Spin-wave interference in magnetic vortex stacks

Spin waves with wavelengths in the nanometre range could serve as data carriers in future magnonic logic or signal processing devices. We investigate the interference of spin waves emitted from magnetic vortices in two exchange-coupled vortex stacks. The spin-wave dynamics are studied using scanning transmission X-ray microscopy and micromagnetic simulations. Stacks of vortices provide an excellent controllability of spin-wave properties including a tunable wavelength in the 100 nm regime and manipulation of their propagation direction via the magnetisation configuration. Furthermore, interference gives rise to amplified or reduced spin-wave amplitudes in distinct areas of the structure providing controlled confinement crucial for future applications of spin waves. Spin waves are promising candidates as a building block for future magnonic devices. The authors present a combined numerical and experimental study of spin-wave interferences in stacks of magnetic vortices that are efficient spin-wave emitters in the nanometre regime.

S pin waves are promising candidates as a building block for future magnonic logic or signal processing devices 1-5 . Having full control over the spin waves is fundamental for the realisation of such devices. Interference phenomena can be used to spatially confine the information that is essential for the application of spin waves [6][7][8][9] . Recently it has been shown that stacked vortex structures are efficient emitters of spin waves with tunable wavelength 10 . Spin waves emitted from these vortex stacks have a wavelength in the order of magnitude of 100 nm and thus open a path to miniaturisation.
The magnetic vortex is the ground-state magnetisation configuration in ferromagnetic nanodisks of fitting geometry 11 . The magnetisation circulates in the plane around the vortex core in the centre region, where the magnetisation points out-of-plane. The orientation of the magnetisation in the vortex core is characterised by the polarisation (p = ±1) 12 , where a positive polarisation signifies an upwards-pointing magnetisation. The circularity describes the sense of the in-plane curling (c = ±1), where a positive circularity represents a counter-clockwise rotation of the magnetisation. When excited by magnetic fields or electric currents, the vortex core gyrates around its centre position 13,14 . The resonance frequency of this gyrotropic mode lies in the sub-gigahertz regime 15,16 . The sense of gyration depends on the polarisation of the core 13 . Besides the gyrotropic mode, the vortex can be excited to higher-order spin-wave modes [17][18][19] . When excited with frequencies in the gigahertz regime, the vortex performs a low-amplitude gyration. A stack of two vortices can then serve as spin-wave emitter with a linear dispersion relation and thus tunable wavelength 10 .
We report a direct observation of interferences of spin waves in two exchange-coupled vortex stacks (see Fig. 1a). We use micromagnetic simulations 20 and scanning transmission X-ray microscopy (STXM) measurements at the MAXYMUS microscope of the BESSY II synchrotron in Berlin, Germany, to observe the spin-wave dynamics temporally and spatially resolved. Different stable magnetisation configurations and their dynamic response to high-frequency magnetic fields are investigated. The transient behaviour of one particularly interesting magnetisation state is discussed in more detail. Spin waves emitted by the vortex stacks interfere with each other, resulting in different interference patterns dependent on their magnetisation state. The interference leads to an amplification or reduction of travelling spin waves in distinct points in space. In the experiments, we directly observe spin-wave interference and also spin-wave diffraction patterns.

Results
Samples and magnetisation configurations. The investigated structures are two coupled stacks of disks with a radius r = 800 nm and an overlap o = 300 nm (Fig. 1b). The stacks consist of two permalloy layers with a thickness t Py = 50 nm that are separated and covered by silicon layers of t Si = 10 nm (Fig. 1c). For the excitation of spin waves, the structures are placed upon a coplanar waveguide. A high-frequency current sent through the waveguide leads to a unidirectional high-frequency magnetic field H rf that excites the magnetisation dynamics (see Fig. 1a). In the simulations, the structures are slightly smaller (r = 640 nm) but have the same aspect ratio t Py r and percentile overlap so that they have equal resonance frequencies 15 . Here, the 10 nm spacing between the layers is modelled using vacuum. Due to the overlap, the disks in one layer are exchange coupled. The spacing between the layers prevents an exchange coupling in z-direction and the interlayer coupling is dipolar.
Each layer of the stack contains two vortices, leading to an overall number of four vortices in the structure. Each disk can have a polarisation p = ±1 and a circularity c = ±1. There are thus four possible (c, p)-configurations per disk and 4 4 = 256 possible (c, p)-configurations of the whole structure. We will concentrate on highly symmetric states with differing circularities for the upper and lower disks, i.e. c 1 c 2 = −1 and c 3 c 4 = −1. Only for such circularity configurations there exist spin-wave modes with short wavelengths in the 100 nm regime 10 . Four highly symmetric (c, p)-configurations are schematically depicted in Fig. 1d-g. The white and black dots represent a positive and negative polarisation and the yellow and pink arrows indicate a positive and negative circularity, respectively. We distinguish between states with all positive polarisations (blue: p 1 p 2 = +1 and p 3 p 4 = +1, Fig. 1d, f) and with opposite polarisations in the upper and lower disks (red: p 1 p 2 = −1 and p 3 p 4 = −1, Fig. 1e, g). For these polarisation configurations, states with equal circularities in a layer (pink: c 1 c 3 = +1 and c 2 c 4 = +1, Fig. 1d  z y x f g Apart from the antivortex state, a magnetisation state is formed that shows different circularities of the disks of the same layer ( Fig. 2c, d). This state is referred to as diamond state. Here, the magnetic moments in the overlap region point in the same direction. This magnetisation state is observed for equal ( Micromagnetic simulations of spin-wave dynamics. We investigate the formation of spin waves in exchange-coupled vortex stacks, first taking the example of a diamond magnetisation state with differing polarisations in the stacks (Fig. 2d). The transient magnetisation dynamics are depicted as snapshots in Fig. 3 at different times after switching on a high-frequency magnetic field in y-direction. The excitation frequency is f rf = 6 GHz and the magnetic field amplitude is μ 0 H rf = 0.5 mT. In the Supplementary Material the corresponding movie (Supplementary Movie 1) is available. In the beginning, the vortices rest and only the black and white contrast of the vortex cores in the centre of the disks is visible (Fig. 3a). After 0.2 ns (Fig. 3b) the emission of spin waves with a wavelength of 170 nm by the left vortex cores is observed. The right cores emit spin waves with a far longer wavelength of approximately 900 nm. In Fig. 3c an inwards-travelling mode with a short wavelength of 170 nm emerges at the edges of the right stack. This is a result of a non-reciprocal reflection of the long-wavelength mode that is emitted by the right vortices. The short-wavelength mode emitted by the left vortices travels to the right stack (Fig. 3d) and interferes with the reflected mode (Fig. 3e, f). A further investigation of this simulation data (see also Supplementary Movie 2) reveals that both stacks emit a long and a short-wavelength mode as observed in ref. 10 . These two modes propagate in opposite directions, depending on the absolute circularity configuration in a stack. The propagation directions of the long and the short spin-wave modes are thus topologically protected.
In the following, the spin-wave interferences are studied for all stable magnetisation configurations introduced in Fig. 2. In Fig. 4a-d snapshots of the steady state spin-wave emission are depicted (see also Supplementary Movie 3). In Fig. 4a the dynamic response of an antivortex state with equal polarisations is depicted. Spiral-shaped, propagating spin waves with a wavelength of 90 nm are emitted by the vortex cores. The spin waves travel outwards as indicated by the white arrows. The  (Fig. 4b).
Here, concentric, outwards-travelling spin waves are emitted by the two vortex stacks. The spin waves have a wavelength of 170 nm, thus approximately twice the wavelength of the spin waves in Fig. 4a. The interference of the spin waves leads to a variation of the spin-wave amplitude and an increase of the overall spin-wave amplitude along the x-direction compared to the y-direction. Thus, in the case of Fig. 4b the spin-wave amplitude is confined in the x-direction (see also Supplementary Movie 3b). Snapshots of the dynamics of the diamond magnetisation state are depicted in Fig. 4c, d. In the case of equal polarisations (Fig. 4c), spiral-shaped spin waves with a wavelength of 90 nm are emitted by the left stack. In the right stack, a short-wavelength mode travels inwards as indicated by the white arrows. Since an outwards propagating long-wavelength mode is superimposed, the identification of the inwards propagating short spin waves requires a close observation of the data in Supplementary Movie 3c. Figure 4d shows the steady state motion of a diamond magnetisation state with opposite polarisations in a stack (p 1 p 2 = −1). As in the antivortex state, the vortices emit concentric spin waves with twice the wavelength compared to the state with equal polarisations. In the left stack a nearly undisturbed outwards propagating spin wave is observed. In the right stack, the spin waves propagate inwards and interfere with the spin waves emitted by the left stack. The dynamics show a pronounced interference pattern, as well as an increase of the overall spinwave amplitude along the x-direction compared to the y-direction.
Despite the complexity of the stacked vortex system, simple rules for the general behaviour of the spin waves apply. Most obvious is the wavelength of the spin waves when considering Fig. 4a-d. A short-wavelength spin wave (90 nm) is generated whenever p 1 p 2 = +1 and a longer wavelength spin wave (170 nm) is generated if p 1 p 2 = −1. A vivid explanation for the emergence of spin waves with different wavelengths can be derived from the source of the spin waves. Two gyrating stacked vortex cores lead to local perturbations of the z-component of the magnetisation M z that give rise to the outwards travelling spin waves 10 . Neglecting that the two vortex cores are not in the same layer we assign possible wavelengths for such perturbations of the zcomponent of the magnetisation M z in Fig. 4e, f. In the p 1 p 2 = +1 case (blue), a first order wave that has half the wavelength compared to the p 1 p 2 = −1 case (red) can be assigned and explains the observed spin-wave wavelengths. The second prominent feature when examining Fig. 4a-d is the appearance of spiral-shaped spin waves for p 1 p 2 = +1 and concentric spin waves for p 1 p 2 = −1. A possible explanation can be derived from the gyration direction of the vortex cores that depends on the polarisation. The spin-wave source has a non-vanishing total angular momentum in the p 1 p 2 = +1 case as the cores gyrate in the same direction, whereas in the p 1 p 2 = −1 case the cores gyrate in opposite directions and the total angular momentum vanishes. Thus, in the p 1 p 2 = −1 case the two motions quench the total angular momentum leading to concentric spin waves while they add up in the case of parallel aligned cores (p 1 p 2 = +1), leading to spiral spin waves (compare also Supplementary  Fig. 3). These two vivid explanations only account for the generation process of the spin waves and do not consider reflection or interference of spin waves.
The micromagnetic simulations lead us to suggest that it is possible to observe spin-wave interferences in two exchangecoupled stacks of magnetic vortices. The diamond magnetisation state with different polarisations in the stacks (Fig. 4d) shows the most pronounced interference. Our system provides an excellent controllability of spin-wave properties. Tuning of the circularity configurations 22,23 can be used to manipulate the propagation direction of the spin waves as most prominently seen in Fig. 4b, d as well as in the corresponding Supplementary Movie 3b, d. Tuning of the polarisation configuration [24][25][26][27] can be used to manipulate the wavelength and shape of the spin waves. The interference offers the possibility to confine the information at distinct points in space. This is crucial for the use of spin waves in future magnonic devices.
Experimental observation of spin-wave interferences. We now compare the micromagnetic simulations to the experimental findings. In Fig. 5a, b snapshots of two exemplary STXM measurements are shown, in Fig. 5c, d the corresponding micromagnetic simulations are depicted (see also Supplementary Movie 4 for the corresponding movies). The stable magnetisation configurations corresponding to the dynamics are illustrated in Fig. 5e, f. In Fig. 5a the vortices are excited with a high-frequency magnetic field with a frequency of f rf = 4.978 GHz and an amplitude of μ 0 H rf = 0.2 mT 28 . The vortices in the left stack emit concentric, outwards-propagating spin waves with a wavelength of 215 nm. The vortices on the right side emit spin waves that propagate inwards with the same wavelength. These spin waves interfere with each other as can be seen from the typical interference pattern, i.e., a cone-like region with high-amplitude spin waves in the right stack that is surrounded by a low-amplitude region. This interference pattern strongly resembles the micromagnetic simulation of a diamond magnetisation state with opposite polarisations in a stack that is shown for comparison in Fig. 5c, e. The propagation direction of the spin waves depends on the absolute orientation of the circularities in the stacks (c 1 , c 2 ) and (c 3 , c 4 ). We make sure that the circularities in the experiments and simulations are equally oriented. The concentric form of the spin waves and their wavelength lead to the conclusion that the polarisations in the stacks are aligned antiparallel in the experiments (p 1 p 2 = −1 and p 3 p 4 = −1). The wavelength of the spin waves is in qualitative and quantitative agreement with the simulations for p 1 p 2 = −1 as can be seen in Supplementary Note 2, Supplementary Fig. 2, where the dispersion relations of theoretically predicted spin-wave modes are compared to the micromagnetic simulations and experiments. Thus, we directly observe spin-wave interference in exchange-coupled stacks of vortices. Due to the interference, the amplitude of the spin waves varies in the structure with distinct areas of low and high spinwave amplitudes. In addition to spin-wave interference, spinwave diffraction is observed in another structure (Fig. 5b). The vortices are excited with a high-frequency magnetic field with a frequency of f rf = 5.891 GHz and an amplitude of μ 0 H rf = 0.33 mT. Here, vortices are only present in the left stack. They emit spin waves that propagate outwards and have a wavelength of 176 nm. The spin waves travel to the right stack that does not contain any vortices where the amplitude of the spin waves is reduced. This spin wave pattern is also reproduced by the micromagnetic simulations: Fig. 5d shows a snapshot of a micromagnetic simulation of the spin-wave dynamics of the stable state depicted in Fig. 5f (see also Supplementary Movie 4d).
The stable magnetisation configuration with vortices in only one stack is found in exchange-coupled permalloy stacks with an increased percentile overlap (25% compared to 18.5%, Fig. 5f). The propagation direction of the spin waves, their concentric form, and their wavelength lead to the conclusion that the magnetisation state in the experiments and simulations coincides. The vortices in the left stack emit spin waves that travel to the right stack (Fig. 5b, d). The spin waves are diffracted at the constriction and a diffraction pattern is visible at the right side of the structure. Due to diffraction, the amplitude of the emitted spin waves decreases in the right side of the structure. The experimentally observed spin-wave dynamics are in excellent agreement with the micromagnetic simulations.

Discussion
We have shown experimentally that spin-wave interference and diffraction can be directly observed and controlled in exchangecoupled stacks of magnetic vortices. These stacks generate spin waves with a wavelength in the 100 nm regime. The experiments are precisely reproduced by micromagnetic simulations and are understood in a simplified vivid explanation. The amplitude of the spin waves varies in the structure thus creating distinct areas of low and high spin-wave amplitudes. This is an important feature for the use of vortices as spin-wave emitters in, e.g. spinwave-based signal-processing devices. A cascade of the investigated structure might be used as an efficient way to carry and process information in such a device.

Methods
Sample preparation. Vortex stacks consisting of two permalloy (Ni 80 Fe 20 ) and two silicon layers are prepared by electron-beam lithography, O 2 plasma etching, thermal evaporation, and lift-off processing on 100-nm-thick silicon-nitride membranes transparent for soft X-rays. The permalloy layers have a height of t Py = 50 nm and are separated and covered by silicon layers of t Si = 10 nm in height. The radius of the disks is r = 800 nm and the overlap is o = 300 nm. An ensemble of 180 structures is deposited on a coplanar waveguide with a thickness of 100 nm of copper and a hydrogen silsesquioxane (HSQ) isolation layer of 40 nm.
STXM measurements. The spin-wave dynamics are directly observed using STXM at the MAXYMUS microscope of the BESSY II synchrotron in Berlin, Germany. The magnetic contrast is provided via the X-ray magnetic circular dichroism at the Ni L 3 absorption edge (E ≈ 854 eV). In the mode of operation used here, the setup provides a spatial resolution of 25 nm and a temporal resolution of 100 ps. The magnetisation dynamics are stroboscopically imaged. If the sample is placed perpendicular to the beam axis, the z-component of the magnetisation M z is measured. If the sample is tilted by 30°with respect to the beam axis, additionally one in-plane component of the magnetisation is visible. Using the tilted setup, the circularity configurations in the stacks are determined to c 1 c 2 = −1 and c 3 c 4 = −1.
Micromagnetic simulations. The micromagnetic simulations are performed using the MicroMagnum micromagnetic simulation software 20 . In the simulations, the structures are slightly smaller than in the experiments in order to comply with memory constraints of the graphics processing units. The disk's radius and overlap are r = 640 nm and o = 240 nm. The height of the two permalloy layers is t Py = 40 nm. In the simulations the 10 nm spacing between the layers is modelled using vacuum. The sample size was chosen so that the disks have the same aspect ratio

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.