Geometrically constrained Skyrmions
Article
Geometrically constrained Skyrmions
Swapneel Amit Pathak and Riccardo Hertel * Université de Strasbourg and CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, F-67000Strasbourg, France * Correspondence: [email protected]: date; Accepted: date; Published: date (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:1) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Abstract:
Skyrmions are chiral swirling magnetization structures with nanoscale size. These structureshave attracted considerable attention due to their topological stability and promising applicability innanodevices, since they can be displaced with spin-polarized currents. However, for the comprehensiveimplementation of skyrmions in devices, it is imperative to also attain control over their geometricalposition. Here we show that, through thickness modulations introduced in the host material, it is possibleto constrain three-dimensional skyrmions to desired regions. We investigate skyrmion structures inrectangular FeGe platelets with micromagnetic finite element element simulations. First, we establisha phase diagram of the minimum-energy magnetic state as a function of the external magnetic fieldstrength and the film thickness. Using this understanding, we generate preferential sites for skyrmions inthe material by introducing dot-like “pockets” of reduced film thickness. We show that these pockets canserve as pinning centers for the skyrmions, thus making it possible to obtain a geometric control of theskyrmion position. This control allows stabilizing skyrmions at positions and in configurations that theywould otherwise not attain. Our findings may have implications for technological applications in whichskyrmions are used as units of information that are displaced along racetrack-type shift register devices.
Keywords:
Skyrmions; Micromagnetic Simulations; Geometric Pinning; Finite-Element Modelling
1. Introduction
Magnetic skyrmions, predicted by theory almost 30 years ago, have advanced to a central topic ofresearch [1–3] in nanoscale magnetism over the last decade following their experimental observation[4,5]. Their particular topological properties [6], which impart them high stability and particle-likebehavior [7–9], combined with their room-temperature availability [10,11], reduced dimensions [12] andtheir unique dynamic properties [13], render these magnetic structures promising candidates for futurespintronic applications [14]. Skyrmions are formed in non-centrosymmetric magnetic materials exhibitinga sufficiently strong Dzyaloshinsky-Moriya Interaction (DMI) [4,5,15], i.e. , an antisymmetric energy termthat favors the arrangement of the magnetization in helical spin structures with a specific handednessand spiral period. In extended thin films, the formation and stability of skyrmions depends sensitivelyon various parameters, such as the strength of an externally applied magnetic field, the film thickness,temperature, and the magnetic history of the sample. Phase diagrams have been reported in the literature[10,16–18], displaying the parameter ranges within which skyrmions are stable and where they maytake different forms. Skyrmions may typically either develop individually or in the form of a hexagonalskyrmion lattice. While the occurrence of individual skyrmions makes them attractive candidates forunits of information that can be displaced in a controlled way by spin-polarized electric currents, theirspontaneous arrangement in the form of a periodic lattice could be interesting for magnonic applications,
Journal Not Specified , a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n ournal Not Specified , , 0 2 of 12 where these point-like magnetic structures could play the role of scattering centers of planar spin waves.One drawback of possible applications of skyrmions is the difficulty of controlling their position. Forinstance, if skyrmions are to be used as units of information in race-track type shift-register devices, it isnot only necessary to be able to displace them in a controlled way, but also to make sure that they areshifted between well-defined positions along the track. In domain-wall based concepts for race-trackmemory devices [19], which preceded the skyrmion-based variants, this control of the position wastypically achieved by inserting indentations (“notches”) into the strips [20,21]. It was shown that suchnotches represent preferential sites for domain walls, making it possible to trap domain walls at thesespecific positions, from which they could only be detached after overcoming a certain depinning energy[22]. Although the possibility to capture skyrmions at specific sites has been addressed in the case ofultrathin films, a geometric control analogous to the pinning of domain walls at notches does not yetseem to be firmly established for skyrmions. In two-dimensional systems, strategies for the pinning ofskyrmions include the insertion of point-like defects [23,24] or atomic-scale vacancies [25]. Remarkably,randomly distributed point defects in ultrathin films have also been reported to have little effect on thecurrent-driven skyrmion dynamics [26]. Motivated to further the discussion on this topic by addressingthe three-dimensional case of “bulk” DMI, we use finite-element micromagnetic simulations to studythe extent to which a geometric control of the skyrmion position in a thin-film element can be achievedby introducing a patterning in the form of local variations of the film thickness. Our simulations showthat, by locally lowering the film thickness in sub-micron sized dot-shaped regions, skyrmions can in factbe “captured” at these geometrically defined sites. We find that, by using geometrical constrains of thistype, skyrmions can be stabilized at positions that they would otherwise not adopt. For instance, thesegeometric manipulations make it possible to generate regular, square lattices of skyrmions which, apartfrom exceptional situations [27], are not observed naturally in non-centrosymmetric ferromagnets. It isargued that this control of skyrmion positions in magnetic thin films can open up new possibilities forskyrmionic devices as well as for concepts of magnonic metamaterials.
2. Results
Before addressing the question of how preferential sites for skyrmions can be generated throughnanopatterning, we first investigate, as a preliminary study, the field- and thickness dependence ofskyrmionic structures forming in rectangular FeGe platelets.
We consider rectangular thin-film elements with a lateral size of 180 nm ×
310 nm and thicknessesranging between 5 nm and 75 nm, and simulate the magnetic structures forming in the presence ofa perpendicular external magnetic field with a flux density varying between 0 mT and 900 mT. Thesimulations yield a large variety of possible magnetization states in this thickness and field range, whichcan be classified into six non-trivial types, summarized in Fig. 1. The three main types are the helical stateshown in panel (a), which is characterized by the presence of regular spin spirals extending over largeparts of the sample, the bimeron state (c), which can be interpreted either as particular a type of skyrmionstructures that is stretched along one axis or, alternatively, as a helical state in which the extension of thehelices is limited, and finally the skyrmion lattice state (e), which is characterized by a regular, hexagonalarrangement of skyrmions. ournal Not Specified , , 0 3 of 12 Figure 1.
Non-trivial magnetization states forming in a rectangular FeGe platelet (310 nm ×
180 nm) ofvarying thickness (between 5 nm and 60 nm) at different external field values. The color code describes theout-of-plane component m z of the normalized magnetization, and the isosurfaces indicate the regions where m z is equal to zero. Structures of this type appear at different film thicknesses as the external magnetic fieldis applied along the negative z direction and is varied between 0 mT and 900 mT. The arrangement of theseconfigurations in the image corresponds, roughly, to the order of preferential configurations appearingwith increasing field strength. In addition to these three fundamental states, there are also hybrid states in which two differenttypes of structures coexists, such as the helical-bimeron state shown in Fig. 1b) and the bimeron-skyrmionstate shown in Fig. 1d). These mixed states can be considered as transitional configurations betweenone fundamental state and another, which appear with changes in the external field value or in the filmthickness. At elevated field values and at larger film thicknesses, a quasi-homogeneous state is formed(not shown), where the magnetization is largely aligned along the external field direction. At fields belowsaturation, chiral bobber (ChB) [28] structures are also observed. These complex configurations of themagnetization can be considered as variants of skyrmions which do not traverse the entire thicknessof the sample. Instead, they have a skyrmion-like structure only on one surface, which evolves into aquasi-saturated configuration on the opposite surface on a path along the film thickness. The apex of theChB contains a Bloch point at which the magnetic structure changes in a discontinuous way. ChB structureshave interesting micromagnetic properties and have recently been discussed as magnetic structures thatcould be attractive in the context of spintronic devices [29], but they are not of primary interest for ourstudy. We display an example of a ChB structure only for completeness in the upper right of Fig. 1f), whereit coexists alongside four ordinary skyrmions. The magnetic structures were obtained by starting from arandom initial configuration and by a subsequent energy minimization. For more details see section 4.
The various magnetic states described in the previous section are possible equilibrium configurationsof the magnetization forming in the FeGe platelets at different values of the external field and the filmthickness. It is important to note that these magnetic structures are not uniquely determined by the filmthickness and the field strength. Because of this, in order to avoid possible misunderstandings, we didnot specify the values of the thickness and the field strength at which the states shown in Fig. 1 occur.In fact, several metastable states that can be significantly different from each other are often possible ournal Not Specified , , 0 4 of 12 under identical conditions, depending only on the magnetic history of the sample or, in a numericalexperiment, on the initial conditions of the simulation. While it is generally not possible to identify aunique magnetization state that develops in the thin-film element, micromagnetic simulations can be usedto determine the type of magnetic structure that has the lowest energy. The results of these calculations aresummarized in the phase diagram shown in Fig.2a). Figure 2. a) Phase diagram displaying the lowest-energy magnetic configuration in the FeGe platelet asa function of the film thickness and the external field strength. At high fields and large film thickness,the sample is in a quasi-saturated state. By lowering the film thickness, the formation of skyrmionstructures tends to become energetically favorable. b) Energy density of the skyrmions state (blue) and thequasi-saturated state (red) as a function of the film thickness at 650 mT field.
Although the magnetic structure at a specific thickness and field value is generally not unique, thephase diagram helps identifying the most preferable structure as far as the total energy is concerned. Whileat lower field values (below about 400 mT) the phase diagram is rather complex, evidencing a multitudeof possible magnetic structures showing neither any clearly dominating state nor a significant thicknessdependence, the situation becomes simpler at larger field strengths (above about 600 mT). Two main statesemerge in these ranges of larger field values: the skyrmion configuration and the quasi-saturated states.Moreover, these states are separated by a clearly defined boundary in the phase diagram, showing adistinct impact of the film thickness. Specifically, if a field of 650 mT is applied, the formation of skyrmionstructures will be energetically favorable if the film thickness is below 50 nm, while a quasi-saturated statewill be the lowest-energy configuration at larger thickness values, as shown in Fig. 2b). This observationrepresents the fundamental of the concept of geometrically constrained skyrmions that we present in thisstudy. The idea is the following. If the film thickness is locally modulated within a small dot-shaped regionsuch that, at a given field, the skyrmion structure is favorable in that thinner part while in the rest of thesample the thickness is large enough to favor a quasi-homogeneous state, these thickness modulations canbe designed to capture skyrmions. As we will show, this patterning makes it possible to generate pinningsites for skyrmions and, to some extent, to achieve a geometric control of the skyrmion position within thethin-film element. ournal Not Specified , , 0 5 of 12 We now consider magnetic structures forming in a FeGe platelet of 60 nm thickness containing dot-likecylindrical cavities within which the thickness is locally reduced to 30 nm. The phase diagram displayedin Fig. 2 suggests that, at external field values of about 650 mT, the insertion of these cavities results in ageometry with specific regions favoring the stability of skyrmion structures in a thin-film element which,without such modulation, would tend to form a quasi-homogeneous magnetic configuration. This can leadto the formation, or the trapping, of skyrmions that are geometrically constrained to the regions in whichthe pockets have been introduced. Fig. 3 shows such a geometrically constrained skyrmion in a 60 nmthick platelet. The skyrmion remains confined to the small region in which the thickness is reduced by50 % through two cylindrical pockets with depth of 15 nm and radius of r =
20 nm, inserted symmetricallyon both the top and the bottom surface of the film.
Figure 3. a) A skyrmion is formed at the base of the cylindrical pocket. At the inner cylinder surface of thecavities, the magnetization circulates on closed loops, thereby facilitating the formation of the skyrmion inthe center. The semitransparent representation of the surfaces shows the formation of the skyrmion in bothpockets, on the top and the bottom surface. The magnetic structure is displayed by arrows on the samplesurfaces. Some of the arrows have been remove in order to improve the visibility of the structure. b) Viewon the simulated skyrmion structure from inside the film. The skyrmion core connects the bases of thecylindrical pockets in the positive z direction, while the surrounding volume is magnetized in the negative z direction. The core of the skyrmion is delimited by a cylindrical isosurface m z =
0, shown here as a weak,transparent contrast in order to preserve the view on the central magnetic structure. Only a small subset ofthe computed arrows of the magnetization direction calculated within the volume is displayed.
The geometrically constrained skyrmion, shown in Fig. 3, is stabilized by the geometry for tworeasons. Firstly, as discussed before, in this field range the skyrmion state is generally favored becauseof the reduced film thickness. Secondly, the vortex-like magnetic configuration forming on the interiorcylinder surfaces of the cavity helps pinning the position of the skyrmion to the center of the pocket. Thiscylindrical flux-closure structure thereby provides boundary conditions, albeit not in a mathematical sense,which constrain the skyrmion to this dot-like geometry. By forming such a cylindrical vortex structure, themagnetization finds a nearly optimal way to adapt to competing micromagnetic interactions. It therebysatisfies both the tendency of the DMI to introduce chiral, swirling patterns as well as the tendencyimposed by the magnetostatic interaction to form flux-closure structures with the magnetization alignedalong the surfaces. Without the geometric modification in the form of pockets on the surface, the magneticstructure would be in a quasi-homogeneous state. The simulations show that a symmetric insertion ofthese pockets on both the top and the bottom surfaces is necessary to obtain the desired stability and ournal Not Specified , , 0 6 of 12 localization of skyrmions. If the thickness variation is introduced only on one of the surfaces, the pinningof skyrmions appears to be much less effective.If geometric modifications of the sample surface as described above can stabilize a skyrmion thatwould otherwise not form, the question arises whether this effect can be used to place skyrmions at specificpositions where they might be generated or removed in a controlled way through external manipulation.This could be of interest, e.g. , for device concepts in which skyrmions are utilized as binary units ofinformation, in a context similar to that of dot-patterned magnetic media for high-density data storage [30].In this case, the skyrmion pockets would take the role of the magnetic nanodots in bit-patterned media.While it is beyond the scope of this study to discuss the technical feasibility of such storage media or toexplore the ability to write and delete individual skyrmion patterns into the pockets, we can show that,indeed, it is possible to stabilize skyrmions in various geometrically predefined locations that could beaddressed individually. Figure 4.
Geometrically constrained skyrmions in FeGe platelets. By introducing circular pockets atspecific positions, skyrmions can be artificially stabilized at positions that they would otherwise not attain.The geometric control, however, is not unlimited. Attempts to pack skyrmions too closely or to place themtoo close to the sample boundary can fail. This is shown in panel f), where skyrmions are stabilized only inthe three central pockets, while the two outermost pockets remain empty.
Fig. 4)a-e) shows several examples of simulations in which the position of skyrmions in a thin-filmelement can be predetermined by introducing several pockets of the type discussed before. As shown inFig. 4e), our simulations predict the possibility to stabilize six skyrmions at well-defined positions, placedon a regular grid, in our sub-micron FeGe platelet. Although the results shown in Fig. 4 may suggesta nearly optimal geometric control of the skyrmion positions, it is important to note that the pocketsdiscussed here merely provide preferential sites for skyrmions. The latter may or may not form or remainpinned at those sites. In particular, it is not sufficient to thin-out a part of the sample in a sample to ensurethe appearance of geometrically constrained skyrmions. The purpose of such pockets could rather be tocapture existing skyrmions and to fix their positions at well-defined positions, similar to the domain-wallpinning role that is played by notches in conventional racetrack-memory devices [19,31]. It should also benoted that the geometric trapping of skyrmions with such pockets does not always work, in particularwhen the pockets are too closely packed. As a rule of thumb, the material must observe a characteristicminimal distance between the skyrmions that is given by the material-dependent long-range helical period l D , which in the case of FeGe is about 70 nm (see section 4). We also found that skyrmions cannot bestabilized at positions too close to the lateral sample boundaries due to repulsion [32]. An example of such ournal Not Specified , , 0 7 of 12 a failed attempt is shown in Fig. 4f). In spite of these limitations, the ability to geometrically constrainskyrmions provides an attractive way to obtain control over the skyrmion position in thin-film elements,which could have important technological implications.
3. Discussion
By means of micromagnetic finite-element simulations we have presented a possibility to controlthe position of magnetic skyrmions at predefined positions within a thin-film element by introducingcylindrical nano-pockets graved into the surface. Our concept to geometrically constrain skyrmions via such dot-like thickness variations is in many ways analogous to the idea of geometrically constraineddomain walls [33] in cylindrical nanowires, or to studies in which indentations have been introducedin rectangular strips in order to capture head-to-head domain walls in race-track type memory devices[21]. In those cases, too, the desired effect of the geometric constraint is to define preferential sites forspecific micromagnetic structures, such that the magnetic structures constrained at those artificial pinningsites require a certain activation energy in order to detach from them. The pockets described in thiswork could effectively play this role in the case of skyrmions driven along magnetic strips by meansof spin-polarized electrical currents. Such a geometric control of their position would allow shiftingskyrmions between well-defined points on the track. Moreover, as mentioned before, the trapping ofskyrmions at dot-like sites could also serve as a principle for skyrmion-based data storage devices, withoutnecessarily involving any displacement or depinning processes. If skyrmions can be selectively generatedand dissolved at such preferential sites, e.g. , by means of the field of a magnetized nano-tip or through alocalized spin-polarized current traversing the film thickness, the geometrically constrained skyrmionscould represent units of information that could be written and erased. Perhaps such skyrmionic dotmaterial could even be stacked in three dimensions for ultra-high density storage purposes. To addresssuch possibilities, future research directions could explore ways to reversibly insert skyrmions in thesegeometrically defined regions. Another potentially interesting use of our concept concerns magnonicapplications [34]. Since skyrmions can act as point-like scattering centers for spin waves, the ability toarrange them at specific sites as described in this study could open up new perspectives, as this couldresult in a new type of magnonic metamaterials in the form of artificial magnon Bragg lattices consistingof skyrmions arranged on regular lattice sites. Such artificial structures could be tailored to yield specificscattering and interference properties for spin waves that could not be obtained otherwise.
4. Materials and Methods
The material modelled in this study is FeGe. Due to its well-known helimagnetic properties, thisB20-type non-centrosymmetric material serves as a prototype for materials hosting chiral magneticstructures that develop due to the “bulk” DMI effect, as opposed to certain systems of ultrathin magneticfilms and substrates that can generate an “interfacial” DMI [1]. The micromagnetic parameters ofFeGe are [18] A = × − J m − , M s =
384 kA m − , and D = × − J m − , where A is theferromagentic exchange constant, M s the saturation magnetization and D the DMI constant. We neglectany magnetocrystalline anisotropy of the material, setting the uniaxial anisotropy to zero, K u = − . Acharacteristic length scale of this material is the long-range helical period l d = π A / | D | (cid:39)
70 nm. Thislength scale describes the typical period length of magnetic spirals forming as a result of the competinginteractions of the ferromagnetic exchange on one hand and the DMI on the other.With these parameters, the total energy E tot of the system is given by the sum of the Zeeman term,the ferromagnetic exchange, the DMI interaction and the magnetostatic energy: ournal Not Specified , , 0 8 of 12 E tot = (cid:90) V (cid:32) µ H ext · M + A · ∑ i = x , y , z ( ∇ m i ) + D m · ( ∇ × m ) − µ M · ∇ u (cid:33) d V (1)Here V is the sample volume, H e xt is the externally applied magnetic field, µ = π × − V s A − m − is the vacuum permeability, m = M / M s is the reduced (normalized) magnetization, and u is the magnetostatic scalar potential. The magnetostatic (demagnetizing) field H d = − ∇ u is the gradientfield of the magnetostatic potential. We calculate the magnetostatic potential u , which accounts for thelong-range dipolar interaction, by using the hybrid finite-element method / boundary element method(FEM/BEM) introduced by Fredkin and Koehler [35,36]. The dense matrix occurring in the boundaryintegral part of this formalism is represented using H type hierarchical matrices [37]. This data-sparserepresentation effectively overcomes size limitations arising from the boundary element method, as ityields a linear scaling of the computational resources required for the calculation of the magnetostatic term,which would otherwise grow quadratically with the number of degrees of freedom on the surface.For each energy term, an effective field H eff is defined as the variational derivative of thecorresponding partial energy E , H eff ( r , t ) = − δ E [ M ( r , t )] µ δ M (2)Specifically, the effective field of the ferromagnetic exchange is µ H ( xc ) eff ( r , t ) = − A ∆ m (3)and the effective field of the DMI is µ H ( DMI ) eff ( r , t ) = − D ( ∇ × m ) (4)Together with the magnetostatic field and the external (Zeeman) field, these effective fields enter theLandau-Lifshitz-Gilbert (LLG) equation [38], which describes the evolution of the magnetization field M ( r , t ) in time, d M d t = − γ ( M × H eff ) + α M s (cid:18) M × d M d t (cid:19) (5)where γ is the gyromagnetic ratio and α is a phenomenological, dimensionless damping constant. We usethe LLG equation to calculate equilibrium structures M ( r ) of the magnetization, by integrating in timeuntil convergence is reached. In the numerical simulations, convergence is achieved when either the totalenergy ceases to change over a long period, or when the torque (magnitude of the right hand side of theLLG equation) drops below a user-defined threshold.The geometry of the samples is designed with FreeCAD [39] and the discretization into lineartetrahedral elements is performed with Netgen [40]. The visalization of the FEM data was done withParaView [41]. The cell size does not exceed 2.5 nm, which is well below the exchange length l ex = (cid:112) A / µ M s (cid:39) M i defined at each node (vertex) i of the finite-element mesh. The magnetostatic field as well as the effectivefields of the ferromagnetic exchange and the DMI are calculated within in each tetrahedral element. Theelement-based data of these fields is then mapped onto the nodes of the mesh, in order to calculatethe effective field acting on the magnetization and thus to calculate the evolution of the magnetization ournal Not Specified , , 0 9 of 12 in time at each node according to the LLG equation. More details on the calculation of the convergedmagnetization states are given in appendix A. Author Contributions:
Conceptualization, S.A.P; methodology, R.H. and S.A.P.; software, R.H.; writing–original draftpreparation, R.H. and S.A.P.; writing–review and editing, R.H. and S.A.P.; visualization, R.H. and S.A.P.; supervision,R.H.; project administration, R.H.; funding acquisition, R.H. All authors have read and agreed to the published versionof the manuscript.
Funding:
This work has benefited from support by the initiative of excellence IDEX-Unistra (ANR-10-IDEX-0002-02)through the French National Research Agency (ANR) as part of the “Investment for the Future” program.
Acknowledgments:
The authors acknowledge the High Performance Computing center of the University ofStrasbourg for supporting this work by providing access to computing resources. Part of the computing resources werefunded by the Equipex Equip@Meso project (Programme Investissements d’Avenir) and the CPER Alsacalcul/BigData.
Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; inthe collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish theresults.
Abbreviations
The following abbreviations are used in this manuscript:DMI Dzyaloshinksii-Moryia interactionLLG Landau-Lifshitz-Gilbert equationFEM Finite Element MethodBEM Boundary Element MethodGPU Graphical Processing Unit3D three-dimensionalChB chiral bobber
Appendix A Energy minimization
Because in this particular study we are not interested in the dynamic evolution of the magnetizationbut only in static, converged magnetic structures, the integration of the LLG equation in the code fulfilsthe practical role of guiding the system along a path of energy-minimization in an iterative way. Since thedynamics of the magnetization during the transition from the initial to the converged state is irrelevant forthis work, we are free to choose a conveniently large damping parameter α = γ = z direction andsubsequently let the system relax in the presence of an external magnetic field aligned along the negative z direction. The z axis is oriented parallel to the surface normal, as shown in Fig. 1. References
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