# Acoustically driven magnon-phonon coupling in a layered antiferromagnet

Thomas P. Lyons  Center for Emergent Matter Science, RIKEN, Wako-shi, Saitama
351-0198, Japan  Jorge Puebla<EMAIL_ADDRESS>Center for
Emergent Matter Science, RIKEN, Wako-shi, Saitama 351-0198, Japan  Kei
Yamamoto  Advanced Science Research Center, Japan Atomic Energy Agency, Tokai,
Ibaraki 319-1195, Japan  Center for Emergent Matter Science, RIKEN, Wako-shi,
Saitama 351-0198, Japan  Russell S. Deacon  Advanced Device Laboratory, RIKEN,
Wako-shi, Saitama 351-0198, Japan  Center for Emergent Matter Science, RIKEN,
Wako-shi, Saitama 351-0198, Japan  Yunyoung Hwang  Center for Emergent Matter
Science, RIKEN, Wako-shi, Saitama 351-0198, Japan  Institute for Solid State
Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan  Koji Ishibashi
Advanced Device Laboratory, RIKEN, Wako-shi, Saitama 351-0198, Japan  Center
for Emergent Matter Science, RIKEN, Wako-shi, Saitama 351-0198, Japan
Sadamichi Maekawa  Center for Emergent Matter Science, RIKEN, Wako-shi,
Saitama 351-0198, Japan  Advanced Science Research Center, Japan Atomic Energy
Agency, Tokai, Ibaraki 319-1195, Japan  Kavli Institute for Theoretical
Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China  Yoshichika Otani  Center for Emergent Matter Science,
RIKEN, Wako-shi, Saitama 351-0198, Japan  Institute for Solid State Physics,
University of Tokyo, Kashiwa, Chiba 277-8581, Japan

###### Abstract

Harnessing the causal relationships between mechanical and magnetic properties
of van der Waals materials presents a wealth of untapped opportunity for
scientific and technological advancement, from precision sensing to novel
memories. This can, however, only be exploited if the means exist to
efficiently interface with the magnetoelastic interaction. Here, we
demonstrate acoustically-driven spin-wave resonance in a crystalline
antiferromagnet, chromium trichloride, via surface acoustic wave irradiation.
The resulting magnon-phonon coupling is found to depend strongly on sample
temperature and external magnetic field orientation, and displays a high
sensitivity to extremely weak magnetic anisotropy fields in the few mT range.
Our work demonstrates a natural pairing between power-efficient strain-wave
technology and the excellent mechanical properties of van der Waals materials,
representing a foothold towards widespread future adoption of dynamic magneto-
acoustics.

Figure 1:  Magnon-phonon coupling in layered CrCl3. (a) Schematic of the
devices used in this work. See text for description. (b) CrCl3 consists of
stacked ferromagnetic layers of alternating in-plane magnetization,
represented by two spin sublattices (green and blue arrows). In the absence of
an external magnetic field, the sublattice magnetizations point away from each
other, while an applied field causes them to cant. In-phase and out-of-phase
precession of the sublattice magnetizations are associated with acoustic and
optical magnon modes, respectively. (c) SAW transmission signal through CrCl3
in Sample 1 as a function of applied magnetic field strength at an angle
$\phi=45\degree$, at various sample temperatures. (d) Extracted resonance
field strengths for the acoustic and optical magnon modes at various Sample 1
temperatures. Overlaid curves are calculated from the model described in the
text. (e) Calculated frequency dependence of the acoustic and optical magnon
modes as a function of applied magnetic field, at $T=4,6$ and 8 K. Figure 2:
Acoustic magnon mode dependence on external field angle and temperature (a-f)
Polar plots of SAW absorption by the acoustic magnon mode in Sample 2, at
various sample temperatures, as a function of applied external magnetic field
orientation in the sample plane. Asymmetry at lower temperatures arises due to
very weak uniaxial anisotropy $\sim 2$ mT. Upon heating, the expected
symmetric response of the magnetoelastic interaction is recovered. Absorption
disappears at $T=14$ K, close to the Néel temperature.  Figure 3:  Theoretical
model for acoustic mode (a) Calculated acoustic mode frequency dependence on
external magnetic field orientation $\phi$. (b, c) Simulated polar plots of
SAW absorption by the acoustic magnon as a function of external magnetic field
orientation, using parameters for Sample 2. The striking difference in
response is largely attributed to a change in anisotropy of only $\sim 1$ mT.

From uncertain beginnings, the technological advantages of antiferromagnets
over ferromagnets are now well known, including fast operation, immunity
against device crosstalk and stray fields, and amenability to low power
control via spin currents or proximitized materials Jungwirth  _et al._
(2016); Fukami  _et al._ (2020). However, these very same advantageous
properties can be a double-edged sword, being partly responsible for a general
lack of understanding of antiferromagnets as compared to ferromagnets. The
high spin-wave frequencies can be prohibitive for probes based on microwave
electronics, while the insensitivity to measurement techniques such as SQUID
magnetometry or the magento-optical Kerr effect limit the effectiveness of
these popular conventional magnetic probes. A less well-known probe, which has
proven itself useful in the study of ferromagnets, relies not on optical or
direct magnetic sensing but instead employs the magnetoelastic interaction
between spin-waves and acoustic waves Puebla  _et al._ (2020); Yamamoto  _et
al._ (2022). When in contact with a piezoelectric material, the magnetic film
can be irradiated with surface acoustic waves (SAWs). Beyond the magnetic
film, the transmitted SAWs can be measured, providing information on the
magnet’s response to external stimuli Puebla  _et al._ (2020); Xu  _et al._
(2020). Aside from the energy efficient generation, inherently low
attenuation, suitability for miniaturization and long distance propagation of
SAWs Dreher  _et al._ (2012); Nie  _et al._ (2022); Puebla  _et al._ (2020), a
particular advantage of this technique is that it does not discriminate
between ferromagnetic and antiferromagnetic order, and indeed may even be
stronger for the latter Yamada  _et al._ (1966).

SAW technology is relatively mature, having found multiple applications in the
microelectronics industry, yet continues to play a key role at the forefront
of fundamental research, with recent notable advances including SAW-driven
transport of single electrons in gallium arsenide Wang  _et al._ (2022),
semiconductor interlayer excitons in van der Waals heterobilayers Peng  _et
al._ (2022), and manipulation of the charge density wave in layered
superconductors Yokoi  _et al._ (2020), amongst other advances Nie  _et al._
(2022). Utilizing SAWs as a probe of ferromagnetism has proven highly
effective, for instance in understanding the fundamentals of magnetoelasticity
and magnetostriction, or more recently in revealing the various mechanisms of
SAW nonreciprocity Xu  _et al._ (2020); Puebla  _et al._ (2020); Sasaki  _et
al._ (2017); Hernández-Mínguez  _et al._ (2020); Verba  _et al._ (2019); Küß
_et al._ (2020, 2021); Shah  _et al._ (2020). Such works have laid the
foundations for the active field of SAW-spintronics, in which dynamically
applied strain can modulate magnetic properties Dreher  _et al._ (2012);
Puebla  _et al._ (2022). This technique is mature for ferromagnets, and has
recently been proven effective for multiferroics Sasaki  _et al._ (2019) and
synthetic antiferromagnets Matsumoto  _et al._ (2022); Verba  _et al._ (2019),
but a demonstration of SAW-driven magnon-phonon coupling in a crystalline
antiferromagnet remains elusive.

Here, we utilize SAWs to drive spin-wave resonance in a layered crystalline
antiferromagnet, chromium trichloride (CrCl3), a material characterized by
layers of alternating magnetization weakly bound by van der Waals attraction
McGuire  _et al._ (2017); Narath and Davis (1965). The antiferromagnetic order
occurs only between adjacent monolayers rather than within them, giving rise
to relatively weak interlayer exchange and associated lower frequency range of
spin excitations in CrCl3 as compared to conventional antiferromagnets
Macneill  _et al._ (2019); McGuire  _et al._ (2017). The combination of easy
flake transfer onto arbitrary substrates, with sub-10 GHz spin excitations, is
advantageous for integration of CrCl3 into SAW devices, where
antiferromagnetic magnetoelasticity can be probed directly. After first
demonstrating acoustic antiferromagnetic resonance, we proceed to study the
influence of temperature and angle of applied external magnetic field on the
magnon-phonon coupling. The sets of experimental data are analyzed by
extending the established theoretical model for SAW-spin wave coupling in
ferromagnetic films Yamamoto  _et al._ (2022); Xu  _et al._ (2020). Combined
with a mean-field calculation of the temperature dependence, our model
reproduces the observed features well, confirming the amenability of SAWs as a
powerful probe to elucidate the dynamics of van der Waals magnets, especially
given their excellent plasticity Cantos-Prieto  _et al._ (2021). Considering
also that acoustic magnetic resonance generates spin currents, which have been
shown to travel over long distances in antiferromagnets, our results offer an
alternative route towards novel spintronic devices with layered crystals
Awschalom  _et al._ (2021); Lebrun  _et al._ (2018); Burch  _et al._ (2018);
Ghiasi  _et al._ (2021).

Two devices are studied in this work. They each consist of lithium niobate
(LiNbO3) substrates with aluminium interdigital transducers (IDTs) either side
of a CrCl3 flake (Fig. 1a). Each IDT, 1 or 2, can generate SAWs at 1.1 GHz and
wavelength 3.2 $\upmu$m, which subsequently propagate along the surface of the
LiNbO3, interact with the CrCl3 flake, and then reach the other IDT where they
are detected. By measuring SAW transmission in this way, any absorption of
acoustic energy by the antiferromagnet can be detected (see methods). Sample 1
is quasi-bulk, at $\sim 4\;\upmu$m thick, while Sample 2 is much thinner at
$\sim 120$ nm (see Supplementary Information (SI)).

Below the Néel temperature of $\sim 14$ K, layered CrCl3 is composed of
stacked ferromagnetic layers ordered antiferromagnetically McGuire  _et al._
(2017); Narath and Davis (1965). Alternate layers belong to one of two spin
sublattices oriented collinearly in the layer plane, owing to easy plane
anisotropy of strength $\sim 250$ mT (Fig. 1b) McGuire  _et al._ (2017). Two
magnon modes arise from in-phase or out-of-phase precession of the two
sublattice macrospins, described as acoustic and optical modes, respectively
Macneill  _et al._ (2019). In our experiments we apply an external magnetic
field perpendicular to the crystal c-axis, inducing the two spin sublattice
magentizations to cant towards the applied field direction (Fig. 1b). Such
noncollinear canting modifies their precession frequency, thereby bringing the
magnon modes into resonance with the acoustic wave.

We first apply an external magnetic field at an angle $\phi=45\degree$ to the
SAW propagation direction in Sample 1, and measure the amplitude of the SAW
transmission. The result is shown in Fig. 1c, where clear transmission dips
can be seen arising from absorption of SAWs by magnons. At $T=6$ K, absorption
is observed at approximately 30 and 150 mT, attributed to the acoustic and
optical modes, respectively. Examples of other external field orientations can
be seen in the SI. Upon heating the sample, the optical mode absorption shifts
to lower resonance field strengths while the acoustic mode stays largely
insensitive to temperature (Fig. 1d). At $T=13$ K, the two modes are no longer
resolved, and at $T=14$ K, close to the Néel temperature McGuire  _et al._
(2017), they have disappeared.

The observed temperature dependence of the resonance field can be modelled by
combining a simple mean-field theory with the known formulae for spin wave
resonance in easy-plane antiferromagnets Macneill  _et al._ (2019)

$H_{\rm res}=\begin{cases}\sqrt{2H_{E}/(2H_{E}+M_{s})}\omega/\gamma&{\rm
acoustic}\\\ \sqrt{4H_{E}^{2}-2H_{E}\omega^{2}/(M_{s}\gamma^{2})}&{\rm
optical}\end{cases}$ (1)

Here $H_{E}$ is the interlayer exchange field, $M_{s}$ is the saturation
magnetization, $\omega$ is the SAW frequency, and $\gamma/2\pi=28$ GHz/T is
the gyromagnetic ratio respectively. We solve the molecular field equation
self-consistently in the macrospin limit $S\rightarrow\infty$ to obtain
$M_{s}(T)$. This approximation also implies $H_{E}(T)\propto M_{s}(T)$, which
predicts the optical mode resonance field tends towards zero as the Néel
temperature is approached while the acoustic mode remains unchanged. The
calculated temperature dependence is plotted in Fig. 1d and agrees well with
the experimental data. The small increase of the observed acoustic mode
resonance field towards higher temperature Zeisner  _et al._ (2020) points to
breakdown of the mean-field approximation near the phase transition. The same
model can be used to calculate the effective magnon frequency evolution as a
function of applied magnetic field strength, as shown in Fig. 1e.

We now consider the coupling between SAWs and the acoustic magnon mode in
greater detail. Figure 2 shows absorption by the acoustic mode as a function
of external magnetic field orientation in the plane of Sample 2, where the
vertical axis (0$\degree$ \- 180$\degree$ line) is the SAW propagation axis.
At $T=4.2$ K, we observe four lobes of strong absorption, seen only when the
external magnetic field is applied at angles smaller than 45$\degree$ to the
SAW propagation axis. As the temperature is increased to $T=12$ K, they
migrate to new positions which are more rotationally symmetric. By $T=14$ K,
close to the Néel temperature, the absorption has disappeared, in agreement
with Sample 1.

To fully understand the results in Fig. 2, we must consider the interplay
between antiferromagnetic resonance and magnon-SAW coupling. Each has its own
dependence on external magnetic field orientation, with the latter defining
the window through which we can observe the former. Firstly we focus on the
magnetic response of CrCl3 itself. Close inspection of Fig. 2a reveals that
not only the magnitude of absorption but also the resonance field depends
strongly on the magnetic field angle $\phi$ at $T=4.2$ K, indicating the
presence of magnetic uniaxial anisotropy. To reproduce this observation, we
calculate the acoustic mode resonance frequency as a function of $\phi$
computed for a model that includes an in-plane uniaxial anisotropy field
$\mu_{0}H_{u}\approx 2.1$ mT, oriented approximately along the line
171$\degree$ \- 351$\degree$. Although this anisotropy is itself very weak, it
induces a sizable zero-field magnon frequency gap of
$\gamma\mu_{0}\sqrt{2H_{u}(2H_{E}+M_{s}+H_{u})}\sim 1.2$ GHz, above the SAW
frequency of 1.1 GHz. As can be seen at $T=4$ K in Fig. 3a, for
$30\degree\lesssim\phi\lesssim 130\degree$ and $210\degree\lesssim\phi\lesssim
310\degree$, the frequency monotonically increases as $H$ increases so that
the acoustic magnon never becomes resonant with the SAWs. Only in the
remaining angular ranges are acoustic spin-wave resonances observable, which
correspond to the lobes in Fig. 2a.

According to the well-known formula $H_{u}(T)\propto M_{s}(T)^{2}$ Callen and
Callen (1966), the uniaxial anisotropy tends to zero as $T$ increases towards
the Néel point. We find it reduces to $\approx 0.6$ mT at $T=12$ K, lowering
the zero-field magnon frequency below the SAW frequency, and thereby allowing
acoustic magnon resonance at 1.1 GHz for all angles at around $25-30$ mT (Fig.
3a). While uniaxial anisotropy of $\sim 1$ mT has been observed before in
CrCl3 Kuhlow (1982), the origin remains ambiguous. Here, we tentatively
ascribe it to negative thermal expansion in CrCl3, in which the $a$-axis
lattice constant gradually increases upon cooling the crystal below $T=50$ K,
owing to magnon induced expansion of the lattice Schneeloch  _et al._ (2022);
Liu  _et al._ (2022). Our results hint at the applicability of SAWs to further
investigate this poorly understood effect, or moreover exploit it for highly
sensitive static strain or force sensing applications.

To complete the picture, we now consider the magnon-SAW coupling dependence on
external field orientation, which has proven the key to accessing various
parameters in ferromagnetic materials Xu  _et al._ (2020). Given that, unlike
ferromagnets, the antiferromagnetic sublattice magnetizations do not simply
align with the external field, we model the magnetoelastic coupling in CrCl3
by a free energy density $F_{\rm
me}=b\epsilon_{ab}(n_{a}^{A}n_{b}^{A}+n_{a}^{B}n_{b}^{B})+2c\epsilon_{ab}n_{a}^{A}n_{b}^{B}$.
Here $\epsilon_{ab}$ is the strain tensor, $n_{a}^{A},n_{a}^{B}$ are
components of the normalized sublattice magnetization vectors, and Einstein’s
summation convention is assumed. $b$ is an intrasublattice magnetoelastic
coefficient, a direct generalization of the ferromagnetic magnetoelasticity.
$c$ is an intersublattice coefficient, unique to antiferromagnets, which was
studied in literature Borovik-Romanov and Rudashevskii (1965). Let
$\phi_{A},\phi_{B}$ be the angles between the SAW propagation direction and
the respective sublattice magnetizations. The corresponding magnon-SAW
couplings $g_{A},g_{B}$ exhibit the following angle dependence (see SI):

$\displaystyle g_{A}\propto$ $\displaystyle\
b\sin\phi_{A}\cos\phi_{A}+c\sin\phi_{A}\cos\phi_{B},$ (2) $\displaystyle
g_{B}\propto$ $\displaystyle\
b\sin\phi_{B}\cos\phi_{B}+c\sin\phi_{B}\cos\phi_{A}.$ (3)

The acoustic and optical modes see $g_{A}\pm g_{B}$ respectively, reflecting
the phase relations between the two sublattices. For acoustic mode resonance,
$H$ is small so that $\phi_{B}\approx\phi_{A}+\pi\approx\phi\pm\pi/2$,
yielding $g_{A}+g_{B}\propto\sin 2\phi$. This acoustic magnon-SAW coupling
filters the nominally observable resonance frequencies shown in Fig. 3a to
give the cumulative responses shown in Fig. 3b, c, in which vanishing
absorption can be seen at $\phi=0\degree,90\degree,180\degree,270\degree$. The
agreement with Fig. 2a, e is satisfactory.

Next, we consider optical magnon-phonon coupling. Figures. 4a, b show the
optical mode absorption in Sample 2, seen to some extent at every angle of
applied field. This isotropic behaviour, in stark contrast to that displayed
by the acoustic mode, arises because the two canted spin sublattices adopt an
almost parallel configuration at the relatively high field strength needed to
reach resonance, i.e.
$\phi_{A}\approx\phi+\delta,\phi_{B}\approx\phi-\delta,|\delta|\ll\pi$.
Equations (2) and (3) therefore yield $g_{A}-g_{B}\propto\left(b\cos
2\phi+c\right)\sin 2\delta$. We note that the intrasublattice coupling $b$
alone gives a vanishing absorption at $\phi=45\degree$, inconsistent with both
Sample 1 (Fig. 1c) and Sample 2 (Fig. 4a, b). Hence we take $b=0,c\sim 10^{6}$
J/m3 with the aforementioned temperature dependent $H_{E},M_{s},H_{u}$ to
generate Figs. 4c, d, which show the simulated optical mode absorption at
$T=4$ K and 13 K, respectively. The agreement with experiment is satisfactory
at $T=4.2$ K, and reasonable at $T=13$ K, given the simplifications to the
model (such as an absence of broadening/disorder) and the expected breakdown
of the mean-field approximation close to the phase transition.

Figure 4:  Optical magnon mode dependence on external field angle and
temperature (a,b) Experimental and (c,d) simulated polar plots of SAW
absorption by the optical mode in Sample 2 as a function of external magnetic
field orientation at $T=4$ K (experimental base temperature $T=4.2$ K) and 13
K.

In conclusion, we demonstrate GHz-range SAW-driven magnon-phonon coupling in a
crystalline antiferromagnet. This demonstration paves the way towards
acoustically driven spintronic devices based on designer van der Waals
heterostructures, which may combine antiferromagnetic, semiconducting,
metallic and insulating layers to realise diverse outcomes in spin conversion
Otani  _et al._ (2017); Ghiasi  _et al._ (2021). Moreover, it has been
proposed that monolayer CrCl3 exhibits true 2D XY-ferromagnetism, allowing
study of the Berezinskii–Kosterlitz–Thouless phase transition Kosterlitz and
Thouless (1973), and predicted to play host to topological spin textures
Bedoya-Pinto  _et al._ (2021). Creation and manipulation of such excitations
by SAWs is a tantalising prospect, as has been recently achieved in
conventional ferromagnetic systems Yokouchi  _et al._ (2020).

##  I Methods

###  I.1 Sample fabrication

First, IDTs (35 nm aluminium) and electrodes (5 nm titanium / 200 nm gold) are
deposited onto 128$\degree$ Y-cut LiNbO3 chips. The IDT fingers are 400 nm
wide with 1.2 $\upmu$m spacing, giving a SAW wavelength 3.2 $\upmu$m and
frequency 1.1 GHz. The distance between IDT1 and IDT2 is approximately 600
$\upmu$m. Next, bulk CrCl3 is exfoliated onto polydimethylsiloxane (PDMS)
sheets using sticky tape (Nitto). Flakes with uniform thickness are
transferred onto LiNbO3 between IDT1 and IDT2 using a conventional PDMS dry
stamping technique. Bulk CrCl3 crystals are obtained from the commercial
suppliers 2D Semiconductors (USA) and HQ Graphene (Netherlands).

###  I.2 Acoustic antiferromagnetic resonance measurements

The LiNbO3 chip is mounted on a radio-frequency compatible chip carrier and
loaded into either a Montana closed-cycle cryostat with external electromagnet
in 1 axis (Sample 1), or a helium bath cryostat with superconducting magnet
coils in 2 axes (Sample 2). The former has a base temperature around 5 K and
the latter around 4.2 K. Both cryostats allow variable sample temperature up
to at least 30 K. Coaxial cables are used to connect the chip carrier to a
vector network analyzer which is capable of measuring SAW transmission at 1.1
GHz. A time gating function is applied to the signal in order to filter out
electromagnetic noise and retrieve the signals S21 and S12 at longer
timescales 150 - 250 ns.

##  II Acknowledgements

The authors would like to thank Joseph Barker, Olena Gomonay, Hidekazu
Kurebayashi, and Sean Stansill for helpful comments. TPL acknowledges support
from the JSPS postdoctoral fellowships for research in Japan scheme, and KY
from JST PRESTO Grant No. JPMJPR20LB, Japan and JSPS KAKENHI (No. 21K13886).
JP is financially supported by Grants-in-Aid for Scientific Research (S) (No.
19H05629) and JSPS KAKENHI (20H01865), from MEXT, Japan. RSD is supported by
Grants-in-Aid for Scientific Research (S) (No. 19H05610), from MEXT, Japan.
Y.H. is supported by the RIKEN Junior Research Associate Program. SM is
financially supported by JST CREST Grant (No.JPMJCR19J4, No.JPMJCR1874 and
No.JPMJCR20C1) and JSPS KAKENH (No.17H02927 and No.20H10865) from MEXT, Japan.
YO is is financially supported by Grants-in-Aid for Scientific Research (S)
(No. 19H05629).

##  III Author contributions

T. P. L., J. P. and R. S. D. performed experiments. T. P. L., J. P. and Y. H.
fabricated samples. All authors contributed to data interpretation and
analysis. K. Y. and S. M. developed the theoretical model. T. P. L. and K. Y.
wrote the paper. J. P. and Y. O. initiated and supervised the project.

## References

  * Jungwirth  _et al._ (2016) T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nature Nanotechnology 11, 231 (2016).
  * Fukami  _et al._ (2020) S. Fukami, V. O. Lorenz, and O. Gomonay, Journal of Applied Physics 128 (2020), 10.1063/5.0023614.
  * Puebla  _et al._ (2020) J. Puebla, M. Xu, B. Rana, K. Yamamoto, S. Maekawa, and Y. Otani, Journal of Physics D: Applied Physics 53 (2020), 10.1088/1361-6463/ab7efe.
  * Yamamoto  _et al._ (2022) K. Yamamoto, M. Xu, J. Puebla, Y. Otani, and S. Maekawa, Journal of Magnetism and Magnetic Materials 545 (2022), 10.1016/j.jmmm.2021.168672.
  * Xu  _et al._ (2020) M. Xu, K. Yamamoto, J. Puebla, K. Baumgaertl, B. Rana, K. Miura, H. Takahashi, D. Grundler, S. Maekawa, and Y. Otani, Sci. Adv 6 (2020), 10.1126/sciadv.abb1724.
  * Dreher  _et al._ (2012) L. Dreher, M. Weiler, M. Pernpeintner, H. Huebl, R. Gross, M. S. Brandt, and S. T. Goennenwein, Physical Review B 86 (2012), 10.1103/PhysRevB.86.134415.
  * Nie  _et al._ (2022) X.-C. Nie, X. Wu, Y. Wang, S. Ban, Z. Lei, J. Yi, Y. Liu, and Y. Liu, Nanoscale Horizons (2022), 10.1039/d2nh00458e.
  * Yamada  _et al._ (1966) T. Yamada, S. Saito, and Y. Shimomura, Journal of the Physical Society of Japan 21, 672 (1966).
  * Wang  _et al._ (2022) J. Wang, S. Ota, H. Edlbauer, B. Jadot, P. A. Mortemousque, A. Richard, Y. Okazaki, S. Nakamura, A. Ludwig, A. D. Wieck, M. Urdampilleta, T. Meunier, T. Kodera, N. H. Kaneko, S. Takada, and C. Bäuerle, Physical Review X 12 (2022), 10.1103/PhysRevX.12.031035.
  * Peng  _et al._ (2022) R. Peng, A. Ripin, Y. Ye, J. Zhu, C. Wu, S. Lee, H. Li, T. Taniguchi, K. Watanabe, T. Cao, X. Xu, and M. Li, Nature Communications 13 (2022), 10.1038/s41467-022-29042-9.
  * Yokoi  _et al._ (2020) M. Yokoi, S. Fujiwara, T. Kawamura, T. Arakawa, K. Aoyama, H. Fukuyama, K. Kobayashi, and Y. Niimi, Sci. Adv 6 (2020), 10.1126/sciadv.aba1377.
  * Sasaki  _et al._ (2017) R. Sasaki, Y. Nii, Y. Iguchi, and Y. Onose, Phys. Rev. B 95, 020407 (2017).
  * Hernández-Mínguez  _et al._ (2020) A. Hernández-Mínguez, F. Macià, J. M. Hernàndez, J. Herfort, and P. V. Santos, Phys. Rev. Appl. 13, 044018 (2020).
  * Verba  _et al._ (2019) R. Verba, V. Tiberkevich, and A. Slavin, Physical Review Applied 12 (2019), 10.1103/PhysRevApplied.12.054061.
  * Küß  _et al._ (2020) M. Küß, M. Heigl, L. Flacke, A. Hörner, M. Weiler, M. Albrecht, and A. Wixforth, Phys. Rev. Lett. 125, 217203 (2020).
  * Küß  _et al._ (2021) M. Küß, M. Heigl, L. Flacke, A. Hörner, M. Weiler, A. Wixforth, and M. Albrecht, Phys. Rev. Appl. 15, 034060 (2021).
  * Shah  _et al._ (2020) P. J. Shah, D. A. Bas, I. Lisenkov, A. Matyushov, N. X. Sun, and M. R. Page, Science Advances 6, eabc5648 (2020).
  * Puebla  _et al._ (2022) J. Puebla, Y. Hwang, S. Maekawa, and Y. Otani, Applied Physics Letters 120 (2022), 10.1063/5.0093654.
  * Sasaki  _et al._ (2019) R. Sasaki, Y. Nii, and Y. Onose, Physical Review B 99 (2019), 10.1103/PhysRevB.99.014418.
  * Matsumoto  _et al._ (2022) H. Matsumoto, T. Kawada, M. Ishibashi, M. Kawaguchi, and M. Hayashi, Applied Physics Express 15 (2022), 10.35848/1882-0786/ac6da1.
  * McGuire  _et al._ (2017) M. A. McGuire, G. Clark, S. Kc, W. M. Chance, G. E. Jellison, V. R. Cooper, X. Xu, and B. C. Sales, Physical Review Materials 1 (2017), 10.1103/PhysRevMaterials.1.014001.
  * Narath and Davis (1965) A. Narath and H. L. Davis, Physical Review 137 (1965), 10.1103/PhysRev.137.A163.
  * Macneill  _et al._ (2019) D. Macneill, J. T. Hou, D. R. Klein, P. Zhang, P. Jarillo-Herrero, and L. Liu, Physical Review Letters 123 (2019), 10.1103/PhysRevLett.123.047204.
  * Cantos-Prieto  _et al._ (2021) F. Cantos-Prieto, A. Falin, M. Alliati, D. Qian, R. Zhang, T. Tao, M. R. Barnett, E. J. Santos, L. H. Li, and E. Navarro-Moratalla, Nano Letters 21, 3379 (2021).
  * Awschalom  _et al._ (2021) D. D. Awschalom, C. R. Du, R. He, F. J. Heremans, A. Hoffmann, J. Hou, H. Kurebayashi, Y. Li, L. Liu, V. Novosad, _et al._ , IEEE Transactions on Quantum Engineering 2, 1 (2021).
  * Lebrun  _et al._ (2018) R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh, L. Baldrati, J. Cramer, A. Brataas, R. A. Duine, and M. Kläui, Nature 561, 222 (2018).
  * Burch  _et al._ (2018) K. S. Burch, D. Mandrus, and J. G. Park, Nature 563, 47 (2018).
  * Ghiasi  _et al._ (2021) T. S. Ghiasi, A. A. Kaverzin, A. H. Dismukes, D. K. de Wal, X. Roy, and B. J. van Wees, Nature Nanotechnology 16, 788 (2021).
  * Zeisner  _et al._ (2020) J. Zeisner, K. Mehlawat, A. Alfonsov, M. Roslova, A. Doert, T.and Isaeva, B. Büchner, and V. Kataev, Physical Review Materials 4, 064406 (2020).
  * Callen and Callen (1966) H. B. Callen and E. Callen, Journal of Physics and Chemistry of Solids 27, 1271 (1966).
  * Kuhlow (1982) B. Kuhlow, physica status solidi (a) 72, 161 (1982).
  * Schneeloch  _et al._ (2022) J. A. Schneeloch, Y. Tao, Y. Cheng, L. Daemen, G. Xu, Q. Zhang, and D. Louca, npj Quantum Materials 7 (2022), 10.1038/s41535-022-00473-3.
  * Liu  _et al._ (2022) S. Liu, M. Q. Long, and Y. P. Wang, Applied Physics Letters 120 (2022), 10.1063/5.0084070.
  * Borovik-Romanov and Rudashevskii (1965) A. S. Borovik-Romanov and E. G. Rudashevskii, Journal of Experimental and Theoretical Physics 20, 1407 (1965).
  * Otani  _et al._ (2017) Y. Otani, M. Shiraishi, A. Oiwa, E. Saitoh, and S. Murakami, Nature Physics 13, 829 (2017).
  * Kosterlitz and Thouless (1973) J. M. Kosterlitz and D. J. Thouless, Journal of Physics C: Solid State Physics 6, 1181 (1973).
  * Bedoya-Pinto  _et al._ (2021) A. Bedoya-Pinto, J.-R. Ji, A. K. Pandeya, P. Gargiani, M. Valvidares, P. Sessi, J. M. Taylor, F. Radu, K. Chang, and S. S. P. Parkin, Science 374, 616 (2021).
  * Yokouchi  _et al._ (2020) T. Yokouchi, S. Sugimoto, B. Rana, S. Seki, N. Ogawa, S. Kasai, and Y. Otani, Nature Nanotechnology 15, 361 (2020).
  * Rose (1999) J. L. Rose,  _Ulstrasonic Waves in Solid Media_ (Cambridge University Press, Cambridge, 1999).

Supplementary Information for: Acoustically driven magnon-phonon coupling in a
layered antiferromagnet

##  IV Supplementary Note 1: Theoretical model

###  IV.1 Temperature dependence

We consider a smiple Heisenberg model of antiferromagnet

$H=-J_{\parallel}\sum_{\langle
m,n\rangle}\left(\bm{S}^{A}_{m}\cdot\bm{S}^{A}_{n}+\bm{S}_{m}^{B}\cdot\bm{S}^{B}_{n}\right)-J_{\perp}\sum_{\\{m,n\\}}\bm{S}^{A}_{m}\cdot\bm{S}_{n}^{B},$
(4)

where $\bm{S}^{A}_{n},\bm{S}^{B}_{n}$ denote spins belonging to $A$ and $B$
sublattices respectively with $n$ labelling the lattice sites. For modeling
CrCl3, we take $J_{\parallel}>0$ the ferromagnetic intra-layer exchange and
$J_{\perp}<0$ the antiferromagnetic inter-layer exchange interactions
respectively while $\langle m,n\rangle$ and $\\{m,n\\}$ denote intra- and
inter-sublattice nearest neighbour links.

We use the mean-field ansatz

$\langle\bm{S}_{n}^{A}\rangle=\langle
S\rangle\hat{\bm{z}},\quad\langle\bm{S}_{n}^{B}\rangle=-\langle
S\rangle\hat{\bm{z}},$ (5)

which gives the molecular fields

$\bm{B}_{A}=\left(z_{\parallel}J_{\parallel}-z_{\perp}J_{\perp}\right)\langle
S\rangle\hat{\bm{z}},\quad\bm{B}_{B}=-\left(z_{\parallel}J_{\parallel}-z_{\perp}J_{\perp}\right)\langle
S\rangle\hat{\bm{z}},$ (6)

where $z_{\parallel}$ and $z_{\perp}$ are the numbers of intra- and inter-
sublattice links per atom. The expectation value $\langle S\rangle$ is
determined by solving

$\frac{\langle S\rangle}{S}=B_{S}\left(q\frac{\langle
S\rangle}{S}\right),\quad
q=\frac{z_{\parallel}J_{\parallel}-z_{\perp}J_{\perp}}{k_{B}T},$ (7)

where $B_{S}\left(x\right)$ is the Brillouin function

$B_{S}\left(x\right)=\frac{2S+1}{2S}\coth\frac{2S+1}{2S}x-\frac{1}{2S}\coth\frac{1}{2S}x.$
(8)

The asymptotic expansion

$B_{S}\left(x\right)=\frac{S+1}{3S}x-\frac{S+1}{3S}\frac{2S^{2}+2S+1}{30S^{2}}x^{3}+\cdots,\quad
x\rightarrow 0$ (9)

implies at $T=T_{N}$

$q_{T_{N}}=\frac{3S}{S+1}=\frac{z_{\parallel}J_{\parallel}-z_{\perp}J_{\perp}}{k_{B}T_{N}}$
(10)

so that one can eliminate the microscopic parameters in favour of $T_{N}$;

$q=\frac{3S}{S+1}\frac{T_{N}}{T}.$ (11)

Throughout all the calculations in this work, we take $T_{N}=14$ K. For the
spin parameter $S$, we have two choices; the nominal spin of Cr3+ $S=3/2$, or
the macrospin approximation $S\rightarrow\infty$. Since past literature report
a two-step phase transition where the 2D honeycomb layers first order
ferromagnetically, which is followed by the antiferromagnetic order in the
out-of-plane direction McGuire  _et al._ (2017), we think the latter is more
appropriate and use it to compute

$M_{s}\left(T\right)=M_{s}\left(0\right)\lim_{S\rightarrow\infty}\frac{\langle
S\rangle}{S},$ (12)

where $\langle S\rangle/S$ on the right-hand-side is taken to be the self-
consistent (numerical) solution of Eq. (7). We note that the other choice
$S=3/2$ gives a similar temperature dependence, and the difference is
irrelevant considering the qualitative nature of our analysis. In this
formulation, $\bm{B}_{A}=-\bm{B}_{B}$ should be proportional to the exchange
field $H_{E}$ appearing in the spin wave analysis, which yields

$H_{E}\left(T\right)=H_{E}\left(0\right)\lim_{S\rightarrow\infty}\frac{\langle
S\rangle}{S}.$ (13)

We fixed the values of $M_{s}\left(0\right)$ and $H_{E}\left(0\right)$ for
Sample 1 such that $M_{s}\left(T=1.56~{}{\rm K}\right)=250$ mT and
$H_{E}\left(T=1.56~{}{\rm K}\right)=100$ mT and used them to generate Fig.
1d,e in the main text. Note that the reference temperature of 1.56 K was
chosen so as to facilitate comparison with Ref. Macneill  _et al._ (2019).

For Sample 2, the uniaxial anisotropy also appears to be important. We model
it by adding the following term to the macroscopic free energy density;

$F_{u}=-K_{u}\left\\{\left(\hat{\bm{u}}\cdot\bm{n}_{A}\right)^{2}+\left(\hat{\bm{u}}\cdot\bm{n}_{B}\right)^{2}\right\\},$
(14)

where $\hat{\bm{u}}$ is the unit vector along the easy axis, and $K_{u}$ is
the strength of anisotropy in units of energy density (for definitions of free
energy density and $\bm{n}_{A,B}$, see the next section). Defined in this
phenomenological way, the temperature dependence of $K_{u}$ is well
established theoretically in a seminal work by Callen Callen and Callen (1966)
to be $K_{u}\propto M_{s}\left(T\right)^{3}$. In computing the transmission
spectra, this form was assumed and resulted in the temperature dependence of
Fig. 3.

###  IV.2 Spin wave resonance fields

Although the focus of the present work is magneto-elastic coupling, a large
part of the experimental results can be understood by considering only
magnetic properties of CrCl3. Let $\bm{n}_{A}=\bm{M}_{A}/M_{s}$ and
$\bm{n}_{B}=\bm{M}_{B}/M_{s}$ be the normalized magnetization vectors for the
respective sublattice, and introduce spherical coordinate variables by

$\bm{n}_{A}=\begin{pmatrix}\sin\theta_{A}\cos\phi_{A}\\\
\sin\theta_{A}\sin\phi_{A}\\\ \cos\theta_{A}\\\
\end{pmatrix},\quad\bm{n}_{B}=\begin{pmatrix}\sin\theta_{B}\cos\phi_{B}\\\
\sin\theta_{B}\sin\phi_{B}\\\ \cos\theta_{B}\\\ \end{pmatrix}.$ (15)

We set our coordinate $z$-axis to be along the crystal $c$-axis of CrCl3. All
the macroscopic magnetic properties should be derivable from an appropriately
constructed free energy density $F$. For our purposes, it is sufficient to
assume the following form:

$\displaystyle F$ $\displaystyle=$ $\displaystyle
J_{E}\left\\{\sin\theta_{A}\sin\theta_{B}\cos\left(\phi_{A}-\phi_{B}\right)+\cos\theta_{A}\cos\theta_{B}\right\\}-K_{\perp}\left(\cos^{2}\theta_{A}+\cos^{2}\theta_{B}\right)$
(16) $\displaystyle-\frac{K_{u}}{2}\left\\{\sin^{2}\theta_{A}\cos
2\left(\phi_{A}-\phi_{u}\right)+\sin^{2}\theta_{B}\cos
2\left(\phi_{B}-\phi_{u}\right)\right\\}-\mu_{0}M_{s}\bm{H}\cdot\left(\bm{n}_{A}+\bm{n}_{B}\right).$

Here $J_{E}>0$ is the antiferromagnetic exchange energy density (not to be
confused with the microscopic exchange energies $J_{\parallel},J_{\perp}$ in
the previous section), $K_{\perp}$ is the out-of-plane uniaxial anisotropy
that arises from the intra-layer demagnetizing field and spin-orbit
interactions, $K_{u}\geq 0$ is an externally induced in-plane uniaxial
anisotropy that breaks the 6-fold rotation symmetry of CrCl3, $\phi_{u}$
represents its easy-axis direction that is taken to be a free parameter, and
$\bm{H}$ is the external magnetic field. While we assume that the out-of-plane
anisotropy is dominated by the demagnetizing field $K_{\perp}=-M_{s}^{2}/2$,
the spin-orbit contribution might not be entirely negligible. Although
including it can change the theoretical temperature dependence, corrections to
the mean-field approximation is far more likely sources of discrepancy so that
we do not pursue this direction any further.

We take $\bm{H}$ to be in the $ab$-plane and parameterize it by

$\bm{H}=H\begin{pmatrix}\cos\phi\\\ -\sin\phi\\\ 0\\\ \end{pmatrix},\quad
H>0.$ (17)

The unusual sign convention for the $y$-component is in accordance with the
clockwise convention of the polar plots in the main text. The equilibrium
magnetization configuration is determined by minimizing $F$. It is clear that
$\theta_{A}=\theta_{B}=\pi/2$. While we speak of spin waves, the wavelength
relevant to our study is of order 1 $\mu$m, for which the effect of exchange
interactions is expected to be subdominant. Therefore, we treat them as if
they were spatially uniform modes. The linearized Landau-Lifshitz equation
reads

$\left\\{i\frac{\omega}{\gamma\mu_{0}}\begin{pmatrix}0&-1&0&0\\\ 1&0&0&0\\\
0&0&0&-1\\\ 0&0&1&0\\\ \end{pmatrix}-\begin{pmatrix}A_{1}&0&C_{1}&0\\\
0&A_{2}&0&C_{2}\\\ C_{1}&0&B_{1}&0\\\ 0&C_{2}&0&B_{2}\\\
\end{pmatrix}\right\\}\begin{pmatrix}\delta\theta_{A}\\\
\delta\phi_{A}\sin\theta_{A}\\\ \delta\theta_{B}\\\
\delta\phi_{B}\sin\theta_{B}\\\ \end{pmatrix}=0,$ (18)

where $\theta_{A},\phi_{A},\theta_{B},\phi_{B}$ now refer to the equilibrium
state and $\delta\theta_{A},\delta\phi_{A},\delta\theta_{B},\delta\phi_{B}$
small perturbations around it, and

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle-
H_{E}\cos\left(\phi_{A}-\phi_{B}\right)+H\cos\left(\phi+\phi_{A}\right)+M_{s}+H_{u}\cos
2\left(\phi_{A}-\phi_{u}\right),$ $\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle-
H_{E}\cos\left(\phi_{A}-\phi_{B}\right)+H\cos\left(\phi+\phi_{A}\right)+2H_{u}\cos
2\left(\phi_{A}-\phi_{u}\right),$ $\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle-
H_{E}\cos\left(\phi_{A}-\phi_{B}\right)+H\cos\left(\phi+\phi_{B}\right)+M_{s}+H_{u}\cos
2\left(\phi_{B}-\phi_{u}\right),$ $\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle-
H_{E}\cos\left(\phi_{A}-\phi_{B}\right)+H\cos\left(\phi+\phi_{B}\right)+2H_{u}\cos
2\left(\phi_{B}-\phi_{u}\right),$ $\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle H_{E},$ $\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle
H_{E}\cos\left(\phi_{A}-\phi_{B}\right).$

The eigenfrequencies are obtained to be

$\displaystyle\frac{\omega^{2}}{\gamma^{2}\mu_{0}^{2}}$ $\displaystyle=$
$\displaystyle\frac{A_{1}A_{2}+B_{1}B_{2}+2C_{1}C_{2}}{2}\pm\Bigg{\\{}\left(\frac{A_{1}A_{2}-B_{1}B_{2}}{2}\right)^{2}+\frac{A_{1}B_{1}+A_{2}B_{2}}{2}\left(C_{1}^{2}+C_{2}^{2}\right)$
(19)
$\displaystyle+\left(A_{1}A_{2}+B_{1}B_{2}\right)C_{1}C_{2}-\frac{A_{1}B_{1}-A_{2}B_{2}}{2}\left(C_{1}^{2}-C_{2}^{2}\right)\Bigg{\\}}^{1/2}$
$\displaystyle=$ $\displaystyle\frac{A_{1}A_{2}+B_{1}B_{2}+2C_{1}C_{2}}{2}$
$\displaystyle\pm\sqrt{\left(\frac{A_{1}A_{2}-B_{1}B_{2}}{2}\right)^{2}+A_{1}B_{1}C_{2}^{2}+A_{2}B_{2}C_{1}^{2}+\left(A_{1}A_{2}+B_{1}B_{2}\right)C_{1}C_{2}}.$

Setting $H_{u}=0$, they reduce to the frequency equivalent of Eq. (1) in the
main text. In generating Fig. 3a in the main text, we minimized $F$ in Eq.
(16) numerically to obtain $\phi_{A},\phi_{B},\theta_{A},\theta_{B}$, and
evaluated Eq. (19) with the minus sign to plot the acoustic mode resonance
frequencies. The parameters for Sample 2, which are temperature dependant via
Eqs. (12) and (13), were set at $T=1.56$ K as $\mu_{0}M_{s}=250$ mT,
$\mu_{0}H_{E}=130$ mT, $\mu_{0}H_{u}=2.4$ mT, and $\phi_{u}=-\pi/20$ rad
$\approx-9\degree$. Note that $H_{E}$ being different from Sample 1 is not
surprising considering that the two samples were taken from crystals grown in
different conditions.

###  IV.3 Magnon-SAW interactions

In this subsection, we discuss the coupling between Rayleigh surface acoustic
wave and the antiferromagnetic spin waves described in the previous section.
Let an isotropic elastic body (the substrate plus the magnetic film on top)
occupy the half space $z<0$ and assume there is no stress applied on the
surface $z=0$. Acoustic waves in an isotropic media are characterized by only
two parameters; longitudinal and transverse sound velocities $c_{P}$ and
$c_{S}$. When Rayleigh surface acoustic wave with frequency $\omega=c_{R}k$ is
propagating along $x$-axis, the displacement vector $\bm{u}$ is given by

$\bm{u}=\Re\left[C\begin{pmatrix}\left(1-\xi_{S}^{2}\right)\left\\{e^{\kappa_{P}z}-\left(1-\xi_{S}^{2}\right)e^{\kappa_{S}z}\right\\}\\\
0\\\
-i\sqrt{1-\xi_{P}^{2}}\left\\{\left(1-\xi_{S}^{2}\right)e^{\kappa_{P}z}-e^{\kappa_{S}z}\right\\}\\\
\end{pmatrix}e^{-i\left(\omega t-kx\right)}\right].$ (20)

Here $c_{R}$ is the Rayleigh wave velocity solely determined by $c_{P},c_{S}$,
$C$ is a constant, and

$\kappa_{P}=k\sqrt{1-\frac{c_{R}^{2}}{c_{P}^{2}}},\quad\kappa_{S}=k\sqrt{1-\frac{c_{R}^{2}}{c_{S}^{2}}},\quad\xi_{P}^{2}=\frac{c_{R}^{2}}{c_{P}^{2}},\quad\xi_{S}^{2}=\frac{c_{R}^{2}}{2c_{S}^{2}}.$
(21)

The physical discussions should be based on the strain tensor
$\epsilon_{ab}=\left(\partial_{b}u_{a}+\partial_{a}u_{b}\right)/2$ instead of
$\bm{u}$ itself. The boundary condition enforces $\epsilon_{zx}=0$ at the
surface and it stays close to zero within $\sim 1/k$ from the surface. In our
setup, for the $\sim 100$ nm films, we can therefore assume $\epsilon_{zx}$ is
subdominant. Then the only nonzero components of the strain tensor to be taken
into account are $\epsilon_{xx}$ and $\epsilon_{zz}$.

We introduce the free energy density of magneto-elasticity to derive magnon-
phonon coupling. Assuming full rotational symmetry, we use the following form:

$F_{\rm
me}=b\epsilon_{ab}\left(n_{a}^{A}n_{b}^{A}+n_{a}^{B}n_{b}^{B}\right)+2c\epsilon_{ab}n_{a}^{A}n_{b}^{B}.$
(22)

$b$ corresponds to the usual magneto-elastic coupling of ferromagnetic
materials while the inter-sublattice coefficient $c$ is peculiar to
antiferromagnetic materials. Before going into the detailed calculation, let
us see what kind of angular dependence one should expect for the magnon-SAW
coupling. First of all, we are interested in the linear dynamics for which the
free energy should be quadratic in the small fluctuations. The strain tensor
$\epsilon_{ab}$ itself is already a small fluctuation, so that we need to keep
only the first order terms in the perturbation of the magnetizations. Denoting
the perturbations $\delta n_{a}^{A},\delta n_{a}^{B}$ and using
$n_{a}^{A},n_{b}^{B}$ to refer to the fixed ground state values, the quadratic
free energy reads

$F_{\rm me}\approx 2b\epsilon_{ab}\left(n_{a}^{A}\delta
n_{b}^{A}+n_{a}^{B}\delta
n_{b}^{B}\right)+2c\epsilon_{ab}\left(n_{a}^{A}\delta
n_{b}^{B}+n_{a}^{B}\delta n_{b}^{A}\right).$

Next, as far as Rayleigh surface acoustic waves coupled to a thin magnetic
field are concerned, as discussed above, we will need to keep only
$\epsilon_{xx}$ and $\epsilon_{zz}$. However, because we consider only the
ground states in the plane, $n_{z}^{A}=n_{z}^{B}=0$. Therefore, one can reduce
the free energy further to obtain

$F_{\rm me}\sim 2\epsilon_{xx}\left\\{b\left(n_{x}^{A}\delta
n_{x}^{A}+n_{x}^{B}\delta n_{x}^{B}\right)+c\left(n_{x}^{A}\delta
n_{x}^{B}+n_{x}^{B}\delta n_{x}^{A}\right)\right\\}.$ (23)

Let us emphasize that $n_{x}^{A},n_{x}^{B}$ are not dynamical variables in
this expression but just fixed coefficients that depend on $H$ and $\phi$,
which is the root cause of angle dependence of the magnon-phonon couplings.
The situation is slightly more complicated, however, since $\bm{n}^{A}$ and
$\delta\bm{n}^{A}$ (and similarly for $B$) are orthogonal to each other so
that some extra angle dependence arises from $\delta n_{x}^{A},\delta
n_{x}^{B}$. To be quantitative, we write the perturbed magnetisation vectors
as

$\displaystyle\bm{n}^{A}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos\phi_{A}\\\ \sin\phi_{A}\\\ 0\\\
\end{pmatrix},\quad\delta\bm{n}^{A}\ =\
\delta\phi_{A}\begin{pmatrix}-\sin\phi_{A}\\\ \cos\phi_{A}\\\ 0\\\
\end{pmatrix}-\delta\theta_{A}\begin{pmatrix}0\\\ 0\\\ 1\\\ \end{pmatrix},$
$\displaystyle\bm{n}^{B}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos\phi_{B}\\\ \sin\phi_{B}\\\ 0\\\
\end{pmatrix},\quad\delta\bm{n}^{B}\ =\
\delta\phi_{B}\begin{pmatrix}-\sin\phi_{B}\\\ \cos\phi_{B}\\\ 0\\\
\end{pmatrix}-\delta\theta_{B}\begin{pmatrix}0\\\ 0\\\ 1\\\ \end{pmatrix}.$

Note that the ground state angles $\phi_{A},\phi_{B}$ appear not only in the
ground state direction but also multiplying the in-plane fluctuations
$\delta\phi_{A},\delta\phi_{B}$. Substituting these into Eq. (23) yields

$\displaystyle F_{\rm me}$ $\displaystyle\sim$
$\displaystyle-2\epsilon_{xx}\Big{\\{}b\left(\delta\phi_{A}\cos\phi_{A}\sin\phi_{A}+\delta\phi_{B}\cos\phi_{B}\sin\phi_{B}\right)$
(24)
$\displaystyle+c\left(\delta\phi_{A}\cos\phi_{B}\sin\phi_{A}+\delta\phi_{B}\cos\phi_{A}\sin\phi_{B}\right)\Big{\\}}.$

If it were a ferromagnet, the direction of magnetization would roughly follow
the magnetic field $\phi_{A}\approx\phi$ so that this expression explains why
the magnon-phonon coupling is proportional to $\sin 2\phi=2\cos\phi\sin\phi$
and maximised at $\phi=45\degree$. Since we are dealing with an
antiferromagnet, $\phi_{A},\phi_{B}$ have more complicated relations with
$\phi$. In addition, the eigenmodes are acoustic and optical (only
approximately if $H_{u}\neq 0$), i.e. in-phase and out-of-phase precessions of
$\delta\bm{n}^{A}$ and $\delta\bm{n}^{B}$. Thus let us introduce new variables

$\delta\phi_{\rm ac}=\delta\phi_{A}+\delta\phi_{B},\quad\delta\phi_{\rm
op}=\delta\phi_{A}-\delta\phi_{B}.$ (25)

In terms of those eigenmode variables, Eq. (24) reads

$\displaystyle F_{\rm me}$ $\displaystyle\sim$
$\displaystyle-\epsilon_{xx}\Big{[}b\left\\{\delta\phi_{\rm
ac}\sin\left(\phi_{A}+\phi_{B}\right)\cos\left(\phi_{A}-\phi_{B}\right)+\delta\phi_{\rm
op}\sin\left(\phi_{A}-\phi_{B}\right)\cos\left(\phi_{A}+\phi_{B}\right)\right\\}$
(26) $\displaystyle+c\left\\{\delta\phi_{\rm
ac}\sin\left(\phi_{A}+\phi_{B}\right)+\delta\phi_{\rm
op}\sin\left(\phi_{A}-\phi_{B}\right)\right\\}\Big{]},$

where we used some trigonometric identities to simplify the result. Therefore,
in order to understand the angular dependence of the magnon-phonon coupling in
antiferromagnets, one needs to know $\phi_{A}+\phi_{B}$ and
$\phi_{A}-\phi_{B}$ as a function of $\phi$. They are in general complicated.
However, if there is no in-plane anisotropy, by symmetry considerations, we
expect (note our convention of $\phi$ in Eq. (17))

$\phi_{A}+\phi=-\left(\phi_{B}+\phi\right)\equiv\frac{\pi}{2}-\phi_{\rm
cant}.$ (27)

The notation is based on the following observation: In the limit of weak field
$H\rightarrow 0$, $\bm{n}^{A}$ and $\bm{n}^{B}$ are antiparallel to each other
and perpendicular to $\bm{H}$ so that the canting angle $\phi_{\rm cant}=0$.
$\phi_{\rm cant}$ should monotonically increase as $H$ increases. With this,
one obtains

$\phi_{A}+\phi_{B}=-2\phi,\quad\phi_{A}-\phi_{B}=\pi-2\phi_{\rm
cant}\left(H\right).$ (28)

Importantly, $\phi_{A}-\phi_{B}$ is independent of $\phi$. Thus, one derives

$\displaystyle g_{\rm ac}$ $\displaystyle\propto$
$\displaystyle-\left\\{b\cos\left(\phi_{A}-\phi_{B}\right)+c\right\\}\sin\left(\phi_{A}+\phi_{B}\right)\
=\ -\left(b\cos 2\phi_{\rm cant}-c\right)\sin 2\phi,$ (29) $\displaystyle
g_{\rm op}$ $\displaystyle\propto$
$\displaystyle-\left\\{b\cos\left(\phi_{A}+\phi_{B}\right)+c\right\\}\sin\left(\phi_{A}-\phi_{B}\right)\
=\ -\left(b\cos 2\phi+c\right)\sin 2\phi_{\rm cant}.$ (30)

Therefore, the acoustic mode coupling is proportional to $\sin 2\phi$ while
the optical coupling is $\cos 2\phi$ for the intra-sublattice term $\propto b$
and angle independent for the inter-sublattice term $\propto c$.

In order to calculate the SAW transmission amplitude, one needs to be more
systematic. Following the approach taken by Refs. Verba  _et al._ (2019);
Yamamoto  _et al._ (2022) for magnon-SAW coupling in ferromagnetic materials,
one may reduce the equations of motion to the following form:

$\displaystyle\left\\{i\frac{\omega}{\gamma\mu_{0}}\begin{pmatrix}0&-1&0&0\\\
1&0&0&0\\\ 0&0&0&-1\\\ 0&0&1&0\\\
\end{pmatrix}-\begin{pmatrix}A_{1}&0&C_{1}&0\\\ 0&A_{2}&0&C_{2}\\\
C_{1}&0&B_{1}&0\\\ 0&C_{2}&0&B_{2}\\\
\end{pmatrix}\right\\}\begin{pmatrix}\delta\theta_{A}\\\
\delta\phi_{A}\sin\theta_{A}\\\ \delta\theta_{B}\\\
\delta\phi_{B}\sin\theta_{B}\\\ \end{pmatrix}=\epsilon_{R}\begin{pmatrix}0\\\
g_{A}\\\ 0\\\ g_{A}\\\ \end{pmatrix},$ (31)
$\displaystyle\left\\{\rho\left(\frac{\omega^{2}}{k^{2}}-c_{R}^{2}\right)+i\eta\omega\right\\}\epsilon_{R}=\sigma+\overline{g_{A}}\delta\phi_{A}\sin\theta_{A}+\overline{g_{B}}\delta\phi_{B}\sin\theta_{B},$
(32)

where $\epsilon_{R}$ is an appropriately normalised amplitude of SAW,
$\rho=4650$ kg/m3 is the mass density of LiNbO3, $c_{R}\sim 3800$ m/s is the
velocity of Rayleigh mode, $\sigma$ is the external stress generated by the
input IDT, and $\eta$ is the coefficient of viscosity in the Kelvin-Voight
model of viscoelasticity Rose (1999). $g_{A}$ and $g_{B}$ are the effective
magnon-phonon coupling coefficients in the thin film limit arising from the
isotropic magneto-elastic interaction (22):

$\displaystyle g_{A}$ $\displaystyle=$ $\displaystyle-
iC_{R}\sqrt{kd}\left[b\sin
2\phi_{A}+c\left\\{\sin\left(\phi_{A}-\phi_{B}\right)+\sin\left(\phi_{A}+\phi_{B}\right)\right\\}\right],$
(33) $\displaystyle g_{B}$ $\displaystyle=$ $\displaystyle-
iC_{R}\sqrt{kd}\left[b\sin
2\phi_{B}+c\left\\{\sin\left(\phi_{B}-\phi_{A}\right)+\sin\left(\phi_{A}+\phi_{B}\right)\right\\}\right],$
(34)

where $C_{R}$ is a constant of order unity Yamamoto2020. We analytically
solved Eqs. (31) and (32) for $\epsilon_{R}$ with $\omega=c_{R}k$,
$b=0,C_{R}\sqrt{kd}c=10^{6}$, numerically computed $\phi_{A},\phi_{B}$ for
given $H,\phi$, and evaluated $\epsilon_{R}$ to generate Figs. 3 and 4.

As a closing remark, we note that the theoretical model here is meant for
capturing qualitative trends. In particular, the theory predicts very sharp
lines for optical modes, while the experimental data point to multiple peak
structure with a large broadening. There are two main factors that may cause
the disagreement:

  1. 1.

The relative height of acoustic and optical peaks depends strongly on the
precise form of magneto-elastic free energy, which may contain many more terms
than included in Eq. (22.

  2. 2.

The broadening arising from fluctuations and inhomogeneity is not accounted
for, which can become important when the optical mode frequency nears zero.

The growth quality of commercially obtained van der Waals magnetic materials
is currently quite poor, but in time the material quality will likely improve,
bringing with it our understanding of the above points. For the sake of
presentation, the color coding in the simulated polar plots of Figs. 3 and 4
is based on a biased normalization. We use dB units (logarithmic scale), and
assign the brightest color to a value of transmission appropriately large
compared with the actual minimum of the data set. This is appropriate given
that the figures intend to display the magnetic resonance positions, rather
than amplitudes.

##  V Supplementary Note 2: Sample details

Two samples were studied in this work. Sample 1 is quasi-bulk, at $\sim
4\;\upmu$m thick (measured by 3D scanning laser microscopy), while sample 2 is
much thinner at $\sim 120$ nm (measured by atomic force microscopy). Both
flakes were exfoliated with Nitto tape and transferred onto piezoelectric
LiNbO3 substrates by polydimethylsiloxane (PDMS) stamping. A laser microscope
image of sample 1 is shown in Fig. 5a and a bright field microscope image of
sample 2 in Fig. 5b.

Figure 5:  Sample images. (a) Scanning laser microscope image of sample 1,
which is $\sim 4\;\upmu$m thick. IDTs can be seen on the left and right of the
image. Oil residue on the device was present only after all measurements
presented this work were completed. (b) Bright field microscope image of
sample 2, which is $\sim 121$ nm thick.

##  VI Supplementary Note 3: Sample 1 response to external magnetic field
orientation

Sample 1 was measured in a Montana magneto-optical cryostat with an
electromagnet supplying an external magnetic field in one axis only. Over
several repeated cooldown cycles, with manual sample rotation each time, the
response of sample 1 to external field orientation can be studied coarsely.
Figs 6 and 7 show the SAW transmission in sample 1 at various field
orientations. In agreement with sample 2, the optical mode is seen to absorb
SAWs at all angles, however, the acoustic mode absorption appears more
isotropic in sample 1 compared to sample 2, being present at all angles except
for $0\degree$.

Figure 6:  Sample 1 transmission data. Sample 1 transmission data including
S21 and S12 (opposite SAW wavevectors) for the external magnetic field
oriented at (a) $-45\degree$ (equivalent to $315\degree$), (b) $-20\degree$
(equivalent to $340\degree$), (c) $0\degree$ to the SAW propagation axis.
Figure 7:  Sample 1 transmission data. Sample 1 transmission data including
S21 and S12 (opposite SAW wavevectors) for the external magnetic field
oriented at (a) $20\degree$, (b) $45\degree$, (c) $90\degree$ to the SAW
propagation axis.