# Reconfigurable spin-wave platform based on interplay between nanodots and
waveguide in hybrid magnonic crystal

Krzysztof Szulc<EMAIL_ADDRESS>Institute of Spintronics and
Quantum Information, Faculty of Physics, Adam Mickiewicz University, Poznań,
Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland  Mateusz Zelent  Institute
of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz
University, Poznań, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland  Maciej
Krawczyk  Institute of Spintronics and Quantum Information, Faculty of
Physics, Adam Mickiewicz University, Poznań, Uniwersytetu Poznańskiego 2,
61-614 Poznań, Poland

(August 28, 2024)

###### Abstract

We present a hybrid magnonic crystal composed of a chain of nanodots with
strong perpendicular magnetic anisotropy and Dzyaloshinskii–Moriya
interaction, positioned above a permalloy waveguide. The study examines two
different magnetization states in the nanodots: a single-domain state and an
egg-shaped skyrmion state. Due to the dipolar coupling between the dot and the
waveguide, a strongly bound hybrid magnetization texture is formed in the
system. Our numerical results show complex spin-wave spectra, combining the
effects of periodicity, magnetization texture, and hybridization of the
propagating waves in the waveguide with the dot/skyrmion modes. The systems
are characterized by different band gap sizes. For the skyrmion state, the
azimuthal modes confined to the skyrmion domain wall lead to the formation of
flat bands at low frequencies, while at higher frequencies we identify among
them modes interacting with the propagating waves, which can introduce
additional non-Bragg band gaps, as well as isolated modes leading to the
formation of bound states. On the other hand, the system with a single-domain
state in nanodots offers a wide range of frequencies where the spin waves are
predominantly in the waveguide. Thus, the study shows that the proposed hybrid
magnonic crystals have many distinct functionalities, highlighting their
reconfigurable potential, magnon–magnon couplings, mode localization, and
bound states overlapping with the propagating waves. This opens up potential
applications in analog and quantum magnonics, spin-wave filtering, and the
establishment of magnonic neural networks.

††preprint: APS/123-QED

##  I Introduction

Over the past decade, spin-wave (SW) computing has been extensively researched
as a potential candidate to complement and surpass CMOS-based technologies [1,
2] for digital [3] or analog signal processing [4, 5] and neural network
implementation [6, 7]. This is because SWs offer high-frequency operation,
even at tens of GHz, miniaturization down to the nanoscale, well below 100 nm,
and most importantly, ultralow power consumption, as low as 1 aJ per
operation. Moreover, they can locally interact with magnetic solitons, i.e.
domain walls in 1D and magnetic vortices or skyrmions in 2D, and thus can
hybridize with, be excited by, and be controlled by soliton dynamics [8, 9,
10].

Magnetic skyrmions are topologically protected 2D magnetization textures,
known for their stability and very small size, especially Néel skyrmions in
thin ferromagnetic films, which are stabilized by Dzyaloshinskii–Moriya
interaction (DMI) [11]. Their dynamics can be driven by external forces such
as magnetic field, electric current, structural stress, thermal fluctuations,
or laser pulses [12], which expands their possible applications also in
magnonics, e.g., to control wave propagation [13], scatter SWs [14], form SW
frequency combs [15], or to excite propagating SWs [16] in thin films. This
makes them promising candidates for information storage and processing [17,
18, 19, 20]. In particular, their potential has attracted considerable
interest in non-Boolean logic and unconventional computing devices [21, 22,
23]. However, due to the high damping of SWs in multilayers possessing DMI
[24, 25], with a few exceptions those effects remain mainly numerical
demonstrations [26].

In a skyrmion within confined geometry, three types of eigenmodes have been
observed [27, 28]: gyroscopic, breathing, and azimuthal modes. The gyroscopic
mode refers to the rotational motion of the skyrmion core [29]. The breathing
mode involves the radial oscillation of the size of the skyrmion [30].
Azimuthal modes are SWs propagating along the skyrmion circumference [31, 32,
33]. Their quantization is described by an azimuthal wave number, with
clockwise (CW) and counterclockwise (CCW) degeneracy lifted by the asymmetric
exchange interaction. When the dots are arranged in a chain or array, bands of
collective skyrmion excitations can be formed [34]. However, the dynamic
dipolar coupling between the skyrmions is rather weak, especially between the
azimuthal modes, and the non-zero bandwidths have been numerically
demonstrated only for gyrotropic or breathing modes [31].

Hybrid structures are commonly used to obtain systems that combines two,
usually mutually exclusive, material properties, such as ferromagnetism and
superconductivity [35]. This is also true for magnonics and skyrmions. The
former requires long propagation distances and thus low damping, while the
latter requires DMI resulting from spin–orbit interactions and heavy
neighboring metals, which is associated with increased damping. The bilayer
structure composed of yttrium iron garnet and the Co/Pt multilayer with the
skyrmion has been proposed as a point source to excite short SWs with tens of
nm wavelength [36].

We propose a hybrid structure consisting of a SW conduit made of a low-damping
material (Py) and a chain of (Ir/Co/Pt) nanodots of 300 nm diameter, forming a
hybrid magnonic crystal (HMC) that serves as a multifunctional platform for SW
applications. Such an HMC structure has shown an extended range of DMI values,
which facilitates Néel-type skyrmion stabilization at comparatively lower DMI
values [37]. Using micromagnetic simulations, we show that this HMC exhibits
several interesting properties arising from the coupling between the
subsystems relevant to control SWs. These include flat magnonic bands both
below and above the bottom of the SW spectrum in Py, and a reprogrammable
magnonic band structure, where the width of the band gaps is modified by the
magnetization texture in the dots at the remanence: skyrmion or single-domain
state. The former property provides a system for the realization of bound
states in the continuum in a magnonic domain [38], the latter for the SW
filtering or Bose–Einstein condensation realization [39]. Moreover, the
dynamic coupling between SWs propagating in the waveguide and SWs confined to
the domain wall of the skyrmion results in band anticrossing, making this
system suitable for exploiting the magnon–magnon coupling, and thus useful for
quantum magnonics applications [40]. Taking into account the
multifunctionality, the proposed HMCs represent a promising platform for
magnonic artificial neural networks, as proposed in Ref. [41], or where the
waveguide serves as synapses connected by propagating SWs, and interacting
resonant neurons, i.e. nanodots on the waveguide with the rich spectra of SW
modes [42]. The complex interactions between propagating SWs and nanodisks
discussed in this paper open avenues for creating tunable artificial neurons.

##  II Methods

Figure 1: The visual representation of the system under consideration. The
Ir/Co/Pt multilayer dot is separated from the 4.5 nm-thick Py strip by a 3 nm-
thick nonmagnetic layer. In the dot, an egg-shaped Néel-type skyrmion is
stabilized by the magnetostatic coupling to the skyrmion imprint on the in-
plane magnetized strip. The arrows and their color (according to the HSL-cone
color scale) indicate the direction of magnetization. Note that the figure is
not to scale.

The system under investigation is presented in Fig. 1. It consists of the
infinitely-long waveguide made of permalloy (Py, Ni80Fe20) with a width of 300
nm and a thickness of 4.5 nm and a chain of Co dots with a 300 nm diameter and
1.5 nm thickness. The dots are laying centrally above the waveguide with a
relative separation of 50 nm, resulting in the periodic structure with a
lattice constant of 350 nm. The waveguide and the dots are separated by a 3
nm-thick nonmagnetic layer.

The magnetization dynamics of the system are described by the
Landau–Lifshitz–Gilbert equation:

$\frac{\partial\mathbf{M}}{\partial
t}=-|\gamma|\mu_{0}\mathbf{M}\times\mathbf{H}_{\mathrm{eff}}+\frac{\alpha}{M_{\mathrm{S}}}\mathbf{M}\times\frac{\partial\mathbf{M}}{\partial
t},$ (1)

where $\gamma$ is the gyromagnetic ratio, $\mu_{0}$ is the magnetic
permeability of vacuum, $\mathbf{H}_{\mathrm{eff}}$ is the effective magnetic
field, $\alpha$ is the damping constant, and $M_{\mathrm{S}}$ is the
saturation magnetization. The effective magnetic field
$\mathbf{H}_{\mathrm{eff}}$ is described as follows:

$\begin{split}\mathbf{H}_{\mathrm{eff}}=\mathbf{H}_{0}+\frac{2A_{\mathrm{ex}}}{\mu_{0}M_{\mathrm{S}}^{2}}\nabla^{2}\mathbf{M}+\frac{2K_{\mathrm{PMA}}}{\mu_{0}M_{\mathrm{S}}^{2}}M_{z}\hat{\bf
z}-\nabla\varphi+\\\ +\frac{2D}{\mu_{0}M_{\mathrm{S}}^{2}}\left(\frac{\partial
M_{z}}{\partial x}\hat{\bf x}+\frac{\partial M_{z}}{\partial y}\hat{\bf
y}-\left(\frac{\partial M_{x}}{\partial x}+\frac{\partial M_{y}}{\partial
y}\right)\hat{\bf z}\right),\end{split}$ (2)

where $\mathbf{H}_{0}$ is the external magnetic field, $A_{\mathrm{ex}}$ is
the exchange stiffness constant, $K_{\mathrm{PMA}}$ is the perpendicular
magnetic anisotropy constant, $D$ is the Dzyaloshinskii–Moriya constant, and
$\varphi$ is the magnetic scalar potential, which can be determined from the
formula

$\nabla^{2}\varphi=\nabla\cdot\mathbf{M},$ (3)

which is derived from Maxwell equations in the magnetostatic approximation.

The system was studied using the finite-element method simulations in COMSOL
Multiphysics [43]. The simulations were performed in the 3D model with the
implementation of Eqs. (1) and (3). The static magnetization configuration was
stabilized in the time-domain simulation with periodic boundary conditions
placed at the ends of the unit cell perpendicular to the $x$-axis to introduce
the periodicity into the system. For the proper calculation of the stray
magnetic field, the condition $\varphi=0$ is applied at a distance of
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the system. As an initial
magnetization configuration, the waveguide is uniformly magnetized along the
$x$-axis while the dots are uniformly magnetized along the $z$-axis (for the
study of a single-domain state configuration) or have a skyrmion inside (for
the study of skyrmion state) [44]. The magnetic state relaxation lasts
$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$. The dispersion relation was
calculated using the eigenfrequency solver. For this purpose, the
Landau–Lifshitz–Gilbert equation is solved in its linearized form, where the
total magnetization vector $\mathbf{M}=\mathbf{M}_{0}+\mathbf{m}\,e^{i\omega
t}$ is splitted to a static component $\mathbf{M}_{0}=(M_{0x},M_{0y},M_{0z})$
and a dynamic component $\mathbf{m}=(m_{x},m_{y},m_{z})$. The equation takes
the form of an eigenvalue equation, where the complex eigenvalues give the
frequencies, the dynamic magnetization $\mathbf{m}$ and the dynamic magnetic
scalar potential are the eigenvectors, and the wavevector is a sweep
parameter. Here, the periodic boundary conditions are replaced by Bloch
boundary conditions. The tetrahedral mesh is used with a maximum size of 5 nm
in the dot and 7 nm in the waveguide. Outside the magnetic material, the mesh
grows with ratio 1.4. On the sides where Bloch boundary condition is applied,
we prepared identical triangular meshes.

The material parameters of Py are
$M_{\mathrm{S}}=$800\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}\mathrm{m}$$,
$A_{\mathrm{ex}}=$13\text{\,}\mathrm{p}\mathrm{J}\mathrm{/}\mathrm{m}$$,
$D=0$, $K_{\mathrm{PMA}}=0$, $\alpha=0.005$. The magnetic dot is defined with
an effective-medium approach [45, 46, 47, 48] as a structure with DMI and PMA,
where the three repetitions of the 0.5 nm-thick Ir/Co/Pt multilayer are
simulated as a single Co layer with an effective thickness. The effective
parameters of the dot are
$M_{\mathrm{S}}=$956\text{\,}\mathrm{k}\mathrm{A}\mathrm{/}\mathrm{m}$$,
$A_{\mathrm{ex}}=$10\text{\,}\mathrm{p}\mathrm{J}\mathrm{/}\mathrm{m}$$,
$D=$-1.6\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$$,
$K_{\mathrm{PMA}}=$717\text{\,}\mathrm{k}\mathrm{J}\mathrm{/}\mathrm{m}^{3}$$,
$\alpha=0.02$. In all calculations, the external magnetic field
$\mathbf{H}_{0}=0$.

##  III Results and discussion

Figure 2: Configuration of the magnetization in the unit cell of the coupled
system of waveguide with the chain of dots in (a,b) the single-domain state
and (c,d) skyrmion state. The magnetization is shown in the $xy$-planes
crossing (a,c) the center of the dot and (b,d) the center of the waveguide.
The color map shows the $M_{0z}$ component of the magnetization, and the in-
plane component $M_{0xy}$ is presented with the arrows.

###  III.1 Static magnetization texture

First, let us consider the waveguide and nanodot subsystems separately,
focusing on the static magnetic configuration of the system in the absence of
an external magnetic field. In the multilayer nanodot, characterized by strong
PMA and DMI, various magnetization states can be stabilized, including an out-
of-plane single-domain state, a Néel-type skyrmion, a double-domain structure,
and a worm-like domains [49]. In this paper, we are focusing on two of the
above-mentioned configurations—a single-domain state (SD) and a skyrmion state
(Sk). While the literature is well-versed in the static and dynamic behavior
of these structures in isolation [50, 33], their static and dynamic properties
become complex in a compact hybrid system [37].

Figure 2 presents the static magnetic configuration of a single unit cell of
the HMC. Here, we consider hybrid systems with two different magnetization
configurations—the waveguide (W) with a chain of dots with the single-domain
out-of-plane magnetization state (W/SD) and the waveguide with a chain of dots
with skyrmions (W/Sk). The magnetization texture is shown on the $xy$-planes
crossing the center of the dot (a,c) and the center of the waveguide (b,d),
respectively. In the HMC, the magnetization configuration differs from that of
isolated subsystems. This change, induced by the dipolar coupling, is mainly
caused by the competition between the strong PMA in the dots, which favors
magnetization along the $z$-axis, and the shape anisotropy inherent in the
waveguide, which induces a preference for magnetization along the $x$-axis.

In the W/SD system, the most pronounced effect of dipolar interaction between
the subsystems manifests just beneath the edges of the dot, as illustrated in
Fig. 2(b). Here, the peak deviation in magnetization reaches
$\max{|M_{0y}|}=216$ kA/m along the $y$-axis and $\max{|M_{0z}|}=23$ kA/m
along the $z$-axis. Notably, within the nanodot itself, the magnetization
deviation of approximately 2% is present close to the dot edge.

The static magnetic texture in the W/Sk system undergoes more significant
modification. Unlike the configurations observed in the individual subsystems,
the skyrmion is not only imprinted in the waveguide, but also takes on an egg-
like shape instead of being circular. This static effect has already been
demonstrated in the system with a single dot and finite strip in Ref. [37].
Please note that the imprint intensity is stronger in the W/Sk system than in
the W/SD system. The average net magnetization along the easy axis in the W/Sk
system decreases by 20 kA/m in comparison with only 5 kA/m for W/SD system.
Also, the maximum deviation reaches $\max{|M_{0y}|}=475$ kA/m and
$\max{|M_{0z}|}=59$ kA/m.

###  III.2 Spin-wave dynamics

Figure 3: Dispersion relation of (a) the W/SD system and (b) the W/Sk system.
Color maps show the intensity measured for the out-of-plane component of the
magnetization $m_{z}$. Intensity is scaled logarithmically. Dashed black lines
show the dispersion relation of waveguide itself without dots. Subfigures (c)
and (d) are zoom-ins of the W/SD and W/Sk systems’ dispersion relations as
marked by blue rectangles in subfigures (a) and (b), respectively. Gray
rectangles mark the Bragg gaps in the dispersion relation. (e) Bar chart
collecting band-gap widths for both W/SD and W/Sk systems.

Following the analysis of the static magnetization configurations of the
system, we performed numerical simulations of the SW dynamics. The dispersion
relations of W/SD and W/Sk systems are depicted in Fig. 3(a) and Fig. 3(b),
respectively, where the color map indicates the intensity $I$ of the out-of-
plane dynamic magnetization component $m_{z}$ across the entire system. The
intensity of each mode is quantified as follows:

$I_{\mathrm{mode}}(k,f_{n}(k))=\left|\iiint_{V}m_{z}(f_{n}(k))e^{ikx}\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\right|^{2},$
(4)

where $V$ denotes the volume of magnetic material within a single unit cell
and $f_{n}(k)$ is the frequency of the $n$th mode at wavevector $k$. Next, the
intensities of all modes are converted into the Lorentzian function and then
summed to give the total intensity $I$

$I(k,f)=\sum_{n}\frac{I_{\mathrm{mode}}(k,f_{n}(k))}{\mathrm{Im}[f_{n}(k)]\left(1+\left(\frac{f-\mathrm{Re}[f_{n}(k)]}{\mathrm{Im}[f_{n}(k)])}\right)^{2}\right)}$
(5)

at wavevector $k$ and frequency $f$. This method of quantifying intensity
makes these results comparable to the Brillouin light scattering measurement
results [51]. For comparison, the dispersion relation of an isolated waveguide
without a chain of dots is illustrated with dashed black lines. The comparison
between the dispersion relation of the isolated waveguide and the frequencies
of the SW modes in the isolated dot in the single-domain and skyrmion states
is presented in Section S1 of the Supplementary Materials.

The dispersion relations of both systems contain complex mode dependencies,
caused by the interaction between the dots and their imprints in the
waveguide. The highest-intensity mode follows the fundamental mode of an
isolated waveguide. The antisymmetric waveguide modes are barely visible in
the plots due to the nature of the intensity calculation. Above the third
waveguide mode, the intensity distribution is unclear and only the fundamental
modes are recognizable. The reflected branches and band gaps are present as a
result of the periodicity induced by the arrangement of the dots.

However, there are significant differences in the band-gap width of the Bragg
gaps among the systems. The zoom-ins of the dispersions of W/SD and W/Sk
systems are shown in Fig. 3(c) and Fig. 3(d), respectively, with gray strips
marking the positions of the band gaps. The widths of the first five band gaps
are listed in Fig. 3(e). The W/SD system is characterized by larger low-order
gaps, with the size exceeding 400 MHz. The size of higher-order gaps is much
smaller, with gap 5 already being similar in size to the linewidth of the
modes (which is 77 MHz), making it barely noticeable. In contrast, the W/Sk
system is characterized by larger sizes of higher-order band gaps, i.e., third
and higher. Interestingly, the first band gap is completely absent. Due to the
backward wave character of the mode at low wavevectors, the edge of the first
Brillouin zone lies close to the frequency minimum. As a result, the first and
second bands share the same character. In the W/SD system, the stronger
interaction between the modes pushes the first band much below the frequency
of the isolated waveguide (see Fig. 3(a)). In the W/Sk system, this
interaction is weaker, which causes the first band maximum to be at a higher
frequency than the second band minimum, leading to the absence of the band
gap, as visible in Fig. 3(d). These properties clearly demonstrate the
reprogrammable nature of the proposed HMC system. By preserving the frequency
positions of the band gaps, we can modify their width and even close or open
the first band gap simply by changing the magnetization state in the dots.

Another difference is the presence of numerous flat bands in the dispersion of
the W/Sk system, which lie below the waveguide modes and begin at frequencies
below 1 GHz. These modes are directly connected to the dynamics of the
skyrmion domain wall in the dots which starts at the level of hundreds of MHz
(see, Fig. S1 in the Supplementary Materials). At higher frequencies, some of
the skyrmion modes hybridize with the waveguide modes. Interestingly, one of
these modes leads to the generation of an additional band gap with a width of
53 MHz, marked as gap 3a in Fig. 3(d). The modified spectra indicate that the
presence of skyrmions in dots can directly affect the dynamics of SWs
propagating in the waveguide. Obviously, such modes are not present in the
W/SD system since the lowest resonant mode of the dot in a single-domain state
is at the frequency of 9 GHz, which is above the third waveguide mode. These
results show that the change of a magnetization configuration of the dots can
induce additional flat bands in the SW spectrum and also magnon–magnon
coupling, effects which are currently under intense investigation and also
important from an application point of view. In the next section, we will
explore these properties, focusing on understanding of the physical mechanisms
involved.

###  III.3 Mode localization

Figure 4: The dispersion relation in the first Brillouin zone presents the
localization of modes in both (b) W/SD and (c) W/Sk systems. Each mode
localization value is indicated by the color of the point on the dispersion.
The corresponding plots with the localization value are shown in (a) for the
W/SD system and in (d) for the W/Sk system. Here, the color of the point marks
the absolute value of the wavevector. Dashed black vertical lines mark the
limits of ranges. (e) SW mode profiles for 5 modes in the W/SD system and 9
modes in the W/Sk system. The modes are marked on the dispersion relations
with a square point and a label. In each mode profile, the left color map
displays the $m_{y}$ magnetization component in the $xy$-plane at the center
of the dot, while the right color map displays $m_{y}$ in the $xy$-plane at
the center of the waveguide. The intensity is normalized so that the maximum
value of $|m_{y}|$ is 1 for each of the mode profiles. All profiles are
labeled and their wavevector and frequency are given. The animated version of
this figure is available in [52].

In order to deepen the analysis of the SW dynamics in both systems, we
calculated the localization of the modes and plotted it on the dispersion
relation folded to the first Brillouin zone as shown in Fig. 4(b) and (c). We
define the mode localization as a measure of how much of the intensity of a
given mode comes from a given subsystem (in this case – a waveguide). It is
calculated as

$L(k,f_{n})=\frac{\mathcal{I}_{\mathrm{w}}}{\mathcal{I}_{\mathrm{w}}+\mathcal{I}_{\mathrm{d}}},$
(6)

where $\mathcal{I}_{\mathrm{w}}$ and $\mathcal{I}_{\mathrm{d}}$ are the
intensity of the mode in the waveguide and dot, respectively. Intensity is
measured as

$\mathcal{I}_{\mathrm{w(d)}}(k,f_{n})=\iiint_{V_{\mathrm{w(d)}}}\left|\mathbf{m}(k,f_{n})\right|^{2}\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$
(7)

where $V_{\mathrm{w(d)}}$ is the volume of the waveguide (dot) in a unit cell.
The mode fully localized in the dot has a value of $L=0$ and is marked in
Figs. 4(b,c) with light blue color, while the mode fully localized in the
waveguide has a value of $L=1$ and is marked with light brown color.

The mode localization for the W/SD system is shown in Fig. 4(b). The SW
profiles of 5 selected modes (W/SD-1–W/SD-5) are presented in the first row of
Fig. 4(e). Additionally, we plot the frequency as a function of localization
in Fig. 4(a). In the range I, the lowest mode has a frequency above 3 GHz, and
all modes up to 8.9 GHz are predominantly confined to the waveguide with
minimal amplitude in the dot. The lowest localization value in this region is
$L=0.89$. This result is consistent with the simulations of eigenstates of an
isolated dot, where the lowest mode was observed at 9 GHz (see Section S1 in
the Supplementary Materials). Therefore, any excitation in the dot below this
frequency is only a forced oscillation. Most of the modes in this area
resemble typical waveguide modes, e.g. mode W/SD-3. However, other modes, such
as W/SD-1 and W/SD-2, are significantly distorted. Despite the Bragg gap being
present only at the edge of the Brillouin zone, their branches are coupled
throughout the entire range of the zone. This results in a non-uniform SW
amplitude, even at zero $k$.

Above 8.9 GHz, the dispersion relation is densely populated with modes of
mixed localization. In this frequency range, marked as range II, the modes
have a mixed character, with a localization value in the range between 0.14
and 0.96, with the upper limit decreasing to 0.83 above 10.5 GHz. Among them,
there are modes localized predominantly in the dot (e.g. mode W/SD-4), which
originate from the resonant modes in the dot. However, there is always
significant energy leakage to the waveguide. On the other hand, the
propagating waveguide modes, such as mode W/SD-5, also have significant
amplitude in the dot. The presence of the dot also strongly modifies the
wavefront of the SW propagating in the waveguide (see mode W/SD-5). This
effect can be used to excite propagating SWs in the waveguide by exciting the
dots themselves, similar to the excitation of short-wavelength SWs with 2D
diffraction couplers [53].

The W/Sk system exhibits different behavior, as presented in Fig. 4(c). The SW
profiles of 9 selected modes are presented in two bottom rows of Fig. 4(e).
Additionally, we plot the frequency as a function of localization in Fig.
4(d).

Range I spans the frequencies from 100 MHz to 3.5 GHz, which is below the
frequency of the lowest waveguide band. It contains 15 flat bands related to
the modes localized in dots, which are clockwise and counterclockwise
azimuthal modes of the skyrmion domain wall, similar to the skyrmion in an
isolated dot (see Fig. S1 in Supplementary Materials) [33]. The lowest
frequency mode, W/Sk-1, is a 3rd order counterclockwise mode. The skyrmion
breathing mode, W/Sk-2, has a frequency of 1.226 GHz and is only the ninth
lowest mode. The largest localization in this range is $L=0.08$. The modes in
this range have very small bandwidths ranging from 9 kHz to 34 MHz (breathing
mode, W/Sk-2). This is because these modes can only interact through dipolar
interactions or evanescent SWs, which are the only ones that can exist in the
waveguide at such low frequencies. The simulation of the dot chain in the
skyrmion state but without the waveguide (see Table 1 in Section S2 of the
Supplementary Materials) shows that the bandwidths of most of the bands are
significantly smaller in the absence of the waveguide, except for the
breathing mode (of comparable bandwidth, 33 MHz) and the fourth
counterclockwise mode, which is larger than in the system with the waveguide
(further details on this comparison can be found in Section S2 of the
Supplementary Materials). This suggests the coupling of the skyrmion modes
from the range I through the evanescent waves in the waveguide. Such an effect
is similar to the enhanced SW transmission in bi-component 1D MC at
frequencies below the FMR frequency of one of the constituent materials [54].
However, here it is theoretically predicted for 1D HMC consisting of a
homogeneous film and a chain of dots in the skyrmion state, the system easily
extendable to a 2D array. The effects described above could be exploited for
the design of frequency-selective magnonic devices, enabling precise control
over signal modulation and processing at the nanoscale. The distinct
localization and narrow bandwidths of these modes offer opportunities to
create highly-efficient filters or oscillators that operate within a
precisely-defined frequency range. Furthermore, these weakly dispersive bands
can be used to exploit the flat-band physics recently demonstrated in 1D MCs
with periodic modulation of a DMI [39, 55].

Starting from 3.5 GHz, similarly to W/SD system, modes localized in the
waveguide start to appear but they coexist with skyrmion modes in this range
(marked as range II). Interestingly, the skyrmion is always slightly excited
even if the mode is strongly localized in the waveguide (see mode W/Sk-3 with
localization 0.925). On the one hand, the clockwise skyrmion modes hybridize
with the waveguide modes, resulting in mixed modes that are confined to both
the waveguide and the dot. In Fig. 4(d), these modes form characteristic
horizontally-aligned points with localization between 0.05 and about 0.55. As
it was shown before, such a hybridization leads to the presence of an
additional band gap marked as gap 3a in Fig. 3(d), whose modes are labeled
W/Sk-4 and W/Sk-5. Thus, it is possible to excite propagating modes in the
waveguide with a specific wavelength by excitation of specific skyrmion modes,
and study the recently intensively explored physics of the dynamically coupled
systems, in particular magnon–magnon coupling [56, 57, 58]. Moreover, the
resonant coupling offers a possibility for the realization of artificial
neural networks [41, 42], where the propagating SWs act as synapses connecting
neurons, playing through the nanodot in skyrmion or single-domain state.
However, this requires the activation of neurons by propagating SWs. Such a
nonlinear property can be achieved by slightly increasing the SW amplitude so
that the static magnetic component decreases, resulting in a change of the
resonance frequency (e.g. around the modes W/Sk-4 and W/Sk-5 or around 4 GHz,
with a change of just about 10 MHz) [59, 60, 61] and, depending on the
realization, activating or deactivating of the resonance effect.

On the other hand, counterclockwise skyrmion modes form a vertical line of
points in Fig. 4(d) and have a strong localization in dots with $L$ not
exceeding 0.02. The interaction between counterclockwise modes and waveguide
modes is negligible. Modes W/Sk-6 and W/Sk-7 differ in frequency only by 1 MHz
but their localization values are 0.97 and 0.01, respectively. Moreover, the
small amplitude in the dot for mode W/Sk-6 is not connected with the skyrmion
mode W/Sk-7, confirming a lack of coupling between them. It points to the
possibility of exploiting these modes, which are strongly localized in the dot
or waveguide but are uncoupled, as bound states in the continuum. This effect
that has been extensively studied in photonics but has yet to be explored in
magnonics [38].

The last range, marked as range III, starts at 10 GHz, from where the
dispersion is densely populated with modes having mixed localization. These
modes include bulk dot modes, where the skyrmion core and the magnetization
outside are excited (modes W/Sk-8 and W/Sk-9) and are coupled with the
waveguide. However, counterclockwise skyrmion modes with very low localization
and bandwidth below 10 kHz still exist in this range. Excluding them, the
localization ranges from 0.13 to 0.96. Above 11.5 GHz, it ranges from 0.23 to
0.82. Range III starts at a higher frequency than the analogical range II in
the W/SD system. This is due to the presence of skyrmion in the dot, which
induces specific confinement of the resonant modes in the dot, leading to an
increase in their frequency.

##  IV Conclusions

We have studied a one-dimensional HMC consisting of an infinitely-long Py
waveguide and a chain of nanodots with PMA and DMI (Ir/Co/Pt), in which we
consider two different magnetic states: a skyrmion and a single-domain state.
The static magnetization configuration in the HMC differs from that of its
isolated subsystems. The configuration of the dot imprints the magnetization
texture upon the waveguide, at the same time, the skyrmion shape becomes
strongly distorted, taking on an egg-like shape. This makes a SW dynamic in an
HMC complex while increasing the skyrmion stability and offering
multifunctional properties for advancing magnonics.

The dispersion relations of both systems exhibit characteristic features of
magnonic crystals, such as folded branches and band gaps. However, there is a
substantial difference in the sizes of the band gaps. Additionally, the W/Sk
system has a large number of flat low-frequency skyrmion modes. These modes
are azimuthal rotating modes, both clockwise and counterclockwise, localized
in the skyrmion domain wall and are characterized by very narrow bandwidths
ranging from single kHz to single MHz. Interestingly, the bandwidths are
significantly larger compared to those of the dot chain without waveguide,
indicating evanescent-wave coupling between the skyrmions in W/Sk system. The
flat bands may also overlap with the waveguide modes at higher frequencies
and, interestingly, depending on their sense of rotation, can hybridize with
them, sometimes even leading to additional band gaps in the spectrum, or be
uncoupled. In the same frequency range in the W/SD system, all modes are
almost exclusively localized in the waveguide.

At frequencies above 9 GHz, the resonant modes of the dots begin to appear and
strongly hybridize with the waveguide modes, causing the localization of the
modes to become mixed. However, in the W/Sk system, some of the modes
(localized in the skyrmion domain wall) still can not interact with the
waveguide at high frequencies, which promises the realization of the bound
state in the continuum in magnonics.

The above-mentioned properties offer several useful functionalities for
magnonics, including reconfigurability, filtering, magnon–magnon
hybridizations, uncoupled SW modes in the band structure, as well as SW-
skyrmion bands together with their evanescence coupling. These functionalities
are suitable for the realization of magnonic artificial neural networks.

## Data availibility

The raw data files that support this study are available via the Zenodo
repository [52].

###### Acknowledgements.

K.S. and M.K. acknowledge the financial support from National Science Centre,
Poland, grants no. UMO-2020/39/I/ST3/02413 and UMO-2021/41/N/ST3/04478. K.S.
acknowledges the financial support from the Foundation for Polish Science.

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