## Abstract

Domain walls naturally arise whenever a symmetry is spontaneously broken. They interconnect regions with different realizations of the broken symmetry, promoting structure formation from cosmological length scales to the atomic level^{1,2}. In ferroelectric and ferromagnetic materials, domain walls with unique functionalities emerge, holding great promise for nanoelectronics and spintronics applications^{3,4,5}. These walls are usually of Ising, Bloch or Néel type and separate homogeneously ordered domains. Here we demonstrate that a wide variety of new domain walls occurs in the presence of spatially modulated domain states. Using magnetic force microscopy and micromagnetic simulations, we show three fundamental classes of domain walls to arise in the near-room-temperature helimagnet iron germanium. In contrast to conventional ferroics, the domain walls exhibit a well-defined inner structure, which—analogous to cholesteric liquid crystals—consists of topological disclination and dislocation defects. Similar to the magnetic skyrmions that form in the same material^{6,7}, the domain walls can carry a finite topological charge, permitting an efficient coupling to spin currents and contributions to a topological Hall effect. Our study establishes a new family of magnetic nano-objects with non-trivial topology, opening the door to innovative device concepts based on helimagnetic domain walls.

## Main

In chiral magnets, the Dzyaloshinskii–Moriya interaction twists the magnetization and leads to a helimagnetic ground state. The magnetic moments periodically wind around a certain axis **Q**, with the wavelength *λ* determined by a competition between the usual exchange interaction and the Dzyaloshinskii–Moriya interaction^{8}. The overall magnetization can be visualized as a stack of equidistant sheets with uniformly oriented moments, where the orientation rotates from one sheet to the next (Fig. 1a,b). The lamellar morphology of the chiral spin texture is thus analogous to cholesteric liquid crystals^{9} and, to some extent, magnetic stripe domains^{10}. However, the length scales involved are strikingly different. In helimagnets, the characteristic periodicity is up to three orders of magnitude smaller, with lamellae having nanoscopic dimensions.

Despite this difference in length scales, one may expect these systems to develop similar types of defect^{11,12}. Smooth spatial variations of the helix axis **Q** simply result in a curvature of the lamellar spin structure^{13}. More pronounced variations, however, may break the periodicity and induce vortices—that is, disclinations (Fig. 1c,d). The strength of such vortices is parametrized by the winding angle of the helix axis on a path encircling the vortex core. As the helix axis is a director (**Q** = −**Q**), half-integer vortices are possible with +π and −π rotations (Fig. 1c,d). Furthermore, disclinations can pair up and form edge dislocations (Fig. 1e), with the Burgers vector *B* quantifying the distance *D* between them.

We reveal the fundamental importance of disclination and dislocation defects for the formation and properties of domain walls in helimagnets, presenting such walls as a magnetic analogue to grain boundaries in liquid crystals. A wide variety of functional helimagnetic domain walls with non-trivial topology is observed, including the vortex domain walls suggested recently^{14}. Using magnetic force microscopy (MFM), we study domain walls in the near-room-temperature chiral magnet iron germanium (FeGe; *T*_{N} = 278 K). FeGe single crystals are grown by chemical vapour transport and their phase purity (B20) is confirmed by powder X-ray diffraction (see Methods and Supplementary Fig. S0 for details). For our MFM studies, (100)- and (110)-oriented samples with a thickness of ≈500 μm are cut and chemo-mechanically polished with silica slurry, yielding flat surfaces with a roughness of about 1 nm. MFM is sensitive to the out-of-plane component of the magnetic stray field, and helimagnetic order manifests as a stripe-like pattern of bright and dark lines in the MFM scans in Fig. 2a–c (corresponding topography images are presented in Supplementary Fig. 4). The periodicity of the pattern is *λ* = 70 nm, matching with previous bulk data^{15}. However, we find equivalent values of *λ* on (100)- and (110)-oriented surfaces, which indicates that **Q** preferentially lies within the surface plane. This orientation is different from the bulk, where **Q** orients along the 〈100〉 direction. The MFM data thus reflect a surface-anchoring of **Q** implying a surface reconstruction of magnetic order^{16}—that is, formation of magnetic surface domains with in-plane **Q**. Furthermore, the measured statistical orientational distribution of **Q** is almost flat with a slight tendency to point along a 〈100〉 direction (if present) within the surface plane. The observations indicate that the surface domains are largely independent of both the crystallographic orientation and bulk magnetism.

Most interestingly, the MFM scans reveal helimagnetic domain walls with a complex, but well-defined inner structure. Their structure crucially depends on the angle ∠(**Q**_{1},**Q**_{2}) enclosed between the helix axes of the adjacent domains as defined and summarized in Fig. 2. Based on our data we distinguish curvature walls (type I, Fig. 2a), zigzag disclination walls (type II, Fig. 2b) and dislocation walls (type III, Fig. 2c).

Walls of type I occur for the smallest angles ∠(**Q**_{1},**Q**_{2}) ⪅85° and exhibit a smooth, continuous rotation of the helix axis from **Q**_{1} to **Q**_{2} (Fig. 2a). Type II walls display a characteristic zigzag pattern of alternating ±π disclinations. They arise for intermediate ∠(**Q**_{1},**Q**_{2}) (Fig. 2b) and induce distortions in the helimagnetic spin structure that extend over micrometre-sized distances away from the wall. Type III walls form for large ∠(**Q**_{1},**Q**_{2}), involving an abrupt transformation from **Q**_{1} to **Q**_{2} (Fig. 2c). On closer inspection, type III walls can be identified as a chain of magnetic edge dislocations.

The length of domain walls (type I to III) can reach several micrometres, exceeding the characteristic length scale of the helical structure, *λ*, by two orders of magnitude. Individual walls can also change their type and thereby adapt to local variations of ∠(**Q**_{1},**Q**_{2}). An example is seen in Fig. 2c, where a dislocation wall (type III) turns into a curvature wall (type I). To evaluate pinning effects, we image domain walls before and after heating above *T*_{N} and also record their response to magnetic fields as exemplified in Supplementary Fig. 3. The data show that domain walls tend to appear in similar positions after cooling, which we attribute to the presence of structural defects at the surface or in surface-near regions. The pinning is overcome by transiently applying a moderate magnetic field (50 mT in Supplementary Fig. 3), reflecting that the effect is weak. Most importantly, we observe that domain walls reappear with a different spin texture after annealing. This change shows that defects act as pinning centres for domain walls, but they do not govern their magnetic structure. Furthermore, we observe an additional constant, non-dipolar background in our MFM images wherever the helimagnetic spin structure is distorted (see bright contrast in Fig. 2). Whether this background originates from, for example, quadrupolar magnetic fields^{17} or other low-symmetry phenomena remains to be demonstrated.

Most importantly for this study, the observation of the three types of domain wall strikingly corroborates the analogy between chiral magnets and cholesteric liquid crystals, where such domain walls have been observed too^{18,19,20}. This universality emphasizes that the domain wall formation is governed by the inherent topology arising from the lamellar structure, being independent of the involved length scales and microscopic properties.

Figure 2d presents a quantitative analysis of more than 90 measured domain walls. The larger angle *α* enclosed by the domain wall and one of the helix axes **Q**_{
i
}, measured within the triangle defined by **Q**_{1} and **Q**_{2} (Fig. 2a–c), is plotted as a function of ∠(**Q**_{1},**Q**_{2}). The data show that, at the surface, **Q** is oriented in all directions, resulting in a broad spectrum of angles (0° ≤ ∠(**Q**_{1},**Q**_{2}) ≤ 180°). Type I walls dominate for angles ⪅85° (red dots), type II walls are realized approximately between 85° and 140° (blue dots) and type III walls for angles ⪆140° (green dots). Figure 2d reflects that structure and orientation of the walls are interlinked. For walls of type I and III, the angle *α* follows the bisecting line (\(\alpha =\frac{1}{2}\angle ({{\bf{Q}}}_{1},{{\bf{Q}}}_{2})\)). In contrast, type II walls have a tendency to orient parallel to one of the **Q**_{
i
}, so that *α* = 90°. Close to the transition regions around 85° and 140°, we occasionally observe a special case of type II walls. This subgroup of walls exhibits a minimal distance *D* = *λ*/2 between ±π disclinations (light-blue dots) as previously observed^{7} and discussed theoretically^{14}. Bisecting domain walls as well as zigzag structures were also anticipated in ref. ^{14}.

To understand the relation between the orientation of a wall and its topological magnetic structure, we perform two-dimensional micromagnetic simulations (see also Supplementary Information). We determine the energy density for various domain walls as function of ∠(**Q**_{1},**Q**_{2}) on a finite-size system extending up to a cutoff distance *L* = 12*λ* away from the wall (Fig. 3). The calculations reveal that type I walls (Fig. 3(i)) are lowest in energy for small angles ∠(**Q**_{1},**Q**_{2}) (red line in Fig. 3). Type II walls with *α* = 90° (Fig. 3(ii)) become energetically favourable as ∠(**Q**_{1},**Q**_{2}) exceeds 85°, which is in excellent agreement with the experimental data (Fig. 2d).

The blue lines in Fig. 3 correspond to the energies of type II walls with varying distance *D* between ±π disclinations. The energy of a pair of +π and −π disclinations grows logarithmically with *D* (ref. ^{9}) so that for an angle ∠(**Q**_{1},**Q**_{2}) = 90° the energy per length of the type II domain wall scales as log*D*/*D*. As a consequence, domain walls with large *D* are in principle preferred energetically (see Supplementary Information). To account for the finite *D* observed in the MFM images, we consider a finite cutoff length *L* in our simulations (*L* = 12λ, that is, *D* = 9*λ*/2 in Fig. 3). For angles ∠(**Q**_{1},**Q**_{2}) that deviate from 90°, the arrangement of disclinations is elastically distorted with an energy cost that grows quadratically with the deviation ∠(**Q**_{1},**Q**_{2})−90°, but linearly with *D*. As the angle ∠(**Q**_{1},**Q**_{2}) increases, type II walls with smaller *D* are therefore energetically favoured until a defect distance of *D* = 4λ/2 is reached, which marks the transition from type II to type III walls. Type II walls with minimal defect distance (*D* = *λ*/2, Fig. 3(iii)) are more costly than other configurations. Their high energy cost is in agreement with their rareness in our MFM data and their tendency to occur in transition regions (light-blue dots in Fig. 2d).

At around 145°, walls of type III become energetically less costly than type II walls (*D* = 3*λ*/2). Different micromagnetic wall structures are possible, all with similar energy (Fig. 3(iv–vii)). Within our numerical accuracy, a chain of serially aligned edge dislocations, first with a Burgers vector *B* = 3*λ* (see Supplementary Information) and then with *B* = 2*λ* (Fig. 3(iv)), is lowest in energy. Towards larger angles, type III walls with *B* = 2λ and *B* = λ (Fig. 3(v)) become almost degenerate. In addition, more complex domain wall structures with slightly higher energy arise (Fig. 3(vi,vii)). Consistent with the small energy difference, a large variety of type III walls is observed experimentally (see also Supplementary Fig. 2), and the micromagnetic structure frequently changes along the wall (Fig. 2c).

Aside from their intriguing magnetic textures, the domain walls may also carry a topological skyrmion number, which fundamentally distinguishes them from classical magnetic stripe domains and other lamellar structures. In general, the topology of a magnetic structure is identified via the topological charge density \({\rho }_{{\rm{top}}}=\frac{1}{4\pi }\hat{M}({\partial }_{x}\hat{M}\times {\partial }_{y}\hat{M})\) for the unit vector field of the magnetization \(\hat{M}\) defined within the (*x*,*y*) plane^{21}. A single skyrmion with the magnetization pointing downwards at its centre and a topological charge \(W=\int {\rm{{d}}}x{\rm{{d}}}y{\rho }_{{\rm{top}}}=-1\) is depicted in Fig. 4a. The same charge is obtained if the structure is embedded in a topologically trivial helimagnetic background as illustrated in Fig. 4b. However, this embedded skyrmion is equivalent to a pair of edge dislocations with a Burgers vector *B* = *λ* (Fig. 4c,d). The dislocation with *B* = *λ* can thus be interpreted as a single meron, that is, a half-skyrmion, which carries charge \(W=-\frac{1}{2}\). The sign reflects the orientation of the magnetization at the meron core. Straightforward generalization of this argument (see Supplementary Information for details) yields a general relation between the topological charge of a dislocation and its Burgers vector

where mod_{2} is the modulo 2 operation, and the sign, *s* = ±1, is determined by the orientation of \(\hat{M}\) at the dislocation centre.

To obtain the topological charge for domain walls, one has to add the charges of dislocations or, alternatively, pairs of ±π disclinations contained within the wall. This leads to the conclusion that type I and type III walls with continuous stripes have zero topological charge (Fig. 3(i,iv,v)). However, the type III walls of Fig. 3(vi,vii) with broken stripes have a finite charge *W*. Similarly, type II walls can have a zero or finite charge *W* (see also Fig. 4e). An odd dimensionless distance 2*D*/*λ* between ±π disclinations leads to *W *≠ 0 and an even distance to *W* = 0. Such topological textures with a finite charge *W *≠ 0 are in general expected^{21,22} to give rise to an emergent electrodynamics for electrons and magnons and to contribute to a topological Hall effect^{23,24,25,26}.

Our results reveal a new class of magnetic spin textures with non-trivial topology and establish a striking analogy between topological domain walls in chiral magnets and defect networks in mesoscopic liquid crystals. The walls naturally arise when cooling the sample below *T*_{N} as domains nucleate and grow, and it can be expected that they also profoundly influence macroscopic properties. Enabled by their chiral magnetic microstructure, the walls can carry a finite topological charge and, hence, couple efficiently to spin currents, which is to be demonstrated in future experiments. The walls and their building blocks, the magnetic disclinations and dislocations, are probably associated with long lifetimes that play a key role for the a.c. susceptibility and general relaxation dynamics in helimagnets^{27,28}. In particular, they might be crucial for previously observed melting processes^{29,30} and the pressure-induced non-Fermi liquid behaviour^{31,32,33}. The observations apply to chiral magnets in general and reveal a large variety of topologically non-trivial magnetic nano-objects—beyond skyrmions^{21,34,35}—extending the field of topology-based spintronics into the realm of helimagnetic domain walls.

## Methods

### Sample growth

FeGe single crystals were grown by chemical vapour transport from FeGe B35 powder with I_{2} (20 mg) in an evacuated quartz tube. The tube was mounted in a heated three-zone furnace for 1 month with a thermal gradient of 560 °C to 500 °C. This leads to the growth of 0.5 × 1 × 1 mm^{3} large B20 FeGe crystals at the lower temperature side. The B20 crystal structure was confirmed by powder X-ray diffraction (see Supplementary Fig. S0) and surfaces for MFM imaging were oriented using Laue diffraction.

### Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon request.

## Additional information

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## Acknowledgements

We thank M. Fiebig for direct financial support. The work at ETH was supported by the Swiss National Science Foundation through grants 200021-149192 and 200021-137520. L.K. and M.G. were supported by SFB 1143 'Correlated Magnetism: From Frustration To Topology' and DFG grant GA 1072/5-1. J.M. and A.R. were supported by SFB 1238 (Kontrolle und Dynamik von Quantenmaterialien). J.M. also acknowledges support from the Deutsche Telekom Stiftung and Bonn-Cologne Graduate School of Physics and Astronomy BCGS. N.K. acknowledges funding through the JSPS Grant-in-Aids for Scientific Research (S) no. 24224009 and for Young Scientists (Start-up) no. 26886005. D.M. thanks the Research Council of Norway (FRINATEK project no. 263228) and the Norwegian University of Science and Technology (NTNU) for financial support.

## Author information

### Affiliations

#### Department of Materials, ETH Zurich, Zürich, Switzerland

- P. Schoenherr
- & D. Meier

#### Institut für Theoretische Physik, Universität zu Köln, Köln, Germany

- J. Müller
- , A. Rosch
- & M. Garst

#### Institut für Theoretische Physik, Technische Universität Dresden, Dresden, Germany

- L. Köhler
- & M. Garst

#### Department of Applied Physics, University of Tokyo, Tokyo, Japan

- N. Kanazawa
- & Y. Tokura

#### RIKEN Center for Emergent Matter Science (CEMS), Wako, Japan

- Y. Tokura

#### Department of Materials Science and Engineering, Norwegian University of Science and Technology (NTNU), Trondheim, Norway

- D. Meier

#### Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology (NTNU), Trondheim, Norway

- D. Meier

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### Contributions

P.S. conducted the MFM experiments supervised by D.M.; J.M. performed the two-dimensional micromagnetic simulations supervised by M.G. and A.R.; N.K. grew the FeGe single crystals under supervision of Y.T.; P.S., J.M., L.K., M.G. and D.M. classified the domain walls; M.G. and L.K. defined the skyrmion charge of edge dislocations; P.S., J.M., L.K., M.G. and D.M. wrote the paper; M.G. and D.M. supervised the work. All authors discussed the results and contributed to their interpretation.

### Competing interests

The authors declare no competing interests.

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## Supplementary information

### Supplementary Materials

Supplementary Text, Supplementary References 36–40, Supplementary Figures S0–S17

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