Quantized transport for a skyrmion moving on a two-dimensional periodic substrate

Abstract

We examine the dynamics of a skyrmion moving over a two-dimensional periodic substrate utilizing simulations of a particle-based skyrmion model. We specifically examine the role of the non-dissipative Magnus term on the driven motion and the resulting skyrmion velocity-force curves. In the overdamped limit, there is a depinning transition into a sliding state in which the skyrmion moves in the same direction as the external drive. When there is a finite Magnus component in the equation of motion, a skyrmion in the absence of a substrate moves at an angle with respect to the direction of the external driving force. When a periodic substrate is added, the direction of motion or Hall angle of the skyrmion is dependent on the amplitude of the external drive, only approaching the substrate-free limit for higher drives. Due to the underlying symmetry of the substrate the direction of skyrmion motion does not change continuously as a function of drive, but rather forms a series of discrete steps corresponding to integer or rational ratios of the velocity components perpendicular ( V ⊥ ) and parallel ( V || ) to the external drive direction: V ⊥ / V || = n/m, where n and m are integers. The skyrmion passes through a series of directional locking phases in which the motion is locked to certain symmetry directions of the substrate for fixed intervals of the drive amplitude. Within a given directionally locked phase, the Hall angle remains constant and the skyrmion moves in an orderly fashion through the sample. Signatures of the transitions into and out of these locked phases take the form of pronounced cusps in the skyrmion velocity versus force curves, as well as regions of negative differential mobility in which the net skyrmion velocity decreases with increasing external driving force. The number of steps in the transport curve increases when the relative strength of the Magnus term is increased. We also observe an overshoot phenomena in the directional locking, where the skyrmion motion can lock to a Hall angle greater than the clean limit value and then jump back to the lower value at higher drives. The skyrmion-substrate interactions can also produce a skyrmion acceleration effect in which, due to the non-dissipative dynamics, the skyrmion velocity exceeds the value expected to be produced by the external drive. We find that these effects are robust for different types of periodic substrates. Using a simple model for a skyrmion interacting with a single pinning site, we can capture the behavior of the change in the Hall angle with increasing external drive. When the skyrmion moves through the pinning site, its trajectory exhibits a side step phenomenon since the Magnus term induces a curvature in the skyrmion orbit. As the drive increases, this curvature is reduced and the side step effect is also reduced. Increasing the strength of the Magnus term reduces the range of impact parameters over which the skyrmion can be captured by a pinning site, which is one of the reasons that strong Magnus force effects reduce the...

Magnus-induced ratchet effects for skyrmions interacting with asymmetric substrates

“…In the presence of pinning, the Magnus term reduces the effectiveness of the pinning by causing the skyrmions to deflect around the edges of attractive pinning sites [26,28,29] rather than being captured by the pinning sites as overdamped vortices are. The Magnus term also produces complex winding orbits for skyrmions moving in confined regions [36] or through pinning sites [26,28,29,[37][38][39]. The ratio of the strength of the Magnus term m α to the dissipative term d α can be 10 or higher for skyrmion systems [17].…”

“…In the overdamped limit, no ratchet effect appears for F ac ⊥ ; however, we find that the Magnus effect produces a novel ratchet effect for F ac ⊥ when it curves the skyrmion trajectories into the asymmetry or x direction. We model the skyrmion dynamics in a sample with periodic boundary conditions using a particle based description [28,39] and vary the ratio of the Magnus to dissipative dynamic terms, the amplitude and frequency of the ac drive, and the strength of the substrate.…”

Direct observation of the skyrmion Hall effect

“…. As these experiments were performed at low current densities ( < 1 × 10 5 A/cm 2 ), the absence of transverse motion can be attributed to the creep motion of skyrmions in the low current density regime, in which the direction of motion was influenced strongly by the pining potential of randomly distributed defects [31][32][33] . By progressively increasing the current density, it should be possible to drive skyrmions from the creep motion regime into the steady flow motion regime, as suggested by recent theoretical studies on the collective transport of skyrmions with random disorders/defects [31][32][33] .…”

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

“…The skyrmions can be set into motion by an applied current and are observed to have a very small depinning threshold [38][39][40][41]46,47 , in part because the effectiveness of the Magnus force can be up to ten times stronger than the dissipative force component. The Magnus force introduces a velocity component of the skyrmion that is perpendicular to the direction of an imposed external force, so a skyrmion deflects from or spirals around an attractive pinning site rather than moving directly toward the potential minimum as would occur for systems governed by overdamped dynamics 40,41,[47][48][49][50][51] . Since skyrmions are particle-like objects, many of their dynamical properties can be captured using a point particle model based on a modified Theile's equation that takes into account repulsive skyrmion-skyrmion interactions, the Magnus force, damping, and substrate interactions 47,52 .…”