# Experimental Observation of Vortex Gyration Excited by Surface Acoustic Waves

R. Lopes Seeger, F. Millo, G. Soares, J.-V. Kim, A. Solignac, G. de Loubens, T. Devolder

► To cite this version:

R. Lopes Seeger, F. Millo, G. Soares, J.-V. Kim, A. Solignac, et al., Experimental Observation of Vortex Gyration Excited by Surface Acoustic Waves. Physical Review Letters, 2025, 134 (17), pp.176704. 10.1103/PhysRevLett.134.176704. hal-05103076

HAL Id: hal-05103076

https://hal.science/hal-05103076v1

Submitted on 8 Jun 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Experimental observation of vortex gyration excited by surface acoustic waves

R. Lopes Seeger,¹,²,* F. Millo,¹ G. Soares,² J.-V. Kim,¹ A. Solignac,² G. de Loubens,² and T. Devolder¹

¹Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, 91120 Palaiseau, France

²SPEC, CEA, CNRS, Université Paris-Saclay, 91191 Gif-sur-Yvette, France

(Dated: March 24, 2025)

We report experiments using magnetic resonance force microscopy to investigate the dynamics of magnetic vortices in sub-micrometer CoFeB disks grown on a piezoelectric substrate. We compare the vortex gyration excited inductively by microwave magnetic fields and through magnetoelastic coupling via surface acoustic waves (SAWs). Based on the device geometry and modeling performed, we find that the dominant mechanism for SAW-driven excitation is through lattice rotations, which generate an effective field localized at the vortex core. This contrasts with the magnetoelastic strains typically assumed in similar experiments. Moreover, we demonstrate that this magnetorotation torque can be tuned by applying a perpendicular magnetic field. Unlike spin wave excitations, nontrivial spin textures, such as vortices, effectively couple to magnetorotation torques, making them valuable probes for studying such phenomena.

Driving magnetization dynamics with surface acous-
tic waves (SAWs) has garnered recent attention due
to promising applications in magnonics and microwave
acoustics [1–4]. Particular focus has been directed toward
coupling spin waves with SAWs, since the nonlinear and
nonreciprocal properties of the former can potentially be
conferred to the latter [5–9], opening up new device pos-
sibilities such as acoustic isolators. Besides ferromagnets,
recent advancements have extended SAW-driven magne-
tization dynamics to multiferroics [10], synthetic antifer-
romagnets [11], and layered antiferromagnets [12].

Most studies to date have only focused on the role
of the magnetoelastic strain induced by SAWs, ϵ<sub>ij</sub> =
(∂u<sub>i</sub>/∂x<sub>j</sub> + ∂u<sub>j</sub>/∂x<sub>i</sub>)/2, where u denotes the lattice dis-
placement and i, j the different Cartesian coordinates.
For example, longitudinal strain of the form ϵ<sub>xx</sub> produces
a magnetoelastic “tickle” field that can induce the motion
of magnetic vortex cores in thin-film disks, but only with
a static magnetic field applied in the film plane which
displaces the equilibrium position of the core [13]. This
approach was recently implemented using a time-varying-
strain in the MHz range to drive the vortex gyrotropic
mode [14]. Alternatively, additional nonzero lattice de-
formations can also appear in real piezoelectric materials
with different symmetries to longitudinal strain [15, 16].
In particular, the rotational motion of the lattice within
the magnetic material causes reorientation of the surface
normal direction, which couples to the magnetization via
magnetic anisotropy fields [17]. This so-called magne-
torotation contribution (resulting in a “rolling” field),
ω<sub>ij</sub> = (∂u<sub>i</sub>/∂x<sub>j</sub> − ∂u<sub>j</sub>/∂x<sub>i</sub>)/2, is active in materials with
vanishing magnetostriction. Here, we demonstrate ex-
perimentally that magnetic vortices are sensitive probes
for magnetorotation torques when the vortex radial sym-
metry is preserved. In this regime, steady-state gyration
can be induced by SAWs without an in-plane applied
fields to displace the vortex core from the disk center.

Figure 1(a) presents a schematic illustration of the ex-
perimental setup. We study the influence of SAW prop-

<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) Sketch of the MRFM experiment to detect the vortex gyration induced by SAWs. An IDT excites SAWs on a piezoelectric substrate towards a disk in the vortex state (the colors encode the in-plane magnetization component). The disk is placed a = 104 µm away from the IDT. (b) The setup includes a top antenna for inductive excitation, shown in an optical image. The inset is a zoomed view of the antenna’s constriction with several CoFeB microdisks.</figcaption></figure>

agation through a CoFeB disk positioned on a Z-cut
LiNbO3 piezoelectric substrate. The disk has a thickness
of d<sub>FM</sub> = 34 nm and a diameter of 2L = 800 nm. Res-
onant SAWs with a frequency of 0.86 GHz and a wave-
length of 4.5 µm, corresponding to the second harmonic
order, are generated by applying an rf voltage on the
interdigital transducer (IDT) with a periodicity 9 µm.
They propagate on the surface of the piezoelectric sub-
strate [18] along the crystalline X-axis of LiNbO3, which
coincides with the x-axis shown in Fig. 1(a). When the
SAWs impinge on the CoFeB disk, they induce lattice vi-
brations in the ferromagnet which give rise to additional
contributions to the effective magnetic field through the
magnetoelastic and magnetorotation coupling. To pro-
vide a point of comparison, a microwave antenna [with a width of $w_{ant} = 4~\mu\text{m}$, Fig. 1(b)] is also placed on top of the disk to excite magnetization dynamics inductively with rf magnetic fields. For both excitation methods, detection of the magnetization dynamics is performed with a magnetic resonance force microscope (MRFM) [18–20], providing unequivocal proof of the magnetoacoustic origin of the detected excitation.

MRFM spectroscopy of the magnetic vortex dynamics induced by an in-plane microwave field is presented in Fig. 2(a). By sweeping the amplitude of the perpendicular magnetic field, $H_z$, at fixed microwave frequency, we observe an erratic variation of the gyration frequency of the vortex, ranging from 0.4 to 1.2 GHz, instead of the expected linear variation [20], from $\nu_G = 5/(9\pi)\gamma\mu_0 M_s (d_{FM}/L) \approx 0.72$ GHz at zero field to 1.44 GHz close to saturation [21]. Similar results were obtained on another CoFeB disk [18]. This behavior is attributed to the strong pinning of the vortex core by grain defects in the CoFeB disk, a phenomenon that has also been reported for other material systems [22, 23]. In Fig. 2(b), we show the MRFM spectrum of the SAW-driven vortex gyration. We observe similar features in gyrotropic response in the two cases, although there are small differences in the relative amplitudes of spectral features that might arise from the distinct physical mechanisms behind the effective fields generated.

We also conducted frequency scans at fixed perpendicular field with both the microwave antenna and the IDT [Figs. 2(c) and 2(d)]. While a broadband excitation is observed for field-driven dynamics with the microwave antenna, SAW-driven excitations are only effective at the resonance of the IDT with the SAWs, namely around $\nu \approx 0.86$ GHz. As such, a large response in the IDT-driven case is only observed when the gyration frequency is appropriately tuned by $H_z$, which for the system studied is around 820 mT. Note that the spectral response under IDT driving is narrower than the response under field excitation, i.e., 15.7 vs. 30.9 MHz, respectively. This is because the measured response under IDT-driving is primarily determined by the SAW resonance, which is narrower than the overall gyrotropic response of the vortex. This observation lends further support to magnetoacoustic torques being responsible for vortex excitation in the IDT-driven case.

To gain further insight into the magnetoacoustic signal, we performed experiments with higher frequency resolution around the SAW resonance, as shown in Fig. 3. For the antenna excitation, the response remains nearly constant over the frequency interval considered, while strong oscillations in the MRFM signal are observed for the IDT excitation. These oscillations arise from standing acoustic modes formed along the SAW propagation direction, which are caused by the reflection from the edge of the substrate [24]. Traveling SAWs along the $LiNbO_3$ surface can indeed propagate over millimeter distances [25].

<figure><img src="image_3.png" /><figcaption>FIG. 2. Comparison of vortex gyrotropic excitation using microwave inductive and magnetoacoustic fields. (a) Vortex gyration frequency versus perpendicular magnetic field using the broadband inductive antenna, with MRFM signal amplitude as contrast. (b) Gyrotropic spectroscopy as a function of perpendicular field with excitation from either the antenna or IDT at 0.86 GHz (2nd harmonics of SAW). (c) Frequency scans comparing inductive and acoustic vortex excitations. (d) Line cuts of the data in (c) under $\mu_0 H_z = 820$ mT.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 3. High-frequency resolution gyrotropic spectroscopy. Perpendicular magnetic field dependence for an excitation employing the antenna (a) or the IDT (b). In the latter case, oscillations with a period of $\approx 2.4$ MHz appear due to standing acoustic modes in the $LiNbO_3$ substrate.</figcaption></figure>

The sample length along the SAW propagation direction is approximately $\ell = 1.8$ mm, which for the typical SAW velocity in $LiNbO_3$ ($v = 3900$ m/s) results in a frequency splitting $v/\ell$ of around 2.2 MHz between consecutive standing modes, in good agreement with the oscillation period observed in Fig. 3(b) [18]. The consequence of standing acoustic modes is that the elastic power injected by the IDT is effectively spread out over the length $\ell$ of the substrate, in contrast to the more
<figure><img src="image_5.png" /><figcaption>FIG. 4. MRFM signal amplitude as a function of the excitation power. Logarithmic-scale dependence as measured with (a) inductive and (b) acoustic excitation as measured for different microwave frequencies. (Inset) Selected MRFM spectra. The experiments are performed in the same range of perpendicular field as in Fig. 3.</figcaption></figure>

localized power of the microwave field generated by the inductive antenna. The observation of such a fine structure in the SAW signal unambiguously points towards the acoustic origin of the vortex gyration induced by the IDT excitation.

Let us now turn to the magnitude of the effective excitation field generated by the magnetoacoustic coupling. Fig. 4 compares the dependence of the MRFM signal amplitude at the gyration frequency, for both inductive and acoustic excitations, on the input power, $P$. We observe a monotonic increase in the gyration signal as a function of $P$. At low excitation powers a symmetric resonance peak is observed, which is consistent with a regime of linear dynamics. At higher powers, we observe hallmarks of nonlinearity in the spectral response, such as a distortion in the peaks [inset in Fig. 4(b)], as reported in previous works [26, 27]. The similar power dependence between the two excitation methods used suggests that the propagating SAWs also exert torques that are capable of exciting large precession amplitudes, albeit requiring 24 dB more electrical power. However, this figure believes a higher efficiency of the magnetoacoustic coupling, since a more accurate comparison should account for the effective magnetic volumes driven by the two sources; we estimate this to be $20 \log_{10}(\ell/w_{\text{ant}}) \approx 53$ dB. The energy efficiency of magnetoacoustic excitation is, in fact, three orders of magnitude greater than that of inductive excitation.

To gain more insight into the role of the magnetoacoustic torques, we performed micromagnetics simulations of the vortex dynamics using the MuMax3 code [28]. We model the SAW impinging on the CoFeB disk as a plane wave with frequency $\nu$ and deformation amplitude $A_{\text{SAW}}$ for the strain components $\epsilon_{xx}$ and $\epsilon_{zz}$, as well as for the rotation component $\omega_{xz}$ [18]. Note that the shear strain $\epsilon_{xz}$ exactly vanishes at the surface between the piezoelectric substrate and the magnetic disk. These torques correspond to the non-zero lattice strain and rotation

components of a Rayleigh SAW traveling in an isotropic substrate mimicking the Z-cut of LiNbO$_3$ [16]. We first examine the effect of the longitudinal strain alone. As discussed in Refs. 29 and 13, $\epsilon_{xx}$ alone cannot displace the vortex core as its corresponding effective field is antisymmetric about the disk center [Fig. 5(a)], resulting in a zero net torque acting on the vortex core. This issue can be circumvented by applying an additional static in-plane magnetic field, which shifts the equilibrium position of the core away from the center. This field breaks the axial symmetry of the static vortex, resulting in a finite torque from the strain component that can subsequently drive the core dynamics [13, 14]. In experiments, if the applied field is not perfectly aligned in the out-of-plane direction, it will result in an in-plane field projection that may shift the equilibrium position away from the disk center. We next consider the magnetorotation term, $\omega_{xz}$, which results in the effective field profile shown in Fig. 5(b). In contrast to the tickle field, the effective rolling field associated with $\omega_{xz}$ is non-vanishing and highly localized to the core. This term therefore exerts a net torque on the core irrespective of its position in the disk, thereby providing a means of initiating gyration from equilibrium and at zero in-plane applied field.

Figure 5(c) shows the simulated core trajectories for selected values of $A_{\text{SAW}}$, with $\mu_0 H_z = 500$ mT. The trajectories are circular for moderate deformation amplitudes but become distorted and elliptical for large values of $A_{\text{SAW}}$. To highlight the role of the tickle and rolling fields, the dashed line in Fig. 5(c) shows the circular trajectory obtained with only the $\omega_{xz}$ component at $A_{\text{SAW}} = 50 \times 10^{-5}$ and $70 \times 10^{-5}$. Fig. 5(d) shows the variation of the average radius of gyration as a function of $A_{\text{SAW}}$ for three different values of $\mu_0 H_z$. The orbital radius increases linearly with $A_{\text{SAW}}$ up to a certain value. Within this range of magnetoacoustic torques, the gyration frequency remains largely independent of $A_{\text{SAW}}$, with nonlinear frequency shifts and noncircular orbits appearing once stronger torques are applied. For $\mu_0 H_z = 500$ mT, for example, the linear behavior is observed for $A_{\text{SAW}} < 25 \times 10^{-5}$. The closed symbols in Fig. 5(d) correspond to cases in which both the tickle and rolling fields are applied simultaneously, which show that the strain terms only provide a negligible correction in the orbital radius (and only beyond linear regime), meaning that the rotational contribution alone is sufficient to induce steady state gyration of the vortex core. For that reason, whether the SAWs couple to the gyrotropic mode is determined by the effective magnetoacoustic fields acting on the core at its equilibrium position. The rotation term acts to displace the vortex core from equilibrium at the disk center, but as it gyrates the couple of the strain terms averages to zero.

Figure 5(e) presents the dependence of the orbital radius on the applied perpendicular field with $A_{\text{SAW}} = 10 \times 10^{-5}$. With this SAW amplitude, the core trajec-
<figure><img src="image_6.png" /><figcaption>FIG. 5. Normalized magnetoacoustic field ($b_{eff}$) distribution over the vortex due to (a) lattice strain $\epsilon_{xx}$ and (b) rotation $\omega_{xz}$ at zero applied field, with color coding indicating the x-direction field. (c) Vortex core trajectories for selected deformation amplitudes $A_{SAW}$ at the gyrotropic frequency ($\mu_0 H_z$ = 500 mT). Bold lines represent trajectories driven by $\epsilon_{xx}$, $\epsilon_{zz}$, and $\omega_{xz}$; dashed lines show those driven by $\omega_{xz}$ alone. (d) Trajectory radius as a function of $A_{SAW}$ for different applied fields, with open symbols showing $\omega_{xz}$-only effects. (e) Trajectory radius versus applied field $\mu_0 H_z$ for $A_{SAW} = 10 \times 10^{-5}$. (Inset) Normalized rolling fields distribution over the vortex with different applied fields, with dashed circles indicating the disk edge.</figcaption></figure>

tory is circular over the full range of studied fields. The perpendicular field acts to tilt the magnetic moments of the curling vortex texture out of the plane toward the field, resulting in a vortex cone state with a larger core [20, 30]. The larger orbital radius with increasing $\mu_0 H_z$ arises from a larger rolling field that appears across the full canted structure, rather than being only localized at the core at $\mu_0 H_z = 0$ [inset in Fig. 5(e)].

Further insight into the role of the different magnetoacoustic torques can be gained through the rigid-core approximation. In this approach, the magnetization dynamics is determined solely by the motion of the core, whose coordinates $\mathbf{X} = [X(t), Y(t)]$ evolve according to the Thiele equation [18, 31],

$$ \mathbf{G} \times \frac{d\mathbf{X}}{dt} + \alpha \mathbf{D} \cdot \frac{d\mathbf{X}}{dt} = -\frac{\partial U}{\partial \mathbf{X}}, \quad (1) $$

where $\mathbf{G}$ is the gyrovector, $\mathbf{D}$ is the damping tensor, and $U$ represents the effective potential energy seen by the vortex core. Without external driving, $U = U_0 (||\mathbf{X}||^2)$ is a central potential governed by the confining effects of

the disk, resulting in damped gyrotropic motion of the vortex core around the disk center. To illustrate the mechanics of this approach, consider first the more familiar example of a spatially uniform in-plane microwave field applied along $x$, which enters the Zeeman term (assuming $\mu_0 H_z = 0$)

$$ U_Z(t) = -\mu_0 M_s \int dV h_x(t) m_x(\mathbf{r}, t). \quad (2) $$

Because the dominant contribution to the integral comes from the region outside the core, i.e., where $m_z(\mathbf{r}) \simeq 0$, we can take $m_x(\mathbf{r}) \simeq \cos \Phi_0$, where $\Phi_0 = \tan^{-1}[y - Y/(x - X)] \pm \pi/2$, which gives $U_Z = \mp \mu_0 M_s \pi d_{FM} L h_x Y$ and results in the time-dependent force

$$ \mathbf{F}_Z(t) = -\frac{\partial U_Z}{\partial \mathbf{X}} = (0, \pm \mu_0 M_s \pi d_{FM} L h_x(t)). \quad (3) $$

This represents a “volumic” contribution and is independent of the core size and position within the approximation used.

We next discuss the time-dependent magnetoelastic term

$$ U_{\text{me}}(t) = B_1 \int dV \, \epsilon_{ii}(t) m_i(\mathbf{r}, t)^2, \quad (4) $$

with $i = x, z$, which acts like a uniaxial anisotropy in the film plane ($\epsilon_{xx}$) or perpendicular to the plane ($\epsilon_{zz}$). The $\epsilon_{xx}$ term is also volumic and depends on the in-plane magnetization outside the core. By using the same approximation as for the Zeeman term, we find to lowest order for small core displacements $U_{\text{me,xx}} = (\pi B_1 d_{FM})(2 - X^2 + Y^2)/4$, which represents a hyperbolic paraboloid potential. In contrast, the $\epsilon_{zz}$ term only depends on the perpendicular component of the magnetization, $m_z(\mathbf{r}, t)$, which is only nonzero in the core region. While this term can excite radial spin wave modes of the vortex state [32–34], it gives no contribution to Eq. (1), since the integral of $m_z(\mathbf{r}, t)^2$ over the disk is constant and therefore independent of the core position, $U_{\text{me,zz}} = \text{const}$. The resulting magnetoelastic force is then

$$ \mathbf{F}_{\text{me}} = -\frac{\partial U_{\text{me,xx}}}{\partial \mathbf{X}} = \frac{\pi B_1 d_{\text{FM}}}{2} \epsilon_{xx}(t) (X, -Y), \quad (5) $$

which vanishes when the core is at the disk center, $\mathbf{X} = \mathbf{0}$, as discussed earlier.

Consider now the time-dependent magnetorotation

$$ U_{\text{mr}}(t) = \mu_0 M_s^2 \int dV \, \omega_{xz}(t) m_x(\mathbf{r}, t) m_z(\mathbf{r}, t), \quad (6) $$

Unlike $\epsilon_{xx}$, which is driven by the in-plane magnetization, or $\epsilon_{zz}$, which is dominated by the core center, the magnetorotation term is driven by the boundary or “waist” of the core at which $\sin \Theta_0 \cos \Theta_0$ is largest, where $\Theta_0$ represents the polar angle of the magnetization. Compared to
the Zeeman term, the core “waist” appears thus like an additional convolution kernel of cos $\Phi_0$ in $U_{mr}$. For small displacements of the core, the force can be expressed in terms of the static core profile $\Theta_0 = \Theta_0(r)$ as

$$ \mathbf{F}_{mr} = -\frac{U_{mr}}{\partial \mathbf{X}} = (0, \mp \pi \mu_0 M_s^2 d_{FM} \omega_{xz} f(\Theta_0)), \quad (7) $$

where $f(\Theta_0)$ is a structure factor

$$ f(\Theta_0) = \int dr \left( r \frac{\partial \Theta_0}{\partial r} \cos 2\Theta_0 + \frac{\sin 2\Theta_0}{2} \right). \quad (8) $$

The integral can be evaluated with the Usov *ansatz* to give $f(\Theta_0) = (1 - \log 2)b$, where $b$ is the core radius. For $\mu_0 H_z > 0$, $m_z$ is nonzero outside the core which further enhances $f(\Theta_0)$.

Our results demonstrate that nonuniform spin textures like magnetic vortices can be useful for probing magnetorotation coupling like $\omega_{xz}$. Beyond offering new avenues towards SAW-based microwave magnetoacoustics, our findings also suggest that other nontrivial textures, such as domain walls and skyrmions, could also exhibit behaviors that are not accessible through magnetoelastic strains alone.

This work was supported by the Agence Nationale de la Recherche (ANR) under Contract No. ANR-20-CE24-0025 (MAXSAW), the “Investissements d’Avenir” program LabEx NanoSaclay, under Contract No. ANR-10-LABX-0035 (SPICY), and the France2030 program PEPR SPIN, under Contract Nos. ANR-22-EXSP-0008 and ANR-22-EXSP-0004.

<footnote>* rafael.lopesseseeeger@cea.fr</footnote>

<footnote>[1] M. Weiler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M. S. Brandt, and S. T. Goennenwein, Elastically driven ferromagnetic resonance in nickel thin films, Physical Review Letters **106**, 10.1103/PhysRevLett.106.117601 (2011).</footnote>

<footnote>[2] L. Thevenard, C. Gourdon, J. Y. Prieur, H. J. von Bardeleben, S. Vincent, L. Becerra, L. Largeau, and J.-Y. Duquesne, Surface-acoustic-wave-driven ferromagnetic resonance in (Ga,Mn)(As,P) epilayers, Phys. Rev. B **90**, 094401 (2014).</footnote>

<footnote>[3] R. Sasaki, Y. Nii, Y. Iguchi, and Y. Onose, Nonreciprocal propagation of surface acoustic wave in ni/linbo$_3$, Phys. Rev. B **95**, 020407 (2017).</footnote>

<footnote>[4] J. Puebla, M. Xu, B. Rana, K. Yamamoto, S. Maekawa, and Y. Otani, Acoustic ferromagnetic resonance and spin pumping induced by surface acoustic waves, Journal of Physics D: Applied Physics **53**, 10.1088/1361-6463/ab7efe (2020).</footnote>

<footnote>[5] R. Verba, V. Tiberkevich, and A. Slavin, Wide-band nonreciprocity of surface acoustic waves induced by magnetoelastic coupling with a synthetic antiferromagnet, Phys. Rev. Appl. **12**, 054061 (2019).</footnote>

<footnote>[6] A. Hernández-Mínguez, F. Macià, J. M. Hernández, J. Herfort, and P. V. Santos, Large nonreciprocal propagation of surface acoustic waves in epitaxial ferromagnetic</footnote>

<footnote>semiconductor hybrid structures, Physical Review Applied **13**, 10.1103/PhysRevApplied.13.044018 (2020).</footnote>

<footnote>[7] M. Xu, K. Yamamoto, J. Puebla, K. Baumgaertl, B. Rana, K. Miura, H. Takahashi, D. Grundler, S. Maekawa, and Y. Otani, Nonreciprocal surface acoustic wave propagation via magneto-rotation coupling, Sci. Adv **6**, https://doi.org/10.1126/sciadv.abb1724 (2020).</footnote>

<footnote>[8] F. Millo, J.-P. Adam, C. Chappert, J.-V. Kim, A. Mouhoub, A. Solignac, and T. Devolder, Unidirectionality of spin waves in synthetic antiferromagnets, Phys. Rev. Appl. **20**, 054051 (2023).</footnote>

<footnote>[9] M. Küß, S. Glamsch, A. Hörner, and M. Albrecht, Wide-band nonreciprocal transmission of surface acoustic waves in synthetic antiferromagnets, ACS Appl. Electron. Mater. **10.1021/acsaelm.3c01709** (2024).</footnote>

<footnote>[10] R. Sasaki, Y. Nii, and Y. Onose, Surface acoustic wave coupled to magnetic resonance on multiferroic cub$_2$O$_4$, Phys. Rev. B **99**, 014418 (2019).</footnote>

<footnote>[11] H. Matsumoto, T. Kawada, M. Ishibashi, M. Kawaguchi, and M. Hayashi, Large surface acoustic wave nonreciprocity in synthetic antiferromagnets, Applied Physics Express **15**, 063003 (2022).</footnote>

<footnote>[12] T. P. Lyons, J. Puebla, K. Yamamoto, R. S. Deacon, Y. Hwang, K. Ishibashi, S. Maekawa, and Y. Otani, Acoustically driven magnon-phonon coupling in a layered antiferromagnet, Phys. Rev. Lett. **131**, 196701 (2023).</footnote>

<footnote>[13] A. Koujok, A. Riveros, D. R. Rodrigues, G. Finocchio, M. Weiler, A. Hamadeh, and P. Pirro, Resonant excitation of vortex gyrotropic mode via surface acoustic waves, Applied Physics Letters **123**, 132403 (2023).</footnote>

<footnote>[14] V. Iurchuk, J. Lindner, J. Fassbender, and A. Kákay, Excitation of the gyrotropic mode in a magnetic vortex by time-varying strain, Phys. Rev. Lett. **133**, 146701 (2024).</footnote>

<footnote>[15] K. Yamamoto and S. Maekawa, Magnetostatic field induced by mechanical deformations, Annalen der Physik **536**, 2300395 (2023).</footnote>

<footnote>[16] R. Lopes Seeger, L. La Spina, V. Laude, F. Millo, A. Bartasyte, S. Margueron, A. Solignac, G. de Loubens, L. Thevenard, C. Gourdon, C. Chappert, and T. Devolder, Symmetry of the coupling between surface acoustic waves and spin waves in synthetic antiferromagnets, Phys. Rev. B **109**, 104416 (2024).</footnote>

<footnote>[17] S. Maekawa and M. Tachiki, Surface acoustic attenuation due to surface spin wave in ferro- and antiferromagnets, AIP Conference Proceedings **29**, 542 (1976).</footnote>

<footnote>[18] See Supplemental Material at [URL] for further details and additional data on sample fabrication, IDT characterization, MRFM measurements, micromagnetics simulations, and the Thiele model.</footnote>

<footnote>[19] O. Klein, G. de Loubens, V. V. Naletov, F. Boust, T. Guillet, H. Hurdequint, A. Leksikov, A. N. Slavin, V. S. Tiberkevich, and N. Vukadinovic, Ferromagnetic resonance force spectroscopy of individual submicron-size samples, Phys. Rev. B **78**, 144410 (2008).</footnote>

<footnote>[20] G. de Loubens, A. Riegler, B. Pigeau, F. Lochner, F. Boust, K. Y. Guslienko, H. Hurdequint, L. W. Molenkamp, G. Schmidt, A. N. Slavin, V. S. Tiberkevich, N. Vukadinovic, and O. Klein, Bistability of vortex core dynamics in a single perpendicularly magnetized nanodisk, Physical Review Letters **102**, 177602 (2009).</footnote>

<footnote>[21] $H_{sat} \simeq M_s$ is the saturation field of the disk, $\mu_0 M_s \simeq 1.7$ T the CoFeB saturation magnetization, and $\gamma/(2\pi) \simeq 28$ GHz/T its gyrotropic ratio.</footnote>

<footnote>[22] R. L. Compton and P. A. Crowell, Dynamics of a pinned</footnote>
magnetic vortex, Phys. Rev. Lett. 97, 137202 (2006).

[23] S. Shreya, A. S. Jenkins, Y. Rezaeiyan, R. Li, T. Böhnert, L. Benetti, R. Ferreira, F. Moradi, and H. Farkhani, Granular vortex spin-torque nano oscillator for reservoir computing, Scientific Reports 13, 16722 (2023).

[24] K. An, A. N. Litvinenko, R. Kohno, A. A. Fuad, V. V. Naletov, L. Vila, U. Ebels, G. de Loubens, H. Hurd- equint, N. Beaulieu, J. Ben Youssef, N. Vukadinovic, G. E. W. Bauer, A. N. Slavin, V. S. Tiberkevich, and O. Klein, Coherent long-range transfer of angular mo- mentum between magnon猫kittel modes by phonons, Phys. Rev. B 101, 060407 (2020).

[25] B. Casals, N. Statuto, M. Foerster, A. Hernández- Mínguez, R. Cichelero, P. Manshausen, A. Mandziak, L. Aballe, J. M. Hernàndez, and F. Macià, Generation and imaging of magnetoacoustic waves over millimeter distances, Phys. Rev. Lett. 124, 137202 (2020).

[26] B. Pigeau, G. de Loubens, O. Klein, A. Riegler, F. Lochner, G. Schmidt, L. W. Molenkamp, V. S. Tiberkevich, and A. N. Slavin, A frequency-controlled magnetic vortex memory, Applied Physics Letters 96, 132506 (2010).

[27] A. Drews, B. Krüger, G. Selke, T. Kamionka, A. Vogel, M. Martens, U. Merkt, D. Möller, and G. Meier, Nonlin- ear magnetic vortex gyration, Phys. Rev. B 85, 144417 (2012).

[28] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, The design and verification of MuMax3, AIP Advances 4, 107133 (2014).

[29] T. A. Ostler, R. Cuadrado, R. W. Chantrell, A. W. Rush- forth, and S. A. Cavill, Strain induced vortex core switch- ing in planar magnetostrictive nanostructures, Phys. Rev. Lett. 115, 067202 (2015).

[30] B. A. Ivanov and G. M. Wysin, Magnon modes for a circular two-dimensional easy-plane ferromagnet in the cone state, Phys. Rev. B 65, 134434 (2002).

[31] K. Y. Guslienko, Magnetic vortex state stability, revers- sal and dynamics in restricted geometries, Journal of Nanoscience and Nanotechnology 8, 2745 (2008).

[32] B. A. Ivanov and C. E. Zaspel, High frequency modes in vortex-state nanomagnets, Phys. Rev. Lett. 94, 027205 (2005).

[33] K. Vogt, O. Sukhostavets, H. Schultheiss, B. Obry, P. Pirro, A. A. Serga, T. Sebastian, J. Gonzalez, K. Y. Guslienko, and B. Hillebrands, Optical detection of vor- tex spin-wave eigenmodes in microstructured ferromag- netic disks, Phys. Rev. B 84, 174401 (2011).

[34] B. Taurel, T. Valet, V. V. Naletov, N. Vukadinovic, G. de Loubens, and O. Klein, Complete mapping of the spin-wave spectrum in a vortex-state nanodisk, Physical Review B 93, 184427 (2016).