Domain structure and domain wall dynamics in topological chiral antiferromagnets from the viewpoint of magnetic octupole

Magnetoelectric transport phenomena such as the anisotropic magnetoresistance (AMR) and the anomalous Hall effect (AHE) have been central transport properties in traditional magnetoelectronics as well as magnetism. These electric responses were first applied for sensors thanks to high-quality thin film growth and micro-fabrication technological development. Furthermore, the discovery of the giant magnetoresistance^1,2 and tunnel magnetoresistance effects^3–6 in the magnetic multilayers triggered the research field, magnetoelectronics, leading to device applications such as sensors, HDD heads, and nonvolatile magnetoresistive memories. In these devices, the “magnetization” of a ferromagnet is an information bit and a magnetic order parameter for the total free energy. The macroscopic ferromagnet often exhibits a multi-domain state that has no net-magnetizations to minimize the magnetic free energy. However, the device application sometimes requires controlling the magnetic structure by using, e.g., the magnetocrystalline, interfacial, and shape anisotropy to realize a single domain structure.

Effective methods to manipulate the magnetic structures and dynamics of magnetic domains and domain walls have been developed. For example, the spin-transfer torque (STT)^7–11 enabled us to electrically control the magnetization in a ferromagnet without applying an external magnetic field. A typical application of the STT is the spin-transfer torque magnetic random access memory (STT-MRAM), advancing rapidly; it could replace the existing semiconductor memory in the future. Recently, in addition to the conventional STT using the ferromagnetic spin injector, the spin–orbit torque (SOT) generated via the spin Hall effect (SHE) or the Edelstein effect (EE) has led to more active research with a focus on the application of “low power consumption” magnetic solid-state memory devices.¹² 

Under such circumstances, antiferromagnets (AFMs) are attractive because of expectations for higher performance of magnetic devices as shown in Table I. (1) The antiferromagnet exhibits a compensated spin configuration in which the net magnetization is zero, which is suitable for high-density memory devices; (2) the spin dynamics due to the anisotropy field, $HA$⁠, and the exchange field, $HE$⁠, is in the THz range (⁠$ωAFM≈γHAHE$⁠), faster than that in the case of a ferromagnet (⁠$ωFM≈γHA$⁠) by about 2–3 orders of magnitude;^13,14 and (3) there is a high degree of freedom in selecting materials with Néel temperature above room temperature regardless of metals, insulators, or semiconductors.^15–17 Here, $γ$ is the gyromagnetic ratio (⁠$=1.76×1011 deg s−1 T−1$⁠), $HA$ is the anisotropy field due to magnetocrystalline anisotropy (< 1 T), and the field $HE$ ranges from about 100–1000 T. From the viewpoint of magnetic domains and domain walls, magnetostatic energy does not affect the formation of domain and domain walls and their dynamics in the AFM materials that have no macroscopic magnetization. Interestingly, they behave differently from ferromagnetic materials.

It has already been demonstrated that ferrimagnets at the angular momentum compensation temperature where the entire system behaves similarly to antiferromagnets with no net angular momentum. The rapid deceleration phenomenon called “Walker breakdown”¹⁸ due to the domain wall structural instability does not occur at this temperature. The domain wall speed could reach as fast as several km/s, an order of magnitude faster than that of the ferromagnetic domain wall.¹⁹ This finding suggests that a similar domain wall propagation could occur in AFMs. Therefore, the domain wall speed in the racetrack memory²⁰ can be increased using the AFMs. Importantly, the property of having no macroscopic magnetization brings the above advantages against ferromagnets. However, one cannot measure and control the electrical, optical, and thermal responses like ferromagnetic materials. This has been an issue in device development using AFMs and research on magnetic domains and domain walls.

The observation of AMR at room temperature in the antiferromagnet FeRh provided a significant opportunity to advance the AFM domain and its domain wall dynamics research. The AMR is magnetoresistance that depends on the relative angle between the magnetization and the current. The scattering of conduction electrons depends on the magnetization direction due to the anisotropic electronic structure caused by spin–orbit interaction. The ferromagnetic AMR varies as an even function of the magnetization direction; it changes in the collinear antiferromagnet with the angle between the current and the Néel vector. However, it has not been demonstrated because of the difficulty in controlling the antiferromagnetic order itself by an external magnetic field. Notably, the antiferromagnetic FeRh was proven to exhibit the AMR accompanying the change in the arrangement of antiferromagnetic order by cooling in the magnetic field from the antiferromagnetic-ferromagnetic transition temperature of 400 K.^21,22

It was demonstrated using CuMnAs²³ and Mn₂Au^24,25 that one could control the antiferromagnetic order electrically instead of using a magnetic field. Although these materials have spatial inversion symmetry as a system, each spin sublattice has a structure that locally breaks the spatial inversion symmetry. Therefore, the torque with opposite sign, “Néel spin–orbit torque”²⁶ exerted on the pair of spin sublattices, rotates the antiferromagnetic order to the direction perpendicular to the electric current. For a more systematic understanding, research carried out from the viewpoint of symmetry, mainly “cluster magnetic multipole,” revealed the AFMs that exhibit the controllable magnetic order via the Néel spin–orbit torque have a ferroic toroidal order.^27,28 Recently, the magnetic order of collinear antiferromagnetic insulator NiO with a simple rock salt NaCl structure appeared controllable using the spin accumulation generated by the SHE of Pt.^29,30 It has also been demonstrated that the resulting AFM state could be determined by spin Hall magnetoresistance (SMR).^29–31 In addition, the observation of antiferromagnetic domains by optical means has also been performed. A well-known method is a combination of x-ray Magnetic Linear Dichroism (XMLD) and Photoemission Electron Microscopy (PEEM).^32,33 Relatively compact table-top measurement systems based on magneto-optical effects such as the magneto-optical Voigt effect³⁴ and magneto-optical birefringence effect³⁵ have also been developed. In this way, the AMR effect, SMR effect, and magneto-optical effect can be used to measure the collinear antiferromagnetic order. We should note that all the effects described above have an even function response to the magnetization or time-reversal operation, that is, the relative angle of the Néel vector to the current or the polarization plane of the light.

Another major trend is the study of non-collinear antiferromagnetic materials.^36–44 One of the representative materials is Mn₃Sn, exhibiting odd-function responses such as the AHE³⁷ and anomalous Nernst effect (ANE)^45,46 at room temperature, similar to ferromagnets. Mn₃Sn has a crystal structure in which Kagome lattices composed of magnetic Mn atoms are stacking in the [0001] direction. Mn magnetic moments form a non-collinear antiferromagnetic order in the Kagome plane with the inverse triangular spin structure. Wherein, an inverse triangular spin structure having a 120° order with a uniform negative vector chirality below ∼430 K, where the Mn moments slightly cant and produce a tiny uncompensated moment of ∼0.003 μ_B/Mn within the Kagome plane, allowing the antiferromagnetic order to be controlled by a magnetic field [Fig. 1(a)].^47,48 Interestingly, the cluster magnetic octupole comprising two sets of three spin sublattices⁴⁹ takes ferroic order on the Kagome lattice, macroscopically breaking the time-reversal symmetry. Previous studies revealed that this cluster magnetic octupole, not the magnetic dipole, plays an essential role as the magnetic order parameter for the AHE and ANE.⁴⁹ 

Generally, the AHE has two responsible mechanisms: extrinsic mechanism such as side jump and skew scattering and intrinsic mechanism derived from the Berry phase created by the band structure.⁵⁰ Since the SHE^51,52 and AHE usually share the mechanism, a SHE could appear in the antiferromagnetic Mn₃Sn. As expected, experiments using micro-fabricated Mn₃Sn samples revealed the conventional SHE⁵³ and also the “Magnetic Spin Hall Effect (MSHE)”⁵⁴ that reflects the broken time-reversal symmetry different from the ordinary SHE. The polarity of spin accumulation at the Mn₃Sn surface via the MSHE changes its sign upon reversal of the magnetic octupole. This spintronics function may lead to a magnetic switching device that breaks through the conventional technique. In addition, it has been clarified that the cluster magnetic octupole is an order parameter that determines the distribution of the giant Berry curvature in the momentum space associated with the Weyl node, i.e., the topology of the electronic structure.⁵⁵ Based on this concept, various behaviors have been discovered in unconventional AFMs.

The magneto-optical Kerr effect (MOKE) is a phenomenon in which the polarization plane of linearly polarized incident light rotates when reflected from a magnetized surface. Therefore, we can study magnetization processes and magnetic domain structures nondestructively. Generally, the magnetization and spin–orbit interaction cause this effect, and thus ferromagnets have been the primary materials studied. On the other hand, it has been challenging to detect the MOKE signal in AFMs with zero or very small magnetization. However, similar to the AHE, the MOKE in AFMs has recently been theoretically discussed in terms of symmetry in a system having a finite Berry curvature in momentum space due to macroscopic time-reversal symmetry breaking. The theory also predicted a giant MOKE in AFMs.⁵⁷ Indeed, a single crystal Mn₃Sn exhibited a MOKE signal comparable to that observed in a conventional ferromagnet at room temperature and zero magnetic field [Fig. 2(a)].⁵⁸ Comparing the Kerr rotation angle spectrum and the first-principles calculation confirmed that this giant Kerr effect appears in association with the cluster magnetic octupole [Fig. 2(b)].

Moreover, the magneto-optical microscope measurement demonstrated that the reversal of the antiferromagnetic domain accompanies the change in contrast of the MOKE image (the positive and negative signs of the Kerr rotation angle) comprising the magnetic octupoles. Figures 2(c) and 2(d) show that the reverse MOD nucleates and propagates with increasing the applied magnetic field. A giant MOKE derived from the ferroic magnetic octupole also occurs in Mn₃Ge,⁶¹ which has a similar crystal and magnetic structure to Mn₃Sn.⁶¹ As shown in the temperature dependence of Fig. 3(a), a large Kerr rotation angle of about 10 mdeg appears below 365 K or less. On the other hand, the thermomagnetic curve shows a sharp decrease at about 370 K. These experimental results indicate that the Néel temperature T_N lies in the range from 365 to 370 K. The discrepancy of 5 K is within an error of temperature controller of the MOKE magnetometer. For the Mn₃Ge single crystal growth, one can use the flux method, the Czochralski, and Bridgman methods. Because the surface of the as-grown crystal obtained by the flux method is sufficiently smooth, optical measurements are possible without surface polishing. In this case, we can expect almost the same characteristics at the surface and inside the sample. It is compatible with the magneto-optical Kerr effect measurement measuring several tens of nanometers from the surface. The Kerr effect is observable on the as-grown surface although it is not on all the surfaces of Mn₃Ge [Fig. 3(b)]. In addition, if the surface is mirror-polished and then appropriately heat-treated, the Kerr effect appears uniformly all over the surface [Fig. 3(c)].

Interestingly, the switching field of the Kerr effect is significantly different between the as-grown surface and the polished surface [Figs. 3(e) and 3(f)]. The octupole domain switching fields for Mn₃Sn and Mn₃Ge single crystals estimated from magnetization and AHE measurements range several to several tens of mT [Fig. 3(d)], smaller than the expected anisotropy field of several T. These results suggest that the reversal of the magnetic octupole domain occurs via the reverse MOD nucleation and propagation of the domain wall, not by collective rotation. Mn₃Ge obtained by the flux method exhibits a tiny switching field because of very few pinning sites that hinder the domain wall displacement inside the sample. More recent studies have also demonstrated magnetic domain observations on the as-grown surface. When considering the domain wall structure as twisting cluster octupoles, this experiment shows that the Néel-type domain wall with the rotating octupole moments in the Kagome plane coexists with the Bloch-type domain wall with the twisting octupole moments in the direction perpendicular to the plane. These two octupole domain walls align almost orthogonally with each other, reflecting the crystal structure. In the ferromagnetic materials, the Bloch and Néel domain walls' magnetostatic energies differ significantly depending on the sample thickness, and thus they rarely coexist. One should note that Mn₃Sn and Mn₃Ge can accommodate both the Néel- and Bloch-type octupole domain walls, unlike the ferromagnetic materials. This is because the domain wall energies of the Néel- and Bloch-type octupole walls are almost the same due to the small magnetostatic energy and the strong in-plane magnetic anisotropy that keeps spins lying in the Kagome plane.

Recently, research on domain wall observation using the ANE has been reported in addition to the MOKE experiments.⁶² The ANE is a phenomenon in which a temperature gradient applied to a ferromagnetic metal induces an electromotive force (anomalous Nernst voltage) in the direction perpendicular to both the temperature gradient and the magnetization. As shown in Figs. 4(a) and 4(b), the sample only needs electrodes for measuring the anomalous Nernst voltage. Therefore, the ANE is useful to study domain wall dynamics in sub micrometer-scale thin wires. Since the anomalous Nernst voltage is proportional to the areal ratio of magnetic domains with positive and negative magnetization, the magnetic domain state can be inferred from the voltage change when the magnetic domain walls propagate in a magnetic field [Fig. 4(c)].

As mentioned above, Mn₃Sn is an antiferromagnet, exhibiting a giant ANE derived from the cluster magnetic octupole.^45,46 Therefore, it is possible to detect the magnetic octupole domain (MOD) information using the ANE. Figure 4(d) shows an ANE measurement device composed of a focused ion beam (FIB) microfabricated slab of the Mn₃Sn single-crystal with dimensions of a few μm in width, 10 μm in length, and 200 nm in thickness and a Ta heater.^63,64 In this device, the heat current generated by the Ta heater flows in the sample plane; the reversal of the MOD perpendicular to the sample plane gives rise to an associated ANE signal. The Nernst voltages, V₁ and V₂, drop abruptly at ±0.5 T and reverse their sign after taking plateaus of about zero voltage in the field up to ±0.7 T [Fig. 4(e)].⁶⁴ The results indicate a magnetic octupole domain wall (MODW) along the short edge of the sample pinned at the center of the slab after nucleation of the reversed MOD at ±0.5 T. The depinning and subsequent propagation of the MODW take place at ±0.7 T.

In addition to the above-described measurement of the ANE by applying a uniform heat flow to the sample using the heater, scanning laser-induced thermal gradient microscopy^65,66 is an effective alternative method to image the multi MOD structure. Figure 5(a) shows the mapped ANE voltage generated in an epitaxial Mn₃Sn thin film by the laser-induced thermal gradient across the thickness with scanning the spot with a diameter of 1.5 μm. This method yields a multi MOD image shown in Fig. 5(b).⁶⁷ The spatial resolution is of the order of the laser spot diameter. In addition, Fig. 5(c) demonstrates that writing a memory bit of MOD is possible by magnetic field cooling processes after laser heating locally above the Néel temperature.

Magnetoresistance and Hall resistance measurements are the standard methods for evaluating transport properties. One should note that the magnetic domain structure and its dynamics can also be studied using these methods. Interestingly, these techniques have already been applied for studying the MOD structure and its dynamics in FIB fabricated Mn₃Sn thin slabs from single-crystals.⁶⁸ As shown in Fig. 6(a), in the sample having eight voltage terminals (1–8) in addition to the horizontal current electrodes, the spike-and dip-shaped resistance changes appear in the magnetic field dependence of the lateral (longitudinal) resistance [Fig. 6(b)]. Such electrical responses were first found in a ferromagnetic multilayer film with perpendicular magnetic anisotropy, coined as asymmetric magnetoresistance (ASMR).⁶⁹ As shown in Figs. 6(c-i) and (c-ii), when the current is applied to the sample, the AHE causes non-equilibrium charge accumulation at the sample edges. In the magnetic domains with up (red) and down (blue) magnetization [Figs. 6(c-iii) and 6(c-iv)], the sign of the accumulated charge is reversed so that a potential difference occurs at both edges across the domain wall. Therefore, when the domain wall enters between the terminals attached to the same side of the sample, the potential difference can be measured as an ASMR signal. Indeed, this effect appears in Mn₃Sn, which has a vanishingly small magnetization but shows a giant AHE due to the ordered MOD [Fig. 6(b)]. Furthermore, the spike-shaped variation in the ASMR signal is in good agreement with the stepwise reversal processes of the MOD in the AHE in Fig. 6(d). The above consideration infers the MODW propagation, as shown in the insets of Fig. 6(b).

Furthermore, as shown in Fig. 7(a), the MODW nucleation and propagation experiments in the FIB microfabricated Mn₃Sn single-crystal wedge have been conducted using current pulses. Here, a magnetic field of 860 Oe is applied perpendicular to the basal plane, and the thickness of the slab is varied from left to right (500 nm–1 μm) along the thin wire. A 50 ms pulse current of 10⁹ A/m² applied to the Hall probe 1 nucleates a reversed MOD; simultaneously, it gives rise to the change in the AHE signal as shown on the left side of Fig. 7(c). When the MODW propagates and reaches at Hall probe 2, as shown in the right inset of Fig. 7(c), the change in the anomalous Hall effect occurs. Importantly, this behavior occurs with the applied current density above 6.7 × 10⁹ A/m² in the [0001] direction of the sample [Fig. 7(b)].

The MODW speed increases with the applied current density, being 0.1–1 m/s at about $109$ A/m², and a fast domain wall motion comparable to that of ferrimagnets can be expected at about 10^11–12 A/m² [Fig. 7(d)]. The MODW propagates from right to left irrespective of the sign of the pulse current. However, as shown in Fig. 7(e), the velocity slightly changes depending on whether the applied current is positive or negative. This behavior's origin is a spin torque associated with magnetic octupole. Still, we have to conduct more detailed experiments to clarify the above-mentioned current polarity-dependent MODW motion. Furthermore, to understand the MOD structure and MODW dynamics observed in this experiment, we performed atomic-scale micromagnetic calculations of the MODW structure [Figs. 7(f) and 7(g)].

The results show that the Mn₃Sn thin wire should have a 180° AFM domain wall structure consisting of four MODs, β+, α−, γ+, and β− [classified in Fig. 1(b)] with three 60° AFM domain walls whose domain wall structure is Bloch like, twisting along the [0001] direction. The simulation also reveals a relatively long domain wall of a net length of ∼800 nm containing two MODs with the size of each being ∼100 nm and three MODWs with the length of each being ∼200 nm. In this experiment, one can only detect the 180° domain wall dynamics because the terminals' distance was several μm. Still, observing the response derived from the 60° domain wall is a challenge for the future.

In the field of spintronics, a physical phenomenon using an antiferromagnet has been attracting attention. Until recently, it has been challenging to detect and control the macroscopic response in antiferromagnetic materials. However, the AMR, SMR effects, and Néel spin–orbit torque could detect and control the magnetic state in collinear antiferromagnets. In addition, the AHE, ANE, and MOKE have occurred in non-collinear (chiral) antiferromagnets. These experimental findings assure the possibility of AFM as a candidate material for spintronic applications. The above macroscopic responses are derived from the symmetry of matter. More importantly, we can use cluster magnetic multipoles as order parameters in chiral antiferromagnets to understand the phenomenon more simply and intuitively.

In this paper, we set our focus on magnetic chiral antiferromagnets Mn₃Sn and Mn₃Ge that show the odd-function response to time-reversal operations such as the giant AHE, ANE, and MOKE that were thought to appear only in ferromagnets. We presented recent studies on the domain structure and domain wall dynamics from the octupole viewpoint. In the first half of the manuscript, we described the “Observation method of magnetic domain composed of the cluster magnetic octupole” using the response to light, heat, and electricity. In the second half, “Control of domain wall using the electrical method and measurement of its dynamics,” we introduced the results by comparing theories such as micromagnetic calculation. Until now, experiments have been conducted mainly on bulk single crystal samples of Mn₃Sn, but recently, Mn₃Sn thin film samples showing a large odd-function response such as the AHE^70,71 and ANE⁶⁷ have also been reported.

There have been observed a variety of thin-film and interface phenomena such as the spin Hall effect in a thin film sample with a thickness of several tens to hundreds of nanometers,⁵³ THz Hall effect determined from the Faraday rotation of linearly polarized THz light,^72,73 and most recently, electrical control of the magnetic octupole in Mn₃Sn/Pt (or W) bilayers by SOT.⁷⁴ Many theoretical studies on the electrical control of chiral antiferromagnetic order^56,75–78 have been conducted, and further understanding will be made from spintronics measurements using thin films and microfabricated samples. Very interestingly, the collinear antiferromagnet was believed to have only an even function response to the time-reversal operation, such as the magnetoresistance effect. However, we now know that, by introducing an appropriate asymmetry into the crystal structure, the anomalous Hall effect can appear.^79,80 The importance of the multipole concept will be applied to a broader range of antiferromagnetic materials. We hope this article can contribute to the future development of topological AFMs studies.

Some of the research introduced in this article is based on the collaboration with S. Nakatsuji, R. Arita, T. Koretsune, M. Suzuki, M. Ikhlas, H. Man, H. Narita, M. Wu, H. Isshiki, T. Chien, S. Sugimoto, E. Nakatani, Y. Yamane, K. Kondo, M. Kimata, T. Tomita, R. D. Shull, D. B. Gopman, Y. P. Kabanov, O. M. J. van't Erve, J. Orenstein, L. Wu, D. Rees, S. Patankar, C. L. Chien, and Y. Li. The authors sincerely appreciate a lot of support through experiments and discussions with collaborators such as D. Qu, P. K. Muduli, N. Leo, A. Kobayashi, H. Tsai, T. Nomoto, A. Sakai, T. Nakano, K. Yakushiji, and S. Miwa. The research studies mentioned above are partly supported by CREST (Grant No. JPMJCR18T3 and JPMJCR15Q5), Grant-in-Aid for Scientific Research on Innovative Areas, “Nano Spin Conversion Science” (Grant Nos. 26103001 and 26103002), “J-Physics” (Grant Nos. 15H05882 and 15H05883), JSPS KAKENHI (Grant Nos. 15H05702 and 16H02209), and Johns Hopkins University Institute for Quantum Matter (DE-FG02-08ER46544).