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For an overdamped particle moving over a two-dimensional periodic substrateunder combined dc and ac drives, a series of steps can appear in the velocityforce curves that are known as Shapiro steps. Here we show that for skyrmionsdriven over a two-dimensional periodic obstacle array with a dc drive and an acdrive that is either parallel or perpendicular to the dc drive, the systemexhibits numerous transverse and longitudinal synchronization dynamics due tothe Magnus force. These phenomena originate in interactions between twodifferent types of phase locking effects: Shapiro steps and directionallocking. In some cases, the skyrmion Hall angle is constant but longitudinalShapiro steps appear, while in other regimes the skyrmion Hall angle can eitherincrease or decrease with increasing dc drive during the phase locking as theskyrmion locks to different symmetry directions of the obstacle lattice. For atransverse ac drive we find that strong Hall angle overshoots can occur incertain locked phases where the skyrmion is moving at an angle that isconsiderably larger than the intrinsic Hall angle. For the strongest Magnusforce, the phase locking effects are reduced and there are larger regions ofdisordered dynamics. We show that the skyrmion Hall angle can be controlled byfixing the dc drive and changing the amplitude of the ac drive.

Shapiro Steps and Nonlinear Skyrmion Hall Angles For dc and ac Driven Skyrmions on a Two Dimensional Periodic Substrate