Electrical nucleation, displacement, and detection of antiferromagnetic domain walls in the chiral antiferromagnet Mn3Sn

Antiferromagnets exhibiting distinctive responses to the electric and magnetic fields have attracted attention as breakthrough materials in spintronics. The current-induced Néel-order spin-orbit torque can manipulate the antiferromagnetic domain wall (AFDW) in a collinear CuMnAs owing to a lack of local inversion symmetry. Here, we demonstrate that the electrical nucleation, displacement, and detection of AFDWs are also possible in a noncollinear antiferromagnet, i.e., chiral Mn3Sn with local inversion symmetry. The asymmetric magnetoresistance measurements reveal that AFDWs align parallel to the kagome planes in the microfabricated wire. Numerical calculation shows these AFDWs consist of stepwise sub-micron size Bloch wall-like spin textures in which the octupole moment gradually rotates over three segments of domain walls. We further observed that the application of a pulse-current drives these octupole based AFDWs along the wire. Our findings could provide a guiding principle for engineering the AFDW structure in the chiral antiferromagnetic materials. Antiferromagnetic systems for application in spintronics are less developed that their ferromagnetic counterparts but offer the prospect of increased stability and speed in their magnetisation dynamics. Here, the authors investigate the electrical detection and displacement of antiferromagnetic domain walls in Mn3Sn single crystals.

lectrical manipulation of the magnetic domain 1-3 is the essential technology for spintronics. In recent years, antiferromagnetic spintronics 4 has attracted considerable interest because of various advantages, such as negligible stray fields and fast spin dynamics in the frequency range of subterahertz. It has, however, been challenging to control magnetic states of antiferromagnets by electrical means due to zero spontaneous magnetization, and weak magnetic susceptibilities in magnetic fields. The recent demonstration of field-like, Néelorder spin-orbit torque in collinear antiferromagnets with local inversion asymmetry 5 , CuMnAs 6,7 , and Mn 2 Au 8 , led to the breakthrough in the antiferromagnetic spintronics. These works have revealed a promising potential for electrical manipulation of antiferromagnetic domain walls (AFDWs) and paved the way for the use of antiferromagnets as well as ferromagnets for memory device applications.
Besides, the new class of noncollinear chiral antiferromagnet, Mn 3 Sn, has currently attracted a great deal of attention owing to its intriguing electronic properties. Mn 3 Sn has a hexagonal kagome lattice structure with a space group, P6 3 /mmc 9 , recently discovered as a magnetic Weyl semimetal 10 . Despite vanishingly small magnetization [11][12][13] , the inverse triangular chiral antiferromagnetic phase of Mn 3 Sn below 430 K shows unusual transport properties, such as a substantial anomalous Hall effect (AHE) 14,15 , giant anomalous Nernst effect 16 , strong anisotropic magnetoconductance 10 , and magnetic spin Hall effect 17 . These properties originate from the macroscopic breaking of timereversal symmetry (TRS) associated with the noncollinear antiferromagnetic order. A recent theory by Suzuki et al. 18 has clarified that it is a ferroic order of a cluster magnetic octupole that breaks TRS. Higo et al. 19 performed the magneto-optical Kerr effect (MOKE) measurements and succeeded in demonstrating the presence of the TRS-broken magnetic octupolar domains and domain walls. Figure 1a, b shows the crystal structure and cluster magnetic octupole in Mn 3 Sn. The cluster octupole moment resides on hexagonally located six Mn atoms between neighboring kagome planes z = 0 and z = 1/2 as enclosed by blue lines in Fig. 1a, and has six spin configurations labeled by α±, β±, and γ± domains indicated in Fig. 1b 18,20 . For instance, the α+ domain contains the magnetic structure characterized by the octupole moment, which breaks TRS, and induces a very small remanent magnetization (~2 mμ B per Mn) along the octupole, 2 1 10 ½ direction (blue arrow in Fig. 1b). In the same manner, the cluster octupole can be defined for all six domains and rotates every 60°within kagome planes. For the use of Mn 3 Sn as antiferromagnetic spintronic materials, we need to manipulate such a TRS-broken domain structure by electrical means.
In sharp contrast with conventional antiferromagnets, there are two significant technical benefits in Mn 3 Sn. First, we can use the external magnetic field to control the direction of the very weak remanent magnetization along the polarization axis of the cluster octupole moment 14,18 . Second, a topological response appears corresponding to the octupole configuration in the transport property, such as AHE 14,15 . Therefore, a similar experimental procedure used for ferromagnets can be applied to Mn 3 Sn.
In this work, we have investigated the method for electrical manipulation of AFDWs in a microfabricated single-crystal Mn 3 Sn. We observed the asymmetric magnetoresistance (AS-MR) effect 21 due to AFDW displacement in Mn 3 Sn, which revealed that a submicrometer Bloch-like octupole domain wall is trapped parallel to kagome planes in-between the two pairs of electrodes. By using the above detection method, we have also demonstrated AFDW injection and subsequent displacement as a response to the local injection of electric current.

Results
Electrical detection of AFDWs. Figure 2a shows the scanning electron microscope image of the sample for detecting AFDW by magnetoresistance measurements. The four pairs of detection electrodes 1-5, 2-6, 3-7, and 4-8 were attached to a μm-scale Mn 3 Sn rectangular plate perpendicular to the [0001] direction, besides current injection pads (I + , I − ) connected parallel to the [0001] direction. The external magnetic field H was applied along the [01 10] direction perpendicular to the sample plane to make the antiferromagnetic single-domain state. Figure 2b, c presents the typical longitudinal magnetoresistance curves performed by using two different pairs of electrodes (6-7 and 2-3), where red (blue) symbols indicate the results for the positive (negative) field sweep. For the measurements using electrodes 6-7 (Fig. 2b), a hysteretic dip (peak) structure appeared in the magnetoresistance ρ 67 for the positive (negative) sweep around H~+3200 Oe (−3100 Oe).
For the measurement by using electrodes 2-3 ( Fig. 2c), on the other hand, a peak (dip) in the magnetoresistance ρ 23 appeared around the positive (negative) coercivity for the positive (negative) sweep. Such odd responses in the longitudinal magnetoresistance originate from the AS-MR effect known for large AHE materials 21 Fig. 1b. Each Mn moment at overlapping two triangular sublattices is labeled with "1-6". The exchange integrals are illustrated by blue arrows between the closest Mn moments in the same sublattice (j 1 ) and the neighboring kagome planes (j 2 ), respectively. b Six types of cluster octupole moments consisting of two kagome triangles between neighboring layers 18,20 . The octupole moment parallel to a very small remnant magnetization is illustrated by blue arrows.  These results, in other words, indicate that AFDW moves along the [0001] direction during the fieldsweeping process. Figure 2d, the transverse Hall resistivities ρ 26 and ρ 37 normalized by the high field saturation value ρ HS (|H| = 1.1 T), supports the above hypothesis. The switching field of the left-hand side ρ 26 is always smaller than that of the right-hand side ρ 37 . Furthermore, the field range where the AS-MR takes place coincides well with the range where the difference in switching field is present between ρ 26 and the above experimental facts, indicating that the domain wall propagates from left to right along the [0001] direction, and the AS-MR behaviors are triggered when the domain wall propagates between the electrode pairs.
The insets of Fig. 2b, c correspond to four kinds of the domain configurations (i)-(iv) in Fig. 2e at a given field, where the red region indicates the region of positive cluster octupole moment (β+ domain in Fig. 1b). At the same time, the blue one presents a negative moment (β− domain). For example, in case of (ii), the dip (peak) under the positive field in the top (middle) panel is attributable to the positive (negative) E CA appearing at the top (bottom) edge across β− and β+ domains. The sign of E CA is presented by a tiny arrow at the top or bottom of the domain illustration. The peak (dip) under the negative field can be understood in the same manner by switching β+ and β− domains. Our setup with 4-μm-separated electrodes indicates that the length of an AFDW between these β− and β+ domains would be shorter than this separation distance. The MOKE measurements 19 also revealed that the AFDW appeared in-between β+ and β− domains, but its precise dimension and the microscopic spin texture have been difficult to be detected experimentally using our μm-scale devices. We then have investigated the atomistic numerical calculation to access the fine details of the AFDW, which will be introduced in the last section of this report.
Electrical nucleation of AFDWs. Based on the above insights, we have next performed experiments using the wedge-shaped Mn 3 Sn device with thickness variation along the [0001] direction, as shown in Fig. 3a, to demonstrate AFDW nucleation by the pulsecurrent injection. Such a thickness profile enables the sharp domain switching by field sweep, as shown in Fig. 3b, in contrast to the multistep switching of Fig. 2d. The five pairs of electrodes along the [0001] direction can monitor longitudinal resistivities, ρ 910 , ρ 45 , and ρ 15 in the same manner as the demonstration in Fig. 2. Figure 3c shows the field dependence of ρ 15 along the 01 10 ½ direction. As is expected from the sharp switching in this sample, we observe no positive or negative hump unique to AS-MR near the coercive field (~1200 Oe), indicating that no AFDW nucleates during the switching process. Next, we injected a current pulse between the electrodes 10-5 with the current density of 3.9 × 10 9 Am −2 and the duration of 50 ms, to induce the local AFDW nucleation. Figure 3d, e exhibits field dependences of ρ 910 and ρ 45 after the current pulse injection. We set the external field at +860 Oe during the pulse injection to assist the AFDW nucleation and then decreased down to +420 Oe in order not to wipe out the injected AFDW during the measurement. The above ρ vs. H curves comprise the ascending (positive) sweep from +420 Oe to +10,000 Oe (red symbols) and the descending (negative) sweep from +420 Oe to −10,000 Oe (blue symbols). The negative jump takes place around 740 Oe ≤ H ≤ 1200 Oe for ρ 910 , while the positive step appears in the same field region for ρ 45 during the positive sweep. Besides, no jump appears in a magnetoresistance during the negative sweep. These odd behaviors in magnetoresistance are unique to AS-MR, assuring a successful injection of AFDW along the kagome planes at the electrodes 10-5.
The typical threshold current density for the AFDW nucleation is j t~1 0 9 A m −2 under H = 100-1000 Oe, which is about two orders of magnitude smaller than the typical values j t;FM $ 10 11À12 A m À2 for ferromagnets 22 and j t;SOT $ 10 11 A m À2 for spin-orbit torque-induced switching in CuMnAs 6 . The small j t maybe characteristic of the antiferromagnetic nature of Mn 3 Sn, where the noncollinear spins may rotate with small cost of additional magnetic energy owing to the spin frustration at kagome triangles 23 , compared with the switching of the collinear ferromagnetic nature. Further investigation is necessary for quantitative comparison with previous results of the spin torque-induced switching.
The schematic insets in the figure correspond to the domain configurations, as in the cases of Fig. 2b, c. The shape of the AS-MR peak/dip in Fig. 3d, e appears more rectangular than multilevel peak/dip structures in Fig. 2b, c. This apparent difference indicates that the confined domain wall in Fig. 3d, e experiences less pinning potentials than in Fig. 2b, c, reflecting the smooth switching process due to the wedge profile, where the thickness variation induces a potential gradient of AFDWs along the [0001] slope direction.
Electrical displacement of AFDWs. We fabricated a device for the current pulse injection measurement as shown in Fig. 4a to nucleate a straight AFDW in a more controlled manner similar to the conventional experiment in ferromagnetic nanowires 24,25 . We also add a wedge-shape variation in the wire thickness to initiate a sharp domain wall propagation identical to the situation in Fig. 3a. The sample has two Hall bars labeled as probes 1 and 2, where probe 1 is for injection, while probe 2 is for detection, respectively. The assisting perpendicular field of +860 Oe is applied along the 01 10 ½ direction during the whole experimental process.
First, we have investigated the AHE response to the currentinduced nucleation process of the AFDW by applying a 50-ms pulse current with j ¼ þ7:2 10 9 A m À2 along the 2 1 10 ½ direction. Figure 4b shows a successful domain-switching process observed as an evolution of the Hall resistivity at probe 1 (ρ H1 ) normalized by the saturation field |H| = 1.1 T (ρ HS1 ), as ρ H1 /ρ HS1 . After injecting the 20th pulse current, the value of ρ H1 /ρ HS1 jumps up to 1.0, indicating that the ½01 10 oriented β− cluster octupole domain at probe 1 entirely flipped to the β+ domain 26 .
Once the switching is complete, there are a pair of AFDWs nucleated at the intersection of probe 1 and a horizontal wire along the [0001] direction. The application of the second pulse current drives the AFDW toward probe 2 that detects the AHE signal as a function of the pulse duration, giving information about the evolution of AFD configuration. Before starting the second pulse experiment, we have checked whether the pulse current could nucleate an additional domain or not. Figure 4c presents the normalized Hall resistivity at probe 2 ρ H2 /ρ HS2 after the application of the second pulse but without the first pulse. The value of ρ H2 /ρ HS2 remained unchanged with the average current densities j a ≤ 9:7 ± 0:1 ð Þ 10 9 A m À2 , assuring that the second pulse itself nucleates no additional AFDW. Therefore, we set the current density below the above value for the following experiments. Figure 4d shows the normalized anomalous Hall resistance ρ 2 / ρ HS2 at probe 2 as a function of the pulse duration time D t , i.e., an AFDW propagation time. We chose three different current densities J a relative to the depinning threshold current density j c ¼ 4:5 10 9 A m À2 , j a ¼ þ 6:7 ± 0:3 ð Þ 10 9 A m À2 > j c j j (solid symbols), j a ¼ À 6:7 ± 0:3 ð Þ 10 9 A m À2 < À j c j j (open symbols), and j a ¼ þ 4:3 ± 0:2 ð Þ 10 9 A m À2 < j c j j (green symbols), respectively. An abrupt ρ 2 jump took place only when injected current density exceeded its depinning threshold (j a > |j c | or j a <−|j c |). These experimental facts assure that the injected AFDW can propagate toward probe 2 in response to the second pulse injection. This behavior is similar to the case for the ferromagnetic nanowires 24,25 . Therefore, the value of D t at the jump must be the traveling time of the AFDW, ΔD t0 , and the propagation velocity v DW can thus be given by v DW ¼ L=ΔD t0 with the separation distance L of two Hall probes. The estimated v DW is plotted as a function of j a in Fig. 4e.
Since the potential energy U of the AFDW scales with the crosssectional area of the Mn 3 Sn wire, the wedge shape induces a potential gradient ΔU along the slope in the [0001] direction, resulting in a unidirectional propagation from the thicker to the thinner region with the positive velocity (v DW > 0), as shown in Fig. 4e. This situation indicates that the potential gradient is larger than the pinning forces of the distributed pinning centers along the slope. The propagation of the AFDW is thus like the creep type 27 Here, U C is the pinning potential due to disorders, f the driving force, and f C the threshold force. Our results are nicely fit to the above equation, as indicated by the black dotted The normalized transverse Hall measurement using the electrodes 5-10 (ρ H /ρ HS ). The red symbols present positive sweep (−10,000 Oe to +10,000 Oe), while the blue ones present negative sweep (+10,000 Oe to −10,000 Oe). c The longitudinal measurements using the electrodes 1-5 (ρ 15 ) without pulse injections. The external field is swept from −10,000 Oe to +10,000 Oe. d, e asymmetric magnetoresistance (AS-MR) measurements after the pulse injection using the electrodes 9-10 (ρ 910 ) and 4-5 (ρ 45 ). The red symbols and arrow present positive sweep from H = +420 Oe to H = +10,000 Oe, while the blue ones present negative sweep from H = +420 Oe to H = −10,000 Oe. The pulse currents with the current density of 3.9 × 10 9 A m −2 and the duration time of 50 ms were injected using the electrodes 10-5 under the assisting field H = +860 Oe along 01 10 ½ direction. Inset illustration presents corresponding domain configurations under each field; the red region indicates β+ domain, whereas the blue region indicates β− domain in Fig. 1b. Bottom panel: the error bars are much smaller than the data point and are not presented. ½ direction, and then driven by the pulse current along [0001] direction. Subsequent anomalous Hall measurement at Hall probe 2 monitors whether the injected domain has reached probe 2 or not. b AFDW injection event detected by the Hall probe. The direction of the octupole moment is identified via normalized Hall resistivity at Hall probe 1, ρ 1 /ρ HS1 . c The driving pulse-current density dependence of the normalized anomalous Hall resistivity at detection probe 2, ρ 2 /ρ HS2 , without the injection pulse current. The pulse duration time is set to be 50 μs. d The total pulse duration time D i dependence of the normalized anomalous Hall resistivity at Hall probe 2, ρ 2 /ρ HS2 , for positive depinning current j a > j c j j (black symbols), negative depinning current j a < À j c j j (open symbols), and below depinning current j a j j< j c j j (green symbols). Bottom insets illustrate domain configurations before ("0") and after propagation ("1"). The positive current j a > 0, (negative current: j a < 0) is set so that an electron e − flows from Hall probe 1 (2) to probe 2 (1). e, f Current-density dependence of the raw AFDW velocity v DW and its directional contribution Δv DW . A dashed line in Fig. 4e indicates the fitting of thermally induced creep component v creep . The error bars are much smaller than the data point and negligible, except for those in Fig. 4e, f. line, yielding μ ≈ 0.53, indicating the contributions of the randomfield disorder by Mn atoms 28 . Besides, we also observe the tiny current-polarity-dependent contribution in v DW obtained from Fig. 4f. The finite Δv DW implies the nontrivial current-induced contribution to AFDW propagation, except for heating and shape effects. The Δv DW of 0.1-1.0 ms −1 under j a $ 10 9 A m À2 is very small, but it could linearly increase up to the order of 10-100 m s −1 at j a $ 10 11 A m À2 . Moreover, the Δv DW vs j a implies that the AFDW moves in the same direction as the electron flow. The physical origin of this nontrivial current-polarity dependence is not clear. We, however, speculate that the origin may be the Dzyaloshinskii-Moriya interaction attributed to weak ferromagnetism of Mn 3 Sn 29 , or a new class of spin torque exerting on the macroscopic cluster octupole moments in Mn 3 Sn 18,30 . Further investigation is indeed necessary to achieve a robust conclusion.
Numerical calculation of AFD structure. For an in-depth understanding of the AFDW structure in the Mn 3 Sn wire, we performed micromagnetic calculations based on an atomistic micromagnetic calculation 31 , using the experimentally reported lattice constant of Mn atoms in kagome triangles. The motion of the spins at triangular sublattices in kagome planes is calculated by Landau-Lifshitz-Gilbert equation 32 solved by a fourth-order Runge-Kutta algorithm. Here, m i is each sublattice moment of six manganese atoms, which consist of an octupole moment in Fig. 1a, and the damping constant α is set α = 1 for quick energy minimization. The effective net field H eff is calculated by integrating the exchange field H exc , the magnetic anisotropy field H K , and Dzyaloshinskii-Moriya interaction (DMI) field H DMI . We ignored the dipolar interaction between spins. We calculated above effective fields as the energy first derivative with respect to the octupole moment M ¼ P 6 i¼1 m i as and The material parameters used in the simulation are the exchange integrals j ij ¼ j 1 ¼ À2:8 meV per link for (i, j) in the same kagome plane, j ij ¼ j 2 ¼ À2:8 meV per link for (i, j) in the neighboring kagome plane (Fig. 1a), the anisotropy constant along [0001] direction K ¼ K ? ¼ À14 meV per atom, and 2110 ½ direction K ¼ K k ¼ 0:187 meV per atom, the Dzyaloshinskii-Moriya constant D 1j ¼ 0:635 meV Å −1 , the gyromagnetic ratio γ ¼ 1:76 10 11 T À1 s À1 , and the lattice constants a = 5.363 Å, b = 4.327 Å 20,23 . The calculation region in [0001], [1210], and 2110 ½ directions are 7.0895 μm, 5.363 Å, and 10.726 Å, respectively. Figure 5 summarizes the calculation of the AFDW structure. Importantly, the calculation reproduces the weak remanent magnetization δm $ 4 mμ B parallel to the octupole moment, resulting from the inverse triangular distribution of six Mn sublattice moments in Fig. 1b. These cluster octupoles with weak magnetization are essential to describe the calculated AFDW structure shown in Fig. 5a. The boundary condition for the calculated AFDW is a 180°rotation of the octupole moment along the [0001] direction. The AFDW consists of four AFDs, β +, α−, γ+, and β− with three 60°AFDWs in-between neighboring AFDs, as shown in Fig. 1a, b. As can be seen in Fig. 5b, the overall AFDW structure exhibits the stepwise rotation over 180°. We call this type of AFDW structure, "Bloch-like 180°A FDW". The thicknesses of each AFD and 60°AFDW are, respectively,~120 and~185 nm, yielding approximately 800 nm for the entire Bloch-like 180°AFDW.

Discussion
Such sub-μm domain wall should be detectable by optical means as already reported in a MOKE study 19 . Our μm-scale devices have enough space resolution to monitor the whole 180°A FDW as discussed in Figs. 2-4, but should be difficult to detect each 60°AFDW separately. Further fine processing down to 100-10-nm scale would enable to discuss such stepwise AFDW profiles by electrical means, to be observed via AHE and AS-MR.
In summary, we have investigated magnetotransport properties for a chiral antiferromagnet Mn 3 Sn and developed an electrical means to control sub-μm size AFDWs. The AFDWs for Mn 3   domains. Interestingly, the introduction of the wedge shape turns out to be a practical means to induce smooth propagation of straight AFDW. These results could provide useful insights for the memory application in noncollinear antiferromagnetic spintronics.

Methods
Bulk preparation. Polycrystalline samples were made by melting the mixtures of manganese and tin in an alumina crucible sealed in an evacuated quartz ampoule in a box furnace at 1050°C for 6 h. In preparation for single-crystal growth, the obtained polycrystalline materials were crushed into powders, compacted into pellets, and inserted into an alumina crucible that was subsequently sealed in an evacuated silica ampoule. Single-crystal growth was performed using a singlezone Bridgman furnace with a maximum temperature of 1080°C and growth speed of 1.5 mm h −1 .
Devise fabrication. Single crystals were polished down to sub-mm pieces. Then, micro-Hall bars were fabricated by using a focused ion beam equipment (Scios DualBeam, Thermo Fisher Scientific Ltd.). The original sub-mm bulk pieces were formed into sub-μm thin plates by Ga+ ion beam accelerated by 5-30 kV with the injection angle of 0-5°. Subsequently, these plates were mounted onto a Si/SiO 2 substrate and fixed by perpendicularly injected Ga+ ion beam. The size of a Mn 3 Sn plate for AS-MR detection in Fig. 2a is 15 μm in its width and 500 nm in its thickness. The Mn 3 Sn wedge-shaped plate for AFDW nucleation in Fig. 3a has a width of 10 μm and a thickness of the wedge varies from 500 nm to 1 μm along the [0001] direction. The Mn 3 Sn Hall bars for current-induced displacement in Fig. 4a have a width of 2 μm, with 5 μm in separation from probe 1 to 2, and 1 μm in their probe widths. In addition, wedge slopes varied sample thickness from 500 nm at Hall probe 1 to 1 μm at probe 2. They were finally embedded in the transport measurement device by merging conventional lift-off techniques for Cu and Ti/Au electrodes and W deposition using a FIB apparatus.
AFDW nucleation process. We performed all the measurements at room temperature and initialized the Mn 3 Sn spin configuration by a strong enough field, H− 1.1T along the 01 10 ½ direction before starting the experiment. We have checked the Joule heating effect by monitoring the current amplitude dependence of AHE conductivity σ H 15 . The estimated temperature increase by pulse-current injections is less than 40 K, indicating that our Mn 3 Sn device is in the AFM phase during measurements.
The initial condition for nucleating an AFDW was j ¼ þ7:6 10 9 A m À2 with H = +860 Oe along 01 10 ½ direction for all the current-induced AFDW displacement experiments. This assisting perpendicular field of +860 Oe was kept constant during propagation process. Small offsets of ρ H 2 =ρ HS 2 $ 0:2 before zeroduration times in Fig. 4d implied that the AFD at probe 2 was slightly reversed by the exchange coupling between each kagome plane. Since these offsets did not change after the pulse-current injection, we assumed that they did not affect the AFDW propagation process.