Field-free spin-orbit switching of perpendicular magnetization enabled by dislocation-induced in-plane symmetry breaking

Current induced spin-orbit torque (SOT) holds great promise for next generation magnetic-memory technology. Field-free SOT switching of perpendicular magnetization requires the breaking of in-plane symmetry, which can be artificially introduced by external magnetic field, exchange coupling or device asymmetry. Recently it has been shown that the exploitation of inherent crystal symmetry offers a simple and potentially efficient route towards field-free switching. However, applying this approach to the benchmark SOT materials such as ferromagnets and heavy metals is challenging. Here, we present a strategy to break the in-plane symmetry of Pt/Co heterostructures by designing the orientation of Burgers vectors of dislocations. We show that the lattice of Pt/Co is tilted by about 1.2° when the Burgers vector has an out-of-plane component. Consequently, a tilted magnetic easy axis is induced and can be tuned from nearly in-plane to out-of-plane, enabling the field-free SOT switching of perpendicular magnetization components at room temperature with a relatively low current density (~1011 A/m2) and excellent stability (> 104 cycles). This strategy is expected to be applicable to engineer a wide range of symmetry-related functionalities for future electronic and magnetic devices.

1. To make the Burgers vectors with an out-of-plane component a useful mechanism to introduce the tilted PMA, the authors should not only evaluate the tilting angle θe of crystalline planes near the dislocation (shown in Supplementary Figure S3 and S4) but also evaluate the density of the dislocations and make sure the mechanism work for the whole stack.In this current version, the HAADF images have showed a zoomed-in area.The wide-range distribution of the desired dislocations should be characterized or estimated, which is important for practical applications.2. The authors mentioned the structure have application potentials.For applicable SOT memory devices, magnetic tunnel junctions fabricated on Si/SiO2 substrate are needed currently.A discussion how to integrate this dislocation mechanism with the sophisticated MRAM technology is helpful.3.In Fig.4(d), why is there a partial switching loop for the blue line as I//[001] compared with 4(c)? 4. The experiments were conducted at room temperature; how about the influence of the heating effect in the SOT switching measurement?What is the switching degree by SOT relative to the Rxy in the Rxy-Hoop curve? 5. How about the damping-like and field-like torque efficiency for the system and compare the measured value with other references?6.To illustrate the influence of the Burgers vectors with an out-of-plane component (the stack grown on (110) MgO), the other samples grown on the (111) and (001) MgO substrates can be ideal control samples.Using these control samples, other entangling factors involved if any during the film or device fabrication process can be clearly ruled out.I suggest the author to conduct the same tilting angle measurement and the SOT-switching measurement for the (111) and (001) samples as well.And further check whether (1) the easy axis tilting is absent and (2) field-free switching is not achievable for the control device.7. The switching performance under different fields for example, (-1000 Oe, +1000 Oe) with a spacing of 100 Oe or even smaller can be given.In this case, readers can estimate an effective inplane field introduced by the tilted easy axis by observing the critical field condition at which SOT switching is disabled.8.It also seems strange why a small tilting angle of about 1o in the atomic structure can introduce such a large tilting PMA angle as 30o (tilted by 60o as shown in Figure 3a&e).So the characterization of the tilting angle is still puzzling for me.In many cases, we also observed a sharp switching in the Rxy vs Hin-plane curves similar with Figure 3c.However, this curve cannot guarantee a tilted PMA.I suggest the authors conduct the sample measurement of Figure 3c for the control devices on (111) and (100) substrates and check the perpendicular magnetization switching by the in-plane field is still possible or not? 9. Regarding the critical current density, it is strongly dependent on the pulse duration.How about the current pulse duration used in this experiment?If the pulse is as long as above microsecond, the current density of 3.0×107A/cm2 is a moderate value.10.There is a typo in Page 5. "tiling""tilting".11.Is there any typo for the current density of ~1015 A/m2 for the Pt/Co/AlOx system?This value seems not physical.12.In the legend of Fig. 4d "for Pt(5)/Co(0.6)/Pt(1.4)/NiO(20)/MgOheterostructure with θM~90o", I suggest a more concise value is used instead of 90o.If the angle is 90o, field-free switching is physically impossible.A more concise value smaller than 90o would appear more realistic and convincing and not misleading.
Reviewer #2: Remarks to the Author: The authors proposed a new strategy of field-free spin-orbit torque (SOT) switching by introducing in-plane symmetry breaking using dislocations with an out-of-plane Burgers vector.The out-ofplane Burgers vector induces a tilted magnetic easy axis that enables field-free SOT switching of perpendicular magnetization.The authors claimed that their method is superior to previously demonstrated field-free SOT switching ones as it can be generally applicable to various material systems.However, I disagree with this argument for the following reasons.
First, to design the orientation of Burgers vectors of dislocations, they use a single crystalline MgO substrate and a NiO underlayer, on which Pt grows hetero-epitaxially enabling control of dislocations.I wonder if this can be applied to other SOT materials such as W or Ta.Can the authors demonstrate the applicability of this method to other material systems, either experimentally or through calculation?Moreover, I suggest that the authors should discuss the merits of their method compared to already reported ones showing field-free SOT switching using tilted magnetic anisotropy [Zhi Li, ACS Appl. Electron. Mater. 2022, H. Kim, Adv. Func. Mater. 2022, L. You, PNAS 2015].
Secondly, I wonder if the magnetization is fully switched by SOT without a magnetic field.It seems odd to me that the change in Hall resistance (deltaR_xy) for the sample with theta_M=16 degrees is much larger than that for the sample with theta_M=~90 degrees.The authors should compare the field-free SOT switching results to the anomalous Hall resistance measured with perpendicular magnetic fields and show how much the magnetization is switched.I don't think it is reasonable to compare critical switching current density with other results unless full switching is achieved.

Reviewer #1:
All-electrical manipulation of perpendicular magnetization has long been an interesting topic for the last decade.Many groups have investigated the Pt/Co heterostructures as a typical PMA system.In this manuscript, it was demonstrated possible to design the orientation of the Burgers vectors of dislocations using different MgO substrates with various orientations.Furthermore, the proper Burgers vector on MgO (110) introduced a tilted PMA easy axis for the Pt/Co system, which assisted the desired field-free SOT switching for the system.This work is physically interesting and may be influential in providing a new proposal to realize practical SOT devices.The layout of the manuscript is also logical and clear.I think the manuscript can be given a further consideration after addressing the following issues.

Response:
We thank this referee's for recognizing our work being "physically interesting and may be influential in providing a new proposal to realize practical SOT devices".Following her/his insightful suggestions, we have now performed additional experiments and analyses in this revised manuscript and supplementary materials.Below we present our responses to specific comments.1.To make the Burgers vectors with an out-of-plane component a useful mechanism to introduce the tilted PMA, the authors should not only evaluate the tilting angle θe of crystalline planes near the dislocation (shown in Supplementary Figure S3 and S4) but also evaluate the density of the dislocations and make sure the mechanism work for the whole stack.In this current version, the HAADF images have showed a zoomed-in area.The wide-range distribution of the desired dislocations should be characterized or estimated, which is important for practical applications.

Response:
We thank the referee for making this insightful comment.The dislocation arrays in (100)-oriented and (110)-oriented heterostructures are confirmed by geometry phase analysis, as shown in Figure R1 below.For (100)-oriented and (110)-oriented heterostructures, the density of dislocation is estimated to be about 4.11×10 6 /cm, and 3.70×10 6 /cm, respectively.
Considering the lattice constant of Pt (3.9 Å), Co(FCC) (3.55 Å), the theory model predicts the density of dislocation to be about 1.44~3.57×10 6/cm, which is very close to our experimental results.In addition, we also utilize synchrotron based XRD to measure the tilting of Pt lattice, which is also confirmed as shown in Fig. 7 of Supplementary Materials.
Thus the presences of lattice tilting induced by dislocation are identified in the whole samples, rather than selected areas.This issue is mentioned on page 5 of revised manuscript.These results are now presented in Fig. 6 and Fig. 7 of Supplementary Materials.The dislocations are denoted by white arrows.
2. The authors mentioned the structure have application potentials.For applicable SOT memory devices, magnetic tunnel junctions fabricated on Si/SiO2 substrate are needed currently.A discussion how to integrate this dislocation mechanism with the sophisticated MRAM technology is helpful.

Response:
We acknowledge this important suggestion.According to J. Appl. Phys. 91, 5728-5734 (2002), the highly textured (110)-oriented MgO film can be deposited on (100)-Si substrate by PLD.Thus, it is possible to fabricate (110)-oriented heterostructures on (100)-Si substrate by combining PLD and magnetron sputtering techniques.Thereby, we believe it is possible to integrate the dislocation mechanism on the silicon-based MRAM technology.This part has been discussed on page 10 of the revised manuscript.We thank this referee for noting this issue.It is noted that the variation of Rxy of the blue line is very tiny (about 3% of magnetic switching) as compared with the other direction.To   Upper panel corresponds to heterostructure with θM = 16°and lower panel corresponds to heterostructure with θM = 84°.
5. What is the switching degree by SOT relative to the Rxy in the Rxy-Hoop curve? Response: For device with θM = 16°, the field-free switching ratio is about 70%, see Figure R4 below.For devices with θM = 84°, measurements on multiple devices reveal that the field-free switching ratio ranges from 40% to about 60%, see Figure R5 below.Note that these switching ratios are similar to those reported results in Nat.Nanotechnol.14, 939-944 (2019) (switching ratio about 20%), and Nat.Mater.20, 800-804 (2021) (switching ratio about 70%).
The incomplete field-free switching can be attributed to the pinning effect and current shunting from Hall voltage arms.For devices with switching ratio about 60%, the endurance test also confirms a good stability, as shown in Figure R6 below.This issue is mentioned on page 9 of revised manuscript.The results are included in Fig. 17 of Supplementary Materials.As the in-plane magnetic field Hx larger than anisotropic field HK, the magnetization vector M is aligned along the x-direction, leading to the SHH resistance to be written as  In addition, the damping-like spin-torque efficiency can be estimated as Here, we take the saturated magnetization of Co as Ms = 1000 emu/cm 3 , and t Co = 0.6 nm for the thickness of Co.By taking the HDL/Je of two directions, the damping-like torque efficiency ξ t along [ 1 10 ] ( ξ DL [1 10] ) is calculated as 0.088, and ξ t along [001] ( ξ DL [001] ) is about 0.078, respectively.
We also apply transverse fields in SHH resistance measurement, e.   001) samples as well.And further check whether (1) the easy axis tilting is absent and (2) field-free switching is not achievable for the control device. Response: We thank this referee for making this constructive comment.Following her/his suggestion, we have now characterized the magnetic anisotropy of (100)-and (111)-oriented heterostructures and its relation with the SOT switching ratio.We utilize the polar angular (γ) dependent Rxy measurement as shown in Fig. 3d.See Figure R8 blow, the polar angular (γ) dependent Rxy shows that the tilted magnetic easy axis is absent in (100)-and (111)-oriented heterostructures, which is also consistent with magnetometer results in Fig. 13 of Supplementary Materials.We then conduct the SOT switching experiments in (100)-and (111)-oriented heterostructures, as shown in Figure R9 below.The field-free SOT switching is also absent.Note that this comment is related to Question 9 below.This issue is mentioned on page 9 of revised manuscript.These results are incorporated into Fig. 13 and Fig. 21 of Supplementary Materials.as shown in Fig. R11b and R11d).It is noted that [1 10] is the magnetic hard axis (see Fig. 2 of revised Manuscript) and the tilted easy axis is within the plane perpendicular to [ 1 10 ].
Therefore, deterministic SOT switching of perpendicular magnetization is not expected without th .For θM = 16°, the magnetic easy axis is very close to the [001]-direction.Thus the th along [001] would strongly favor magnetization along this direction, resulting in the absence of SOT switching, see Figure R11b below.For θM = 84°, the easy axis is close to out-of-plane [110]-direction and the application of th would assist SOT switching (Figure R11d).This issue is mentioned on page 10 of revised manuscript, and these new results and discussions are now included in the Note 6 and Fig. 23 in Supplementary Materials.
Figure R11.The mechanism of field-dependent SOT switching.a-b, the effect of th in heterostructure with θM = 16°.a, the current pulse is applied along [ 1 10 ], the hard axis is away from sample plane and the th is insufficient to compensate SOT switching.b, the current pulse is applied along [001], the magnetization is strongly favored by th , thus the SOT switching is absent.c-d, the effect of th in heterostructure with θM = 84°.c, the current pulse is applied along [1 10], the th could force the magnetization align along hard axis, and the SOT switching is compensated.d, the current pulse is applied along [001], the th strongly favors the magnetization and assists the SOT switching.The easy axis (E.A.) is denoted by brown lines, and hard axis (H.A.) is denoted by green dash lines.9.It also seems strange why a small tilting angle of about 1 o in the atomic structure can introduce such a large tilting PMA angle as 30 o (tilted by 60 o as shown in Figure 3a&e).So the characterization of the tilting angle is still puzzling for me.In many cases, we also observed a sharp switching in the Rxy vs Hin-plane curves similar with Figure 3c.However, this curve cannot guarantee a tilted PMA.I suggest the authors conduct the sample measurement of Figure 3c for the control devices on ( 111) and (100) substrates and check the perpendicular magnetization switching by the in-plane field is still possible or not?

Response:
We appreciate this important suggestion.Firstly, as the reviewer suggested, we have characterized the switching of perpendicular magnetization by in-plane field in both (100)and (111)-oriented heterostructures by using use the same method in Manuscript Fig. 3c.As shown in Figure R12, no switching of polar MOKE signal is observed, which is consistent with previous analysis that suggests the tilted magnetic easy axis can only exist in (110)-oriented heterostructures.We also incorporate these new results into Supplementary Fig. 13 of the Supplementary Materials.
In addition, as discussed in Supplementary Note 3 and Supplementary Fig. 14, the tilted anisotropy can be understood as the competition among perpendicular magnetic anisotropy, crystalline anisotropy and shape anisotropy.As shown in our simulation results in Supplementary Fig. 14b of Supplementary Materials, when the effective perpendicular anisotropy energy is close to the crystalline anisotropy (ξ=  10.Regarding the critical current density, it is strongly dependent on the pulse duration.How about the current pulse duration used in this experiment?If the pulse is as long as above microsecond, the current density of 3.0×10 7 A/cm 2 is a moderate value.

Response:
We have now characterized the critical current density with different pulse durations.As shown in Figure R13, the critical switching current density of field-free SOT switching gradually increases as the pulsed widths reduce from 2 ms to 300 μs .This issue is mentioned on page 9 of revised manuscript.These results are also incorporated into Fig.20 of Supplementary Materials.11.There is a typo in Page 5. "tiling" "tilting".

Response:
This typo has now been corrected.In addition, we have now performed a careful proof reading thorough this revised manuscript.
12. Is there any typo for the current density of ~10 15 A/m 2 for the Pt/Co/AlOx system?This value seems not physical.

Response:
We are sorry for this confusion.This value is discussed in Supplementary Materials of Science 363, 1435-1439 (2019), noted as 'we find that the critical current density required to switch the OOP-IP element is around 4.7×10 11 A/cm 2 '.However, as compared with previous studies in Pt/Co/AlOx system, this value is relatively higher, and may be ascribed to a typo from the authors of Science 363, 1435-1439 (2019).Therefore, we have deleted it in the revised manuscript.
13.In the legend of Fig. 4d "for Pt(5)/Co(0.6)/Pt(1.4)/NiO(20)/MgOheterostructure with θM~90 o ", I suggest a more concise value is used instead of 90 o .If the angle is 90 o , field-free switching is physically impossible.A more concise value smaller than 90 o would appear more realistic and convincing and not misleading.

Response:
We thank the reviewer for this suggestion.The easy axis is actually close to 84 o as determined by anomalous Hall resistance, see Fig. 15 of Supplementary Materials.In the initial manuscript, we used 90 o to state that it is close to out-of-plane axis.We agree with the reviewer that this might cause confusion.Therefore, we have modified it as 84 o .In summary, the results in Pd-based multilayers demonstrate that our design strategy can be extended to other material systems.We have included these results on page 10 of revised manuscript and Fig. 25 of Supplementary Materials.
2. Moreover, I suggest that the authors should discuss the merits of their method compared to already reported ones showing field-free SOT switching using tilted magnetic anisotropy [Zhi Li, ACS Appl. Electron. Mater. 2022, H. Kim, Adv. Func. Mater. 2022, L. You, PNAS 2015]. Response: We are particularly thankful for this important question.As the referee suggested, discussions on tilted anisotropy and comparison with other methods are now included on the page 10 of the revised manuscript and Table 1 of Supplementary Materials.
The referee has listed three main methods to induce titled anisotropy.We will compare our strategy with these methods in the following.Firstly, in the reference [L.You, PNAS 2015], the tilted anisotropy is induced by fabricating a wedge-like nanomagnet device with lateral size of a few hundred nanometers, which requires complex fabrication process.
Secondly, in the reference [H.Kim, Adv.Func.Mater.2022], the tilted anisotropy is induced by interlayer magnetic coupling in the multilayer structures, which demands a very precise control of thickness in each layer.In addition, the additional layers also impair the inherent advantages of a single layer magnet.Thirdly, in the reference [Zhi Li, ACS Appl.Electron.
Mater.2022], an easy cone anisotropy is induced, and the deterministic switching is induced by the remnant in-plane magnetization component.However, due to the special type of anisotropy, the zero-field SOT switching ratio determined by ΔRxy is very small (less than 2%) as shown in the reference [Zhi Li, ACS Appl.Electron.Mater.2022].Furthermore, none of these studies have carefully checked the endurance of SOT switching.
By contrast, in our proposed method by designing crystal symmetry and dislocation, we can realize the tilted anisotropy in the single-layer ferromagnetic film in the whole sample, without the use of complex fabrication or interlayer coupling.The field-free SOT switching is achieved with relatively low critical current density (10 11 A/m 2 ), high switching ratio (70%), and good endurance (10 4 cycles).These highlight the advantages of our method that could be beneficial for future applications.The important parameters are tabulated below:  Electron. Mater. 4, 4033-4041 (2022).
3. Secondly, I wonder if the magnetization is fully switched by SOT without a magnetic field.
It seems odd to me that the change in Hall resistance (deltaR_xy) for the sample with theta_M=16 degrees is much larger than that for the sample with theta_M=~90 degrees.The authors should compare the field-free SOT switching results to the anomalous Hall resistance measured with perpendicular magnetic fields and show how much the magnetization is switched.I don't think it is reasonable to compare critical switching current density with other results unless full switching is achieved. Response: We thank the referee for pointing out this issue.Firstly, we want to clarify that the AHE resistance are different for the θM = 16°and θM = 84°heterostructures, due to the shunting current in additional Pt layers.As shown in the hysteresis loop of Rxy (red curves) in Fig. R16 and R17, the AHE resistance at zero field is approximately 0.25 Ω and 0.17 Ω for the θM = 16°and θM = 84°heterostructures.
By comparing the hysteresis loop of Rxy and current-driven SOT switching, we find that the field-free SOT switching ratio of heterostructure with θM = 16°is about 70%, as shown in    Reviewer #1: Remarks to the Author: Liang, et al. has substantially revised the manuscript according to the comments and questions from referees.They have added (1) the ( 100) and ( 111) control samples to show the uniqueness of the (110) substrates, (2) the SOT efficiency characterizations, (3) analysis of the heating effect and (4) larger range estimation of the dislocation density, which can ease the reproducing complexity of others and guide the later use of this technique.However, I am still questioning the buildup of the tiled PMA easy axis by the M-H loop or MOKE measurement.Logically, if the easy axis of a PMA film is tilted from the film normal (the nominal one), its magnetization behavior should be in principle measurable following the coming logic.A tilted magnetization can be deemed equivalently in some sense as the chiral coupling between an in-plane one and a PMA one.
In this case, an in-plane bias field (a fixed small magnetic field applied in-plane) should influence the magnetizing process of the PMA component by Hz.And moreover, this influence should be maximized (totally absent) as the applied in-plane field is colinear (perpendicular) with the direction of the in-plane component.In this case, one would expect a similar magnetization behavior as observed in an interlayer-DMI-coupled system [Han D. S. et al.Long-range chiral exchange interaction in synthetic antiferromagnets.Nat. Mater. 2019, 18, 703−708.].
Here should be Figure 2 of the reference paper Fig. 1 Abnormal magnetization behavior of an interlayer-DMI coupled system whose perpendicular magnetization process Mz-Hz (a) can be affected by an in-plane field Hin as Hin is applied in a proper direction (b and c).
This measurement needs both Hz and Hx, which is not so unique.I think the manuscript should take this measurement into account to straightforwardly evidence the existence of the tilted PMA easy axis for the (110) system but absent for the (100) and ( 111) counterparts.
Reviewer #2: Remarks to the Author: I appreciate the authors'efforts to responding to the reviewers' comments with additional experiments and explanations.This alleviates my concerns and the revised manuscript is much improved.Therefore, I recommend this manuscript for publication in Nature Communications.

Reviewer #1:
Liang, et al. has substantially revised the manuscript according to the comments and questions from referees.They have added (1) the ( 100) and ( 111) control samples to show the uniqueness of the (110) substrates, (2) the SOT efficiency characterizations, (3) analysis of the heating effect and (4) larger range estimation of the dislocation density, which can ease the reproducing complexity of others and guide the later use of this technique.

Response:
We thank this referee's for recognizing our revised manuscript being "ease the reproducing complexity of others and guide the later use of this technique".Following her/his insightful suggestions, we have now performed additional experiments and analyses in this revised manuscript and supplementary materials.Below we present our responses to specific comments.
1.However, I am still questioning the buildup of the tiled PMA easy axis by the M-H loop or MOKE measurement.Logically, if the easy axis of a PMA film is tilted from the film normal (the nominal one), its magnetization behavior should be in principle measurable following the coming logic.A tilted magnetization can be deemed equivalently in some sense as the chiral coupling between an in-plane one and a PMA one.In this case, an in-plane bias field (a fixed small magnetic field applied in-plane) should influence the magnetizing process of the PMA component by Hz.And moreover, this influence should be maximized (totally absent) as the applied in-plane field is colinear (perpendicular) with the direction of the in-plane component.
In this case, one would expect a similar magnetization behavior as observed in an interlayer-DMI-coupled system [Han D. S. et al.Long-range chiral exchange interaction in synthetic antiferromagnets.Nat. Mater. 2019, 18, 703−708.].
Here should be Figure 2 of the reference paper.This measurement needs both Hz and Hx, which is not so unique.I think the manuscript should take this measurement into account to straightforwardly evidence the existence of the tilted PMA easy axis for the (110) system but absent for the (100) and ( 111) counterparts.

Response:
We thank the referee for making this insightful comment.As the referee suggested, we   41247B).We really appreciate the referees' valuable and helpful comments.Following their stimulating comments, we have revised our manuscript accordingly and made necessary changes which were marked in blue.
We believe that we have clarified all concerns from referees, both the clarity and robustness of our manuscript have been enhanced.This revised manuscript could attract broad interest from the spintronic community, and meet the publishing criteria of Nature Communications.

Below is a list of major changes:
1: The manuscript has been checked and revised according to the guidance from author checklist.

Yours Sincerely
Yuan-Huan Lin on behalf of all coauthors.
3. InFig.4(d),why is there a partial switching loop for the blue line as I//[001] compared with 4(c)?Response: further clarify this issue, we also conduct the SOT switching experiments again and change the duration of pulse (from 2 ms to 500 us), see Figure R2 below.It is likely that the change of Rxy as I//[001] can be ascribed as artifact that were introduced by the thermal effect.To avoid ambiguity, we also incorporate these results in Fig. 20 of Supplementary Materials.

Figure R3 .
Figure R3.The estimated Joule heating induced by current pulses.a, the temperature dependent Rxx with dc current of 0.1 mA.b, estimated temperature increase by monitoring the Rxx under various ac current.The red circles represent the experimental data, and the blue lines represent the parabolic fitting curve.c, the temperature-dependent AHE resistance.

Figure R4 .
Figure R4.The SOT switching ratio in sample with θM = 16°.The dash lines denote the remnant Rxy in the zero field.

Figure R6 .
Figure R6.Endurance test of field-free SOT switching with ±Jpulse of 5.7×10 11 A/m 2 along 1 10 , which consistently confirms the switching ratio about 60%.a, the pulse cycle for more than 10 4 cycles.b, the field-free SOT switching loop before and after conducting the endurance test.The pulse width is 300 μs.
Figure R7b and R7c below.By changing the amplitude of ac current and fitting the SHH resistance, as shown in Figure R7d below.The HDL/Je along [001] is estimated to be 4.25 mT per 10 11 A/m 2 , and the HDL/Je along [1 10] is about 4.83 mT per 10 11 A/m 2 .These values are close to the recorded value in similar Pt/Co heterostructures (Phys.Rev. B 93, 144409 (2016)).
g. current along [1 10] ([001]) as field is along [001] ([ 1 10 ]), which can characterize the field-like effective field (Phys.Rev. Appl.11, 034018 (2019).).However, the R xy 2ω measured by transverse magnetic field is about 1 or 2 order lower than the R xy 2ω measured by longitude magnetic field (see the insets of FigureR7b and R7c).Thus, the field-like torque can be neglected and the damping-like torque plays the main role in SOT switching, which is consistent with previous studies (Nat.Commun.12, 4555 (2021); Phys.Rev.Lett.109, 096602 (2012)).This issue is mentioned on page 9 of revised manuscript.These results are discussed and included in Note 4 and Fig.18of Supplementary Materials.

Figure
Figure R7.SOT efficiency measured by harmonic Hall measurements.a, the schematic diagram of the measurement geometry.The magnetization M is oscillated by H DL .The yellow arrows indicate the direction of magnetization M. The experimental results of R xy 2ω

Figure R9 .
Figure R9.The absence of field-free SOT switching of perpendicular magnetization in (100)and (111)-oriented heterostructures.The results are shifted for better visualization.

Figure R10 .
Figure R10.The field-dependent SOT switching.The external magnetic field H is parallel to , a small tilting of crystalline lattice is enough to induced a tilted magnetic easy axis.

Figure R12 .
Figure R12.The absence of perpendicular magnetization switching in the presence of in-plane magnetic field for a, (100)-oriented heterostructure and b, (111)-oriented heterostructure.
/Pd(1.4)/Co(0.6)/Pd(1.4)/NiO(20)/MgO(110)and Pt(5)/Pd(1.4)/Co(0.6)/Pd(1.4)/MgO(110)heterostructures, as shown in Figure R14.It is noted that the field-free SOT switching is observed in both heterostructures.Moreover, the dependences of the field-free SOT switching behaviors on crystalline orientations are consistent with the results in manuscript.These results highlight that our design strategy can be extended to other material systems.

Figure R15 .
Figure R15.The absence of remnant perpendicular magnetization in the W-based heterostructures.

Figure R16 .
Figure R16.The spin-orbit torque switching ratio of device with sample θM = 16°.The dash lines denote the remnant Rxy in the zero field.

Figure R17 .
Figure R17.The field-free SOT switching ratio of sample θM = 84°in various devices.The current pulse is applied along the [1 10] direction.

Figure R18 .
Figure R18.The polar-MOKE images of field-free SOT switching.a-b, the results of sample θM = 16°with current pulse of 3.6 × 10 11 A/m 2 .c-d, the results of sample θM = 84°with current pulse of 3.9 × 10 11 A/m 2 .The pulse width is 2 ms.The initial magnetization direction was set by a magnetic field μ0H, as shown in the upper Panel (for sample θM = 16°, μ0H is 50 mT and along [001].For sample θM = 84°, μ0H is 10 mT and along [110]).The field-free SOT switching results are shown in lower panel.The current channel is 20 μm wide.

Fig. 1
Fig.1 Abnormal magnetization behavior of an interlayer-DMI coupled system whose perpendicular magnetization process Mz-Hz (a) can be affected by an in-plane field Hin as Hin is applied in a proper direction (b and c).
have confirmed that the in-plane bias field HIP indeed affects the perpendicular magnetization switching (Mz-Hz) in our heterostructures, which thereby further confirmed the tilted magnetic anisotropy.Here, the Mz component is detected by the polar MOKE as sweeping out-of-plane field Hz.As shown in FigureR1a and 1c, when the HIP is applied along[001]    direction, the hysteresis loops of polar MOKE signal as a function of Hz show a clear shift, for (110)-oriented heterostructures with θM=84°and θM=16°.Moreover, the loop shift changes to the opposite direction as we reverse the HIP direction.By contrast, the loop shift is absent when the HIP is applied along [1 10] direction, as shown in FigureR1b and 1d.These results further verify the existence of the tilted magnetic easy axis in our (110)-oriented heterostructures.

Figure R1 .
Figure R1.The shift of polar MOKE hysteresis loop induced by in-plane bias field HIP in (110)-orientated heterostructures.a and b show the polar MOKE hysteresis as sweeping out-of-plane field Hz of (110)-oriented heterostructure with θM=84°, when HIP is applied along a, [001] or b, [1 10] direction.c and d show the polar MOKE hysteresis as sweeping out-of-plane field Hz of (110)-oriented heterostructure with θM=16°, when HIP is applied along c, [001] or d, [1 10 ] direction.The upper panels show the schematic diagrams of HIP and tilted magnetic easy axis.The magnitude of HIP is 10 mT.

Table 1 .
The comparison with other studies based on tilted magnetic anisotropy.