Fig. 5: Experimental investigation of magnetic wireframe geometries with high-order vorticity.

a Side-view scanning electron microscopy (SEM) image of a magnetic semi-pyramid and corresponding XMCD-PEEM shadow contrasts taken at different azimuthal angles of (b) ϕ = 0^∘ and (c) ϕ = − 25^∘. Homogeneously magnetized line segments provide a monochrome shadow contrast being blue for the magnetization pointing parallel to the beam and red when it is antiparallel. These segments have diameter of around 130 nm and obtain homogeneous magnetic state. The formation of dipolar contrast in a line segments indicate on the appearance of vortex-type magnetization rotation, which is typical for the nanowires with a diameter of around 180 nm used for the wireframe formation. d Schematics of the magnetization distribution in the semi-pyramid derived from the set of XMCD-PEEM images. Black arrows indicate the magnetization direction in each line segment of the wireframe geometry. This object is homeomorphic to a 2-torus. Hence, it accommodates 2 surface antivortices. e Side-view SEM image of a magnetic diamond wireframe geometry and corresponding XMCD-PEEM shadow contrasts taken at different azimuthal angles of (f) ϕ = 0^∘ and (g) ϕ = − 25^∘. h Schematics of the magnetization distribution in the diamond wireframe derived from the set of XMCD-PEEM images. This object contains two bulk antivortices at the top and bottom vertices of the diamond. These two bulk antivortices correspond to 4 surface antivortices, which is in line with the Euler characteristic of this geometry, χ_3−torus = − 4. i Side-view SEM image of a magnetic cube wireframe geometry and corresponding XMCD-PEEM shadow contrasts taken at different azimuthal angles of (j) ϕ = 0^∘ and (k) ϕ = − 25^∘. l Schematics of the magnetization distribution in the cube wireframe derived from the set of XMCD-PEEM images. This object accommodates 8 surface antivortices, which agrees with its Euler characteristics, χ_5−torus = − 8. Each of the magnetic wireframes shown in this figure is based on Co₃Fe alloy written by FEBID. To enable full magnetic characterization using XMCD-PEEM, the objects are fabricated on a non-magnetic PtC pillar, which is also prepared by FEBID.