Stability of Hopfions in Bulk Magnets with Competing Exchange Interactions

Magnetic hopfions are string-like three-dimensional topological solitons, characterised by the Hopf invariant. They serve as a fundamental prototype for three-dimensional magnetic quasi-particles and are an inspiration for novel device concepts in the field of spintronics. Based on a micromagnetic model and without considering temperature, the existence of such hopfions has been predicted in certain magnets with competing exchange interactions. However, physical realisation of freely moving hopfions in bulk magnets have so far been elusive. Here, we consider an effective Heisenberg model with competing exchange interactions and study the stability of small toroidal hopfions with Hopf number $Q_\text{H}=1$ by finding first-order saddle points on the energy surface representing the transition state for the decay of hopfions via the formation of two coupled Bloch points. We combine the geodesic nudged elastic band method and an adapted implementation of the dimer method to resolve the sharp energy profile of the reaction path near the saddle point. Our analysis reveals that the energy barrier can reach substantial height and is largely determined by the size of the hopfion relative to the lattice constant.Comment: 13 pages, 8 figure

Material systems for FM-/AFM-coupled skyrmions in Co/Pt-based multilayers

By means of systematic first-principles calculations based on density functional theory we search for suitable materials that can host antiferromagnetically coupled skyrmions. We concentrate on fcc-stacked (111)-oriented metallic $Z$/Co/Pt ($Z=4d$ series: Y$-$Pd, the noble metals: Cu, Ag, Au, post noble metals: Zn and Cd) magnetic multilayers of films of monatomic thickness. We present quantitative trends of magnetic properties: Magnetic moments, interlayer exchange coupling, spin-stiffness, Dzyaloshinskii-Moriya interaction, magnetic anisotropy, and the critical temperature. We show that some of the $Z$ elements (Zn, Y, Zr, Nb, Tc, Ru, Rh, and Cd) can induce antiferromagnetic interlayer coupling between the magnetic Co layers, and that they influence the easy magnetization axis. Employing a multiscale approach, we transfer the micromagnetic parameters determined from $ab$ $initio$ to a micromagnetic energy functional and search for one-dimensional spin-spiral solutions and two-dimensional skyrmions. We determine the skyrmion radius by numerically solving the equation of the skyrmion profile. We found an analytical expression for the skyrmion radius that covers our numerical results and is valid for a large regime of micromagnetic parameters. Based on this expression we have proposed a model that allows to extrapolate from the $ab$ $initio$ results of monatomic films to multilayers with Co films consisting of several atomic layers containing $10\,$nm skyrmions. We found thickness regimes where tiny changes of the film thickness may alter the skyrmion radius by orders of magnitude. We estimated the skyrmion size as function of temperature and found that the size can easily double going from cryogenic to room temperature. We suggest promising material systems for ferromagnetically and antiferromagnetically coupled spin-spiral and skyrmion systems.Comment: 18 pages, 7 figure

Spirit: Multifunctional Framework for Atomistic Spin Simulations

The \textit{Spirit} framework is designed for atomic scale spin simulations of magnetic systems of arbitrary geometry and magnetic structure, providing a graphical user interface with powerful visualizations and an easy to use scripting interface. An extended Heisenberg type spin-lattice Hamiltonian including competing exchange interactions between neighbors at arbitrary distance, higher-order exchange, Dzyaloshinskii-Moriya and dipole-dipole interactions is used to describe the energetics of a system of classical spins localised at atom positions. A variety of common simulations methods are implemented including Monte Carlo and various time evolution algorithms based on the Landau-Lifshitz-Gilbert equation of motion, which can be used to determine static ground state and metastable spin configurations, sample equilibrium and finite temperature thermodynamical properties of magnetic materials and nanostructures or calculate dynamical trajectories including spin torques induced by stochastic temperature or electric current. Methods for finding the mechanism and rate of thermally assisted transitions include the geodesic nudged elastic band method, which can be applied when both initial and final states are specified, and the minimum mode following method when only the initial state is given. The lifetime of magnetic states and rate of transitions can be evaluated within the harmonic approximation of transition-state theory. The framework offers performant CPU and GPU parallelizations. All methods are verified and applications to several systems, such as vortices, domain walls, skyrmions and bobbers are described

A spin model for intrinsic antiferromagnetic skyrmions on a triangular lattice

Skyrmions are prospected as the potential future of data storage due to their topologically protected spin structures. However, traditional ferromagnetic (FM) skyrmions experience deflection when driven with an electric current, hindering their usage in spintronics. Antiferromagnetic (AFM) skyrmions, consisting of two FM solitons coupled antiferromagnetically, are predicted to have a zero Magnus force, making them promising candidates for spintronic racetrack memories. Currently, they have been stabilized in synthetic AFM structures, i.e. multilayers hosting FM skyrmions, which couple antiferromagnetically through a non-magnetic spacer, while recent first-principles simulations predict their emergence in an intrinsic form, within an row-wise AFM single monolayer of Cr deposited on PdFe bilayer grown on Ir(111) surfaces. The latter material forms a triangular lattice, where single and interlinked AFM skyrmions can be stabilized. Here, we explore the minimal Heisenberg model enabling the occurrence of such AFM solitons and the underlying phase diagrams by accounting for the interplay between the Dzyaloshinskii-Moriya and Heisenberg exchange interactions, as well as the magnetic anisotropy and impact of magnetic field. By providing the fundamental basis to identify and understand the behavior of intrinsic AFM skyrmions, we anticipate our model to become a powerful tool for exploring and designing new topological magnetic materials to conceptualize devices for AFM spintronics

Intrinsic Néel Antiferromagnetic Multimeronic Spin Textures in Ultrathin Films

Topological antiferromagnetism is a vibrant and captivating research field, generating considerable enthusiasm with the aim of identifying topologically protected magnetic states of key importance in the hybrid realm of topology, magnetism, and spintronics. While topological antiferromagnetic (AFM) solitons bear various advantages with respect to their ferromagnetic cousins, their observation is scarce. Utilizing first-principles simulations, here we predict new chiral particles in the realm of AFM topological magnetism, exchange-frustrated multimeronic spin textures hosted by a Néel magnetic state, arising universally in single Mn layers directly grown on an Ir(111) surface or interfaced with Pd-based films. These nanoscale topological structures are intrinsic; i.e. they form in a single AFM material, can carry distinct topological charges, and can combine in various multimeronic sequences with enhanced stability against external magnetic fields. We envision the frustrated Néel AFM multimerons as exciting highly sought after AFM solitons having the potential to be utilized in novel spintronic devices hinging on nonsynthetic AFM quantum materials, further advancing the frontiers of nanotechnology and nanophysic

A spin model for intrinsic antiferromagnetic skyrmions on a triangular lattice

Skyrmions are prospected as the potential future of data storage due to their topologically protected spin structures. However, traditional ferromagnetic (FM) skyrmions experience deflection when driven with an electric current, hindering their usage in spintronics. Antiferromagnetic (AFM) skyrmions, consisting of two FM solitons coupled antiferromagnetically, are predicted to have a zero Magnus force, making them promising candidates for spintronic racetrack memories. Currently, they have been stabilized in synthetic AFM structures, i.e. multilayers hosting FM skyrmions, which couple antiferromagnetically through a non-magnetic spacer, while recent first-principles simulations predict their emergence in an intrinsic form, within an row-wise AFM single monolayer of Cr deposited on PdFe bilayer grown on Ir(111) surfaces. The latter material forms a triangular lattice, where single and interlinked AFM skyrmions can be stabilized. Here, we explore the minimal Heisenberg model enabling the occurrence of such AFM solitons and the underlying phase diagrams by accounting for the interplay between the Dzyaloshinskii-Moriya and Heisenberg exchange interactions, as well as the magnetic anisotropy and impact of magnetic field. By providing the fundamental basis to identify and understand the behavior of intrinsic AFM skyrmions, we anticipate our model to become a powerful tool for exploring and designing new topological magnetic materials to conceptualize devices for AFM spintronics.Comment: 21 pages, 3 figure