𝑧 𝛼𝛼 = 0 𝑑𝑑𝑥𝑥𝑧𝑧 −𝑑𝑑𝑥𝑥𝑥𝑥\n2 and orbital hybridization generates OAM ∘ .\n𝑑𝑑𝑧𝑧 −𝑑𝑑𝑧𝑧𝑥𝑥 𝐿𝐿𝑥𝑥 𝛼𝛼 = 45 (𝑑𝑑𝑥𝑥𝑧𝑧 +\n2\n𝑑𝑑𝑧𝑧𝑥𝑥)−𝑑𝑑𝑥𝑥𝑥𝑥 𝑑𝑑𝑧𝑧 −(𝑑𝑑𝑥𝑥𝑧𝑧 +𝑑𝑑𝑧𝑧𝑥𝑥) −𝐿𝐿𝑥𝑥 +𝐿𝐿𝑥𝑥\n* Crystal symmetry claims:\n2. The abstract and text make strong statements about the absence of direct evidence\nlinking OAM to crystal symmetry; either provide supporting references or soften these\nclaims.\nResponse: We thank the referee’s comment about the connection of OAM generation\nand crystal symmetry. Firstly, we agree with the reviewer that the current claims on\nrelationship of OAM with crystal symmetry might be strong. In our work, the origin of\nthe orbital angular momentum (OAM) is the orbital Rashba effect, whose existence\ndoes not depend on crystal symmetry. We did not intend to claim that crystal symmetry\nis a prerequisite for the generation of OAM. What we aim to emphasize is that the\nanisotropy of the OAM response is linked to crystal symmetry, which is supported by\nour experimental observations of the linking of a clear four-fold symmetry in the OAM-\nrelated phenomena, including both orbital-torque-induced switching and orbital-Hall-\ninduced orbital torque, to the underlying 4-fold rotational crystal symmetry in CuO.\nThis symmetry-depending OAM is also further corroborated by detailed first-principles\ncalculations. To avoid ambiguity, we have revised the abstract and the relevant sections\nof the main text to explicitly clarify that crystal symmetry governs the anisotropy of\nOAM, rather than its existence, and we have softened the original wording accordingly.\n3. Why is the OREE anisotropic while the spin Rashba-Edelstein effect (SREE) is\nisotropic? Both are forms of angular momentum, and their cartesian components may\ndepend on crystal symmetry.\nResponse: First of all, we apologize for showing the inaccurate calculation for OREE\nand SREE. The CuO (111) film has not only the anisotropic REE but also the\nanisotropic charge conductivity. The CuO [111] film is deposited on MgO [001] film\nwith its mirror counterpart and four-fold rotational part [Fig. R2a]. Since the charge\nconductivity is anisotropic, each part of CuO (111) film undergoes different intensity\nelectric field even if the same intensity of charge current is applied. The anisotropy of\nthe inverse of the longitudinal charge conductivity is plotted in Fig. R2b. In order to\ninclude the anisotropy of the longitudinal charge conductivity, we recalculate the OREE\nand SREE susceptibilities, the OREE and the SREE divided by the longitudinal charge\nconductivity, to reproduce the experimental results. From these re-calculation results,\nwe observe that both OAM and spin are anisotropic with respect to the charge current\ndirection.\nFig. R2. (a) Crystalline structure of an CuO (111) film deposited on MgO (001) film,\nblue and red lines indicate mirror symmetries of the CuO (111) film. (b) Averaged the\ninverse of charge conductivity of CuO (111) film. (c) The OAM-projected Fermi\nsurface of the CuO (111) part A. (d) and the SAM-projected Fermi surface of the CuO\n(111) part A.\nNevertheless, we emphasize that our calculation shows the first example for the\nanisotropic OREE dependent on the crystal field. Note that the previous work by A. El\nHamdi, et al. showed the isotropic OREE, which is independent from the crystal field\n[A. El Hamdi, et al. Nat. Phys. 19, 1855-1860 (2023)]. The main difference of our work\nfrom the previous work by A. El Hamdi, et al. is the anisotropic OAM texture of the\nFermi surface.\nFig. R2c and Fig.R2d show the OAM and SAM-projected Fermi surfaces of the CuO\npart, respectively. The Fermi surface is elongated along direction and shrunken\nalong direction. This Fermi surface anisotropy affects the orbital hybridization and\n𝑘𝑘𝑥𝑥\nthe OAM anisotropy. Fig. 2c shows that both inner and outer bands on the Fermi surface\n𝑘𝑘𝑥𝑥\nof CuO (111) part have strong y-direction OAM and weak x-direction OAM .\nThis anisotropic OAM texture of CuO (111) film yields the anisotropic angular\n±𝐿𝐿𝑥𝑥 ±𝐿𝐿𝑥𝑥\ndependence of the OREE susceptibility. In contrast, the Fermi surface, especially the\nstrongly OAM-polarized bands, from the previous work [A. El Hamdi, et al. Nat. Phys.\n19, 1855-1860 (2023)], is isotropic, so that the OAM is also isotropically polarized.\nThis isotropic OAM texture generates the isotropic angular dependence of the OREE\nresponse.\nOur calculations show the anisotropy both in the OREE and the SREE susceptibilities,\nbut the angular dependence of the SREE susceptibility is quantitatively different from\nthe OREE susceptibility [Fig. 4e]. This difference between the OAM and the SAM\ncomes from the difference between the OAM and the SAM polarization. In Fig. 2d, the\ninner band and outer band on the Fermi surface are SAM-polarized opposite to each\nother, and each band has strong x-direction SAM and weak y-direction SAM .\nThe SAM polarization depends not only on the Fermi surface anisotropy but also the\n±𝑆𝑆𝑥𝑥 ±𝑆𝑆𝑥𝑥\nspin-orbit coupling of each band [S. R. Park et al. PRL 107, 156803 (2011)]. Thus, the\nSAM is polarized differently from the OAM, which leads to the difference between the\nOREE and the SREE susceptibilities.\n* Experimental evidence for OAM:\n4. Spin torque may arise from both OAM and SAM. Can the OREE contribution be\ndistinguished, e.g., via SARPES or another experimental method?\nResponse: We thank the referee for this question. We agree that spin–orbit torques may\nin general originate from both spin and orbital angular momentum contributions. In our\ncase, CuO is a nonmagnetic oxide composed of relatively light elements which exhibit\nweak spin-orbit coupling, as proved in several previous works, [Go, Dongwook, et al.\nPhysical Review B 103.12 (2021), Ding S, et al. Physical Review Letters 125, 177201\n(2020).]These results are also consistent with our calculation results, hence we believe\nthe dominant origin of spin orbital torque in our system is from OREE.\nExperimentally disentangling OAM and SAM contributions is highly nontrivial once\nspin–orbit coupling is present, since OAM and SAM are no longer independent\nquantities. Orbital angular momentum is generally can only be accessed directly\nthrough approaches such as orbital pumping [Nat Electron 7, 646–652 (2024)] and\norbital MOKE [Nature 619, 52–56 (2023)], which are beyond the scope of the present\nwork. While SARPES can directly probe spin textures in momentum space, it does not\nprovide a direct measurement of orbital angular momentum. Moreover, SARPES are\nparticularly challenging for non-in-situ grown thin film.\n5. Clarify whether OAM generation by OREE produces an OAM current, and specify\nits direction and polarization relative to the Ni magnetization.\nResponse: We thank the reviewer for raising this important point and giving us the\nopportunity to clarify the nature of orbital angular momentum (OAM) generation in our\nsystem. Strictly speaking, we do not consider the orbital Rashba–Edelstein effect\n(OREE) to directly generate an orbital angular momentum current in the same sense as\nthe orbital Hall effect. Instead, in the present CuO/Ni heterostructure, the OREE leads\nto an interfacial orbital angular momentum accumulation induced by the applied\nelectric field. To avoid ambiguity, we have revised the manuscript and replaced\nprevious descriptions of an “OAM current” with the more precise term “orbital\naccumulation.”\nThe accumulated orbital angular momentum at interface can be transferred across the\nCuO/Ni interface and injected into the Ni layer. Following the reviewer’s suggestion,\nwe further clarify the geometric relation between the electric field, the direction of\norbital accumulation transfer, and the orbital polarization. When the electric field is\napplied along the 45° direction defined in the manuscript, the sign of OREE in CuO is\npositive, as confirmed by our first-principles calculations. In this configuration, the\ndirection of orbital accumulation transfer , the orbital polarization , and the\nelectric field form a mutually orthogonal vector triad, consistent with ,\n𝐽𝐽𝑂𝑂 𝑝𝑝𝑂𝑂\nas illustrated in Fig. R3. Importantly, it should be stressed that the direction of the\n𝐽𝐽𝑐𝑐 𝑝𝑝𝑂𝑂 ∥ 𝐽𝐽𝑐𝑐 ×𝐽𝐽𝑂𝑂\norbital accumulation transfer and the orbital polarization are independent of the Ni\nmagnetization. The magnetic layer enters only at the subsequent conversion stage:\nwithin the Ni layer, the injected orbital polarization is converted into spin polarization\nvia spin–orbit coupling, with the sign of the conversion determined by the sign of the\nSOC in Ni. In Ni, where the spin–orbit coupling is positive, the resulting spin\npolarization is aligned with the orbital polarization . Consequently, the direction\nof the damping-like torque is governed by the conventional spin–orbit torque symmetry\n𝑆𝑆 𝑂𝑂\nand depends on the magnetization direction, , satisfying as\nschematically shown in Fig. R3.\n𝑀𝑀 𝜏𝜏𝐷𝐷𝐷𝐷𝐷𝐷 ∥ −𝑀𝑀×(𝑀𝑀×𝑆𝑆)\nFig. R3 schematic of the relative directions for orbital polarization, electrical field\nmagnetization of Ni and direction of damping-like torque\n* Current-induced switching:\n*6 Data gaps from $-10$ mA to $+10$ mA (Fig. 3) and points outside the hysteresis\ncurve need explanation.\nResponse: We thank the reviewer for pointing out these issues and for giving us the\nopportunity to clarify them. Regarding the data gap between −10 mA and +10 mA in\nFig. 3, we emphasize that the measurements were performed continuously over the full\ncurrent range. In response to the reviewer’s comment, we have now revised the\nmanuscript and included the complete data set in the revised Fig. 3 to avoid any\nambiguity. The data in the low-current region were originally omitted solely to improve\nthe visual clarity of the switching behavior. The dashed line is only for guidance; we\napologize for the misguiding.\n7. Ensure the CuO crystal structure remains unaffected by Pt; has this been verified by\nXRD or LEED?\nResponse: We thank the reviewer for raising concern regarding the possible influence\nof the Pt layer on the CuO crystal structure.\nFrom the perspective of thin-film growth kinetics, the Pt layer in our samples is\ndeposited by sputtering only after the CuO film has been fully cooled down to room\ntemperature. Under such temperature conditions, CuO does not undergo\nrecrystallization or structural reconstruction. While sputtering and the vacuum\nenvironment may slightly affect the interfacial oxidation state locally, they are not\nexpected to modify the crystal structure or symmetry of the CuO film, which are\ndetermined during the high-temperature growth stage. Moreover, experimentally, we\nnote that all structural characterizations presented in the main text were performed on\nCuO films without a Pt capping layer. To further provide a direct experimental\nverification, we have prepared additional CuO/Pt samples and compared their structural\nproperties with those of uncapped CuO films. As shown in Fig. R4a, the θ–2θ XRD\npeaks of both pure CuO and CuO/Pt samples are located at 38.3°, and no noticeable\nchange in the half width is observed. This indicates that the Pt deposition does not affect\nthe lattice structure of the CuO film. We added this discussion to S5 in the revised\nsupporting materials.\nTo further clarify the four-fold symmetry of the CuO crystal structure, we additionally\nperformed XRD pole figure measurements. A clear four-fold rotational symmetry is\nobserved (highlighted by the yellow markers) as shown in Fig.R4b, consistent with the\nφ-scan results discussed previously. In addition, two mirror planes along the MgO [110]\nand [−110] directions are also observed. These mirror symmetries do not change the\nfour-fold rotational symmetry relevant to our analysis; nevertheless, this structural\nfeature has been explicitly included in our subsequent theoretical calculations (Please\nsee our response to Reviewer #1’s 3rd, 10th, and 12th comments).\nFig. R4 The XRD results for CuO structure. (a) comparison of XRD results for CuO\nsample (black line) and CuO/Pt sample (red line). The peak of the CuO layer remains\nthe same. (b) Pole figure of the MgO/CuO. The red square is MgO <110> and the\nyellow squares are from CuO indicating a 4-fold rotational symmetry.\n8. Causality vs. consistency: Statements linking the four-fold anisotropy of $\\xi$ to CuO\ncrystal symmetry are strong; consistency is not proof of causality and should be\ncarefully justified.\nResponse: We agree with the reviewer that consistency alone does not constitute proof\nof causality, and we appreciate the opportunity to further clarify the causal role of\ncrystal symmetry in the observed four-fold anisotropy of orbital angular momentum\n(OAM). The orbital Rashba–Edelstein effect originates from current-induced orbital\nhybridization, which is explicitly constrained by the crystal symmetry. In crystalline\nCuO, the four-fold rotational symmetry dictates the allowed orbital hopping paths and\norbital textures on the Fermi surface. Our first-principles calculations directly\ndemonstrate that this crystal-field-induced orbital anisotropy leads to a four-fold\nangular dependence of the orbital Rashba susceptibility, providing a clear physical\nmechanism linking crystal symmetry to the anisotropic OAM response. Moreover, we\nexclude alternative source of anisotropy from Ni layer. Our Ni layer is polycrystalline\nand exhibits no in-plane angular dependence on its magnetic or transport properties.\nThis part of discussion is added in the supplementary section S7. Therefore, these\nresults establish a physically well-founded causal link between the crystal symmetry of\nCuO and the observed anisotropic orbital Rashba–Edelstein effect.\nTo further strengthen the reliability of our conclusions, we have also reanalyzed the\nharmonic Hall data by including error bars corresponding to the 95% confidence\nintervals of the damping-like torque obtained from the fitting procedure, as shown in\nFig. R5. These error bars confirm that the anisotropic behavior observed in crystalline\nCuO is statistically significant and not a consequence of fitting uncertainty. The\ndamping-like torque includes error bars are added in the supplementary section S6.\nFig. R5 Harmonic Hall voltage measurement results with error bars for (a)\nCuO/Ni(8nm) and (b) CuO/Ni(10nm).\n* Theoretical calculations:\n9. Why is OAM much larger than SAM (in absolute value, Fig. 4)?\nResponse: The main reason that the OAM is larger than SAM is the weak spin-orbit\ncoupling (SOC) of Cu and O atoms. The OREE could be generated even in the absence\nof SOC, while the SREE requires the SOC [S. R. Park et al. PRL 107, 156803 (2011)].\nThe SOC converts the OAM to SAM so that the OREE is converted into the SREE.\nIf the SOC is weak, the OAM is barely converted into the SAM. Cu and O atoms are light\nelements, so they exhibit small SOC. Thus, the SAM is much smaller than the OAM.\n10. Explain the necessity of the averaging procedure in the Supplemental Material;\ncould selecting and along and be sufficient?\n∘ ∘\n𝑣𝑣 𝛼𝛼 = 0 45\nFig. R6. (a) Crystalline structure of an CuO (111) film deposited on MgO (001) film,\nblue and red lines indicate mirror symmetries of the CuO (111) film. (b) When a charge\ncurrent is applied along direction, each CuO part A-D undergoes the charge current\nalong , , , and , respectively.\n𝛂𝛂\n𝛂𝛂 𝛂𝛂+𝟗𝟗𝟗𝟗 ° 𝛂𝛂+𝟏𝟏𝟏𝟏𝟗𝟗 ° 𝛂𝛂+𝟐𝟐𝟐𝟐𝟗𝟗 °\nResponse: For simple explanation, we describe the crystalline structure of CuO (111)\nfilms deposited on MgO (001) film in Fig. R6. A CuO (111) film has no rotational\nsymmetry, but MgO (001) film has four-fold rotational symmetry with two mirror\nsymmetries. Thus, a CuO (111) film with its mirror counterpart are aligned on MgO\n(001) film with conserving the four-fold rotational symmetry [Fig. R9a].\nSuppose that a charge current is applied onto this structure along the direction\n. Since the CuO film is deposited along four-fold rotational symmetry,\neach CuO part undergoes the charge current with the direction , , ,\ncos α 𝑥𝑥 +sin α 𝑦𝑦\nand , respectively [Fig. R6b]. Each film generates the OREE and the SREE\nα α+90 ° α+180 °\nfrom these different directions of the charge current. Note that the CuO (111) film is so\nα+270 °\nanisotropic that the OREE and the SREE strongly depends on the direction of the charge\ncurrent. Thus, the OREE and the SREE response from each CuO film are different from\neach other. Thus, we have to consider all of these contributions from each CuO part and\naverage these responses to obtain the total OREE and the SREE.\nIn the previous manuscript, we plotted the REE response only for and ,\nsince the CuO (111) film has mirror symmetries along these directions∘. As you∘r\nα = 𝟗𝟗 𝟒𝟒𝟒𝟒\ncomment, the angular dependence of the REE response could become clear by plotting\nthe results with denser choices of α. For different α, we consider the REE contribution\nfrom the mirror counterpart of the CuO part, since the CuO (111) film has the mirror\nsymmetries only along and .\n∘ ∘\nα = 𝟗𝟗 𝟒𝟒𝟒𝟒\nFig. R7. The total REE response with respect to the charge current direction .\n𝛂𝛂\nWe plot the charge current direction dependence of the OREE, the SREE, and the total\nREE responses in Fig, R7. The angular dependence of the REE response is well-\nmatched to the experimental results in Fig. 4e; the total REE has the sign change from\nnegative at to positive at .\n∘ ∘\nα = 𝟗𝟗 α = 𝟒𝟒𝟒𝟒\n11. Reference energies in Figs. S1, S2, and Fig. 4 seem inconsistent; consider using a\nsingle reference, e.g., the VBM.\nResponse: We thank the first referee for this important comment. We have revised Fig4\nfor clarity and consistent with Fig. S1. Fig. S2 represents the OREE and the SREE in\nthe wider energy range, which is the same as Fig. 4. Thus, we have replaced Fig. S2\nwith the OREE, the SREE, and the total REE responses with respect to the various\nangle .\nα\n12. Did direct calculations S = ψ | S | ψ and L = ψ | r × p | ψ yield the same\ntrends as the Berry-curvature-type approach?\n⟨⟩ ⟨ ⟩ ⟨⟩ ⟨ ⟩\nResponse: In non-centrosymmetric system, the Berry curvature\ncan be accumulated on the Fermi surface like as the OAM or the spin REE. This\n〈Ω〉 = ⟨𝜕𝜕𝜕𝜕|×|𝜕𝜕𝜕𝜕⟩\naccumulation is the same as the Berry curvature dipole [I. Sodemann et al. PRL 115,\n216806 (2015)]. Thus, we calculate the Berry curvature accumulation on the CuO (111)\nfilm with the same calculation method for the OAM and the spin.\nFig. R8. Berry curvature accumulation on the CuO (111) film with respect to the\n𝑶𝑶\nFermi energy for (red) and 𝝌𝝌 𝒚𝒚(𝒛𝒛blue) (a) and the charge current direction (b).\n∘ ∘\n𝜶𝜶 = 𝟗𝟗 𝟒𝟒𝟒𝟒\nFig. R8 shows the Berry curvature accumulation on the CuO (111) film. In Fig. R8a,\nwe plot the Berry curvature accumulation with respect to the Fermi energy for\n(red) and (blue), respectively. Like the OREE and the SREE, the Berry curvature∘\n𝛼𝛼 = 0\nis accumulat∘ed with the opposite signs for and . The Berry curvature\n45\naccumulation for is six times larger tha∘n ∘. We plot the angular\n𝛼𝛼 = 0 45\ndependence of the Berry c∘urvature accumulation in Fig R7b. ∘The angular dependence\nα = 45 𝛼𝛼 = 0\nof the Berry curvature accumulation has the same trends as the OREE and the SREE\nsusceptibilities. Thus, the direct calculation of the OAM and the spin yield the same\ntrends as the Berry curvature accumulation.\n# Minor Comments\n13. Terminology: Clarify what is meant by “unconventional four-fold symmetry” (page\n2).\nResponse: We want to use the “unconventional” to highlight the difference with the\nprevious “conventional isotropic SOT”. To avoid misunderstanding for readers, we\ndelete the “unconventional” in the manuscript.\n* Sample preparation and characterization:\n14. Substrate description is inconsistent: text states MgO(110) (page 3), Figure 1\ncaption states MgO(001).\nResponse: Thanks for the attentive comment, the typo is revised.\n15. Fig. 1b (top view of CuO): four-fold symmetry is not clearly visible. LEED imaging\nis recommended to confirm symmetry.\nResponse: The CuO in Figure 1b is the single domain of the CuO, and it is a schematic\nof the growing mode for CuO thin film on MgO (001) substrate.\n* Measurement details:\n16. How is the current angle α varied in steps of in the Hall bar?\n°\nvarious MgO orientations with 15° steps. For clarity, we added the related explanation\nin the method section.\n17. Explain the connection between and OREE.\n𝐵𝐵𝐷𝐷𝐷𝐷𝐷𝐷\nResponse: OREE generates orbital angular momentum and diffuses into the Ni layer.\nWith SOC in the Ni, orbital angular momentum converts into spin angular momentum\nand applies a damping-like torque on Ni through SOT process. Thus, in second\nharmonic Hall voltage measurement, represent the total torque which originates\n𝐵𝐵DLT\nfrom both OREE and SREE.\n𝐵𝐵DLT\nReviewer #2 (Remarks to the Author):\nIn this work, the authors report the observation of a four-fold anisotropy in the Rashba\nEdelstein effect in single-crystalline CuO thin films. By combining with the first-\nprinciple calculations, the authors attribute the anisotropic Rashba Edelstein effect to\nthe orbital transport. Research on orbital transport is pivotal, as it paves the way for\nthe development of orbitronics. However, the distinction between spin and orbital\ncontributions in this work relies exclusively on theoretical calculations. Experimental\nvalidation or corroborating evidence is needed to firmly establish this interpretation.\nBelow I list several questions.\nResponse: We thank the referee’s feedback for our work. We addressed the referee’s\nconcerns in the following.\n1. The XRD θ-2θ scan of the CuO thin film exhibits a broad peak, indicating reduced\ncrystalline quality that may be attributed to sample inhomogeneity and small grain size.\nIt is essential to discuss whether these structural imperfections could influence the\nobserved anisotropic Rashba-Edelstein effect, as they might affect spin/orbital\ntransport or torque generation mechanisms.\nResponse: We thank the second referee for this comment. We agree that the θ–2θ peak\nwidth can be influenced by factors such as finite coherence length and structural\ninhomogeneity. In our case, the θ–2θ peak mainly reflects the finite out-of-plane\ncoherent length associated with the limited thin film thickness. Therefore, the broad\npeak does not indicate imperfections of the crystalline structure. Here we show a\nschematic of the θ-2θ peaks with different thickness thin film sample according to\nScherrer equation in Figure R9a. When the thickness decreases, the peak tends to be\nmore broadening.\nMoreover, the crystalline quality of CuO thin film is further confirmed by the rocking\ncurve in Figure R9b. The sharp rocking curve indicates good out-of-plane alignment,\nand suggests the long-range crystalline orientation is well preserved despite the finite\nthickness induced θ–2θ peak broadening. Hence, the broadening θ-2θ peak is mainly\ndue to the thickness not the imperfection of the structure. The structural imperfection\naffects the spin/orbital transport or torque generation mechanisms does not exist in our\nCuO thin film.\nFig. R9. (a) Schematics of the θ-2θ scan for thin film samples with different thickness.\n(b) Rocking curve of CuO thin film sample.\n2. The authors report a four-fold anisotropy in the current-induced in-plane damping-\nlike torque. Their first-principles calculations attribute this symmetry-dependent torque\nprimarily to the anisotropic orbital Rashba-Edelstein effect, in contrast to the expected\nisotropy of the spin Rashba-Edelstein effect. However, the experimental methodology\nfor disentangling the individual contributions from spin and orbital moments remains\nelusive. It would be valuable if the authors could address this point, perhaps by\nproposing an experimental approach to clearly distinguish the spin and orbital\ncontributions.\nResponse: We thank the second referee for this valuable comment. We agree that the\nmeasuring damping-like torque alone cannot distinguish the contribution of spin\nRashba-Edelstein effect or orbital Rashba-Edelstein effect. Currently, the distinction\nbetween spin- and orbital-originated torques is therefore typically made based on their\ndifferent material dependences and conversion pathways. Orbital torques rely on orbital\ncurrent generation followed by orbital-to-spin conversion, whose efficiency depends\nsensitively on the spin–orbit coupling strength of the adjacent ferromagnetic or\nconversion layer. In contrast, spin Rashba–Edelstein torques are expected to be\ncomparatively insensitive to this factor. Experimentally, an indirect approach to further\ndistinguish spin and orbital contributions is to vary the ferromagnetic layer, as the\nefficiency of orbital-to-spin angular momentum conversion depends on the spin–orbit\ncoupling of the ferromagnet [Lee D, et al. Nature communications 12, 6710 (2021)].\nIn our work CuO exhibits relatively weak spin–orbit coupling, this has been proved by\nprevious work [Go, Dongwook, et al. Physical Review B 103.12 (2021), Ding S, et al.\nPhysical Review Letters 125, 177201 (2020).] and our first-principles calculation (refer\nto our response for Reviewer 1 comment 9th). Therefore, the main contribution to the\ndamping-like torque is the orbital angular momentum generated from OREE.\n3. As shown in Fig. 2e, the current-induced in-plane damping-like torque exhibits an\napproximate, but not perfect, four-fold symmetry. Notably, the values at α = 45° and\n315° are not equivalent. What could account for this deviation from the ideal symmetry?\nResponse: We thank the second referee for this question. As the referee points out, the\ndamping-like torque data show small deviations from an ideal four-fold symmetry, and\nthe values at α = 45° and 315° are not exactly identical.\nTo better reflect these uncertainties, we have revised Fig. 2e by adding error bars and\nshown in Fig S7a. Importantly, these minor deviations do not affect the main conclusion\nof our manuscript: within experimental uncertainty, the damping-like torque exhibits a\nclear dominant four-fold symmetry that follows the crystal symmetry of CuO.\n4. The current-induced magnetic switching is presented in Fig. 3 for two devices with\nα angles of 0° and 45°. Perpendicular switching was achieved in both cases through\nthe application of an external magnetic field. It should be noted that the direction of the\napplied field differed between the two devices (along 0° and 45°, respectively). The\neffect of the magnetic field orientation, therefore, merits a systematic investigation.\nResponse: We thank the second referee for this comment. We agree that the applied\nmagnetic field can influence current-induced magnetic switching. Firstly, an external\nmagnetic field may directly affect the generation of spin or orbital angular momentum,\nfor example by modifying the spin Hall angle (SHA) or orbital Hall angle (OHA).\nExisting reports on magnetic-field-induced changes of SHA or OHA are predominantly\nfound in magnetic materials, where the magnetic field directly couples to the magnetic\norder [Zayets, Vadym, and Andrey S. Mishchenko. Physical Review B 102.10 (2020):\n100404.]. In contrast, CuO in our study is nonmagnetic at room temperature, the\ngeneration of OREE is based on the hybridization between O p states and Cu d states\n[Go, Dongwook, et al. Physical Review B 103.12 (2021): L121113.], which can hardly\naffected by the external magnetic field.\nSecondly, the magnetic field can influence the switching process of the magnetic layer\nby affecting domain-wall configuration. During current-induced switching, the domain-\nwall orientation is determined by the direction of the applied in-plane magnetic field,\nand together with the spin (or orbital) polarization, it sets the switching polarity. In our\nexperimental setup, however, the positive in-plane magnetic field is always applied\ncollinearly with the current direction for all device geometries. As a result, the domain-\nwall chirality during the switching process is expected to be identical for devices\nmeasured at α = 0° and 45°. Under this condition, the switching polarity should be the\nsame if it were determined solely by the magnetic field.\nThe experimentally observed reversal of switching polarity between α = 0° and 45°\ntherefore cannot be attributed to the orientation of the applied magnetic field. Instead,\nit reflects a reversal of the current-induced angular-momentum polarization, consistent\nwith an intrinsic change of the orbital angular momentum associated with crystal-\nsymmetry-dependent OAM generation. Based on these considerations, we conclude\nthat the effect of magnetic field orientation does not merit a systematic investigation in\nthe context of the present work.\n5. Unlike the perpendicular switching in Fig. 3, which requires external magnetic field\nassistance, the CuO thin film may be exploited to achieve field-free in-plane magnetic\nswitching. This approach completely obviates the need for an external magnetic field,\nthus circumventing related effects.\nResponse: We thank the second referee for this valuable comment. We agree that\nachieving field-free in-plane magnetic switching can be a possible experimental\napproach to prove the generation of opposite direction of orbital torque. However, as\nwe discussed in comment 4, our perpendicular magnetic anisotropy devices provide a\nclear and well-established platform for this purpose, where the switching polarity\ndirectly reflects the sign of the damping-like torque.\n6. In first-principle calculations, a parameter *c* is introduced to represent the orbital-\nto-spin conversion coefficient. In my opinion, the value of *c* should depend on the\nmaterial used as the orbital-to-spin conversion layer (e.g., Pt or Ni). Is this correct?\nResponse: We thank the second referee for this valuable question. As the referee has\npointed out, the orbital-to-spin conversion coefficient *c* depends on materials. This\ncoefficient mainly depends on the spin-orbit coupling strength of the material [D. Lee\net al. Nat. Commun. 12, 6710 (2021)]. Materials with small spin-orbit coupling such as\nFe or Co have small orbital-to-spin conversion coefficients. In contrast, the materials\nwith large spin-orbit coupling such as Ni or Pt have large orbital-to-spin conversion\ncoefficients.\nIn the revised manuscript, we re-evaluate the OREE and the SREE response considering\nthe angular dependence of the longitudinal charge conductivity, since the anisotropic\nCuO (111) film has anisotropic longitudinal charge conductivity. Through the updated\ncalculation, we found that the material dependence of the orbital-to-spin conversion\ncoefficient does not affect the sign change of the REE response of CuO film. In the\nupdated Fig. 4, both the OREE and SREE are anisotropic so that they have the opposite\nsigns for and . That is, the total REE response also has the same trend like\nas the OREE and∘ the SRE∘E. The orbital-to-spin conversion coefficient only affects the\n𝛼𝛼 = 0 45\nrelative intensities of the REE response for and . This first-principles\ncalculation supports the observation of the sig∘n change∘ of CuO/Pt/[Co/Ni]\nn\n𝛼𝛼 = 0 45\nheterostructure [Fig. 3] from to , even if the orbital-to-spin conversion\nlayer is replaced by Pt. ∘ ∘\n𝛼𝛼 = 0 45\n7. Have the authors considered the potential influence of the substrate? The observed\nRashba-Edelstein effect in this study may also originate from the interface between the\nCuO film and the MgO substrate, analogous to the interfacial effect reported in a\nprevious work (e.g., Adv. Mater. 34, 2109449 (2022)). It would be important to include\na discussion on this aspect to better clarify the origin of the observed phenomena.\nResponse: We thank the second referee for this point. We agree that the substrate can\ninfluence spin–orbit-related responses, primarily by affecting the crystalline symmetry,\nepitaxial orientation, or strain state of the film.\nThe interfacial Rashba–Edelstein effect reported in Adv. Mater. 34, 2109449 (2022) is\ndiscussed in STO/SRO/h-BN. In that work, the Rashba effect is generated at the SRO/h-\nBN interface and directly acts on the adjacent magnetic SRO layer, leading to\nmagnetization switching. The role of the substrate in that system is indirect and does\nnot serve as the source of the Rashba effect.\nIn our CuO/Ni heterostructures, even if an orbital Rashba–Edelstein effect presents at\nthe MgO/CuO interface, it would be spatially separated from the Ni layer and therefore\ncannot directly contribute to the torque acting on Ni. The observed switching and torque\nresponses are instead governed by the angular-momentum generation within the CuO\nlayer and its transfer across the CuO/Ni interface.\nWe therefore conclude that while the substrate may influence the crystalline properties\nof CuO, it is not the primary origin of the observed Rashba–Edelstein effect or the\ncrystal-symmetry-dependent torque reported in this work. (also refer to our response to\nReviewer 1 8th comment).\nReviewer #3 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports.\nThis is part of the Nature Communications initiative to facilitate training in peer review\nand to provide appropriate recognition for Early Career Researchers who co-review\nmanuscripts.\nResponse: We thank the third referee for participating in the peer-review process and\nfor their contribution.\nReviewer #4 (Remarks to the Author):\nR. Xiao et al. presents a compelling study on the crystal-symmetry-dependent Orbital\nRashba-Edelstein Effect (OREE) in epitaxial CuO thin films. The authors demonstrate,\nfor the first time, that orbital angular momentum generation can be directly governed\nby the underlying crystal symmetry, leading to a four-fold anisotropy and sign reversal\nof the spin–orbit torque efficiency. The work is highly innovative, filling a key gap in\ncurrent orbitronic research where OREE has been treated as isotropic. Overall, the\nresults are novel and technically rigorous. Before recommending publication, I would\nsuggest that the authors address a few minor comments for further improvement.\nResponse: We thank the referee’s positive feedback for our work. We addressed the\nreferee’s concerns in the following.\n1. The authors prepared sample as Ni over CuO layer. The properties of the Ni layer\nmight change due to oxidization and induce angular dependence. Did the authors rule\nout this possibility?\nResponse: We thank the fourth referee for this important question. We agree that it is\nnecessary to exclude the possibility that the Ni layer itself introduces angular-dependent\neffects. In our samples, the Ni layer is deposited at room temperature and does not grow\nepitaxially on CuO, resulting in a polycrystalline film without a well-defined in-plane\ncrystalline symmetry. Under such conditions, the intrinsic magnetic and transport\nproperties of Ni are expected to be isotropic in the film plane. This isotropy is confirmed\nby angle-dependent anomalous Hall effect (AHE) performed on multiple devices,\nwhich show no detectable angular dependence within experimental uncertainty as\nshown in Fig.R10. These results indicate that neither oxidation nor structural variations\nin the Ni layer induce a measurable angular anisotropy. We therefore conclude that the\nobserved four-fold anisotropy originates from the CuO layer and its crystal symmetry,\nrather than from the Ni layer. We added this discussion to S7 in the revised supporting\nmaterials.\nFig. R10. AHE amplitude of devices with different fabricated angles.\n2. Discussing the efficiency of CuO OREE, the author demonstrates the efficiency of\nOREE by using an effective field per current density. It would be better if they provide\nor discuss a comparison between the CuO OREE efficiency and the spin–orbit torque\nefficiency of Pt, to better benchmark the effectiveness of CuO as an orbital torque\nsource.\nResponse: We thank the fourth referee for this valuable question. As the referee has\npointed out, OREE efficiency is important for further applications for this phenomenon.\nIn previous work, the damping-like torque efficiency is comparable to Pt [An, Hongyu,\net al. Nature communications 7.1 (2016): 13069.]. The naturally oxidized CuO is\nx\nconductive in previous work, but our CuO thin film is an insulator. In our CuO/Ni\nheterostructure, the applied current is strongly shunted by the Ni layer, so that the OREE\noriginates from the interfacial electric field rather than bulk current flow in CuO,\nmaking a direct comparison of spin–orbit torque efficiency with conductive CuO or Pt\nx\nsystems not meaningful; the main focus of this work is therefore the unique crystal-\nsymmetry-governed anisotropy of the OREE rather than its absolute strength.\n3. The authors mentioned this finding can be applied to advanced spintronic devices,\ncan the author elaborate on a specific picture or scenario of how the anisotropic OREE\ncould be utilized in practical applications\nResponse: We thank the fourth referee for this valuable question. Since the anisotropic\norbital torque can be generated from the crystalline CuO/FM structure. The logic device\ncan refer to Zhao, T., Zheng, Z., Wang, J. et al. Nat Commun 16, 2988 (2025). In\nneuromorphic computing, unlike conventional approaches [Marrows, Christopher H.,\net al. npj Spintronics 2.1 (2024): 12.] that rely on layer thickness or current amplitude\nto modulate weights, crystal-symmetry-controlled anisotropic OREE offers a route to\nencode direction-dependent and nonvolatile weight states within a single device.\nMinor comments:\n1. The manuscript refers to “Ref. 38” when discussing the LaAlO₃/SrTiO₃ interface\nOREE, but this reference number does not appear in the reference list. Please verify\nand correct the numbering or add the missing citation to ensure consistency.\nResponse: Thanks for the detailed comment, the arrangement of reference is updated.\n2. Throughout the manuscript, the term “first-principal calculation” is repeatedly used.\nThe correct expression should be “first-principles calculation”.\nResponse: Thanks for the detailed comment, the typo is corrected.\nReviewer #5 (Remarks to the Author):\nRui Xiao et al. investigated the crystal symmetry dependent orbital Rashba Edelstein\neffect in four-fold crystalline CuO/Ni heterostructure. Moreover, the crystal-symmetry-\ndependent spin torque efficiency sign reversal was obtained in the heterostructure.\nCrystal symmetry dependent orbital Rashba Edelstein effect provides an interesting\nway to control orbital current and improve torque efficiency in orbitronic devices.\nBefore recommending its acceptance, several issues need to be clarified as following:\nResponse: We thank the referee’s positive feedback for our work. We addressed the\nreferee’s concerns in the following.\n1) The generation of orbital current inducing magnetization switching in CuO-based\nheterostructures was reported through orbital Rashba Edelstein effect at CuO/Cu\ninterface (such as PRL 2020, 125 177201; APL 2022, 121 072404; Adv. Mater. 2023,\n35, 2300858). In this manuscript, the authors reported that the orbital Rashba Edelstein\neffect in a CuO/Ni film without CuO/Cu interface. How to explain the source of orbital\nRashba Edelstein effect?\nResponse: We thank the fifth referee for this valuable question. In previous studies, the\nCuO layers were typically formed by natural oxidation of Cu in ambient conditions.\nx\nAs a result, the system consists of a CuO layer with a gradual oxygen concentration.\nx\nIn our manuscript, the source of the orbital Rashba–Edelstein effect is the intrinsic\norbital hybridization between different orbitals in crystalline CuO, which is governed\nby its crystal symmetry. This current-induced orbital hybridization leads to an\naccumulation of orbital angular momentum at the CuO/Ni interface. The generated\norbital angular momentum is subsequently converted into a spin current through the\nfinite spin–orbit coupling provided by the Ni layer, giving rise to the observed spin–\norbit torque. Therefore, the orbital Rashba–Edelstein effect reported here originates\nfrom the orbital hybridization within the CuO layer.\n2) In the case of the CuO/Ni film, the Ni film is single crystal, polycrystalline, or\namorphous? The crystal-symmetry-dependent orbital Rashba Edelstein effect may be\nrelated to the crystal structure of Ni. How to rule out this possibility?\nResponse: We thank the fifth referee for this important question. In our CuO/Ni\nheterostructures, the Ni layer is deposited at room temperature and does not grow\nepitaxially on CuO. Under these conditions, the Ni film is expected to be polycrystalline,\nwithout a well-defined crystalline symmetry. Experimentally, no diffraction peak from\nthe Ni layer is observed in the XRD θ–2θ scan (Fig.R11), which is consistent with a\nthin (≈8 nm), polycrystalline film lacking long-range crystallographic order rather than\na single-crystalline or textured Ni layer. As a result, the Ni layer cannot impose a\nspecific crystal symmetry that would give rise to a four-fold anisotropic response.\nMoreover, self-induced spin–orbit torques originating from the ferromagnetic layer\nitself generally require much thicker magnetic layers (on the order of tens of nanometers)\nto become significant. In our devices, the Ni thickness is below 10 nm, rendering any\nNi self-torque contribution negligible. We therefore conclude that the observed crystal-\nsymmetry-dependent orbital Rashba–Edelstein effect does not originate from the Ni\nlayer, but from the CuO layer and its intrinsic crystal symmetry.\nFig. R11. XRD results for CuO/Ni samples. 2θ scans from 30° to 60°.\n3) In Fig. 3b, c, the Rxy data are quite sparse, could the authors have lost some of them?\nResponse: We thank the fifth referee for this comment. For directly showing the\nswitching polarity, the currents where the magnetization remaining stable are not shown.\nIn response to the reviewer’s comment, we have revised the manuscript and included\nthe complete data set in the revised Fig. 3 to avoid any ambiguity. The data in the low-\ncurrent region were originally omitted solely to improve the visual clarity of the\nswitching behavior.\n4) Breaking the mirror symmetry can realize the current-driven magnetization\nswitching without magnetic field. This artcle fucus on the crystal symmetry dependent\nswitching, thus the authors should provide field-free current-induced magnetization\nswitching, which will improve the quality of the manuscript.\nResponse: We thank the fifth referee for this comment. We agree that field-free current-\ninduced magnetization switching can be achieved by deliberately breaking mirror\nsymmetry in appropriate material systems. However, the CuO-based heterostructures\nstudied in this work preserve mirror symmetry and are therefore not expected to support\ndeterministic field-free switching. The focus of the present manuscript is to investigate\nthe crystal-symmetry-dependent anisotropy and sign reversal of the orbital Rashba–\nEdelstein effect, rather than to engineer field-free switching geometries. Introducing\nadditional symmetry breaking to realize field-free switching would go beyond the scope\nof this study and is not required to support our main conclusions.\n5) The spin torque efficiency of CuO/Ni(10nm) sample (Fig. S5) is markedly improved\ncompared to the CuO/Ni(8nm) (Fig. 2e). Please elaborate on this difference.\nResponse: We thank the fifth referee for this valuable question. We apologize for this\ntypo error; the correct data is revised in Fig. 2e. And it turns out the efficiency between\nthese two samples is almost the same.\n6) Regarding the CuO/Pt[CoNi]n with 2 nm Pt, I suggest to add some discussions about\nthinner Pt (such as 1 nm) to reduce spin current from the spin Hall effect.\nResponse: We thank the fifth referee for this suggestion. We agree that reducing the Pt\nthickness is an effective way to suppress spin Hall current contributions. In our devices,\nhowever, a Pt thickness of approximately 2 nm is intentionally chosen, as it serves both\nas a suitable seed layer to establish perpendicular magnetic anisotropy in the\nferromagnetic multilayer and as an efficient orbital-to-spin angular momentum\nconversion layer. Previous studies have shown that an ultrathin Pt layer around 2 nm\ncan significantly enhance orbital torque efficiency by providing strong spin–orbit\ncoupling for orbital-to-spin conversion [Ding et al., Phys. Rev. Lett. 125, 177201\n(2020)]. In contrast, when the Pt thickness is reduced to ~1 nm under our experimental\nconditions, the adjacent ferromagnetic layer does not exhibit stable perpendicular\nmagnetic anisotropy, preventing reliable current-induced switching measurements.\nTherefore, thinner Pt layers do not provide a meaningful comparison in the present\ndevice geometry, and the observed switching behavior cannot be attributed to a\ndominant spin Hall effect from the Pt layer."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I greatly appreciate the detailed and convincing response provided by the authors, which satisfactorily addresses my previous comments. I strongly suggest including Figures R1 and R2 (or at least panels (c), and (d) of Figure R2), together with a brief discussion as presented in the authors’ reply, in Section S1 of the Supplemental Material, as this would significantly enhance the paper’s value for the community. In light of these clarifications and improvements, it is my pleasure to recommend the paper for publication in Nature Communications.", "reviewer_2": "Although the experimental approach to clearly distinguish spin and orbital contributions remains elusive in the revised manuscript, this work on the anisotropic orbital Rashba-Edelstein effect in a crystalline CuO thin film is nevertheless timely and significant. I therefore recommend its publication in Nature Communications. Below I list certain minor comments for further improvement.\n1. The manuscript states that the epitaxial CuO thin films were deposited by PLD at a fixed temperature of 500°C. Could the authors comment on the significance of this temperature for the crystalline quality of the films? Furthermore, have films been deposited at different temperatures to explore this parameter? If so, is the observed anisotropic orbital Rashba-Edelstein effect consistently present across different deposition temperatures?\n2. Regarding comment 4 from the fifth reviewer, I agree with the authors that field‑free perpendicular switching is beyond the scope of this work. Nevertheless, achieving such switching is of significant importance in spintronics. To my knowledge, there are very few reports on field‑free perpendicular switching realized via orbital torque. Could the authors comment on whether crystal symmetry engineering could provide a possible route to achieve this effect in CuO thin films? It would be valuable to include a brief discussion on this point in the conclusion section of the main text.\n3. The incomplete switching in Fig. 3 requires clarification. Please discuss its origin and its implications for the efficiency and applicability of the observed orbital torque.\n4. While the authors have provided a reasonable explanation based on material properties and existing literature to support the dominance of the orbital contribution, this remains an indirect argument. To strengthen the claim of an anisotropic orbital Rashba-Edelstein effect more directly, could the authors suggest a feasible experimental strategy or measurable signature—beyond varying the ferromagnetic layer—that could unambiguously differentiate an anisotropic orbital torque from a potentially anisotropic spin torque in future studies? For instance, are there predictions regarding distinct dynamic responses between the two mechanisms that could be targeted in future advanced measurements?", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "The authors have comprehensively addressed all of my concerns. I am pleased to recommend this paper for publication.", "reviewer_5": "The authors have addressed properly my concerns. I now recommend the acceptance of the paper."}, "author_response": "Response to reviewers’ comments\nThe referee’s comments are quoted in blue and our responses in black. Revision\nrelated parts are highlighted.\nReviewer #1 (Remarks to the Author):\nI greatly appreciate the detailed and convincing response provided by the authors,\nwhich satisfactorily addresses my previous comments. I strongly suggest including\nFigures R1 and R2 (or at least panels (c), and (d) of Figure R2), together with a brief\ndiscussion as presented in the authors’ reply, in Section S1 of the Supplemental Material,\nas this would significantly enhance the paper’s value for the community. In light of these\nclarifications and improvements, it is my pleasure to recommend the paper for\npublication in Nature Communications.\nResponse: We thank the referee’s positive feedback for our work. We have added\nFigures R1 and R2 with related discussion into our supplementary information.\nReviewer #2 (Remarks to the Author):\nAlthough the experimental approach to clearly distinguish spin and orbital\ncontributions remains elusive in the revised manuscript, this work on the anisotropic\norbital Rashba-Edelstein effect in a crystalline CuO thin film is nevertheless timely and\nsignificant. I therefore recommend its publication in Nature Communications. Below I\nlist certain minor comments for further improvement.\nResponse: We thank the referee’s positive feedback for our work. We addressed the\nreferee’s concerns in the following.\n1. The manuscript states that the epitaxial CuO thin films were deposited by PLD at a\nfixed temperature of 500°C. Could the authors comment on the significance of this\ntemperature for the crystalline quality of the films? Furthermore, have films been\ndeposited at different temperatures to explore this parameter? If so, is the observed\nanisotropic orbital Rashba-Edelstein effect consistently present across different\ndeposition temperatures?\nResponse: We thank the referee for this comment. The deposition temperature of\n500 °C was chosen based on prior studies on epitaxial CuO thin films, which identify\nthis temperature range as favorable for obtaining phase-pure, well-oriented CuO with\ngood crystalline quality under moderate oxygen pressures [Yang et al., J. Phys. D 50,\n035303 (2017); Kita et al., J. Cryst. Growth 115, 752–757 (1991)]. In addition, the Cu–\nO phase diagram reported by Ito et al. [J. Mater. Sci. 33, 3555–3566 (1998)] indicates\nthat conditions around 500 °C and our chosen oxygen pressure lie within the stability\nwindow of stoichiometric CuO, minimizing the formation of secondary Cu₂O or\nmetallic Cu phases. Consistent with this, our films grown at 500 °C exhibit sharp\nrocking curves and clear CuO (111) diffraction peaks, confirming high crystalline\nquality.\nWe have also prepared a SiO₂/CuO/Ni control sample, in which the CuO layer is\nfound to be amorphous. The corresponding XRD θ–2θ scans of both the SiO₂/CuO/Ni\nsample and the bare SiO₂ substrate are shown in Fig. R1.a. The strong diffraction peak\naround 69° is from the Si/SiO substrate. And the second harmonic Hall voltage\n2\nmeasurements result is shown in Fig.R1.b. As we can see, there are no obvious\nanisotropic properties of damping-like torque efficiency.\nFig. R1. a, XRD 2θ scan for SiO /CuO sample and SiO substrate. b, Damping-like\n2 2\ntorque efficiency of SiO2/CuO sample on different devices with various current\ndirections applied.\n2. Regarding comment 4 from the fifth reviewer, I agree with the authors that field free\nperpendicular switching is beyond the scope of this work. Nevertheless, achieving such\nswitching is of significant importance in spintronics. To my knowledge, there are very\nfew reports on field free perpendicular switching realized via orbital torque. Could the\nauthors comment on whether crystal symmetry engineering could provide a possible\nroute to achieve this effect in CuO thin films? It would be valuable to include a brief\ndiscussion on this point in the conclusion section of the main text.\nResponse: We thank the referee’s comment. We agree that field free perpendicular\nswitching by orbital torque is significant and gives a bright view for further application\nof orbital torque driven spintronics devices. From our perspective, crystal symmetry\nengineering could provide a viable route toward this goal in CuO-based systems.\nSpecifically, lowering the in-plane rotational symmetry of crystalline CuO may enable\nthe generation of unconventional orbital angular momentum components with an out-\nof-plane character. After orbital-to-spin conversion in the adjacent ferromagnet, such\ncomponents could in principle give rise to a field-free perpendicular damping-like\ntorque. We have added a brief discussion of this part in our conclusion.\n3. The incomplete switching in Fig. 3 requires clarification. Please discuss its origin\nand its implications for the efficiency and applicability of the observed orbital torque.\nResponse: We thank the referee for this valuable comment. The incomplete switching\nobserved in Fig. 3 can originate from various possible reasons like domain wall pinning\neffect or thermal induced demagnetization. Our switching measurements were\nperformed on Hall-bar devices with a transverse bar for electrical detection, which\nintroduces additional shape anisotropy and local pinning sites that tend to stabilize\nmulti-domain configurations during current-induced switching. This geometric\nconstraint is known to hinder complete magnetization reversal in Hall-bar structures\nand has been reported in previous studies [Zhao et al., J. Appl. Phys. 93, 7753–7755\n(2003)]. Also, Joule heating during high-current pulses can induce thermal\ndemagnetization in the CoNi layer, reducing and , promoting stochastic domain\nnucleation, and creating spatially nonuniform switching. This can hinder complete\n𝑀𝑀𝑠𝑠 𝐾𝐾eff\ndomain propagation in Hall-bar devices and lead to the observed incomplete switching\n[Wang, Xiao, et al. Science advances 5.8 (2019): eaaw8904.]. In practical devices, these\nparameters can be significantly improved by engineering the magnetic layer properties\n(e.g., coercivity, saturation magnetization) or device geometry, which is beyond the\nscope of the present study.\n4. While the authors have provided a reasonable explanation based on material\nproperties and existing literature to support the dominance of the orbital contribution,\nthis remains an indirect argument. To strengthen the claim of an anisotropic orbital\nRashba-Edelstein effect more directly, could the authors suggest a feasible\nexperimental strategy or measurable signature—beyond varying the ferromagnetic\nlayer—that could unambiguously differentiate an anisotropic orbital torque from a\npotentially anisotropic spin torque in future studies? For instance, are there predictions\nregarding distinct dynamic responses between the two mechanisms that could be\ntargeted in future advanced measurements?\nResponse: We thank the referee for this thoughtful and forward-looking question. At\npresent, the broader community has not yet established a directly electrical method that\ncan unambiguously disentangle spin and orbital torque contributions in systems with\nfinite spin–orbit coupling. This difficulty is intrinsic to most solid-state materials,\nwhere spin and orbital degrees of freedom are inherently entangled. We are aware that\nseveral emerging optical [Choi, YG., Jo, D., Ko, KH. et al. Nature 619, 52–56 (2023)]\nand THz-based techniques have recently been explored to probe orbital dynamics more\ndirectly [Xu, Y., Zhang, F., Fert, A. et al. Nat Commun 15, 2043 (2024).]. However,\nthese measurements require specialized instrumentation and analysis frameworks that\nlie outside our current experimental expertise, and we therefore refrain from offering\ndetailed technical prescriptions that we cannot fully substantiate."}, {"round": 3, "reviewer_comments": {"reviewer_2": "The authors have addressed all my concerns.I recommend its publication in Nature Communications."}, "author_response": ""}]

Decision letter