Scattering of spin waves in a multimode waveguide under the influence of confined magnetic skyrmion

The magnetic configuration of a skyrmion is an intriguing research object.^1,2 Because of the topologically protected stability of their magnetization texture at the nanoscale, skyrmions are also promising for storing and processing information.^3,4 In ultrathin ferromagnetic films and multilayers, Néel-type skyrmions with a defined sense of magnetization rotation going in the radial direction from inside out can be stabilized, and they can have a diameter as small as tens of nanometers. Their multilayered configuration gives us an opportunity to design the Dzyaloshinskii–Moriya interaction (DMI) and perpendicular magnetic anisotropy, the two crucial parameters for skyrmion stabilization that also determine skyrmion size.⁵ As magnetic solitons, skyrmions have a rich spectrum of dynamics, from gyrotropic motion to spin wave (SW) type oscillations confined to the skyrmion domain wall.^6–10 This offers us a possibility for exploring propagating SW–skyrmion interactions.^11–13 

These two elementary excitations in magnetic systems can interact as classical particles. In particular, skew and rainbow scattering as well as side jumps were theoretically predicted for SWs passing through a skyrmion in thin ferromagnetic films.^14,15 The first type was associated with skyrmion topology, i.e., the emergent magnetic field generated by the Berry curvature of the skyrmion. The rainbow type of scattering that resulted in multiple peaks in the scattering spectra had a similar source, as it originated in the singularity of the Aharonov–Bohm flux density in the skyrmion center.¹⁴ The side jump was correlated with the total magnetization of the skyrmion.¹⁶ A back action is also possible in the scattering process, in which SWs can transfer momentum to the skyrmion and generate its motion.¹⁴ With the high amplitude of the driving SWs scattered across the skyrmion, nonlinear three-magnon scattering may become dominant, resulting in the magnonic frequency comb generation.¹³ Here, the interaction with the skyrmion breathing mode was responsible for the observed effects.

SW refraction control by a periodic skyrmion chain in a thin ferromagnetic film was numerically demonstrated in Ref. 17. Note that the interaction of SWs with a periodic lattice of skyrmions in the waveguide or nanodot chain geometry may result in the formation of a magnonic band structure, where the group velocity of the SW bands can have positive and negative sign, and they can be designed by the skyrmion–skyrmion separation.^18–20 In the waveguide geometry, the propagating SWs can induce the skyrmion motion as well.¹² However, such a possibility was indicated only for SWs at very high frequencies that are above 50 GHz, so the results are difficult to verify experimentally.

Despite a number of theoretical investigations of the SW–skyrmion interaction, experimental studies are not available. The difficulty in realization of such studies lies in the stabilization and measurements of the skyrmion itself. We propose to use a hybrid system that is useful for studying SW–skyrmion coupling and scattering processes, which can also provide a possibility for experimental realization. The system consists of an in-plane magnetized soft ferromagnetic conduit suitable for SW propagation and a nanodot with a Néel-type skyrmion placed just above this multimode waveguide. Due to the development of a code for the excitation of propagating SWs with a precisely defined wave number and frequency, we performed a detailed analysis of the scattering of various SW modes on the hybrid structure formed by the skyrmion and the imprint created upon the waveguide. We show that the scattering process strongly depends on the SW frequency and width-quantization number of SW. We correlate the scattering effectiveness with the SW-spectra of the skyrmion and its imprint. We believe that the proposed hybrid structure can be beneficial for both spintronics and magnonics applications.

We consider a 9.23 µm long and 2 nm thick waveguide made of Permalloy (Py) of width w = 768 nm and magnetized along its axis. Centrally, 2 nm above the waveguide, a 2-nm-thick circular Co nanodot of diameter 600 nm is placed, which hosts a Néel-type skyrmion; see Fig. 1. In the waveguide, we assume a saturation magnetization M_s = 800 kA/m, an exchange stiffness A_ex = 13 pJ/m, and a uniaxial in-plane anisotropy constant K_IP = 8.042 J/m³ with the easy-axis along the x axis (see the coordinate system in Fig. 1). In the case of the nanodot, we use M_s = 956 kA/m, A_ex = 10 pJ/m, the perpendicular magnetic anisotropy constant K_PMA = 717 kJ/m³ with the easy-axis along the z-axis, and Dzyaloshinskii–Moriya exchange constant D_int = 0.001 J/m². This set of parameters corresponds to the ultrathin layer of Pt/Co/Ir,^21–23 which presents favorable conditions for skyrmion stabilization. In the whole system, we assume a very low value of damping, α = 0.0001, and zero external magnetic field.

We use micromagnetic simulations (MuMax3²⁴) to study SW propagation through the system. In the simulations, we discretize the system by 3072 × 256 × 3 cells with a grid cell size of 3 × 3 × 3 nm³. To avoid SW reflections from the waveguide ends, we employ a 1.2 µm wide absorbing layer at the sides of the waveguide [see Fig. 1(a)] where damping increases quadratically up to 0.15. Each simulation starts with the determination of an equilibrium magnetic configuration; next, we study different aspects of magnetization dynamics.

The excitation of selected width-quantized SW modes in the backward-volume geometry was recently studied.²⁵ Here, to investigate the impact of nanodots on the propagation of various quantized SW modes in a waveguide, we use a self-developed code for the generation of a microwave magnetic field that allows unidirectional excitation²⁶ of the selected SW modes with well-defined wavenumbers k_x,n. To excite the nth mode at the frequency f_n, we select from the dispersion relation the corresponding value of the wavenumber k_x,n [see the exemplary points marked in Fig. 2(a)]. Then, we use the microwave field with the spatial and temporal distribution mimicking the profile of the excited SWs with a proper wavelength, frequency, and quantization (see animations in the supplementary material), i.e.,

where μ₀h₀ = 0.01 mT is a microwave field amplitude and x₀ is the center of the excitation area. $Dt=0.5+arctan(ft−P/2)×10/P/π$ with P = 4 introduces a gradual increase of the microwave field amplitude saturating after ca. P periods (D(t = P/f) = 0.94). $Gξ=exp−ξ2/0.4$ is the Gauss function, with L = 4π/k_x,n and E_n(y) mimicking the SW profile along the y axis,

where w_eff,n is the effective waveguide’s width for the nth mode. For simplicity, in all simulations we use w_eff,n = w. We assume an excitation region of two wavelengths wide along the x axis; therefore, if multiple bands are very close to each other, the efficiency of selective excitation may be reduced. According to Ref. 27, due to dipolar interactions and a small stripe width to thickness aspect ratio, the SWs are pinned at the lateral edges of the waveguide; but for the purpose of simplifying the analysis and owing to larger waveguide width than dot diameter, its contribution to the scattering process will not be discussed in our paper.

The equilibrium magnetization configuration is shown in Figs. 1(b)–1(d). Although the waveguide is magnetized along the x axis, just below the nanodot hosting Néel-type skyrmion, the homogeneous magnetic configuration is disturbed, i.e., we observe skyrmion’s imprint upon the waveguide. Moreover, due to the static dipolar interaction between the waveguide and the nanodot, the skyrmion loses its circular symmetry and takes on the shape of an egg.²³ 

The dispersion relation of the isolate Py stripe (see details in the supplementary material), shown in Fig. 2(a), reveals a series of bands that represent the propagation of successive modes quantized by the waveguide width and indexed by mode number n. The band of the fundamental mode (n = 1) starts at $≈4.25$ GHz, indicating the frequency range available for propagating waves. The two-mode part of the spectra (n = 2) starts at 4.75 GHz, and with increasing frequency, additional new channels for scattering of SWs are found to open. There are also two low-frequency peaks in the spectra, at 0.6 GHz is a gyroscopic mode of the skyrmion and at 3.75 GHz is a resonance of the imprint. Both have frequencies below the ferromagnetic resonance (FMR) frequency of the waveguide; thus, they do not contribute to the linear scattering process of the propagating SWs considered in this paper.

In the hybrid system (Fig. 1), in addition to the SW bands [Fig. 2(a)], there are resonances related to excitation confined to the skyrmion and the imprint, marked in Fig. 2(b) with the red and black lines, respectively (supplementary material). On comparison of Figs. 2(a) and 2(b), we found that for frequencies larger than the bottom of the waveguide’s dispersion, the amplitudes in the spectrum do not descend to zero. Resonance amplitudes are higher in the imprint at lower frequencies, while they are more pronounced in the nanodot at higher frequencies. This opens the possibility for dynamic interactions of the propagating SWs with the skyrmion and the imprint.

To study the scattering of SWs on the skyrmion–imprint hybrid structure, we perform steady-state simulations (see the supplementary material), where we utilize the SW source described by Eq. (1). Due to filtering of the excitation frequency, we observe only the linear scattering process. Typical steady-state SW amplitude distributions are shown in Fig. 3. In each subplot (a)–(f), a single mode of a well-defined quantization across the waveguide width is excited by the antenna placed on the left part of the structure. We show in separate figures the amplitude distributions for waves propagating to the left (k_x < 0, reflected waves) and to the right (k_x > 0, incident and transmitted waves), i.e., the top and bottom panels of each subplot, respectively.

Figure 3(a) illustrates the impact of the nanodot on the propagation of the n = 1 mode at f = 4.5 GHz with k_x = 11.4 rad/μm. The amplitude of the reflected SWs is greater than that of the transmitted SWs. The wavelength is similar to the size of the skyrmion, whereas at higher frequencies, the wavelength is smaller. In Fig. 3(b), the steady state for the n = 3 mode at f = 6.1 GHz with k_x = 23.0 rad/μm is shown. The transmitted (k_x > 0) and reflected (k_x < 0) SWs have a comparable amplitude. However, a small modulation of the reflected and transmitted waves is noticeable, which indicates scattering of the incident mode to other modes, and disturbance of the symmetry in the reflection can be observed.

A comparison of the exemplary SW amplitude distributions for modes with odd and even numbers n at f = 6.8 GHz is shown in Figs. 3(c) and 3(d), which illustrate the results for n = 4, k_x = 25.6 rad/μm and n = 5, k_x = 18.0 rad/μm, respectively. For the incident mode n = 4, we observe that the transmitted and reflected SWs retain the mode character. For n = 5 at the same frequency, due to interaction with the skyrmion and its imprint, the spatial distributions of the transmitted and reflected waves are complex and represent the superposition of different modes. At this frequency, the transmitted SWs’ amplitude exceeds the reflected SWs’ amplitude.

Figure 3(e) presents results obtained for f = 7.9 GHz and n = 7 for k_x = 16.0 rad/μm. Here, both reflected and transmitted waves form complex interference patterns that cannot be associated with a simple superposition of two modes. Interestingly, similar results for complex transmission were presented in Ref. 28 for Damon–Eshbach modes, where only even or only odd modes interfere with each other. In Fig. 3(f), the profile of the n = 2 mode at frequency f = 9.6 GHz with k_x = 51.6 rad/μm is presented. Again, the transmitted and reflected waves are represented by complex interference patterns. Finally, at higher frequencies, the low-n modes are not perfectly excited [Fig. 3(f)]. This is a result of small differences in wavenumbers between successive bands (see, Fig. 2). This limitation can be resolved by using a wider SW excitation region. However, for the sake of simplicity, we used the same SW source definition in all the simulations.

The results presented above show different scattering processes that occur at different frequencies. To obtain deeper insight into the modes participating in the scattering process, we propose the analysis in the wavenumber space. Therefore, the two-dimensional Fast Fourier Transform (FFT) over the x- and y-coordinates of the complex amplitude of m_y(x, y), calculated separately for the incident (between the antenna and the nanodot, with k_x > 0), transmitted (to the right of nanodot, k_x > 0), and reflected (between the antenna and the nanodot, k_x < 0) waves, is averaged over the k_x-coordinate. This operation can be represented as $myp(ky)=⟨FFTx,ymy±(x,y)⟩kx$⁠, where the complex SW amplitude distributions with k_x > 0 are represented by $my+$ and those with k < 0 by $my−$⁠. The superscripts p ∈ {i, t, r} represent incident, transmitted, and reflected waves, respectively. To obtain $myi$ and $myr$⁠, we calculate the two-dimensional FFT for the region −1.95 µm $<x<−0.45$ μm, whereas to calculate $myt$⁠, we use 1.2 µm $<x<2.7$ μm. Finally, for the selected frequencies and all the available modes, we present $myp(ky)$ in the form of bar plots, see Fig. 4. Due to the finite width of the waveguide (w) and the properties of FFT, the spectral resolution is equal 2π/w. Therefore, it is more convenient to represent k_y as a dimensionless wavenumber κ = k_yw/(2π). In these units, the resolution in the inverse space is equal to 1. Due to the properties of FFT, only for the antisymmetric modes (n = 2, 4, 6, …), a discrete (for full pinning) or almost discrete peaks (for partial pinning) will be obtained. For symmetric modes (n = 1, 3, 5, …), the situation is more complicated. First, the spectral resolution misrepresents k_y = nπ/w for odd n (the spectral resolution is 2π/w). Furthermore, when calculating FFT, we assume that the signal m_y(y) is repeated along the y axis, but this periodicity does not coincide with integer multiples of the SW wavelength along the y axis, λ_y = 2π/k_y = (2/n)w, for the symmetric modes. Thus, for these modes, the profiles $myp(ky)$ are blurred in the inverse space, especially for the modes with low n values. This effect is clearly visible in Fig. 4, where for incident antisymmetric modes, we see almost discrete peaks (single bars for positive and negative κ), whereas for the symmetric modes, the SW amplitude is blurred among more bars. Therefore, we assume the criterion of noticeable scattering when the pattern of wide red or green bars corresponding to different κ changes significantly with respect to the incident waves, represented by narrow black bars.

In Figs. 4(e) and 4(f) are shown results for f = 5.1 GHz and f = 4.5 GHz, respectively. For f = 4.5 GHz, we see that the transmitted and reflected modes preserve the character of the incident mode since the proportions between the heights of the individual bars are maintained. Moreover, the bar amplitudes for the transmitted waves are lower than those for the reflected waves. The same situation occurs for f = 5.1 GHz and n = 1. However, for f = 5.1 GHz and n = 2, the bars corresponding to an increase in the transmitted waves. Moreover, here we have almost discrete peaks for the incident, transmitted, and reflected waves. In this frequency range, we do not observe scattering to other modes.

Figures 4(c) and 4(d) illustrate the results for f = 6.8 GHz and f = 6.1 GHz, respectively. The reflected wave amplitude is still significant. However, as the frequency and mode number increase, the amplitude of the transmitted waves also grows. We can see that selected modes are scattered. In transmission and reflection, it is visible for n = 3 at 6.1 GHz and n = 1, 3, 5 at 6.8 GHz.

For higher frequencies, illustrated in Figs. 4(a) and 4(b), f = 9.6 GHz and f = 7.9 GHz, respectively, all incident modes after passing under the skyrmion as well as the reflected modes are scattered (the amplitudes are distributed over many κ values). For all the cases, scattered SWs are usually visible as a superposition of modes with the same mode number parity as the parity of incident mode. However, there are some visible exceptions, cf. Fig. 4(a) n = 3, 6, 7, (b) n = 4, and (d) n = 1.

A summary of the different scattering efficiencies of the SW modes on the hybrid skyrmion–imprint structure at different frequencies is shown in Fig. 5. To obtain these graphs, we use normalized to one dependencies $m̃yi(κ)$⁠, $m̃yt(κ)$⁠, and $m̃yr(κ)$⁠, representing the functions of the SW amplitude of the incident, transmitted, and reflected waves, respectively, see examples in Fig. 4. Note that here we normalize these dependencies to one independently for the incident, transmitted, and reflected waves. The values in the first matrix [cf. Fig. 5(a)] represent the averaged absolute value of the difference between $m̃yt$ and $m̃yi$⁠, i.e., $et=12N∫−NN|m̃yt(κ)−m̃yi(κ)|dκ$⁠, where N = 15 (the value N is chosen arbitrarily and must be greater than the mode number used in simulations, here, we neglected intensities for |κ| > 15). Similarly, the values in the second matrix e_r [Fig. 5(b)] depict the differences between $m̃yr$ and $m̃yi$⁠. Therefore, for e_t = 0 (e_r = 0), we have the same profile of the incident and the transmitted (reflected) SW. The greater the value, the more pronounced the scattering to other modes is.

The analysis of Fig. 5 allows us to distinguish three frequency regimes: (i) up to 6 GHz—the nature of the transmitted and reflected modes is the same as that of the incident, the conversion of modes does not appear; (ii) from 6 to 8 GHz—some transmitted (reflected) modes are scattered, the others retain the nature of the incident wave; (iii) 8 GHz and higher frequencies—the incident wave is scattered and usually preserves its symmetry (see Fig. 4). These frequency regimes coincide well with the resonance spectrum of the hybrid structure presented in Fig. 2. The efficiency for scattering to other modes increases with the frequency for both the reflected and transmitted waves. Interestingly, even for higher frequencies, we can observe some scenarios where the values of e_t are relatively small; cf. n = 1 and n = 6 for f = 9.8 GHz. This means that although the values of e_t and e_r generally increase with increasing frequency, this increase is rather irregular in nature and exceptions can be observed. Moreover, the efficiency of this scattering process depends not only on the SW frequency but also on the mode number. Interestingly, regimes (i) and (ii) fall into the regimes where the excitations of the imprint dominates in the spectrum, while in regime (iii), the dynamics of the skyrmion prevails. This clearly indicates the effect of the dynamic magnetostatic coupling of propagating SWs with the dynamic excitations of the skyrmion.

We have shown that the placement of nanodot with Néel-type skyrmion above the waveguide affects SW propagation. To demonstrate that, we have developed a method suitable for the unidirectional excitation of selected SW modes and tools to process the simulation results. Our results show the influence of the skyrmion–imprint hybrid on the transmission, reflection, and scattering of different quantized SW modes. We distinguished three frequency regimes for the behavior of reflected and transmitted modes. In the first, each of the modes is transmitted and reflected with the preserved symmetry of the incident SW. In the second regime, selected modes are scattered into other modes; sometimes, scattering occurs for reflected or transmitted SWs. In the third regime, almost all modes (with only a few exceptions) are scattered; however, the resulting interference pattern is usually constructed of modes with the same parity of mode numbers as the incident SW. Furthermore, reflection dominates at low frequencies; however, as the frequency increases, the transmission mode starts to dominate. The results indicate that wave propagation is affected by the presence of either the skyrmion or its imprint upon the waveguide.

Overall, we have demonstrated that the presence of an additional element with magnetic texture significantly affects the propagation of SWs in the waveguide. This influence strongly depends on both the frequency of SWs and their mode number, and it is dependent on the resonance spectra of the skyrmion–imprint hybrid structure. Therefore, utilizing nanoelements hosting different magnetization textures placed above the conduit is a promising way to control SW flow in multimodal waveguides. These results open a new path in the development of nonplanar magnonic circuits for signal processing, with the control of propagating SWs by exploring the third dimension.

Seethe supplementary material for the details of micromagnetic simulations used to perform the results presented.

The author would like to thank Wojciech Śmigaj for stimulating discussions. The research leading to these results was funded by the National Science Centre of Poland, Project Nos. 2019/35/D/ST3/03729 and 2018/30/Q/ST3/00416. Simulations were partially performed at the Poznan Supercomputing and Networking Center (Grant No. 398).

The authors have no conflicts to disclose.

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