Commensuration Effects on Skyrmion Hall Angle and Drag for Manipulation of Skyrmions on Two-Dimensional Periodic Substrates

We examine the dynamics of an individually driven skyrmion moving through a background lattice of skyrmions coupled to a 2D periodic substrate as we vary the ratio of the number of skyrmions to the number of pinning sites across commensurate and incommensurate conditions. As the skyrmion density increases, the skyrmion Hall angle is nonmonotonic, dropping to low or zero values in commensurate states and rising to an enhanced value in incommensurate states. Under commensuration, the driven skyrmion is channeled by a symmetry direction of the pinning array and exhibits an increased velocity. The velocity fluctuations have a narrow band signature at fillings where the skyrmion Hall angle is zero, while for incommensurate fillings, the skyrmion motion is disordered and the velocity noise is broad band. Under commensurate conditions, multi-step depinning transitions appear and the skyrmion Hall angle is zero at low drives but becomes finite at higher drives, while incommensurate fillings have a single depinning transition. As the Magnus force increases, commensuration velocity peaks cross over to dips, and new directional locking angles appear. At large Magnus forces, particularly at commensurate fillings, a velocity boost can occur in which the skyrmion moves faster than the applied drive due to the alignment of the Magnus-induced velocity with the driving direction. Increase the Magnus force can produce regimes of enhanced pinning when the skyrmion is forced to move along a non-symmetry direction of the periodic pinning array. This is in contrast to systems with random pinning, where increasing the Magnus force generally reduces the pinning effect. We demonstrate these dynamics for both square and triangular substrates, and map out the different regimes as a function of filling fraction, pinning force, and the strength of the Magnus force in a series of dynamic phase diagrams.