Creation and observation of Hopfions in magnetic multilayer systems

Abstract

Among topological solitons, magnetic skyrmions are two-dimensional particle-like objects with a continuous winding of the magnetization, and magnetic Hopfions are three-dimensional objects that can be formed from a closed loop of twisted skyrmion strings. Theoretical models suggest that magnetic Hopfions can be stabilized in frustrated or chiral magnetic systems, and target skymions can be transformed into Hopfions by adapting their perpendicular magnetic anisotropy, but their experimental verification has been elusive so far. Here, we present an experimental study of magnetic Hopfions that are created in Ir/Co/Pt multilayers shaped into nanoscale disks, known to host target skyrmions. To characterize three-dimensional spin textures that distinguish Hopfions from target skyrmions magnetic images are recorded with surface-sensitive X-ray photoemission electron microscopy and bulk-sensitive soft X-ray transmission microscopy using element-specific X-ray magnetic circular dichroism effects as magnetic contrast. These results could stimulate further investigations of Hopfions and their potential application in three-dimensional spintronics devices.

Exploring instantons in nonlinear sigma models with spin-lattice systems

“…In principle, these can now include ring-and knotlike objects [27][28][29][30], so-called Hopfions. Similar to skyrmions, the latter have also been proposed to exist as the lowest energy eigenstates of realistic condensed matter systems [31][32][33][34]. In our example, however, as we will argue, these structures are not stabilized in a chiral magnet in (1 + 2) dimensions.…”

“…Stable Hopfion configurations have first been proposed in the Skyrme-Faddeev model [27,28] and were later discussed in two-condensate superconducting systems [47,48]. Furthermore, they have also been studied as static minimal-energy solutions in threedimensional chiral magnets [31][32][33][34]. In our example, a truly topological instanton would therefore need to carry a nonvanishing Hopf charge.…”

Magnetic skyrmion braids

“…In contrast to magnetic hopfions 38 – 40 , skyrmion braids cannot be localized completely in three dimensions without singularities. In other words, skyrmion strings start on one surface of a sample and end on the other surface.…”

The AiiDA-Spirit Plugin for Automated Spin-Dynamics Simulations and Multi-Scale Modeling Based on First-Principles Calculations

“…Spin-dynamics calculations based on the Landau-Lifshitz-Gilbert (LLG) equation (Landau and Lifshitz, 1935;Gilbert, 2004) are a widely used tool for this multi-scale modeling of magnetic materials (Dupé et al, 2014;Hoffmann et al, 2017;Hoffmann et al, 2021;Weißenhofer et al, 2021), providing access to the collective behavior of millions of spins (Müller et al, 2019). This approach allows to find, for instance, the (non-collinear) magnetic ground state based on an energy minimization of the extended Heisenberg Hamiltonian or to study the dynamics of magnetic solitons such as skyrmions (Mu€hlbauer et al, 2009;Yu et al, 2010;Heinze et al, 2011;Back et al, 2020) or hopfions (Bogolubsky, 1988;Sutcliffe, 2018;Kent et al, 2021) at finite temperature. In combination with the geodesic nudged elastic band method and the harmonic transition state theory (Bessarab et al, 2012;Bessarab et al, 2015) it furthermore gives insight into the stability of aforementioned objects (Müller et al, 2019).…”