Ahmad Abboud

Rémi Garcia

Abdelkader Lahmadi

Michaël Rusinowitch

Adel Bouhoula

R2-D2: Filter Rule set Decomposition and Distribution in Software Defined Networks

come

Background and Motivation

Context -Large number of filtering rules due to the increasing number of hosts and applications.

-Increase in number of attacks that affect entries in accesscontrol lists (ACL).

-Expensive and power-hungry ternary content-addressable memory (TCAM).

Research question

How to decompose and distribute filtering rules on a set of limited size switch tables ?

Overview of Longest prefix matching (LPM) representation

Rule Address field Action

1 0 0 * * A 1 2 0 0 0 * A 2

Rule Representation

-Single field filtering.

-Sufficient for blacklists.

-Rules represented in a binary tree.

-One rule at most on each tree node.

Decomposition Algorithm

-Input a set of rules and switches.

-Search the binary tree in order to find the best candidates.

-All rules present in the chosen node and all nodes below it, will be added to the switch.

-Minimize the number of generated rules by merging old forward rules with the ones from the best candidates.

-No rule duplication.

Forward Rule Generation

-Forward rules avoid processing packets filtered by previous switches multiple times.

-Rules with deny action does not require a forward rule to be generated.

IP Action

Decomposition over a graph

-Series-parallels graphs.

-Build the binary tree from a parallel and series composition.

-Simplify the binary tree using S-components.

-Packets with different sources will be processed by the same rule table of an intersection switch.

Evaluation

-12 sets of data generated using ClassBench.

-Percentage of forward action field between 0 and 100%.

-Rules with same action type have zero overhead.

-Around 15% overhead on 8 switches path length.

Figure :

 : Figure: Compact representation of rules prefixes (00*,01*,0*,1*,0*) in a binary tree.

Figure :

 : Figure: Illustration of forward rule generation with a rule set and two successive switches.

Figure

  Figure: S-component.

Figure :

 : Figure: Tree representation of a series-parallel graph.

  Effect of action field on overhead OH and C min using acl1 rule set. Rule space overhead while distributing a rule set with a 50% of rules having Forward actions.

Table :

 : Example of a rule set in a switch table.

	If a switch receives a packet with 0001 as address
	Prioritized list strategy : Rule 1 is first, A 1 is applied.
	LP M strategy : Rule 2 is most specific, A 2 is applied.

Acknowledgement

This work is supported by a CIFRE convention between the ANRT (National Association of Research and Technology) and the company NUMERYX Technologies.


Liang Liu

Chenghang Zhou

Xinyu Shu

Changjian Li

Tieyang Zhao

Weinan Lin

Jinyu Deng

Qidong Xie

Shaohai Chen

Jing Zhou

Rui Guo

Han Wang

Jihang Yu

Shu Shi

Ping Yang

Stephen Pennycook

Aurelien Manchon
email: manchon@cinam.univ-mrs.fr

Jingsheng Chen

Symmetry-dependent field-free switching of perpendicular magnetization

 . Instead, an outof-plane spin-orbit torque 15 could directly switch perpendicular magnetization. Here we

with the wide variety of SOT bilayers with low-symmetry point groups [START_REF] Laref | Elusive Dzyaloshinskii-Moriya interaction in monolayer Fe 3 GeTe 2[END_REF][START_REF] Johansen | Current control of magnetism in two-dimensional Fe 3 GeTe 2[END_REF] at the interface may uncover further unconventional spin-torques in future.

According to the spin Hall scenario in an HM/FM bilayer, a charge current flowing in the HM layer with in-plane direction (x) generates a spin current in the out-of-plane direction (z) with spin polarization along the y-direction. When flowing into the FM layer, the spin current generates a damping-like torque whose direction is set by the spin-polarization direction (y).

This in-plane damping-like torque cannot deterministically switch the perpendicular magnetization. So far, two major solutions have been used to achieve the field-free perpendicular magnetization switching. One is by using the in-plane damping-like torque and a method to break the torque-symmetry, which includes engineering the exchange interaction 7-9 /interlayer coupling 6 along x-direction and tilting the magnetization [START_REF] You | Switching of perpendicularly polarized nanomagnets with spin orbit torque without an external magnetic field by engineering a tilted anisotropy[END_REF][START_REF] Liu | Current-induced magnetization switching in all-oxide heterostructures[END_REF] along ydirection. The other is by generating an out-of-plane spin-torque along z-direction using an extra ferromagnetic layer [START_REF] Baek | Spin currents and spin-orbit torques in ferromagnetic trilayers[END_REF] . However, all these designs complicate the device structure and may introduce extrinsic effects. Therefore, it is highly pursued to have an out-of-plane SOT in the HM/FM bilayer itself.

We deposited the L1 1 -ordered CuPt (10 nm)/CoPt (4 nm) bilayer on the SrTiO 3 (111) substrate (see Methods). Compared with the interfacial magnetic anisotropy in conventional (see Methods) and measured the Hall resistance change for Hall bars with different θ I (see Fig. 2e). No external magnetic field was applied for the measurement. With a current density of about 2.4×10 7 A/cm 2 , the magnetization can be reversibly switched at angles θ I = 0°, 60°, 120°, and 180°. The switching polarity for θ I = 0° and 120° is anti-clockwise while that for θ I = 60° and 180° is clockwise. For θ I = 30°, 90°, and 150°, the switching loop vanishes, which means the magnetization cannot be switched by the current. We define ΔR I as the difference in Hall resistance at zero current after current pulsing with +20 mA and -20 mA, respectively.

We found the θ I dependence of ΔR I shows a 3-fold symmetry (Fig. 2d). The field-free switching occurs when the current is along the low-symmetry axes, whereas it is prohibited when the current is along the high-symmetry axes. It is worth noting that the field-free switching (with smaller ΔR I ) also exists at some other values of θ I (15°, 45°, 75°, 105°, 135°, 165°) that are neither along the high-symmetry axes nor along the low-symmetry axes (see Supplementary Fig. 6). We can use a cosine function to fit the θ I dependence of ΔR I with a period of 120°, as shown in Fig. 2d. To understand this symmetry-dependent field-free switching, we measure the current-induced effective fields.

We first estimated the in-plane damping-like effective field of the CuPt/CoPt bilayer by harmonic Hall voltage analysis [START_REF] Garello | Symmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures[END_REF][START_REF] Hayashi | Quantitative characterization of the spin-orbit torque using harmonic Hall voltage measurements[END_REF][START_REF] Pi | Tilting of the spin orientation induced by Rashba effect in ferromagnetic metal layer[END_REF][START_REF] Kim | Layer thickness dependence of the current-induced effective field vector in Ta|CoFeB|MgO[END_REF] . Fig. 3a shows the first (𝑉 ! ) and second (𝑉 !! ) harmonic Hall signals for the Hall bar with θ I = 0° in a small in-plane field (H) range. The harmonic Hall signals for other θ I are shown in Supplementary Fig. 7. After subtracting thermoelectric contribution (Supplementary Fig. 8), we obtain the damping-like effective field by using the

formula 23 ∆𝐻 !" = -2 ! ! !!!! ! !!!! ! , (1)
where B L(T) is defined as

!! !! !" / ! ! ! ! !! ! !(!)
. ξ~0.07 is the ratio of planar Hall voltage to anomalous Hall voltage. The current dependence of ΔH DL is shown in Fig. 3b. The results for 0° and 60° (low-symmetry axes) are almost the same as that for 30 and 90° (high-symmetry axes). According to the linear fit, the spin torque efficiency (µ 0 ΔH DL /J ac ) is calculated to be 9 mT/(1×10 8 A/cm 2 ) and remains almost constant with a variation smaller than 1 mT/(1×10 8 A/cm 2 ) for varying θ I , as shown in Fig. 3c. In general, the in-plane damping-like torque in the HM/FM bilayer was considered to mainly originate from the spin Hall effect of the HM layer.

Our result indicates an isotropic spin Hall angle, consistent with a previous report [START_REF] Guillemard | Charge-spin current conversion in high quality epitaxial Fe/Pt systems: Isotropic spin Hall angle along different in-plane crystalline directions[END_REF] . Then we characterized the out-of-plane SOT effective field (ΔH OOP ) by measuring the anomalous Hall loop under applying different d.c. currents [START_REF] Yu | Switching of perpendicular magnetization by spin-orbit torques in the absence of external magnetic fields[END_REF][START_REF] Pai | Determination of spin torque efficiencies in heterostructures with perpendicular magnetic anisotropy[END_REF] . As shown in Fig. 3d, when we apply a positive (+18 mA) and a negative (-18 mA) pulsed currents (30 µs duration) into the Hall bar with θ I = 0°, the centers of the anomalous Hall loop are shifted to the left and the right, respectively.

Based on the loop shift, a ΔH OOP of +0.8 mT (-0.8 mT) is estimated for +18 mA (-18 mA).

While for θ I = 60°, the loop shifts are opposite to that for 0°, which give a negative (positive) ΔH OOP for a positive (negative) current (see Fig. 3f). For θ I = 30° (Fig. 3e) and θ I = 90° (Fig. 3g), the loop shift vanishes, which gives a ΔH OOP of around 0. Fig. 3h shows the current dependence of ΔH OOP (see Supplementary Fig. 91011). We found that ΔH OOP for θ I = 30° and θ I = 90° (high-symmetry axes) almost remains at 0 for all current values. In contrast, for the low-symmetry axis of θ I = 0° (θ I = 60°), ΔH OOP is close to 0 when the current is below 15 mA, and then abruptly increases (decreases) with the current. We will discuss the possible origin of this threshold current observed in the CuPt/CoPt bilayer further below. We estimated the out-of-plane spin-torque efficiency (µ 0 ΔH OOP /J pulse ) by using ΔH OOP measured at ±18 mA for different θ I (Supplementary Fig. 12). In Fig. 3i, µ 0 ΔH OOP /J pulse shows a 3-fold angular dependence on θ I , with an amplitude of 3.7 mT/(1×10 8 A/cm 2 ). The 3-fold angular dependence of the out-of-plane effective field is consistent with the 3-fold field-free switching (Supplementary Fig. 13). In previous reports, the out-of-plane SOT effective field could be obtained after inducing an in-plane magnetization component (M x ), through applying an in-plane magnetic field [START_REF] Pai | Determination of spin torque efficiencies in heterostructures with perpendicular magnetic anisotropy[END_REF] or by introducing an exchange bias [START_REF] Fukami | Magnetization switching by spin-orbit torque in an antiferromagnet-ferromagnet bilayer system[END_REF] . These unidirectional controls should give a 1-fold angular dependent SOT switching. In contrast, the 3-fold angular dependent field-free switching in L1 1 CuPt/CoPt bilayer should have an intrinsic origin related to the crystal structure. As shown in Supplementary Fig. 14, after two thousand of repeated switching loops in a 26-days period, the switching behavior remains stable. We observed ΔR I almost keep unchanged (degradation<4%) with switching cycles, indicating an excellent switching performance. We also performed an endurance test for the L1 1 CuPt/CoPt bilayer with a shorter pulse width (30 ns) and obtained an endurance up to 5×10 5 cycles (Supplementary Fig. 16).

The L1 1 phase adopts the point group 𝑅3𝑚. Due to inversion symmetry breaking at the interface, the point group of the CuPt/CoPt heterostructure reduces to 3𝑚1, i.e., a triad around [111] direction and a mirror plane normal to [1-10] direction. Applying these two symmetry operations to the current-driven spin-torque tensor, we obtain 28

𝑻 = 𝜒 !" 𝒎× 𝒛×𝑬 ×𝒎 + 𝜒 !! 𝒎× 𝑚 ! 𝐸 ! + 𝑚 ! 𝐸 ! 𝒙 + 𝑚 ! 𝐸 ! -𝑚 ! 𝐸 ! 𝒚 . (2)
For short handedness, the Cartesian coordinates (x, y, z) refer to the crystallographic directions ([1-10], [11-2], [111]). The first term is the damping-like torque expected in any HM/FM heterostructures lacking inversion symmetry. The second term (3m torque) is specific to the 3𝑚1 point group. In Supplementary Section 9, we found that the magnetization reversal of the field-free switching in CuPt/CoPt is dominated by the nucleation of small magnetic domains. To understand how the damping-like and the 3m torques cooperate and lead to deterministic switching of small magnetic domains, we consider a ferromagnetic macrospin with uniaxial perpendicular anisotropy, subjected to these two torque components. The magnetic energy reads [START_REF] Johansen | Current control of magnetism in two-dimensional Fe 3 GeTe 2[END_REF] (see Supplementary Section 10)

𝑊 = 𝐾 ! sin ! 𝜃 + ! !! ! ! ! sin(2𝜑 + 𝜑 ! ) sin ! 𝜃, (3)
where (𝜃, 𝜑) are the polar and azimuthal angles of the magnetization direction in the (x, y, z) frame. Depending on the direction and sign of the current, the second term either competes or cooperates with the perpendicular anisotropy. Setting 𝜒 !! = 𝜂𝜒 !" and using a Gaussian

pulse, 𝜒 !" = 𝜒 !" ! 𝑒 ! !!! ! ! !
, we compute the final state of this macrospin (see Supplementary Fig. 22-24). Fig. 4a shows the calculated stability diagram using this model. This figure

shows that for 𝜑 ! = 0°, 60°, and 120°, reversible switching is obtained. Noticeably, the hysteresis obtained for 𝜑 ! =0° and 𝜑 ! = 120° have the same polarity, while it is opposite for

𝜑 ! = 60°.
Finally, along 𝜑 ! = 30°, 90°, and 150°, no reversible switching is obtained (see Supplementary Fig. 23). This stability diagram is consistent with the experimental data in Fig. 2e. In order to further clarify the mechanism, we propose a scenario for the field-free switching. The scenario is depicted in Fig. 4b-f. At the initial state, the magnetization is aligned along -z (Fig. 4b) and the only torque that is active is the damping-like torque

𝑻 ∼ 𝜒 !" 𝒎× 𝒛×𝑬 ×𝒎 ≡ 𝒛×𝑬. (4)
The 3m torque vanishes because 𝑚 ! , 𝑚 ! = 0. Under the damping-like torque, the magnetization is brought close to the plane, along the direction perpendicular to the electric field, 𝒎||𝒛×𝑬. In this case, the damping-like torque vanishes by definition, and only the 3m torque is active, so

𝑻~𝜒 !! 𝒎× 𝑚 ! 𝐸 ! + 𝑚 ! 𝐸 ! 𝒙 + 𝑚 ! 𝐸 ! -𝑚 ! 𝐸 ! 𝒚 . ( 5
)
Now the influence of this torque on the final magnetization state depends on the direction of the electric field, as displayed in Fig. 4c-e. If the electric field is applied along 𝜑 ! = 0°. 𝑬 = 𝐸𝒙 (Fig. 4c), the magnetization lies along 𝒎||𝒚, and the 3m torque reads

𝑻~𝜒 !! 𝒎×𝑬 ≡ -𝜒 !! 𝒛, (6)
and pushes the magnetization towards +z (based on the experimental results, 𝜒 !! < 0). In

𝑻~-𝜒 !! 𝒎×𝑬 ≡ 𝜒 !! 𝒛, (7)
and pushes the magnetization towards -z. As a result, reversible magnetization reversal is favored but the hysteresis has an opposite sign compared to the case 𝜑 ! = 0°. The above scenario well interprets the symmetry-dependent field-free switching in the L1 1 CuPt/CoPt system, and it may also explain the threshold current for the ΔH OOP in Fig. 3h since the 3m torque can only act on the magnetization when the injected current is large enough to bring the magnetization close to the plane.

As the 3m torque originates from a peculiar symmetry (3-fold rotation and mirror symmetry) of the CuPt crystal, its strength is governed not only by the spin-orbit coupling but also the bulk crystal field. Therefore, engineering the chemical bonds within the material to enhance the crystal field could increase the torque magnitude. This observation calls for further materials exploration. In our experiments, the strength of ΔH OOP in L1 switching (Supplementary Section 13). These above experimental results imply that the good crystallinity and the good chemical ordering of the CuPt layer are prerequisites for the generation of the 3m torque. We note that the field-free switching in Fig. 2 is incomplete if we compare ΔR I (Fig. 2e) with the anomalous Hall loop (Fig. 2c). This could be due to the pinning effect from the Hall voltage arms. Therefore, we fabricated the CoPt layer on CuPt into a round pillar (Fig. 5a). Fig. 5b and Fig. 5c show the anomalous Hall loop and the current-induced switching loop of the pillar device for θ I =0°, where we observed a complete switching. For further verification, we used the magneto-optical Kerr effect (MOKE) microscopy to check the magnetization states before and after SOT switching (see Supplementary Section 14). As shown in Fig. 5d, we firstly applied a saturation magnetic field along -z direction to initialize the magnetization of the pillar to -M z . Then we removed the external magnetic field and applied a +22 mA pulsed current into the channel, and we found the pillar area turned black, representing a +M z state (Fig. 5e). Next, by successively applying -22 mA and +22 mA pulse, the color in the pillar area become grey (-M z , Fig. 5f)

and black (+M z , Fig. 5g), respectively. Comparing the colors in Fig. 5f with Fig. 5d, we found that a -22 mA pulse can fully recover the saturated magnetization (-M z ). On the other side, we also confirmed a +22 mA pulse can fully recover the +M z state by comparing Fig. 5j with Fig. The red lines in (a) and (e) outline the pillar edges. For processes (d-g), we firstly applied a negative magnetic field along the -z direction to initialize the magnetization to -M z . To enhance the contrast, the background in the whole image was subtracted such that the colors within and outside the pillar region are the same (grey). However, only the grey color in the pillar area represents a -M z state (d), since there is no CoPt outside the pillar region. After that, we successively applied +22 mA (e), -22 mA (f), and +22 mA (g) into the CuPt channel to check the magnetization evolution. The dark color in (e) and (g) represent a +M z state. For processes (h-k), we firstly applied a positive magnetic field along the +z direction to initialize the magnetization to +M z . Then the whole background was subtracted and the grey color in the pillar area represents +M z (h). After that, we successively applied -22 mA (i), +22 mA (j), MOKE measurement. We use a MOKE from Evico Magnetics to image the magnetic domains during the SOT switching. In order to enhance the contrast, we subtract the background signal for the whole view after each initialization with an out-of-plane magnetic field. Then we removed the magnetic field and applied a pulsed current with a duration of 30 µs. The MOKE image was then captured after a stabilization time longer than 2 s.

  observe an out-of-plane spin-orbit torque in an HM/FM bilayer of L1 1 -ordered CuPt/CoPt and demonstrate field-free switching of the perpendicular magnetization of the CoPt layer. The low symmetry point group (3m1) at the CuPt/CoPt interface gives rise to this spin torque, herein after referred as 3m torque, which strongly depends on the relative orientation of current flow and crystal symmetry. We observe a 3-fold angular dependence in both the field-free switching and the current-induced out-ofplane effective field. Because of the intrinsic nature of the 3m torque, the field-free switching in CuPt/CoPt shows good endurance in cycling experiments. Experiments

!

  other words, if the magnetization was initially along -z, the 3m torque favors magnetization reversal. If the electric field is applied along𝜑 ! = 30°, 𝑬 = 𝐸( ! ! 𝒙 + ! ! 𝒚), seeFig. 4d, the magnetization lies along 𝒎||(-! ! 𝒙 + ! ! 𝒚) and the 3m torque vanishes 𝑻~0. In other words, there is no preferential direction and no reversible switching occurs. Finally, if the electric field is applied along 𝜑 ! = 60°, 𝑬 = 𝐸( ! ! 𝒙 + ! ! 𝒚), see Fig. 4e, the magnetization lies along 𝒚) and the 3m torque reads

5h.

  The above MOKE measurement further confirms the full SOT switching in CuPt/CoPt. In summary, our work demonstrates the observation of the out-of-plane spin-torque in the L1 1 CuPt/CoPt bilayer due to the low-symmetry property at the interface. Since there are different kinds of low-symmetry point groups at various SOT interfaces, it is expected that corresponding "low-symmetry" spin-torques could exist. Compared with previous field-free switching of perpendicular switching with extra functional layers or additional asymmetry designs, the field-free switching in L1 1 CuPt/CoPt is simpler in design, and possess good thermal stability and good endurance, which makes it a promising candidate for magnetic memories.

Figure

  Figure 1| Crystal structure and symmetry analysis. a, The crystal structure of L1 1 CuPt

Figure

  Figure 2| Symmetry-dependent magnetic field-free magnetization switching. a,

Figure

  Figure 3| Symmetry dependence of current-induced effective fields. a, First and second

Figure

  Figure 4| Theoretical model for current-induced spin-torque in CuPt/CoPt bilayer. a,

Figure

  Figure 5| Current-induced field-free magnetization switching in CuPt (13 nm)/CoPt (4

and - 22 5 ( 4

 2254 mA (k) into the CuPt channel to check the magnetization evolution. The white color in (i) and (k) represents a -M z state. The pulse width of the current for the measurement is 30 µs.MethodsSample growth and device fabrication. CuPt (10 nm)/CoPt (4 nm) bilayer was epitaxially deposited on SrTiO 3 (111) single-crystal substrate by d.c. magnetron sputtering (AJA). For the deposition of CuPt, the Ar pressure is 5 mTorr and the temperature is 500 °C. The sputtering powers for Cu target and Pt target are both 50 W. For the deposition of CoPt, the Ar pressure is 6 mTorr and the temperature is 300 °C. The sputtering powers for Co target and Pt target are 32 W and 21 W, respectively. The nominal deposition rates for CuPt and CoPt are 0.14 nm/s and 0.051 nm/s, respectively. Then the bilayer was left to cool down to room temperature and a 2 nm SiO 2 protection layer was deposited. After deposition, the films were patterned into 5 µm Hall bars with different in-plane orientations by using laser writer and Ar ion milling. Then the contact electrode pattern was defined by laser writer and followed by the deposition of Ti (5 nm)/Cu (100 nm) and the lift-off process. For varying CuPt deposition temperature measurements, CuPt (13 nm)/CoPt (4 nm) samples with CuPt deposited at 200 °C, 300 °C, 400 °C, and 500 °C are prepared. For the CuPt (10 nm)/[Co/Ni] nm), the growth conditions of the CuPt are the same as that for the CuPt (10 nm)/CoPt (4 nm) sample, expect that the CuPt for room temperature-CuPt (10 nm)/[Co/Ni] 5 (4 nm) sample is deposited at room temperature. The [Co/Ni] 5 layer is deposited at room temperature.Current-induced switching measurement. In current-induced magnetization switching measurement, for each data point, a pulsed d.c. electrical current with a duration of 30 µs was applied. After 8 seconds, the Hall resistance was recorded by using a small a.c. excitation current (50 µA).STEM sample preparation and characterization. CuPt/CoPt STEM sample was fabricated by a focused ion beam machine (FEI Versa 3D system). The sample was thinned down using a Ga ion beam first with an accelerating voltage of 30 kV and then with an accelerating voltage of 8 kV. After that, the sample was polished by a 2 kV ion beam to minimize the beam damaged region. The structure characterization was conducted with a JEOL ARM200 STEM which is equipped with an ASCOR probe corrector operating at an accelerating voltage of 200 kV. The HAADF image was acquired with a probe forming angle of 30 mrad and collection inner and outer semiangles of 68 and 280 mrad, respectively.

Acknowledgements

The research is supported by the Singapore National Research Foundation under CRP Award No. NRF-CRP10-2012-02 and the Singapore Ministry of Education MOE2018-T2-1-019, MOE2018-T2-2-043, A*STAR Grant No. A1983C0036, A*STAR IAF-ICP 11801E0036, MOE Tier1 R-284-000-195-114 and the King Abdullah University of Science and Technology (KAUST). J.S.C. is a member of the Singapore Spintronics Consortium (SG-SPIN).

Data availability

The authors declare that the main data supporting the findings of this study are available within the Letter and its Supplementary Information. Extra data are available from the corresponding author upon reasonable request. Soure Data are provided with this paper.

HM/FM bilayers, the bulk magnetic anisotropy in L1 1 CoPt enables higher memory density with an excellent thermal stability [START_REF] Iwata | Perpendicular magnetic anisotropy and magneto-optical Kerr spectra of MBE-grown PtCo alloy films[END_REF][START_REF] Dannenberg | Surface energies of stoichiometric FePt and CoPt alloys and their implications for nanoparticle morphologies[END_REF] . The crystal structure of L1 1 CuPt (or L1 1 CoPt) is shown in Fig. 1a, where Cu (Co) and Pt atomic layers alternatively stack along the [111] direction in a rhombohedral structure. All crystallographic directions used in this study are denoted with the rhombohedral lattices. Therefore, in the (111) plane, we can define three high-symmetry axes ([11-2] and equivalent) and three low-symmetry axes ([1-10] and equivalent) (see Supplementary Fig. 1).

The symmetry in Fig. 1b is similar to that for WTe 2 , where the out-of-plane SOT was suggested to be allowed when the current is applied along the low-symmetry axis 20 . Supplementary Fig. 2 shows the θ-2θ X-ray diffraction spectra of our CuPt/CoPt bilayer, where the (111) peak indicates the L1 1 phase. The epitaxial growth of the bilayer was confirmed by the cross-sectional scanning transmission electron microscopy (STEM) in Fig. 1c, where we observed a sharp CuPt/CoPt interface. The CuPt layer has a well-arranged atomically layered structure where the brighter atoms are Pt and the darker ones are Cu (see Supplementary Fig. 3b for the schematic drawing). In contrast, for the CoPt layer, the intensity contrast of the elements is not sufficient for clearly distinguishing the Co and Pt atoms, which indicates a lower chemical ordering parameter in CoPt, compared with that in

The L1 1 CuPt/CoPt bilayer was fabricated into 5 µm Hall bar devices (Fig. 2a) with varying azimuth angles (θ I ) to the [1-10] direction (Fig. 2b). The anomalous Hall loops for θ I = 0° and other angles are shown in Fig. 2c and Supplementary Fig. 4, respectively. Supplementary Fig. 5 shows the magnetic hysteresis loops for the bilayer before patterning,

Competing financial interests' statement

The authors declare no competing financial interests.

Correspondence and requests for materials should be addressed to J.S.C. (msecj@nus.edu.sg) and A.M. (manchon@cinam.univ-mrs.fr)