# Exciting high-frequency short-wavelength spin waves using high harmonics of
a magnonic cavity mode

Nikhil Kumar<EMAIL_ADDRESS>Electronics and Communication
Engineering Department, National Institute of Technology Calicut, 673 601
India  Paweł Gruszecki<EMAIL_ADDRESS>Institute of Spintronics and
Quantum Information, Faculty of Physics, Adam Mickiewicz University, Poznań,
Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland  Mateusz Gołębiewski
<EMAIL_ADDRESS>Institute of Spintronics and Quantum
Information, Faculty of Physics, Adam Mickiewicz University, Poznań,
Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland  Jarosław W. Kłos
<EMAIL_ADDRESS>Institute of Spintronics and Quantum Information, Faculty of
Physics, Adam Mickiewicz University, Poznań, Uniwersytetu Poznańskiego 2,
61-614 Poznań, Poland  Maciej Krawczyk<EMAIL_ADDRESS>Institute of
Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz
University, Poznań, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland

###### Abstract

Confined spin-wave modes are a promising object for studying nonlinear effects
and future quantum technologies. Here, using micromagnetic simulations, we use
a microwave magnetic field from a coplanar waveguide (CPW) to pump a standing
spin-wave confined in the cavity of magnonic crystal. We find that the
frequency of the fundamental cavity mode is equal to the ferromagnetic
resonance frequency of the plane film and overlaps with the magnonic bandgap,
allowing high magnetic field tunability. Multi-frequency harmonics of the
cavity mode are generated once the microwave amplitude surpasses a certain
threshold. Specifically, the second and third harmonics at 0.5 T equate to
48.6 and 72.9 GHz with wavelengths of 44 and 22 nm respectively, which
propagate into the crystal. This effect reaches saturation when the CPW covers
the entire cavity, making the system feasible for realization. These processes
show potential for the advancement of magnonics at high-frequencies and very
short-wavelengths.

††preprint: AIP/123-QED

Magnonics, which uses spin waves (SWs) to transmit and process information, is
one of the fastest developing areas of research in modern magnetismKruglyak,
Demokritov, and Grundler (2010); Mahmoud  _et al._ (2020); Nikitov  _et al._
(2015); Chumak and Schultheiss (2017); Barman  _et al._ (2021); Flebus  _et
al._ (2023). It covers a wide frequency range from sub-GHz to tens of THz,
with corresponding wavelengths from micrometers to nanometers, and is free of
translational electron motion and Joule heating. This makes it a promising
alternative to electronics and photonics in terms of high operating frequency,
low power consumption and miniaturizationCsaba, Papp, and Porod (2017);
Mahmoud  _et al._ (2020); Dieny  _et al._ (2020); Hirohata  _et al._ (2020);
Pirro  _et al._ (2021); Chumak  _et al._ (2022). Magnonic crystals (MCs) are
the magnetic equivalent of photonic or phononic crystals, which are important
elements of photonic Joannopoulos  _et al._ (2008); Soljačić and Joannopoulos
(2004) and phononic Maldovan (2013); Vasileiadis  _et al._ (2021)
technologies, respectively, and are also considered to be an important element
of future magnonic circuitsKrawczyk and Puszkarski (2008); Lenk  _et al._
(2011); Krawczyk and Grundler (2014); Chumak, Serga, and Hillebrands (2017).
The structures formed from the magnetic thin film by periodic patterning are
so-called planar MCs. The characteristic in-plane sizes of the considered
structures are in the range of tens or single hundreds of nanometers, while
the thicknesses of the magnetic films are about a few or tens of nanometers
Sokolovskyy and Krawczyk (2011); Mamica, Krawczyk, and Grundler (2019); Kłos
_et al._ (2012, 2014); Gubbiotti  _et al._ (2010). In this context, the SW
modes in antidot lattices (ADLs), i.e, thin ferromagnetic films periodically
structured with holes, have attracted considerable interest Vysotskii,
Nikitov, and Filimonov (2005); Yu  _et al._ (2007); Pechan  _et al._ (2005);
McPhail  _et al._ (2005); Kostylev  _et al._ (2008); Ulrichs, Lenk, and
Münzenberg (2010); Hu  _et al._ (2011); Tacchi  _et al._ (2015) and are useful
for controlling SW velocityNeusser  _et al._ (2011), exploiting magnonic band
gaps and diffractionGieniusz  _et al._ (2017); Gołębiewski  _et al._ (2020);
Riedel  _et al._ (2023).

However, the fundamental obstacles in the design of magnonic devices are,
among others, difficulties related to the excitation of SWs, especially of
high frequency and short wavelengths. To excite short SWs, i.e. with
wavelengths below 100 nm, the use of microwave stripe lines Ciubotaru  _et
al._ (2016), magnetic solitons (domain wall Woo, Delaney, and Beach (2017) or
vortex Dieterle  _et al._ (2019)), grating couplers Yu  _et al._ (2016) or
inhomogeneity of the internal magnetic field Mushenok  _et al._ (2017) has
been proposed. The first approach requires ultra-narrow metallic strips that
limit the transmitted power, the latter two are used at low frequencies that
are limited by the natural oscillations of the magnetic solitons or
ferromagnetic nanoelements. In the latter case, the frequencies are usually
below or around the ferromagnetic resonance frequency of the respective bulk
material.

By introducing a defect into the ADL, e.g. in the form of a filled hole (or a
few filled holes), we can trap SW energy in this region whenever a resonant
frequency of cavity falls into the magnonic band gap of the MC Kruglyak  _et
al._ (2006); Klos and Tkachenko (2013); Di  _et al._ (2014); Morozova  _et
al._ (2015). Such a magnonic cavity allows building-up of SW amplitude Kumar
and Prabhakar (2018); Kumar (2022), similar to electromagnetic waves in a
photonic crystal cavity Joannopoulos  _et al._ (2008), when externally pumped
at the cavity resonant frequency. Finally, at sufficiently high wave
amplitudes, nonlinear processes are initiated. This effect is widely studied
and exploited in photonics cavities, in particular, to enhance the generation
of 2${}^{\text{nd}}$ and 3${}^{\text{rd}}$ harmonics Soljačić and Joannopoulos
(2004); Lalanne, Sauvan, and Hugonin (2008).

Recently, multi-frequency generation in magnonics has also been demonstrated
Schultheiss  _et al._ (2019); Koerner  _et al._ (2022); Hula  _et al._ (2022);
Groß  _et al._ (2022). However, the frequency of the generated SW modes hardly
exceeds 10 GHz. In particular, the results of time-resolved scanning
transmission X-ray microscopy measurements demonstrate the generation of SW
high harmonics, up to $7^{\text{th}}$ order, in ADL based on 50 nm thick Py
(antidot diameter 450 nm and lattice constant 900 nm) Groß  _et al._ (2022).
The observed conversion efficiency is between 12.4% and 33.3%, while the
pumping microwave field frequency is only 0.93 GHz, so the $7^{\text{th}}$
order harmonic frequency is $6.51$ GHz. However, the modes involved in the
process are not clearly identified.

This paper presents micromagnetic simulations of an ADL in a Py thin film with
a L3 cavity (3 antidots removed), pumped by a standard coplanar waveguide
(CPW) (Fig. 1). The SW amplitude of the fundamental cavity mode can easily
exceed the nonlinearity threshold, resulting in high frequency harmonics. The
fundamental cavity mode frequency corresponds to the ferromagnetic resonance
(FMR) frequency of the uniform Py layer and this frequency is located within
the magnonic bandgap of the ADL. This allows wide frequency tunability by the
magnetic field: changing it from 200 mT to 1 T results in the
2${}^{\text{nd}}$ and 3${}^{\text{rd}}$ harmonics in the frequency ranges
28-80 GHz and 42-120 GHz, while keeping the SW wavelength below 62 and 40 nm,
respectively. The presented mechanism of high-frequency SW generation
demonstrates its general character, the feasibility of its experimental
realization, and its usefulness for the generation of high-frequency short-
wavelength SWs for a wide range of magnonic applications.

##  I Device model

Figure 1:  (a) Schematic of the device – an antidot lattice with a three-hole
defect cavity (L3) centered at $x=0$ and $y=0$. Magnonic crystal is made of a
5 nm thick Py-film with magnetization saturated by the magnetic field $H_{0}$
directed along the $y$-axis, the diameter of the antidots is 80 nm and the
lattice constant is 150 nm. A CPW antenna is placed just above the film,
centered at $x_{0}$ (in the plot $x_{0}=0$), and consists of a source line of
width $w$ and two ground lines, which are separated from the source line by
gaps of width $s$. (b) The SWs are excited by the $x$ and $z$ components of
the microwave frequency magnetic field ${\bf h}_{\text{mf}}$ of amplitude
$h_{\text{mf}}$ produced by CPW. The calculated field distribution from CPW
with $w=40$ nm and $s=20$ nm is shown.

The device geometry used in this study is shown in Fig. 1(a). It is an ADL
with a square lattice of antidots in a 5 nm thick Py film, with an antidot
diameter of 80 nm and a lattice constant of $a=150$ nm. A three-antidot defect
L3 (i.e., the area of 3 unit cells without the holes) is introduced in this
structure. The system is uniformly magnetized by an in-plane bias magnetic
field $\mu_{0}H_{0}>0.2$ T directed along the $y$-axis ($\mu_{0}$ is the
permeability of the vacuum). The SWs are excited by a microwave frequency
magnetic field ($\textbf{h}_{\text{mf}}$) generated by a CPW transducer
consisting of a signal line of width $w$ separated by $s=20$ nm gaps from two
identical ground lines (much wider than $w$), and with centered at $x_{0}$. To
enhance the excitation of SWs, the CPW is aligned along the $y$-axis.

Two simulation methods are used to study the SW dynamics. Comsol Multiphysics
package is used to calculate the magnonic band structure and cavity modes.
Here, we use the finite element method (FEM) to solve the eigenproblem
obtained from the linearized Landau-Lifshitz equation in real space and the
frequency domain. The nonlinear effects are simulated in the real space and
time domain (with finite difference time domain method, FDTD) using the Mumax3
micromagnetic solverVansteenkiste  _et al._ (2014), which solves the full
Landau-Lifshitz-Gilbert equation. The methods are described in detail in the
Methods section. We use typical material parameters and typical values for Py,
i.e., the saturation magnetization $M_{s}=800$ kA/m, the exchange constant
$A=13$ pJ/m, and in FDTD simulations the damping $\alpha=0.01$. The
magnetization dynamics inside the cavity are studied by probing the
magnetization over $450\times 150$ nm area, i.e., three unit cells along the
$x$-axis and 1 along the $y$-axis.

Figure 2:  (a) Dispersion relation of SWs over the Y-$\Gamma$-X-M-Y path in
the first Brillouin zone (as indicated in the inset) in the 2D ADL calculated
with FEM. The magnonic band gap is marked in transparent red. (b) Spin-wave
spectra obtained using FDTD using broadband excitation
($\mu_{0}h_{\text{mf}}=1$ mT) by a spatially uniform microwave field for a
uniform Py film (the blue line), an ADL (the black line), and an ADL with the
L3 cavity (the thick green line). The cyan stars indicate the mode profiles
displayed in (c)-(h). Eigenmodes in the ADL are demonstrated in (c)-(e), while
mode profiles for the ADL with L3 cavity are presented in (f)-(h). The color
utilized represents the amplitude of the out-of-the-plane component of the
magnetization in arbitrary units. The external magnetic field
$\mu_{0}H_{0}=0.5$ T is applied along the $y$-axis.

##  II Results

###  II.1 Band structure and the resonance spectra

The first step in our investigation is examining the magnonic band structure
of the ADL (without the defect). It is shown in Fig. 2(a) and is obtained by
FEM at 0.5 T external magnetic field. The dispersion relation is over the path
Y-$\Gamma$-X-M-Y in the first Brillouin zone (see the inset showing the first
Brillouin zone and the high-symmetry path). Importantly, we see a well-defined
full bandgap, ranging from 23.05 GHz to 24.95 GHz. The two bands below the
band gap originate in the edge modes, i.e., the modes with the amplitude
concentrated near the antidot edges, while the bands above are bulk modes,
with the first one a fundamental mode of the ADL Tacchi  _et al._ (2015).

The resonance response of the ADL, ADL with the L3 cavity, and as a reference
of the plain Py film are shown in Fig. 2(b). The results are obtained with
FDTD method by applying a spatially uniform broadband microwave magnetic field
linearly polarized along the $z$-axis:

$\mathbf{h}_{\text{mf}}=h_{\text{mf}}\text{sinc}\left(2\pi
f_{\text{cut}}\left(t-t_{0}\right)\right)[0,0,1],$ (1)

where, the amplitude of microwave field: $\mu_{0}h_{\text{mf}}=1$ mT, the cut-
off frequency: $f_{\mathrm{cut}}=40$ GHz, and $t_{0}=8/f_{\mathrm{cut}}$. The
intensity is calculated by integrating the normalized $z$ component of the
magnetization vector ($m_{z}\equiv M_{z}/M_{\text{S}}$) over the cavity region
(i.e., the area of three unit cells in ADL and in the plane film) in a steady
state. For presenting intensity, we use a logarithmic scale:
$20\log_{10}(|m_{z}|)$ in dB units.

For a plain Py film, we have only one distinctive peak at 24.3 GHz,
corresponding to its FMR frequency. In the ADL there are several intense peaks
Tacchi  _et al._ (2015), the most intense are shown in Fig. 2(c-d): (c) the
symmetric edge mode at 22.5 GHz ($1^{\text{st}}$ band), (d) the fundamental
excitation of the ADL at 25.8 GHz, and (e) a higher order SW at 28.5 GHz. This
is in full agreement with the dispersion relation presented in Fig. 2(a). For
the ADL with the cavity, we obtained the spectrum (green line in Fig.2(b))
which is quite similar to the undetected ADL, but with a well-defined peak at
$f_{0}=24.3$ GHz, i.e. in the bandgap of the ADL. The peak represents the
fundamental cavity mode (see Fig. 2(f)) which oscillates in phase, and its
amplitude is confined inside the cavity. Interestingly, the frequency of this
mode is the same as the FMR of a uniform Py film (see green and blue lines in
Fig. 2(b)). There are also high-frequency resonant modes in the cavity with
enhanced SW amplitude inside the cavity, as shown in Fig. 2(g) and (h).
However, their frequencies overlap with the continuous bands of the ADL, and
for that reason, the SW amplitude of these modes spread over the entire ADL.

Learned from photonics, the enhancement of the pumping power on the cavity
mode shall result in entering into nonlinear regime whenever the cavity is a
medium possessing some kind of nonlinearity Soljačić and Joannopoulos (2004);
Lalanne, Sauvan, and Hugonin (2008). In magnetism, nonlinearity is its
inherent property Gurevich and Melkov (1996). Therefore, in the following part
of the manuscript, we will focus on the fundamental cavity mode pumped by the
microwave magnetic field generated by CPW at 24.3 GHz and at high amplitudes.

###  II.2 Nonlinear dynamics

Figure 3:  (a) The SW spectra ($\propto m_{z}^{2}$) in the L3 cavity excited
with CPW of $w=400$ nm, located at $x_{0}=0$ at frequency 24.3 GHz and
amplitude $\mu_{0}h_{\text{mf}}$: 1 µT (the black-dashed line), 0.1 mT (the
red line), 1 mT (the green line), and 4 mT (the orange line). (b) Intensity
($\propto m_{z}^{2}$) of the fundamental cavity mode ($f_{0}=24.3$ GHz), and
its second ($f_{2}\equiv 2f_{0}=48.6$ GHz), and third ($f_{3}\equiv
3f_{0}=72.9$ GHz) harmonics on the microwave excitation field strength
$h_{\text{mf}}$ from CPWs with signal line widths $w=40$ nm and 400 nm
represented as open and closed points, respectively. The black dashed lines
represent the fitted dependencies from Eq. (2), whereas the vertical narrow
green line represents the field of value 0.1 mT used in Fig. 6. Simulation
with FDTD was performed at $\mu_{0}H_{0}=0.5$ T.

Frequency spectra of the ADL cavity system are shown in Fig. 3(a). The
calculations were performed for 0.5 T magnetic field and the CPW of $w=400$ nm
wide, located at the center of the cavity ($x_{0}=0$), which generate the
fields $h_{\text{mf}}=1$ µT, 100 µT, 1 mT, and 4 mT. Three peaks,
corresponding to the frequencies $f_{0}=24.3$ GHz, $f_{2}=48.6$ GHz, and
$f_{3}=72.9$ GHz, are clearly visible at higher amplitudes of the microwave
field, except the lowest considered value $h_{\text{ms}}=1$ µT, where $f_{2}$
and $f_{3}$ are at the base signal level below $-150$ dB. These frequencies
correspond to the excitation of the fundamental cavity mode ($f_{0}$) and its
second ($f_{2}\equiv 2f_{0}$) and third ($f_{3}\equiv 3f_{0}$) harmonics,
respectively. The higher the amplitude of the microwave field, the higher the
intensities of the modes. However, there are significant changes in the
relative peak amplitudes between the fundamental cavity mode and its
harmonics. For the fundamental cavity mode, there is an increase of 70 dB in
the peak as we increase the microwave amplitude from 1 µT to 4 mT, whereas for
the second and third harmonics, there is an increase of approximately 100 dB
and 80 dB, respectively.

To shed more light on the efficiency of the SW excitations we perform
systematic simulations with increased $h_{\text{mf}}$ from 1 µT to 0.1 T using
the same CPW antenna. The results are shown in Fig. 3(b) with full diamond,
square, and circle dots for $f_{0}$, $f_{2}$, and $f_{3}$, respectively. The
intensity of the fundamental cavity mode increases from $-80$ dB at $10^{-3}$
mT till $-10$ dB at 10 mT, and saturates with a further increase of
$h_{\text{mf}}$. The intensities of the 2${}^{\text{nd}}$ and
3${}^{\text{rd}}$ harmonics start to grow at about 0.01 mT and 0.1 mT,
respectively, grow almost linearly in this double logarithmic scale, and both
reach an intensity only 20 dB less than the $f_{0}$ mode at 40 mT.

This means that the effectiveness of the nonlinear pumping of the second and
third harmonic increases according to the square and cubic function of
$h_{\text{mf}}$, with different threshold amplitudes of $h_{\text{mf}}$. This
is according to the theoretical predictions for the plane film and plane waves
Gurevich and Melkov (1996) indicating that the intensity of the
$n^{\text{th}}$ harmonics:

$I^{(n)}\propto\left(\frac{h_{\text{mf}}}{H_{0}}\right)^{n}.$ (2)

Similar behaviour has already been observed experimentally in the microwave
pumping of the elliptical Py nanodot (10 nm thick and 1 µm $\times$ 0.5 µm
lateral dimensions) saturated along the long axis Demidov  _et al._ (2011).
However, in Ref. [Demidov  _et al._ , 2011] the excitation was off-resonant,
i.e., the generation of the 2${}^{\text{nd}}$ and 3${}^{\text{rd}}$ harmonic
was only observed at $2f_{\text{ex,1}}$ and $3f_{\text{ex,2}}$, with the
pumping field frequency $f_{\text{ex,1}}$ and $f_{\text{ex,2}}$, respectively.
Here, $2f_{\text{ex,1}}$ and $3f_{\text{ex,2}}$ correspond to the frequency of
the antisymmetric (with the nodal line along the long axis of the ellipse) and
symmetric (fundamental) eigenmode of the nanodot, respectively. This means
that for a given frequency of the microwave field $f_{\text{ex,1}}$ or
$f_{\text{ex,2}}$ only the 2${}^{\text{nd}}$ or 3${}^{\text{rd}}$ harmonic can
be observed.

Figure 4:  (a-c) Spatial distributions of amplitudes of the perpendicular
component of reduced magnetization $m_{z}$ of the SW modes corresponding to
the (a) fundamental cavity mode (24.3 GHz), and its (b) second (48.6 GHz), and
(c) third (72.9 GHz) harmonics. FDTD results are obtained with the CPW of
$w=400$ nm, $x_{0}=0$, $h_{\text{mf}}=4$ mT, and $\mu_{0}H_{0}=0.5$ T. (d-f)
Representation of the modes from (a)-(c) in the reciprocal space. The size of
the Brillouin zone is ca. 42 rad/µm. The green lines in (e) and (f) represent
analyticalKalinikos and Slavin (1986) isofrequency contours for uniform film
at frequencies 48.6 GHz 72.9 GHz, respectively.

The spatial distributions of the fundamental cavity mode ($f_{0}$), mode
$f_{2}$ and $f_{3}$ at $h_{\text{mf}}=4.0$ mT microwave excitation field are
shown in Fig. 4(a), (b) and (c), respectively. The clear fine wavy pattern is
seen at 48.6 and 72.9 GHz indicating a short wavelength of excited SWs. The
amplitude of these harmonics spreads to the ADL, because, the accessible bands
of the SWs in the ADL at these high frequencies are very dense with all the
modes being propagating. There are slightly larger amplitudes along the
$y$-axis, which may be advantageous to use these modes to exploit the
propagation of waves at these high frequencies. Indeed, Fig. 4(e)-(f) shows
that in reciprocal space the 2${}^{\text{nd}}$ and 3${}^{\text{rd}}$ harmonics
have discrete spectra at high values of wavenumber. There is a fine regularly
spaced arrangement of the spots in Fig. 4(e)-(f) in the $k$-space, which is
related to the periodicity of the structure and the discrete values of the
reciprocal lattice vectors. These bright spots are located around the almost
circular isofrequency contours of SWs in the plain Py film at 48.6 and 72.9
GHz. Their radius is 180 and 220 rad/µm for $f_{2}$ and $f_{3}$, and
correspond to the SWs at wavelengths as short as 35 and 29 nm, respectively.
The spots along the direction of $H_{0}$ have larger intensities that indicate
the preferred direction of SW propagation, accordingly with observation in
Fig. 4(b) and (c). In the future, the design of isofrequency contours of the
ADL at relevant frequencies may allow the propagation direction of nonlinearly
excited high-frequency SWs to be tailored to specific requirements. This can
be achieved by proper design of the ADL geometry and the magnonic band
structure Klos  _et al._ (2015); Sadovnikov  _et al._ (2016).

The spatial distributions of the fundamental cavity mode ($f_{0}$), mode
$f_{2}$ and $f_{3}$ at a microwave excitation field of $h_{\text{mf}}=4.0$ mT
are displayed in Fig. 4(a), (b) and (c). The figures present a neat,
undulating pattern at 48.6 and 72.9 GHz indicating a short wavelength of
excited SWs. Notably, the amplitude of these harmonics extends from the cavity
to the ADL as the magnonic bands in the ADL at these high frequencies are very
dense with all the modes being propagating. Slight amplitude increases occur
along the $y$-axis, suggesting that utilizing these modes could be useful for
wave propagation at high frequencies. In fact, as seen in Fig. 4(e)-(f), the
2${}^{\text{nd}}$ and 3${}^{\text{rd}}$ harmonics possess discrete spectra in
reciprocal space with regularly spaced spots at high wavenumbers. This pattern
is related to the structure’s periodicity and the discrete values of its
reciprocal lattice vectors. The bright spots are located around the almost
circular isofrequency contours of SWs in the plain Py film at 48.6 and 72.9
GHz. For $f_{2}$ and $f_{3}$, their radius is 180 and 220 rad/µm,
respectively, which corresponds to the SWs at wavelengths as short as 35 and
29 nm. The spots have larger intensities along the direction of $H_{0}$ that
indicate the preferred direction of SW propagation in the ADL, accordingly
with observation in Fig. 4(b) and (c). In the future, designing isofrequency
contours of the ADL at pertinent frequencies could enable tailoring of the
propagation direction of nonlinearly excited high-frequency SWs towards
specific needs. This objective can be accomplished through appropriate ADL
geometry design and tailored magnonic band structure Klos  _et al._ (2015);
Sadovnikov  _et al._ (2016).

Figure 5: (a) Dependence of the fundamental cavity mode frequency on external
static magnetic field magnitude, $H_{0}$. The crosses mark the frequency of
the cavity mode obtained from FEM in the supercell approach, while the red
line represents the FMR frequency of the plain Py film from the Kittel’s
formulae Kittel (1946). The green dotted lines indicate the edges of the band
gap obtained from FEM. (b) Intensity of the fundamental cavity mode ($f_{0}$)
and its second ($2f_{0}$) and third harmonics ($3f_{0}$) as a function of
$H_{0}$ obtained by FDTD. The markers indicate the intensities obtained by
excitation with a CPW line $w=400$ nm wide at $\mu_{0}h_{\text{mf}}=0.1$ mT at
frequencies following the red line in (a).

According to Eq. (2), the effectiveness of the nonlinear process can also be
increased by decreasing $H_{0}$. However, to maintain similar conditions for
the ADL cavity system at different values of $H_{0}$, the frequency of the
fundamental cavity mode must follow the magnonic band gap as the magnetic
field is varied. In Fig. 5(a), we show that the frequency of the fundamental
cavity mode follows the FMR frequency of the plain Py film, and in a wide
range of fields, i.e. from 0.2 to 1 T, the frequency of this mode (increasing
from 13.99 to 39.56 GHz, respectively) falls into the magnonic band gap.
Taking the frequencies of the FMR mode with increasing $H_{0}$, assuming a
pumped microwave magnetic field amplitude $h_{\text{mf}}=0.1$ mT, we perform
simulations. The results are shown in Fig. 5(b). We see that as the magnetic
field decreases from 1 T to 0.3 T, the intensity of the 2${}^{\text{nd}}$ and
3${}^{\text{rd}}$ harmonics increases by 21.7 dB and 34.5 dB respectively.
However, there is a downside to decreasing $H_{0}$, it is that a lower
frequency means longer wavelengths. In particular, at 1 T the
2${}^{\text{nd}}$ and 3${}^{\text{rd}}$ harmonics are at 79.12 and 118.68 GHz
with wavelengths of 31.1 and 21.6 nm respectively, while at 0.2 T they are at
27.98 and 41.97 GHz and wavelengths of 61.4 and 39.8 nm, respectively.

###  II.3 Getting ready to experiment

The type of the ADL considered in this paper has been broadly investigated in
the last few years Neusser  _et al._ (2008, 2011); Tacchi  _et al._ (2015);
Zelent  _et al._ (2017); Groß  _et al._ (2022), so the nonlinear effects
presented above should be feasible for experimental realization. To make our
idea closer to realization we studied the influence of the CPW width and
position with respect to the cavity center on the frequency multiplication.

The dependence of the SW intensity at $f_{0},f_{2},f_{3}$ on the position of
the CPW ($w=400$ nm emitting a microwave magnetic field $h_{\text{mf}}=0.1$
mT) at $\mu_{0}H_{0}=0.5$ T is shown in Fig. 6(a). Since $h_{\text{mf}}$ is
below the threshold for $f_{3}$, the discussion below relates directly to
$f_{0}$ and $f_{2}$, but similar dependencies were also found at higher values
of $h_{\text{mf}}$ for $f_{3}$. As expected, the intensities of the SWs
decrease as the CPW signal line is moved away from the centre of the cavity.
The rate of decrease accelerates significantly at $x_{0}>425$ nm. This is due
to the fact that, from this position, the emission occurs outside the cavity,
and the excitation of the cavity mode is only via evanescent SW modes in the
ADL at the frequency of 24.3 GHz, which is in the bandgap. The study shows
that the $x_{0}\approx 0$ is the most appropriate for excitation of SWs inside
the cavity for CPW narrower than the cavity length.

Figure 6:  The dependence of the SW intensities ($\propto m_{z}^{2}$) at
$f_{0}=24.3$ GHz, $f_{2}=2\times 24.3$ GHz, and third $f_{3}=3\times 24.3$ GHz
on (a) the position of the CPW antenna $x_{0}$ with respect of cavity center
($w=400$ nm and $h_{\text{mf}}=0.1$ mT), and (b) the CPW signal line width $w$
($x_{0}=0$ and $h_{\text{mf}}=0.1$ mT). The gray area in (a) denotes region
with $x_{0}\in[0,300~{}\mathrm{nm}]$, whereas in (b) are with $w>150$ nm.

The influence of the width of the CPW signal line on the intensity of the
excited SW modes is shown in Fig. 3(b). We place the CPW centrally over the
cavity and increase $w$ starting at $w=40$ nm, we keep $h_{\text{mf}}=0.1$ mT.
The intensity of $f_{0}$ and $f_{2}$ mode increases with $w$, and as expected
saturates when the signal line covers the entire cavity. We conclude that the
optimal conditions for multi-frequency SW mode generation are when the cavity
is completely covered by the signal line, regardless of its position relative
to the cavity centre. However, if the signal line is narrower than the cavity,
the intensity of the excited SW modes becomes sensitive to $x_{0}$ and $w$.
Moreover, their values influence also the threshold field values for the
generation of high-frequency harmonics. As shown in Fig. 3(b) the threshold
fields are 0.04 and 0.4 mT for CPW with $w=40$ nm (empty symbols), which are
higher than for $w=400$ nm, (solid symbols), 0.01 and 0.1 T, respectively.

##  III Summary

In this article, we describe a method to enhance the SW energy confined in a
magnonic crystal cavity created in a thin Py-film based ADL to enter the
nonlinear regime. We have shown that the microwave-pumped fundamental cavity
mode, which has the frequency of the plain film FMR fitting into a complete
bandgap, can easily reach the nonlinearity threshold leading to the generation
of modes at multiple frequencies. Thus, we propose a way to generate high
frequency SWs in the range of 28 to 79 GHz and 42 to 119 GHz with very short
wavelengths of 62 to 31 nm and 40 to 22 nm for the 2${}^{\text{nd}}$ and
3${}^{\text{rd}}$ harmonics, respectively, by varying the external magnetic
field from 1 to 0.3 T. The harmonics follow known dependencies on the
amplitude of the pumped microwave field and the magnitude of the static
magnetic fieldGurevich and Melkov (1996), allowing the effectiveness of multi-
frequency mode generation to be significantly enhanced. The effects
demonstrated in this paper can be achieved in standard ADLs using a simple CPW
antenna, whose main limitation is the overlap of the signal line with the
cavity surface. This provides an opportunity to investigate magnonics
applications at high frequencies, which overlap with 6G microwave bands and
have very short wavelengths, up to 100,000 times shorter than the wavelength
of microwaves at these frequencies.

Further optimization of the magnonic crystal structure can even increase the
demonstrated conversion efficiency of FMR oscillations to high-frequency
oscillations. This can be achieved if, instead of using a cavity to confine
only the SW at the input frequency (fundamental cavity mode), the structure
also allows the mode to be confined at the output frequency(s) (high-frequency
harmonics). Learning from photonics, not only can the power consumption be
greatly reduced, but in principle 100% conversion can be achieved Bravo-Abad
_et al._ (2007); Rodriguez  _et al._ (2007).

### Acknowledgements

This work was supported in part by the National Science Center Poland project
OPUS-LAP no 2020/39/I/ST3/02413. The simulations were partially performed in
Poznań Supercomputing and Networking Center and at the National Institute of
Technology Calicut, India.

## Methods

### Micromagnetic simulations

To achieve precise outcomes for the ADL’s dispersion relations and to gain a
better understanding of the complexity of the SW modes inside the crystal
cavity, we utilized FEM and FDTD method to solve the Landau-Lifshitz and
Maxwell equations. Within simulations, every magnetic moment in the predefined
unit cells is given by normalized unit vectors
$\textbf{m}=\textbf{M}/M_{\text{S}},$ where M is the total magnetization, and
$M_{\text{S}}$ is the saturation magnetization of the ferromagnetic material.
Our approach then focuses on solving the Landau-Lifshitz-Gilbert (LLG)
equation:

$\frac{d\textbf{m}}{dt}=\gamma\frac{1}{1+\alpha^{2}}\left(\textbf{m}\times\textbf{B}_{\text{eff}}+\alpha\left(\textbf{m}\times\left(\textbf{m}\times\textbf{B}_{\text{eff}}\right)\right)\right),$
(3)

where $\gamma$ is the gyromagnetic ratio and $\alpha$ is the dimensionless
damping coefficient. The effective magnetic flux density field,
$\textbf{B}_{\mathrm{eff}}$ includes externally applied field,
$\textbf{B}_{\mathrm{0}}\equiv\mu_{0}\textbf{H}_{0}$, together with the
magnetostatic demagnetization, $\textbf{B}_{\mathrm{d}}$, and the exchange
field, $\textbf{B}_{\mathrm{exch}}$:

$\textbf{B}_{\text{eff}}=\textbf{B}_{\text{ext}}+\textbf{B}_{\text{d}}+\textbf{B}_{\text{exch}}.$
(4)

#### Finite element method

In the FEM analysis implemented with Comsol Multiphysics, we addressed the
eigenproblem derived from Eqs. 3 and 4, neglecting damping, i.e., $\alpha=0$,
and defining the demagnetizing field amplitude, $\textbf{H}_{\text{d}}$
($\equiv\textbf{B}_{\mathrm{d}}/\mu_{\text{0}}$), as the gradient of the
magnetic scalar potential, $U_{\text{m}}$: $\textbf{H}_{\text{d}}=-\nabla
U_{\text{m}}$, which inside a magnetic body fulfill the equation:
$\nabla^{2}U_{\text{m}}=\nabla\cdot\textbf{M}$. By considering full
magnetization saturation along the direction $\hat{i}$ and utilizing a linear
approximation, we could separate the magnetization vector into its static and
dynamic (dependent on time $t$ and position r) components as
$\textbf{m}(\textbf{r},t)=m_{i}\hat{i}+\delta\textbf{m}(\textbf{r},t)\;\forall\;(\delta\textbf{m}\perp\hat{i})$,
neglecting all nonlinear terms in the dynamic magnetization
$\delta\textbf{m}(\textbf{r},t)$. Additional information on this methodology
can be found in Refs. [Mruczkiewicz  _et al._ , 2013; Rychły  _et al._ ,
2018].

Using Bloch-Floquet boundary conditions (BC) on the unit cell boundaries:

$\textbf{m}_{\text{dst}}=\textbf{m}_{\text{src}}e^{-i\textbf{k}\cdot(\textbf{r}_{\text{dst}}-\textbf{r}_{\text{src}})},$
(5)

we model an infinite MC layer. Here, k is the 2D Bloch wavenumber, m is the
normalized magnetic moment defined both at the target (dst) and source (src),
r are the spatial coordinates of the boundaries where the BCs are applied, and
$i$ is the imaginary unit. By parametrically sweeping the wavenumber,
eigenfrequencies are determined at each interval. The resulting wavenumber
versus frequency plots yields the dispersion curves for the periodic structure
Hakoda  _et al._ (2018); Collet  _et al._ (2011).

For planes perpendicular to the plane of the MC layer, we have used Dirichlet
BCs to suppress the scalar magnetic potential ($U_{\text{m}}$) at its out-of-
plane boundaries. To ensure the physical accuracy and convergence of the
simulation, it is crucial to position the conditions sufficiently far from the
specimen. In our simulations, we set the cell’s height to be 10 times the
layer’s unit cell width (1.5 µm). The average quality of the tetrahedral
discretization mesh, as determined by the volume-to-length parameter, is 0.91.
This results in a mesh comprising 13630 prisms for the MCs unit cell, with one
element per thickness and excluding the surrounding environment. The cavity
mode simulations in Comsol Multiphysics (Fig. 5) used the unit supercell
approach, which consists of a cavity surrounded by 6 (parallel to the long
side) and 8 (perpendicular) antidotes. This results in a symmetrical size of
the supercell with a side equal to 1.35 µm. In this case, the height of the
environment is 20 times this width (27 µm), and the amount of mesh elements of
this supercell is 37330 with an average quality of 0.82 (and with one prism
per thickness).

#### Finite difference time domain method

The time domain simulations in the paper have been obtained with the open-
source software MuMax3 employing the finite difference time domain method
Vansteenkiste  _et al._ (2014) with the RK45 solver (based on Dormand-Prince
method) Dormand and Prince (1980) to solve Eq. (3). The micromagnetic cell
size was taken as $5\times 5\times 5$ nm3 with cell dimensions smaller than
the exchange length $\left(l_{\text{ex}}=5.7\,\text{nm}\ \text{for
Py}\right)$. We used the in-plane periodic boundary condition (PBC), i.e., the
supercell of dimension 4.5 µm $\times$ 3 µm has been multiplied 10 and 50
times along the $x$ and $y$ directions, respectively.

To get the mode profiles, we calculate each magnetization component’s
pointwise FFT over time and then visualize the real part corresponding to a
particular resonance frequency, see details in Ref. [Gruszecki and
Kisielewski, 2023].

The nonlinear response of the system excited by a single frequency microwave
source was analyzed accordingly with the methodology used in Ref. [Gruszecki
_et al._ , 2022].

The mf current transmitted along the $y$ axis generates a magnetic field
$\mathbf{h}_{\mathrm{mf}}$, which exerts a torque on the magnetization in Py.
The dependence of $\mathbf{h}_{\mathrm{mf}}$ with peak amplitude
$h_{\text{mf}}$ on $x$ coordinate can be approximated by the equationGruszecki
_et al._ (2016):

$\mathbf{h}_{\mathrm{mf}}(x)=h_{\mathrm{mf}}\begin{bmatrix}\frac{1}{2}+\frac{1}{2}\mathrm{cos}\left[\frac{2\pi}{2s}(|x|-\frac{w}{2})\right]\\\
0\\\
\frac{1}{5}\mathrm{sign}(x)\left\\{\frac{1}{2}+\frac{1}{2}\mathrm{cos}\left[\frac{2\pi}{s}(|x|-\frac{w}{2}-\frac{s}{2}\right]\right\\}\end{bmatrix}.$
(6)

### DATA AVAILABILITY

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

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