Studies of the coupling of elastic waves with other wave excitations, including spin waves, in nanostructures are experiencing a revival, which is mostly due to the progress in experimental techniques.¹⁻¹² Magnetostrictive systems are promising for switching the magnetic configuration and controlling spin-wave propagation by nonmagnetic means.^5,13−15 The strategy to use strain (elastic waves) or an electric field (electromagnetic waves) is technically less problematic than controlling the magnetic system by magnetic field for the realization of logic gates,⁵ magnetic recording,¹³ or signal processing.^16,17 The combined impact of elastic deformation and electric field on magnetization dynamics can be used in cavity magnonics¹⁸ to realize quantum information processing.

The mechanism of magnetoelastic interaction is complex and difficult to understand. An efficient way to get an insight into it is to first consider the simplest structures, which constitute a platform for more sophisticated magnetoelastic nanodevices. Here, we consider the generic structure, a magnetostrictive layer deposited on a nonmagnetic substrate, with the magnetization dynamics in the layer coupled to the elastic excitations concentrated on the surface of the structure.

For propagating waves the coupling is known to require the matching of both the frequencies and the wavelengths.¹⁹ This condition can be fulfilled for spin waves (SWs)^20,21 and surface acoustic waves (SAWs)^22,23 existing in the same range of frequencies and wave vectors. However, the strength of the magnetoelastic interaction depends on the orientation of the wave vector with respect to the direction of the static magnetic field. Moreover, this interaction is different for different types of SAWs, specifically, Rayleigh-SAWs (R-SAWs) and Love-SAWs (L-SAWs), which differ in the orientation of dynamic component displacement. Thus, the coupling is strongly anisotropic^4,24 and cannot be observed for arbitrary SAWs and SWs, even if their frequencies and wave vectors match.

Another fundamental limiting factor is the cross-section of the SAWs and SWs. Although the profiles of SAWs and SWs can relatively easily agree in the in-plane direction, which is ensured by the wave-vector matching, the net interaction can be minimized or even suppressed for some particular profiles in the out-of-plane direction. The role of the latter factor is not fully recognized in the scientific community.

Using finite element calculations based on a theoretical model (see Supporting Information for details), we interpret our Brillouin light scattering (BLS) experimental data concerning the magnetoelastic dispersion relation of thermally excited SAWs and SWs.^25,26 To explain the existence (or absence) of magnetoelastic interaction, manifested as anticrossings of magnonic and phononic dispersion branches, we need to take into account not only the anisotropy of the interaction but also the profiles of the SAWs and SWs in the direction normal to the surface. We report the observation of magnetoelastic interaction between L-SAWs and SWs for the backward volume geometry (wave vector parallel to the static magnetic field), in which the coupling between R-SAWs and SWs is weak because of the anisotropy of this interaction. Finally, we show that the lack of expected interaction between R-SAWs and SWs for the oblique geometry (field at 45° with respect to the direction of the wave vector) can only be explained based on the spatial distribution of the displacement (and the related strain) within the magnetic layer. Our results demonstrate an interplay between the anisotropy of interaction and the spatial cross-section of excitations of both types.

The considered system, presented in Figure 1, consists of a magnetostrictive CoFeB(2.1 nm)/Au(0.9 nm)x20 multilayer deposited on a thick nonmagnetic Si substrate with an Au(15 nm)/Ti(4 nm) buffer bilayer (see Supporting Information for details). The multilayer has an in-plane-oriented saturation magnetization ensured by the sufficient thickness of the CoFeB layers.²⁷ For the fundamental spin-wave (F-SW) mode and the lowest perpendicularly standing spin waves (PS-SWs),^20,28 the multilayer can be regarded as an effective medium of reduced magnetization (due to the surface anisotropy at the Au/CoFeB interface) gained at the expense of a slight increase in the spin-wave damping.²⁷ The SWs are confined to the magnetic part of the multilayer and precess with in-plane and out-of-plane components of magnetization, m_∥ and m_⊥, respectively. The orientation of m_∥ can be changed with the direction ϕ of the external field H₀. The orientation of the plane of incidence and the angle of incidence θ in Brillouin light scattering (BLS) measurements determine the direction and magnitude of the wave vector k for the SAWs and SWs observed in the experiment (in our study the wave-vector direction is fixed, k = kx̂).

Potentially, three types of SAWs can be observed, with a different polarization of the displacement, i.e., R-SAWs and S-SAWs (with elliptical polarization of the displacement u_x/|u_x| = ±iu_z/|u_z|) and L-SAWs (transfers with linearly polarized component u_y). The strength of the magnetoelastic interaction depends on the direction of the external magnetic field and is different for different types (polarization) of SAWs. Shown in Figure 1, the angular plots of the magnetoelastic field magnitude have lobes at ϕ = 0° or ϕ = 45° for L-SAWs and R-/S-SAWs, respectively; this illustrates the differences in anisotropy of the SAW/SW interaction.

A magnetic field of appropriate magnitude is necessary for interaction between SWs and SAWs to occur in the wave-vector range accessible for BLS. An increase in the magnetic field pushes up the SW frequencies, while the SAW frequencies remain unaffected. As a result, the (anti)crossings between magnonic and phononic dispersion branches f(k) are shifted toward larger values of wave vector k.

Figure 2 shows the experimental and theoretical magnetoelastic dispersion relations in the system presented in Figure 1 for two different values of the magnetic field applied along the direction of the wave vector k, i.e., for backward volume geometry. As a result of increasing the field from 30 mT (Figure 2a) to 50 mT (Figure 2b), the crossing of the fundamental spin-wave (F-SW) mode shifts up; see also the plot in Figure 3. Figure 2 presents BLS spectrum sweeps for successive wave-vector values in the form of maps where the color intensity represents the BLS intensity. The blue-red lines in the frequency and wave-vector domain are the experimental magnetoelastic dispersion branches. The spectra of phonons (blue lines) and magnons (red lines) are measured separately (see Supporting Information) and then superimposed in one plot. The experimental data are supplemented by the results of finite element method calculations (see Supporting Information). The calculated dispersion relations are plotted as dotted lines. The dispersion branches are identified as those of SAWs or SWs of specific types by the inspection of numerically calculated profiles of the dynamic displacement and magnetization. A striking finding is the lack of an L-SAW branch in the experimental data, except for a interaction region (anticrossing) with the F-SW branch, observed just below the R-SAW branch. This anticrossing cannot be identified as R-SAW/F-SW interaction, as it is observed clearly below the high-intensity R-SAW dispersion branch. Moreover, the theory does not predict R-SAW/F-SW interaction for the magnetic field applied along the direction of the wave vector (for ϕ = 0, i.e., for the backward volume geometry for SWs).

The details and mechanism of this interaction are presented in Figure 3. The main plots show the anticrossing of the F-SW mode with the R-SAW and L-SAW for two external field values, 30 mT and 50 mT, in the backward volume geometry. Indicating the frequencies that correspond to the maximums of the BLS peaks for successive wave-vector values, the experimental results (points) are distributed quite precisely along the numerical dispersion (dashed lines). By comparing the experimental and numerical data we can attribute with certainty the approximately observed anticrossing to the interaction between the L-SAW and the F-SW. Note that the L-SAW is barely visible in the BLS measurements beyond the anticrossing region (this type of SAW must be measured for cross-polarized incident and refracted beams, which gives a much weaker BLS signal).

The full width at half width of magnonic BLS peaks (fwhm ≈ 0.4 GHz) is slightly larger than the frequency splitting at the F-SW/L-SAW anticrossing (Δf ≈ 0.25 GHz), which still allows for resolving them in the spectrum (see Supporting Information). The measured phonons are damped with a similar rate (fwhm ≈ 0.3 GHz for R-SAW, close to F-SW/L-SAW anticrossing). Taking this value as an upper bound of FWHW for L-SAW peak (invisible in our spectra), we can estimate the cooperativity of F-SW/L-SAW interaction as C = Δf²/FWHW² ≈ 0.4 < 1, which demonstrates that the considered magnon–phonon coupling is relatively weak.

The interaction between SAWs and SWs can be analyzed by considering the magnetoelastic components of the effective field h_me and the stress tensor σ_me in the Landau–Lifshitz equation coupled to the elastodynamic equation^20,29,30

where m(r, t) = m_⊥ + m_∥ and M₀(r) are the dynamic and static components of the magnetization vector M = M₀ + m (note that in our case M₀ ∥ H₀ and |M₀| ≈ M_S, where M_S is the saturation magnetization and m_∥ = |m_∥|(−x̂ cos ϕ + ŷ sin ϕ)); ρ(r) is the mass density, h(r, t) the dynamic effective magnetic field, and $$ are the elements of the stress tensor in the absence of magnetoelastic coupling. The symbol H₀ represents the external magnetic field (aligned in-plane). The parameters γ and μ₀ are the gyromagnetic ratio and the vacuum permeability, respectively, and x_i, x_k are the coordinates x, y, z for i, k = 1, 2, 3. For the geometry considered in Figure 3 (ϕ = 0) the magnetoelastic field and the derivatives $$ of the magnetoelastic stress are:^20,30where b₂ is one of the magnetoelastic coupling constants and $$ are the elements of the strain tensor. From eqs 2 it follows that the R-SAW (u_x ≠ 0, u_y = 0, u_z ≠ 0) affects only the SW via h_z (which depends solely on ε_x,z), whereas the impact of the SW on the R-SAW is ensured only by the derivatives ∂_x and ∂_z of m_z. On the other hand, the L-SAW (u_x = 0, u_y ≠ 0, u_z = 0) is coupled to the SW exclusively by h_y (which only depends on ε_x,y). Reverse coupling (i.e., SW to L-SAW) is guaranteed by ∂_xm_y.

Figure 3 presents color maps of the discussed components of the stress tensor (left columns) for the R-SAW and L-SAW and the derivatives of the dynamic components of the magnetization (right column) for the F-SW. The L-SAW/F-SW interaction results from the large ε_x,y value and the significant values of ∂_xm_y in the magnetic multilayer (0 < z < 60 nm). For R-SAW and F-SW the parameters ε_x,z, ∂_xm_z, and ∂_zm_z in the magnetic multilayer are substantially smaller. The profiles of the L-SAW and F-SW do not change significantly due to the interaction, except for the phase flip between higher and lower breach at anticrossing.

Let us discuss how the L-SAW/F-SW and R-SAW/F-SW interactions vary with changing direction of the magnetic field with respect to the wave vector (oriented along the x-direction). Figure 4(b) shows the magnetoelastic dispersion relation for the oblique orientation of the magnetic field (ϕ = 45°). The dispersion is plotted in the same frequency range as in Figure 3, but the wave-vector range had to be extended to find the crossing of F-SW with L-SAW and R-SAW because of a significantly changed slope of the F-SW dispersion branch.²⁸

Both the L-SAW/F-SW and R-SAW/F-SW couplings are practically negligible. This is not surprising as far as the L-SAW/F-SW coupling is concerned, since in this case the in-plane component of the magnetoelastic field h_me,∥ ∝ ε_xy  cos 2ϕ = 0 (for ϕ = 45°) and the out-of-plane component h_me,⊥ ∝ ε_yz  sin ϕ are very small (ε_yz ≪ε_xy ∝ k, for 1/k much smaller than the thickness of the magnetic multilayer); (see the Supporting Information for details). This effect of angular dependence of the L-SAW/F-SW coupling is illustrated in Figure 4(a) by a polar plot of the frequency splitting Δf(ϕ) at the anticrossing of the L-SAW and F-SW dispersion branches superimposed on the h_me,∥(ϕ) dependence. The good match between these two polar plots (Δf(ϕ) ∝ h_me,∥(ϕ)) confirms that the strong reduction of the frequency splitting at ϕ = 45° is solely related to the anisotropy of magnetoelastic interaction (i.e., to the relative orientation of the wave vector and the external magnetic field).

A similar reasoning based on the anisotropy of magnetoelastic coupling would lead to the prediction of a R-SAW/F-SW anticrossing, since h_me,∥ ∝ ε_xx  sin 2ϕ ∝ k sin 2ϕ is maximal for ϕ = 45° and has significant values for larger k. However, this expectation is not confirmed in our study; the R-SAW/F-SW anticrossing is hardly observed (Δf ≈ 0). To explain this result we must analyze the SAW profile in the direction normal to the surface. For R-SAW the tangential component of the displacement u_x flips phase in the normal direction (z-direction), see Figure 1. For the considered structure we observe a nodal line for u_x and the related component of the strain tensor ε_xx within the magnetic multilayer. As a result the tangential component of the magnetoelastic field h_me,∥ ∝ ε_xx = iku_x is close to zero.

We have shown that not only the anisotropy of magnetoelastic interaction (related to the orientation of the magnetic field) but also the SAW profile within the magnetostrictive layer is of importance for the observation of coupling between SWs and SAWs. For R-SAWs the tangential component of displacement u_x can have nodes within the magnetic layer, resulting in a reduction of the net strength of magnetoelastic interaction even if the strain ε_xx is locally significant. A similar effect is expected for S-SAWs, where u_z flips phase in the normal direction. In an L-SAW the displacement u_y does not have any nodes (u_y changes monotonously in the normal direction). It is worthy of notice that the spatial changes in the SAW profile (e.g., the distance of the node from the surface) are small compared to the wavelength λ = 2π/k. The location of R/S-SAW nodes depends on elastic properties of the system, specifically, the elastic material parameters of the multilayer and the substrate, and the thickness of the multilayer.

Our study reveals an additional factor limiting the interaction between surface acoustic waves and spin waves. The SAW/SW coupling proves to require an appropriate profile of the elastic wave near the surface of the magnetostrictve structure, at distances much smaller than the wavelength. We have shown that this additional factor plays a role for some types of surface acoustic waves (Rayleigh and Sezawa waves), while other types (Love waves) are insensitive to it. We believe that the studies on magnon–phonon interaction in confined geometries (surfaces, cavities) are very promising and can reveal unusual interaction mechanisms similar to those found between cavity photons and magnons.³¹