# Coupling of ferromagnetic and antiferromagnetic spin dynamics in Mn2Au/NiFe
thin-film bilayers

Hassan Al-Hamdo<EMAIL_ADDRESS>Fachbereich Physik and
Landesforschungszentrum OPTIMAS, Rheinland-Pfälzische Technische Universität
Kaiserslautern-Landau, 67663 Kaiserslautern, Germany  Tobias Wagner  Institute
of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz, Germany  Yaryna
Lytvynenko  Institute of Physics, Johannes Gutenberg-University Mainz, 55099
Mainz, Germany  Institute of Magnetism of the NAS of Ukraine and MES of
Ukraine, 03142 Kyiv, Ukraine  Gutenberg Kendzo  Fachbereich Physik and
Landesforschungszentrum OPTIMAS, Rheinland-Pfälzische Technische Universität
Kaiserslautern-Landau, 67663 Kaiserslautern, Germany  Sonka Reimers  Institute
of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz, Germany  Moritz
Ruhwedel  Fachbereich Physik and Landesforschungszentrum OPTIMAS, Rheinland-
Pfälzische Technische Universität Kaiserslautern-Landau, 67663 Kaiserslautern,
Germany  Misbah Yaqoob  Fachbereich Physik and Landesforschungszentrum
OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau,
67663 Kaiserslautern, Germany  Vitaliy I. Vasyuchka  Fachbereich Physik and
Landesforschungszentrum OPTIMAS, Rheinland-Pfälzische Technische Universität
Kaiserslautern-Landau, 67663 Kaiserslautern, Germany  Philipp Pirro
Fachbereich Physik and Landesforschungszentrum OPTIMAS, Rheinland-Pfälzische
Technische Universität Kaiserslautern-Landau, 67663 Kaiserslautern, Germany
Jairo Sinova  Institute of Physics, Johannes Gutenberg-University Mainz, 55099
Mainz, Germany  Mathias Kläui  Institute of Physics, Johannes Gutenberg-
University Mainz, 55099 Mainz, Germany  Martin Jourdan  Institute of Physics,
Johannes Gutenberg-University Mainz, 55099 Mainz, Germany  Olena Gomonay
Institute of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz,
Germany  Mathias Weiler  Fachbereich Physik and Landesforschungszentrum
OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau,
67663 Kaiserslautern, Germany

###### Abstract

We investigate magnetization dynamics of Mn2Au/Py (Ni80Fe20) thin film
bilayers using broadband ferromagnetic resonance (FMR) and Brillouin light
scattering spectroscopy. Our bilayers exhibit two resonant modes with zero-
field frequencies up to almost $40\text{\,}\mathrm{GHz}$, far above the
single-layer Py FMR. Our model calculations attribute these modes to the
coupling of the Py FMR and the two antiferromagnetic resonance (AFMR) modes of
Mn2Au. The coupling-strength is in the order of
$1.6\text{\,}\mathrm{T}\text{\,}\mathrm{nm}$ at room temperature for nm-thick
Py. Our model reveals the dependence of the hybrid modes on the AFMR
frequencies and interfacial coupling as well as the evanescent character of
the spin waves that extend across the Mn2Au/Py interface.

Ferromagnets have a net magnetic moment and uniform spin dynamics in the GHz
range [1]. In contrast, collinear antiferromagnets have two equal but opposite
sublattice magnetizations with vanishing net magnetic moment [2, 3, 3] and
spin dynamics that can reach the THz range [4, 5]. In addition to THz spin
dynamics, antiferromagnets can exhibit significant stability of their magnetic
moments against external magnetic perturbations [6]. The difference in the
magnetization dynamics of ferromagnetic (FM) and antiferromagnetic (AFM)
materials could potentially be exploited in applications that integrate AFM
materials in high-frequency spintronic devices. A promising approach to
enhance the FM spin dynamics frequencies and control FM spin-wave dispersions
might be the combination of FM and AFM thin-film layers with interfacial
exchange coupling. As a result of interfacial exchange coupling, a pronounced
increase in the coercivity of the FM layer [7, 8, 9, 10, 11] and exchange bias
[12, 13, 14, 15, 16, 17, 18, 19] can be observed. Interfacial coupling also
modifies GHz spin dynamics in FM/FM, chiral FM/FM and AFM/FM heterostructures,
in particular the magnetic damping and anisotropy [20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. However, the role of
the THz frequency spin dynamics of AFMs for the hybrid spin dynamics in AFM/FM
bilayers has not been revealed so far. Consequently, AFM/FM bilayers have so
far not been leveraged to control hybrid mode frequencies or study AFM
dynamics without requiring THz spectroscopy tools.

Here we show that by coupling AFM and FM modes, we can make the AFM dynamics
visible in the GHz range and use the exchange-enhancement [41] of AFM modes to
elevate the FM spin dynamics frequencies. The existence of such a coupling
between FM and AFM spin dynamics opens new possibilities to design next
generation magneto-electronic devices that exploit AFM materials beyond
exchange bias. The Mn2Au/Py system [42] is particularly promising for studying
the fundamental properties of interfacial coupling between FM and AFM thin
films, as this system has a well-defined interface termination of AFM moments
[43]. The coupling of the static Py magnetization to the Mn2Au Néel vector can
be used to control the AFM Néel vector orientation by magnetic fields even
below $1\text{\,}\mathrm{T}$ [43, 44].

We investigate the resonant magnetic dynamics of a hybrid system consisting of
thin-film poly-crystalline FM layers (Py) deposited on single-crystalline
thin-film antiferromagnets (Mn2Au). We study the quasi-uniform dynamics by
vector-network-analyzer ferromagnetic resonance (VNA-FMR) and we study the
spin-wave response by Brillouin Light Scattering (BLS). We model the observed
two eigenmodes of the hybrid system in the context of evanescent spin-wave
modes that extend from the Py layer into the Mn2Au layer and which are coupled
to the two non-degenerate modes of the easy-plane antiferromagnet.

Figure 1: (a) Real part of the background corrected VNA signal of a Mn2Au (40
nm)/Py (10 nm) sample. The dashed line indicates the expected FMR for a bare
Py thin film. (b) Resonance frequencies $f_{1}$ and $f_{2}$ vs. external
magnetic field $H_{0}$ obtained from fitting data in panel (a). The lines are
fits to the modified Kittel equation from our model (see text). (c) Same as
(b) but for a Mn2Au (40 nm)/Py (5 nm) sample. (d) The zero-field mode
frequencies $f_{1}(H_{0}=0)$ and $f_{2}(H_{0}=0)$ increase with increasing
$d_{\mathrm{FM}}^{-1}$.

To carry out this investigation, epitaxial Mn2Au thin films with a thickness
of 40 nm, and Py with variable thicknesses $$2\text{\,}\mathrm{nm}$\leq
d_{\mathrm{FM}}\leq$30\text{\,}\mathrm{nm}$$ are deposited on Al2O3 substrate
[42, 43, 45]. To determine the resonance frequency of the uniform modes in the
samples we performed VNA-FMR in the frequency sweep mode. The external
magnetic field is applied in the sample plane (for more details see [46]).

Figure 1(a) shows the background-corrected
$\mathrm{Re}(d\textsubscript{D}S\textsubscript{21}/dH)$ VNA-FMR data [47]
obtained for a Mn2Au (40 nm)/Py (10 nm) sample. We observe two distinct
resonance modes, a faint mode with frequency $f_{1}$ and a stronger mode with
frequency $f_{2}>f_{1}$. The mode frequencies are enhanced by about 10 GHz
compared to the uncoupled FMR frequency for in-plane isotropic polycrystalline
Py (dashed line). In contrast to earlier studies that found an enhancement of
the FMR frequency of a similar magnitude in FM/AFM bilayers [30, 48], we
observe two distinct modes. We fit the obtained complex-valued
$d\textsubscript{D}S\textsubscript{21}$ spectra as a function of frequency for
a series of $H_{0}$-values by the sum of two magnetic resonances as described
in the SI [46]. From the fits we obtain the two resonance frequencies $f_{1}$
and $f_{2}$ as well as the corresponding linewidths and spectral weights. We
show the fitted $f_{1}$ and $f_{2}$ in Fig. 1(b). We perform identical
measurements for a series of
Mn2Au($40\text{\,}\mathrm{nm}$)/Py($d_{\mathrm{FM}}$) samples, where the Py
thickness $$4\text{\,}\mathrm{nm}$\leq
d_{\mathrm{FM}}\leq$30\text{\,}\mathrm{nm}$$ is varied. When changing the Py
thickness from $d_{\mathrm{FM}}=$10\text{\,}\mathrm{nm}$$ to
$d_{\mathrm{FM}}=$5\text{\,}\mathrm{nm}$$, we observe that both, $f_{1}$ and
$f_{2}$, are clearly enhanced further, with $f_{2}$ reaching a zero-field
resonance frequency of about $28\text{\,}\mathrm{GHz}$ as shown in Fig. 1(c).
The lines in Fig. 1(b) and (c) are fits to Eq. (1) as explained in the
following. From the full dataset of VNA-FMR measurements performed on a series
of Mn2Au (40 nm)/Py ($d_{\mathrm{FM}}$) bilayers (see SI [46]), we extract the
zero-field resonance frequencies shown in Fig. 1(d). Both resonance mode
frequencies increase when decreasing $d_{\mathrm{FM}}$, suggesting an
interfacial origin of the resonance frequency enhancement.

Figure 2: (a) Sketch of the sample structure and coupling of modes. The FMR
with frequency $f_{\mathrm{FM}}$ couples to the two antiferromagnetic modes in
the AFM. This results in hybrid modes with frequencies $f_{1}$ and $f_{2}$
with perpendicular wave-vector $k_{i,\perp}$. The Néel vector excitations
decay within $1/\kappa_{i}$ (see text). (b) The effective stiffness fields
resulting from the mode coupling exceed $0.6\text{\,}\mathrm{T}$ for Py
thickness $d_{\mathrm{FM}}\leq$5\text{\,}\mathrm{nm}$$.

We model the observed dynamics by calculating the magnon spectra of coupled
ferro- and antiferromagnetic layers. The essential results of our model are
sketched in Fig. 2(a), while a figure depicting the quantitative calculation
is shown in the SI [46]. The FMR of the ferromagnet becomes evanescent into
the AFM layer due to a change of boundary conditions resulting from the
coupling of FM and AFM spin dynamics. This results in an effective
modification of the FM wavevector $k_{\perp}$ along the film normal. The
derivation of an expression for $k_{\perp}$ is the main result of our
theoretical model calculation explained in the following.

We start from the standard dipolar-exchange spin-wave dispersion in a
tangentially magnetized ferromagnetic thin film given by [1, 49]

$f(k_{\perp},k_{\|})=\frac{\gamma}{2\pi}\sqrt{B_{\perp}B_{\parallel}}$ (1)

with the effective out-of-plane stiffness field

$B_{\perp}=\mu_{0}H_{0}+J_{\mathrm{FM}}(k_{\perp}^{2}+k_{\|}^{2})+\mu_{0}M_{\mathrm{FM}}G_{0}$
(2)

and in-plane stiffness field

$B_{\parallel}=\mu_{0}H_{0}+J_{\mathrm{FM}}(k_{\perp}^{2}+k_{\|}^{2})+\mu_{0}M_{\mathrm{FM}}(1-G_{0})\;.$
(3)

Here, $M_{\mathrm{FM}}$ is the saturation magnetization of the ferromagnet.
The external magnetic field $\mathbf{H}_{0}$ is applied in the film plane
parallel to the magnetization $\mathbf{M}$.
$J_{\mathrm{FM}}=A_{\mathrm{FM}}/M_{\mathrm{FM}}$ with exchange stiffness
$A_{\mathrm{FM}}$. The factor $G_{0}=(1-\exp(-k_{\|}d))/(k_{\|}d)$ accounts
for the effects of dipole-dipole interactions [49], where $k_{\|}$ is the in-
plane wave vector. In the VNA-FMR experiments discussed above, $k_{\|}=0$, but
$k_{\perp}\neq 0$ as sketched in Fig. 2(a).

We assume exchange coupling between the magnetization of the ferromagnetic
layer, $\mathbf{M}=M_{\mathrm{FM}}\mathbf{m}$, and the Néel vector,
$\mathbf{N}=M_{\mathrm{AFM}}\mathbf{n}$, in a thin layer near the ferro-
antiferromagnetic interface. Here $M_{\mathrm{FM}}$ and $M_{\mathrm{AFM}}$ are
saturation magnetizations of ferro- and antiferromagnetic layer, and both
$\mathbf{m}$ and $\mathbf{n}$ are unit vectors. The corresponding contribution
to the energy density of the bilayer due to the interfacial exchange coupling
is modelled as $w_{\mathrm{int}}=-\xi
J_{\mathrm{exch}}M_{\mathrm{AFM}}M_{\mathrm{FM}}\mathbf{m}\cdot\mathbf{n}\delta(z)$,
where $J_{\mathrm{exch}}$ is the exchange coupling between FM and AFM, $\xi$
is the thickness of the interfacial region in which the coupling is nonzero,
and $\delta(z)$ is the Dirac function. The Py layer of thickness
$d_{\mathrm{FM}}$ is modelled as an easy-plane ferromagnet with negligibly
small in-plane anisotropy. Mn2Au is treated as a two-sublattice easy-plane
antiferromagnet with tetragonal magnetic anistropy that sets two equivalent
orthogonal easy directions within the film plane. We can assume that due to
the strong interfacial exchange coupling $J_{\mathrm{exch}}$,
$\mathbf{M}\|\mathbf{N}$ in equilibrium [43]. The magnon spectra are
calculated based on coupled Landau-Lifshitz equations for ferro-and
antiferromagnetic layers (see SI [46]). The spectra of the antiferromagnetic
layer include two magnon branches corresponding to oscillations of the Néel
vector either in-plane ($f_{\|0}$) or out-of plane ($f_{\perp 0}$) with the
frequencies
$f_{\|\mathrm{AFM}}=\sqrt{f_{\|0}^{2}+c^{2}(k_{\perp}^{2}+k_{\|}^{2})}$ and
$f_{\perp\mathrm{AFM}}=\sqrt{f_{\perp
0}^{2}+c^{2}(k_{\perp}^{2}+k_{\|}^{2})}$. Here, $2\pi c$ is the magnon
velocity in the antiferromagnet which depends on the AFM exchange stiffness,
$A_{\mathrm{AFM}}$, and the exchange field $B_{\mathrm{ex}}$ that keeps the
antiparallel alignment of magnetic sublattices in the antiferromagnet:
$c=(\gamma/2\pi)\sqrt{A_{\mathrm{AFM}}B_{\mathrm{ex}}/M_{\mathrm{AFM}}}$. The
hierarchy of frequencies is $f_{\perp 0}\gg f_{\|0}\gg f_{\mathrm{FM}}$, and
the effect of the dc magnetic field $\mathbf{H}_{0}$ on the AFM spectra can be
neglected. Next, we calculate the magnon spectra assuming nonzero coupling
between the ferro- and antiferromagnetic layers. The coupling modifies the
boundary conditions at the interface (at $z=0$) as follows:

$\displaystyle-A_{\mathrm{FM}}\partial_{z}m_{\alpha}+\xi
J_{\mathrm{exch}}M_{\mathrm{AFM}}M_{\mathrm{FM}}(m_{\alpha}-n_{\alpha})=0,$
(4) $\displaystyle A_{\mathrm{AFM}}\partial_{z}n_{\alpha}+\xi
J_{\mathrm{exch}}M_{\mathrm{AFM}}M_{\mathrm{FM}}(n_{\alpha}-m_{\alpha})=0,$

where $\alpha$ denotes out-of-plane, $\perp$, or in-plane, $\|$, components of
$\mathbf{m}$ and $\mathbf{n}$. Our calculations show that the eigenmodes
consist of the superposition of propagating ferromagnetic magnons and
evanescent oscillations of the Néel vector in the near-interface region that
decay as $\propto\exp\left(-\kappa z\right)$, where $\kappa$ is the decay
constant. Moreover, the spectra also split into two branches, $f_{1}$ and
$f_{2}$, corresponding to excitation of either in-plane or out-of-plane
oscillations in the antiferromagnetic layer. The decay length $\kappa$ depends
on the eigenfrequency of the mode and is different for in-plane and out-of-
plane branches: $\kappa_{1}=\sqrt{f_{\|0}^{2}-f_{1}^{2}}/c\approx f_{\|0}/c$
and $\kappa_{2}=\sqrt{f_{\perp 0}^{2}-f_{2}^{2}}/c\approx f_{\perp 0}/c$. The
eigenfrequencies $f_{1}\equiv f(k_{\|},k_{1\perp})$ and $f_{2}\equiv
f(k_{\|},k_{2\perp})$ are calculated from Eq. (1) by substituting those values
of wave-vectors $k_{1\perp}$ and $k_{2\perp}$ that satisfy the boundary
conditions in Eq. (4) (for more details see SI [46]). For estimating
$k_{1\perp}$ and $k_{2\perp}$, we focus on the lowest modes with
$k_{1\perp},k_{2\perp}\ll\pi/d_{\mathrm{FM}}$ that have larger overlap with
the homogeneous rf magnetic field than the other modes. We fit Eq. (1) to the
experimentally determined resonance frequencies as a function of $H_{0}$ for
each FM layer thickness with $k_{1\,\perp}$ and $k_{2\,\perp}$ as fitting
parameters. We determine the FM k-vectors for each resonance frequency branch
$f_{1}$ and $f_{2}$. From the boundary conditions in Eq. (4) and Eq. (1) we
obtain the approximate expression

$k_{1/2\perp}=\sqrt{\frac{1}{J_{\mathrm{FM}}}\frac{\xi\kappa_{1,2}J_{\mathrm{exch}}A_{\mathrm{AFM}}}{\kappa_{1,2}J_{\mathrm{AFM}}+\xi
J_{\mathrm{exch}}M_{\mathrm{FM}}}}\frac{1}{\sqrt{d_{\mathrm{FM}}}},$ (5)

where we neglected the difference between $M_{\mathrm{FM}}$ and
$M_{\mathrm{AFM}}$. The results of accurate numerical calculations of
$k_{1/2\perp}$ are provided in the SI [46]. The combinations
$B_{1}^{\mathrm{eff}}\equiv J_{\mathrm{FM}}k_{1\perp}^{2}$ and
$B_{2}^{\mathrm{eff}}\equiv J_{\mathrm{FM}}k_{2\perp}^{2}$ that appear in Eq.
(1) represent the effective stiffness field induced by the coupled
ferromagnetic-antiferromagnetic spin dynamics. According to Eq. (5), the
stiffness fields scale linearly as $1/d_{\mathrm{FM}}$, and they are
proportional to the exchange coupling $J_{\mathrm{exch}}$ between ferro- and
antiferromagnetic layers. We show the stiffness fields of both modes obtained
from fitting $f_{1}$ and $f_{2}$ vs. $H_{0}$ by Eq. (1) (data points) in Fig.
2(b). The lines are linear fits in accordance with the scaling expected from
Eq. (5).

We note that the two different values of the stiffness fields correspond to
the coupling with different antiferromagnetic modes and have a dynamic origin.
These values can be related with the effective anisotropy field estimated in
[43] from coercivity measurements. Because the coercive field
$\mu_{0}H_{\mathrm{c}}$ of our sample series agrees within experimental
uncertainty to $B_{1}^{\mathrm{eff}}$ (see SI [46]) we can estimate

$B_{1}^{\mathrm{eff}}\approx\mu_{0}H_{\mathrm{c}}=\frac{4\xi
B_{\mathrm{an}}J_{\mathrm{exch}}M_{\mathrm{FM}}d_{\mathrm{AFM}}}{4B_{\mathrm{an}}d_{\mathrm{AFM}}+\xi
J_{\mathrm{exch}}M_{\mathrm{FM}}}\frac{1}{d_{\mathrm{FM}}},$ (6)

where $B_{\mathrm{an}}$ is the AFM in-plane anisotropy field and
$d_{\mathrm{AFM}}$ is the AFM layer thickness. This expression corresponds to
Eq. (2) of Ref. [43]. The stiffness field $B_{\mathrm{2}}^{\mathrm{eff}}$ is
related to the out-of-plane anisotropy of Mn2Au in a similar way. However,
this field can be observed only in the magnetic dynamics explored here and not
in the static measurements previously performed in [43].

Using Eq. (6) for fitting the experimentally determined
$\mu_{0}H_{\mathrm{c}}$ [46, 43], we determine $\xi
J_{\mathrm{exch}}M_{\mathrm{FM}}\approx 1.6\,\mathrm{T\,nm}$. Using fitting of
the effective fields $B_{1}^{\mathrm{eff}}\equiv
J_{\mathrm{FM}}k_{1\perp}^{2}$ and $B_{2}^{\mathrm{eff}}\equiv
J_{\mathrm{FM}}k_{2\perp}^{2}$ as a function of the inverse FM thickness, as
shown in Fig. 2(b), we further determine
$B_{2}^{\mathrm{eff}}/B_{1}^{\mathrm{eff}}\approx 2.4$. This ratio is inserted
into Eq. (5), from which we obtain

$\frac{B_{2}^{\mathrm{eff}}}{B_{1}^{\mathrm{eff}}}=\frac{\kappa_{2}}{\kappa_{1}}\cdot\frac{{\kappa_{1}J_{\mathrm{AFM}}}+\xi
J_{\mathrm{exch}}M_{\mathrm{FM}}}{\kappa_{2}J_{\mathrm{AFM}}+\xi
J_{\mathrm{exch}}M_{\mathrm{FM}}}.$ (7)

From Eq. (7) we can estimate

$\frac{1}{\kappa_{1}}-\frac{3}{\kappa_{2}}=\left(\frac{B_{2}^{\mathrm{eff}}}{B_{1}^{\mathrm{eff}}}-1\right)\frac{J_{\mathrm{AFM}}}{\xi
J_{\mathrm{exch}}M_{\mathrm{FM}}}\approx 11\,\mathrm{nm}.$ (8)

Taking into account that $\kappa_{1}\ll\kappa_{2}$ and using the value of
magnon velocity $2\pi c=$22.49\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$
(see SI [46]), we estimate $f_{\|0}\approx$0.3\text{\,}\mathrm{THz}$$, which
is about a factor two larger than previous direct THz measurements of magnons
in Mn2Au films grown by a different technique [50].

Figure 3:  (a) BLS spectrum. The solid lines are Lorentzian fits used to
determine $f_{2}$, the dashed line represents the background signal. (b) The
spin-wave dispersion of samples with $5\text{\,}\mathrm{nm}$ and
$30\text{\,}\mathrm{nm}$ Py obtained by fitting the BLS spectra (points) and
fit to Eq. (1) (lines). Spectra are shifted on the y-axis by the
$f_{2}(k_{\parallel}=0)$ mode frequency for clarity. (c) The stiffness field
$B_{2}^{\mathrm{eff}}$ extracted from fitting the BLS measurements is in full
agreement with that observed in FMR measurements.

To separate the impact of $k_{\parallel}$ and $k_{\perp}$ on the hybrid mode
dynamics, we carried out additional wave-vector resolved BLS measurements of
thermal magnons [51, 52, 53, 54, 55].

Fig. 3(a) shows an exemplary BLS spectrum. The peaks at $\pm f_{2}$ in the BLS
spectra are fitted by Lorentzian functions to obtain the dependence of $f_{2}$
on in-plane wave-vector shown in Fig. 3(b) for samples with
$5\text{\,}\mathrm{nm}$ and $30\text{\,}\mathrm{nm}$ thick Py layers. The
signal-to-noise ratio only allowed us to measure the $f_{2}$ mode by BLS. We
fit the spin-wave dispersions to Eq. (1) and obtain the stiffness fields
$B_{2}^{\mathrm{eff}}$ shown in Fig. 3(c) that are in excellent agreement with
our FMR results. The agreement of the stiffness fields observed in BLS and FMR
demonstrates that the coupling of AFM and FM modes impacts $k_{\perp}$ and not
$k_{\parallel}$ in accordance with the assumptions in our model.

In summary, we have made a detailed investigation of the magnetization
dynamics in exchange-coupled bilayers of Mn2Au/Py. We demonstrate that the
interfacial exchange coupling in the ferro/antiferromagnetic bilayer system
enables the control of the hybrid-mode resonance frequency and spin-wave
dispersion by variation of the thickness of the ferromagnetic layer. The
splitting of the unperturbed ferromagnetic resonance frequency of Py into two
non-degenerate modes with strongly enhanced frequency can be understood in the
context of coupling of Py dynamics to the in-plane and out-of-plane modes of
the easy-plane antiferromagnet. The same modes are observed also in the
thermal magnon spectrum recorded by BLS spectroscopy, demonstrating that the
coupling is independent of the in-plane $k$-vector of the spin waves in the
hybrid system. The magnitude of the frequency enhancement depends on the
strength of the interlayer exchange coupling and the magnon frequencies of the
AFM. By independent determination of the interlayer exchange coupling
strength, our method allows to estimate the AFM magnon frequencies even though
they lie far above the regime experimentally accessible to us. Exploiting THz
AFM dynamics to control sub-THz hybrid spin dynamics may find applications in
high-frequency devices such as spin-torque oscillators.

###### Acknowledgements.

We acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) within the Transregional Collaborative Research
Center TRR 173/2–268565370 “Spin+X” (Projects B13, A01, A03, A05, A11, B02,
B11, and B12). M.J. and M. K. acknowledge support by EU HORIZON-
CL4-2021-DIGITAL-EMERGING-01-14 programme under grant agreement No. 101070287
(SWAN-on-Chip). O.G. and T.W. also acknowledge support from Deutsche
Forschungsgemeinschaft (DFG) SPP 2137, project number 403233384. J.S.
additionally acknowledges funding from the Grant Agency of the Czech Republic
grant no. 19-28375X. M.K. acknowledges support from the Horizon 2020 and
Horizon Europe Framework of the European Commission under Grant No.863155
(S-NEBULA). M.W. acknowledges support from the European Research Council (ERC)
under the European Union’s Horizon Europe research and innovation programme
(Grant agreement No. 101044526).

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