Noise fluctuations and drive dependence of the skyrmion Hall effect in disordered systems

Skyrmions in magnetic systems are particle-like objects predicted to occur in materials with chiral interactions [1]. The existence of a hexagonal skyrmion lattice in chiral magnets was subsequently confirmed in neutron scattering experiments [2] and in direct imaging experiments [3]. Since then, skyrmion states have been found in an increasing number of compounds [4–8], including materials in which skyrmions are stable at room temperature [9–14]. Skyrmions can be set into motion by applying an external current [15, 16], and effective skyrmion velocity versus driving force curves can be calculated from changes in the Hall resistance [17, 18] or by direct imaging of the skyrmion motion [9, 14]. Additionally, transport curves can be studied numerically with continuum and particle based models [19–23]. Both experiments and simulations show that there is a finite depinning threshold for skyrmion motion similar to that found for the depinning of current-driven vortex lattices in type-II superconductors [24–26]. Since skyrmions have particle like properties and can be moved with very low driving currents, they are promising candidates for spintronic applications [27, 28], so an understanding of skyrmion motion and depinning is of paramount importance. Additionally, skyrmions represent an interesting dynamical system to study due to the strong non-dissipative effect of the Magnus force they experience, which is generally very weak or absent altogether in other systems where depinning and sliding phenomena occur.

For particle-based representations of the motion of objects such as superconducting vortices, a damping term of strength ${\alpha }_{{\rm{d}}}$ aligns the particle velocity in the direction of the net force acting on the particle, while a Magnus term of strength ${\alpha }_{{\rm{m}}}$ rotates the velocity component in the direction perpendicular to the net force. In most systems studied to date, the Magnus term is very weak compared to the damping term, but in skyrmion systems the ratio of the Magnus and damping terms can be as large as ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}\sim 10$ [17, 19, 21, 29]. One consequence of the dominance of the Magnus term is that under an external driving force, skyrmions develop velocity components both parallel (${V}_{| | }$) and perpendicular (V_⊥) to the external drive, producing a skyrmion Hall angle of ${\theta }_{{\rm{sk}}}={\tan }^{-1}(R)$, where $R=| {V}_{\perp }/{V}_{| | }|$. In a completely pin-free system, the intrinsic skyrmion Hall angle has a constant value ${\theta }_{{\rm{sk}}}^{\mathrm{int}}={\tan }^{-1}({\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}});$ however, in the presence of pinning a moving skyrmion exhibits a side jump phenomenon in the direction of the drive so that the measured Hall angle is smaller than the clean value [22, 23, 30]. In studies of these side jumps using both continuum and particle based models for a skyrmion interacting with a single pinning site [22] and a periodic array of pinning sites [30], R increases with increasing external drive until the skyrmions are moving fast enough that the pinning becomes ineffective and the side jump effect is minimized.

Particle-based studies of skyrmions with an intrinsic Hall angle of ${\theta }_{{\rm{sk}}}^{\mathrm{int}}=84^\circ$ moving through random pinning arrays show that ${\theta }_{{\rm{sk}}}=40^\circ$ at small drives and that ${\theta }_{{\rm{sk}}}$ increases with increasing drive until saturating at ${\theta }_{{\rm{sk}}}={\theta }_{{\rm{sk}}}^{\mathrm{int}}$ for high drives [23]. In recent imaging experiments performed in the single skyrmion limit [31] it was shown that R = 0 and ${\theta }_{{\rm{sk}}}=0$ at depinning and that both quantities increase linearly with increasing drive; however, the range of accessible driving forces was too low to permit observation of a saturation effect. These experiments were performed in a regime of relatively strong pinning, where upper limits of $R\sim 0.4$ and ${\theta }_{{\rm{sk}}}=20^\circ$ are expected. A natural question is how universal the linear behavior of R and ${\theta }_{{\rm{sk}}}$ is as a function of drive, and whether the results remain robust for larger intrinsic values of ${\theta }_{{\rm{sk}}}$. It is also interesting to ask what happens in the weak pinning limit where the skyrmions form a hexagonal lattice and depin elastically. In studies of overdamped systems such as superconducting vortices, it is known that the strong and weak pinning limits are separated by a transition from elastic to plastic depinning and have very different transport curve characteristics [24, 26], so a similar phenomenon could arise in the skyrmion Hall effect. Noise fluctuations have also been used as another method to study the dynamics of magnetic systems [32]. In superconducting vortex systems, the plastic flow regime is associated with large voltage noise fluctuations of $1/{f}^{\alpha }$ form [33–36], while when the system dynamically orders at higher drives, narrow band noise features appear and the noise power is strongly reduced [26, 37–39]. Here we show that changes in the skyrmion Hall angle are correlated with changes in the skyrmion velocity fluctuations and the shape of the velocity noise spectrum. In the plastic flow region where R increases linearly with drive, there is a $1/{f}^{\alpha }$ velocity noise signal with $\alpha =1.0$, while when R reaches its saturation value, there is a crossover to white noise or weak narrow band noise, indicating that noise measures could provide another way to probe skyrmion dynamics. In general, we find that the narrow band noise features are much weaker in the skyrmion case than in the superconducting vortex case due to the Magnus effect.

Simulation and system— We consider a 2D simulation with periodic boundary conditions in the x and y-directions using a particle-based model of a modified Thiele equation recently developed for skyrmions interacting with random [21, 23] and periodic [30, 40] pinning substrates. The simulated region contains N skyrmions, and the time evolution of a single skyrmion i at position ${{\bf{r}}}_{i}$ is governed by the following equation:

$$\begin{eqnarray}&&{\alpha }_{{\rm{d}}}{{\bf{v}}}_{i}+{\alpha }_{{\rm{m}}}\hat{{\bf{z}}}\times {{\bf{v}}}_{i}={{\bf{F}}}_{i}^{{\rm{ss}}}+{{\bf{F}}}_{i}^{{\rm{sp}}}+{{\bf{F}}}^{D}.\end{eqnarray} \tag{ 1 }$$

Here, the skyrmion velocity is ${{\bf{v}}}_{i}={\rm{d}}{{\bf{r}}}_{i}/{\rm{d}}{t}$, ${\alpha }_{{\rm{d}}}$ is the damping term, and ${\alpha }_{{\rm{m}}}$ is the Magnus term. We impose the condition ${\alpha }_{{\rm{d}}}^{2}+{\alpha }_{{\rm{m}}}^{2}=1$ to maintain a constant magnitude of the skyrmion velocity for varied ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}$. The repulsive skyrmion–skyrmion interaction force is given by ${{\bf{F}}}_{i}^{{\rm{ss}}}={\sum }_{j=1}^{N}{K}_{1}({r}_{{ij}}){\hat{{\bf{r}}}}_{{ij}}$ where ${r}_{{ij}}=| {{\bf{r}}}_{i}-{{\bf{r}}}_{j}|$, ${\hat{{\bf{r}}}}_{{ij}}=({{\bf{r}}}_{i}-{{\bf{r}}}_{j})/{r}_{{ij}}$, and K₁ is a modified Bessel function that falls off exponentially for large r_ij. The pinning force ${{\bf{F}}}_{i}^{{\rm{sp}}}$ arises from non-overlapping randomly placed pinning sites modeled as harmonic traps with an amplitude of F_p and a radius of ${R}_{p}=0.3$ as used in previous studies [23]. The driving force ${{\bf{F}}}^{D}={F}_{D}\hat{{\bf{x}}}$ is from an applied current interacting with the emergent magnetic flux carried by the skyrmion [17, 29]. We increase ${{\bf{F}}}^{D}$ slowly to avoid transient effects. In order to match the experiments, we take the driving force to be in the positive x-direction so that the Hall effect is in the negative y-direction. We measure the average skyrmion velocity ${V}_{| | }=\langle {N}^{-1}{\sum }_{i}^{N}{{\bf{v}}}_{i}\cdot \hat{{\bf{x}}}\rangle$ (${V}_{\perp }=\langle {N}^{-1}{\sum }_{i}^{N}{{\bf{v}}}_{i}\cdot \hat{{\bf{y}}}\rangle$) in the direction parallel (perpendicular) to the applied drive, and we characterize the Hall effect by measuring $R=| {V}_{\perp }/{V}_{| | }|$ for varied F^D. The skyrmion Hall angle is ${\theta }_{{\rm{sk}}}={\tan }^{-1}R$. We consider a system of size L = 36 with a fixed skyrmion density of ${\rho }_{{\rm{sk}}}=0.16$ and pinning densities ranging from n_p = 0.006 25 to n_p = 0.2.

In figures 1(a) and (b) we plot $| {V}_{\perp }|$, $| {V}_{| | }|$, and R versus F_D for a system with ${F}_{p}=1.0$, ${n}_{p}=0.1$, and ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}=5.708$. In this regime, plastic depinning occurs, meaning that at the depinning threshold some skyrmions can be temporarily trapped at pinning sites while other skyrmions move around them. The velocity–force curves are nonlinear, and $| {V}_{\perp }|$ increases more rapidly with increasing F_D than $| {V}_{| | }|$. The inset of figure 1(a) shows that $| {V}_{| | }| \gt | {V}_{\perp }|$ for ${F}_{D}\lt 0.1$, indicating that just above the depinning transition the skyrmions are moving predominantly in the direction of the driving force. In figure 1(b), R increases linearly with increasing F_D for $0.04\lt {F}_{D}\lt 0.74$, as indicated by the linear fit, while for ${F}_{D}\gt 0.74$ R saturates to the intrinsic value of R = 5.708 marked with a dashed line. The inset of figure 1(b) shows the corresponding ${\theta }_{{\rm{sk}}}$ versus F_D. From an initial value of 0°, ${\theta }_{{\rm{sk}}}$ increases with increasing F_D before saturating at the clean limit value of ${\theta }_{{\rm{sk}}}=80.06^\circ$. Although the linear increase in R with F_D is similar to the behavior observed in the experiments of [31], ${\theta }_{{\rm{sk}}}$ does not show the same linear behavior as in the experiments; however, we show later that when the intrinsic skyrmion Hall angle is small, ${\theta }_{{\rm{sk}}}$ varies linearly with drive. We note that the experiments in [31] were performed in the single skyrmion limit rather than in the many skyrmion plastic flow limit we study. This could impact the behavior of the Hall angle, making it difficult to directly compare our results with these experiments.

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In figure 2 we illustrate the skyrmion positions and trajectories obtained during a fixed period of time at different drives for the system in figure 1. At ${F}_{D}=0.02$ in figure 2(a), R = 0.15 and the average drift is predominantly along the x-direction parallel to the drive, taking the form of riverlike channels along which individual skyrmions intermittently switch between pinned and moving states. In figure 2(b), for ${F}_{D}=0.05$ we find R = 0.6, and observe wider channels that begin to tilt along the negative y-direction. At ${F}_{D}=0.2$ in figure 2(c), R = 1.64 and ${\theta }_{{sk}}=58.6^\circ$. The skyrmion trajectories are more strongly tilted along the $-y$ direction, and there are still regions of temporarily pinned skyrmions coexisting with moving skyrmions. As the drive increases, individual skyrmions spend less time in the pinned state. Figure 2(d) shows a snapshot of the trajectories over a shorter time scale at ${F}_{D}=1.05$ where R = 5.59. Here the plastic motion is lost and the skyrmions form a moving crystal translating at an angle of $-79.8^\circ$ with respect to the external driving direction, which is close to the clean value limit of ${\theta }_{{\rm{sk}}}$. In general, the deviations from linear behavior that appear as R reaches its saturation value in figure 1(b) coincide with the loss of coexisting pinned and moving skyrmions, and are thus correlated with the end of plastic flow.

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In figure 3(a) we show R versus F_D for the system from figure 1 at varied ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}$. In all cases, between the depinning transition and the free flowing phase there is a plastic flow phase in which R increases linearly with F_D with a slope that increases with increasing ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}$. In contrast to the nonlinear dependence of ${\theta }_{{\rm{sk}}}$ on F_D at ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}=5.71$ illustrated in the inset of figure 1(b), figure 3(b) shows that for ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}=0.3737$, ${\theta }_{{\rm{sk}}}$ increases linearly with F_D and ${\theta }_{{\rm{sk}}}^{\mathrm{int}}=20.5$. To understand the linear behavior, consider the expansion of ${\tan }^{-1}(x)=x-{x}^{3}/3\,+\,{x}^{5}/5...$ For small ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}$, as in the experiments, ${\tan }^{-1}(R)\sim R$, and since R increases linearly with F_D, ${\theta }_{{sk}}$ also increases linearly with F_D. In general, for ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}\lt 1.0$ we find an extended region over which ${\theta }_{{\rm{sk}}}$ grows linearly with F_D, while for ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}\gt 1.0$, the dependence of ${\theta }_{{\rm{sk}}}$ on F_D has nonlinear features similar to those shown in the inset of figure 1(b). In figure 3(c) we plot R versus F_D for a system with ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}=5.708$ for varied F_p. In all cases R increases linearly with F_D before saturating; however, for increasing F_p, the slope of R decreases while the saturation of R shifts to higher values of F_D. In general, the linear behavior in R is present whenever F_p is strong enough to produce plastic flow. In figure 3(d) we show R versus F_D at ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}=5.708$ for varied pinning densities n_p. In each case, there is a region in which R increases linearly with F_D, with a slope that increases with increasing n_p. As n_p becomes small, the nonlinear region just above depinning where R increases very rapidly with drive becomes more prominent.

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For weak pinning, the skyrmions form a triangular lattice and exhibit elastic depinning, in which each skyrmion maintains the same neighbors over time. In figure 4(a) we plot the critical depinning force F_c and the fraction P₆ of sixfold-coordinated skyrmions versus F_p for a system with ${n}_{p}=0.1$ and ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}=5.708$. For $0\lt {F}_{p}\lt 0.04$, the skyrmions depin elastically. In this regime, ${P}_{6}=1.0$ and F_c increases as ${F}_{c}\propto {F}_{p}^{2}$ as expected for the collective depinning of elastic lattices [25]. For ${F}_{p}\geqslant 0.04$, P₆ drops due to the appearance of topological defects in the lattice, and the system depins plastically, with ${F}_{c}\propto {F}_{p}$ as expected for single particle depinning or plastic flow.

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In figure 4(b) we plot R versus F_D in samples with ${F}_{p}=0.01$ and n_p = 0.1 in the elastic depinning regime for varied ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}$. We highlight the nonlinear behavior for the ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}=5.708$ case by a fit of the form $R\propto {({F}_{D}-{F}_{c})}^{\beta }$ with $\beta =0.26$ and ${F}_{c}=0.000\,184$. The dotted line indicates the corresponding clean limit value of R = 5.708. We find that R is always nonlinear within the elastic flow regime, but that there is no universal value of β, which ranges from $\beta =0.15$ to $\beta =0.5$ with varying ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}$. The change in the Hall angle with drive is most pronounced just above the depinning threshold, as indicated by the rapid change in R at small F_D. This results from the elastic stiffness of the skyrmion lattice which prevents individual skyrmions from occupying the most favorable substrate locations. In contrast, R changes more slowly at small F_D in the plastic flow regime, where the softer skyrmion lattice can adapt to the disordered pinning sites. In figure 4(c) we plot R versus F_D at ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}=5.708$ and n_p = 0.1 for varied F_p, showing a reduction in R with increasing F_p. A fit of the ${F}_{p}=0.04$ curve in the plastic depinning regime shows a linear increase of R with F_D, while for ${F}_{p}\lt 0.04$ in the elastic regime, the dependence of R on F_D is nonlinear. Just above depinning in the elastic regime, the skyrmion flow direction rotates with increasing drive.

Correlations between noise fluctuations and the skyrmion Hall effect.

The power spectrum of the velocity noise fluctuations at different applied drives represents another method that can be used to probe the dynamics of driven condensed matter systems. In the superconducting vortex case, the total noise power over a particular frequency range or the overall shape of the noise power spectrum can be determined by measuring the voltage time series at a particular current. Both experiments and simulations have shown that in the plastic flow regime, where the vortex flow is disordered and consists of a combination of pinned and flowing particles, the low frequency noise power is large and the voltage noise spectrum has a $1/{f}^{\alpha }$ character with $1.0\leqslant \alpha \leqslant 2.0$. In contrast, the low frequency noise power is considerably reduced in the elastic or ordered flow regime, where the noise is either white with $\alpha =0$ or exhibits a characteristic washboard frequency associated with narrow band noise. Based on these changes in the noise characteristics, it is possible to map out a dynamical phase diagram for the vortex system.

In the skyrmion system, the Hall resistance can be used to detect the motion of skyrmions, and therefore, in analogy with the voltage response in a superconductor, fluctuations in the Hall resistance at a specific applied current should reflect fluctuations in the skyrmion velocity. Since the recent experiments of [31] used imaging techniques to measure R, it is desirable to understand whether changes in R are correlated with changes in the fluctuations of other quantities. We measure the time series of ${V}_{| | }(t)$ and ${V}_{\perp }(t)$ in samples with ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}=5.71$, ${F}_{p}=1.0$, and ${n}_{p}=0.1$, the same parameters used in figures 1 and 2. For these values, R increases linearly with increasing F_D over the range $0.01\lt {F}_{D}\lt 0.8$ before saturating close to the clean value. For each value of F_D, we then construct the power spectrum

$$\begin{eqnarray}&&S(\omega )={\left|\int {V}_{| | ,\perp }(t){{\rm{e}}}^{-{\rm{i}}\omega t}{\rm{d}}{t}\right|}^{2},\end{eqnarray} \tag{ 2 }$$

where F_D is held constant during an interval of $1.7\times {10}^{5}$ simulation time steps.

In figure 5(a) we plot $S(\omega )$ for ${V}_{| | }(t)$ and ${V}_{\perp }(t)$ at ${F}_{D}=0.05$ in the plastic flow regime. Here, the spectral shape is very similar in each case, while the noise power at low frequencies is slightly higher for ${V}_{| | }$ than for V_⊥. At F_D = 0.4 in figure 5(b), deep in the plastic flow phase, the noise power is much higher for V_⊥ than for ${V}_{| | }$ and can be fit reasonably well to a ${\omega }^{-1}$ form, while ${V}_{| | }$ also has an ${\omega }^{-1}$ shape over a less extended region. The two spectra have equal power only for high ω. Figure 5(c) shows $S(\omega )$ at ${F}_{D}=1.05$, which corresponds to the saturation region of R. The V_⊥ signal still has the highest spectral power, but both spectra now exhibit a white or ${\omega }^{0}$ shape. There is a small bump at low frequency which is more prominent in ${V}_{| | }$ that may correspond to a narrow band noise feature. We note that in the overdamped limit of ${\alpha }_{{\rm{m}}}/{\alpha }_{{\rm{d}}}=0$ at this same drive, where the particles have formed a moving lattice, there is a strong narrow band noise feature, suggesting that the Magnus term is responsible for the lack of a strong narrow band noise peak in figure 5(c). In general, we find that the power spectrum for the skyrmions shows $1/\omega$ noise in the plastic flow regime and white noise in the saturation regime.

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Using the power spectrum, we can calculate the noise power S₀ at a specific value of ω. In figure 6 we plot ${S}_{0}=S(\omega =50)$ for ${V}_{| | }$ and V_⊥ versus F_D, along with the corresponding R curve from figure 1. At low F_D, the value of S₀ is nearly the same for both ${V}_{| | }$ and V_⊥. The noise power for V_⊥ increases more rapidly with increasing F_D and both S₀ curves reach a maximum near ${F}_{D}=0.5$ before decreasing as R reaches its saturation value. In general S₀ is large whenever the spectrum has a $1/\omega$ shape. This result shows that noise power fluctuations could be used to probe changes in the skyrmion Hall effect and even dynamical transitions from plastic to elastic skyrmion flow.

We note that in real skyrmion systems, the skyrmions can also have internal modes of motion that could affect the noise power. Such internal modes are not captured by the particle model, and would likely occur at much higher frequencies than those of the skyrmion center of of mass motion that we analyze here. It would be interesting to see if such modes arise in experiment and to determine whether they can also modify the skyrmion Hall angle.

We have investigated the skyrmion Hall effect by measuring the ratio R of the skyrmion velocity perpendicular and parallel to an applied driving force. In the disorder-free limit, R and the skyrmion Hall angle take constant values independent of the applied drive; however, in the presence of pinning these quantities become drive-dependent, and in the strong pinning regime R increases linearly from zero with increasing drive, in agreement with recent experiments. For large intrinsic Hall angles, the current-dependent Hall angle increases nonlinearly with increasing drive; however, for small intrinsic Hall angles such as in recent experiments, both the current-dependent Hall angle and R increase linearly with drive as found experimentally. The linear dependence of R on drive is robust for a wide range of intrinsic Hall angle values, pinning strengths, and pinning densities, and appears whenever the system exhibits plastic flow. For weaker pinning where the skyrmions depin elastically, R has a nonlinear drive dependence and increases very rapidly just above depinning. We observe a crossover from nonlinear to linear drive dependence of R as a function of the pinning strength, which coincides with the transition from elastic to plastic depinning. We also show how R correlates with changes in the power spectra of the velocity noise fluctuations both parallel and perpendicular to the drive. In the plastic flow regime where R increases linearly with increasing F_D, we find $1/f$ noise that crosses over to white noise at higher drives. The noise power drops dramatically as R saturates at high drives.

We gratefully acknowledge the support of the US Department of Energy through the LANL/LDRD program for this work. This work was carried out under the auspices of the NNSA of the US DoE at LANL under Contract No. DE-AC52-06NA25396.