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[Satoshi Sugimoto](https://orcid.org/0000-0002-7148-2372), [Yasufumi Araki](https://orcid.org/0000-0001-5873-7695), [Yukiko K. Takahashi](https://orcid.org/0000-0001-9197-7236), [Jun’ichi Ieda](https://orcid.org/0000-0001-6069-8533), [Shinya Kasai](https://orcid.org/0000-0001-7149-4800)

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[Symmetry-reduced topological interface for unleashing a multidirectional spin-orbit torque](https://mdr.nims.go.jp/datasets/c23b733e-61fe-46c6-b350-ec2ba7a365f2)

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Symmetry-reduced topological interface for unleashing a multidirectional spin-orbit torquecommunications physics ArticleA Nature Portfolio journalhttps://doi.org/10.1038/s42005-025-02024-1Symmetry-reduced topological interfacefor unleashing a multidirectional spin-orbit torqueCheck for updatesSatoshi Sugimoto1, Yasufumi Araki 2, Yukiko K. Takahashi 1, Jun’ichi Ieda 2 & Shinya Kasai 1Current-induced spin-orbit torqueat heavymetal/ferromagnet interfacesplays a critical role inmodernspintronics. The challenges in relevant research are to enhance the conversion efficiency andgenerateunconventional torque components. Here, we propose a simple protocol to meet such demands byengineering symmetrical structures, specifically by a wedge-shaped topological material interface inBi2Te3/CoFeB heterostructures. The symmetry of the topological interface is manipulated to lead tounconventional emergences of both the out-of-plane and in-plane spin polarizations. Torquemeasurements show that these polarizations have conversion efficiencies up to 10%, characterizedby the nonreciprocal ferromagnetic resonance spectra and the distinctive torque profile. The observedenhancement in the unconventional spin-orbit torque is attributed to the wedge-oriented symmetryreduction of the spin-momentum locking structure of interfacial Dirac fermions. This advance opensavenues for designing geometrically characteristic spintronics devices, demonstrating the potential ofusing multidirectional spin-orbit torques for forthcoming nanotechnological applications.A trend of all-electric operation of spintronics devices has resulted from thedevelopment of spin-orbit torque (SOT) applications over the last decade1–4,wherehigh-speedoperationswithhigh endurance areprincipally achievablefor memory and logic devices5 with simple thin-film heterostructures. Thedevelopment of SOT devices is linked to progress in studies on spin-to-charge conversion. In addition to its conversion efficiency, the direction ofspin polarization also plays a crucial role in designing SOT deviceconfigurations6,7. Manipulation of the spin polarization direction is parti-cularly important in solving the problem of nondeterministic nature in theswitching process of perpendicular magnetization3. For the conventionalspin Hall or Rashba SOT, it vanishes once the magnetization tilts down tothe in-plane direction, resulting in uncontrollable final states in the absenceof an applied external field. This vanishing of the SOT is caused by thesymmetry of the system because rotational and mirror symmetries deter-mine the relation between the current injection and spin polarizationdirections. Therefore, reduction of symmetry could be used to overcomegeometric constraints in SOTapplications.At an early stage of development,convoluted device engineering is used to exploit exchange interactions8,9,interlayer coupling10, structural asymmetry11–13, and magnetic anisotropy14.The use of noncollinear antiferromagnets (including Mn) in spin-current injection layers has been investigated, where the unconventionalexchange coupling at the interface gives rise to an out-of-plane spinpolarization15–17.There has been considerable interest in using alternative approaches toproduce unconventional SOTs by decreasing structural symmetry whilemaintaining the original heterostructures. A decrease in the rotational andmirror symmetries of the crystalline structure results in a warping of spin-orbit coupling (SOC) pattern in momentum space, leading to a rich varia-tion in spin polarization that is not expected from the conventional SOTfrom isotropic spin Hall or Rashba effects18,19. To date, threefold symmetryhas been experimentally observed in several heavy metal compounds,namely,WTe220,NbSe221, andL11-CuPt/CoPt22, where an SOT fromout-of-plane spin polarization has been recently reported. The emergence of such anontrivial torque component has enabled field-free SOToperation of heavymetal/ferromagnet (HM/FM) heterostructures in several experiments22–25.However, the typical amplitude of this SOT remains one order ofmagnitudesmaller than that of the conventional SOT15–17,20, which hinders the devel-opment of SOT for various applications. Therefore, we need to designunconventional SOTs by reducing the symmetries of system, with theirmagnitudes reaching up to that of the conventional SOT.In this study, we propose an alternative protocol to generate andenhanceunconventional SOTbasedon symmetry reduction in a topologicalinsulator (TI) material Bi2Te3. Bi2Te3 shows the spin-momentum lockedelectronic states on the surface, with the hexagonal warping structure due tothe discrete rotational symmetry of crystal26–28. From such a warped spin-momentum locking structure, we may expect both the enhancement of the1Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki, 305-0047, Japan. 2AdvancedScience Research Center, Japan Atomic Energy Agency, 2-4 Shirakata, Tokai-mura, Ibaraki, 319-1195, Japan. e-mail: KASAI.Shinya@nims.go.jpCommunications Physics |           (2025) 8:100 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s42005-025-02024-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s42005-025-02024-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s42005-025-02024-1&domain=pdfhttp://orcid.org/0000-0001-5873-7695http://orcid.org/0000-0001-5873-7695http://orcid.org/0000-0001-5873-7695http://orcid.org/0000-0001-5873-7695http://orcid.org/0000-0001-5873-7695http://orcid.org/0000-0001-9197-7236http://orcid.org/0000-0001-9197-7236http://orcid.org/0000-0001-9197-7236http://orcid.org/0000-0001-9197-7236http://orcid.org/0000-0001-9197-7236http://orcid.org/0000-0001-6069-8533http://orcid.org/0000-0001-6069-8533http://orcid.org/0000-0001-6069-8533http://orcid.org/0000-0001-6069-8533http://orcid.org/0000-0001-6069-8533http://orcid.org/0000-0001-7149-4800http://orcid.org/0000-0001-7149-4800http://orcid.org/0000-0001-7149-4800http://orcid.org/0000-0001-7149-4800http://orcid.org/0000-0001-7149-4800mailto:KASAI.Shinya@nims.go.jpwww.nature.com/commsphysSOT efficiency29–31 and the emergence of the unconventional SOTcomponent32–34. The pioneering experimental works35,36 with Bi-basedchalcogenides recordedmore than 100% efficiencies in their charge-to-spinconversion process, which later enabled the ultralow power SOT switchingwith double-digit decrease in the critical current density37. Numerous stu-dies with TIs were conducted to improve the conversion efficiencies so far,by adjusting TI crystallinity38 or positioning the Fermi level closer to theDirac point39. On the other hand, its capability as a platform for theunconventional SOTs,whichwas proposed by several theoreticalworks, hasnot been paid attention to in experiments so far.The essence of this work is to adopt the TI not simply as an efficientsource of SOT, but rather as a source of multi-directional SOT componentsby manipulating its unique symmetry structure. Since the effect of spin-momentum locking in the surfaceDirac electrons ismuch stronger than theconventional Rashba SOC40,41, electron spin polarization on the TI surface ishighly sensitive to symmetry reduction. Therefore, we may expect that thecombination of the warping effect from the crystalline structure and theasymmetry of the sample structure can generate unconventional spinpolarization of a sizeable magnitude. Here, wemanipulate the (001) surfaceof the TI material Bi2Te3, which is originally in the threefold point groupC3v42–45. We introduce an atomic-scale thickness gradient by synthesizing awedge-shaped film to reduce the rotational and mirror symmetries of thesurface. Filmwedginghas amarked effecton theTI surface state, resulting infurther distortion in the spin-momentum locking structure. By comparingthe measurements using Bi2Te3/CoFeB heterostructures without and withthe film wedging, we report the emergence of multidirectional SOT arisingfrom the film wedging. The unconventional parallel and out-of-plane spinpolarizations therein reach up to 50% the conventional component ortho-gonal to the current flow. In particular, a spin polarization parallel to thecurrent direction is observed in such symmetry-reduced systems, which areprincipally prohibited in highly symmetric crystals.Results and discussionSymmetry reduction protocol for C3v systemsWebriefly discuss the effect of symmetry reduction on the SOT structure inmagnetic heterostructures based on group theory analysis in two dimen-sions (2D). A spin current of a spin polarization S that is injected into amagnetic layer in response to an electric field E can exert a field-like (FL)SOT τFL / m× S and a damping-like (DL) SOT τDL / m× ðS×mÞ on themagnetization m. The possible form of the induced S is an axial vectorbelonging to the irreducible representation in common with the angularmomentum, which can be constructed from combinations of E,m, and anyother perturbations applied to the system. Symmetry reduction results in arich variation in the SOT by decreasing the number of classes of irreduciblerepresentations4,18. Below, we consider a fixed Cartesian coordinate system(X,Y,Z) for the crystal, where theZ-axis is the rotational symmetry axis (thefilm normal). We extract some SOT components characteristic of eachsymmetry group (all possible SOT structures are listed in SupplementaryNote 1: Group-theory analysis of spin-orbit torque under wedging).We start from the largest symmetry possible at a 2D interface, the C∞vgroup in the continuum, which is symmetric under an arbitrary rotationRϕaround the Z-axis and a reflection σv by arbitrary mirror planes containingthe Z-axis (Fig. 1a). This situation principally applies to polycrystal andamorphous, where the crystal orientations are randomly distributed andintegrated out in total. For instance, a Rashba-type SOCwith in-plane spinpolarization, / ðkXσY � kYσXÞ, is permitted for this symmetric structure.The linear Dirac-cone dispersion on the TI surface is the typical example inthis class. The possible form of S induced solely by an in-plane E is limitedto S1v / Ẑ×E ¼ ð�EY ; EX ; 0Þ, which is expected from the conventionalspin Hall and Rashba-Edelstein effects, as shown in Fig. 1d.The crystalline structure decreases C∞v at the interface to a discretegroup. Among the discrete point groups in 2D, the C3v structure commonlyappears in cubic and hexagonal crystals, including at the (001) topologicalFig. 1 | Schematics of bands and spin distributions under different symmetrygroup structures at the interface and corresponding current-induced spinpolarizations. Schematized structures of the interfacial bands and Fermi surface.a Isotropic heterostructures that belong to the C∞v point group. b Heterostructurewith threefold rotational symmetry (C3v). c C3v heterostructure with film wedging(in the direction F), where the point group is reduced to the trivial C1 group.Schematics of the sample geometry and corresponding spin polarization S of thespin-orbit torque (SOT). dW/CoFeB heterostructure belonging to C∞v point groupexhibiting only in-plane polarization S1v orthogonal to the current. e ferromagnet(FM) and topological insulator (TI) (Bi2Te3) heterostructure beonging to C3v pointgroup permitting out-of-plane polarization S3v. f FM and wedged TI (Bi2Te3) het-erostructure beonging toC1 point group inducing both out-of-plane polarization S3vand in-plane polarization S1 parallel to the current.https://doi.org/10.1038/s42005-025-02024-1 ArticleCommunications Physics |           (2025) 8:100 2www.nature.com/commsphysinterface of Bi2Te3. This point group hosts the threefold rotationalsymmetry R3 around the Z-axis and the reflection symmetry σv by thethree mirror planes. We here fix the X- and Y-axes of the Cartesiancoordinate system such that the XZ-plane coincides with one of themirror planes of σv. The C3v structure allows a hexagonal warping for theSOC. For the 2D Dirac fermions on the TI surface, the warping effect isparametrized as / Re kX þ ikY� �3σZ, producing an unconventional out-of-plane spin polarization on the Fermi surface, as shown in Fig. 1b4,18,46.This term can produce an out-of-plane spin component of the SOT,SZ3v / E �mð ÞẐ, which is allowed in C∞v but prohibited for the in-planeRashba SOC. The C3v structure also permits in-plane components of theSOT, S3v / EXmX � EYmY� �X̂ þ �EXmY � EYmX� �Ŷ, in addition toS1v produced by the combination of E and m. This S3v makes theemergence of field-like τ3m ¼ τ3m m× S3v� �and damping-like τPH ¼τPH m× S3v ×m� �� �possible in C3v heterostructures, which have beenclassified as the “3m torque” and the “planar Hall torque” in precedingtheories, respectively19,32–34. These types of SOTs have been experimen-tally reported for the L11-CuPt/CoPt interface22.On this C3v interface, we consider the effect of film wedging. Filmwedging can be regarded as an in-plane vector-type perturbation F in thedirection of the thickness gradient because it transforms under both the R3rotations and σv reflections but does not break time-reversal symmetry. Thisperturbation reduces the symmetry group down to the trivial C1 group. Itconsiderably distorts the structure of the fully spin-momentum-lockedDirac fermions on the TI surface, as shown in Fig. 1c (see SupplementalNote 1 for a detailed discussion). Combinations of E and F yield uncon-ventional forms of the in-plane and out-of-plane components Sk1 and S?1 ,that are not captured in S1v and S3v:Sk1 / EXFY þ EYFX� �X̂ þ EXFX � EYFY� �Ŷ; S?1 / EXFY � EYFX� �Ẑ;ð1Þwhere all three spin components are permitted (Fig. 1f). The out-of-planecomponent S?1 arises if the wedgingF is designed to be perpendicular to thecurrent injection direction E. For instance, if the current is injected in theX-direction, wedging should be introduced in the Y-direction to obtain theout-of-plane component. In this case, the in-plane component Sk1 becomesparallel to the current injection direction, Sk1 / EXFY X̂. Such a componentwith the spin polarization along the current direction is not permitted atO(m0) in either Sv or S3v. These unconventional SOT components result inthe extinction of the geometrically invariant point (τ = 0) during themagnetization switching process. Therefore, the symmetry engineeringdiscussed here could be used to realize deterministic field-free switching.The F effect considered here could be applied to model any other in-planeinhomogeneity, e.g., an in-plane temperature gradient or asymmetrybetween electrodes, although the corresponding effects are not ascontrollable as those of film wedging.Measurements of multidirectional SOT at a symmetry-reducedinterfaceThe aforementioned protocols were demonstrated experimentally for thetopological surface state of Bi2Te3.AsBi2Te3 forms a rhombohedral unit cell(Fig. 2a), its (001) surface belongs to the C3v group (Fig. 2b). Figure 2d is aschematic of an atomic-scale thickness gradient in awedge structure that weprepared in the epitaxial heterostructure of [001]-Bi2Te3 and amorphousCoFeB (Fig. 2c). The film thickness in the heterostructure varies by 4t ¼ð3 ± 1ÞA over a channel width of 12 μm. This wedging acts as an in-planevector perturbation F, which reduces the symmetry of the system down tothe trivial C1 group, as discussed in the previous section. We usedFig. 2 | Ferromagnetic resonance-based torque measurements obtained undersymmetry reduction of the rhombohedral Bi2Te3 (001) surface. a The crystalstructure of the rhombohedral Bi2Te3. b Plane view of the hexagon projected alongthe [001] direction, where two Bi atomic layers (orange) are sandwiched betweenfour Te atomic layers (the top and bottom layers are shown in gray and red,respectively). For simplicity, only the atoms in the hexagonal region (shaded red) areshown. Themirror symmetryM is present for the (�110) plane (black solid line) but isabsent for the (110) plane (red dashed line). c Cross-sectional high-angle annulardark-field scanning transmission electronmicroscope (STEM) image of the epitaxialBi2Te3/CoFeB bilayer gazed from Bi2Te3-½100� direction. d Schematic of the spintorque ferromagnetic resonance (ST-FMR) measurement process for heavy metal/ferromagnet (HM/FM) heterostructures. The device geometry is presented byscanning electron microscope (SEM) image of a Bi2Te3/CoFeB bilayer sample. Thethickness gradient 4t ¼ ð3 ± 1ÞA is set along the y-axis, orthogonal to the currentdirection (x-axis). An in-planemagnetic field μ0H is applied to 45° from the x-axis tomaximize the anisotropic magnetoresistance (AMR) projection. Both the current(|| x) and thickness gradient (|| y) are applied at the (001) plane of Bi2Te3 (thecrystalline structure of the sample is detailed in Supplementary Note 3). Measure-ments with the uniformly thick W/CoFeB and Bi2Te3/CoFeB heterostructures aredone with the same configuration.https://doi.org/10.1038/s42005-025-02024-1 ArticleCommunications Physics |           (2025) 8:100 3www.nature.com/commsphysspin-torque ferromagnetic resonance (ST-FMR) to evaluate the SOTcomponents in this heterostructure2,47. Distinct from the crystalline coor-dinate system (X, Y, Z), we describe the laboratory frame by the Cartesiancoordinate system (x, y, z), where the x- and y-axes correspond to thedirections of the current injection (j, or E) and wedging (F), respectively. Toidentify the effect of symmetry reduction on the SOTs, we performed thesame measurements on two different samples: polycrystalline W/CoFeB(belonging to C∞v) and Bi2Te3/CoFeB with a uniform thickness (belongingto C3v).Figure 3a–c presents the ST-FMR voltage spectra, VST�FMR, recordedfor the three types of prepared heterostructures, where the x-component ofthe magnetic field is either parallel (μ0H>0) or antiparallel (μ0H<0) to thecurrent direction. The ST-FMR spectrum of the W/CoFeB heterostructurehas a reciprocal structure, where inversion of thefield directionflips the signof the signal, indicating a conventional SOT structure (Fig. 3a). By contrast,the spectra of the Bi2Te3/CoFeB heterostructures exhibit clear non-reciprocity against field inversion, where an asymmetric large peak inVST�FMR for μ0H > 0 switches to a tiny dip for μ0H < 0 (Fig. 3b, c). More-over, these nonreciprocal FMR spectra vary depending on whether there iswedging in the Bi2Te3 film. The antisymmetric component under μ0H < 0almost vanishes at the wedge-shaped interface (see the cyan line in Fig. 3f)but remains finite in the uniformly thick heterostructure (Fig. 3e). Thenonreciprocities observed here can be ascribed to the emergence of anasymmetric SOT unique to the Bi2Te3/CoFeB heterostructures. To quantifythe thermal effects from the Joule heating, we also performed the ST-FMRmeasurements with lower power excitations. They showed the spectra withtheir shapes almost consistent with those in Fig. 3, which implies that thethermal effects giveonlyquantitativedifferences (see SupplementaryNote6:Effects of thermal components).To obtain a full picture of the SOT, we performed ST-FMR mea-surementswith varying the direction of the in-planemagneticfield. Figure 4shows the profiles of the symmetric and antisymmetric ST-FMR voltagecomponents, VSym and VAsym, as functions of the in-plane angle θ of thefield direction measured from the current-injection direction (x-axis), asshown in the inset of Fig. 2d. These plots are shown for the conventionalW/CoFeB system in Fig. 4a, b, for the uniformly thick Bi2Te3/CoFeB system inFig. 4c, d, and for the wedge-shaped Bi2Te3/CoFeB system in Fig. 4e, f. Theconventional SOT for the C∞v point group results in simple forms48, suchthat VSymðAsymÞðθÞ / sin 2θ cos θ. These simple forms are shown as blackdashed lines in Fig. 4a–f. For W/CoFeB, the simple forms are in excellentagreement with the observed profiles, as shown in Fig. 4a, b. However, theobserved profiles of VSymðAsymÞðθÞ both the uniformly thick and wedge-shaped Bi2Te3/CoFeB heterostructures are considerably more complexand do not correspond to simple sin 2θ cos θ functions, as shown inFig. 4c–f. The sin 2θ cos θ function is symmetric for θ ¼ 90°; 270° andantisymmetric for θ ¼ 0°; 180°. By comparison, the observed profiles ofVSym exhibit minor deviations from this behavior, and those of VAsym areentirely different. Suchdiscrepancies are considerably greater for thewedge-shaped samples (Fig. 4e, f) than for the uniformly thick samples (Fig. 4c, d).These differences can be attributed to the unconventional spin polarizationswhich are permitted by the hexagonal warping and film wedging, as hasbeen discussed in the previous section.To understand the unconventional spin polarizations from thereduced symmetries, we here decompose the observed profiles ofVSymðAsymÞðθÞ into the components of SOT. Incorporating FL and DLcomponents τi;FL; τi;DL� �in each spin direction (i = x, y, z) and 3m andplanar Hall components ðτ3m; τPHÞ, the field-angle dependence of the tor-que components becomes, τSOT θð Þ ¼ τθSOT θð Þθ̂ þ τzSOT θð Þẑ, whereFig. 3 | Spin torque ferromagnetic resonance voltage spectra for different pointgroup structures. ST-FMRvoltage spectra of the in-planemagneticfield μ0H for (a),W (3 nm)/CoFeB (5 nm), b uniformly thick Bi2Te3 (8 nm)/CoFeB (5 nm), andcwedge-shapedBi2Te3 (8 nm+4t)/CoFeB (5 nm) heterostructures.4t denoted thefilm thickness variation as 4t ¼ ð3 ± 1ÞA. The corresponding fitted curves ofindividual heterostructures are plotted at (d, e, and f), respectively. The experimentaldata are shown as closed (μ0H>0) and open (μ0H<0) black symbols, and the fittedcurves for μ0H>0 (μ0H<0) are shown in red (magenta) for the symmetric part VSymand blue (cyan) for the antisymmetric part VAsym. The sign of μ0H is defined to beconsistent with the sign of the x-component of μ0H.https://doi.org/10.1038/s42005-025-02024-1 ArticleCommunications Physics |           (2025) 8:100 4www.nature.com/commsphysFig. 4 | Angular dependence of spin-torque ferromagnetic resonance signalsobserved for heterostructures in different point groups. The symmetric ðVSymÞand antisymmetric ðVAsymÞ components of the rectified voltage are plotted asfunctions of the in-plane magnetic field angle θ measured from the current injectiondirection, as determined from ST-FMR measurements for (a, b), W (3 nm)/CoFeB(5 nm), (c, d), uniformly thick Bi2Te3 (8 nm)/CoFeB (5 nm) and (e, f), wedge-shaped Bi2Te3 (8 nm + 4t)/CoFeB (5 nm) heterostructures. The black dashed linesin (a–f) are fitted curves including only the conventional spin polarization com-ponents (τy;DLðFLÞ). The extended fitting results are based on Eqs. (2) and (3),including both the conventional and unconventional components of SOT(τx;DLðFLÞ; τy;DLðFLÞ; τz;DLðFLÞ), are shown as blue (c, d) and red (e, f) solid lines.https://doi.org/10.1038/s42005-025-02024-1 ArticleCommunications Physics |           (2025) 8:100 5www.nature.com/commsphysτθSOT θð Þ ¼ τx;DL sin θ � τy;DL cos θ � τz;FL þ τPH sin 2θ and τzSOT θð Þ ¼�τx;FL sin θ þ τy;FL cos θ � τz;DL � τ3m sin 2θ (the derivation is given inSupplementaryNote 2: Field-direction dependence of the ST-FMR signals).Substituting these expressions into VSymðAsymÞðθÞ yields17,49VSym / sin 2θτθSOT θð Þ; ð2ÞVAsym / sin 2θτzSOT θð Þ: ð3ÞThe measured VSym Asymð Þ θð Þ can be consistently fitted by theseequations. We note that τSOT θð Þ is also affected by the back reaction fromthe spin pumping and the inverse Edelstein effect. Once their contributionsare taken into account, the estimated value of τy;DL will be quantitativelyshifted (see Supplementary Note 4: Contributions of spin pumping and theinverse Edelstein effect), while the estimations of the other unconventionalSOT terms are left unchanged. The fitted results are shown by the solid linesin Fig. 3a–d, with the coefficients provided in Table SIII in SupplementaryNote5:Coefficients for the anisotropic SOTcomponents.Thesefits produceτxðzÞ;DL and τxðzÞ;FL with sizable magnitudes, which are approximately10–50% of that of τy;DLðFLÞ, which implies that the unconventional SOTcomponents are significantly induced by the symmetry reduction. Thestrong out-of-plane torques jτz;DLðFLÞj comparable to jτy;DLj for both theuniformly thick and wedged heterostructures are attributed to the violationof the continuous rotational symmetry at the Bi2Te3/CoFeB interface. Wenote that the uniform thickness Bi2Te3 also shows a finite jτz;DLðFLÞj≠0,typically at film thickness thinner than 10 nm (See Supplementary Note 3:Microstructure analysis for Bi2Te3/CoFeB heterostructures). The sputteredthickerfilms includemore crystalline imperfections, twinning, and/or c-axismisalignments, which tend to obscure jτz;DLðFLÞj arising from the bulk C3Vstructure. Our 8 nm thickness samples are designed to be thin enough torelatively avoid those obstacles.The SOTs from the spin polarizations parallel to the current direction,τx;DL and τx;FL, are only observed in the wedge-shaped sample. This type ofparallel spin polarization is principally prohibited under threefold rotationaland mirror symmetries, as discussed in the previous section. The emergenceof these two distinct SOT components was confirmed by measurements ofcurrent-inducedmagnetization switching processes and the second-harmonicHall resistivity (details are provided in Supplementary Note 7: In-planemagnetization switching process by nonreciprocal SOT in Bi2Te3/CoFeBheterostructure, Supplementary Note 8: Domain nucleation process under-lying in-plane magnetization switching in Bi2Te3/CoFeB heterostructure, andSupplementary Note 9: Second-harmonic Hall measurements for Bi2Te3/CoFeB heterostructures). These results show that the symmetry reduction byatomic-scale modulation of the thickness of Bi2Te3/CoFeB producesunconventional parallel and out-of-plane spin polarizations for the SOT.These unconventional spin polarization components lead to anisotropyin the SOT, which eliminates the invariant points satisfying τSOT θð Þ ¼ 0. Tovisualize these invariant points, we present vector plots of the angular profilesof the in-plane τθSOT θð Þ and out-of-plane τzSOT θð Þ components of the SOTs inFig. 5a, b, respectively. The SOT profile is symmetric for the W/CoFeB het-erostructure (shown as black dashed arrows) but highly asymmetric for boththe uniform and wedged heterostructures of Bi2Te3/CoFeB (as shown by theblue and red solid arrows, respectively). The magnitude of the in-planecomponent τθSOT θð Þ for the wedged heterostructure (Fig. 5a) is almost twice aslarge for Hx> 0 (�90°≤ θ ≤ 90°) as for Hx< 0 (90°≤ θ ≤ 270°). The out-of-plane component τzSOT θð Þ also exhibits large anisotropy (Fig. 5b), whichevidences the emergence of a sizable out-of-plane spin component induced atthe microwedged topological interface. As a result of these asymmetries, theFig. 5 | Angular profiles of multidirectional spin-orbit torque componentsobserved for heterostructures in different point groups.Angular profiles of (a) in-plane SOT τθSOTðθÞ and b out-of-plane SOT τzSOTðθÞ for W/CoFeB (black dashedarrows), uniformly thick Bi2Te3/CoFeB (blue solid arrows), and wedge-shapedBi2Te3/CoFeB heterostructures (red solid arrows). Each arrow indicates the direc-tion of each SOT component at the angle θ. The gray plane in (b) representsτzSOT θð Þ ¼ 0. c Angular profile of the SOT norm τSOT θð Þ�� �� in polar coordinates forthe three heterostructures shown in (a, b). The inner black solid circle representsτSOTðθÞ�� �� ¼ 0. Only the W/CoFeB heterostructure shows τSOTðθÞ�� �� ¼ 0 where theblack dashed curve touches the inner circle (shown by the positions of the blackdots), corresponding to the invariant points of the SOT.https://doi.org/10.1038/s42005-025-02024-1 ArticleCommunications Physics |           (2025) 8:100 6www.nature.com/commsphysSOT magnitude jτSOT θð Þj also becomes anisotropic, as shown by the polarplot in Fig. 5c. To clearly show this result, jτSOT θð Þj for each heterostructure isrescaled by a certain factor, where the origin jτSOTðθÞj ¼ 0 is represented bythe inner black solid circle. Whereas the plot for theW/CoFeB heterostructureincludes invariant points touching the jτSOTðθÞj ¼ 0 circle at θ ¼ 90° and270°, both the uniformly thick and wedge-shaped Bi2Te3/CoFeB hetero-structures exhibit nonzero τSOTðθÞ�� ��≠0 for any θ, indicating the elimination ofthe invariant points. This behavior facilitates the deterministic switching ofmagnetization because the SOT remains finite anywhere during the switchingprocess. This capability is attributed to the emergence of unconventional spinpolarization components, which are the direct consequence of the symmetryreduction of the topological surface state.Differences between unconventional and conventional SOTsFor themultidirectional SOT in the symmetry-reduced heterostructure, thespin-to-charge conversion efficiency6,7,49 should be extended to the multi-component quantities ξs;DL and ξs;FL asξs;DL ¼eMstFMtTI_ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Meffμ0H0s×τx;DL=τy;FLτy;DL=τy;FLτz;DL=τy;FL0B@1CA; ð4Þξs;FL ¼eMstFMtTI_ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Meffμ0H0s×τx;FL=τy;FL1τz;FL=τy;FL0B@1CA; ð5Þwhere e is the elementary charge,Ms is the saturationmagnetizationof theFMlayer, tFM and tTI are the thicknesses of the FMandBi2Te3 layers, respectively,and Meff is the effective magnetization (see Methods). Charge-to-spinconversion occurs mainly at the 2D interface in the Bi2Te3/CoFeBheterostructures. The efficiency of the y-component of the FL SOT (theleadingcomponentarising in theC∞v group), ξy;FL, isnormalized tounity.Theextended conversion efficiencies from the wedged heterostructure arequantified relative to ξy;FL as ðξx;DL; ξy;DL; ξz;DLÞ ¼ ð�0:07 ± 0:01; 0:24 ± 0:02;�0:10 ± 0:01Þ and ðξx;FL; ξz;FLÞ ¼ ð�0:02 ± 0:00; 0:08 ± 0:01Þ.The scale of the multidirectional spin-to-charge conversion efficiencyobtained above is considerably large because of the symmetry-reduced spin-momentum locking on the Dirac surface states. This result is demonstratedin Table 1, in which the conversion efficiencies ξi ði ¼ x; y; zÞ obtained forthe wedged Bi2Te3/CoFeB heterostructure herein are compared with thosereported in previous experimental studies on unconventional SOT com-ponents. Except for the conventional component along the y direction50,51(jξy;DLj≫ 0:2), there are considerable differences in the efficiencies of thein-plane (x) and out-of-plane (z) components. We find a notable in-planeDL torque efficiency jξx;DLj � 0:1, corresponding to the spin componentparallel to the current direction, which has never been reported in ordinaryHM/FM heterostructures due to the limitation by the rotational symmetryat the interface. Since this effect is not present in our uniform-thicknesssample, we identify its origin not as sample imperfections, but solely as oursymmetry reduction protocol. The highest values for this componentreported to date have been achieved in structurally engineered devices, i.e.,field-annealed W/CoFeB heterostructures with canted easy axes52 andultrathin [Pt/Co] multilayers53. In contrast, our wedged Bi2Te3/CoFeBheterostructure exhibits jξx;DLj comparable to those champion records ofstructurally engineered devices, even though its structural modulation ismuch simple with an atomic-scale thickness gradient. In addition to thex-component, the out-of-plane DL and FL torque efficiencies,jξz;DLj � jξz;FLj � 0:1, are also greater than those in previous reports usingnoncolinear antiferromagnets15–17 and structural engineering devices49,53(jξz;FLj � 0:01).We speculate that apossible origin of suchanenhancementof jξz;DLðFLÞj is the atomic-scale thickness modulation that is not exactly intheC3v structure. As given by Eq. (1), the thicknessmodulation, representedby the perturbation F, also yields an additional contribution to the out-of-plane spin component. Our measurement results imply that the com-bination of the atomic-scale thickness modulation with the strongspin-momentum locking structureunique to theDirac surface state of aTI isa powerful tool for inducing multidirectional SOT.Our protocol for reducing the symmetry of the topological surface statecan be regarded as producing an ultimate form of the low-symmetry spintorques with sizable magnitudes, by breaking the crystalline symmetries inmaterials with strong SOC. This strategy has been developed over the lastseveral years18,20, and the results of this study demonstrate an alternativeroute for manipulating the spin polarization in SOTs. Compared withprevious engineering approaches, our approach to generate the multi-directional SOT is implemented with a rather simple design with a micro-wedge structure, which can operate with good stability and endurance indevices. Moreover, (00l)-oriented Bi-chalcogenide thin films are formed bythe weak van der Waals interactions between the adjacent the quintuplelayers, and hence they can be deposited onto various substrates with dif-ferent fabrication techniques54–56. From these features, our approachmay becompatible toward the future industrial applications combined withdevelopments of in-plane perturbation application methodologies, a pro-nounced wedge designing, or the glancing angle deposition (GLAD)57, etc.The concept of symmetry reduction is innovative and can be adaptedwithout degrading themagneto-transport properties of the original devices.Thus, symmetry reduction widens the range of approaches that can beapplied to forthcoming magnetic-memory technology and logic devices.MethodsSample preparationThe Bi2Te3 (8.0–8.3 nm)/Co20Fe60B20 (5 nm)/Ru (2 nm) multilayer wasgrown on a (001)-oriented sapphire Al2O3 substrate by DC magnetronsputtering. The Al2O3(001) substrate was first annealed above 1000°C for2 hours to obtain an atomically flat surface. The substrate was then quicklymoved to the sputtering chamber, and a rhombohedral Bi2Te3 layer wassputtered at 2.1 As�1 using a Bi2Te3.8 alloy target. An atomic thicknessgradient was created using a moving shutter during the sputtering process.Subsequent in situ annealing was performed at 360°C for 1 hour. It wasconfirmed that excess Te atoms had evaporated during the sputteringprocess. The formation of the stoichiometric Bi2Te3 film was confirmed byX-ray diffraction (XRD) and inductively coupled plasma‒mass spectro-metry analysis. The remaining ferromagnetic CoFeB layer and Ru cappinglayer were sputtered at room temperature.The Bi2Te3/CoFeB heterostructures were patterned into bars withwidths of 2–40 μm, and Ta/Au electrodes for RF injection and electrodeswere subsequently fabricated using electron beam lithography (EBL) andphotolithography.Film characterizationTransport in and the microstructure of the film were evaluated using XRD,resistivity measurements, atomic force microscopy, scanning electrodeTable 1 | Spin-to-charge conversion efficiencies of variousHM/FM heterostructures with unconventional polarizationMaterial ξx;DL ξy;DL ξz;DL ξx;FL ξz;FLBi2Te3/CFB (this study) −0.07 0.24 −0.10 −0.02 0.08WTe2 ref. 20 0.03 0.013 0.0325Mn3SnN ref. 15 0.003 0.053Mn3GaN ref. 17 −0.013 0.025 0.019 −0.15MnPd3 ref. 16 0.017 0.41 0.011[Pt/Co]5/Mg/CoFeB ref. 53 −0.083 −0.102 −0.033Ta/CoFeB/TaOx ref. 11 0.58FM/Ti/CoFeB/MgO ref. 13 −0.048MgO/Py/MgO ref. 49 0.018 0.047https://doi.org/10.1038/s42005-025-02024-1 ArticleCommunications Physics |           (2025) 8:100 7www.nature.com/commsphysmicroscopy (SEM), and high-angle annular dark-field scanning transmis-sion electron microscopy. The detailed results are presented in Supple-mentary Note 3.The resistivity of the sputtered Bi2Te3 8-nm-thick film used for theexperiments was measured as 320 ± 20 μΩ cm. The resistivity of theamorphous CoFeB in our devices exceeded 100 μΩ cm, indicating thatsufficient RF current flowed through the bottomBi2Te3 layer. Themagneticproperties of the Bi2Te3/CoFeB heterostructures were estimated from FMRspectra using theKittelmodel. An effective saturationmagnetizationMeff of120:2 ± 2:4ð ÞmT was determined by fitting the Kittel model to the FMRspectra. The field misalignment was estimated to be less than 0.3 mT fromthe symmetry of the spectra.MeasurementsThe ST-FMR measurements were carried out by applying a microwavecurrent at a fixed frequency (in the 7–15GHz range) to the Bi2Te3/CoFeBbilayer. The lock-in technique with amplitude modulation at 1317Hz wasutilized to improve the signal sensitivity. A rotatable electromagnet wasprepared and used to determine the angle-dependent torque presented inFig. 3a–d. Magnetic fields were swept from+0.28 T to−0.27 T to drive theCoFeB through its resonance state. The transmission and reflection coef-ficients for the RF cables and devices were calibrated by using vector net-work analyzermeasurements in the relevant frequency range. Amicrowavefrequencyof 15 GHzand an appliedmicrowave powerof 15 dBmwere usedto obtain the results shown in Figs. 2 and 3. The detected voltage VST�FMRwas divided into symmetric and antisymmetric components (VSym;VAsym)via resonance spectral fitting as follows:VST�FMR ¼ VSym4H24H2 þ H �H0� �2 þ VAsymΔH H � H0� �4H2 þ H �H0� �2 ;where 4H denotes the peak width and H0 represents the resonance field.Data availabilityThe data used to generate the plots presented in this paper and other resultsare available from the corresponding author upon reasonable request.Received: 27 May 2024; Accepted: 27 February 2025;References1. 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A15, 1460–1465 (1997).AcknowledgementsThisworkwas partially supported by the Japan Society for the Promotion ofScience (JSPS)KAKENHI (Grantnos.20K14419,22K03538, 23K17882, and24K00952), and JST and PRESTO (Grant No. JPMJPR18L3), Japan.Author contributionsY.T. conceived the project. S.S. planned and performed the experiments.Y.A. developed the theoretical models with input from J.I. S.S. analyzed thedatawith input fromS.K.All the authors discussed the results.S.S., Y.A., J.I.,and S.K. wrote the manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s42005-025-02024-1.Correspondence and requests for materials should be addressed toShinya Kasai.Peer review information Communications Physics thanks EmanueleLongo and the other, anonymous, reviewer(s) for their contribution to thepeer review of this work.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License,which permits any non-commercial use, sharing, distribution andreproduction in any medium or format, as long as you give appropriatecredit to the original author(s) and the source, provide a link to the CreativeCommons licence, and indicate if you modified the licensed material. 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Toview a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2025https://doi.org/10.1038/s42005-025-02024-1 ArticleCommunications Physics |           (2025) 8:100 9https://doi.org/10.1038/s42005-025-02024-1http://www.nature.com/reprintshttp://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/commsphys Symmetry-reduced topological interface for unleashing a multidirectional spin-orbit torque Results and discussion Symmetry reduction protocol for C3v systems Measurements of multidirectional SOT at a symmetry-reduced interface Differences between unconventional and conventional SOTs Methods Sample preparation Film characterization Measurements Data availability References Acknowledgements Author contributions Competing interests Additional information