Topological transitions, pinning and ratchets for driven magnetic hopfions in nanostructures.

Using atomistic simulations, we examine the dynamics of three-dimensional magnetic hopfions interacting with an array of line defects or posts as a function of defect spacing, defect strength, and current. We find a pinned phase, a sliding phase where a hopfion can move through the posts or hurdles by distorting, and a regime where the hopfion becomes compressed and transforms into a toron that is half the size of the hopfion and moves at a lower velocity. The toron states occur when the defects are strong; however, in the toron regime, it is possible to stabilize sliding hopfions by increasing the applied current. Hopfions move without a Hall angle, while the toron moves with a finite Hall angle. We also show that when a hopfion interacts with an asymmetric array of planar defects, a ratchet effect consisting of a net dc motion can be realized under purely ac driving.