Experimental observation of the curvature-induced asymmetric spin-wave dispersion in hexagonal nanotubes
Lukas Körber, Michael Zimmermann, Sebastian Wintz, Simone Finizio, Markus Weigand, Jörg Raabe, Jorge A. Otálora, Helmut Schultheiss, Elisabeth Josten, Jürgen Lindner, Christian H. Back, Attila Kákay
EExperimental observation of the curvature-induced asymmetric spin-wave dispersionin hexagonal nanotubes
Lukas K¨orber,
1, 2, ∗ Michael Zimmermann, Sebastian Wintz,
4, 5
Simone Finizio, Markus Weigand,
5, 6
J¨org Raabe, Jorge A. Ot´alora, Helmut Schultheiss,
1, 2
Elisabeth Josten, J¨urgen Lindner, Christian H. Back,
9, 3 and Attila K´akay Helmholtz-Zentrum Dresden - Rossendorf, Institute of Ion Beam Physics and Materials Research,Bautzner Landstraße 400, 01328 Dresden, Germany Fakultt Physik, Technische Universit¨at Dresden, D-01062 Dresden, Germany Fakultt fr Physik, Universit¨at Regensburg, Universit¨atsstraße 31, D-93053 Regensburg, Germany Swiss Light Source, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland Max-Planck-Institut fr Intelligente Systeme, 70569 Stuttgart, Germany Helmholtz-Zentrum Berlin, 12489 Berlin, Germany Departamento de Fsica, Universidad Catlica del Norte,Avenida Angamos 0610, Casilla 1280, Antofagasta, Chile Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons (ER-C) and Peter Gr¨unberg Institute (PGI),Forschungszentrum J¨ulich, 52425 J¨ulich, Germany Physik-Department, Technische Universit¨at M¨unchen, 85748 Garching b. M¨unchen, Germany (Dated: September 7, 2020)Theoretical and numerical studies on curved magnetic nano-objects predict numerous excitingeffects that can be referred to as magneto-chiral effects, which do not originate from the intrin-sic Dzyaloshinskii-Moriya interaction or surface-induced anisotropies. The origin of these chiraleffects is the isotropic exchange or the dipole-dipole interaction present in all magnetic materialsbut renormalized by the curvature. Here, we demonstrate experimentally that curvature inducedeffects originating from the dipole-dipole interaction are directly observable by measuring spin-wavepropagation in magnetic nanotubes with hexagonal cross section using time resolved scanning trans-mission X-ray microscopy. We show that the dispersion relation is asymmetric upon reversal of thewave vector when the propagation direction is perpendicular to the static magnetization. Thereforecounter-propagating spin waves of the same frequency exhibit different wavelenghts. Hexagonalnanotubes have a complex dispersion, resulting from spin-wave modes localised to the flat facets orto the extremely curved regions between the facets. The dispersion relations obtained experimen-tally and from micromagnetic simulations are in good agreement. These results show that spin-wavetransport is possible in 3D, and that the dipole-dipole induced magneto-chiral effects are significant.
After having been proposed by Bloch in the 1930s [1],the propagation of spin waves (or magnons) – the ele-mentary excitations in magnetically ordered systems –has been studied extensively in the past. Because oftheir peculiar linear and nonlinear characteristics, spinwaves promise great potential in information transportand processing as, e.g. , the magnon transistor [2] and themagnonic diode [3] for multifunctional spin-wave logicapplications. Spin waves (including the spatially uni-form ferromagnetic resonance precession) have also beenproven to be an excellent tool to probe the magnetic char-acteristics of solids as they are sensitive to spin currents[4–6], impurities [7–9], crystal anisotropies [10] or asym-metric exchange interactions, among others. For exam-ple, the presence of an asymmetric interaction such asthe DzyaloshinskiiMoriya interaction (DMI) leads to anasymmetric dispersion and consequently to a nonrecip-rocal propagation of spin waves, therein [11–16]. Similarnon-reciprocal spin-wave propagation is observed in mag-netic bilayers [17, 18]. Therefore, the study of spin-wavepropagation is both of a technological as well as a funda- ∗ Corresponding author: [email protected] mental interest.While many of the aforementioned effects have beeninvestigated mostly in bulk or in flat thin-film samples,over the last decade, curvature induced effects have beenuncovered as a new way to manipulate magnetic equi-libria as well as spin dynamics. Numerous theoreti-cal and numerical works have already shown that thesurface curvature of magnetic membranes leads to phe-nomena which are not present in flat geometries of thesame material [19–28]. As a result of the bending, inconventional soft magnetic materials exotic non-collinearmagnetic textures such as skyrmions [29] may be stabi-lized or magnetisation dynamics can be influenced, lead-ing to left-right symmetry breaking of domain wall mo-tion [19, 20], asymmetric spin-wave transport [25, 30] orthe emergence of a topological Berry phase [31]. So far,experimental evidence for curvature-induced effects wasshown in [32], in which the surface curvature resulted inthe bending of the domain walls in a flux-closure mag-netic structure and more recently in the work by Volkov et al. [33]. In the latter, it is shown that due to thecurvature-induced effect originating from the exchangeinteraction, domain walls in parabolic permalloy wiresare pinned by the curvature gradient and that by measur- a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p ing the de-pinning of these domain walls the magnitude ofthe exchange-induced curvilinear effect can be indirectlydetermined. Similar to the microscopic DMI, curvature-induced effects are associated to the class of magneto-chiral interactions as they often break chiral symmetryin magnetic systems.The influence of curvature on magnetic equilibria hasbeen shown to be mainly due to a renormalization of themagnetic exchange interaction [23, 34]. Magnetizationdynamics, on the other hand, are mostly perturbed bycurvature-induced magnetic charges [19, 20, 22], i.e. , bya renormalization of the dipolar interaction. In our re-cent works [25, 26], it was predicted that this leads to anasymmetric dispersion of spin waves in round magneticnanotubes being in the vortex state.In this Letter, we demonstrate experimentally thatcurvature induced effects originating from the dipole-dipole interaction are directly observable by measuringspin-wave transport in magnetic nanotubes with hexago-nal cross section in the vortex state, using time-resolvedscanning transmission X-ray microscopy (TR-STXM)[35–37]. These results are a direct qualitative confirma-tion of our theoretical and numerical prediction for roundnanotubes [25, 26] with the advantage that the hexagonalcore/shell structures can be grown with high quality bymolecular beam epitaxy. Moreover, our micromagneticsimulations and TR-STXM experiments show that themode spectrum of hexagonal tubes, accessible by a sim-ple microwave antenna is much more diverse than thatof round tubes and splits into several branches. Besidesthe modes localized and propagating along the top- andbottom facets of the tube, spin waves propagating on theside facets or localized to the highly curved corners atthe junctions of the facets of the hexagonal cross-sectionare found. At the same time, modes with a similar col-lective behaviour as the spin waves found in round tubescould also be addressed. The dispersion is predicted tobe asymmetric for all branches and for those branchesaccessible by the TR-STXM frequency range it is foundto be the case, which directly demonstrates the presenceof curvature-induced magneto-chiral effects.Despite the remarkable achievements on novel ap-proaches for magnetic nano-membranes preparation,such as stretchable [38], rolled-up [39, 40] and flexiblemagnetic membranes, as well as flexible displays [41],shapeable nanoelectronics [42] and even printable spin-tronic devices [43], to our knowledge, there was no successin the preparation of high-quality round ferromagnetictubes with sufficiently low damping to allow for exper-imental studies of spin-wave transport. The closest ap-proach was recently reported by Zimmermann et al. [44]who succeeded in fabrication of hexagonal ferromagneticnanotubes with sub-nanometer surface roughness, usinga GaAs single crystal rod coated in-situ with Ni Fe (permalloy) capped with Al for protection. The used in-cident angle growth conditions during the evaporationprocess lead to an easy-plane magnetic anisotropy per-pendicular to the symmetry axis of the nanotube, result- FIG. 1. (a) Transmission-electron-microscopy image of ahexagonal permalloy nanotube with 250 nm outer diameter,30 nm thickness and 12 µ m length on a GaAs wire. The goldstrip-like antenna (here, colored for visual purposes) was pat-terned on a SiN membrane and the nanotube was placed onthe top using a focused ion beam (FIB) tool and a micromanipulator. The Oersted field of an rf-current is used to ex-cite spin waves. (b) Cross-sectional sketch of the hexagonalnanotube showing the layer structure. The permalloy layeris directly evaporated on the GaAs wire in situ and cappedwith Al to avoid oxidation. A relative angle of 60 degree wasused between the X-ray beam and the symmetry axis of thenanotube. This configuration allows for being sensitive tothe in-plane dynamic magnetization of the top- and bottomsurfaces of the tube, which is expected to be larger than theout-of-plane dynamic magnetization component. ing in a stable vortex state in the hexagonal nanotube, asconfirmed by STXM measurements [44]. This finally al-lowed us to experimentally study the curvature-inducedeffects on spin-wave dynamics in nanotubes. For thispurpose, the previously manufactured hexagonal tubewith an outer diameter of 250 nm, a thickness of 30 nmand a length of 12 µ m was placed on a silicon membranewith a gold antenna underneath to excite spin waves (seeFig. 1). Spin waves propagating away from the antennaare measured using TR-STXM. The experimental resultsare complemented by micromagnetic simulations usinga custom version of the GPU-accelerated code MuMax [45] (see Supplemental Materials for details).In the experiments the spin waves were excited at fixedfrequencies. In Fig. 2(a-c), we show exemplary snap-shots of two counter-propagating spin-wave modes at5 .
571 GHz, 8 .
571 GHz and 9 .
571 GHz, obtained by mi-cromagnetic simulation and TR-STXM, respectively. Asseen from the profiles, the modes exhibit different local-ization within the cross section of the hexagonal tube, e.g. , there are modes more localized in the corners ofthe tube (5 .
571 GHz in Fig. 2(a)) or on the facets of thetube (8 .
571 GHz and 9 .
571 GHz in Fig. 2(b,c)). In thiscontext, we also would like to refer to the experimentalmovies of these modes, provided in the supplemental ma-terial, which show nicely the localization of the modes at5 .
571 GHz in the corners of the hexagonal cross section.We observe an intensity asymmetry of the modes at largefrequencies. This is a commonly known effect for Damon-Eshbach spin waves (with k ⊥ M eq ) that are excitedwith a strip-line antenna in magnetic thin films [46], andis also present here as our tubes are in the vortex state( k ⊥ M eq ). Moreover, in the numerical mode snapshotat 5 .
571 GHz (Fig. 2(a)), one can already clearly see awave-vector asymmetry for the two counter-propagatingmodes which is the evidence for an asymmetric spin-wavedispersion.
FIG. 2. (a)-(c) Numerical (left) and experimental (right)snapshots of the dynamical magnetization at a fixed timefor modes propagating in the corners of the hexagonal tube(5 .
571 GHz) and for modes mostly propagating on the top andbottom facets (8 .
571 GHz and 9 .
571 GHz). For the numericalprofiles, a side view is shown next to the projection of the up-per half the nanotube. For better illustration, they have beenstretched in the width direction. Above of the experimentalprofiles, we show averaged linescans along the tube as a visualaid. The position of the antenna is in all cases marked witha translucent gold-colored patch.
To obtain the full dispersion we extracted the wave vec-tors for different excitation frequencies. For this, we av-erage the experimentally measured data along the widthdirection of the tube (shown on top of the STXM snap-shots in Fig. 2) and fit these curves for individual timeframes with decaying sinusoidal functions along the longaxis of the tube (propagation direction). This methodwas chosen because only a few wavelengths are observedin the measurements, and a Fourier analysis to obtainthe wave vectors at a given frequency was not conclu-sive. In order to interpret the experimental results we additionally calculate the dispersion using micromagneticsimulations. The antenna geometry and material param-eters including the easy-plane anisotropy, are the sameas in the experiments. In the simulations we use a tubelength of 10 µ m. After having excited all frequencies upto a cut-off frequency of f c = ω c / π = 50 GHz homo-geneously using a sin( ω c t ) / ( ω c t ) pulse and letting themagnetization dynamics evolve for 25 ns, the wave-vectordependent frequencies are obtained using a fast Fouriertransform (FFT) in time and along the tube’s main ( z )axis. The resulting dispersion is shown in Fig. 3(a). Theexperimental values are added as dots on the numericaldispersion in Fig. 3(a) and show a very good agreementwith the simulation.For comparison, in Fig. 3(c), we show the numericallyobtained dispersion for a round tube with the same an-tenna geometry, material parameters and more impor-tantly, equal excitation conditions. As seen in the dis-persion calculated with our finite element code Tetra-Mag [47], the field pulse in this geometry couples to themodes propagating on the top- and bottom sides of theround tube (which may be referred to as modes withan azimuthal index of m = ± k z = 0; the correspondingFourier spectrum is plotted in Fig. 3(b). Performing awindowed inverse Fourier transform at the frequenciesof the three lowest peaks reveals the spatial mode pro-files. Cross-sections of these profiles at the center of thetube are shown as insets in Fig. 3(b). We find that thepeak lowest in frequency at 2 .
80 GHz corresponds to spinwaves which are localized in the corners of the hexagonaltube. In accordance to this, recall, that e.g. the modeexcited at f = 5 .
571 GHz (shown in Fig. 2(a)) exhibitedthe same localization as it belongs to the this particu-lar branch of the dispersion. Above in frequency, thepeak at 6 .
45 GHz in Fig. 3(b) corresponds to the modeslocalized on the side facets of the tube. This explainswhy the second branch in the dispersion is much weakerin intensity, because the antenna at the bottom of thetube is not able to efficiently couple to these modes. Asa consequence, in the experiments we can neglect anycontribution of the side-facet modes. Finally, the thirdbranch in the dispersion, in Fig. 3(b) represented by thepeak at 7 .
64 GHz corresponds to modes propagating onthe top-and-bottom facets of the hexagonal tube. Themode snapshots presented earlier in Fig. 2(b,c) belongto this dispersion branch. These modes are most simi-lar to those in the Damon-Eshbach geometry [48] in thin
FIG. 3. (a) Numerically and experimentally obtained dis-persion of the spin waves in a hexagonal permalloy nanotubewith 250 nm outer diameter and 30 nm thickness, excited witha strip-line microwave antenna at the center of the tube. Thespectra were obtained by micromagnetic simulations (usingMuMax [45]), shown as heat map, and TR-STXM, shown asdots on top of the map, respectively. (b) The branches of thedispersion are categorized by their cross-sectional mode pro-files at k z = 0 which show waves propagating at the corners ofthe hexagonal cross-section (2 .
80 GHz), along the side facets(6 .
45 GHz) and along the top and bottom facets (7 .
64 GHz).The dispersion is found to be asymmetric with respect to in-version of the wave vector k z for all branches. For comparison,(c) shows the numerically obtained asymmetric spin-wave dis-persion relation of a round vortex-state nanotube with equaldimensions, material parameters and excitation scheme (notethe different scales in frequency and wave vector). Panel (d)shows the corresponding power spectrum of the dynamicalmagnetization at wave number k z = 0. The main peak at f = 5 .
40 GHz corresponds to the dynamics of modes travel-ing on the top and bottom half of the tube. films. Additionally, there is a fourth branch which doesnot have any intensity at k z = 0 and is, therefore, notseen in the power spectrum in Fig. 3(b) but only in thefull dispersion Fig. 3(a). Indeed, these modes correspondto the azimuthal ( m = ±
1) modes known from roundnanotubes (Fig. 3(d)). The different branches of the dis- persion exhibit multiple (avoided and unavoided) levelcrossings. Note, that a thorough discussion of the modalspectrum presented here would go beyond the scope ofthis Letter and will be published elsewhere.
FIG. 4. Absolute and relative wave-vector asymmetry,∆ k = k + − k − and ∆ k rel = ∆ k/ ( k + + k − ), obtained by mi-cromagnetic simulation and TR-STXM using the dispersiondata presented in Fig. 3(a). Finally, we discuss the asymmetry of the dispersion.Although from the snapshots obtained by micromag-netic simulations it can clearly be seen that the counter-propagating waves at the same frequency inherit a dif-ferent wave number. This might not be immediatelyobvious from the experimental data. In order to quan-tify the left-right asymmetry of the spin-wave propaga-tion, we define the absolute difference in wave vector∆ k for counter-propagating waves at the same frequency,∆ k ( f ) = k + ( f ) − k − ( f ), as the asymmetry of the disper-sion. From the dispersion relation shown in Fig. 3(a)one can already conclude that the asymmetry is presentfor all branches of the dispersion. In Fig. 4 the abso-lute asymmetry for the corner as well as the top-bottomfacet modes obtained from simulations (solid lines) andfrom experiments (hollow dots) is plotted together withthe relative asymmetry ∆ k rel = ∆ k/ ( k + + k − ). Unfortu-nately, from the experiments we were not always able toobtain satisfactory fits for both propagation directions forall the measured frequencies, thus ∆ k could not alwaysbe extracted. However, as seen in Fig. 4, the experimen-tal results are still in a qualitative agreement with themicromagnetic simulations.The overall sign of the asymmetry ∆ k depends on thedispersion branch. This feature is also present for roundnanotubes for which the sign can also change for brancheswith different azimuthal indices m . Notably, not onlythe corner modes propagating in the highly curved re-gions of the hexagonal nanotube are strongly influencedby the curvature-induced magnetochiral effects. Thereis also a significant influence on the modes that prop-agate localized to the facets which are, for themselves,flat objects. But still they constitute to a net curvatureif combined into a tube. Next to the overall dispersionasymmetry, one can clearly see from Fig. 3(a) that thesaddle points or bottoms of the dispersion branches arealready fairly shifted with respect to k z = 0 for the sidefacet and the top-bottom-facet modes. In case of spin-wave transport in the Damon-Eshbach configuration, asin the case of the current manuscript, this is an indicationthat an analytical expression for the dispersion containsa term which is, in first order approximation, linear inthe wave number k z . This is common for the spin wavesin systems with a magneto-chiral interaction, e.g. thin-film samples with intrinsic Dzyaloshinskii-Moriya inter-action [49], and, as recently shown, round nanotubes inthe vortex state [25, 26].Let us note, that the antiparallel alignment of theequilibrium magnetization in opposite facets would alonelead to an asymmetry, namely to a linear shift of thedispersion in the small k limit [18]. This effect con-tributes as an approximately hyperbolic decay in the rel-ative wave-vector asymmetry with respect to the excita-tion frequency f (see Supplemental Material). We see inFig. 4(b), that this is not the case for the hexagonal nan-otube meaning that the asymmetry is strongly influencedby the curvature. Discussing the relation between thislong-range dipolar coupling of the two opposite facets andthe curvature-related effects would go beyond the scopeof this Letter. However, both effects have their origin inthe nontrivial spatial distribution of the dynamic dipolarfields in the tube.In conclusion, we showed spin-wave propagation inthree-dimensional permalloy nanotubes of hexagonal cross section using TR-STXM measurements. Wedemonstrate experimentally, that the curvature-inducedmagneto-chiral effects originating from the dipole-dipoleinteraction are directly observable in agreement with nu-merical and analytical predictions. We find that the dis-persion relation is asymmetric regarding the wave vector,therefore spin waves of the same frequency propagat-ing into opposite directions have different wavelengths.Moreover, the magneto-chiral effect and therefore theasymmetric spin-wave transport originating from thedipole-dipole interaction is present for all modes, evenfor those localized to the flat facets. Since hexagonalnanotubes can be manufactured with a very high qualityand precision, we believe, that they are a perfect modelsystem for the emerging field of spin-wave transport incurvilinear geometries. Therewith, hexagonal tubes pro-vide an opportunity to unveil novel functionalities, suchas unidirectional spin-wave transport in non-reciprocalwave guides induced by the curvature.The experiments were mainly performed at theMAXYMUS endstation of BESSY II at Helmholtz-Zentrum Berlin, Germany. We thank HZB for the al-location of synchrotron radiation beam time. Some ex-periments were performed at the PolLux endstation ofthe Swiss Light Source. We acknowledge the Paul Scher-rer Institut, Villigen PSI, Switzerland for provision ofsynchrotron radiation beamtime. The PolLux end sta-tion was financed by the German Ministerium fr Bildungund Forschung (BMBF) through contracts 05K16WEDand 05K19WE2. 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