Table Of Content

Magnetic domain wall dynamics in wide permalloy strips Virginia Estévez∗ and Lasse Laurson COMP Centre of Excellence and Helsinki Institute of Physics, Department of Applied Physics, Aalto University, P.O.Box 11100, FI-00076 Aalto, Espoo, Finland. Domain walls in soft permalloy strips may exhibit various equilibrium micromagnetic structures depending on the width and thickness of the strip, ranging from the well-known transverse and vortex walls in narrow and thin strips to double and triple vortex walls recently reported in wider strips[V.EstévezandL.Laurson,Phys. Rev. B91,054407(2015)]. Hereweanalyzethefielddriven 6 dynamics of such domain walls in permalloy strips of widths from 240 nm up to 6 µm, using the 1 knownequilibriumdomainwallstructuresasinitialconfigurations. Ourmicromagneticsimulations 0 show that the domain wall dynamics in wide strips is very complex, and depends strongly on the 2 geometryofthesystem,aswellasonthemagnitudeofthedrivingfield. Wediscussindetailtherich variety of the dynamical behaviors found, including dynamic transitions between different domain n wallstructures,periodicdynamicsofavortexcoreclosetothestripedge,transitionstowardssimpler a domain wall structures of the multi-vortex domain walls controlled by vortex polarity, and the fact J thatforsomecombinationsofthestripgeometryandthedrivingfieldthesystemcannotsupporta 9 compact domain wall. 2 PACSnumbers: 75.60.Ch,75.78.Fg,75.78.Cd ] l l a h I. INTRODUCTION strips with either TW or VW as the equilibrium DW - structure8–13,36–41. In both cases, the DW dynamics ex- s hibits a Walker breakdown42, an instability occurring e Thestudyofstaticanddynamicpropertiesofthemag- m when the DW internal degrees of freedom are excited netic domain walls (DWs) in ferromagnetic nanostruc- by a strong enough external driving force. This force . tures has recently attracted a lot of attention not only t can be a magnetic field B > B or a spin-polarized a due to the related fundamental physics aspects, but also W m because of the emergence of technological applications electric current J > JW, with BW and JW the Walker field and current density, respectively8,11,15,22. For small - based on DWs and their dynamics, such as memory1,2 d and logic devices3–5. Consequently, DW dynamics in driving forces, the DW velocity increases up to BW or n J (steady, or viscous regime). At the onset of the nanostrips and wires has been extensively studied, us- W o Walker breakdown, the DW velocity decreases abruptly ing both applied magnetic fields6–15 and spin-polarized c asaconsequenceofperiodictransitionsbetweendifferent [ electric currents16–21 as the driving force. DW structures. In narrow and thin strips with TW as One of the typical materials used for the study of 1 the equilibrium structure, repeated transitions between v DWs and their dynamics in strip geometry is permal- TWs of different polarities (or signs of the TW internal 0 loy, a soft magnetic material with a negligible magne- magnetization) take place via nucleation and propaga- 6 tocrystalline anisotropy23,24. Such systems exhibit in- tion of an antivortex across the strip width; we will refer 0 plane domains along the long axis of the strip, induced to the transient state as the antivortex wall, or AVW10. 8 by shape anisotropy. Previous works have shown that 0 For wider strips with a VW equilibrium DW, periodic the structures of the DWs separating such domains may . transitionsbetweenVWandTWstructuresaretypically 1 be relatively complex, and depend strongly on the sam- observed10,36. However, so far little is known about the 0 ple geometry9,10,25–27. For nanostrip geometries, with 6 DW dynamics in even wider strips, especially ones with the strip width and thickness of the order of 100 nm 1 DVW or TVW as the equilibrium DW structure. and 10 nm or less, respectively, the known equilibrium : v DW structures are the transverse DW (TW) and the In this paper we analyze numerically field driven DW Xi asymmetric transverse DW (ATW)27–33. For somewhat dynamicsinpermalloystripsconsideringawiderangeof wider and/or thicker strips, the vortex DW (VW) is strip widths w from 240 nm up to 6 µm. We focus on r a thestablestructure27,30–34. Arelated,three-dimensional relatively thin strips with the thickness ∆ in the range z vortex-like flux-closing DW structure has been reported of 5 to 25 nm. Depending on w and ∆ , the equilibrium z in permalloy strips of thickness exceeding 60 nm.35 Re- DW structure in such strips can be either VW, DVW cently we have shown that in the case of even wider or TVW26 (see Fig. 1 for examples of these structures). permalloy strips than those with VW as the equilibrium Since the space of parameters spanned by w, ∆ and z DWstructure,twoadditionalequilibriumDWstructures B is quite large, we focus on specific example cases, ext appear,namelythedoublevortex(DVW)andtriplevor- covering each of the three above-mentioned equilibrium tex walls (TVW)26. DW structures, and illustrating the complex nature of Studies of field and current driven dynamics of DWs theDWdynamicsinwidestrips,exhibitingalsoastrong in permalloy strips have focused mostly on narrow dependence on both the geometry of the system, and on 2 the magnitude of the driving field. Already for rather (a) thickness  confined geometries with VW as the equilibrium DW z structure, several possible dynamical behaviors are en- width w countered,inparticularforlargeenoughB (i.e. above ext length l thesmallfieldsteady/viscousregime). Inadditiontovar- iousperiodictransitionsoccurringbetweendifferentDW (b) structures, we also report on localizedoscillatory motion of the vortex core close to the strip edge. This novel behavior is characteristic of wide enough strips, giving VW DVW TVW rise to an extended plateau in the velocity vs field curve. AnothersignatureofDWdynamicsinwidestripsisthat for some combinations of w, ∆ and B , DWs are un- z ext stable: the DW’s width grows without limit as different components of the DW assume a different propagation velocity14. DVWs are found to be structurally stable DW against small applied fields depending on the polarity of the vortices. In the stable case (parallel polarity), B thedynamicsexhibitsauniquesmallfieldsteady/viscous regime, but larger fields tend to induce a transition into a simpler VW structure, or lead to complex, turbulent- like dynamics with repeated transitions between numer- FIG. 1. (color online) (a) Geometry of the permalloy strip. ous different DW structures. For antiparallel polarity, (b)Atopviewofthemagnetizationintheinitialstate. Mag- the DVW transforms in a VW even for very small fields. netization points along the long axis of the strip within the For larger fields the dynamics observed is the same in- two domains (as indicated by the arrows) forming a head- dependently of the polarity. The same is true also for to-head configuration. The domain wall separating these TVWs, whose stability at low fields depends on the po- domains has a geometry-depedendent equilibrium structure. laritiesofthethreevortices. IngeneraltheTVWstendto Here, we consider systems where the latter is either a vortex transform into simpler DW structures during their field- wall(VW),doublevortexwall(DVW),oratriplevortexwall drivendynamics. Thepaperisorganizedasfollows: The (TVW). To study the DW dynamics, an external magnetic field B is applied along the long axis of the strip. following section (Section II) describes the details of our ext micromagneticsimulations,whileinthethreesectionsaf- ter that (Sections III, IV and V), DW dynamics starting The micromagnetic simulations have been performed fromVW,DVWandTVWequilibriuminitialconfigura- usingtheGPU-basedmicromagneticcodeMuMax343–45. tions, respectively, are described and discussed. Section To calculate the magnetization dynamics of the system, VI finishes the paper with a summary and conclusions. the Landau-Lifshitz-Gilbert equation46,47, ∂m/∂t=γH ×m+αm×∂m/∂t, (1) eff II. MICROMAGNETIC SIMULATIONS issolvednumerically. Here,misthemagnetization,γthe The system under study consists of a permalloy strip gyromagneticratio,andHeff theeffectivefield,withcon- of width w and thickness ∆ , satisfying ∆ (cid:28) w, see tributions due to exchange, Zeeman, and demagnetizing z z Fig.1(a). Tomimicaninfinitelylongstrip,themagnetic energies. The size of the discretization cell used depends charges at the two ends of the strip are compensated. In onthesystemsize,rangingfrom3to5nm,butisalways allthecasesconsideredtheactualsimulatedlengthl,i.e. boundedbytheexchangelength,Λ=(2A/µ0Ms2)1/2 ≈5 the length of the computional window which is centered nm, in the in-plane directions, and equals ∆z in the out- at and moving with the DW, satisfies l ≥ 4w. The ini- of-plane direction. The DW velocity is calculated once tial state is an in-plane head-to-head domain structure, the DW structure is in the steady state; thus, the total with the equilibrium DW structure26 corresponding to simulation time considered is at least 300 ns. the w and ∆ of the strip in the middle of the sample, z see Fig. 1 (b). To analyze the DW dynamics, a constant externalmagneticfieldB isappliedalongthelongaxis III. VORTEX WALL DYNAMICS ext of the strip. All the results presented in this work have been obtained using the typical material parameters of ThevortexwallisaDWstructurecharacterizedbythe permalloy, i.e. saturation magnetization M =860×103 chirality or sense of rotation of the vortex, and the core s A/m, exchange constant A = 13 × 10−12 J/m, and polarity p. The latter may assume two different values, ex theGilbertdampingconstantα=0.01. Wefocusonthe p = ±1, corresponding to the two possible out-of-plane idealcaseofperfectstripswithoutanykindofdisorderor magnetization directions of the core. Vortex wall is the impurities, and consider a temperature T equal to zero. equilibrium DW structure within a wide range of w and 3 600 0 ns 7.5 ns (a) ∆= 10 nm z ∆= 20 nm 500 z 400 12 ns 20 ns w = 240 nm ] s m/300 [ v 200 25 ns 32 ns 100 0 37 ns 45 ns 0 1 2 3 4 B [mT] ext (b) w = 768 nm  = 15 nm z 300 U N FIG. 3. (color online) Vortex wall dynamics in a strip with ] S w =240 nm and ∆ =10 nm for B =1.5 mT (i.e. above s z ext m/ T the Walker field BW = 1.2 mT). The DW dynamics repeats [200 A a cycle where the vortex core first moves out of the strip v B through the top strip edge, forming an ATW, followed by L the injection of a vortex core from the top edge with the E opposite polarity, which subsequently moves across the strip 100 tothebottomedge,reversingtheATWmagnetization. Then avortex,againwithapolarityoppositetothatoftheprevious one,isnucleatedfromthebottomedge,andmovestothetop edge,againreversingtheATWmagnetization. Thenthesame 0 process repeats. 0 1 2 3 4 B [mT] ext FIG.2. (coloronline)v(B )curvesfordifferentstripgeome- to clear differences in the v(B ) curves already below ext ext tries,wheretheVWistheequilibriumDWstructure. (a)For the ∆ dependent Walker field. In the case of ∆ = 10 z z two strips, both having the same width of w = 240 nm and nm, for very small fields v increases linearly with B , a ext in-planecellsize5nm,anddifferentthicknesses∆z =10nm behaviorarisingfromsteadyVWmotionwiththevortex and∆z =20nm. (b)Thesameforastripwithw=768nm, core assuming an off-centre position within the strip10. ∆ =15 nm and in-plane cell size 3 nm. For 3 mT ≤B ≤ z ext For B ≥ 0.4 mT the vortex core is expelled out of 3.1 mT the DW is unstable; the DW width grows without ext the strip due to the gyrotropic force48, leading to a TW, bound. andtoasub-linearv(B )relation10. Incontrast,forthe ext thickerstripwith∆ =20nm,thelineardependenceofv z ∆z26, and thus it is a pertinent question to what extent on Bext, originating from steady VW dynamics with the thedynamicsexhibitedbyitdependsonthegeometryof vortex core within the strip, persists up to the Walker the strip. To this end, in what follows, we consider some field. Notice also that for Bext such that the VW has example cases, demonstrating that already in relatively transformed into a TW for ∆z = 10 nm, the TW ve- narrow strips, several different kinds of field-driven VW locity for a given Bext exceeds the corresponding VW dynamics may be observed. velocity in the system with ∆z = 20 nm. This is likely due to the large energy dissipation associated with the dynamics of the vortex core48. A. Vortex wall dynamics in narrow strips AbovetheWalkerbreakdown, v dropsdramaticallyas a consequence of the onset of periodic transformations To illustrate this, we start by considering the DW ve- between different types of DW structures. An example locity v as a function of the applied field B . In Fig. 2 of this behavior is shown in Fig. 3, where the VW dy- ext we show that quite different v(B ) curves are obtained namics for B = 1.5 mT, exceeding the Walker field ext ext depending on w and ∆ . Fig. 2 (a) shows that for a rel- B , is shown for a strip with w = 240 nm and ∆ = z W z atively narrow strip width w = 240 nm, considering two 10nm(seealsoSupplementalMaterialMovie149). Peri- differentthicknesses(∆ =10nmand∆ =20nm)leads odic transformations between VW and ATW structures z z 4 0 ns 2 ns 400 300 m] n 5 ns 7 ns [yc200 100 0 10 ns 12 ns 0 5 10 15 20 25 t [ns] FIG. 5. (color online) The trajectory y of the vortex core in c thevicinityoftheedgeofthesystemforw=768nm,∆ =15 z 13 ns 16 ns nm and Bext = 1.5 mT, illustrating the attraction-repulsion effect. The red line at y = 0 corresponds to the strip edge. c TheinsetshowsanexampleoftheDWconfigurationwiththe vortex core close to the edge of the strip. 18 ns 21 ns theravortexoranantivortexcore(seealsoSupplemental 1 Material Movie 249). Thicker strips appear to lead to a smaller number of different periodicities. For instance, for w =240 nm and ∆ =20 nm, only two periodicities 23 ns 25 ns z are observed for B > B , consisting of transforma- ext W tionsbetweenVWandATWstructures. WhiletheATW reversesitsmagnetization,theVWstructuresexhibital- ternatingpolaritiesbutnochangeofchiralitytakesplace (not shown). For some fields, also a nucleation of an antivortex in the edge of the strip is observed, followed by an vortex-antivortex annihilation process. Thus, already FIG. 4. (color online) Vortex wall dynamics in a strip with for relatively confined geometries, the field driven VW w = 240 nm and ∆ = 10 nm for B = 4 mT (above z ext the Walker breakdown, B = 1.2 mT). Now the dynamics dynamics is quite complex, given that different geome- W alternatesbetweenATWsofdifferentmagnetization,withits tries and applied fields may lead to numerous different reversal mediated by either a vortex or an antivortex core dynamical behaviors. moving across the strip width. B. Vortex wall dynamics in wide strips are observed, mediated by the transverse motion of the vortex core across the strip width. Notice that here the In the case of wider strips, the VW dynamics is even vortex core dynamics alternates between injection from more complex. Fig. 2(b) shows the v(B ) curve for ext the top and bottom edges, followed by its top-to-bottom a strip with w = 768 nm and ∆ = 15 nm. As can z and bottom-to-top propagation across the strip, respec- be seen, the resulting v(B ) curve is very different to ext tively. InRef.10,thesamegeometryandBext werefound those observed for more narrow strips [Fig. 2 (a)], and to always lead to vortex core injection from the top strip consists of six different regimes. Analogously to more edge, followed by its “downwards” motion towards the narrow strips, at low fields the velocity increases linearly bottomedge. Wehavecheckedthatthisisduetoslightly with B , with the vortex core assuming again a steady, ext differentvaluesofMsandAex(i.e. Ms=8×105A/mand Bext dependent off-centre position. Aex = 10−11 J/m, respectively) used in the simulations For 0.6 mT ≤ Bext ≤ 2.4 mT, the v(Bext) curve ini- of Ref.10. Given such sensitivity to small differences in tially exhibits a small, negative gradient, but looks oth- thesimulationconditions,onemayexpectthatalsoother erwise essentially like a plateau. Thus, the DW is able kinds of VW dynamics may be observed. to maintain a relatively high velocity of approximately An example thereof is provided in Fig. 4, where the 250 m/s, while for more narrow strips [Fig. 2(a)] Walker same geometry as above (i.e. w = 240 nm and ∆ breakdown leads to significantly smaller DW velocities z = 10 nm) is considered for a larger driving field of for most of this range of B . This behavior arises as a ext B = 4 mT. Now, the DW exhibits transformations consequence of the vortex core moving towards the strip ext between VW, ATW and AVW structures, with the re- edge due to the gyrotropic force, but contrary to the be- peated ATW magnetization reversal mediated by alter- havior observed in more narrow strips, it is not able to nating nucleation and propagation across the strip of ei- leavethestrip. Instead,thecompetitionbetweenthegy- 5 rotropic force and a repulsive interaction of the vortex (a) core with the half-antivortex (HAV) edge defect14 leads to periodic oscillations of the vortex core y-position in 0 ns 7 ns the vicinity of the strip edge. Fig. 5 shows an example of the trajectory (y coordinate y as a function of time c t) of the vortex core in a strip with w = 768 nm and ∆ = 15 nm with the DW subject to an applied field z of B = 1.5 mT. A snapshot of the DW configuration 12 ns 15 ns ext with the vortex core close to the HAV edge defect is in- cluded. An example of this process is also shown in the Supplemental Material Movie 349. The apparent emis- sionsofspinwavesobservedinthemoviemayberelated to short-lived (faster than the frame rate) transient dy- 18 ns 21 ns namicsinvolvingnucleation/annihilationprocesses. Such behavior arises in wide strips due to the interplay be- tweenthegyrotropicforceandtheinteractionofthevor- tex core with the HAV being different to the one in narrow strips. Since in permalloy everything is dominated (b) byshapeanisotropyoriginatingfromthestripedges,the energy needed to displace the vortex core from the middle of the strip towards the edge by the gyrotropic force 0 ns is smaller than in narrow strips. This is evidenced also by the steep increase of v(B ) in Fig. 2(b) for small ext B . At the same time, the HAV edge defect is more ext 3 ns localizedinrelativetermsinwiderstrips,thusleadingto a relatively strong short-range vortex core-HAV interaction, preventingtheformerfrombeingexpelledfromthe 5 ns strip. In other words, the energy necessary to move the vortex core from its equilibrium position in the middle of the strip towards the strip edge is smaller than the energy needed to push the core out of the strip. As a 10 ns result, thecoreexhibitsoscillatorydynamicsclosetothe strip edge, and the DW assumes a relatively high propagation velocity. In what follows, we shall refer to this 25 ns phenomenon as the attraction-repulsion effect. Thisbehaviorischaracteristicofallfieldvalueswithin 1 therange0.6mT≤B ≤2.4mT,withtheexceptionof ext 50 ns B =2.1 mT. Here, v is a little smaller than elsewhere ext in the plateau region visible in Fig. 2 (b). For that case, although most of the time the usual attraction-repulsion effect takes place, in rare instances the repulsion effect is strong enough to displace the vortex core all the way to FIG.6. (coloronline)(a)Walkerbreakdown-typeofdynamics the other edge of the strip, leading to a structural DW forw=768nm,∆ =15nmandB =2.5mT.Thevortex z ext transformation (VW to ATW), and to a reduced DW coreexhibitsasymmetriclateraloscillations,leavingthestrip propagation velocity. at the top edge, but reversing the direction of motion and polarity of the core well inside the strip when closer to the For 2.5 mT ≤ B ≤ 2.7 mT, v decreases abruptly ext bottom edge. (b) The instability occuring for B = 3 mT, [Fig.2(b)]. Inthisregime,theDWdynamicsexhibitsre- ext wheretheDWwidthgrowswithoutlimitastheleadingHAV peatedtransitionsbetweenaVWandakindofstretched edgedefectmovesfasterthanthecombinationofthetrailing TW (or ATW) structure, see Fig. 6(a). The peculiar HAV and the vortex core. feature of this Walker breakdown -like dynamics is that while the vortex core is able to reach the top edge of thestrip, transformingtheDWmomentarilyintoaTW- cillationsissignificantlylarger,leadingtoahigherrateof like structure, the vortex core never reaches the opposite energy dissipation associated with the internal dynamics (bottom) edge of the strip. Thus, the transient TW-like of the DW, resulting also in a significantly reduced DW structurealwayshasthesamemagnetization. Theresult- propagation velocity. ingoscillatorydynamicsofthevortexcoreresemblesthat observedaboveinthecontextoftheattraction-repulsion For B = 2.8 mT and B = 2.9 mT, the velocity ext ext effect. However, the amplitude of the transverse core os- increasesastheattraction-repulsioneffectappearsagain. 6 InthecaseofB =2.8mTalthoughalsoanattraction- ext repulsioneffectisobserved, thetransformationsbetween 0 ns 45 ns different DW structures (VW and ATW) are predomi- nant. For B = 2.9 mT only the attraction-repulsion ext effect is observed, leading to a higher DW propagation velocity [Fig. 2 (b)]. 60 ns 60.5 ns For B = 3 mT and B = 3.1 mT, the leading ext ext and trailing edges of the DW start to move at different velocities, implying that the DW width grows without limit, and thus the system is unable to support a compact DW structure. The leading HAV moves faster than the trailing one, and the vortex core exhibits oscillatory 63 ns 65 ns transverse motion in the vicinity of the latter, see Fig. 6 (b). Eventually the trailing edge of the DW exits the computationalwindow. SuchunstableDWdynamicshas previouslybeenreportedinRef.14, andrendersthemea- surement of a well-defined DW velocity impossible. Curiously,forlargerfields(3.2mT(cid:54)Bext (cid:54)4mT;we FIG. 7. (color online) VW dynamics for Bext =3.3 mT in a focushereonfieldsupto4mT)theDWstructureissta- strip with w = 768 nm and ∆ = 15 nm. The trailing HAV z ble again, and the v(B ) curve displays a second high- edge defect repeatedly emits a vortex-antivortex pair, which ext velocity plateau with v exceeding 250 m/s. There, after then annihilates, and the process is repeated. a relatively long initial transient, the DW dynamics pro- ceedsasfollows: InthevicinityoftheleadingedgeHAV, thevortexcoreexhibitssimilaroscillatorydynamicsasin result, emission of spin waves occurs after the annihila- theattraction-repulsioneffect. ThetrailingHAVrepeat- tion process51–53. Here, we have observed both parallel edly emits a vortex-antivortex pair, which subsequently and antiparallel processes of annihilation of the vortex- annihilates. Fig. 7andtheSupplementalMaterialMovie antivortex pair. 449 illustratethisdynamics. Noticethatwhiletheabove- ForevenlargerstripstheVWdynamicsisagainfound mentioned sequence is the dominating one, sometimes to be different from that observed in more narrow sys- thevortex-antivortexpairfailstoannihilateinthevicin- tems. Forexample,letusconsiderastripwithw =1536 ity of the trailing HAV, and is able to move to the oppo- nm and ∆ =15 nm, for which both VW and DVW are z site edge, leading to a short period of more complex dy- equilibriumstructures(i.e. theyhavethesameenergy26). namicsbeforetheabove-mentionedsequencestartsagain Thev(B )curveobtainedwhenstartingfromaVWini- ext (see Supplemental Material Movie 449). Similar vortex- tial state is shown as red squares in Fig. 8. As before, antivortex annihilation processes within DWs have pre- the low field steady/viscous regime (for B ≤ 0.4 mT) ext viously been reported in strips with much more confined is followed by a plateau-like regime where the velocity lateral dimensions, and applied fields roughly 5 times as decreases slowly with the field (for 0.5 mT ≤B ≤ 1.4 ext high as here50. mT), due to the attraction-repulsion effect. Contrary to Vortex-antivortex annihilation processes can be paral- the case of the more narrow strip considered in Fig. 2 lelorantiparalleldependingonthepolarityofthevortex- (b), here no Walker breakdown is observed. Instead, antivortex pair51–53. When the vortex and the antivor- the attraction-repulsion regime is terminated by an in- tex have the same polarity (parallel case), the process stability where the DW width grows without bound for is continuous, whereas for the antiparallel one there is Bext =1.5 mT and Bext =1.6 mT. emission of spin waves as a result of the annihilation. Above this unstable regime, the velocity again de- This is related to the topological charges (winding num- creases linearly with the field (1.7 mT ≤ B ≤ 2.1 ext ber n) and the polarities of the defects. DW structures mT). An example of the type of DW dynamics observed are composed of topological defects, which have associ- in this regime is shown in Fig. 9 for B = 2.1 mT (see ext ated winding numbers n = +1 for vortices, -1 for an- alsoSupplementalMaterialMovie549). Whenthefieldis tivortices, and ± 1/2 for edge defects54. In DW struc- applied the vortex core is displaced from its equilibrium turesthetopologicaldefectsarecompensated,leadingto position, and an antivortex enters into the strip. After zero total winding number, see Fig. 12. However, there that, a vortex nucleates into the strip. The antivortex are also other “topological charges” associated with the moves towards the previous vortex, and an annihilation winding number and the polarity of the core of vortices process between them takes places (snapshot at 34 ns and antivortices, e.g. the skyrmion charge q = np/255. showstheDWstructurejustaftertheannihilation). Un- Forparallelpolarity,vortexandantivortexhaveopposite like in the case observed for the strip with w = 768nm skyrmion numbers, leading to a continuous annihilation and∆ =15nm,wheretheprocessesofannihilationwere z process. However, for the antiparallel case the skyrmion betweenthevortexandantivortexthatnucleateintothe numbers are equal, and there is no compensation. As a strip,nowtheannihilationisbetweentheantivortexand 7 600 w = 1536 nm 0 ns  = 15 nm 500 z ] DVW U s [ m/400 VW NS 10 ns v T UNSTABLE 300 A B L 200 E 19 ns 100 0 30 ns 0 1 2 3 4 B [mT] ext FIG. 8. (color online) v(Bext) curves for a VW and a DVW 34 ns initial state in a strip with w = 1536 nm and ∆ = 15 nm, z wherebothstructuresareequilibriumDWs. In-planecellsize equal to 3 nm. The gray areas indicate unstable regions (no compact DWs). 58 ns the vortex that was inside the strip previously. Once the annihilationprocessoccurs,thispartoftheDWstructure starts to move towards the other vortex, reducing the 63 ns width of the DW structure. Again two parts of the DW structure are moving with different velocities. The vor- texthatdoesnotparticipateintheannihilationisfeeling an attraction-repulsion effect. Although this is the main dynamicalsequence,sometimesothermorecomplexpro- cesses are observed, such as one involving a vortex and two antivortices. FIG. 9. (color online) DW dynamics starting from a VW initial state for w = 1536 nm, ∆ = 15 nm and B = 2.1 For 2.2 ≤ B ≤ 3.6 mT, an extended region where z ext ext mT.TheDWdynamicsshowsthenucleationofanantivortex the system is again unable to support compact DWs is into the strip from the top edge, followed by the injection of encountered. Twodifferenttypesofinstabilityhavebeen a vortex. The antivortex moves towards the vortex that was observed. For 2.2≤B ≤ 3.4 mT the instability is like ext previouslyinthestrip,leadingtoanannihilationprocessbe- the one shown in Fig. 6 (b), where the DW width grows tweenthem. Afterthat,theDWstructurereducesitswidth, without limit. For Bext = 3.5 mT and Bext = 3.6mT, resulting a VW. Then the process is repeated. although the DW width also increases without limit, at thesametimeseveralvortex-antivortexannihilationpro- cesses take place. We have also analyzed the dependence of the VW ve- For even larger fields, compact DWs are observed locity within the low-field steady/viscous regime as a again, with their large velocity increasing linearly with function of w and ∆ , within a range of w and ∆ such z z B . Notice that for B = 4 mT, v exceeds 550 m/s. that VW is the equilibrium structure. The main panel ext ext In this regime, the DW dynamics is quite chaotic with of Fig. 11 shows the VW velocity as a function of w for transformations between complex dynamical states. In ∆ = 15 nm and B = 0.3 mT, displaying an increas- z ext Fig. 10 the DW dynamics is shown for B = 3.8 mT ing trend of v with w. Since the width of VWs in wider ext (see also Supplemental Material Movie 649). Repeated strips is larger, such a trend is in agreement with the 1d annihilation processes between vortices and antivortices model, predicting a linear dependence of v on the DW are observed. Note that in this regime, creation of vor- width10. This can also be rationalized by the fact that a ticesandantivorticesoccursnotonlyattheedges(asfor givenamountofspinrotationleadstoDWdisplacements lower fields), but also within the strip (see snapshots at proportionaltotheDWwidth. InsetofFig.11displaysv 6, 12 and 21 ns in Fig. 10). Similar processes of vortex- as a function of ∆ for w =768 nm and B =0.3 mT, z ext antivortex pair generation within the strip have been re- showing that the VW velocity decreases with the strip ported before56. thickness. Thismaybeattributedtothefactthatinthe 8 300 0 ns 400 w = 768nm 300s] m/ 250 200v [ 100 3 ns 200 0 5 10∆ 1[n5m]20 25 30 s] z m/ v [150 6 ns ∆ = 15nm 100 z 50 100 1000 9 ns w [nm] FIG. 11. (color online) Main figure: VW velocity v as a function of the width of the strip w when the thickness and fieldarefixedto∆ =15nmandB =0.3mT,respectively. z ext 12 ns w is in logarithmic scale. Inset: VW velocity v as a function of ∆ for w=768 nm and B =0.3 mT. z ext 15 ns dynamics exhibits a unique low-field regime. However, the topology of the DVW, consisting of two vortices and 4HAVedgedefects54 [seeFig. 12(b)]impliesthattheen- ergybarrierfortransformingtheDVWintoaVWisnot very high. Thus, already for relatively low fields, DVWs 21 ns tend to transform into VWs, via a process in which one of the vortices annihilates with two edge defects at the edge of the strip. Since the total winding number of a vortexandtwoedgedefectsiszero, thisprocesshappens easily, i.e. it does not require a large applied field. Thiscanbeillustratedbyconsideringe.g. astripwith FIG. 10. (color online) DW dynamics starting from a VW w = 1536 nm and ∆ = 15 nm, corresponding again initial state for w = 1536 nm, ∆ = 15 nm and B = 3.8 z z ext to the phase boundary between VW and DVW equilib- mT. Very complex, possibly chaotic dynamics is observed. riumstructureswheretheyhavethesameenergy26. The v(B ) curve obtained by starting from a DVW initial ext steady state, the DW propagation occurs by a preces- DW with equal polarity for both vortices (parallel case) sional motion around an out-of-plane demagnetizing or is represented in Fig. 8, together with the corresponding stray field, which is created by precession in the external data for the VW initial state in the same strip geometry field9. As the thickness of the strip increases, this de- (see the previous Section). For low fields (Bext (cid:54) 0.6 magnetizing field decreases, leading to a decreasing DW mT), the v(Bext) curves are quite different for the two velocity. structures, indicating that the DVW structure is stable against small applied fields, and thus exhibits a unique small field steady/viscous regime. In this regime, the IV. DOUBLE VORTEX WALL DYNAMICS moving DVW structure initially exhibits transient oscillatory relative motion of the two vortex cores, with the For larger strips with widths around 1µm and above, oscillation amplitude decreasing with time and eventu- theDVWistheequilibriumDWstructure26. Thisstruc- allyvanishingwhentheregimewithsteadyDVWmotion tureincludestwovorticeswithoppositesenseofrotation, is reached [see Fig. 13 (a), and Supplemental Material see Fig. 1(b). As occurs for the VW, the DVW presents Movie749]. Inthislow-fieldregimetheVWmovesfaster different dynamical behaviors depending on the lateral than the DVW. This is due to the increased energy dis- dimensions of the strip, and the magnitude of the driv- sipation related to the second vortex core of the DVW. ing field. Moreover, in the case of DVW the polarities However, if the polarities of the vortices are opposite of the two vortices are an important factor that strongly (antiparallel case), the dynamics at low fields is very dif- affects the dynamics. For small enough B the DVW ferent. DVW is unstable and transforms into a VW, see ext structureisstabledependingonthepolaritiesofthevor- Fig.13 (b). Contrary to the static case where the polar- tices. For the case in which the DVW is stable, the DW ity barely contributes to the energy of the equilibrium 9 (a) (b) (a) -1/2 . . . . x +1 0 ns 0 ns -1/2 (b) 10 ns 10 ns -1/2 20 ns 25 ns +1 +1 35 ns 30 ns -1/2 -1/2 -1/2 (c) 60 ns 60 ns -1/2 -1/2 -1/2 +1 FIG. 13. (color online) DVW dynamics in a strip with +1 w=1536 nm and ∆ =15 nm for B =0.3 mT and differ- +1 z ext ent polarities of the vortices. (a) Parallel polarity; dashed lines help to see the transient relative motion of the vor- texcores,beforethesteadystatemotionwithanon-evolving DVWstructureisreached. (b)Antiparallelpolarity;DVWis -1/2 -1/2 -1/2 not stable, and becomes a VW. In the case of DVW with antiparallel polarity [see top FIG. 12. (color online) The equilibrium DW structures in inFig. 13(b)], thegyrotropicforceononeofthevortices terms of their constituent elementary topological defects, i.e. is opposite to the one exerted on the other. This means vortices with a winding number +1 (full circles) and edge that one vortex will move to the top edge, and the other defectswithwindingnumber-1/2(emptycircles). (a)Vortex wall, (b) double vortex wall, and (c) triple vortex wall. to the bottom one. As a consequence one of the vortices will be expelled, i.e. the one located closer to the edge in its direction of movement. Fig. 13 (b) shows how the DVW transforms into a VW. state26, in the dynamics it plays an important role. This Returning to the parallel case, for B = 0.7 mT, the ext is because when an external field is applied, the vortex externalfieldisstrongenoughtopushoneofthevortices corefeelsagyrotropicforce48Fg =pGz×v,thatdepends out of the system (a process that takes place via anni- (cid:98) onthepolarity; G=2πJtisthegyrotropicconstantand hilation of the vortex with the two HAV edge defects), v the velocity. In the case of a VW this force tries to and the DW transforms into a VW. Then, the steady expel the vortex out through the top or bottom edge de- state velocity equals that of the corresponding VW ini- pendingonthepolarity, andindependentlyofthevortex tial state, a behavior that persists up to B = 1.1 mT ext chirality. For DVWs with parallel polarity [see the panel (see Fig. 8). Thus, also here the attraction-repulsion topofFig.13(a)],bothvorticeswilltrytoleavethestrip effect is observed. However, the DW dynamics exhibits through the same edge due to an equal gyrotropic force some dependence on the initial state also for fields ex- exerted on them. As in a DW structure the topological ceeding1mT.ForB =1.2mT,theDVWinitialstate ext defectshavetobecompensatedfor,i.e. thetotalwinding leads to a Walker breakdown-type of behavior, with re- number must be equal to zero, only one of the vortices peatedtransitionsbetweendifferentDWstructures(VW canbeexpelled. Asaresult,acompetitionbetweenthem and ATW), similarly to the observations in Fig. 6 (a). appears, which tends to keep both vortices in the strip, However, the corresponding system with a VW initial see Fig. 13 (a). state displays the attraction-repulsion effect, and a sig- 10 nificantly larger v for the same field. Also, while for a VW initial state Bext = 1.3 mT and 1.4 mT lead to sta- ((aa)) (b) bleattraction-repulsiontypeofbehavior,startingfroma . DVW initial state leads to unstable behavior where the . . x DWwidthgrowswithoutlimit. ForB =1.5mT,such ext an instability is encountered also when starting from the . VWinitialconfiguration. ForB =1.6mTtheinstabil- x ext ity occurs when the VW is the initial state, but not for the case in which DVW is the initial configuration. For larger fields, the two initial conditions lead to the same steady state dynamics, described in the previous section (c) (d) for the VW initial state, see Figs. 9 and 10. Similar DVW dynamics is observed also for other w . x x and ∆ values with DVW as the equilibrium structure. x z However, in the parallel case the “critical” field magnitude at which one of the two vortices is expelled from x x the strip, leading to a VW, depends on w and ∆ in a z non-trivial fashion. For some systems belonging to the DVW equilibrium phase this field is quite small, around 0.1 mT. A possible explanation is that the energy re- (e) (f) quiredtomoveoneofthevorticestowardsthestripedge issmallerinwiderstrips,giventhatshapeanisotropy(an . x edge effect) is weaker in wider strips. x x . . V. TRIPLE VORTEX WALL DYNAMICS Forstripswithevenlargerlateraldimensions,TVWis theequilibriumDWstructure26,seeFig. 1(b). Asinthe (h) (g) caseofDVW,alsoTVWtendstotransformintosimpler DW structures under applied fields for topological rea- . . sons. TVWs are composed of three vortices and 6 HAV . x edge defects, see Fig. 12 (c). Thus, one or two vortices can be expelled from the strip with relative ease, given . x thattheycanannihilatewithtwoHAVsatthestripedge inaprocesswherethetotalwindingnumberisconserved. Only in the case of very small fields the TVWs maintain their structure during motion depending on the polarities of the vortices. As in the DVW case, the polarity is fundamental for the stability of the TVW structure. As for TVWs there are three different vortices, the sit- FIG. 14. (color online) Top view sketch of the different polarity configurations for a TVW structure. uation is more complex than in the case of DVWs. Fig. 14 shows the different combinations of the polarities of the TVW structure. Contrary to the DVWs, where for parallel polarity of the vortices the DW structure is sta- tial in the TVW dynamics. In Figs. 14 (e) and (g) two ble at low fields, for TVW the sign of the polarity is also different polarity configurations are shown, where two important. For a strip with w = 5120 nm and ∆ = 25 vortices have polarity p = -1 and the third one has p = z nm under B = 0.05 mT, we have observed that when +1. ThesetwoconfigurationsshowdifferentDWdynam- ext the three vortices are parallel with polarity p = -1 [Fig. ics. To illustrate this we consider a strip with w = 5120 14 (d)], the TVW is stable, whereas for polarity p= +1 nm and ∆ = 25 nm, where TVW is one of the equilib- z [Fig. 14 (h)] TVW transforms into a DVW (not shown). rium DW structures26, considering both polarity config- Thisisrelatedtothegyrotropicforceexertedonthevor- urations. ForB =0.05mTandoneoftheabovemen- ext tices and the interactions between them. Depending on tioned configuration, TVW is stable, keeping its struc- the polarity the gyrotropic force acts in one direction or ture during the simulation time, see Fig. 15 (a). For the the other, facilitating or preventing the expulsion of the second one, the initial TVW transforms into a DVW as vortex out of the strip. one of the three vortices is expelled from the strip. The On the other hand, for antiparallel polarity also the resultingDVWexhibitssteadydynamicswithinthesim- position of the vortices with a given polarity is essen- ulation time scale, see Fig. 15 (b). This is because the