Revealing the higher-order spin nature of the Hall effect in non-collinear antiferromagnet $\mathrm{Mn_{3}Ni_{0.35}Cu_{0.65}N}$

Broken time reversal symmetry ($\mathcal{T}$) and its interplay with the spin-orbit coupling (SOC) result in the transverse flow of electrons generating an anomalous Hall voltage, a signature that distinguishes a ferromagnet (FM) from conventional antiferromagnets (AFM) [1, 2]. However, it has been predicted that a certain class of AFMs with noncollinear spin textures can lead to an unusual type of anomalous Hall effect (AHE) that can exist even in the absence of a net magnetization due to spin-lattice coupling [3, 4]. Whilst in noncollinear antiferromagnets (NC-AFM) it has been shown that their weak magnetization is insufficient to account for the AHE [5, 6, 7, 8, 9], further work to experimentally substantiate the fundamental mechanism responsible for the observed AHE signals has still not been undertaken. This is because any studies of the AHE have, so far, only been performed with the electrical measurements in plane while driving a magnetic field out of the plane, entangling the magnetization signal to any novel contribution to the AHE. In order to understand the origin of the AHE signal, one must go beyond the perpendicular field geometry and apply the field in different directions in space to identify the symmetry, and from this the mechanism leading to the AHE signal. Here we show the full dependence of the AHE in the NC-AFM Mn₃Ni_0.35Cu_0.65N ($\mathrm{Mn_{3}NiCuN}$) when the magnetic field is swept not only out-of-plane but also in the plane. The in-plane field experimental configuration, by construction, does not allow for the conventional dipole (magnetization) component of the AHE signal to contribute. We can therefore show, using a variety of theoretical techniques, that at high fields the in-plane AHE comes purely from the octupole moment and, surprisingly, an additional topological Hall-like (THE) signal occurs at low fields. Our results demonstrate that in NC-AFMs one must expand beyond the dipole contribution to the AHE to explain their rich, coexisting orders. Such coexisting orders go beyond the magnetization dynamics achievable in conventional FMs and AFMs. Harnessing these coexisting orders paves the way for spintronic devices that go beyond the state-of-the-art.

The antisymmetric AHE in the NC-AFM arises from the octupole moment produced by the noncollinear frustrated spin structure in the kagome plane [3, 10] which is in the (111) plane in the case of Mn₃NiCuN, the material we are investigating in this work (Fig. 1(a)). This can be visualized as an emergent octupole moment [11, 12, 10, 13, 14, 15, 16], $\vec{Q}_{T_{1g}}$ (the first term in equation (1)) that purely originates from the coplanar orientation of the spin texture lying in the kagome plane (Fig. 1(a)). In $\mathrm{Mn_{3}NiCuN}$, the direction of $\vec{Q}_{T_{1g}}$ points out of the kagome plane, along [111] direction. Upon expanding the AHE signal in the local spins, beyond the conventional ferromagnetic case, the octupole and topological Hall effects [17, 18, 19, 20] emerge as higher order contributions to the Hall effect in noncollinear compounds. Hence, the total contributions to the AHE are,

where equation (1) is obtained by using representation theory to derive the irreducible representations of $\mathrm{Mn_{3}NiCuN}$ as described in section ‘2 Representation theory’ of the supplementary material [21]. The second term of equation (1) represents the conventional AHE, proportional to the net magnetic dipole moment, $\vec{M}_{T_{1g}}$, as observed in a regular ferromagnet such as Fe [1], and that can arise in our system due to the canting of the spins upon the application of an external magnetic field. Generally, in this type of NC-AFM [22], including our system (see Fig. S2(a) in ‘1.1 Sample characterization’ [21]), the induced moment is very small [22] and therefore has a very small effect on the AHE.

Equation (1) predicts that $\vec{Q}_{T_{1g}}$ produces the maximum AHE when the magnetic field is swept out of the plane while the electrical measurements are performed in the kagome plane. This arrangement is similar to the measurement of AHE in a regular ferromagnet (FM) where the direction of the applied electric current ($J_{xx}$), voltage measurement ($V_{xy}$) and magnetic moment ($M_{z}$) are orthogonal to each other ($V_{xy}\propto(J_{xx}M_{z})$). Alternatively, in NC-AFMs with a vanishing magnetic moment, $\vec{Q}_{T_{1g}}$ plays the role of a fictitious magnetization, akin to $M_{z}$ in ferromagnets. Nonetheless our symmetry analysis reveals a fundamental difference between the octupole driven AHE and the conventional dipole driven AHE: the magnetic octupole supports a specific finite AHE response when the external field is rotated in the plane of the electrical measurements, whereas the dipole driven AHE remains unaffected. Thus, in plane fields have to be used to reveal this contribution.

The octupole vector $\vec{Q}_{T_{1g}}$ hosts a total of eight poles in $\mathrm{Mn_{3}NiCuN}$ as shown with the grey vectors in Fig. 1(b), two of which point out of the plane, along the [111] direction and six others have projections both in the plane and out of the plane with 120^∘ in-plane rotational symmetry enforced by the crystal structure (space group $Pm\overline{3}m$ (No. 221), point group $m\overline{3}m$ ($O_{h}$)). It is evident that when an external field is applied out-of-plane, along the [111] axis, one expects to obtain an AHE from the combination of octupole ($\vec{Q}_{T_{1g}}$) and dipole ($\vec{M}_{T_{1g}}$) contributions. However, our symmetry analysis additionally shows (see ‘2.6 Approximate ground state’ [21]) that when the in-plane applied magnetic field is collinear to the in-plane projections of any one of the six components of these $\vec{Q}_{T_{1g}}$ vectors, it immediately couples to the octupoles by reorienting the spin-configuration into one of the equivalent (111) planes as shown in Fig. 1(c). This coupling provides direct access to the out-of-plane projection of $\vec{Q}_{T_{1g}}$ which can lead to an AHE as per equation (1) when electrical measurements are performed in the plane. Due to this, one expects a 120^∘ angular dependence of the measured AHE as theoretically calculated in Fig. 1(d). We point out here that we plot all components of the $\vec{Q}_{T_{1g}}$ vector in Fig. 1(d) and find that its dependence on the magnetic field resembles that of the magnetization, $\vec{M}_{T_{1g}}$, which illustrates its role as a fictitious magnetization; however, the key difference is that the octupole produces finite contributions to the in-plane AHE, unlike $\vec{M}_{T_{1g}}$. This defines a distinct difference from the conventional (dipole-driven) AHE in a FM as the AHE is not allowed to occur when electrical measurements and applied magnetic field are coplanar.

In addition, our theoretical calculations suggest a third term in equation (1) that predicts a nontrivial Hall effect originating from the scalar chirality of the spin textures. Such a quantity was first discussed in high temperature superconductors [23, 24] when considering contributions to the Hall conductivity, whose underlying physics can similarly be attributed to the THE in a skyrmion [25, 17, 18, 19, 20]. This can be observed in the low field regime when the spins reorient, maximizing the scalar spin chirality.

To test these theoretical predictions, $\mathrm{Mn_{3}NiCuN}$ (111) thin film of 20 nm thickness are grown on an MgO (111) substrate, and then capped with a 3 nm thin Pt layer to prevent oxidation. After the thin film growth, Hall bar devices were patterned on the (111) kagome plane as shown in Fig. 2(g). The details of the sample preparation can be found in the method section and in reference [22]. First, we verify the presence of the $\vec{Q}_{T_{1g}}$ component’s contribution to the AHE signal by measuring the standard Hall effect in the kagome plane while sweeping the magnetic field out-of-plane field along the [111] direction (z-axis) at 100 K, well below the Néel temperature ($T_{N}\sim$ 200 K) [22]. The longitudinal resistivity of the film is $\sim 150~{}\mu\Omega\mathrm{cm}^{-1}$ at a temperature 100 K. Fig. 2(a) shows the measured AHE data for the z-field sweep that can be fitted by using $R_{xy}=A\tanh(B)$. We also verify that this AHE signal disappears above the Néel temperature (see Fig. S2(b) in ‘1.1 Sample characterization’ [21]). This result is consistent within the framework of the octupole moment ($\vec{Q}_{T_{1g}}$) induced AHE (equation (1), first term), but additionally a small magnetic moment is also present in our system (see Fig. S2(a) in ‘1.1 Sample characterization’ [21]), meaning that the magnetic dipole component ($\vec{M}_{T_{1g}}$) also contributes to the AHE. The measured anomalous Hall resistance ($R_{xy}$) is comparable to the previous reports [26, 9, 27, 8].

Now we perform the measurements of transverse resistance on the (111) plane while sweeping the magnetic field in the plane along different in-plane crystallographic directions (Figs. 2(b-c)). We observe two important features: (1) AHE-like signals, representing a step in the measured resistance, and (2) topological Hall-like (THE) signals, which are additional features only appearing in the low field regime which we will discuss later. Our data can be fitted well to equation (2),

where $R^{\mathrm{Oct}}_{xy}$ represents the maximum contribution coming from the octupole moment, and $R^{\mathrm{SSC}}_{xy}$ the maximum contribution of the scalar spin chirality, $B$ is the magnitude of the magnetic field, $B_{c}$ is the coercivity, $a$ is related to the slope of $R^{\mathrm{Oct}}_{xy}$ switching, and c is related to the width of the scalar spin chirality signal.

The observation of AHE-like signal in such coplanar geometry is a nontrivial finding as it is strictly prohibited in regular FMs. However, it is a signature that applied magnetic fields can access one of the components of $\vec{Q}_{T_{1g}}$ depending on the direction of the magnetic field-sweep with respect to the crystal axis (Figs. 1(c-d)) which is further evident following the red curve in Fig. 2(h) that exhibits a roughly 120^∘ angular dependence, consistent with our theoretical predictions (Fig. 1). The error-bars in the data points represent the standard deviation in five different measurements.

In these experiments surprising additional features appear at low field values which can be fitted by a Gaussian function that changes sign for positive and negative applied field cycling direction. Features like these are often attributed to THE as they occur in the presence of skyrmions whose nontrivial topology can induce an emergent effective magnetic field [25, 18]. The spin textures in our system exhibit no such topology [25], however it is possible that in the low field regime the spins rotate out of their coplanar orientation and maximize the scalar spin chirality ($\vec{S}_{i}\cdot(\vec{S}_{j}\times\vec{S}_{k})$) [17] thereby producing THE-like signals as predicted by the third term in equation (1). A consequence of this is that for certain $\phi$ where the THE-like component is larger than the $\vec{Q}_{T_{1g}}$ component, $R_{xy}$ switches before B changes polarity as shown in Fig. 2(b) for $\phi=-75^{\circ}$. This would be thermodynamically prohibited if the AHE has contributions only from $\vec{Q}_{T_{1g}}$ or $\vec{M}_{T_{1g}}$. To verify the source of the additional contribution, we compute the components of the scalar spin chirality by parameterizing a free energy expression of the irreducible representations with first principles calculations and solve a Landau-Lifshitz-Gilbert equation to provide the dependence of the scalar spin chirality ($\vec{S}_{i}\cdot(\vec{S}_{j}\times\vec{S}_{k})$) and octupole ($\vec{Q}_{T_{1g}}$) moment as a function of in-plane magnetic field angle $\phi$. The simulation suggests that the non-coplanar spin-textures are possible and predicts that both the octupole signal (arising from $\vec{Q}_{T_{1g}}$) and the THE-like signal (arising from $\vec{S}_{i}\cdot(\vec{S}_{j}\times\vec{S}_{k})$) possess an angular dependence with 120^∘ rotational symmetry (Fig. 2(i)). This result is qualitatively consistent with the angular dependence of the both the signals in our experiment (Fig. 2(h)) complete with their opposing signs.

Whilst attempts have been made to quantify the behavior of the complex spin structures present in NC-AFMs using simple pictures to explain the sign change, for example, in the AHE [5, 6, 8, 9, 28, 29, 30, 31, 32, 33, 7, 34, 35, 36], our work provides a full understanding of the signal of the AHE as a function of the magnetic field. Our analysis of the underlying spin dynamics elucidates a clear interplay of coexisting orders within a noncollinear antiferromagnet, which sets this class of magnetic order apart from conventional ferromagnets and antiferromagnets. These collinear spin structures host only one order parameter apiece, the magnetization and the Néel vector, whereas noncollinear antiferromagnets exhibit further complexities that have direct consequences on the AHE that can only be understood through expansion in local spins beyond the ferromagnetic case. For $\mathrm{Mn_{3}NiCuN}$, and subsequently the whole class of $\mathrm{Mn_{3}XY}$ compounds, one can uncover the coexistence of three distinct orders: the magnetization, a quantity in part responsible for the conventional AHE in this NC-AFM, the octupole contribution, that provides both conventional and also in-plane AHE contributions and finally the THE-like component generated when the spins are non-coplanar. Although this particular material is merely a prototypical example of the vast array of other NC-AFM orders, we present a very clear paradigm shift in the way these materials are to be viewed: that they have multiple unconventional order parameters which behave like a magnetization, yet they affect the Hall transport in unique ways.

In summary, we systematically study the symmetry of the transverse Hall resistance in $\mathrm{Mn_{3}NiCuN}$ (111) by applying magnetic fields in the kagome plane along related crystallographic directions. We find that whenever the external applied magnetic field is collinear to the in-plane projections of any one of the octupoles, an AHE can be detected even in the limit of vanishing magnetization. The six octupoles that have in-plane components lead to a 120^∘ symmetric AHE signal even when magnetic field and electrical measurements are coplanar, a remarkable contrast from the conventional AHE in FMs. We observe THE-like features and decompose the signal into octupole and THE-like components, finding the THE-like components to behave with a similar 120^∘ angular dependence as the octupole component while sweeping the magnetic field in the kagome plane. We attribute this effect to a scalar-chirality emerging during the spin-reorientation at low magnetic fields. Our experimental results are well supported by theoretical predictions. Our work represents a significant step in uncovering the complex, non-trivial phenomena in NC-AFMs, greatly enhancing the understanding of the octupole moments, and distilling the complex spin textures into clear, coexisting orders. This understanding opens the possibility to explore multiple means to harness NC-AFMs for novel spintronic technological applications.

Acknowledgments

We acknowledge funding from King Abdullah University of Science and Technology (KAUST) under award 2020 $-$ CRG8 $-$ 4048. A.M was supported by the Excellence Initiative of Aix-Marseille University - A*MIdex, a French ”Investissements d’Avenir program”. In addition, the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) $-$ Grant No. TRR 173/2 $-$ 268565370 Spin+X (projects A01, BO2 and A11). A.R, B.B, L.L.D and M.K acknowledge funding from the European Union’s Framework Programme for Research and Innovation Horizon 2020 (2014 $-$ 2020) under the Marie Skłodowska-Curie Grant Agreement No. 860060 (ITN MagnEFi). Y.M., J.S., W.F., and Y.Y. acknowledge the funding under the Joint Sino-German Research Projects (Chinese Grant No. 12061131002 and German Grant No. 1731/10-1) and the Sino-German Mobility Programme (Grant No. M-0142). R.Y.D. and L.L-D. acknowledge funding from Spanish ”Ministerio de Ciencia e Innovacion” under project PID2020-117024GD-C41. T.G.S., F.R.L., D.G. and Y.M. gratefully acknowledge the Jülich Supercomputing Centre for providing computational resources under project jiff40. T.H and H.A acknowledge funding from the Japan Society for the Promotion of Science (KAKENHI Grant Nos. 20H02602 and 19K15445).