Chiral Skyrmions Interacting with Chiral Flowers

The chiral nature of active matter plays an important role in the dynamics of active matter interacting with chiral structures. Skyrmions are chiral objects, and their interactions with chiral nanostructures can lead to intriguing phenomena. Here, we explore the random-walk dynamics of a thermally activated chiral skyrmion interacting with a chiral flower-like obstacle in a ferromagnetic layer, which could create topology-dependent outcomes. It is a spontaneous mesoscopic order-from-disorder phenomenon driven by the thermal fluctuations and topological nature of skyrmions that exists only in ferromagnetic and ferrimagnetic systems. The interactions between the skyrmions and chiral flowers at finite temperatures can be utilized to control the skyrmion position and distribution without applying any external driving force or temperature gradient. The phenomenon that thermally activated skyrmions are dynamically coupled to chiral flowers may provide a new way to design topological sorting devices.

In 2013, Mijalkov and Volpe demonstrated the possibility that particle-like chiral microswimmers performing circular active Brownian motion can be sorted in a chiral environment formed by using some static obstacle patterns on the substrate [69,70], where the chirality of circular Brownian motion couples to chiral features present in the environment.As skyrmions also show circular Brownian motion due to their nontrivial topology [36-38, 40, 41, 43, 47-50, 56], it is therefore envisioned that the thermally activated random-walk dynamics of skyrmions may also be modified in a chiral environment due to the skyrmion-substrate interactions, which is the focus of this work.However, it should be noted that active matter systems have some form of selfpropulsion [66,67,69,70], while the Brownian skyrmions are only undergoing thermal motion and are not self-propelled.
The square and chiral flower-like obstacles considered in this work are schematically depicted in Figure 1.A real example of a chiral flower is given in Figure 1 [71,72].Micro-and nanostructures mimicking chiral flowers may control the dynamics of active chiral matter [66,67,69,70] as well as thermally activated chiral skyrmions.In Figure 1(c), a thermally activated skyrmion shows clockwise or counterclockwise Brownian gyromotion, which depends on the sign of Q.If a skyrmion is initially placed within a chiral flower, then it may escape from or be confined by the chiral flower.The outcome depends on the chirality of the flower and the sign of Q, which could result in the topological sorting and create an order (sorting)-from-disorder (Brownian motion) phenomenon.However, if a skyrmion is initially placed within a square obstacle, then it will be confined by the square.
We first show the typical Brownian gyromotion of a ferromagnetic skyrmion within a square obstacle in a two-dimensional model, which results in the confinement of the skyrmion [48,50,73].
The length, width, and thickness of the ferromagnetic layer equal 256, 256, and 1 nm, respectively.The square pattern is made of four obstacle bars, which are rectangle regions locally modified to have enhanced PMA K o .We assume that K o /K = 10 in order to make sure that the skyrmions cannot penetrate the obstacle boundary [74].Such a square pattern on the ferromagnetic substrate can, in principle, be fabricated in experiments [50,[75][76][77].The width of each obstacle bar is 10 nm, and the distance between two parallel inner edges of the square pattern is 100 nm.The distance between the two parallel outer edges is thus 120 nm.The square center overlaps the ferromagnetic layer center, as indicated in Figure 2 Initially, a skyrmion is placed and relaxed at the center of the ferromagnetic layer.We then simulate the thermal random-walk dynamics (i.e., the Brownian motion) of the skyrmion at a temperature of T = 150 K for 500 ns.The trajectories of the skyrmions with Q = −1 and Q = +1 are given in Figures 2(a  The confinement leads to a square shape of the overlapped skyrmion position distribution for 500 ns of simulation.The skyrmion motion guided by the square edges is similar to that guided by grain boundaries [47], which may enhance the skyrmion diffusion.The Brownian gyromotion of a skyrmion is a feature of its topological nature, which is due to the Magnus force associated with the net skyrmion number [36-38, 40, 41, 43, 47-50, 56].We note that the Magnus force is absent in the antiferromagnetic system, where a skyrmion may not show Brownian gyromotion [38,56]. The time-dependent velocities of the skyrmions with Q = −1 and Q = +1 interacting with the The opening width between the two orthogonal obstacle bars is set to 30 nm.The chiral flower center overlaps the ferromagnetic layer center, as shown in Figure 3(a).The opening width should be wider but not much wider than the skyrmion diameter, and the area within the chiral flower should not be too large.Otherwise, the skyrmion may not interact with the chiral flower in an effective way depending on its diffusion at a given temperature.If the opening width is much larger than the skyrmion diameter, the skyrmion should be easier to escape and enter the flower, and travel in all possible directions equally often during long times, leading to achiral results.
A skyrmion is initially placed and relaxed at the ferromagnetic layer center.We then simulate the thermal random-walk dynamics of a skyrmion at a given temperature for 500 ns.We first show typical desired outcomes of the skyrmion with Q = ±1 interacting with a left-or right-handed flower at T = 150 K in Figures 3(a The time-dependent velocity and skyrmion radius during the skyrmion-flower interaction are given in Figure S2 (see Supporting Information), and the time-dependent total energy of the system is given in Figure S3 (see Supporting Information).
The desired outcomes may be achieved when the skyrmion interacts effectively with the chiral flower for a long enough time.However, as a skyrmion has a certain lifetime at finite temperature, it may collapse before or after achievement of the desired outcomes.With reasonable computational workload, we carry out 18 simulations with different random seeds for each temperature and skyrmion-flower configuration and summarize both the desired and undesired outcomes.For the skyrmion with Q = −1 interacting with a left-handed flower, we obtain four events of desired outcome in 18 simulations [Figure 3(i)].For the skyrmion with Q = +1 interacting with a left-handed flower, we obtain 15 events of desired outcome [Figure 3(j)], including two cases in which the skyrmion collapses within the chiral flower.We also note that three undesired "OUT" events happen, which may be due to the fact that the skyrmion size is transiently much smaller than the opening width when it moves to an exit of the left-handed flower along the inner edge of the obstacle bar (see Movie S7).Such a situation may be avoided by slightly reducing the opening width or increasing the skyrmion size; however, it also indicates that the skyrmion is able to travel along the path unfavored by the skyrmion-flower interaction, especially during long times or when the skyrmion-flower interaction is ineffective.For the skyrmion with Q = −1 interacting with a right-handed flower, we obtain 17 events of the desired outcome [Figure 3(k)].For the skyrmion with Q = +1 interacting with a right-handed flower, we obtain three events of desired outcome [Figure 3(l)].We note that when the desired outcome is the "IN" event, both effective and ineffective skyrmion-flower interactions may result in the desired outcome.We also show the skyrmion interacting with an achiral square with corner gaps in the Supporting Information (see Figure S4), where it is expected that both skyrmions with Q = ±1 can escape easily to explore the whole sample, and the outcomes are independent of Q.
In Figure 4, we further show that the counts of achieving the desired and undesired outcomes within the 500 ns-long simulation of the skyrmion-flower interaction depend on the temperature.
When the temperature is too low [Figure 4(a); T = 100 K], the skyrmion diffusion is weak and it cannot interact with the chiral flower effectively.In such a case, we obtain 18 "IN" events in 18 simulations for the skyrmions with Q = ±1 within a right-handed flower.When T = 150 K [Figure 4(b)], we obtain 17 desired "IN" events for the skyrmion with Q = −1 interacting a right-handed flower, and three desired "OUT" events for the skyrmion with Q = +1 interacting with a right-handed flower.When T = 180 K [Figure 4(c)], we obtain 17 desired "IN" events for the skyrmion with Q = −1 interacting a right-handed flower, and four desired "OUT" events for the skyrmion with Q = +1 interacting with a right-handed flower.It suggests that the skyrmionflower interaction could be more effective due to more active skyrmion at elevated temperature.However, when the temperature is too high [Figure 4(d); T = 200 K], the thermal fluctuations may result in the collapse of the skyrmion in more simulations due to the significantly reduced skyrmion lifetime.The mean skyrmion size may also increase with the temperature, while the size and geometry of the chiral flower are fixed.Therefore, the skyrmion may not interact with the chiral flower effectively when the temperature is too high.
In conclusion, we have studied the thermal random-walk dynamics of a ferromagnetic skyrmion in a chiral environment, where the interactions between skyrmions and chiral obstacles (i.e., the chiral flowers) could lead to topology-dependent spontaneous sorting of skyrmions.The position of a skyrmion can be manipulated by using a simple chiral flower-like obstacle pattern at finite temperature in the absence of an external drive if the skyrmion is placed initially at the center of the flower, which is a state of artificially low entropy.Namely, an effective interaction between the chiral flower (either left or right) and the skyrmion (either Q = −1 or +1) could result in the escape or confinement of the skyrmion.For both outcomes, as the skyrmion tends to explore the whole space (i.e., inside and outside the flower) during long times due to its thermal diffusion, the disorder and entropy of the system should increase with time, while the total energy is conserved over time despite fluctuations due to the thermal effect.Thus, the skyrmion behaviors in such a closed system are in line with the first and second laws of thermodynamics.However, we point out that some systems are more ordered even when there is increased entropy in the ordered state [78].
Our results reveal the unique thermal dynamics of chiral topological spin textures interacting with chiral structures.Our results also suggest that it is possible to build a topological sorting device based on chiral flower-like structures, in which skyrmions with opposite signs of topological charges could generate different dynamic outcomes.

METHODS
Computational Simulations.The simulations are performed by using the micromagnetic simulator MUMAX 3 [79,80] on several commercial graphics processing units, including NVIDIA GeForce RTX 3070 and RTX 3060 Ti.The magnetization dynamics at finite temperature is governed by the stochastic Landau-Lifshitz-Gilbert (LLG) equation [79,80], where m = M /M S = 1 is the reduced magnetization, M S is the saturation magnetization, t is the time, γ 0 is the absolute gyromagnetic ratio, α is the Gilbert damping parameter, is the effective field with µ 0 and ε being the vacuum permeability constant and average energy density, respectively.h f is a thermal fluctuating field satisfying [79,80] < h i (x, t) >= 0, where i and j are Cartesian components, k B is the Boltzmann constant, T is the temperature, and V is the volume of a single mesh cell.δ ij and δ(. . . ) denote the Kronecker and Dirac delta symbols, respectively.The energy terms considered in the model include the ferromagnetic exchange energy, interface-induced chiral exchange energy, perpendicular magnetic anisotropy (PMA) energy, and demagnetization energy.Thus, the average energy density is given as [79,80] ε =A (∇m where A, D, and K are the ferromagnetic exchange, DM interaction, and PMA constants, respectively.B d is the demagnetization field.n is the unit surface normal vector.m z is the out-of-plane component of m.The default magnetic parameters are 26-28, and 74: The mesh size is 2 × 2 × 1 nm 3 , which ensures good computational accuracy and efficiency.The finite-temperature simulation is performed with a fixed integration time step of 10 fs and a given random seed.

ASSOCIATED CONTENT Supporting Information
The Supporting Information is available free of charge at The random seed equals 777.The skyrmion escapes from the left flower as an undesired outcome.
In order to show both the skyrmion and obstacle pattern, the movie shows the time-dependent magnetic anisotropy energy density instead of the magnetization.(MP4) (a), which is a left-handed flower showing a fixed left-contort corolla.The chirality of a flower, either left-handed or right-handed [Figure 1(b)], is an important property of floral symmetry.Some flowers have a contort petal aestivation, which is most pronounced in floral buds and may be less prominent in open flowers [71, 72].In chiral flowers, two morphs are possible as shown in Figure 1(b): contorted to the left and contorted to the right

Figure 1 .
Figure 1.A thermally activated chiral skyrmion interacting with a chiral flower-like obstacle.(a) An exemplary chiral flower (Vinca minor) found in Shinjuku City by the authors, which shows a fixed leftcontort corolla.(b) Diagrams showing the left-right asymmetry in flowers with fixed corolla contortion, i.e., the left-contort and right-contort corollas.(c) Typical desired outcomes of a chiral skyrmion interacting with a chiral flower or a square: skyrmion confined (i.e., "IN") and skyrmion escaped (i.e, "OUT").The outcomes depend on the skyrmion number Q as well as the left-right asymmetry of the chiral flower.The skyrmions with Q = −1 and Q = +1 are denoted by blue and red dots, respectively.

Figure 2 .
Figure 2. A thermally activated skyrmion confined by a square obstacle.Typical trajectories of (a) a skyrmion with Q = −1 and (b) a skyrmion with Q = +1 confined by a square are given.Three-dimensional illustrations show the time-dependent skyrmion position and the skyrmion texture with (c) Q = −1 or (d) Q = +1.The skyrmions with Q = −1 and Q = +1 show clockwise and counterclockwise circular motion along the inner edges of the square obstacle, respectively.Time-dependent velocities v of the skyrmions with (e) Q = −1 and (f) Q = +1 are given.Time-dependent skyrmion Hall angles of the skyrmions with (g) Q = −1 and (h) Q = +1 are also shown for selected time ranges, indicating the clockwise and counterclockwise circular motion, respectively.(i) Time-dependent skyrmion radius.The skyrmion dynamics is simulated at T = 150 K for 500 ns with a time step of 0.5 ns.The time step is small enough to show the Brownian motion with a reasonable precision (see Figure S1 in the Supporting Information).

Figure 3 .
Figure 3.A thermally activated skyrmion interacting with a left-handed or right-handed flower.(a) Typical trajectory of a skyrmion with Q = −1 interacting with a left-handed flower.The skyrmion escapes from the left-handed flower due to its clockwise Brownian gyromotion, i.e, the outcome is a desired "OUT".(b) Typical trajectory of a skyrmion with Q = +1 interacting with a left-handed flower.The skyrmion is confined by the left-handed flower due to its counterclockwise Brownian gyromotion, i.e, the outcome is a desired "IN".(c) Typical trajectory of a skyrmion with Q = −1 interacting with a right-handed flower.The skyrmion is confined by the right-handed flower due to its clockwise Brownian gyromotion, i.e, the outcome is a desired "IN".(d) Typical trajectory of a skyrmion with Q = +1 interacting with a righthanded flower.The skyrmion escapes from the right-handed flower due to its counterclockwise Brownian gyromotion, i.e, the outcome is a desired "OUT".(e)-(h) Time-dependent skyrmion Hall angles for selected time ranges, corresponding to (a) and (b), respectively.(i)-(l) The outcome counts for a skyrmion with Q = ±1 interacting with a left-or right-handed flower.Eighteen simulations are done with different random seeds for each skyrmion-flower configuration.Both the desired and undesired outcomes are counted.The skyrmion dynamics is simulated at T = 150 K for 500 ns with a time step of 0.5 ns.

Figure 4 .
Figure 4. Outcome counts for a skyrmion interacting with a right-handed flower at different temperatures.A skyrmion with Q = ±1 interacting with a right-handed flower at (a) T = 100 K, (b) T = 150 K, (c) T = 180 K, and (d) T = 200 K. Eighteen simulations are done with different random seeds for each temperature.Both desired and undesired outcomes are counted.The skyrmion dynamics is simulated for 500 ns with a time step of 0.5 ns.
[https://doi.org/10.1021/acs.nanolett.-3c03792].Additional simulation results, including the time-dependent velocity and radius of a skyrmion interacting with a left or right chiral flower, the time-dependent total energy of the system, the interaction between a skyrmion with a square with corner gaps, and the time-step-dependent skyrmion trajectories.(PDF) Movie S1: A thermally activated skyrmion with Q = −1 interacting with a square obstacle pattern.The skyrmion dynamics is simulated at T = 150 K for 500 ns with a time step of 0.5 ns.The random seed equals 111.It shows the confinement of the skyrmion by the square.In order to show both the skyrmion and obstacle pattern, the movie shows the time-dependent magnetic anisotropy energy density instead of the magnetization.(MP4) Movie S2: A thermally activated skyrmion with Q = +1 interacting with a square obstacle pattern.The skyrmion dynamics is simulated at T = 150 K for 500 ns with a time step of 0.5 ns.The random seed equals 111.It shows the confinement of the skyrmion by the square.In order to show both the skyrmion and obstacle pattern, the movie shows the time-dependent magnetic anisotropy energy density instead of the magnetization.(MP4) Movie S3: A thermally activated skyrmion with Q = −1 interacting with a left flower-like obstacle pattern.The skyrmion dynamics is simulated at T = 150 K for 500 ns with a time step of 0.5 ns.The random seed equals 222.The skyrmion escapes from the left flower as a desired outcome.In order to show both the skyrmion and obstacle pattern, the movie shows the time-dependent magnetic anisotropy energy density instead of the magnetization.(MP4) Movie S4: A thermally activated skyrmion with Q = +1 interacting with a left flower-like obstacle pattern.The skyrmion dynamics is simulated at T = 150 K for 500 ns with a time step of 0.5 ns.The random seed equals 111.The skyrmion is confined by the left flower as a desired outcome.In order to show both the skyrmion and obstacle pattern, the movie shows the time-dependent magnetic anisotropy energy density instead of the magnetization.(MP4) Movie S5: A thermally activated skyrmion with Q = −1 interacting with a right flower-like obstacle pattern.The skyrmion dynamics is simulated at T = 150 K for 500 ns with a time step of 0.5 ns.The random seed equals 222.The skyrmion is confined by the right flower as a desired outcome.In order to show both the skyrmion and obstacle pattern, the movie shows the time-dependent magnetic anisotropy energy density instead of the magnetization.(MP4) Movie S6: A thermally activated skyrmion with Q = +1 interacting with a right flower-like obstacle pattern.The skyrmion dynamics is simulated at T = 150 K for 500 ns with a time step of 0.5 ns.The random seed equals 222.The skyrmion escapes from the right flower as a desired outcome.In order to show both the skyrmion and obstacle pattern, the movie shows the time-dependent magnetic anisotropy energy density instead of the magnetization.(MP4) Movie S7: A thermally activated skyrmion with Q = +1 interacting with a left flower-like obstacle pattern.The skyrmion dynamics is simulated at T = 150 K for 500 ns with a time step of 0.5 ns.