# Electrical spectroscopy of the spin-wave dispersion and bistability in
gallium-doped yttrium iron garnet

Joris J. Carmiggelt1, Olaf C. Dreijer1, Carsten Dubs2, Oleksii Surzhenko2,
Toeno van der Sar1,∗

(  1Department of Quantum Nanoscience, Kavli Institute of Nanoscience, Delft
University of Technology, 2628 CJ Delft, The Netherlands
2INNOVENT e.V. Technologieentwicklung, D-07745 Jena, Germany
∗ Corresponding author. Email<EMAIL_ADDRESS>)

###### Abstract

Yttrium iron garnet (YIG) is a magnetic insulator with record-low damping,
allowing spin-wave transport over macroscopic distances. Doping YIG with
gallium ions greatly reduces the demagnetizing field and introduces a
perpendicular magnetic anisotropy, which leads to an isotropic spin-wave
dispersion that facilitates spin-wave optics and spin-wave steering. Here, we
characterize the dispersion of a gallium-doped YIG (Ga:YIG) thin film using
electrical spectroscopy. We determine the magnetic anisotropy parameters from
the ferromagnetic resonance frequency and use propagating spin wave
spectroscopy in the Damon-Eshbach configuration to detect the small spin-wave
magnetic fields of this ultrathin weak magnet over a wide range of
wavevectors, enabling the extraction of the exchange constant
$\alpha=1.3(2)\times 10^{-12}$ J/m. The frequencies of the spin waves shift
with increasing drive power, which eventually leads to the foldover of the
spin-wave modes. Our results shed light on isotropic spin-wave transport in
Ga:YIG and highlight the potential of electrical spectroscopy to map out the
dispersion and bistability of propagating spin waves in magnets with a low
saturation magnetization.

tocsectionMain Text Yttrium iron garnet (YIG) is a magnetic insulator that is
famous for its low Gilbert damping and long-range spin-wave propagation [1].
At low bias fields the YIG magnetization is typically pushed in the plane by
the demagnetizing field [2], leading to an anisotropic spin-wave dispersion at
microwave frequencies. For applications that rely on spin-wave optics and
spin-wave steering an isotropic spin-wave dispersion is desirable [3], which
can be achieved by introducing gallium dopants in the YIG: The presence of the
dopants reduces the saturation magnetization and thereby the demagnetizing
field [4], and induces a perpendicular magnetic anisotropy (PMA) [5, 6], such
that the magnetization points out-of-plane. Isotropic transport of forward-
volume spin waves has been observed even at zero bias field [7], opening the
door for spin-wave logic devices [8, 9, 10].

To harness isotropic spin waves it is essential to know the spin-wave
dispersion, which is dominated by the exchange interaction for magnets with a
low saturation magnetization [11]. Here, we use all-electrical spectroscopy of
propagating spin waves to characterize the spin-wave dispersion of a 45-nm-
thick film of gallium-doped YIG (Ga:YIG). Rather than looking at the discrete
mode numbers of perpendicular standing spin waves [12], this method enables
extracting the exchange constant by monitoring the spin-wave transmission for
a continuously-tunable range of wavevectors. We show that this technique has
sufficient sensitivity to characterize spin waves in nanometer-thick Ga:YIG
films, where perpendicular modes may be challenging to detect due to their
high frequencies and small mode overlap with the stripline drive field.

We extract the anisotropy parameters from the field dependence of the
ferromagnetic resonance (FMR) frequency at different bias field orientations
and find that the PMA is strong enough to lift the magnetization out of the
plane. Next, we characterize the spin-wave dispersion from electrically-
detected spin-wave spectra. We measure in the Damon-Eshbach configuration to
boost the inductive coupling of the spin waves to the striplines [13],
allowing the extraction of the spin-wave group velocity over a wide range of
wavevectors from which we determine the exchange constant. When increasing the
microwave excitation power, we observe clear frequency shifts of the spin-wave
modes. The shifts result in the foldover of spin waves, which we verify by
comparing upward and downward frequency sweeps. These results benchmark
propagating spin wave spectroscopy as an accessible tool to characterize the
exchange constant and spin-wave foldover in technologically attractive thin-
film magnetic insulators with low saturation magnetization and PMA.

We use liquid phase epitaxy to grow a 45-nm-thick film of Ga:YIG on an
(111)-oriented gadolinium gallium garnet (GGG) substrate (supplementary
material section 1). Using vibrating sample magnetometry (VSM) we determine
the saturation magnetization $M_{\text{s}}=1.52(6)\times 10^{4}$ A/m (Fig. 1a,
the number in parentheses denotes the 95% confidence interval), which is
approximately an order of magnitude smaller than undoped YIG films of similar
thicknesses [36].

Figure 1: The saturation magnetization and magnetic anisotropies of Ga:YIG.
(a) Hysteresis loop of the magnetization of a 45-nm-thick Ga:YIG film as a
function of out-of-plane magnetic field $B_{0}$ measured using vibrating
sample magnetometry and corrected for magnetic background. The arrows denote
the sweep direction of the magnetic field. (b) FMR measurements using an out-
of-plane (green) and in-plane (red) magnetic field $B_{0}$. From the fits of
the FMR frequencies (solid lines) we determine the perpendicular and cubic
anisotropy fields (see text).

In addition to PMA, Ga:YIG films also have a cubic magnetic anisotropy due to
a cubic unit cell. We start by determining the cubic and perpendicular
anisotropy fields from the ferromagnetic resonance (FMR) frequencies
$\omega_{\text{FMR}}/2\pi$ using an out-of-plane ($\perp$) and in-plane ($||$)
magnetic bias field $B_{0}$. For (111)-oriented films the out-of-plane and in-
plane Kittel relations are given by [36, 37]

$\omega_{\text{FMR($\perp$)}}=\gamma_{\perp}(B_{0}-\mu_{0}M_{\text{s}}+\frac{2K_{2\perp}}{M_{\text{s}}}-\frac{4}{3}\frac{K_{4}}{M_{\text{s}}}),$
(1)
$\omega_{\text{FMR($||$)}}=\gamma_{||}\sqrt{B_{0}(B_{0}+\mu_{0}M_{\text{s}}-\frac{2K_{2\perp}}{M_{\text{s}}}-\frac{K_{4}}{M_{\text{s}}})}.$
(2)

Here $\gamma_{\perp,||}=g_{\perp,||}\mu_{\text{B}}/\hbar$ is the gyromagnetic
ratio with $g_{\perp,||}$ the anisotropic g-factor, $\mu_{\text{B}}$ the Bohr
magneton and $\hbar$ the reduced Planck constant, $\mu_{0}$ is the magnetic
permeability of free space, $K_{2\perp}$ is the uniaxial out-of-plane
anisotropy (e.g. PMA) constant and $K_{4}$ the cubic anisotropy constant.
During the in-plane FMR measurement we apply the magnetic field along the
$[1\overline{1}0]$ crystallographic axis to minimize the out-of-plane
component of the magnetization (supplementary material section 2). We neglect
any uniaxial in-plane anisotropy as it is known to be small in YIG samples
[36].

By substituting the value of $M_{\text{s}}$ that we obtained with VSM into
equations 1 and 2, we can determine $K_{2\perp}$ and $K_{4}$ from the FMR
frequencies (Fig. 1b) [35]. From the fits (solid lines) we extract the
uniaxial out-of-plane anisotropy field $2K_{2\perp}/M_{\text{s}}=104.7(8)$ mT
and the cubic anisotropy field $2K_{4}/M_{\text{s}}=-8.2(5)$ mT (supplementary
material section 3). Undoped YIG films of similar thicknesses have comparable
cubic anisotropy fields [36], which agrees with previous work on micrometer-
scale films showing that the cubic anisotropy of YIG does not depend on
gallium concentration [17]. We determine the in-plane and out-of-plane
g-factors to be $g_{||}=2.041(4)$ and $g_{\perp}=2.101(3)$ [34].

Figure 2: All-electrical propagating spin wave spectroscopy. (a) Optical
micrograph of a Ga:YIG film with two gold striplines that are connected to the
ports of a vector network analyser (VNA). Port 1 applies a microwave current
(typical excitation power: $-35$ dBm) that induces a radio-frequency magnetic
field $B^{\text{RF}}$ at the injector stripline. This field excites
propagating spin waves that couple inductively to the detector stripline at a
distance $s$. The generated microwave current is amplified and detected at
port 2. A static magnetic field $B_{0}$ is applied in the Damon-Eshbach
configuration and is oriented such that the chirality of $B^{\text{RF}}$
favours the excitation of spin waves propagating towards the detector
stripline [19]. (b) Field-derivative of the microwave transmission
$|\text{S}_{\text{21}}|$ between two striplines ($w=1$ $\upmu$m, $s=6$
$\upmu$m) as a function of $B_{0}$ and microwave frequency. The colormap is
squeezed, such that also fringes corresponding to low-amplitude spin waves are
visible. A masked background was subtracted to highlight the signal attributed
to spin waves (supplementary material section 4).

We now use propagating spin wave spectroscopy to characterize the spin-wave
dispersion in Ga:YIG. We measure the microwave transmission
$|\text{S}_{\text{21}}|$ between two microstrips fabricated directly on the
Ga:YIG as a function of static magnetic field $B_{0}$ and frequency $f$ (Fig.
2a). The magnetic field is applied in the Damon-Eshbach geometry to maximize
the inductive coupling between the spin waves and the striplines [13]. We
measure a clear Damon-Eshbach spin-wave signal in the microwave transmission
spectrum when $B_{0}$ overcomes the PMA and pushes the spins in the plane
(Fig. 2b, supplementary material section 4). The signal appears at a finite
frequency, because the bias field $B_{0}$ is applied along the
$[11\overline{2}]$ crystallographic axis with a finite out-of-plane angle of
$\sim 1^{\circ}$ (supplementary material section 2).

The fringes in the transmission spectra result from the interference between
the spin waves and the microwave excitation field [38, 21]. Each fringe
indicates an extra spin-wavelength $\lambda$ that fits between the striplines.
We can thus use the fringes to determine the group velocity $v_{\text{g}}$ of
the spin waves via [22]

$v_{\text{g}}=\frac{\partial\omega_{\text{SW}}}{\partial
k}\approx\frac{2\pi\Delta f}{2\pi/s}=\Delta fs.$ (3)

Here $\omega_{\text{SW}}=2\pi f$ and $k=2\pi/\lambda$ are the spin wave’s
angular frequency and wavevector, $\Delta f$ is the frequency difference
between two consecutive maxima or minima of the fringes (Fig. 3a) and $s$ is
the center-to-center distance between both microstrips.

We extract the exchange constant of our Ga:YIG film by fitting the measured
group velocity to an analytical expression derived from the spin-wave
dispersion. The Damon-Eshbach spin-wave dispersion for magnetic thin films
with cubic and perpendicular anisotropy is given by [37] (supplementary
material section 5)

$\omega_{\text{SW}}(k)=\sqrt{\omega_{B}(\omega_{B}+\omega_{M}-\omega_{K})+\frac{\omega_{M}t}{2}(\omega_{M}-\omega_{K})k+\gamma_{||}D(2\omega_{B}+\omega_{M}-\omega_{K})k^{2}+\gamma_{||}^{2}D^{2}k^{4}}.$
(4)

Here we defined for notational convenience
$\omega_{B}=\gamma_{||}B_{\text{0}}$,
$\omega_{M}=\gamma_{||}\mu_{0}M_{\text{s}}$,
$\omega_{D}=\frac{\gamma_{||}D}{M_{\text{s}}}$, and
$\omega_{K}=\gamma_{||}(2K_{2\perp}/M_{\text{s}}+K_{4}/M_{\text{s}})$, $t$ is
the thickness of the film and $D=2\alpha/M_{\text{s}}$ is the spin stiffness,
with $\alpha$ the exchange constant. Differentiating with respect to $k$ gives
an analytical expression for the group velocity

$v_{\text{g}}(k)=\frac{1}{2\sqrt{\omega_{\text{SW}}(k)}}\big{(}\frac{\omega_{M}t}{2}(\omega_{M}-\omega_{K})+2\gamma_{||}D(2\omega_{B}+\omega_{M}-\omega_{K})k+4\gamma_{||}^{2}D^{2}k^{3}\big{)}.$
(5)

Since we determined $M_{\text{s}}$ and the anisotropy constants from the VSM
and FMR measurements, the exchange constant is the only unknown variable in
the dispersion. We determine the exchange constant from spin-wave spectra
measured using two sets of striplines with different widths and line-to-line
distances ($w=1$ $\upmu$m, $s=6$ $\upmu$m and $w=2.5$ $\upmu$m, $s=12.5$
$\upmu$m) at the same static field (Fig. 3a,b). First we extract
$v_{\text{g}}$ as a function of frequency from the extrema in the spin-wave
spectra using equation 3 (Fig. 3c). By then fitting the measured
$v_{\text{g}}(f)$ using equations 4 and 5 (solid line in Fig. 3c), we find
$\alpha=1.3(2)\times 10^{-12}$ J/m and $B_{0}=117.5(3)$ mT (supplementary
material section 3). The determined exchange constant is about 3 times smaller
compared to undoped YIG [12], which is in line with earlier observations of a
decreasing exchange constant with increasing gallium concentration in
micrometer-thick YIG films [23]. Simultaneously the spin stiffness is
increased by about 3 times compared to undoped YIG [12] due to the strong
reduction of the saturation magnetization. For large wavelengths the group
velocity is negative as a result of the PMA in the sample.

Figure 3: Extracting the exchange constant from spin-wave transmission
spectra. (a,b) Background-subtracted linetraces of $|\text{S}_{\text{21}}|$
for two sets of striplines (a: $w=1$ $\upmu$m, $s=6$ $\upmu$m, b: $w=2.5$
$\upmu$m, $s=12.5$ $\upmu$m, excitation power: $-35$ dBm). The red circles (a)
and green squares (inset of b) mark the extrema of the spin-wave fringes. (c)
From the frequency difference between the extrema $\Delta f$ we determine the
group velocity $v_{\text{g}}$ of the spin waves at the center frequency
between the extrema. The blue line fits the data with an analytical expression
for $v_{\text{g}}$, extracting the exchange constant $\alpha=1.3(2)\times
10^{-12}$ J/m. (d,e) Normalized Fourier amplitude of the $y$ and $z$
components of the microwave excitation field $B^{\text{RF}}$ for striplines
with widths $w=1$ $\upmu$m (d) and $w=2.5$ $\upmu$m (e). (f) Reconstructed
spin-wave dispersion based on the fit in (c). The shaded areas correspond to
the frequencies of the extrema in (a,b). The dashed lines are the same as in
(d,e) and indicate the nodes in $|B^{\text{RF}}(k)|$ of the striplines. Only
spin waves that are efficiently excited and detected by the striplines are
observed in (a,b).

The spin-wave excitation and detection efficiency depends on the absolute
value of the Fourier amplitude of the radio-frequency magnetic field
$B^{\text{RF}}$ generated by a stripline, which oscillates in $k$ with a
period given by $\Delta k=2\pi/w$ (Fig. 3e) [38, 21]. To verify that the spin
waves we observe are efficiently excited and detected by our striplines, we
substitute the extracted exchange constant into equation 4 and plot the spin-
wave dispersion (Fig. 3f). The shaded areas correspond to the frequencies of
the spin-wave fringes (Fig. 3a,b) and the dashed lines indicate the nodes in
$|B^{\text{RF}}(k)|$ of both striplines (Fig. 3d,e). We conclude that the
fringes in Fig. 3a correspond to spin waves excited by the first maximum of
$|B^{\text{RF}}(k)|$ and that the fringes in Fig. 3b correspond to spin waves
excited by the second maximum.

Surprisingly, we do not observe fringes in Fig. 3b corresponding to the first
maximum of $|B^{\text{RF}}(k)|$, but rather see a dip in this frequency range
(arrows in Fig. 3b,f). This can be understood by noting that the average
frequency difference between the fringes would be smaller than the spin-wave
linewidth (supplementary material section 6). Low-amplitude fringes
corresponding to small-wavelength spin waves excited by the second k-space
maximum of the 1-$\upmu$m-wide stripline are also visible (Fig. 2b,
supplementary material section 7). These results demonstrate that the spin-
wave dispersion in weak magnets can be reliably extracted using propagating
spin wave spectroscopy by combining measurements on striplines with different
widths and spacings.

When strongly driven to large amplitudes, the FMR behaves like a Duffing
oscillator with a bistable response. Such bistability could potentially be
harnessed for microwave switching [24]. Foldover of the FMR and standing spin-
wave modes has been studied for several decades [24, 25, 26], but foldover of
propagating spin waves was only observed before in active feedback rings [27],
spin-pumped systems [28] and magnonic ring resonators [29]. We show that we
can characterize the foldover of propagating spin waves in Ga:YIG thin films
using our spectroscopy technique.

When increasing the drive power we observe frequency shifts of the spin waves
(Fig. 4a,c). These non-linear shifts result from the four-magnon self-
interaction term in the spin-wave Hamiltonian. For an in-plane magnetized thin
film, the shifts are given by [39]

$\tilde{\omega}_{k}=\omega_{k}+W_{kk,kk}|a_{k}|^{2}.$ (6)

Here $\tilde{\omega}_{k}$ ($\omega_{k}$) is the non-linear (linear) spin-wave
angular frequency, $W_{kk,kk}$ is the four-wave frequency-shift parameter and
$a_{k}$ is the spin-wave amplitude. In our case $W_{kk,kk}$ is positive as a
result of the PMA in the sample, leading to positive frequency shifts of the
spin-wave modes (supplementary material section 8). The low-frequency spin
waves start shifting first, because the stripline is the most efficient in
exciting spin waves with small wavenumbers (Fig. 3d,e). The spin-wave modes
start shifting at a surprisingly low drive power of $\sim-30$ dBm, potentially
caused by reduced spin-wave scattering [26] due to the low density of states
associated with the increased spin stiffness and reduced saturation
magnetization of our sample.

Figure 4: Observation of spin-wave frequency shifts and foldover. (a) Spin-
wave spectra at different excitation powers ($w=1$ $\upmu$m, $s=6$ $\upmu$m).
Low-frequency spin waves shift to higher frequencies when the microwave
excitation power is increased. The markers indicate the sharp transition at
which the spin waves fall back to their unshifted frequencies and serve as a
guide to the eye. (b) Sketch of the normalized spin-wave amplitude vs
frequency for increasing drive power $P$, showing the upward frequency shift
away from the low-power resonance frequency $f_{0}$. Above a critical power
$P_{\text{c}}$ the frequency shift results in the foldover of the spin-wave
mode. As a result, the spin waves fall back to their unshifted dispersion at
higher frequencies for upward frequency sweeps (red arrows, a) than downward
sweeps (pink arrows, c).

In the high-power microwave spectra we observe an abrupt transition at which
the spin waves fall back to their unshifted low-power frequencies, indicating
the foldover of the spin waves. As the spin-wave amplitude increases the spin-
wave modes shift to higher frequencies, until the maximum amplitude is reached
and the spin waves fall back to their low-amplitude dispersion (Fig. 4b).

To demonstrate the foldover behaviour, we compare upward and downward
frequency sweeps (Fig. 4a,c). As expected the spin waves fall back to their
unshifted dispersion earlier when sweeping against the frequency shift
direction than when the sweep is in the same direction. The spin-wave
amplitude and wavevector is thus bistable for the frequencies at which the
foldover occurs. For these frequencies the stripline can excite two different
wavelengths of spin waves at the same excitation power depending on the sweep
direction that was used in the past.

The observed frequency shifts provide an extra knob for tuning the dispersion
of spin waves. They give rise to strong non-linear microwave transmission
between the striplines as a function of excitation power, which may provide
opportunities for neuromorphic computing devices that simulate the spiking of
artificial neurons above a certain input threshold [29, 31].

In summary, we used propagating spin wave spectroscopy to characterize the
spin-wave dispersion in a 45-nm-thick film of Ga:YIG. The gallium doping
reduces the saturation magnetization of the YIG and introduces a small PMA
that lifts the magnetization out of the plane and causes the dispersion to be
dominated by the exchange constant. We extract the exchange constant by
fitting the group velocity at different frequencies and demonstrate that the
detected spin waves are efficiently excited by the excitation fields of the
striplines. Finally, we observe pronounced power-dependent frequency shifts of
the spin waves that lead to foldover and mode bistability. Our results
highlight the potential of all-electrical spectroscopy to shed light on the
dispersion and nonlinear response of propagating spin waves in weakly-magnetic
thin films.

Supplementary material: See the supplementary material for methods, details on
the data analysis and error estimations, additional measurements and
calculations of the FMR frequency, spin-wave dispersion and non-linear
frequency-shift parameter.
Author contributions: J.J.C. and T.v.d.S. conceived the experiment. J.J.C. and
O.D. built the experimental setup, performed the experiments and analyzed the
data. C.D. grew the Ga:YIG film and O.S. performed the VSM measurement. J.J.C.
fabricated the striplines. J.J.C. and T.v.d.S. wrote the manuscript with
contributions from all coauthors. T.v.d.S. supervised the project.
Acknowledgements: This work was supported by the Netherlands Organisation for
Scientific Research (NWO/OCW), as part of the Frontiers of Nanoscience program
and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
-271741898. The authors thank A.V. Chumak for reviewing the manuscript, A.
Katan, E. Lesne for useful discussions and C.C. Pothoven for fabricating the
magnet holders used in the experimental setup. We also thank the staff of the
TU Delft electronic support division and the Kavli Nanolab Delft for their
support in soldering the printed circuit board and fabricating the microwave
striplines.
Competing interests: The authors declare that they have no competing
interests.
Data availability: All data contained in the figures are available in
Zenodo.org at http://doi.org/10.5281/zenodo.5494466, reference number [32].
Additional data related to this paper are available from the corresponding
author upon reasonable request.

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center Supplementary material Electrical spectroscopy of the spin-wave
dispersion and bistability in gallium-doped yttrium iron garnetJoris J.
Carmiggelt1, Olaf C. Dreijer1, Carsten Dubs2, Oleksii Surzhenko2,Toeno van der
Sar1,∗

1Department of Quantum Nanoscience, Kavli Institute of Nanoscience, Delft
University of Technology, 2628 CJ Delft, The Netherlands
2INNOVENT e.V. Technologieentwicklung, D-07745 Jena, Germany
∗ Corresponding author. Email<EMAIL_ADDRESS>
##  1 Ga:YIG sample and experimental setup

A 45-nm-thick film of gallium-doped yttrium iron garnet (Ga:YIG) was grown
using liquid phase epitaxy on a one-inch (111) gadolinium gallium garnet (GGG)
substrate and cut into chips of 5x5x0.5 mm3. Striplines were fabricated on top
of the Ga:YIG by e-beam lithography using PMMA(A8 495)/PMMA(A3 950) bilayer
resist with an Elektra92 coating to avoid charging, and subsequent evaporation
of Ti/Au (10 nm/190 nm). We wirebond the striplines to a printed circuit board
and connect them to our vector network analyser (VNA, Keysight, P9372A) via
small, non-magnetic SMPM connectors (Amphenol RF, 925-169J-51PT) to minimize
spurious magnetic-field dependent signals and maximize the dynamic range of
the bias field. Before reaching the VNA, the signals are amplified by a low
noise +20 dB amplifier (Minicircuits, ZX60-83LN-S+) to avoid detection noise
on the order of our signals. We place the sample between two large cylindrical
permanent magnets (Supermagnete, S35-20-N) to apply a strong and homogeneous
bias field. The magnets sit in home-built magnet holders that are mounted on
computer-controlled translation stages (Thorlabs, MTS25-Z8, 25 mm range),
which allow sweeping the field. We calibrate the magnetic field using a Hall
probe (Hirst Magnetic Instruments, GM08). All measurements were performed at
room temperature.

##  2 Effect of the magnetic field alignment on the FMR frequency

In this section we show that for (111)-oriented lattices with cubic anisotropy
the in-plane Kittel relation holds when a strong magnetic field $B_{0}$ is
applied along the in-plane $[1\overline{1}0]$ crystallographic axis. We also
investigate the effect of a $\sim 1^{\circ}$ out-of-plane angle of $B_{0}$ on
the FMR frequency and the magnetization direction. Such a small angle may be
present due to the manual placement of the sample in our setup (section 1).
The FMR frequency is calculated according to [33]

$\omega_{\text{FMR}}^{2}=\frac{\gamma^{2}}{\sin(\theta_{M})^{2}}\cdot\Big{(}\frac{\partial^{2}F}{\partial\theta_{M}^{2}}\frac{\partial^{2}F}{\partial\phi_{M}^{2}}-\big{(}\frac{\partial^{2}F}{\partial\theta_{M}\partial\phi_{M}}\big{)}^{2}\Big{)}.$
(S1)

Here $\theta_{M}$ is the angle of the magnetization with respect to the film’s
normal, $\phi_{M}$ is the in-plane angle of the magnetization with respect to
the $[1\overline{1}0]$ crystallographic axis and
$F=\frac{F^{\prime}}{M_{\text{s}}}$, with $F^{\prime}$ the free energy density
and $M_{\text{s}}$ the saturation magnetization (Fig. S1).
$\gamma=\frac{g\mu_{\text{B}}}{\hbar}$ is the gyromagnetic ratio, with
$\mu_{\text{B}}$ the Bohr magneton and $\hbar$ the reduced Planck constant.
The anisotropic g-factor is given by
$g=\sqrt{g_{\perp}^{2}\cos(\theta_{M})^{2}+g_{||}^{2}\sin(\theta_{M})^{2}}$,
with $g_{||}$ and $g_{\perp}$ respectively the in-plane and out-of-plane
g-factors [34].
For (111)-oriented films with cubic and uniaxial out-of-plane magnetic
anisotropies the normalized free energy density is given by [35, 36]

$\begin{split}&F=-B_{0}\Big{(}\sin(\theta_{M})\sin(\theta_{B})\cos(\theta_{M}-\theta_{B})+\cos(\theta_{M})\cos(\theta_{B})\Big{)}+\frac{1}{2}\big{(}\mu_{0}M_{\text{s}}-\frac{2K_{2\perp}}{M_{\text{s}}}\big{)}\cos^{2}(\theta_{M})\\\
&+\frac{1}{2}\cdot\frac{2K_{4}}{M_{\text{s}}}\Big{(}\frac{1}{3}\cos^{4}(\theta_{M})+\frac{1}{4}\sin^{4}(\theta_{M})-\frac{\sqrt{2}}{3}\sin^{3}(\theta_{M})\cos(\theta_{M})\sin(3\phi_{M})\Big{)},\end{split}$
(S2)

with $\theta_{B}$ and $\phi_{B}$ the angles of $B_{0}$ with respect to
respectively the film’s normal and the in-plane $[1\overline{1}0]$
crystallographic axis (Fig. S1) and $\mu_{0}$ the vacuum permeability.
$\frac{2K_{2\perp}}{M_{\text{s}}}$ and $\frac{2K_{4}}{M_{\text{s}}}$ are
respectively the uniaxial out-of-plane and cubic anisotropy fields, with
$K_{2\perp}$ and $K_{4}$ the perpendicular and cubic anisotropy constants.
Note that to calculate the FMR frequency using equation S1 at a certain
$B_{0}$, $\theta_{B}$ and $\phi_{B}$, we first need to find $\theta_{M}$ and
$\phi_{M}$ that minimize the free energy by numerically solving
$\frac{\partial F}{\partial\theta_{M}}(\theta_{M},\phi_{M})=0$ and
$\frac{\partial F}{\partial\phi_{M}}(\theta_{M},\phi_{M})=0$.
Using equations S1 and S2 we can calculate the FMR frequency for an out-of-
plane magnetic field and magnetization ($\theta_{B}=\theta_{M}=0^{\circ}$),
which gives

$\omega_{\text{FMR($\perp$)}}=\gamma_{\perp}(B_{0}-\mu_{0}M_{\text{s}}+\frac{2K_{2\perp}}{M_{\text{s}}}-\frac{4K_{4}}{3M_{\text{s}}}).$
(S3)

For an in-plane magnetic field and magnetization
($\theta_{B}=\theta_{M}=90^{\circ}$), we find

$\omega_{\text{FMR($||$)}}=\gamma_{||}\sqrt{B_{0}\cdot\big{(}B_{0}+\mu_{0}M_{\text{s}}-\frac{2K_{2\perp}}{M_{\text{s}}}-\frac{K_{4}}{M_{\text{s}}}\big{)}-2\big{(}\frac{K_{4}}{M_{\text{s}}}\cos(3\phi_{M})\big{)}^{2}}.$
(S4)

The factor 3 in the cosine arises from the triangular in-plane symmetry of a
cubic unit cell with its normal along the $[111]$ direction (Fig. S1). In our
measurements a large in-plane magnetic field is needed to overcome the
perpendicular anisotropy and push the magnetization in the plane, such that
generally $B_{0}\gg|\frac{2K_{4}}{M_{\text{s}}}|=8.2$ mT and we can ignore the
last term [35]

$\omega_{\text{FMR($||$)}}=\gamma_{||}\sqrt{B_{0}\cdot(B_{0}+\mu_{0}M_{\text{s}}-\frac{2K_{2\perp}}{M_{\text{s}}}-\frac{K_{4}}{M_{\text{s}}})}.$
(S5)

Equations S3 and S5 are the same as equations 1 and 2 in the main text.

Figure S1: Coordinate frame and crystallographic axes in Ga:YIG. The
$[1\overline{1}0]$ axis is slightly displaced to highlight the triangular
symmetry plane (light blue) of the $(111)$-oriented cubic unit cell. After
[35].  Figure S2: Dependence of the FMR frequency on the direction of the
external magnetic field $\bm{B_{0}}$. (a) Flipchip FMR measurement with
$B_{0}$ applied parallel to a 180-$\upmu$m-wide excitation stripline and along
the in-plane $[1\overline{1}0]$ crystallographic direction
($\phi_{B}=0^{\circ}$). The FMR is extracted from the field-derivative of the
microwave transmission $|\text{S}_{21}|$. The solid white line shows a fit to
the Kittel relation (equation S5). Using the extracted anisotropy fields and
gyromagnetic ratio, the FMR frequency for the entire $B_{0}$-range was
numerically calculated assuming $\theta_{B}=90^{\circ}$ (black dashed line)
and $\theta_{B}=89^{\circ}$ (red dashed line). (b) The minimum FMR frequency
is raised at $\theta_{B}=89^{\circ}$ because the magnetization does not
abruptly turn into the plane. (c) Similar FMR measurement, but with $B_{0}$
applied along the $[11\overline{2}]$ direction ($\phi_{B}=90^{\circ}$). The
white line is the same as in (a). The black and red dashed lines are the
calculated FMR frequencies for $\phi_{B}=90^{\circ}$ at respectively
$\theta_{B}=90^{\circ}$ and $\theta_{B}=89^{\circ}$ using the parameters
extracted in (a). (d) The magnetization maintains a finite out-of-plane
component even when $\theta_{B}=90^{\circ}$. The blue dashed line indicates
the field at which the exchange constant was determined from the spin-wave
spectra. In (a) and (c) a similar background subtraction was performed as in
Fig. S4.

###  2.1 FMR frequency and magnetization direction at $\phi_{B}=0^{\circ}$

Fig. S2a shows a flipchip FMR measurement with the magnetic field applied
along the $[1\overline{1}0]$ direction ($\theta_{B}=90^{\circ}$,
$\phi_{B}=0^{\circ}$). The solid white line shows a fit to equation S5 for
magnetic fields at which the FMR frequency is increasing. Together with the
fitted out-of-plane FMR, we extract $\frac{2K_{2\perp}}{M_{\text{s}}}=104.7$
mT, $\frac{2K_{4}}{M_{\text{s}}}=-8.2$ mT and $\frac{\gamma_{||}}{2\pi}=28.56$
MHz/mT. The same fit and data are presented in Fig. 1b of the main text.
We can calculate the FMR frequency also for low bias fields by substituting
the extracted parameters into equations S1 and S2. We obtain the black dashed
line, which fits reasonably well to the measured FMR, even when the FMR
frequency is decreasing with field. The red dashed line shows the calculated
FMR frequency when $B_{0}$ has an $1^{\circ}$ out-of-plane angle
($\theta_{B}=89^{\circ}$), which dramatically increases the minimum FMR
frequency. This is because the magnetization turns only asymptotically into
the plane when the angle is offset, instead of abruptly (Fig. S2b, black line:
$\theta_{B}=90^{\circ}$, red line: $\theta_{B}=89^{\circ}$).
We note that in Fig. S2a at large bias fields both the black and red dashed
lines overlap with the white fit. Therefore, we conclude that the in-plane FMR
at $\phi_{B}=0^{\circ}$ is quite robust to any small out-of-plane component of
the static field that might be present in our experimental setup, validating
the white fit using equation S5 [35].

###  2.2 FMR frequency and magnetization direction at $\phi_{B}=90^{\circ}$

Fig. S2c shows a similar flipchip FMR measurement as in Fig. S2a, but now with
the field applied along the $[11\overline{2}]$ direction
($\theta_{B}=90^{\circ}$, $\phi_{B}=90^{\circ}$, the white line is the same as
in Fig. S2a and is added as a reference). The FMR reaches a minimum frequency
of about 1 GHz, which is significantly larger than the minimum in the
$\phi_{B}=0^{\circ}$ geometry. We reproduce this enhanced frequency minimum by
calculating the expected FMR frequency using the parameters extracted in
section 2.1 (black dashed line, we ignore any potential in-plane anisotropy of
the g-factor). The calculated FMR frequency matches the measured FMR
remarkably well for all magnetic field values, demonstrating the accuracy of
the white fit.
Again we attribute the enhanced FMR minumum to the fact that the magnetization
only slowly turns into the plane, even for a perfect in-plane magnetic field
$\theta_{B}=90^{\circ}$ (Fig. S2d, black line). As a result the FMR frequency
asymptotically approaches the in-plane Kittel relation (equation S5, white
line). Similar to before, a change of $1^{\circ}$ in $\theta_{B}$ lifts the
minimum FMR frequency, explaining the minimum FMR frequency of about 1.25 GHz
observed in Fig. 2b in the main text. Variations on the order of $1^{\circ}$
in $\theta_{B}$ are expected in our measurement setup since we manually place
the sample between two permanent magnets (section 1).
Fig. S2d shows that the magnetization does not point exactly in the plane
during our propagating spin wave spectroscopy measurements, even though this
is assumed in the data analysis. We derived the exchange constant from spin-
wave spectra taken at approximately $B_{0}=117.5$ mT, at which the
magnetization points $\sim$3-6 degrees out of the plane (blue dashed line in
Fig. S2d). We neglect this small out-of-plane angle, because we expect the
induced error to be negligible compared to the $\sim 15\%$ error obtained from
the fit in Fig. 3c in the main text.

##  3 Systematic error in the applied bias field

In this section we calculate how a systematic error in the applied bias field
affects the error of the anisotropy fields, which we extracted from the FMR
frequency (Fig. 1b of the main text). From the fits of the FMR frequency we
obtain $\gamma_{\perp}=29.40(3)$ MHz/mT and
$\alpha=-\mu_{0}M_{\text{s}}+\frac{2K_{2\perp}}{M_{\text{s}}}-\frac{2}{3}\frac{2K_{4}}{M_{\text{s}}}=91.1(2)$
mT (out-of-plane bias field), $\gamma_{||}=28.56(4)$ MHz/mT and
$\beta=\mu_{0}M_{\text{s}}-\frac{2K_{2\perp}}{M_{\text{s}}}-\frac{1}{2}\frac{2K_{4}}{M_{\text{s}}}=-81.5(1)$
mT (in-plane bias field). Since we know from vibrating sample magnetometry
(VSM) that $M_{\text{s}}=1.52(6)\cdot 10^{4}$ A/m, we can calculate the
magnetic anisotropy fields

$\begin{split}&\frac{2K_{4}}{M_{\text{s}}}=-\frac{6}{7}(\alpha+\beta)=8.2(2)\text{
mT},\\\
&\frac{2K_{2\perp}}{M_{\text{s}}}=\mu_{0}M_{\text{s}}+\frac{3}{7}\alpha-\frac{4}{7}\beta=104.7(8)\text{
mT}.\end{split}$ (S6)

Since we manually place our sample between the magnets (section 1), it may
have a small offset of $\sim 1$ mm with respect to the center position. Such
an offset would cause a systematic error in the applied magnetic field
$B_{0}$, which enhances the error of the anisotropy fields. To obtain a
conservative estimate of these errors, we determine the systematic error in
the applied magnetic field via

$\Delta B_{0}(x)=B_{0}(x+1)+B_{0}(x-1)-2\cdot B_{0}(x).$ (S7)

$B_{0}(x)$ is the magnetic field of a cylindrical magnet at a distance of $x$
mm along its symmetry axis

$B_{0}(x)=\frac{B_{r}}{2}\Big{(}\frac{x+L}{\sqrt{r^{2}+(x+L)^{2}}}-\frac{x}{\sqrt{r^{2}+x^{2}}}\Big{)}.$
(S8)

Here $B_{r}=1320$ mT is the remanence, $L=20$ mm and $r=17.5$ mm are the
length and radius of the magnet. Fig. S3 shows the calculated error $\Delta
B_{0}(x)$ for a 1-mm-offset against the magnetic field $B_{0}$ at the center
position between the magnets. Including this error in the fit of Fig. 1b in
the main text gives $\alpha=91.1(3)$ mT and $\beta=-81.5(5)$ mT, resulting in
a slight increase in the error of the cubic anisotropy field
$\frac{2K_{4}}{M_{\text{s}}}=8.2(5)\text{ mT}$. The errors in the gyromagnetic
ratios and perpendicular anisotropy field do not change significantly. At the
magnetic field $B_{0}=117.5$ mT at which we took the spin-wave spectra the
error in the field $\Delta B_{0}$ is $\sim 0.3$ mT.

Figure S3: Error in the static magnetic field as a result of a 1 mm offset of
the sample with respect to the center position between the magnets. For
magnetic fields between 100 mT and 150 mT an error of $\sim$ 0.3-0.5 mT is
expected.

##  4 Background-subtraction procedures of the spin-wave spectra

For the spin-wave spectra in Fig. 3a,b and Fig. 4a,c in the main text a
background spectrum was subtracted consisting of the mean $|\text{S}_{21}|$
transmission at 100 mT and 138 mT, for which there are no spin waves in the
frequency range of interest. In Fig. 2b in the main text a background was
subtracted using Gwyddion (Fig. S4).

Figure S4: Background-subtraction procedure of the microwave spectrum in Fig.
2b of the main text. The measured data (left figure) contains spurious signals
attributed to small changes in the microwave transmission of the cables and
connectors that attach the VNA to the striplines as a function of magnetic
field. We filter these signals by first masking the high-curvature part of
measured data that contains the spin-wave fringes. Then we fit a fifth-order
polynomial through each horizontal line, excluding the masked data, and
subtract it as a background (middle figure). The resulting spectrum only
contains the spin-wave fringes (right figure, same as Fig. 2b in the main
text). The image processing was performed using Gwyddion (version 2.58).

##  5 The spin-wave dispersion of a magnetic thin film with perpendicular and
cubic magnetic anisotropy

The spin-wave dispersion for magnetic thin films with perpendicular magnetic
anisotropy (PMA) and cubic anisotropy was derived in reference [37]. Equation
30 of this work states the dispersion for an (111)-oriented film with in-plane
magnetization, similar as in our experiment

$\begin{split}&\omega_{\text{SW}}(k)=\\\
&\gamma_{||}\sqrt{\big{(}B_{0}+Dk^{2}+\mu_{0}M_{\text{s}}(1-f)-\frac{2K_{2\perp}}{M_{\text{s}}}-\frac{K_{4}}{M_{\text{s}}}\big{)}\cdot\big{(}B_{0}+Dk^{2}+\mu_{0}M_{\text{s}}f\sin^{2}(\phi)\big{)}-2\big{(}\frac{K_{4}}{M_{\text{s}}}\cos(3\phi_{M})\big{)}^{2}}.\end{split}$
(S9)

Here $\omega_{\text{SW}}$ is the angular frequency of a spin wave with
wavevector $k$ that propagates at an angle $\phi$ with respect to the
magnetization. $D=2\alpha/M_{\text{s}}$ is the spin stiffness, with $\alpha$
the exchange constant, and $f=1-(1-e^{-kt})/kt$ with $t$ the thickness of the
film and $\phi_{M}$ is the angle of the magnetization with respect to
$[1\overline{1}0]$ crystallographic direction. We note that if we set $k=0$ in
equation S9, we obtain the in-plane FMR frequency derived before (equation
S4).
In our experiment we measure spin waves in the Damon-Eshbach configuration
($\phi=\pi/2$), we apply the external field $B_{0}$ along $[11\overline{2}]$
($\phi_{M}=\pi/2$) and the wavelengths of the detected spin waves are much
smaller than the thickness of the film ($kt\ll 1$), such that we can
approximate $f\approx kt/2$. This gives

$\omega_{\text{SW}}(k)=\sqrt{\big{(}\omega_{B}+\gamma_{||}Dk^{2}-\omega_{K}+\omega_{M}(1-kt/2)\big{)}\big{(}\omega_{B}+\gamma_{||}Dk^{2}+\omega_{M}kt/2\big{)}},$
(S10)

where we defined $\omega_{B}=\gamma_{||}B_{\text{0}}$,
$\omega_{M}=\gamma_{||}\mu_{0}M_{\text{s}}$, and
$\omega_{K}=\gamma_{||}(\frac{2K_{2\perp}}{M_{\text{s}}}+\frac{K_{4}}{M_{\text{s}}})$
for convenience of notation. Working out the brackets and rearranging the
terms in orders of $k$ gives

$\omega_{\text{SW}}=\sqrt{\omega_{B}\big{(}\omega_{B}+\omega_{M}-\omega_{K}\big{)}+\frac{\omega_{M}t}{2}\big{(}\omega_{M}-\omega_{K}\big{)}k+\gamma_{||}D\big{(}2\omega_{B}+\omega_{M}-\omega_{K}-(\frac{\omega_{M}t}{2})^{2}\big{)}k^{2}+\gamma_{||}^{2}D^{2}k^{4}}.$
(S11)

For the spin-wave spectra taken at $B_{0}=117.5$ mT we find
($\frac{\omega_{M}t}{2})^{2}\ll 2\omega_{B}+\omega_{M}-\omega_{K}$ due to the
low saturation magnetisation and thickness of our film, such that we can
further approximate

$\omega_{\text{SW}}=\sqrt{\omega_{B}(\omega_{B}+\omega_{M}-\omega_{K})+\frac{\omega_{M}t}{2}(\omega_{M}-\omega_{K})k+\gamma_{||}D(2\omega_{B}+\omega_{M}-\omega_{K})k^{2}+\gamma_{||}^{2}D^{2}k^{4}},$
(S12)

which is equation 4 in the main text.
We derive the group velocity $v_{\text{g}}$ by differentiating with respect to
$k$

$v_{\text{g}}=\frac{\partial\omega_{\text{SW}}}{\partial
k}=\frac{1}{2\sqrt{\omega_{\text{SW}}}}\big{(}\frac{\omega_{M}t}{2}(\omega_{M}-\omega_{K})+2\gamma_{||}D(2\omega_{B}+\omega_{M}-\omega_{K})k+4\gamma_{||}^{2}D^{2}k^{3}\big{)},$
(S13)

which is equation 5 in the main text.

##  6 Comparing the frequency difference between fringes to the spin-wave
linewidth

In this section we calculate the expected average frequency difference $\Delta
f$ between spin-wave fringes excited by the first maximum of the microwave
driving field Fourier amplitude ($|B^{\text{RF}}(k)|$) in Fig. 3b of the main
text. The stripline has a width $w=2.5$ $\upmu$m, such that
$|B^{\text{RF}}(k)|$ has its first node at $k_{\text{min}}=\frac{2\pi}{2.5}$
$\upmu\text{m}^{-1}$ [38]. Everytime another wavelength fits within the
center-to-center distance $s$ between both striplines another fringe is
observed in the signal. Therefore the condition $s=n\lambda$ applies for every
$n$th fringe, with $\lambda$ the spin-wave wavelength. This means that fringes
occur every $\Delta k=\frac{2\pi}{s}=\frac{2\pi}{12.5}$ $\upmu\text{m}^{-1}$
in k-space. In the first maximum of the excitation spectrum we would thus
expect $\frac{k_{\text{min}}}{\Delta k}=5$ fringes. According to the
reconstructed dispersion (Fig. 3f of the main text) the frequency difference
between spin waves with wavevector $k_{\text{min}}$ and the minimum of the
band is about $20$ MHz, leading to an average frequency difference of
$\frac{20}{5}=4$ MHz between consecutive fringes. This is on the order of the
FMR linewidth of undoped YIG films of similar thicknesses [36]. Assuming that
Ga:YIG has a similar or larger linewidth, we argue that we cannot resolve
fringes in the first maximum of the excitation field’s Fourier amplitude
because they are too narrow compared to the intrinsic spin-wave linewidth.

##  7 Zoomed-in spin-wave spectra displaying low-amplitude fringes

Figure S5: Detailed microwave spectrum zoomed-in on the spin-wave fringes.
Low-amplitude fringes excited by the second maximum of the excitation field’s
Fourier amplitude are visible at high frequencies. The actual measured data
without any background-subtraction is presented ($w=1$ $\upmu$m, $s=6$
$\upmu$m, excitation power -35 dBm).

##  8 Calculation of the non-linear frequency-shift coefficient

For Damon-Eshbach spin waves with wavevector $k$ and frequency
$\omega_{k}/2\pi$ the non-linear four-magnon frequency-shift coefficient
$W_{kk,kk}$ is given by [39]

$\begin{split}&W_{kk,kk}=\frac{1}{2}\big{(}\frac{2\omega_{B}+\omega_{M}(N_{xx,k}+N_{yy,k})}{2\omega_{k}}\big{)}^{2}\cdot\big{(}3\omega_{B}+\omega_{M}(2N_{zz,0}+N_{zz,2k})\big{)}\\\
&-\frac{1}{2}\big{(}3\omega_{B}+\omega_{M}(N_{xx,k}+N_{yy,k}+N_{zz,2k})\big{)},\end{split}$
(S14)

with $N_{ij,k}$ the $(i,j)$th index of the spin-wave tensor $N_{k}$. The
three-wave correction term vanishes since the spin waves propagate
perpendicular to the magnetization. The precessional $xyz$-frame is defined
such that $z$ points in the plane along the magnetization, $x$ along the film
normal and $y$ points in-plane perpendicular to $z$ and parallel to the
wavevector of the spin waves.
$N_{k}$ is the Fourier transform of the tensorial Green’s function
$N(r,r^{\prime})=N(r,r^{\prime})_{\text{dip}}+N(r,r^{\prime})_{\text{ex}}+N(r,r^{\prime})_{\text{ani}}$,
which has components due to uniaxial anisotropy and the dipolar and exchange
interactions

$N_{k}e^{ikr}=\int
N(r,r^{\prime})e^{ikr^{\prime}}d^{3}r^{\prime}=\int\big{(}N(r,r^{\prime})_{\text{dip}}+N(r,r^{\prime})_{\text{ex}}\big{)}e^{ikr^{\prime}}d^{3}r^{\prime}+\int
N(r,r^{\prime})_{\text{ani}}e^{ikr^{\prime}}d^{3}r^{\prime}.$ (S15)

The contribution to $N_{k}$ from the $N(r,r^{\prime})_{\text{dip}}$ and
$N(r,r^{\prime})_{\text{ex}}$ components in the thin-film limit were derived
earlier [39]. Following this work, $N(r,r^{\prime})_{\text{ani}}$ due to
uniaxial anisotropy in the out-of-plane $x$-direction is given by

$N(r,r^{\prime})_{\text{ani}}=-\frac{B_{2\perp}}{\mu_{0}M_{\text{s}}}\delta(r-r^{\prime})\hat{x}\otimes\hat{x}.$
(S16)

Here $B_{2\perp}=\frac{2K_{2\perp}}{M_{\text{s}}}$ is the uniaxial out-of-
plane anisotropy field, $\otimes$ denotes a dyadic unit vector product and
$\delta(r-r^{\prime})$ is the Dirac delta function. As a result of the dyadic
product only the $(x,x)$ index of $N(r,r^{\prime})_{\text{ani}}$ is non-zero,
leading to a contribution on $N_{xx,k}$

$N_{xx,k}e^{ikr}=\int-\frac{B_{2\perp}}{\mu_{0}M_{\text{s}}}\delta(r-r^{\prime})e^{ikr^{\prime}}d^{3}r^{\prime}=-\frac{B_{2\perp}}{\mu_{0}M_{\text{s}}}e^{ikr}.$
(S17)

By adding this contribution to the other components, we find that the diagonal
elements of $N_{k}$ in the Damon-Eshbach configuration are given by

$\begin{split}&N_{xx,k}=\frac{D}{\mu_{0}M_{\text{s}}}k^{2}+1-f-\frac{B_{2\perp}}{\mu_{0}M_{\text{s}}},\\\
&N_{yy,k}=\frac{D}{\mu_{0}M_{\text{s}}}k^{2}+f,\\\
&N_{xx,k}=\frac{D}{\mu_{0}M_{\text{s}}}k^{2},\\\ \end{split}$ (S18)

with $f=1-(1-e^{-kt})/kt$ and $t$ the thickness of the film as before. We
neglected the cubic anisotropy since it is small relative to the uniaxial
anisotropy.

Figure S6: Non-linear frequency-shift coefficient $\bm{W_{kk,kk}}$ for Damon-
Eshbach spin waves in Ga:YIG. We used the dispersion in Fig. 3f of the main
text as an input, together with the extracted parameters $B_{0}=117.5$ mT,
$\alpha=1.3\cdot 10^{-12}$ J/m, $\frac{2K_{2\perp}}{M_{\text{s}}}=104.7$ mT,
$\frac{\gamma_{||}}{2\pi}=28.56$ MHz/mT, $M_{\text{s}}=1.52\cdot 10^{-4}$ A/m
and $t=45$ nm. The cubic anisotropy is neglected. The positive sign of the
calculated frequency-shift coefficient matches the positive frequency shifts
observed in the experiment.

By substituting equations S18 into equation S14 we can calculate $W_{kk,kk}$
for the wavevectors relevant for this work (Fig. S6). For all these
wavevectors $W_{kk,kk}$ is positive, explaining the positive frequency shifts
of the spin waves that we observe when increasing the drive power. This is in
contrast to the frequency shift caused by the reduction of the saturation
magnetization as a result of strong driving or heating. In this simple picture
a downward frequency shift is expected for in-plane magnetization (Fig. S7),
highlighting the value of the Hamiltonian formalism that was used to calculate
the non-linear frequency-shift coefficient [39].

Figure S7: Expected downward frequency shift upon reduction of the saturation
magnetization. Field dependence of the FMR frequency of Ga:YIG for unreduced
saturation magnetization ($M_{\text{s}}=1.52\cdot 10^{4}$ A/m, black line) and
for 10%-reduced saturation magnetization ($M_{\text{s}}=1.37\cdot 10^{4}$ A/m,
red line). The bias field is applied in the $[11\overline{2}]$ direction and
the magnetic anisotropy fields are the same for both curves. The dashed line
indicates the field at which we performed our spin-wave spectroscopy
measurements. Clearly a negative frequency shift is expected upon decreasing
the saturation magnetization, which is in contrast to the positive frequency
shifts we observe.


# Towards Edge-Cloud Architectures for Personal Protective Equipment Detection

Jarosław Legierski<EMAIL_ADDRESS>Orange Innovation, Orange
Polska S.A.WarsawPoland00-001 ,  Kajetan Rachwał
<EMAIL_ADDRESS>,  Piotr Sowinski<EMAIL_ADDRESS>Systems Research Institute Polish Academy of SciencesPoland Warsaw Univesity
of TechnologyWarsawPoland ,  Wojciech Niewolski<EMAIL_ADDRESS>Orange Innovation, Orange Polska S.A.Poland Warsaw Univesity of
TechnologyWarsawPoland ,  Przemysław Ratuszek<EMAIL_ADDRESS>,  Zbigniew Kopertowski<EMAIL_ADDRESS>Orange Innovation,
Orange Polska S.A.WarsawPoland ,  Marcin Paprzycki
<EMAIL_ADDRESS>and  Maria Ganzha<EMAIL_ADDRESS>Systems Research Institute Polish Academy of SciencesWarsawPoland

###### Abstract.

Detecting Personal Protective Equipment in images and video streams is a
relevant problem in ensuring the safety of construction workers. In this
contribution, an architecture enabling live image recognition of such
equipment is proposed. The solution is deployable in two settings – edge-cloud
and edge-only. The system was tested on an active construction site, as a part
of a larger scenario, within the scope of the ASSIST-IoT H2020 project. To
determine the feasibility of the edge-only variant, a model for counting
people wearing safety helmets was developed using the YOLOX method. It was
found that an edge-only deployment is possible for this use case, given the
hardware infrastructure available on site. In the preliminary evaluation,
several important observations were made, that are crucial to the further
development and deployment of the system. Future work will include an in-depth
investigation of performance aspects of the two architecture variants.

edge-cloud continuum architectures, PPE detection, image recognition, worker
safety

††conference: International Conference on Information Management & Machine
Intelligence; December 23–24, 2022; Jaipur, India

##  1\. Introduction

Nowadays, the demand for intelligent video analytics is growing across a wide
spectrum of application areas (Zhang et al., 2019). The key part of such
systems is usually an image recognition (IR) component. However, as of today,
the IR subsystem is, most commonly, deployed in the cloud. This approach
offers multiple benefits, such as availability of large and scalable
computational resources, reliable APIs, and shifting the burden of system
maintenance to the cloud service provider. However, this comes at a cost.
Sending data to the cloud raises both security and privacy concerns. Moreover,
communicating with the cloud always induces network latency, which may be
significant in time-critical applications. To address issues brought about by
cloud-centric solutions, edge computing has been proposed. Here, the core of
the approach is processing the data as close to the source as possible. This
allows for latency reduction, and helps ensure the security and privacy of
data, which remains within the local network. However, edge computing has its
own set of issues. Typically, the computational resources, which are available
at the edge are considerably smaller. A possible solution to addressing the
downsides of both these options is a combined approach – an _edge-cloud
continuum_ , where data is partially processed on the edge and partially in
the cloud. However, this raises the obvious question: at which point(s),
within the continuum, individual parts of the system should be deployed.

Here, this question is considered within a real-world scenario of monitoring
the entrance to an active construction site. Specifically, the system is
tasked with ensuring that (1) no unauthorized people enter the worksite, and
(2) everybody is wearing appropriate Personal Protective Equipment (PPE), i.e.
helmets and safety vests. The scenario is evaluated as part of the ASSIST-IoT
project, on a construction site in Warsaw, Poland, managed by the construction
company Mostostal Warszawa. Here, the edge versus cloud discussion becomes
particularly relevant. On the one hand, the privacy of workers is of paramount
importance, while latencies must be minimized, to ensure a quick reaction,
which hints at an edge deployment. On the other hand, given the limited
hardware resources available on the edge, and the extremely harsh conditions
of the construction site, a cloud deployment seems attractive.

Given the possible benefits of both solutions, in this contribution, a
solution is proposed for an edge-cloud continuum video analytics architecture.
The architecture can be deployed in two variants (edge-only, and edge-cloud),
described in the _Architecture_ section. Moreover, to determine the viability
of the solution, an initial experimental study was performed. Here, an IR
model was developed and integrated with the edge-only variant of the
architecture. Next, it was tasked with detecting when personnel wearing PPE
entered and exited the work site.

##  2\. Background

To provide a context for this study, the state of the art of (1) IR system
architectures and (2) machine learning models for PPE detection is summarized.

### System architectures

The most obvious benefit of deploying IR systems on the edge is the decreased
latency. This was demonstrated in (Zhou and Keoh, 2020), where facial
recognition models were deployed on the edge. The authors found that deploying
the models on the edge resulted in significantly better response speeds, as
compared to a cloud deployment. In other studies (Liu et al., 2018, 2019), the
viability of deploying deep convolutional neural networks (CNNs) in the edge-
only scenario was investigated. CNNs are characterized by high resource
utilization, and thus are typically deployed in the cloud. The studies found
that deploying CNNs is viable on mobile devices, when parts of the computation
can be offloaded to other edge devices. Edge deployment allowed to achieve a
consistently low latency of 2.24 ms while using CNNs to perform real-time
object tracking in augmented reality (Liu et al., 2019). Both studies showed
that the edge deployment distributing the workload increased the inference
capabilities of the system, as the models could not be run on the disconnected
mobile devices alone.

One study (Huang et al., 2017) investigated an edge-cloud architecture, where
data preprocessing servers were deployed close to the data source. The
preprocessed data was then sent to a cloud-based deep-learning platform. This
resulted in decreasing network latency and traffic. It also increased the
security and privacy of the raw data.

On the other hand, edge deployments are more limited in terms of the available
hardware. Low computational resources naturally limit the size of models and
inference speed. A study compared different implementations (based on
TensorFlow, TensorRT, and TFLite) of the same video processing model (Koubaa
et al., 2022), and found them to differ in their resource utilization. The
choice of implementation influenced the energy consumption of the model, as
well as its inference speed. Interestingly, the slowest implementation
(TFLite) was the most energy efficient. It was also found that TFLite managed
to remain on par with the other implementations in terms of speed, when
processing low-resolution video. In the case of high-resolution video, more
resource-intensive models were needed to maintain the speed, suggesting that a
cloud deployment could be more beneficial in low-resource settings.
Nevertheless, some resource-intensive models can be deployed on the edge, if
resources available there are sufficient. The deployment proposed in a
different study required all nodes to be equipped with a GPU (Liu et al.,
2018). This allowed the authors to use CNNs on the edge. A similar result was
reported in (Wu et al., 2021), were IR models deployed on a Raspberry Pi 4B,
equipped with a camera, and an Intel Neural Compute Stick 2 (a USB device for
deep learning inference on the edge) were studied. These devices were chosen
for their low power consumption and good computing capabilities. Overall, a
model tasked with detecting PPE in the form of helmets and safety vests
achieved precision on the order of $99.5\%$.

### Models for PPE detection

Effective video analytics-based methods for detecting the presence of
protective helmets, worn by workers, due to its health and safety importance,
is currently a hot research topic.

The usage of existing, unmodified machine learning models for detecting
protective head covers does not provide sufficient detection accuracy, as
proven in a recent study (Zhou et al., 2021a). In said article, several
versions of the popular YOLO algorithm (Bochkovskiy et al., 2020) were
compared. It was shown that the most effective version of YOLO for helmet
detection is the v4. After improving the loss function, it achieved more than
93% accuracy during tests. A similar study (Liu et al., 2022) focused on
improving the YOLOv5 algorithm. The system achieved results close to 97%
accuracy, thanks to the improvement of the structure of the neural network.
Another study (Zhou et al., 2021b), also investigated improving YOLOv5.
However, instead of the algorithm itself, work was focused on processing of
input data by applying filters on the input image. This allowed to improve the
accuracy to above 95%. Yet another study (Xu et al., 2022) presented an
approach for improving the detection speed and accuracy by designing a multi-
level pyramidal feature fusion network based on the ConCaNet attention
mechanism. Here, YOLOv3 was applied and a dataset with 6000 images was used.
The results demonstrate the effectiveness of this approach, which managed to
reduce the number of necessary parameters.

Helmet detection can also be done using the SSD-MobileNet algorithm (Howard et
al., 2017), which is based on yet another variant of CNN. An analysis of this
method, reported in (Li et al., 2020), tested its effectiveness and managed to
reach 80% accuracy during tests. In a wider comparison of algorithm types (Nan
et al., 2022), the authors proposed a helmet detection method based on a
dynamically changing neural network – SHDDM (Safety Helmet Detection Dynamic
Model). The developed model analyzes the human posture and defines the area
where the helmet should be located, to eliminate the detection of the helmet
outside the head area and thus reduce the false positive rate. There are also
other approaches to helmet detection, such as methods based on color and shape
used to to locate the face, and the proper wearing of a helmet (Zhang et al.,
2017). Another solution used low-resolution images, captured from a video
stream, using the Local Binary Pattern (LBP) and gray-level co-occurrence
matrix (GLCM) methods along with a back-propagation neural network (Xinhua et
al., 2021).

Another study (di Benedetto et al., 2019) investigated the usefulness of
artificially created images in the training of CNNs for PPE detection. The
paper presented the results achieved with YOLOv3, trained on artificial images
generated by the Rockstar Advanced Game Engine (RAGE) from the Grand Theft
Auto V video game. This approach achieved a mean average precision (mAP) of
only 55.11% on a test dataset consisting of real-world images. The mAP for
synthetic images was much higher at 87.24%. It should be noted that the poor
results for the real-world images are most likely caused by the RAGE engine
being unable to generate a sufficient amount of head, welding mask, ear
protection, and chest object variations.

As can be seen, there are many possibilities for detecting protective helmets.
Here, the SHDDM is particularly noteworthy, as it has an important feature of
checking whether the helmet is worn properly, and not only detecting its
presence. This, in turn, is particularly relevant in real-world applications.

##  3\.  Proposed Architecture

The proposed video analytics system can be deployed in two architecture
variants: edge-cloud (Fig. 1) and edge-only (Fig. 2). As outlined above, there
are reasons to believe that both variants may be appropriate for the
considered scenario. Both architectures share a common core deployed on the
edge, consisting of: a camera, the Image Processor (IP) component, and the OSH
(Occupational Safety and Health) manager’s mobile device.

Figure 1. Edge-cloud deployment

Figure 2. Edge deployment

The camera (in the reported experiments the Dahua IPC-HFW5449T-ASE-LED was
used) provides a live RTSP video stream, which is directed to the Image
Processor. The IP is a service written in Python, which can optionally perform
preliminary image analysis. Using configurable methods such as motion
detection and brightness thresholding, the IP is able to discard image frames
that do not contain moving people, reducing network traffic to components
involved in actual image analysis. It is also responsible for communicating
with the rest of the system, designed in accordance with the ASSIST-IoT
reference architecture (Fornés-Leal et al., 2022). IP communicates with the
rest of the system publishing alerts to an MQTT topic. This design allows
other components and devices in the ASSIST-IoT deployment to be notified in a
streaming manner of any OSH violations, such as workers not wearing protective
helmets.

In the first version of the architecture – the edge-cloud deployment – the IP
is configured to use the cloud-based AWS Rekognition platform, with its PPE
detection service.

In the edge-only variant, the video analysis is performed by the Orange AI&ML
Platform, which is deployed on a server on the construction site. This edge
deployment allows for maintaining lower network latency, and ensures the
privacy of worker data. The AI&ML Platform’s services are written as Python
runnable modules that provide their own APIs and GUIs. The services can reuse
the APIs and GUIs provided by the platform, or build them from scratch. A
service collects frames from a video source, processes them in an ML pipeline
specific to the service, and adapts or interprets the results. The inference
results from the Platform are forwarded to external services, with the use of
provided connectors. As the Orange AI&ML Platform operates on the edge, all
video processing takes place on the client’s site, ensuring full security of
customer data (video) and compliance with appropriate regulations, such as
GDPR.

##  4\. Methodology

As part of this study, a preliminary version of the edge-only variant of the
architecture was deployed on an active construction site. Using the Orange
AI&ML Platform, a model was trained to count people wearing helmets entering
and exiting a specific area. The system counts people in helmets in defined
recognition areas (bounding boxes), crossing the yellow and green lines
visible in Figs. 3 and 4. People entering the construction site are counted
after crossing the green line, while people leaving are counted after crossing
the yellow line. The machine learning pipeline consists of a YOLOX object
detection model, trained for detecting heads in helmets, and a DeepSORT (Wojke
et al., 2017) multi-object tracking algorithm. The YOLOX model was trained
using a dataset provided by the Northeastern University of China
(https://public.roboflow.com/object-detection/hard-hat-workers).

The system’s results were compared to those obtained from an algorithm built
into the Dahua camera. It should be noted that the camera counted _all_ people
entering and leaving, including those without protective helmets. However,
this should not impact the results much, as the safety regulations on this
particular site forbid entering it without a helmet and the rule is strictly
enforced before workers reach the counting location.

The measurements were performed in two series – each using a different
bounding box definition. A single series spanned the length of one workday on
the construction site. The number of entering and leaving people was counted
in hourly intervals (between 5 AM and 7 PM).

Figure 3. Bounding boxes location on November 22 (before modification)
Figure 4. Bounding boxes location on November 24 (after modification)

##  5\. Results

The Tables 1 and 2 present the results of the performed experiments. The Table
1 contains measurements made on 22nd November 2022, with the bounding box set
as presented in Fig. 3. The average difference between the number of people
entering, as measured by the camera and the model was equal to $-6.21$, with
the standard deviation of $\sigma=5.08$, whereas for people exiting it was
$1.35$ and $\sigma=2.73$ respectively. The correlation between entrances
detected by the camera and the model deployed on the AI&ML platform, expressed
by the Pearson coefficient is $0.988$, whereas for exits $0.995$. The
correlations were found to be statistically significant ($p\leq 0.05$). Table
2 contains measurements from 24th November 2022 (for modified detection areas,
depicted in Fig. 4). On that day, the average difference for entering was
$-4.93$ with $\sigma=4.25$ and for exiting $3.92$ with $\sigma=4.92$. For
these measurements the Pearson coefficient for people entering is equal to
$0.993$ and exiting $0.989$. The correlations were found to be statistically
significant ($p\leq 0.05$).

The tables also present differences in the number of people detected by the
camera and the AI&ML platform and the sum of these differences calculated for
both movement directions: entries and exits.

During the experiments, several unexpected events took place, which had a
significant impact on the reported results. Workers were observed acting in an
unexpected manner – lingering or walking around the detection area (Fig. 5).
It was also noticed that sometimes the workers put on their helmets after
having passed the detection area (Fig. 6). These behaviors present a challenge
to the future system, as they significantly affect its accuracy.

Hour | 05:00 | 06:00 | 07:00 | 08:00 | 09:00 | 10:00 | 11:00 | 12:00 | 13:00 | 14:00 | 15:00 | 16:00 | 17:00 | 18:00 | Total
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Dahua In | 12 | 65 | 84 | 47 | 26 | 84 | 50 | 51 | 28 | 70 | 28 | 8 | 9 | 0 | 562
Dahua Out | 2 | 15 | 21 | 35 | 81 | 44 | 61 | 31 | 63 | 32 | 59 | 66 | 26 | 8 | 544
AI&ML In | 11 | 78 | 87 | 52 | 28 | 96 | 60 | 58 | 43 | 75 | 33 | 18 | 10 | 0 | 649
AI&ML Out | 2 | 13 | 23 | 33 | 73 | 44 | 63 | 31 | 58 | 31 | 59 | 62 | 25 | 8 | 525
Diff. In | 1 | -13 | -3 | -5 | -2 | -12 | -10 | -7 | -15 | -5 | -5 | -10 | -1 | 0 | -87
Diff. Out | 0 | 2 | -2 | 2 | 8 | 0 | -2 | 0 | 5 | 1 | 0 | 4 | 1 | 0 | 19

Table 1. Entries and exits to the construction site, 22 November 2022.

Hour | 05:00 | 06:00 | 07:00 | 08:00 | 09:00 | 10:00 | 11:00 | 12:00 | 13:00 | 14:00 | 15:00 | 16:00 | 17:00 | 18:00 | Total
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Dahua In | 4 | 57 | 113 | 62 | 34 | 73 | 75 | 65 | 56 | 93 | 27 | 10 | 9 | 0 | 678
Dahua Out | 0 | 10 | 29 | 57 | 80 | 53 | 74 | 41 | 82 | 52 | 49 | 84 | 20 | 8 | 639
AI&ML In | 3 | 61 | 113 | 68 | 43 | 76 | 79 | 73 | 69 | 98 | 33 | 21 | 10 | 0 | 747
AI&ML Out | 0 | 11 | 26 | 46 | 64 | 48 | 72 | 38 | 73 | 49 | 47 | 82 | 22 | 6 | 584
Diff. In | 1 | -4 | 0 | -6 | -9 | -3 | -4 | -8 | -13 | -5 | -6 | -11 | -1 | 0 | -69
Diff. Out | 0 | -1 | 3 | 11 | 16 | 5 | 2 | 3 | 9 | 3 | 2 | 2 | -2 | 2 | 55

Table 2. Entries and exits to the construction site, 24 November 2022.

Figure 5. Unexpected worker behavior – staying in the detection area for
longer
Figure 6. Unexpected worker behavior – putting the helmet on behind the
detection line

##  6\. Concluding remarks

The tested model demonstrated relatively good performance in the investigated
scenario. Its accuracy when tasked with counting people wearing protective
helmets was found to be sufficient, and was validated against a different
system. A number of discrepancies between the counts of the model and the
camera can be attributed to unexpected situations (Figs. 5 and 6) and the fact
that the Dahua camera did not differentiate people wearing and not wearing
helmets. The high correlation coefficient between the camera and the Orange
AI&ML Platform’s model allows to conclude that the two solutions perform
comparably well.

It should be noted that there were changes in the correlation between the days
of experiments. These differences are explained by the changes to the bounding
box. This is one of the parameters that have to be investigated further.

Both variants of the proposed architecture can be used in the investigated
scenario of PPE detection on a construction site. The feasibility of using an
edge-deployment was confirmed – the server’s computational capabilities were
sufficient to maintain satisfactory inference accuracy. Therefore, it can be
concluded that the construction site is equipped with sufficient hardware to
warrant further experiments with the deployment.

In the future, the two proposed architecture variants will be compared in
terms of network latencies, resource utilization, and their accuracy. The
presented model will also be tested further, which will include manually
annotating the videos to obtain a ground truth for comparison. This will allow
for determining the actual accuracy of the developed model. Further
optimization of bounding box locations is also planned.

###### Acknowledgements.

Work supported by ASSIST-IoT project funded from the European Union’s H2020
RIA program under grant 957258.

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