Gyrotropic resonance of individual Néel skyrmions in Ir/Fe/Co/Pt multilayers

Bhartendu Satywali,1, ∗ Fusheng Ma,1, ∗ Shikun He,2, 1 M. Raju,1 Volodymyr P.Kravchuk,3, 4 Markus Garst,5 Anjan Soumyanarayanan,2, 1, † and C. Panagopoulos1, †

1Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,Nanyang Technological University, 637371 Singapore

2Data Storage Institute, Agency for Science, Technology, and Research (A*STAR), 138634 Singapore3Leibniz-Institut für Festkörper- und Werkstoffforschung, IFW Dresden, D-01171 Dresden, Germany

4Bogolyubov Institute for Theoretical Physics of National Academy of Sciences of Ukraine, 03680 Kyiv, Ukraine5Institut für Theoretische Physik, TU Dresden, 01062 Dresden, Germany

(Dated: February 2018)

Magnetic skyrmions are nanoscale spin structures recently discovered at room temperature (RT) in multi-layer films. Employing their novel topological properties towards exciting technological prospects requiresa mechanistic understanding of the excitation and relaxation mechanisms governing their stability and dy-namics. Here we report on the magnetization dynamics of RT Néel skyrmions in Ir/Fe/Co/Pt multilayerfilms. We observe a ubiquitous excitation mode in the microwave absorption spectrum, arising from thegyrotropic resonance of topological skyrmions, that is robust over a wide range of temperatures and sam-ple compositions. A combination of simulations and analytical calculations establish that the spectrumis shaped by the interplay of interlayer and interfacial magnetic interactions unique to multilayers, yield-ing skyrmion resonances strongly renormalized to lower frequencies. Our work provides fundamentalspectroscopic insights on the spatiotemporal dynamics of topological spin structures, and crucial directionstowards their functionalization in nanoscale devices.

Magnetic skyrmions are topologically protected spintextures with solitonic, i.e. particle-like properties1,2.Their room temperature (RT) realization in mul-tilayer films possessing interfacial Dzyaloshinskii-Moriya interaction (DMI)3–7 promises imminentspintronic applications8,9, due to the expected plia-bility to charge currents and spin torques10, coupledwith topological stability2,8. Harnessing the poten-tial of these magnetic quasiparticles requires a com-prehensive understanding of the excitation and re-laxation mechanisms governing their spatiotemporaldynamics11.

The dynamic response of a skyrmion to electro-magnetic potentials includes a characteristic trans-verse, or gyrotropic component which derives fromits topological charge12,13, and is described by dis-tinct rotational modes11–13. Meanwhile, extrinsicperturbations encountered by a moving skyrmion,e.g. thermal fluctuations or disorder, may deformits spin structure to an extent governed by internalexcitation modes12,14. Conversely, similar excitationsmay be engineered to enable deterministic skyrmionnucleation in devices15. Finally, the efficiency of itselectrodynamic response is governed by spin relax-ation and scattering mechanisms, and characterizedby the damping parameter8. A panoramic picture ofskyrmion magnetization dynamics in multilayers isessential for their efficient manipulation in devicesutilizing their mobility in racetracks16,17, switchingin dots15, and microwave response in detectors andoscillators18,19.

Skyrmion excitations have thus far been inves-tigated primarily in crystalline magnets at cryo-genic temperatures11,20–24. Such ”helimagnets” hostordered, dense lattices of identical predominantly

Bloch-textured skyrmions over a narrow tempera-ture range2,21. In contrast, multilayer skyrmionswould likely have a considerably distinct behavior,due to their columnar Néel texture3, sensitivity tointerlayer and interfacial magnetic interactions ab-sent in helimagnets8, and marked inhomogeneity intheir individual and collective entity due to inher-ent magnetic granularity4. Moreover, the stabilityof skyrmions over a wide range of thermodynamicparameters in multilayers7 would enable unprece-dented investigations of topological spin excitations,including their thermal stability and sensitivity tomagnetic interactions.

Here we present a detailed study of the magnetiza-tion dynamics of RT Néel skyrmions in Ir/Fe/Co/Ptmultilayers. A robust excitation mode is found in fer-romagnetic resonance (FMR) spectroscopy measure-ments, arising from gyrotropic motion of individualNéel skyrmions, and persisting over a wide range oftemperatures and sample compositions. Micromag-netic simulations and analytical calculations demon-strate the roles of interfacial and interlayer interac-tions in shaping the distinctive resonance spectrumof skyrmions, and enabling its detection across thegranular magnetic landscape.

Magnetization DynamicsThe sputtered [Ir(10)/Fe(x)/Co(y)/Pt(10)]20 multi-layer films (thickness in angstroms in parentheses)studied in this work are known to host RT Néelskyrmions with smoothly tunable properties that canbe modulated by the Fe/Co composition7. Herewe focus on two 1 nm Fe(x)/Co(y) compositions –Fe(4)/Co(6) and Fe(5)/Co(5) – which show typicalskyrmion densities ∼ 30 μm−2 (i.e. mean distance

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FIG. 1. Magnetization Dynamics. (a) Schematic of theFMR setup. The coplanar waveguide (CPW, brown) isused to measure the transmitted signal (S12, blue) in re-sponse to an in-plane (IP) excitation field (hrf, red) at fre-quency f with a static out-of-plane (OP) field H. Inset:schematic of the Ir/Fe/Co/Pt multilayer stack. (b) FMRspectra above saturation (H > HS) for sample Fe(4)/Co(6)at several frequencies between 15-20 GHz, correspondingto uniform FM precession25. (c) Dispersion of resonancefield with frequency, determined from the Lorentzian fitsin (b), and used to extract the anisotropy field, µ0HK. (d)Resonance linewidth ∆H plotted against frequency, andused to determine the effective damping, αeff.

aSk ∼ 200 nm c.f. typical diameter, dSk ∼ 50 nm,see §SI 1). Notably, the large magnitude of DMI(D ∼ 2.0 mJ/m2) relative to the exchange stiffness(A ∼ 11 pJ/m) and out-of-plane (OP) anisotropy(Keff ∼ 0.0− 0.05 MJ/m3) results in a definitive Néeltexture (see §SI 3). Recent studies have further evi-denced the persistence of their texture and stabilityover a large temperature range26.

Broadband FMR spectroscopy measurementswere performed using a home-built coplanar waveg-uide (CPW) setup tailor-made for ultrathin magneticfilms (Fig. 1a, see Methods)27. FMR spectra wererecorded in transmission mode (S12) over a fre-quency ( f ) range of 5-20 GHz, with the OP magneticfield (H) swept over ±1 T, following establishedrecipes for multilayer films23,24,27,28. Fig. 1b showsrepresentative spectra recorded above saturation(H > HS): the resonant lineshape correspondingto uniform (Kittel) precession of the saturatedferromagnetic (FM) moment, MS. Lorentzian fitsto these spectra give the resonance field-frequencydispersion (Fig. 1c) and the linewidth (Fig. 1d),which are used to extract the anisotropy field,µ0HK

25, and the effective damping, αeff, respectively(see Methods). The considerable magnitude of αeff(0.105 ± 0.001) in our multilayer stack, comparedto bulk crystals and thin films, is due to the sum-mation of various Gilbert-like damping effects29,

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FIG. 2. Microwave Resonances of Magnetic Textures. (a)Normalized OP magnetization M/MS (blue) and dM/dH(purple) against H/HS (H swept to zero) for Fe(5)/Co(5)at RT. Top: MFM images (scale bar: 500 nm) at fields in-dicated by dashed black lines, showing labyrinthine stripe(LS, left) and skyrmion (SK, right) phases. (b) Color plotof normalized FMR spectra plotted against reduced field(H/HS) and frequency (2π f /γHS, γ is the gyromagneticratio). Overlaid points show resonances corresponding tothree distinct modes, identified by Lorentzian fitting (see(d)). Top: horizontal bar delineates approximate extents ofLS, SK, and FM phases. (c) Representative linecuts show-ing raw FMR spectra at selected frequencies. Dashed linesshow the dispersion of observed minima, corresponding toresonances in (b). (d) Sample Lorentizan superposition fitsto FMR spectra, used to determine the resonances (points:raw data, lines: fits).

as well as interfacial effects that include spin wavescattering, spin pumping, and interface defectscattering27–29. The experimental determination ofαeff for this multilayer skyrmion host is a crucialstep towards modeling skyrmion dynamics andswitching characteristics8,15.

Resonant Magnetic TexturesWe now turn to the microwave response of mag-netic textures that emerge at fields below saturation(H < HS), and form disordered skyrmion lattice(SK, Fig. 2a: right inset) and labyrinthine stripe (LS,Fig. 2a: left inset) phases respectively. Fig. 2c shows awaterfall plot of representative high-resolution FMRspectra which exhibit a visible dichotomy in reso-nance characteristics. While high frequency spec-tra (e.g. 13 GHz) exhibit two prominent minima athigh (H > HS) and low (H � HS) fields respec-tively, low frequency spectra have only one observ-able dip at intermediate fields (H ∼ 0.8 − 1 HS).Notably, these minima correspond to distinctly dis-

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persing resonances that together form a character-istic “Y”-shape (Fig. 2c). We use Lorentzian super-position fits to determine and delineate these reso-nances (e.g. Fig. 2d). The dispersion of the fitted res-onances, overlaid in Fig. 2b on a normalized spectralplot, is used to establish their relationship with mag-netic phases (Fig. 2a).

The resonances in Fig. 2b can be delineatedinto three distinct modes – each corresponding toa unique magnetic phase. First, the FM phase(H > HS) exhibits the familiar positively dis-persing Kittel mode (blue circles) arising from uni-form precession25. Next, the LS phase (H �HS) harbors a high frequency mode (magenta trian-gles) with negative dispersion, analogous to well-studied helimagnonic resonances in Bloch-texturedcompounds21. Finally, the SK phase (H/HS ∼0.8 − 1) hosts a distinct, low frequency resonancemode (green squares) with positive dispersion andconsiderable spectral weight. This is reminis-cent of gyrotropic excitations reported in Bloch-textured magnets11,20,21,23, and can indeed be iden-tified as a magnon-skyrmion bound state arisingfrom counter-clockwise (CCW) gyration of individ-ual Néel skyrmions as we show below. Surprisingly,this SK resonance exhibits a pronounced renormal-ization towards lower frequencies with respect to theKittel mode, a phenomenon hitherto unobserved inany skyrmion material.

Systematic multilayer micromagnetic simulationswere performed to establish the relationship betweenthe magnetic textures and the observed resonances(details in Methods). The equilibrium magnetic con-figurations exhibit consistently Néel texture (see §SI3) with the expected field evolution (FM, SK, and LS:Fig. 3a). Their magnetization dynamics were simu-lated from their temporal response to in-plane (IP)excitation fields (Fig. 3c), yielding a spectral diagram(Fig. 3b) with varying f and H. In agreement withexperiments (Fig. 2b), the resonances in FM and LSphases are at higher frequencies, and disperse withpositive and negative slopes respectively. Mean-while, the SK phase exhibits two positively dispers-ing resonances: one, at higher frequencies (yellow di-amonds), appears similar to the FM mode; while theother, at lower frequencies (green squares), is notice-ably steeper, consistent with experiment (Fig. 2b).

Gyrotropic Skyrmion ResonanceThe robust nature of the observed SK resonance(Fig. 2b), characteristic of phase coherence, is partic-ularly compelling given that it arises from a dilute,disordered ensemble of inhomogenous skyrmions(∼ 25% size variation, see §SI 1). Furthermore,the qualitative agreement of its distinguished phe-nomenology with multilayer simulations (Fig. 3b)raises several intriguing questions: (1) what is themicroscopic mechanism for the low-frequency SKresonance? (2) what are its quantum numbers? (3)

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FIG. 3. Micromagnetic Simulation of Microwave Res-onances. (a) Simulated equilibrium M/MS (blue), anddM/dH (purple) against H/HS for Fe(5)/Co(5) at RT (c.f.Fig. 2, details in Methods). Top: representative OP magne-tization (mz) images (scale bar: 500 nm) at fields indicatedby dashed black lines for LS (left), SK (centre), and FM(right) phases. (b) Color plot of simulated microwave res-onance intensity in reduced f , H units (c.f. Fig. 2b). Over-laid points show dispersing modes within LS, SK, and FMphases. Top: horizontal bar delineates approximate extentsof these phases. (c) Linecuts extracted at dashed black linesin (b) showing typical spectra from the respective phases.(d) Simulated dispersion comparison for the multilayer (b)with a single layer with similar magnetic parameters. Thesteepening of the SK mode, unique to multilayers, requiresinterlayer dipolar coupling.

what causes the idiosyncratic dispersion of the SKmode? We turn to spatiotemporal Fourier analysesof simulations and analytical calculations to addressthese compelling issues.

First, we examine temporal snapshots of the nor-malized magnetization, m = M/MS (Fig. 4a), andtopological charge density, qT = 1

4π m (∂xm × ∂ym)(Fig. 4b), of a resonant skyrmion (see Methods). Dueto the momentum conservation30, the center-of-massof qT, RqT (Fig. 4b, filled circle) remains static, how-ever its maximum demonstrates CCW rotation. No-tably, the center-of-mass of mz, Rmz (Fig. 4a, opencircle) is not static and also rotates in a CCW di-rection. In this case, as in more involved types ofskyrmion dynamics, RqT and Rmz behave like mass-less and massive particles respectively13,31. Theseanalyses (details in §SI 5) conclusively inform thatthe observed SK resonance arises from the gyrotropicCCW excitation of single Néel skyrmions.

Second, for a dilute ensemble as here (typicalaSk/dSk ∼ 4 � 1), skyrmions should respondindividually to the excitation field, in contrast tocollective excitations of well-studied helimagneticskyrmion lattices11,21. It is noteworthy that the de-velopment of collective coherence is not essential for

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FIG. 4. Gyrotropic CCW Origin of Skyrmion Resonance.(a-b) Temporal snapshots of normalized skyrmion magne-tization m (a: arrows/color-scale denote IP/OP compo-nents) and its topological charge density qT (b, defined intext), showing CCW rotation through one period (τ). Filledand open green circles denote the center-of-mass of qT andmz respectively. (c) Spin-wave function (SWF) of a resonat-ing skyrmion at a given instant, shown as the deviation,∆m, from the static configuration (mz: OP, mϕ: IP), usingFourier analysis of simulations. (d) Radial cuts through themaxima of ∆mz and ∆mϕ, the latter possessing a noticeablylong tail. Vertical dashed line shows the skyrmion radius,dSk/2 for comparison.

experimental detection: the incoherent response ofa finite density of skyrmions could produce a largeenough signal to explain their observed signature.Such an excitation can therefore be described by awell-localized spin-wave function (SWF). The SWF(Fig. 4c, §SI 6), determined from the temporal devia-tions in magnetization ∆m from the static skyrmionconfiguration, displays profiles characteristic of anindividual CCW excitation with appropriate quan-tum numbers (details in Methods). Next, radiallinecuts through SWF maxima for OP (∆mz) and IP(∆mϕ) components (Fig. 4d) peak as expected at theskyrmion radius (dSk/2). In contrast to the single-layer theory (§SI 5), the long-range dipolar field inthe multilayer structure results in a long tail in ∆mϕ.Interestingly, this long tail might give rise to a resid-ual coupling between adjacent skyrmions that couldlead to a synchronization of excitations, a subject forfuture investigation.

Third, we investigate the renormalization of themultilayer CCW mode to lower frequencies with re-spect to the Kittel mode. Analytical calculations ofthe Néel skyrmion spectrum for a comparable singlelayer (§SI 5)32 suggest that the CCW mode can corre-

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FIG. 5. Thermodynamic Evolution of Resonances. (a-b)Dispersions of LS, SK (CCW), and FM resonance modesfor Fe(5)/Co(5) at selected temperatures (T =100, 200, 300K) from (a) experiment, and (b) simulations (multilayers,0 K with T-dependent parameters, see Methods). (c) T-dependence of experimentally determined values for satu-ration magnetization MS (top) and Keff (bottom) for sam-ples Fe(5)/Co(5) and Fe(4)/Co(6). (d) Measured resonancedispersions, d f /dH over 100-300 K for both samples, ob-tained from linear fits (e.g. to a), plotted against Keff.

spond only to a shallow bound state, with frequencypractically indistinguishable from the Kittel mode, asfor Bloch systems14,33. Indeed, this is verified by sin-gle layer simulations (Fig. 3d, §SI 4), which do not ex-hibit renormalization, in line with literature34,35. To-gether, these conclusively point to interlayer dipolarcoupling, uniquely present in multilayers, as beingresponsible for the CCW renormalization. The dipo-lar coupling aligns layer-wise moments within mul-tilayer skyrmions, producing an attractive interac-tion that may lower the gyrotropic frequency well be-low the magnon continuum. Meanwhile, the higherfrequency SK mode, uniquely observed in simula-tions (Fig. 3b: FM residual), arises from a delocal-ized excitation of the background, reminiscent of theuniform Kittel mode25. It is spectrally distinguishedin multilayers due to the stronger dispersion of theCCW, while its substantially lower weight is likelythe reason for its absence within experimental reso-lution.

Finally, the persistence of Ir/Fe/Co/Pt skyrmionsover a wide range of thermodynamic parameters7,26

offers an unprecedented window into the evolutionof skyrmion resonances. To this end, we examinethe temperature variation (100-300 K) of Fe(5)/Co(5)resonances within experiments (Fig. 5a) and sim-ulations (Fig. 5b, parameters in Methods) respec-tively. The FM and LS modes do not vary substan-tially, however the CCW mode is noticeably steeperat lower temperatures (Fig. 5a). While zero Kelvin

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simulations cannot establish full quantitative consis-tency with experiment, they do reproduce the CCWtrend (Fig. 5b), thereby explicitly discounting ther-mal activation effects. Therefore, we investigateits thermodynamic origin by studying the evolutionof resonance mode dispersions, d f /dH, across twosamples over these temperatures. From Fig. 5d, theCCW dispersion steepens by a factor of 5, and no-tably exhibits a near-linear variation with Keff – thelatter also increases considerably (0.01–0.2 MJ/m3)across our measurements. Indeed, theoretical re-ports point to easy-axis anisotropy favouring thegyrotropic CCW mode and lowering its resonancefrequency31,32, consistent with our findings.

OutlookWe have reported a systematic magnetization dy-namics study of Néel skyrmions that form dilute, in-homogeneous ensembles in Ir/Fe/Co/Pt multilay-ers. We have identified a strong microwave res-onance arising from CCW gyration of individualskyrmions, and note its persistence over a widerange of thermodynamic parameters. Multilayersimulations and analytical calculations unveil theroles of interlayer dipolar coupling and easy-axisanisotropy in renormalizing the skyrmion resonancespectrum against the magnon continuum. These re-sults establish a comprehensive foundational scaf-fold between multilayer magnetic interactions andmagnetization dynamics of Néel skyrmions, andprovide quantifiable insights on their stability anddynamics.

The excitation of RT skyrmions in multilayers isremarkable in light of inherent skyrmionic gran-ularity and inhomogeneity, and is promising to-wards their exploitation for new physics and tech-nology. Such excitations enable deterministic nu-cleation and manipulation of skyrmions in deviceconfigurations8, and can be used to realize skyrmion-based microwave detectors and oscillators18,19. Therenormalization of the CCW mode due to the inter-layer coupling clearly separates this resonance fromthe continuum, which is key for an interference-freefunctionalization involving skyrmion-based datatransmission9 and magnon-based logic36 in a singledevice. Our work forms a cornerstone for mechanis-tic tailoring of skyrmion dynamics in technologicallyrelevant systems8, and opens a new chapter on topo-logical magnonics11,36.

Acknowledgments. We acknowledge Anthony Tan andOphir Auslaender for experimental inputs, and UlrikeNitzsche for technical assistance. We also acknowledge thesupport of the National Supercomputing Centre (NSCC),Singapore, the A*STAR Computational Resource Center(A*CRC), Singapore for computational work. This workwas supported by the the A*STAR Pharos Fund of Sin-gapore (Ref. No. 1527400026), the Ministry of Educa-tion (MoE), Academic Research Fund Tier 2 (Ref. No.

MOE2014-T2-1-050) of Singapore, and the National Re-search Foundation (NRF) of Singapore, NRF - Investiga-torship (Ref. No.: NRF-NRFI2015-04). V.K. acknowledgessupport from the Alexander von Humboldt Foundationand the National Academy of Sciences of Ukraine (ProjectNo. 0116U003192). M.G. acknowledges financial supportfrom DFG CRC 1143 and DFG Grant 1072/5-1.

Author Contributions. A.S. and C.P. conceived the re-search and coordinated the project. M.R. deposited andcharacterized the films with A.S. S.H. and C.P. designedthe FMR experiments. B.S. and S.H. performed the experi-ments, and F.M., B.S., and S.H. analyzed the data. F.M. per-formed the micromagnetic simulations. V.K. and M.G. per-formed the theoretical calculations. All authors discussedthe results and contributed to the manuscript.

∗ These authors contributed equally to the work.† Correspondence should be addressed to

A.S. (souma@dsi.a-star.edu.sg) or C.P. (chris-tos@ntu.edu.sg).

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MethodsFilm Deposition & CharacterizationMultilayer films of:Ta(30)/Pt(100)/[Ir(10)/Fe(x)/Co(y)/Pt(10)]20/Pt(20)(numbers in parentheses denote nominal layer thickness inangstroms) were sputtered on thermally oxidized 100 mmSi wafers. The deposition parameters, and structural andmagnetic properties of the multilayer stacks have been re-ported previously7.

The results reported here correspond to two composi-tions: Fe(4)/Co(6) and Fe(5)/Co(5). Magnetization M(H)were measured using a Quantum Design™ Magnetic Prop-erties Measurement System (MPMS) in OP/IP configura-tion to determine HS, MS, and Keff (§SI 1). MFM im-ages were acquired using a D3100 AFM manufactured byBruker™, mounted on a vibration-isolated platform, usingsharp, ultra-low moment SSS-MFMR™ tips. The reportedskyrmion properties are consistent with previous RT re-sults (see §SI 1)7, recently extended to temperatures downto 5 K26.

FMR MeasurementsBroadband FMR measurements were performed using ahome-built setup37, previously used for high-resolutionspectroscopy of ultrathin magnetic films27. A 4×8 mmsample was inductively coupled to a U-shaped CPW us-ing a spring-loaded sample holder. Microwave excitationswere sourced parallel to the sample plane (Fig. 1a), and thetransmitted signal (S12) was measured using a KeysightPNA N5222 vector network analyzer (VNA). External OPmagnetic fields up to ±1 T were applied to saturate thesample, and data were recorded at each frequency in fieldsweep mode. The results presented here correspond to Hswept from above +HS to zero (full sweep in §SI 2). Lowtemperature measurements were performed with the sam-ple holder mounted on a stage attached to a continuousflow cryostat27.

For uniform Kittel precession (H > HS) with frequencyfr, the resonance field Hr and linewidth ∆Hr were ex-tracted by fitting the spectra to the expected form of theOP dynamic susceptibility27. The Hr − fr dispersion of theuniform mode (Fig. 1c) can be described by the OP Kittelequation25:

2π fr = γ(µ0Hr)− γµ0(HK −MS) (M1)

The gyromagnetic ratio γ = gµB/h and the anisotropyfield, µ0HK can be extracted from Eq. M1 (Fig. 1c, see §SI 1).Meanwhile, the linewidth ∆Hr (Fig. 1d) is described by28:

∆Hr = ∆H0 +4παeff

µ0γfr (M2)

A linear fit to ∆Hr − fr (Fig. 1d) determines the inhomo-geneous broadening, ∆H0, and importantly, the effectivedamping, αeff

27.For data acquired at or below HS (Fig. 2), the resonances

modes for LS, SK, and FM phases were determined byfitting the spectra to a superposition of Lorentzian line-shapes. Typical fit results are shown in Fig. 2b. The spec-tra were plotted with reduced field (H/HS) and frequency

7

(2π f /γHS) units (Fig. 2b) for comparison with simulations(Fig. 3b), and across samples (Fig. 5).

Micromagnetic SimulationsMicromagnetic simulations of equilibrium magnetic tex-tures and their microwave response were performed us-ing the mumax³ package38, which accounts for interfacialDMI. The multilayers were modeled with a mesh size of4 × 4 × 1 nm over a 2 × 2 μm area for comparison withexperiments. The magnetic (Fe, Co) layers were mod-eled as a 1 nm FM layer, while the non-magnetic (Pt, Ir)layers were represented as 1 nm spacers. The magneticparameters MS and Keff were determined from SQUIDmagnetometry, α from FMR measurements, and A and Dfrom micromagnetic fits to MFM measurements using es-tablished techniques3,4. The RT parameters have been re-ported previously7, and those for lower temperatures weredetermined similarly (see Fig. 5c, §SI 3). Notably, all simu-lations were performed at 0 K.

First, a set of simulations were performed using the20 stack repeats configuration consistent with experiment(Full Stack). The magnetization was randomized at H = 0,and relaxed to obtain the equilibrium configuration. ThenH was increased progressively up to HS to simulate thefield evolution of magnetic textures. Layer-wise analysesof spin configurations in the SK phase reveal uniformlyNéel-textured skyrmions with size variations of up to 10%across layers (detailed analysis in §SI 3). The consistencyof spin configuration across layers is in line with the largemagnitude of DMI (∼2 mJ/m2) in our stacks.

Subsequently, the simulations were repeated using pe-riodic boundary conditions (PBC) in the z direction (9 re-peats each along ±z) to mimic the Full Stack behavior, i.e.to incorporate interlayer dipolar coupling without the ad-ditional ∼ 20× computational cost. The PBC results wereconsistent with Full Stack (within ∼ 10%, see §SI 3), andwith magnetometry and MFM experiments. Computation-ally expensive magnetization dynamics work followed asimilar protocol: quantitative comparison of Full Stack andPBC for one set of parameters (Fe(5)/Co(5) at 300 K, see §SI4), followed by PBC for the remainder.

The magnetization dynamics were simulated by exam-ining the response of the equilibrium magnetic textures toa spatially uniform IP excitation field (see §SI 4),

h(t) = h0 sin (2π fct) / (2π fct) (M3)

Here, we set h0 to 10 mT, and fc to 50 GHz – the latterbeing above the experimental bandwidth. The transientdynamics were computed over 10 ns, and the power spec-tral density was determined from the Fourier transform ofthe spatially averaged magnetization response. This proce-dure was repeated at all relevant H values to obtain Fig. 3b.

Systematics associated with damping, grid size, and inter-layer interactions are detailed in §SI 4.

Fourier Analysis of Skyrmion ResonancesThe skyrmion excitations at a given (H, f ) was visual-ized applying a sinusoidal IP excitation field, h(t) =h0 sin (2π f t), and acquiring snapshots of the m(r) (Fig. 4a)and qT(r) (Fig. 4b) at 10 ps intervals (Fig. 4a-b). Thecentres-of-mass (circles in Fig. 4a-b) are determined for mzto be Rmz =

∫dr r(1−mz)/

∫dr(1−mz), and for qT to be

RqT =∫

dr rqT(r)/QSk, where the net topological charge,QSk =

∫dr qT(r) = −1 for a skyrmion.

To analytically establish the origin of these resonances,the m(r) for each temporal snapshot was first projectedonto a cylindrical reference frame {er, eϕ, ez} centered onRqT , which remains static through the period of the simu-lation (τ). Next, the deviations of OP (∆mz = mz − mz,0)and IP (∆mϕ = mϕ − mϕ,0) components relative to thestatic skyrmion configuration m0 were determined (note:mϕ,0 = 0 for a Néel skyrmion). The spatiotemporal de-viation thus obtained, ∆m (t, r, ϕ) was then Fourier trans-formed for azimuthal separation of resonance modes

F(z,ϕ)µ (ω) =

1πR2τ

τ∫0

dtR∫

0

dr r2π∫0

dϕ ∆mz,ϕ(t, r, ϕ)ei(ωt+µϕ),

(M4)where R is radius of the disk-shaped sample and impor-tantly µ ∈ Z is the azimuthal wave number. Spectra

F(z,ϕ)µ (ω) are calculated for |µ| ≤ 4 (i.e. nine values of µ).

Both ∆mz and ∆mϕ exhibit the most intense response for

µ = −1 (CCW mode). The Fourier spectra, F(z,ϕ)−1 (ω), have

a well-pronounced discrete structure for the resonances{ωi}. The lowest of these, ωccw = mini{ωi}, is recognizedas the eigen-frequency if the CCW mode.

To extract the radial profiles of the CCW mode, we cal-culate:

F (z,ϕ)(r) =1

2πτ

τ∫0

dt2π∫0

dϕ ∆mz,ϕ(t, r, ϕ)ei(ωt+µϕ), (M5)

where µ = −1, and ω = ωCCW is determined above.The radial profiles mz(r) and mϕ(r) can then be obtainedby accounting for the general symmetry31 ∆mz(t, r, ϕ) =mz(r) cos(ωt+µϕ+ η), and ∆mϕ(t, r, ϕ) = mϕ(r) sin(ωt+µϕ+ η), where η is an arbitrary phase. In this caseF z(r) ≈12 mz(r)e−iη andF ϕ(r) ≈ i

2 mϕ(r)e−iη . Thus, one can recon-struct the CCW mode as

∆mz,ϕ(t, r, ϕ) = 2 · Re[F (z,ϕ)(r)] cos(ωt + µϕ) (M6)

+ 2 · Im[F (z,ϕ)(r)] sin(ωt + µϕ)

Results of this reconstruction are shown in Fig. 4c-d.