Hopfion rings in a cubic chiral magnet

Abstract

Magnetic skyrmions and hopfions are topological solitons1, well-localized field configurations that have gained significant attention over the past decade due to their unique particle-like properties, making them promising objects for spintronic applications. Skyrmions2,3 are two-dimensional solitons resembling vortex-like string structures that can penetrate an entire sample. Hopfions4-9 are three-dimensional solitons confined within a magnetic sample volume and can be considered as closed twisted skyrmion strings that take the shape of a ring in the simplest case. Despite extensive research on magnetic skyrmions, the direct observation of magnetic hopfions has remained challenging. Here, we present the first direct observations of coupled states of hopfions and skyrmion strings in FeGe plates using transmission electron microscopy. Furthermore, we provide a protocol for nucleating hopfion rings around skyrmion strings in confined samples, which we verify using Lorentz imaging and electron holography. Our results are highly reproducible, in full agreement with micromagnetic simulations, and offer new insight into the diversity of topological solitons in three-dimensional chiral magnets and their potential for practical applications.