# Probing spin dynamics of ultra-thin van der Waals magnets via photon-magnon
coupling

Christoph W. Zollitsch<EMAIL_ADDRESS>London Centre for
Nanotechnology, University College London, 17-19 Gordon Street, London, WCH1
0AH, UK  Safe Khan  London Centre for Nanotechnology, University College
London, 17-19 Gordon Street, London, WCH1 0AH, UK  Vu Thanh Trung Nam
Department of Physics, Faculty of Science, National University of Singapore, 2
Science Drive 3, Singapore 117542, Singapore  Ivan A. Verzhbitskiy  Department
of Physics, Faculty of Science, National University of Singapore, 2 Science
Drive 3, Singapore 117542, Singapore  Dimitrios Sagkovits  London Centre for
Nanotechnology, University College London, 17-19 Gordon Street, London, WCH1
0AH, UK  National Physical Laboratory, Hampton Road, Teddington TW11 0LW, UK
James O’Sullivan  London Centre for Nanotechnology, University College London,
17-19 Gordon Street, London, WCH1 0AH, UK  Oscar W. Kennedy  London Centre for
Nanotechnology, University College London, 17-19 Gordon Street, London, WCH1
0AH, UK  Mara Strungaru  Institute for Condensed Matter Physics and Complex
Systems, School of Physics and Astronomy, The University of Edinburgh,
Edinburgh EH9 3FD, UK  Elton J. G. Santos  Institute for Condensed Matter
Physics and Complex Systems, School of Physics and Astronomy, The University
of Edinburgh, Edinburgh EH9 3FD, UK  Higgs Centre for Theoretical Physics, The
University of Edinburgh, Edinburgh EH9 3FD, UK  John J. L. Morton  London
Centre for Nanotechnology, University College London, 17-19 Gordon Street,
London, WCH1 0AH, UK  Department of Electronic & Electrical Engineering, UCL,
London WC1E 7JE, United Kingdom  Goki Eda  Centre for Advanced 2D Materials,
National University of Singapore, 6 Science Drive 2, Singapore 117546,
Singapore  Department of Physics, Faculty of Science, National University of
Singapore, 2 Science Drive 3, Singapore 117542, Singapore  Department of
Chemistry, Faculty of Science, National University of Singapore, 3 Science
Drive 3, Singapore 117543, Singapore  Hidekazu Kurebayashi  London Centre for
Nanotechnology, University College London, 17-19 Gordon Street, London, WCH1
0AH, UK  Department of Electronic & Electrical Engineering, UCL, London WC1E
7JE, United Kingdom

Layered van der Waals (vdW) magnets can maintain a magnetic order even down to
the single-layer regimeGong  _et al._ (2017); Huang  _et al._ (2017); Lee  _et
al._ (2016)and hold promise for integrated spintronic devices Huang  _et al._
(2020a); Wang  _et al._ (2022). While the magnetic ground state of vdW magnets
was extensively studied, key parameters of spin dynamics, like the Gilbert
damping, crucial for designing ultra-fast spintronic devices, remains largely
unexplored. Despite recent studies covering the optical frequency range Zhang
_et al._ (2020a); Cenker  _et al._ (2021), achieving spin wave control at
microwave frequencies is highly desirable, a regime where modern integrated
information technologies predominantly operate Chumak  _et al._ (2015);
Rameshti  _et al._ (2021). The intrinsically small numbers of spins, however,
poses a major challenge to this. Here, we present a novel approach to detect
spin dynamics mediated by photon-magnon coupling between high-Q
superconducting resonators and ultra-thin flakes of Cr2Ge2Te6 (CGT) as thin as
11 nm. We test and benchmark our technique with 23 individual CGT flakes and
extract an upper limit for the Gilbert damping parameter. These results are
crucial in designing on-chip integrated circuits using vdW magnets and offer
prospects for probing spin dynamics of monolayer vdW magnets.

van der Waals (vdW) materials Geim and Grigorieva (2013); Novoselov  _et al._
(2016); Wang  _et al._ (2022) consist of individual atomic layers bonded by
vdW forces and can host different types of collective excitations such as
plasmons, phonons and magnons. Strong coupling between these excitation modes
and electromagnetic waves (i.e. photonic modes) creates confined light-matter
hybrid modes, termed polaritons. Polaritons in vdW materials are an ideal
model system to explore a variety of polaritonic states Basov, Fogler, and de
Abajo (2016); Low  _et al._ (2017), e.g. surface plasmon polaritons in
graphene Fei  _et al._ (2012); Chen  _et al._ (2012) and exciton polaritons in
a monolayer MoS2 embedded inside a dielectric microcavity Liu  _et al._
(2015). These states can be further modified by electrostatic gating, as well
as by hetero-structuring with dissimilar vdW layers.

Numerous studies on magnon polaritons (MPs) Rameshti  _et al._ (2021);
Awschalom  _et al._ (2021) have been using macroscopic yttrium iron garnet
(YIG) coupled to either three-dimensional cavities Zhang  _et al._ (2014) or
to on-chip resonators Huebl  _et al._ (2013); Wang  _et al._ (2019), with
potential applications in ultra-fast information processing, non-reciprocity
or microwave to optical transduction. By reducing the number of excitations,
MPs find application in the quantum regime e.g., magnon number counting via an
electromagnetically coupled superconducting qubit Tabuchi  _et al._ (2015);
Lachance-Quirion  _et al._ (2019) or as a building block for Bell state
generation Yuan  _et al._ (2022).

The rapidly developing research around polaritons and specifically MPs has so
far, been little studied in magnetic vdW materials due to the relatively
recent discoveries of long-range magnetic order in vdW systems at the few
monolayer regime Gong  _et al._ (2017); Huang  _et al._ (2017), in addition to
its technically challenging realization. Stable MP states are formed by
strongly coupling the magnetic field oscillation of a resonant photon to the
collective magnetization oscillation in a magnetic material. In a simplified
picture, the collective coupling strength $g_{\mathrm{eff}}$ scales linearly
with the strength of the oscillating magnetic field of a resonator and the
square root number of spins Huebl  _et al._ (2013). For studies involving bulk
magnetic materials and low quality and large microwave resonators, strong
coupling is achieved when $g_{\mathrm{eff}}/2\pi$ is in the MHz range, which
is accomplished with relative ease due to the abundance of spins in bulk
magnetic materials. A reduction of the bulk dimensions down from mm to $\mu$m
and nm scales, the typical lateral dimensions and thickness of vdW material
monolayers, results in a decrease of the coupling strength by at least 6
orders of magnitude. Commonly used microwave resonators are not able to
produce strong enough oscillating magnetic fields to compensate for such a
reduction in absolute number of spins. Only by advanced resonator design and
engineering the regime of strongly coupled MPs in monolayer vdW magnetic
materials can be accomplished, granting access to spin dynamic physics at a
true 2d monolayer limit and research on MPs in nano-scale devices where the
whole range of on-chip tuning and engineering tools, such as electric fields
or device design, are available.

Magnons or magnon polaritons have been observed in magnetic vdW materials, but
it had been restricted to either to the optical frequency range Zhang  _et
al._ (2020a); Cenker  _et al._ (2021) or a large thickness limit Mandal  _et
al._ (2020); Zhang  _et al._ (2021), respectively. Here, we address directly
the challenge of detecting spin dynamics in ultra-thin vdW magnetic materials
and the creation of MPs by magnon-photon coupling in the microwave frequency
range, using superconducting resonators optimized for increased magnon-photon
coupling. By using microwave resonators with a small mode volume, we not only
increased its oscillating magnetic field strength but also matched it more
efficiently to the size of nanoscale vdW flakes. Our work presents a
fundamental cornerstone for a general blueprint for designing and developing
magnon-photon hybrids for any type of ultra-thin or monolayer vdW magnetic
material, enabling research on on-chip microwave applications for (quantum)
information processing.

Figure 1: Magnon-photon coupling between thin CGT and a superconducting
resonator. a Schematic of a resonator shows the design in detail, indicating
the areas of high $E_{1}$-field (yellow) and $B_{1}$-field (green)
intensities, as well as the orientation of the externally applied field
$B_{0}$. Finally, a schematic zoom in of the section loaded with a CGT flake
is shown. The collective coupling between a microwave photon and the
magnetization of the CGT is illustrated. b Micrograph image of a CGT flake
transferred onto the narrow section of a resonator. c AFM image of the CGT
flake together with a height profile along the blue solid line in the AFM
image. The red solid line is a fit to the flake thickness. The results of this
resonator are presented in Fig. 2.  Figure 2: Magnon-photon coupling observed
in resonator microwave transmission. a $\left|S_{21}\right|^{2}$ as a function
static magnetic field $B_{0}$ and frequency, with the microwave transmission
encoded in the color. The results are obtained from the resonator shown in Fig
1 (b) and (c). b $\left|S_{21}\right|^{2}$ as a function of frequency at fixed
magnetic fields, indicated in a by dashed vertical lines. c and d Resonance
frequency $\omega_{\mathrm{res}}$ and effective loss rate
$\kappa_{\mathrm{eff}}$ as a function of magnetic field. Note the multiple
resonance peaks, indicating multiple CGT FMRs. The dashed orange lines are
results from the semi-optimized fit.

In this article, we report on the observation of spin dynamics and the
creation of MPs in the high cooperativity regime with the vdW ferromagnet CGT
of $\mathrm{n}\mathrm{m}$ scale thickness. We combine a precise transfer
process of exfoliated CGT flakes and high sensitivity superconducting
resonators, to access and study the dynamical response of coupled photon-
magnon states in a small-volume ($\mathrm{n}\mathrm{m}$-thick and
$\mathrm{\SIUnitSymbolMicro}\mathrm{m}$-sized) CGT flake (illustrated in Fig.
1 (a)). High-quality-factor superconducting lumped element resonators are
chosen to be the counterpart due to their extremely small mode volume
($\approx 6000\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{3}$) and consequently
strong oscillating magnetic fields ($\approx 25\,\mathrm{n}\mathrm{T}$, see
SI), resulting in high spin sensitivities Eichler  _et al._ (2017);
Weichselbaumer  _et al._ (2019). At cryogenic temperatures, we perform low-
power microwave spectroscopy on multiple resonator-vdW-flake hybrids, covering
a frequency range from $12\,\mathrm{G}\mathrm{Hz}$ to
$18\,\mathrm{G}\mathrm{Hz}$ for a variety of thickness. Multiple peaks of
spin-wave resonances are observed for each CGT flake measured. By taking the
main peak of each spectrum, we find that all measured points can be fitted
very well by a single curve calculated by the Kittel formula with bulk CGT
parameters. Furthermore, we extracted the linewidth for the thinnest CGT flake
investigated, $11\,\mathrm{n}\mathrm{m}$ or 15 monolayers (ML), and determined
an upper limit of the Gilbert damping parameter of $0.02$. This value is
comparable to the damping reported for 3d transition metal ferromagnets,
suggesting that magnetic vdW flakes have the potential for the fabrication of
functional spintronic devices.

We investigate the dynamics of $\mathrm{n}\mathrm{m}$-thick CGT flakes, using
superconducting lumped element resonators made of NbN (see methods for
fabrication details and SI and Ref. [Zollitsch  _et al._ , 2019] for more
performance details). The advantages of a lumped element design are the
spatial separation of the oscillating magnetic field $B_{1}$ and electric
field $E_{1}$ and the concentration of $B_{1}$ within a narrow wire section of
the resonators, as indicated in Fig. 1 (a). Additionally, the $B_{1}$ field
distribution is homogeneous along the length of the narrow wire section (see
finite element simulations in SI). This magnetic-field concentration is our
primary reason to use this type of resonator in order to reduce the photon
mode volume as well as achieve a considerable mode overlap between the
resonator photon mode and CGT magnon mode, and consequently, a large coupling
strength. We therefore transfer CGT flakes onto these narrow sections (Fig. 1
(b)). Details of CGT flake transfers are described in the methods section.
Optical imaging and atomic force microscopy (AFM) measurements are used to
characterise the size and thickness of the CGT flakes (see Fig. 1 (c)).
Measured thicknesses range from $153\pm 23\,\mathrm{n}\mathrm{m}$ down to
$11\pm 1.8\,\mathrm{n}\mathrm{m}$ (15 ML), enabling a thickness dependent
study of CGT flakes and their coupling to the resonators.

We measured the microwave transmission $\left|S_{21}\right|^{2}$ as a function
of frequency and externally applied magnetic field $B_{0}$ for each resonator
at a temperature of $1.8\,\mathrm{K}$, using a microwave power of
approximately $-80\,\mathrm{d}\mathrm{B}\mathrm{m}$ at the resonator chip.
Figure 2 (a) shows the resulting 2D plot of $\left|S_{21}\right|^{2}$ for a
resonator loaded with a $17\,\mathrm{n}\mathrm{m}\pm
0.8\,\mathrm{n}\mathrm{m}$ thick CGT flake (see Fig. 1 (b) and (c) for the
respective micrograph and AFM images). A resonator peak can be clearly
observed for each magnetic field, with its resonance frequency
$\omega_{\mathrm{res}}$ decreasing with increasing magnetic field. The
reduction of the frequency is a result of a slow degradation of the
superconductivity by $B_{0}$, which in general exhibits a parabolic dependence
Healey  _et al._ (2008). For $580\,\mathrm{m}\mathrm{T}\leq B_{0}\leq
630\,\mathrm{m}\mathrm{T}$ the resonator prominence is reduced, highlighted by
$\left|S_{21}\right|^{2}$ as a function of frequency for four constant $B_{0}$
values in Fig. 2 (b). Within this field range, the mode resonance has been
modified due to its hybridization with the magnetic modes of the CGT flake. To
further quantify the interaction, we fit each $\left|S_{21}\right|^{2}$
profile by a Fano resonance lineshape (solid orange lines in Fig. 2 (b)) to
account for an asymmetric resonance peak due to additional microwave
interference in the circuitry Fano (1961); Khalil  _et al._ (2012),

$\left|S_{21}\right|^{2}=S_{0}+A\frac{\left(q\nicefrac{{\kappa_{\mathrm{eff}}}}{{2}}+\omega-\omega_{\mathrm{res}}\right)^{2}}{(\nicefrac{{\kappa_{\mathrm{eff}}}}{{2}})^{2}+\left(\omega-\omega_{\mathrm{res}}\right)^{2}}.$
(1)

Here, $S_{0}$ is the microwave transmission baseline, $A$ the peak amplitude,
$q$ describes the asymmetry of the lineshape and $\kappa_{\mathrm{eff}}$
represents the effective loss rate of the hybrid system. Figure 2 (c) shows
$\omega_{\mathrm{res}}$ of the hybrid system as a function of $B_{0}$.
$\omega_{\mathrm{res}}$ experiences a dispersive shift when the photon mode
and the magnon mode hybridize, indicating an onset of a strong interaction
between the two individual systems Herskind  _et al._ (2009); Bushev  _et al._
(2011); Huebl  _et al._ (2013); Lachance-Quirion  _et al._ (2019); Khan  _et
al._ (2021). The observation of a continuous dispersive shift with no clear
mode separation suggests that the hybrid system resides in a regime of high
cooperativity Bushev  _et al._ (2011); Huebl  _et al._ (2013) (we discuss a
quantitative analysis later). We observe multiple shifts in
$\omega_{\mathrm{res}}$, suggesting an interaction of several magnon modes
with the resonator in our experiment.

Signatures of the resonator–CGT-flake coupling are also characterised by
$\kappa_{\mathrm{eff}}$ of the hybrid system (Fig. 2 (d)).
$\kappa_{\mathrm{eff}}$ is enhanced from the value of the resonator loss rate
$\kappa_{0}$ due to an additional loss introduced by the magnon system
characterized by the loss rate $\gamma$ Herskind  _et al._ (2009); Huebl  _et
al._ (2013); Tabuchi  _et al._ (2014). Consistent with the $B_{0}$ dependence
of $\omega_{\mathrm{res}}$, $\kappa_{\mathrm{eff}}$ shows a rich structure,
having its main peak at $598\,\mathrm{m}\mathrm{T}$, together with less
prominent peaks distributed around it. Based on a formalism for coupled-
harmonic-oscillator systems in the high cooperativity regime Herskind  _et
al._ (2009); Bushev  _et al._ (2011); Khan  _et al._ (2021), we use the
following to analyse our experimental results with multiple peaks:

$\omega_{\mathrm{res}}=\omega_{\mathrm{res,0}}+mB_{0}^{2}+\sum_{k=-n}^{+n}\frac{g_{\mathrm{eff},k}^{2}\Delta_{k}}{\Delta_{k}^{2}+\gamma^{2}},$
(2)

$\kappa_{\mathrm{eff}}=\kappa_{0}+\sum_{k=-n}^{+n}\frac{g_{\mathrm{eff},k}^{2}\gamma}{\Delta_{k}^{2}+\gamma^{2}}.$
(3)

with the detuning factor for each resonance as
$\Delta_{k}=\frac{g_{\mathrm{CGT}}\mu_{\mathrm{B}}}{\hbar}\left(B_{0}-B_{\mathrm{FMR},k}\right)$.

Figure 3: Summary of CGT-FMR conditions. a Extracted CGT resonance fields and
frequencies from the set of resonators loaded with CGT flakes of different
thickness. Resonance values are taken from the most prominent peaks in
$\kappa_{\mathrm{eff}}$. The solid curve is calculated using the Kittel
formalism presented in Khan  _et al._ (2019), using same parameters, with
$g_{\mathrm{CGT}}=2.18$, $\mu_{0}M_{\mathrm{s}}=211.4\,\mathrm{m}\mathrm{T}$
and $K_{\mathrm{u}}=3.84\times
10^{4}\,\nicefrac{{\mathrm{J}}}{{\mathrm{m}^{3}}}$. b Wider magnetic field
range of a where the CGT flake thickness for the different symbols is
indicated by the color gradient given in a.

Here, $\omega_{\mathrm{res,0}}$ is the resonator resonance frequency at
$B_{0}=0\,\mathrm{T}$ and $m$ represents the curvature of the resonance
frequency decrease due to the applied magnetic field. $B_{\mathrm{FMR},k}$ is
the CGT FMR field, $g_{\mathrm{CGT}}$ the g-factor of CGT and
$g_{\mathrm{eff},k}$ gives the collective coupling strength between photon and
magnon mode. The summation is over all resonance modes $k$ present on the low
or high field (frequency) side of the main resonance mode, where $n$ gives the
number of modes on one side. For simplicity, we assume a symmetric
distribution of modes about the main mode. The large number of multiple modes
and their strong overlap prevent a reliable application of a fully optimized
fit to the data, due to the large number of free parameters required. In an
effort to gain a good estimate of the model parameters we apply the model
functions Eq. (2) and (3) in a two-step semi-optimized fashion (see SI for
details). With this approach, we arrive at a model in good agreement with
$\omega_{\mathrm{res}}$ and $\kappa_{\mathrm{eff}}$ (see dashed lines in Fig.
2 (c), (d) and the SI for additional results). We can reproduce the data using
$\gamma/2\pi=94.03\pm 5.95\,\mathrm{M}\mathrm{Hz}$ and a collective coupling
strength of the main mode of $13.25\pm 1\,\mathrm{M}\mathrm{Hz}$. Together
with $\kappa_{0}/2\pi=1.4\pm 0.02\,\mathrm{M}\mathrm{Hz}$ the system resides
in the high cooperativity regime, classified by the cooperativity
$C=\nicefrac{{g_{\mathrm{eff}}^{2}}}{{\kappa_{0}\gamma}}=1.3>1$ Herskind  _et
al._ (2009); Zhang  _et al._ (2014). In this regime, magnon polaritons are
created and coherently exchange excitations between magnons and resonator
photons on a rate given by $g_{\mathrm{eff}}$. The created MPs are, however,
short lived and the excitations predominately dissipate in the magnonic
system, as $g_{\mathrm{eff}}<\gamma$.

Our analysis suggests that the separation of the different FMR modes is of the
same order of magnitude as the loss rate (see SI for additional data). We
consider that these are from standing spin wave resonances, commonly observed
for thin magnetic films Serga, Chumak, and Hillebrands (2010) and with one
reported observation in bulk of the vdW material CrI3 Kapoor  _et al._ (2021).
In thin-film magnets under a static magnetic field applied in-plane, the
magnetic-dipole interaction generates two prominent spin wave branches for an
in-plane momentum, the backward volume spin wave (BVMSW) and magnetostatic
surface spin wave (MSSW) modes Kalinikos and Slavin (1986); Bhaskar  _et al._
(2020). These spin wave modes have different dispersion relations, having
higher (MSSWs) and lower (BVMSWs) resonance frequencies with respect to that
of the uniform FMR mode. We calculate the distance of these standing spin-wave
modes based on magnetic parameters of bulk CGT as well as the lateral
dimensions of the flakes (see SI for more details). We can find spin waves
having a frequency separation within $100\,\mathrm{M}\mathrm{Hz}$ and
$200\,\mathrm{M}\mathrm{Hz}$ ($3.3\,\mathrm{m}\mathrm{T}$ to
$6.6\,\mathrm{m}\mathrm{T}$ in magnetic field units), which are consistent
with our experimental observation in terms of its mode separation. We also
considered a possibility that each layer of CGT might have different magnetic
parameters, and thus producing different individual resonance modes. Our
numerical simulations based on atomistic spin dynamicsWahab  _et al._ (2021);
Kartsev  _et al._ (2020) rule out this possibility, as resonance modes from
individual layers average to a single mode as soon as a fraction of $10\%$ of
inter-layer exchange coupling is introduced (see SI for more details).
Therefore, we speculate that the multiple mode nature we observe in our
experiments is likely originating from intrinsic properties of the CGT flakes.

Figure 3 shows the extracted $\omega_{\mathrm{FRM}}$ as a function of
$B_{\mathrm{FMR}}$ for each resonator–CGT-flake hybrid. The experimental
values are in excellent agreement with a curve calculated by the Kittel
equation with magnetic parameters for bulk CGT Khan  _et al._ (2019), from
which the data exhibits a standard deviation of less than $5\,\%$. This
agreement, achieved by independent characterizations of 23 CGT flakes measured
by superconducting resonators, is experimental evidence that the magnetic
parameters that determine the dispersion of
$\omega_{\mathrm{FRM}}\left(B_{\mathrm{FMR}}\right)$, i.e. the CGT g-factor
$g_{\mathrm{CGT}}$, saturation magnetization $M_{\mathrm{s}}$ and uniaxial
anisotropy $K_{\mathrm{u}}$, exhibit little thickness dependence in exfoliated
CGT flakes, and are not disturbed by the transfer onto the resonator
structure. We note, that this demonstrates that vdW magnetic materials are
particularly attractive for device applications, as they are less prone for
contamination from exfoliation.

Finally, we present our analysis of $\kappa_{\mathrm{eff}}$ for a resonator
with a $11\pm 1.8\,\mathrm{n}\mathrm{m}$ CGT flake in Fig.4. With the
thickness of a single layer of CGT being $0.7\,\mathrm{n}\mathrm{m}$ Gong  _et
al._ (2017), this flake consists of 15 monolayers and is the thinnest in our
series. Figure 4 (a) and (b) show $\omega_{\mathrm{res}}$ and
$\kappa_{\mathrm{eff}}$ as a function of $B_{0}$, respectively. While the
response of the CGT flake shows a prominent signature in
$\kappa_{\mathrm{eff}}$, the CGT FMR is considerably more subtle in
$\omega_{\mathrm{res}}$. This highlights the excellent sensitivity of the
high-Q superconducting resonators in our study. $\kappa_{\mathrm{eff}}$
features five well-separated peaks with the main peak at
$B_{0}=547\,\mathrm{m}\mathrm{T}$, which enables us to perform a single-peak
fully optimized analysis for each, in contrast to our multi-step analysis for
the remainder of the devices. From the main peak profile, we extract
$g_{\mathrm{eff}}/2\pi=3.61\pm 0.09\,\mathrm{M}\mathrm{Hz}$,
$\gamma/2\pi=126.26\pm 8.5\,\mathrm{M}\mathrm{Hz}$ and
$\kappa_{0}/2\pi=0.92\pm 0.05\,\mathrm{M}\mathrm{Hz}$. We compare the
experimental value of $g_{\mathrm{eff}}$ with a numerically calculated
$g_{\mathrm{eff,simu}}$, using the dimensions of the CGT flake determined by
AFM measurements (see SI for details). The calculation yields
$g_{\mathrm{eff,simu}}/2\pi=8.94\,\mathrm{M}\mathrm{Hz}$, lying within the
same order of magnitude. The overestimation is likely due to in-perfect
experimental conditions, like non-optimal placement of the flake,
uncertainties in the thickness and dimension determination as well as
excluding the additional modes in the calculation (see SI). With $\gamma\gg
g_{\mathrm{eff}}$ and $C=0.11$, the hybrid system is in the weak coupling
regime Zhang  _et al._ (2014), but due to the highly sensitive resonator with
its small $\kappa_{0}$ the response from the magnon system can still be
detected. With the extracted $\gamma/2\pi$ we can give an upper limit of the
Gilbert damping in CGT, by calculating
$\alpha_{\mathrm{upper}}=\gamma/\omega_{\mathrm{FMR}}$. We find
$\alpha_{\mathrm{upper}}$ as $0.021\pm 0.002$, which is comparable to other
transition metal magnetic materials Mankovsky  _et al._ (2013), and is in very
good agreement with a previously reported effective Gilbert damping parameter
determined by laser induced magnetization dynamics Zhang  _et al._ (2020b).
Here, we emphasise that the actual Gilbert damping value is lower due to a
finite, extrinsic inhomogeneous broadening contribution.

We further use these results to benchmark the sensitivity of our measurement
techniques. The detection limit is given by comparing the main peak height
characterised by $g_{\mathrm{eff}}^{2}/\gamma$ and the median noise amplitude
which is $18\,\mathrm{k}\mathrm{Hz}$ in Fig. 4 (b) where
$g_{\mathrm{eff}}^{2}/2\pi\gamma$ = 103 kHz. By assuming the same lateral
dimensions and scale the thickness down to a single monolayer, while keeping
$\gamma$ constant, we calculate the expected signal reduction numerically by
$g_{\mathrm{eff,simu,1ML}}/g_{\mathrm{eff,simu,15ML}}$ to 0.26. We obtain
$(0.26g_{\mathrm{eff}})^{2}/2\pi\gamma=7\,\mathrm{k}\mathrm{Hz}$ for the
monolayer limit. Although this suggests the noise amplitude is greater than
the expected peak amplitude, we can overturn this condition by improving the
coupling strength by optimising the resonator design as well as by reducing
the noise level by averaging a number of multiple scans.

Figure 4: Magnon-photon coupling for the thinnest CGT flake. a Resonance
frequency $\omega_{\mathrm{res}}$ and b effective loss rate
$\kappa_{\mathrm{eff}}$ as a function of magnetic field of a resonator loaded
with the thinnest CGT, consisting of 15 ML. The solid orange lines are results
a fit to Eq.(2) and (3), respectively.

In summary, we provide the first demonstration of photon-magnon coupling
between a superconducting resonator and $\mathrm{n}\mathrm{m}$-thick vdW
flakes of CGT, using a total of 23 devices with different CGT flakes of
thickness from $153\,\mathrm{n}\mathrm{m}$ down to $11\,\mathrm{n}\mathrm{m}$.
By employing a coupled-harmonic-oscillator model, we extract the coupling
strength, magnetic resonance field and relaxation rates for both photon and
magnon modes in our devices. From our semi-broadband experiments, we find that
the magnetic properties of exfoliated CGT flakes are robust against the
transfer process, with a standard deviation of less than $5\,\%$ to expected
resonance values from bulk parameters. The upper limit of the Gilbert damping
in the 15 ML thick CGT flake is determined to be $0.021$, which is comparable
to commonly used ferromagnetic thin-films such as NiFe and CoFeB and thus
making CGT attractive for similar device applications. We highlight that the
damping parameter is key in precessional magnetisation switching Rowlands  _et
al._ (2019); Meo  _et al._ (2022), auto-oscillations by dc currents Wagner
_et al._ (2018); Haidar  _et al._ (2019), and comprehensive spin-orbit
transport in vdW magnetic systems Kurebayashi  _et al._ (2022). The presented
techniques are readily transferable to other vdW magnetic systems to study
spin dynamics in atomically-thin crystalline materials. While creating stable
magnon polaritons is still an open challenge due to the large loss rate
$\gamma$ of the CGT magnon system, this work offers an important approach
towards its achievement as there are still potential improvements to the
measurement sensitivity such as resonator mode volume reduction by introducing
$\mathrm{n}\mathrm{m}$ scale constrictionsMcKenzie-Sell  _et al._ (2019);
Gimeno  _et al._ (2020) and use of exfoliation/transfer techniques to produce
larger flakes to enhance the mode overlap (hence coupling strength)Huang  _et
al._ (2020b); Zhou  _et al._ (2022). With concerted efforts, the formation of
magnon polaritons in few layers vdW materials will become feasible.

## Author Contribution

C.W.Z, S.K. and H.K. conceived the experimental project. Resonator design and
optimization was done by J.O’S., O.W.K, C.W.Z and supervised by J.J.L.M.
Resonator fabrication and characterization was done by C.W.Z. CGT crystals
were grown by I.A.V. and exfoliated and transferred by I.A.V. and N.V.T.T. and
supervised by E.G.. D.S. measured AFM on the CGT flakes on the resonators.
C.W.Z. performed the experiments and the data analysis with input from S.K.
and H.K. Atomistic spin dynamics simulations were carried out by M.S.
supervised by E.J.G.S.. C.W.Z., M.S., I.A.V. and H.K. wrote the manuscript
with input from all authors.

## Methods

Superconducting Resonators: The resonators were fabricated by direct laser
writing and a metal lift-off process. The individual
$5\,\mathrm{m}\mathrm{m}\times 5\,\mathrm{m}\mathrm{m}$ chips are scribed from
an intrinsic, high resistivity
($\rho>5000\,\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$) n-type silicon
wafer of $250\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ thickness. For a well
defined lift-off, we use a double photoresist layer of LOR and SR1805. The
resonator structures are transferred into the resist by a Heidelberg Direct
Writer system. After development, $\sim 50\,\mathrm{n}\mathrm{m}$ NbN are
deposited by magnetron sputtering in a SVS6000 chamber, at a base pressure of
$7\times 10^{-7}\,$mbar, using a sputter power of $200\,\mathrm{W}$ in an
50:50 Ar/N atmosphere held at $5\times 10^{-3}\,$mbar, with both gas flows set
to 50 SCCM Zollitsch  _et al._ (2019). Finally, the lift-off is done in a 1165
solvent to release the resonator structures.

CGT Crystal Growth: CGT crystals used in this study were grown via chemical
vapour transport. To this end, high-purity elemental precursors of Cr (chips,
$\geq\,99.995\%$), Ge (powder, $\geq\,99.999\%$), and Te (shots, $99.999\%$)
were mixed in the molar weight ratio Cr:Ge:Te = 10:13.5:76.5, loaded into a
thick-wall quartz ampule and sealed under the vacuum of $\sim 10^{-5}\,$mbar.
Then, the ampule was loaded into a two-zone furnace, heated up and kept at
$950\,\degreecelsius$ for 1 week to homogenize the precursors. To ensure high-
quality growth, the ampule was slowly cooled
($0.4\,\degreecelsius/\mathrm{h}$) maintaining a small temperature gradient
between the opposite ends of the ampule. Once the ampule reached
$500\,\degreecelsius$, the furnace was turned off allowing the ampule to cool
down to room temperature naturally. The large ($\sim 1\,\mathrm{c}\mathrm{m}$)
single-crystalline flakes were extracted from the excess tellurium and stored
in the inert environment.

CGT Flake Transfer: Devices for this study were made via transfer of single-
crystalline thin flakes on top of the superconducting resonators. The flakes
were first exfoliated from bulk crystals on the clean surface of a home-cured
PDMS (polydimethylsiloxane, Sylgard 184) substrate. The thickness of the CGT
flakes on PDMS was estimated through the contrast variation with transmission
optical microscopy. Then, the selected flake was transferred to a resonator.
The transfer was performed in air at room temperature. To minimize the air
exposure, the entire process of exfoliation, inspection and transfer was
reduced to 10-15 min per resonator. For the flakes thicker than
$50\,\mathrm{n}\mathrm{m}$, the strong optical absorption of CGT prevented the
accurate thickness estimation with optical contrast. For those flakes, the
thickness was estimated via a quick AFM scan performed on the PDMS substrate
before the transfer step. Ready devices were stored in inert conditions.

## Acknowledgment

This study is supported by EPSRC on EP/T006749/1 and EP/V035630/1. G.E.
acknowledges support from the Ministry of Education (MOE), Singapore, under
AcRF Tier 3 (MOE2018-T3-1-005) and the Singapore National Research Foundation
for funding the research under medium-sized centre program. E.J.G.S.
acknowledges computational resources through CIRRUS Tier-2 HPC Service (ec131
Cirrus Project) at EPCC (http://www.cirrus.ac.uk) funded by the University of
Edinburgh and EPSRC (EP/P020267/1); ARCHER UK National Supercomputing Service
(http://www.archer.ac.uk) via Project d429. EJGS acknowledges the Spanish
Ministry of Science’s grant program “Europa-Excelencia” under grant number
EUR2020-112238, the EPSRC Early Career Fellowship (EP/T021578/1), and the
University of Edinburgh for funding support. D.S. acknowledges EPSRC funding
through the Centre for Doctoral Training in Advanced Characterisation of
Materials (EP/L015277/1).

## Data Availability

The data that support the findings of this study are available within the
paper, Supplemental Material and upon reasonable request.

## Competing interests

The Authors declare no conflict of interests.

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Supplemental Material - Probing spin dynamics of ultra-thin van der Waals
magnets via photon-magnon coupling

##  I Microwave Setup

Figure S1: Microwave delivery and detection setup. Schematic of the microwave
delivery and detection circuit. The image shows the coplanar waveguide
transmission line. A resonator chip is placed on top of the transmission line
for read out. On the right, a schematic layout of the resonators on a single
chip is shown.

Figure S1 shows a schematic of the used microwave measurement setup. We are
using a Keysight E5071C vector network analyzer (VNA) to deliver and detect
microwaves. The VNA is connected to a low temperature probe, fitted into a
closed cycle helium cryostat and cooled to a base temperature of about
$1.8\,\mathrm{K}$. The microwave signal is transmitted into the cryostat and
is attenuated by $-20\,\mathrm{d}\mathrm{B}$. The attenuator is positioned
just before the sample box and provides a thermal anchoring for the center
conductor of the coaxial cable to minimize the thermal load onto the sample.
The output line is equipped with a Low Noise Factory LNC6_20C cryogenic
amplifier, operating between $6-20\,\mathrm{G}\mathrm{Hz}$ with an average
amplification of $+32\,\mathrm{d}\mathrm{B}$. The transmitted and amplified
signal is finally detected by the VNA. Figure S1 also shows an image of the
coplanar waveguide transmission line PCB, loaded with a resonator ship, of
which a schematic shows the resonator layout on a single chip. The cryostat is
equipped with a mechanical rotation stage and prior to the measurements the
superconducting resonators are carefully aligned to the externally applied
static magnetic field $B_{0}$, such that the field is in the plane of the
superconductor and along the narrow section of the resonators.

##  II Resonator Characterization

In this study, we fabricate twelve superconducting lumped element resonators
on each of three resonator chips were fabricated using the same design (see
schematic Fig. 1 (a) in the main text). Prior to transfer of the CGT flakes,
we characterized the resonators at a temperature of $1.8\,\mathrm{K}$ and zero
applied magnetic field, using microwave powers of about
$-80\,\mathrm{d}\mathrm{B}\mathrm{m}$ at the resonators, which is well below
the bifurcation limit starting above $-60\,\mathrm{d}\mathrm{B}\mathrm{m}$.
Due to finite fabrication tolerances the resonator parameters have some
variation, while some didn’t work at all. However, the targeted resonance
frequencies are well reproducible and very similar for the 3 different chips.
We compare the resonator parameters before and after transfer of the CGT
flakes and collate the parameters in Tab. 1. Note, the resonator parameters
with the CGT flakes on were obtained with a static magnetic field applied in
the plane of the superconductor, but far detuned from the CGT FMR. In
addition, we add the respective thickness of the flake on each resonator,
acquired from AFM measurements. Here, we give the values of the thickest
region of a given flake on a resonator, as the thickest region will dominate
the FMR signal. Due to the arbitrary shape of exfoliated flakes, some exhibit
regions of different thickness, as seen e.g. in Fig. S4 (h) and (i).

$\omega_{\mathrm{res,before}}$ ($\mathrm{M}\mathrm{Hz}$) | $Q_{\mathrm{L,before}}$ |  $\omega_{\mathrm{res,after}}$ ($\mathrm{M}\mathrm{Hz}$) | $Q_{\mathrm{L,after}}$ | CGT Thickness ($\mathrm{n}\mathrm{m}$)
---|---|---|---|---
12165 | 1978 | 12063 | 5733 | 16
13303 | 7357 | 13177 | 4950 | -
13968 | 5575 | 13860 | 4679 | 49
14184 | 6492 | 14048 | 5627 | 153
16648 | 6606 | 16470 | 5021 | 23
17431 | 3215 | 17237 | 6826 | 24
17959 | 7595 | 17790 | 3963 | 26
Table 1: Resonator Chip 1 Parameters $\omega_{\mathrm{res,before}}$ ($\mathrm{M}\mathrm{Hz}$) | $Q_{\mathrm{L,before}}$ |  $\omega_{\mathrm{res,after}}$ ($\mathrm{M}\mathrm{Hz}$) | $Q_{\mathrm{L,after}}$ | CGT Thickness ($\mathrm{n}\mathrm{m}$)
---|---|---|---|---
12285 | 360 | 12153 | 7135 | 49
12669 | 3600 | 12548 | 6693 | 103
12782 | 3448 | 12648 | 6557 | 106
13393 | 4643 | 13244 | 4501 | 34
13760 | 6858 | 13620 | 5488 | 96
14395 | 9048 | 14201 | 4139 | 37
16075 | 7283 | - | - | -
17048 | 6541 | 16811 | 4241 | 75
Table 2: Resonator Chip 2 Parameters $\omega_{\mathrm{res,before}}$ ($\mathrm{M}\mathrm{Hz}$) | $Q_{\mathrm{L,before}}$ |  $\omega_{\mathrm{res,after}}$ ($\mathrm{M}\mathrm{Hz}$) | $Q_{\mathrm{L,after}}$ | CGT Thickness ($\mathrm{n}\mathrm{m}$)
---|---|---|---|---
12043 | 6114 | 11899 | 6044 | 60
12456 | 2716 | 12314 | 6938 | 11
12996 | 5828 | 12848 | 4600 | 17
13422 | 6517 | 13272 | 5461 | 90
13719 | 6800 | 13582 | 6608 | -
14238 | 9184 | 14064 | 5420 | 74
15390 | 8680 | 15219 | 6030 | 31
15821 | 2386 | 15604 | 4769 | 33
16430 | 7518 | 16193 | 5780 | 30
17308 | 6521 | 17054 | 5569 | 138
18111 | 3542 | 17870 | 4643 | 50
Table 3: Resonator Chip 3 Parameters

##  III Resonator and Coupling Simulation

We use finite element and numerical simulations to optimize our resonator
design. Key requirements of our resonators are a strong resilience to
externally applied static magnetic fields and a small mode volume. To achieve
a large field resilience we reduced the area of the resonator to minimize
effects of the magnetic field on the superconducting film. Further, we
designed the resonators such that they act as lumped element resonators. Here,
the resonance frequency is given by the total capacitance and inductance of
the structure, with $\omega_{\mathrm{res}}=\nicefrac{{1}}{{\sqrt{LC}}}$,
analogues to a parallel LC circuit. This allows us to locally separate
oscillating electric and magnetic fields and also to concentrate the magnetic
fields in more confined regions, resulting in very small mode volumes. To
verify the lumped element nature of our resonators we performed finite element
simulations, using CST Microwave Studio. Figure S2 shows the resulting
magnitude of the E-field (left side) and H-field (right side) distribution
along the resonator structure for the resonator design producing the results
shown in Fig. 2 in the main text. The E-field is concentrated along the
parallel running wire sections, with its strength approaching zero along the
narrow wire section. The opposite is the case for the H-field, where it is
zero along the parallel wire sections and strongly concentrated along the
narrow wire section. Note, that the H-field magnitude is homogeneous along the
whole of the narrow wire section.

Figure S2: Finite element simulations of resonator. CST Microwave Studio
simulation of the distribution of E-fields and H-fields across the resonator
structure. The color encoded fields represent the magnitude values.

The CST Microwave Studio at hand allowed us a simulation with perfect electric
conductors. This is sufficient to model the general electric and magnetic
energy distributions and resonance frequencies, however, not to simulate the
corresponding oscillating magnetic field distribution, created by a
superconducting rectangular wire. To this end, we numerically solve the Biot-
Savart law for a rectangular wire cross-section Primenko, Osipov, and Rudnev
(2017), assuming a superconducting current distribution
$\textbf{J}_{\mathrm{x,z}}$ Lee, Orlando, and Lyons (1994),

$\textbf{B}_{1,\mathrm{x,z}}=\frac{\mu_{0}}{2\pi}\int_{-w/2}^{w/2}\int_{-d/2}^{d/2}\frac{\textbf{J}\times\textbf{r}}{\left(x-x^{\prime}\right)^{2}+\left(z-z^{\prime}\right)^{2}}dx^{\prime}dz^{\prime},$
(S1)

with the vectors as $\textbf{J}=\left(0,J(x,z),0\right)^{T}$ and
$\textbf{r}=\left(x-x^{\prime},0,z-z^{\prime}\right)^{T}$ and $\mu_{0}$ being
the magnetic constant. The integration is performed over the cross-section of
the wire, of width $w$ and thickness $d$. We define the wire cross-section in
the x-z-plane, with $w$ in x-direction and $d$ in z-direction. The length of
the wire is along the y-direction. For a superconducting wire, the current is
not homogeneously distributed over the cross-section of the wire. Current is
only flowing on the surface and is exponentially decaying towards the center
of the wire. The characteristic length scale is given by the London
penetration depth $\lambda_{\mathrm{L}}$. We use the following expression for
the current distribution Lee, Orlando, and Lyons (1994)

$J(x,z)=J_{1}\left(\frac{\cosh{\nicefrac{{z^{\prime}}}{{\lambda_{\mathrm{L}}}}}}{\cosh{\nicefrac{{d}}{{\lambda_{\mathrm{L}}}}}}\left[C\frac{\cosh{\nicefrac{{x^{\prime}}}{{l_{1}}}}}{\cosh{\nicefrac{{w}}{{l_{1}}}}}+\frac{1-\nicefrac{{\cosh{\nicefrac{{x^{\prime}}}{{l_{2}}}}}}{{\cosh{\nicefrac{{w}}{{l_{2}}}}}}}{\sqrt{1-(\nicefrac{{x^{\prime}}}{{w}})^{2}}}\right]+\frac{J_{2}}{J_{1}}\frac{\cosh{\nicefrac{{x^{\prime}}}{{\lambda_{\mathrm{L}}}}}}{\cosh{\nicefrac{{w}}{{\lambda_{\mathrm{L}}}}}}\right),$
(S2)

where

$\displaystyle\frac{J_{2}}{J_{1}}$
$\displaystyle=\frac{1.008}{\cosh{\nicefrac{{d}}{{\lambda_{\mathrm{L}}}}}}\sqrt{\frac{\nicefrac{{w}}{{\lambda_{\bot}}}}{\nicefrac{{4*\lambda_{\bot}}}{{\lambda_{\mathrm{L}}}}}-0.08301\nicefrac{{\lambda_{\mathrm{L}}}}{{\lambda_{\bot}}}},$
$\displaystyle C$
$\displaystyle=\left(0.506\sqrt{\nicefrac{{w}}{{2\lambda_{\bot}}}}\right)^{0.75},$
$\displaystyle l_{1}$
$\displaystyle=\lambda_{\mathrm{L}}\sqrt{\nicefrac{{2\lambda_{\mathrm{L}}}}{{\lambda_{\bot}}}},$
$\displaystyle l_{2}$
$\displaystyle=0.774\nicefrac{{\lambda_{\mathrm{L}}^{2}}}{{\lambda_{\bot}}}+0.5152\lambda_{\bot},$
$\displaystyle\lambda_{\bot}$
$\displaystyle=\nicefrac{{\lambda_{\mathrm{L}}}}{{2d}}.$

The prefactors $J_{1}$ and $J_{2}$ define the amplitude of the current density
and hence the absolute value of the oscillating magnetic field $B_{1}$. We
define $J_{1}$ by normalizing the vacuum $B_{1}$ field to the energy density
stored in the resonator Zollitsch  _et al._ (2015); Weichselbaumer  _et al._
(2019)

$\frac{1}{2}\frac{\hbar\omega_{\mathrm{res}}}{2}=\frac{1}{2\mu_{0}}\int\textbf{B}_{1}^{2}dV=\frac{1}{2\mu_{0}}B_{1}^{2}V_{\mathrm{m}},$
(S3)

with $V_{\mathrm{m}}$ representing the resonator mode volume. The additional
factor of $\nicefrac{{1}}{{2}}$ on the left hand side of S3 takes into account
that only half of the total energy is stored in the magnetic field Schoelkopf
and Girvin (2008). As our resonator design is a quasi 1-dimensional structure
we have to define boundaries for the mode volume in the x- and z-direction. A
common assumption is to use the width of the conductor wire $w$ Schuster  _et
al._ (2007). For simplicity, we approximate the x-z-area of the mode
distribution with the area of an ellipse. For the last dimension we use the
length of the narrow wire section, supported by the CST Microwave Studio
simulations (see Fig. S2). In total we find the mode volume to be
$V_{\mathrm{m}}=((\pi 3.0\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}\times
2.025\,\mathrm{\SIUnitSymbolMicro}\mathrm{m})-w\times d)\times
300\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}=5696\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{3}$.
Figure S3 shows the resulting distribution of the oscillating magnetic field
for the cross-section of the rectangular wire of width
$w=2\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ and thickness
$d=50\,\mathrm{n}\mathrm{m}$. The magnitude
$\left|\textbf{B}_{1,\mathrm{x,z}}\right|$ is encoded in the color and the
arrows indicate the ${B}_{1,\mathrm{x}}$ and ${B}_{1,\mathrm{z}}$ components
of the oscillating field.

Figure S3: Cross-section of resonator magnetic field distribution. Calculated
magnitude of the magnetic field distribution around the cross-section of a
rectangular superconducting wire. The wire cross-section lies in the center,
indicated by the grey rectangular. The red arrows show the direction of the
magnetic field.

With the simulated $\textbf{B}_{1}$ field distribution we can calculate the
position dependent single photon - single spin coupling strength
$g_{0}(\textbf{r})$ Zollitsch  _et al._ (2015); Weichselbaumer  _et al._
(2019) for each magnetic moment per unit cell of CGT (ab-plane
$0.68\,\mathrm{n}\mathrm{m}$ Li  _et al._ (2018); Sun  _et al._ (2018), along
the c-axis $0.7\,\mathrm{n}\mathrm{m}$ Gong  _et al._ (2017)). Summation over
all CGT unit cells $N$ within the mode volume of the resonator results in the
collective coupling strength

$g_{\mathrm{eff}}=\sqrt{\sum_{i=1}^{\mathrm{N}}\left|g_{0}(\textbf{r}_{i})\right|^{2}}=\frac{g_{\mathrm{CGT}}\mu_{\mathrm{B}}}{2\hbar}\sqrt{\sum_{i=1}^{\mathrm{N}}\left|B_{1}(\textbf{r}_{i})\right|^{2}}=\frac{g_{\mathrm{CGT}}\mu_{\mathrm{B}}}{2\hbar}N_{\mathrm{y}}\sqrt{\sum_{i=1}^{\mathrm{N}}\left[(B_{\mathrm{x,}i}^{2}+B_{\mathrm{z,}i}^{2})\right]}.$
(S4)

Here, $\mu_{\mathrm{B}}$ is the Bohr magneton, $N_{\mathrm{y}}$ is the number
of unit cells along the y-direction and $g_{\mathrm{CGT}}$ is the g-factor for
CGT for which a value of $2.18$ Khan  _et al._ (2019) is used. Note, we give
the collective coupling strength for spin $\nicefrac{{1}}{{2}}$ and for linear
polarized microwaves Zollitsch  _et al._ (2015). For the calculation of
$g_{\mathrm{eff}}$ for the resonator loaded with 15 monolayers of CGT we
extracted its lateral dimension from the AFM measurements (see Fig. S3 (g)) to
$2\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ along the x-direction and
$12\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ along the y-direction. The flake
is assumed to lie directly on top of the superconducting wire without any gap
in between. For these values the simulation yields
$g_{\mathrm{eff}}/2\pi=8.94\,\mathrm{M}\mathrm{Hz}$, which is about a factor
$2.5$ larger than the experimentally determined value of
$3.61\,\mathrm{M}\mathrm{Hz}$. The overestimation of the simulation most
likely results from non-ideal conditions in the experiment. The corresponding
flake lies at the top end of the resonators narrow wire section (see Fig. S4
(g)), where $\textbf{B}_{1}$ is concentrated. The finite element simulations
show that in this area the field strength is already declining, resulting in a
reduced coupling strength. Further, AFM can overestimate the thickness of a
flake slightly for when there is a gap between resonator surface and flake
Gong  _et al._ (2017). The calculation also not includes the multiple peaks
observed in the experiment, which - depending on their real nature - can
distribute the magnon density over all resonant peaks. Nevertheless, we can
use the simulation to estimate the signal reduction by scaling down the
thickness of the flake to a single monolayer. Reducing the simulation to a
single monolayer, while keeping the lateral dimensions, results in
$g_{\mathrm{eff}}/2\pi=2.33\,\mathrm{M}\mathrm{Hz}$, a reduction by a factor
of $0.26$.

##  IV AFM Measurements on CGT Flakes

Figure S4: AFM measurements. AFM profile images with respective height profile
(above) along the resonator inductor wire (blue and purple lines in profile
images, with the arrow indicating scan direction). a \- c figures for
resonator chip 1 (refer to Tab. 1), having resonance frequencies with CGT of
$17237\,\mathrm{M}\mathrm{Hz}$, $17790\,\mathrm{M}\mathrm{Hz}$ and
$16470\,\mathrm{M}\mathrm{Hz}$, respectively. d \- f figures for resonator
chip 2 (refer to Tab. 2), having resonance frequencies with CGT of
$13244\,\mathrm{M}\mathrm{Hz}$, $12063\,\mathrm{M}\mathrm{Hz}$ and
$13620\,\mathrm{M}\mathrm{Hz}$, respectively. g \- i images for resonator chip
3 (refer to Tab. 3), having resonance frequencies with CGT of
$12314\,\mathrm{M}\mathrm{Hz}$, $13272\,\mathrm{M}\mathrm{Hz}$ and
$11899\,\mathrm{M}\mathrm{Hz}$, respectively. The red and green solid lines
are fits to the height profiles.

After the transfer of the CGT flakes onto the individual resonators and after
measuring FMR, we characterized the thickness of the flakes by AFM. Figure S4
shows a selection of height profile maps from the three resonator chips,
including a height profile along the inductor wire of the resonator (blue line
in the AFM profile images in Fig. S4). To extract the thickness we fit the
steps in the height profile (red or green lines in the height profiles in Fig.
S4). Note, the height values are relative values with an arbitrary offset.
Figure S4 (g) shows the thinnest flake of this study, where the processed FMR
data is shown in Fig. 4 in the main text.

##  V Analysis and Additional FMR Data

We analyze our experimental data, using the model functions (2) and (3) from
the main text in a two-step semi-optimized fashion. The main intention for
this approach is to minimize the number of free parameters in our model
functions. In a first coarse step, we match the collective coupling strength
$g_{\mathrm{eff},k}$ to fit the experimental data, assume a constant
separation between the individual magnon modes at $B_{\mathrm{FMR,k}}$ and the
same magnon loss rate $\gamma$ for all modes and determine the resonator loss
rate $\kappa_{0}$ from the resonator transmission far detuned from the FMR
with the CGT flakes. This results in 3 free parameters for the first stage of
our analysis, the magnon loss rate $\gamma$, $B_{\mathrm{FMR}}$ of the main
mode and the constant separation between the $B_{\mathrm{FMR,k}}$. After this
first step we arrive at a best fit to the envelope of the experimental data,
however with not matching amplitudes. In a consecutive second step, we
manually optimize the $g_{\mathrm{eff},k}$ to arrive at a model in good
agreement with $\omega_{\mathrm{res}}$ and $\kappa_{\mathrm{eff}}$ (see dashed
lines in Fig. S5).

Fig. S5 shows additional results from the corresponding FMR measurements
performed on the in Fig. S4 showed resonators. As described in the main text,
the measurements were performed at a temperature of $1.8\,\mathrm{K}$ and
recording the microwave transmission $\left|S_{21}\right|^{2}$ as a function
of the static magnetic field. Analyzing the microwave transmission by fitting
a Fano resonance lineshape to it we extract the effective loss rate of the
resonator, interacting with the CGT $\kappa_{\mathrm{eff}}$. Figure S5 shows
the resulting $\kappa_{\mathrm{eff}}$ as a function of the magnetic field. In
general, the response of the CGT FMR is complex and varies for the different
resonators. The resonance lineshape is not well described by just a single
Lorentzian and requires multiple peaks to produce a good agreement. For some
resonators, $\kappa_{\mathrm{eff}}$ exhibits obvious peaks, residing on a
broader spectrum (see Fig. S5 (c), (f) and (i)). Together with the observation
of well and clearly separated peaks for the resonator loaded with the thinnest
CGT flake of $11\,\mathrm{n}\mathrm{m}$, we motivating the multiple peak
analysis as presented in the main text. However, as the individual peaks are
overlapping for the remainder of the resonators we only applied a qualitative
analysis.

Figure S5: Additional data on magnon-photon coupling of CGT-resonator devices.
Results from FMR measurements with effective loss rate
$\kappa_{\mathrm{eff}}/2\pi$ as a function of the static magnetic field. a \-
c results for resonator chip 1 (refer to Tab. 1), having resonance frequencies
with CGT of $17237\,\mathrm{M}\mathrm{Hz}$, $17790\,\mathrm{M}\mathrm{Hz}$ and
$16470\,\mathrm{M}\mathrm{Hz}$, respectively. d \- f results for resonator
chip 2 (refer to Tab. 2), having resonance frequencies with CGT of
$13244\,\mathrm{M}\mathrm{Hz}$, $12063\,\mathrm{M}\mathrm{Hz}$ and
$13620\,\mathrm{M}\mathrm{Hz}$, respectively. g \- i results for resonator
chip 3 (refer to Tab. 3), having resonance frequencies with CGT of
$12314\,\mathrm{M}\mathrm{Hz}$, $13272\,\mathrm{M}\mathrm{Hz}$ and
$11899\,\mathrm{M}\mathrm{Hz}$, respectively. The orange solid lines are semi-
optimized fits, as described in the main text.

##  VI Magneto-Static Spin-Wave Dispersion in Thin-Film Magnets with
Perpendicular Anisotropy

Here we describe the spin-wave mode frequency in a thin-film magnet with
perpendicular anisotropy along the film normal. We consider this at the
magnetic-dipole limit where the wavelength is relatively large and the
exchange interaction contribution to the spin-wave dispersion is neglected.
Furthermore, standing spin-wave modes along the thickness direction are also
ruled out since these modes only appear at much higher frequencies than the
main mode, where we consistently observe additional peaks at both higher and
lower frequencies from the main mode. The mode (angular) frequency ($\omega$)
for wavevector $k=0$ when we apply a magnetic field $B$ along one of the film
plane directions can be given by Eq. 3d in Ref.Farle (1998) as:

$\left(\frac{\omega}{\gamma}\right)^{2}=B\left(B+\mu_{0}M_{\mathrm{s}}-\frac{2K_{\mathrm{u}}}{M_{\mathrm{s}}}\right).$
(S5)

Here, $\gamma$, $M_{\mathrm{s}}$ and $K_{\mathrm{u}}$ are the gyromagnetic
ratio, saturation magnetization and the perpendicular anisotropy energy
density, respectively. Note, that the total field within
$\mu_{0}M_{\mathrm{s}}-\frac{2K_{\mathrm{u}}}{M_{\mathrm{s}}}$ is negative for
perpendicularly-magnetized films which we consider in this section. Within the
magnetic-dipole limit, the demagnetization term $\mu_{0}M_{\mathrm{s}}$ is
modified for spin-waves with finite $k$, depending on the relative orientation
between the $M_{\mathrm{s}}$ and $k$ directions. Here we follow the expression
given in Serga et al. Serga, Chumak, and Hillebrands (2010). For pure backward
volume magnetostatic modes where $k\parallel M_{\mathrm{s}}$ (illustrated in
Fig. S6), the mode frequency becomes:

$\left(\frac{\omega_{\mathrm{BVMSW}}}{\gamma}\right)^{2}=B\left(B+\mu_{0}M_{\mathrm{s}}\left(\frac{1-e^{-kt}}{kt}\right)-\frac{2K_{\mathrm{u}}}{M_{\mathrm{s}}}\right),$
(S6)

where $t$ is the thickness of the magnet. Note, that this expression is only
valid for the case where $M_{\mathrm{s}}$ is colinear to $B$, meaning that
$|B|>|\mu_{0}M_{\mathrm{s}}-\frac{2K_{\mathrm{u}}}{M_{\mathrm{s}}}|$. To the
limit of $k\rightarrow 0$, the term $(1-e^{-kt})/kt$ is reduced to unity,
consistent to Eq. (S5). When $k$ is nonzero, we can observe that
$\omega_{\mathrm{BVMSW}}$ becomes smaller than that for $k$ = 0, exhibiting a
negative group velocity for this spin-wave mode. As the opposite extreme where
$k\perp M_{\mathrm{s}}$ (illustrated in Fig. S6), the resonance frequency
becomes larger than that for $k$ = 0 and is called magneto-static surface
spin-wave mode. The mode frequency expression for this mode is given by:

$\left(\frac{\omega_{\mathrm{MSSW}}}{\gamma}\right)^{2}=B\left(B+\mu_{0}M_{\mathrm{s}}-\frac{2K_{\mathrm{u}}}{M_{\mathrm{s}}}\right)+\mu_{0}^{2}M_{\mathrm{s}}^{2}\left(1-e^{-2kt}\right).$
(S7)

Here, $\mu_{0}^{2}M_{\mathrm{s}}^{2}\left(1-e^{-2kt}\right)$ is the spin-wave
correction term which goes to zero for $k\rightarrow 0$ (hence consistent to
Eq. (S5)) and becomes positive for $k>0$, meaning that
$\omega_{\mathrm{MSSW}}$ becomes larger as soon as spin-waves gain momentum
along this direction. We use these two expressions to in an effort to explain
the origin of the multiple peaks in our experiments.

Figure S6: Spin-wave dispersion. Spin-wave resonance frequency for BVMSW
(green solid line) and MSSW (yellow solid line) as a function of wavevector.
The dashed blue line is the resonance frequency of the $k=0$ main mode. The
parameters used are $B_{0}=598\,\mathrm{m}\mathrm{T}$,
$g_{\mathrm{CGT}}=2.18$, $\mu_{0}M_{\mathrm{s}}=194.3\,\mathrm{m}\mathrm{T}$
and $K_{\mathrm{u}}=3.84\times
10^{4}\,\nicefrac{{\mathrm{J}}}{{\mathrm{m}^{3}}}$. The grey area highlights a
$100\,\mathrm{M}\mathrm{Hz}$ margin relative to the main mode, indicating the
order of magnitude of the mode splitting observed in the experiment. The
arrows on the right hand side illustrate the relative wavevector orientations
of the BVMSW and MSSW spin-wave modes with respect to the static magnetic
field.

Figure S6 plots the calculated $\omega_{\mathrm{BVMSW}}/2\pi$ and
$\omega_{\mathrm{MSSW}}/2\pi$ as a function of wavevector $k$. The range of
wavevector is chosen such that the resulting resonance frequencies are within
the same order of magnitude as the observed mode splittings in the experiment
($\propto 100\,\mathrm{M}\mathrm{Hz}$). The corresponding wavelength to a
$100\,\mathrm{M}\mathrm{Hz}$ resonance offset to the main mode are about
$2.2\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ and $620\,\mathrm{n}\mathrm{m}$
for $\omega_{\mathrm{BVMSW}}$ and $\omega_{\mathrm{MSSW}}$, respectively.
These values are within a reasonable scale for our different lateral CGT flake
dimensions under investigation. This suggests that spin-wave modes are likely
the origin of the multiple resonance peaks observed.

##  VII Atomistic Spin Dynamics Simulations of FMR

To study the ferromagnetic resonance in CGT we perform atomistic spin dynamics
simulations Wahab  _et al._ (2021); Kartsev  _et al._ (2020). The magnetic
Hamiltonian employed in the simulations is given by:

$\mathcal{H}=-\frac{1}{2}\,\sum_{i,j}\mathbf{S}_{i}\mathcal{J}_{ij}\mathbf{S}_{j}\>-\sum_{i}D_{i}(\mathbf{S}_{i}\,\cdot\mathbf{e})^{2}-\sum_{i}\mu_{i}\mathbf{S}_{i}\,\cdot(\mathbf{B_{0}+B_{1}})$
(S8)

where $i$, $j$ represent the atoms index, $\mathcal{J}_{ij}$ represents the
exchange interaction tensor, $D_{i}$ the uniaxial anisotropy, which for CGT is
orientated out of plane ($\mathbf{e}=(0,0,1)$) and $\mathbf{B_{0}}$ the
external static magnetic field applied in-plane during the ferromagnetic
resonance simulations and $\mathbf{B_{1}}=B_{1}\sin(2\pi\nu t)$ the
oscillating field applied perpendicular with respect to $\mathbf{B_{0}}$. The
CGT system has been parameterized from first principle methodsGong  _et al._
(2017), up to the third nearest neighbor intralayer and interlayer exchange.
The exchange values have also been re-scaled by Gong et al. Gong  _et al._
(2017) with a 0.72 factor to obtain the experimental $T_{C}$ and multiplied by
$S^{2}$ to match the magnetic Hamiltonian. The magnetic moment or Cr is
considered 3.26 $\mu_{\mathrm{B}}$ Verzhbitskiy  _et al._ (2020) and the
uniaxial anisotropy has a value of $0.05$ meV as extracted from first
principle methods Gong  _et al._ (2017). The parameters used in the
simulations are given in Table 4.

Quantity | Symbol | quantity | units
---|---|---|---
Timestep | $ts$ | 0.1 | fs
Thermal bath coupling | $\alpha$ | 0.02 |
Gyromagnetic ratio | $\gamma_{e}$ | 1.760859 $\times$ $10^{11}$ | rad s-1T-1
Magnetic moment | $\mu_{B}$ | 3.26Verzhbitskiy  _et al._ (2020) | $\mu_{B}$
Uniaxial anisotropy | $D_{i}$ | 0.05 Gong  _et al._ (2017) | meV/link
Simulation temperature | T | 0.001 | K
Static magnetic field | $B_{0}$ | 0.9, 0.7 | T
Oscillating magnetic field amplitude | $B_{0}$ | 0.001 | T
FMR frequency | $\nu$ | varied | GHz
Intralayer exchange, NN | $J_{1}$ | 2.71 Gong  _et al._ (2017) | meV/link
Intralayer exchange, 2NN | $J_{2}$ | \- 0.058 Gong  _et al._ (2017) | meV/link
Intralayer exchange, 3NN | $J_{3}$ | 0.115 Gong  _et al._ (2017) | meV/link
Interlayer exchange, NN | $J^{z}_{1}$ | -0.036 Gong  _et al._ (2017) | meV/link
Interlayer exchange, 2NN | $J^{z}_{2}$ | 0.086 Gong  _et al._ (2017) | meV/link
Interlayer exchange, 3NN | $J^{z}_{3}$ | 0.27 Gong  _et al._ (2017) | meV/link

Table 4: Simulation parameters for FMR on CGT system

.

FMR calculations have previously been employed for atomistic models, and can
reproduced well the variation of linewidth with temperature, for example, in
recording media systems Strungaru  _et al._ (2020). Hence, in the current
simulations we use the same setup of frequency swept FMR Strungaru  _et al._
(2020) and we obtain the spectra by performing a Fourier transform of the
magnetisation component parallel to the oscillating field. Since these
calculations are done close to 0K, no averaging is require to reduce the
thermal noise. To excite the FMR mode, we apply a DC field in-plane of 0.9 T
on x-direction and an AC field perpendicular to the DC field, on y-direction.
The Fourier transform has been performed for the y-component of magnetisation
for $5\,\mathrm{n}\mathrm{s}$ after an initial $1\,\mathrm{n}\mathrm{s}$
equilibration time. A thermal bath coupling has been chosen in agreement with
the upper limit of the Gilbert damping observed in experiments.

The system size we performed FMR on is a 4-layer CGT system, with lateral size
of $6.91\,\mathrm{n}\mathrm{m}\times 11.97\,\mathrm{n}\mathrm{m}$, periodic
boundary conditions in xy and total of 1600 atoms. The small system size has
been used to reduce the computational cost associated with FMR simulations.
Experiments have showed modified g-factors due to photon-magnon coupling hence
hereby we propose a simple model where the properties of the individual layers
have been modified to include different gyromagnetic ratio, as illustrated in
Fig. S7a.

We can define the resonance frequencies for each magnetic layer using the
Kittel equation in the case of in-plane applied field with perpendicular
anisotropy $B_{\perp\mathrm{u}}$:

$\omega=\gamma\sqrt{B_{0}(B_{0}-B_{\perp\mathrm{u}})}$ (S9)

We next investigate the FMR signal for a few cases assuming the CGT monolayers
at low or strong interlayer exchange couplings
$J_{z}^{\prime}=0,0.1\%,10\%,100\%J_{z}$, where $J_{z}$ corresponds to the
pristine interlayer exchange (Fig. S7b-c). In the low interlayer exchange
regime ($J_{z}^{\prime}=0,0.1\%J_{z}$), the CGT presents multiple peaks with
each frequency corresponding to the layer dependent gyromagnetic ratio,
$\gamma$ \- $\nu(\gamma_{1})=16.81\,\mathrm{G}\mathrm{Hz}$,
$\nu(\gamma_{2})=25.22\,\mathrm{G}\mathrm{Hz}$,
$\nu(\gamma_{3})=33.62\,\mathrm{G}\mathrm{Hz}$. At
$J^{\prime}_{z}=0.1\%J_{z}^{\prime}$ (Fig. S7b) we can still observe resonance
peaks corresponding to each individual layer. However by increasing the
exchange coupling to $10\%J_{z}^{\prime}$ or higher (Fig. S7c) there is a
single FMR peak indicating that the system behave coherently with all layers
having the same FMR frequency. The single FMR frequency corresponds to the
average magnetic properties of the CGT layers. Small variations of the
resonance frequency as function of the inter-layer exchange coupling can be
observed which these being correlated to the transition of the system from the
multi-peaks regime to a coherent excitation. By calculating the damping of the
highest resonance peaks from a Lorenzian fit, we reobtain the damping
corresponding to the input thermal bath coupling, $0.02$ with a relative tinny
error $\sim 5\%$. Overall, the interlayer exchange coupling locks the dynamics
of individual layers coherently together without allowing multiple frequencies
at the FMR signaldat .

Figure S7: Atomistic simulations. a, Schematic of the crystal structure of CGT
with atoms defined by different colours. b, FMR spectra of 4 layer CGT where
the layers are low interayer exchange coupled ($0,0.1\%~{}J_{z}^{\prime}$,
where $J_{z}^{\prime}$ is the pristine CGT interlayer exchange). c, Similar as
b, but with the layers at a strong exchange coupling
($10\%,100\%~{}J_{z}^{\prime}$). The solid lines in b-c represent a Lorenzian
fit to the numerical data.

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