The Hall effect is one of the best known effects in (solid-state) physics. Conventionally, this phenomenon describes the occurrence of charge currents that are perpendicular to an externally applied electric field due to a time-reversal symmetry breaking magnetic field. Besides, in ferromagnetic systems, the net magnetization can break time-reversal symmetry even in the absence of a magnetic field which allows the so-called anomalous Hall effect. This effect originates from extrinsic and intrinsic contributions that are both related to the existence of spin-orbit coupling [1]. Moreover, another contribution to the Hall effect, which is known as the topological Hall effect, can exist even if spin-orbit coupling is negligible. It may occur in certain noncollinear noncoplanar magnetic textures with a nonzero scalar spin chirality like skyrmions [2, 3]. However, recent works [4, 5] reported the occurrence of an anomalous Hall effect in several compensated kagome magnets (cf. Fig. 1). These materials are coplanar antiferromagnets with vanishing net magnetization, and still, a group theoretical analysis allows the existence of the effect. The large conductivities obtained via first-principle calculations have been confirmed in experiments for Mn3Sn [6] and Mn3Ge [7]. However, a straightforward microscopic picture for this phenomenon was still missing. In this talk, we present an explanation on a microscopic level based on tight-binding calculations and analytical considerations [8]. For coplanar kagome magnets, we show the equivalence of spin-orbit coupling and an out-of-plane tilting of the magnetic moments. The existence of spin-orbit interaction does not only break a combined time-reversal and mirror symmetry of the Hamiltonian but can be transformed to a magnetic texture that is virtually canted, whereas, the original texture remains coplanar [cf. Fig. 2(a)]. Consequently, the ‘new’ anomalous Hall effect can be interpreted as a combination of an effective anomalous and topological Hall effect due to the net magnetic moment and the net scalar spin chirality of this virtual magnetic texture, respectively. Furthermore, as we demonstrate, a noncoplanar kagome magnet with spin-orbit coupling is able to behave like a system that is virtually coplanar and with compensated spin-orbit coupling [cf. Fig. 2(b)]. In this case, the combination of mirror and time-reversal symmetry of the Hamiltonian that was broken before has been restored. A critical out-of-plane tilting angle of the real texture can be found, where the virtual texture is coplanar and the Hall effect is absent for all energies. As we show in detail, the electronic properties are determined by this virtual texture that is hidden in the Hamiltonian. In consequent investigations, the calculations have been repeated for other transport quantities like the spin Hall effect where charge currents are converted into spin currents. These results can again be related to the virtual spin texture which has, however, different consequences for the spin Hall effect. Besides, in order to simulate the experimental situation, the investigated model was extended from a two-dimensional kagome lattice, as considered here, to a more realistic model including d-orbitals and kagome planes that are stacked along the out-of-plane direction.

**References:**

[1] N. Nagaosa, J. Sinova, S. Onoda, et al. "Anomalous Hall effect". Rev. Mod. Phys. 82, 1539 (2010). [2] P. Bruno, V. K. Dugaev, and M. Taillefumier. "Topological Hall Effect and Berry Phase in Magnetic Nanostructures". Phys. Rev. Lett. 93, 096806 (2004). [3] N. Nagaosa and Y. Tokura. "Topological properties and dynamics of magnetic skyrmions". Nat. Nanotechnol. 8, 899 (2013). [4] H. Chen, Q. Niu, and A. H. MacDonald. "Anomalous Hall Effect Arising from Noncollinear Antiferromagnetism". Phys. Rev. Lett. 112, 017205 (2014). [5] Y. Zhang, Y. Sun, H. Yang, et al. "Strong anisotropic anomalous Hall effect and spin Hall effect in the chiral antiferromagnetic compounds Mn3X (X=Ge, Sn, Ga, Ir, Rh, and Pt)". Phys. Rev. B 95, 075128 (2017). [6] S. Nakatsuji, N. Kiyohara, and T. Higo. "Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature". Nature (London) 527, 212 (2015). [7] A. K. Nayak, J. E. Fischer, Y. Sun, et al. "Large anomalous Hall effect driven by a nonvanishing Berry curvature in the noncolinear antiferromagnet Mn3Ge". Sci. Adv. 2, e1501870 (2016). [8] O. Busch, B. Göbel, and I. Mertig. "Microscopic origin of the anomalous Hall effect in noncollinear kagome magnets". Phys. Rev. Research 2, 033112 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/5ba5815bd43591c766a0d90db120a54b.jpg)
Fig. 1. Overview of the discussed coplanar kagome magnets. The arrows indicate the magnetic moments that are located at the basis atoms of the unit cell. The configurations can be characterized by their vector spin chiralities [(a), (b) positive, and (c), (d) negative]. Besides, each magnetic texture has different symmetry planes (time reversal T, mirror M or a combination of both) as indicated by the red lines.

![](https://assets.underline.io/uploads/markdown_image/1/image/93336328e4e3193aa019ed2b5bbc3832.jpg)
 
Fig. 2. Within the considered model, spin-orbit coupling gives rise to a virtual magnetic texture (red arrows) that is tilted out of the kagome plane with respect to the original magnetic texture (black arrows). (a) The virtual texture of a coplanar configuration can be noncoplanar resulting in the occurrence of an effective anomalous and topological Hall effect. (b) At a critical out-of-plane canting angle of the real magnetic texture, spin-orbit coupling is compensated and the virtual magnetic texture is coplanar. In this case, the anomalous Hall effect is absent.

 Interpretation of the anomalous Hall effect as an effective topological Hall effect in noncollinear kagome magnets

Magnetic skyrmions have attracted enormous research interest since their discovery a decade ago. Especially the non-trivial real-space topology of these nano-whirls leads to fundamentally interesting and technologically relevant effects – the skyrmion Hall effect of the texture and the topological Hall effect of the electrons. Furthermore, it grants skyrmions in a ferromagnetic surrounding great stability even at small sizes, making skyrmions aspirants to become the carriers of information in the future. Still, the utilization of skyrmions in spintronic devices has not been achieved yet, among other reasons, due to shortcomings in their current-driven motion. In this talk, we present our recent advances in the field of topological spin textures that go beyond skyrmions [1]. The majority of the discussed objects can be considered the combination of multiple skyrmions or the skyrmion analogues in different magnetic surroundings, as well as three-dimensional generalizations. We classify the alternative magnetic quasiparticles – some of them observed experimentally, others theoretical predictions – and present the most relevant and auspicious advantages of this emerging field. A special focus is on magnetic antiskyrmions [2,3,4], bimerons [5], antiferromagnetic skyrmions [6] and hopfions [7]. These objects (shown in Fig. 1) exhibit advantageous emergent electrodynamic effects compared to conventional skyrmions, either due to their lower symmetry or due to a compensated topological charge. As we will show, all four of these objects can be driven parallel to the current, without a skyrmion Hall effect which makes them the ideal bits in data storage devices. Furthermore, we show that the antiferromagnetic skyrmions exhibit a topological version of the spin Hall effect which could help their detection. For bimerons and hopfions we show that the topologically induced Hall response can be distinguished much more easily from the anomalous and conventional Hall effect. This is because their emergent field is perpendicular to the net magnetization and the stabilizing magnetic field; all three quantities are (anti)parallel for skyrmions and most other magnetic nano-objects. Also, we show that some of these objects can even coexist, allowing for an advanced version of the racetrack memory data storage, where a bit sequence could, for example, be encoded by a sequence of skyrmions (‘1’ bit) and antiskyrmions (‘0’ bit); cf. Fig. 2. This concept would be more reliable than conventional racetracks: When the information is encoded by the presence or absence of a single type of topological object, the information is susceptible to thermal motion and interactions between the carriers of information. An unintended change in the bit sequence would lead to a loss of the stored information. Only when the information is encoded by at least two distinct carriers, the bit sequence can become irregular without any harm to the stored data.

**References:**

 [1] B. Göbel, I. Mertig, O. Tretiakov. “Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles”. Physics Reports in press (2020), arXiv preprint: 2005.01390 [2] J. Jena*, B. Göbel*, S. Parkin. et al. “Elliptical Bloch skyrmion chiral twins in an antiskyrmion system”. Nature Communications 11, 1115 (2020). [3] J. Jena*, B. Göbel*, S. Parkin. et al. “ Evolution and competition between chiral spin textures in nanostripes with D2d symmetry”. Science Advances 6, eabc0723 (2020). [4] P. Sivakumar, B. Göbel, S. Parkin. et al. “ Topological Hall Signatures of Two Chiral Spin Textures Hosted in a Single Tetragonal Inverse Heusler Thin Film”. ACS Nano 14, 13463 (2020). [5] B. Göbel, I. Mertig, O. Tretiakov. et al. “Magnetic bimerons as skyrmion analogues in in-plane magnets”. Phys. Rev. B. 99, 060407(R) (2019). [6] B. Göbel, A. Mook, J. Henk, I. Mertig. “Antiferromagnetic skyrmion crystals: Generation, topological Hall, and topological spin Hall effect”. Phys. Rev. B. 96, 060406(R) (2017). [7] B. Göbel, C. Akosa, G. Tatara, I. Mertig. “Topological Hall signatures of magnetic hopfions”. Phys. Rev. Research 2, 013315 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/1df6edc367627cf9506b431bbb84b52b.jpg)
Overview of the discussed magnetic nano-objects. (a) Magnetic antiskyrmion, (b) magnetic bimeron, (c) antiferromagnetic skyrmion, (d) magnetic hopfion. The arrows correspond to magnetic moments whose color is determined by their orientation.

![](https://assets.underline.io/uploads/markdown_image/1/image/fa4bb03806ecc363261983a4f2311ee5.jpg)
In Heusler materials the anisotropic DMI stabilizes antiskyrmions (red) and elliptically deformed skyrmions (blue). These coexisting types of topologically distinct objects may encode data as ‘0’ and ‘1’ bits, respectively.

 Going Beyond Skyrmions: Alternative Magnetic Nano-Objects for Spintronics

Magnetic skyrmions (Sks) and antiskyrmions (aSks) are topologically protected non-trivial spin textures1, 2. Those objects can be used as magnetic bits in spintronics devices3, 4. However, the presence of the skyrmion Hall effect in the Sk always hinders its movement along the straight path, and the transverse defection of it will cause a problem on reading data in devices. The unwanted Hall effect can be eliminated in aSk host systems2, 5 such that it can be moved in a straight path along the current direction6. Therefore, it is highly desired to prepare nano-stripes along specific crystallographic directions of the aSk host compound. Here we stabilize a single-chain of aSks in nano-stripes of Mn1.4Pt0.9Pd0.1Sn prepared along different crystallographic directions within the tetragonal basal plane7. We observe the enhanced stability of aSks compared to bulk systems without providing in-plane magnetic field components. Additionally, the presence of dipole-dipole interactions with the anisotropic Dzyaloshinskii-Moriya exchange interaction (DMI) in the Heusler compound Mn1.4Pt0.9Pd0.1Sn allows the stability of distinct spin textures called elliptical Bloch skyrmions7,8. The co-existence of elliptical skyrmions (eSks) and aSks is found in nano-stripes at room temperature7. The simultaneous existence of different spin textures in a racetrack has great advantages. It is not needed to write the distances between the magnetic bits anymore, unlike the racetrack of only single type magnetic objects. Our results provide a gateway for the remaining steps towards realizing an advanced version of racetrack storage devices based on magnetic aSks and eSks.

**References:**

1. Mühlbauer et. al., Science 323, 915 (2009). 2. Nayak et. al., Nature 548, 561 (2017). 3. Parkin et. al., Science 320, 190 (2008). 4. Fert et. al., Nat. Nanotechnol. 8, 152 (2013). 5. Jena et. al., Nano Lett. 20, 59 (2019). 6. Huang et. al., Phys. Rev. B 96, 144412 (2017). 7. Jena et. al., Sci. Adv. 6, 49, eabc 0723 (2020). 8. Jena et. al., Nat. Commun. 11, 1115 (2020).

**Images:**

 Evolution and competition between chiral spin textures in nano-stripes with D2d symmetry

Magnetic skyrmions have attracted enormous research interest since their discovery a decade ago. Especially the non-trivial real-space topology of these nano-whirls leads to fundamentally interesting and technologically relevant effects – the skyrmion Hall effect of the texture and the topological Hall effect of the electrons. Furthermore, it grants skyrmions in a ferromagnetic surrounding great stability even at small sizes, making skyrmions aspirants to become the carriers of information in the future. Still, the utilization of skyrmions in spintronic devices has not been achieved yet, among others, due to shortcomings in their current-driven motion. In this talk, we present recent trends in the field of topological spin textures that go beyond skyrmions [1]. The majority of these objects can be considered the combination of multiple skyrmions or the skyrmion analogues in different magnetic surroundings, as well as three-dimensional generalizations. We classify the alternative magnetic quasiparticles – some of them observed experimentally, others theoretical predictions – and present the most relevant and auspicious advantages of this emerging field. A special focus is on magnetic antiskyrmions [2], bimerons [3], antiferromagnetic skyrmions [4] and hopfions [5]. These objects (shown in Fig. 1) exhibit advantageous emergent electrodynamic effects compared to skyrmions, either due to their lower symmetry or due to a compensated topological charge.References: [1] B. Göbel, I. Mertig, O. Tretiakov “Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles” arXiv preprint: 2005.01390 [2] J. Jena*, B. Göbel*, T. Ma, V. Kumar, R. Saha, I. Mertig, C. Felser, S. Parkin. “Elliptical Bloch skyrmion chiral twins in an antiskyrmion system” Nature Communications 11, 1115 (2020). [3] B. Göbel, A. Mook, J. Henk, I. Mertig, O. Tretiakov. “Magnetic bimerons as skyrmion analogues in in-plane magnets” Phys. Rev. B. 99, 060407(R) (2019). [4] B. Göbel, A. Mook, J. Henk, I. Mertig, “Antiferromagnetic skyrmion crystals: Generation, topological Hall, and topological spin Hall effect” Phys. Rev. B. 96, 060406(R) (2017). [5] B. Göbel, C. Akosa, G. Tatara, I. Mertig. “Topological Hall signatures of magnetic hopfions” Phys. Rev. Research 2, 013315 (2020).

Beyond skyrmions: alternative magnetic quasiparticles INVITED

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Börge Göbel

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SHORT BIO

Presentations

Interpretation of the anomalous Hall effect as an effective topological Hall effect in noncollinear kagome magnets

Going Beyond Skyrmions: Alternative Magnetic Nano-Objects for Spintronics

Evolution and competition between chiral spin textures in nano-stripes with D2d symmetry

Beyond skyrmions: alternative magnetic quasiparticles INVITED

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