#  Nonreciprocal collective magnetostatic wave modes in geometrically
asymmetric bilayer structure with nonmagnetic spacer

P. I. Gerevenkov<EMAIL_ADDRESS>http://www.ioffe.ru/ferrolab/
Ioffe Institute, 194021 St. Petersburg, Russia  V. D. Bessonov  M.N. Mikheev
Institute of Metal Physics, UB of RAS, 620108 Ekaterinburg, Russia  V. S.
Teplov  M.N. Mikheev Institute of Metal Physics, UB of RAS, 620108
Ekaterinburg, Russia  A. V. Telegin  M.N. Mikheev Institute of Metal Physics,
UB of RAS, 620108 Ekaterinburg, Russia  A. M. Kalashnikova  Ioffe Institute,
194021 St. Petersburg, Russia  N. E. Khokhlov  Ioffe Institute, 194021 St.
Petersburg, Russia

###### Abstract

Nonreciprocity, i.e. inequivalence in amplitudes and frequencies of spin waves
propagating in opposite directions, is a key property underlying functionality
in prospective magnonic devices. Here we demonstrate experimentally and
theoretically a simple approach to induce frequency nonreciprocity in a
magnetostatically coupled ferromagnetic bilayer structure with a nonmagnetic
spacer by its geometrical asymmetry. Using Brillouin light scattering, we show
the formation of two collective spin wave modes in Fe81Ga19/Cu/Fe81Ga19
structure with different thicknesses of ferromagnetic layers. Experimental
reconstruction and theoretical modeling of the dispersions of acoustic and
optical collective spin wave modes reveal that both possess nonreciprocity
reaching several percent at the wavenumber of $22~{}\cdot~{}10^{4}$ rad/cm.
The analysis demonstrates that the shift of the amplitudes of counter-
propagating coupled modes towards either of the layers is responsible for the
nonreciprocity because of the pronounced dependence of spin wave frequency on
the layers thickness. The proposed approach enables the design of multilayered
ferromagnetic structures with a given spin wave dispersion for magnonic logic
gates.

††preprint: APS/123-QED

## Introduction

Nonreciprocity, i.e. inequivalence of a medium properties sensed by a
(quasi)particle traversing it in positive and negative directions, underlies
the operation of indispensable elements of modern electronic, microwave
techniques, and optics. Limitations of transistor density and power
consumption associated with conventional electron-charge technologies promote
other quesiparticles as information carriers for next generation of computing.
An emerging competitor of electronics, magnonics [1, 2] employs spin waves and
their quasi-particles, magnons, to encode, carry, and process the information.
Thus, magnonics is eager for the development of basic nonreciprocal elements,
such as spin wave diode [3, 4], circulator [3], half-adder [5], etc. The clear
advantage of magnonics is that nonreciprocity of spin waves’ amplitudes is
intrinsic for magnetic thin films, as was recognized already in the 1960s [6].
However, the functionality and tunability of nonreciprocal magnonic elements
[7, 8] broaden significantly if they support frequency nonreciprocity as well,
when waves of the same wavenumber traveling in opposite directions possess
different frequencies [9]. These require finding media or designing artificial
structures possessing strong frequency nonreciprocity for spin waves while
being compatible with miniaturization and maintaining low losses. Intrinsic
frequency nonreciprocity of the spin waves requires that the medium possess
chirality [10], which is lacking in most materials relevant for magnonics
applications. Therefore, symmetry breaking of various origins in thin films
and heterostructures is now seen as the most promising way to realize non-
reciprocity of spin waves [11, 12, 13, 14, 15, 16].

It has recently been recognized that the collective behavior of spin waves in
coupled waveguides, magnonic crystals, etc. offers extended possibility to
design and tune dispersion properties. This owns to the fact that even small
change in properties of one of the element of such a structure may result in
drastic changes in behavior of coupled spin wave modes [17, 18]. In this
respect, heterostructures with two coupled magnetic layers have recently
attracted great interest due to their ability to support collective
magnetization dynamics and to the possibility of controlling it by
independently adjusting characteristics of individual constituents [19] and
coupling between them [20, 21]. In contrast to the widely considered
nonreciprocity associated with interfacial anisotropy [22] and / or the
Dzyaloshinskii-Moriya interaction [23, 24, 25], the difference in the
dispersion of collective magnetostatic spin wave (MSW) modes propagating in
opposite directions in multilayered structures also relies on the interaction
between layers. In [26, 27, 28, 29], nonreciprocity is demonstrated in
synthetic antiferromagnets, systems with antiferromagnetic ordering of
ferromagnetic layers due to indirect exchange interaction. Nonreciprocity has
been demonstrated in systems of exchange-coupled layers with a parallel
orientation of magnetization [4, 30, 31]. On the other hand, in [30, 32], it
was shown that multilayered systems characterized by differences in the
magnetic parameters of their constituents, e.g. saturation magnetization and
anisotropy, also support the nonreciprocity of MSW, which is based on
asymmetry in magnetic parameters. The antiparallel orientation of
magnetizations in dipole-coupled layers also serves as a source of
nonreciprocity [28, 24].

In this article, we propose to realize nonreciprocal MSW propagation in a
ferromagnetic bilayer structure, where the two dipole-coupled ferromagnetic
layers possess similar magnetic properties, but have different thicknesses.
The structure exhibits a parallel orientation of the magnetizations of the
layers. Using Brillioun light scattering, we examine the dispersion of thermal
MSWs in the bilayer structure and reveal the formation of collective acoustic
(in-phase) and optical (out-of-phase) modes. Both modes demonstrate
considerable nonreciprocity reaching several percent at a wavenumber of
$22\cdot 10^{4}$ rad cm-1. Using a theoretical model which takes into account
dipolar, exchange, and anisotropy contributions to spin wave dispersion, we
show that, along with the difference in layers’ magnetic parameters, the
leading contribution to the nonreciprocity comes from the difference in the
spin wave frequencies of the layers due to their different thicknesses. The
thickness effect is pronounced in films with a thickness of the order of 10 nm
and can be additionally enhanced due to interfacial spin pinning. We further
suggest the conditions for maximizing the nonreciprocity, which gives an
optimal geometrical asymmetry defined by the relation between the layer
thicknesses and the pinning conditions at the interfaces. Such artificially
induced nonreciprocity gives the opportunity to design magnonic logic gates
based on their geometrical parameters rather than on the properties of the
materials.

## Sample and methods

To study the possibility of noreciprocal MSWs propagation in a magnetic
multilayered structure with asymmetry in the geometrical characteristics of
the magnetic constituents, we have chosen a bilayer structure consisting of
two dipolarly coupled layers of the ferromagnetic alloy galfenol separated by
a copper layer, specifically Fe81Ga19(7 nm)/Cu(5 nm)/Fe81Ga19(4 nm), grown by
sputter deposition on a (100)-GaAs substrate. In the following, the 7-nm
galfenol layer is referred to as the top layer, and the 4-nm layer adjacent to
the substrate – as the bottom layer (inset in Fig. 1 a). The top layer is
protected by Al (3 nm) and SiO2(120 nm) capping layers. The preparation and
characterization of the structure are described in details elsewhere [33].
Furthermore, a single 20 nm film of galfenol on a (100)-GaAs substrate was
deposited with the same capped layers as in the case of bilayer structure
[34]. The single film has a thickness comparable to the total thickness of the
layered structure, and thus it is used as a reference to determine the MSW
dispersion in the case of a single film. We choose galfenol-based structures
because they exhibit low Gilbert damping [35], high values of magnetization
precession lifetime [36, 37] and long MSW propagation length [38], which makes
galfenol a prospective material for magnonic applications. According to
ferromagnetic resonance measurements, the saturation magnetization of the
ferromagnetic layers is estimated as $\mu_{0}M_{S}=1.7$ T at layer thicknesses
of 7 and 20 nm [38, 37]. The saturation magnetization of the 4 nm layer is
slightly reduced due to the interface with the GaAs substrate [39] and is
$\mu_{0}M_{S}=1.6$ T. All galfenol layers demonstrate effective cubic
magnetocrystalline anisotropy with the parameter $K_{C}=2.7\cdot 10^{4}$ J m-3
and an additional uniaxial growth-induced anisotropy in the film’s plane [40].
The parameters of the uniaxial anisotropy are $K_{U\,t}=-0.3\cdot 10^{4}$ J
m-3 for the top layer and $K_{U\,b}=-1.1\cdot 10^{4}$ J m-3 for the bottom
layer in the bilayer structure. $K_{U}=-1\cdot 10^{4}$ J/m3 for the 20-nm
single film [38].

The dispersion of thermal MSWs was studied by the Brillouin light scattering
(BLS) technique in the backscattering geometry [41, 42, 43] (inset in Fig. 1
a). A single mode laser with a wavelength of 532 nm is focused on the sample
surface into a spot with a diameter of 50 $\mu$m using an objective lens.
Frequency resolution is achieved using a six-pass Fabry-Perot interferometer.
The resolution in the wave vectors is achieved by tilting the optical axis of
the detection beam from the normal to the film’s plane in the range of angles
$\theta=0-70$ ∘. The corresponding range of wavenumbers is $k_{y}=0-22\cdot
10^{4}$ rad cm-1. All experiments are carried out at room temperature in an
external magnetic field of $\mu_{0}H_{\mathrm{ext}}=100$ mT applied in the
sample plane and perpendicularly to the registered wavevector. The orientation
of $\mathbf{H}_{ext}$ is chosen either along the easy (EA) or hard (HA)
magnetization axes in the film’s plane (see details in Sec. II Suppl.
Materials [44]).

Figure 1:  (a) Experimental BLS signal (diamonds) for Stokes and anti-Stokes
parts of the spectrum for bilayer sample at $k_{y}=13.5\cdot 10^{4}$ rad cm-1
(tilted angle $\theta=35$ ∘). Solid line shows best fit with Lorenz peaks.
Insert: geometry of the experiment. (b and c) Thermal magnons dispersions in
bilayer sample for $\mathrm{{\bf H}_{ext}}$ applied along easy and hard
magnetization directions, respectively. (d) Thermal magnons dispersions for
magnetization along easy and hard axes in the single galfenol film. (e and f)
Nonreciprocity parameter $\delta$ of MSWs vs. wavenumber $k_{y}$ determined
using data from (b and c), respectively. Lines in (b-f) show the data from the
model (see Sec. I Supp.Mat. [44]). The color area on panel (e and f) separates
the wavenumber ranges at which the resolution of the peaks in the spectrum is
achieved or not (see sketch at the top of the panel).

## Results and discussion

Figure 1 a presents a typical BLS spectrum of the bilayer structure obtained
at $\theta=35\,^{\circ}$, $k_{y}~{}=~{}13.5~{}\cdot~{}10^{4}$ rad cm-1. Two
peaks are visible in the Stokes and anti-Stokes parts of the spectrum with
central frequencies $f$ of 11 and 18 GHz. The experimental dependencies of $f$
on the wavenumber $k_{y}$ are obtained by approximating the peaks in the BLS
spectra using Lorentz functions (shown by symbols in Fig. 1 b and c). $f$ of
the high frequency peak increases with wavenumber, while the lower one
demonstrates only a weak dependence of its position on $k_{y}$. This behavior
is observed for both directions of $\textbf{H}_{\mathrm{ext}}$, parallel to HA
and EA. To reveal the origin of the two peaks in the BLS spectra, we compare
the dispersions obtained with those in the 20-nm single galfenol film shown in
Fig. 1 d. For the single film, the BLS spectra also contain two peaks. The
frequency of one peak increases with $k_{y}$, and thus can be readily assigned
to the magnetostatic surface spin wave (MSSW). The frequency of the second
peak is weakly dependent on the wavenumber, and corresponds to the spin wave
resonance (SWR). In the studied range of $k_{y}$, the frequencies of the SWR
are higher than those of the MSSW, according to the typical properties of
these waves in single magnetic layers [45]. In striking contrast, in the
bilayer structure, the high frequency peak shows the stronger dependence on
$k_{y}$. Furthermore, at $k_{y}=0$, the frequencies of both dispersion
branches coincide, while they differ in the single layer. Therefore, the two
dispersion branches in the bilayer structure cannot be explained in terms of
MSSWs and SWRs. Instead, the observed dispersions indicate the formation of
the collective magnetization dynamics of two ferromagnetic layers and are
described in terms of the collective acoustic and optical modes (Fig. 1 b and
c). The formation of these modes was demonstrated in [46, 47] and a
theoretical description is given in [48, 49]. It should be noted, that the
dispersions of MSW in the multilayered structure do not coincide with the
dispersions of its individual ferromagnetic constituents (see Sec. IV Suppl.
Materials [44]).

Figure 2:  Nonreciprocity parameter $\delta$ from the model vs. structure
asymmetry $\alpha$ (a) and interlayer thickness $d_{\mathrm{Cu}}$ (b) for
acoustic (red line) and optical (green line) branches of dispersion. (c)
Distributions of the amplitudes of in-plane $M_{y}$ and out-of-plane $M_{z}$
magnetization components over the thickness of the structure for EA the
acoustic and optical branches for structures with minimal (I), maximal (III)
$\delta$ and parameters of bilayer structure under the experiment (II). Data
are given for the value of $|k_{y}|=22\cdot 10^{4}$ rad cm-1.

To substantiate the attribution of the two peaks in the BLS spectra to
scattering by the acoustic and optical modes, we used the theoretical approach
developed in [50, 51, 18]. The details of the theoretical calculations are
presented in Sec. I-III Suppl. Materials [44]. The fit for the single film
using the model is shown by lines in Fig. 1 d. Because of the equality of
frequencies for waves propagating in the opposite directions, i.e., zero
nonreciprocity in frequencies (see Sec. V Suppl. Materials [44]) for the
single galfenol layer, we use the same boundary conditions for all interfaces
in the subsequent analysis. The boundary conditions in the model introduced in
the form of in-plane $\zeta$ and out-of-plane $\xi$ pinning parameters. For
the single film sample, the best fit is achieved at values of the parameters
$\xi=-46.6$ and 48.6 nm, $\zeta=6.2$ and 11.8 nm for $\mathbf{H}_{ext}$ along
EA and HA, respectively. The parameter of exchange stiffness of galfenol from
the fit is $A=8\cdot 10^{-12}$ J m-1, which is comparable to the value for
pure iron [52].

The obtained exchange stiffness parameter makes it possible to approximate the
dispersion dependencies of the bilayer structure (lines in Fig. 1 b and c).
Good agreement with the experimental results supporting assignment of the
peaks in the BLS spectra to the coupled modes is achieved using a model that
takes into account only the dipolar coupling between the layers [18].
Following the single layer case, for the bilayer structure, we assume equal
pinning parameters for the external interfaces of the whole bilayer structure.
The calculation results (see Sec. VI Suppl. Materials [44]) show that a change
in the pinning parameters at the interfaces with the Cu interlayer leads only
to a quantitative change in the results. Since a change in the pinning at the
interfaces with copper does not change the simulation results qualitatively,
we assume the same pinning parameters for all interfaces further for
simplicity. The values of the pinning parameters are $\xi=-1.5\cdot 10^{5}$
and $7.5\cdot 10^{3}$ nm, $\zeta=88$ and $3.6\cdot 10^{3}$ nm for
$\mathbf{H}_{ext}$ along EA and HA, respectively. Changes in pinning
parameters $\xi$ and $\zeta$ compared to the single-film case can be
associated with significantly thinner layers in the multilayered structure and
the greater influence of interfaces on spin dynamics [53].

Having identified the origin of the peaks in the BLS spectra of the bilayer
structure, we examine do the coupled modes possess nonreciprocity. Indeed,
comparing the frequencies at $\pm k_{y}$, one notices differences between
them. To quantify nonreciprocity, we introduce the dimensionless parameter
$\delta(k_{y})=2[f(k_{y})-f(-k_{y})]/[f(k_{y})+f(-k_{y})]$. In the single
galfenol film $\delta$ is equal to zero over the range of studied $k_{y}$ both
in experiments and in the model, confirming that the interfaces alone do not
introduce nonreciprocity for such a film (see Sec. V Suppl. Materials [44] for
details of extracting $\delta$ from the experimental data). In the bilayer
structure, in contrast, nonzero $\delta$ is found for both the acoustic and
the optical modes, as shown by dots in Fig. 1 e and f for magnetization along
EA and HA. The nonreciprocity $\delta(k_{y})$ increasing with $k_{y}$ and
possessing the opposite slope for the acoustic and optical branches is clearly
seen when the two spectral peaks in the BLS signal are fully resolved, i.e.
when $k_{y}$ exceeds $\approx 10^{4}$ rad cm-1 (see non-shadowed areas in Fig.
1 e and f). Importantly, the model also predicts the presence of
nonreciprocity in the bilayer structure. The results of the model are shown by
lines and are in qualitative agreement with the experiment.

Such a dependence of nonreciprocity on the wavenumber has been previously
found for the case of identical dipole-coupled Co layers with antiparallel
magnetizations [24]. In our experiments, however, the magnetizations of the
top and bottom layers are parallel to each other, and thus a different source
of nonreciprocity should be identified. Interface-induced nonreciprocity can
be excluded as well, as the single film does not demonstrate nonreciprocity in
experiments, and identical boundary conditions for the bilayer structure were
used in the model, as discussed above. No exchange coupling between the
layers, which is another possible source of nonreciprocity, is present. Thus,
we argue that the nonreciprocity observed in the experiments and reproduced in
the model is caused by the geometric asymmetry of the structure, i.e. by the
difference in the thicknesses of ferromagetic layers, and by the difference in
magnetic parameters of the layer.

### Theoretical analysis

Figure 3:  (a and b) Relationship in the amplitudes of the magnetization
dynamics integrated over the layer thicknesses between two ferromagnetic
layers as a function of the geometric asymmetry for the (a) acoustic and (b)
optical modes. Data for a wave propagating in the positive and negative
directions are shown by solid and dashed lines, respectively. (c) Frequency of
the magnetostatic surface mode vs. thickness of the single galfenol layer at
$k_{y}=22\cdot 10^{4}$ rad cm-1. Line 1 shows frequency behavior without
considering exchange interaction and interface anisotropy obtained from the
theory [1]. Lines 2 and 3 show frequency behavior obtained from model using in
this work at pining parameters $\xi=\zeta=-100$ nm and $\xi=\zeta=10$ nm,
respectively. (d) Asymmetry between the BLS peak amplitudes (markers) and
between the $M_{z}$ values on the top of the structure at $k_{y}=\pm 22\cdot
10^{4}$ rad cm-1 (lines). Data for the acoustic branch are shown with square
markers and solid line, and data for the optical branch are shown with circles
and dotted line.

To determine the impact of geometric asymmetry, we independently vary in the
model the thickness of the nonmagnetic spacer $d_{\mathrm{Cu}}$ and the
parameter of the geometrical asymmetry of the structure
$\alpha=(d_{t}-d_{b})/(d_{t}+d_{b})$, where $d_{t}$ and $d_{b}$ are the
thicknesses of the top and bottom ferromagnetic layers, respectively. Figure 2
a shows the dependence $\delta(\alpha)$ for both collective spin wave modes at
$|k_{y}|=22\cdot 10^{4}$ rad cm-1, $d_{\mathrm{Cu}}=5$ nm, and
$\mathbf{H}_{ext}$ directed along EA. The values of $\delta$ for the acoustic
and optical branches of the dispersion are of opposite signs for all $\alpha$.
The modulus of $\delta$ for the optical branch is somewhat higher than for the
acoustic one for the entire range of $\alpha$. At a certain value of $\alpha$
the nonreciprocity inverts its sign for both branches (see the point I in Fig.
2 a). We note, that point I is shifted relative to the completely symmetric
structure with $\alpha=0$. For the structure with equal magnetic parameters of
the two ferromagnetic layers, no nonreciprocity is found in a symmetric
structure for both dispersion branches (see Sec. VII in Suppl. Material [44])
in agreement with previous reports [54]. In this case, the asymmetric
structures with opposite sign of $\alpha$ show equal absolute values of
$\delta$. $\delta$ can reach tens of percent, if the structure with
appropriate pinning parameters on interfaces and geometrical asymmetry will be
prepared (Fig. S.5 a in Sec. VIII in Suppl. Material [44]). Figure 2 a shows
that the geometric asymmetry gives a substantial contribution to the
nonreciprocity, and it is at least comparable to the contribution of the
difference in the magnetic parameters. As the asymmetry is tuned from the
point I towards -1 or +1, $\delta(\alpha)$ increases and reaches the extreme
points marked as III in Fig. 2 a. The values of asymmetry at which maxima of
$|\delta|$ are observed, are the same for both branches, and depend on the
pinning parameters $\zeta$ and $\xi$. The asymmetry of the structure under
study is marked by point II in Fig. 2 a.

Figure 2 b shows the dependence of $\delta$ on $d_{\mathrm{Cu}}$ for the
asymmetry value at which the nonreciprocity is maximal ($\alpha=-0.56$). The
dependence of the nonreciprocity on the nonmagnetic interlayer thickness has
the same character for all values of $\alpha$. Decreasing $d_{\mathrm{Cu}}$ in
the range from 10 to 2 nm gives only a weak increase of the nonreciprocity
confirming the dominating role of structural asymmetry in the emergence of
nonreciprocity. With a further decrease of $d_{\mathrm{Cu}}$, it would be
necessary to take into account the exchange interaction between the
ferromagnetic layers [21], which is discussed by other authors [26, 24] and is
beyond the scope of this work.

To illustrate the physical picture underlying the observed and predicted
nonreciprocity, we show in Fig. 2 c the distributions of the amplitudes of the
dynamic in-plane $M_{y}$ and out-of-plane $M_{z}$ magnetization components
over the thickness of the structure for various values of nonreciprocity
(points I – III in Fig. 2 a). Distributions are given for the optical and
acoustic branches for waves propagating in opposite directions. Based on these
distributions, we compute the shift
$\Delta\tilde{M}_{t-b}=(\tilde{M}_{t}-\tilde{M}_{b})/(\tilde{M}_{t}+\tilde{M}_{b})$
of the amplitudes of the modes
$\tilde{M}_{t,b}=\sqrt{\tilde{M}^{2}_{y(t,b)}+\tilde{M}^{2}_{z(t,b)}}$
integrated over the thickness of the layer towards the top or the bottom
layer. Figures 3 a and b show the shift as a function of $\alpha$ for the
parameters of the structure under study. The following features should be
noted:

  * •

Positive slops of $\Delta\tilde{M}_{t-b}(\alpha)$ indicate that the modes show
localization in a thicker layer, except for the case of large $|\alpha|$ for
the acoustic branch.

  * •

The vertical relative displacement of $\Delta\tilde{M}_{t-b}(\alpha)$ for
positive and negative $k_{y}$ (compare the solid and dashed lines in Fig. 3 a
and b) shows that the maximum of the mode amplitude is additionally shifted to
the upper or the lower ferromagnetic layer, depending on the sign of $k_{y}$.

  * •

This shift of the amplitude maxima dependent of the sign of $k_{y}$ appears to
be opposite for the acoustic and optical branches (compare Figs. 3 a and b).

The combination of the first two features suggest that the nonreciprocity
induced by the geometrical asymmetry $\alpha$ originates from a stronger or
weaker localization in the thicker layer, depending on the $k_{y}$ sign. The
third feature explains the opposite sign of nonreciprocity observed for the
acoustic and optical branches.

Fig. 3 c shows the frequency change with thickness for a single galfenol
layer. Line 1 shows the frequency of the pure surface magnetostatic mode
without considering exchange interaction and interface anisotropy obtained
from the theory [1]. One sees that, even in this simple case, there is a
nonzero change in frequency with the thickness of the thin films. Thus, the
different degree of localization of the coupled spin waves mode in thicker and
thinner layer controlled by the sign of $k_{y}$ (Fig. 3 a and b) affects the
frequency of this mode. This effect is enhanced if one accounts for the
interfacial pinning and exchange interaction within the ferromagnetic layer.
Lines 2 and 3 in Fig. 3 c demonstrate thickness dependences of the MSW
frequency for different pining parameters. As one can see, depending on the
pinning parameters, the variation of the frequency with thickness becomes more
potent pronounced and can exhibit different slopes. Such a behavior has been
experimentally demonstrated for iron films on various substrates [55, 18].
Importantly, for layers thicker than 20 nm, the frequency is almost
independent of the thickness (Fig. 3 c). In the model, an increase in the
total thickness of two ferromagnetic layers leads to a decrease in
nonreciprocity (Sec. IX in Suppl. Material [44]). As the layer thickness
decreases below $\approx$5 nm, the change of the frequency with the thickness
becomes drastic, which can prevent the formation of coupled modes in
structures with a high value of $\alpha$, as observed in our calculations for
$|\alpha|>0.8$, indeed.

Finally, we address the Stokes / anti-Stokes amplitude asymmetry introduced as
$\Delta A=(A_{S}-A_{aS})/(A_{S}+A_{aS})$, where $A_{S}$ and $A_{aS}$ are the
peak amplitudes in the Stokes and anti-Stokes parts of the BLS spectrum,
obtained from the fit. The Stokes / anti-Stokes amplitude asymmetry $\Delta A$
for the acoustic and optical branches both increase with increasing $|k_{y}|$
as shown in Fig. 3 d. To explain this fact, we analyzed amplitudes and related
Stokes / anti-Stokes asymmetry $\Delta A$ for both branches at the top
interface of the structure predicted by the model, as the magnetization
dynamics at this interface gives the dominating contribution to the BLS
signal, especially for the large $k_{y}$. The results for the acoustic and
optical modes are shown in Fig. 3 d by solid and dashed lines, respectively.
The data from the model demonstrate qualitative agreement with the experiment.
This behavior can be explained by the distribution of the amplitudes within
each layer. Within one layer, both modes have an amplitude distribution
typical for a surface MSW, that is, the amplitude’s maximum shifts to the
bottom interface of the layer for positive values of $k_{y}$ and to the top
interface for negative ones. As a result, the Stokes / anti-Stokes asymmetry
$\Delta A$ in the BLS experiment appears to be of the same sign for both
modes. Thus, while the nonreciprocity $\delta$ stems from the predominant
localization of the modes in one of the layers of the bilayer structure, the
Stokes / anti-Stokes asymmetry is governed by the shift of the spin wave
amplitudes within its top layer.

## Conclusions

In conclusion, we have experimentally demonstrated the nonreciprocal behavior
of the collective spin wave modes in the system of two parallel magnetized
ferromagnetic layers of galfenol separated by a nonmagnetic copper spacer with
pure dipolar coupling between them. The analysis shows that the geometric
asymmetry of the structure, i.e. difference in thicknesses of the
ferromagnetic layers, gives a pronounced contribution to the observed
nonreciprocity, which appears to be comparable to the contribution because of
the differences in the magnetic parameters of the magnetic layers. Modeling
shows that the nonreciprocity in the geometrically asymmetric structure stems
from shift of the spin wave mode maxima towards a thicker or thinner layer
controlled by the sign and value of $k_{y}$, and from the dependence of the
spin waves frequencies on the thin layers thickness. Total nonreciprocity
achieved in the experiments reaches 2 % for $k=22\cdot 10^{4}$ rad cm-1.
However, it can be increased up to tens of percent in the structure with
appropriate pinning parameters on interfaces and geometrical asymmetry.

Although this nonreciprocity is somewhat lower than the one achieved because
of the difference in saturation magnetization between the layers of
multilayered structures [4, 30], its value is sufficient to have an impact on
the total spin waves’ nonreciprocity in multilayered structures. Our analysis
further suggests that there is an optimum ratio between the layer thicknesses
supporting both the formation of the coupled modes and their nonreciprocity,
which is also controlled by the spins pinning at interfaces. Thus, tailored
dispersion and nonreciprocity of the coupled modes can be obtained by
fabricating a structure with certain values of the pining parameters [56, 57]
and geometrical asymmetry. Additionally, as shown in recent work [58], the
pinning parameters and thus the behavior of spin waves can be controlled at
ultrashort time scales. Furthermore, the considered bilayer structure supports
optical excitation of coherent coupled modes [59], in which case mutual spin
pumping may contribute to nonreciprocity. Therefore, the demonstrated
geometrical control of spin wave nonreciprocity paves the way to the design of
magnonic logic gates based on predicted and controllable dispersion of MSWs
[1].

###### Acknowledgements.

The authors thank S. O. Demokritov for insightful discussions. The work of
P.I.G., A.M.K. and N.E.Kh. is supported by RSF (project 20-12-00309,
https://rscf.ru/en/project/20-12-00309/). V.D.B. acknowledges RFBR (Grant
19-32-50141). V. D. B., V. S. T. and A. V. T. acknowledge the Russian
Government Program “Spin” no. 122021000036-3.

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