Micromagnetic calculations are compared with faster model calculations of interacting nanoscopic magnetic islands representing an element of a shakti spin ice lattice. Several pathways for transitions between equivalent ground states are studied. The model calculations describe the interaction between the islands either with the point dipole approximation, or with a dumbbell approximation where the distance between the two poles is optimized to match the micromagnetic results. The closest agreement in the energy of both local minima as well as transition state configurations where one macrospin has rotated by 90° is obtained with a dumbbell model where the distance between the poles is ca. 20 % smaller than the island length.

Theoretical calculations of optimal control paths minimizing the energy cost of the magnetization reversal in 1D magnetic nanowires are presented.The energy-efficient reversal mechanism is studied as a function of the nanowire length and Gilbert damping parameter. For short nanowires, theoptimal reversal mechanism corresponds to a uniform rotation of magnetization. If the length of the wire exceeds a certain critical length definedby the material parameters, switching time and damping, a standing spin wave emerges during magnetization switching. Comparison between thecalculated optimal control paths and minimum energy paths reveals that realization of high energy efficiency of switching does not necessarilytranslate to the minimization of the energy barrier between the target magnetic states.

Energy-efficient switching of nanoscale magnets requires application of a time-varying magnetic field characterized by microwave frequency. At finite temperatures, even weak thermal fluctuations induce perturbations in the magnetization that can accumulate in time, disrupt the phase locking between the magnetization and the applied field, and eventually compromise magnetization switching. It is demonstrated here that the magnetization reversal is mostly disturbed by unstable perturbations arising in a certain domain of the configuration space of a nanomagnet. The instabilities can be suppressed and the probability of magnetization switching enhanced by applying an additional stimulus such as a weak longitudinal magnetic field that ensures bounded dynamics of the perturbations. Application of the stabilizing longitudinal field to a uniaxial nanomagnet makes it possible to reach a desired probability of magnetization switching even at elevated temperatures. The principle of suppressing instabilities provides a general approach to enhancing thermal stability of magnetization dynamics.

A solution to energy-efficient magnetization switching in a nanoparticle with biaxial anisotropy is presented. Optimal control paths minimizing the energy cost of magnetization reversal are calculated numerically as functions of the switching time and materials properties, and used to derive energy-efficient switching pulses of external magnetic field. Hard-axis anisotropy reduces the minimum energy cost of magnetization switching due to the internal torque in the desired switching direction. Analytical estimates quantifying this effect are obtained based on the perturbation theory. The optimal switching time providing a tradeoff between fast switching and energy efficiency is obtained. The energy cost of switching and the energy barrier between the stable states can be controlled independently in a biaxial nanomagnet. This provides a solution to the dilemma between energy-efficient writability and good thermal stability of magnetic memory elements.

Recent experimental data demonstrate emerging magnetic order in platinum atomically thin nanowires. Furthermore, an unusual form of magnetic anisotropy-colossal magnetic anisotropy (CMA)-was earlier predicted to exist in atomically thin platinum nanowires. Using spin dynamics simulations based on first-principles calculations, we here explore the spin dynamics of atomically thin platinum wires to reveal the spin relaxation signature of colossal magnetic anisotropy, comparing it with other types of anisotropy such as uniaxial magnetic anisotropy (UMA). We find that the CMA alters the spin relaxation process distinctly and, most importantly, causes a large speed-up of the magnetic relaxation compared to uniaxial magnetic anisotropy. The magnetic behavior of the nanowire exhibiting CMA should be possible to identify experimentally at the nanosecond time scale for temperatures below 5 K. This time-scale is accessible in e.g., soft x-ray free electron laser experiments.

The skyrmion racetrack is a promising concept for future information technology. There, binary bits are carried by nanoscale spin swirls-skyrmions-driven along magnetic strips. Stability of the skyrmions is a critical issue for realising this technology. Here we demonstrate that the racetrack skyrmion lifetime can be calculated from first principles as a function of temperature, magnetic field and track width. Our method combines harmonic transition state theory extended to include Goldstone modes, with an atomistic spin Hamiltonian parametrized from density functional theory calculations. We demonstrate that two annihilation mechanisms contribute to the skyrmion stability: At low external magnetic field, escape through the track boundary prevails, but a crossover field exists, above which the collapse in the interior becomes dominant. Considering a Pd/Fe bilayer on an Ir(111) substrate as a well-established model system, the calculated skyrmion lifetime is found to be consistent with reported experimental measurements. Our simulations also show that the Arrhenius pre-exponential factor of escape depends only weakly on the external magnetic field, whereas the pre-exponential factor for collapse is strongly field dependent. Our results open the door for predictive simulations, free from empirical parameters, to aid the design of skyrmion-based information technology.

The formation of layer-heterogeneous periodic magnetic states in metallic systems is explained in terms of the dependence of the energy of itinerant electrons on the magnetic ordering of localized spins. The interaction of the local magnetic moments with conduction electrons with a certain spin projection is described by a set of spin-dependent delta-function potentials. A matrix method is developed permitting one to calculate the energy spectrum and the density of states of itinerant electrons in the presence of a layered magnetic heterogeneity. This method is used to explain oscillations of the interlayer exchange interaction in metallic magnetic superlattices and the stabilization of spin-density wave structures in transition metals and alloys.

A rate theory for thermally activated transitions in spin systems is presented. It is based on a transition-state approximation derived from Landau-Lifshitz equations of motion and quadratic expansion of the energy surface at minima and first order saddle points. While the flux out of the initial state vanishes at first order saddle points, the integrated flux over the hyperplanar transition state is nonzero and gives a rate estimate in good agreement with direct dynamical simulations of test systems over a range in damping constant. The preexponential factor obtained for transitions in model systems representing nanoclusters with 3 to 139 transition metal adatoms is on the order of 1011 to 1013s -1, similar to that of atomic rearrangements.

Calculations of stable and metastable magnetic states as well as minimum energy paths for transitions between states are carried out using a noncollinear extension of the multiple-impurity Alexander-Anderson model and a magnetic force theorem which is derived and used to evaluate the total energy gradient with respect to orientation of magnetic moments-an important tool for efficient navigation on the energy surface. By using this force theorem, the search for stable and metastable magnetic states as well as minimum energy paths revealing the mechanism and activation energy of transitions can be carried out efficiently. For Fe monolayer on W(110) surface, the model gives magnetic moment as well as exchange coupling between nearest and next-nearest neighbors that are in good agreement with previous density functional theory calculations. When applied to nanoscale Fe islands on this surface, the magnetic moment is predicted to be 10% larger for atoms at the island rim, explaining in part an experimentally observed trend in the energy barrier for magnetization reversal in small islands. Surprisingly, the magnetic moment of the atoms does not change much along the minimum energy path for the transitions, which for islands containing more than 15 atom rows along either [001] or [1 (1) over bar0] directions involves the formation of a thin, temporary domain wall. A noncollinear magnetic state is identified in a 7 x 7 atomic row Fe island where the magnetic moments are arranged in an antivortex configuration with the central ones pointing out of the (110) plane. This illustrates how the model can describe complicated exchange interactions even though it contains only a few parameters. The minimum energy path between this antivortex state and the collinear ground state is also calculated and the thermal stability of the antivortex state estimated.

The effect of hydrogen adsorption on the magnetic properties of an Fe 3 cluster immersed in a Cu(111) surface has been calculated using density functional theory and the results used to parametrize an Alexander-Anderson model which takes into account the interaction of d electrons with itinerant electrons. A number of adatom configurations containing one to seven H atoms were analyzed. The sequential addition of hydrogen atoms is found to monotonically reduce the total magnetic moment of the cluster with the effect being strongest when the H atoms sit at low-coordinated sites. Decomposition of the charge density indicates a transfer of 0.3 electrons to each of the H atoms from both the Fe atoms and from the copper substrate, irrespective of adsorption site and coverage. The magnetic moment of only the nearest neighbor Fe atoms is reduced, mainly due to increased population of minority spin d states. This can be modeled by increased indirect coupling of d states via the conduction s band in the Alexander-Anderson model. 

A method for finding minimum energy paths of transitions in magnetic systems is presented. The path is optimized with respect to orientation of the magnetic vectors while their magnitudes are fixed or obtained from separate calculations. The curvature of the configuration space is taken into account by: (1) using geodesics to evaluate distances and displacements of the system during the optimization, and (2) projecting the path tangent and the magnetic force on the tangent space of the manifold defined by all possible orientations of the magnetic vectors. The method, named geodesic nudged elastic band (GNEB), and its implementation are illustrated with calculations of complex transitions involving annihilation and creation of skyrmion and antivortex states. The lifetime of the latter was determined within harmonic transition state theory using a noncollinear extension of the Alexander-Anderson model.

The stability of magnetic states and the mechanism for magnetic transitions can be analyzed in terms of the shape of the energy surface, which gives the energy as a function of the angles determining the orientation of the magnetic moments. Minima on the energy surface correspond to stable or metastable magnetic states and can represent parallel, antiparallel or, more generally, non-collinear arrangements. A rate theory has been developed for systems with arbitrary number, N, of magnetic moments, to estimate the thermal stability of magnetic states and the mechanism for magnetic transitions based on a transition state theory approach. The minimum energy path on the 2N-dimensional energy surface is determined to identify the transition mechanism and estimate the activation energy barrier. A pre-exponential factor in the rate expression is obtained from the Landau-Lifshitz-Gilbert equation for spin dynamics. The velocity is zero at saddle points so it is particularly important in this context to realize that the transition state is a dividing surface with 2N -1 degrees of freedom, not just a saddle point. An application of this rate theory to nanoscale Fe islands on W(110) has revealed how the transition mechanism and rate depend on island shape and size. Qualitative agreement is obtained with experimental measurements both for the activation energy and the pre-exponential factor. In particular, a distinct maximum is observed in the pre-exponential factor for islands where two possible transition mechanisms are competing: Uniform rotation and the formation of a temporary domain wall. The entropy of the transition state is enhanced for those islands making the pre-exponential factor more than an order of magnitude larger than for islands were only the uniform rotation is viable.

Theoretical calculations of thermal spin transitions in nanoscale clusters on a surface are presented. The mechanisms for magnetization reversal are identified and the activation energy and pre-exponential factor for the rate are evaluated using a recently developed harmonic transition state theory and a Heisenberg-type Hamiltonian. A maximum is found in the pre-exponential factor as a function of cluster size corresponding to a crossover from a uniform rotation mechanism to temporary domain wall formation. As the islands grow, the energy barrier increases up to a limit where the domain wall is fully established. For larger islands, the minimum energy path becomes flat resulting in a significant recrossing correction to the transition state theory estimate of the rate. The parameters of the Hamiltonian are chosen to mimic Fe clusters on a W(110) surface, a system that has previously been studied extensively in the laboratory and the calculated results are found to be in close agreement with the reported measurements.

The two-dimensional Heisenberg exchange model with out-of-plane anisotropy and a Dzyaloshinskii-Moriya interaction is employed to investigate the lifetime and stability of antiferromagnetic (AFM) skyrmions as a function of temperature and external magnetic field. An isolated AFM skyrmion is metastable at zero temperature in a certain parameter range set by two boundaries separating the skyrmion state from the uniform AFM phase and a stripe domain phase. The distribution of the energy barriers for the AFM skyrmion decay into the uniform AFM state complements the zero-temperature stability diagram and demonstrates that the skyrmion stability region is significantly narrowed at finite temperatures. We show that the AFM skyrmion stability can be enhanced by an application of magnetic field, whose strength is comparable to the spin-flop field. This stabilization of AFM skyrmions in external magnetic fields is in sharp contrast to the behavior of their ferromagnetic counterparts. Furthermore, we demonstrate that the AFM skyrmions are stable on timescales of milliseconds below 50 K for realistic material parameters, making it feasible to observe them in modern experiments.

A coupled atomistic spin and lattice dynamics approach is developed which merges the dynamics of these two degrees of freedom into a single set of coupled equations of motion. The underlying microscopic model comprises local exchange interactions between the electron spin and magnetic moment and the local couplings between the electronic charge and lattice displacements. An effective action for the spin and lattice variables is constructed in which the interactions among the spin and lattice components are determined by the underlying electronic structure. In this way, expressions are obtained for the electronically mediated couplings between the spin and lattice degrees of freedom, besides the well known interatomic force constants and spin-spin interactions. These former susceptibilities provide an atomistic ab initio description for the coupled spin and lattice dynamics. It is important to notice that this theory is strictly bilinear in the spin and lattice variables and provides a minimal model for the coupled dynamics of these subsystems and that the two subsystems are treated on the same footing. Questions concerning time-reversal and inversion symmetry are rigorously addressed and it is shown how these aspects are absorbed in the tensor structure of the interaction fields. By means of these results regarding the spin-lattice coupling, simple explanations of ionic dimerization in double-antiferromagnetic materials, as well as charge density waves induced by a nonuniform spin structure, are given. In the final parts, coupled equations of motion for the combined spin and lattice dynamics are constructed, which subsequently can be reduced to a form which is analogous to the Landau-Lifshitz-Gilbert equations for spin dynamics and a damped driven mechanical oscillator for the ionic motion. It is important to notice, however, that these equations comprise contributions that couple these descriptions into one unified formulation. Finally, Kubo-like expressions for the discussed exchanges in terms of integrals over the electronic structure and, moreover, analogous expressions for the damping within and between the subsystems are provided. The proposed formalism and types of couplings enable a step forward in the microscopic first principles modeling of coupled spin and lattice quantities in a consistent format.

We provide a theoretical background for electric-field-assisted thermally activated writing and deleting of magnetic skyrmions in ultrathin transition-metal films. We apply an atomistic spin model, which includes the exchange interaction, the Dzyaloshinskii-Moriya interaction, and the magnetocrystalline anisotropy energy. The strengths of the magnetic interactions are taken from density functional theory (DFT) calculations for a Pd/Fe bilayer on the Ir(111) surface. We systematically vary all magnetic interactions up to ±10% treating the magnetoelectric effect in linear response. The critical magnetic fields marking the onset of the skyrmion phase and the field-polarized phase shift considerably upon varying the interaction constants due to the electric field. Based on harmonic transition state theory, we calculate the transition rates for skyrmion nucleation and annihilation, which are in good agreement with experimental values for Pd/Fe/Ir(111). The field-dependent variation of energy barriers and preexponential factors leads to large changes of the transition rates, which are accompanied by changes in skyrmion radii. Finally, we simulate the electric-field-dependent writing and deleting of magnetic skyrmions in Pd/Fe/Ir(111) based on the master equation and transition rates obtained using the magnetic interactions calculated via DFT for electric fields of E=±0.5 V/Å. The magnetic-field-dependent skyrmion probability follows a Fermi-Dirac distribution function of the free energy difference of the skyrmion state and the ferromagnetic (FM) state. The probability function for the opposite electric field directions is in striking agreement with experimental results [Romming, Science 341, 636 (2013)0036-807510.1126/science.1240573]. 

Magnetic skyrmions have raised high hopes for future spintronic devices. For many applications, it would be of great advantage to have more than one metastable particle-like texture available. The coexistence of skyrmions and antiskyrmions has been proposed in inversion-symmetric magnets with exchange frustration. However, so far only model systems have been studied and the lifetime of coexisting metastable topological spin structures has not been obtained. Here, we predict that skyrmions and antiskyrmions with diameters below 10 nm can coexist at zero magnetic field in a Rh/Co bilayer on the Ir(111) surface—an experimentally feasible system. We show that the lifetimes of metastable skyrmions and antiskyrmions in the ferromagnetic ground state are above one hour for temperatures up to 75 and 48 K, respectively. The entropic contribution to the nucleation and annihilation rates differs for skyrmions and antiskyrmions. This opens the route to the thermally activated creation of coexisting skyrmions and antiskyrmions in frustrated magnets with Dzyaloshinskii–Moriya interaction.

We show that stable skyrmions with diameters of a few nanometers can emerge in atomic Pd/Fe bilayers on the Rh(111) surface. Based on density functional theory we calculate the exchange and the Dzyaloshinskii-Moriya interaction as well as the magnetocrystalline anisotropy energy. The latter two terms are driven by spin-orbit coupling and significantly reduced compared to Pd/Fe bilayers on Ir(111) as expected since Rh and Ir are isoelectronic 4d and 5d transition metals. However, there is still a spin spiral ground state at zero magnetic field. Atomistic spin dynamics simulations show that a skyrmion phase occurs at small magnetic fields of ∼1 T. Skyrmion diameters amount to 2-8 nm and skyrmion lifetimes are up to 1 h at temperatures of 25-45 K. 

The multidimensional energy surface of a cholesteric liquid crystal in a planar cell is investigated as a function of spherical coordinates determining the director orientation. Minima on the energy surface correspond to the stable states with particular director distribution. External electric and magnetic fields deform the energy surface and positions of minima. It can lead to the transitions between states, known as the Fréedericksz effect. Transitions can be continuous or discontinuous depending on parameters of the liquid crystal which determine an energy surface. In a case of discontinuous transition when a barrier between stable states is comparable with the thermal energy, the activation transitions may occur, and it leads to the modification of characteristics of the Fréedericksz effect with temperature without explicit temperature dependencies of liquid crystal parameters. A minimum energy path between stable states on the energy surface for the Fréedericksz transition is found using the geodesic nudged elastic band method. Knowledge of this path, which has maximal statistical weight among all other paths, gives the information about a barrier between stable states and configuration of director orientation during the transition. It also allows one to estimate the stability of states with respect to the thermal fluctuations and their lifetime when the system is close to the Fréedericksz transition.

An implementation of the non-collinear Alexander-Anderson model for itinerant electrons in magnetic systems is presented where self-consistency is reached for specified directions of the magnetic moments. This is achieved by means of Lagrange multipliers and a variational principle for determining the transverse and longitudinal components of the magnetic moments as well as the average number of d-electrons using direct optimisation. Various optimisation algorithms are compared and the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm is found to give the best performance. An application to antiferromagnetic Cr crystal is presented where spin-dynamics and curvature of the energy surface are calculated to compare results obtained with and without the constraints on the orientation of the magnetic moments. 

In magnetic exchange force microscopy a magnetic tip is scanned over the surface of a solid and an image representing the exchange interaction recorded. Sudden changes in the image corresponding to magnetization switching can be monitored as a function of the tip-surface distance thereby giving important information about the lifetime of metastable magnetic states and how it is affected by the exchange interaction. Here, theoretical calculations are carried out to study the tip-surface interaction and determine the mechanism and rate of transitions in a magnetic exchange force microscopy experiment, and comparison made with reported experimental data on an Fe cluster interacting with an antiferromagnetic Fe overlayer on a W(001) surface. The activation energy was determined from calculations of minimum energy paths and the pre-exponential factor in the Arrhenius rate expression evaluated from harmonic transition state theory, extended to account for zero modes. A noncollinear extension of the Alexander-Anderson model was used to describe the magnetic properties of an atomic scale representation of the system. The calculations reveal how the tip size, tip-surface distance and magnetic field affect the lifetime of the magnetic states.

The potential for manipulating characteristics of skyrmions in a CrI₃ monolayer using circularly polarised light is explored. The effective skyrmion-light interaction is mediated by bright excitons whose magnetization is selectively influenced by the polarization of photons. The light-induced skyrmion dynamics is illustrated by the dependencies of the skyrmion size and the skyrmion lifetime on the intensity and polarization of the incident light pulse. Two-dimensional magnets hosting excitons thus represent a promising platform for the control of topological magnetic structures by light.

Novel 2D material CrI3 reveals unique combination of 2D ferromagnetism and robust excitonic response. We demonstrate that the possibility of the formation of magnetic topological defects, such as Neel skyrmions, together with large excitonic Zeeman splitting, leads to giant scattering asymmetry, which is the necessary prerequisite for the excitonic anomalous Hall effect. In addition, the diamagnetic effect breaks the inversion symmetry, and in certain cases can result in exciton localization on the skyrmion. This enables the formation of magnetoexcitonic quantum dots with tunable parameters.

Controlled creation of localized magnetic textures beyond conventional π-skyrmions is an important problem in the field of magnetism. Here, by means of spin dynamics simulations, Monte Carlo simulations, and harmonic transition state theory we demonstrate that an elementary chiral magnetic soliton with zero topological charge - the chiral droplet - can be created by thermal fluctuations in the presence of the tilted magnetic field. The proposed protocol relies on an unusual kinetics combining the effects of the entropic stabilization and low-energy barrier for the nucleation of a topologically trivial state. Following this protocol by varying temperature and the tilt of the external magnetic field, one can selectively generate chiral droplets or π-skyrmions in a single system. The coexistence of two distinct magnetic solitons establishes a basis for a rich magnetization dynamics and opens up the possibility for the construction of more complex magnetic textures such as skyrmion bags and skyrmions with chiral kinks. 

We report tailed skyrmions—a new class of stable soliton solutions of the 2D chiral magnet model. Tailed skyrmions have elongated shapes and emerge in a narrow range of fields near the transition between the spin spirals and the saturated state. We analyze the stability range of these solutions in terms of external magnetic field and magnetocrystalline anisotropy. We calculate minimum energy paths and homotopies (continuous transitions) between tailed skyrmions of the same topological charge. The discovery of tailed skyrmions extends the diversity of already known solutions. This is illustrated by solitons with complex morphology, such as tailed skyrmion bags with and without chiral kinks.

Bright excitons in ferromagnetic monolayers of CrI₃ efficiently interact with lattice magnetization, which makes all-optical resonant magnetization control possible in this material. Using the combination of ab initio simulations within the Bethe-Salpeter approach, semiconductor Bloch equations, and Landau-Lifshitz equations, we construct a microscopic theory of this effect. By solving numerically the resulting set of coupled equations describing the dynamics of atomic spins and spins of the excitons, we demonstrate the possibility of tunable control of the macroscopic magnetization of a sample.

A complete analytical solution to the optimal reversal of a macrospin with easy-axis anisotropy is presented. An optimal control path minimizing the energy cost of the reversal is identified and used to derive the time-dependent direction and amplitude of the optimal switching field. The minimum energy cost of the reversal scales inversely with the switching time for fast switching, follows exponential asymptotics for slow switching, and reaches the lower limit proportional to the energy barrier between the target states and to the damping parameter at infinitely long switching time. For a given switching time, the energy cost is never smaller than that for a free macrospin. This limitation can be bypassed by adding a hard anisotropy axis that activates the internal torque in the desired switching direction, thereby significantly reducing the energy cost. A comparison between the calculated optimal control path and minimum energy path reveals that optimal control does not translate to the minimization of the energy barrier but signifies effective use of the system's internal dynamics to aid the desired magnetic transition.

The temperature dependence of the response of a magnetic system to an applied field can be understood qualitatively by considering variations in the energy surface characterizing the system and estimated quantitatively with rate theory. In the system analysed here, Fe/Sm-Co spring magnet, the width of the hysteresis loop is reduced to a half when temperature is raised from 25 K to 300 K. This narrowing can be explained and reproduced quantitatively without invoking temperature dependence of model parameters as has typically been done in previous data analysis. The applied magnetic field lowers the energy barrier for reorientation of the magnetization but thermal activation brings the system over the barrier. A 2-dimensional representation of the energy surface is developed and used to gain insight into the transition mechanism and to demonstrate how the applied field alters the transition path. Our results show the importance of explicitly including the effect of thermal activation when interpreting experiments involving the manipulation of magnetic systems at finite temperature.

Magnetic skyrmions are key candidates for applications in memory, logic and neuromorphic computing. An essential property is their topological protection that is caused by the swirling spin texture and described by a robust integer winding number. However, this protection is strictly enforced only in the continuum, and so the atomic lattice present in all real materials leaves a loophole for switching the winding number. Hence, understanding the microscopic mechanism of this unwinding is crucial for enhancing the stability of skyrmions. Here we use spin-polarized scanning tunnelling microscopy to locally probe skyrmion annihilation by individual hot electrons. We tune the collapse rate by up to four orders of magnitude by using an in-plane magnetic field, and observe distinct transition rate maps that either are radial symmetric or exhibit an excentric hotspot. We compare these maps to atomistic spin simulations based on parameters obtained from first-principles calculations and find that the maps are explained by a radial symmetric collapse at zero in-plane magnetic field and a transition to the recently predicted chimera collapse at finite in-plane magnetic fields. These insights into the transient state of the skyrmion collapse will enable future enhancement of skyrmion stability and designs for intentional skyrmion switches.

Various transitions that a magnetic Skyrmion can undergo are found in calculations using a method for climbing up the energy surface and converging onto first order saddle points. In addition to collapse and escape through a boundary, the method identifies a transition where the Skyrmion divides and forms two Skyrmions. The activation energy for this duplication process can be similar to that of collapse and escape. A tilting of the external magnetic field for a certain time interval is found to induce the duplication process in a dynamical simulation. Such a process could turn out to be an important avenue for the creation of Skyrmions in future magnetic devices.

Fundamentals of the method of transverse displacements for calculating the HF radio-wave propagation paths are presented. The method is based on the direct variational principle for theoptical path functional, but is not reduced to solving the Euler— Lagrange equations. Instead,the initial guess given by an ordered set of points is transformed successively into a ray path,while its endpoints corresponding to the positions of the transmitter and the receiver are keptfixed throughout the entire iteration process. The results of calculation by the method of transverse displacements are compared with known analytical solutions. The importance of using onlytransverse displacements of the ray path in the optimization procedure is also demonstrated.

Point-to-point ray tracing is an important problem in many fields of science. While direct variational methods where some trajectory is transformed to an optimal one are routinely used in calculations of pathways of seismic waves, chemical reactions, diffusion processes, etc., this approach is not widely known in ionospheric point-to-point ray tracing. We apply the Nudged Elastic Band (NEB) method to a radio wave propagation problem. In the NEB Method, a chain of points which gives a discrete representation of the radio wave ray is adjusted iteratively to an optimal configuration satisfying the Fermat's principle, while the endpoints of the trajectory are kept fixed according to the boundary conditions. Transverse displacements define the radio ray trajectory, while springs between the points control their distribution along the ray. The method is applied to a study of point-to-point ionospheric ray tracing, where the propagation medium is obtained with the International Reference Ionosphere model taking into account traveling ionospheric disturbances. A 2 dimensional representation of the optical path functional is developed and used to gain insight into the fundamental difference between high and low rays. We conclude that high and low rays are minima and saddle points of the optical path functional, respectively. 

We apply two promising approaches, namely, the shooting method based on global optimization and the generalized force method, to the problem of point-to-point ionospheric ray tracing where the ionospheric modes tend to the minimum frequency. In most cases, both methods are capable of finding all rays connecting the given transmitter and receiver. However, the generalized force approach is more stable in locating highly divergent high rays. Application of the methods to a long-range single-hop radio channel demonstrates their potential for expanding the simulated frequency range, refining the model ionograms in the ranges of the minimum frequency of reflection from ionospheric layers, and significantly improving the overall forecast of the lowest usable frequency. The results of numerical simulations were also compared with the experimental data. Joint application of the presented methods provides valuable means for solving applied problems of radio communication forecast.

A variant of the direct optimization method for point-to-point ionospheric ray tracing is presented. The method is well suited for applications where the launch direction of the radio wave ray is unknown, but the position of the receiver is specified instead. Iterative transformation of a candidate path to the sought-for ray is guided by a generalized force, where the definition of the force depends on the ray type. For high rays, the negative gradient of the optical path functional is used. For low rays, the transformation of the gradient is applied, converting the neighborhood of a saddle point to that of a local minimum. Knowledge about the character of the rays is used to establish a scheme for systematic identification of all relevant rays between the given points, without the need to provide an accurate initial estimate for each solution. Various applications of the method to isotropic ionosphere demonstrate its ability to resolve complex ray configurations including 3-D propagation and multi-path propagation where rays are close in the launch direction. Results of the application of the method to ray tracing between Khabarovsk and Tory show good quantitative agreement with the measured oblique ionograms.

Theoretical calculations of thermally activated decay of skyrmions in systems comprising several magnetic monolayers are presented, with a special focus on bilayer systems. Mechanisms of skyrmion collapse are identified and corresponding energy barriers and thermal collapse rates are evaluated as functions of the interlayer exchange coupling and mutual stacking of the monolayers using transition state theory and an atomistic spin Hamiltonian. In order to contrast the results to monolayer systems, the magnetic interactions within each layer are chosen so as to mimic the well-established Pd/Fe/Ir(111) system. Even bilayer systems demonstrate a rich diversity of skyrmion collapse mechanisms that sometimes coexist. For very weakly coupled layers, the skyrmions in each layer decay successively via radially symmetric shrinking. Slightly larger coupling leads to an asymmetric chimera collapse stabilized by the interlayer exchange. When the interlayer exchange coupling reaches a certain critical value, the skyrmions collapse simultaneously. Interestingly, the overall energy barrier for the skyrmion collapse does not always converge to a multiple of that for a monolayer system in the strongly coupled regime. For a certain stacking of the magnetic layers, the energy barrier as a function of the interlayer exchange coupling features a maximum and then decreases with the coupling strength in the strong coupling regime. Calculated mechanisms of skyrmion collapse are used to ultimately predict the skyrmion lifetime. Our results reveal a comprehensive picture of the thermal stability of skyrmions in magnetic multilayers and provide a perspective for realizing skyrmions with controlled properties. 

We show for a simple noncollinear configuration of the atomistic spins (in particular, where one spin is rotated by a finite angle in a ferromagnetic background) that the pairwise energy variation computed in terms of multiple-scattering formalism cannot be fully mapped onto a bilinear Heisenberg spin model even in the absence of spin-orbit coupling. The non-Heisenberg terms induced by the spin-polarized host appear in leading orders in the expansion of the infinitesimal angle variations. However, an Eg-T2g symmetry analysis based on the orbital decomposition of the exchange parameters in bcc Fe leads to the conclusion that the nearest-neighbor exchange parameters related to the T2g orbitals are essentially Heisenberg-like: they do not depend on the spin configuration, and can, in this case, be mapped onto a Heisenberg spin model even in extreme noncollinear cases.

The stability of skyrmions in various environments is estimated by analyzing the multidimensional surface describing the energy of the system as a function of the directions of the magnetic moments in the system. The energy is given by a Heisenberg-like Hamiltonian including terms representing Dzyaloshinskii-Moriya interaction, anisotropy energy and interaction with an external magnetic field. Local minima on this surface correspond to the ferromagnetic and skyrmion states. Minimum energy paths (MEP) between the minima are calculated using the geodesic nudged elastic band method. The maximum energy along an MEP corresponds to a first order saddle point on the energy surface and gives an estimate of the activation energy for the magnetic transition, such as creation and annihilation of a skyrmion. The pre-exponential factor in the Arrhenius law for the rate, the so-called attempt frequency, is estimated within harmonic transition state theory where the eigenvalues of the Hessian at the saddle point and the local minima are used to characterize the shape of the energy surface. For some degrees of freedom, so-called “zero modes”, the energy of the system remains invariant. They need to be treated separately and give rise to temperature dependence of the attempt frequency. As an example application of this general theory, the lifetime of a skyrmion in a track of finite width for a PdFe overlayer on a Ir(1 1 1) substrate is calculated as a function of track width and external magnetic field. Also, the effect of non-magnetic impurities is studied. Various MEPs for annihilation inside a track, via the boundary of a track and at an impurity are presented. The attempt frequency as well as the activation energy has been calculated for each mechanism to estimate the transition rate as a function of temperature.

The stability of magnetic skyrmions against thermal fluctuations and external perturbations is investigated within the framework of harmonic transition state theory for magnetic degrees of freedom. The influence of confined geometry and atomic scale non-magnetic defects on the skyrmion lifetime is estimated. It is shown that a skyrmion on a track has lower activation energy for annihilation and higher energy for nucleation if the size of the skyrmion is comparable with the width of the track. Two mechanisms of skyrmion annihilation are considered: inside the track and escape through the boundary. For both mechanisms, the dependence of activation energy on the track width is calculated. Non-magnetic defects are found to localize skyrmions in their neighborhood and strongly decrease the activation energy for creation and annihilation. This is in agreement with experimental measurements that have found nucleation of skyrmions in presence of spin-polarized current preferably occurring near structural defects.

Breaking the dilemma between small size and room-temperature stability is a necessary prerequisite for skyrmion-based information technology. Here we demonstrate by means of rate theory and an atomistic spin Hamiltonian that the stability of isolated skyrmions in ultrathin ferromagnetic films can be enhanced by the concerted variation of magnetic interactions while keeping the skyrmion size unchanged. We predict film systems where the lifetime of sub-10 nm skyrmions can reach years at ambient conditions. The long lifetime of such small skyrmions is due to exceptionally large Arrhenius pre-exponential factor and the stabilizing effect of the energy barrier is insignificant at room temperature. A dramatic increase in the pre-exponential factor is achieved thanks to the softening of magnon modes of the skyrmion, thereby increasing the entropy of the skyrmion with respect to the transition state for collapse. Increasing the number of skyrmion deformation modes should be a guiding principle for the realization of nanoscale, room-temperature stable skyrmions.

The relationship between the size and stability of isolated skyrmions in a magnetic monolayer is analyzed based on minimum energy path calculations and atomistic spin Hamiltonian. It is demonstrated that the energy barrier protecting the skyrmion from collapse to the ferromagnetic state is not uniquely defined by the skyrmion size, although these two properties as functions of relevant material parameters follow similar trends. Stability of nanoscale skyrmions can be enhanced by a concerted adjustment of material parameters. The proposed parameter transformation conserves the skyrmion size, but does not conserve the skyrmion shape which changes from an arrow-like pattern to a profile that resembles magnetic bubbles. This transformation of the skyrmion shape is accompanied by an increase in the collapse energy barrier and thus enhancement of skyrmion stability.

Transitions between states of a magnetic system can occur by jumps over an energy barrier or by quantum mechanical tunneling through the energy barrier. The rate of such transitions is an important consideration when the stability of magnetic states is assessed for example for nanoscale candidates for data storage devices. The shift in transition mechanism from jumps to tunneling as the temperature is lowered is analyzed and a general expression derived for the crossover temperature. The jump rate is evaluated using a harmonic approximation to transition state theory. First, the minimum energy path for the transition is found with the geodesic nudged elastic band method. The activation energy for the jumps is obtained from the maximum along the path, a saddle point on the energy surface, and the eigenvalues of the Hessian matrix at that point as well as at the initial state minimum used to estimate the entropic pre-exponential factor. The crossover temperature for quantum mechanical tunneling is evaluated from the second derivatives of the energy with respect to orientation of the spin vector at the saddle point. The resulting expression is applied to test problems where analytical results have previously been derived, namely uniaxial and biaxial spin systems with two-fold anisotropy. The effect of adding four-fold anisotropy on the crossover temperature is demonstrated. Calculations of the jump rate and crossover temperature for tunneling are also made for a molecular magnet containing an Mn-4 group. The results are in excellent agreement with previously reported experimental measurements on this system.

Transitions between magnetic states of a system coupled to a heat bath can occur by exceeding the energy barrier, but as temperature is lowered quantum mechanical tunneling through the barrier becomes the dominant transition mechanism. A method is presented for estimating the onset temperature for tunneling in a system with an arbitrary number of spins using the second derivatives of the energy with respect to the orientation of the magnetic vectors at the first order saddle point on the energy surface characterizing the over-the-barrier mechanism. An application to a monomer and a dimer of molecular magnets containing a Mn 4 group is presented and the result found to be in excellent agreement with reported experimental measurements.

A method is presented for finding instantons in magnetic systems – optimal paths corresponding to tunneling from one magnetic state to another at a finite temperature. The method involves analytical continuation of the energy to allow for complex values of the angle variables. First, a set of discretization points are placed equally spaced on a chosen energy contour. Then, an estimate of the corresponding temperature is obtained using Landau-Lifshitz dynamics in imaginary time along the contour. Finally, the distribution of the discretization points as well as the energy are systematically refined by converging on the nearest stationary point of the Euclidean action, thereby obtaining a discrete representation of the closest instanton at the given temperature. The method is illustrated with an application to a system consisting of a single spin subject to uniaxial anisotropy and transverse external magnetic field. First-order and second-order crossovers from over-the-barrier mechanism to tunneling are found depending on the applied field, and the difference in the dependence of the instanton temperature on the energy illustrated for the two cases. By comparing the Boltzmann factors for over-the-barrier and tunneling transitions, the cross-over temperature between the two mechanisms is estimated for both first-and second-order cross-over.

Magnetic skyrmions are nano-scale magnetic states that could be used in various spintronics devices. A central issue is the mechanism and rate of various possible annihilation processes and the lifetime of metastable skyrmions. While most studies have focused on classical over-the-barrier mechanism for annihilation, it is also possible that quantum mechanical tunneling through the energy barrier takes place. Calculations of the lifetime of magnetic skyrmions in a two-dimensional lattice are presented and the rate of tunneling compared with the classical annihilation rate. A remarkably strong variation in the onset temperature for tunneling and the lifetime of the skyrmion is found as a function of the values of parameters in the extended Heisenberg Hamiltonian, i.e. the out-of-plane anisotropy, Dzyaloshinskii–Moriya interaction and applied magnetic field. Materials parameters and conditions are identified where the onset of tunneling could be observed on a laboratory time scale. In particular, it is predicted that skyrmion tunneling could be observed in the PdFe/Ir(111) system when an external magnetic field on the order of 6T is applied.

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 fieldlike and dampinglike 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. 

Skyrmions are localized, topologically non-trivial spin structures which have raised high hopes for future spintronic applications. A key issue is skyrmion stability with respect to annihilation into the ferromagnetic state. Energy barriers for this collapse have been calculated taking only nearest neighbor exchange interactions into account. Here, we demonstrate that exchange frustration can greatly enhance skyrmion stability. We focus on the prototypical film system Pd/Fe/Ir(111) and use an atomistic spin model parametrized from first-principles calculations. We show that energy barriers and critical fields of skyrmion collapse as well as skyrmion lifetimes are drastically enhanced due to frustrated exchange and that antiskyrmions are metastable. In contrast an effective nearest-neighbor exchange model can only account for equilibrium properties of skyrmions such as their magnetic field dependent profile or the zero temperature phase diagram. Our work shows that frustration of long range exchange interactions – a typical feature in itinerant electron magnets – is a route towards enhanced skyrmion stability even in systems with a ferromagnetic ground state.

We show that thermal stability of skyrmions due to entropic effects can be strongly affected by external control parameters such as magnetic field and interface composition. The lifetimes of isolated skyrmions in atomic Pd/Fe bilayers on Ir(111) and on Rh(111) are calculated in the framework of harmonic transition state theory based on an atomistic spin model parametrized from density functional theory. Depending on the system the attempt frequency for skyrmion collapse can change by up to nine orders of magnitude with the strength of the applied magnetic field. We demonstrate that this effect is due to a drastic change of entropy with skyrmion radius which opens a route toward stabilizing sub-10-nm skyrmions at room temperature.

We present a metaheuristic conditional neural-network-based method aimed at identifying physically interest-ing metastable states in a potential energy surface of high rugosity. To demonstrate how this method works, we identify and analyze spin textures with topological charge Q ranging from 1 to -13 (where antiskyrmions have Q < 0) in the Pd/Fe/Ir(111) system, which we model using a classical atomistic spin Hamiltonian based on parameters computed from density functional theory. To facilitate the harvest of relevant spin textures, we make use of the newly developed segment anything model. Spin textures with Q ranging from -3 to -6 are further analyzed using finite-temperature spin-dynamics simulations. We observe that for temperatures up to around 20 K, lifetimes longer than 200 ps are predicted, and that when these textures decay, new topological spin textures are formed. We also find that the relative stability of the spin textures depend linearly on the topological charge, but only when comparing the most stable antiskyrmions for each topological charge. In general, the number of holes (i.e., non-self-intersecting curves that define closed domain walls in the structure) in the spin texture is an important predictor of stability-the more holes, the less stable the texture. Methods for systematic identification and characterization of complex metastable skyrmionic textures-such as the one demonstrated here-are highly relevant for advancements in the field of topological spintronics.