Annihilation mechanisms for interacting skyrmions in magnetic nanowire

Magnetic skyrmions are considered potential candidates for spintronics-based memory and logic devices. For achieving high-density and high-speed devices, it is essential to study their interactions. In this paper, the interaction, dynamics and annihilation mechanisms of Néel skyrmions in nanowire confinement under the influence of spin-transfer torque (STT) and edge forces have been studied. Initially isolated, two Néel skyrmions are brought into proximity, leading to distinct interaction scenarios characterized by varying current densities. We explore the impact of these interactions on skyrmion trajectories, size evolution, and annihilation phenomena. Our findings reveal the interplay of skyrmion–skyrmion repulsive forces, edge effects, and the influence of STT, shedding light on the rich dynamics of these topological magnetic textures. Furthermore, we unveil the distinct annihilation mechanisms of the leading and trailing skyrmions under different forces, providing valuable insights into the fundamental physics of skyrmion behavior in confined geometries.


Introduction
In recent years, magnetic skyrmions have attracted much attention due to their non-collinear spin configuration and topological stability [1][2][3][4][5].Their nanoscale size makes them a strong candidate for future magnetic memory devices [6][7][8][9], logic devices [10,11], and neuromorphic computing applications [12][13][14][15].The skyrmions can also be created and displaced by different schemes such as magnetic field [16][17][18][19][20], spin-transfer torque (STT) [21][22][23], electric voltage [24][25][26], spin-orbit torque [27], surface acoustic wave [28,29], or combinations Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.thereof, which makes them well-suited for energy-efficient applications.On the other hand, the second key requirement for a functional device is the sensing or detection that can be achieved by the Hall effect and tunneling magnetoresistance [30].Compared to devices based on magnetic domain walls, they can be moved from one state to another using significantly lower current densities [31][32][33][34][35].While skyrmions were initially observed in asymmetric chiral magnet MnSi at low temperatures [2], the surge in interest in skyrmions occurred when they were discovered in magnetic multilayers, primarily due to the interfacial Dzyaloshinskii-Moriya interaction [36,37].
In the context of functional devices, the necessity arises to manipulate not just a single skyrmion, but rather multiple skyrmions.In such cases, the number of skyrmions and the way they interact with one another significantly influence their dynamics.In this paper, we investigate the interaction between two skyrmions within a magnetic nanowire subjected to STT.A comparison of their dynamics to the case of a single skyrmion is also presented.

Model
In this study, micromagnetic simulation using Mumax3 was employed to explore the dynamic of two interacting Néel skyrmions exhibiting left-handed chirality.The two skyrmions are initially separated from each within constrained magnetic nanowires [38].The magnetization at each grid point was calculated over time, following Landau-Lifshitz-Gilbert equation in the presence of STT: This equation describes the change of magnetization unit vector m i over time, where γ and α represent the gyromagnetic ratio and Gilbert damping constant which accounts for energy dissipation due to spin relaxation, respectively.The effective field H e acting on local magnetization is composed of several energy terms, including magnetostatic, exchange, Dzyaloshinskii-Moriya, and anisotropy energy terms.The equation also incorporates terms related to the interaction between the magnetization and a spin-polarized electric current j(r) through STT, as well as non-adiabatic effects.Parameters such as spin polarization (P), lattice constant (a), and non-adiabatic torque strength (β J ) are taken into consideration.We incorporated intrinsic material parameters, including saturation magnetization (M S ), exchange stiffness constant (A), Dzyaloshinskii-Moriya strength (D), magnetocrystalline anisotropy energy (K u ), and damping constant (α).The nanowire has dimensions of 2 nm thickness, 300 nm length (L), and 70 nm width (W).We divided the device into 2 nm cubes using a grid size of 256 × 128 × 1.Our simulations were conducted without considering thermal fluctuations or the presence of an external magnetic field.It is worthy to note that magnetic skyrmions have been observed at zero field as reported in [39][40][41][42].Non-periodic boundary conditions were applied, and magnetic charge on the edges was considered to realistically capture edge effects in the nanowire.This study primarily focused on comprehending the behavior of interacting skyrmions when subjected to a spin-polarized current, aiming to observe their dynamic responses.

Results and discussion
In this study, we examined the interactions between two Néel skyrmions, each approximately 23 nm in diameter.Initially, the skyrmions are positioned at a non-interacting separation distance of about 65 nm.These skyrmions were confined within a geometric space of 300 nm × 70 nm and exhibited negligible mutual interactions, with the repulsive force between them measuring less than 10 −13 N m −1 [6].To explore various facets of skyrmion interactions, we applied an in-plane STT with variable current density values.Initially, a current density value of J = 9 × 10 11 A m −2 was selected causing the trailing skyrmion (S T ) to experience a direct interaction with the leading skyrmion (S L ).Subsequently, after the collapse of S L , the S T encountered interactions only with the edge.This scenario occurred within a range of J values.At lower J values, S T exclusively interacted with S L while both skyrmions coexisted.For intermediate J values, only one skyrmion (S T ) survived, whereas for higher J values, both skyrmions underwent sequential annihilations, with S L collapsing first, followed by S T .In figure 1(a), both the paths of the trailing skyrmion, denoted as S T (solid blue arrow), and the leading skyrmion, labeled as S L (dotted red arrow), are observed for a current density of 9 × 10 11 A m −2 .Initially, these skyrmions are positioned at the center of a nanowire measuring 300 nm in length and 70 nm in width.As the STT drives S L toward the right edge, it follows a compressed counter-clockwise helical trajectory due to a complex interplay of several factors.The core polarization of a Néel skyrmion plays a significant role in its dynamic behavior under the influence of STT.The torque exerted by the spin-polarized current tends to drive the skyrmion in a certain direction, and the orientation of the core polarization contributes to this motion.The helical trajectory observed in the motion of Néel skyrmions is influenced by the core polarization.It is found that the core polarization of the Néel skyrmion significantly influences the direction of its helical path.When it is set to +1, the skyrmion follows a counterclockwise helical trajectory.Conversely, when the core polarization is set to −1, the skyrmion exhibits a clockwise helical path.Figure 1(a) illustrates the influence of the repulsive force exerted by S L on S T , leading to a change in S T 's trajectory.This results in S T shifting towards the top edge while its radius decreases from 10 nm to 6 nm.The key factors contributing to this rotation include the Magnus force (F g ), the repulsive force (Fs -s) between the interacting skyrmions [43], and the dissipative force, which is proportional to the velocity of skyrmion and acts in the opposite direction to its motion [44].The Magnus force does not perform work but affects the trajectory of skyrmions, explaining their weak pinning [45].Simultaneously, the repulsive force between the skyrmions amplifies this effect, causing counterclockwise rotation during their interaction.Moreover, edge interactions originating from the right and transverse edges of the nanowire significantly shape the skyrmions' trajectories.These edge interactions introduce additional forces that compress and divert the skyrmion paths, contributing to the observed counterclockwise rotation and the complex dynamics of skyrmions in confined magnetic systems.Around t ≈ 3.4 ns, S L undergoes significant compression towards the right, resulting in a substantial reduction in its size.In figure 1(a), only the paths of the two interacting skyrmions are depicted.Figure 1(b) presents S T path after S L annihilation.It is worth mentioning that S T resumes its anticipated helical path while maintaining a radius of approximately 8 nm, still smaller than its initial radius of about 10 nm.The trajectory shown for S T post-annihilation is a continuation of the path displayed in figure 1(a) when the two skyrmions interacted.The S L annihilation occurs when its size reduces to around 5 nm, allowing S T to expand in size as the skyrmion-skyrmion interaction dissipates.
Figures 1(c)-(h) are snapshots of the positions and dimensions of the two interacting skyrmions at different times.The initial configuration is shown in figure 1(c) where the two skyrmions are located at the center of the nanowire.In figures 1(d)-(f) taken at t = 1.25 ns, 2.9 ns and 3.3 ns, respectively, it can be seen that the leading skyrmion S L is contracted and S T is driven toward the top edge.After S L is annihilated, only S T is present as shown in figures 1(g) and (h) for 4.4 ns and 7.3 ns, respectively.For current densities below 9 × 10 11 A m −2 , and considering the specific calculation conditions employed in this study, it is observed that the STTdriven force alone is insufficient to annihilate S L , even when coupled with the repulsive force from S T .More intriguingly, when J = 9 × 10 11 A m −2 , a compelling situation with three distinct phases emerges.Initially, S T experiences the dissipative force from STT and the repulsive force Fs -s from S L , without the influence of edge forces, for t ≲ 3.4 ns.During the same time frame, S L is simultaneously subjected to the force from the STT, the Fs -s from S T , and the repulsive forces originating from both the right edge (F R ) and the top edge (F T ).
Subsequently, following the annihilation of S L at t > 3.4 ns, S T encounters repulsive forces originating from F R and F T .The dependency of the nanowire size in skyrmions dynamics was studied, particularly with respect to the width rather than the length.The nanowire's dimensions play a significant role in shaping the dynamics of interacting Néel skyrmions under the influence of the STT effect and edge forces.
In devices with smaller widths, the skyrmions were confined within the restricted space.Under the same current density (J = 9 × 10 11 A m −2 ), we observed that narrower widths play a crucial role in maintaining the stability of the skyrmions, preventing premature annihilation.This confinement effect is conducive to the preservation of individual skyrmions and their controlled interactions, the required current density for annihilation is higher, offering a protective threshold for the skyrmions.For the case of devices with larger widths the annihilation of skyrmions was observed under the same current density.This phenomenon is attributed to the reduced confinement effect, allowing for a more significant influence of edge forces.The larger width facilitates the interaction of skyrmions with the edges, leading to successive annihilation events.It was discovered that the helical motion and the annihilation process were not significantly affected by the nanowire thickness.Across various thickness values, the skyrmions dynamics and stability were not much changed.
The behavior of skyrmions within the nanowire is influenced by several forces arising from different physical mechanisms.These forces have an impact on the skyrmion's stability and mobility as can be seen in figure 2(a).These forces include contributions from the driving force, Magnus force, damping force, gyrotropic force, edge force and skyrmioninteraction force.The application of a spin-polarized current in the +x direction induces the driving force, which affects the skyrmion's translational motion along the applied current direction.The Magnus force is due to the non-uniform rotation of the spin structure around the core of the skyrmion.This force is directed perpendicular to both the velocity and the direction of the spin-polarized current.The Magnus force contributes to the lateral motion of the skyrmion, influencing its trajectory.While the damping force is associated with the dissipative nature of the skyrmion's motion.It acts opposite to the velocity of the skyrmion, tending to slow down its translational and rotational motions.The gyrotropic force arises from the gyration of the skyrmion's magnetic moments.It is perpendicular to both the velocity and the local magnetization dynamics.The gyrotropic force contributes to the circular motion of the skyrmion, creating a rotational component in its trajectory.The edge force is associated with the nanowire, it influences the path of the skyrmion near the edges of the magnetic nanowire and contributes to the overall forces acting on it.The repulsive force arises from the topological nature of skyrmions, as they share the same topological charge.Figures 2(b) and (c) depict the positional evolution of skyrmions concerning the right edge, illustrating two distinct scenarios: one involving the interaction between skyrmions S L and S T and the other involving a single skyrmion (unaffected by interactions).In figure 2(b), it is evident that skyrmion S L , located very close to the right edge (within a range of 10 nm-30 nm), undergoes significant compression before its annihilation at t ≈ 3.4 ns.This annihilation is driven by the counteracting force Fs -s, which opposes the edge force and prevents S L from being expelled far from the edge, as seen in the case of a single skyrmion (figure 2(c)).
By examining the change in the size of each skyrmion, we present in figure 2(d) a substantial reduction in the radius R L of S L , from an initial size of approximately 12 nm to below 6 nm.This reduction in R L is a response to the strength of the forces acting on S L .It is important to note that for current densities below 9 × 10 11 A m −2 , S L maintains a similar configuration, with the acting forces not reaching a critical threshold for annihilation.This suggests that when the skyrmion's radius falls below a critical value, it becomes unstable and eventually collapses.Thus, the reduction in a skyrmion's radius serves as an indicator of the net force's strength acting on it and whether it remains stable or undergoes annihilation.Moreover, for smaller initial sizes, both skyrmions were observed to annihilate, whereas larger initial sizes exhibited resilience, requiring a larger current density to initiate annihilation.This highlights the intricate relationship between the initial size of the skyrmion and the required amount of current to initiate annihilation.
Figure 2(c) provides a comparison of the distance from the right edge X R for S T , subjected to the force Fs -s, and an isolated skyrmion unaffected by interactions with other skyrmions.In the latter case, the single skyrmion experiences only the repulsive force from the right edge.The difference in X R before the annihilation of S L (t ≲ 3.4 ns) is due to Fs -s, and the single skyrmion can be driven closer to the edge of the nanowire (approximately 10 nm) before rebounding in a helical path.For t ≳ 4 ns, X R for S L remains the same, regardless of whether it was alone or in the presence of S T (figure 2(b)).The dashed region in figure 2(c) illustrates the effect of Fs -s originating from S L , exerting a repulsive force that pushes S L away from the edge.This result has also been verified through the calculation of radius R T , as shown in figure 2(e).The reduction in R T due to Fs -s is found to be smaller than the force acting on a single skyrmion due to the edge force.After the annihilation of S L , we computed the x-component of S T velocity, V x , as depicted in the inset of figure 3. Notably, V x displays a damped oscillatory behavior, serving as an indicator of the helical motion described in figure 1(b).In this context, we assume that the skyrmion possesses an effective mass m * that depends on its magnetic properties [46][47][48][49][50]. Furthermore, the repulsive force is proportional to the acceleration a.To explore this relationship, we calculated the time derivative of V x , thereby obtaining the x-component of acceleration a x , which is presented in figure 3.By considering the exponential decay dependence of the repulsive force due to the edge f on the skyrmionedge distance, as reported by [51][52][53], it can be observed that f /f o exhibits similar behavior to a x .Here, f o is a physical parameter expressed as in [51].The force f and the acceleration a x are due to the effect of the right edge on S T .To understand the force between the two skyrmions, we calculated the two components of acceleration for both S L and S T in the timeframe just before S L annihilation, as shown in figure 4. As illustrated in figure 1(a), it is crucial to consider the y-component of acceleration for both S L and S T because Fs -s induces a significant motion toward the top edge of the device.
The acceleration experienced by S T as a result of its interaction with the leading skyrmion S L reveals two distinct components, as evident from the data presented in figure 4. It is important to consider both of these components as both skyrmions are strongly propelled towards the upper edge due to the influence of Fs -s, as depicted in figure 1(a).The acceleration experienced by S T in response to its interaction with the leading skyrmion S L exhibits dual components, as illustrated in figure 4. The y-component cannot be disregarded, as the motion of both skyrmions drifts towards the upper edge due to the influence of Fs -s (figure 1(a)).When we track the movement of S T , we observe a deceleration in its motion, as indicated by the negative x-component of acceleration (proportional to the x-component of Fs -s).For skyrmion S T , the repulsive force from the right edge on S L is induced by the Fs -s originating from S L .A significant motion of S T towards the upper edge initiates at approximately 250 nm along the device and continues until S L is annihilated.The two components of S T acceleration are graphed in figure 4(a).Similarly, there is a rapid change in the path of S L towards the top of the device due to the combined effects of Fs -s and the net force from the two edges, F R and F T .By examining the two graphs in figure 4, it becomes evident that the acceleration components for each skyrmion differ due to the variations in the forces acting upon them.
To elucidate the distinct forces acting on each skyrmion, figure 5(a) offers a comprehensive overview of the forces influencing S L and S T .In this configuration, S L is subject to the skyrmion-skyrmion repulsive force (Fs -s), which arises due to the presence of two skyrmions with the same chirality.This force tends to push each skyrmion away from each other.Additionally, S L experiences supplementary forces, F R and F T , arising from the right and transverse edges, respectively.This analysis corresponds to a specific moment in time, just prior to 3.4 ns, marking the annihilation of skyrmion S L .Since S T is being pushed away from the edges due to its interaction with S L , we can disregard the forces F R and F T .Figure 5(a) clearly illustrates that S T path is significantly compressed by Fs -s from S T , along with the forces F R , F T , and the driving force from the STT.The cumulative effect of these forces results in a reduction in S L size, like a hedgehog under imminent threat.For J values below 9 × 10 11 A m −2 , the skyrmion S L could still bounce back, expand to recover its size, and stabilize once the current is removed.However, for J ⩾ 11 × 10 11 A m −2 , S L fate takes a different course, with S L observed to undergo core annihilation, a phenomenon that will be discussed in figure 6.In figure 5(b), it is evident that the significant reduction in S T radius is associated with a decrease in its distance (d) from S L .This data is taken before the annihilation of S L when S T is solely influenced by Fs -s, as indicated in figure 1.To confirm the collapse of the leading skyrmion from the core and to explore the conditions for annihilating the trailing skyrmion while understanding its mechanism, we increased the current density J. Remarkably, a consistent behavior was observed in the range of 9 × 10 11 A m −2 to 11 × 10 11 A m −2 , wherein S L underwent annihilation while S T remained within the device.For J ⩾ 11 × 10 11 A m −2 , the driving force from the STT effect proved strong enough to cause the annihilation of S L , followed by S T .Because there are different forces acting on each skyrmion, we were interested in understanding their annihilation mechanisms.In figure 6(a), we present a plot of the paths of the two skyrmions within the nanowire for J = 13 × 10 11 A m −2 .Because the STT driving force is relatively high, the leading skyrmion is annihilated first, near the top corner of the device, followed by the annihilation of the trailing skyrmion approximately 1 ns later.The blue dotted arrow indicates the path of S L , which closely resembles that of S T .The positions (r) of each skyrmion relative to the bottom edge of the device are calculated and plotted in figure 6(b).Both S L and S T maintain almost constant radial velocities, indicated by the straightline, until they encounter a slight disturbance near the edge, resulting in their eventual collapse.Figure 6(c) illustrates the two skyrmions at t = 1.5 ns, with S L moving closer to the top corner of the device and undergoing significant shrinkage compared to S L , primarily due to the combined effect of forces from the two edges and Fs -s.As S T is propelled closer to S L by the STT driving force, S L size continues to diminish until it reaches a critical point and eventually collapses (figures 6(d) and (e)).For t ≳ 2.1 ns, S T is isolated and subject to both the STT driving force and the edge repulsive force, leading to its eventual annihilation (figure 6(f)).
To develop a comprehensive understanding of the annihilation process, especially when two skyrmions are annihilated sequentially, our focus was elucidating the distinct forces experienced by each skyrmion during this process.As can be seen in figure 7, the enlarged image of the skyrmion S L at t S = 1.7 ns is shown in figure 7(a) when it is in close proximity to the corner.A significant reduction in the size of SL can be seen until its collapse from the core in approximately 70 ps (figures 7(e) and (f)).The parameter ∆t, displayed in the inset of figure 7, represents the time lag relative to the starting time t S .Figures 7(f)-(h) serve to validate the swift core collapse of S L , which occurs within an extremely brief timeframe of less than 50 ps.Following the annihilation of the leading skyrmion, S L , the trailing skyrmion, S T , found itself isolated and moved towards the edge along the trajectory illustrated in figure 6(a) due to the significant driving forces of STT and the Magnus effect.In this case, there is no skyrmionskyrmion interaction, and it is anticipated that the annihilation process for S T will differ from that of S L .To confirm this hypothesis, we examined the evolution of the skyrmion structures within a very brief time frame as in figure 7 for the S L case.At t S ∼ 2.5 ns, S T inevitably approached the right edge, as evident in figure 8(a).This deformation of the magnetic structure of the skyrmion led to an expansion resembling a bubble and a disruption of symmetry from the right side (figures 8(b)-(e)).Surprisingly, instead of a continuous expansion resulting in the skyrmion growing larger, S T began to contract until it completely collapsed (figures 8(f)-(j)).The interaction with the boundary disrupted the topological structure of the skyrmion, causing it to transform into a different magnetic configuration.
This research has promising implications involving magnetic information processing and storage, due to the complex dynamics of interacting Néel skyrmions inside magnetic nanowires, such as helical motions, path compressions, and successive annihilation events, present a rich palette of phenomena that can be harnessed for novel information carrier devices.A single Néel skyrmion represents a unique and  stable magnetic configuration with a lifetime of 500 ns [54], making it a robust candidate for encoding binary information around ensures that a skyrmion can persist for a significant duration without undergoing annihilation.This characteristic is vital for the reliable representation of digital bits.The helical motions and path compressions observed during the interaction of two skyrmions can be leveraged for efficient information processing.Controlled manipulation of skyrmions through external stimuli, such as STT, allows for the creation, transportation, and logical operations of skyrmions.The successive annihilation events observed at higher current densities provide an intriguing mechanism for information erasure.By applying an external spin current to induce annihilation, the information encoded in the skyrmion can be effectively erased.This feature is crucial for applications where dynamic read and write operations are required, to a rewritable information carrier.It is also important to know the threshold current value to avoid unwanted erasure of skyrmions when they are interacting (devices with more than one skyrmion).The inherent stability of skyrmions and the controllable dynamics of interacting skyrmions are important characteristics for the development of high-density and high-speed magnetic memory devices.

Conclusion
In conclusion, our study provides a comprehensive exploration of Néel skyrmion behavior within confined magnetic nanowires under the influence of STT and edge forces.We have uncovered several key insights into their intricate dynamics.The trajectories of interacting skyrmions exhibit complex interplays of forces, resulting in helical motions and path compression.Importantly, the reduction in the radius of the leading skyrmion acts as a predictive indicator of its annihilation.Notably, higher current densities induce successive annihilation events, with the leading skyrmion typically collapsing before the trailing one.Edge interactions play a pivotal role in shaping skyrmion paths and facilitating annihilation.

Figure 1 .
Figure 1.(a) Trajectories of interacting skyrmions with a solid blue arrow representing the trailing skyrmion (S T ), and a dotted red arrow indicating the path of the leading skyrmion (S L ).The color-coded scheme illustrates changes in the skyrmion radius.(b) Trajectory persistence of the trailing skyrmion (S T ) after the annihilation of the leading skyrmion (S L ).Despite the annihilation event, S T maintains its helical trajectory.(c)-(h) snapshots of the positions of the two skyrmions at t = 0 ns, 1.25 ns, 2.9 ns, 3.3 ns, 4.4 ns and 7.3 ns, respectively.The current density was fixed to 9 × 10 11 A m −2 .

Figure 2 .
Figure 2. (a) Visualization of forces acting on a Néel skyrmion.(b) and (c) Positional evolution of skyrmions with respect to the right edge, depicting interactions between two skyrmions (S L and S T ) versus an isolated skyrmion.X R denotes the distance from the right edge.(d) and (e) Time dependence of the skyrmions radius R for S T and S L and compared to the case of a single skyrmion.

Figure 3 .
Figure 3.The relationship between the repulsive force ( f /f o) and ax, along as a function of the time.The inset is the calculated x-component of S T velocity (Vx) versus time.

Figure 4 .
Figure 4. Acceleration components of S T due to its interaction with S L , highlighting the importance of the y-component.The x-and y-components of acceleration for both S L and S T are plotted in the context of their motion toward the top edge under the influence of Fs-s.

Figure 5 .
Figure 5. (a) Visualization of forces acting on S L and S T , with a focus on S L 's compression due to multiple forces.(b) Changes in radius R T for trailing skyrmion S T and its distance d from S L as a function of the distance x, demonstrating the impact of interaction forces.

Figure 6 .
Figure 6.(a) Skyrmion paths in the nanowire for high current density (J = 13 × 10 11 A m −2 ), depicting annihilation of S L followed by S T .(b) Skyrmion radius with respect to the time.(c) Skyrmions at t = 1.5 ns, highlighting the S L significant shrinkage due to edge forces and Fs-s.(d)-(e) Further shrinking of S L and subsequent annihilation as S T approaches.(f) Isolation of S T and its eventual annihilation under STT driving force and edge repulsion.

Figure 7 .
Figure 7. (a)-(e) Comprehensive exploration of annihilation processes for successive skyrmion collapse.The evolution of S L 's size and its rapid core collapse within a short timeframe are depicted.(f)-(h) confirmation of S L 's core collapse in less than 50 ps.

Figure 8 .
Figure 8. Visualization of S T behavior post-S L annihilation, illustrating its approach to the right edge, expansion, symmetry breakage, and ultimate collapse.