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

Abstract

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.

Introduction

Electrical manipulation of the magnetic domain^1,2,3 is the essential technology for spintronics. In recent years, antiferromagnetic spintronics⁴ 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, Néel-order spin–orbit torque in collinear antiferromagnets with local inversion asymmetry⁵, CuMnAs^6,7, and Mn₂Au⁸, 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, Mn₃Sn, has currently attracted a great deal of attention owing to its intriguing electronic properties. Mn₃Sn has a hexagonal kagome lattice structure with a space group, P6₃/mmc⁹, recently discovered as a magnetic Weyl semimetal¹⁰. Despite vanishingly small magnetization^11,12,13, the inverse triangular chiral antiferromagnetic phase of Mn₃Sn below 430 K shows unusual transport properties, such as a substantial anomalous Hall effect (AHE)^14,15, giant anomalous Nernst effect¹⁶, strong anisotropic magnetoconductance¹⁰, and magnetic spin Hall effect¹⁷. These properties originate from the macroscopic breaking of time-reversal 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 Mn₃Sn. 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, $\left[ {2\bar 1\bar 10} \right]$ 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 Mn₃Sn as antiferromagnetic spintronic materials, we need to manipulate such a TRS-broken domain structure by electrical means.

Fig. 1: Crystal and domain structure of Mn₃Sn.

a Top view of the lattice structure for Mn₃Sn. Both Mn and Sn atoms are depicted with different colors between those in the z = 0 and 1/2 plane. Blue-colored hexagonal areas indicate units of octupole moments shown in 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 (j₁) and the neighboring kagome planes (j₂), 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.

In sharp contrast with conventional antiferromagnets, there are two significant technical benefits in Mn₃Sn. 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 Mn₃Sn.

In this work, we have investigated the method for electrical manipulation of AFDWs in a microfabricated single-crystal Mn₃Sn. We observed the asymmetric magnetoresistance (AS-MR) effect²¹ due to AFDW displacement in Mn₃Sn, 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 Mn₃Sn 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 [$01\bar 10$] 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 ρ₆₇ for the positive (negative) sweep around H ~+3200 Oe (−3100 Oe).

Fig. 2: Electrical detection of antiferromagnetic domain walls (AFDWs) for Mn₃Sn.

a The scanning electron microscope (SEM) image of the sample used for detecting AFDWs by the asymmetric magnetoresistance (AS-MR). The detection current flows to the [0001] direction and the external field is applied to the [$01\bar 10$] direction. b, c The longitudinal measurements using the electrodes 6–7 (ρ₆₇) and 2–3 (ρ₂₃). d The transverse Hall measurements using the electrodes 2–6 (ρ₂₆) and 3–7 (ρ₃₇) normalized by the high field saturation value ρ_HS (|H| =1.1 T). The red symbols and arrow present positive sweep, while the blue ones present negative sweep. Inset illustrations (i)–(iv) present the corresponding domain configurations under each field; the red region indicates β+ domain, whereas the blue region indicates β− domain in Fig. 1b. e The schematic of AS-MR emerged across the domain wall for materials with large anomalous Hall conductivity, for given domain configurations of (i)–(iv) in Fig. 2b, c. The error bars are much smaller than the data point and are not presented.

For the measurement by using electrodes 2–3 (Fig. 2c), on the other hand, a peak (dip) in the magnetoresistance ρ₂₃ 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 materials²¹. The schematic image of the AS-MR is given in cases (i) and (ii) in Fig. 2e. The oppositely aligned cluster octupole moment M in the antiferromagnetic Mn₃Sn, across the domain wall, causes opposite Hall conductance. In such a case, the AHE gives rise to the steady charge accumulation in the vicinity of the domain walls, resulting in effective electrical fields E_CA across the domain wall. The sign of E_CA is opposite to each other between the upper and lower edges, causing an odd response in the magnetoresistance effect between the top (6–7) and bottom (2–3) electrodes.

These results, in other words, indicate that AFDW moves along the [0001] direction during the field- sweeping process. Figure 2d, the transverse Hall resistivities ρ₂₆ and ρ₃₇ 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 ρ₂₆ is always smaller than that of the right-hand side ρ₃₇. Furthermore, the field range where the AS-MR takes place coincides well with the range where the difference in switching field is present between ρ₂₆ 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 measurements¹⁹ 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 Mn₃Sn device with thickness variation along the [0001] direction, as shown in Fig. 3a, to demonstrate AFDW nucleation by the pulse-current 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, ρ₉₁₀, ρ₄₅, and ρ₁₅ in the same manner as the demonstration in Fig. 2.

Fig. 3: Current-induced injections of antiferromagnetic domain walls (AFDWs) for Mn₃Sn.

a The scanning electron microscope (SEM) image of the sample used for AFDW injections by pulse currents. The detection current flows to the [0001] direction and the external field is applied normal to the plane. b 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 (ρ₁₅) 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 (ρ₉₁₀) and 4–5 (ρ₄₅). 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 × 10⁹ 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 $\left[ {01\bar 10} \right]$ 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.

Figure 3c shows the field dependence of ρ₁₅ along the $\left[ {01\bar 10} \right]$ 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 × 10⁹ Am⁻² and the duration of 50 ms, to induce the local AFDW nucleation. Figure 3d, e exhibits field dependences of ρ₉₁₀ and ρ₄₅ 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 ρ₉₁₀, while the positive step appears in the same field region for ρ₄₅ 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 ~ 10⁹ A m⁻² under H = 100–1000 Oe, which is about two orders of magnitude smaller than the typical values $j_{{\mathrm{t}},{\mathrm{FM}}}\sim 10^{11 - 12}\,{\mathrm{A}}\,{\mathrm{m}}^{ - 2}$ for ferromagnets²² and $j_{{\mathrm{t}},{\mathrm{SOT}}}\sim 10^{11}\,{\mathrm{A}}\,{\mathrm{m}}^{ - 2}$ for spin–orbit torque-induced switching in CuMnAs⁶. The small j_t maybe characteristic of the antiferromagnetic nature of Mn₃Sn, where the noncollinear spins may rotate with small cost of additional magnetic energy owing to the spin frustration at kagome triangles²³, 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 $\left[ {01\bar 10} \right]$ direction during the whole experimental process.

Fig. 4: Current-induced propagation of antiferromagnetic domain walls (AFDWs) for Mn₃Sn.

a The scanning electron microscope (SEM) image together with the schematic diagram for domain wall injection/driving experiments. An antiferromagnetic domain is generated at the right Hall probe 1 by the injection pulse current along $\left[ {2\bar 1\bar 10} \right]$ 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, ρ₁/ρ_HS1. c The driving pulse–current density dependence of the normalized anomalous Hall resistivity at detection probe 2, ρ₂/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, ρ₂/ρ_HS2, for positive depinning current $j_{\mathrm{a}} \, > \, \left| {j_{\mathrm{c}}} \right|$ (black symbols), negative depinning current $j_{\mathrm{a}} < - \left| {j_{\mathrm{c}}} \right|$ (open symbols), and below depinning current $\left| {j_{\mathrm{a}}} \right| < \left| {j_{\mathrm{c}}} \right|$ (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.

First, we have investigated the AHE response to the current-induced nucleation process of the AFDW by applying a 50-ms pulse current with $j = + 7.2 \times 10^9\,{\mathrm{A}}\,{\mathrm{m}}^{ - 2}$ along the $\left[ {2\bar 1\bar 10} \right]$ 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 $[01\bar 10]$ oriented β− cluster octupole domain at probe 1 entirely flipped to the β+ domain²⁶.

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 $j_{\mathrm{a}} \le \left( {9.7 \pm 0.1} \right) \times 10^9\,{\mathrm{A}}\,{\mathrm{m}}^{ - 2}$, 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 ρ₂/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 $j_{\mathrm{c}} = 4.5 \times 10^9\,{\mathrm{A}}\,{\mathrm{m}}^{ - 2}$, $j_{\mathrm{a}} = + \left( {6.7 \pm 0.3} \right) \times 10^9{\mathrm{A}}\,{\mathrm{m}}^{ - 2} \, > \, \left| {j_{\mathrm{c}}} \right|$ (solid symbols), $j_{\mathrm{a}} = - \left( {6.7 \pm 0.3} \right) \times 10^9{\mathrm{A}}\,{\mathrm{m}}^{ - 2} \, {<} - \left| {j_{\mathrm{c}}} \right|$ (open symbols), and $j_{\mathrm{a}} = + \left( {4.3 \pm 0.2} \right) \times 10^9{\mathrm{A}}\,{\mathrm{m}}^{ - 2} \, < \, \left| {j_{\mathrm{c}}} \right|$ (green symbols), respectively. An abrupt ρ₂ 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 $v_{{\mathrm{DW}}} = L/\Delta D_{t0}$ 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 cross-sectional area of the Mn₃Sn 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 type²⁷ whose propagation velocity $v_{{\mathrm{DW}}}\sim v_{{\mathrm{creep}}} = v_0{\mathrm{exp}}\left[ { - \frac{{U_C}}{{kT}}\left( {\frac{{f_C}}{f}} \right)^{ - \mu }} \right]$. 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 line, yielding μ ≈ 0.53, indicating the contributions of the random-field disorder by Mn atoms²⁸. Besides, we also observe the tiny current-polarity-dependent contribution in v_DW obtained from $\Delta v_{{\mathrm{DW}}} = v_{{\mathrm{DW}}} - v_{{\mathrm{creep}}} \, \ne \, 0$, plotted in 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 $j_{\mathrm{a}}\sim 10^9\,{\mathrm{A}}\,{\mathrm{m}}^{ - 2}$ is very small, but it could linearly increase up to the order of 10–100 m s⁻¹ at $j_{\mathrm{a}}\sim 10^{11}\,{\mathrm{A}}\,{\mathrm{m}}^{ - 2}$. Moreover, the $\Delta v_{{\mathrm{DW}}}\,vs\,j_{\mathrm{a}}$ 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 Mn₃Sn²⁹, or a new class of spin torque exerting on the macroscopic cluster octupole moments in Mn₃Sn^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 Mn₃Sn wire, we performed micromagnetic calculations based on an atomistic micromagnetic calculation³¹, 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 equation³²

$$\partial _t{\mathbf{m}}_i = - {\upgamma}{\mathbf{m}}_i \times {\mathbf{H}}_{{\mathrm{eff}}} + \alpha {\mathbf{m}}_i \times \partial _t{\mathbf{m}}_i\left( {i = 1,2, \ldots ,6} \right)$$

(1)

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 ${\mathbf{M}} = \sum_{i = 1}^6 {\mathbf{m}}_i$ as

$$H^{{\mathrm{exc}}} = - \frac{{\delta \varepsilon ^{{\mathrm{exc}}}}}{{\delta {\mathbf{M}}}}\,{\mathrm{with}}\,\varepsilon ^{{\mathrm{exc}}} = - \mathop {\sum }\limits_{i \ne j} j_{ij}{\mathbf{m}}_i \cdot {\mathbf{m}}_j,$$

(2)

$$H_i^{\mathrm{K}} = - \frac{{\delta \varepsilon _i^{\mathrm{K}}}}{{\delta {\mathbf{m}}_{\boldsymbol{i}}}}\,{\mathrm{with}}\,\varepsilon _i^{\mathrm{K}} = - K\left( {{\mathbf{n}}_i \cdot {\mathbf{m}}_i} \right)^2,$$

(3)

and

$$H^{{\mathrm{DMI}}} = - \frac{{\delta \varepsilon _1^{{\mathrm{DMI}}}}}{{\delta {\mathbf{M}}}}\,{\mathrm{with}}\,\varepsilon _1^{{\mathrm{DMI}}} = \mathop {\sum}\limits_j {D_{1j}} \left( {m_{1x} \cdot m_{jy} - m_{1y} \cdot m_{jx}} \right).$$

(4)

The material parameters used in the simulation are the exchange integrals $j_{ij} = j_1 = - 2.8\,{\mathrm{meV}}\,{\mathrm{per}}\,{\mathrm{link}}$ for (i, j) in the same kagome plane, $j_{ij} = j_2 = - 2.8\,{\mathrm{meV}}\,{\mathrm{per}}\,{\mathrm{link}}$ for (i, j) in the neighboring kagome plane (Fig. 1a), the anisotropy constant along [0001] direction $K = K_ \bot = - 14\,{\mathrm{meV}}\,{\mathrm{per}}\,{\mathrm{atom}}$, and $\left[ {2\overline {11} 0} \right]$ direction $K = K_\parallel = 0.187\,{\mathrm{meV}}\,{\mathrm{per}}\,{\mathrm{atom}}$, the Dzyaloshinskii–Moriya constant $D_{1j} = 0.635\,{\mathrm{meV}}$ Å⁻¹, the gyromagnetic ratio $\gamma = 1.76 \times 10^{11}\,{\mathrm{T}}^{ - 1}{\mathrm{s}}^{ - 1}$, and the lattice constants a = 5.363 Å, b = 4.327 Å^20,23. The calculation region in [0001], [1210], and $\left[ {2\overline {11} 0} \right]$ 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 $\delta {\mathbf{m}}\sim 4\,m\mu _{\mathrm{B}}$ 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° AFDW”. 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.

Fig. 5: Atomistic calculation results of antiferromagnetic domain walls (AFDWs) structures confined in Mn₃Sn.

a An illustration of a calculated 180° antiferromagnetic octupole domain wall for Mn₃Sn. The entire 180° AFDW structure can be divided into three segments of 60° AFDWs sandwiched by four antiferromagnetic domains, those octupole moments rotating along the x-axis ([0001] direction) with the rotation angle φ. b Calculated profile of the octupole moment φ along the x-axis. The octupole moments gradually rotate between neighboring kagome planes, forming stepwise Bloch domain wall configuration.

Discussion

Such sub-μm domain wall should be detectable by optical means as already reported in a MOKE study¹⁹. Our μm-scale devices have enough space resolution to monitor the whole 180° AFDW 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 Mn₃Sn and developed an electrical means to control sub-μm size AFDWs. The AFDWs for Mn₃Sn consist of three Bloch-type domain walls sandwiched by four different domains, and their positions can be detected by the AS-MR effect. We have demonstrated that the current pulse injections can nucleate and displace these 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 single-zone 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/SiO₂ substrate and fixed by perpendicularly injected Ga+ ion beam. The size of a Mn₃Sn plate for AS-MR detection in Fig. 2a is 15 μm in its width and 500 nm in its thickness. The Mn₃Sn 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 Mn₃Sn 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 Mn₃Sn spin configuration by a strong enough field, H ~−1.1T along the $\left[ {01\bar 10} \right]$ direction before starting the experiment. We have checked the Joule heating effect by monitoring the current amplitude dependence of AHE conductivity σ_H¹⁵. The estimated temperature increase by pulse-current injections is less than 40 K, indicating that our Mn₃Sn device is in the AFM phase during measurements.

The initial condition for nucleating an AFDW was $j = + 7.6 \times 10^9\,{\mathrm{A}}\,{\mathrm{m}}^{ - 2}$ with H = +860 Oe along $\left[ {01\bar 10} \right]$ 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 $\rho _{{\mathrm{H}}_2}/\rho _{{\mathrm{HS}}_2}\sim 0.2$ before zero-duration 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.