Pavel F. Bessarab

Method for finding mechanism and activation energy of magnetic transitions, applied to skyrmion and antivortex annihilation

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.

Potential Energy Surfaces and Rates of Spin Transitions

Abstract 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.

Calculations of magnetic states and minimum energy paths of transitions using a noncollinear extension of the Alexander-Anderson model and a magnetic force theorem

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¯ 10] directions involves the formation of a thin, temporary domain wall. A noncollinear magnetic state is identified in a 7 × 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.

Lifetime of racetrack skyrmions

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.

Enhanced skyrmion stability due to exchange frustration

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.

The effect of confinement and defects on the thermal stability of skyrmions

Abstract 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.

Effect of hydrogen adsorption on the magnetic properties of a surface nanocluster of iron

The effect of hydrogen adsorption on the magnetic properties of an Fe

Energy surface and lifetime of magnetic skyrmions

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Qualitative insight and quantitative analysis of the effect of temperature on the coercivity of a magnetic system

cluster immersed in a Cu(111) surface has been calculated using densifty 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 0.4 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 and 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.

Spin relaxation signature of colossal magnetic anisotropy in platinum atomic chains

Abstract 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.