The method based on umbilics that expose line-like organization of complex director fields is used to introduce umbilic surfaces as a numerically robust probe of three-dimensional (3D) topological solitons in frustrated cholesteric liquid crystals. We present a coordinate-free analytical formulation of the umbilic-line approach that ensures reliable detection of umbilics on discrete simulation grids and thus avoids the problems caused by instabilities and sensitivity to coordinate choices. By using our method we introduce the laboratory-referenced phase field giving a natural tool for intuitive surface colorization. In addition, we employ this field to define the two fundamental integer invariants of umbilic loops: the transverse index (the strength) and the longitudinal winding (the profile twist). These invariants directly link the umbilic geometry to the topological characteristics of textures, thus enabling soliton identification and a comparison of solitons by topological content. We apply the technique to the three canonical solitons obtained by the free-energy minimization: the toron and the looped cholesteric fingers of the first and second types with the Hopf indices equal to zero and unity, respectively. It is found that the umbilic-surface representation clearly exposes defect structures, discriminates between visually similar but topologically distinct textures and provides a tool for quantifying and visualizing 3D solitons from director field data.

We show that optimal control can turn the skyrmion Hall effect from a source of typically unwanted transverse motion into a mechanism for reducing Ohmic losses in skyrmion transport. We consider a pair of antiferromagnetically coupled skyrmions confined to a tubular geometry, where their relative transverse displacement becomes a periodic internal coordinate. The skyrmion Hall effect drives this coordinate, while finite interlayer coupling makes the resulting relation between applied current and longitudinal velocity nonlinear, allowing different current protocols to produce the same prescribed average velocity. We determine which of these protocols minimizes Joule heating. Depending on the interlayer coupling and spin-transfer-torque parameters, the optimal current is either constant, with the skyrmion pair maintaining a fixed relative position, or time dependent, with the pair undergoing periodic relative motion around the tube. In either case, the additional internal degree of freedom created by the skyrmion Hall effect and interlayer coupling enables skyrmion transport at a lower power than in the uncoupled limit.

We investigate the internal structure and dynamics of transverse domain walls in amorphous, stressed ferromagnetic microwires by comparing two magnetoelastic anisotropy models. In the complete model, all three principal stress components (axial, radial, circumferential) extracted from a realistic stress profile are converted into spatially varying anisotropies; in the reduced model, only the dominant stress component in each radial region is retained. Micromagnetic simulations reveal that the reduced model produces exaggerated peripheral deviations-stronger radial magnetization projections and deeper penetration of the disturbed layer-compared to the complete model. Energy analysis show that omitting non-dominant anisotropy leads to underestimation of domain wall-defect interactions and a sharp, shell-like radial ordering at higher values of surface anisotropy. Furthermore, dissipation calculations based on the Thiele approach indicate that the reduced model overestimates domain wall velocity by up to 50%. These results demonstrate that incorporating the full stress tensor is essential for accurate prediction of both static domain wall profiles and their dynamic response in stressed microwires.

We present an optimal field-free protocol for current-induced switching of a perpendicularly magnetized ferromagnetic insulator nanoelement on the surface of a topological insulator. The time dependence of in-plane components of the surface current, which drives the magnetization reversal via the Dirac spin-orbit torque with minimal Joule heating, is derived analytically as a function of the switching time and material properties. Our analysis identifies that energy-efficient switching is achieved for vanishing damping-like torque. The optimal reversal time that balances switching speed and energy efficiency is determined. When we compare topological insulators to heavy-metal systems, we find similar switching costs for the optimal ratio between the spin-orbit torque coefficients. However, topological insulators offer the advantage of tunable material properties. Finally, we propose a robust and efficient simplified switching protocol using a down-chirped rotating current pulse, tailored to realistic ferromagnetic/topological insulator systems.

The structure of domain walls in cylindrical nano-and microwires with a non-uniform anisotropy distribution in the transverse-radial direction has been studied.This distribution can be controlled by mechanical stresses associated with specific wire manufacturing methods as well as with the glass coating in some types of microwires.Our calculations have shown that in the presence of axial anisotropy in the core of the wire and radial anisotropy near its surface, various configurations of domain walls can be stabilized.A diagram of magnetic states has been calculated depending on the radial anisotropy values.The stability of various types of domain walls and their possible transformation under the excitation of thermal fluctuations and external perturbations are discussed.

A BSTRACT For magnetic wires and other systems with cylindrical symmetry, algorithms have been proposed for constructing a multidimensional energy surface, searching for minimal energy paths between locally stable states and the activation energies of transitions between such states. The mechanisms of nucleation and transformation of domain walls of various types in amorphous ferromagnetic nanowires with a nonuniform anisotropy distribution have been studied. The stability of the domain walls structure with respect to thermal fluctuations and random external perturbations has been assessed.

A generalized Thiele equation, which includes a given set of low-energy excitations of an equilibrium magnetic structure, has been derived to describe the dynamics of chiral topological systems. A “breathing” mode of magnetic skyrmions corresponding to a change in their size is included. The relaxation of the magnetic structure under the variation of a magnetic field has been studied. The importance of keeping the Hamiltonian form of the equations of motion has been demonstrated. The radius and helicity of a skyrmion are canonically conjugate variables, and only their simultaneous inclusion allows the reproduction of the main features of the magnetic relaxation after the magnetic field is switched-on/off, which is accompanied by oscillations of the radius.

The dependence of the lifetimes and rates of spontaneous nucleation of topological magnetic soli -tons on the external magnetic field is calculated within the framework of the harmonic transition state theory for magnetic degrees of freedom. For two-dimensional magnetic skyrmions, the influence of the magnetic field on the collapse rate was found to be greater than on the nucleation rate. This is explained by the weaker dependence of the energy of the transition state on the external field compared to the energy of the metastable skyrmion. The balance of the nucleation and collapse of skyrmion rates makes it possible to determine the average equilibrium concentration of skyrmions in a thin film as a function of the external field and temperature. It is shown that skyrmion and antiskyrmion states can exist simultaneously in quasi-two-dimensional thin films in tilted external magnetic field. The minimum energy paths for the collapse of these topological solitons and magnetic configurations in the vicinity of saddle point have been found and compared.

The lifetimes of magnetic hopfions on a discrete lattice with competing exchange interactions are calculated within the framework of the transition state theory for magnetic degrees of freedom. Three sets of discrete model parameters corresponding to the same continuous micromagnetic model are considered. Minimal energy paths for hopfion collapses were found on the multidimensional energy surface of the system. The activation energies of the collapse processes have been calculated. It turned out that the activation energy differs significantly for the three considered values of the parameters, which indicates the importance of lattice effects, when the hopfion radius equals several lattice constants. Along with the collapse, the hopfion escape process through the sample boundary is studied. It is shown that this process does not require an activation energy. The lifetimes of hopfions are found and it is shown that they can exist only at temperatures of a few kelvins and practically cannot be generated due to thermal fluctuations.

EDITORIAL article Front. Phys., 23 August 2023Sec. Condensed Matter Physics Volume 11 - 2023 | https://doi.org/10.3389/fphy.2023.1275990

We consider a twisted magnetic bilayer subject to the perpendicular electric field. The interplay of induced Dzyaloshinskii - Moriya interaction and spatially varying moir\'e exchange potential results in complex non-collinear magnetic phases in these structures. We numerically demonstrate the coexistence of intralayer skyrmions and bound interlayer skyrmion pairs and show that they are characterized by distinct dynamics under the action of external in-plane electric field. Specifically we demonstrate the railing behaviour of skyrmions along the domain walls which could find applications in spintronic devices based on van der Waals magnets.

The lifetime of a magnetic skyrmion in a two-dimensional lattice is calculated as a function of size, from nanometer scale to microns, while the shape of the skyrmion remains the same. Values of the parameters in the extended Heisenberg Hamiltonian that includes exchange, anisotropy, Dzyaloshinskii-Moriya interaction, and the effect of an external field, are scaled in accordance with a continuous micromagnetic description. The lifetime increases dramatically with increased skyrmion size both because of a decrease in the entropy of the transition state with respect to that of the skyrmion state, resulting in an increase in the pre-exponential factor by four orders of magnitude, and because the activation energy increases towards an upper bound consistent with the energy of the Belavin-Polyakov soliton. While the calculated lifetime of the skyrmion for the Hamiltonian parameter values chosen here is less than a microsecond when the radius is 5 lattice constants, it reaches years when the radius is 100 lattice constants at a temperature of 200 K and an hour at room temperature. The calculations of the largest skyrmion studied here explicitly include over 20 000 000 spins.

Cubic chiral magnets exhibit a remarkable diversity of two-dimensional topological magnetic textures, including skyrmions. However, the experimental confirmation of topological states localized in all three spatial dimensions remains challenging. In this paper, we investigate a three-dimensional topological state called a heliknoton, which is a hopfion embedded into a helix or conic background. We explore the range of parameters at which the heliknoton can be stabilized under realistic conditions using micromagnetic modeling, harmonic transition state theory, and stochastic spin dynamics simulations. We present theoretical Lorentz TEM images of the heliknoton, which can be used for experimental comparison. Additionally, we discuss the stability of the heliknoton at finite temperatures and the mechanism of its collapse. Our study offers a pathway for future experimental investigations of three-dimensional topological solitons in magnetic crystals.

We study pairwise interactions between localized topological structures in chiral magnetic and cholesteric liquid crystal (CLC) systems confined in the planar geometry. Our calculations for magnetics are based on the lattice model that takes into account the bulk and surface anisotropies along with the exchange and the Dzyaloshinskii-Moriya interactions. In CLC cells, these anisotropies describe the energy of interaction with an external magnetic or electric field and the anchoring energy assuming that the magnetic or electric anisotropy is negative and the boundary conditions are homeotropic. We have selected the region of the phase diagram, where various localized solitonlike structures, including skyrmion tubes, torons, and leeches, embedded in the ground state of the z-cone (conical phase) coexist, and carried out numerical analysis of the distance dependencies of the effective intersoliton interaction potentials. For skyrmions and torons, the potentials are found to be attractive in the large separation region. It turned out that for these potentials, the effects of axial asymmetry are negligible. By contrast, it turned out that for the intermediate structures between the skyrmions and torons known as the leeches, the leech-leech potentials generally depend on the orientation of the intersoliton separation vector and their large distance parts may become repulsive at certain directions of the vector. All the potentials have the short distance repulsive parts and the local minima located at the equilibrium separations. It is found that the skyrmion-skyrmion potential has an additional metastable configuration shifted towards the short-distance region.

Topological protection of chiral magnetic structures is investigated by taking a two-dimensional magnetic skyrmion as an example. The skyrmion lifetime is calculated based on harmonic transition state theory for a discrete lattice model using various values of the ratio of the lattice constant and the skyrmion size. Parameters of the system corresponding to exchange, anisotropy and Dzyaloshinskii–Moriya interaction are chosen in such a way as to keep the energy and size of the skyrmion unchanged for small values of the lattice constant, using scaling relations derived from continuous micromagnetic description. The number of magnetic moments included in the calculations reaches more than a million. The results indicate that in the limit of infinitesimal lattice constant, the energy barrier for skyrmion collapse approaches the Belavin–Polyakov lower bound of the energy of a topological soliton in the σ-model, the entropy contribution to the pre-exponential factor in the Arrhenius rate expression for collapse approaches a constant and the skyrmion lifetime can, for large enough number of spins, correspond to thermally stable skyrmion at room temperature even without magnetic dipole–dipole interaction.

The mechanisms of the collapse of skyrmion structures in synthetic antiferromagnets and the activation energy of such processes are studied within the transition state theory based on the analysis of the multidimensional energy surface of the system and the construction of minimum energy paths between the corresponding states. Synthetic antiferromagnets consist of two thin ferromagnetic films separated by a nonmagnetic metal spacer, the conduction electrons of which provide antiferromagnetic interlayer exchange interaction. A discrete Heisenberg-type model is used, which includes symmetric and antisymmetric exchange in each layer, interaction with the applied magnetic field, and the aforementioned interlayer exchange interaction. The experimentally observed magnetic structures are reproduced. It is shown that the most probable mechanism for the collapse of skyrmion pairs involves an asymmetric state with a skyrmion in one layer. The activation energy for such a process is calculated. It is 16% lower than the numerical estimates based on the micromagnetic ansatz, but is a factor of 1.4 higher than that corresponding to the annihilation of a skyrmion of the same size in one layer.

We combine numerical modeling and analytical design techniques to study several of the most common localized topological structures in frustrated chiral nematic liquid crystal cells. An energy minimization procedure is applied to the lattice model to simulate the director field distributions. These distributions are also approximated using the suitably designed analytical ansatz. We present both simulated and approximated results for optical polarizing microscopy textures and different visualizations of director field structure such as distributions of the azimuthal director angle and isolines for the normal component of the director in coordinate planes. The ansatz correctly mimicked the geometry and optical properties of the solitonic structures under consideration.

It is demonstrated by means of the optimal control theory that the energy cost of the spin-orbit torque induced reversal of a nanomagnet with perpendicular anisotropy can be strongly reduced by proper shaping of both in-plane components of the current pulse. The time-dependence of the optimal switching pulse that minimizes the energy cost associated with Joule heating is derived analytically in terms of the required reversal time and material properties. The optimal reversal time providing a tradeoff between the switching speed and energy efficiency is obtained. A sweet-spot balance between the field-like and damping-like components of the spin-orbit torque is discovered; it permits for a particularly efficient switching by a down-chirped rotating current pulse whose duration does not need to be adjusted precisely.

We theoretically study orientational structures in chiral magnetics and cholesteric liquid crystal (CLC) nanosystems confined in the slab geometry. Our analysis is based on the model that, in addition to the exchange and the Dzyaloshinskii-Moriya interactions, takes into account the bulk and surface anisotropies. In CLC films, these anisotropies describe the energy of interaction with external magnetic/electric field and the anchoring energy assuming that magnetic/electric anisotropy is negative and the boundary conditions are homeotropic. We have computed the phase diagram and found that the ground state of the film is represented by various delocalized structures depending on the bulk and surface anisotropy parameters, κ^{b} and κ^{s}. These include the z helix and the z cone states, the oblique, and the x helicoids. The minimum energy paths connecting the ground state and metastable helicoids and the energy barriers separating these states are evaluated. We have shown that there is a variety of localized topological structures such as the skyrmion tube, the toron, and the bobber that can be embedded in different ground states including the z cone (conical phase) and tilted fingerprint states. We have also found the structure called the leech that can be viewed as an intermediate state between the toron and the skyrmion tube.