Physical Review B 96, 085106 (2017)

Fluctuations and Noise Signatures of Driven Magnetic Skyrmions

Sebastián A. Díaz, 1,2 C. J. O. Reichhardt, 1 Daniel P. Arovas, 2 Avadh Saxena, 1 and C. Reichhardt 1

1Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
2Department of Physics, University of California, San Diego, La Jolla, California 92093, USA

(Received 13 April 2017; revised manuscript received 7 July 2017; published 2 August 2017)

Magnetic skyrmions are particle-like objects with topologically-protected stability which can be set into motion with an applied current. Using a particle-based model we simulate current-driven magnetic skyrmions interacting with random quenched disorder and examine the skyrmion velocity fluctuations parallel and perpendicular to the direction of motion as a function of increasing drive. We show that the Magnus force contribution to skyrmion dynamics combined with the random pinning produces an isotropic effective shaking temperature. As a result, the skyrmions form a moving crystal at large drives instead of the moving smectic state observed in systems with a negligible Magnus force where the effective shaking temperature is anisotropic. We demonstrate that spectral analysis of the velocity noise fluctuations can be used to identify dynamical phase transitions and to extract information about the different dynamic phases, and show how the velocity noise fluctuations are correlated with changes in the skyrmion Hall angle, transport features, and skyrmion lattice structure.
I. INTRODUCTION
II. MODEL AND SIMULATION
III. DYNAMIC PHASES
A. Moving lattice structure
B. Diffusion
C. Small intrinsic skyrmion Hall angles
IV. NOISE SIGNATURES
V. DISCUSSION
VI. SUMMARY
REFERENCES

I.  INTRODUCTION

Skyrmions are particle-like objects that emerge due to collective interactions of atomic-scale magnetic degrees of freedom, and they are stabilized by their topological properties [1,2,3]. An increasing number of magnetic systems capable of supporting skyrmions have been identified, including recent work on materials in which the skyrmions are stable at room temperature [4,5,6,7]. Skyrmions can also be readily set into motion with an applied current [8,9,10,11,12,13,14,15,16], and this property, along with their size scale, makes them promising candidates as information carriers for applications in high-density magnetic storage [17,18,19,20] and novel logic devices [21,22].
Similar to other particle-like objects such as vortices in type-II superconductors [23,24,25,26], electrons in Wigner crystals [27,28,29], and colloids moving on a rough substrate [30,31,32], skyrmions exhibit depinning and collective dynamics when driven over quenched disorder [33]. All of these systems can exhibit dynamical transitions between distinct types of motion as a function of the strength of the driving force. Such dynamic phases include a low drive pinned phase, a plastic or disordered flow phase in which there is a combination of moving and pinned particles [33,23,30,31,34,35], moving liquid phases in which all the particles are in motion but the overall moving structure is disordered [33,23], and high driving dynamically ordered or quasi-ordered states including moving smectic states and moving anisotropic crystals [33,23,25,26,28,32,36,37,38,39,40,41,42,43]. These different phases can be identified by direct visualization [39,41], changes in the structure factor of the particle configurations [25,37,38,40,42,43] and features in the transport curves or differential conductivity [33,23,26,28,34,36,38,40]. The dynamic states can also be characterized by measuring changes in the noise fluctuations of the moving particles as a function of increasing drive [33]. One of the best examples of a system in which noise fluctuations can be used to identify the dynamical states is flux motion in type-II superconductors, where the voltage noise generated at fixed current or drive is produced by the velocity fluctuations of the vortices. Both experiments and simulations show that in the plastic flow regime the noise fluctuations typically exhibit a large low frequency component with a 1/fα characteristic [24,38,43] where α = 1.0 to 2.0, while in the moving liquid phase the noise is white with α = 0 [24,38]. At higher drives when a moving ordered state forms, a narrow band noise signature appears with peaks at frequencies that are correlated with the periodicity of the moving structure [38,41,43,44,45,46].
Koshelev and Vinokur [36] argued that the transition from the disordered flow state to the ordered flow state at higher drives in vortex systems can be understood by representing the fluctuating forces exerted by the pinning sites on the moving vortices as an effective shaking temperature Tsh. As the driving force Fd increases, Tsh decreases according to Tsh ∼ 1/Fd since the more rapidly moving vortices have less time to respond to the pinning forces. At high drives the pinning sites act only as a weak perturbation of magnitude Tsh on the rapidly moving vortices, and the mutual repulsion between vortices dominates over the pinning forces so that the vortices form an ordered lattice. In contrast, at lower drives the fluctuations created by the pinning sites are stronger and Tsh becomes large enough to melt the vortex lattice, in analogy to the melting of an equilibrium system under ordinary thermal fluctuations. Further theoretical work [47,48] showed that due to the direction of the external driving force, the shaking temperature from the pinning-induced dynamical perturbations is anisotropic, and thus the dynamically ordered state is also anisotropic, taking the form of a moving smectic or a moving anisotropic crystal. Such states have been observed in experiment [39] and simulations [37,38,40]. Computational studies have also directly shown that the dynamic fluctuations in the different phases are anisotropic [40] and that the shaking temperature decreases with increasing drive [49].
Skyrmions share many features with vortices in type-II superconductors. Both are particle-like objects with repulsive interactions, both form triangular lattices in the absence of quenched disorder, and both can be set into motion with an applied current. Thus, analyzing the noise fluctuations of moving skyrmions should provide a valuable method for characterizing the skyrmion dynamics. One key difference between vortices and skyrmions is that skyrmion motion typically has a large Magnus component, while vortex systems are generally in the overdamped limit where the Magnus term is weak or absent. The Magnus term generates a skyrmion velocity component that is perpendicular to the external force, and it is not known what effect this has on the noise fluctuations. Due to the Magnus term, the skyrmions move at an angle, called the skyrmion Hall angle θSk, with respect to the driving force. In the absence of pinning θSk has a fixed intrinsic value of θintSk = arctan(αm/αd) where αm is the strength of the Magnus term and αd is the damping coefficient. Both simulations [16,50,51,52,53,54] and experiments [14,15] have shown that when pinning is present, θSk is strongly drive dependent, taking the value θSk ≈ 0 just above depinning and increasing to θSk ≈ θintSk at high drives where the dynamic fluctuations induced by the pinning are reduced. This drive dependence is a result of the side-jump motions the skyrmions experience when they interact with pinning sites, which diminishes at higher drives [50,51,52,55]. Previous simulation work showed that in the presence of pinning the skyrmions form a pinned skyrmion glass that depins plastically into a disordered flowing state followed by a transition to an ordered flowing state at high drives [51]. Unlike the overdamped vortices, which form a moving smectic state, the skyrmions form a moving isotropic crystal, suggesting that the dynamic fluctuations and shaking temperature experienced by the moving skyrmions are different in nature from those in the vortex system. Previous numerical studies indicate that skyrmions show a broad band noise signature near depinning and narrow band noise at high drives [53]; however, these studies were limited in scope and did not include diffusive measures that could indicate how the fluctuations correlate with the structure of the moving skyrmions.
Here we examine skyrmions moving over random quenched disorder and measure the velocity fluctuations both parallel and perpendicular to the direction of skyrmion motion for varied intrinsic skyrmion Hall angles and drives. We use a particle-based model [16,50,51,52,53,55] in which the skyrmion dynamics is described by a modified Thiele equation [55,56]. In the overdamped limit the Hall angle is zero with respect to the driving direction, the velocity fluctuations are highly anisotropic at all drives, and the system forms a moving smectic state at large drives as indicated by the presence of two dominant peaks in the structure factor. When the Magnus term is finite, we find that the velocity fluctuations are anisotropic in the plastic flow regime but become isotropic at the transition to a moving ordered phase in which the skyrmions form a moving triangular crystal with equal weight in the six Bragg peaks of the structure factor. The skyrmion Hall angle is zero just at depinning within the plastic flow regime and gradually increases with increasing drive before saturating to a value close to θSkint at the transition to the moving crystal state. We show that the velocity noise fluctuations undergo a crossover from a broad band noise signature in the plastic flow phase to narrow band noise in the moving crystal. In general the skyrmion system exhibits a richer variety of narrow band noise than the vortex system, such as switching events that are associated with small rotations in the moving lattice. We also show that at small but finite intrinsic skyrmion Hall angles, multiple dynamical transitions can occur in the moving state, including a transition from disordered flow to a moving smectic followed by a transition to a moving crystal. These transitions can be detected through changes in the velocity noise signal and jumps in the skyrmion Hall angle.
The paper is organized as follows. In Section II we describe the system and our simulation method. Section III covers the dynamic phases observed for varied intrinsic skyrmion Hall angles, the behavior of the velocity fluctuations parallel and perpendicular to the skyrmion motion, and the correlation of these fluctuations with changes in transport curves, particle structure, and θSk. In Section III A we show how the different dynamic phases produce distinct structure factor signatures, while in Section III B we examine the diffusive behavior in the moving frame and find that the shaking temperature in the dynamically ordered phase is generally isotropic for skyrmions and anisotropic in the overdamped limit. We consider the limit of small intrinsic skyrmion Hall angles in Section III C and show that the system exhibits both a moving smectic and a moving crystal phase. In Section IV we measure the velocity noise signals both parallel and perpendicular to the direction of skyrmion motion and show that there is a transition from broad band noise in the plastic flow region to a narrow band noise signal in the moving crystal state. We discuss possible future directions in Section V, and in Section VI we provide a summary of our results.

II.  MODEL AND SIMULATION

We consider a two-dimensional system in which the skyrmions are modeled as point particles obeying dynamics that are described by a modified Thiele's equation which includes quenched disorder and skyrmion-skyrmion interactions [55,56]. The particle-based model is applicable under conditions where excitation of internal modes in individual skyrmions can be neglected and where the distance between skyrmions is comparable to or larger than the size of an individual skyrmion. We simulate N = 480 skyrmions moving in a system of size Lx ×Ly with Lx=34.64 and Ly=36 containing Np = 259 randomly distributed pinning sites. The equation of motion of skyrmion i is given by
αd vi − αm
^
z
 
 ×vi = Fiss + Fip + FD.
(1)
Here vi=dri/dt is the instantaneous velocity and ri is the position of skyrmion i, while αd=1.0 is the damping coefficient for the dissipative viscous force. The second term on the left hand side is the Magnus force which originates from the topological charge of each skyrmion. It produces no work and causes the skyrmions to move in the direction perpendicular to the net force. The skyrmion-skyrmion interaction force [51,55] is given by Fiss = ∑j = 1N K1(rij) ∧rij, where rij = |ri − rj|, ∧rij = (ri − rj)/rij , and K1 is the modified Bessel function of the second kind. The randomly distributed pinning sites are modeled as finite range parabolic traps that produce a pinning force described by Fip = ∑k = 1Np (Fp/rp) (ri − rk(p)) Θ( rp − |ri − rk(p)| ), where Fp=1.5 is the maximum pinning force, rp=0.35 is the radius of the pinning sites, rk(p) is the location of the k-th pinning site, and Θ is the Heaviside step function. The external driving force, FD = FD ∧x, represents the Lorentz force exerted on the emergent quantized magnetic flux carried by each skyrmion by an applied electric current [3,9]. The dynamics of superconducting vortices are also described by Eq. (1) in the limit where the coefficient αm of the Magnus term vanishes, since the vortex dynamics are dominated by damping. We refer to the case αm=0 as the overdamped vortex limit.
We measure V=N−1∑i=1Nvi, the instantaneous velocity averaged over all of the skyrmions, as well as its time-averaged value 〈V〉. In the absence of pinning sites, the angle θ between V and the driving force is equal to θSkint, the intrinsic skyrmion Hall angle. When pinning is present, however, θ fluctuates around the drive-dependent value of θSk. We resolve V into components V||=V ·∧xcosθSk+V ·∧ysinθSk and V⊥=V ·∧ycosθSk−V ·∧xsinθSk that are parallel and perpendicular, respectively, to the direction of θSk. Figure 1 shows an image of our system as well as a diagram illustrating the relationship between V, V||, V⊥, θSk, and the driving force FD which is applied along the x direction.
Fig1.png
Figure 1: (a) Real-space snapshot of the simulated system. Open black circles indicate pinning sites and filled blue dots indicate the instantaneous skyrmion positions. (b) Diagram illustrating the relationship between the driving force FD ∧x applied along the x direction and the instantaneous skyrmion velocity V averaged over all skyrmions. The skyrmion Hall angle θSk varies with FD, and the time-averaged skyrmion velocity 〈V〉 is aligned with θSk. The direction of V fluctuates around θSk and we resolve V into its components V|| and V⊥ that are parallel and perpendicular, respectively, to the direction defined by θSk.
To investigate the effect of the Magnus term, we perform simulations for different values of αm corresponding to intrinsic skyrmion Hall angles of θintSk=0°, 10°, 20°, 30°, 45°, 60°, 70°, and 80°. We initialize the skyrmions in a triangular lattice and slowly increase FD from FD=0 to FD=8.0 in increments of ∆FD=0.01 every 5 ×105 simulation time steps. For each drive increment, we allow the system to equilibrate during 5 ×104 simulation time steps, compute θSk for the equilibrated system based on the direction of 〈V〉, and then use θSk to measure V|| and V⊥ during the remaining 4.5 ×105 simulation time steps. From the resulting time series we construct σ|| and σ⊥, the standard deviations of V|| and V⊥, respectively. We also obtain S||(ω) and S⊥(ω), the power spectral densities of the fluctuations in the parallel and perpendicular velocity components. Using the instantaneous positions of all the skyrmions, we calculate the mean squared displacements ∆|| and ∆⊥ parallel and perpendicular to 〈V〉, as well as the static structure factor S(q) = | N−1 ∑i = 1N exp[ i q·ri ] |2. We determine Pn, the fraction of n-fold coordinated skyrmions, according to Pn=N−1∑i=1N δ(n−zi) where zi is the coordination number of skyrmion i obtained from a Voronoi tessellation. We focus on P6, which measures the amount of six-fold or hexagonal ordering of the moving skyrmions, as well as P5 and P7, which are correlated with the appearance of dislocations composed of 5-7 paired dislocations. There is also some finite weight in P4 due to the shearing of a triangular lattice structure. All other coordination numbers have vanishing weight and are not shown.

III.  DYNAMIC PHASES

Fig2.png
Figure 2: Results from the overdamped limit αm=0 with θintSk = 0°. (a) Magnitude of the time-averaged velocity 〈|V|〉 vs driving force FD. The inset shows θSk vs FD, where there is no driving force dependence and θSk = 0°. (b) Standard deviation σ|| (upper blue curve) and σ⊥ (lower red curve) of the parallel and perpendicular instantaneous velocity, respectively, vs FD showing a strong anisotropy. The labels a, b, and c indicate the values of FD at which the structure factors in Fig. 5(a,b,c) were obtained. (c) P4 (lower green curve), P5 (central light orange curve), P6 (upper black curve), and P7 (central dark red curve), the fraction of 4-, 5-, 6-, and 7-fold coordinated particles, respectively, vs FD showing a transition to an ordered state near FD = 2.0.
In Fig. 2(a) we plot the magnitude of the time-averaged velocity 〈|V|〉 versus driving force FD for the overdamped vortex limit with αm=0 and θSkint=0. For FD < 2.0, 〈|V|〉 has a nonlinear shape consistent with a plastic depinning process, and crosses over to a linear dependence on driving force for FD ≥ 2.0. As shown in the inset of Fig. 2(a), θSk = 0 for all FD, indicating that the particles are moving in the direction of the applied drive, as expected for the vortex limit. In Fig. 2(b), the plot of σ|| and σ⊥ versus FD indicates that the velocity fluctuations are strongly anisotropic, with σ|| > σ⊥. Both σ|| and σ⊥ reach peak values near FD=1.0, indicating a maximum in the plasticity of the flow. This is followed by a decrease in σ|| to a saturation value of σ|| ≈ 0.27 at high drives, and a dip in σ⊥ followed by a gradual increase to a saturation value of σ⊥ ≈ 0.19 at high drives. The plots of P4, P5, P6, and P7 versus FD in Fig 2(c) show that for 0.0 < FD < 1.2, P6 ≈ 0.4 and the vortex positions are strongly disordered. For FD > 1.2, P6 increases to a value of P6 ≈ 0.9 indicating that dynamical reordering of the vortices has occurred, while P5 and P7 drop to P5,7 ≈ 0.05 and track each other as the defects form paired 5-7 gliding dislocations.
Fig3.png
Figure 3: Results from a system with θintSk=45°. (a) 〈|V|〉 vs FD. The inset shows θSk vs FD where θSk = 0° at FD=0 and increases to θSk=θSkint when the system enters the moving crystal phase. (b) σ|| (upper blue curve) and σ⊥ (lower red curve) vs FD. σ|| ≈ σ⊥ when the system enters the moving crystal state, where the velocity fluctuations are isotropic. The labels d, e, and f indicate the values of FD at which the structure factors in Fig. 5(d,e,f) were obtained. (c) P4 (lower green curve), P5 (central light orange curve), P6 (upper black curve), and P7 (central dark red curve) vs FD showing a transition to the moving crystal state near FD = 2.5. Inset: P6 vs FD for systems with θintSk=0° (gray curve) and θintSk=45° (black curve) over the range 2.0 < FD < 4.0. The θSkint=45° system reaches a higher saturation value of P6=0.96 at FD=2.5.
Figure 3 illustrates the same quantities as above in a system with a finite Magnus term where θintSk=45°. In Fig. 3(a), the shape of 〈|V|〉 versus FD is nonlinear for FD < 2.0, indicating a plastic depinning process similar to that of the θintSk = 0° case shown in Fig. 2(a), but with a lower overall magnitude of 〈|V|〉. The plot of σ|| and σ⊥ versus FD in Fig. 3(b) shows that σ|| > σ⊥ only for FD < 2.5 in the disordered flow regime, while for FD ≥ 2.5 when the system has dynamically ordered, σ|| ≈ σ⊥, indicating that the velocity fluctuations are isotropic unlike in the vortex case. In Fig. 3(c), we plot P4, P5, P6, and P7 versus FD. The disordered plastic flow with low P6 and high σ|| persists out to slightly higher drives in the θSkint=45° system compared to the vortex system, as indicated by the fact that the peak in σ|| occurs at FD=1.1 in Fig. 3(b) but at FD=1.0 in Fig. 2(b). The dynamic reordering also shifts to higher drives, falling at FD=2.0 in Fig. 2(c) but at FD=2.5 in Fig. 3(c); however, although P6 continues to slowly increase with FD in the vortex system above the dynamic reordering transition, in the θSkint = 45° system P6 reaches a higher saturation value of P6=0.96 at FD=2.5 and does not evolve as FD further increases, as shown in the inset of Fig. 3(c). This difference is due to the anisotropic velocity fluctuations in the vortex case, where the dislocations remaining after the dynamic reordering process has occurred undergo gliding motion and can only slowly annihilate each other, and the isotropic velocity fluctuations in the skyrmion case, where the remaining dislocations can climb under effectively thermal excitations and annihilate each other much more efficiently. The inset of Fig. 3(a) shows that θSk ≈ 0 at small FD and gradually increases to θSk=θintSk=45°, reaching its saturation value at FD=1.5 where the difference between σ|| and σ⊥ in Fig. 3(b) has become small. This indicates that the direction of skyrmion motion in the plastic flow state gradually rotates until the drive is high enough that all the skyrmions have depinned. The drive dependence of θSk is in agreement with previous simulation studies [51] and experiments [14] which show an increase in θSk with driving force and a saturation to the intrinsic value θSkint at high drives.
Fig4.png
Figure 4: Results from a system with θSkint=70°. (a) 〈|V|〉 vs FD. The inset shows θSk vs FD where θSk=0 at FD=0 and increases to θSk=θSkint when the system enters the moving crystal phase. (b) σ|| (upper left blue curve) and σ⊥ (lower left red curve) vs FD showing that the velocity fluctuations become isotropic at high drives. The labels g, h, and i indicate the values of FD at which the structure factors in Fig. 5(g,h,i) were obtained. Inset: A zoom of the main panel highlighting the small jump in σ⊥ at the transition from the moving liquid to the moving crystal. (c) P4 (lower green curve), P5 (central light orange curve), P6 (upper black curve), and P7 (central dark red curve) vs FD showing a multi-step transition to the moving crystal state near FD=2.75.
The same quantities as above for a system with θintSk=70° appear in Fig. 4. 〈|V|〉 versus FD in Fig. 4(a) has a similar nonlinear shape for FD < 2 as found for the other values of θintSk, and at higher drives the magnitude of 〈|V|〉 is further decreased compared to the vortex system. The inset of Fig. 4(a) indicates that at low drives θSk=0, while the skyrmion Hall angle saturates to the value θSk=θSkint = 70° at FD=1.5. In Fig. 4(b), the plot of σ|| and σ⊥ versus FD shows that there is a sharp decrease in both quantities when the system enters the moving liquid phase at FD=1.5, and that at high drives σ⊥ is slightly larger than σ||. The plot of P4, P5, P6, and P7 versus FD in Fig. 4(c) indicates that there are two distinct disordered flowing phases. For 0.3 < FD < 1.5, the system exhibits plastic flow (PF) where not all of the skyrmions are moving, σ|| > σ⊥, P6 = 0.4, P4 > 0, and P5 > P7. In contrast, for 1.5 < FD < 2.5 the system forms what we call a moving liquid (ML) phase in which all of the skyrmions have depinned and have gained some partial order, with P6 ≈ 0.625, P4=0, and P5=P7 indicating that all the remaining dislocations have formed pairs. For FD > 2.5 the system dynamically orders into a moving crystal as shown by the jump in P6 to P6 ≈ 0.98 and the drop in P5 and P7 to P5 = P7 ≈ 0.02. At the transition from the moving liquid to the moving crystal there is also a small jump in σ⊥, as shown in the inset of Fig. 4(b).
Fig5.png
Figure 5: Static structure factors S(q). (a,b,c) The system in Fig. 2 with θSk = 0° at drives of (a) FD=1.3 in the plastic flow state, (b) FD=2.0 in the moving smectic state, and (c) FD=8.0 in the moving anisotropic crystal state. (d,e,f) The system in Fig. 3 with θSk = 45° at drives of (d) FD=1.6 in the moving liquid state, (e) FD=2.4 in a slightly anisotropic moving crystal state, and (f) FD=8.0 in the moving crystal state. (g,h,i) The system in Fig. 4 with θSk = 70° at drives of (g) FD=2.0 in the moving liquid state, (h) FD=2.8 in a slightly anisotropic moving crystal state, and (i) FD=8.0 in the moving crystal state.

A.  Moving lattice structure

The different features in the transport curves and velocity noise fluctuations correlate with changes in the static structure factor S(q). In Fig. 5(a,b,c) we plot S(q) for the overdamped system with θintSk = 0° from Fig. 2 at drives of FD = 1.3, 2.0, and 8.0, respectively. At FD = 1.3 the system is undergoing plastic flow and S(q) in Fig. 5(a) exhibits a ring feature indicative of an amorphous structure, while at FD = 2.0 in Fig. 5(b) the system is in a moving smectic state in which the particles travel in one-dimensional chains that slide past one another, producing two peaks in S(q) along the qx = 0 axis. At FD = 8.0 in Fig. 5(c) there is more order in the system as indicated by the additional smeared peaks appearing in S(q); however, the system is still in a moving smectic or a strongly anisotropic moving crystal state as shown by the anisotropy in σ⊥ and σ|| in Fig. 2(b). This result is in agreement with previous simulations and experiments in the overdamped limit for vortex systems [37,38,39].
In Fig. 5(d,e,f) we plot S(q) for the system in Fig. 3 with θintSk = 45° at drives of FD = 1.6, 2.4 and 8.0. At FD = 1.6 in Fig. 5(d) the system is in a moving liquid phase, while at FD = 2.4 in Fig. 5(e) the system has dynamically ordered into a slightly anisotropic moving crystal phase. Here S(q) is rotated compared to the θintSk = 0° case in Fig. 5(b) since the skyrmion lattice is moving at an angle with respect to the drive direction. There is weak anisotropy in the moving structure as indicated by the smearing of four of the peaks in S(q), which is consistent with the small anisotropy observed in σ⊥ and σ|| for this drive value in Fig.  3(b). At FD = 8.0 in Fig. 5(f) where σ⊥ = σ||, the system forms a moving crystal state in which all six peaks in S(q) have almost equal weight. In Fig. 5(g,h,i) we plot S(q) for the system in Fig. 4 with θintSk = 70° at drives of FD = 2.0, 2.8, and 8.0, respectively. Similar to the θintSk=45° case, at FD=2.0 in Fig. 5(g) the system forms a disordered moving liquid state, at FD=2.8 in Fig. 5(h) a slightly anisotropic moving crystal state appears, and at high drives of FD=8.0 in Fig. 5(i), an isotropic moving crystal forms as indicated by the equal weight of the six peaks in S(q).
Fig6.png
Figure 6: Dynamic phase diagram as a function of the intrinsic skyrmion Hall angle θintSk vs the drive strength FD. PF: plastic flow phase; ML: moving liquid phase; MS: moving smectic phase; and MC: moving crystal phase. The line separating PF from ML is at FD=Fp. For FD < Fp, only a portion of the skyrmions are flowing, while for FD ≥ Fp, all of the skyrmions are moving. At high θSkint the MC phase is lost due to a Magnus melting effect. Black squares indicate the ML-MC boundary, green circles mark the ML-MS boundary, and blue triangles show the location of the MS-MC boundary.
In Fig. 6 we plot a dynamic phase diagram as a function of θintSk versus FD. For FD/Fp < 1.0 there is a mixture of pinned and moving skyrmions and we find a plastic flow (PF) phase with a disordered structure. In this regime, the velocity fluctuations are anisotropic with σ|| > σ⊥, and θSk increases from zero as FD increases. Just above FD/Fp = 1.0, all the skyrmions are moving but the structure is still disordered, placing the system in a moving liquid (ML) phase. The value of θSk saturates to θSk=θSkint at the onset of the ML phase. For θintSk < 10°, at higher drives the system transitions into the same moving smectic (MS) state found in the vortex system. At θintSk = 10° we find that the ML-MS transition is followed at slightly higher FD by a transition into a moving crystal (MC) phase. In the MC phase the fluctuations are isotropic or only slightly anisotropic with σ⊥ >~σ||. For 10° < θintSk ≤ 70° we find no MS phase and the system transitions directly from the ML phase to the MC phase. For θintSk = 80°, above the PF-ML transition the system remains in the ML phase out to the highest values of FD we have considered; however, in principle it is possible that the system could dynamically order at much higher drives. At θ = 0°, the ML phase extends from FD=1.5, when all the particles depin, to FD=1.8. Both P5 and P7 decrease throughout the ML regime before reaching a partial saturation at the onset of the MS phase, as shown in Fig. 2.
The loss of the MC phase at large values of θintSk is due to the nondissipative nature of the Magnus term which becomes increasingly dominant as the intrinsic skyrmion Hall angle increases, creating larger dynamical fluctuations that prevent the skyrmions from reordering due to what we term a Magnus melting effect.

B.  Diffusion

For vortex systems, the dynamic fluctuations of the vortices induced by interactions with the pinning sites are argued to act as an anisotropic effective shaking temperature, with the largest fluctuations occurring along the driving direction. In order to test the nature of the fluctuations we measure the mean squared displacement (MSD) of the skyrmions by projecting the individual skyrmion displacements into the directions parallel and perpendicular to the drive-dependent θSk direction. We define the parallel MSD in the center-of-mass frame of reference at time t as ∆||(t) = N−1 ∑i = 1N [ ~ri,||(t) − ~ri,||(0) ]2, where ~ri,||(t) = ri,||(t) − R\scriptscriptstyleCM||(t). Similarly, the perpendicular MSD in the center-of-mass frame of reference is given by ∆⊥(t) = N−1 ∑i = 1N [ ~ri,⊥(t) − ~ri,⊥(0) ]2, where ~ri,⊥(t) = ri,⊥(t) − R\scriptscriptstyleCM⊥(t). Here we have used the parallel and perpendicular components of the position of the i-th skyrmion and of the center of mass R\scriptscriptstyleCM. These components are given by ri,||=ri ·∧xcosθSk+ri ·∧ysinθSk, ri,⊥=ri ·∧ycosθSk−ri ·∧xsinθSk, and similarly for R\scriptscriptstyleCM. The position of the center of mass is R\scriptscriptstyleCM = N−1 ∑i = 1N ri.
At long times, ∆||(t) and ∆⊥(t) exhibit a power law behavior and are proportional to tα, where α is the diffusion exponent. We write ∆||(t) ∝ tα|| and ∆⊥(t) ∝ tα⊥.
Fig7.png
Figure 7: (a,c,e) The parallel and perpendicular mean square displacements, ∆|| (blue) and ∆⊥ (red), vs time in simulation steps obtained at FD=4.0. (a) At θSkint = 0° in the moving smectic phase, the diffusion exponents are α||=2.01 and α⊥=0.04. (c) At θSkint=45° in the moving crystal phase, α||=0.41 and α⊥=0.47. (e) At θSkint=70° in the moving crystal phase, α||=0.27 and α⊥=0.19. (b,d,f) M||=∆||(τ) and M⊥=∆⊥(τ) vs FD with τ = 5 ×105 simulation time steps. (b) θSkint=0°, (d) θSkint=45°, and (f) θSkint=70°. As the intrinsic skyrmion Hall angle increases, the Magnus term becomes dominant over the dissipation and the anisotropy in ∆|| and ∆⊥, which serve as measures of the effective temperatures parallel and perpendicular to θSk, is greatly reduced.
In Fig. 7(a) we plot ∆|| and ∆⊥ versus time for the overdamped case of θintSk = 0° at a drive of FD = 4.0 where the system forms a moving smectic state. Here ∆|| >> ∆⊥ and power law fits of the long time behavior give α|| ≈ 2, consistent with ballistic motion in the parallel direction, and α⊥=0.04. This result is in agreement with previous simulation studies measuring the diffusive behavior in the moving smectic phase for vortices in type-II superconductors [40]. The superdiffusive behavior of ∆|| arises due to the fact that in the moving smectic state the particles move in one-dimensional chains that slip past each other due to their slightly different velocities. As a result, the distance between any two particles in different chains in the moving center of mass frame grows linearly with time. Since there is no transverse hopping of particles from chain to chain, diffusion perpendicular to the driving direction is strongly suppressed. In Fig. 7(b) we plot M|| and M⊥, the value of ∆|| and ∆⊥ at a time of τ = 5 ×105 simulation time steps, versus FD for the θSkint=0° system in Fig. 7(a). Here M|| > M⊥ for all FD, and there is a drop in M⊥ at the ML-MS transition. Within the ML phase, ∆⊥ behaves diffusively while in the MS phase it has subdiffusive behavior (not shown).
In Fig. 7(c) we plot ∆|| and ∆⊥ versus time for a system with θintSk = 45° at FD = 4.0 in the moving crystal phase. Here the displacements are isotropic and we find subdiffusive behavior with α|| ≈ α⊥ ≈ 0.45. In general, in the moving crystal phase a small number of dislocations are present that can slowly climb or glide, producing the weak subdiffusive behavior in both ∆|| and ∆⊥. The plot of M|| and M⊥ versus FD in Fig. 7(d) indicates that the displacements are anisotropic in the PF phase for FD < 2.0, and then become isotropic in the MC phase for FD ≥ 2.0. These results indicate that the isotropic nature of the effective shaking temperature in the moving skyrmion system is responsible for the formation of an isotropic moving crystal phase. Figure 7(e) shows ∆|| and ∆⊥ versus time in the moving crystal phase at FD=4.0 for a system with θintSk = 70°. We find subdiffusive behavior in both directions with α|| ≈ α⊥ ≈ 0.22, and ∆⊥ is slightly larger than ∆||. In Fig. 7(f) we show M|| and M⊥ versus FD for the same system, where we observe a transition to isotropic diffusion at the higher drives. At θintSk = 80° where the moving liquid phase persists up to high drives, there is still a transition from anisotropic diffusion in the plastic flow phase to isotropic diffusion in the ML phase; however, within the ML phase α|| and α⊥ have much higher values than what we observe in the MC phase for smaller θintSk. This indicates that even within the disordered flow regime, the Magnus dominated dynamics modify the diffusive behavior compared to what is observed in the overdamped case.
Fig8.png
Figure 8: The transition from a moving smectic to a moving crystal for a system with θintSk = 10°. (a) ∆|| (upper blue line) and ∆⊥ (lower red line) vs time in simulation time steps at FD=2.0 in the moving smectic phase. α||=1.54, indicating superdiffusive behavior, while α⊥=0.24, indicating subdiffusive behavior. (b) The corresponding structure factor S(q). (c) Voronoi construction of the instantaneous skyrmion positions at FD=2.0. Black dots indicate sixfold-coordinated skyrmions, red dots indicate fivefold-coordinated skyrmions, and blue dots indicate sevenfold-coordinated skyrmions. In the smectic state, the defects combine into 5-7 pairs that glide along the driving direction. (d) ∆|| (lower blue line) and ∆⊥ (upper red line) vs time in simulation time steps at FD = 4.0 in the moving crystal phase. (e) The corresponding structure factor S(q). (f) The corresponding Voronoi construction of the instantaneous skyrmion positions at FD=4.0 shows an almost completely ordered lattice.

C.  Small intrinsic skyrmion Hall angles

The small intrinsic skyrmion Hall angle case of θintSk = 10° is particularly interesting since it exhibits both a MS and a MC phase. In Fig. 8(a) we plot ∆⊥ and ∆|| as a function of time for the θintSk = 10° system at FD = 2.0 in the MS phase. Here ∆|| is superdiffusive with α||=1.54 while ∆⊥ is subdiffusive with α⊥=0.24. Figure 8(b) shows that the structure factor at FD=2.0 contains prominent peaks indicative of a smectic ordering. The moving smectic is tilted with respect to the x axis since the skyrmions are moving at an angle θSk ≈ 9.8° relative to the driving direction. This differs from the MS that forms for θintSk = 0°, shown in Fig. 5(b), which remains aligned with the drive direction. In Fig. 8(c), a Voronoi construction of the instantaneous skyrmion positions in the MS phase indicates that the defects assemble into 5-7 pairs that glide along the direction in which the skyrmions are moving.
At FD=2.75, a transition occurs into the MC state where the skyrmions move at an angle θSk = 10° to the driving direction, indicating that the MS-MC transition is also associated with a lattice rotation. In Fig. 8(d) the plot of ∆⊥ and ∆|| versus time in simulation steps at FD=4.0 in the MC state shows that the system is much more isotropic and that both ∆⊥ and ∆|| exhibit subdiffusive behavior with α|| ≈ α⊥ ≈ 0.425. The corresponding structure factor in Fig. 8(e) contains six peaks with equal weight. In general the moving crystal phases are considerably more ordered than the moving smectic phase, as illustrated by the Voronoi construction of the instantaneous skyrmion positions in the MC phase shown in Fig. 8(f), where there are significantly fewer 5-7 dislocation pairs compared to the MS phase. Additionally, the dislocations in the MC are not aligned with the direction of motion and can move slowly in any of the six symmetry directions of the skyrmion lattice, leading to the subdiffusive behavior. It is possible that for even smaller but finite values of θintSk, MS-MC transitions could occur that are pushed to higher values of FD as θintSk decreases. Although skyrmion systems generally have θintSk > 25°, a finite but small Magnus term can still arise in some vortex systems, so it may be possible to observe MS-MC transitions in superconducting systems, particularly for weak pinning where large vortex velocities could be realized.

IV.  NOISE SIGNATURES

Fig9.png
Figure 9: Noise power spectral density plots S||(ω) (blue) for velocity fluctuations parallel to θSk and S⊥(ω) (red) for velocity fluctuations perpendicular to θSk. (a,b,c) A sample with θSkint=0° at (a) FD = 1.0 in the disordered flow state, (b) FD = 2.0 in the moving smectic phase, and (c) FD = 4.0 in moving smectic phase. (d,e,f) A sample with θSkint=45° at (d) FD = 1.0 in the disordered flow state, (e) FD = 2.0 in the moving crystal phase, and (f) FD = 4.0 in moving crystal phase. (g,h,i) A sample with θSkint=70° at (g) FD = 1.0 in the disordered flow state, (h) FD = 2.0 in the moving liquid phase, and (i) FD = 4.0 in the moving crystal phase.
We next analyze the velocity noise power spectral densities, S||(ω) and S⊥(ω), computed from time series taken at fixed FD of the velocity fluctuations parallel and perpendicular to θSk, respectively. In Fig. 9(a,b,c) we plot S||(ω) and S⊥(ω) for the overdamped case of θintSk = 0° at FD = 1.0 in the plastic flow regime, FD = 2.0 in the moving smectic phase, and at FD = 4.0 in the moving smectic phase. At FD = 1.0, we find broad band noise in both directions with the largest noise power in the parallel direction. S||(ω) has a 1/ω feature at low frequencies along with a 1/ω2 tail at high frequencies, while S⊥(ω) becomes white at low frequencies. At FD = 2.0 in the moving smectic state, characteristic frequencies emerge which become more pronounced at FD=4.0. We find a washboard frequency of ω ≈ 14.5 at FD=4.0 that is produced by the perturbations of the periodic vortex lattice by the underlying pinning sites. This frequency corresponds to ω = 2π〈|V| 〉/a, where a is the characteristic spacing between the vortices in the driving direction. We also observe a time-of-flight signal at ω = 2π〈|V| 〉/Lx which produces weaker peaks with ω < 1. In superconducting systems both of these frequencies have been measured in experiments and simulations. In Fig. 9(d,e,f) we plot S||(ω) and S⊥(ω) for a system with θintSk = 45° at FD = 1.0, 2.0, and 4.0, respectively. At FD = 1.0 in the plastic flow regime we again find a broad band noise signal, but the anisotropy between S||(ω) and S⊥(ω) is smaller than in the overdamped case. In the moving crystal phase, at FD = 2.0 the noise is isotropic and a narrow band frequency emerges which becomes sharper at FD = 4.0. In Fig. 9(g,h,i) we show S||(ω) and S⊥(ω) for a system with θintSk = 70° at FD=1.0, 2.0, and 4.0, respectively. At FD = 2.0 the system is in a ML phase and the noise power is flat at low frequencies, while at FD = 4.0 we find a strong washboard signal in both directions with slightly higher signal strength at higher frequencies in the perpendicular direction. These results indicate that moving skyrmion lattices can exhibit a narrow band noise signature similar to that found in the vortex system, and that analysis of this noise could be used to extract the lattice constant and skyrmion velocity.
Fig10.png
Figure 10: (a,b,c,d) Heightfield plots of the power spectral density S|| as a function of FD and ω. (a) θSkint = 0°. (b) θSkint = 45°. (c) θSkint = 70°, where the two dashed lines indicate the switching events illustrated in panel (g). (d) θSkint = 10°, where there is a MS phase in the region between the two dashed lines. (e) θSk vs FD at θintSk = 10°. (f) The structure factor peak heights h1 (red), h2 (black), and h3 (blue) vs FD in a system with θSkint = 10°. Pairs of peaks 180° apart are indistinguishable in the plotted range, shown are those corresponding to the first two quadrants. At large drives, peak h1 is at an angle ϕ1 = 40° with respect to the x axis, peak h2 is at ϕ2 = 100°, and peak h3 is at ϕ3 = 160°. (g) The angles ϕ1, ϕ2, and ϕ3 at which three of the structure factor peaks appear vs FD for the system in panel (c) with θSkint = 70°. The two dashed lines indicate switching events that are associated with a lattice rotation.
Another feature produced in the noise signatures by a finite Magnus term is switching events that are associated with global rotations in the moving skyrmion structures. These switching events occur due to the drive dependence of the flow direction generated by the Magnus term. Such switching events do not occur in an overdamped system since the lattice remains locked to the driving direction once it has entered the MS phase. In Fig. 10(a) we plot a heightfield showing the magnitude of the S|| noise component as a function of FD versus ω in the overdamped case of θintSk=0°. The plastic flow regime for FD < 1.75 is distinguished by large amplitude low frequency noise, while the onset of the narrow band noise at FD=1.75 is indicated by a series of lines that shift to higher frequencies with increasing FD. For θintSk = 45°, the heightfield plot of S|| in Fig. 10(b) shows several switching events that occur near FD = 2.25 as indicated by the horizontal features. Figure 10(c) shows a heightfield plot of S|| for θintSk = 70°. Here the plastic phase appears as a strong signal at low frequency for FD < 1.5, while for 1.5 < FD < 2.8 the system is in the moving liquid phase. The dashed line at FD = 2.85 denotes the transition into the MC phase, while the dashed line at FD = 4.85 indicates a switching event. We measure ϕn, the angle between the nth structure factor peak and the x axis, for the θintSk=70° system, and plot ϕ1, ϕ2, and ϕ3 for the peaks in the first two quadrants versus FD in Fig. 10(g). The two dashed lines are at the same values of FD as the dashed lines in Fig. 10(c). The first dashed line at FD = 2.85 indicates that just after the moving crystal forms, there is a switching event in the lattice orientation, while the dashed line at FD = 4.85 shows that there is a second switching event associated with a lattice rotation. In Fig. 10(c), several of the narrow band noise peaks disappear above the second switching event.
Figure 10(d) shows a heightfield plot of S||(ω) versus FD at θintSk = 10° where a frequency shift is associated with the MS-MC transition. The region in which the MS phase appears is bounded by the two dashed lines. Once the system enters the MC phase, several additional narrow band noise peaks arise. The plot of θSk versus FD for the θintSk=10° system in Fig. 10(e) shows that there is a jump in θSk to θSk=9.8° at the transition to the MC phase. We measure the magnitude hn of individual peaks in the structure factor S(q) and plot h1, h2, and h3 versus FD for the three peaks in the first and second quadrants in Fig. 10(f) for the same system. In the MS phase, only h2 is large, indicating the strong asymmetry of the weight of the peaks due to the smectic ordering. At the transition to the MC phase, all the peaks in the structure factor have similar weights. These results indicate that switching events in the noise spectral power measurements can be used to deduce information about the lattice orientation.

V.  DISCUSSION

We have only considered the case of rigid skyrmions; however, it is possible that internal modes or distortions could arise that would modify the transport signatures. In Ref. [15] it is argued that distortions in the skyrmion shape as a function of current or drive can change the skyrmion Hall effect. For example, if the skyrmions become elongated, the fluctuations would be more anisotropic and this could make the skyrmion-skyrmion interactions anisotropic, which would tend to stabilize moving smectic phases rather than moving crystal phases at higher drives. Since narrow band noise appears in both the moving smectic and moving crystal phases, velocity noise measurements are a robust probe of changes in the ordering of the moving skyrmions. Additionally, the switching events in the noise spectral power should be a robust feature. It is also possible for breathing modes to appear in the moving skyrmion state which are not included in the particle-based model. These could introduce additional dynamical modes which could couple to the velocity noise modes; alternatively, the breathing mode frequencies could be much higher than those of the velocity noise and would appear as additional noise peaks.
Future studies could examine such skyrmion dynamics using continuum models, which are computationally expensive and more difficult to use for probing low frequency effects. Another approach would be to add correlated fluctuations mimicking the dynamical effects to the continuum models in order to reduce the computational resource requirements.
Although the diffusive behavior we observe is generated dynamically, we can compare our results to equilibrium thermal studies of skyrmions [57,58], which show that the Magnus force reduces the diffusion since the particles move in more circular patterns rather than diffusing outward as in the overdamped case. In our system, under high drives the fluctuations experienced by the skyrmions produce a reduced diffusion, leading to the appearance of a moving crystal, whereas in an overdamped system, there is still strong diffusion along the direction of drive, causing a moving smectic state to form.

VI.  SUMMARY

We have numerically examined the velocity fluctuations parallel and perpendicular to the direction of motion for skyrmions driven over random disorder for different values of the Magnus term or intrinsic skyrmion Hall angle. In the overdamped limit, the system undergoes a transition from a disordered phase to a moving smectic state in which the velocity fluctuations are strongly anisotropic and have the largest magnitude parallel to the direction of motion. In the center of mass frame of the smectic state, the particle displacements are either diffusive or superdiffusive in the direction of motion but subdiffusive in the perpendicular direction. For a finite intrinsic skyrmion Hall angle, the system is disordered at low drives, while at higher drives it transitions into a moving crystal state which exhibits isotropic velocity fluctuations and has subdiffusive particle displacements in the center of mass frame both parallel and perpendicular to the direction of motion. The isotropic nature of the moving skyrmion crystal is a result of the Magnus term, which generates velocity fluctuations perpendicular to the force fluctuations experienced by the moving skyrmions due to the pinning sites. In general, moving skyrmion lattices are more ordered than moving vortex lattices and the dynamic shaking temperature is isotropic for skyrmions but anisotropic in vortex systems. We show that the velocity noise power spectra can be used to identify the transition from the plastic flow or moving liquid state to the moving crystal state in skyrmion systems since it changes from a broad band noise signal to a narrow band noise signal. In the moving crystal state, velocity noise power peaks appear at the washboard frequency, which permits the calculation of the velocity and lattice spacing of the moving skyrmion lattice. We find that the moving skyrmion lattice can exhibit discrete switching events associated with global lattice reorientations due to the dependence of the skyrmion Hall angle on the magnitude of the external drive. These switching events produce changes in the noise fluctuations as well as rotations in the structure factor. For small but finite intrinsic skyrmion Hall angles, the system exhibits a mixture of both vortex-like and skyrmion-like dynamical behavior. At lower drives the system first dynamically orders into a moving smectic state, but at higher drives there is a transition into a moving crystal which is associated with both a change in the direction of skyrmion motion as well as pronounced changes in the velocity noise signal. Skyrmion velocity fluctuations could be measured by direct imaging of moving skyrmions, measuring fluctuations in the topological Hall resistance, performing magnetic noise measurements with Hall probes, or measuring the time dependence of the structure factor.

ACKNOWLEDGMENTS

This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396.

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