# “Polymerization” of Bimerons in Quasi-Two-Dimensional Chiral Magnets with Easy-Plane Anisotropy

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}OSeO

_{3}[12], where they represent three-dimensional (3D) filaments along the field direction [13,14]. Afterwards, 3D isolated Skyrmions (IS) [15,16] have been microscopically spotted in thin layers of the cubic helimagnets (Fe,Co)Si [17] and FeGe [18], where they undergo an additional screw towards the confining surfaces and thereby gain stability in a broad range of temperatures and magnetic fields [19].

_{4}S

_{8}and GaV

_{4}Se

_{8}[20,21] (see the exact phenomenological form of DMIs in [1] for chiral magnets with different crystal symmetries). In these Néel skyrmions, the magnetization rotates radially from the Skyrmion center to the outskirt, as shown in Figure 1a. Alternatively, thin-film multilayer structures represent a 2D arena for Néel Skyrmions, where they can be manipulated as particle-like entities. The breaking of the inversion symmetry and the resulting DMI both originate from the interfaces between a heavy metal layer and Skyrmion-hosting magnetic layer, such as occurs in PdFe/Ir (111) bilayers [22]. Such systems are extremely versatile as regards the choice of the magnetic, non-magnetic, and capping layers as well as the possibility of stacking.

#### 1.1. Bimerons within the Spiral States of Chiral Magnets

#### 1.2. Bimerons within the Tilted Ferromagnetic States

_{0.5}Co

_{0.5}Ge was recently reported in [40]. Chains of bimerons dubbed “schools” were observed in thin layers of chiral liquid crystals [41] with thickness slightly smaller than the spiral pitch. Bimerons have been predicted by first-principle calculations and atomistic simulations as applied to van der Waals magnetoelectric CrISe/In

_{2}Se

_{3}heterostructures [42]. Moreover, merons have been introduced in the context of quark confinement in the nonlinear O(3) $\sigma $ model [43,44]. A loosely bound collection of parallel Skyrmion chains was demonstrated in nematic superconductors for low fields in [45]. Such nematic systems are two-component superconductors that break rotational symmetry but exhibit a mixed symmetry that couples spatial rotations and phase difference rotations. The chiral p-wave superconducting state supports a rich spectrum of topological excitations different from those in conventional superconducting states [46]. With the appropriate sign of the phase winding, two-quanta vortices were shown to always be energetically preferred over two isolated single quanta vortices.

## 2. Phenomenological Theory of Bimerons in Two-Dimensional Helimagnets

#### Energy Minimization

## 3. The Properties of Isolated Bimerons

#### 3.1. Internal Structure of Bimerons

#### 3.2. Bimeron–Bimeron Attraction

#### 3.3. Examples of Inter-Skyrmion Attraction

## 4. Bimeron “Macromolecules”

#### 4.1. Linear Bimeron Macromolecules—Chains

#### 4.2. Ring-Shaped Bimeron Macromolecules—“$(\pm )$Roundabouts”

_{3}. It was highlighted that 2D vdW magnets with an additional twist between layers open a unique avenue for investigating the properties of bimerons with remarkable flexibility, either through external stimuli or through the creation of heterostructures.

#### 4.3. Ring-Shaped Bimeron Macromolecules—“Crossings”

#### 4.4. Stability of Bimeron Macromolecules

## 5. Bimeron “Polymers”

#### 5.1. Combination of “Roundabouts” and “Crossings”

_{6}H

_{6}. For only three bimerons attached to the $(+)$roundabout (Figure 13b), the resulting macromolecule deforms, i.e., the mutual inter-bimeron angle varies around the ring, with interstitial bimerons being slightly drawn into the interior of the ring (the ideal tripod is shown by the black solid lines). For other combinations of crossings around the $(+)$roundabout, the macromolecules become unstable and break apart (see Supplementary Video S4).

#### 5.2. Periodic Tessellations

#### 5.3. Interconnection between Bimeron Polymers and the Hexagonal Skyrmion Lattice

_{0.5}Co

_{0.5}Ge [40], where the shape anisotropy fulfilled the role of the easy-plane anisotropy. The uniaxial anisotropy can be tuned by applying perpendicular strain in van der Waals magnetoelectric heterostructures, and varies in a wide range from $-3.5$ to $1.6$ meV, that is, from the easy-plane to the easy-axis characteristic [42]. The creation and annihilation of bimerons has been achieved via perpendicular strain and an electric field without an external magnetic field [42].

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Bogdanov, A.N.; Yablonsky, D.A. Thermodynamically stable vortices in magnetically ordered crystals. Mixed state of magnetics. Zh. Eksp. Teor. Fiz.
**1989**, 95, 178, [Sov. Phys. JETP 1989, 68, 101]. [Google Scholar] - Nagaosa, N.; Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol.
**2013**, 8, 899. [Google Scholar] [CrossRef] [PubMed] - Roessler, U.K.; Leonov, A.A.; Bogdanov, A.N. Skyrmionic textures in chiral magnets. J. Phys.
**2010**, 200, 022029. [Google Scholar] - Zhang, X.; Zhao, G.P.; Fangohr, H.; Liu, J.P.; Xia, W.X.; Xia, J.; Morvan, F.J. Skyrmion-skyrmion and skyrmion-edge repulsions in skyrmion-based racetrack memory. Sci. Rep.
**2015**, 5, 7643. [Google Scholar] [CrossRef] [PubMed] - Bogdanov, A.; Hubert, A. Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magn. Magn. Mater.
**1994**, 138, 255. [Google Scholar] [CrossRef] - Bogdanov, A.; Hubert, A. The stability of vortex-like structures in uniaxial ferromagnets. J. Magn. Magn. Mater.
**1999**, 195, 182. [Google Scholar] [CrossRef] - Dzyaloshinskii, I.E. A thermodynamic theory of weak ferromagnetism of antiferromagnetics. J. Phys. Chem. Sol.
**1958**, 4, 241. [Google Scholar] [CrossRef] - Moriya, T. Anisotropic Superexchange Interaction and Weak Ferromagnetism. Phys. Rev.
**1960**, 120, 91. [Google Scholar] [CrossRef] - Wiesendanger, R. Nanoscale magnetic skyrmions in metallic films and multilayers: A new twist for spintronics. Nat. Rev. Mater.
**2016**, 1, 16044. [Google Scholar] [CrossRef] - Müehlbauer, S.; Binz, B.; Jonietz, F.; Pfleiderer, C.; Rosch, A.; Neubauer, A.; Georgii, R.; Böni, P. Skyrmion lattice in a chiral magnet. Science
**2009**, 323, 915. [Google Scholar] [CrossRef] - Wilhelm, H.; Baenitz, M.; Schmidt, M.; Roessler, U.K.; Leonov, A.A.; Bogdanov, A.N. Precursor phenomena at the magnetic ordering of the cubic helimagnet FeGe. Phys. Rev. Lett.
**2011**, 107, 127203. [Google Scholar] [CrossRef] [PubMed] - Seki, S.; Yu, X.Z.; Ishiwata, S.; Tokura, Y.; Bogdanov, A.N. Observation of Skyrmions in a Multiferroic Material. Science
**2012**, 336, 198. [Google Scholar] [CrossRef] [PubMed] - McGrouther, D.; Binz, B.; Lamb, R.J.; Krajnak, M.; McFadzean, S.; McVitie, S.; Stamps, R.L.; Leonov, A.O.; Bogdanov, A.N.; Togawa, Y. Internal structure of hexagonal skyrmion lattices in cubic helimagnets. New J. Phys.
**2016**, 18, 095004. [Google Scholar] [CrossRef] - Birch, M.T.; Cortés-Ortuño, D.; Turnbull, L.A.; Wilson, M.N.; Groß, F.; Träger, N.; Laurenson, A.; Bukin, N.; Moody, S.H.; Weigand, M.; et al. Real-space imaging of confined magnetic skyrmion tubes. Nat. Commun.
**2020**, 11, 1726. [Google Scholar] [CrossRef] - Wolf, D.; Schneider, S.; Rößler, U.K.; Kovacs, A.; Schmidt, M.; Dunin-Borkowski, R.E.; Büchner, B.; Rellinghaus, B.; Lubk, A. Unveiling the three-dimensional spin texture of skyrmion tubes. Nat. Nanotechnol.
**2022**, 17, 250. [Google Scholar] [CrossRef] [PubMed] - Schneider, S.; Wolf, D.; Stolt, M.J.; Jin, S.; Pohl, D.; Rellinghaus, B.; Schmidt, M.; Büchner, B.; Goennenwein, S.; Nielsch, K.; et al. Induction Mapping of the 3D-Modulated Spin Texture of Skyrmions in Thin Helimagnets. Phys. Rev. Lett.
**2018**, 120, 217201. [Google Scholar] [CrossRef] [PubMed] - Yu, X.Z.; Onose, Y.; Kanazawa, N.; Park, J.H.; Han, J.H.; Matsui, Y.; Nagaosa, N.; Tokura, Y. Real-space observation of a two-dimensional skyrmion crystal. Nature
**2010**, 465, 901. [Google Scholar] [CrossRef] - Yu, X.Z.; Kanazawa, N.; Onose, Y.; Kimoto, K.; Zhang, W.Z.; Ishiwata, S.; Matsui, Y.; Tokura, Y. Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGe. Nat. Mater.
**2011**, 10, 106. [Google Scholar] [CrossRef] - Leonov, A.O.; Togawa, Y.; Monchesky, T.L.; Bogdanov, A.N.; Kishine, J.; Kousaka, Y.; Miyagawa, M.; Koyama, T.; Akimitsu, J.; Koyama, T.S.; et al. Chiral surface twists and skyrmion stability in nanolayers of cubic helimagnets. Phys. Rev. Lett.
**2016**, 117, 087202. [Google Scholar] [CrossRef] - Bordacs, S.; Butykai, A.; Szigeti, B.G.; White, J.S.; Cubitt, R.; Leonov, A.O.; Widmann, S.; Ehlers, D.; Krug von Nidda, H.-A.; Tsurkan, V.; et al. Equilibrium Skyrmion Lattice Ground State in a Polar Easy-plane Magnet. Sci. Rep.
**2017**, 7, 7584. [Google Scholar] [CrossRef] - Fujima, Y.; Abe, N.; Tokunaga, Y.; Arima, T. Thermodynamically stable skyrmion lattice at low temperatures in a bulk crystal of lacunar spinel GaV
_{4}S_{8}. Phys. Rev. B**2017**, 95, 180410. [Google Scholar] [CrossRef] - Romming, N.; Hanneken, C.; Menzel, M.; Bickel, J.E.; Wolter, B.; von Bergmann, K.; Kubetzka, A.; Wiesendanger, R. Writing and Deleting Single Magnetic Skyrmions. Science
**2013**, 341, 636. [Google Scholar] [CrossRef] - Sampaio, J.; Cros, V.; Rohart, S.; Thiaville, A.; Fert, A. Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructures. Nat. Nanotechnol.
**2013**, 8, 839844. [Google Scholar] [CrossRef] - Tomasello, E.M.R.; Zivieri, R.; Torres, L.; Carpentieri, M.; Finocchio, G. A strategy for the design of skyrmion racetrack memories. Sci. Rep.
**2014**, 4, 6784. [Google Scholar] [CrossRef] - Shigenaga, T.; Leonov, A.O. Harnessing Skyrmion Hall Effect by Thickness Gradients in Wedge-Shaped Samples of Cubic Helimagnets. Nanomaterials
**2023**, 13, 2073. [Google Scholar] [CrossRef] - Cortes-Ortuno, D.; Wang, W.; Beg, M.; Pepper, R.A.; Bisotti, M.-A.; Carey, R.; Vousden, M.; Kluyver, T.; Hovorka, O.; Fangohr, H. Thermal stability and topological protection of skyrmions in nanotracks. Sci. Rep.
**2017**, 7, 1. [Google Scholar] [CrossRef] - Schulz, T.; Ritz, R.; Bauer, A.; Halder, M.; Wagner, M.; Franz, C.; Pfleiderer, C.; Everschor, K.; Garst, M.; Rosch, A. Emergent electrodynamics of skyrmions in a chiral magnet. Nat. Phys.
**2012**, 8, 301–304. [Google Scholar] [CrossRef] - Jonietz, F.; Mühlbauer, S.; Pfleiderer, C.; Neubauer, A.; Münzer, W.; Bauer, A.; Adams, T.; Georgii, R.; Böni, P.; Duine, R.A.; et al. Spin Transfer Torques in MnSi at Ultralow Current Densities. Science
**2010**, 330, 1648–1651. [Google Scholar] [CrossRef] - Reichhardt, C.; Reichhardt, C.J.O.; Milosevic, M.V. Statics and dynamics of skyrmions interacting with disorder and nanostructures. Rev. Mod. Phys.
**2022**, 94, 035005. [Google Scholar] [CrossRef] - Kovalev, A.A.; Sandhoefner, S. Skyrmions and Antiskyrmions in Quasi-Two-Dimensional Magnets. Front. Phys.
**2018**, 6, 98. [Google Scholar] [CrossRef] - Ezawa, M. Compact merons and skyrmions in thin chiral magnetic films. Phys. Rev. B
**2011**, 83, 100408. [Google Scholar] [CrossRef] - Mukai, N.; Leonov, A.O. Skyrmion and meron ordering in quasi-two-dimensional chiral magnets. Phys. Rev. B
**2022**, 106, 224428. [Google Scholar] [CrossRef] - Shinjo, T.; Okuno, T.; Hassdorf, R.; Shigeto, K.; Ono, T. Magnetic Vortex Core Observation in Circular Dots of Permalloy. Science
**2000**, 289, 930. [Google Scholar] [CrossRef] - Butenko, A.B.; Leonov, A.A.; Bogdanov, A.N.; Roessler, U.K. Theory of vortex states in magnetic nanodisks with induced Dzyaloshinskii-Moriya interactions. Phys. Rev. B
**2009**, 80, 134410. [Google Scholar] [CrossRef] - Kharkov, Y.A.; Sushkov, O.P.; Mostovoy, M. Bound States of Skyrmions and Merons near the Lifshitz Point. Phys. Rev. Lett.
**2017**, 119, 207201. [Google Scholar] [CrossRef] - Li, X.; Shen, L.; Bai, Y.; Wang, J.; Zhang, X.; Xia, J.; Ezawa, M.; Tretiakov, O.A.; Xu, X.; Mruczkiewicz, M.; et al. Bimeron clusters in chiral antiferromagnets. NPJ Comput. Mater.
**2020**, 6, 169. [Google Scholar] [CrossRef] - Leonov, A.O.; Kézsmárki, I. Asymmetric isolated skyrmions in polar magnets with easy-plane anisotropy. Phys. Rev. B
**2017**, 96, 014423. [Google Scholar] [CrossRef] - Jani, H.; Lin, J.C.; Chen, J.; Harrison, J.; Maccherozzi, F.; Schad, J.; Prakash, S.; Eom, C.-B.; Ariando, A.; Venkatesan, T.; et al. Antiferromagnetic half-skyrmions and bimerons at room temperature. Nature
**2021**, 590, 74. [Google Scholar] [CrossRef] - Ohara, K.; Zhang, X.; Chen, Y.; Kato, S.; Xia, J.; Ezawa, M.; Tretiakov, O.A.; Hou, Z.; Zhou, Y.; Zhao, G.; et al. Reversible Transformation between Isolated Skyrmions and Bimerons. Nano Lett.
**2022**, 22, 8559. [Google Scholar] [CrossRef] [PubMed] - Yu, X.; Kanazawa, N.; Zhang, X.; Takahashi, Y.; Iakoubovskii, K.V.; Nakajima, K.; Tanigaki, T.; Mochizuki, M.; Tokura, Y. Spontaneous Vortex-Antivortex Pairs and Their Topological Transitions in a Chiral-Lattice Magnet. Adv. Mater.
**2024**, 36, 2306441. [Google Scholar] [CrossRef] - Sohn, H.R.O.; Liu, C.D.; Smalyukh, I.I. Schools of skyrmions with electrically tunable elastic interactions. Nat. Commun.
**2019**, 10, 4744. [Google Scholar] [CrossRef] [PubMed] - Shen, Z.; Dong, S.; Yao, X. Manipulation of magnetic topological textures via perpendicular strain and polarization in van der Waals magnetoelectric heterostructures. Phys. Rev. B
**2023**, 108, L140412. [Google Scholar] [CrossRef] - Bachmann, D.; Lianeris, M.; Komineas, S. Meron configurations in easy-plane chiral magnets. Phys. Rev. B
**2023**, 108, 014402. [Google Scholar] [CrossRef] - Gross, D.J. Meron configurations in the two-dimensional O(3) σ-model. Nucl. Phys. B
**1978**, 132, 439. [Google Scholar] [CrossRef] - Speight, M.; Winyard, T.; Babaev, E. Symmetries, length scales, magnetic response, and skyrmion chains in nematic superconductors. Phys. Rev. B
**2023**, 107, 195204. [Google Scholar] [CrossRef] - Garaud, J.; Babaev, E. Properties of skyrmions and multiquanta vortices in chiral p-wave superconductors. Sci. Rep.
**2015**, 5, 17540. [Google Scholar] [CrossRef] [PubMed] - Goebel, B.; Mook, A.; Henk, J.; Mertig, I.; Tretiakov, O.A. Magnetic bimerons as skyrmion analogues in in-plane magnets. Phys. Rev. B
**2019**, 99, 060407(R). [Google Scholar] [CrossRef] - Udalov, O.G.; Beloborodov, I.S.; Sapozhnikov, M.V. Magnetic skyrmions and bimerons in films with anisotropic interfacial Dzyaloshinskii-Moriya interaction. Phys. Rev. B
**2019**, 103, 174416. [Google Scholar] [CrossRef] - Barton-Singer, B.; Barton-Singer, B.J. Stability and asymptotic interactions of chiral magnetic skyrmions in a tilted magnetic field. SciPost Phys.
**2023**, 15, 011. [Google Scholar] [CrossRef] - Xia, J.; Zhang, X.; Liu, X.; Zhou, Y.; Ezawa, M. Qubits based on merons in magnetic nanodisks. Commun. Mater.
**2022**, 3, 88. [Google Scholar] [CrossRef] - Murooka, R.; Leonov, A.O.; Inoue, K.; Ohe, J. Current-induced shuttlecock-like movement of non-axisymmetric chiral skyrmions. Sci. Rep.
**2020**, 10, 396. [Google Scholar] [CrossRef] [PubMed] - Vansteenkiste, A.; Leliaert, J.; Dvornik, M.; Helsen, M.; Garcia-Sanchez, M.; Van Waeyenberge, B. The design and verification of MuMax3. AIP Adv.
**2014**, 4, 107133. [Google Scholar] [CrossRef] - Leliaert, J.; Dvornik, M.; Mulkers, J.; De Clercq, J.; Milosevic, M.V.; Van Waeyenberge, B. Fast micromagnetic simulations on GPU—recent advances made with mumax
^{3}. J. Phys. D Appl. Phys.**2018**, 51, 123002. [Google Scholar] [CrossRef] - Leonov, A.O.; Pappas, C.; Smalyukh, I.I. Field-driven metamorphoses of isolated skyrmions within the conical state of cubic helimagnets. Phys. Rev. B
**2021**, 104, 064432. [Google Scholar] [CrossRef] - Leonov, A.O.; Monchesky, T.L.; Romming, N.; Kubetzka, A.; Bogdanov, A.N.; Wiesendanger, R. The properties of isolated chiral skyrmions in thin magnetic films. New J. Phys.
**2016**, 18, 065003. [Google Scholar] [CrossRef] - Leonov, A.O. Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets. Magnetism
**2023**, 3, 297–307. [Google Scholar] [CrossRef] - Leonov, A.O.; Roessler, U.K. Mechanism of Skyrmion Attraction in Chiral Magnets near the Ordering Temperatures. Nanomaterials
**2023**, 13, 891. [Google Scholar] [CrossRef] [PubMed] - Leonov, A.O.; Mostovoy, M. Multiply periodic states and isolated skyrmions in an anisotropic frustrated magnet. Nat. Commun.
**2015**, 6, 8275. [Google Scholar] [CrossRef] - Rozsa, L.; Deak, A.; Simon, E.; Yanes, R.; Udvardi, L.; Szunyogh, L.; Nowak, U. Skyrmions with Attractive Interactions in an Ultrathin Magnetic Film. Phys. Rev. Lett.
**2016**, 117, 157205. [Google Scholar] [CrossRef] - Leonov, A.O. Skyrmion clusters and chains in bulk and thin-layered cubic helimagnets. Phys. Rev. B
**2022**, 105, 094404. [Google Scholar] [CrossRef] - Kim, K.-M.; Go, G.; Park, M.J.; Kim, S.K. Emergence of Stable Meron Quartets in Twisted Magnets. Nano Lett.
**2024**, 24, 74. [Google Scholar] [CrossRef] [PubMed] - Leask, P. Baby Skyrmion crystals. Phys. Rev. D
**2022**, 105, 025010. [Google Scholar] [CrossRef] - Kobayashi, M.; Nitta, M. Fractional vortex molecules and vortex polygons in a baby Skyrme model. Phys. Rev. D
**2013**, 87, 125013. [Google Scholar] [CrossRef] - Bessarab, P.F.; Uzdin, V.; Jonsson, H. Method for finding mechanism and activation energy of magnetic transitions, applied to skyrmion and antivortex annihilation. Comp. Phys. Commun.
**2015**, 196, 335. [Google Scholar] [CrossRef] - Lin, S.-Z.; Saxena, A.; Batista, C.D. Skyrmion fractionalization and merons in chiral magnets with easy-plane anisotropy. Phys. Rev. B
**2015**, 91, 224407. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Schematics of isolated Néel Skyrmions in polar magnets with ${\mathrm{C}}_{nv}$ symmetry (or in multilayers with the induced DMI). (

**b**,

**c**) Schematics of bimerons formed as ruptures of the cycloidal spiral according to (3). The magnetization field wraps only the corresponding half of the S

_{2}sphere (

**b**). The upper anti-meron within the blue circle in (

**c**) and the lower meron within the red circle can mutually annihilate, leaving a bimeron pair with total charge $Q=0$. (

**d**,

**e**) Schematics of bimerons formed as a result of wrapping the upper or lower hemisphere. There are two varieties, merons and anti-merons, with positive and negative vorticity $m=\pm 1$. (

**f**) Schematics of two bimerons with opposite polarity obtained from the axisymmetric Skyrmion (middle panel) by magnetization rotation around the y axis with the angle $\pi /2$. The magnetization in the center of the isolated Skyrmion (the middle panel in (

**f**), point O) now points horizontally along the dipole moment of the bimeron in the upper and lower panels of (

**f**). Two points A and B with ${m}_{z}=0$ within the isolated Skyrmions (the middle panel in (

**f**)) now become the centers of circular and crescent-shaped merons (the upper and lower panels of (

**f**)).

**Figure 2.**(

**a**) Magnetic phase diagram of the solutions for model (7) with easy-plane uniaxial anisotropy (EPA). The filled areas designate regions of thermodynamic stability of the corresponding phases: blue shading—cycloidal spiral (

**b**); green shading—elliptical cone (

**c**); white shading—polarized ferromagnetic state (

**d**); yellow shading—hexagonal Skyrmion lattice; red shading—tilted ferromagnetic state (

**e**). The thick black lines indicate the first-order phase transitions between corresponding phases, while the thin black lines indicate the second-order phase transitions. The field is measured in units of ${H}_{0}={D}^{2}/A\left|\mathbf{M}\right|$, i.e., $\mathbf{h}=\mathbf{H}/{H}_{0}$, and ${k}_{u}={K}_{u}{M}^{2}A/{D}^{2}$ is the non-dimensional anisotropy constant.

**Figure 3.**Internal magnetic structure of an isolated bimeron characterized by the $xy$-color plots of the ${m}_{z}$-component of the magnetization (

**a**), the DMI and total energy densities (

**c**,

**d**), and the topological charge density (

**e**); $h=0$, ${k}_{u}=-1.5$. The cross-cut (

**b**) along the meron center shows the part with the opposite rotational sense, the center of which is located at distance ${p}^{\prime}$ from the center of the circular meron, with p being the distance between the centers of the vortex and antivortex. The insets in (

**b**) show corresponding zoomed 2D (

**i**) and 1D (

**ii**) magnetization distributions, i.e., in (

**i**) the color indicates the magnetization value in the range $[-0.2;0.2]$. The corresponding color plots for the exchange energy density and the anisotropy energy densities are shown in (

**f**,

**g**). The Zeeman energy is 0, as $h=0$, while the field is measured in the units of ${H}_{0}={D}^{2}/A\left|\mathbf{M}\right|$, i.e., $\mathbf{h}=\mathbf{H}/{H}_{0}$. The magnetization vector $\mathbf{m}(x,y)=\mathbf{M}/\left|\mathbf{M}\right|$ is normalized to unity; ${k}_{u}={K}_{u}{M}^{2}A/{D}^{2}$ is the nondimensional anisotropy constant; the spatial coordinates $\mathbf{x}$ are measured in units of the characteristic length of modulated states ${L}_{D}=A/D$; and the value $\lambda =4\pi {L}_{D}$ for zero magnetic field is the period of the cycloid.

**Figure 4.**(

**a**,

**b**) Field- and anisotropy-driven evolution of the parameters p and ${p}^{\prime}$ for a fixed value of the anisotropy (${k}_{u}=-1.5$, (

**a**)) and field ($h=0$, (

**b**)), correspondingly; p is the distance between the centers of the vortex and the antivortex, while ${p}^{\prime}$ is the distance from the center of the circular meron to the center of the circular area with the wrong rotational sense. For the field-driven transformation of bimerons, the parameters p and ${p}^{\prime}$ as well as their ratio (

**c**) all increase. This means that the parameter ${p}^{\prime}$ increases faster than the parameter p, with both processes indicating the transformation into an isolated Skyrmion surrounded by a homogeneous state with ${m}_{z}=1$. The depth of the region with the wrong twist $\Delta {m}_{z}$ tends to zero (

**c**). Qualitatively, the same process occurs with growing anisotropy (

**b**,

**d**). At some critical anisotropy value, the bimerons become too small to be addressed using the chosen cell sizes of the numerical grids. The color plots of the magnetization in (

**e**,

**f**) reflect the above-mentioned field- and anisotropy-driven transformations. The black arrows in all color plots are the projections of the magnetization onto the $xy$ plane.

**Figure 5.**(

**a**) The inter-bimeron potential $\Phi $ versus the distance d between the centers of circular anti-merons for two bimerons oriented head-to-tail (blue curve and left inset) and/or side-by-side (red curve and right inset). The color plots for the total energy density (

**b**) are shown for bimerons located at larger distances (point A, right panel), in the minimum (point B, middle panel), and at shorter distances (point C, left panel). These color plots indicate the underlying reason for the attracting interaction and the minimum of $\Phi $; at some optimal inter-meron distance, the first bimeron covers the circular region with the positive energy density of the subsequent bimeron. The color plots of the total energy and DMI energy density (

**c**) do not demonstrate any energy benefit from coupling two bimerons. On the contrary, overlapping “wings” lead to an energy increase (red curve in (

**a**)). Here, $h=0,{k}_{u}=-1.5$.

**Figure 6.**(

**a**) The interaction potential for two bimerons with the opposite polarities exhibits only the inter-particle repulsion. Here, we orient two bimerons with either their circular merons (the red curve and left panels in (

**b**,

**c**)) or with their crescents (the blue curve and right panels in (

**b**,

**c**)) facing the inter-meron area. The insets show color plots of the magnetization at the indicated points of the curves. To plot both curves, we pinned the magnetization in the centers of the circular anti-merons. Obviously, being unpinned, the two bimerons find a path to annihilate, as they have the opposite topological charges. The corresponding DMI and full energy density distributions are plotted in (

**c**). $h=0$, ${k}_{u}=-1.5$.

**Figure 7.**Internal structure of a bimeron chain with six constituent bimerons. (

**a**) Color plot for the ${m}_{z}$-component of the magnetization. (

**b**) Color plot for the DMI energy density. (

**c**) Color plot for the total energy density. (

**d**) Color plot for the energy density of the easy-plane anisotropy. (

**e**) Color plot for the exchange energy density. $h=0$, ${k}_{u}=-1.5$. The field is measured in units of ${H}_{0}={D}^{2}/A\left|\mathbf{M}\right|$, i.e., $\mathbf{h}=\mathbf{H}/{H}_{0}$. The magnetization vector $\mathbf{m}(x,y)=\mathbf{M}/\left|\mathbf{M}\right|$ is normalized to unity. ${k}_{u}={K}_{u}{M}^{2}A/{D}^{2}$ is the nondimensional anisotropy constant. Spatial coordinates $\mathbf{x}$ are measured in units of the characteristic length of modulated states ${L}_{D}=A/D$. The value $\lambda =4\pi {L}_{D}$ for zero magnetic field is the period of the cycloid.

**Figure 8.**Internal structure of a $(-)$roundabout with counterclockwise circling of bimerons. (

**a**) Color plot for the ${m}_{z}$-component of the magnetization. (

**b**) Color plot for the DMI energy density. (

**c**) Color plot for the total energy density. (

**d**) Color plot for the energy density of the easy-plane anisotropy. (

**e**) Color plot for the exchange energy density. $h=0$, ${k}_{u}=-1.5$. The field is measured in units of ${H}_{0}={D}^{2}/A\left|\mathbf{M}\right|$, i.e., $\mathbf{h}=\mathbf{H}/{H}_{0}$. The magnetization vector $\mathbf{m}(x,y)=\mathbf{M}/\left|\mathbf{M}\right|$ is normalized to unity. ${k}_{u}={K}_{u}{M}^{2}A/{D}^{2}$ is the nondimensional anisotropy constant. Spatial coordinates $\mathbf{x}$ are measured in units of the characteristic length of modulated states ${L}_{D}=A/D$. The value $\lambda =4\pi {L}_{D}$ for zero magnetic field is the period of the cycloid.

**Figure 9.**Internal structure of a $(-)$roundabout with clockwise circling of bimerons. (

**a**) Color plot for the ${m}_{z}$-component of the magnetization. (

**b**) Color plot for the DMI energy density. (

**c**) Color plot for the total energy density. (

**d**) Color plot for the energy density of the easy-plane anisotropy. (

**e**) Color plot for the exchange energy density. $h=0$, ${k}_{u}=-1.5$. The field is measured in units of ${H}_{0}={D}^{2}/A\left|\mathbf{M}\right|$, i.e., $\mathbf{h}=\mathbf{H}/{H}_{0}$. The magnetization vector $\mathbf{m}(x,y)=\mathbf{M}/\left|\mathbf{M}\right|$ is normalized to unity. ${k}_{u}={K}_{u}{M}^{2}A/{D}^{2}$ is the nondimensional anisotropy constant. Spatial coordinates $\mathbf{x}$ are measured in units of the characteristic length of modulated states ${L}_{D}=A/D$. The value $\lambda =4\pi {L}_{D}$ for zero magnetic field is the period of the cycloid.

**Figure 10.**Internal structure of a $(+)$roundabout. (

**a**) Color plot for the ${m}_{z}$-component of the magnetization. (

**b**) Color plot for the DMI energy density. (

**c**) Color plot for the total energy density. (

**d**) Color plot for the energy density of the easy-plane anisotropy. (

**e**) Color plot for the exchange energy density. $h=0$, ${k}_{u}=-1.5$. The field is measured in units of ${H}_{0}={D}^{2}/A\left|\mathbf{M}\right|$, i.e., $\mathbf{h}=\mathbf{H}/{H}_{0}$. The magnetization vector $\mathbf{m}(x,y)=\mathbf{M}/\left|\mathbf{M}\right|$ is normalized to unity. ${k}_{u}={K}_{u}{M}^{2}A/{D}^{2}$ is the nondimensional anisotropy constant. Spatial coordinates $\mathbf{x}$ are measured in units of the characteristic length of modulated states ${L}_{D}=A/D$. The value $\lambda =4\pi {L}_{D}$ for zero magnetic field is the period of the cycloid.

**Figure 11.**Internal structure of a bimeron “crossing”. (

**a**) Color plot for the ${m}_{z}$-component of the magnetization. (

**b**) Color plot for the DMI energy density. (

**c**) Color plot for the total energy density. (

**d**) Color plot for the energy density of the easy-plane anisotropy. (

**e**) Color plot for the exchange energy density. $h=0$, ${k}_{u}=-1.5$. The field is measured in units of ${H}_{0}={D}^{2}/A\left|\mathbf{M}\right|$, i.e., $\mathbf{h}=\mathbf{H}/{H}_{0}$. The magnetization vector $\mathbf{m}(x,y)=\mathbf{M}/\left|\mathbf{M}\right|$ is normalized to unity. ${k}_{u}={K}_{u}{M}^{2}A/{D}^{2}$ is the nondimensional anisotropy constant. Spatial coordinates $\mathbf{x}$ are measured in units of the characteristic length of modulated states ${L}_{D}=A/D$. The value $\lambda =4\pi {L}_{D}$ for zero magnetic field is the period of the cycloid.

**Figure 12.**(

**a**) Total energy of different bimeron macromolecules in dependence on the number N of constituent bimerons. The energy of edge states formed at the specimen boundary is excluded. The inset shows the topological charge of the central meron formed in circular macromolecules, computed as $Q-N$, where Q is the total charge of the magnetization distribution excluding the edge states and N is the number of exterior bimerons. The color coding is the same as used in the main graph: red for crossings, blue for $(-)$roundabouts, green for $(+)$roundabouts, and black for chains. For crossings and $(-)$roundabouts, the charges of the central merons are the same, $Q=-1/2$. Solid lines connecting points with a fixed number N indicate macromolecules which are robust against transformation into chains. Other macromolecules, indicated by dotted lines, can be wrapped into chains by the displacement of the central meron, i.e., such macromolecules are stable for symmetrical scaling of inter-meron distances but loose their stability while being deformed. (

**b**) Total energy of bimeron macromolecules taking into account the energy of the edge states, which clearly favor $(+)$roundabouts. The edge states bear their own topological charges $Q=0.16$, which uniformly shift the dependencies in the inset. (

**c**,

**d**) Transformation of a $(+)$roundabout with $N=5$ into a buckled chain (see Supplementary Video S1). (

**e**,

**f**) Transformation of a $(-)$roundabout with $N=7$ into a looped chain (see Supplementary Video S2). (

**g**,

**h**) Transformation of a crossing with $N=5$ into a crossing with $N=3$ and fragments of attached chains (see Supplementary Video S3). $h=0,{k}_{u}=-1.5$.

**Figure 13.**Stable bimeron macromolecules obtained by combinations of “crossings” and “roundabouts” for $N=6$ (

**a**,

**b**) and $N=8$ (

**c**–

**e**) in $(+)$roundabouts. In “benzene” (

**a**), all angles between bimerons are $2\pi /3$, as dictated by both ring varieties. In its $(N=8)$ counterpart (

**c**), however, the angle between the bimerons within the ring is $3\pi /4$, as specified by the “roundabout”. Less symmetric macromolecules (

**b**,

**d**,

**e**) exhibit structural deformations and the inter-bimeron angles vary around the rings, which may lead to instability (see Supplementary Video S4). In addition, $(+)$“roundabouts” with eight bimerons can alternatively be connected into a stripe—an analogue of the chain (

**f**). $h=0,{k}_{u}=-1.5$.

**Figure 14.**(

**a**) The linking process of three chains with the same number of bimerons. The red cross highlights the $(N=4)$ crossing with the inter-bimeron angles $\pi /2$, which facilitates the creation of a bimeron cluster with a square arrangement of bimerons (

**b**). (

**c**) Stable two-dimensional periodic tessellation with hexagonal ordering of bimerons. Such a state is possible only if the sides of the hexagonal cells contain several bimerons (two bimerons in the present case). $h=0,{k}_{u}=-1.5$.

**Figure 15.**If the cell side of the periodic tessellations in Figure 14c contains just one bimeron, such a hexagonal order transforms into a random bimeron distribution. As an initial state in the first panel, we use a hexagonal bimeron lattice with a periodic mixture of $(+)$roundabouts and $(-)$roundabouts. Therefore, periodic boundary conditions can be used at both sides of the numerical grid. The subsequent panels demonstrate the disintegration process toward a random bimeron polymer (see Supplementary Video S5). $h=0,{k}_{u}=-1.5$.

**Figure 16.**(

**a**–

**d**) The internal structure of the hexagonal SkL, shown as color plots of the topological charge density within the yellow-shaded region of the phase diagram in Figure 2a. With increasing easy-plane anisotropy (from left to right in the first row), the nuclei of merons with the opposite topological charges emerge within the cell boundaries. Mutual transformation between the hexagonal SkL and a disordered bimeron polymer is achieved by changing the uniaxial anisotropy from $-0.5$ to $-1.5$ (second row) and back from $-1.5$ to $-0.5$ (third row). The ordered SkL (

**e**) is a local energy minimum for ${k}_{u}=-0.5,\phantom{\rule{0.166667em}{0ex}}h=0$. If the anisotropy is suddenly switched to $-1.5$, then the SkL undergoes the following transformation: (

**f**) merons and anti-merons with positive polarity nucleate pairwise within the cell boundary (highlighted by black circles); (

**g**) the hexagonal cell becomes distorted, enabling merons and anti-merons to approach each other; (

**h**) merons and anti-merons merge and annihilate (highlighted by white circles); and (

**i**) only bimerons are left. The vortex originates from the Skyrmion in the center of the SkL, whereas the anti-vortex is a remainder of the boundary (see Supplementary Video S6). A disordered bimeron polymer (

**j**) is a metastable cluster formed at ${k}_{u}=-1.5,\phantom{\rule{0.166667em}{0ex}}h=0.5$. If the anisotropy switches to $-0.5$, then the SkL represents the global minimum of the energy functional (7). This means that the circular merons with polarity against the field become energetically favorable. These anti-merons rearrange and fill the whole space, whereas the anti-vortices with polarity along the field squeeze into the boundary regions (

**k**,

**l**). In (

**m**), the Skyrmions form a disordered state but eventually manage to form an almost perfectly hexagonal arrangement (

**n**) (see Supplementary Video S7).

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## Share and Cite

**MDPI and ACS Style**

Mukai, N.; Leonov, A.O.
“Polymerization” of Bimerons in Quasi-Two-Dimensional Chiral Magnets with Easy-Plane Anisotropy. *Nanomaterials* **2024**, *14*, 504.
https://doi.org/10.3390/nano14060504

**AMA Style**

Mukai N, Leonov AO.
“Polymerization” of Bimerons in Quasi-Two-Dimensional Chiral Magnets with Easy-Plane Anisotropy. *Nanomaterials*. 2024; 14(6):504.
https://doi.org/10.3390/nano14060504

**Chicago/Turabian Style**

Mukai, Natsuki, and Andrey O. Leonov.
2024. "“Polymerization” of Bimerons in Quasi-Two-Dimensional Chiral Magnets with Easy-Plane Anisotropy" *Nanomaterials* 14, no. 6: 504.
https://doi.org/10.3390/nano14060504