# Spin wave confinement in hybrid superconductor-ferrimagnet nanostructure

Julia Kharlan1,2, Krzysztof Sobucki1, Krzysztof Szulc1, Sara Memarzadeh1,
Jarosław W. Kłos1<EMAIL_ADDRESS>1ISQI, Faculty of Physics, Adam Mickiewicz
University, Poznań, Poland
2Institute of Magnetism NASU and MESU, Kyiv, Ukraine

###### Abstract

Eddy currents in a superconductor shield the magnetic field in its interior
and are responsible for the formation of a magnetic stray field outside of the
superconducting structure. The stray field can be controlled by the external
magnetic field and affect the magnetization dynamics in the magnetic system
placed in its range. In the case of a hybrid system consisting of a
superconducting strip placed over a magnetic layer, we predict theoretically
the confinement of spin waves in the well of the static stray field. The
number of bound states and their frequencies can be controlled by an external
magnetic field. We have presented the results of semi-analytical calculations
complemented by numerical modeling.

##  I Introduction

The states of superconductivity and ferromagnetism are very rarely observed in
a single material. Their intrinsic co-existence (i.e. in one uniform phase,
for the same electrons) was found for triplet pairing in a proximity to a
magnetic quantum critical point (e.g. for ${\rm UGe_{2}}$) [1, 2]. The other
possibility, known for long time and more conventional [3, 4], is the
coexistence of two phases where large and localized moments of 4f electrons
(Er, Gd) provide long-range strong ferromagnetism whereas 3d conduction
electrons are responsible for superconductivity.

The hybrid systems [5, 6, 7, 8], where superconductor and ferromagnet are part
of the same structure and interact with each other, usually offer much more
flexibility both in design and implementation of new features. The
superconductor/ferromagnet hybrids can be divided into two categories – where
both subsystems are in direct contact [9] or separated by non-magnetic non-
conducting material. In the latter case, the coupling at a distance results
from the fact that both the eddy currents in the superconductor and the
magnetic moments in the ferromagnet generate a magnetic field. The coupling
provided by the magnetic field can be controlled by the external magnetic
field as well as be tailored by the geometry, since both the distribution of
the eddy currents and the magnetization configuration depend on these factors.

In electromagnetically-coupled hybrids, the ferromagnet can modify the
properties of the superconductor, e.g. the magnetic screening can increase the
value of critical current density in the superconductor [10], or presence of
the stray field produced by ferromagnetic nanoelements can affect the
nucleation of vortices [11] or pin and guide the vortices in the
superconductor [12]. Similarly, the presence of the superconductor can have
influence on the ferromagnet, e.g. by controlling the magnetization dynamics
[13]. In this research field, we can find the reports about induction of
magnonic crystals and nonreciprocal spin wave (SW) transmission in uniform
magnetic layer due to the screening of dynamic demagnetizing field by a
superconductor [14], enhancement of nonlinear SW dynamics [15], Bragg
scattering of SWs on the field produced by Abrikosov vortex lattice [16, 17],
or SW generation by moving vortices [18, 19, 20]. Undoubtedly,
superconductor/ferromagnet hybrids offer many possibilities for controlling
the dynamics of SWs. One of the topics not fully explored is the problem of
localization of SWs in these systems.

In this study, we conduct a theoretical and numerical investigation of the SW
confinement induced in a uniform ferrimagnetic (FM) layer by the stray field
of a superconducting (SC) strip. We demonstrate that by adjusting the applied
field, we can impact on the depth of the stray field well and thereby control
the number and frequencies of the confined SW modes.

##  II Model

The considered hybrid system consists of gallium-doped yttrium iron garnet
(Ga:YIG) FM thin film and Nb SC strip in the Meissner state, electrically
isolated from each other by thin non-magnetic spacer (see Fig. 1). According
to the Meissner effect, a SC strip expels a magnetic field from its volume by
means of eddy currents. These currents create a non-uniform distribution of
the magnetic field throughout the entire space, including the FM film. Such
geometry enables the investigation of the coupling between FM and SC
subsystems as purely classical, where the stray magnetic field from the eddy
currents in SC strip impacts the magnetization in FM layer.

Figure 1: (a) A thin FM film ($a=20$ nm) is exposed to the stray field of a
rectangular SC strip ($w=400$ nm, $t=100$ nm). The ferrimagnet (Ga:YIG) and
the superconductor (Nb) are separated by a small gap ($g=10$ nm). The magnetic
film has low $M_{\rm s}$ and exhibits PMA, so it is magnetized out of plane in
the absence of the external magnetic field. The sketches (b,c) illustrate the
mechanism of SW localization. (b) The static internal magnetic field is
lowered in the region of the ferrimagnet underneath the SC strip. This leads
to the confinement of SW modes (c), which are quantized in the quasi-parabolic
well of the internal field.

The system is placed in an external magnetic field perpendicular to the FM
layer. In Ga:YIG, the shape anisotropy is overcome by the perpendicular
magnetic anisotropy (PMA), leading to the magnetization being directed out of
plane even in the absence of external magnetic field. Operating with the
relatively small external field, we can sustain the Meissner state in confined
geometry of the strip, and observe the impact of the stray field of SC strip
on the magnetization dynamics in FM layer. Then, the stray field of SC strip
is tuned by the external field and induces the well of static effective field
of controllable depth in the FM layer. The well can confine the SWs of the
frequencies lower than the ferromagnetic resonance (FMR) frequency of FM layer
in the absence of SC strip.

A uniformly-magnetized infinite FM layer does not produce outside any static
stray field – only the internal demagnetizing field ($-M_{\rm
s}\hat{\mathbf{y}}$) is present. However, the stray field of SC strip induces
a weak magnetization texture in FM layer, close to SC strip edges. It can
produce a static stray field, although it is small and can be neglected – see
Section S1 in Supplemental Material. The dynamic stray field coming from the
SW modes in FM layer can be also considered as negligible. For FM layer
magnetized in the out-of-plane direction, the dynamic components of
magnetization are tangential to the surfaces and do not produce any surface
magnetic charges. On the other hand, the dynamic stray field can be generated
by the volume magnetic charges but it requires a strong non-uniformity of the
SW profile, which is observed for the higher SW modes (not investigated here).
Therefore, we assume that eddy currents are unaffected by SWs and remain
constant. To sum up, both static and dynamic stray fields induced in FM layer
are negligible for the SC strip. Therefore, there is no need to solve self-
consistent problem for our system and take into account the mutual interaction
between the FM layer and the SC strip.

Taking these assumptions into account, our studies were carried out in two
stages. We first calculated the static stray field generated by the SC strip.
It was determined from the distribution of SC currents, which was found by a
semi-analytical solution of the London equations. The static field generated
by SC strip was then included as a correction to the effective field to
Landau-Lifshitz (LL) equation. The LL equation was solved both semi-
analytically and numerically and was used to find the confined SW modes.

###  II.1 The static magnetic field produced by superconducting strip

The London equation has been solved analytically for a number of geometries,
such as a film [21], a cylindrical wire [22] and an infinitely-thin
cylindrical dot [23]. However, the analytical solution of the London equation
presents difficulties, even for such a simple structure as a strip, due to the
impossibility of variable separation. We have therefore used the semi-
analytical method developed by Brandt [24, 25]. For the SC strip in an
external field applied along the $y$-direction
$\mathbf{B}_{0}=B_{0}\hat{\mathbf{y}}$, the Meissner effect induces a current
of density $\mathbf{J}=J(x,y)\hat{\mathbf{z}}$ flowing along the $z$-axis and
generating a stray field $\mathbf{B}_{\rm sc}(x,y)$, which lacks $z$-component
– see Fig. 1(b). According to the Ampere law, we can relate the current
density to the total field $\mathbf{B}$ and vector potential $\mathbf{A}$:

$\mu_{0}\mathbf{J}=\nabla\times\mathbf{B}=\nabla\times\nabla\times\mathbf{A}=-\Delta\mathbf{A},$
(1)

where we used the Coulomb gauge ($\nabla\cdot\mathbf{A}=0$), and the symbol
$\mu_{0}$ denotes the permeability of vacuum. Eq. (1) can be interpreted as
two-dimensional Poisson equation for vector potential
$\mathbf{A}=A(x,y)\hat{\mathbf{z}}$ with a current
$\mathbf{J}=J(x,y)\hat{\mathbf{z}}$ being a source:

$\Delta A(x,y)=-\mu_{0}J(x,y),$ (2)

where $\Delta$ is a two-dimensional Laplacian. Let’s decompose the vector
potential $\mathbf{A}=(xB_{0}+A_{\rm sc}(x,y))\hat{\mathbf{z}}$ into the
contribution related to uniform external field: $\mu_{0}xH_{0}$ and the field
produced by SC: $A_{\rm sc}(x,y)$ and then relate the latter inside SC to a
current $J(x,y)$ by London equation with Coulomb gauge:

$A_{\rm sc}(x,y)=-\mu_{0}\lambda^{2}J(x,y),$ (3)

where $\lambda$ is London penetration depth. Then, using Eq. (3), we can write
Eq. (2) in the following integral form:

$\lambda^{2}J(x,y)\\!=\\!\\!\\!\iint_{S}dx^{\prime}dy^{\prime}Q(x,y,x^{\prime},y^{\prime})J(x^{\prime},y^{\prime})+\\!xH_{0},$
(4)

where $S$ is the cross-section of the SC strip in $xy$-plane and the function
$Q=\tfrac{1}{2\pi}\ln(\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}})$ is the
integral kernel for two-dimensional Laplace operator.

Equation (4) can be integrated numerically by sampling the function $J(x,y)$
on a square grid of equidistant points $(x_{i},y_{j})$:
$x_{i}=((i-\tfrac{1}{2})/N_{x}-\tfrac{1}{2})w$,
$y_{j}=((j-\tfrac{1}{2})/N_{y}-\tfrac{1}{2})t$, where $w$ and $t$ are the
strip width and thickness respectively (see Fig. 1) and $N_{x}$ and $N_{y}$
are number of points in $x$\- and $y$-direction, where $N_{x}/w\approx
N_{y}/t$ and $i=1,\ldots,N_{x}$, $j=1,\ldots,N_{y}$. By marking the points
$\mathbf{r}_{l}=(x_{i},y_{j})$ with an index $l=1,\ldots,N\\!=\\!N_{x}N_{y}$,
the function $J(x,y)$ becomes a vector $J_{l}$ with $N$ components, and the
integral kernel is transformed into the $N\times N$ matrix $Q_{l,l^{\prime}}$.
Note that $Q_{l,l^{\prime}}$ diverges for $l=l^{\prime}$, which should be
avoided. Eq. (4) for the density current distribution $J(x,y)$ can then be
rewritten in a matrix form:

$\lambda^{2}J_{l}=\sum_{l^{\prime}}\tilde{Q}_{l,l^{\prime}}J_{l^{\prime}}+\frac{1}{\mu_{0}}x_{l^{\prime}}B_{0},$
(5)

where
$\tilde{Q}_{l,l^{\prime}}=Q_{l,l^{\prime}}\frac{w}{N_{x}}\frac{t}{N_{y}}$. The
solution of Eq. (5) for $J_{l}$ can be found when we invert the matrix
$P_{l,l^{\prime}}=\lambda^{2}\delta_{l,l^{\prime}}-\tilde{Q}_{l,l^{\prime}}$:

$J_{l}=\frac{1}{\mu_{0}}B_{0}\sum_{l^{\prime}}(P^{-1})_{l,l^{\prime}}x_{l^{\prime}}.$
(6)

Finally, the distribution of the field $B_{\rm sc}$ produced by eddy currents
can be determined using the Biot-Savart law:

$\begin{split}B_{{\rm
sc},x}(x,y)=\frac{\mu_{0}}{2\pi}\iint_{S}dx^{\prime}dy^{\prime}\frac{J(x^{\prime},y^{\prime})(y^{\prime}-y)}{(x^{\prime}-x)^{2}+(y^{\prime}-y)^{2}},\\\
B_{{\rm
sc},y}(x,y)=-\frac{\mu_{0}}{2\pi}\iint_{S}dx^{\prime}dy^{\prime}\frac{J(x^{\prime},y^{\prime})(x^{\prime}-x)}{(x^{\prime}-x)^{2}+(y^{\prime}-y)^{2}},\end{split}$
(7)

where the integrals in Eq. (7) can be calculated numerically by sampling the
current density $J$ on the cross-section $S$ of SC strip.

###  II.2 The spin-wave modes confined in the well of magnetic field induced
by superconducting strip

After calculating the stray field generated by the SC strip, the next step is
to investigate the SW localization in the FM layer placed under the strip. Two
assumptions were made in our theoretical model: (i) our calculations have
shown that the magnetic field generated by the SC strip does not vary
significantly across the thin FM layer, so we assume that the SC field does
not depend on the $y$-coordinate and is equal to the value at the film center;
(ii) we have neglected the tangential component of the SC field and,
consequently, the deviation of the magnetization from the film normal, so we
have included only the normal component of the SC field, which is responsible
for the formation of the well for localized SW modes, and thus considered the
uniform magnetization directed perpendicular to the film plane.

The magnetization dynamics in continuous medium is described semi-classically
by the LL equation:

$\frac{\partial\mathbf{M}(\mathbf{r},t)}{\partial
t}=-|\gamma|\mu_{0}\mathbf{M}(\mathbf{r},t)\times\mathbf{H}_{\rm
eff}(\mathbf{r},t),$ (8)

where $\gamma$ is gyromagnetic ratio and $\mathbf{H}_{\rm eff}$ is effective
magnetic field, calculated from the free energy density $F$ [26]:

$\begin{split}F=-(\mathbf{B}_{0}+\mathbf{B}_{\rm
sc})\cdot\mathbf{M}+\frac{A_{\rm ex}}{M_{\rm s}^{2}}(\nabla\mathbf{M})^{2}\\\
-\frac{1}{2}\mu_{0}\mathbf{H}_{\rm d}\cdot\mathbf{M}-\frac{K_{\rm u}}{M_{\rm
s}^{2}}(\mathbf{M}\cdot\hat{\mathbf{n}})^{2}\end{split}$ (9)

with respect to magnetization: $\mathbf{H}_{\rm
eff}=-\frac{1}{\mu_{0}}\frac{\delta F}{\delta\mathbf{M}}$. Also, in Eq. (8),
we neglected the damping. The material parameters in Eq. (9): $M_{\rm s}$,
$A_{\rm ex}$, $K_{\rm u}$ denote the saturation magnetization, exchange
stiffness constant and uniaxial anisotropy, respectively. In our case, the
magnetocrystalline anisotropy is easy-axis anisotropy ($K_{\rm u}>0$),
oriented in the out-of-plane direction ($\hat{\mathbf{n}}=\hat{\mathbf{y}}$)
which overcome the shape anisotropy ($K_{\rm u}>\mu_{0}M^{2}_{\rm s}/2$). The
demagnetizing field is non-local (which means that $\mathbf{H}_{\rm
d}(\mathbf{r},t)$ at specific point $\mathbf{r}$ depends on the magnetization
distribution in the whole magnetic body). We use the linear approximation,
where the magnetization precesses harmonically around the equilibrium position
$\mathbf{M}_{0}(\mathbf{r})=M_{\rm s}\hat{\mathbf{y}}$ with the amplitude
$\mathbf{m}(\mathbf{r})=m_{x}\hat{\mathbf{x}}+m_{z}\hat{\mathbf{z}}$ much
smaller than the static magnetization component ($|\mathbf{m}(\mathbf{r})|\ll
M_{\rm s}$), giving the total magnetization vector
$\mathbf{M}(\mathbf{r},t)\approx\mathbf{M}_{0}(\mathbf{r})+\mathbf{m}(\mathbf{r})e^{i\omega
t}$. Then, we can decompose the demagnetizing field to static and dynamic
component $\mathbf{H}_{\rm d}(\mathbf{r},t)\approx-M_{\rm
s}\hat{\mathbf{y}}+\mathbf{h}_{\rm d}(\mathbf{r})e^{i\omega t}$, where the
dynamic component $\mathbf{h}_{\rm
d}=h_{\mathrm{d},x}\hat{\mathbf{x}}+h_{\mathrm{d},z}\hat{\mathbf{z}}$ can be
written in a general form:

$\mathbf{h}_{\rm
d}(\mathbf{r})=-\nabla\int_{V}dv^{\prime}\mathbf{m}(\mathbf{r}^{\prime})\cdot\nabla^{\prime}\frac{1}{4\pi|\mathbf{r}-\mathbf{r}^{\prime}|}.$
(10)

The LL equation (8) can be then written in the linearized form:

$\begin{split}\;\;i\omega\;m_{x}(x)=&|\gamma|\left[\left(B_{0}-\mu_{0}M_{\rm
s}+\frac{2K_{u}}{M_{\rm s}}+B_{{\rm sc},y}(x)\right)m_{z}(x)\right.\\\
&-\left.\frac{2A_{\rm ex}}{M_{\rm
s}}\frac{d^{2}}{dx^{2}}m_{z}(x)-\mu_{0}M_{\rm s}h_{{\rm d},z}(x)\right],\\\
-i\omega\;m_{z}(x)=&|\gamma|\left[\left(B_{0}-\mu_{0}M_{\rm
s}+\frac{2K_{u}}{M_{\rm s}}+B_{{\rm sc},y}(x)\right)m_{x}(x)\right.\\\
&-\left.\frac{2A_{\rm ex}}{M_{\rm
s}}\frac{d^{2}}{dx^{2}}m_{x}(x)-\mu_{0}M_{\rm s}h_{{\rm
d},x}(x)\right],\end{split}$ (11)

where we used the assumption that inside the thin FM layer both magnetization
and field are constant in an out-of-plane direction, i.e. do not depend on
$y$-coordinate, and the tangential component of the stray field of SC can be
neglected: $B_{{\rm sc},x}\approx~{}0$.

In most cases, it is impossible to obtain a rigorous analytical solution of LL
equation for the SWs in nanostructures. Usually the eigenfunctions of the
exchange field operator, which, however, are not eigenfunctions of the
operator for demagnetizing field, are used as trial functions in Ritz method.
Such an approach, when applied to planar nanoelements with relatively uniform
magnetization distribution, allows to obtain the solutions which are close to
the exact ones [27, 28, 29, 30]. In these systems, the demagnetizing field is
almost constant in the whole volume (due to the small magnetic volume
charges), except the vicinity of the edges (where surface magnetic charges
play role). However, such an approach does not work in a case of nanoelements
with strongly inhomogeneous demagnetizing or external fields. In such cases,
it is necessary to consider trial solutions being the eigenfunctions of the
operator, which is a sum of the exchange operator and the operators which
express the impact of demagnetizing effects or external fields. For instance,
the SW localization in tangentially-magnetized thin layer induced by the
dipolar stray field of magnetic sphere was considered in [31]. In this system,
the static stray field of the sphere, perceptible by the SW in FM layer, was
approximated by parabolic functions. In the framework of this approximation,
the linearized LL equation was reduced to the system of coupled Schrödinger-
like equations for quantum harmonic oscillator with dynamic demagnetizing
field of FM layer considered as a perturbation [31, 32, 33]. Therefore, the
eigenfunctions in the form of Hermite polynomials could be chosen as trial
solutions for SW modes.

Considering the assumptions made at the beginning of Sec. II, we can use the
approach proposed in Ref. [31] for investigating SC-FM hybrid system. The
linearized LL equation can be then written in a compact form:

$\begin{split}\;\;i\omega\;m_{x}(x)&=|\gamma|\big{(}\Xi(x)m_{z}(x)-\mu_{0}M_{\rm
s}h_{{\rm d},z}(x)\big{)},\\\
-i\omega\;m_{z}(x)&=|\gamma|\big{(}\Xi(x)m_{x}(x)-\mu_{0}M_{\rm s}h_{{\rm
d},x}(x)\big{)}.\end{split}$ (12)

The operator $\Xi(x)=-\tfrac{2A_{\rm ex}}{M_{\rm
s}}\tfrac{d^{2}}{dx^{2}}+B_{y}(x)$ describes the impact of dynamic exchange
field and total static field:

$B_{y}(x)=\\!B_{0}-\mu_{0}M_{\rm s}+\frac{2K_{\rm u}}{M_{\rm
s}}+\underbrace{B_{0}(\beta_{0}+\beta_{2}x^{2}+O(x^{2}))}_{B_{{\rm
sc},y}(x)},$ (13)

on magnetization dynamics. It is worth noting that, according to Eqs. (6) and
(7), the stray field $B_{{\rm sc},y}(x)$ is linearly scaled by external field
$B_{0}$.

By introducing the circular polarization for dynamic component of
magnetization: $m=m_{z}+im_{x}$ and neglecting the dynamic demagnetizing field
$\mathbf{h}_{\rm d}$, we can transform the set of equations (12) into the form
equivalent to the stationary Schrödinger equation:

$\Xi(x)\,m_{n}(x)=\frac{1}{|\gamma|}\omega_{0,n}m_{n}(x).$ (14)

Equation (14) can be solved analytically after neglecting higher-order terms
in the expansion in Eq. (13). The problem is then reduced to the magnonic
counterpart of quantum harmonic oscillator with equidistant eigenfrequencies
$\omega_{0,n}$ and corresponding eigenfunctions $m_{n}(x)$ expressed by
Hermite polynomials
$H_{n}(\xi)~{}=~{}(-1)^{n}\exp(\xi^{2})\tfrac{d^{n}}{d\xi^{n}}\exp(-\xi^{2})$:

$m_{n}(\xi=Kx)=CM_{\rm
s}\underbrace{\frac{1}{\sqrt{2^{n}n!\sqrt{\pi}}}\exp(-\xi^{2})H_{n}(\xi)}_{\psi_{n}(\xi)}$
(15)

for every $n^{\rm th}$ SW mode. The scaling factor
$K=\sqrt[4]{B_{0}\,\beta_{2}\,\tfrac{M_{\rm s}}{2A_{\rm ex}}}$ expresses the
interplay between the strength of exchange interaction and the curvature
(strength) of the well of SC stray field, in which the SW modes are confined.
The real function $\psi_{n}(\xi)$ is normalized:
$\int_{-\infty}^{\infty}\psi_{n}(\xi)=1$ and dimensionless constant $C$
expresses the complex SW amplitude.

The approximation $B_{{\rm sc},y}(x)\approx B_{0}(\beta_{0}+\beta_{2}x^{2})$
is not valid for wide SC strips: $w\gg\lambda$ (where the well $B_{{\rm
sc},y}(x)$ becomes relatively flat at the bottom) and narrow strips:
$w<\lambda$ or small $B_{0}$ (where the well is shallow and distortion of its
parabolic shape, related to the presence of the barriers on its sides, affects
the mode of the lowest frequency). Therefore, we solve Eq. (14) numerically
for the stray field $B_{{\rm sc},y}$ in a full form given by Eq. (7) – this
task is relatively simple and does not demand extensive computations. The
obtained frequencies $\tilde{\omega}_{0,n}\neq{\omega}_{0,n}$ of bound SW
modes were calculated for realistic (i.e. non-parabolic) stray field of SC
strip but with dynamic dipolar interaction neglected. However, we used the
approximated (by Hermite polynomials) profiles of the eigenmodes from Eq. (15)
to include the role of dynamic dipolar interactions.

The next step in our computational procedure is to include the impact of the
dynamic demagnetizing field $\mathbf{h}_{\rm d}$ and correct the frequencies
of individual modes $\tilde{\omega}_{0,n}$ calculated in the absence of
$\mathbf{h}_{\rm d}$. We are going to limit our calculations to so-called
diagonal approximation [33], which was satisfactory for many similar cases
[32, 31], where the difference between successive frequencies
$\tilde{\omega}_{0,n}$ is large with respect to frequency shift due to dynamic
dipolar interaction. This approximation neglects also the inter-mode dipolar
coupling. According to Eq. (10), the dynamic demagnetizing field produced by a
single mode can be written in a form:

$h_{{\rm
d},p,n}(x)=-\int_{V}dv^{\prime}\sum_{q}m_{q,n}(x^{\prime})\frac{d^{2}}{dpdq^{\prime}}\frac{1}{4\pi|\mathbf{r}-\mathbf{r}^{\prime}|},$
(16)

were the indices $p$ and $q$ run over the coordinates $x$ and $z$, and
derivative $\frac{d}{dq^{\prime}}$ means $\frac{d}{dx^{\prime}}$ (or
$\frac{d}{dz^{\prime}}$) for $q=x$ (or $q=z$). The symbol $m_{q,n}(x)$ denotes
the components of dynamic magnetization for $n^{\rm th}$ mode.

The spectrum of eigenmodes, which includes the correction resulting from the
presence of dynamic dipolar interaction, has the following form (see Appendix
A for derivation):

$\omega_{n}\\!=\\!\sqrt{\big{(}\tilde{\omega}_{0,n}\\!-\\!|\gamma|\mu_{0}M_{\rm
s}\langle\tilde{h}_{{\rm
d},x}\rangle_{n}\big{)}\big{(}\tilde{\omega}_{0,n}\\!-\\!|\gamma|\mu_{0}M_{\rm
s}\langle\tilde{h}_{{\rm d},z}\rangle_{n}\big{)}},$ (17)

where $\langle\tilde{h}_{{\rm d},p}\rangle_{n}$ are averaged and normalized
(dimensionless) components of $\mathbf{h}_{\rm d}$. The averaged field
$\langle\tilde{h}_{{\rm d},z}\rangle_{n}$ is equal to zero in considered
system, whereas $\langle\tilde{h}_{{\rm d},x}\rangle_{n}$ can be expressed by
the compact formula (see Ref. [31] and Appendix B):

$\langle\tilde{h}_{{\rm
d},x}\rangle_{n}\\!\approx-\frac{1}{K}\int_{-\infty}^{\infty}\\!dk_{x}\left(1\\!-\\!\frac{1-e^{-k_{x}a}}{k_{x}a}\right)\tilde{\psi}_{n}\left(\frac{k_{x}}{K}\right),$
(18)

where
$\tilde{\psi}_{n}(\mathcal{K})=\tfrac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-i\mathcal{K}\xi}\psi_{n}(\xi)d\xi$
is the Fourier transform of $\psi(\xi)$. The one-dimensional integral in Eq.
(18) is relatively simple to compute in comparison with the calculations of
$\langle\tilde{h}_{{\rm d},u}\rangle_{n}$ directly from Eq. (16). It is worth
noting that we approximate $m_{p,n}(x)$ by taking $m_{p,n}(x)\approx
C_{p}M_{\rm s}\psi_{n}(Kx)$ where $\psi_{n}(Kx)$ are eigenfunctions of
$\Xi(x)$. The amplitudes $C_{p}$ are canceled during the normalization – see
Appendix B.

#### Micromagnetic simulations

To perform micromagnetic simulations we used the open-source environment
Mumax3 [34] that solves LL equation using finite-difference method in the time
domain. We discretized the system with a regular mesh with unit cell $2\times
20\times 20$ nm3 (along the $x$, $y$, and $z$-axis respectively). The system
was placed in an inhomogeneous external magnetic field
$\mathbf{B}_{0}+\mathbf{B}_{\rm sc}(\mathbf{r})$ that includes the full
contribution of the stray field of SC strip. The field profile was imported
from an external file containing the results of semi-analytic calculations of
$\mathbf{B}_{\rm sc}(\mathbf{r})$. We applied the periodic boundary conditions
along the $z$-axis (1024 repetitions) to simulate an infinitely-long strip. We
increased the damping constant close to the edges of the domain perpendicular
to $x$-axis from $\alpha_{\mathrm{edge}}=0$ to $0.5$ in order to eliminate the
reflections along $x$-axis edges. The SWs in the system were numerically
excited by a rectangular model-antenna of width $12$ nm, placed in the center
of the film. The antenna generated a local dynamic magnetic field described
with the function
$H_{\mathrm{ant}}(t)=H_{0}\mathrm{sinc}(2\mathrm{\pi}f_{\mathrm{cut}}(t-t_{0}))\hat{x}$,
where $\mu_{0}H_{0}=0.024B_{0}$, $f_{\mathrm{cut}}$ is a cutoff frequency that
varies in the simulations depending on the depth of the well of stray field
$\Delta B_{{\rm sc},y}$ – see Fig. 2(a). In each case of different $\Delta
B_{{\rm sc},y}$, the simulation was run for the timespan
$1000/f_{\mathrm{cut}}$. The results of simulations were saved continuously as
the snapshots of magnetic configuration with time sampling
$t_{\mathrm{sample}}=0.5/(1.5f_{\mathrm{cut}})$. In the postprocessing, all
saved magnetic configurations were used to calculate the spectra by employing
the Fourier transform. The visualizations of particular modes in the system
were done by calculating the inverse Fourier transform of frequencies
corresponding to the modes.

The micromagnetic simulations were also used to calculate the stray field
produced by magnetization texture in FM layer resulting from the action of a
stray field from SC strip – see Section S1 in Supplemental Material.

#### Finite-element method

Numerical simulations were performed in COMSOL Multiphysics using finite-
element method in order to calculate the stray field coming from the SC strip
subjected to the external magnetic field. To be consistent with the semi-
analytical calculations, we used London equation

$\Delta A=\frac{1}{\lambda^{2}}A$ (19)

inside the SC material and Ampere law

$\Delta A=0$ (20)

outside of the SC material, both modified to the form for magnetic vector
potential $A$. We investigated the system in two-dimensional model in
$xy$-plane. In this case, the Coulomb gauge $\nabla\cdot\mathbf{A}=0$ and the
condition on the SC boundary $\mathbf{A}\cdot\hat{n}=0$ are automatically
fulfilled. External magnetic field $B_{0}$ was applied as a boundary condition
$-\partial_{x}A=B_{0}$ far away from the SC material. Here, the SC strip was
placed in the center of 100 $\mu$m square-shaped cell, on which boundary this
condition is applied.

We have checked the validity of London model for considered SC structure by
finite-element method calculation of the stray field based on Ginzburg-Landau
model – see Section S2 in Supplemental Material.

##  III Results

We considered SC strip of the thickness $t=100$ nm and width $w=400$ (or $800$
nm, for comparison). The strip was made of Nb for which we assumed the London
penetration depth $\lambda=50$ nm [35]. The SC strip was placed over the FM
layer of thickness $a=20$ nm and separated from its top surface by the non-
magnetic and non-conducting spacer of thickness $g=10$ nm – see Fig. 1. The FM
layer, made of Ga:YIG, was characterized by the following values of material
parameters: saturation magnetization $M_{\rm s}=16$ kA/m, exchange stiffness
$A_{\mathrm{ex}}=1.37$ pJ/m and gyromagnetic ratio $\gamma=179$ GHz/T. The
Ga:YIG layer can have a strong PMA [36]. We assumed the realistic value of the
uniaxial anisotropy $K_{\mathrm{u}}=756$ J/$\mathrm{m}^{3}$, which ensures the
out-of-plane orientation of magnetization even in the absence of external
magnetic field. For micromagnetic simulations, we included the magnetization
damping constant for Ga:YIG $\alpha=0.001$. For semi-analytical calculations
the damping was neglected.

###  III.1 Stray field of superconducting strip

Figure 2: The profiles of the magnetic field components (a) $B_{{\rm sc},y}$
and (b) $B_{{\rm sc},x}$ produced by SC strip, calculated using Eq. (7) (solid
lines) and using numerical simulations (square dots). (c) The deflection
$\theta_{M}$ of the magnetization from the direction of external field
$\mathbf{B}_{0}=B_{0}\hat{\mathbf{y}}$. The inset plot shows the evolution of
the maximum angle deviation with the external magnetic field value. All
profiles (a,b,c) are plotted along the line $y=0$, passing through the center
of the strip: $(x,y)=(0,0)$. Dashed vertical lines mark the edges of SC strip.
Black and red lines correspond to the external field $B_{0}$ equal to 30 mT
and 90 mT, respectively. Please note that to calculate the total magnetic
field, we have to add $\mathbf{B}_{0}$ and $\mathbf{B}_{{\rm sc}}$.

Figure 2(a,b) presents the profiles of the $y$-component (Fig. 2(a)) and
$x$-component (Fig. 2(b)) of the stray field $\mathbf{B}_{\rm sc}$ produced by
SC strip of the width $w=400$ nm for two selected values of out-of-plane
applied field: $B_{0}=30$ mT (black lines) and $90$ mT (red lines). The
profiles are the $x$-coordinate dependencies calculated for fixed
$y=y_{0}=-(t/2+g+a/2)=-70$ nm, i.e. in the middle of the FM film. The
dependencies $B_{{\rm sc},y}(x)$, $B_{{\rm sc},x}(x)$ are consistent with the
expected onion-like shape of the field around SC strip (see Fig. 1(b)). The
well of $B_{{\rm sc},y}(x)$ is a signature of the screening of external field
$\mathbf{B}_{0}=B_{0}\hat{\mathbf{y}}$ by the cost of the increase of the
field close to the edges of the strip (vertical dashed lines in Fig. 2). On
the other hand, the in-plane component of the stray field $B_{{\rm sc},x}$
reflects the deflection of the magnetic field lines by-passing the SC strip.
This effect is the strongest in the proximity of the strip edges. The stray
field was calculated semi-analytically using Eq. (7) – (red and black lines in
Fig. 2(a,b)), and obtained results were successfully cross-checked using
finite-element method calculations – (red and black square dots in Fig.
2(a,b)). Eq. (7) and the plots in Figs. 2(a,b) show that the stray field is
linearly scalable with the external field $B_{0}$. While we increase $B_{0}$
from 30 mT to 90 mT, the magnitude of the profiles $B_{{\rm sc},y}(x)$,
$B_{{\rm sc},x}(x)$ are increasing exactly three times – see Eqs. (6) and (7).

In the absence of the SC strip, the static magnetization is oriented out of
plane due to the PMA in Ga:YIG layer but the in-plane component $B_{{\rm
sc},x}$ tilts magnetization from the layer’s normal – see Fig.2(c).
Surprisingly, the angle between the magnetization and the applied field’s
direction increases with the increase of the magnitude of applied field.
However, it is understandable if we keep in mind that the in-plane component
of the stray field, which is responsible for the magnetization tilting,
increases with the applied field: $B_{{\rm sc},x}\propto B_{0}$ – see Fig.
2(b). The angle, at which magnetization is deflected from out-of-plane
direction $\theta_{M}(x)$ (Fig.2(c)), can be calculated by finding minimum of
the free energy density (9) for successive positions $x$. Our calculations
have shown that the inhomogeneous exchange interaction caused by non-collinear
magnetization texture does not significantly influence the value of the
magnetization deviation angle. Therefore, to estimate angle $\theta_{M}$, we
considered the free magnetic energy density of the film in an external
magnetic field applied at an angle $\theta_{B}$ neglecting exchange
interaction:

$\begin{split}F(\theta)=&-BM_{s}\cos(\theta-\theta_{B})\\\
&+\tfrac{1}{2}\mu_{0}M_{\rm s}\cos^{2}(\theta)-K_{\rm
u}\cos^{2}(\theta)\end{split}$ (21)

where the angle $\theta$ is trial orientation of the magnetization for which
we look for a minimum of $F(\theta)$ at every position $x$ independently. The
angle $\theta=\theta_{M}$ corresponding to the minimum: $F_{\rm
min}=F(\theta_{M})$ determines the equilibrium orientation of the
magnetization. The first term in Eq. (21) corresponds to the Zeeman
interaction with total magnetic field, the second one describes the
demagnetizing energy (shape anisotropy), and the third one corresponds to the
energy related to the out-of-plane uniaxial anisotropy. It is worth noting
that both the magnitude $B(x)=|\mathbf{B}_{0}+\mathbf{B}_{\rm sc}(x)|$ and the
angle $\theta_{B}(x)={\rm arctan}[B_{{\rm sc},x}(x)/(B_{0}+B_{{\rm
sc},y}(x))]$ of total static magnetic field are position-dependent.

The minimization of Eq. (21) results in an equation for the local
magnetization angle $\theta_{M}(x)$ as a function of the $x$-coordinate. For
relatively small values of the angles $\theta-\theta_{B}$ and $\theta$, the
trigonometric functions in Eq. (21) can be expanded in Taylor series which
results in the polynomial equation for $\theta_{M}$:

$\begin{split}\frac{1}{\mu_{0}}B\left[(\theta_{M}-\theta_{B})-\frac{1}{6}(\theta_{M}-\theta_{B})^{3}\right]\\\
+\left(\frac{2K_{u}}{M_{s}}-M_{s}\right)\left[\theta_{M}-\frac{2}{3}\theta^{3}\right]=0,\end{split}$
(22)

which is relatively easy to solve. For larger values of the applied field
$B_{0}$ and related deflection angles, the equilibrium orientation
$\theta_{M}(x)$ must be found numerically. The inset plot in Fig. 2(c) shows
evolution of the maximum angle deviation $\theta_{M,\rm{max}}$ with the value
of applied magnetic field $B_{0}$.

It should be noted that the formation of non-collinear magnetization texture
allows the SW excitation by an alternating magnetic field applied along the
external static field $\mathbf{B}_{0}$. In this case, the SW modes will only
be excited in the regions below the edges of the SC strips, where the static
magnetization is tilted.

Two-dimensional maps of the stray field $\mathbf{B}_{{\rm sc}}$ are shown in
Section S2 in Supplemental Material, comparing maps calculated from the London
model to those obtained through the solution of Ginzburg-Landau equations. Our
results show that for the assumed value of $\lambda$ = 50 nm, we are able to
keep the Meissner state for the field $B_{0}=30$ mT, and moreover, the results
of the London model are almost the same as the results for the Ginzburg-Landau
model.

Figure 3: The depth of the well of SC field $\Delta B_{{\rm sc},y}$ (see Fig.
2(a)) as a function of London penetration depth $\lambda$ calculated using Eq.
(7) (solid lines) and numerical simulations (square dots). The black and red
lines correspond to two different values of the external field $B_{0}=30$ mT
and 90 mT, respectively. Dashed vertical lines show critical values of
$\lambda$ for Meissner ($\lambda_{C1}$) and mixed ($\lambda_{C2}$) states for
$B_{0}=30$ mT and 90 mT in the Ginzburg-Landau model assuming $\kappa=1$.

London penetration depth $\lambda$ is a crucial parameter, which determines
efficiency of the magnetic field screening by SC material. To understand how
$\lambda$ influences on the condition for the SW localization, we have
calculated the depth of the well $B_{{\rm sc},y}(x)$ (see Fig. 2(a)) as a
function of $\lambda$ for the width of SC strip $w=400$ nm and two selected
values of the external field $B_{0}=30$ mT (black lines and squares) and 90 mT
(red line and squares) – see Fig. 3. Semi-analytical calculations performed
with the help of Eq. (7) (solid lines) are in perfect agreement with results
of numerical simulations (square dots) for $\lambda>20$ nm, while for smaller
value of $\lambda$ semi-analytical computations faces difficulties since they
require the bulk cell size to be much smaller than $\lambda$, what demands a
significant increase of the number of cells and time-consuming computations.
The dependencies $\Delta B_{{\rm sc},y}(\lambda)$ were presented in Fig. 3. It
is evident that the decrease of $\Delta B_{{\rm sc},y}$ with increasing
$\lambda$ is a manifestation of the gradual disappearance of SC properties,
which is revealed as a weakening stray field that allows the external field to
penetrate deeper into the SC strip. However, this picture becomes more
complicated when we consider the more general theory of superconductivity
described by Ginzburg-Landau model which allows the phase transitions to mixed
and normal phase at the critical fields $B_{C1}$ and $B_{C2}$, respectively,
for type-II superconductors like Nd. Taking into account the critical fields
dependencies on material parameters: penetration depth $\lambda$ and
correlation length $\xi$, we should look for the critical values of material
parameters for which the Meissner state or superconductivity at all is
destroyed. To estimate critical values of $\lambda$, we have used the formulas
for critical fields of a bulk SC: $B_{C2}=\Phi_{0}/(2\pi\xi)$ and
$B_{C1}=B_{C2}\kappa/\sqrt{2}$ (where $\Phi_{0}$ is magnetic flux quantum) and
we set $\kappa=\xi/\lambda=1$ (which is in the range of experimental values
for Nb $0.85<\kappa<1.3$ [37]). For $B_{0}=30$ mT, $\lambda_{C1}=88$ nm and
$\lambda_{C2}=105$ nm (black dashed lines in Fig. 3). It guarantees the
Meissner state and justify the usage London’s theory, which were confirmed by
numerical calculations according to the Ginzburg-Landau theory (see Section S2
in Supplemental Material). For $B_{0}=90$ mT, $\lambda_{C1}=51$ nm and
$\lambda_{C2}=90$ nm (red dashed lines in Fig. 3). Taking into account that
these values are approximate, there is no guarantee that obtained results for
$B_{0}=90$ mT can be observed experimentally.

In general, the London penetration depth is not only material-specific but
depends also on the sample shape and dimensions. It was shown in Ref. 35 that
a decrease of the sample size leads to an increase of $\lambda$. In our
theoretical calculations we took $\lambda=50$ nm, which is not far from the
experimental value [35].

###  III.2 The spin-wave modes confined in the well of magnetic field induced
by superconducting strip

Figure 4 presents the SW profiles of the modes localized in the well of
effective field produced by the SC strip, for the external fields: $B_{0}=10$
mT (Fig. 4(c)), 30 mT (Fig. 4(a,d)) and 90 mT (Fig. 4(b,e)). The profiles were
computed numerically with the help of Mumax3 software (light red solid lines)
and calculated using the semi-analytical model (red dashes lines). The blue
lines show FMR frequency of homogeneous film at certain $B_{0}$ fields in the
absence of the SC strip. Left and right columns in Fig. 4 correspond to the
width of SC strip $w=400$ nm (Fig. 4(a,b)) and 800 nm (Fig. 4(c,d,e)),
respectively. For larger value of external field $B_{0}$, the well $B_{{\rm
sc},y}(x)$ is deeper (see Fig. 2(a)) and can accommodate more SW modes.
However, there is an upper limit for $B_{0}$ caused by phase transition at
$B_{C1}$. Our model, based on London’s theory, does not include the existence
of mixed state, which can appear in the considered range of external field
$B_{0}$. The presence of vortices will modify the stray field produced by the
SC strip which will be important in the experimental implementation of
investigated system. However, the ultimate limit is given by the second
critical field[38]. On the other hand, the field $B_{0}=10$ mT is too small
for the strip of width $400$ nm to create a well which is deep enough to
confine SW modes. For the external fields $B_{0}=30$ mT (Fig. 4(a)) or $90$ mT
(Fig. 4(b)) the well is sufficiently deep to bond one or two SW modes (solid
and dashed lines) of the frequencies lower than FMR frequency of homogeneous
film (horizontal blue line). Another strategy for SW binding is to widen the
SC strip and thus widen the related well of the stray field.

Figure 4: The SW profiles of the modes localized in the effective field well
produced by SC strip for the external field values $B_{0}=10$ mT (c), $30$ mT
(a,d) and $90$ mT (b,e) obtained from micromagnetic simulations using Mumax3
environment (light red solid lines) and the results of the semi-analytical
model (red dashed lines). The blue lines show FMR frequency of a homogenous
film for given external magnetic field $B_{0}$ value in the absence of the SC
strip. The left and right columns correspond to the SC width $w=400$ nm and
$800$ nm, respectively. Vertical black dashed lines indicate the position of
the SC strip edges. All the figures share the same frequency and space scales
and the amplitude of modes is normalized to enhance the readability of the
figure (the difference between the results obtained by the semi-analytical
approach and micromagnetic simulations is shown by the offset in the figure).

In Fig. 4(c-e), for $w=800$ nm, we observe a larger number of localized SW
modes, what is caused by two factors: (i) the frequency difference between SW
mode energy levels becomes smaller due to the widening of the well (the main
factor), and (ii) the depth of the well is slightly increased with the
widening of SC strip (e.g., for the $w=800$ nm $\Delta B_{sc,y}$ is about 1.4
times larger than for $w=400$ nm). Thus, for $w=800$ nm, there are 1, 2, and 4
localized modes for $B_{0}=10$ mT (Fig.. 4(c)), $30$ mT (Fig. 4(d)), and $90$
mT (Fig. 4(e)), respectively. It should be noted that the approximation of the
well $B_{{\rm sc},y}(x)$ by a parabolic well (13) is valid in the considered
system only for finite range of the widths $w$ (around $350$ nm), while for
much smaller or larger width, the terms higher than quadratic must be included
in the expansion (13). For example, for $w=400$ nm and $800$ nm, the
approximation of $B_{sc,y}$ requires including up to $4^{\rm th}$ and $6^{\rm
th}$ order terms, respectively. The semi-analytical results (dashed lines) are
in relatively good agreement with numerical solution of LL equation obtained
using the MuMax3 environment (solid lines). Due to the presence of higher
order terms in $B_{{\rm sc},y}$, the SW modes are not equidistant in contrast
to the eigenfunctions of the quantum harmonic oscillator, i.e., Hermitian
functions. However, our mode profiles can be approximated by Hermitian
functions with fitting parameter $\beta_{2}$ in (15). This trick allowed us to
simplify the averaged demagnetizing field to a compact form (18) (see Appendix
B) and easily find the eigenfrequencies (17).

Figure 5: The dependence of the localized SW modes on the external magnetic
field $B_{0}$ for the systems with SC strip width $w=400$ nm (a) and $w=800$
nm (b). The solid lines and square dots correspond to the semi-analytical
calculations and micromagnetic simulations respectively. Dashed lines show FMR
frequency of the uniformly-magnetized film.

The dependence of frequencies of the localized SW modes on the external
magnetic field is presented in Fig. 5 for the SC strip width $w=400$ nm (Fig.
5(a)) and $800$ nm (Fig. 5(b)). The results of semi-analytical calculations
and micromagnetic simulations are marked by solid lines and dots,
respectively. As mentioned in the previous paragraph, the number of localized
modes rises with the increase of the external field value and the width of the
SC strip. In general, the results of theoretical calculations and
micromagnetic simulations are in good agreement. The discrepancies are larger
for the strip width $w=800$ nm than for $400$ nm and they increase with
increasing external magnetic field. This can be related to the simplifications
underlying our theoretical model, according to which we have neglected the
tangential component of the field produced by the SC strip $B_{{\rm sc},x}$,
which slightly deflects the static magnetization from the normal to the FM
layer and forms a non-collinear magnetization structure in the vicinity of the
SC strip edges (see Fig. 2(c)). In our model, we have considered the uniform
magnetization ground state where the FM layer is magnetized out of plane. This
assumption is reasonable because the magnetization deviation angle
$\theta_{M}$ is relatively small for a narrow strip, while for a wider strip
the deviation angle is larger and increases significantly for large external
field values. To prove that this is the main reason for the discrepancies
between micromagnetic simulations and the semi-analytical model, we have
performed micromagnetic simulations with $B_{{\rm sc},x}=0$, which showed
results much closer to theoretical calculations. Small differences can also be
caused by the use of the diagonal approximation in our theory. However, the
diagonal elements of the dynamical demagnetization field are quite small,
which means that non-diagonal elements can really be neglected because they
are usually smaller than diagonal elements.

##  IV Conclusions

We examined a superconductor-ferrimagnet hybrid planar nanostructure wherein a
flat SC strip is situated above a uniform FM layer while the external field is
applied out of plane. In this configuration, the flat SC strip, being in a
Meissner state, produces a stray field that reduces the static effective field
inside the thin FM layer placed below. When the FM layer is made of soft
magnetic material with PMA, the SW modes can be confined within the region of
lowered field (i.e. in the well of internal field). These modes have
frequencies lower than the FMR frequency of the FM layer, which explains their
exponential decay outside the region of the well. By adjusting the value of
the uniform external field, we can control the depth of the well and modify
the number and frequencies of confined SW modes. This outcome is difficult to
achieve in the conventional magnonic nanostructures where the stray field, as
a result of demagnetization, depends on the geometry and magnetic
configuration. The first is determined at the fabrication stage and the latter
can be modified by an external field, but its full control requires a strong
fields due to large demagnetization effects in planar structures.

Our work integrates the numerical and semi-analytical studies. The analytical
approach is computationally undemanding and provides deep insight into the
magnetization dynamics and the mechanism of SW localization in dipolar-coupled
superconductor-ferrimagnet systems. The presented ideas can be used to design
superconductor-ferrimagnet hybrid devices for on-demand control of SW
localization and propagation.

###### Acknowledgements.

The authors would like to thank M. Silaev, O. Dobrovolskiy, O. Tartakivska and
M. Zelent for fruitful discussions. The work was supported from the grants of
the National Science Center – Poland, Nos. UMO-2019/35/D/ST3/03729,
UMO-2021/43/I/ST3/00550, and UMO-2021/41/N/ST3/04478. The numerical
simulations were performed at Poznań Supercomputing and Networking Center
(Grant No. pl0095-01). Krzysztof Sobucki is a scholarship recipient of the
Adam Mickiewicz University Foundation for the academic year 2023/2024.

## Data availability

Data supporting this study are openly available from the repository at
https://zenodo.org/records/10405886.

##  Appendix A Dispersion relation – derivation

The dispersion relation can be derived from Eq. (12) after introducing the
average of the dipolar field (dipolar operator) in the way similar to the
approach known in quantum mechanics. Let us express each SW mode by the
function from orthonormal base: $m_{p,n}(x)\approx C_{p}M_{\rm s}\psi_{n}(Kx)$
as in Eq. (15). When we introduce the norm
$\langle\psi^{2}_{n}\rangle:=\int_{V}dv|\psi_{n}(Kx)|^{2}\\!=\\!\frac{ab}{K}$,
and then multiply both equations in (12) by $\psi^{*}_{n}(Kx)$ and integrate
them over the whole volume of magnetic system, we obtain:

$\begin{split}\;\;i\omega\;C_{x}\langle\psi^{2}_{n}\rangle&=C_{z}\langle\psi_{n}|\Xi(x)|\psi_{n}\rangle-|\gamma|\mu_{0}\langle
h_{{\rm d},z}\rangle_{n},\\\
-i\omega\;C_{z}\langle\psi^{2}_{n}\rangle&=C_{x}\langle\psi_{n}|\Xi(x)|\psi_{n}\rangle-|\gamma|\mu_{0}\langle
h_{{\rm d},x}\rangle_{n},\end{split}$ (23)

where $\langle h_{{\rm d},p}\rangle_{n}:=\int_{V}dv\psi^{*}_{n}(Kx)h_{{\rm
d},p}(x)$ for $p=\\{x,y\\}$ can be considered as an averaged component of the
dynamic demagnetizing field. This definition is clear when we notice that the
field $h_{{\rm d},p,n}$ is the result of action of the integral operator on
$\psi_{n}(Kx)$ (strictly related to the magnetization $m_{p,n}(x)$) – see Eq.
(16). Taking into account that $\psi_{n}(Kx)$ are eigenfunctions of the
operator $\Xi(x)$ with the eigenvalues $\omega_{0,n}$, but not of the
dynamical demagnetizing field operator, Eq. (23) can be rewritten as:

$\begin{split}\;\;i\omega\;C_{x}&=C_{z}\big{(}\omega_{0,n}-|\gamma|\mu_{0}M_{\rm
s}\langle\tilde{h}_{{\rm d},z}\rangle_{n}\big{)},\\\
-i\omega\;C_{z}&=C_{x}\big{(}\omega_{0,n}-|\gamma|\mu_{0}M_{\rm
s}\langle\tilde{h}_{{\rm d},x}\rangle_{n}\big{)},\end{split}$ (24)

where $\langle\tilde{h}_{{\rm d},p}\rangle_{n}=1/(C_{p}M_{\rm s})\langle
h_{{\rm d},p}\rangle$ is a dimensionless averaged field (see Appendix B). The
homogeneous set of equations (24) gives non-trivial solution for the resonance
frequency $\omega_{n}$ when

$\begin{split}\begin{vmatrix}\;\;i\omega\;&\omega_{0,n}-|\gamma|\mu_{0}M_{\rm
s}\langle\tilde{h}_{{\rm d},z}\rangle_{n}\\\
\omega_{0,n}-|\gamma|\mu_{0}M_{\rm s}\langle\tilde{h}_{{\rm
d},x}\rangle_{n}&-i\omega\end{vmatrix}=0\end{split},$ (25)

which leads to dispersion relation shown in Eq. (17).

##  Appendix B Averaged dipolar field – derivation

For every $n^{\rm th}$ SW mode $\mathbf{m}_{n}(x)$ (confined in the well of
stray field), the dynamical demagnetizing field $\mathbf{h}_{{\rm d},n}(x)$
(related to own dynamics of $\mathbf{m}_{n}(x)$) is given by Eq. (16). The
field $\mathbf{h}_{{\rm d},n}(x)$ can be considered as a result of action of
integral operator on magnetization $\mathbf{m}_{n}$. When we approximate
$\mathbf{m}_{n}$ by the function $\psi_{n}(Kx)$ from orthonormal set:
$m_{p,n}(x)\approx C_{p}M_{\rm s}\psi_{n}(Kx)$ (see Eq. (15)) and integrate
$\mathbf{h}_{{\rm d},n(x)}$ over the whole volume with $\psi_{n}(Kx)$, we
obtain average value of the dipolar field operator:

$\begin{split}\langle h_{{\rm d},p}\rangle_{n}\\!=\\!-M_{\rm
s}\sum_{q}C_{q}\int_{V}\int_{V^{\prime}}\\!dvdv^{\prime}\;&\psi_{n}(Kx)\psi_{n}(Kx^{\prime})\times\\\
&\frac{d^{2}}{dp\,dq^{\prime}}\frac{1}{4\pi|\mathbf{r}-\mathbf{r^{\prime}}|},\end{split}$
(26)

where the indices $p$ and $q$ run over the coordinates $x$ and $z$, and
derivative $\frac{d}{dq^{\prime}}$ means $\frac{d}{dx^{\prime}}$ (or
$\frac{d}{dz^{\prime}}$) for $q=x$ (or $q=z$). Each of the integrals in (26)
is actually a three-fold integral:
$\int_{-\infty}^{\infty}dx\int_{y_{0}-a/2}^{y_{0}+a/2}dy\int_{-b/2}^{b/2}dz$,
where $y_{0}=-(t/2+g+a/2)$. This problem can be simplified when we express the
function $\frac{1}{4\pi|\mathbf{r}-\mathbf{r^{\prime}}|}$ in the Fourier space
for the wave vector $\mathbf{k}=k_{x}\hat{\mathbf{x}}+k_{z}\hat{\mathbf{z}}$
[39]:

$\begin{split}\frac{1}{4\pi|\mathbf{r}-\mathbf{r^{\prime}}|}=\frac{1}{2\pi}\\!\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\\!dk_{x}dk_{z}\;\frac{1}{k}e^{-k|y-y^{\prime}|}\times\\\
e^{ik_{x}(x-x^{\prime})}e^{ik_{z}(z-z^{\prime})},\end{split}$ (27)

where $k=\sqrt{k_{x}^{2}+k_{y}^{2}}$, and then use the identities:

$\begin{split}\int_{y_{0}-a/2}^{y_{0}+a/2}\int_{y_{0}-a/2}^{y_{0}+a/2}\\!\\!dydy^{\prime}\;\frac{1}{k}e^{-k|y-y^{\prime}|}=\frac{2a}{k}\left(\\!1-\frac{1-e^{-ka}}{ka}\\!\right),\\\
\int_{-b/2}^{b/2}\int_{-b/2}^{b/2}\\!\\!dzdz^{\prime}e^{ik_{z}(z-z^{\prime})}=2\pi
b\;\delta(k_{z}),\;{\rm for}\;b\\!\rightarrow\\!\infty.\end{split}$ (28)

This allows to express (26) in the form:

$\begin{split}\langle h_{{\rm d},p}\rangle_{n}\\!=\\!-M_{\rm
s}\frac{ab}{K^{2}}\sum_{q}C_{q}\int dk_{x}\left(\int
dk_{z}\frac{k_{p}k_{q}}{k^{2}}\times\right.\\\
\left.\left(1-\frac{1-e^{ka}}{ka}\right)\delta(k_{z})\,\tilde{\psi}_{n}^{2}\\!\left(\frac{k_{x}}{K}\right)\right).\end{split}$
(29)

One can notice that $\langle h_{{\rm d},z}\rangle_{n}=0$ and $\langle h_{{\rm
d},x}\rangle_{n}$ contains only the term for $p=x$. In the linear regime, the
dynamic demagnetizing field scales linearly with the SW amplitude, therefore,
it is useful to introduce the normalized dimensionless average field:

$\langle\tilde{h}_{{\rm d},x}\rangle_{n}=\frac{1}{C_{x}M_{\rm s}}\langle
h_{{\rm d},x}\rangle_{n},$ (30)

where $C_{x}M_{\rm s}$ is a SW amplitude in A/m. Finally, Eqs. (29) and (30)
lead to Eq. (18).

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