Magnetic octupole domain evolution and domain-wall structure in the noncollinear Weyl antiferromagnet Mn₃Ge

Information storage technologies with high density and fast operation are in great demand for modern digital transformation. Researchers have been exploring recording technology innovations such as MRAM (magnetoresistive random access memory) using modern magnetic recording methods.^1–3 However, one of the obstacles is that magnetic crosstalk between adjacent modules becomes considerable with increasing device integration.^4,5 Under such circumstances, antiferromagnets (AFMs) have been considered potential alternative candidates for next-generation memory materials.^6,7 The advantage is that AFMs are immune to magnetic perturbation, which is favorable for large-scale integration within a small device size. Besides, AFMs exhibit terahertz spin dynamics, which promise a much faster data recording speed than ferromagnets (FMs).⁸ Nevertheless, AFMs generally have small responses to the external magnetic field due to near-zero magnetization. It is challenging to manipulate an antiferromagnetic bit using a recording head inductively, likewise with FMs.

Pioneering progress has demonstrated the electrical manipulation of the antiferromagnetic state in both collinear and noncollinear AFMs. The antiferromagnetic order can be switched via self-induced alternating Rashba field in the inversion symmetry broken system (CuMnAs⁹ and Mn₂Au¹⁰) or via spin–orbit torque in bilayers consisting of AFMs and spin Hall materials, such as NiO/Pt,¹¹ α-Fe₂O₃/Pt,¹² Mn₃Sn/Pt,¹³ or α-Fe₂O₃/(Bi, Sb)₂Te₃.¹⁴ The necessary switching current density is typically about 10¹⁰ A/m², an order of magnitude smaller than in the FM case.^15,16 However, the obstacle is the small readout signals, which rely on anisotropic magnetoresistance^9,10 or spin Hall magnetoresistance.^11,12 With this perspective, the noncollinear Weyl AFMs Mn₃X (X = Ge, Sn)^17–19 become more attractive than their conventional collinear AFM counterparts.²⁰ The magnetic Weyl states lead to giant magnetic responses as significant as FMs, promising efficient readout without losing the advantages of AFMs.²¹ 

Mn₃X has a hexagonal crystal structure with six-fold symmetry, as an example of Mn₃Ge shown in Fig. 1(a). An antichiral triangular spin texture is stabilized by the Dzyaloshinskii–Moriya interaction.²² It has mirror symmetry and time reversal symmetry (TRS) along the x- and z-directions respectively, whereas the TRS is broken in the y-direction. Such TRS breaking can be characterized using a macroscopic order parameter, i.e., the magnetic octupole, responsible for the spontaneous anomalous Hall effect (AHE).^23,24 The octupole represents a magnetic cluster containing six neighboring Mn atoms in z = 0 and z = 1/2 kagome planes, as shown in Fig. 1(a). The magnetic octupole states depicted in Fig. 1(b) with the order parameters of α_±, β_±, and γ_± form the magnetic octupole domain (MOD) and MOD wall (MODW),²⁴ detectable via magneto-optical Kerr effect (MOKE).^25,26 Interestingly, a weak ferromagnetism (∼7 mμ_B/Mn)¹⁸ associated with octupolarization enables tuning octupoles by a small magnetic field, which is promising for spintronic applications. However, to design MODW-based spintronic devices, it is indispensable to thoroughly figure out the MOD and MODW properties, whereas this has been merely reported. Here, we report a systematic study on MOD evolution involving the nucleation, propagation, and pinning processes with a choice of Mn₃Ge single crystal. Besides, we elucidate the detailed MODW structure based on experimental observation and micromagnetic simulation. We aim to provide comprehensive information on MOD and MODW to support spintronic device engineering.

To study the MOD evolution process, we chose a Mn₃Ge single crystal grown by the Bismuth flux method.²⁶ The obtained single crystal sample is a nearly perfect hexagonal column with a clean surface, enabling us to observe MOKE without additional surface polishing. We carried out a simultaneous measurement of polar MOKE and AHE to study the MOD evolution with the set-up of Fig. 2(a) under a sweeping magnetic field along the $011̄0$ direction. Figure 2(b) shows the hysteresis loops of MOKE and AHE. The obtained Kerr angle in Mn₃Ge is as large as that in Mn₃Sn,²⁵ while the coercivity is much smaller. The small coercivities (∼6 Oe for MOKE and ∼15 Oe for AHE) indicate uniform and high crystallinity in Mn₃Ge. The positive and negative values correspond, respectively, to oppositely aligned MODs. Furthermore, stepwise switching appeared in the hysteresis loop, which may be related to the MODW intrinsic depinning process.²⁷ The small depinning field (∼25 Oe) indicates the high crystallinity of the Mn₃Ge single crystal. The MOD evolution process was captured using a high-resolution CCD Kerr microscope, as shown in Fig. 2(c) (also see the supplementary material video for more detail). A magnetic field was applied to reset the magnetization in the crystal down to −100 Oe, the magnitude of which is larger than the saturation field according to the MOKE hysteresis loop in Fig. 2(b). We subtracted the image background at this state. MOD images were then continuously captured while sweeping the magnetic field from −20 Oe up to 30 Oe with a sweep rate of 2.5 Oe/s. The exposure time of the CCD camera was 50 ms. The gray area at −100 Oe and the white one at 25 Oe represent downward- and upward-aligned MODs, respectively. Generally, the domain nucleation in FMs occurs at the edge with a higher demagnetizing energy.²⁸ However, the MOKE image shows the MOD droplet nucleating in the central region in contrast to FMs. The nucleation site is no longer necessary to be the edge in Mn₃Ge because of the vanishing net magnetization.

The nucleated droplet MOD with a considerable volume gradually changes its contrast from gray to white with a fixed boundary, ascribed to the coherent rotation of the magnetic octupole moments. It is a clear contrast to the FM case, where the domain evolution occurs via nanometer-sized domain nucleation followed by domain-wall propagation.^28,29 As the grayscale in the CCD camera is proportional to the out-of-plane component of the magnetic octupole moment M_z, we can extract the octupole rotation angle β from cos β = M_z/M, as shown in Fig. 3(a) and Fig. S1. It is clear that the magnetic octupoles continuously rotate from downward to upward under a narrow magnetic field window of ∼2 Oe. As a result, a π-MODW forms between the reversed droplet and the initially saturated region, from which one can estimate the MODW energy density. As shown in Fig. 3(b), the first droplet of the reversed domain emerges in the magnetic field range from H₁ = 1.875 Oe to H₂ = 3 Oe. The magnetic potential energy compensates for the energy cost due to the new MODW creation. We have $μ0M⋅H2−μ0M⋅H1⋅V=γw⋅ΔS$⁠, where M and γ_w denote the saturation magnetization (7 mμ_B/Mn) and the energy density of a π-MODW, respectively. V is the volume of the reversed MOD, and ΔS is the area of the created MODW. We estimated the V and ΔS by assuming the hemi-ellipsoidal MOD domain beneath the surface, which is energetically stable, as illustrated in Fig. 3(c). Ultimately, the obtained π-γ_w is ∼0.0198 mJ/m² in Mn₃Ge. Interestingly, domain-wall repulsion is observable with increasing magnetic fields, where some nucleated MODs grow while others shrink (see the supplementary material video for details). This phenomenon could happen in a twisted 2π domain-wall structure with the same chirality for each sub-wall.³⁰ The two sub-walls with the same chirality cannot annihilate each other but repel each other under the pressure of the magnetic field. The previous report shows that the chirality of MODW depends on the prior history of initialization.³¹ Additionally, the exchange repulsion of twisted 2π domain walls in FMs is only observable in sufficiently thin films with small domain-wall separations.³² However, the separation between two walls, in our case, is surprisingly large—about 10 µm. It may be due to the micrometer-size MODW width and the large exchange strength of AFMs because the exchange energy pulls the twisted domain walls apart.³⁰ 

The MOD expands via MODW propagation, which the extrinsic pinning effect could influence. The origin of extrinsic pining is usually complicated, e.g., by inevitable crystallographic defects and surface roughness.³³ Here, we discuss the influence of extrinsic pinning on the MODW propagation in the unpolished, polished, and FIB-fabricated Mn₃Ge samples with different surface conditions. It is well-known that the MOKE measurement is only sensitive to the surface magnetic state within tens of nanometers of penetration depth. We, therefore, examined the MODW propagation by comparing the simultaneous surface-sensitive MOKE and bulk-sensitive AHE measurements. Figures 3(d)–3(f) show the hysteresis loops of the MOKE and AHE in unpolished, polished, and FIB-fabricated Mn₃Ge. In unpolished Mn₃Ge [Fig. 3(d)], the MOKE coercivity is ∼5 Oe, comparable to the AHE coercivity (∼10 Oe). The polished Mn₃Ge exhibits a much larger MOKE coercivity (∼1200 Oe) than the AHE coercivity, as shown in Fig. 3(e). Regarding the FIB-fabricated Mn₃Ge platelet, the MOKE coercivity remains unchanged from the polished Mn₃Ge, as shown in Fig. 3(f). Instead, the AHE coercivity increases up to ∼1200 Oe, comparable to the MOKE coercivity.

The coercivity change is ascribed to the extrinsic surface pinning effect. Based on the above-mentioned simultaneous measurement results, we speculate an the following scenario of MODW propagation. The depinning field is small in the unpolished sample (∼10 Oe), with a few inevitable defects during the crystal growth. The nucleation of MOD occurs near the surface region shown in Fig. 2(c), followed by the expansion into the bulk region. Therefore, the MOKE and AHE hysteresis loops acquire comparably small coercivities in Fig. 3(d). In contrast, the polished sample surface contains polish-induced pinning sites with a much stronger depinning field (∼1200 Oe). Therefore, the nucleated MOD alternatively expands from the bulk region, giving rise to a small AHE coercivity. When the magnetic field is sufficiently large to overcome the surface pinning, the surface region reversal completes, leading to a large MOKE coercivity. In the FIB fabricated sample, the sample thickness (∼0.5 µm) is smaller than the MODW width (∼1.4 µm shown in Sec. III), thus resulting in no MODW formation along the thickness direction. Instead, the magnetization reversal takes place via a MOD nucleation with coherent octupole rotation, followed by MODW propagation along the longitudinal direction. Therefore, the MOKE and AHE hysteresis loops show comparable coercivities when the MODW passes through the Hall bar position, as shown in Fig. 3(f).

Finally, we discuss the MODW structure. For bulk AFMs in general, due to the absence of net magnetization, there is no preference for either Bloch- or Néel-walls; they are energetically equivalent.^34,35 It differs from the case in bulk FMs, where magnetostatic energy impedes the Néel walls.²⁹ Interestingly, in Mn₃Ge, the octupole moments can only rotate in the kagome plane due to the easy plane anisotropy. Therefore, the kagome plane orientation distinguishes between Bloch- and Néel-like MODWs (hereinafter referred to as Bloch- and Néel-walls). The directional change of the MOD across the MODW occurs via inter-kagome-plane rotation for the Bloch-wall and intra-kagome-plane rotation for the Néel-wall. Figure 4(a) shows a MOKE image, including the Bloch- and Néel-walls. The intensity profiles indicating the [01$1̄$0] components of the magnetic octupole moment for the Bloch- [Fig. 4(b)] and the Néel-wall [Fig. 4(c)] were converted from the corresponding grayscale images. They follow the tanh x functions likewise FMs.³⁶ The widths of MODWs are extracted ∼2.3 µm for Bloch-wall and ∼1.6 µm for Néel-wall. Unfortunately, the signal-to-noise ratio of the CCD Kerr microscope is not high enough to visualize the more detailed profiles. Hence, we performed the atomistic numerical simulation^33,37 to elucidate the confined MODW structure using the material parameters extracted from the magnetization measurements (Fig. S2). Figures 4(d) and 4(e) show the stepwise π-Bloch- and Néel-wall profiles. It consists of four domain segments along easy axes separated by three π/3-walls. The width of each π/3-Bloch-wall is 0.178 µm, yielding 1.303 µm for a complete π-Bloch-wall. Similarly, the widths of π/3- and π-Néel-walls are respectively 0.199 and 1.447 µm, 1.1 times larger than the Bloch-walls. Moreover, the π/3-Bloch-wall energy density is 0.0055 mJ/m², and the π/3-Néel-wall energy density is 0.0051 mJ/m². With comparable energy densities, we do not expect the preferred propagation of either Néel-wall or Bloch-wall under the magnetic field, as shown in Fig. 2(c). One can quantify a π-Bloch- or Néel-wall energy density of ∼0.016 mJ/m², in good agreement with the above-mentioned experimental estimation. The MODW energy density of Mn₃Ge is three orders of magnitude smaller than the hard magnets like CoPt and Nd₂Fe₁₄B while being comparable to that of soft magnet Ni₈₀Fe₂₀.²⁹ 

In conclusion, we systematically studied the MOD evolution and the MODW structure in the noncollinear Weyl AFM Mn₃Ge. Our optical and transport measurements revealed new features in Mn₃Ge that differ from conventional FMs, e.g., the large MOD nucleation volume in the central region, the extrinsic surface pinning influences on the MODW propagation, and the coexistence of Bloch- and Néel-walls in bulk with comparable sizes and energy densities. In addition, our micromagnetic simulation elucidated the MODW energy density consistent with the experimental observations. Our findings provide significant insights for understanding the MODs and MODWs, thus supporting MODW engineering, such as electrical control of MODW motion.

The Supplementary material, including the supplementary video and texts, is available for additional information.

We acknowledge Dr. T. Yokouchi for his support on the CCD Kerr microscope set-up. This work was financially supported by JST-CREST (Grant No. JPMJCR18T3) and JST-Mirai Program (Grant No. JPMJMI20A1), Japan Science and Technology Agency, and Grants-in-Aid for Scientific Research on Innovative Areas “Nano Spin Conversion Science” (Grant No. JP26103002) and “J-Physics” (Grant Nos. 15H05882 and 15H05883) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. M. Wu. would like to acknowledge support from the JSPS “Research Program for Young Scientists” (Grant No. 21J21461).

The authors have no conflicts to disclose.

Mingxing Wu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal). Kouta Kondou: Formal analysis (equal); Funding acquisition (equal); Supervision (supporting); Writing – review & editing (equal). Yoshinobu Nakatani: Methodology (equal); Software (lead). Taishi Chen: Methodology (equal); Resources (equal). Hironari Isshiki: Funding acquisition (supporting); Supervision (supporting). Tomoya Higo: Investigation (supporting). Satoru Nakatsuji: Conceptualization (equal); Funding acquisition (lead); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Yoshichika Otani: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.