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Emission and Propagation of Multi-Dimensional Spin Waves with nanoscale wavelengths in Anisotropic Spin Textures

Spin waves offer intriguing novel perspectives for computing and signal processing, since their damping can be lower than the Ohmic losses in conventional CMOS circuits. For controlling the spatial extent and propagation of spin waves on the actual chip, magnetic domain walls show considerable potential as magnonic waveguides. However, low-loss guidance of spin waves with nanoscale wavelengths, in particular around angled tracks, remains to be shown. Here we experimentally demonstrate that such advanced control of propagating spin waves can be obtained using natural features of magnetic order in an interlayer exchange-coupled, anisotropic ferromagnetic bilayer. Using Scanning Transmission X-Ray Microscopy, we image generation of spin waves and their propagation across distances exceeding multiple times the wavelength, in extended planar geometries as well as along one-dimensional domain walls, which can be straight and curved. The observed range of wavelengths is between 1 {\mu}m and 150 nm, at corresponding excitation frequencies from 250 MHz to 3 GHz. Our results show routes towards practical implementation of magnonic waveguides employing domain walls in future spin wave logic and computational circuits.

Spin waves are the elementary excitations of the order parameter in ferromagnetic materials [ Fig. 1a)] [1][2][3]. Also referred to as magnons, they can be used similarly to electrons in CMOS circuitry, but with lower losses, to transmit information, and, therefore, are currently attracting a lot of interest as possible information carriers in alternative computing schemes [4][5][6][7]. One of the most pressing issues in present-day high-performance computing are the high power requirements and the necessary heat removal associated with the Ohmic losses in conventional electronic CMOS circuits -latest generation supercomputers easily consume power in the order of ten Megawatts. On a wider societal scale, the reduction of signalprocessing losses, in particular in personal mobile communication devices may create substantial benefits due to reduced power consumption, resulting in extended battery life and improved environmental sustainability. Another substantial advantage of spin-wave technology is the fact, that in the GHz range, magnon wavelengths are several orders of magnitude shorter than those of electromagnetic waves [8]. Thus, a significant device miniaturisation can be achieved for applications where the wavelength imposes a critical constraint on the device footprint. For such purposes it will be crucial to utilise spin waves with wavelengths in the subµm range, where both magnetostatic and exchange effects are relevant (dipole-exchange waves) [5]. While surface acoustic waves are already present as short-wavelength signal carrier in today's communication technology, spin waves offer a superior scalability of wave excitation and propagation at frequencies above 2 GHz as well as a much wider frequency tunability [9,10].
While in Ref. 31, the lateral position of the excited magnetisation amplitudes could be well controlled by tuning the lateral domain wall position, these modes however quickly decayed along the domain wall coordinate with increasing distance from the microwave antenna within subwavelength length scales.
These key issues -short wavelength-spin wave generation and spin wave guidance -are the points we address in this work, where we make use of naturally formed anisotropic spin textures. First, we demonstrate the excitation and propagation of two-dimensional planar spinwaves [ Fig. 1c)] excited by the oscillation of straight domain walls. We observe that these excitations can travel distances spanning multiples of the wavelength. Second, we observe excitation and propagation of spin wave modes confined to quasi-one-dimensional natural waveguides (straight or curved) formed by domain walls embedded in a two-dimensional host medium [ Fig. 1d)].
Our samples are Co40Fe40B20/Ru/Ni81Fe19 multilayers with (46.6/0.8/44.9) nm thickness, patterned into disc-and square-shaped elements, having lateral sizes of several microns [ Fig. 2a)]. Each ferromagnetic layer exhibits an in-plane uniaxial anisotropy. The Ru interlayer causes antiferromagnetic coupling between the two ferromagnetic layers [44] (see methods part for further details).
The magnetic ground state configuration stabilized in this system is a pair of stacked vortices, with opposite vorticity due to the antiferromagnetic interlayer exchange coupling. The As Fig. 2c) indicates, the out-of-plane magnetization components of the respective layers couple ferromagnetically to each other by their stray field. In particular, this is true for the polarizations of the vortex cores. Micromagnetic simulations confirm this and reveal that the domain wall formed in the sample is, in fact, a mixture between Néel and Bloch types of domain walls [45], where the in-plane components couple antiferromagnetically across the Ru interlayer, as in the domains. The complex ground state magnetic pattern is illustrated in panels d) and e) of Fig. 2. Figure 2d) displays a schematic top view of the domain wall structure in the CoFeB layer, showing the mixed Bloch and Néel components. In Fig. 2e) a cross-section of the bilayer system is shown, which can be imagined as resulting from a cut along the blue lines in panel c) of the Figure, revealing the out-of-plane magnetization components in the domain wall to follow a flux-closing distribution between the two layers.
Spin waves can be excited in such anisotropic spin textures by applying an alternating magnetic field, as shown in Figure 3. The corresponding measurements were made by means of time-resolved (TR)-STXM imaging, allowing for a stroboscopic time-resolution of approximately 100 ps. Figure 3 (a) is a snapshot of the magnetic excitations at an Oersted field frequency of 1.11 GHz, taken at the Ni absorption edge, displaying the out-of-plane contrast.
Plane spin waves are visible, with wave fronts parallel to the domain wall, and propagating away from the domain wall towards the rim of the elliptical element, as indicated by the green arrow. The oscillating Oersted field in-plane component is oriented along the minor axis of the ellipse, perpendicular to the domain wall. The main effect of the Oersted field is to excite dynamics of the domain wall, and that the excited domain wall acts as a confined perpendicular source for the observed spin waves [24,25,29]. In more detail, by acting on the full sample volume, the field excites a non-resonant antiphase width oscillation of the walls in the two different layers which is coherent over the wall length, causing highly localised out-of-plane torques in the wall vicinity. Thereby spin waves are essentially excited via a linear and coherent coupling of the discrete wall mode to the spin-wave continuum, similar to the situation of vortex core driven spin waves reported earlier [18,27].
The periodic-in-time nature of the waves allows capturing the wave motion at discrete, equispaced phases in each scanned pixel, and composing the recorded data into movie-like arrangements, which impressively show the propagation of these spin-excitations [see movies in the SI (1)]. A comparison of the absorption data taken along the green arrow in Fig. 3a) at different time slices yields the wavelength of the wave, and in particular its speed of propagation. Three of these time slices are shown in Fig. 3b). Notably, the spin wave amplitude does not visibly decrease across the distance of two micrometers, corresponding to about 7.5 times the wavelength. Increasing the excitation frequency to 1.46 GHz results in a similar wave pattern, but with shorter wavelength [ Fig. 3c)]. By comparison with the contrast of the vortex core, the spin-wave amplitudes are estimated to reach a precession angle of beyond 5 degrees, which can be considered as very high when compared to standard spin-wave excitation techniques.
Around the vortex center, as shown in the magnified image Fig. 3d), in addition to the plane waves generated by the oscillating wall, there exist radial wave fronts which arise from the motion of the vortex core, which acts as a point source [18]. In comparison to the plane waves excited by the domain wall, these radially symmetric waves must decay faster in power density, with a factor of 1/r in addition to the exponential decay induced by the Gilbert damping, r corresponding to the distance from the vortex center. This difference can be regarded as a consequence of the fact that the plane waves are excited by a one-dimensional source (the domain wall), while in the case of the radial waves, the source is zero-dimensional (the vortex core). As these two wave forms are excited simultaneously, patterns of interference arise which are also visible in Fig. 3d).
In this manner, we can excite planar spin waves for a broad range of frequencies up to 3 GHz. Yet, one can expect this process to scale to even higher frequencies, if the magnetisation gradient of the exciting source (domain wall or vortex core) was enhanced as for domain walls in systems with strong perpendicular magnetic anisotropy or if the spin-wave dispersion relation was tuned to longer wavelengths by modifying the magnetic layer stack. Remarkable effects, however, appear when going to rather low excitation frequencies, as shown in Fig. 4, displaying excitations at 0.52 GHz and 0.26 GHz [panel a) and b), respectively]. At these low frequencies, no visible excitations exist in the domains, yet the data clearly shows spin waves propagating confined to the domain wall in the directions away from the vortex cores. The wavelength of these waves can be controlled in the same way as in the previous cases, i.e. by tuning the excitation frequency. Note that there is a directional asymmetry in the spin-wave emission at 0.52 GHz, which, however can be attributed to sample imperfections in the exciting core region as simulations indicate a symmetric emission pattern. Irrespective of this, the wave amplitude is again still significant even after a propagation distance extending from the vortex core to the rim of the ellipse. This is made possible in a way that is analogue to the above described case of planar waves in a two-dimensional medium excited by the one-dimensional domain wall: For the waves propagating along the domain wall, the source is of dimension zero; however, due to the confinement to the domain wall, the propagation medium is effectively onedimensional. As a result, geometrical decay of the amplitude is avoided, making the domain wall act as a low-loss waveguide (c.f. SI movie M4 of propagating spin waves in the domain walls). The confined waves excited here can be considered a bilayer analog of the spin-wave mode predicted by Winter for a single domain wall [33], while also the coherent domain wall resonance (or infinite wavelength k=0) case of such a bilayer has been theoretically studied in Ref. [46] [cf. SI (2)].
In order to shed light on the physics underlying these observations, we followed a twofold strategy: First, both observed phenomena -the excitation and propagation of planar spin waves in the domain, and one-dimensional waves confined to the domain wall -were investigated and qualitatively confirmed with micromagnetic simulations. For that purpose, the experimental static magnetization distribution was reproduced prior to excitation by an ac magnetic field. We refer to SI movie M7 for a qualitative confirmation of the gapless onedimensional spin-wave mode in the domain wall. In order to obtain the details of the dispersion relation for the planar waves in the domains within reasonable computation time, the system was modeled by two continuous, homogeneously magnetized coupled layers (further details can be found in the methods section). The experimental plane spin-wave dispersion was quantitatively reproduced by these simulations.
In addition to the simulations, we developed a theory (see SI for in-depth technical details) for the propagation of spin waves in two exchange-coupled extended ferromagnetic films. The core of the theory considers spin wave modes in thin magnetic films, where the magnetization along the coordinate perpendicular to the film plane can be considered homogeneous. The case of thicker films as in the experiment is accounted for by splitting each ferromagnetic layer into a number N of thin films of equal thickness, so that for each of these films the thin-film approximation holds. The N thin films of each layer are then coupled to each other by an effective ferromagnetic intralayer exchange coupling, whose strength is determined by estimating the energy of a magnetization distribution subject to homogeneous torsion and by requiring consistency with the continuum limit. The theory thus enables us to quickly compute dispersion relations for spin waves in the interlayer exchange-coupled bilayer system with ferromagnetic layers whose thicknesses exceed the exchange lengths of the respective material. In we find good agreement with the experimental data. The elevated value of Ku(CoFeB) is reasonable, since we expect the CoFeB to react sensitively to strain exerted by the patterned waveguide microstructure onto the elliptical element [47]. Using the same parameters as in the theory, we also compute the dispersion using micromagnetic simulations [grey dots in Fig. 5)].
Note, that the antiparallel bilayer system considered here can in principle host two separate spin-wave modes of acoustic and optical character [cf. SI (2)]. Since the optical mode however resides at much higher frequencies and is not accessed by our experiments, we only consider the acoustic mode in the following.
A striking feature of the acoustic plane wave dispersion is the existence of a local minimum at low k around 5 rad/µm, and accordingly, a frequency gap, below which no spin wave excitations are possible. The local minimum at some finite value of the wave vector in Fig. 5 can be understood due to a combination of the non-reciprocity induced by the dipolar coupling between the two antiferromagnetically coupled magnetic layers [18,48] [SI (2)] and the uniaxial magnetic anisotropy. Namely, when the anisotropy is null, the collective dispersion in Damon-Eshbach geometry (k M0) [49] has a minimum of zero frequency at k = 0, while at finite anisotropies, this minimum is shifted to finite values of both wave vector and frequency.
Such a k-shifting of the dispersion minimum is somewhat analogous to that induced by the Dzyaloshinskii-Moriya interaction on ferromagnetic/heavy-metal alloys, where the minimum of the dispersion is also shifted. [50]. Note that we only observe the slow branch (k +5 rad/µm) of the non-reciprocal dispersion relation in our experiment since the wavelengths of the fast branch are of the order or exceeding the sample size for the frequencies given [cf. SI Remarkably, even after the turn, the wave packet maintains a considerable amplitude. While the spatial distribution of the domain wall in our experiment is solely a consequence of both dipolar sample confinement fields and magnetostrictive anisotropies, it was shown that further control of the domain wall position can be achieved e.g. by exchange bias patterning [34,39], ferroelectric coupling [21,22], or external magnetic fields [31].
To summarize, the work presented here addresses several key aspects of magnonic computing by exploiting magnetic anisotropy. The first aspect is related to energy and signal range. We demonstrated that textures in a magnetization distribution, like domain walls and vortex cores, can serve as sources for the generation of short-wavelength, dipole-exchange spin waves of directional nature, that is, planar waves in magnetic domains and waves confined to domain walls, which due to their geometry, are not subject to the reduction of amplitude due to the geometric dilution of the energy flow. This is an important result, since such waves minimize the losses occurring during propagation. Indeed, we found that the resulting excitations can travel distances easily spanning several microns, i.e., significantly exceeding multiples of the nanoscale wavelengths -a necessary condition, e.g., for magnon interferencebased applications. The second aspect is to identify possible waveguides for magnonic chips.
Here, we showed that domain walls can serve as such waveguides, combining several useful properties. First, due to their inherent symmetry, and consequently their near-gapless dispersion relation, spin waves can selectively be excited in these structures. In addition, we showed that spin wave packets can travel along angled domain walls while largely maintaining their amplitude. Such possibility of angled signal guidance is vital for chip design, and therefore our result may enable new solutions to the development of magnonic circuits.

(µm)
Redraw profiles with Origin/CorelDraw 3 Different green lines (as previously paper) Green in (b) can be more dark (text&graph) Maybe mask ellipse with oval? 2 7.5 µm +470 ps +235 ps an enlarged image of the center region. In addition to the plane waves originating from the domain wall, spin waves with circular wave fronts can be seen. The latter are emitted from the vortex core (c.f. [18]). Using such images, we obtain the dispersion relation for the various types of waves.

µm
Maybe mask ellipse with oval?  The samples were prepared on x-ray transparent silicon-nitride membrane substrates with a thickness of 200 nm. Multilayer films of Ni81Fe19/Ru/Co40Fe40B20/Al were deposited by magnetron sputtering onto these, where Al (5 nm) is serving as a capping layer for oxidation protection. The thicknesses of the ferromagnetic layers, NiFe and CoFeB, were determined by transmission electron microscopy to be 44.9 nm and 46.6 nm, respectively. The Ru spacer (0.8 nm nominal) between them mediates an antiferromagnetic interlayer exchange coupling [44] according to the hard axis magnetization reversal of extended multilayer stacks, which was measured by magneto-optic Kerr effect (MOKE) [52]. Additional MOKE measurements on corresponding single layer films revealed collinear uniaxial magnetic anisotropies of 0.2 kJ/m 3 for NiFe and 1.1 kJ/m 3 for CoFeB. However, in order to reproduce the experimentally found static magnetic configuration in the micromagnetic simulations, a significantly larger value around 3 kJ/m 3 is required for CoFeB. This elevated value for the in-plane uniaxial anisotropy in CoFeB can be attributed to strain, which in our case is caused by the contact with the waveguide. In fact, CoFeB is known for its sensitivity to strain and the orientation of the experimentally observed magnetic pattern with respect to the waveguide is consistent with this interpretation [47]. The patterning of the microelements was realized by electron beam lithography (EBL) and consecutive ion beam etching. Upon an initial oxygen plasma treatment for adhesive purposes, a negative resist (MA-N 2910) was spun onto the multilayer films. In a second step, the microelements were exposed by EBL. The samples were then developed for 300 s in MA-D 525 and rinsed in de-ionized water. Finally, the samples were exposed to an argon ion beam at two different angles (85° and 5°) for physically etching the magnetic microelements out of the continuous films. Remaining resist was removed by aceton and a second oxygen plasma treating. For magnetic field excitation, a copper strip of 200 nm thickness was fabricated on top of the microelements by means of EBL, electron beam evaporation deposition, and lift-off processing [18]. The patterned microstrip has a width of 5 µm, hence the resulting magnetic Oersted field from a flowing electric current of one mA can be estimated to µ0H = 4π * 10 -2 mT.

Time-resolved STXM.
The magnetic orientation in the multilayer microelement investigated was imaged by means of synchrotron based scanning transmission x-ray microscopy (STXM) [53]. Here, a Fresnel zone plate is used to focus a monochromatic x-ray beam onto the sample. The locally transmitted xray intensity is then measured by a single pixel detector, hence raster scanning the sample yields a two-dimensional absorption image with approximately 25 nm lateral resolution. Using furthermore circularly polarized x-rays allows for exploiting x-ray magnetic circular dichroism (XMCD) [54] leading to a magnetic contrast. As XMCD only occurs at the element specific resonant absorption edges, the magnetic signal from both ferromagnetic layers, NiFe and CoFeB, can be separated by tuning the incident x-rays to the corresponding L3 energies, Ni L3 ~853 eV and Co L3 ~778 eV, respectively. On the other hand, a collective signal from both layers can be collected from the Fe L3 edge at ~708 eV since both layers contain Fe. The magnetic contrast acquired is proportional to the projection of the magnetic orientation m = M/M on the x-ray propagation direction ek. Therefore, in normal incidence, the STXM setup is sensitive to the perpendicular magnetization component, while an inclined sample mounting also allows for detecting in-plane magnetization components.
The magnetization dynamics of the multilayer microelements was imaged stroboscopically by means of time-resolved STXM. This method utilizes the specific time structure of the incident x-ray pulses, i.e. 2 ns repetition rate at ~100 ps effective pulse length.
Each incoming signal (photon or no photon transmitted) is routed after every pulse to a periodic counting register of a field programmable gate array. Here the number of registers (Q) sets the maximum non-stroboscopic observation period (Q*2 ns), while the number of excitation repetitions in this period (J) sets the nominal time resolution as well as the excitation frequency in case of a continuous sinusoidal excitation. The excitation current was measured both in front of and behind the sample by means of -20 dB pick-off tees through an oscilloscope.

Micromagnetic simulations.
Micromagnetic simulations based on the time integration of the Landau-Lifshitz-Gilbert [55,56] equation were carried out using the code MuMax 3 [57]. The simulations were performed to compute the spin wave dispersion relations in the coupled layer system. The ferromagnetic layers are homogeneously magnetised and the dispersion relations are calculated in a thin film approach. Therefore, the system was discretised into (4096, 16

Data availability.
The datasets generated and/or analysed during the current study are available from the corresponding authors on reasonable request.

Time-resolved scanning transmission x-ray microscopy
Selected time-resolved scanning transmission X-ray microscopy measurements are shown as movies (M1-M6) with perpendicular magnetic sensitivity for both different samples (#1 and #2) and excitation schemes (continuous wave and pulsed). These movies display the absolute (~mz) and normalised (~∆mz) magnetic contrast at different absorption edges (Fe, Co, Ni L3) as described below: M1_s1_Ni_8um-5um_cw1.11GHz_absolute-normalised

(2) Spin-wave modes in antiparallel magnetic bilayers
The bilayer system with antiparallel magnetic layers considered in the present work can host spin-wave excitations of acoustic and optical character. This applies both to spin waves in the bulk domains and to waves in the domain wall. The acoustic mode spin waves in the domain wall can be considered as the bilayer-analogue of the modes proposed by Winter [33]. The magnetization precession associated with that mode for finite k results in a string-like inphase oscillation of both individual domain walls. In the limit ! → 0 this mode corresponds to the coherent domain wall oscillations as described by Stamps et al. [46]. GHz) for both modes, and that the frequency minimum of the accoustic mode is shifted to finite k to the slow branch side.
In our experiment, however, we only observe waves belonging to the slow branch of the accoustic mode. One reason for this is that the optical mode only exists at frequencies above those addressed in our experiment. Secondly, for the accoustic mode, the fast branch is not observed as the expected amplitudes are relatively low and because the wavelengths at the experimental frequencies are of the order of or beyond the sample size itself.
For the one-dimensional waves confined to the domain walls, the situation is similar.
The accoustic and optical spin-wave modes are also non-reciprocal. As mentioned above, the acoustic mode is gapless due to the Goldstone theorem and in agreement with earlier predictions, while the domain wall optical mode still exhibits a frequency gap. Also for the spin waves confined to the domain walls, we only abserve the slow branch of the accoustic mode.  For the first case, it can be seen that 8 4 (% 4 , -9 ) falls off drastically from low to high kvectors, with an oscillatory k-dependence superimposed, so that at wave lengths such as observed in the experiment (% 4 > 10 rad/µm and higher), the excitation amplitude is only a fraction of its maximum value (that of the spatially homogeneous component). In the second case of the narrower antenna, the fall-off towards higher k is less dramatic, but the overall amplitude is so low that around % 4 = 10 rad/µm, the two cases are actually comparable (c.f. the inset Figure S2). The excitation by spin textures as in our experiment is more efficient because it uses the very strong spatially inhomogeneous internal fields arising from dynamics of the spin textures (driven by the homogeneous external field), which in turn excite the highk spin waves.

I. SPIN WAVES IN COUPLED FERROMAGNETIC BILAYERS
A schematic diagram of the bilayer structure is shown in Fig. 1. The system is composed of two exchange and dipolarly coupled ferromagnetic (FM) layers (1) and (2) with in-plane homogenous magnetization M (1) and M (2) , respectively. The layers have di↵erent magnetic parameters and thicknesses, and s denotes the separation between them. A local coordinates system (X ⌫ ,Y ⌫ ,Z ⌫ ) is defined for each layer (⌫ = 1, 2) [see Fig. 1 x y X 1 The temporal evolution of the system is described by the Landau-Lifshitz (LL) equatioṅ where the dot denotes time derivative, is the absolute value of the gyromagnetic ratio, and H e(⌫) (r, t) is the e↵ective field acting on layer ⌫. In the linear response regime, both magnetization and e↵ective field can be written as M (⌫) (r, t) = M (⌫) sX⌫ + m(r, t) and H e(⌫) (r, t) = H e0 X⌫ +h e (r, t) (see Fig. 1). Thus, the linearized equations of motion (1) where an harmonic dynamic response has been assumed m(r, t) = m(r)e i!t . Also, according to the equilibrium condition H e0 Y⌫ = H e0 Z⌫ = 0. On the other side, for monochromatic spin waves propagating along x, the dynamic magnetization components of both FM layers can be written as m(x) = m(k)e ikx , where k is the wave vector. In the same way, the dynamic e↵ective field is given by h e (x) = h e (k)e ikx . Therefore, the terms M and Thus, Eq. (2) can be expressed in matrix form, this is where the transpose of matrix m(k) is given by m T (k) = [m Y 1 (k), m Z 1 (k), m Y 2 (k), m Z 2 (k)] The matrix elements are given by