Emergence of topological bimerons in monolayer CrSBr

doi:10.21203/rs.3.rs-4236383/v1

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Emergence of topological bimerons in monolayer CrSBr

https://doi.org/10.21203/rs.3.rs-4236383/v1

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The rich and fascinating physics of topological spin textures in van der Waals two-dimensional magnets has motivated recent growing interests, though a comprehensive understanding remains elusive. Here, by atomistic simulations on monolayer CrSBr, we find two different magnetic phases emerging under non-equilibrium conditions at a standard ferromagnetic transition T_c and at a distinct lower temperature T*. Moreover, the real-space analysis of the spin texture reveals the emergence of metastable topological bimeron defects below T*, showing an algebraic-like decaying spin-spin correlation function. The Dzyaloshinskii-Moriya interaction, induced by the local site asymmetry in the centrosymmetric CrSBr monolayer, is proved to be the origin of the bimerons formation. Furthermore, the increasing bimerons density upon increasing the cooling rate follows a Kibble-Zurek behavior, suggesting a handle to drive and control topological bimerons below T*. Our results put forward CrSBr as an important candidate for the investigation of dynamical behavior of bimerons in vdW magnets.

Physical sciences/Physics/Condensed-matter physics/Magnetic properties and materials

Physical sciences/Physics/Condensed-matter physics/Spintronics

Physical sciences/Materials science/Theory and computation/Electronic structure

Van der Waals (vdW) magnets have been identified as an emerging class of materials, providing useful insights in fundamental condensed matter physics and potential applications in spintronics^1–6. In particular, some nontrivial topological spin phases, such as skyrmions, merons, bubbles, etc. have been observed or predicted at equilibrium^7–9. However, out-of-equilibrium dynamical processes, where some unconventional topological defects and domain walls might emerge^10–12, are much less explored.

Among the recently synthesized vdW magnets, the A-type antiferromagnetic (AFM) vdW CrSBr has received great attention^13–21. CrSBr shows high stability under ambient conditions and its magnetic transition temperature is quite high (i.e. T_N = 132 K for the bulk¹³ and T_c = 146 K for monolayer¹⁴). The orthorhombic structure of CrSBr makes this material anisotropic and results in large differences in magneto-transport along the three different a, b, c directions^13,16,19. Even more interestingly, besides the magnetic transition temperature T_N, many experiments found that the magnetic susceptibility, when measured as a function of temperature, shows a kink at T* near 30 ~ 40 K^13,16,18,19. Moreover, due to the strong coupling between magnetic order and charge transport, negative magnetoresistance effects were observed at T < T*¹³. The unusual changes in magnetic susceptibility and magnetoresistance suggest that an unconventional magnetic order at T* might be present, thus attracting a lot of attention. Some studies suggested that this “hidden” order comes from magnetic defects and impurities¹⁹ or magnetic fluctuations under field¹⁶. Another research associated the phase transition to the slowing down of magnetic fluctuations, accompanied by a continuous reorientation of internal fields¹⁸. Interestingly, such an unconventional magnetic order is not observed in all experiments^20,21. In any case, the unresolved controversy surrounding the unconventional order in CrSBr at T* highlights the complexity of the magnetic phases of the system and urgently calls for further investigations.

In this work, we study the magnetism of monolayer CrSBr using first principles calculations, combined with Monte-Carlo and spin dynamics simulations. Our results show that, at equilibrium, a standard FM transition occurs at 112 K. However, under non-equilibrium conditions, when decreasing the temperature, the CrSBr monolayer undergoes a transition to a phase where topological bimeron-like defects form. These observations are supported by the calculations of spin-spin correlation functions and real-space spin textures. Moreover, by analyzing the phase diagrams obtained by artificially tuning the relevant parameters of the model, we found that the bimerons formation is induced by the Dzyaloshinskii-Moriya (DM) interaction, arising from the local site asymmetry in the CrSBr monolayer. Finally, we show that the density of bimerons as a function of the temperature-quenching rate follows the Kibble-Zurek mechanism and can be further modulated by the competition between DM and exchange interactions.

Spin Hamiltonian and DFT Results

The crystal structure of monolayer CrSBr is illustrated in Fig. 1a. The lattice constants, optimized by Density Functional Theory (DFT) (see Methods), are a = 3.506 Å, b = 4.698 Å, respectively. Monolayer CrSBr belongs to the centrosymmetric Pmmn space group, the inversion center being located at the center of second-nearest (2nd, label 0 and 2 in Fig. 1a) neighbor Cr-Cr pairs. However, due to the local site inversion breaking, no inversion center occurs for the first-nearest (1st, 0 and 1) and third-nearest (3rd, 0 and 2) neighbor Cr-Cr pairs. Therefore, according to Moriya rules²², the DM interaction^22,23 is along the b (a) direction and labeled as D_y (D_x) for 1st (3rd ) Cr-Cr pair in the following.

The magnetic interactions are modeled by using the spin Hamiltonian with the following general form:

$$H=\frac{1}{2}{\sum }_{i\ne j}{S}_{i}{J}_{ij}{S}_{j}+{\sum }_{i}{S}_{i}{A}_{i}{S}_{i}$$

where J_ij and A_i represent the exchange coupling interaction tensor and single ion anisotropy (SIA), respectively, where all the parameters can be obtained from DFT calculations using the four-state method. The J_ij results are shown in Supplementary Table 1. The diagonal terms J_xx, J_yy, and J_zz of the three nearest neighbors are all positive, showing ferromagnetism and preventing magnetic frustration in monolayer CrSBr. The antisymmetric part of J_ij corresponds to the DM interaction and can be expressed as ${D}_{\gamma }={\frac{1}{2}\epsilon }_{\alpha \beta \gamma }({J}_{\alpha \beta }-{J}_{\beta \alpha })$, where ${\epsilon }_{\alpha \beta \gamma }$ denotes the Levi-Civita symbol. The DM interactions of 1st and 3rd nearest Cr-Cr pairs in CrSBr are D_y = 0.16 meV and D_x = -0.29 meV, respectively, in agreement with the symmetry analysis by Moriya rules. Moreover, the third nearest-neighbor exchange constant, ${J}_{03}=({J}_{xx}+{J}_{yy}+{J}_{zz})/3$, is 2.96 meV and D_x/J₀₃ is ~ 0.1, i.e., in the typical range for the formation of topological spin textures²⁴. In addition, due to the orthorhombic structure of CrSBr, the A_xx, A_yy, and A_zz parameters are not equal to each other; rather, the calculated A_xx-A_zz and A_yy-A_zz are − 0.01 meV and − 0.02 meV, respectively, consistent with experiments suggesting the easy, hard, and intermediate axis of CrSBr to be along the b, c, and a direction, respectively ¹⁵.

Equilibrium Monte-Carlo Simulations

Based on Eq. (1), we performed atomistic spin simulations via Monte-Carlo (MC) (see Methods) for a 100 nm × 100 nm monolayer CrSBr, focusing on its magnetic transition temperature. On one hand, a classic FM M-T curve is shown in Fig. 1b, indicating a FM ground state of monolayer CrSBr with Curie temperature T_c ~ 112 K. The T_c is slightly lower than experiments, possibly due to computational parameters (such as the Hubbard U value within DFT + U) or to the underestimated lattice constant²⁵. To determine the magnetic order universality class, we fit the curve using the relation $M={M}_{0}{[1-\frac{T}{{T}_{c}}]}^{\beta }$, where β is the critical exponent^26,27. The 2D-Ising and 2D-XY models are also plotted in Fig. 1b for comparison. The fitted β is 0.22, suggesting that, although the monolayer CrSBr shows a (weakly) uniaxial in-plane magnetic anisotropy, its magnetic order closely follows the 2D-XY model. On the other hand, the experimentally observed phase transition at T* around 30 ~ 40 K is not observed in our MC simulations. This suggests that T* might be related to some metastable states, not detected by means of equilibrium MC simulations.

Non-equilibrium Spin Dynamics Simulations

In order to probe the possible emergence of an unconventional order in monolayer CrSBr at T < T*, we perform non-equilibrium spin dynamics simulations via the Landau-Lifshitz-Gilbert (LLG) equation, with 10-times averaged results shown in Fig. 1c. The in-plane magnetization, ${M}_{xy}=<\sqrt{{\left({\sum }_{i=1}^{N}{S}_{i}^{x}\right)}^{2}+{\left({\sum }_{i=1}^{N}{S}_{i}^{y}\right)}^{2}}>/N$, is much higher than the out-of-plane component ${M}_{z}=<{\sum }_{i=1}^{N}{S}_{i}^{z}>/N$, indicating the easy-plane magnetism of CrSBr. However, the results of non-equilibrium spin dynamics are qualitatively different from equilibrium MC simulations, especially for the behavior at low T where M/M_s is far from 1. To have some further insights, the spin-spin correlation function, $C\left(r\right)=<{S}_{1}\left({r}_{1}\right)\bullet {S}_{2}\left({ r}_{2}\right)>/{\left|S\right|}^{2}$, at different temperatures is evaluated by both MC (Fig. 1d) and spin dynamics (Fig. 1e) methods. For MC calculations, the correlation function sharply decays from 1 at short-range to a constant at long-range for different temperatures, with increasing magnitude upon decreasing T. For spin dynamics simulations, the behavior of the correlation function at high T (100 and 150 K) is similar to that obtained via MC. However, at intermediate T (40 and 80 K), the correlation function exponentially decays to nonzero values. Interestingly, at low T (0, 5, 20 K and T < 28 K in Supplementary Fig. 1), an algebraic-like decaying correlation function is observed, suggestive of a BKT behavior where “quasi-order” dominated by vortex-antivortex pairs may arise. We will focus on the spin dynamics results in the following, while the comparison between equilibrium MC and non-equilibrium spin dynamics calculations will be discussed at length below.

To gain a more comprehensive understanding of the magnetism at different temperatures, we further investigate the spin textures in real-space obtained with spin dynamics. The in-plane and out-of-plane magnetization are shown in the upper and lower panels of Fig. 2a, respectively. It is clear that, upon cooling, the long-range random disordered state at 150 K (see also zoom in Fig. 2b) changes to a nearly short-range ordered magnetic-domains state at 50 K; subsequently, some magnetic defects emerge at 10 K. Some of the defects annihilate during the cooling process and, at last, some clear magnetic defects appear to be randomly distributed at domain wall at very low temperatures (the detailed spin evolution process is shown in Supplementary Fig. 2). The representative defects in the green and orange circular regions are shown enlarged in Fig. 2c, where in total four different topological spin textures are observed. Unlike the original 2D-XY model, where all the spins are constrained to be in-plane, our topological spin textures exhibit a 3D distribution, with spins at the core pointing out-of-plane and spins at the boundary lying in-plane. Consistent with what is expected for in-plane anisotropy, such topological defects are merons or antimerons, each having a ± 1/2 topological number²⁸. Interestingly, due to the conserved winding number in in-plane magnets, the core-up (core-down) antivortex and core-down (core-up) vortex always occur in pair²⁹. As a result, topological defect pairs form “bimerons”³⁰ in monolayer CrSBr. We note here that after a long-time evolution, the bimerons disappear and the phase collapses to a trivial FM ground state. The metastable magnetic defect state is estimated to be of the order of only 0.15 meV/nm² higher in energy than the FM state. Moreover, within spin dynamics simulations, the lifetime of the defects is ~ 1.5 ns, i.e., of the same order of magnitude as other magnetic defects experimentally observed^31–33. The bimerons also occasionally emerge when using MC methods (see Supplementary Fig. 3).

Influence of Different Exchange Interactions

In order to achieve long-range magnetic order, anisotropic magnetic interactions must be included to overcome the Mermin-Wagner theorem. To further investigate the role of the different anisotropic magnetic interactions, we artificially change the value of some interactions, with respect to the DFT-derived values considered so far. First, artificial magnetic parameters that are similar to those calculated for CrSBr (i.e., D_x = -0.4 meV, D_y/D_x = -0.5, J₁ = J₂ = J₃ = J, J/D_x = -6) are used to clarify the role of SIA under an out-of-plane external magnetic field B_z, as shown in Fig. 3. For magnets with easy-plane anisotropy (SIA < 0), the bimerons are present under low B_z. It is noteworthy that these bimerons exist even at SIA = 0 (see Fig. 3a and c). When simply setting the SIA = 0 and keep other parameters as derived from DFT, the bimerons in the domain wall region are also observed (see Supplementary Fig. 4). Under a suitably applied external B_z, the spins will tend to rotate toward the B_z direction, destroying the bimerons and forming the bubbles (cf. Figure 3b). When B_z > 1T, all the spins are along the z direction and a trivial FM state occurs. Interestingly, as the conditions change from SIA < 0 to SIA > 0 (i.e., easy-axis anisotropy), some domains occur at a smaller B_z (see Fig. 3). Furthermore, a suitable B_z (~ 0.1 T) can change a “trivial-domain” state to a “nontrivial skyrmion bubble”³⁴ state (see Fig. 3d). As expected, all spins rotate to the z direction when B_z is strong enough (here B_z > 0.2 T). Therefore, within the phase diagram shown in Fig. 3, we can conclude that the bimerons emerge at SIA ≤ 0 at low B_z. We also remark that the dipole-dipole interaction favors the in-plane spin direction in magnetic films and is reported to contribute to topological defects in CrCl₃^35,36. We note that the tiny dipole-dipole interaction in CrSBr may increase the in-plane magnetic anisotropy; however, our results show that the bimerons always occur, no matter whether the dipole-dipole interaction is explicitly considered or not (see Supplementary Fig. 5). Therefore, up to this point, neither the SIA nor the dipole-dipole interaction represent the main reason for driving the bimerons formation.

Let’s now turn to another anisotropic magnetic interaction in CrSBr, i.e., the antisymmetric DM interaction. Interestingly, as we artificially remove the DM interactions in CrSBr, the topological bimerons disappear, only some domains are present (see Supplementary Fig. 6). To further illustrate the role of DM interaction, we fix D_x = -0.4 meV, J₁ = J₂ = J₃ = J, SIA = 0 and focus on the variation of bimeron states (meron-antimeron pairs) as a function of |D_y/D_x| and |J/D_x|, as shown in Fig. 4. The bimerons emerge in all conditions in this phase diagram. Furthermore, the DM interaction will favor orthogonal spins, therefore resulting in an increase of the density of bimerons, as |J/D_x| (|D_y/D_x|) decreases (increases). Moreover, not only the number but also the lifetime of bimerons is increased for a higher D_x and D_y. For example, the lifetime of bimerons in the red circle region in Fig. 4 is ~ 75 ns and the metastable bimeron state is 0.13 meV/nm² higher in energy than the FM state. These results demonstrate that the antisymmetric DM interaction is crucial for the formation of topological bimerons defects in monolayer CrSBr.

Kibble-Zurek Mechanism

As shown in Supplementary Fig. 2, the bimerons occur at the domain walls and annihilate one by one if, after cooling down the temperature, the system is allowed to evolve for a long time. Interestingly, as shown in Fig. 5a, the density of bimerons becomes higher as the cooling rate increases. The nearly power-law relation between the density of topological defects and the cooling rate in monolayer CrSBr is suggestive of the Kibble-Zurek (KZ) mechanism^11,12. Though the KZ mechanism was first predicted in the early Universe and was later tested in liquid crystals, superfluid helium, superconductors, ultracold gases, etc., its investigation in magnets is still limited^37–39. The KZ mechanism allows for the formation of defects near the phase transition; however, the cooling rate has a large influence on the evolution of the defects, suggesting that the presence of bimerons might depend on the “history” of the sample. T* might be even changed under different cooling rates. Although we are not able to precisely determine the numerical T* value, as the cooling rate becomes slower, there seems to be a tendency for the transition from an exponential-like to an algebraic-like decaying curve for spin-spin correlations to occur at a lower temperature (see Supplementary Fig. 7).

Finally, finite-size effects are considered. Figure 5b illustrates that at ~ 0 K, when the simulation size is smaller than 30 nm, both the in-plane and out-of-plane magnetizations, M_xy and M_z, show a nearly constant value and without large fluctuations, even after averaging the results 10 times. The M_xy ~ 1 indicates monolayer CrSBr to resemble an in-plane ferromagnet for a small system size. When the system size reaches ~ 50 nm, the magnetization drops and, importantly, large magnetic fluctuations occur. This is attributed to the size of the system being close to the bimeron size (~ 38 nm, see Supplementary Note 1), and both the bimerons and domains have the possibility to emerge when considering different initial random states (Supplementary Fig. 8). At last, when the system size is large enough, the bimerons emerge and can be uniformly distributed at the domain wall regions, leading both the averaged M_xy and M_z values to gradually decrease as the system size increases. Meanwhile, due to the KZ influence, the bimeron size changes to ~ 28nm and ~ 60nm when the cooling rate becomes two times faster and slower, respectively (see Supplementary Figs. 9 and 10).

The equilibrium MC simulation can be approximately viewed as a non-equilibrium spin dynamics simulation with a “super-slow” cooling rate. To verify this idea, we used the output spin textures at intermediate temperature (T_inter) obtained from MC calculation as input to do the spin dynamics calculations. The total energy and evolution of the spin textures demonstrated the above idea (see Supplementary Note 2, Supplementary Figs. 11 and 12). As a result, near-to-ground states at each temperature are obtained and the metastable bimeron state is hardly observed in the MC calculations (see Supplementary Fig. 3). In order to get the metastable state in MC calculations, although not following a very rigorous procedure, one can artificially decrease the equilibration steps in Monte-Carlo simulations; in that case, anomalies in the magnetic susceptibility could be observed both at T_c and T* in monolayer CrSBr (see Supplementary Fig. 13), a behavior which is compatible with the experimental results. This might also explain why, besides the T_N, some experiments find another phase transition at T*^13,16,18,19, whereas some other experiments do not^20,21. Moreover, we note that a different magnetic behavior is probed via neutron powder diffraction as for average magnetism and via muon spin resonance for its dynamics¹⁸. Although no topological bimeron defects were directly discovered, these results indicate that the equilibrium and non-equilibrium magnetic behaviors on CrSBr seem quite different.

Based on the above analysis on critical exponents, spin-spin correlation, real-space spin textures, and Kibble-Zurek-like behavior, one could speculate T* to be related to the BKT phase transition^40,41. We note however that our spin-Hamiltonian is markedly different from the 2D-XY model, for which the BKT physics is usually discussed. In any case, our predictions call for experimental measurements, in particular focused on detecting the above physical parameters and to shed light on the nature of T*.

In summary, within the equilibrium Monte-Carlo simulations, monolayer CrSBr follows rather closely a 2D-XY model with T_c = 112K. Interestingly, under non-equilibrium spin dynamics simulations, topological bimeron defects emerge at T*, the density of bimerons following the KZ behaviour upon temperature quenching. Finally, the spin textures of monolayer CrSBr can be tuned by modifying the SIA and/or the competition between J and DM interactions. While possibly shedding light on the complex magnetization behaviour observed in experiments, our results provide a picture based on topological bimeron defects possibly indicative of an unconventional BKT phase transition in low-dimensional magnets, where DM interactions play a remarkable role.

First principles calculations were performed using the Vienna Ab-initio Simulations Package (VASP)⁴². The exchange-correlation potential was treated using the generalized gradient approximation (GGA) within the Perdew-Burke-Ernzerhof (PBE) scheme⁴³. The Dudarev’s method⁴⁴ with an effective Hubbard U = 3.0 eV was used to account for correlation effects of Cr-3d orbitals. A plane-wave cutoff energy of 360 eV and a 11 × 9 × 1 Γ-centered k-mesh are adopted in all the calculations. A vacuum region of 15 Å was used to avoid the artificial interaction caused by periodic boundary conditions. All the atoms are fully relaxed until the forces and the total energy are lower than 0.001 eV/Å and 10^− 6 eV, respectively.

The atomistic spin dynamics and Monte-Carlo simulations were performed using the VAMPIRE package⁴⁵. The magnetic transition temperature was calculated by Monte-Carlo simulations with a 50 nm × 50 nm system, with both equilibration and averaging phases done using 50,000 steps. If not mentioned, other simulations were described by solving the spin dynamics Landau-Lifshitz-Gilbert (LLG) equation and integrated numerically using the Heun numerical scheme with a 100 nm × 100 nm system size under cooling-time 0.1 ns. Since no experimental data are available for the damping constant, we selected it as 1 in the simulations. Periodic boundary conditions were used to perform the calculations. A randomly distributed spin texture was selected as initial state. All the simulations were performed using the field cooling method, where the temperature starts from 160 K and decreases to near 0 K in steps of 1 K. For each simulation, 3×10⁶ total calculation steps were performed at an increment time step of 0.1 fs.

Competing interests

The authors declare no competing interests.

Author contributions

B.Y. conceived the idea. S. P. supervised the project. B. Y and S. P prepared the manuscript. B.Y. did all the simulations. All authors discussed the results and contributed to the manuscript. Correspondence should be addressed to B. Y. at [email protected] and S. P. at [email protected].

Acknowledgements

S.P acknowledges support by the Next/Generation EU program, PRIN-2022 project “SORBET: Spin-ORBit Effects in Two-dimensional magnets” (IT-MIUR Grant No. 2022ZY8HJY). X.H. acknowledges support by the National Natural Science Foundation of China [NSFC, Grant No. 12134017]. Calculations were performed at Beijing PARATERA Technology Co., LTD and at CINECA via the ISCRA C initiative (grant HP10CJ2EW2).

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