Octupole-driven magnetoresistance in an antiferromagnetic tunnel junction

The tunnelling electric current passing through a magnetic tunnel junction (MTJ) is strongly dependent on the relative orientation of magnetizations in ferromagnetic electrodes sandwiching an insulating barrier, rendering efficient readout of spintronics devices1–5. Thus, tunnelling magnetoresistance (TMR) is considered to be proportional to spin polarization at the interface1 and, to date, has been studied primarily in ferromagnets. Here we report observation of TMR in an all-antiferromagnetic tunnel junction consisting of Mn3Sn/MgO/Mn3Sn (ref. 6). We measured a TMR ratio of around 2% at room temperature, which arises between the parallel and antiparallel configurations of the cluster magnetic octupoles in the chiral antiferromagnetic state. Moreover, we carried out measurements using a Fe/MgO/Mn3Sn MTJ and show that the sign and direction of anisotropic longitudinal spin-polarized current in the antiferromagnet7 can be controlled by octupole direction. Strikingly, the TMR ratio (about 2%) of the all-antiferromagnetic MTJ is much larger than that estimated using the observed spin polarization. Theoretically, we found that the chiral antiferromagnetic MTJ may produce a substantially large TMR ratio as a result of the time-reversal, symmetry-breaking polarization characteristic of cluster magnetic octupoles. Our work lays the foundation for the development of ultrafast and efficient spintronic devices using antiferromagnets8–10.

The tunnelling electric current passing through a magnetic tunnel junction (MTJ) is strongly dependent on the relative orientation of magnetizations in ferromagnetic electrodes sandwiching an insulating barrier, rendering efficient readout of spintronics devices [1][2][3][4][5] . Thus, tunnelling magnetoresistance (TMR) is considered to be proportional to spin polarization at the interface 1 and, to date, has been studied primarily in ferromagnets. Here we report observation of TMR in an all-antiferromagnetic tunnel junction consisting of Mn 3 Sn/MgO/Mn 3 Sn (ref. 6 ). We measured a TMR ratio of around 2% at room temperature, which arises between the parallel and antiparallel configurations of the cluster magnetic octupoles in the chiral antiferromagnetic state. Moreover, we carried out measurements using a Fe/MgO/Mn 3 Sn MTJ and show that the sign and direction of anisotropic longitudinal spin-polarized current in the antiferromagnet 7 can be controlled by octupole direction. Strikingly, the TMR ratio (about 2%) of the all-antiferromagnetic MTJ is much larger than that estimated using the observed spin polarization. Theoretically, we found that the chiral antiferromagnetic MTJ may produce a substantially large TMR ratio as a result of the time-reversal, symmetry-breaking polarization characteristic of cluster magnetic octupoles. Our work lays the foundation for the development of ultrafast and efficient spintronic devices using antiferromagnets [8][9][10] .
Regarding the perspective of future applications, it is highly important to develop TMR using all-antiferromagnetic MTJs, which potentially generates a high magnetoresistance ratio. On the other hand, research and development of the TMR effect has been restricted primarily to ferromagnetic MTJs because the effect is considered to derive from finite spin polarization at the interface between the magnetic electrodes and insulating barrier [1][2][3][4][5] . Thus, no reports have been published to date on the TMR effect using an all-antiferromagnetic MTJ. In addition, as the basis for designing the STT-MRAM based solely on antiferromagnets, observation and manipulation of longitudinal spin-polarized current is necessary but, again, has never been performed for antiferromagnets. Here we report our discovery of a finite TMR of approximately 2% using a MTJ solely comprising an antiferromagnet. Moreover, our experiment has further clarified the existence of the anisotropic, longitudinal spin-polarized current in an antiferromagnet. Strikingly, the size of TMR in the all-antiferromagnetic MTJ is much larger than the conventional estimate based on spin polarization and thus should derive from a new mechanism.
Ferromagnetic TMR produces the binary signals '0' and '1', respectively, via parallel and antiparallel arrangements of spin polarization in a pair of ferromagnetic electrodes (Fig. 1a). In principle, analogous binary states can be defined for an antiferromagnetic MTJ providing that the antiferromagnetic states break time-reversal symmetry (TRS) macroscopically. Such antiferromagnets have recently been discovered and intensively studied for the development of antiferromagnetic spintronics, because they produce large transverse effects despite vanishingly small magnetization 6,[23][24][25][26]31,32 . Breaking of TRS in these antiferromagnetic states leads to characteristic electronic structures such as magnetic Weyl semimetal state 33,34 , spin-splitting bands 7,31,32,35,36 and polarization of cluster magnetic multipoles 37 .
Here we focus on a specific example, the chiral antiferromagnet Mn 3 Sn. This hexagonal kagome metal has attracted much interest because it exhibits large transverse responses normally absent in antiferromagnets 6,[38][39][40] . Below the Néel temperature T N ≈ 430 K, all Mn moments of approximately 3 μ B lying in the kagome plane form an antichiral 120° spin order. The magnetic texture can be viewed as a ferroic order of cluster magnetic octupole lying in the kagome plane ( Fig. 1b and Methods) 37,39 . This ferroic octupole order breaks the TRS macroscopically and allows us to define states 0 and 1 as the parallel and antiparallel configurations, respectively, of octupole polarization (Fig. 1b, orange arrows and Methods). In addition, it is ferroic octupole order that stabilizes the magnetic Weyl semimetal state 33,34 and drives large transverse responses such as the anomalous Hall effect 6 , anomalous Nernst effect 38 and magneto-optical Kerr effect (MOKE) 39 .
To further demonstrate the analogy between spin polarization in ferromagnets, and octupole polarization in antiferromagnets, we drew the projected density of states (DOS) for the positive and negative polarization of spins (Fig. 1c) and magnetic octupoles (Fig. 1d) in the chiral antiferromagnetic state of Mn 3 Sn. In sharp contrast with the symmetric DOS projected onto spin-up and -down states (Fig. 1c), we found a clear energy shift between the majority and minority bands of the octupole moment ( Fig. 1d and Methods), closely resembling the shift in the majority and minority bands of the spin moment in ferromagnetic iron (Extended Data Fig. 2a). For conventional ferromagnetic TMR, the imbalance in spin polarization is the key origin according to the basic model for TMR proposed by Julliere 1 (Fig. 1a, Extended Data Fig. 2a,b and Methods). This suggests that similar TRS breaking due to the imbalance seen in octupole polarization might lead to antiferromagnetic TMR (Fig. 1b,d).
To examine whether a finite TMR effect can be generated between the parallel and antiparallel arrangements of the cluster magnetic octupole moments in Mn 3 Sn, we carried out a theoretical simulation from first principles (Methods and Extended Data Fig. 3). We used specific geometry for calculations-namely, the Mn 3 Sn/vacuum/Mn 3 Sn (0001) structure, in which octupole moments are inside the plane perpendicular to the conducting path. We calculated tunnelling conductances for the parallel and antiparallel arrangements of the octupole moments at the Fermi level, which give the finite positive TMR ratio (Extended Data Fig. 3d and Methods). Although the net magnetic moments of Mn 3 Sn were vanishingly small, the calculated TMR ratio was as large as that obtained in a ferromagnetic counterpart, Fe/vacuum/Fe MTJ (Extended Data Fig. 3i). This indicates that Mn 3 Sn has great potential for use in TMR-based devices such as MRAM, because of the macroscopic TRS breaking captured by cluster octupole moments (Methods). In addition, we found that the transmission properties are robust against both interfacial disorder and the relative lateral shift of the atomic layer at the interface, being useful for TMR applications (Methods, Extended Data Motivated by this possibility, we fabricated the Mn 3 Sn-based MTJ to demonstrate the TMR effect corresponding to binary states 0 and 1. To observe reliable tunnelling conduction, preparation of a continuous and smooth tunnelling barrier is essential. As shown in Fig. 2a, we fabricated the stack consisting of W (9 nm)/Mn 3 Sn (12 nm)/MgO (3.3 nm)/ Mn 3 Sn (42 nm)/Ta (5 nm) from the MgO substrate side (Methods). According to theory, tunnelling conductance can be potentially anisotropic ( Supplementary Information and Supplementary Fig. 1). a, Schematics of ferromagnetic (FM) tunnel junction sandwiching an insulating barrier (middle layer, darker colour); the parallel 0 and antiparallel 1 configurations of ferromagnetic moments (purple arrows) generate two different resistance states. b, Schematics of a tunnel junction made solely of antiferromagnetic (AFM) electrodes with a magnetic order parameterbreaking TRS, such as the cluster magnetic octupole in Mn 3 Sn. The antiferromagnet Mn 3 Sn has a hexagonal Ni 3 Sn-type structure (space group P63/mmc) 6 , and Mn moments (red and blue arrows) form a cluster magnetic octupole (orange arrows) consisting of six spins on the kagome bilayer. States 0 and 1 arise when the top and bottom octupole polarizations are parallel and antiparallel, respectively. c, Projected density of states (pDOS) onto 3d orbitals computed using DFT for two opposite spin states with σ + and σ − in Mn 3 Sn. The pDOS is symmetric in terms of spin polarization (red and blue arrows). Here the direction of spin polarization is parallel to the a axis of the crystal. d, Calculated pDOS with projection onto octupolar ordered states with opposite octupole moments τ+ and τ−. Here the polarization direction of the cluster magnetic octupole is parallel to the a axis of the crystal and lies in the kagome plane. Inset, corresponding octupole-majority and -minority states used for calculations. Black and red lines correspond to octupole-majority (pDOS maj ) and octupole-minority (pDOS min ) DOSs, respectively. The absolute value of pDOS maj is higher when energy level E is lower than Fermi energy E F , whereas that of pDOS min dominates at higher E. The polarity of octupole polarization changes sign by tuning the Fermi energy of Mn 3 Sn (Extended Data Fig. 2c).

Article
On the other hand, characterization of films by Hall and Kerr effects may be possible only when octupole polarization has the out-of-plane component. Given these two constraints, we fabricated an epitaxial film of Mn 3 Sn with the (0111) orientation aligned close to the out-of-plane direction by taking advantage of the fact that the MgO barrier tends to grow along the [001] direction (Extended Data Fig. 10 and Methods). Transmission electron microscopy (TEM) imaging confirmed the presence of stack films with a sharp and continuous interface between Mn 3 Sn and MgO (Fig. 2a, Extended Data Fig. 9 and Methods). In addition, the c axis of Mn 3 Sn was found to be approximately 30° off the normal direction of the film (Extended Data Fig. 10 and Methods), allowing us to characterize the films and to measure tunnelling conduction in the Mn 3 Sn/MgO/Mn 3 Sn trilayer, as discussed below.
For observation of the TMR effect, the top and bottom Mn 3 Sn electrodes are required to have two distinctive coercivities so that the parallel and antiparallel configurations of octupole polarizations arise as a function of magnetic field (Fig. 1b). We characterized the coercivities of the two Mn 3 Sn layers in the MTJ by field-sweep measurements of polar MOKE at room temperature, where a polarized light beam was injected into both top and bottom layers ( Fig. 2b and Methods). Figure 2c presents polar MOKE loops measured in magnetic field B applied perpendicular to the plane. The top and bottom Mn 3 Sn layers produced clear hysteresis, with square loops as a function of field B, and yielded sizeable changes in Kerr rotation angles |θ K | (Fig. 2c). It should be noted that the coercive field of B c , at approximately 0.5 T for the top Mn 3 Sn (Fig. 2c, red), is twofold that of B c at 0.25 T for the bottom layer ( Fig. 2c, blue). Therefore, these two coercivities should generate state 1 when top and bottom octupole moments are antiparallel.
We further examined the formation of binary states via Hall measurements in a magnetic field perpendicular to the film plane. To stabilize state 0, in which octupole moments in both the top and bottom Mn 3 Sn point upward, we first applied the magnetic field of +2 T, which is larger than all coercive fields. By sweeping the field from +2 to −2 T (Fig. 2d) we found two transitions at B c (approximately −0.2 and approximately −0.6 T) close to the coercivities seen in the MOKE measurements above, supporting the premise that coercive fields are nearly identical in the bulk and interface detected by the Hall effect and MOKE measurements, respectively. When the field passed through the smaller coercivity at B c (approximately -0.2 T), the octupole moment in the bottom Mn 3 Sn layer reversed, producing state 1 with antiparallel configuration (red regions in Fig. 2d,e). When the field passed through the larger coercivity of B c (approximately −0.6 T), the octupole moment in the top layer also began rotating downwards, establishing state 0 in which octupole moments were again parallel. When the field was scanned back from −2 to +2 T, these two transitions also occurred at B c (approximately 0.2 and approximately 0.6 T), stabilizing a further state 1 between 0.2 and 0.6 T (blue regions in Fig. 2d,e), as well as a state 0 greater than 0.6 T.
We then moved our focus to the field dependence of the TMR in the antiferromagnetic MTJ. By scanning the field applied perpendicularly to the film plane from +1.5 to −1.5 T, we found that resistance exhibited a clear drop when B was between −0.2 and −0.6 T. This low-resistance state (LRS) appeared again between 0.2 and 0.6 T when we swept the field backwards from −1.5 to +1.5 T. The LRS and the remaining high-resistance state (HRS) correspond to states 1 and 0 in the MOKE and Hall measurements (Fig. 2c,d)-that is, the antiparallel and parallel configurations of the top and bottom octupole moments, respectively. By adapting the definition TMR = (HRS - LRS)/LRS × 100%, we estimated the TMR ratio to be around 0.6% at room temperature in this device. The inset in Fig. 2e shows tunnelling resistance as a function of MgO thickness; the exponential increase in resistance confirmed the reliability of tunnelling conduction.
Our first observation of TMR in the all-antiferromagnetic tunnel junction Mn 3 Sn/MgO/Mn 3 Sn is outstanding. On the other hand, the magnitude is still small, raising the question of whether this arose from the TRS-breaking octupole polarization-as our theory suggests-or from spin polarization, which is widely accepted to generate the conventional TMR effects [1][2][3][4][5] . Previous theoretical work indicates that Mn 3 Sn may host the longitudinal spin-polarized current 7 because the non-collinear antiferromagnetic state breaks the TRS macroscopically 6,7 . In the following, we examine the latter scenario by investigating spin-polarized current in Mn 3 Sn through the conventional TMR scheme 1 .
Theoretically, spin-polarized current is described by a spinconductivity tensor σ ij k , in which electric field E along axis j drives the spin current flowing along direction i with spin polarization aligned along the k axis 7 . The non-collinear antichiral ordering produces momentum-dependent spin polarization on the Fermi surface, and thus the application of an electric field may drive net longitudinal spin-polarized current with the spin-polarization direction along the   Article octupole polarization ( Supplementary Fig. 2). To evaluate the magnitude and sign of the longitudinal spin-polarized current generated in Mn 3 Sn, we fabricated a Fe/MgO/Mn 3 Sn MTJ with an in-plane magnetized Fe layer using, respectively, the antiferromagnet Mn 3 Sn and the typical ferromagnet Fe as the source and detector of the spin-polarized current. We can use the magnetic field to orient spin polarization by directing octupole polarization along the magnetic field. First, for detection of σ ii j we applied the in-plane field so that the directions of electric field E and magnetic field B (i and j, respectively) were perpendicular (Fig. 3a). By sweeping the magnetic field along the in-plane direction j from +1.5 to −1.5 T and back to +1.5 T, a negative TMR ratio of 0.2% was observed (Fig. 3c). Assuming the spin polarization of Fe/MgO to be around 0.6, analysis based on the Julliere model yields a spin polarization of Mn 3 Sn of about −2 × 10 −3 (Methods) 1 . Here, the spin-polarization direction (solid blue arrow in Fig. 3a) is perpendicular to the electric field (solid blue arrow) and the negative sign of spin polarization is consistent with the theoretical analysis of σ ii j (ref. 7 ).
Second, we measured the TMR in another series of Fe/MgO/Mn 3 Sn MTJs with a perpendicular magnetized Fe layer as a function of magnetic field B along the out-of-plane direction parallel to electric field E, for detection of σ ii i (Methods and Fig. 3b). In contrast with the in-plane geometry above, a positive ratio of 0.5% (Fig. 3d) was observed in the out-of-plane magnetized Fe/MgO/Mn 3 Sn. Our analysis found the spin polarization of Mn 3 Sn to be approximately +4 × 10 −3 , consistent with the predicted positive sign of σ ii i for the case in which the spin polarization direction (small blue arrow in Fig. 3b) is parallel to the electric field (blue dashed arrow). The bias dependence of TMR (Extended Data Fig. 6), which may give useful information about the energy dependence of polarization 41 , clarified the symmetric bias dependence of the TMR ratio for the perpendicular (E⊥B) and parallel (E//B) configurations (Extended Data Fig. 7), confirming the anisotropy of the longitudinal spin-polarized current in Mn 3 Sn.
To further confirm the TRS-breaking character of the longitudinal spin-polarized current, we examined the TMR in the configuration shown in Fig. 3a after polarizion of the octupoles in Mn 3 Sn with an in-plane magnetic field, B of +1.5 T and −1.5 T. For each polarized state we measured the TMR by sweeping the in-plane magnetic field within 30 mT in which only Fe moments were switched. The two different polarities of TMR were clearly observed, corresponding to polarization in +1.5 T (Fig. 3e, top) and −1.5 T (Fig. 3e, bottom), in accordance with the TRS-breaking character of the spin current causing the hysteresis shown in Fig. 3c. Moreover, the TMR in Fe/MgO/Mn 3 Sn exhibited temperature dependence characteristic of Mn 3 Sn (Fig. 3f); it is only in the antichiral non-collinear phase in which the octupoles form a ferroic order that can generate the spin-polarized current 7 . Thus, the observed temperature dependence confirms that the TMR effect in Fe/MgO/Mn 3 Sn is driven by the spin-polarized current generated in Mn 3 Sn.
Finally, the spin polarization experimentally obtained for Mn 3 Sn allowed us to estimate the TMR ratio of Mn 3 Sn/MgO/Mn 3 Sn based on the Julliere model 1 (Methods). Namely, if we assume that the TMR is based on tunnelling between spin-polarized states as in conventional ferromagnets, the ratio should be about 0.002%, which is more than two orders of magnitude lower than the value of around 0.6% shown in Fig. 2e. In addition, with increased barrier thickness and tunnelling resistance area we found that the ratio was increased up to 1.6% (Extended Data Fig. 8 and Methods). This indicates that the TMR observed in Mn 3 Sn/MgO/Mn 3 Sn originated not from the spin-polarized current or weak ferromagnetic moments due to canting, but from TRS breaking in the antiferromagnet-that is, the momentum-dependent, spin-splitting bands 7,32,35,36 caused by the polarization of magnetic octupoles 37 .
In conclusion, we have demonstrated that the all-antiferromagnetic tunnel junction Mn 3 Sn/MgO/Mn 3 Sn shows the TMR effect at a ratio of about 2% at room temperature between the parallel and antiparallel alignments of octupole moments. The TMR ratio is far larger than its estimate based on the anisotropic spin polarization found for Mn 3 Sn. Besides, our theory predicts that the antiferromagnetic TMR effect originating from the TRS-breaking magnetic octupole may become as large as conventional TMR commonly seen in ferromagnets. Thus, our first observations of the all-antiferromagnetic TMR effect, as well as the anisotropic spin-polarized current, provide new and useful functionality, opening new directions in the research and development of antiferromagnetic spintronics and magnetic memory technology.

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Sample growth Mn 3 Sn/MgO/Mn 3 Sn MTJs. A W (9 nm)/Mn 3 Sn (12 nm)/MgO (3.3 nm)/
Mn 3 Sn (42 nm)/Ta (5 nm) (left: substrate side, right: surface side) multilayer was grown on a MgO(001) substrate. The W (9 nm)/Mn 3 Sn (12 nm)/ MgO (3.3 nm) layer was fabricated by the molecular beam epitaxy (MBE) method under ultrahigh vacuum (UHV) at a base pressure of 2 × 10 −8 Pa. The MgO(001) substrate was annealed at 800 °C for 10 min in the MBE chamber before deposition. The W layer (9 nm) was deposited at a rate of 0.1 Å s −1 at 300 °C and subsequent annealment at 800 °C for 10 min. The Mn 3 Sn layer (12 nm) was fabricated at a rate of 0.25 Å s −1 with coevaporation of Mn and Sn, in which the deposition rate of Mn and Sn was set for the stoichiometric composition Mn 3 Sn. The Mn 3 Sn layer (3 nm) was first deposited at room temperature and then annealed at 320 °C. The extra Mn 3 Sn layer (9 nm) was deposited at approximately 260 °C. Subsequently the MgO layer (3.3 nm) was fabricated at a rate of 0.1 Å s −1 at room temperature. The stack was later annealed at 600 °C for 30 min. As shown in Extended Data Fig. 1a, streak patterns were observed by reflection high-energy electron diffraction (RHEED), confirming the formation of epitaxial growth of flat interfaces in the W (9 nm)/Mn 3 Sn (12 nm)/MgO (3.3 nm) layer. The incident electron beam is parallel to the MgO [100] direction. The stack was transferred to a magnetron sputtering chamber with base pressure greater than 5 × 10 −7 Pa. In the sputtering chamber, the Mn 3 Sn (42 nm)/Ta (5 nm) layer was grown at room temperature by magnetron sputtering at a rate of 0.1 nm s −1 and a power of 60 W and Ar gas pressure of 0.5 Pa. After deposition, the entire stack was annealed at 450 °C to crystallize the Mn 3 Sn layer (42 nm), similar to our previous work for polycrystalline Mn 3 Sn 42 .
To investigate crystallinity and orientation, cross-sectional transmission electron microscopy (TEM) images for the W (9 nm)/Mn 3 Sn (12 nm)/MgO (3.3 nm)/Mn 3 Sn (42 nm)/Ta (5 nm) multilayer were taken at room temperature using a commercial TEM system ( JEOL, JEM-ARM200F). The maximum operating voltage was 200 kV. Samples for TEM observation were prepared from the TMR device, consisting of the multilayer. using a focused ion beam (Hitachi High-Tech NX2000, Ga 2-30 kV, Ar 1 kV). Before processing, a protective film (C (100 nm)/W (100 nm)) was deposited on an area of 10 × 3 μm 2 on the sample surface and the sample subsequently thinned by focused Ga and Ar ion beams. The TEM images presented in Fig. 2a and Extended Data Fig. 10a show the sharp interface between the Mn 3 Sn layers and the MgO layer. Nanobeam electron diffraction patterns of the Mn 3 Sn (top), MgO, Mn 3 Sn (bottom) and W layers show epitaxial growth from the W layer to the MgO barrier (Extended Data Fig. 10b-e). As shown in Extended Data Fig. 10a, we fabricated the epitaxial Mn 3 Sn layer on the MgO substrate(001)[010]||W(001) [110] stacks that have the (0111) orientation aligned close to the direction of thickness. In this orientation the kagome plane, which is the magnetic easy plane for the cluster magnetic octupole, is oriented along nearly 60° off the normal direction of the film-that is, the c axis of Mn 3 Sn is about 30° off the normal direction of the film, as shown by the transparent green plane in Extended Data Fig. 10f. Here the size of the (0111)-oriented Mn 3 Sn crystallite was confirmed to be approximately 100 nm. The MgO barrier on the epitaxial Mn 3 Sn (bottom) layer has around 10 nm crystallites and shows the (001) orientation with a mosaicness of about 10°. An atomic arrangement of each layer shows a potential epitaxial relationship (Extended Data Fig. 10g) in which the lattice constants of MgO, W and Mn 3 Sn are considered to be a = 4.21 Å for MgO, a = 3.17 Å for W 43 and a = 5.66 Å and c = 4.53 Å for Mn 3 Sn 6 . Whereas the Mn arrangement is not perfectly square and may thus have 90°-rotated variants in the film plane, the out-of-plane orientation for all variants should be similar to that observed in the present study (Extended Data Fig. 10a).  After deposition, the entire stack was annealed at 450 °C for 30 min. In this stack the V layer was used as the seed layer, which can help to induce the strong perpendicular magnetic anisotropy of ultrathin Fe (under 1 nm) 44,45 .
We carried out all fabrication processes in situ, including sample transfer from the MBE chamber to the sputtering chamber. The sputtering-grown Mn 3 Sn layer (42 nm) on the MgO layer was used for all samples. The composition of this Mn 3 Sn layer was determined to be Mn 3.15 Sn 0.85 by scanning electron microscopy-energy-dispersive X-ray spectroscopy. Although this composition is Mn rich, it is also a stable composition for a single phase of the D0 19 Mn 3 Sn 42 , in which excess Mn randomly occupies the Sn site 6,46 .

MTJ fabrication and magnetic and transport measurement
We encapsulated MTJ devices in SiO x , with electrical contacts formed from 60 nm Pt. Hall measurements were conducted at 300 K in a commercial physical property measurement system (Quantum Design). The field dependence of Hall resistivity was obtained after subtracting the longitudinal resistivity contribution, which was found to be constant as a function of magnetic field. Zero-field Hall resistivity, ρ H (B = 0) was estimated as (ρ H (B = +0) − ρ H (B = −0)/2. Here +0 and −0 were used to indicate zero magnetic field approached from +2 and −2 T, respectively. Tunnelling resistance was measured with a two-probe method in a probe system with an electromagnet at room temperature. A commercial source measurement unit (Keithley 2400, Tektronix) was used to measure resistance for the microfabricated magnetic tunnel junctions. Because a fixed resistor of 1 kΩ was series connected with the MTJ to protect it during measurements, MTJ data include 1 kΩ from the fixed resistor. Electrical measurements recorded below room temperature were performed under 10 −4 Pa in a vacuum chamber cooled by a helium compressor. The sample substrate was fixed on a Cu sample stage and temperature measured by a thermometer inside the sample stage.

MOKE magnetometry
The magnetic field dependence of MOKE was measured using a commercial system (NanoMOKE3, Quantum Design). The top and bottom Mn 3 Sn layers in the W (9 nm)/Mn 3 Sn (12 nm)/MgO (3.3 nm)/Mn 3 Sn (42 nm)/Ta (5 nm) stacks were used for measurement in a polar MOKE configuration under out-of-plane applied magnetic fields between -1.3 and +1.3 T at room temperature. MOKE loops were acquired using a 660 nm semiconductor laser and a spatial light modulator enabling acquisition of 20 hysteresis loops at a rate of 0.1 Hz. To obtain the MOKE signal, we subtracted the B-linear part originating from the extrinsic contribution (for example, a Faraday effect of the optical lenses). Our MOKE measurements for the W (9 nm)/Mn 3 Sn (12 nm)/MgO (3.3 nm)/ Mn 3 Sn (42 nm)/Ta (5 nm) stacks showed that the coercive field of B c (approximately 0.5 T) for the top Mn 3 Sn (Fig. 2c, red) was twice that of B c (about 0.25 T) for the bottom layer (Fig. 2c, blue), probably due to the difference in buffer layers facing each Mn 3 Sn (W for bottom and MgO for top), as well as to variation in the thickness of Mn 3 Sn.

Calculation of the projected density of states
The projected density of states (pDOS) of Mn 3 Sn and body-centred cubic Fe (bcc-Fe) was calculated with the Wannier functions obtained using the WANNIER90 package [47][48][49] , in which localized Wannier functions are constructed by projection of Bloch wave functions onto atomic orbitals. Bloch wave functions were obtained by density functional theory (DFT) calculations using the QUANTUM ESPRESSO (QE) packages 50,51 . In DFT calculations, the projector-augmented wave pseudopotential with spin-orbit couplings 52 was used, with exchange correlation taken into account by Perdew-Burke-Ernzerhof-type generalized gradient approximation 53 . For Mn 3 Sn, the lattice constants a = 5.665 Å and c = 4.531 Å were used 54 and k-point meshes were 7 × 7 × 7 and 8 × 8 × 8 for self-consistent field (scf) and non-scf calculations, respectively. The cut-off energies of wave function and charge density were 80 and 320 Ry, respectively. Bloch wave functions were projected onto the s-, p-and d-orbitals of Mn ions and the s-and p-orbitals of Sn ions. For bcc-Fe we used the lattice constant a = 2.87 Å, and 8 × 8 × 8 and 12 × 12 × 12 k-point grids were used for scf and non-scf calculations, respectively. We set the energy cut-offs of wave function and charge density as 80 and 500 Ry, respectively. Bloch states were projected onto the s-, p-and d-orbitals of the Fe ion. In Wannierization we set the k-point mesh as 8 × 8 × 8 and 12 × 12 × 12 for Mn 3 Sn and bcc-Fe, respectively. Using Wannier functions we calculated pDOS with the 64 × 64 × 64 k-point grid onto cluster magnetic octupolar ordered states of Mn 3 Sn and the magnetic ordered states of bcc-Fe.

Cluster magnetic octupole and its polarization
For the symmetry operation of the structural D 6h point group, the noncollinear magnetic order of Mn 3 Sn has the same transformation properties as the cluster magnetic octupole moment, with 1−1 , and thus its ferroic order breaks time-reversal symmetry macroscopically. This also indicates that the cluster magnetic octupole is parallel to the weak ferromagnetic moment (approximately 7 mμ B /Mn) due to spin canting, which arises as a result of the competition between Dzyaloshinskii-Moriya interaction, exchange coupling and single-ion anisotropy 56 . Therefore, the driving mechanism of the anomalous Hall effect of Mn 3 Sn can be interpreted as the ferroic order of the octupole moment of Mn 3 Sn, in the same way as the ferromagnetic order of the dipole moment of Fe. The cluster multipole theory is thus useful for understanding the underlying physics on non-collinear antiferromagnets such as Mn 3 Sn 37 . In the calculation of octupole polarization, we estimated the expectation value of the following operator with p = 3 and q = ±1 for the Bloch wave functions obtained by generalized gradient approximation calculation: ( ) is the number of atoms of the μth cluster, σ i is the Pauli matrices defined for the spin degrees of freedom of the ith atom, is the position of the ith atom, Y pq are the spherical harmonics and R i , θ i and ϕ i are the distance, polar angle and azimuthal angle, respectively, of the ith atom.

Estimation of spin polarization of Fe and Mn 3 Sn using the Julliere model
Based on the Julliere model 1 , the relative conduction change, ΔG between states 0 and 1 is described as ΔG = G × 2P 1 P 2 /(1 + P 1 P 2 ), where G, P 1 and P 2 correspond, respectively to the tunnelling conduction and spin-polarization ratio of the effective tunnelling density of states of magnetic electrodes 1 and 2. The spin polarization of Fe/MgO in our device is assumed to be about 0.6 (ref. 45 ).

Fermi level shift and resultant negative TMR
Given the strong Fermi energy dependence on the sign of octupole polarization (Extended Data Fig. 2c), the sign of TMR should be very sensitive to the Fermi energy of Mn 3 Sn. Therefore, interfacial engineering may influence the Fermi energy of Mn 3 Sn. Extended Data Fig. 4a shows the Hall resistivity of Mn 3 Sn as a function of temperature. When decreasing the temperature, Hall resistivity vanishes due to the phase transition from non-collinear antichiral to spin-spiral. Both the anomalous Hall effect and non-zero spin polarization exist only in the non-collinear antichiral phase of Mn 3 Sn rather than in the spinspiral phase. Nevertheless, phase transition temperature detected through TMR (black circle) was around 100 K lower than that (blue square) found in Hall resistivity (Extended Data Fig. 4a). This is most probably because the Fermi energy of the interfacial Mn 3 Sn shifted as compared with that of the Mn 3 Sn film, due to its contact with MgO. In addition, the distinct Fermi level shift of Mn 3 Sn at the bottom and top may have arisen due to variation in the thermal annealing process (Methods). Here we roughly simulated normalized TMR via octupole polarizations from bottom τ bottom and top Mn 3 Sn τ top , normalized TMR = τ bottom × τ top , assuming that the difference in Fermi energy shift is 1 eV (Extended Data Fig. 4b).

First-principles calculation of the tunnelling magnetoresistance effect with Mn 3 Sn electrodes
To theoretically simulate the TMR effect from first principles, we use the PWCOND package in the QE package [57][58][59] in which ballistic transport along the z direction is calculated by solving the scattering problem on the Bloch wave function obtained by DFT calculation 60 . Generally for TMR, it is important to have in-depth understanding of the barrier material [61][62][63] . On the other hand, before investigating the role of the barrier material it is crucial to determine whether there is any TMR effect in the all-antiferromagnetic tunnel junction. Thus, in our calculation, for simplicity we use Mn 3 Sn for the electrodes and the vacuum for the barrier. In practice, we calculate the tunnelling conductance in the Mn 3 Sn/vacuum/Mn 3 Sn MTJ. The MTJ in our calculations is made by stacking Mn 3 Sn along the c axis: the conducting path is perpendicular to the ab plane of Mn 3 Sn. The schematics of the MTJ are shown in Extended Data Fig. 3a. We calculated transmissions for both parallel and antiparallel configurations; the cluster magnetic octupole moments of the two electrodes point in the same and opposite directions in the parallel and antiparallel configurations, respectively. First, we separate the entire MTJ system into three parts: left and the right leads consisting of the bulk Mn 3 Sn and the scattering region, comprising a pair of two monolayers of Mn 3 Sn and the vacuum region between. We performed the DFT calculation with QE for each of the three parts. We set the k-point mesh as 7 × 7 × 7 for the leads and 7 × 7 × 1 for the scattering region. In calculation of the scattering region, the constraint on magnetic moments was imposed to stabilize the magnetic structure. When we calculated the electronic structure of the scattering region in the antiparallel configuration, to smoothly connect the leads and the scattering region in the transmission calculation we treated the doubled scattering region, which consists of the original scattering region and its copy, attached to the original one with its magnetic configuration inverted. In this way we calculated the electronic structure of the Mn 3 Sn/vacuum/Mn 3 Sn/Mn 3 Sn/vacuum/Mn 3 Sn system. The doubled scattering region was cut in half and only the original was considered in transmission calculations.
Then, connecting the leads and scattering region, we calculated transmission. We set the k ⊥ = (k x , k y ) point mesh in the xy plane as 32 × 32. We obtained transmissions at each k ⊥ -point for the parallel/antiparallel configurations, T P/AP (k ⊥ ); conductances for parallel/antiparallel configurations, G P/AP , were determined by the Landauer-Büttiker formula [64][65][66][67] The TMR ratio was calculated as (G P − G AP )/G AP .
The vacuum thickness, d, dependence of the total transmission at the Fermi level for the parallel and antiparallel configurations, is shown in Extended Data Fig. 3b. We also plotted the resistance-area product (RA) for each configuration, which is the normalized resistance given as (RA) = A/G, where A is the cross-section area. We found that G P was larger than G AP in all cases of d, and both G P and G AP almost exponentially decayed with d, as shown in Extended Data Fig. 3b,c. We thus obtained the positive TMR ratio (Extended Data Fig. 3d). Transmissions resolved by k ⊥ -points are shown in Extended Data Fig. 3e,f, which indicates that transmission behaviour does differ between the parallel and antiparallel configurations.
For comparison with the Mn 3 Sn/vacuum/Mn 3 Sn MTJ, we also investigated the transmission properties of a ferromagnetic MTJ with the vacuum barrier, the Fe/vacuum/Fe system. In the same manner as for the Mn 3 Sn/vacuum/Mn 3 Sn MTJ, we calculated electronic structures without spin-orbit couplings of the leads composed by the bulk bcc-Fe with 8 × 8 × 8 k-mesh, and of the scattering region which has the vacuum sandwiched between a pair of four monolayers of Fe with 8 × 8 × 1 k-mesh. We performed the transmission calculation using 100 × 100 k ⊥ -point grids. Extended Data Fig. 3g,h shows the d-dependence of total transmissions and RA values at the Fermi level for both parallel and antiparallel arrangements. The TMR ratio with respect to d is plotted in Extended Data Fig. 3i, taking as large a value as that in the Mn 3 Sn/vacuum/Mn 3 Sn MTJ. The results should be sufficient to serve as a qualitative reference for the Mn 3 Sn/vacuum/Mn 3 Sn MTJ, whereas a small non-monotonic change in the TMR ratio was observed, which could converge by the calculation with higher accuracy.
The electronic states that dominate tunnelling transport correspond to electrons tunnelling in the normal direction to the interface, and thus the polarization of such states is important for discussion of tunnelling physics 68 . In fact, the tunnelling conductance of the Fe/MgO/Fe MTJ has the peak at k ⊥ approximately 0, supporting this concept 61,62 . In a more complex system such as Mn 3 Sn, however, the states whose group velocity carries only normal incidence components exist not only at k ⊥ approximately 0 but also at general k points 69 , and such states should largely contribute to tunnelling conductance following the concepts proposed by Slonczewski 68 . Extended Data Fig. 3e,f shows the in-plane momentum (k x , k y ) dependence of transmission integrated over momentum along the tunnelling direction. Notably, our results clarify that transmission involves not only those states at k ⊥ approximately 0, but also of the widely extended momentum region in the Brillouin zone. Given that the results in the figures are those projected to the in-plane momentum, tunnelling electrons arise not only from momentum k ⊥ of approximately 0 related to normal incidence, but rather from entire region of the Brillouin zone. Thus, to qualitatively understand the mechanism of tunnelling conductance, we should focus not only on the states with k ⊥ approximately 0 but rather use the measure reflecting contributions from the entire Brillouin zone. Given the fact that the antiferromagnetic state of Mn 3 Sn can be viewed as the ferroic order of cluster magnetic octupole, such a measure should be the summation of octupole polarization over the entire momentum space-the density of states projected onto octupole polarization.

Effect of interfacial structures on transmission properties
We investigated the robustness of transmission properties against variation in interfacial structures in the Mn 3 Sn/vacuum/Mn 3 Sn MTJ. We examined two types of variation of the interface: interfacial disorder and lateral shift.
First we studied the effect of disorders. We incorporated interfacial disorders into the Mn 3 Sn/vacuum/Mn 3 Sn MTJ by artificially moving some atoms facing the vacuum barriers; we shifted upwards one of the Mn atoms in the lower layer of Mn 3 Sn at the interface, and shifted downwards one of the Mn atoms in the upper layer of Mn 3 Sn. These atoms were shifted by 0.113 Å-that is, 2.5% of the c axis length of Mn 3 Sn, which does not qualitatively change the electronic and magnetic properties of the system. Extended Data Fig. 5a-c shows the results of calculations against the disorder with barrier thickness 4.531 Å. Whereas shifting of atoms decreased transmission in the parallel configuration and increased it in the antiparallel configuration, overall transmission properties did not change. We also examined the case in which atoms move inversely to the case above: the atom in the lower layer moved downwards whereas that in the upper layer moved upwards. The results of these calculations (Extended Data Fig. 5d-f) suggest that transmission properties did not qualitatively change in this case either. Whereas it is better to use a larger supercell for more precise evaluation of interfacial disorder, we expect that TMR properties may not largely change from those shown here.
Second, we examined the effect of lateral shift. Mn 3 Sn has two layers in a unit cell along the c axis-say, A and B. In the calculations whose results are shown in Fig. 3, layers A and B are at the interface, which we call geometry-I. Here we also consider the case in which two B-layers face each other (see the inset in Extended Data Fig. 5i), which we call geometry-II. We show the results of the calculations with geometries-I and II in Fig. 5g-i. These results indicate that the properties in tunnelling conductance do not largely change, whereas total transmissions in geometry-II take larger values than those in geometry-I. We note that for geometry-II we take the doubled unit cell both for the parallel and antiparallel configurations, due to its geometry. We confirmed that the finite TMR effect is observed for other lattice-matching configurations-that is, when two electrodes have differently oriented easy axes of cluster magnetic octupoles by 120° in geometry-I ( Supplementary Fig. 4). We also examined the TMR effect by shifting the upper layer of Mn 3 Sn along the a axis by half of the lattice constant a, and confirmed that this shift qualitatively maintained the TMR effect ( Supplementary Fig. 5).

Bias-dependent TMR measurements
The bias dependence of TMR can provide useful information about the energy dependence of polarization 41 . Thus, we performed TMR measurements as a function of bias voltage in the Fe/MgO/Mn 3 Sn tunnel junction for both the perpendicular (E⊥B) and parallel (E//B) configurations for the electric (E) and magnetic fields (B), corresponding to Fig. 3c,d, respectively. In the following, we show the experimental summary for the perpendicular (E⊥B) case as a representative example (Extended Data Fig. 6). Extended Data Fig. 6a,b shows the minor loops under bias voltage of +0.6 and −0.5 V, in which Fe moments are switched while Mn 3 Sn moments remain fixed.
The key observation is the symmetric bias dependence of the TMR ratio on the perpendicular and parallel configurations (Extended Data Fig. 7). This becomes clear if we consider a bias-independent term of the order of around 0.14 % (horizontal dashed line). According to our measurement configuration, the hot electron arises from Mn 3 Sn and thus MR should reflect the unoccupied DOS of Fe for positive bias (Extended Data Fig. 7a). In this regime, the spin polarization of Mn 3 Sn should be determined by the DOS of Mn 3 Sn in the vicinity of the Fermi level. Spin polarization around the Fermi level in Mn 3 Sn is anisotropic and can be characterized by the tensor, as discussed above. Thus, MR should have negative and positive signs for perpendicular and parallel configurations, respectively (Extended Data Fig. 7b). This explains the following observations of our experiment. Namely, after subtracting the constant background MR term of around 0.14%, we find that MR has magnitudes similar to the opposite signs for perpendicular and parallel configurations. Bias dependence must arise from the unoccupied DOS of Fe and should be similar to that observed in the Fe/MgO/Fe MTJ 5 .
On the other hand, negative bias should provide the hot electron from Fe to Mn 3 Sn. Thus, MR would be determined by the spin polarizations of unoccupied DOS for Mn 3 Sn and DOS for Fe in the vicinity of the Fermi level. As compared with the positive bias regime, MR in the negative bias is strongly bias dependent (Extended Data Fig. 7b). This would be due to the bias voltage dependence of the anisotropic spin polarization in Mn 3 Sn. Interestingly, the MRs for both perpendicular and parallel MTJs converge at around 0.1% at bias V below −0.5. Most probably, this is a result of isotropic spin polarization of Mn 3 Sn from spin canting.

Measurement of the thickness dependence of TMR
The TMR in Mn 3 Sn/MgO/Mn 3 Sn MTJ was measured by variation in MgO thickness. When the barrier became thicker, TMR value increased, consistent with the theoretical calculations shown in Extended Data Fig. 2. We also performed a comparison between experiments and calculations, as shown in Extended Data Fig. 8. Whereas these calculations use the vacuum barrier, we set the x axis to be the resistance area rather than MgO thickness itself, for clarity.

Data availability
The data that support the findings of this study are available from the corresponding authors on reasonable request.