Valery M. Uzdin

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.

Spin-density wave in Cr without the nesting property of the Fermi surface

Recent ab initio calculations have shown that the theory of the spin density wave (SDW) based on the nesting property of the chromium Fermi surface is not able to explain solely the stability of the incommensurate SDW. Using the model Hamiltonian approach, we develop a method which allows self-consistent calculation of SDWs with different wavelengths. The magnetic profile and the energy of the system agree well with ab initio results. To check the stability of the SDW we introduce a local external magnetic field which induces transitions towards SDW or antiferromagnetic (AF) states. A very small energy which is enough to create the nodes of the SDW determines the new mechanism of SDW formation without supposition about the nesting property of the Fermi surface. The individual nodes of the SDW are found to be quite stable. They can be considered as low-energy excitations, which do not interact at large distance and annihilate when they approach close to one another. About 10% of non-magnetic Cr atoms randomly distributed in a few (100) layers leads to the formation of the SDW state with the nodal plane inside these layers. A similar trend arises when non-magnetic V impurities are introduced in bulk Cr. The increase of SDW wavelength with temperature and the transition from an SDW to a layered AF state when magnetic impurities (Fe, Mn) are introduced into bulk Cr are discussed on the basis of the developed theory.

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.

Remote control of the Fe magnetic moment in magnetic heterostructures

In magnetic superlattices with interlayer exchange coupling, not only the coupling strength but also the magnetic moment of the ferromagnetic layer can be altered non-locally by modifying the electronic structure of the non-magnetic spacer layer. Specifically, changes of the electronic structure of the V spacer layers in Fe/V (001) superlattices are seen to affect the adjacent Fe layers. By reversibly loading the V layer with hydrogen, the magnetic moment of Fe increases, whereas the induced magnetic moment in V remains unchanged. The nature of this remote and non-local control of the magnetic moment is connected with a d-electron charge transfer and effective shift of the Fermi level relative to the d-bands of Fe and V, as elucidated on the basis of self-consistent electronic structure calculations.

Evolution of atomic magnetic moments in Fe/V multilayers with hydrogen loading

Abstract The change of the magnetic saturation magnetisation of Fe/V superlattices upon in situ loading with hydrogen was investigated using a highly sensitive Faraday balance. Without hydrogen the magnetic moments in the superlattice were found to be considerably smaller than those obtained by ab initio calculations, whereas the moment increase with hydrogen was notably larger than predicted by ab initio calculations. Calculations of the Fe/V and Fe/V–H superlattices with intermixing at the interface were performed within the periodic Anderson model. They demonstrate the key role of interface alloying for the formation of magnetic properties.

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.

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

Atomic scale interface structure in metallic superlattices

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