Fluctuations and noise signatures of driven magnetic skyrmions

Magnetic skyrmions are particlelike 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.

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 $F_D\widehat{\mathit{x}}$ applied along the $x$ direction and the instantaneous skyrmion velocity $\mathbf{V}$ averaged over all skyrmions. The skyrmion Hall angle ${\theta }_{\text{Sk}}$ varies with $F_D$, and the time-averaged skyrmion velocity $\langle \mathbf{V}\rangle$ is aligned with ${\theta }_{\text{Sk}}$. The direction of $\mathbf{V}$ fluctuates around ${\theta }_{\text{Sk}}$ and we resolve $\mathbf{V}$ into its components $V_{\left|\right|}$ and $V_{\perp }$ that are parallel and perpendicular, respectively, to the direction defined by ${\theta }_{\text{Sk}}$.

Figure 2

Results from the overdamped limit ${\alpha }_m=0$ with ${\theta }_{\text{Sk}}^{\text{int}}=0^{\circ }$. (a) Magnitude of the time-averaged velocity $\langle \left|\mathbf{V}\right|\rangle$ vs driving force $F_D$. The inset shows ${\theta }_{\text{Sk}}$ vs $F_D$, where there is no driving force dependence and ${\theta }_{\text{Sk}}=0^{\circ }$. (b) Standard deviation ${\sigma }_{\left|\right|}$ (upper blue curve) and ${\sigma }_{\perp }$ (lower red curve) of the parallel and perpendicular instantaneous velocity, respectively, vs $F_D$ showing a strong anisotropy. The labels a, b, and c indicate the values of $F_D$ at which the structure factors in Figs. 5–5 were obtained. (c) $P_4$ (lower green curve), $P_5$ (central light orange curve), $P_6$ (upper black curve), and $P_7$ (central dark red curve), the fraction of four-, five-, six-, and sevenfold coordinated particles, respectively, vs $F_D$ showing a transition to an ordered state near $F_D=2.0$.

Figure 3

Results from a system with ${\theta }_{\text{Sk}}^{\text{int}}={45}^{\circ }$. (a) $\langle \left|\mathbf{V}\right|\rangle$ vs $F_D$. The inset shows ${\theta }_{\text{Sk}}$ vs $F_D$ where ${\theta }_{\text{Sk}}=0^{\circ }$ at $F_D=0$ and increases to ${\theta }_{\text{Sk}}={\theta }_{\text{Sk}}^{\text{int}}$ when the system enters the moving crystal phase. (b) ${\sigma }_{\left|\right|}$ (upper blue curve) and ${\sigma }_{\perp }$ (lower red curve) vs $F_D$. ${\sigma }_{\left|\right|}\approx {\sigma }_{\perp }$ when the system enters the moving crystal state, where the velocity fluctuations are isotropic. The labels d, e, and f indicate the values of $F_D$ at which the structure factors in Figs. 5–5 were obtained. (c) $P_4$ (lower green curve), $P_5$ (central light orange curve), $P_6$ (upper black curve), and $P_7$ (central dark red curve) vs $F_D$ showing a transition to the moving crystal state near $F_D=2.5$. Inset: $P_6$ vs $F_D$ for systems with ${\theta }_{\text{Sk}}^{\text{int}}=0^{\circ }$ (magneta curve) and ${\theta }_{\text{Sk}}^{\text{int}}={45}^{\circ }$ (black curve) over the range $2.0<F_D<4.0$. The ${\theta }_{\text{Sk}}^{\text{int}}={45}^{\circ }$ system reaches a higher saturation value of $P_6=0.96$ at $F_D=2.5$.

Figure 4

Results from a system with ${\theta }_{\text{Sk}}^{\text{int}}={70}^{\circ }$. (a) $\langle \left|\mathbf{V}\right|\rangle$ vs $F_D$. The inset shows ${\theta }_{\text{Sk}}$ vs $F_D$ where ${\theta }_{\text{Sk}}=0^{\circ }$ at $F_D=0$ and increases to ${\theta }_{\text{Sk}}={\theta }_{\text{Sk}}^{\text{int}}$ when the system enters the moving crystal phase. (b) ${\sigma }_{\left|\right|}$ (upper left blue curve) and ${\sigma }_{\perp }$ (lower left red curve) vs $F_D$ showing that the velocity fluctuations become isotropic at high drives. The labels g, h, and i indicate the values of $F_D$ at which the structure factors in Figs. 5–5 were obtained. Inset: A zoom of the main panel highlighting the small jump in ${\sigma }_{\perp }$ at the transition from the moving liquid to the moving crystal. (c) $P_4$ (lower green curve), $P_5$ (central light orange curve), $P_6$ (upper black curve), and $P_7$ (central dark red curve) vs $F_D$ showing a multistep transition to the moving crystal state near $F_D=2.75$.

Figure 5

Static structure factors $S\left(\mathit{q}\right)$. (a)–(c) The system in Fig. 2 with ${\theta }_{\text{Sk}}^{\mathrm{int}}=0^{\circ }$ at drives of (a) $F_D=1.3$ in the plastic flow state, (b) $F_D=2.0$ in the moving smectic state, and (c) $F_D=8.0$ in the moving anisotropic crystal state. (d)–(f) The system in Fig. 3 with ${\theta }_{\text{Sk}}^{\mathrm{int}}={45}^{\circ }$ at drives of (d) $F_D=1.6$ in the moving liquid state, (e) $F_D=2.4$ in a slightly anisotropic moving crystal state, and (f) $F_D=8.0$ in the moving crystal state. (g)–(i) The system in Fig. 4 with ${\theta }_{\text{Sk}}^{\mathrm{int}}={70}^{\circ }$ at drives of (g) $F_D=2.0$ in the moving liquid state, (h) $F_D=2.8$ in a slightly anisotropic moving crystal state, and (i) $F_D=8.0$ in the moving crystal state.

Figure 6

Dynamic phase diagram as a function of the intrinsic skyrmion Hall angle ${\theta }_{\text{Sk}}^{\text{int}}$ vs the drive strength $F_D$. 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 $F_D=F_p$. For $F_D<F_p$, only a portion of the skyrmions are flowing, while for $F_D\ge F_p$, all of the skyrmions are moving. At high ${\theta }_{\text{Sk}}^{\text{int}}$ 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.

Figure 7

(a), (c), and (e) The parallel and perpendicular mean square displacements ${\mathrm{\Delta }}_{\left|\right|}$ (blue) and ${\mathrm{\Delta }}_{\perp }$ (red) vs time in simulation steps obtained at $F_D=4.0$. (a) At ${\theta }_{\text{Sk}}^{\text{int}}=0^{\circ }$ in the moving smectic phase, the diffusion exponents are ${\alpha }_{\left|\right|}=2.01$ and ${\alpha }_{\perp }=0.04$. (c) At ${\theta }_{\text{Sk}}^{\text{int}}={45}^{\circ }$ in the moving crystal phase, ${\alpha }_{\left|\right|}=0.41$ and ${\alpha }_{\perp }=0.47$. (e) At ${\theta }_{\text{Sk}}^{\text{int}}={70}^{\circ }$ in the moving crystal phase, ${\alpha }_{\left|\right|}=0.27$ and ${\alpha }_{\perp }=0.19$. (b), (d), and (f) $M_{\left|\right|}={\mathrm{\Delta }}_{\left|\right|}\left(\tau \right)$ and $M_{\perp }={\mathrm{\Delta }}_{\perp }\left(\tau \right)$ vs $F_D$ with $\tau =5\times {10}^5$ simulation time steps. (b) ${\theta }_{\text{Sk}}^{\text{int}}=0^{\circ }$, (d) ${\theta }_{\text{Sk}}^{\text{int}}={45}^{\circ }$, and (f) ${\theta }_{\text{Sk}}^{\text{int}}={70}^{\circ }$. As the intrinsic skyrmion Hall angle increases, the Magnus term becomes dominant over the dissipation and the anisotropy in ${\mathrm{\Delta }}_{\left|\right|}$ and ${\mathrm{\Delta }}_{\perp }$, which serve as measures of the effective temperatures parallel and perpendicular to ${\theta }_{\text{Sk}}$, is greatly reduced.

Figure 8

The transition from a moving smectic to a moving crystal for a system with ${\theta }_{\text{Sk}}^{\text{int}}={10}^{\circ }$. (a) ${\mathrm{\Delta }}_{\left|\right|}$ (upper blue line) and ${\mathrm{\Delta }}_{\perp }$ (lower red line) vs time in simulation time steps at $F_D=2.0$ in the moving smectic phase. ${\alpha }_{\left|\right|}=1.54$, indicating superdiffusive behavior, while ${\alpha }_{\perp }=0.24$, indicating subdiffusive behavior. (b) The corresponding structure factor $S\left(\mathit{q}\right)$. (c) Voronoi construction of the instantaneous skyrmion positions at $F_D=2.0$. 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) ${\mathrm{\Delta }}_{\left|\right|}$ (lower blue line) and ${\mathrm{\Delta }}_{\perp }$ (upper red line) vs time in simulation time steps at $F_D=4.0$ in the moving crystal phase. (e) The corresponding structure factor $S\left(\mathit{q}\right)$. (f) The corresponding Voronoi construction of the instantaneous skyrmion positions at $F_D=4.0$ shows an almost completely ordered lattice.

Figure 9

Noise power spectral density plots $S_{\left|\right|}\left(\omega \right)$ (blue) for velocity fluctuations parallel to ${\theta }_{\text{Sk}}$ and $S_{\perp }\left(\omega \right)$ (red) for velocity fluctuations perpendicular to ${\theta }_{\text{Sk}}$. (a)–(c) A sample with ${\theta }_{\text{Sk}}^{\text{int}}=0^{\circ }$ at (a) $F_D=1.0$ in the disordered flow state, (b) $F_D=2.0$ in the moving smectic phase, and (c) $F_D=4.0$ in the moving smectic phase. (d)–(f) A sample with ${\theta }_{\text{Sk}}^{\text{int}}={45}^{\circ }$ at (d) $F_D=1.0$ in the disordered flow state, (e) $F_D=2.0$ in the moving liquid phase, and (f) $F_D=4.0$ in the moving crystal phase. (g)–(i) A sample with ${\theta }_{\text{Sk}}^{\text{int}}={70}^{\circ }$ at (g) $F_D=1.0$ in the disordered flow state, (h) $F_D=2.0$ in the moving liquid phase, and (i) $F_D=4.0$ in the moving crystal phase.

Figure 10

(a)–(d) Height-field plots of the power spectral density $S_{\left|\right|}$ as a function of $F_D$ and $\omega$. (a) ${\theta }_{\text{Sk}}^{\text{int}}=0^{\circ }$. (b) ${\theta }_{\text{Sk}}^{\text{int}}={45}^{\circ }$. (c) ${\theta }_{\text{Sk}}^{\text{int}}={70}^{\circ }$, where the two dashed lines indicate the switching events illustrated in (g). (d) ${\theta }_{\text{Sk}}^{\text{int}}={10}^{\circ }$, where there is a MS phase in the region between the two dashed lines. (e) ${\theta }_{\text{Sk}}$ vs $F_D$ at ${\theta }_{\text{Sk}}^{\text{int}}={10}^{\circ }$. (f) The structure factor peak heights $h_1$ (red), $h_2$ (black), and $h_3$ (blue) vs $F_D$ in a system with ${\theta }_{\text{Sk}}^{\text{int}}={10}^{\circ }$. Pairs of peaks ${180}^{\circ }$ apart are indistinguishable in the plotted range, shown are those corresponding to the first two quadrants. At large drives, peak $h_1$ is at an angle ${\varphi }_1={40}^{\circ }$ with respect to the $x$ axis, peak $h_2$ is at ${\varphi }_2={100}^{\circ }$, and peak $h_3$ is at ${\varphi }_3={160}^{\circ }$. (g) The angles ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$ at which three of the structure factor peaks appear vs $F_D$ for the system in (c) with ${\theta }_{\text{Sk}}^{\text{int}}={70}^{\circ }$. The two dashed lines indicate switching events that are associated with a lattice rotation.