Complex free-space magnetic field textures induced by three-dimensional magnetic nanostructures

The design of complex, competing effects in magnetic systems-be it via the introduction of nonlinear interactions^1-4, or the patterning of three-dimensional geometries^5,6-is an emerging route to achieve new functionalities. In particular, through the design of three-dimensional geometries and curvature, intrastructure properties such as anisotropy and chirality, both geometry-induced and intrinsic, can be directly controlled, leading to a host of new physics and functionalities, such as three-dimensional chiral spin states⁷, ultrafast chiral domain wall dynamics^8-10 and spin textures with new spin topologies^7,11. Here, we advance beyond the control of intrastructure properties in three dimensions and tailor the magnetostatic coupling of neighbouring magnetic structures, an interstructure property that allows us to generate complex textures in the magnetic stray field. For this, we harness direct write nanofabrication techniques, creating intertwined nanomagnetic cobalt double helices, where curvature, torsion, chirality and magnetic coupling are jointly exploited. By reconstructing the three-dimensional vectorial magnetic state of the double helices with soft-X-ray magnetic laminography^12,13, we identify the presence of a regular array of highly coupled locked domain wall pairs in neighbouring helices. Micromagnetic simulations reveal that the magnetization configuration leads to the formation of an array of complex textures in the magnetic induction, consisting of vortices in the magnetization and antivortices in free space, which together form an effective B field cross-tie wall¹⁴. The design and creation of complex three-dimensional magnetic field nanotextures opens new possibilities for smart materials¹⁵, unconventional computing^2,16, particle trapping^17,18 and magnetic imaging¹⁹.