Shapiro steps for skyrmion motion on a washboard potential with longitudinal and transverse ac drives

Abstract

We numerically study the behavior of two-dimensional skyrmions in the presence of a quasi-onedimensional sinusoidal substrate under the influence of externally applied dc and ac drives. In the overdamped limit, when both dc and ac drives are aligned in the longitudinal direction parallel to the direction of the substrate modulation, the velocity-force curves exhibit classic Shapiro step features when the frequency of the ac drive matches the washboard frequency that is dynamically generated by the motion of the skyrmions over the substrate, similar to previous observations in superconducting vortex systems. In the case of skyrmions, the additional contribution to the skyrmion motion from a non-dissipative Magnus force shifts the location of the locking steps to higher dc drives, and we find that the skyrmions move at an angle with respect to the direction of the dc drive. For a longitudinal dc drive and a perpendicular or transverse ac drive, the overdamped system exhibits no Shapiro steps; however, when a finite Magnus force is present we find pronounced transverse Shapiro steps along with complex two-dimensional periodic orbits of the skyrmions in the phaselocked regimes. Both the longitudinal and transverse ac drives produce locking steps whose widths oscillate with increasing ac drive amplitude. We examine the role of collective skyrmion interactions and find that additional fractional locking steps occur for both longitudinal and transverse ac drives. At higher skyrmion densities, the system undergoes a series of dynamical order-disorder transitions, with the skyrmions forming a moving solid on the phase locking steps and a fluctuating dynamical liquid in regimes between the steps.

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

“…Simulation and system-We consider a 2D simulation with periodic boundary conditions in the x and ydirections 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 r i is governed by the following equation:…”

Motion of skyrmions in nanowires driven by magnonic momentum-transfer forces

“…The position of an isolated magnetic skyrmion can be manipulated by an external driving force. A spinpolarized electric current has been reported to be an effective driving force for the motion of magnetic skyrmions in confined geometries [14,21,22,[39][40][41]. In addition, the scattering of a propagating spin wave has been demonstrated to generate a momentum transfer, resulting in a skyrmion or domain wall motion [42][43][44][45][46][47][48][49][50].…”

Shapiro spikes and negative mobility for skyrmion motion on quasi-one-dimensional periodic substrates

“…Magnus effects are known to be important for skyrmions in chiral magnets, where the ratio of the Magnus term to the damping term can be ten or higher [12][13][14][15][16][17]. Skyrmions can be set into motion by an applied spin-polarized current, and the Magnus term has been shown to strongly affect the interaction of the moving skyrmions with pinning sites, leading to reduced depinning thresholds [14-16, 18, 19], a drive dependent skyrmion Hall angle [19][20][21][22], and skyrmion speed up effects [20,21].Recent studies of skyrmions driven over a periodic q1D substrate by a dc drive that is parallel to the substrate periodicity direction combined with a perpendicular ac drive showed that a new class of Magnus-induced Shapiro steps arises due to an effective coupling by the Magnus term of the perpendicular and parallel particle motion, whereas in the overdamped limit no Shapiro steps occur for this drive configuration [23]. Here we examine skyrmions confined to a 2D plane containing a q1D periodic substrate and moving under the influence of a dc drive applied perpendicular to the substrate periodicity direction along with a parallel or perpendicular ac drive, and we find that a rich variety of dynamical phases can occur.…”

“…We map the evolution of the dynamic phases as a function of ac drive amplitude and the ratio of the Magnus to the damping term. In addition to their interest as signatures of a new dynamical system, these results could be important in providing a new way to precisely control the direction of motion of skyrmions in order to realize skyrmion-based memory or logic devices [28].Simulation-We model a 2D system with periodic boundary conditions in the x and y directions containing a q1D substrate and a skyrmion treated with a particlebased model that has previously been used to examine driven skyrmion motion in random [18,19], 2D periodic [21], and 1D periodic substrates [23,29]. The skyrmion dynamics are determined using the following equation of…”