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Tomohiro Uchimura, Jiahao Han, Ping Tang, Ju-Young Yoon, [Yutaro Takeuchi](https://orcid.org/0000-0002-5031-1347), Yuta Yamane, Shun Kanai, Gerrit E. W. Bauer, Hideo Ohno, Shunsuke Fukami

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[Unconventional Spin Hall Magnetoresistance in Noncollinear Antiferromagnet/Heavy-Metal Stacks](https://mdr.nims.go.jp/datasets/8e036e28-c58a-4411-b8b8-59cde6808960)

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Unconventional Spin Hall Magnetoresistance in Non-Collinear Antiferromagnet/Heavy Metal StacksTomohiro Uchimura1,2, Jiahao Han1,3*, Ping Tang3*, Ju-Young Yoon1, Yutaro Takeuchi3,4, Yuta Yamane1,5, Shun Kanai1,2,3,6,7,8,9, Gerrit E. W. Bauer3,8,10,11, Hideo Ohno1,2,3,8,12, and Shunsuke Fukami1,2,3,8,12,13*1Laboratory for Nanoelectronics and Spintronics, Research Institute of Electrical Communication, Tohoku University, Sendai, Japan.2Graduate School of Engineering, Tohoku University, Sendai, Japan.3Advanced Institute for Materials Research, Tohoku University, Sendai, Japan.4International Center for Young Scientists, National Institute for Materials Science, Tsukuba, Japan.5Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai, Japan.6PRESTO, Japan Science and Technology Agency, Kawaguchi, Japan.7Division for the Establishment of Frontier Sciences of Organization for Advanced Studies at Tohoku University, Tohoku University, Sendai, Japan.8Center for Science and Innovation in Spintronics, Tohoku University, Sendai, Japan.9National Institutes for Quantum Science and Technology, Takasaki, Japan.10Institute for Materials Research, Tohoku University, Sendai, Japan.11Kavli Institute for Theoretical Sciences, University of the Chinese Academy of Sciences, Beijing, China.12Center for Innovative Integrated Electronic Systems, Tohoku University, Sendai, Japan.13Inamori Research Institute for Science, Kyoto, Japan.*Corresponding authors: jiahao.han.c8@tohoku.ac.jp; tang.ping.a2@tohoku.ac.jp; s-fukami@riec.tohoku.ac.jpAbstract: We study the spin Hall magnetoresistance (SMR) in non-collinear antiferromagnet Mn3Sn/heavy metal stacks. The measured SMR exhibits peculiar magnetic field angle and magnitude dependence that sharply deviates from the conventional SMR theory based on the damping-like spin-transfer torque. An alternative model based on a coherent field-like torque reproduces the observations well. Our work reveals a previously unrecognized mechanism of interfacial exchange that indicates a precession of the conduction-electron spins due to the collective local exchange fields of the non-collinear antiferromagnetic order. The unraveled physics is essential to understand and control spin transport in unconventional magnetic materials.Antiferromagnets (AFMs) have been enthusiastically studied in spintronics due to their potential towards high-efficiency, high-density, and ultrafast devices for memory and computing [1,2,3]. When an AFM is attached to a normal conductor with significant spin-orbit interaction, an applied charge current  in the latter can switch [4,5,6,7,8,9] or excite the antiferromagnetic order [10,11,12] by a spin Hall current  and a concomitant non-equilibrium spin accumulation  at the interface [Fig. 1(a)]. Meanwhile, a reflected spin current  at the interface is subsequently converted to a feedback charge current that modulates the resistance of the normal conductor. This effect is known as the spin Hall magnetoresistance (SMR) [13]. Since its first observation in ferromagnets (FMs), the studies on the SMR and related effects have advanced the understanding of spin dynamics and transport induced by the spin Hall effect [13,14,15], the Rashba-Edelstein effect [16], the spin-momentum locking of topological surface states [17], the orbital Hall/Rashba-Edelstein effect [18], etc., and to study the interplay between spin currents and complex magnetic textures such as antiferromagnetic orders [19,20,21,22,23,24], spiral spin structures [25], and skyrmions [26,27].At a resistive magnet/normal conductor interface with dominant exchange interaction, the SMR is determined by the spin current that flows along the film normal and is absorbed at the interface, formulated by scattering theory as  [28,29]. The unit vectors  represent the set of sublattice magnetic moments with index  and  denotes the polarization of the incident spin current  in the coordinate system of Fig. 1(a).  is the spin-mixing conductance in terms of the reflection amplitudes  of spin-up and spin-down electrons, where  is the magnitude of the electron charge and  is the reduced Planck constant. The real and imaginary parts of  quantify the contributions of individual damping-like and field-like torques by the sum over , dissipating  and rotating it from the  direction. In the conventional theory, only the  term contributes to the spin current component  parallel to , thereby modulating the resistivity ρ of the normal conductor in the form of a feedback charge current induced by the inverse spin Hall effect. Combining scattering with diffusion theory in the normal conductor leads to , where  is a constant and  is the SMR magnitude that increases with [30,31]. This picture reflects the current understanding of the SMR in FMs and AFMs [13,20,21,22,23] and shares similar physics with the spin pumping and the spin Seebeck effect [32]. In the particular case of collinear AFMs with two sublattice moments and a Néel vector ,  reduces to , leading to  [21,22,23]. To first order the field-like torque due to  contributes an anomalous Hall-like effect in ferromagnetic systems [33,34,35]. However, it remains elusive whether the field-like torque can play a fundamental role in the antiferromagnetic SMR. The recently studied non-collinear AFMs, represented by D019-Mn3Sn [36], offer an intriguing platform to tackle the above questions by its unique magnetic order. Mn3Sn exhibits an inverse-triangular chiral-spin structure containing three magnetic sublattices  in the (0001) kagome plane [Fig. 1(b)]. A slight canting of the sublattice moments results in a small net moment . The absence of a common spin quantization axis that defines spin-up and spin-down eigenstates in such a non-collinear spin structure [37] is beyond the conventional scattering-diffusion theory for the SMR [28]. The spin structure of Mn3Sn sources a variety of unusual phenomena, including the antiferromagnetic anomalous Hall effect [36], the octupole-driven tunneling magnetoresistance [37], magnetic engineering of the electronic quantum-metric structure [38], electric manipulation of topological antiferromagnetic states [9], and characteristic chiral-spin dynamics [12,39].Here, through a combined experimental and theoretical study, we report unexpected SMR features in Mn3Sn/heavy metal stacks beyond current theories. By rotating the chiral-spin structure via a magnetic field, we find that the field angle dependence of the measured SMR follows the net magnetic moment rather than that of individual sublattices. This observation sharply differs from the modeling based on the damping-like spin-transfer torque on each sublattice, and is well reproduced by a phenomenological model based on a coherent field-like torque that stems from a precession of the conduction-electron spins around the collective interface exchange fields. We find here that such a process modulates the spin accumulation in the heavy metal and contributes to the SMR to second order in the “field-like” spin-mixing conductance. This model is further supported by field strength-dependent experiments. We deposited stacks of Ru (5 nm)/Mn3Sn (30 nm)/MgO (1.3 nm)/Ru (1 nm) on MgO (111) single-crystal substrates by magnetron sputtering, followed by annealing at 500 °C for an hour. The light metal Ru serves as a buffer layer to support the epitaxial growth of (0001)-oriented (or C-plane oriented) D019-Mn3Sn [40]. The out-of-plane X-ray diffraction (XRD) pattern of the stack is shown in Fig. 1(c). The existence of (0002) and (0004) peaks of D019-Mn3Sn confirm the (0001) epitaxy with the kagome plane lying in the film plane. The scanning transmission electron microscopy (STEM) image in Fig. 1(d) visualizes the (0001) epitaxy and the single-crystal ordering. We next fabricated the AFM/heavy metal stacks of Ru (5 nm)/Mn3Sn (15 nm)/Pt (5 nm) (Mn3Sn/Pt in short) by etching Mn3Sn (15 nm)/MgO/Ru and sputter-depositing Pt at room temperature without breaking the vacuum. This protocol prevents the formation of secondary phases such as a ferromagnetic MnPt alloy at the Mn3Sn/Pt interface, see Supplemental Section 1 of Supplemental Material [41], which includes references [42,43,44,45,46,47,48]. A control sample of Ru (5 nm)/Mn3Sn (15 nm)/MgO (1.3 nm)/Ru (1 nm) (Mn3Sn/MgO in short) was prepared by the same route as the Pt-capped sample, i.e., etching Mn3Sn (15 nm)/MgO/Ru and sputter-depositing MgO/Ru at room temperature, which minimizes differences due to etching with the main experiment. We carried out measurements at room temperature unless specified.The magnetic hysteresis loops of Mn3Sn/Pt in Supplemental Section 1 [41] show a coercivity <30 mT and a weak net magnetization of 18 mT at the alignment field of 0.5 T, as observed in previous studies on Mn3Sn thin films with the same orientation [48]. An applied field in the kagome plane of several teslas therefore fully rotates the chiral-spin structure. Moreover, nearly identical hysteresis loops after zero-field and field cooling protocols are consistent with the absence of secondary ferromagnetic phases at the Mn3Sn/Pt interface.Figure 2(a) depicts the scheme for measuring the longitudinal resistance R of Mn3Sn/Pt in a patterned stripe device with a rotating magnetic field H of 9 T in the kagome plane. The magnetic field is strong enough to fully rotate the chiral-spin structure and make. The MR ratio  is negative and reaches a peak value of  (or 0) when m is parallel (or perpendicular) to the electron spin [Fig. 2(b)]. This trend is inconsistent with reports on the SMR of collinear AFM/Pt bilayers, but agrees with the phenomenology of the conventional SMR in FM/Pt. We argue that it should not be attributed to an extra ferromagnetic order of Mn3Sn, if any. The first reason is the angle-dependent MR of Mn3Sn/MgO without Pt capping [Fig. 2(c)] that strongly differs from the MR of Mn3Sn/Pt and the anisotropic magnetoresistance (AMR) of a ferromagnetic order. Instead, it is consistent with the chiral anomaly-induced transport in the Weyl semimetal states [9,47]. To the best of our knowledge, ferromagnetic phases hardly occur at the surface or interfaces of high-quality Mn3Sn, as indicated in Supplemental Section 1 [41]. The second reason is that the MR ratio keeps increasing with the magnetic field up to 9 T (to be shown later in Fig. 4), while a hypothetical residual ferromagnetic phase and its induced MR should saturate at a moderate field [49]. The ordinary MR in a Pt sample without Mn3Sn is very small [Fig. 2(d)]. These observations are consistent with earlier MR measurements in Mn3Sn-based heterostructures in terms of magnitude and angle dependence [50,51]. Moving forward, this study will reveal distinct features and origins of the spin transport subject to the non-collinear antiferromagnetic order, to be discussed in Figs. 3 and 4.In FM/Pt, an interfacial proximity effect may contribute to the AMR by magnetizing the Pt atoms at the interface [52]. A common approach to assess such contribution is to rotate the magnetization in the orthogonal planes of xy, yz, and zx. As a key feature, the SMR (AMR) should dominate (disappear) in the yz scan, as the angle between the electron spin (charge current) and the magnetization varies (does not change) [13,53]. We perform the yz scan in a -oriented (or M-plane oriented) Mn3Sn sample grown on heavy metal W/Ta buffer layers (the epitaxial growth follows the methods in references [12,40]), where the kagome plane aligns in the yz plane so that the magnetic field can fully rotate the chiral-spin structure [39] [Fig. 2(e)]. The measured MR in Fig. 2(f) shows the same trend as that in Fig. 2(b), which supports the SMR mechanism. The SMR decreases with the thickness of Mn3Sn, which can be explained by the shunting of the current in the W/Ta heavy metal layers by the metallic AFM (Supplemental Section 2 [41]). Furthermore, given that the magnetization of Mn3Sn is much smaller than that of the ferrimagnetic Y3Fe5O12 and the ferromagnetic Co, the proximity-induced AMR is unlikely to contribute by >10–5 to the total MR ratio of Mn3Sn/Pt. Thus, we attribute the observed MR in the (0001) and -oriented Mn3Sn/heavy metal samples to the SMR. The unequal SMR magnitudes of these samples should be due to the different heavy metals, interfacial structures, and magnetic orientations. The angle-dependent MR with all rotational planes of (0001) and -oriented Mn3Sn/heavy metal samples in Supplemental Section 3 [41] confirms our SMR scenario.We now move on to explain the origin of the observed exceptional SMR for the non-collinear AFM Mn3Sn. We start with calculating the orientations of the unit vectors  () [Fig. 3(a)] by minimizing the magnetic free energy density , with exchange constant , bulk Dzyaloshinskii-Moriya interaction constant , anisotropy constant , spontaneous magnetization of one magnetic sublattice , and unit vector  connecting two Sn atoms as the closest neighbors of a Mn atom, which gives rise to the six-fold anisotropy [54]. Figure 3(b) plots the y-component of  as a function of the angle of a 9 T in-plane magnetic field. We first calculate the angle dependence of the SMR in the conventional formula of  [Fig. 3(c)], where each  interacts individually with the electron spin through the damping-like transfer of spin angular momentum. However, it fails to reproduce the measured result in Fig. 2(b). On the other hand, the plot  in Fig. 3(d) agrees well with the observations, suggesting the emergence of a predominantly field-like torque. This calculation disregards the interfacial Dzyaloshinskii-Moriya interaction from Pt and the resultant out-of-kagome-plane magnetic moments that alternate among Mn atoms but do not modify the net moment of Mn3Sn [38]. Given the finite SMR of W/Ta/-oriented Mn3Sn in which such tilting vanishes, this approximation should not affect our interpretation of the SMR.Next we show that non-collinear spin-diffusion theory explains the unconventional SMR in terms of the emergence of a field-like torque. The spin current generated by the spin Hall effect of Pt (with thickness d) establishes a non-uniform spin density s along the thickness (z) direction and experiences a torque  by the three sublattice moments at the Mn3Sn/Pt interface (z = 0). The stationary-state spin continuity equation in Pt reads . We take , with  the density of states per unit volume at the Fermi level and  the spin chemical potential.  is the spin current density that varies along z, with  and  the ordinary resistivity and the spin Hall angle of Pt, and E the applied electric field along x.  is the spin relaxation time [28]. The interfacial s-d exchange interaction from the chiral-spin structure of Mn3Sn precesses the electron spin [Fig. 4(a)] by the torquewhere  is the interfacial s-d exchange integral in units of eVm and the delta function  indicates the interface torque [55]. While Eq. (1) has the same form as the field-like torque in the existing scattering theory, calculating the rotation of s from the incident spin polarization  poses a self-consistency problem that has been overlooked in the conventional scattering-diffusion theory for the SMR. The summation in Eq. (1) implies that the electron spin precession is a collective consequence of the local exchange fields: the torque arises coherently from the chiral-spin structure. With boundary conditions  and , we calculate  and its feedback charge current density  in the electric field direction (x) through the inverse spin Hall effect. Accordingly, the total resistivity is given by (see derivations in Supplemental Section 4 [41])where  is the spin diffusion length of Pt,  is a dimensionless parameter, and  is a “field-like” spin-mixing conductance. Eq. (2) has been simplified from the general expression in Supplemental Section 4 for  [41]. It reflects the second-order contribution in  and reproduces the angle-dependent SMR of the non-collinear AFM. Taking , ,  [49], and the measured SMR ratio of  that solely counts the relative resistance change of Pt (deduced from Figs. 2(b) and 2(c), after considering the current shunting and the opposite MR in Ru/Mn3Sn), we arrive at  and  at 9 T, which is comparable to the real-part spin-mixing conductance in FMs and collinear AFMs [13,49,56] (for a direct comparison, one needs to multiply  by ). Differences with the full version of Eq. (2) [41] do not significantly influence the magnitudes determined here. Our  is proportional to  that measures the distortion from the perfect inverse-triangular chiral-spin structure with zero net moment. In FMs and collinear AFMs, the orientation of a unit-vector magnetization or Néel vector is sufficient to determine the SMR from the damping-like torque [22,49]. In contrast, the perfect chiral-spin structure cancels the effect of field-like torque on the electron spin. Hence, observing the angle-dependent SMR in our system requires a finite , which is encoded in . To verify this hypothesis, we measure the MR in Mn3Sn/Pt with a rotating magnetic field from 2 to 9 T and find that the MR magnitude indeed increases with field strength [Fig. 4(b)]. By substituting the calculated orientations of  for different field strengths into Eq. (2) for the SMR [Fig. 4(c)] and considering the field-dependent chiral anomaly in the full stack (Supplemental Section 5 [41]), the agreement of experiments and calculations in Fig. 4(d) supports our model and interpretation. Using the values of  and  and =Pt, we obtain , matching the interface exchange constant of FM/normal metal systems [55]. Our observations should not be attributed to the recently reported small paramagnetic SMR of –10–5 at low temperatures [57] that does not reflect the antiferromagnetic order essential to obtain the agreement in Fig. 4(d). The low-temperature MR in the chiral-spin phase of Mn3Sn can be explained by our SMR model (Supplemental Section 6 [41]).One may ask why the current-induced manipulation of non-collinear AFMs is well described by the damping-like torque even without considering the field-like torque. From previous work [12,54] we know that the field-like torque for a typical threshold current density of 107 A/cm2 is too weak to overcome the coercivity of Mn3Sn. In contrast, the damping-like torque from the perpendicular-to-kagome-plane component of s triggers the internal exchange interaction of Mn3Sn, which is strong enough to rotate the chiral-spin structure. But why is “field-like” spin-mixing conductance in Mn3Sn two orders of magnitude larger than that of FMs, either calculated [58] or measured in terms of an interface-induced anomalous Hall-like effect [35]? In wave-vector k resolved calculation  approximates , but  has both signs, leading to an effective cancellation when integrating over the complicated Fermi surface of transition-metal FMs [35]. Here the spins of all electrons scattered at the interface appear to precess in an effective magnetic field of the same sign, which could be confirmed by first-principles calculations or dedicated measurements of the anomalous Hall effect. In summary, by comparing experiments and modeling, we reveal and interpret an unconventional SMR in non-collinear AFM Mn3Sn/heavy metal stacks. The SMR can be explained by an interfacial field-like torque, which arises from the conduction-electron spin precession around the collective local exchange fields of the chiral-spin structure and thus modulates the spin current that causes the SMR in conjunction with the spin Hall effects. 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(b) Atomic and spin structures in a kagome plane of Mn3Sn. The shadowed atoms locate in the neighboring kagome plane. (c) Out-of-plane XRD plot and (d) Cross-section STEM image of a Ru (5 nm)/Mn3Sn (30 nm)/MgO (1.3 nm)/Ru (1 nm) sample on a MgO (111) single-crystal substrate, in which Mn3Sn has a (0001) epitaxy.Fig. 2. Measured SMR in the Mn3Sn/heavy metal samples. (a) Schematic of the (0001)-oriented Mn3Sn/Pt sample with a charge current along  and a magnetic field H in the xy plane. (b) to (d) MR ratio measured in the samples of Mn3Sn/Pt, Mn3Sn/MgO, and just Pt in the configuration of (a). (e) Schematic of the W/Ta/-oriented Mn3Sn sample with a charge current along  and a magnetic field H in the yz plane. (f) MR ratio measured in the sample of (e). The lines in (b) and (f) are fits to . The magnetic field is kept at 9 T in this figure.Fig. 3. Calculated orientations of the sublattice magnetic moments  in Mn3Sn. (a) Orientations of  and the net magnetic moment m under a magnetic field of 9 T along x The canting of  and  from the edges of an equilateral triangle is exaggerated for a clear view. (b) y-component of , (c) , and (d)  as a function of the magnetic field angle.Fig. 4. Origin of the unconventional SMR. (a) Precession of conduction-electron spin due to the collective local exchange fields of the chiral-spin structure. The incident spin polarization is along y. (b) Measured MR and (c) calculated MR [SMR of Eq. (2)] in the (0001)-oriented Mn3Sn/Pt sample at different magnetic field strengths. The lines in (b) are fits to . (d) Comparison of the measured and calculated MR magnitudes (absolute values). The data are normalized by the values at 9 T.17image4.pngimage1.jpegimage2.pngimage3.png