**footnotetext: These authors contributed equally to this work

# Hybrid magnetization dynamics in
$\text{Cu}_{\text{2}}\text{OSeO}_{\text{3}}$/NiFe heterostructures

Carolina Lüthi∗ Walther-Meißner-Institut, Bayerische Akademie der
Wissenschaften, Garching, Germany  Physics Department, Technical University of
Munich, Garching, Germany  Luis Flacke∗ Walther-Meißner-Institut, Bayerische
Akademie der Wissenschaften, Garching, Germany  Physics Department, Technical
University of Munich, Garching, Germany  Aisha Aqeel  Physics Department,
Technical University of Munich, Garching, Germany  Munich Center for Quantum
Science and Technology (MCQST), Munich, Germany  Akashdeep Kamra  Condensed
Matter Physics Center (IFIMAC) and Departamento de Física Teórica de la
Materia Condensada, Universidad Autónoma de Madrid, Madrid, Spain  Rudolf
Gross  Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften,
Garching, Germany  Physics Department, Technical University of Munich,
Garching, Germany  Munich Center for Quantum Science and Technology (MCQST),
Munich, Germany  Christian Back  Physics Department, Technical University of
Munich, Garching, Germany  Munich Center for Quantum Science and Technology
(MCQST), Munich, Germany  Mathias Weiler<EMAIL_ADDRESS>Walther-
Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany
Physics Department, Technical University of Munich, Garching, Germany
Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technical University
of Kaiserslautern, Kaiserslautern, Germany

###### Abstract

We investigate the coupled magnetization dynamics in heterostructures of a
single crystal of the chiral magnet $\mathrm{Cu_{2}OSeO_{3}}$ (CSO) and a
polycrystalline ferromagnet $\mathrm{NiFe}$ (Py) thin film using broadband
ferromagnetic resonance (FMR) at cryogenic temperatures. We observe the
excitation of a hybrid mode (HM) below the helimagnetic transition temperature
of CSO. This HM is attributed to the spin dynamics at the CSO/Py interface. We
study the HM by measuring its resonance frequencies for in plane rotations of
the external magnetic field. We find that the HM exhibits dominantly four-fold
anisotropy, in contrast to the FMR of CSO and Py.

††preprint: AIP/123-QED

Chiral magnets exhibit non-collinear spin structures such as spin helices and
magnetic skyrmions below their critical temperature $T_{c}$ and critical field
$H_{c2}$ Stasinopoulos  _et al._ (2017a). Skyrmions are topologically
protected non-coplanar magnetization configurations that can behave as
particle-like objects. Furthermore, they are small yet stable making them
suitable to become the carriers of information in future devices Back  _et
al._ (2020); Okuyama  _et al._ (2019); Legrand  _et al._ (2017); Woo  _et al._
(2016); Jiang  _et al._ (2015); Fert, Cros, and Sampaio (2013); Sampaio  _et
al._ (2013); Jonietz  _et al._ (2010); Yu  _et al._ (2012, 2010); Muhlbauer
_et al._ (2009). The non-collinear spin structure of chiral magnets gives rise
to intriguing magnetization dynamics, in particular in their skyrmion lattice
phase Schwarze  _et al._ (2015). The recently discovered low-temperature
skyrmion phase Chacon  _et al._ (2018) leads to additional striking spin
dynamical signatures Aqeel  _et al._ (2021) in the low-damping Stasinopoulos
_et al._ (2017b) chiral magnet CSO. The periodicity of the magnetic lattice
leads to naturally formed magnonic crystals Weiler  _et al._ (2017) and the
skyrmion eigenmodes can be coupled to photonic resonators with high
cooperativity Liensberger  _et al._ (2021). The chiral properties of skyrmions
give rise to non-reciprocal spin-wave dynamics Seki  _et al._ (2020). Thus,
the emerging field of spin dynamics of chiral magnets has already revealed
important fundamental insights with perspectives for practical applications.

In topologically trivial magnets, the now well studied coupling between
multiple magnetic layers Bruno (1995); Crew  _et al._ (2003); Gonzalez-Chavez
_et al._ (2013); Heinrich  _et al._ (2003); McMichael  _et al._ (1998);
Schäfer  _et al._ (2012); Woltersdorf  _et al._ (2007) resulted in the
discovery of some of the most technologically relevant effects such as
tunneling magnetoresistance Julliere (1975) or giant magnetoresistance Baibich
_et al._ (1988); Binasch  _et al._ (1989). Heterostructures of topologically
trivial magnets can exhibit coupled spin dynamics that can lead to, e.g.,
excitation of nanoscale spin waves Klingler  _et al._ (2018). Much less is
known about spin dynamics in heterostructures of collinear and chiral magnets.
The coupling between distinct order parameters across interfaces has explained
important phenomena such as proximity effects, exchange bias or exchange
spring-induced hard magnets Hellman  _et al._ (2017). However, the studies of
excitations in chiral magnets are so far limited to a single magnetically
ordered constituent Takeuchi, Mizushima, and Mochizuki (2019). Even though the
formation of novel topological order at the chiral magnet/ferromagnet
interface was predicted by theory it has not yet been observed in experiment
Kawaguchi, Tanaka, and Nagaosa (2016).

In this work, we investigate the hybrid magnetization dynamics of
heterosturctures of thin film metallic ferromagnets and bulk chiral magnets,
in this case the ferrimagnetic insulator $\mathrm{Cu_{2}OSeO_{3}}$ (CSO). To
study the magnetization dynamics of the chiral magnet/ferromagnet
heterostructures in the GHz frequency regime we use broadband ferromagnetic
resonance spectroscopy. We experimentally determine and phenomenologically
model the resonance frequencies in such heterostructures. Thereby, we find
that a hybrid mode of the chiral magnet/ferromagnet heterostructure is
excited, which we attribute to the spin dynamics at the interface of the two
magnetic layers.

Figure 1:  (a) Experimental setup: The (111)-oriented
$\mathrm{Cu_{2}OSeO_{3}}/\mathrm{NiFe}$ heterostructure is placed on the CPW,
which generates a magnetic field ($\mathbf{h}$) within the sample due to the
application of an ac current flowing from port 1 (P1) to port 2 (P2) of the
center conductor. (b) Schematic phase diagram of the $\mathrm{Cu_{2}OSeO_{3}}$
crystal defining the helimagnetic transition temperature $T_{c}=58.2$ K Qian
_et al._ (2016) as well as the critical fields $\mu_{0}H_{c1}$ and
$\mu_{0}H_{c2}$ (H: helical state, C: conical state, S: skyrmionic state, F:
ferrimagnetic state). The vertical dashed line indicates the phases of the CSO
at 5 K in dependence of the external field.

We investigate a $\mathrm{CSO/Ni_{80}Fe_{20}}$ (CSO/Py) sample, where the Py
thin film has a thickness of 40 nm. The CSO crystal is (111)-oriented and cut
to a cuboid shape with dimensions $L_{x}=2.5$ mm, $L_{y}=1.5$ mm, and
$L_{z}=0.8$ mm. It was grown by a chemical vapor transport method Aqeel  _et
al._ (2021)(for details on the crystal orientation of the CSO and the sample
preparation see Supplemental Material I and II . ). We place the CSO/Py hybrid
on top of a coplanar waveguide (CPW) with a center conductor width of
$w=127\mu$m as shown in Fig. 1 (a). The CPW is connected to two ports P1 and
P2 of a vector network analyzer (VNA), which measures the change of
transmission from P1 to P2 defined as the complex transmission parameter
$S_{21}$ as a function of frequency $f$ and external magnetic field $\mu_{0}H$
at a fixed microwave power of 1 mW (0 dBm). We then place the CPW/CSO/Py
assembly into the variable temperature insert of a superconducting 3D-vector
magnet. By applying a static external magnetic field $\mu_{0}\mathbf{H}$ in
the plane of the Py thin film and setting the temperature to 5 K we can access
the helical $(H)$, conical $(C)$, and ferrimagnetic $(F)$ phases of the CSO as
schematically depicted in Fig. 1 (b).

Figure 2:  Measured broadband ferromagnetic resonance spectrum of the CSO/Py
sample at 5 K as a function of the frequency $f$. The fixed external magnetic
field $\mu_{0}\mathbf{H}=120$ mT is applied along the $\mathbf{x}$-axis
($\phi_{H}=0^{\circ}$) as indicated in the sample sketches. (a) The CSO
crystal faces the CPW. Thus, only ferrimagnetic CSO modes apppear (grey marked
frequency range). (b) The Py thin film faces the CPW. In addition to the CSO
modes the Py FMR mode appears at high frequencies as well as a hybrid mode
(HM) at medium frequencies (inset). This HM is attributed to the spin dynamics
at the interface of the CSO/Py sample, indicated as an interfacial layer in
the sample sketch.

As a reference measurement, we first place the CSO/Py hybrid sample on the CPW
with the Py facing away from the CPW. Due to the large thickness of the CSO
layer, the oscillating magnetic field generated by the CPW does not reach the
Py layer. In this way, we only excite the magnetization in CSO itself with no
influence of the Py layer. We apply a fixed external magnetic field
$\mu_{0}H~{}=~{}120$ mT along the $\mathbf{x}$-axis
($\phi_{H}~{}=~{}0^{\circ}$) at 5 K, as illustrated schematically in the top
panel of Fig. 2 (a). For this temperature and external magnetic field strength
the CSO magnetization is in the field polarized phase. To correct for the
microwave background of the complex transmission parameter $S_{21}$ we use the
derivative divide method Maier-Flaig  _et al._ (2018), to obtain the field
derivative of the complex transmission parameter
$\frac{\mathrm{\partial}S_{\mathrm{21}}}{\mathrm{\partial}H}$ devided by
$S_{21}$. On the bottom panel of Fig. 2 (a)
$\mathrm{Re}(\mathrm{\partial}_{\mathrm{D}}S_{\mathrm{21}}/\mathrm{\partial}\mathrm{H})$
for the CPW/CSO/Py assembly at 5 K is shown as a function of the frequency
$f$. In the grey marked frequency range several resonances appear. These are
attributed to the excitation of magnetostatic modes of the cuboid-shaped CSO
crystal. In the frequency range 7 GHz < $f$ < 12 GHz no additional modes are
observed (inset). After determining the response of the isolated CSO
magnetization dynamics, we place the CSO/Py hybrid on the CPW with the Py
facing the CPW and again apply a fixed magnetic field $\mu_{0}H=120$ mT along
the $\mathbf{x}$-axis as schematically shown in the top panel of Fig. 2 (b).
Now, the field generated by the CPW interacts with the Py layer as well as the
CSO as the Py layer is a thin film. In the bottom panel of Fig. 2 (b),
$\mathrm{Re}(\mathrm{\partial}_{\mathrm{D}}S_{\mathrm{21}}/\mathrm{\partial}\mathrm{H})$
at 5 K is shown for the CPW/Py/CSO assembly as a function of the frequency
$f$. In addition to the resonance lines of CSO (grey marked frequency range)
also the Py FMR line appears close to 10 GHz as expected. The change in the
CSO mode spectrum is attributed to the presence of the metallic film and
concomitant shielding of the microwave field in the bulk of CSO. Furthermore,
we observe an additional medium frequency mode, which is shifted by about 1
GHz to lower frequencies than the Py FMR mode. This hybrid mode (HM) is also
observed in the CSO/Py hybrid in the conical phase of CSO (see Fig. 6 in
Supplemental Material . ). For the dependence of the HM on the magnitude of
the external magnetic field see Supplemental Material III . . We attribute the
appearance of this additional mode to the spin dynamics at the CSO/Py
interface. Thus, in a simplified macrospin picture, we may treat our bilayer
as a trilayer with a new interfacial layer inheriting properties from both
sides. The HM is then modelled as a result of macrospin dynamics of the
interlayer as discussed in the following.

Figure 3: Fits of the frequency spectrum for a fixed magnitude of the external
field $\mu_{0}\mathrm{H}~{}=~{}120$ mT at 5 K. The magnetic field direction is
rotated in the Py thin film plane and defined by the angle $\phi_{H}$ with
respect to the $\mathbf{x}$-axis as indicated in the sketches of the sample.
The blue dots correspond to the measurement data and the orange line to the
fit. (a) $\phi_{H}~{}=~{}0^{\circ}$. The peak of the HM at 9.2 GHz and the Py
FMR peak-dip at 10.4 GHz are clearly separable. (b)
$\phi_{H}~{}=~{}30^{\circ}$. The two resonance frequencies of the Py mode and
the HM are less separated. (c) $\phi_{H}=60^{\circ}$. The two resonance
frequencies of the Py mode and the HM are not separable any more. Figure 4:
(a) and (b) Dependence of the Py (blue dots), CSO (green dots), and HM (orange
dots) resonance frequencies on the external magnetic field direction for a
360° rotation in the Py thin film plane and a fixed magnitude
$\mathrm{\mu_{0}H}~{}=~{}120$ mT at 5 K (a) and 50 K (b). The green shaded
range around the CSO resonance frequencies indicates the frequency
distribution of the magnetostatic CSO modes. The fit errors are smaller than
the symbol size. The solid lines show the simulation result according to Eq.
(1) of the Py (blue line), CSO (green line), and HM (orange line) resonance
frequencies which are in good agreement with the measurement data. (c) Fit
result of the Py (red dots) and HM (blue dots) resonance frequencies in
dependence of the temperature with fixed external field
$\mathrm{\mu_{0}H}~{}=~{}120$ mT applied along the $\mathbf{x}$-axis
($\phi_{H}~{}=~{}0$). Above the critical temperature $T_{c}=58.2$ K of the CSO
the HM mode vanishes.

To investigate the dependence of the HM on the direction of the external
magnetic field we apply a field with a fixed magnitude of $\mu_{0}H~{}=~{}120$
mT and rotate the field direction by $360^{\circ}$ in the Py film plane. For a
quantitative analysis of the HM we simultaneously fit the Py FMR peak and the
HM peak in the frequency domain of the transmission parameter $S_{21}$ for
each fixed external field direction (for detailed informations on the fit
model see Supplemental Material IV . ). In Fig. 3 three exemplary fits of the
frequency spectrum are shown for a fixed magnitude of the field $\mu_{0}H=120$
mT and for different directions under which it was applied. In Fig. 3 (a) the
external field is applied in the Py thin film plane along the
$\mathbf{x}$-axis as schematically depicted at the top of Fig. 3 (a). We find
the peak of the hybrid mode at 9.2 GHz, which has a small amplitude compared
to the Py FMR peak-dip at 10.4 GHz. In Fig. 3 (b) we show the fit result for
the field applied under an angle $\phi_{H}~{}=~{}30^{\circ}$ with respect to
the $\mathbf{x}$-axis as depicted at the top of Fig. 3 (b). Now, the two
resonance frequencies of the Py mode and the HM are less separated than in
Fig. 3 (a), as the hybrid mode moved to higher frequencies and the Py mode to
lower frequencies. In Fig. 3 (c) the field is applied under an angle
$\phi_{H}~{}=~{}60^{\circ}$. The two resonance frequencies of the Py mode and
the HM are not separable anymore as the HM resonance frequency is presumably
superimposed on the Py resonance frequency. Furthermore, the amplitude of the
FMR signal decreases in Fig. 3 (b) and (c). As the applied field shifts away
from the $\mathbf{x}$-axis, so does the equilibrium magnetization. As a
result, the CPW oscillating field component transverse to the static
magnetization also diminishes. This results in a reduction of the recorded FMR
signal amplitude.

To better understand the behaviour of the spin dynamics in the sample in
dependence of the external magnetic field and temperature we fit the resonance
frequencies of each layer for a full rotation of the external field applied in
the Py thin film plane and we evaluate the dependence of the HM resonance
frequency on temperature by fitting its resonance frequency for a fixed field
direction at different temperatures. In Fig. 4 (a) the fitted resonance
frequencies at 5 K of the Py FMR mode (blue dots), the CSO modes (green
shading indicates the frequency range over which the CSO magnetostatic modes
are observed) and the hybrid mode (orange dots) are plotted against the angle
$\phi_{H}$. In the fit model, we only consider the CSO mode with the strongest
amplitude (green dots). Furthermore, for $\phi_{H}\simeq$ 90° we are not able
to fit the HM, as it either vanishes or merges with the Py FMR mode. Due to
the demagnetization field in the CSO crystal the CSO resonance frequencies
show a cosine like dependence on the angle under which the external field is
applied. Furthermore, we observe an uniaxial anisotropy in the Py resonance
frequencies. This uniaxial anisotropy is strongest for 5 K and weakens for
increasing temperatures (See Fig. 4 (b) and Supplemental Material VI . ).
Additionally, a broadening of the Py resonance minima and a narrowing of the
respective peaks is observed. The HM shows a striking feature with its angle
dependence being inverted with respect to the CSO and Py modes. Its presence
and frequency difference with respect to the Py mode at $\phi_{H}=0$ vanishes
when surpassing the CSO ordering temperature as shown in Fig. 4 (c).

The angle and temperature dependence of the observed modes reveals a complex
interaction within the heterostructure. The following explanation attempts
expose the non-trivial nature of the system by leading to major discrepancies
when assuming simple models with isotropic interaction parameters.

First, we start by noting the effect of CSO on the system. By increasing the
temperature a reduced uniaxial anisotropy not only of the CSO but also of the
Py and HM mode becomes evident (see Fig. 4 (a) and (b) and Supplemental
Material VI . ). This reduction agrees with a decreased spin ordering in CSO
at higher temperatures, such that the effect of magnetic stray field of the
bulk CSO crystal on the Py layer can be assumed to be the dominant effect seen
by the uniaxial anisotropy in the Py mode (see calculation of the anisotropy
field according to Ref. Engel-Herbert and Hesjedal, 2005 in the Supplemental
Material VI).

It also supports our inference that CSO plays the main role in determining the
frequency of the HM. In the following, we argue that the HM is indeed the
result of a coupling of the multilayer system. Figure 4 (c) visualizes the
dependence of the Py and the HM resonance frequencies on temperature where the
external field has a magnitude of 120 mT and is applied under an angle
$\phi_{H}~{}=~{}0$°. With increasing temperature the HM mode approaches the Py
mode until it vanishes above 58 K. Thus, above the critical temperature of
CSO, the coupling of the CSO and Py dynamics vanishes and the HM cannot be
excited any more. This demonstrates that the existence of the HM is a direct
consequence of the magnetic ordering of CSO. We thus exclude non-uniformity of
the Py layer as its origin. Its inverted angular dependence further
substantiates this conclusion as otherwise equal symmetry with respect to the
Py and CSO modes can be expected.

Assuming exchange interaction and spin torques at the interface similar as in
Ref. Klingler  _et al._ , 2018 and including such terms in the Landau-
Lifshitz-Gilbert (LLG) equation would result in four solutions for resonance
frequencies. Namely, the uncoupled CSO and Py layer, as well as two coupled
modes. It might be possible that the second coupled mode is hidden in the
series of magnetostatic modes of the CSO. But in this case, again, the
solutions for the hybrid modes would only lead to a constant shift of the HM
frequencies compared to the Py resonance frequencies without an angle
dependent gap between the Py and the HM resonance frequencies. Nonetheless, if
the exchange interaction was to be anisotropic, the additional degree of
freedom could result in the observed symmetry of the HM.

CSO is known for its intrinsic symmetry breaking and the existence of
Dzyaloshinskii-Moryia interaction (DMI). The Py thin film experiences a
symmetry breaking at the interface with CSO and a residual effect of the DMI
could influence the magnetization dynamics of Py at the interface resulting in
a virtual additional layer. As the DMI favors a perpendicular alignment the
excitation of magnetization is favoured and its interaction would reduce the
resonance frequency due to reduced total energy Garst, Waizner, and Grundler
(2017). However, also in this case, an anisotropic interaction parameter has
to be assumed to explain the angle dependence as otherwise a constant
frequency shift is to be expected again Garst, Waizner, and Grundler (2017).
On the other hand, if we consider the HM to possess a nonzero inplane
wavevector, it would not be influenced by the uniaxial shape anisotropy that
appears to dominate the Py and CSO resonance frequencies. The angle-dependence
of the HM is still not fully captured in that case, and an additional origin
needs to be found for it.

Another observation which makes a clear interpretation even more difficult is
the broadening/narrowing of the Py resonance minima/maxima. Such a behaviour
is usually seen in material systems with a cubic, i.e., four-fold symmetry
superimposed with the previously described uniaxial anisotropy. This
contradicts the three-fold symmetry that is expected from the
(111)-orientation of the CSO crystal. Naively, a cubic magnetocrystalline
anisotropy in the Py layer could explain the observation Liensberger  _et al._
(2019) . Yet it is not clear why such a crystalline structure should have
developed when considering either the (111)-oriented substrate or the
continuous rotation of the sample during the deposition process.

The chiral magnet/ferromagnet heterostructure proves to be a highly complex
material system raising further questions about anisotropic interaction
parameters and fundamental symmetry manifestations that go beyond the scope of
this work. To phenomenologically model the data in Fig. 4 (a) and (b) we use a
Landau-Lifshitz-Gilbert (LLG) approach, where we treat the magnetization
dynamics in the sample as a three-layer system consisting of the uncoupled Py
and CSO layer as well as an interfacial layer as shown at the top of Fig. 2
(b). Thereby, the magnetization dynamics of each of these three layers are
treated as macrospins. The equation of motion for magnetization
$\mathbf{M}_{i}$ then reads

$\displaystyle\frac{\mathrm{d}\mathbf{M}_{i}}{\mathrm{d}t}=$
$\displaystyle-\gamma_{i}\mathbf{M}_{i}\times\mu_{0}\mathbf{H}_{\mathrm{eff}_{i}}(\phi_{H})+\frac{\alpha_{i}}{M_{s_{i}}}\mathbf{M}_{i}\times\frac{\mathrm{d}\mathbf{M}_{i}}{\mathrm{d}t}.$
(1)

Here, $\gamma_{i}$ is the gyromagnetic ratio and $\alpha_{i}$ the Gilbert
damping parameter of layer $i$. For all three magnetic layers the effective
field $\mathbf{H}_{\mathrm{eff}_{i}}$ accounts for the external field, the
driving field, and the demagnetization field. In case of the Py layer
$\mathbf{H}_{\mathrm{eff}}$ additionally includes a phenomenological uniaxial
magnetic anisotropy, which is expected due to the demagnetization fields of
the CSO and can be regarded effectively as a shape anisotropy of the CSO.
However, also a finite contribution due to the trapped flux in the
superconducting coils is to be expected (See Supplemental Material VI) . . For
both the Py layer and the interfacial layer we also include a phenomenological
anisotropy with a four-fold symmetry in accordance with our experimental
observation. We note, again, that the four-fold symmetry does not correspond
to the cubic anisotropy of CSO, because the projection of the cubic anisotropy
to the (111) plane would result in a three-fold symmetry. The effective field
of layer $i$ is given by the Eqs. (7), (8), and (12) in Supplemental Material
VII . . In Fig. 4 (a) and (b) the Py (blue line), HM (orange line), and CSO
(green line) resonance frequencies simulated with Eq. (1) are in good
agreement with the measurement data (symbols).

In conclusion, we investigated the coupled magnetization dynamics in a CSO/Py
heterostructure by broadband ferromagnetic resonance experiments at cryogenic
temperatures. We found that a hybrid mode at the CSO/Py interface is excited.
While a microscopic picture for the formation of the HM is so far missing, our
experimental findings pave the way for future experiments on coupled spin
dynamics in topologically non trivial magnetic bilayer systems, which have
recently attracted great interest from the viewpoint of possible applications
in high-performance memory devices Takeuchi, Mizushima, and Mochizuki (2019);
Soumyanarayanan  _et al._ (2016); Fert, Cros, and Sampaio (2013).

###### Acknowledgements.

We gratefully acknowledge the financial support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) via WE 5386/5-1 and
BA 2181/19-1 and financial support from the Spanish Ministry for Science and
Innovation – AEI Grant CEX2018-000805-M (through the “Maria de Maeztu”
Programme for Units of Excellence in R&D). We would also like to thank M.
Müller for support in carrying out some of the measurements.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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## Supplementary information for the article
Hybrid magnetization dynamics in
$\text{Cu}_{\text{2}}\text{OSeO}_{\text{3}}$/NiFe heterostructures

##  I CSO crystal orientation

In Fig. 5 the back-reflection Laue image of the (111)-oriented CSO crystal is
shown. The red dashed lines indicate the threefold symmetry of the Laue
pattern.

Figure 5: Laue pattern from back-reflection topograph of the (111)-oriented
CSO crystal.

##  II Sample preparation

Before sputtering the Py layer on top of the CSO crystal, we polished its
surface and cleaned it in an acetone bath. After sputtering, we annealed the
sample in vacuum at $T=250^{\circ}$ for 5 minutes. The parameters used during
the sputter process are listed in table 1.

Table 1: Sputtering parameters of the NiFe grown on CSO. The sample was grown at a pressure $p=5\times 10^{-3}$ mbar. Target material | power (W) | rate ($$\mathrm{\SIUnitSymbolAngstrom}$/$s) | time (s) | type
---|---|---|---|---
NiFe | 15 | 1.7 | 235 | face-to-face

##  III Dependence of the hybrid mode on the external magnetic field and the
temperature

In Fig. 6 the background-corrected field-derivative [Maier-Flaig  _et al._ ,
2018] of the VNA transmission spectra
$\mathrm{Re}(\mathrm{\partial}_{\mathrm{D}}S_{\mathrm{21}}/\mathrm{\partial}\mathrm{H})$
at 5 K is shown as a function of the frequency $f$ and the external magnetic
field amplitude $H$. The external field $\mathbf{H}$ is applied along the
$\mathbf{x}$-axis ($\phi_{H}~{}=~{}0$). The helical (H), conical (C), and
ferrimagnetic (F) phases of the CSO can be observed. The hybrid mode (HM)
clearly appears in the conical and ferrimagnetic phases of the CSO.

Figure 6:
$\mathrm{Re}(\mathrm{\partial}_{\mathrm{D}}S_{\mathrm{21}}/\mathrm{\partial}\mathrm{H})$
at 5 K as a function of the frequency $f$ and the external magnetic field $H$.
$H$ is applied along the $\mathbf{x}$-axis ($\phi_{H}~{}=~{}0$). Approximate
phase transitions of the helical (H), conical (C), and ferrimagnetic (F)
phases are marked by the dashed vertical lines. The hybrid mode (HM) appears
in the conical and ferrimagnetic phase of the CSO. Figure 7: Background-
corrected field-derivative Maier-Flaig  _et al._ (2018) of the VNA
transmission spectra
$\mathrm{Re}(\mathrm{\partial}_{\mathrm{D}}S_{\mathrm{21}}/\mathrm{\partial}\mathrm{H})$
at different temperatures as a function of the frequency $f$ and the external
magnetic field $H$ in the ferrimagnetic phase of the CSO. The external field
is applied along the $\mathbf{x}$-axis. (a) and (b) The Py FMR mode, the HM
mode and the CSO modes appear for temperatures beneath the critical
temperature $T_{c}$ of the CSO. (c) For temperatures higher than $T_{c}$ the
HM does not appear as CSO is not in the ferrimagnetic phase any more.

In Fig. 7
$\mathrm{Re}(\mathrm{\partial}_{\mathrm{D}}S_{\mathrm{21}}/\mathrm{\partial}\mathrm{H})$
for different temperatures is shown as a function of the frequency $f$ and the
external field $H$ in the ferrimagnetic phase of the CSO. Fig. 7 (a) shows the
transmission spectra at 5 K. The variation of the HM mode frequency with
$\mathbf{H}$ is parallel to that of the Py FMR mode. It is shifted about 1 GHz
to lower frequencies than the Py FMR mode. The low frequency modes are
attributed to magnetostatic modes of the CSO crystal. Fig. 7 (b) shows the
transmission spectra at 20 K. The HM still appears with a dispersion parallel
to the Py FMR mode. Fig. 7 (c) shows the transmission spectra at 70 K. For
temperatures above the critical temperature of the CSO the HM does not appear.

##  IV Fitting method of the FMR peaks

The Py FMR peak and the HM peak were simultaneously fitted in the frequency
domain of the background corrected field-derivative of the transmission
parameter $S_{21}$ for each fixed external field direction. As a fit model we
used the central difference quotient $S_{21}$ given by Maier-Flaig  _et al._
(2018)

$d_{D}S_{21}=i\omega
Ae^{-i\phi}\frac{\chi(\omega+\Delta\omega_{\pm})-\chi(\omega-\Delta\omega_{\pm})}{2\Delta\omega_{\pm}}+S_{21_{b}},$
(2)

where $S_{21_{b}}$ corrects for the offset and the slope of the background of
the $S_{21}$ signal, $A$ is the modified signal amplitude, which also takes
the partial derivative $\frac{\partial\omega}{\mathrm{d}H}$ into account and
$\Delta\omega_{\pm}=\Delta H_{\pm}\frac{\partial\omega}{\partial
H_{\mathrm{ext}}}\approx\partial H_{\mathrm{ext}}\gamma\mu_{0}$ is the
modulation amplitude with the field step width $\Delta H_{\pm}$. The high
frequency magnetic susceptibility $\chi$ is given by Maier-Flaig  _et al._
(2018)

$\chi(\omega,H_{\mathrm{ext}})=\frac{\omega_{M}(\gamma\mu_{0}H_{\mathrm{ext}}+i\Delta\omega)}{(\omega_{\mathrm{res}}(H_{\mathrm{ext}}))^{2}-\omega^{2}+i\omega\Delta\omega},$
(3)

where $\omega_{\mathrm{res}}$ is the resonance frequency, $\Delta\omega$ the
full width at half maximum linewidth, and
$\omega_{M}~{}=~{}\gamma~{}\mu_{0}~{}M_{s}$. The ferrimagnetic CSO modes were
simultaneously fitted as one mode only with the fit model given in Eq. (2). To
determine the boundaries for the frequency range of the CSO modes in
dependence of the angle under which the external magnetic field was applied,
we only took modes into account with an amplitude bigger than a threshold of
about $1/8$ of the maximum CSO resonance amplitude. This was done to avoid
designating random noise as part of the resonance frequencies. Two exemplary
fits of the CSO modes where the external magnetic field is applied under an
angle $\phi_{H}~{}=~{}0$° and $\phi_{H}~{}=~{}90$° with respect to the
$\mathbf{x}$-axis are shown in Fig. 8.

Figure 8: The solid lines (orange) show the fit of the CSO data (blue dots).
(a) The external field is applied along the $\mathbf{x}$-axis
$(\phi_{H}~{}=~{}0)$° in the Py thin film plane. (b) The external field is
applied under an angle $\phi_{H}~{}=~{}90$°.

##  V Determination of the material parameters

To determine the g-factor $g$ and the saturation magnetization $M_{s}$ of the
Py layer at 5 K, the Py FMR line was fitted by the in-plane Kittel equation

$\mu_{0}H_{\mathrm{ext}}=-\mu_{0}H_{\mathrm{ani}}-\frac{\mu_{0}M_{s}}{2}+\sqrt{\bigg{(}\frac{fh}{g\mu_{B}}\bigg{)}^{2}+\bigg{(}\frac{\mu_{0}M_{s}}{2}\bigg{)}^{2}}.$
(4)

The damping parameter $\alpha$ was fitted by

$\mu_{0}H_{\mathrm{ext}}=\mu_{0}H_{\mathrm{inh}}+\frac{4\pi\alpha\hbar}{g\mu_{B}}f.$
(5)

The thereby obtained values $\mu_{0}M_{s}~{}=~{}1$ T, $\alpha~{}=~{}0.01$,
$g~{}=~{}2.07$ are in good agreement with previous reports Li and Bailey
(2016).

##  VI Uniaxial anisotropy of the Py resonance frequencies

In Fig. 9 the fitted Py resonance frequencies are shown for a 360° angle
rotation of the external magnetic field applied in the Py thin film plane at 5
K, 50 K, and 70 K. The uniaxial anisotropy in the Py resonance frequencies
decreases considerable for temperatures above $T_{c}~{}=~{}58.2$ K. To
estimate the effect of the CSO magnetic stray field on the Py magnetization
dynamics we calculated the difference of the magnetic stray field on top of
the CSO crystal for an external field applied in the (111)-plane of the CSO
crystal along the long side of the crystal and along the short side of the
crystal according to Ref. Engel-Herbert and Hesjedal, 2005. This resulted in
an anisotropy field $H_{ani}~{}=~{}50$ mT at 5 K, which corresponds to a
frequency shift of the Py resonance of about 2 GHz when the external magnetic
field with a magnitude of 120 mT is rotated from $\phi_{H}~{}=~{}0$° to
$\phi_{H}=90$°. This is in the order of magnitude of the observed uniaxial
anisotropy of the Py resonance frequencies. Thus, we attribute the observed
uniaxial anisotropy of the Py resonance frequencies to the effect of the
magnetic stray field coming from the CSO below the Py layer. Above $T_{c}$ the
CSO is not magnetically ordered and the anisotropy in the resonance
frequencies decreases a lot. We still observe a maximal frequency shift of the
Py resonance frequencies of about 0.36 GHz at 70 K, which corresponds to an
anisotropy field of 7.5 mT. This might be caused by the trapped flux in the
superconducting coils of the 3D-vector magnet, as approximately the same shift
appears at 300 K (data not shown). The setup frequently exhibits trapped flux
in the range of 10 mT which substantiates the assumption. Furthermore, small
contributions can be induced by the not perfect alignment of the sample in the
3D vectormagnet.

Figure 9: Fitted Py resonance frequencies plotted against the angle under
which the external magnetic field is applied at 5 K (green dots), 50 K (orange
dots), and 70 K (blue dots). The uniaxial anisotropy is strongest for low
temperatures. Above $T_{c}~{}=~{}58.2$ K it decreases considerable.

##  VII Simulation method

To simulate the resonance frequencies of the Py, CSO, and HM we solve the
Landau-Lifshitz-Gilbert (LLG) equation

$\displaystyle\frac{\mathrm{d}\mathbf{M}_{i}}{\mathrm{d}t}=$
$\displaystyle-\gamma_{i}\mathbf{M}_{i}\times\mu_{0}\mathbf{H}_{\mathrm{eff}_{i}}(\phi_{H})+\frac{\alpha_{i}}{M_{s_{i}}}\mathbf{M}_{i}\times\frac{\mathrm{d}\mathbf{M}_{i}}{\mathrm{d}t}$
(6)

in a coordinate system where $\mathbf{x}$ corresponds to the in-plane
equilibrium orientation of all the macrospins, $\mathbf{z}$ is the oop
direction, and $\mathbf{y}$ the in-plane direction perpendicular to
$\mathbf{x}$. We consider a rotation of the external magnetic field in the
film plane with an angle $\phi_{H}$ to the long side of the CSO crystal (along
the CPW center conductor). Furthermore, we assume that the magnetization in
the sample is fully aligned with the external magnetic field in equilibrium.
The effective magnetic field $\mathbf{H_{\mathrm{eff_{py}}}}$ of the Py
accounts for the external magnetic field $\mathrm{\mathbf{H_{\mathrm{ext}}}}$,
the uniaxial anisotropy field $B_{\mathrm{u}}$, the cubic anisotropy field
$B_{\mathrm{c}}$ that we assume to exist at the interface to explain the data,
and the demagnetization field of a thin film.

$\displaystyle\begin{split}\mathbf{H_{\mathrm{eff_{py}}}}&=(\mu_{0}\mathrm{H_{ext}}+\mathrm{B_{\mathrm{u}}}\sin{(\phi_{H})}^{2}+\mathrm{B_{\mathrm{c}}}\sin{(2\phi_{H})}^{2})\hat{\textbf{x}}\\\
&-\mu_{0}m_{z}(t)\hat{\textbf{z}}.\end{split}$ (7)

The effective magnetic field $\mathbf{H_{\mathrm{eff_{HM}}}}$ of the HM
accounts for the external magnetic field $\mathrm{\mathbf{H_{\mathrm{ext}}}}$,
the cubic anisotropy field $B_{\mathrm{c}}$, and the demagnetization field of
a thin film.

$\displaystyle\begin{split}\mathbf{H_{\mathrm{eff_{HM}}}}&=(\mu_{0}\mathrm{H_{ext}}+\mathrm{B_{\mathrm{c}}}\sin{(2\phi_{H})}^{2})\hat{\textbf{x}}\\\
&-\mu_{0}m_{z}(t)\hat{\textbf{z}}.\end{split}$ (8)

The effective magnetic field $\mathbf{H_{\mathrm{eff_{CSO}}}}$ of the CSO
accounts for the external magnetic field $\mathrm{\mathbf{H_{\mathrm{ext}}}}$
and the demagnetization field of a bulk crystal.

$\displaystyle\mathbf{H_{\mathrm{eff_{CSO}}}}=\mu_{0}\mathrm{H_{ext}}\hat{\textbf{x}}-\mu_{0}\left(\begin{array}[]{c}N_{x}(\phi_{H})\cdot
M_{s}\\\ N_{y}(\phi_{H})\cdot m_{y_{j}}(t)\\\ N_{z}\cdot m_{z_{j}}(t)\\\
\end{array}\right),$ (12)

with the demagnetization factors
$N_{x}(\phi_{H})=N_{x_{0}}\cos(\phi_{H})^{2}+N_{y_{0}}\sin(\phi_{H})^{2}$ and
$N_{y}(\phi_{H})=N_{y_{0}}\cos(\phi_{H})^{2}+N_{x_{0}}\sin(\phi_{H})^{2}$,
with $N_{x_{0}}~{}=~{}0.145$, $N_{y_{0}}~{}=~{}0.285$, and
$N_{z}~{}=~{}0.570$, which we calculated from the crystal geometry according
to Ref. Osborn, 1945. Furthermore, we use for the dynamic magnetization the
ansatz $m_{y_{i},z_{i}}(t)~{}=~{}m_{y_{i},z_{i}}~{}\cdot~{}e^{-i\omega t}$.
The temperature independent material parameters for the simulation of the Py
are $\alpha~{}=~{}0.01$, $g~{}=~{}2.07$, as extracted in Supplemental Material
V. For CSO we use $\alpha~{}=~{}0.003$, $g~{}=~{}2$, which are in good
agreement with previous reports Weiler  _et al._ (2017); Li and Bailey (2016);
Belesi  _et al._ (2011). For the HM we use a saturation magnetization which is
lower than the Py saturation magnetization
$M_{s_{\mathrm{HM}}}~{}=~{}0.75~{}M_{s_{\mathrm{Py}}}$. We motivate this
assumption by the lower saturation magnetization of the CSO as well as the
findings in Refs [Hellman  _et al._ , 2017] and [Vaz, Bland, and Lauhoff,
2008], where it was shown that the saturation magnetization at the interface
of two magnetic materials can be varied due to interfacial effects.
Furthermore, the temperature dependent parameters used in our numerical
calculations are given in Tab. 2.

Table 2: Temperature dependent parameter sets used for the numerical solving of Eq. (6) at $T~{}=~{}5$K and $T~{}=~{}50$K. Layer |  $\mathrm{B_{\mathrm{u}}}$ (mT) |  $\mathrm{B_{\mathrm{c}}}$ (mT) |  $\mathrm{M_{s}}$ (A/m)
---|---|---|---
NiFe | -23 (T = 5 K) | -3 (T = 5 K) | 1/$\mu_{0}$ (T = 5 K)
NiFe | -16 (T = 50 K) | -1.5 (T = 50 K) | 0.98/$\mu_{0}$ (T = 50 K)
HM | - | -6 (T = 5 K) | 0.755/$\mu_{0}$ (T = 5 K)
HM | - | -3.4 (T = 50 K) | 0.82/$\mu_{0}$ (T = 50 K)
CSO | - | - | 0.4/$\mu_{0}$ (T = 5 K)
CSO | - | - | 0.21/$\mu_{0}$ (T = 50 K)

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