# Giant magneto-elasticity in FeRh thin films evidenced via strong phonon-magnon coupling

D. Ourdani, Catherine Gourdon, Jon Ander Arregi, Vojtěch Uhlíř, Y. Roussigné, L. Thevenard

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D. Ourdani, Catherine Gourdon, Jon Ander Arregi, Vojtěch Uhlíř, Y. Roussigné, et al. Giant magneto-elasticity in FeRh thin films evidenced via strong phonon-magnon coupling. 2025. hal-05356780

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Distributed under a Creative Commons Attribution 4.0 International License
Giant magneto-elasticity in FeRh thin films
evidenced via strong phonon-magnon coupling

D. Ourdani,†,‡ C. Gourdon,† J. A. Arregi,¶ V. Uhlíř,¶,§ Y. Roussigné,‡ and L.

Thevenard*,†

†Sorbonne Université, CNRS, Institut des Nanosciences de Paris, 4 place Jussieu, 75252

Paris, France

‡LSPM, Université Paris 13, Sorbonne Paris Cité, 99 avenue Jean-Baptiste Clément,

93430 Villetaneuse, France

¶CEITEC BUT, Brno University of Technology, Purkyňova 123, 612 00 Brno, Czech

Republic

§Institute of Physical Engineering, Brno University of Technology, Technická 2, 616 69

Brno, Czech Republic

E-mail: thevenard@insp.jussieu.fr

Abstract

We report a strong coupling between a Love surface acoustic wave (SAW) and a spin wave in epitaxial FeRh as analyzed from their anti-crossing dispersion relationships obtained by Brillouin light scattering. Reproducing the results using a numerical model that solves the coupled Landau-Lifshitz and Christoffel equations reveals that this effect stems from both the large magneto-elastic coupling coefficient in this material, $|B_2|=17$ MJ/m$^3$, and the strength of the Love-wave magneto-acoustic effective field when the magnetization vector **M** is parallel to the SAW wavevector **k**. We also observe a substantial, albeit weaker, coupling in the unusual configuration of the Sezawa
surface wave crossing the pseudo Damon-Eshbach mode, i.e. when the angle between **M** and **k** is $\pi/4$. These findings identify the optimal acoustic modes and magnetic geometries for achieving strong phonon-magnon hybridization, paving the way for coherent information transfer via elastic waves in magneto-acoustic nanodevices.

Direct interconnection between structural and magnetic degrees of freedom is a powerful tool for creating materials with novel physical properties. Since the theoretical predictions of Charles Kittel in 1958, the effect of magnon-phonon interactions on either the magnetic or elastic sub-systems has been demonstrated in various magnetic materials,¹⁻⁵ generally considering a weak coupling regime where acoustic waves drive (or are being driven by) the ferromagnetic resonance (FMR). Over the past decade, a variety of physical phenomena offered by the strong coupling regime has been evidenced.³,⁶⁻⁸ The latter requires (i) a large magneto-elastic coupling (parametrized by $B_1$ and $B_2$ constants), (ii) optimal frequency and wavevector matching of phonons and magnons, as well as (iii) exceptionally long lifetimes - resulting from either very weak magnetic damping or from cavity geometries. This may lead to a modification of the eigenfrequencies of phonons and magnons, a gap opening in their crossing dispersions,³,⁶,⁷ the appearance of over- and undertones of the FMR frequency⁹⁻¹¹ or even the precessional switching of magnetization using acoustic waves.¹²

In this article, we report on the remarkably large anisotropic magneto-elastic coefficient $B_2$ of an equiatomic epitaxial FeRh¹³ thin film. This material is usually considered for the giant magneto-caloric effect,¹⁴ elastocaloric effect,¹⁵ magneto-resistive effects¹⁶,¹⁷ and volume magnetostriction¹⁸⁻²⁰ accompanying the first order transition from a low-temperature antiferromagnetic (AFM) state to a ferromagnetic (FM) state, but we focus here on its uniform, high temperature, FM phase. While the $B_2$ coefficient is often challenging to measure on thin magnetic films,²¹,²² we are able to estimate it from the gap opening of the intersecting magnon and surface phonon branches, using a combination of Brillouin light scattering (BLS) experiments and calculations of the phonon/magnon dynamics coupled via both magneto-elastic and magneto-rotation energies. The latter is a $B_2$-independent
term that captures the modification of the shape anisotropy through the surface acoustic
wave-induced rotation of the lattice.²³⁻²⁵

By looking at three different acoustic modes crossing two distinct spin wave (SW) modes,
we moreover identify the geometry leading to the strongest hybridization: a Love wave trav-
eling parallel to the (backward volume) SW. We understand this to stem from both : (i)
how the different strain components are weighted by magneto-elastic and magneto-rotation
terms, and (ii) the strong ellipticity of the magnetization dynamics and how it affects the
magneto-elastic energy.

The 36 nm thick epitaxial FeRh film was grown on a MgO (001) substrate via DC mag-
netron sputtering from an equiatomic target. The growth of the magnetic layer was per-
formed in high vacuum, followed by a 45 min anneal at 780°C, and the deposition of a 2 nm
Pt capping layer (see details in the Supplementary Material, SM). X-ray diffraction confirms
a high-quality FeRh(001) out-of-plane texture and a homogeneous CsCl-type structure, ex-
hibiting cube-on-cube epitaxy with a π/4 in-plane rotation relative to the MgO substrate²⁶
(see schematics in the SM). The layer is weakly strained, with an out-of-plane lattice param-
eter *c* = 3.0118 Å, yielding a strain value of +0.17% with respect to the bulk FeRh lattice
parameter²⁶ (*T*=147°C values).

The magnetic characterization by vibrating sample magnetometry (VSM) reveals a sharp AFM to FM transition upon heating, with a saturation magnetization $M_s$ reaching 1220 kA/m in the FM phase. The transition temperature, defined as the midpoint of the $M_s(T)$ hysteresis, is found at $\sim$ 96°C.

We now focus on the uniform FM phase, and present experiments taken at T=125°C in all
that follows. BLS measurements in the Damon–Eshbach (DE) geometry (wavevector perpen-
dicular to the static magnetization) yield a reduced gyromagnetic ratio γ/2π = 30 GHz/T,
consistent with metallic ferromagnets. Magnetic linear dichroism (MLD) experiments on
a very similar sample evidence the expected easy magnetic axes of FeRh along its in-plane
⟨100⟩ axes. Finally, the elastic properties of the FeRh film were determined using the phonon lines of the BLS spectra (see Table S-1 in the SM). These parameters are critical for the understanding of the magneto-elastic coupling in FeRh, as they govern the dispersion relationship of acoustic waves. Experimental details on these characterizations can be found in Section 1 of the SM.

<figure><img src="image_2.png" /><figcaption>Figure 1: (a) Schematics of the geometry used for the BLS experiment and the calculations. In all the experiments, the field is applied along an easy in-plane <100> axis of the film. **k**<sub>i</sub>, **k**<sub>s</sub>, **k**<sub>r</sub> are the incident, back-scattered and reflected wavevectors. The in-plane wavevector **k** is selected by adjusting the incidence angle θ. (b-c) Two different geometries exploring the interaction between a surface acoustic wave and a spin wave traveling along **k**, either along the static magnetization, or at α = π/4 from it.</figcaption></figure>

The magneto-elastic coupling was investigated in two geometries (Fig. 1(b,c)). The first corresponds to the Backward Volume Wave (BW) configuration (α = 0 in Fig. 1), in which the magnetization is aligned along the [100] crystallographic direction of FeRh by a 100 mT field, and SWs propagate parallel to it (k || M). In the second configuration ("pseudo DE" geometry), the magnetization remains aligned along [100] thanks to a 35 mT field, while the wave vector is oriented along the [110] direction, forming an angle of α = π/4 with respect to M. BLS spectra were measured for both geometries, with the wave vector *k* varied from 8 to 17 µm⁻¹.

The corresponding frequency shifts due to inelastic scattering by phonons and magnons are shown in Fig. 2. Due to the distinct group velocities of magnons and phonons, tuning *k*
leads to intersections between their dispersion branches, with strong magneto-elastic inter-
actions revealed by a clear mode splitting. We give below a detailed analysis of the magnon
spectra in the SW/SAW interaction region.

We begin with the BW geometry, showing in Fig. 2(a) three representative spectra
recorded for different wavevectors. The BW mode can be clearly identified around 11.4
and 10.88 GHz for wave vectors k = 9.98 and 16.99 µm⁻¹. However, within an intermediate
range of wave vectors notably at k = 14.86 µm⁻¹, a second mode emerges alongside the
BW mode. The first perpendicular standing mode, not presented here, is also observed at
a higher frequency, around 20.45 GHz, and was used to find the exchange constant A<sub>ex</sub>=13
pJ/m of the film (see Fig. S-2 of the SM). The magnon measurements taken in this ge-
ometry are summarized in the spectral map shown in Fig. 2(c). A key observation is the
opening of a frequency gap of Δ=1.1 GHz near k = 14.86 µm⁻¹, which is a clear signature
of coupling between two modes, as previously reported in other studies, working either at
fixed field/varying wavevector,³ or at fixed wavevector/varying field.⁶,⁷ The evolution of the
linewidth of the BLS magnon lines is shown in Fig. 3. The linewidth is around κ<sub>m,0</sub> ≈ 1.2
GHz at low vector, undergoes a weak initial drop around k = 9 µm⁻¹, followed by a much
sharper drop down to 0.6 GHz in the anticrossing region. Outside of these two regions, the
ratio of linewidth to frequency is around 0.1, *i.e.* very similar to that found by Jin *et. al*²⁷
in epitaxial FeRh.

To further identify the nature of the additional mode involved in the anticrossing, we
performed complementary measurements targeting elastic excitations. These were obtained
using a noncrossed-analyzer setup designed to suppress the magnon contribution. The re-
sults, plotted as blue symbols in Fig. 2(c), reveal the presence of two acoustic modes: a
Rayleigh surface acoustic wave (R-SAW) and a Sezawa wave (S-SAW), the more guided
higher order mode that appears when the layer velocity is lower than that of the substrate.
Note that both waves exhibit in-plane (along **k**), and out-of-plane atomic displacements (blue
arrows in Fig. 4(a)), the latter being the only one detectable by BLS, via the ripple effect.²⁸
However, neither of these modes intersect the anticrossing region, ruling out their direct involvement. We show in the following section that the BW dispersion curve actually crosses the Love wave dispersion. When uncoupled to a magnon, the purely transverse nature of this mode makes it unobservable by BLS in opaque materials. Using the out-of-resonance phonon linewidth of around $\kappa_{ph,0} \approx 0.4$ GHz seen in the elastic spectra, we compute a cooperativity of $C = \frac{\Delta^2}{\kappa_{m,0}\kappa_{ph,0}} = 2.5$. It should be noted that other authors have defined it differently and/or in forced excitation geometry where the phonon linewidth is given by the rf generator rather than by intrinsic acoustic damping.6,7,29,30 Comparing to the experiment of Babu et. al³ who, like us, monitor the thermal excitation of both phonons and magnons and find $C = 0.4$, one can conclude on the strong coupling nature of the interaction in our system.

A distinct behavior emerges when the SW wave vector is aligned along [110], while the magnetization remains along [100], corresponding to $\alpha = \pi/4$ in Fig. 1. Fig. 2(b) displays once again three representative spectra. The first perpendicular standing wave now comes out at around 18 GHz, with a position varying very weakly with wavevector as expected. The pseudo DE wave is clearly observed at 14.12 and 14.5 GHz for wavevectors $k=11.81$ and $15.18 \ \mu m^{-1}$, respectively. An anticrossing is observed around $k = 14.9 \ \mu m^{-1}$, but without the clear mode splitting observed in the backward configuration. The spectrum at $k = 14.54 \ \mu m^{-1}$ shows the propagating mode at 14.87 GHz though, evidencing the non-monotonous evolution of the magnon frequency around the anti-crossing. Using the same procedure as above we overlay the phonon frequencies in this second configuration as blue squares in Fig. 2(d). The dispersion of the Sezawa surface acoustic wave (S-SAW) now closely matches the position of the anticrossing, in contrast to the BW case, suggesting a coupling between the SW and the S-SAW.

To interpret the experimental BLS spectra and confirm the nature of the observed hybridization between spin and acoustic modes, we developed a theoretical model that captures the dynamic magneto-elastic coupling in the FeRh/MgO system. The model combines the
<figure><img src="image_3.png" /><figcaption>Figure 2: Two different experimental configurations investigated using BLS: (a,c) Backward Volume geometry ($\alpha = 0$), $B=100$ mT, and (b,d) Pseudo Damon-Eshbach (DE) geometry ($\alpha = \pi/4$) with $\mathbf{k} \parallel [110]$ and $\mathbf{M} \parallel [100]$, $B=35$ mT. (a,b) Representative BLS spectra recorded for different wave vectors in each configuration. In (b), the dashed straight (wavy) line highlights the weak (non-monotonous) variation of the standing wave (resp. pseudo DE) mode frequency. (c-d) Dispersion relations: the colored spectral maps show the experimental magnon dispersion, while symbols correspond to the measured phonon modes (elastic waves). Theoretical calculations are indicated by lines: magnons in black, and phonons in blue/purple. The dashed lines correspond to the calculation assuming no magneto-elastic coupling, i.e. $B_2=0$ and $\zeta=0$. See Section 2 of the SM for details on how the simulated curves were obtained.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3: Full width at half maximum (FWHM) of BW ($\alpha$=0 configuration of Fig. 1) and mixed BW-Love waves in the anti-crossing region (near $k$=15.03 $\mu m^{-1}$). Note that the pure Love wave does not appear outside the anti-crossing region.</figcaption></figure>
linearized Landau-Lifshitz (LL) equation for spin dynamics and the Christoffel equation for elastic wave propagation, extended to include magneto-acoustics through both conventional magneto-elastic interactions and magneto-rotational coupling.²³ The two core equations are:

$$ \left\{ \begin{aligned} -\frac{1}{\gamma} \frac{\partial \mathbf{M}(t)}{\partial t} &= \mathbf{M}_s \times (\mathbf{B}_d(t) + \mathbf{B}_{\text{ex}}(t) + \mathbf{b}_{\text{ma}}(t)) + \mathbf{M}(t) \times \mathbf{B}_s, \\ \rho \frac{\partial^2 \mathbf{u}(t)}{\partial t^2} &= \nabla \cdot \boldsymbol{\sigma}_e(t) + \nabla \cdot \boldsymbol{\sigma}_{\text{ma}}(t) \end{aligned} \right. \qquad (1) $$

where $\mathbf{M}(t)$ and $\mathbf{u}(t)$ are the dynamic magnetization and displacement fields, respectively, and $\mathbf{M}_s$ and $\mathbf{B}_s$ are the static magnetization and effective field. $\boldsymbol{\sigma}_e(t)$ is the purely elastic stress. $\mathbf{b}_{\text{ma}}(t)$ and $\boldsymbol{\sigma}_{\text{ma}}(t)$ represent the magneto-acoustic contributions to the effective field and stress, respectively. The amplitude of the former is explicit for Rayleigh/Sezawa and Love waves in Eqs. (2) in the case of the backward configuration (see details of the derivation in the SM):

$$ \left\{ \begin{aligned} |b_{Ray/Sez}| &= \frac{2B_2\varepsilon_{yx}}{M_s} + \zeta\mu_0 M_s \omega_{yx}, \\ |b_{Love}| &= \frac{B_2\varepsilon_{zy}}{M_s}, \end{aligned} \right. \qquad (2) $$

where $B_2$ is the magneto-elastic coupling constant and $\varepsilon_{ij}$ ($\omega_{ij}$) are the amplitudes of the strain (rotation) tensor elements. The coefficient $\zeta$ allows us to turn the magneto-rotation effect on/off in the calculations. The model includes the stiffness tensors and mass densities of both the FeRh film and the MgO substrate. With the magnetization fixed along an easy axis, it allows for an arbitrary in-plane spin wave propagation direction. Both the Gilbert damping of the spin waves and the acoustic damping are neglected in this eigenproblem analysis. This approach offers a robust framework for analyzing and predicting spin-phonon interactions in hybrid magnetic-elastic systems. Additional theoretical details, including the full expression of the energy terms, as well as exchange and dipolar fields $\mathbf{B}_{\text{ex}}(t)$ and $\mathbf{B}_d(t)$,
are provided in the SM (Section 2).

The eigenfrequencies found for the coupled system are shown in Fig. 2(c,d). The numerical calculation yields the dispersion of the BW and pseudo-DE waves (black lines), and of the Rayleigh and Sezawa SAWs (blue lines). The non-hybridized Love SAW dispersion is shown by the dashed purple lines. The model reproduces the hybridized dispersion relations and the anti-crossings observed experimentally in both geometries with a single adjustable parameter: the magneto-elastic coupling constant, found to be $|B_2|=17 \text{ MJ m}^{-3}$. The uncoupled dispersion relationships are shown in dashed lines (setting $B_2$ to zero). We now understand the anticrossing in the backward geometry to result from the interaction with the Love waves, which cannot be observed by the ripple effect in BLS, since the atomic displacements lie purely in-plane (red arrows in Fig. 4(a)). Note that this type of anticrossing had already been seen in CoFeB by Babu *et al.*,³ albeit with a much smaller gap opening of around 0.5 GHz.

The anisotropic magneto-elastic constant found by this method, $|B_2|=17 \text{ MJ m}^{-3}$, is significantly higher than those reported for other magnetostrictive materials such as Co or Fe ($7 \text{ MJ m}^{-3}$),³ and comparable to the highest values found for CoFeB, e.g. $|B_2|=16 \text{ MJ m}^{-3}$ in Hwang *et al.*⁷ It is moreover much larger than the values found on polycrystalline FeRh: $|B_s|=5 \text{ MJ m}^{-3}$ when fitting SAW-FMR data,³¹ or $|B_2| \approx 3 \text{ MJ m}^{-3}$ estimated<fnref n="1" /> from the high field magnetostriction coefficient measurement of Ibarra *et al.*¹⁸ This highlights the exceptional strength of the magneto-elastic interaction in epitaxial FeRh, possibly resulting from the strained state of this thin layer. The increased sensitivity of FeRh to strain in the film form has indeed been evidenced in the past, e.g. through large variations of the transition temperature,²⁶ or the observation of large phase domains along the directions set by the substrate.³² The tetragonal distortion of the film induced by the clamping on the

<footnote n="1">¹To compare our estimation of $|B_2|$ to that of Ibarra *et al.*,¹⁸ we have converted his magnetostriction coefficient $\lambda_{111}=10\text{ppm}$ using $-B_2=3C_{44}\lambda_{111}$, and elastic stiffness constant $C_{44}=76 \text{ GPa}$ for polycrystalline FeRh estimated and validated in Vythelingum *et al*.³¹</footnote>
substrate could indeed be at the source of this magneto-elasticity, very much like what has
been observed on thin cobalt or nickel films whose magneto-elasticity differs substantially
from their bulk counterparts.³³ To explore this hypothesis, we performed BLS measurements
on a 195 nm thick, completely relaxed film.³⁴ Whereas the Damon-Eshbach mode was clearly
observed, the backward wave signal could barely be detected, prohibiting a similar study to
be performed.

Lastly, we discuss why the BW is more strongly coupled with Love waves than with
Rayleigh and Sezawa waves, since only the former leads to a clear gap opening (Fig. 2(c)).
We first compare the amplitudes of their respective magneto-acoustic fields. In the reference
frame of the FeRh film, the effective magneto-acoustic field is obtained from the derivative of
the magneto-elastic and magneto-rotation energies (detailed in Section 2 of the SM³,²³). Its
absolute value is given in Eqs. 2 for both types of acoustic waves in the backward geometry.
In these expressions, we have introduced the coefficient ζ to (de)activate simply the magneto-
rotation effect in our calculation. For a Love wave, the only non-zero component of the
magneto-acoustic field depends solely on the absolute value of B₂, which can thus be directly
obtained by fitting the anticrossing in this geometry. For Rayleigh and Sezawa waves, both
magneto-elastic and magneto-rotation terms intervene in the ratio of their corresponding
energies, the former strongly dominating the latter: $\frac{2B_2}{\mu_0 M_s^2} \approx 19$, since $\mu_0 M_s = 1.5$ T and
$\frac{2B_2}{M_s} \approx 28$ T assuming $B_2=17$ MJ m⁻³.

We now evaluate the weight of shear strain versus lattice rotation components in $b_{Ray/Sez}$
(Eq. 2). From the calculation of the elastic eigenmodes without magneto-acoustic coupling
($B_2=0, \zeta=0$), we compute the depth-profile of the strain and rotation terms of each mode,
Rayleigh, Sezawa and Love (Fig. 4(b-d)), for the wavevector at which they intersect the
backward mode (respectively $k=10.28, 15.03$ and $17.3$ µm⁻¹). For a Rayleigh wave (Fig.
4(d)), the depth-averaged shear strain is roughly 100 times smaller than the rotation term,
but for the more guided Sezawa wave (Fig. 4(b)), it is only in a ratio of 1 to 4. From this,
<figure><img src="image_5.png" /><figcaption>Figure 4: (a) The displacements of Rayleigh/Sezawa waves are indicated by blue arrows ($u_y, u_x$ displacements) and yield $\varepsilon_{yx}$, $\varepsilon_{xx}$, $\varepsilon_{yy}$ strains and $\omega_{yx}$ rotation terms. Love waves have a unique $u_z$ displacement component (red arrow), yielding the strain/rotation terms $\varepsilon_{zy}=\omega_{zy}$ and $\varepsilon_{zx}=\omega_{zx}$ (which does not play any role in the coupling). (b-d) Depth profile of the non-hybridized ($B_2$=0, $\zeta$=0) strain/rotation amplitudes intervening in Eqs. (2) for the three acoustic modes crossing the backward volume wave. Note that the y-scales are not comparable. (b) Sezawa mode at $k$=10.28 $\mu$m$^{-1}$ ($f$=11.6 GHz) (c) Love mode at $k$=15.03 $\mu$m$^{-1}$ (where it antrosses the magnon dispersion, $f$=11.3 GHz) and (d) Rayleigh mode at $k$=17.3 $\mu$m$^{-1}$ ($f$=11.2 GHz). The black line is the (essentially flat) profile of the amplitude of the in-plane dynamical component of the non-hybridized ($B_2$=0, $\zeta$=0) backward volume mode. The FeRh layer lies between $x$=0 and 36 nm, the MgO substrate corresponds to $x$ >36 nm.</figcaption></figure>
we conclude that the magneto-rotation term dominates in Rayleigh waves, but that the magneto-elastic term takes over for the Sezawa mode, with $|b_{Ray}| \ll |b_{Sez}|$.

Finally, to compare the absolute efficiency of Sezawa and Love effective fields, we present in Fig. 5 the thermal squared displacement spectra²⁸ $|u_i|^2$ calculated at : $k=10.28 \ \mu\text{m}^{-1}$ (dotted lines) where the Sezawa mode intersects the BW mode, and $k=15.03 \ \mu\text{m}^{-1}$ (continuous line), where the Love mode intersects the BW mode (details in the SM). We notice that the $u_x$ displacement involved in the Sezawa mode responds with much weaker intensity than the $u_z$ displacement of the Love mode. This is likely due to the fact that at low wavevector, the Sezawa mode wavevector acquires a complex perpendicular component which propagates into the substrate. As a result, the term $\varepsilon_{zy} \approx \frac{k}{2}u_z$ (in $|b_{Love}|$) is much larger than $\omega_{yx} \approx ku_x$ appearing in $|b_{Sez}|$. We conclude that in the BW configuration, the amplitude of the magneto-acoustic fields for the three types of waves range as $|b_{Ray}| \ll |b_{Sez}| < |b_{Love}|$. The consequence is that the coupling with the Sezawa wave is not strong enough to open a gap in the dispersion, but only appears as a weak drop in magnon linewidth observed where they cross, around $k=10.28 \ \mu\text{m}^{-1}$ (Fig. 3).

We now consider the involved energies. In the BW configuration, the coupling between SW and SAW involves terms of the form $\frac{B_2}{M_s^2}[2M_x(t)M_s\varepsilon_{yx} + M_sM_z(t)\varepsilon_{zy}] - \mu_0M_x(t)M_s\omega_{yx}$ (see section 2 of the SM). Interestingly, $M_x(t)$ terms only couple to Rayleigh strain/rotation components, whereas $M_z(t)$ couples to Love wave components. The out-of-plane $M_x(t)$ dynamical component is small compared to the in-plane $M_z(t)$ term (typically thermally excited in a ratio of about 1/9) due to the demagnetizing effect. This effect is amplified by the relative weight of magneto-elastic and magneto-rotation energies, as well as that of strain and rotation components as discussed above. For these reasons, one expects a larger coupling of the backward volume mode with the Love wave (involving $\varepsilon_{zy}$) than with the Rayleigh or Sezawa modes (involving $\varepsilon_{yx}$). In the pseudo-DE geometry instead, only the Sezawa wave crosses the spin wave dispersion in the explored window (Fig. 2(d)). The magneto-acoustic field in this case is $b_{Sez}(t)=\frac{B_2}{M_s}(\varepsilon_{yy}(t)-\varepsilon_{xx}(t))$. Because these two longitudinal strain
<figure><img src="image_6.png" /><figcaption>Figure 5: (a) Calculated phonon dispersion in the absence of magneto-acoustic coupling ($B_2$=0, $\zeta$=0). We have highlighted two wavevectors in particular: $k$=10.28 $\mu$m$^{-1}$ where the Sezawa wave intersects the BW mode, and $k$=15.03 $\mu$m$^{-1}$, where the Love mode intersects the BW mode. (b) Calculated thermal squared displacements involved in Love (L) or Rayleigh (R)/Sezawa (S) modes according to equation 30 in ref$^{28}$ (the coefficient $k_B T / (2\pi^2)$ is omitted, see SM for details).</figcaption></figure>

components are large$^{35}$ and $\pi$-dephased, this quantity is now substantial enough to modify
the magnon linewidth, as observed experimentally (Fig. 3).

To summarize, working in two geometries on a 36 nm thick film of FeRh using Brillouin
light scattering, we evidence the anti-crossing of magnons and phonons, which we fit by a
unique magneto-elastic constant $|B_2|=17$ MJ/m$^3$. This value is close to the largest values
found in CoFeB, and much larger than in bulk Fe ($|B_2|=7$ MJ/m$^3$). While this could be due
to the varying moment on the Rh site, and thus have an origin similar to that of the volume
striction of FeRh,$^{36}$ the strained nature of the layer could also be involved. An ab-initio
treatment of the magneto-elasticity in FeRh would likely provide more definitive answers
than can be explored experimentally at this stage.

Importantly, we demonstrate that the optimal coupling geometry is that of Love waves
traveling parallel to the acoustic wavevector, a configuration already exploited in SAW-based
magnetic field sensors.³⁷,³⁸ We also reveal that Sezawa waves are an interesting alternative
when traveling at π/4 from the magnetization. The non-monotonous behavior of the fre-
quency and/or the linewidth that ensues is a signature of hybridized elastic and magnetic
modes which promises rich physics whereby one particle could be manipulated by the other,
taking the best of both worlds: the weak damping of phonons, and the tunability of magnons
by magnetic fields. We therefore emphasize the necessity of exploring the conditions leading
to strong magneto-acoustic coupling, whether via magneto-elasticity or magneto-rotation
effects.

Acknowledgement

This work has been partly supported by the French Agence Nationale de la Recherche (ANR
ACAF 20-CE30-0027). Access to the CEITEC Nano Research Infrastructure was supported
by the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic under the
project CzechNanoLab (LM2023051). We acknowledge V. Laude for fruitful discussions, and
A. K. Vythelingum and M. Hider for technical assistance.

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