Hybrid chiral domain walls and skyrmions in magnetic multilayers
William Legrand, Jean-Yves Chauleau, Davide Maccariello, Nicolas Reyren, Sophie Collin, Karim Bouzehouane, Nicolas Jaouen, Vincent Cros, Albert Fert
HHybrid chiral domain walls and skyrmions in magnetic multilayers
William Legrand, Jean-Yves Chauleau,
1, 2
Davide Maccariello, Nicolas Reyren, SophieCollin, Karim Bouzehouane, Nicolas Jaouen, Vincent Cros, ∗ and Albert Fert Unit´e Mixte de Physique, CNRS, Thales, Univ. Paris-Sud,Universit´e Paris-Saclay, Palaiseau 91767, France Synchrotron SOLEIL, L’Orme des Merisiers, 91192, Gif-sur-Yvette, France (Dated: January 8, 2018)
Abstract
Noncollinear spin textures in ferromagnetic ultrathin films are currently the subject of renewedinterest since the discovery of the interfacial Dzyaloshinskii-Moriya interaction (DMI). This anti-symmetric exchange interaction selects a given chirality for the spin textures and allows stabilisingconfigurations with nontrivial topology. Moreover, it has many crucial consequences on the dynam-ical properties of these topological structures, including chiral domain walls (DWs) and magneticskyrmions. In the recent years the study of noncollinear spin textures has been extended fromsingle ultrathin layers to magnetic multilayers with broken inversion symmetry. This extensionof the structures in the vertical dimension allows very efficient current-induced motion and room-temperature stability for both N´eel DWs and skyrmions. Here we show how in such multilayeredsystems the interlayer interactions can actually lead to more complex, hybrid chiral magnetisa-tion arrangements. The described thickness-dependent reorientation of DWs is experimentallyconfirmed by studying demagnetised multilayers through circular dichroism in x-ray resonant mag-netic scattering. We also demonstrate a simple yet reliable method for determining the magnitudeof the DMI from static domains measurements even in the presence of these hybrid chiral struc-tures, by taking into account the actual profile of the DWs. The advent of these novel hybridchiral textures has far-reaching implications on how to stabilise and manipulate DWs as well asskymionic structures in magnetic multilayers. a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n he DMI is a form of antisymmetric exchange that promotes canting between neigh-bouring magnetic moments. As the DMI in ultrathin films originates from the strong spin-orbit coupling of interfacial atoms neighbouring the magnetic layer , it is notably foundin a ferromagnet (FM) interfaced with two different heavy metals, such as Pt/Co/Ir , orin magnetic layers inserted between a heavy metal and an oxide, such as Pt/CoFe/MgO .This interfacial DMI has a direct influence on the orientation and chirality of the magneti-sation textures in these systems, in particular on DWs and magnetic bubbles in perpen-dicularly magnetised layers. Due to the energy lowering of the N´eel configuration (internalmagnetisation perpendicular to the DW) associated to the interfacial DMI, the demagnetis-ing effects inside the DW that usually favour the Bloch configuration (internal magneti-sation along the DW) can be overcome and the N´eel orientation favoured . Dependingon the types of interfacial atoms combined with the FM and on the stacking order, theDMI can also change sign, which determines whether clockwise (CW) or counter-clockwise(CCW) chirality is preferred . Such stabilisation of chiral magnetic textures also helpsto stabilise quasi-punctual solitonic structures called skyrmions, in the present case N´eel(hedgehog) skyrmions . In order to obtain stable and compact individual skyrmions atroom-temperature, a successful approach has been to stack up several repeats of an asym-metric combination of ultrathin layers . In this way, it is possible to stabilise columnar-shaped skyrmions, in which the increased magnetic volume reinforces their stability againstthermal fluctuations up to room-temperature, while still preserving both interfacial perpen-dicular magnetic anisotropy (PMA) in each ultrathin FM, and required interfacial DMI dueto the absence of inversion symmetry of the structure.Here our purpose is to investigate the thickness-dependent internal profile of chiral mag-netic DWs and chiral skyrmions in Pt/Co/Ir and Pt/Co/AlO x based multilayers with PMA.The studied multilayers are stackings made of 5 up to 20 repetitions of such trilayers withvarious magnetic layer thicknesses. Our study combines theoretical predictions, togetherwith a direct experimental observation, of a vertical position dependent reorientation of thechirality inside the magnetisation texture in multilayers. Focusing first on the simplest caseof DWs, we predict through micromagnetic simulations not only a significant variation ofthe DW width through the thickness of the multilayers but, most importantly, a twisting ofthe internal DW texture along their thickness. In case of a large number of repetitions inthe stacking, top and bottom layers then host N´eel orientations of opposite chiralities (CW2nd CCW), one in accordance with and the other one opposing the chirality favoured by theinterfacial DMI. This reorientation thus results in an uncommon type of chiral compositeN´eel-Bloch DW. Then, we provide an experimental demonstration of the described reorien-tation of the DW texture in Pt/Co/AlO x multilayers by directly observing the DW chiralityclose to the top surface through circular dichroism in x-ray resonant magnetic scattering.These results involve reconsidering the common implicit assumption of a uniform magneti-sation through the thickness of the multilayers. From the better understanding of the actualdepth-profile of the DW structure through the thickness, we then describe a novel approachto measure the DMI in such multilayers. Using the experimentally measured size of parallelstripe domains, we show how the DMI strength can be quantitatively extracted for eachsystem by comparing the energies of stripe domains with different sizes in micromagneticsimulations. With the support of analytical calculations, we next propose a simple modelwhich allows to predict the occurrence of DW twisting. We finally discuss how our findingsopen new opportunities in order to manipulate more efficiently this new type of compositechiral skyrmions in multilayers, through the engineering of the interfacial spin-orbit torques(SOTs). DIPOLAR-FIELD-INDUCED REORIENTATION OF MAGNETIC DWS
In ultrathin magnetic films, the interfacial DMI can be strong enough to stabilise chiralN´eel DWs. Let us consider a single DW separating two perpendicularly magnetised domainsof magnetisation pointing down ( m · z = −
1) for x < m · z = +1) for x > θ ( x ) = arccos ( m · z ) the polar angle and ψ the (fixed) azimuthal angle of themagnetisation inside the DW, the DW energy is given by σ dw = A (cid:90) + ∞−∞ (cid:18) dθdx (cid:19) dx + D cos ψ (cid:90) + ∞−∞ dθdx dx + K (cid:90) + ∞−∞ sin θdx + σ dem ( ψ ) (1)where A is the Heisenberg exchange interaction amplitude, D the interfacial Dzyaloshinskii-Moriya interaction, K the uniaxial anisotropy, and σ dem the demagnetising energy associatedto dipolar fields. Without DMI, the ψ dependence of σ dem favours Bloch type DWs (cos ψ =0). However it is now well known that in the presence of DMI, as comes out from thisequation, the energy of the DW can be lower for a N´eel orientation (cos ψ = ±
1) which causesthe reorientation of the internal magnetisation of the DW and explains the stabilisation of3´eel structures . Such N´eel DWs in ultrathin films exhibit a fixed chirality controlled bythe sign of the DMI and, notably, have proven to present extremely interesting current-drivendynamics .In the recent years, the study of such chiral spin textures has become a very importantresearch field both theoretically and experimentally. Most analyses of the spin configura-tions in single layers (for DWs and skyrmions) assume a pure N´eel profile and, for DWs,an energy independent of the DW spacing (uncoupled DWs). However, most often in mul-tilayered stacks the DWs are packed close to each other, meaning that the uncoupled DWsapproximation does not apply. Another complexity is that if the demagnetising fields arestrong (i. e. for a large number of repetitions of magnetic layers), a DW structure betweenBloch and N´eel may arise (0 < ψ < π/ , who have derived a precise analytical model of the demagnetising ener-gies for standard 1D magnetisation profiles. In this model, a classical DW profile with onefree internal angle ψ is assumed and the magnetic configuration is parametrised by ∆ (DWwidth), λ (domain periodicity) and ψ only. Knowing the material parameters, a set of threenonlinear equations needs to be solved in order to get the equilibrium value of (∆, λ , ψ ).As we will demonstrate later, this model is very accurate for pure N´eel walls and allowsto estimate the value of D , as well as to determine a threshold minimum value of D thatprevents internal tilting of the magnetisation inside the DWs towards Bloch configuration.This (∆, λ , ψ ) model is based on two strong assumptions: (i) the DW has a classicalarctan profile without any x dependence for ψ and (ii) all layers share the same DW profile,that is, without any z dependence. However, it is to be suspected that spin structures notuniform in the z direction can be stabilised in magnetic multilayers, as was suggested bymicromagnetic simulations and imaging of the emergent field of skyrmions, with con-straints on the stability in order to reconstruct the magnetic profile . We aim here todemonstrate that in most of the multilayered material systems considered in recent exper-iments on skyrmions at room-temperature, the actual magnetisation structure through thethickness of the multilayer is indeed more complex. As a result of the competition betweenthe different interactions, a novel type of chiral hybrid magnetic texture is stabilised withsome profound consequences on their spin-torque-driven dynamics.To support this statement, a first step has been to perform a series of micromagnetic sim-ulations of the DW structure in multilayers. We take the example of a multilayer comprising40 repetitions of X(1 nm)/Co(0 . ,and perform the micromagnetic simulation of the stripe domain configuration in the full ge-ometry including all magnetic layers and non-magnetic spacers (see Methods). In order tostudy the influence of the DMI on the DW configuration, we vary D to be -1.0, 0.0, 1.0 and2 . − , for each of which the DW configuration has been computed. The results of thesesimulations are compiled in Fig. 1, in which we present the evolution of the magnetisationprofile as a function of the layer position inside the stacking in three different ways. The firstcolumn displays cross-sectional views of the magnetisation profile (Figs. 1a–d), the secondcolumn displays the polar angle ( θ angle) of the DW magnetisation across the DW in eachlayer (Figs. 1e–h) and the third column displays its azimuthal angle (Figs. 1i–l). Each linethen corresponds to the different DMI values.From these simulations we find that, for a large range of DMI values, (i) the DW internalmagnetisation is twisted between N´eel and Bloch-type along the z direction (i.e. the positionof the magnetic layer in the stacking), leading to the formation of what we later call ahybrid chiral DW structure, (ii) the DW width varies through the multilayer thickness and(iii) the z component of the DW magnetisation m z cannot be simply assumed to follow anarctan function in all layers and none of the usual models allow to fit correctly the DWmagnetisation structure. We now proceed with a more detailed description of the resultsshown in Fig. 1 with regard to these three points.First, a clear output from these simulations is that the stabilised DW structures are notcorresponding, whatever the value of D , neither with pure Bloch walls (cos ψ = 0), as usuallyexpected for D = 0 in single layers (Fig. 1b), nor with pure N´eel walls (cos ψ = ±
1) evenfor D as large as 2 . − (Fig. 1d). We further notice that even for intermediate D = ± . − (Figs. 1a,c), the DWs do not exhibit an intermediate structure between N´eeland Bloch with a constant ψ angle of the magnetisation inside the DW, but instead show acontinuous variation of this ψ angle between all the magnetic layers of the stacking. Suchrather complex internal magnetisation profiles, different from the single layer case, arise inthese many-repeats multilayers from the competition between the interfacial DMI and thedipolar interactions between each layers. The resulting configuration can be described bythe presence of a mostly Bloch (cos ψ close to zero) wall part in some of the intermediatemagnetic layers, whereas the combined action of the demagnetising fields and DMI fields5 igure 1. a–d. Cross-sectional view of a half simulation volume for [X(1)/Co(0.8)/Z(1)] multi-layer with D = -1.0, 0.0, 1.0 and 2 . − , respectively. Arrows point in the direction of themagnetisation, m z is given by the colour of the arrows from red (-1) to blue (+1), while m y is dis-played by the colour of the grid from black (-1) to white (+1). e–h. Polar angle θ inside the DW ineach layer for D = -1.0–2 . − . The blue to red lines correspond to layers from bottom to top(see colour scale in e. ), while the orange line is the average θ across the thickness. Green and blacklines are the profiles as given by the (∆, λ , ψ ) and K eff models, respectively. i–l. Azimuthal angle ψ inside the DW in each layer for D = -1.0–2 . − . The blue to red lines again correspond tolayers from bottom to top. and bubbles in magnetic thick layers. Here, we do observeit for a large range of values of the DMI. Interestingly, we notice that the position of theBloch wall part with respect to the central layer depends on the strength and sign of theDMI. For increasing | D | values, the preference for a single chirality shifts more and morethe position of the most Bloch DW (minimal | cos ψ | ) up or down depending on the sign of D . For example, for D = 2 . − , only the two top-most layers contain N´eel-like DWswith a sense of rotation opposite to all others (for a large negative D , the two bottom-mostDWs will have an opposite rotation). Only for a large enough value of D ( | D | ≥ . − in the present case), the Bloch-like DWs are excluded and the DW chirality is the samethrough all layers.Second, as can be seen from the θ ( x ) profiles of Figs. 1e–h, the DW width varies signifi-cantly among the different layers. Again due to dipolar effects, the central, Bloch wall partis more compact than the N´eel wall parts at top and bottom which extend over a largerwidth. Overall, the DW widths in the different layers vary significantly, by more than afactor of 2, as can be seen from the slopes at the center of the θ ( x ) profiles.Third, we find that in any individual layer of the stack, the DW profile cannot be properlydescribed by the classical models. On the graphs of Figs. 1e–h, we add the average over alllayers of the θ ( x ) profile of the micromagnetic model (orange lines), the DW profile (uniformalong z ) predicted by the so-called K eff model (black lines), a model in which the multilayeris treated as an effective magnetic media and the DWs as non-interacting (see Methods fordetails), and the DW profile (also uniform along z ) predicted by the (∆, λ , ψ ) model (greenlines). It can be clearly seen that the two latter profiles fail to reproduce both the exactshape of the DW, notably in the region of the tail, as well as the DW width. Notably, thisdifference demonstrates that the actual DW shapes are different from the commonly usedarctan profile. Moreover, as can be seen from the strongly x dependent ψ ( x ) profiles inFigs. 1i–l, the Bloch-N´eel character of the DW in each layer varies across the DW (along x ), instead of being fixed. As a consequence of the DW being partially Bloch, and onlyin the intermediate layers (combined x and z dependence of ψ ), with varying DW widths,the average transverse magnetisation of the DWs is altered and DW energies turn to be7ignificantly different from any simple model predictions. We note that when D exceeds2 . − (not shown here), the DWs resemble pure N´eel DWs, and classical models getaccurate, with a good agreement between micromagnetic and classical models.The main reasons for the discrepancy between the more elaborate micromagnetic simu-lations and the described analytical models are thus that for the latter no dependence of ∆along z is considered and that they assume a fixed DW arctan profile uniform along z , thusmissing the longer tails in the external layers and the strong variation of ψ along x and z .As already described, we instead predict a twisted DW magnetisation through the thickness,constituting a new kind of hybrid chiral DW in DMI systems. The complexity of the DW(or skyrmion, as will be seen later on) profiles in multilayers is well depicted in the curvesshown in Figs. 1 i–l. Note that the magnetisation configuration resembles the flux-closureDW arrangements recently observed in magnetic bilayers . However, in our multilayercase, the magnetisation evolves more continuously along the x direction, as indicated by thevariation of the ψ angle through each layer (see Fig. 1k for example).Therefore, the important difference of hybrid chiral DWs with previous models originatesfrom the competition between the interlayer magnetostatic interactions and DMI, whichresults in these flux-closed DW configurations that never correspond to DWs with a fixedinternal magnetisation angle in between pure Bloch and N´eel types. We will see later that inour multilayers (but more generally in all the multilayered systems experimentally studied sofar), the actual value of D is inferior to 2 . − , which is the threshold value that allowspure N´eel DWs through all layers in the present case. In consequence, it becomes essential toconsider that the actual DW (or skyrmion boundary) textures are similar to the one we havejust described. As detailed in the end of this study, this finding has profound implicationsfor the spin-torque-induced manipulation of these composite DWs or skyrmions . DIRECT OBSERVATION OF THE INTERNAL TWISTING OF THE DWS
In order to verify experimentally the formation of the above-described composite DWs no-tably in presence of a significant DMI, we have grown several series of magnetic multilayeredstackings by sputtering deposition (see Methods). The structures of the studied multilayersare reported in Table I, starting from the substrate side //, where all thicknesses are givenin nanometers and the numbers indexing the brackets are the number of repetitions of the8 able I. List of the studied magnetic multilayers investigated. The saturation magnetisation M s ,the uniaxial anisotropy H k , the measured domains periodicity, the estimated DW width, and thecomparison of estimations of D with a fixed DW energy ( K eff model), (∆, λ , ψ ) model and thepresent full micromagnetic model are given for each multilayer. Two multilayers are not labelledas they have not been studied by CD-XRMS but only characterised by MFM in order to determine D with the different models. M s H k λ ∆ ∆ ,λ,ψ D σ,K eff D ∆ ,λ,ψ D full (kA m − ) (mT) (nm) (nm) (mJ m − ) (mJ m − ) (mJ m − )I //Pt 10/[ Ir 1/Co 0.6/Pt 1 ] /Pt 3 840 330 167 7.27 -1.46 -1.43 -2.30II //Pt 10/[ Ir 1/Co 0.8/Pt 1 ] /Pt 3 1229 640 150 4.13 -2.13 -2.00 -2.00III //Pt 11/[ Co 0.8/Ir 1/Pt 1 ] /Pt 3 637 516 278 7.08 1.48 1.45 1.37IV //Ta 5/Pt 10/[ Co 0.8/Ir 1/Pt 1 ] /Pt 3 683 748 488 5.77 1.69 1.64 1.63V //Pt 11/[ Co 0.8/Ir 1/Pt 1 ] /Pt 3 637 516 244 6.03 1.53 1.50 1.52VI //Ta 5/Pt 10/[ Co 0.8/Ir 1/Pt 1 ] /Pt 3 847 1088 250 4.11 2.11 2.08 2.06VII //Ta 15/Co 0.8/[ Pt 1/Ir 1/Co 0.8 ] /Pt 3 957 500 256 5.00 -1.26 -1.21 -1.14- //Ta 10/Pt 7/[ Pt 1/Co 0.6/Al O ] /Pt 3 1344 469 190 4.06 1.40 0.88 1.29VIII //Ta 10/Pt 7/[ Pt 1/Co 0.8/Al O ] /Pt 3 1373 358 175 3.72 1.20 1.20 1.01- //Ta 10/[ Al O ] /Pt 7 1120 332 131 4.79 -1.91 -1.93 -1.94IX //Ta 10/[ Al O ] /Pt 7 1245 228 131 4.33 -1.76 -1.78 -1.69 base multilayer. The use of different buffer layers allows us to control the growth conditionsand hence the amplitude of the magnetic anisotropy H k and M s values for similar stacks.The choice of opposite orders of the stackings allows us to get opposite signs of the DMI,while increasing the number of repetitions of the same trilayer base allows us to increase theinfluence of interlayer dipolar interactions.To get evidence of the existence of the predicted DW twisting we have measured thecircular dichroism in X-ray resonant magnetic scattering (CD-XRMS), allowing us to di-rectly probe the magnetisation sense of rotation of the top layers of the stackings . Thisexperimental approach has proven to be a relevant and straightforward approach to probethe chiral aspect of closure magnetic domains in FePd as well as the topological windingand chirality in Cu OSeO . In particular, we have recently demonstrated that the DWchirality (CW or CCW) could be unambiguously determined by assessing the dichroic pat-tern sign in opposite stackings of Pt/Co/Ir multilayers . For the present experiments, theenergy is set at the L Co edge and the angle of incidence of the X-rays has been chosen to be18 . ◦ , corresponding to the first diffraction peak of the multilayer. Under these conditions,X-rays are mainly sensitive to the first 15 nm, which corresponds to only 4 to 5 repetitions9rom the top of the multilayered structure that predominate in the measured XRMS sig-nal (see Methods). Before performing the XRMS experiments, all the samples have beendemagnetised in order to reach a magnetic ground state composed either of labyrinthinealternating up and down magnetised domains or parallel stripe domains depending on thechosen demagnetisation procedure (see Methods).In Figs. 2a–c, we display the XRMS dichroism patterns (diffraction patterns with un-polarised X-rays are shown as insets) recorded for the multilayers labelled (II), (VII) and(IX) (with 5, 10 and 20 repetitions, respectively). These three multilayers are the oneshaving the Pt layer on top of the FM ( D <
0, see Table I). In the second line (see Figs.2d–f), the dichroism patterns are the ones obtained for the multilayers labelled (III), (V)and (VIII) (also with 5, 10 and 20 repetitions, respectively) that have the Pt layer below theFM (
D >
0, see Table I). We emphasise that for the first two columns, the multilayers havebeen demagnetised with an oscillating out-of-plane field, leading to a labyrinthine domainconfiguration that generates rings in the diffraction patterns. On the contrary, for multilay-ers (IX) and (VIII) shown in Figs. 2c,f, the demagnetisation procedure was done applyingan in-plane field, leading to parallel stripe domains configuration and resulting in diffractionspots appearing on both sides of the specular peak. The corresponding MFM images areshown in the Supplementary Material.Important conclusions can be drawn from this series of XRMS patterns. First, because thedichroic signal is in all cases maximum along the 90 ◦ -270 ◦ axis, it can be concluded that allthe DWs in these different multilayers share the same type of DW in surface, correspondingto N´eel DWs as we demonstrated recently , instead of tilted DWs in between N´eel andBloch configurations. Second, we clearly observe that for all samples with Pt on top ofthe FM (Figs. 2a–c), a positive (red colour in our convention) dichroism at 90 ◦ is found.This means that the top few layers exhibit a CW N´eel DW chirality whatever the numberof repetitions in the multilayers, which is the DW configuration that can be expected forthis sign of the DMI. On the opposite, for the first two multilayers with Pt below the FM(Figs. 2d,e) the dichroic signal is reversed (maximum positive at 270 ◦ ). It indicates anopposite, CCW N´eel DW chirality for 5 and 10 repetitions, again as expected from thesign of the DMI. A striking observation is then that the dichroism pattern recorded for the20-repeats multilayer with Pt below the FM (shown in Fig. 2f) is again positive at 90 ◦ andthus indicates a CW N´eel chirality in surface, which is opposite to what is expected for a10ositive-DMI-driven chirality. As we have explained previously, this apparent discrepancyis a direct fingerprint of the competition between interlayer dipolar fields and DMI: due tothe 20 repeats in the multilayer, the impact of the dipolar fields is large enough to imposea reversal of the chirality of the DWs in the top layers. Note that we are able to observethis dipolar-field induced chirality reversal only for sample (VIII) and not for multilayer(IX), because for D <
D >
ESTIMATION OF D WITH HYBRID CHIRAL DW TEXTURES
Being able to get a reliable quantitative estimation of the interfacial DMI amplitudehas been the subject of numerous studies in the last couple of years as this parameter iscrucial to understand and control both the statics and dynamics of chiral DWs and/orskyrmions. Different approaches have been proposed. In single magnetic layers, the interfa-cial DMI has been experimentally accessed using Brillouin light spectroscopy , spin-wavespectroscopy , chirality-induced asymmetric DW propagation or asymmetric magneti-sation reversal . However, these different methods appear to be not reliable in case ofinterfacial DMI in multilayered systems. A first reason is that dipolar fields and magneticcouplings between layers shall influence significantly the spin-wave propagation and thus theanalysis of BLS spectra. Moreover, because dense labyrinthic magnetic domains are formedat low fields, it thus prevents the observation of the asymmetric reversal needed to estimatethe DMI. This is why in multilayered systems, analysis of domain spacing in the demagne-tised state or as a function of the perpendicular magnetic field has been proposed to estimatethe magnitude of DMI . This approach is based on the fact that the DW periodicity λ is the result of the balance between domains demagnetisation energy and DW energy , the11 cd e Dichroism
Neg. Pos.
0° 90°180°270° ψ
0° 90°180°270° ψ b fx5 Co(0.8nm)Pt(1nm) x10
Co(0.8nm)Pt(1nm) x20
Co(0.8nm)Pt(1nm)Co(0.8nm)Pt(1nm) Co(0.8nm)Pt(1nm) Co(0.8nm)Pt(1nm)
Dichroism
Neg. Pos. x5 x10 x20(VII)(V) (IX) (VIII)(II) (III)
Figure 2. Circular dichroism analysis of different multilayer stacking configurations with
D <
D > latter being dependent on D , as described above. Assuming that the observed demagnetisedstate is very close to the state of minimum energy given parallel stripe domains, we can thendetermine the DW energy σ dw and deduce a rough estimation of the DMI magnitude | D | with the so-called K eff model (see Methods). This characterisation method has led to theevidence of a significant DMI in magnetic multilayers with broken inversion symmetry .It has however been pointed out recently that such measurements can be largely erroneouswhen neglecting stray-field effects on the DW size and spacing, so that a more comprehensivemodel is required for multilayers . When dipolar interactions become significant but theDW internal configuration remains uniform, a more detailed model such as the (∆, λ , ψ ) ismore accurate . Nevertheless, we now suspect that complex DW or skyrmion structures12an arise in magnetic multilayers depending of the relative strengths of dipolar interactionsand DMI, which calls for a more careful analysis of their spin textures .We show here how the existence of hybrid chiral DWs induced by dipolar fields has alarge impact on the evaluation of D in multilayers. To get an accurate estimation even inthe presence of the hybrid chiral DW structures we have identified, our procedure is then torely on the complete micromagnetic simulations as exemplified above. We vary the domainsperiod in the simulation by changing the number of cells around the measured λ of theparallel stripe domains and then relax the system for different values of D . The extractedvalue D full is D for which the energy density of the simulated system is minimum at λ (seeMethods). The extracted D values for the different multilayers are listed in Table I, in whichwe also compare D full with the values of D estimated with the K eff and (∆, λ , ψ ) models.Although all models give consistent values for pure N´eel DWs, we note significant differencesas soon as at least one layer has a reversed chirality.We emphasise that the measurement of the domain periodicity λ is error-free from thesize of the MFM imaging probe , and that the sensitivity to local defects is reduced dueto the averaging effect of imaging over large sample areas. Thus, we believe that the moststraightforward mean to quantify the DMI in multilayers remains so far to find it from themeasurement of λ in the ground state. Finally, another interesting aspect of this approachusing the periodicity of stripe domains is that aligning parallel stripe domains with varyingdirections allows to measure the DMI along different directions, which can give informationon a potential anisotropic DMI in materials which have a crystalline structure allowingdifferent DMI vectors along their different crystalline axes. We did not find anisotropic DMIin our samples. CRITERION FOR DIPOLAR-FIELD INDUCED TWISTING OF DW CHIRAL-ITY
In the following, our objective is to establish a simple criterion describing the occurrenceof twisted chiral DWs in magnetic multilayered systems. For that we follow an approachopposite to what we have just described as we now study the DW structure considering aknown value of D . Considering parallel domains and by making the assumption that all DWsare N´eel (with D > A dip and A dmi , over a total width of 6∆ aroundthe center of the DW (see Methods). As an example, we show these deduced field profiles(green and blue lines for DMI fields and interlayer dipolar interaction fields, respectively,right scale) for our multilayer (III) [Pt(1)/Co(0.8)/Ir(1)] in Fig. 3, together with the DWhorizontal magnetisation component m x ( x ) profile (black curve, left scale). The dipolarinteraction field B dip (solid, blue line) is obtained by summing surface (dashed, blue line) andvolume (dotted, blue line) magnetic charges contributions. The small difference between thetotal field B dip , t and the dipolar interaction field with the other layers B dip = B dip , S + B dip , V comes from the horizontal component of the self-demagnetising field of the top layer itself,which has no influence on the reorientation and that we do not include in the comparison offield actions. These analytically calculated field and magnetisation profiles match the onesobtained from micromagnetic simulations (shown as squares of the corresponding colours inFig. 3) under the assumptions mentioned just above.When |A dip | > |A dmi | , for D >
D <
0) the top layer (bottom layer) in-plane componentof the magnetisation inside the DW will reverse due to dipolar fields, so that a pure N´eel DWthroughout the whole stack is definitively not stable, whereas for |A dip | < |A dmi | N´eel DWsof the same chirality can be stabilised in all layers. We can thus predict whether the DWswill reorientate or not, as shown in Fig. 4. The areas filled with a uniform colour correspondto |A dip | < |A dmi | in which case pure N´eel-like DWs are stabilised. The colour indicatesthe preferred DW chirality imposed by the DMI. When this condition is not fulfilled, i. e. , |A dip | > |A dmi | , corresponding to the gradient areas, the DW reorientation into flux-closureDWs leading to the stabilisation of hybrid chiral DWs shall occur. The twisting is moreand more pronounced when |A dip | becomes stronger relative to |A dmi | , which is signified bythe progressively lighter background colour. In order to compare these predictions with ourexperiments, we also include in Fig. 4 the experimental observation of the DW chirality ofthe top layers determined by CD-XRMS in all our different multilayers. The colour of thesquares labelled (I-IX) indicate the top-surface DW sense of rotation, blue correspondingto CCW and red to CW chiralities. As described above, the DW surface chirality formultilayer (VIII) is found in agreement with the prediction, i. e. , the corresponding squareis red in a blue gradient area, meaning that the observed chirality is CW, due to dipolar14 igure 3. DW profile (black, left scale) and analytical model for estimation of DMI (green, rightscale) and dipolar (red, right scale) fields for the top layer of (III) [Pt(1)/Co(0.8)/Ir(1)] . Thesquares are the result of micromagnetic minimisation of the energy, while the lines are the fieldsobtained from the model. The dashed and dotted blue lines are respectively surface and volumecharge contributions to the interaction dipolar field (solid blue line). The red line is the total dipolarfield from the model (obtained adding the intralayer demagnetising field to the interlayer interactiondipolar field). The parameter ∆ has been adjusted for the magnetisation arctan analytical profileto fit the micromagnetic profile. fields, whereas the DMI favours a CCW chirality. We further notice that for multilayers(VII) and (IX), the DW reorientation that shall occur in the bottom layers is indeed notobserved by our enhanced surface sensitive technique. This can be seen in Fig. 4, as thecorresponding squares are red in a red gradient area. We believe that these predictions areimportant as they allow to easily obtain for a given set of magnetic parameters M s , K u , A and multilayer geometry, an approximation of the threshold DMI value D u that ensures aunique DW or skyrmion chirality inside the whole stacking. We will see in the last sectionthat knowing whether twisting of the magnetisation chirality occurs or not is crucial as theDW or skyrmion dynamics are strongly modified depending on their actual spin textures.15 igure 4. Diagram comparing A dip and A dmi for each multilayer that has been characterisedby CD-XRMS. When |A dip | < |A dmi | pure N´eel DWs are stabilised, and red (blue) indicates CW(CCW) chirality. The gradient areas correspond to |A dip | > |A dmi | with more and more pronouncedreorientation into flux-closure DWs. Coloured squares indicate the chirality as it has been observedby CD-XRMS for each sample labelled by roman numbers, where red stands for CW chirality andblue stands for CCW chirality. CONSEQUENCE ON DYNAMICS OF SKYRMIONS IN MULTILAYERS
Stackings of ultrathin magnetic layers have been the subject of a large research effortin the last years for demonstration of room-temperature stabilisation of small magneticskyrmions. Moreover, it has already been shown that the interfacial spin-orbit torquescan be used in order to move efficiently these skyrmions. As aforementioned, the existenceof a complex spin texture through the different magnetic layers of a stacking due to thecompetition between interlayer dipolar fields and DMI fields is not only expected for DWs(and observed as we have demonstrated here) but should be equivalent for skyrmions inmultilayers, as it has been also pointed out in other very recent works . The fact thatcolumnar skyrmions stabilised in multilayers, which is the favourite strategy to make themstable at room temperature, possess a twisting of their chiral spin texture through theirthickness shall strongly alter their dynamics. However, to our knowledge this crucial issue16as never been properly considered so far. We show in Fig. 5a an example of the actualskyrmion profile in a multilayer of structure [X(1)/Co(1)/Z(1)] with D = 0 . − , ob-tained by micromagnetic modeling similar as before (see Methods). Equivalently to whathappens for DWs, the skyrmion profile exhibits a progressive reorientation from CCW N´eelto Bloch through its thickness, and finally to CW N´eel chirality in the topmost magneticlayers. Moreover the skyrmion diameter also evolves depending on the layer position in thestack, being larger in the central layers and smaller in the external layers. Such skyrmionswith N´eel caps were recently predicted and thoroughly modeled by micromagnetic simula-tions for amorphous, 50 nm Gd/Fe thick layers . With our findings, we demonstrate thatthe stabilisation of such hybrid chiral skyrmions occurs in multilayered stackings of ultra-thin magnetic and non-magnetic layers with strong interfacial DMI. In our multilayers, thedifferent magnetic layers are indeed exchange decoupled due to Pt/Ir and Pt/Al O spacers,which enhance reorientation effects as compared to exchange-coupled bulk materials likeGd/Fe . In the following, we show that through a precise engineering of the interfacialspin torques in stacked multilayers, new possibilities can be anticipated for achieving a goodcontrol of the current-driven skyrmion dynamics.To this aim, we have simulated the current-induced dynamics of isolated hybrid chiralskyrmions in an extended multilayer similar to the one shown in Fig. 5a, for different valuesof the DMI and different spin injection geometries. Note that we only consider here damping-like torques originating from vertical spin currents, which can be provided for example bythe spin Hall effect and a current flowing along the x direction. The resulting skyrmionvelocities are reported in Fig. 5b-d. In Fig. 5b, we first present the case in which the torqueis applied only in the first bottom and the first top layers and the injected spin polarisation isopposite in these two layers, for example, with the multilayer enclosed between two layers ofthe same heavy metal. We find that both the longitudinal and transverse velocities remainconstant for moderate values of D (with transverse velocity larger than longitudinal velocity,as it is already known, due to the gyrotropic motion of the topological skyrmions ) but thendrops to zero when | D | > D u , corresponding to the critical minimal value of D above whichskyrmions with a unique chirality across all layers are stabilised. The velocity drops becauseskyrmions of opposite chiralities are driven in opposite directions for identical polarisationsof injected spins, but in the same direction for opposite spin injections. As a consequencethe driving forces on the bottom CCW and top CW N´eel skyrmion layers add up in the17ange of D for which the chirality twist is present and cancel out for a uniform skyrmionchirality. In Fig. 5c, we present the results for the opposite case where the injected spinshave identical polarisations in the bottom-most and top-most layers, for example, with themultilayer enclosed between heavy metals of opposite spin Hall angles. In that case wefind that the skyrmion motion is completely canceled up to D u because the hybrid complexspin texture leads to opposite skyrmion chiralities in bottom and top layers. The motionoccurs only for | D | > D u , in which case top and bottom layers chiralities and injectedspins are identical. Finally, for a uniform current injection (Fig. 5d, for example, with a Ptlayer adjacent to the bottom of each ferromagnetic layer), we find that the transverse ( v y )velocity is roughly proportional to D up to D u (note the reduced velocity in cases of onlyexternal layers injection as compared to uniform injection within all layers, as fewer spins areinjected in total). Again it is because skyrmions of opposite chiralities are driven in oppositedirections, which implies that the global motion is related to the balance of CCW and CWN´eel skyrmions layers. Even if D does not directly affect the velocity of a given structure, as D controls the amount of layers with CCW and CW N´eel orientations, it also controls theoverall velocity. We also notice that in this case, the driving forces compensate at D = 0,leading to a zero global v y . The small v x component arises from the Bloch part of thehybrid chiral skyrmion. Interestingly, this case corresponds to the experimental studies ofskyrmion dynamics reported in Refs. 12 and 20 and provide a simple explanation why a verylarge (and hard to achieve) interfacial DMI is mandatory to achieve fast skyrmion motion instacked multilayered systems in comparison to the case of single layers or multilayers withfew repeats. The high value of D u ≈ . − for 20 repetitions of X/Co(1 nm)/Z is aboveall reported values in multilayers. This shows that pure N´eel skyrmions in such structuresare very unlikely to occur. However, D u can be lowered to realistic values in 20 repetitionsof a thinner ferromagnetic layer such as in Pt/Co(0 . x or in structures for which M s is lower. Finally, this series of simulations thus provides guidelines on how to designmultilayers with many repetitions hosting skyrmions, and how to engineer the interfacialspin torques in order to achieve a fast motion when hybrid chiral skyrmions are present inthese multilayers. 18 igure 5. a. Cut view of the simulation volume for a [X(1)/Co(1)/Z(1)] multilayer with D =0 . − . Arrows point in the direction of the magnetisation, m z is given by the colour of thearrows from red (-1) to blue (1), while m y is displayed by the colour of the grid from black (-1) towhite (+1). The m z component in the top layer is represented in perspective view by the colourfrom red (-1) to blue (1) b-d. Skyrmion velocities for different values of D and geometries. Right/uppointing blue/red triangles stand for horizontal/transverse velocity components, obtained for b.opposite injection in bottom-most/top-most layers, c. identical injection in bottom-most/top-mostlayers and d. uniform injection. The injection geometry is depicted by the inset in each case. OUTLOOK
The presented skyrmion dynamics simulations open new perspectives for the engineeringof multilayered magnetic materials. Using such techniques, it will be possible to bettercontrol DW and skyrmion motion with spin-orbit torques in asymmetric multilayers and19ven in most common PMA symmetric stacks, such as Pt/Co and Pd/Co, which exhibitonly low DMI or even no significant DMI. Notably, injecting spin Hall effect spin currentsfrom the same material below and above the multilayer will lead to motion, as the chiralityof the structure reverses from bottom to top. Moreover, as we have shown with the systemdisplayed in Fig. 1, the balance between the number of layers and the value of D in thestructure actually controls the position of the Bloch DW within the multilayer. It is thuspossible to control the overall chirality of the structures by engineering the base multilayerrepetition number, thus tuning the ratio of CCW N´eel, Bloch, and CW N´eel DWs. Thiswould allow to control the direction of the motion in the case of a uniform spin currentinjection as well as the chirality-related properties of DW and skyrmion motion. Moreover,our simulations of current-induced motion show the dynamical stability of the twisted chiralskyrmions as they are robust even under current injection, which can thus be used forspin-current-induced motion.With the help of our description, it is possible to understand and predict the occurrenceof twisting of DWs and skyrmions chirality in multilayers. Notably, the common picture ofa constant internal tilt inside the DW or constant chirality inside the skyrmion and throughthe different layers is revealed not to be valid. We have experimentally demonstrated thisreorientation effect with a surface-sensitive X-ray diffraction technique, allowing to observethe surface chirality of the DW configurations. Such complex structures strongly modify theway the DMI should be quantified in magnetic multilayers.These results highlight the importance of advanced engineering of spin-orbit related in-terfacial properties, combining PMA, SOT and DMI in multilayered systems to promote thestabilisation and the fast dynamics of ultrasmall skyrmions at room-temperature, which areneeded for any type of potential applications. METHODSSamples fabrication and characterisation
All series of multilayers were grown by dc and rf magnetron sputtering at room-temperature on oxidised silicon substrates, after deposition of a Ta or Pt buffer as describedin Table I and capped with Pt( ≥ M S and effective perpendicular anisotropy field H eff = H K − M S / Demagnetisation procedeure and MFM imaging
We have first compared the domain periodicity in the labyrinthine and aligned paralleldomains configurations (obtained after out-of-plane and in-plane demagnetisation, respec-tively), measured by magnetic force microscopy (MFM, see Methods) in lift mode for ourmultilayer (VIII) [Pt(1)/Co(0.8)/AlO x (1)] . For both in-plane and out-of plane demagneti-sation procedures, the maximum oscillating field was set higher than the anisotropy fieldin the multilayers, i.e. B ≥ . . ∼ ◦ ) with the field tofavour multiple reversals of the magnetisation everywhere in order to get closer to a parallelstripes ground state. The extraction of the mean domain periodicity λ after Fourier trans-form reveals that it can differ significantly, by up to 20%, between the two demagnetisationprocedures. All rigorous models of magnetic domains in multilayers have however been de-rived for parallel stripe domains . Even if the difference in the domain width for thetwo demagnetised state configurations may be very small in the ideal case, this deviationprecludes the use of perpendicularly demagnetised domains for the estimation of the DMI,as only the values derived from the parallel stripe domains periodicity provide consistentresults. The multilayers were imaged with Asylum Low Moment tips in double pass, tappingmode followed by lift mode at 20 nm height, at room-temperature. XRMS measurements
XRMS experiments have been carried out at the SEXTANTS beamline of the SOLEILsynchrotron. The diffraction was measured in reflectivity conditions for circularly left (CL)and right (CR) polarisations of the incident X-ray beam. The photon energy was set at Co L edge (778.2 eV) using the RESOXS diffractometer. The diffracted X-rays were imagedon a square CCD detector covering 6 . ◦ at the working distance of this study. All theimages have been geometrically corrected along the Q x -direction in order to account for the21rojection effect related to the photon incidence angle of 18 . ◦ . The sum of the imagesobtained with CL and CR polarised light gives rise to a diffraction pattern around thespecular beam (which was blocked by a beamstop to avoid saturation of the CCD) in thereciprocal plane ( Q x , Q y ). Denoting I CL and I CR the intensities collected by the CCD forCL and CR polarised incident lights, the circular dichroism (CD) of the scattering signal isdefined as ( I CL − I CR ) / ( I CL + I CR ). Domain wall energies - K eff model For uncoupled, independent DWs, separating domains of size λ/ (cid:112) A/K , there exist two straightforward approximations for K and thus forthe DW energy . For single and ultrathin layers (of thickness t (cid:28) ∆), the demagnetisingfields favour in-plane magnetisation inside the DW so that the anisotropy affecting the DWis the effective perpendicular anisotropy K = K u − µ M s /
2. On the contrary, for thicklayers ( t (cid:29) ∆), the in-plane alignment inside the DW is disfavoured and K = K u + µ M s / t are in the order of 10 nm. Following Ref. 45, we can refine the formula ofthe DW anisotropy K by assuming that for calculating demagnetising fields roughly, theDW can be considered as a monodomain magnetic body of width 2∆, height t and infinitelength. Due to the arctan profile of the DW, an elliptic shape is a good approximation.Inserting the demagnetising factors of the elliptic cylinder N x = tt + 2∆ N z = 2∆ t + 2∆in the effective anisotropy K gives a simple expression which allows to find the DW widthby solving ∆ = (cid:115) AK u + µ M s t − t +2∆) By summing the contributions of exchange, DMI, anisotropy and demagnetising fields in theelliptic body , the DW energy is then σ dw = 2 A/ ∆ + 2 K u ∆ − π | D | + µ M s (cid:18) t − t + 2∆ (cid:19) λ (cid:29) ∆ ∼
0) model we get (cid:15) demag = 2 µ M s λt π ∞ (cid:88) n (cid:62) , odd n (cid:0) − e − πnt/λ (cid:1) which allows one to find σ dw from the observed λ . Indeed, by minimising the total energy (cid:15) tot = 2 σ dw /λ + (cid:15) demag relative to λ we get σ dw = µ M s λ t π ∞ (cid:88) n (cid:62) , odd n (cid:18) − e − πnt/λ − πntλ e − πnt/λ (cid:19) which allows one to find D by equaling it to the previous expression of σ dw as we know M s , K u , and A estimated to be 10 pJ m − . A was obtained by determining the Curietemperature from temperature-dependent SQUID measurements. To apply this K eff model,the multilayer is treated as an effective magnetic medium filled with diluted moments. Notethat this assumption is valid as long as the periodicity of the stack is not significantly largerthan the DW size , which is always verified for the samples considered here. Domain wall energies - micromagnetic simulations
In order to find the strength of the DMI in the stripe domains configuration withoutmaking assumptions on the DW profiles, we performed micromagnetic simulations with theMumax3 solver in a 3D mesh accounting for the full geometry of the multilayers. Thesimulation volume is λ ± dλ ×
32 nm × N p , respectively, in the x , y and z directions. TwoDWs separating up, down and up domains are initialised at ( − λ ± dλ ) / λ ± dλ ) / λ ± dλ periodicity of the stripes. Periodic boundary conditionsinclusive of the periodic stray fields calculated for 64 ×
64 identical neighbors in the x and y directions were introduced. The x cell size was 0 .
25 nm for the XRMS multilayers series.The z cell size was 0 . . p andthickness t . Simulations are performed at 0 K.Given the experimental value of λ , the simulation is initialised with its DWs having a ψ = 45 ◦ in-plane tilt of internal moments for x sizes λ − λ − λ , λ +1 nm, λ +2 nm.Each system is relaxed in order to find the ground state energy density (cid:15) ( λ ), so that we getthe local value of d(cid:15)/dλ at λ . Performing this operation for various values of D allows tofind D full such that d(cid:15)/dλ = 0 by interpolation.23 omain wall fields - Analytical derivation To find the dipolar field in the top layer of the multilayer, we have to find the solutionfor the potential φ of Laplace equation ∇ φ = ρ V with ρ V the volume charges and boundaryconditions related to surface charges ρ S . As was done in Lemesh et al. we separate volumeand surface magnetic charges contributions. We consider λ -periodic stripe domains in the x direction (uniform along y ) and approximate the DW profile by the arctan profile. Wenote t the magnetic layer thickness, p the multilayer periodicity and N the total number oflayers. Assuming a perfectly N´eel DW in all layers, we can obtain the DW width ∆ fromthe (∆, λ , ψ ) model.We first solve for a single layer ∇ φ = 0 with ∂φ/∂z ( x, ± t/ − ) = ∂φ/∂z ( x, ± t/ + ) ± ρ S ( x ).As ρ S ( x ) corresponds to the charges of two opposite, alternate DW profiles (up to down anddown to up) localised every λ/
2, we can write ρ S ( x ) = ∞ (cid:88) k = −∞ f S ( x ) ∗ δ ( x − kλ ) + ∞ (cid:88) k −∞ − f S ( x ) ∗ δ ( x − kλ − λ/ f corresponds to a single DW profile f S ( x ) = M s m z ( x ) = M s tanh ( x/ ∆), that is, bycombining all Dirac functions and swapping the derivatives in the convolution product, ρ S ( x ) = f (cid:48) S ( x )2 ∗ g ( x )with g ( x ) = x ∈ [ kλ ; λ/ kλ [ − x ∈ [ λ/ kλ ; λ + kλ [ . As the magnetic charge distribution is λ periodic we can decompose it in Fourier series ρ S ( x ) = ∞ (cid:88) k = −∞ ¯ ρ S ( k )e − πkxλ and solve ∇ ¯ φ ( k, z ) = ∂ ¯ φ ( k, z ) ∂z − (cid:18) πkλ (cid:19) ¯ φ ( k, z ) = 0with ∂ ¯ φ/∂z ( k, ± t/ − ) = ∂ ¯ φ/∂z ( ± t/ + ) ± ¯ ρ S . By properties of the Fourier transform definedas ¯ f ( ξ ) = 1 √ π (cid:90) f ( x )e − iξx dx
24e know that ¯ ρ S = f (cid:48) S / ∗ g = √ π (cid:0) f (cid:48) S ¯ g (cid:1) / √ πiξf S ¯ g/ ρ S ( 2 πkλ ) = √ πiπkλ (cid:20) − iM s ∆ (cid:114) π (cid:18) π ∆ kλ (cid:19)(cid:21) (cid:20) − i √ πk sin (cid:18) kπ (cid:19)(cid:21) = − i (cid:114) π πM s ∆ λ sin (cid:18) kπ (cid:19) csch (cid:18) π ∆ kλ (cid:19) . Using the boundary conditions to solve Laplace equation above the layer we find for z > t/ φ ( x, z ) = ∞ (cid:88) k =1 M s ∆ k sin (cid:18) πk (cid:19) csch (cid:18) π ∆ kλ (cid:19) sinh (cid:18) πktλ (cid:19) sin (cid:18) πkxλ (cid:19) e − πkzλ with x = 0 in the center of the DW between down and up domains. For a multilayer thereare N layers located at p/ kp with 0 ≤ k ≤ N −
1. In the top layer, the interlayerinteraction field will then be the sum of all other layers stray fields B dip , S ( x ) = − µ ∂∂x (cid:34) N − (cid:88) n =1 φ ( x, np ) (cid:35) that is by grouping the exponent sum B dip , S ( x ) = − µ ∂∂x (cid:34) ∞ (cid:88) k =1 , odd M s ∆ k csch (cid:18) π ∆ kλ (cid:19) sinh (cid:18) πktλ (cid:19) sin (cid:18) πkxλ (cid:19) e − πkpλ − e − πkNpλ − e − πkpλ (cid:35) = − µ ∞ (cid:88) k =1 , odd πM s ∆ λ csch (cid:18) π ∆ kλ (cid:19) sinh (cid:18) πktλ (cid:19) cos (cid:18) πkxλ (cid:19) e − πkpλ − e − πkNpλ − e − πkpλ which can be determined numerically.We can now solve for a single layer ∇ φ = ρ V ( x ) for | z | < t/ ∇ φ = 0 for | z | > t/ ∂φ/∂z at z = ± t/
2. We have again ρ V ( x ) = ∞ (cid:88) k = −∞ f V ( x ) ∗ δ ( x − kλ ) + ∞ (cid:88) k −∞ − f V ( x ) ∗ δ ( x − kλ − λ/ f V ( x ) = − M s ∇ · m = − M s ∂m x ∂x = M s ∆ tanh ( x/ ∆)cosh ( x/ ∆)25he volume charges for a single DW. Similar to ρ S we get ρ V ( x ) = [ f (cid:48) V ( x ) ∗ g ( x )] / ρ V ( 2 πkλ ) = f (cid:48) V / ∗ g ( k ) = √ πi (2 πk/λ ) f V ¯ g/ √ πiπkλ (cid:20) iM s ∆ 2 πkλ (cid:114) π (cid:18) π ∆ kλ (cid:19)(cid:21) (cid:20) − i √ πk sin (cid:18) kπ (cid:19)(cid:21) = − i (cid:114) π − π M s ∆ kλ sech (cid:18) π ∆ kλ (cid:19) sin (cid:18) kπ (cid:19) . Using the boundary continuity to solve Laplace equation above the layer we find for z > t/ φ ( x, z ) = ∞ (cid:88) k =1 M s ∆ k sin (cid:18) πk (cid:19) sech (cid:18) π ∆ kλ (cid:19) sinh (cid:18) πktλ (cid:19) sin (cid:18) πkxλ (cid:19) e − πkzλ so that B dip , V ( x ) = − µ ∞ (cid:88) k =1 , odd πM s ∆ λ sech (cid:18) π ∆ kλ (cid:19) sinh (cid:18) πktλ (cid:19) cos (cid:18) πkxλ (cid:19) e − πkpλ − e − πkNpλ − e − πkpλ One still has to consider the self-demagnetising field of the top layer itself. The surfacecharges distribution does not contribute to the z -average of the field as it generates a fieldantisymmetric in z. However the volume charges contribution must be considered. Solvingagain the Laplace equation but inside the magnetic layer ( | z | < t/
2) we find B dip ( x, z ) = µ ∞ (cid:88) k =1 , odd πM s ∆ λ sech (cid:18) π ∆ kλ (cid:19) cos (cid:18) πkxλ (cid:19) (cid:20) cosh (cid:18) πkzλ (cid:19) e πktλ − (cid:21) , that we can average between z = − t/ z = t/ B dip , self ( x ) = µ ∞ (cid:88) k =1 , odd πM s ∆ λ sech (cid:18) π ∆ kλ (cid:19) cos (cid:18) πkxλ (cid:19) (cid:34) − e − πktλ λπkt − (cid:35) . The dipolar interlayer interaction pushing to reverse the DW is then B dip ( x ) = B dip , S ( x ) + B dip , V ( x )while the total dipolar field is finally B dip , t ( x ) = B dip , S ( x ) + B dip , V ( x ) + B dip , self ( x ) . The N´eel internal field along x of the DW can be described by B dmi ( x ) = 2 DM s ∂m z ∂x = 2 DM s ∆ sech (cid:16) x ∆ (cid:17)
26s we approximate it to a classical tan profile.In order to compare the strengths of B dip and B dmi , we evaluate their actions locally by A = 16∆ (cid:90) − B ( x ) m x ( x ) dx that corresponds to the in-plane rotation driving force. Skyrmion velocities - micromagnetic simulations
In order to find the potential current-induced velocities for different skyrmion chiralityconditions and spin current geometries, we performed micromagnetic simulations in the fullgeometry for the multilayer [X(1)/Co(1)/Z(1)] . Here, Co(1) was chosen in order to reducethe simulation grid size. The simulation volume is 256 nm ×
256 nm × N p , respectively, inthe x , y and z directions. Periodic boundary conditions inclusive of the periodic stray fieldscalculated for 3 × x and y directions were introduced. All cellsizes were 1 nm. Before current is applied the configuration is relaxed from an initial Blochskyrmion with CCW wall internal magnetisation. The current density is then modeled as afully polarised (along y ) vertical spin current of current density J = 2 . × A m − . TheGilbert damping α was set to 0.1 and the out-of-plane field to B = 200, or 300 mT (for D > D u ). Due to this required increase of external field to confine the skyrmion to a stablecircular shape for high D > D u , the skyrmion size is reduced and thus velocity as well ,without affecting the discussed quantitative behavior. Simulations were performed at 0 K. AUTHOR CONTRIBUTIONS
W.L., J.-Y.C., N.R., V.C. and N.J. conceived the project; W.L. deposited the multilayerswith the help of S.C.; W.L performed the magnetic characterisation, and together withD.M. and K.B., performed the MFM measurements of domains spacing; J.-Y.C., N.J., N.R.and V.C. performed the XRMS experiments; W.L. developed the analytical derivation part;W.L. and N.R. performed the micromagnetic simulations; all authors contributed to theanalysis and interpretation of the experimental results and to the writing of the manuscript.27
CKNOWLEDGMENTS
We gratefully acknowledge I. Lemesh and F. B¨uttner for sharing their manuscript whileit was still in press. Financial support from FLAG-ERA SoGraph (ANR-15-GRFL-0005)and European Union grant MAGicSky No. FET-Open-665095 is acknowledged. ∗ [email protected] I. Dzyaloshinsky, J. Phys. Chem. Solids , 241 (1958). T. Moriya, Phys. Rev. , 91 (1960). A. Fert,
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