# Study of the magnetoelastic effect in nickel and cobalt thin films at GHz
range using X-ray microscopy

Marc Rovirola<EMAIL_ADDRESS>Dept. of Condensed Matter Physics,
University of Barcelona, 08028 Barcelona, Spain  Institute of Nanoscience and
Nanotechnology (IN2UB), University of Barcelona, 08028 Barcelona, Spain  M.
Waqas Khaliq<EMAIL_ADDRESS>Dept. of Condensed Matter Physics, University
of Barcelona, 08028 Barcelona, Spain  ALBA Synchrotron Light Source, 08290
Cerdanyola del Vallès, Spain  Travis Gustafson  Department of Physics,
Norwegian University of Science and Technology (NTNU), 7034 Trondheim, Norway
Fiona Sosa  Dept. of Condensed Matter Physics, University of Barcelona, 08028
Barcelona, Spain  Blai Casals  Institute of Nanoscience and Nanotechnology
(IN2UB), University of Barcelona, 08028 Barcelona, Spain  Dept. of Applied
Physics, University of Barcelona, 08028 Barcelona, Spain  Joan Manel Hernàndez
Dept. of Condensed Matter Physics, University of Barcelona, 08028 Barcelona,
Spain  Institute of Nanoscience and Nanotechnology (IN2UB), University of
Barcelona, 08028 Barcelona, Spain  Sandra Ruiz-Gómez  Max Planck Institute for
Chemical Physics of Solids, 01187-Dresden, Germany  Miguel Angel Niño  ALBA
Synchrotron Light Source, 08290 Cerdanyola del Vallès, Spain  Lucía Aballe
ALBA Synchrotron Light Source, 08290 Cerdanyola del Vallès, Spain  Alberto
Hernández-Mínguez  Paul-Drude-Institut für Festkörperelektronik, Leibniz-
Institut im Forschungsverbund Berlin e.V., 10117 Berlin, Germany  Michael
Foerster  ALBA Synchrotron Light Source, 08290 Cerdanyola del Vallès, Spain
Ferran Macià<EMAIL_ADDRESS>Dept. of Condensed Matter Physics,
University of Barcelona, 08028 Barcelona, Spain  Institute of Nanoscience and
Nanotechnology (IN2UB), University of Barcelona, 08028 Barcelona, Spain

###### Abstract

We use surface acoustic waves of 1 and 3 GHz in hybrid piezoelectric-magnetic
systems with either cobalt or nickel as a magnetic layer to generate
magnetoacoustic waves and directly image them using stroboscopic X-ray
magnetic circular dichroism imaging. Our measurements visualize and quantify
the amplitudes of both acoustic and magnetic components of the magnetoacoustic
waves, which are generated in the ferromagnetic layer and can propagate over
millimeter distances. Additionally, we quantifiedy the magnetoelastic strain
component for cobalt and nickel through micromagnetic simulations. We obtained
a drop in the magnetoacoustic signal at 3 GHz suggesting a speed limit for the
efficient magnetoelastic coupling in our hybrid devices.

††preprint: APS/123-QED

##  I  Introduction

There is an increasing interest in manipulating magnetization dynamics to
create low-power consumption devices capable of transmitting and encoding
information efficiently [1]. A challenge lies in effectively creating and
controlling magnetic excitations in nanodevices. The most common way to
manipulate magnetization typically involves the use of magnetic fields.
However, this approach has several downsides, including issues with non-
locality, and challenges with integrating controllable magnetic fields into
micrometer-scale devices. Other approaches explore interactions of conducting
electron’s spin [2, 3], lattice phonons [4, 5], or ultrashort light pulses [6]
with magnetic excitations. Promising alternatives have been proposed, such as
the use of strain. The field of straintronics [4] studies strain-induced
effects in solids and special attention is directed to magnetic materials as a
promising alternative for reduced energy-consumption devices [7]. Strain
couples to magnetic states through the magnetoelastic (ME) effect, which is
the change of magnetic properties due to mechanical deformation. This effect
has already been proposed and used in many applications [5] including the
reversing magnetization of patterned nano-magnets [8, 9] or inducing a phase
transition from antiferromagnetic ordering to ferromagnetic ordering [10].

The ME effect can be used to generate spin waves—collective excitations of
magnetic order—through surface acoustic waves (SAWs) [11, 12], which are
strain waves propagating at the surface of a solid. SAWs generate an
oscillating strain in both space and time which can create an oscillating
magnetic anisotropy field on a magnetostrictive material [13, 14]. This
dynamic anisotropy field may induce a magnetization variation with the same
wavelength and frequency as the SAWs and the excitation may propagate up to
millimeter distances [15]. These hybrid waves are referred to as
magnetoacoustic waves (MAWs) and have been studied in a wide range of
materials by measuring acoustic attenuation as a function of the magnetic
state [16, 17, 18, 19, 20] or by direct imaging using magneto-optic Kerr
effect (MOKE) [20, 21] or X-ray magnetic circular dichroism (XMCD) [15, 22].
Using the same principles, more recently also Néel Vector waves have been
observed in antiferromagnetic CuMnAs [23] by X-ray linear dichroism (XMLD).

SAWs are typically generated by means of interdigital transducers (IDTs)
deposited on piezoelectric substrates. IDTs are interlocking arrays of
metallic electrodes that can excite SAWs up to GHz frequencies. IDTs are used
in electronics as delay lines and filters and their integration into
micrometric devices is well-established [24]. SAW-based technology can be
efficiently integrated into low-power devices since the SAW amplitude depends
on the amplitude of the oscillating voltage applied to the IDT instead of the
current.

Magnetic imaging techniques based on MOKE [25, 26] or XMCD [27, 15, 22] bring
the opportunity to directly measure MAWs and thus study the effect of the
coupling between strain and spin waves both in space and in time. Recent
experiments have shown that the ME effect is as efficient in dynamic processes
up to 500 MHz as is in static processes [27], thus fueling the idea of using
SAWs in magnonic applications. One of the key questions in this field is
whether the ME remains efficient at higher frequencies, which are
characteristic of magnetic resonances.

In this paper, we study MAWs at 1 and 3 GHz in nickel and cobalt thin films
using stroboscopic X-ray photoemission electron microscopy (XPEEM) combined
with XMCD imaging. Our analysis reveals that both nickel and cobalt exhibit
MAWs across a wide range of magnetic fields, each material displaying
different profiles and amplitudes. With the help of micromagnetic simulations,
we are able to quantify the strength of the magnetoacoustic coupling for both
magnetic materials at the studied frequencies.

##  II Experimental methods

Figure 1:  (a) Schematic of the experimental setup. The thin film of either
nickel or cobalt is grown in the middle of the acoustic path. The cobalt
sample can only be excited at 1 GHz by either IDT. The microscope (XPEEM
Objective) is operated at 10 kV with respect to the sample to accelerate the
electrons excited by the X-rays. From this setup, we can obtain XPEEM and XMCD
images. The magnetic field is applied in-plane at an angle
$\varphi_{H}=45^{\circ}$ relative to the SAW propagation direction. (b)
Normalized schematics of SAW strain phases, compressive and tensile. The
difference in piezoelectric voltage between each phase is labeled as $V_{pp}$,
from which the strain components can be determined. (c) Representation of the
direction of SAW, magnetic field, and X-ray. In this experiment, the SAW and
X-ray are perpendicular to each other. The horizontal dashed lines represent
the change in magnetization projection captured with the X-rays.  Figure 2:
(a) Nickel $S_{12}$ attenuation at 1 GHz at several angles between the SAW and
the magnetic field showing different absorption values with a maximum close to
45∘ and minimums at 0∘ and above 60∘. The inset shows the S12 attenuation at
positive magnetic fields of the angles 40∘ (black curve) and 30∘ (blue curve).
(b) Cobalt $S_{12}$ attenuation at 1 GHz for several angles between the SAW
and the magnetic field. Cobalt shows absorption when the SAW and the magnetic
field are at an angle between -30∘ and 30∘. The inset shows the S${}_{1}2$
attenuation at positive magnetic fields of the angles 40∘ (black curve) and
30∘ (blue curve).

Thin films of nickel (Ni) and cobalt (Co) with a thickness of 20 nm were grown
on the acoustic path of several SAW delay lines by e-beam evaporation. The
delay lines consist of pairs of unidirectional IDTs deposited on piezoelectric
lithium niobate (128Y-cut LiNbO3) substrates, see Fig. 1 [28]. The IDTs are
designed to generate SAWs with frequencies of 1 GHz and 3 GHz for the Ni
samples and 1 GHz for the Co ones. These frequencies are conveniently selected
to match the synchrotron repetition rate (499.654 MHz), thus allowing for
stroboscopic images. To generate SAWs, we excite the IDTs with an oscillating
electrical signal of the appropriate frequency that is transformed into strain
through the inverse piezoelectric effect [11]. Notice that SAWs are limited to
a substrate depth on the order of the SAW wavelength [29], corresponding to 4
um and 1.32 um for 1 and 3 GHz, respectively. Within this depth region, the
SAW causes a periodic modulation of the crystal lattice in the substrate and
on any thin film deposited on its surface. The SAW propagating through the
hybrid device generates a periodic effective magnetic field in the
ferromagnetic film as a result of the ME effect.

The experiment was performed at the XPEEM experimental station of the ALBA
Synchrotron Light Facility [30]. A schematic representation of the setup is
shown in Fig. 1(a). Details on synchronized SAW stroboscopic XPEEM
measurements and the upgraded setup to support $>500$ MHz excitations can be
found elsewhere [31, 32]. Since the metallicity of Ni and Co shields the
piezoelectric field of the LiNbO3 substrate, only changes in magnetization can
be observed in the areas covered by the ferromagnetic films. To resolve these
changes, two XPEEM images with different light helicity, circular right and
circular left, were subtracted to obtain an XMCD image, in which the contrast
variations are directly related to the in-plane magnetization of the samples.
The insets of Fig. 1(a) display the three imaging modalities of the setup. The
top-left corner depicts an XPEEM image with clear SAWs contrast on the LiNbO3,
accompanied by a weak contrast of magnetic domains on the Co. The top-center
depicts an XMCD image that highlights the magnetic contrast on the Co magnetic
domains, accompanied by faint traces of residual SAWs on the LiNbO3 due to
thermal drifts in subsequent images (and thus different SAW propagation
velocities). The top-right corner depicts a 2-phase-XMCD image, which consists
of capturing two XPEEM images with a 180-phase shift in SAW for each X-ray
helicity, followed by a subtraction between both phases and helicities. This
method enhanced both the magnetic and piezoelectric contrast while eliminating
static contrast such as magnetic domains, enabling a clearer measurement of
MAWs. Figure 1(b) shows the two opposite phases of the SAW, tensile and
compressive. Figure 1(c) shows the directions of the SAW, the magnetic field,
and the X-rays. The SAW and the magnetic field are at 45∘ with respect to each
other to exert maximum magnetoelastic torque, whereas the SAW and X-rays are
perpendicular to each other in order to detect the magnetization changes in
the $y$-direction as represented by the dashed lines.

##  III Results

Figure 3:  (a) XMCD image of Ni showing the MAW driven at 1 GHz. (b) XMCD
image of Ni showing the MAW driven at 3 GHz. (c) Experimentally determined MAW
profiles at 1 GHz (black curve) and 3 GHz (green curve). the red and blue
curves are fittings to a sinusoidal function. (d) Dependence of the MAW
amplitude on the external magnetic field. The left axis represents the
excitation at 1 GHz, while the right axis represents the excitation at 3 GHz.

An initial characterization of the samples was performed to study the angular
dependence of their acoustic-ferromagnetic resonance (a-FMR) at 1 GHz. The
experiment consists in placing our hybrid device (see, Fig. 1(a)) between the
poles of an electromagnet and measuring the transmitted acoustic signal,
$S_{12}$, with a vector network analyzer at different angles between the SAWs
and the in-plane magnetic field. In contrast to FMR, where an rf-magnetic
field is used to drive the magnetization into resonance, a-FMR uses the
periodic strain field of the SAW to induce magnetization precession by
modulating the effective magnetic field of the sample. Additionally, the non-
zero wave vector of the SAW has to be taken into account in the determination
of the resonant magnetic fields and frequencies [17].

The coupling between magnetization and SAWs can generate MAWs under certain
conditions of the external magnetic field. At resonance, significant energy is
transferred from the SAW to the magnetic system, resulting in SAW attenuation
and phase shift that are detected in the S21 measurement [16]. The SAW
attenuation results are presented in Fig. 2 as a function of the applied field
magnitude and angle (in both cases, the magnetic field was swept from negative
to positive values). In both samples, a large attenuation is observed,
regardless of the angle, at a fixed magnetic field that corresponds to the
magnetization-switching field of the film. In addition, Ni (Fig. 2(a)) shows
the typical four-fold shape with large attenuation between $\pm$30∘ and
$\pm$50∘, and a maximum around 45∘ where the ME torque is the largest [33, 16,
34]. As the angle between SAW and the magnetic field exceeds 50∘ or drops
below 30∘, the attenuation decreases, reaching the minimum ME torque at 0∘ and
90∘. In contrast, the SAW attenuation for Co (Fig. 2(b)) is strongest at
magnetic field angles -30∘ and 30∘, above which the attenuation rapidly
decays. Similar results for Co can be found in Ref. [35].

Prior to capturing XMCD images of the ferromagnetic sample, the IDTs are
tested by imaging the piezoelectric component of SAWs on the LiNbO3 substrate
using XPEEM. By adjusting the bias voltage applied to the sample, a secondary
photoelectron energy spectrum for each SAW phase (compressive and tensile) can
be obtained, as done in Ref. [31]. The resulting difference between spectrums
corresponds to the peak-to-peak piezoelectric voltage, $V_{\textrm{pp}}$. By
solving the coupled elastic and electromagnetic equations, we can obtain the
amplitude of the strain components, $e_{\textrm{xx}}$, $e_{\textrm{xz}}$ and
$e_{\textrm{zz}}$, associated with the SAW images, which are listed in Table
1. It is important to note that the strain components may vary depending on
SAW frequency. Thus, we measured the strain component in each sample at the
same SAW excitation frequency and power that are used later on for
magnetization imaging (see Table 1)

Table 1: Amplitudes of the SAW strain components for each ample and SAW frequency Sample | $e_{xx}(10^{-5})$ | $e_{xy}(10^{-5})$ | $e_{zz}(10^{-5})$ |
---|---|---|---|---
Ni (1 GHz) | 14.3 | 2.95 | 4.23 |
Ni (3 GHz) | 19.0 | 3.93 | 5.64 |
Co (1 GHz) | 16.6 | 3.44 | 4.93 |

In the following sections, we show the results obtained from the XMCD imaging,
and discuss the amplitude of the MAWs for the different samples and excitation
frequencies.

###  III.1 MAWs in Nickel

The Ni sample was studied for two SAW frequencies: 1 and 3 GHz. Figures 3(a)
and (b) show 2-phase XMCD images of excitations of both signals (1 and 3 GHz
respectively). The bottom part of the images correspond to LiNbO3 and displays
the SAWs and the top part corresponds to Ni and shows the MAWs as indicated.
Both images are taken with the same field-of-view ($20\times 20$ $\mu$m) and
the same color scale, to be able to distinguish the wavelength and the
contrast difference. Fig. 3(c) shows the MAW profile of both frequencies with
their corresponding sinusoidal fit to showcase the large amplitude difference
(as well as the difference in wavelength). A summary of the normalized MAW
amplitudes at different magnetic fields is presented in Fig. 3(d) for 1 GHz
(black symbols, left axis) and 3 GHz (orange symbols, right axis). The overall
magnetization precession amplitude at 1 GHz is one order of magnitude larger
than that of 3 GHz while the measured strain amplitude at the substrate was
similar for the two frequencies; this will be explored in more detail in Sec.
IV.

###  III.2 MAWs in Cobalt

Figure 4:  (a) XMCD image showing the piezoelectric, LiNbO3, and the
ferromagnetic sample, cobalt, as indicated at an applied magnetic field of 5
mT. The Co displays two big magnetic domains. (b) Experimentally determined
MAW profile of the 5 mT XMCD image of a 1 GHz excitation (black curve) with
the corresponding sinusoidal fit (red curve). (c) MAW amplitude as a function
of the applied magnetic field excited by a 1 GHz SAW.

The Co sample was only studied at 1 GHz. Figure 4(a) displays a representative
2-phase XMCD image with a field-of-view of $50\times 50$ $\mu$m where clear
SAWs in the LiNbO3 and MAWs in the Co at 1 GHz are observed. The profile of
this MAW is shown in Fig. 4(b) with the corresponding sinusoidal fit. Figure
4(c) collects results from different magnetic field values, with no clear
trend and with an observable difference in amplitude for opposite direction of
magnetic field, which could be the result of the magnetorotation coupling [36,
37, 38]. We notice here that the studied range of fields is small and one may
hint a slight amplitude increase close to $\mu_{0}H=\pm 7$ mT. The amplitude
is two orders of magnitude smaller than Ni at 1 GHz (and even one order
smaller than Ni at 3 GHz). We notice here that in both experiments the applied
magnetic field formed an angle of $\sim 30-45^{\circ}$ with respect to the
SAWs which could be outside the large attenuation region for Co (see Fig. 2b);
we will discuss it in Sec. IV.

##  IV Discussion

The experimental techniques employed here measured different quantities; a-FMR
determines the attenuation of SAW energy under MAW resonance, while XMCD is a
direct representation of the magnetization dynamics in the ferromagnetic film.
By comparing our results with micromagnetic simulations we can determine the
efficiency of converting strain into a magnetization variation, thus,
obtaining the magnetoelastic constants.

The experimental results demonstrate a notable consistency in the behavior of
Ni and Co at 1 GHz in both a-FMR and XMCD. Starting with Ni, when placed under
a magnetic field at $\varphi_{H}=$ 30-45∘ relative to the SAWs, the effect of
the ME coupling is the strongest at approximately $\pm$4 mT and it decays as
the magnetic field strength increases. However, while in a-FMR minimal
attenuation is found beyond a magnetic field of 10 mT, XMCD still shows the
presence of MAWs up to 20 mT. In the case of Co, the magnetic field that we
could apply in XMCD imaging did not exceed 7 mT, limiting our knowledge at
higher fields. However, a peak close to 5 mT is observed in both a-FMR and
XMCD experiments, with a decrease in amplitude when approaching 0 mT. Note
that the amplitude of the MAWs of Co is one and two orders of magnitude lower
than nickel at 3 and 1 GHz, respectively (also seen in acoustic attenuation in
Fig. 2). This phenomenon could be due to the high magnetization saturation of
Co, which is three times larger than that of nickel. At 3 GHz frequency, XMCD
data of Ni displays a similar trend with applied magnetic field as for the 1
GHz data, with a maximum around 10 mT.

As already discussed in Section III, we determined the strain amplitude for
each frequency and sample using XPEEM images of SAW on LiNbO3 and employing
the same power settings as in the MAW imaging, see Table 1. We observed
similar relative values between the strain components for both samples at 1
GHz since identical IDTs were used on the same LiNbO3 substrates. Notably, we
also found that the strain at the sample surface measured at 3 GHz in the Ni
sample were comparable and, even, slightly larger than those at 1 GHz. Thus,
the observed differences in MAW amplitudes cannot be explained with strain
values. Therefore, we performed micromagnetic simulations to evaluate the
overall internal magnetic fields and determine the expected MAW amplitudes at
each frequency for both materials. We used MuMax3 [39], which takes into
account the free energy of different magnetic contributions, e.g. Zeeman,
dipolar, exchange, and magnetoelastic to calculate an effective magnetic
field. The contributions of each effect depend on the parameters of the
experiment (SAW frequency and wavelength, external magnetic field,
magnetization saturation). In particular, for the ME effect, the following
free-energy expression is used.

$\displaystyle F_{\text{me}}=$
$\displaystyle\,\,B_{1}\left[\varepsilon_{xx}m_{x}^{2}+\varepsilon_{yy}m_{y}^{2}+\varepsilon_{zz}m_{z}^{2}\right]$
$\displaystyle+2B_{2}\left[\varepsilon_{xy}m_{x}m_{y}+\varepsilon_{xz}m_{x}m_{z}+\varepsilon_{yz}m_{y}m_{z}\right],$
(1)

where $B_{1}$ and $B_{2}$ are the magnetoelastic constants and are taken as
parameters in our simulations, $m_{i}$ are the normalized magnetization
components and $\varepsilon_{ij}$ are the strain components simulated as plane
waves with the following expression:

$\varepsilon_{ij}=e_{ij}\exp(kx-\omega t),$ (2)

where $e_{ij}$ are the strain amplitudes of Table 1, $k$ is the wave vector of
the SAW traveling in the x direction, and $\omega=2\pi f$ is the angular
frequency of the SAWs. Since the SAW excited in the LiNbO3 substrate is a
Rayleigh mode, only the strain amplitudes $e_{xx}$, $e_{zz}$, and $e_{xz}$ are
non-zero in our experiment.

Sample |  $M_{s}$ (kA/m) | $A_{ex}(J/m)$ | $\alpha$ |
---|---|---|---|---
Ni | 490 | $10^{-11}$ | 0.03 |
Co | 1510 | $10^{-11}$ | 0.007 |
Table 2: Sample properties used in the micromagnetic simulations. $M_{s}$ is
the saturation magnetization, $A_{ex}$ the exchange and $\alpha$ the Gilbert
damping. Figure 5: (a) Amplitude of MAW in Ni for excitation frequencies of 1
and 3 GHz with $\varphi_{H}=45^{\circ}$ (solid black and orange curves,
respectively). The dashed lines provide the same information for a
$\varphi_{H}=30^{\circ}$. The magnetoelastic constants used are written in the
same color as the simulation curves. (b) Amplitude of the MAW in Co for an
excitation frequency of 1 GHz. Each color represents a different magnetic
field angle, with the arrow indicating the increase in the angle.

The simulations consist of sweeping the magnetic field gradually from 20 to 0
mT, mirroring the experimental procedure, and recording the magnetization
values in time and space in presence of SAW. The choice of larger to lower
magnetic fields is made to avoid magnetic switching and the formation of
magnetic domains. The results in Fig. 5 show the amplitude of the MAWs
determined by calculating the mean oscillating amplitude over time for each
magnetic field. The results for Ni are presented in Fig. 5 (a) with MAW-
amplitude for 1 GHz (black curves) represented on the left $y$-axis, and for 3
GHz (orange curves) represented on the right $y$-axis. Both frequencies were
simulated at two different angles between the SAWs and the magnetic field
($\varphi_{H}$), at 45∘ (continuous lines) and 30∘ (dashed lines), to account
for possible misalignment errors, and different magnetoelastic constants to
match the experimental results. A first observation is that the maximum
amplitude of MAWs shifts towards larger fields with increasing frequency—which
has to do with ferromagnetic resonance fields that increase with excitation
frequency. Another noticeable effect is the shift to lower fields of MAW
maximum amplitude with increasing $\varphi_{H}$, which has previously been
observed in a-FMR experiments [16, 17], and can be clearly seen in Fig. 5 (a)
for the 3 GHz curves. Figure 5 (b) illustrates the MAW amplitude of the Co
sample when subjected to a 1 GHz excitation for different angles,
$\varphi_{H}$, ranging from 10∘ to 40∘ (from blue to green curves). The
misalignment between MAW and applied field has a more drastic impact on Co
reducing the overall MAW amplitude. This occurs because the resonant frequency
is shifted above the SAW frequency as $\varphi_{H}>10^{\circ}$, resulting in
no resonance.

We now focus on the experimental results showing a drop in efficiency at 3 GHz
for Ni (see, Fig. 3). To account for the lower MAW amplitude, we varied the
magnetoelastic constant, $B_{1}$, until the MAW amplitude matched the
experimental values. For a 1 GHz excitation, a magnetoelastic constant of
$B_{1}=4$ MJ/m3 matched the experimental amplitude, while for a 3 GHz
excitation, a reduced magnetoelastic constant of $B_{1}=0.3$ MJ/m3 was needed.
This loss in efficiency appeared unexpected to us, given that experiments are
done in the same Ni layer and that physical processes involved in magneto-
elasticity are expected to respond easily at the nanosecond scale. We notice
here that the measured strain amplitudes were obtained at the piezoelectric
surface and not at the ferromagnetic film. It could be possible that the
reduced magnetic response at 3 GHz is related to a lower transmission of the
SAW strain to the polycristalline thin layer under large SAW frequencies and
short wavelengths. We measured AFM in our Ni films and the observed grains at
the top of the film were not larger than a few nanometers, a value much
shorter than the SAW wavelengths used in our experiment. However, the
interface between LiNbO3 and Ni has not been characterized.

Table 3: Magnetoelastic constant values for each sample. Sample |  $B_{1}$ (MJ/m3) at $\varphi_{H}$=45∘ |
---|---|---
Ni (1 GHz) | 4 |
Ni (3 GHz) | 0.3 |
Co (1 GHz) | 2 |

Overall, the results of our simulations agree quantitatively well with XMCD
and a-FMR experiments for both Ni and Co. At 1 GHz SAW-excitation we obtained
a value of $B_{1}$ for Ni of $\sim 4$ MJ/m3, comparable to previous
experimental studies [40, 41, 27, 42, 15]. In Co, we obtained a magnetoelastic
constant of $\sim 2$ MJ/m3, which is 2 times smaller than the Ni one at 1 GHz.
However, the simulations indicate that Co at smaller angles may achieve
amplitudes approximately equal to 0.1, comparable to Ni, which could make Co
an interesting material for straintronics, due to its low damping and high
magnetization saturation. At 3 GHz SAW excitation, we found a reduction in the
efficiency for the MAWs in Ni that requires decreasing the magnetoelastic
constant (to $\sim 0.3$ MJ/m3) approximately a factor of 10 compared with 1
GHz excitation. Although our model does not account for direct transfer of
phonon angular momentum to magnetization [33, 43], which might be dependent on
frequency, we consider that a plausible explanation for the drop in efficiency
at 3 GHz could be related to a frequency dependence of the strain transfer
efficiency between LiNbO3 and the Ni film.

##  V Conclusions

We have studied the magnetoelastic effect in Ni and Co at 1 and 3 GHz using
hybrid piezoelectric/ferromagnetic devices and SAWs. Our direct XPEEM-XMCD
imaging shows that large-amplitude MAWs in Ni and Co at frequencies of 1 and 3
GHz can be directly imaged and quantified. We found a significant difference
in MAW amplitude between Ni and Co at 1 GHz excitations; a difference that
cannot be explained only by the larger magnetization saturation value of Co
compared with the value of Ni and thus a lower value of magnetoelastic
constant is obtained for Co. The found values for the magnetoelastic constants
at 1 GHz are similar to those reported in the literature [27, 42, 15, 41, 40].
Additional measurements at 3 GHz SAW excitations show a clear decay in MAW
efficiency suggesting a drop in the phonon coupling between SAW at the
piezoelectric substrate and SAW at the ferromagnetic thin film. Our study
provides valuable insights into the coupling between strain and magnetization
dynamics at the GHz frequency and may guide the development of more efficient
acoustic spintronic (straintronic) devices.

###### Acknowledgements.

The authors would like to thank N. Volkmer for their technical support in the
preparation of the samples. MWK acknowledges Marie Skłodowska-Curie grant
agreement No. 754397 (DOC-FAM) from EU Horizon 2020. MR, BC, JMH, and FM
acknowledge funding from MCIN/AEI/10.13039/501100011033 through grant number:
PID2020-113024GB-100. MWK, MF, and MAN acknowledge funding from MCIN through
grant number: PID2021-122980OB-C5. LA, MF and SRG acknowledge funding from
MCIN through grant number: RTI2018-095303. SRG also acknowledges funding from
Marie Sklodowska-Curie grant number 101061612.

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