Properties and dynamics of meron topological spin textures in the two-dimensional magnet CrCl3

Merons are nontrivial topological spin textures highly relevant for many phenomena in solid state physics. Despite their importance, direct observation of such vortex quasiparticles is scarce and has been limited to a few complex materials. Here, we show the emergence of merons and antimerons in recently discovered two-dimensional (2D) CrCl3 at zero magnetic field. We show their entire evolution from pair creation, their diffusion over metastable domain walls, and collision leading to large magnetic monodomains. Both quasiparticles are stabilized spontaneously during cooling at regions where in-plane magnetic frustration takes place. Their dynamics is determined by the interplay between the strong in-plane dipolar interactions and the weak out-of-plane magnetic anisotropy stabilising a vortex core within a radius of 8–10 nm. Our results push the boundary to what is currently known about non-trivial spin structures in 2D magnets and open exciting opportunities to control magnetic domains via topological quasiparticles.

The strict 2D nature of the layers leads to unique physical properties ranging from stacking dependent interlayer magnetism 5,6 , giant tunneling magnetoresistance 7,8 and second harmonic generation 9 , up to electric field control of magnetic properties 10 . Of particular interest are topological spin excitations 11,12 , e.g. merons, which are crucial to understand fundamental problems of chiral magnetic order 13 and development of novel spintronic devices for information technologies 11 . Layered magnetic materials provide an ideal platform to investigate and harness this critical spin phenomenon as the genuine character of meron systems is intrinsically 2D and the integration of magnetic sheets in device heterostructures is a reality 14 .
Here we demonstrate that monolayer CrCl 3 hosts merons and antimerons in its magnetic structure. Both quasiparticles are created naturally during zero-field cooling at low temperatures.
We find that both spin textures are directly associated with the metastability of the magnetic do-mains on CrCl 3 induced by spin fluctuations. The merons and antimerons assume a random distribution throughout the surface creating a network of topological spin textures with no apparent lattice-order as observed in other materials 13 . The different sites of the network can interact to each other leading to different type of collisions involving meron and antimerons occurring within a nanosecond time-scale. Our results indicate that the control of vortex and antivortex in CrCl 3 is also a driving force for the manipulation of magnetic domains which follows closely the annihilation process of the spin textures.

Results
Our starting point is the following spin Hamiltonian: where S i and S j are the localized magnetic moments on Cr atomic sites i and j which are coupled by pair-wise exchange interactions. J i j and λ i j = J xx,yy l − J zz l (where l sets the nearest neighbours taken into account) are the isotropic and anisotropic bilinear (BL) exchanges, respectively, and D i is the single ion magnetic anisotropy. B i and B dp i represent external and dipole magnetic field sources respectively. We considered in Eq.1 up to third nearest neighbours for J i j (J 1 − J 2 − J 3 ) and λ i j (λ 1 − λ 2 − λ 3 ) in the description of CrCl 3 . The fourth term in Eq.1 represents the biquadratic (BQ) exchange which involves the hopping of two or more electrons between two adjacent sites 15,16 .
Its strength is given by the constant K i j , which is the simplest and most natural form of non-Heisenberg coupling. We recently found that several 2D magnets develop substantial BQ exchange in their magnetic properties which is critical to quantitatively describe important features such as Curie temperatures, thermal stability and magnon spectra 16 . The magnitude of K i j for CrCl 3 is 0.22 meV which is slightly smaller than the BL exchange for the first-nearest neighbors J 1 = 1.28 meV but too large to be ignored. In our implementation the BQ exchange is quite general and can be applied to any pair-wise exchange interaction of arbitrary range. All the parameters in Eq.1 are extracted from non-collinear ab initio simulations including spin-orbit coupling to determine the different components of J i j , λ i j and K i j (i j = xx, yy, zz) as described in Ref. 16 . For atomistic spin dynamics we calculate the effective magnetic field B i arising from the BQ exchange interactions with components: This effective field is then included within the total field B eff describing the time evolution of each atomic spin using the stochastic Landau-Lifshitz-Gilbert (LLG) equation: where γ is the gyromagnetic ratio. See Supplementary Section 1 for details. The dynamics of merons and antimerons is determined at different temperatures T and magnetic fields B eff over a time scale of more than 40 ns since the cooling starts at T >> T c , where T c is the Curie temperature, until it reaches 0 K within 2 ns (Supplementary Section 2). Figure 1 shows that as this process occurs, the nucleation of several small areas with an out-of-plane spin polarization S z perpendicular to the easy-plane of CrCl 3 appeared naturally at T < T c (see Supplementary Figure S1 and Supplementary Movie S1). As the temperature reached 0 K, there is a clear formation of several notch structures which are created randomly all over the crystal without following any apparent pattern or preferential nucleation site (Fig. 1c). Indeed, the number of dark and bright spin textures formed during the cooling at zero field is even indicating some equilibrium on the different signs of  Figure S3 and Supplementary Movie S3). We performed a partial hysteresis calculation to estimate the critical magnetic field to switch all the spins in the system including those at the distinct textures (Supplementary Figure S4). We found a magnitude of 200 mT which is surprisingly large for the small single ion anisotropy but consistent with recent magnetometry measurements 17 .
In order to identify the nature of the dark and bright spin structures, we analysed closely the different patterns formed on CrCl 3 at zero field and low temperatures ( Figure 2). We could identify four main spin textures by looking at the spatial distribution of S z as labelled B1−B4 (Fig. 2a).
Even though they show similar magnitudes of S z along x or y (Fig. 2b) the same does not apply for components S x and S y . That is, some shape anisotropy is observed at the in-plane distribution of the magnetization. We estimated an average radius of around 14.5 nm and 15.7 nm along of x− and y−displacement, respectively, despite of the spin notches considered (Supplementary Section 3). This indicates a more elliptical pattern where the main distinction between the spin textures B1−B4 is the orientation of the spins from the core (out-of-plane) to in-plane away from the centre forming a vortex structure. Such vortex structures are typical of non-trivial topological spin textures such skyrmions and anti-skyrmions 18 . To unveil their true nature we can calculate the topological number N given by [19][20][21][22] :   Figure S6). We can understand this type of magnetic frustration in terms of the competing exchange interactions between the different nearest neigh-bours (1st, 2nd, 3rd) in CrCl 3 16 . In terms of the isotropic exchange (J 1st , J 2nd , J 3rd ), we observe that the frustration takes places due to the third nearest-neighbour which has an anti-ferromagnetic exchange (J 3rd = −0.025 meV) relative to the first-(J 1st = 1.28 meV) and the second-nearest neighbours (J 2nd = 0.072 meV). In terms of the anisotropic exchange (λ 1st , λ 2nd , λ 3rd ), the competition between second-and third-nearest neighbours (λ 2nd = -0.0097 meV , λ 3rd = −0.0051 meV) with the first-nearest neighbours (λ 1st = 0.020 meV) induced that the in-plane spins (S x , S y ) become negligible. In addition, a full in-plane spin polarization along S x and S y would lead to a singularity of the exchange energy which is avoided as S z becomes non-negligible 28  There are no structural constraints in the stabilization of the magnetic domains and the low magnetic anisotropy would orientate the spins more freely without a preferential orientation within the easy-plane 33 . Therefore, such spontaneous formation of merons and antimerons is an astonishing, previously unreported phenomena in the magnetism of any 2D vdW magnet.
An intriguing question that raised by the presence of these topological spin textures in CrCl 3 is what their physical origin. It is known that strongly inhomogeneous magnetic textures can be created due to the competition between local interactions, e.g. exchange and magnetic anisotropy, and long-range interactions mediated by demagnetizing fields and magnetic dipoles 18,[34][35][36][37] . In the case of CrCl 3 , the interplay between dipolar interactions and magnetic anisotropy is one of the main ingredients in the creation of merons and antimerons. Figure 4a-b show that as the cooling process takes place, the different spin textures intrinsically carry a large component of the dipole moment perpendicular to the surface (±300 mT). This is mainly localized at the core of the quasiparticle and assists in stabilizing a strong component of S z . Indeed, there is almost no difference between the projection of the magnetization perpendicular to the surface and that for the dipole-field along z ( Fig. 4a-b). This indicates a close relationship between dipole-dipole interactions and magnetism in CrCl 3 . The distinct spin polarization of the core of the merons and antimerons does not give any significant variation on the magnitude of the dipole-field, which is larger than those within the in-plane components (Fig. 4c-d). Surprisingly, both components of the dipole-field (x and y) reach smaller magnitudes (±200 mT) than those along z throughout the surface, and are strictly zero at the position of the spin textures.
In this context, the in-plane dipole fields favour an in-plane magnetization whereas the single ion anisotropy orients the spins perpendicular to the surface. In CrCl 3 the magnetic anisotropy is small which allows most of the spins to follow the dipolar directions except those at the core of the merons and antimerons. In such particular locations, the stronger z component of the dipole-field pushes the spins out-of-plane enhancing the magnetic anisotropy. In any other part of the surface without the topological spin textures, the magnetization rotates parallel to the surface as an effect of the dipolar interactions 33 . It is worth mentioning that no transition between a previously stabilized magnetic configuration, such as stripes, into the non-trivial magnetic textures is noticed in the spin dynamics with or without an applied field as it has been suggested as a potential origin of bubbles or skyrmions [38][39][40] . Moreover, we do not take into account asymmetric exchange (Dzyaloshinskii-Moriya interactions) into our simulations which was initially checked to have no effect on the dynamics of the merons and antimerons (Supplementary Figure S14). This excludes additional mechanisms based on relativistic effects 18 . We also considered simulations without the inclusion of dipolar fields (Supplementary Figure S7) for CrCl 3 with a two-fold implication. First, there is no appearance of non-trivial topological spin textures as the magnetization is consistently out-ofplane over the entire crystal. Second, there is the stabilization of an easy-axis perpendicular to the surface following the single ion anisotropy. That is, there is a suppression of the easy-plane experimentally observed for CrCl 3 33 . Even though other models 36, 37 utilized for magnetic thin films assumed that the dipole-dipole directions can be effectively replaced by an easy-plane (XY), our calculations indicate that the inclusion of dipolar interactions plays a key role in the description of the magnetic properties of CrCl 3 (Supplementary Figure S11). The spins textures formed at such artificial easy-plane 36, 37 looked more chaotic than those computed without such restriction being more complex to assign any clear feature or to determine a topological number N.
We also noticed that the inclusion of biquadratic exchange and next-nearest neighbours is critical for the stabilization of non-trivial topological spin textures in 2D magnets (see Supplemen-tary Figures S12, S13). For the former, the higher-order exchange between the spins gives further stability to Eq.1 since the sign of K i j = 0.22 meV is positive 16 . For the latter, the non-inclusion which suggests that the system may not be in a local minimum but rather at a flat energy landscape.
As a matter of fact, an increment of D will not cease the fluctuations as our calculations showed that they may be intrinsic to CrCl 3 (Supplementary Figure S15).

Discussion
The discovery of non-trivial topological spin-textures (merons and antimerons) in a non-chiral 2D magnetic material (CrCl 3 ) opens the possibility for other layered materials display such behaviour.
Some guidelines for looking into materials that may develop such quasiparticles would be i) a subsequent control will be achieved for such non-trivial spin topology. In this sense, the emergent electrodynamics initially established for skyrmions 46 can be explored further at a more fundamental level using merons in a truly 2D magnet.

Methods
All methods are included in Supplementary Information which includes Supplementary Sections S1−S5, Supplementary Movies S1−S9 and Supplementary Figures S1−S14.

Data Availability
The data that support the findings of this study are available within the paper and its Supplementary Information.

Competing interests
The Authors declare no conflict of interests.