# Space Charge Effects for Collective Instabilities in Circular Machines

Alexey Burov<EMAIL_ADDRESS>Fermilab, PO Box 500, Batavia, IL 60510-5011

###### Abstract

Coulomb fields of charged particle beams in circular machines determine,
together with wake fields, modes of the collective beam oscillations, both for
transverse and longitudinal degrees of freedom. Recent progress in these two
areas of beam dynamics is discussed.

Suggested keywords

###### pacs:

00.00.Aa , 00.00.Aa , 00.00.Aa , 00.00.Aa

##  I Introduction

This paper discusses recent progress in the area of collective linear dynamics
of charged particle beams, when their space charge (SC) is important; both
transverse and longitudinal planes are considered. Since such a task cannot be
free from some subjectivity and arbitrariness, I beg a pardon for the
inevitable omission of some valuable results.

More than fifty years ago, the first significant publication was presented on
transverse collective instabilities of space-charge-dominated beams in
circular machines; it was a preprint of G. Merle and D. Möhl “The stabilizing
influence of nonlinear space charge on transverse coherent oscillations” Merle
and Möhl (1969). A relatively simple equation of motion was suggested there as
something obvious. Although it was, strictly speaking, neither obvious nor
necessarily justified, it played and continues to play an extraordinarily
important role. This equation implicitly relied on rigidity of the beam
slices; four decades later it was shown that this approximation requires
strong SC, i. e. the SC tune shift to exceed the lattice tune shifts and the
synchrotron tune Burov and Lebedev (2009). Soon after that, the concept of
strong SC was used for bunched beams Burov (2009a), including computation of
Landau damping (LD) with space charge and octupoles taken into account. It was
also demonstrated there, that SC can help to avoid the transverse mode
coupling instability (TMCI). Paradoxical features of this stabilization were
realized few years ago, leading to discovery of the transverse convective
instabilities Burov (2019a), with exponential dependence of the head-to-tail
amplification on the SC tune shift. These and related ideas are discussed in
Sec. II, devoted to the transverse collective oscillations.

Longitudinal modes are discussed in Sec. III, for the bunched beams only,
since the coasting beam theory seems to be well established long ago, see e.g.
Refs. Chao (1993); Ng (2006). For bunched beams, the key intensity limitation
is associated with the loss of LD, when a single-bunch impedance drives a
collective mode outside the incoherent spectrum, making the beam defenseless
against any coupled-bunch instability, no matter how weak it might be. First
time this issue was noted and analyzed in this context by F. Sacherer almost
half a century ago Sacherer (1973); a certain threshold for SC impedance below
transition was estimated. Later this problem was addressed by other authors,
who got the same result, up to a numerical factor; the smallest threshold was
reported in Ref. Burov (2011); it was about three times below the Sacherer’s
one. These discrepancies were related to the model imperfections and to high
sensitivity of the result on the distribution function. However, a recently
published article of Karpov et al. Karpov  _et al._ (2021) has shown that all
those results are incorrect; for the pure SC impedance the LLD intensity
threshold is equal to zero. Soon after that, the entire collective spectrum
for SC impedance below transition was described in Ref. Burov (2021), with
generalization to any multipolarity and to other impedances.The single-bunch
eigensystem problem was reduced to a parameter-less equation; also the
coupled-bunch growth rates were analytically derived. These and related ideas
are considered in Sec. III.

##  II Transverse Oscillations

###  II.1  Coasting Beams

To analyze the beam stability with SC, a linear equation of motion was
suggested by G. Merle and D. Möhl in 1969 Merle and Möhl (1969):

$\frac{d^{2}x_{i}}{dt^{2}}+\omega_{xi}^{2}x_{i}+2\omega_{x}\omega^{\mathrm{sc}}_{i}(x_{i}-\bar{x})+2\omega_{x}\omega^{\mathrm{c}}\bar{x}=0\,.$
(1)

Here $x_{i}=x_{i}(t)$ is a transverse offset of a particle $i$, $\omega_{xi}$
is the betatron frequency of the particle $i$, $\omega_{x}$ is the average
betatron frequency, $\bar{x}=\bar{x}(t)$ is an average offset of that beam
slice where the particle $i$ is located at the given moment of time $t$,
$\omega^{\mathrm{sc}}_{i}<0$ is the SC frequency shift of the particle, and
$\omega^{\mathrm{c}}$ is the coherent frequency shift parameter, proportional
to the ring impedance. The full time derivative $d/dt$ is expressed through
the partial ones, $d/dt=\partial/\partial
t+\omega_{i}\partial/\partial\theta$, where $\omega_{i}$ is the revolution
frequency of the particle, and $\theta=s/R$ is the azimuthal angle, with $s$
as the conventional longitudinal coordinate and $R$ as the average ring
radius. The term ‘slice’ refers to the group of beam particles which Coulomb
fields affect the given particle number $i$, i.e. the particles with positions
somewhere between $s_{i}-a/\gamma$ and $s_{i}+a/\gamma$, where $a$ is the beam
transverse size and $\gamma$ is the Lorentz factor.

Equation (1) implies two important things.

First, it implies that $x_{i}$ relates to the driven part of the single-
particle oscillations, excited by the collective motion of the slices’
centroids $\bar{x}$, while constant amplitudes of free oscillations determine
the space charge frequency shifts. That is why the offset $x_{i}$ is of the
order of centroid offsets, $x_{i}\sim\bar{x}$, so it can be considered
infinitesimally small, while the incoherent amplitudes are of the order of the
beam transverse size.

Second, this equation assumes that each beam slice oscillates as a rigid body,
allowing for a representation of the SC force in the simple way it is done
there. Because of this assumption, Merle-Möhl approach is sometimes addressed
as the rigid-slice or frozen-field model. Possible incorrectness of this
assumption, as well as its very existence, was realized much later, when some
strange features of Eq. (1) were discovered.

For a coasting beam, eigenfunctions of Eq. (1) have the form

$x_{i}\,,\bar{x}\propto\exp\left[-i\,(\omega_{x}+n\omega_{0}+\omega)t+in\theta\right]\,$
(2)

where $n$ is an arbitrary integer, $\omega_{0}$ is the average revolution
frequency, and $\omega$ is a frequency shift of the mode $n$. Substitution of
this form into Eq. (1) yields for the complex amplitudes

$x_{i}=\bar{x}\frac{\omega^{\mathrm{c}}-\omega^{\mathrm{sc}}_{i}}{\omega-\omega^{\mathrm{sc}}_{i}-\delta\omega_{xi}-n\delta\omega_{i}}\,,$
(3)

with the lattice frequency shifts $\delta\omega_{xi}=\omega_{xi}-\omega_{x}$
and $\delta\omega_{i}=\omega_{i}-\omega_{0}$. Note that without lattice
frequency spread, $\delta\omega_{xi}=\delta\omega_{i}=0$, there is always the
rigid-bunch solution, $x_{i}=\bar{x}$, with $\omega=\omega^{\mathrm{c}}$,
independently of the SC tune shifts $\omega^{\mathrm{sc}}_{i}$. This important
physical property of Eq. (1) is a consequence of its SC representation by
means of the term $\propto x_{i}-\bar{x}$. In fact, the Merle-Möhl equation is
the only possible linear dynamic equation consistent with the given incoherent
spectrum, its lattice and SC parts, which represents the coherent SC term by
means of $\propto\bar{x}$ term only, preserving the rigid-bunch mode for zero
lattice tune spread, as it must be from the first principles.

By averaging over the particles, writing the sums as the phase space integrals
with the distribution function, one gets the dispersion relation, i.e. the
equation for the sought-after eigenfrequency $\omega$,

$1=-\int d\Gamma\,J_{x}\frac{\partial f}{\partial
J_{x}}\frac{\omega^{\mathrm{c}}-\omega^{\mathrm{sc}}_{i}}{\omega-\omega^{\mathrm{sc}}_{i}-\delta\omega_{xi}-n\delta\omega_{i}+io}$
(4)

Here $f=f(J_{x},J_{y},\delta p/p)$ is the unperturbed distribution density as
a function of transverse actions $J_{x,y}$ and the relative momentum offset
$\delta p/p$, normalized to unity, $\int d\Gamma f=1$, where
$d\Gamma=dJ_{x}\,dJ_{y}\,d\delta p/p$; the single-particle subscript $i$ has
to be understood as indication to related functional dependences, i.e.
$\delta\omega_{xi}\rightarrow\delta\omega_{x}(J_{x},J_{y},\delta p/p)$, etc.
To get Eq. (4) from Eq. (3), the Hereward rule Hereward (1969) was used,

$\sum_{i}(...)\rightarrow-\int d\Gamma\,J_{x}\,\partial f/\partial
J_{x}\,(...)\,,$

and the Landau rule of going around the pole is explicitly marked,
$\omega\rightarrow\omega+io$, where $o$ is an infinitesimally small positive
number.

It is straightforward to see from the dispersion relation (4) that without
lattice frequency spreads, at $\delta\omega_{xi}=\delta\omega_{i}=0$, the
eigenfrequency $\omega=\omega^{\mathrm{c}}$, independently of the SC tune
shift. Thus, even if the phase space density of the resonant particles were
not zero, i.e. there were particles with the same tune as the coherent mode,
still there would be no Landau damping (LD), irrespectively to nonlinearity of
SC distribution.

The dispersion relation in the form (4) was first derived by D. Möhl and H.
Schönauer in 1974 Möhl and Schönauer (1975), not in the original preprint of
Merle and Möhl. In the latter, some mathematical mistakes were adopted, so the
dispersion relation was derived incorrectly. Due to this, it was mistakenly
concluded there that SC nonlinearities may contribute to Landau damping of
coasting beams even without the lattice frequency spread. This mistake was
later repeated in Ref. Reich and Schönauer (1972) and corrected by Möhl and
Schönauer Möhl and Schönauer (1975). That is why it seems fair to call Eq. (1)
Merle-Möhl equation of motion and Eq. (4) Möhl-Schönauer dispersion relation.

After the simplest case of no lattice frequency spreads, the next by
simplicity is a two-stream beam, $\delta\omega_{i}=\pm\delta\omega_{0}$. For
the Kapchinsky-Vladimirsky transverse distribution with a constant SC
frequency shift, $\omega^{\mathrm{sc}}_{i}=\omega^{\mathrm{sc}}$, Eq. (4)
yields the following spectrum:

$\omega=\frac{\omega^{\mathrm{c}}+\omega^{\mathrm{sc}}}{2}\pm\sqrt{\frac{(\omega^{\mathrm{c}}-\omega^{\mathrm{sc}})^{2}}{4}+n^{2}\delta\omega_{0}^{2}}$
(5)

The instability is driven by the coherent tune shift $\omega^{\mathrm{c}}$, so
the most unstable modes are positive ones, i.e. those associated with the sign
$+$ in Eq. (5). A more detailed analysis shows that for them the two streams
of the beam oscillate approximately in phase, so their wakes add to each
other. For strong SC,
$|\omega^{\mathrm{sc}}|\gg\max(|n|\delta\omega_{0},\,|\omega^{\mathrm{c}}|)$,
the spectrum of positive modes reduces to

$\omega=\omega^{\mathrm{c}}+n^{2}\delta\omega_{0}^{2}/\omega^{\mathrm{sc}}\,,$
(6)

being far away from the incoherent spectrum localized around
$\omega^{\mathrm{sc}}$. Thus, the negative modes not only are barely excited
by the wake, but also stay close to the incoherent spectrum, so their
stability would be provided automatically when the positive modes are stable.

A very general method of analysis of integral dispersion relations like Eq.
(4) was presented in Ref. Ruggiero and Vaccaro (1968). The idea was to reverse
the problem: instead of finding the eigenfrequency $\omega$ when the coherent
tune shift $\omega^{\mathrm{c}}$ is given, let us do the opposite, find the
coherent tune shift $\omega^{\mathrm{c}}$ for a given eigenfrequency $\omega$
for the same dispersion relation. If to run the eigenfrequency along the real
axis, the corresponding coherent tune shift will follow a certain line in its
complex plane, the conformal map of the real axis in the complex plane of
$\omega$ to the complex plane of $\omega^{\mathrm{c}}$. This line,
$\omega^{\mathrm{c}}(\omega)$, is conventionally called the stability diagram;
the beam is stable if and only if its actual coherent tune shift is below the
diagram.

Certain historical investigations Vac (2018) convinced the author of this
paper that it would be fair to call the stability diagram the Vaccaro diagram
(VD), by name of Vittorio Vaccaro, who found how Nyquist’s stability plots can
be modified to become an effective tool for the collective beam dynamics. With
Möhl-Schönauer dispersion relation (4), VD is determined by the beam
distribution function $f$, the lattice frequency shifts and SC tune shifts,
being independent of wakes.

In 2001, M. Blaskiewicz suggested an original method of solving the Jeans-
Vlasov equation with SC and lattice tune spread Blaskiewicz (2001); with
respect to the equation naming, see Ref. Hénon (1982). The method was free
from hidden assumptions of Merle-Möhl approach, being, arguably, more
complicated and less transparent in the computations. The method allowed to
make conclusions regarding SC effects for the stability diagram. In case of
the chromatic lattice tune spread, the diagram of a Gaussian beam essentially
shifted to the left by one half of the maximal SC tune shift. In case of the
octupole nonlinearity M. Blaskiewicz found that focusing octupoles are much
more beneficial for LD than defocusing ones, confirming the same conclusion by
D. Möhl Möhl (1995).

The first analytical attempt to build VD for Eq. (4) was presented in 2004 by
E. Metral and F. Ruggiero Metral and Ruggiero (2004). Namely, they suggested a
solution of the dispersion relation with SC and octupolar nonlinearity, where,
instead of the coasting beam term $n\delta\omega_{i}$, the bunched beam term
$k\omega_{\mathrm{s}}$ was put, with $\omega_{\mathrm{s}}$ as the synchrotron
frequency and $k$ as the head-tail mode number. Such extension of the coasting
beam theory to the bunched case was, however, left unexplained both in the
paper itself and the references it suggested for that matter, including Ref.
Möhl (1995). As it became more clear later, this extension works reasonably
well only when the SC term could be safely omitted. However, the actual merit
of the paper was not in its applicability to bunched beams with SC, but in its
analytical building of VD for coasting beams with octupoles, nonlinear SC, and
insignificant revolution frequency spread, $n\delta\omega_{i}=0$. It was
confirmed, in particular, that the octupole sign becomes crucial for strong
SC; namely, the focusing octupoles are much preferable. The reason is that the
octupoles affect mostly the tail particles, so the collective frequency is
barely touched by them. Landau damping requires resonant particles, i.e. those
which individual tunes are the same as the collective one. Space charge moves
the incoherent tunes down, and does almost nothing for the collective tunes,
thus killing LD. Thus, to restore the latter, the incoherent tunes have to be
moved up to provide higher population of the resonant particles, so the
octupoles have to be focusing. It was also shown in this reference that SC can
be beneficial if it shifts VD on top of the coherent tune, which would be
outside (on the left) of VD without space charge. For very strong SC, it meant
that it is detrimental since it shifts the stability diagram far on the left.

Among multiple reasonable features, Vaccaro diagrams of Ref. Metral and
Ruggiero (2004) showed a strange one: for defocusing octupoles, there was a
kink point of the curve at the real axis, which prevented the line from going
to the lower half-plane, $\Im\omega^{\mathrm{c}}<0$. The kink point looked
strange, since VD should be analytical by the definition.

In 2006, D. Pestrikov published an article Pestrikov (2006) where a similar
problem was solved, but instead of the kink point, the diagram smoothly
continued to the lower half-plain, thus demonstrating Landau antidamping.
Later that year Landau antidamping was confirmed by K.Y. Ng for the same model
as Metral and Ruggiero proposed Metral (2006). On the ground of these
findings, the kink point of Ref. Metral and Ruggiero (2004) was dismissed as a
mistake of the sign. Due to this, however, another problem appeared: at
certain conditions, a Gaussian-like beam with time-independent Hamiltonian
started looking unstable even when the coherent tune shift
$\omega^{\mathrm{c}}$ suggested a decay of the mode. Next year Pestrikov
published another paper Pestrikov (2007), presenting “a self-consistent model”
which showed no antidamping, contrary to the Merle-Möhl model; he expressed
doubt in the validity of the latter.

This doubt was enhanced to a stronger claim by V. Kornilov, O. Boine-
Frankenheim, and I. Hofmann in their publication of 2008 Kornilov  _et al._
(2008). First, they confirmed that VD of Eq. (4) indeed yields Landau
antidamping for defocusing octupoles. Second, they supported this confirmation
by macroparticle simulations within the frozen field model, equivalent to
Merle-Möhl approach. Third, they ran self-consistent macroparticle simulations
for the same conditions, and saw no antidamping. From this, they concluded
that “antidamping can be related to the non-self-consistent treatment of
nonlinear space charge in the simulations and also in the dispersion
relation.”

At that stage, several issues remained unresolved for coasting beams. First,
it was not clear if Landau antidamping is ever possible for Gaussian-like
beams with SC, octupoles and chromaticity. Second, with evidence of
incorrectness of Merle-Möhl analytical approach at certain cases, it was not
clear if their equation could be ever used at all, and under what conditions.
Third, no analytical formulas for the instability thresholds were yet
obtained. These issues were addressed in Ref. Burov and Lebedev (2009).

A possibility of Landau antidamping was denied there as contradicting to the
Second Law of Thermodynamics. Indeed, a beam with real coherent tune shift
$\omega^{\mathrm{c}}$, corresponding to imaginary transverse impedance, i.e.
to zero energy losses, can be described by energy-preserving time-independent
Hamiltonian, so the growing coherent oscillations might take energy from the
incoherent degrees of freedom only. For a Gaussian beam it would mean a
perpetuum mobile of the second kind, forbidden by the Second Law. Landau
antidamping, demonstrated for some parameters by Merle-Möhl dynamic system
(1), is caused by the non-Hamiltonian character of its SC term. Specifically,
the term $\propto\omega^{\mathrm{sc}}_{i}\bar{x}$ is non-Hamiltonian unless
all the SC frequency shifts are identical within the beam slice. Having said
that, it is important to stress that the Merle-Möhl equation of motion with
real coherent tune shift, $\Im\omega^{\mathrm{c}}=0$, may lead to Landau
antidamping only if the incoherent spectrum
$\omega^{\mathrm{sc}}_{i}+\delta\omega_{xi}$ reaches a local maximum in the
action space, which may happen for a defocusing octupole. Although the
equation is not Hamiltonian, for monotonic spectra
$\omega^{\mathrm{sc}}_{i}+\delta\omega_{xi}$ all its van Kampen
eigenfrequencies with real coherent tune shift are real as well, no unphysical
dissipation is introduced.

How reliable is Eq. (1) for LD computation for the monotonic spectra? When the
SC tune shifts depend on the transverse actions, as they normally are, the
defect of the model still should not play a role, if the slices were
sufficiently rigid in their transverse oscillations. In this case only the
tail particles would be responsible for LD, so the energy transfer to them
could be reasonably approximated with the rigid core model. To see when the
core is really rigid, note that if the lattice tune shifts are small with
respect to the tune separation,

$|\delta\omega_{xi}+n\delta\omega_{i}|\ll|\omega^{\mathrm{c}}-\omega^{\mathrm{sc}}_{i}|\,,$
(7)

the particles move together with the related centroids, $x_{i}\approx\bar{x}$,
as it follows from Eq. (3). Thus, if the SC is so strong that this condition
is satisfied for the majority of particles, the slices oscillate almost
without distortions, since almost all the particles oscillate almost
identically to their centroids; so the rigid-slice approximation is justified.
Luckily, for many low- and medium-energy machines, typical SC tune shifts are
much larger than the imaginary part of the coherent tune shifts,
$|\omega^{\mathrm{sc}}_{i}|\gg\Im\omega^{\mathrm{c}}$, so stabilization is
achieved at such a small lattice nonlinearity that Eq. (7) is satisfied,
justifying Merle-Möhl equation. In this case of strong SC, instability
thresholds were explicitly found in Ref. Burov and Lebedev (2009) for Gaussian
beam, both for octupolar and chromatic frequency spreads. Recently, this
method was extended to electron lenses; Landau damping rate introduced by a
Gaussian e-lens for a coasting beam with SC was analytically estimated and
presented in Ref. Lebedev  _et al._ (2021).

###  II.2  Bunched Beams

The coherent spectrum of bunched beam with SC was presented for the first time
by M. Blaskiewicz in 1998 Blaskiewicz (1998) within a simple model of an air-
bag bunch in a square potential well. For a delta-wake, the eigenfrequencies
were found to be the same as Eq. (5) for the two-stream coasting beam, with
the substitution $n\delta\omega_{0}\rightarrow k\omega_{\mathrm{s}}$, where
$k=0,1,2,...$ is the mode counter and $\omega_{\mathrm{s}}$ is the synchrotron
frequency. A new and rather surprising mathematical result of M. Blaskiewicz
Blaskiewicz (1998) showed suppression of the transverse mode coupling
instability (TMCI) by SC; the wake threshold was demonstrated to grow with SC
tune shift, linearly at the strong SC limit,
$|\omega^{\mathrm{sc}}|\gg\omega_{\mathrm{s}}$. This result was obtained for
exponential wakes and the ABS model (Air-Bag, Square-well), so a question was
raised about the sensitivity of this unexpected result to the details of the
wake, potential well and bunch distribution. Also, it was not clear if there
was any limit to this growth of the instability threshold. An explanation of
this growth at moderate SC was suggested to the author by V. Danilov Danilov
(1998) and reproduced in Ref. Ng and Burov (1999). Without SC, TMCI typically
results from crossing of the head-tail mode $0$, shifted down by the wake, and
the mode $-1$, not shifted as much. Space charge, on the contrary, does not
influence the mode $0$ and shifts down the mode $-1$, thus moving their
coupling point to higher intensity.

In the year of 2009, when it was understood that Merle-Möhl approach of rigid
slices is justified for sufficiently strong space charge, it was applied to
bunched beams by the author Burov (2009a). Under the condition of SC tune
shift being much stronger than all other tune shifts and spreads, as well as
the synchrotron tune (strong space charge, SSC), an ordinary linear integro-
differential equation was derived for the bunch modes for an arbitrary
potential well, driving and detuning wakes, longitudinal and transverse bunch
distribution functions. Later that same year V. Balbekov published a paper
Balbekov (2009) with an alternative derivation of the SSC equation, whose
result differed from mine. After checking his derivation and rechecking mine,
I found an algebraic error in my calculations, and derived my ultimate form of
the SSC mode equation, which agreed with Balbekov’s result, suggesting a
slightly more compact form in the erratum Burov (2009b),

$i\frac{\partial\bar{x}}{\partial
t}+\frac{1}{\omega^{\mathrm{sc}}}\frac{\partial}{\partial
s}\left(u^{2}\frac{\partial\bar{x}}{\partial
s}\right)=\mathbb{W}\bar{x}+\mathbb{D}\bar{x}\,.$ (8)

Here $\omega^{\mathrm{sc}}=\omega^{\mathrm{sc}}(s)$ is the SC frequency shift
averaged over the transverse actions at every position $s$, $u^{2}=u^{2}(s)$
is the local rms spread of the longitudinal velocities,
$u^{2}\equiv\left<R^{2}\delta\omega_{i}^{2}\right>$, while $\mathbb{W}$ and
$\mathbb{D}$ are conventional driving and detuning wake linear integral
operators Burov and Danilov (1999); for more details see Burov (2009b). The
equation is complemented by zero-derivative boundary conditions,
$\partial\bar{x}/\partial s=0$ at the bunch edges or at $s=\pm\infty$. For the
eigenfunctions, the time derivative has to be substituted by the sought-for
eigenfrequency $\nu$, i.e. $i\partial/\partial t\rightarrow\nu$. Without
wakes, this equation leads to the Blaskiewicz-type collective spectrum,
$\nu_{k}\simeq k^{2}\omega_{\mathrm{s}}^{2}/\omega^{\mathrm{sc}}$. The
mathematical elegance of Eq. (8) has its price: missing is the Landau damping,
which required additional ideas and computations.

Analytical estimations for LD at SSC were also suggested in Refs. Burov
(2009a, b) for weak head-tail cases, when the wake does not influence the
eigenfunction much. Contrary to coasting beams, it was found that there is an
intrinsic LD, caused by the longitudinal variation of the SC tune shift only,
even without any lattice tune spreads. The physical mechanism of the
dissipation was associated with a break of the slice rigidity at the bunch
edges, where the SC is not strong any more. The slice softening at the bunch
edges opens a way for the energy transfer to the incoherent degrees of
freedom. According to the related estimation, the intrinsic LD rate
$\Lambda_{k}$ at SSC was found to be a steep function of the SC parameter
$q\equiv\omega^{\mathrm{sc}}/\omega_{\mathrm{s}}$ and the positive mode number
$k$, $\Lambda_{k}\simeq k^{4}\omega_{\mathrm{s}}/q^{3}$; the SSC assumes $q\gg
2k$. Six years later these analytical results for SSC eigenfunctions and LD
rates were fully confirmed in Synergia macroparticle simulations by A.
Macridin et al. Macridin  _et al._ (2015), where the intrinsic LD rates were
shown to have their maxima at $q\simeq 2k$. A more subtle case of parametric
Landau damping was treated by A. Macridin et al. in Ref. Macridin  _et al._
(2018) by means of analytical modeling and macroparticle simulations.
Analytical estimations of octupoles-related LD suggested in Refs. Burov
(2009a, b) are still waiting for at least numerical verifications; nothing yet
has been published in that matter. Octupoles, however, are rather inefficient
for LD, which requires significant nonlinearity inside the beam, not far
outside, as octupoles provide. That is why a better instrument for LD is an
electron lens, at least as thin as the beam. Such e-lenses are able to provide
LD without deterioration of the dynamic aperture, as it was pointed out by V.
Shiltsev et al. Shiltsev  _et al._ (2017). Estimations of e-lens-caused LD
rates for bunches with SC were suggested by Y. Alexahin, A. Burov and V.
Shiltsev in 2017 Alexahin  _et al._ (2017).

With the wake taken into account, the Blaskiewicz’ result of linear growth of
the TMCI wake threshold was confirmed in a series of publications, see Refs.
Zolkin  _et al._ (2018); Balbekov (2019) and references therein. A hidden
obstacle with this problem, sometimes caused misleading results, was realized
by V. Balbekov Balbekov (2017), who showed that convergence of the expansion
of the sought-for eigenfunction over the zero-wake basis degrades with SC,
requiring more and more terms for higher SC parameter $q$. The physical reason
of the convergence worsening was recently found by the author of this paper;
it is associated with the head-to-tail amplification, or the convective
instabilities driven by wakes at SSC. When eigenfunctions are significantly
amplified, their expansion over any even basis cannot be of a good
convergence. Clearly manifest subsiding of the instability with SC was
presented in the two-particle model of Ref. Chin  _et al._ (2016).

With the theoretical proof of TMCI vanishing with SC, two problems became
rather obvious, one experimental and the other theoretical. The former
consisted in a reasonable agreement of the transverse instability at CERN SPS
with no-SC theory, while SC tune shift was very strong there, especially with
the old lattice Bartosik (0 23); Metral . The latter problem was related to
the linear growth of the wake threshold with SC. Due to this feature, the
bunch should be stable up to infinite intensity, as soon as its emittance is
low enough, which did not sound as a reasonable statement. The resolution of
both problems was presented by the author few years ago Burov (2019a). The
main idea, already mentioned above, was that, while moving out TMCI, SC moves
another instability in its place, a convective one. Contrary to TMCI, which is
an absolute instability, i.e. has nonzero growth rate, the convective
instabilities grow not in time, but in space, from head to tail Lifshitz and
Pitaevskii (1981). This head-to-tail amplification increases exponentially
with bunch intensity, resulting in one or another physical limit, set by
lattice nonlinearity, beam loss or feedback. When the amplification is large,
even a tiny feedback from tail to head may be sufficient to close the loop and
turn the convective instability into an absolute-convective one, like those
with a microphone close to its loudspeaker. Such a feedback may be presented
with a bunch-by-bunch damper, coupled-bunch wakes, or a halo of the same
bunch. Here a question may be asked, why is the halo needed for the feedback?
Why cannot core particles play this role, when they move to the bunch head
with their high transverse amplitudes acquired at the tail? The answer is that
due to strong SC, the bunch slices are rigid, as it was discussed in the
previous section. Strong SC means that all tune shifts are small compared with
the SC tune shift, so intra-slice degrees of freedom cannot be excited, and
thus the tail particles do not preserve their large amplitudes while moving to
the bunch head; instead, they just follow the existing spacial pattern of the
rigid-slice oscillations.

However, what is impossible for the bunch’s core, might work for its halo,
whose SC tune shift is smaller. That is why the halo particles traveling from
the tail to the head may remember their tail offsets, providing a tail-to-head
feedback. At strong SC, this feedback would be small due to the low population
of that halo, but, if the convective amplification is large enough, even a
small feedback could be sufficient to ignite the absolute-convective
instability, as it was suggested and modeled in Ref. Burov (2018a).
Apparently, the same effect is responsible for the non-monotonic behavior of
the wake instability threshold on the SC parameter reported by Y. Alexahin
Alexahin . A good agreement of his analytically calculated highly convective
mode with the pattern of oscillations seen by A. Oeftiger in macroparticle
simulations for the same conditions deserves to be mentioned. Alas, it was the
last lifetime publication of Yuri, who suddenly passed away in a year after
that Shiltsev and Valishev (2020).

In a recent Ref. Buffat  _et al._ (2021), the joint action of wake and SC
forces was examined by means of a numerical circulant matrix model (CMM)
applied to a bunch represented by several thin rings, from one to fifty, in
the longitudinal phase space with a parabolic potential well; no transverse
variation of the SC tune shifts was assumed. For a single ring, i.e. for the
conventional air-bag, all the conclusions of Ref. Burov (2019a) with its ABS
model were essentially confirmed. For many rings, a dense sequence of
couplings-decouplings of the amplified radial modes were seen for the case of
Gaussian longitudinal distribution, well below the air-bag TMCI threshold at
the given strong SC. The authors note that ”the coupling of the radial modes
seems in contradiction with the results in Balbekov (2019). Nevertheless, the
model used in the latter was obtained with the strong space charge
approximation, which is not met at the edge of a longitudinal Gaussian
distribution. Since the CMM is not based on such an approximation, the
difference can likely be attributed to that.” In other words, this
contradiction suggests that the longitudinal halo is an important part of the
absolute-convective instability observed. The reason for this longitudinal
core-halo instability could be in wider frequency range of the forces the halo
particles see when they pass through the oscillating tail of the core, and
this might be sufficient for them to bring the tail signal to the head. More
detailed studies with this model are certainly needed for definite
conclusions.

It is clear now, that convective instabilities constitute a common obstacle
for high intensity circular machines of low and medium energy, where SC is
significant; they definitely take place at CERN Booster, PS and SPS rings, as
well as at the Fermilab Booster. That is why it is important to understand how
they behave together with other factors of beam dynamics. Transverse
instabilities of a bunch with SC, wake and damper were considered in Ref.
Burov (2018b). In Ref. Kornilov  _et al._ (2013), measurements of a microwave
instability at transition crossing in PS were reported; the instability was
characterized as convective. Recently an analytical model for it was proposed
Burov (2019b) by means of Eq. (8). A simple threshold formula derived there
was found to be in good agreement with the data of Refs. Kornilov  _et al._
(2013); Migliorati  _et al._ (2018). A statement made in Ref. Kornilov  _et
al._ (2013) that “The bunch parameter measurements demonstrate… that the
space-charge effect does not affect the instability thresholds” does not
actually contradict to the rather weak dependence of the threshold bunch
intensity $N_{\mathrm{th}}$ on the transverse emittance,
$N_{\mathrm{th}}\propto\epsilon_{\perp}^{1/4}$ of Ref. Burov (2019b), since
the limited range of the emittances examined in Ref. Kornilov  _et al._ (2013)
and the measurement errors do not allow to resolve the rather weak dependence
on the emittance on the ground of this set of measurements alone Kornilov
(2020).

##  III  Longitudinal Oscillations

In this review of the SC effects in the linear longitudinal collective
oscillations, we limit ourselves by bunched beams, since the coasting beam
theory seems to have been completed long ago Chao (1993); Ng (2006). The only
relatively new contribution to this theory, as far as I know, relates to the
modification of the negative mass instability by the SC shift of the slippage
factor, reported in Ref. Pozdeyev  _et al._ (2009).

Description of bunch dynamics generally requires solving of two consecutive
problems: the bunch steady state has to be found, and, after that, dynamics of
its small perturbations is to be analyzed. The phase space density $F(I)$, the
wake function $W(z)$, and the RF potential $U_{\mathrm{rf}}(z)$ have to be
provided as input functions, where $I$ and $z$ are the action variable and the
longitudinal position along the bunch. The steady state Hamiltonian $H(z,p)$,
with $p$ as the momentum variable associated with the coordinate $z$, full
potential $U(z)$, action $I(H)$ and line density $\lambda(z)$ have to be found
then as solutions of self-consistent equations Burov (2010). From that, the
incoherent frequency spectrum $\Omega(I)$ can be obtained as
$\Omega(I)=\mathrm{d}H/\mathrm{d}I$. For SC or inductive impedances, the
steady-state is determined by the distribution function $F(I)$ and the
dimensionless intensity parameter

$k=-\frac{2Nr_{0}\eta\,\omega_{\mathrm{rf}}^{3}}{\gamma\,c\,\Omega_{0}^{2}}\,\frac{\Im
Z_{n}}{n\,Z_{0}}\,.$ (9)

Here $N$ is the number of protons per bunch; $r_{0}$ is the classical radius
of the beam particles; $c$ is the speed of light; $\gamma$ is the Lorentz
factor; $\eta$ is the slippage factor, $\omega_{\mathrm{rf}}$ is the RF
frequency, $\Omega_{0}$ is the bare RF synchrotron frequncy, $Z_{n}$ is the
impedance at the azimuthal harmonic number $n$; for a pure SC case,
$Z_{n}/(nZ_{0})=i\ln(b/a)/\gamma^{2}$, where $b$ and $a$ are the aperture and
the beam radii (corrections to the logarithmic factor are discussed in Ref.
Baartman (2015)). In particular, for a Gaussian bunch, the incoherent spectrum
for the core particles at the first order of the intensity parameter $k$ is
found as

$\frac{\Omega(I)}{\Omega_{0}}=1-\frac{k}{2\sqrt{2\pi}\sigma^{3}}-\frac{I}{8}\left(1-\frac{3k}{\sqrt{2\pi}\sigma^{5}}\right)\,,$
(10)

where the bunch length $\sigma$ is measured in the RF radians, and the
dimensionless units for the action $I$ are such that its maximal value in the
undistorted RF bucket is $8/\pi$.

When the steady-state problem is solved, the collective dynamics of this state
can be analyzed. Usually it is done by means of linearized Jeans-Vlasov
equation; the collective modes are found as its eigenfunctions. The main
concern here relates to possible instabilities, when at least one of the
eigenfrequencies has a positive imaginary part. However, one should keep in
mind a possibility of convective instabilities with significant amplification
along the bunch. Even if the growth rate is zero, large amplification may lead
to beam loss or emittance degradation. A possibility of longitudinal
convective instabilities for a bunched beam was demonstrated in Ref. Boine-
Frankenheim and Hofmann (2003) for a barrier bucket and when both SC and
resonator impedances are large enough.

At sufficiently low beam intensity, all the collective frequencies normally
supposed to lie within the incoherent spectrum, thus providing their damping
by means of the Landau mechanism. As the intensity increases, some collective
modes move outside the incoherent spectrum. Thus, at certain intensity, Landau
damping becomes either insufficient or totally lost. The latter happens if the
coherent tune is shifted by the collective interaction so far that it cannot
meet resonant particles at all there; this case is termed loss of Landau
damping, LLD. At the rest of this review, we concentrate on this issue,
because of its high experimental importance and a recent breakthrough in this
direction. Undamped collective longitudinal oscillations, interpreted as LLD-
caused, were observed at RHIC Blaskiewicz  _et al._ (2004), Tevatron Moore
_et al._ (2003), SPS Shaposhnikova (2006) and LHC Shaposhnikova  _et al._
(2011). A special role in such phenomena is played by repulsive inductance
above transition energy or space charge (SC) Coulomb forces below it. Being
conservative, such forces cannot drive instabilities by themselves. However,
they can move the coherent frequency outside the incoherent spectrum, above
the maximal incoherent frequency, in which case even a tiny wake field of the
preceding bunches causes an instability. If to liken Landau damping to the
beam immune system, the SC force below transition would play a role of an
immunodeficiency factor, while the wake fields of other bunches would be
similar to all possible viruses. In this situation, the wall inductance plays
the role of a guard of Landau damping, since it behave in same way as SC, but
with the opposite sign; such a guard is not necessarily sufficient, of course.
Above the transition, SC and the chamber inductance switch their roles: the SC
becomes the guard of Landau damping, and the inductance?the thief. Since SC
normally dominates below transition, and sufficiently above it dominates the
chamber inductance, the thief typically overcomes the guard. In this review,
we limit ourselves by this situation of the repulsive inductance, where the
force is proportional to the line density derivative, taken with the negative
sign. For the vacuum chamber, this law is effective with wave numbers no
higher than the inverse aperture; for hadron beams it is typically one or two
orders of magnitude above the inverse bunch length. The SC impedance starts to
roll off $\gamma$ times further than that.

The possibility of LLD for a bunch in the inductive vacuum chamber above
transition was first noted by F. Sacherer Sacherer (1973); he evaluated the
threshold number of particles per bunch, $N_{\mathrm{th}}\propto\sigma^{5}$.
Later this result was essentially confirmed, up to numerical factors, by many
authors Hofmann and Pedersen (1979); Balbekov and Ivanov (1991); Boine-
Frankenheim and Shukla (2005); Zimmermann and Santiago Gonzalez (2008); Burov
(2011). Some discrepancies between these results were explained by different
distributions and model imperfections. The problem of the LLD threshold
calculation looked to be essentially solved until a recent article of I.
Karpov, T. Argyropoulos and E. Shaposhnikova Karpov  _et al._ (2021)
demonstrated that all the previous results were, in fact, incorrect: actually,
there is no LLD threshold for such impedance, and the previous claims were all
based in the insufficiency of the accepted limits on wave numbers $q$ of the
perturbations or insufficient number of the mesh points. This conclusion was
demonstrated in several independent ways, leaving no doubt of its correctness.
It was also shown that if the inductance $iZ(q)/q$ rolls off at certain wave
number $q=q_{c}$, the LLD threshold would be inversely proportional to that
value, $N_{\mathrm{th}}\propto\sigma^{4}/q_{c}$. When the intensity increased,
emergence of a second mode of the discrete spectrum was demonstrated. The
results of this breakthrough article were soon explained and generalized in
Ref Burov (2021). Below we reproduce this explanation.

Qualitatively, the leading collective mode can be approximated as oscillations
of a central part of size $a$ with a small amplitude $\tilde{z}\ll a$. Such
oscillations result in the phase space density perturbation $f\simeq
F^{\prime}\,a\,\tilde{z}$ and the line density perturbation $\rho\simeq
f\,a\simeq F^{\prime}a^{2}\,\tilde{z}$, where
$F^{\prime}=\mathrm{d}F/\mathrm{d}I$ at zero action, $I=0$. For the inductive
impedance, the related collective force is $E\simeq k\rho/a\simeq
kF^{\prime}a\tilde{z}$. For the case under study, $k>0$, this corresponds to
an additional focusing seen by the collective mode, taking it above the
incoherent spectrum. To avoid confusion, let us note that at the same time the
incoherent frequencies are depressed by this wake. The related coherent tune
shift is thus estimated as $\Delta\omega\simeq-kF^{\prime}\,a/2$, in the units
of $\Omega_{0}$. For the same central cluster of particles, the incoherent
tune spread is $\delta\Omega\simeq\pm|\Omega^{\prime}|a^{2}/4$, where
$\Omega^{\prime}=\mathrm{d}\Omega/\mathrm{d}I$ at $I=0$, in the total
potential well. The coherent motion dominates over the incoherent as soon as
$\Delta\omega\geq\delta\Omega$, or

$a\leq 2k|F^{\prime}/\Omega^{\prime}|\equiv\alpha\,.$ (11)

Thus, all the perturbations shorter than $\alpha$ are of relatively small
incoherent tune spread; in other words, there is no Landau damping for them.
The maximal coherent tune shift for them is $\Delta\omega\simeq-
kF^{\prime}\alpha/2\simeq k^{2}F^{\prime 2}/|\Omega^{\prime}|$. The suggested
estimates assume a weak space charge, when the mode size is small,
$\alpha<\sigma$, leading to

$k\leq 0.2\,\sigma^{5}\,.$ (12)

Note that under this assumption the relative depression of the tune derivative
$|\mathrm{d}\Omega/\mathrm{d}I|$ (10) is not large.

The weak SC condition (12) can be compared with the condition of separated
multipoles, or weak headtail condition, requiring for the relative tune shift
to be small. Thanks to Eq. (10), the latter can be presented as

$k\ll 2\sqrt{2\pi}\,\sigma^{3}\,.$ (13)

Thus, we come to a conclusion that there are four SC regimes:

  * •

insignificant, $\alpha\leq q_{c}^{-1}$ or $k<(\pi/8)\sigma^{4}q_{c}^{-1}$;

  * •

weak, $(\pi/8)\sigma^{4}q_{c}^{-1}\leq k\leq 0.2\,\sigma^{5}$;

  * •

medium, $0.2\,\sigma^{5}<k\ll 2\sqrt{2\pi}\,\sigma^{3}$;

  * •

and strong, $k\simeq 2\sqrt{2\pi}\,\sigma^{3}$.

For the first of them, the one with insignificant SC, all the modes are Landau
damped due to the impedance roll-off or due to the intrabeam scattering at
wave numbers $q>q_{c}$. For the second, with weak SC, there is at least one
discrete undamped mode, associated with oscillations of the relatively small
central portion of the bunch, with the size $a\simeq\alpha$. This size is
determined by the equilibrium between SC tune shift and nonlinearity of the RF
force. For the third regime, the medium one, the RF nonlinearity already does
not play a role; all bunch particles are effectively involved into the
undamped collective oscillations, but the synchrotron multipoles are still
well-separated, the coherent and incoherent tune shifts are relatively small,
and the bunch length is mostly determined by the given emittance and RF
potential. For the fourth regime, the one with the strong SC, the potential
well is significantly flattened by the SC forces, and the bunch length is
determined by this condition, $\sigma\simeq k^{1/3}$ Burov (1986); Nagaitsev
_et al._ (1997). For all the regimes, except one of the insignificant SC, the
Landau damping is lost, and even a tiny coupled-bunch (CB) wake may drive an
instability.

The demonstrated back-of-the-envelope estimations essentially explain the
findings of Ref. Karpov  _et al._ (2021) that there is no LLD threshold for
the pure inductive repulsive impedance, being in agreement with the leading
mode character seen in that reference. It is also instructive to note that for
the attractive wake, $k<0$, the collective modes are shifted down with respect
to the incoherent ones. Thus, LLD can happen only for the modes mostly
associated with the high-amplitude particles, not the central ones. At high
amplitudes, however, the derivative of the phase space density $F^{\prime}(I)$
normally tends to zero; thus, the nonzero LLD threshold has to be expected
there, which also agrees with the analysis of Ref. Karpov  _et al._ (2021).

It is worth noting that the method of estimations described in this section
can be applied to any impedance. To show that, let us assume a repulsive
impedance $Z(q)=\zeta(-iq)^{\kappa}$, with some constant parameters $\zeta$
and $\kappa$. With that, the intensity parameter $k$ is modified by a
substitution $-\Im Z/q\rightarrow\zeta$, yielding the coherent tune shift
$\Delta\omega\simeq-kF^{\prime}\,a^{2-\kappa}/2$. Compared with the incoherent
tune spread $\delta\Omega=\pm|\Omega^{\prime}|a^{2}/4$, it leads to a
conclusion that at $\kappa>0$ all the central perturbations with
$a\leq|2kF^{\prime}/\Omega^{\prime}|^{1/\kappa}$ are not damped. Thus, for all
such impedances with $\kappa>0$ the spectral properties should be
qualitatively the same as for the inductive impedance. In particular, they all
must correspond to zero LLD threshold above transition. Note that the
resistive wall impedance belongs to this class; it is a case of $\kappa=1/2$.

For the SC impedance below transition, a special interest is presented by the
weak SC, in terms of the suggested classification, since it relates to lowest
intensities where the stability problems may already appear. In this case,
Jeans-Vlasov dynamic equation reduces, after a proper scaling transformation,
to a parameter-less Hermitian integral equation for the eigenfunctions and
eigenvalues. The eigenvalues which lie above the upper bound of the incoherent
spectrum, constitute a discrete spectrum, relating to the modes without Landau
damping. The tune of the leading mode sits at
$\Delta\omega=0.22|\Omega^{\prime}|\alpha^{2}$ above the maximal incoherent
frequency $\Omega(0)$. Formally speaking, the number of such modes is
infinite; the eigenvalues have their limit point at $\Omega(0)$. When the
impedance roll-ff or intra-beam scattering Blaskiewicz  _et al._ (2004) is
taken into account, it determines the highest-frequency discrete mode. In
numerical computations, the number of discrete modes is also limited by the
mesh size.

In case the inter-bunch wakes are smooth within a single bunch, the coupled-
bunch growth rates can be either analytically expressed or found by means of
the stability diagram Sacherer (1973), as it is shown in Ref. Burov (2021).
Also demonstrated there is how all these results are generalized for arbitrary
multipolarity of the single-bunch modes.

###### Acknowledgements.

I express my deep gratitude to the memory of my long-term friend and wise
colleague Yuri Alexahin (1948-2020), discussions with whom shed light on many
issues mentioned here. Fermilab is operated by Fermi Research Alliance, LLC
under Contract No. DE-AC02-07CH11359 with the United States Department of
Energy.

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# Magnetization Reversal of 50-nm-wide Ni81Fe19 Nanostripes by Ultrashort
Magnons in Yttrium Iron Garnet for Memory-Enhanced Magnonic Circuits

Shreyas S. Joglekar  0000-0001-9354-2666 Korbinian Baumgaertl
0000-0002-9410-1931 Andrea Mucchietto  0000-0001-5562-3916 Francis Berger
Dirk Grundler  0000-0002-4966-9712<EMAIL_ADDRESS>[

###### Abstract

Spin waves (magnons) can enable wave-based neuromorphic computing by which one
aims at overcoming limitations inherent to conventional electronics and the
von Neumann architecture. In this study, we explore the storage of magnon
signals and the magnetization switching of periodic and aperiodic arrays of
Ni81Fe19 (Py) nanostripes with widths $w$ between 50 nm and 200 nm. Spin waves
excited with low microwave power in yttrium iron garnet induce the reversal of
the nanostripes of different $w$ in a small opposing field. Exploiting
microwave-to-magnon transducers for magnon modes with ultrashort wavelengths
$\lambda$, we demonstrate the reversal of 50-nm-wide Py nanostripes by magnons
with $\lambda\approx 100~{}$nm after they have propagated over 25 $\mu$m in
YIG. The findings are important for designing a magnon-based in-memory
computing device.

Equal contribution Equal contribution Ecole Polytechnique Fédérale de Lausanne
(EPFL)] Laboratory of Nanoscale Magnetic Materials and Magnonics, Institute of
Materials (IMX), Ecole Polytechnique Fédérale de Lausanne (EPFL), 1015
Lausanne, Switzerland [Second University] Institute of Electrical and Micro
Engineering (IEM), EPFL, 1015 Lausanne, Switzerland

##  1 Keywords

magnons, ferromagnet, ferrimagnet, magnetization reversal, microwaves

##  2 Introduction

The use of various Artificial Intelligence platforms like Dall-E2 1, ChatGPT 2
has skyrocketed in recent months. Such platforms use Machine Learning
algorithms to generate a tremendous amount of data in computers’ processors
and store it in separate memory units leading to enormous data trafficking,
reduced computing performance and Joule heating losses. These drawbacks of the
conventional technology motivate the search for alternatives such as in-memory
and neuromorphic computing 3. These alternatives require two essential
components: nonlinearity in signal processing and nonvolatile memory.

Recently, magnonic neural networks have been proposed which exploit wave-based
signal processing in a ferrimagnetic thin film combined with an array of
nanostructured ferromagnets. Exploiting propagating spin waves (magnons) in
yttrium iron garnet (YIG) and their interference underneath nanomagnets,
neural network functionality was demonstrated in that the magnetic states of
bistable nanomagnets were iteratively modified and programmed 4, 5, 6. The
spin waves are collective excitations of spins that transfer angular momentum
in a magnetically ordered material. In a low-damping material like YIG they
can propagate over long distances up to the mm length scale 7. They possess
engineered wavelengths down to a few tens of nanometers when excited on-chip
by microwaves at GHz frequencies 8. The interaction with distributed magnets
controls their scattering, phases and interference 5, 9. A similar hybrid
system gives rise to the magnonic holographic memory (MHM) in which coherent
spin waves read out stored data via interference over several unit cells
(magnetic bits) 10. However, for both the magnonic neural network and MHM, it
is not yet clear how to reprogram the magnets and encode the data. Beyond
applying global magnetic 11 or electrical fields 12, local laser heating 13 or
the tip of a magnetic force microscope 14 might be used to reverse
nanomagnets, but magnetization reversal induced by spin waves avoids
additional equipment and infrastructure.

Magnetization switching attributed to magnons in antiferromagnetic NiO was
reported by Wang et al. 15 and Guo et al. 16. The type and wavelengths of spin
waves leading to magnet reversal were not explored however. Baumgaertl et al.
17 demonstrated that spin waves propagating in YIG induced magnetization
reversal of 100 nm wide and 25 to 27 $\mu$m long Permalloy (Ni81Fe19 or Py)
nanostripes in a small opposing field $H$. They were separated from the spin-
wave emitting coplanar waveguide (CPW) by a distance of $\geq 25~{}\mu$m. The
relevant mode had a wavelength $\lambda$ of about 7 $\mu$m. It is now timely
to investigate the magnon-induced reversal of nanostripes of different widths
and spin waves of shorter wavelengths.

In this work we report magnon-induced reversal by propagating spin waves with
$\lambda$ down to 100 nm. Nanostripe widths are varied from 50 to 200 nm.
Going beyond the earlier experiments, we explore the reversal in periodic and
aperiodic arrays of nanostripes of different widths, thereby avoiding a
commensurability effect between $\lambda$ and period $a$. Spin waves with
$\lambda$ close to 100 nm reverse nanostripes at a precessional power $P_{\rm
prec}$ of 10.8 nW in a small opposing field. The value is of the same order as
the one found for long-wavelength spin waves of $\lambda=7~{}\mu$m explored
under similar conditions in Ref. 17. Our findings are promising for enhanced
functionalities of magnonic neural networks and MHMs operating on the
nanoscale.

##  3 Results and discussion

###  3.1 Magnon mode dependent switching of 50-nm-wide nanostripes

Figure 1(a) illustrates the experiment. Arrays of Py nanostripes (gratings)
were fabricated on 100-nm-thick YIG grown on a Gadolinium Gallium Garnet GGG
(111) substrate. In periodic gratings the 20-nm-thick stripes exhibited widths
$w$ of either 50, 100 or 200 nm. The periods amounted to $a=2w$. The sample D1
with stripes of width $w=50~{}$nm is shown in Fig. 1 (b) and sketched in (c).
The period $a$ was 100 nm. In addition, we prepared and investigated a sample
containing an aperiodic grating. Stimulated by Ref. 18, we arranged
nanostripes on a Fibonacci sequence. Thereby the stripe positions were not
commensurate with a propagating magnon wavelength. CPWs were fabricated above
the nanostripes at a signal-to-signal line (edge-to-edge) distance of 35
$\mu$m (25 $\mu$m). Spin waves were excited at the emitter CPW (CPW1) with
port 1 of a Vector Network Analyzer (VNA) and their propagation was analyzed
at the detector CPW (CPW 2) using VNA port 2. An in-plane magnetic field
$\mu_{0}$H was applied using current-controlled solenoids.

Figure 1: (a) Sketch of the experiment involving Py nanostripes fabricated on
100 nm-thin YIG (111) beneath CPWs separated by a signal-to-signal line
distance of 35 $\mu$m. (b) Scanning Electron Micrograph of sample D1 showing
the ground (G) and signal (S) lines of a CPW covering the Py nanostripes and
the YIG film. The lengths of nanowires alternated between 25 and 27 $\mu$m
consistent with Ref. 17. (c) Illustration of one of the nanostripe arrays
(gratings) used in sample D1 with 50-nm-wide stripes and period $a=100~{}$nm.
Scattering parameter Magnitude of $\Delta S21$ measured in transmission
configuration between CPW1 and CPW2 using (d) $P_{\rm irr}=-25~{}$dBm, (e)
$-15~{}$dBm and (f) $-5~{}$dBm. We extract fields $H_{\rm C1}$ and $H_{\rm
C2}$ (indicated by vertical dashed lines) denoting the power-dependent onset
and completion, respectively, of the nanostripes’ reversal.

We applied the following procedures to perform broadband spin-wave
spectroscopy at different VNA powers $P_{\rm irr}$ between -30 and +15 dBm.
The nanostripes were saturated in $-x$ direction using $\mu_{0}H=-90$ mT. Spin
waves were excited at the given $P_{\rm irr}$ for frequencies $f_{\rm irr}$
ranging from 10 MHz to 20 GHz with a step of 2.5 MHz. The magnetic field was
increased to +40 mT in steps of 1 mT. Transmission spectra $S21$ were recorded
at each value of $\mu_{0}H$. The median value of the $S21$ spectra measured
over the range of magnetic field was subtracted from the raw data to remove
the background signal. Figure 1(d-f) show the corresponding spectra $\Delta
S21$ for three different values $P_{\rm irr}$ as indicated in the panels. The
dark branches correspond to different spin wave modes propagating from CPW1 to
CPW2. With increasing power, the high-frequency branches (grating-coupler
modes) labelled by P start at smaller and smaller field $H$ (red dashed
lines). At the same time, the branches AP end at smaller and smaller $H$ (blue
dashed lines) or do not appear in panel (f). We attribute the P (AP) branches
to a magnetic configuration in which Py nanostripes are aligned with (are
anti-parallel to) the magnetization of the YIG. YIG reverses its magnetization
at a field below 2 mT. In an intermediate field regime the high frequency
branches AP and P are not resolved indicating that the gratings are not
uniformly magnetized. This field regime reflects the switching field
distribution of Py nanostripes. Following Ref. 17, the disappearance and
reappearance of high-frequency branches at positive fields define critical
field values $H_{\rm C1}$ and $H_{\rm C2}$, respectively, which quantify the
switching field distribution of the nanostripe arrays. The critical field
$H_{\rm C1}$ ($H_{\rm C2}$) is given by the value of applied magnetic field
$H$ at which the signal strength of the branch AP (P) has decreased
(increased) to 50% of its maximum signal strength. The two critical fields are
attributed to the onset and completion of nanostripes’ reversal. In sample D1,
we extracted $H_{\rm C1}$ and $H_{\rm C2}$ considering the spin wave mode with
wave vector $k=2\pi/\lambda=k_{1}+1G^{\rm D1}$ and $G^{\rm
D1}=2\pi/a=2\pi/(100~{}{\rm nm})$. $k_{1}$ corresponds to the most prominent
wave vector provided by the CPW and $G^{\rm D1}$ to the reciprocal lattice
vector of sample D1. With increasing $P_{\rm irr}$, $H_{\rm C1}$ and $H_{\rm
C2}$ decrease in Fig. 1(d-f). In Fig. 1(f), branch AP is no longer observed.
This observation indicates that the reversal of 50-nm-wide nanostripes starts
at an applied field as low as 1 mT for $P_{\rm irr}=-5~{}$dBm.

Figure 2: (a) Switching yield diagram of D1. The symbols interconnected by
lines indicate the frequency-dependent critical power attributed to the
reversal of about 50% of the nanostripes below CPW1 at +10 mT. (b) Magnitude
of $\Delta S11$ obtained at +10 mT with a low power $P_{\rm sens}$ of -25 dBm.
(c) Switching yield diagram reflecting nanostripes’ reversal below CPW2. (d)
Magnitude of $\Delta S21$ measured at +10 mT using $P_{\rm sens}$ of -5 dBm.
The spectrum differs from (b). A clear shift of mode $|G^{\rm D1}-k_{1}|$ from
6.5 GHz in (b) to 8 GHz in (d) is resolved. For (d), we assume nanostripes
beneath both CPWs to be switched by mode $k_{1}$ which is excited at a power
larger than $P_{\rm C1}$ and $P_{\rm C2}$ at 1.5 GHz. (e) Precessional power
values $P_{\rm prec}$ at frequencies corresponding to modes $k_{1}$ and
$|G^{\rm D1}-k_{1}|$.

Before discussing the power dependence of critical fields in different samples
it is instructive to identify the magnon modes in sample D1 which induce most
efficiently nanostripe reversal in a fixed opposing field of +10 mT. This
field value is smaller than $\mu_{0}H_{\rm C1}$ determined at -30 dBm. For
different $f_{\rm irr}$ the power $P_{\rm irr}$ was increased until the branch
AP (P) in $\Delta S21$ reduced (increased) to 50% of its maximum signal
strength (see methods and supporting information S1). These critical power
values $P_{\rm C1}$ ($P_{\rm C2}$) are summarized in the switching yield
diagrams of Fig. 2(a) [(c)]. We find that the mode $k_{1}$ ($\lambda$ = 7.2
$\mu$m) which is directly excited by CPW1 at 1.5 GHz [Fig. 2(b)] induces
reversal of 50-nm-wide Py stripes at a power $P_{\rm C1}=P_{\rm irr}=-20$ dBm
underneath CPW1. At $P_{\rm C2}=-11~{}$dBm, mode $k_{1}$ [Fig. 2(d)] reversed
about 50% of nanostripes underneath CPW2. These observations are qualitatively
consistent with Ref. 17. For the grating coupler mode with $k=|G^{\rm
D1}-k_{1}|$ ($\lambda$ = 101.4 nm) the corresponding power values read $P_{\rm
C1}=+2$ dBm and $P_{\rm C2}=+8$ dBm, respectively. We note that this magnon
mode with ultrashort wavelength propagated over 25 $\mu$m through bare YIG and
induced stripe reversal underneath CPW2. This was not resolved in Ref. 17.

To compare the efficiency in magnon-induced stripe reversal between different
samples, we evaluate the power $P_{\rm prec}$ transferred to the spin
precession (prec) in YIG 17. This parameter is independent of the individual
microwave-to-magnon transduction of CPWs. The precessional power values
$P_{\rm prec}$ are evaluated by multiplying the critical power $P_{\rm C1}$ or
$P_{\rm C2}$ obtained from the switching yield diagrams with the square of the
magnitude of $\Delta S11$ (see supporting information S2) measured at -30 dBm
and -10 dBm, respectively. Figure 2(e) summarizes precessional power values
$P_{\rm prec,1}$ and $P_{\rm prec,2}$, respectively, for frequencies at which
the magnon modes $k_{1}$ and $k=|G^{\rm D1}-k_{1}|$ are excited. We observe
that mode $k_{1}$ provides $P_{\rm prec,2}$ of 68.9 nW. This value is similar
to the one found in Ref. 17. Strikingly, the grating coupler mode $|G^{\rm
D1}-k_{1}|$ with an ultrashort wavelength of about 100 nm requires $P_{\rm
prec,2}$ of only 10.6 nW for reversing stripes. The data demonstrate that
short-wavelength magnon modes are more efficient than long-wavelength modes in
terms of precessional power needed for reversal.

###  3.2 Magnon-induced switching of nanostripes in an aperiodic grating

In sample D3 nanostripes were arranged in a quasi-crystalline Fibonacci
sequence beneath both the CPWs [Fig. 3(a)]. The nanostripes were 100 nm and
200 nm wide separated by a 100-nm-wide air gap.

Figure 3: (a) Schematic of sample D3 with a Fibonacci grating consisting 100
nm and 200 nm wide Py nanowires separated by 100 nm, (b-d) plots of magnitude
of $\Delta S21$ spectra measured for the device at increasing applied
microwave powers ($P_{\rm irr}$), (e) Switching yield diagram (in blue)
showing a trend of the critical power that corresponds to switching of the
gratings beneath CPW1 by $k_{1}$ mode (f) Magnitude of the reflection spectrum
$\Delta S11$ measured at +14 mT showing the $k_{1}$ mode (g) Switching yield
diagram (in red) of the grating beneath CPW2 which corresponds to switching by
$k_{1}$ mode that propagates from CPW1 to CPW2 as observed from (h) magnitude
of $\Delta S21$ measured at +14 mT. Dotted lines in (g) are a guide to the
eye. They indicate that the critical power $P_{\rm C2}$ for the modes excited
at $f_{\rm irr}<1.5~{}$GHz and $>2.5~{}$GHz is beyond +6 dBm

Figure 3 (b) shows the magnitude of $\Delta S21$ between 10 MHz to 10 GHz
measured at $P_{\rm irr}=-25$ dBm. Dark branches correspond to the spin wave
modes propagating in YIG from CPW1 to CPW2. Multiple spin wave modes are
resolved. These modes are excited due to the CPW (mode $k_{1}$) as well as
reciprocal vectors 19 provided by the Fibonacci grating beneath CPW. Again,
the high-frequency modes can be divided into low-field AP branches and high-
field P branches. Similar to Fig. 1(d-e), branches vanish and reappear when
the applied magnetic field is swept from $-30$ mT to $+40$ mT. Branches chosen
for the analysis of the nanowires’ switching fields are marked by a dashed
blue line (AP mode) for the onset of switching and by a dashed red line (P
mode) for the completion of switching. Figure 3(b-d) shows spectra for step-
wise increased microwave power $P_{\rm irr}$. $H_{\rm C1}$ (blue dashed line)
and $H_{\rm C2}$ (red dashed line) decrease with $P_{\rm irr}$. The AP branch
approaches zero field for $P_{\rm irr}$ above -15 dBm, i.e., at a smaller
power compared to D1, but similar to D2 (see below). We attribute this
observation to a smaller reversal field of 200-nm-wide nanostripes compared to
the narrower ones in the aperiodic array.

Switching yield experiments were performed on D3 similar to D1 but at an
opposing field of +14 mT. The sample D3 was saturated at $\mu_{0}H$= -90 mT
and the field was swept to +14 mT, when the grating was in AP configuration.
We used intentionally the same field value like in Ref. 17 to compare directly
with the reversal reported for a periodic grating of 100-nm-wide nanostripes.
Then, the spin waves were excited at irradiation frequency $f_{\rm irr}$
ranging from 1 GHz to 10 GHz with a 0.25 GHz step. At each $f_{\rm irr}$, the
microwave power $P_{\rm irr}$ was increased from -25 dBm to +6 dBm. Switching
events were recorded by measuring the magnitude of $\Delta S21$ in the
frequency window of 2.5 to 5.5 GHz. The critical powers at which spin wave
signals first reduced and then increased to 50% of the maximum signal strength
were denoted as $P_{\rm C1}$ and $P_{\rm C2}$, respectively. The symbols
displayed as a function of $f_{\rm irr}$ in Fig. 3(e) [3(g)] reflect the onset
(completion) of reversal of stripes beneath CPW1 (CPW2). The switching yield
diagrams show that mode $k_{1}$ induces reversal of nanostripes below CPW1 at
15.8 $\mu$W and below CPW2 at 25.1 $\mu$W. The latter value is a factor of two
smaller compared to Ref. 17 where periodically arranged stripes with
$w=100~{}$nm were reversed underneath CPW2 by mode $k_{1}$ in a field of
$+14~{}$mT. Grating coupler modes in D3 did not allow us to reverse
nanostripes beneath CPW2 up to the value of 50% of the branch’s signal
strength. This is unlike the observation in case of sample D1 with
periodically arranged 50-nm-wide nanowires [see Fig. 2(c)] where a grating
coupler mode switched the remotely placed stripes at $P_{\rm C2}$.

###  3.3 Magnon-induced switching dependent on nanostripe width

In this section, we compare the critical switching fields for samples
incorporating different nanostripe widths. Figure 4(a) to (c) show the
parameters and spin-wave spectra obaianed on sample D2 with $a=200$ nm and
$w=400~{}$nm. The switching fields $H_{\rm C1}$ and $H_{\rm C2}$ reduce with
power $P_{\rm irr}$. However, their values and variation are much smaller than
for D1. In Fig. 4(d) and (e) we summarize power-dependent critical fields
extracted for three different samples D1, D2 (hollow symbols), D3 (solid
rectangle) and compare them to sample D4 reported in Ref. 17 (star). A
decreasing trend in $H_{\rm C1}$ and $H_{\rm C2}$ with $P_{\rm irr}$ is
observed for all samples substantiating magnon-induced reversal for
nanostripes of different widths between 50 and 200 nm.

Figure 4: (a) Sketch of sample D2 with periodic gratings consisting of 200-nm-
wide Py nanostripes arranged with $a=400$ nm. $\Delta S21$ spectra measured on
D2 at an applied power $P_{\rm irr}$ of (b) -25 dBm and (c) -5 dBm. (d)
Critical field $H_{\rm C1}$ and (e) $H_{\rm C2}$ extracted for nanostripes
with widths of 50 nm (D1), 100 nm (D4 adopted from Ref 17), and 200 nm (D2)
denoted by hollow symbols as well as the aperiodic grating (D3) denoted by
solid rectangle.

A detailed analysis provides two key observations:

  1. 1.

D1, D3 and D4 show a sharp reduction in $H_{\rm C1}$ and $H_{\rm C2}$ as
compared to D2. The net reduction in $H_{\rm C2}$ due to spin wave excitation
between -25 dBm to -5 dBm in D1, D3 and D4 is about five times larger than in
D2 (see supporting information S3).

  2. 2.

In Fig. 4(d) and (e), sample D1 with $w=50~{}$nm behaves similarly to D4 with
$w=100~{}$nm presented in Ref. 17 (star). Only D2 with $w=200~{}$nm shows
critical fields which are much lower at low power than all other samples. Topp
et al. in Ref. 20 presented an analytical model allowing us to calculate the
effective transverse demagnetization factors $N_{\rm eff}$ for nanostripes
arranged in periodic lattices. The model quantifies the effect of dipolar
interaction which is known to modify magnetic ansiotropies 21. We find 0.2 for
D1, 0.1 for D4 and 0.05 for D2. Assuming coherent rotation of spins, the
magnetic anisotropy fields should scale accordingly 22, 23, suggesting a
clearly larger reversal field for D1 compared to D4 and D2. We find however,
that samples D1 and D4 show very similar critical fields $H_{\rm C2}$ at the
lowest VNA power. This observation suggests that the reversal mechanism
involves incoherent processes (nonuniform rotation) like spin curling 24 or
domain wall motion 25, 26.

Figure 4 shows that magnon-induced reversal occurred for both the narrowest
nanostripe with a large reversal field of up to about 35 mT and the wide ones
with a reversal field of 10 mT and smaller. The absolute reversal fields of
the different samples do not scale according to their nanostripe widths and
effective demagnetization factors indicating a reversal mechanism involving
incoherent spin rotation. The microscopic mechanism and relevant torque
leading most efficiently to magnon-induced reversal of nanomagnets are still
under debate and not decided by the experiments presented here. Importantly,
our experiments showed that one and the same 50-nm-wide Py nanostripes were
reversed by propagating spin waves in YIG which exhibited wavelengths
$\lambda$ varying by a factor of 70. In case of $\lambda\approx 100$ nm, the
propagation distance was $\geq 250\times\lambda$. Such a macroscopically large
distance allows for both coherent scattering in a nanostructured holographic
memory 10 and computation across several unit cells of a neural network 5
before inducing nonvolatile storage of the computational result in a separate
magnetic bit.

###  3.4 Conclusion

Power dependent spectroscopy performed on thin YIG containing arrays of Py
nanostripes with widths $w$ ranging from 50 to 200 nm evidenced their magnon-
induced reversal in small opposing fields. We observed a sharp decrease in the
reversal fields of stripes with $w=50~{}$nm after applying a VNA power of
about -20 dBm (10 $\mu$W) to the spin-wave emitting coplanar waveguide. The
sharp decrease in switching field was observed also in the sample with an
aperiodic arrangement of nanostripes with $w=100~{}$nm and 200 nm. On the
other hand, a more steady decrease of reversal field was seen in case of
periodic arrays with 200-nm-wide nanowires. However, their initial reversal
field was already small. Spin waves excited directly by the CPW exhibited
$\lambda\approx 7~{}\mu$m and reversed nanostripes in all the four
investigated samples. Magnons with $\lambda\approx 100$ nm induced efficiently
the reversal of 50-nm-wide Py stripes. The required precessional power was as
low as 10 nW in a small opposing field of +10 mT. We highlight that switching
of all the different Py nanostripes was achieved by spin waves which
propagated over 25 $\mu$m in YIG away from the spin wave emitter. Our findings
are important for nanoscale magnonic in-memory computation, the holographic
memory and reconfigurable spin wave-based computing devices.

##  4 Methods

###  4.1 Sample fabrication

We used a 100-nm-thick YIG thin film epitaxially grown on a GGG (111)
substrate. It was provided by the Matesy GmbH in Jena, Germany. A 20-nm-thick
Py (Ni81Fe19) film was deposited on the YIG film using electron beam
evaporation. Two groups of gratings of nanostripes separated by 25 $\mu$m were
etched using a resist mask prepared via electron beam lithography applied to
hydrogen silsesquioxane. The sample D1 (D2) was patterned with 50 nm (200 nm)
wide nanostripes periodically arranged in a 100 nm (400 nm) period. The
lengths of nanostripes were alternately varied between 25 and 27 $\mu$m. D3
was fabricated with 25 $\mu$m long 100 nm and 200 nm-wide nanostripes arranged
in Fibonacci sequence with a gap of 100 nm. The samples were etched with an Ar
ion beam considering an etch stop at the YIG film. Two Coplanar Waveguides
(CPWs) were patterned above the nanostripe gratings with signal-to-signal line
separation of 35 $\mu$m using electron beam lithography with an MMA-PMMA
double layer positive resist, electron beam evaporation of 5 nm thick Ti and
110 nm thick Cu followed by lift-off processing. Ti was deposited for
adhesion.

###  4.2 Broadband Spin Wave Spectroscopy

The magnon-induced reversal was investigated using Keysight Vector Network
Analyzers (VNAs) N5222A and N5222B. The probe station of the VNA setup
consisted of electromagnetic pole shoes which provided a bipolar magnetic
field of up to 90 mT at arbitrary in-plane directions. The CPW1 or emitter CPW
of the sample was connected to the VNA port 1 with a microprobe. CPW2 is
connected to the detector port 2 with an identical microprobe. Initially, the
sample was saturated at -90 mT (in $-y$ direction) and spin waves were excited
with microwaves of frequencies ranging from 10 MHz to 20 GHz with a step of
2.5 MHz. The scattering parameters $S11$ and $S21$ in reflection and
transmission configuration, respectively, were recorded. The static field was
brought to -30 mT and increased to +40 mT in a step-wise manner. At each step,
the scattering parameters were recorded. The measurements were carried out for
spin wave modes excited with applied microwave powers from -25 dBm to +6 dBm.
These all-electrical spin-wave spectroscopy (AESWS) measurements were repeated
for devices with different grating periods and nanostripe widths and also for
gratings with nanowires arranged in Fibonacci sequence. After every
measurement, the median subtracted magnitude of scattering parameters as a
function of static magnetic field were plotted and analyzed for different spin
wave modes. The critical switching fields, i.e. static magnetic fields at
which the gratings started ($H_{\rm C1}$) and completed reversal ($H_{\rm
C2}$) were decided based on the disappearance and reappearance of a specific
spin wave branch for all devices between 8-10 GHz at magnetic fields swept up
to $+40~{}$mT.

###  4.3 Switching yield diagram

We performed switching yield diagram measurements as follows. A positive
opposing field $\mu_{0}H$ was applied to a sample after saturation at negative
fields. A transmission spectrum $S21$ was measured within a frequency window
of 6-9.5 GHz for D1 and 2.5-6.5 GHz for D3 ($f_{\rm sens}$) excited at -25 dBm
($P_{\rm sens}$). We call this operation as sensing of the nanostripes’
magnetic configuration being either anti-parallel (before) or parallel to the
small bias field (after magnon-induced reversal). Spin waves were excited with
irradiation frequency ($f_{\rm irr}$) range of 1-1.25 GHz at an irradiation
power of -25 dBm ($P_{\rm irr}$). After the irradiation, a $S{21}$
transmission spectrum was measured in the sensing window ($f_{\rm sens}$) to
sense the effect of the irradiation on the nanostripes. This measurement
sequence was repeated for the spin wave mode excited at an increasing $P_{\rm
irr}$ from -25 dBm to the maximum of +6 dBm in case of the D3 and up to +15
dBm in case of the D1. The magnetic field protocol was reset and the
measurements were repeated at the next interval of irradiation frequency
($f_{\rm irr}$) up to 10 GHz. As an example, median subtracted magnitude
$\Delta S21$ transmission spectra plotted for D1 measured at $f_{\rm irr}$
from 1.75- 2 GHz across the range of $P_{\rm irr}$ are provided in the
supporting information S1. The spin wave irradiation powers inducing a
disappearance and reappearance of spin wave branches in these plots
corresponded to critical powers needed to start ($P_{\rm C1}$) and complete
($P_{\rm C2}$), respectively, the switching of nanostripes below CPW1 and
CPW2, respectively.

The authors thank for support provided by the Centre of MicroNanoTechnology
(CMi) during the fabrication of the devices. Authors thank Mingran Xu and
Mohammad Hamdi for valuable discussions. The AESWS spectra were visualized
using colourmaps provided by Fabio Crameri 27. The scientific colour map devon
is used to prevent visual distortion of the data. The research was supported
by the SNSF via grant number 197360.

##  5 Data availability statement

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

##  6 Conflict of interest disclosure

The authors declare no conflict of interest.

### Author contributions

S.J., K.B., and D.G. planned the experiments. K.B. and D.G. designed the
samples. K.B. prepared the samples. S.J. performed the experiments together
with K.B., F.B. and A.M. S.J., F.B., A.M. and D.G. analyzed and interpreted
the data. S.J. and D.G. wrote the manuscript. All authors commented on the
manuscript.

####  6.0.1 S1: Criteria for extraction of critical powers $P_{\rm C1}$ and
$P_{\rm C2}$ from switching yield diagram experiment for device D1

The switching yield diagram is a plot of critical power needed to start and
complete the switching of gratings beneath both the CPWs. This is calculated
with the following strategy:

  * •

The device is saturated at -90 mT and applied magnetic field is swept up to
+10 mT (for D1). Median subtracted magnitude of $\Delta S21$ is measured at
$P_{\rm sens}=-25~{}$ dBm and in certain frequency window ($f_{\rm sens}$: 6
to 9.5 GHz), where the magnon modes are detected [black branches in Fig.
5(a)]. Then, a specific magnon mode between certain frequency window $f_{\rm
irr}$ (here, 1.75 to 2 GHz) and for $P_{\rm irr}$ from $-25~{}$dBm to
$+15~{}$dBm is irradiated. After every irradiation step at $P_{\rm irr}$,
magnitude of $\Delta S21$ is measured at $P_{\rm sens}=-25~{}$ dBm and
frequency window $f_{\rm sens}$.

  * •

The criteria of extracting $P_{\rm C1}$ and $P_{\rm C2}$ is explained based on
the disappearance of magnon modes and reappearance at a new frequency
highlighted with blue and yellow brackets respectively.

Figure 5: Procedure for determining the critical switching powers ($P_{\rm
C1}$ and $P_{\rm C2}$) from the $S$21 transmission measurement after exciting
a magnon mode from 1.75 to 2 GHz at an increasing irradiation power ($P_{\rm
irr}$). The two black branches at between 6.5-7 GHz and 8.5-9 GHz correspond
to the spin wave modes sensed for $P_{\rm irr}$ below $-15~{}$dBm. The
integrated signal strength from $f_{\rm sens}$ between 6.4 to 7 GHz (marked
with blue dotted lines) was extracted and plotted as a function of $P_{\rm
irr}$ in Fig. (b) showing the disappearance of the mode between $P_{\rm
irr}:-25$ to $-10~{}$ dBm. $P_{\rm C1}$ corresponds to $P_{\rm irr}$ at which
the mode is at the average of its maximum signal strength and minimum signal
strength or noise floor. For the given $\Delta S21$ spectra, it is $-14~{}$dBm
with an error bar of $\pm 1~{}$ dBm. On the contrary, a spin wave mode appears
at 8 GHz for $P_{\rm irr}>0~{}$dBm. We apply similar criteria but now for the
increase in the signal strength for the branch highlighted in yellow dotted
lines. The signal strength was extracted between 7.8 to 8.04 GHz and plotted
in Fig. (c) as a function of $P_{\rm irr}$. The critical switching power at
which a new mode appears is denoted as $P_{\rm C2}$ and is given by $50\%$ of
maximum signal strength of the branch which corresponds to 0 dBm with an error
bar of $\pm 1~{}$ dBm.

####  6.0.2 S2: Calculation of precessional power ($P_{\rm prec}$) at $P_{\rm
C1}$ and $P_{\rm C2}$ for device D1

Precessional power for a magnon mode with frequency $f_{\rm irr}$ excited at
microwave power ($P_{\rm irr}$) is given by:

$P_{\rm prec}=(\rm MAG(\Delta\textit{S}11)~{}\rm at~{}\textit{f}_{\rm
irr})^{2}\times(\textit{P}_{\rm C}~{}at~{}\textit{P}_{\rm irr});$

Where, $P_{\rm C1}$ and $P_{\rm C2}$ are obtained from switching yield diagram
measurement as shown in the previous section. To calculate magnitude of
$\Delta S$11 [MAG ($\Delta S11$)], we subtract the RAW data using reference
subtraction from the spectra measured at zero field. The following plot shows
reference subtracted $\Delta S11$ spectra.

Figure 6: Magnitude of $\Delta S11$ spectra at 10 mT measured at $-30~{}$dBm
and $-10~{}$dBm after reference subtraction from measurement 0 mT. The signal
strength for both $k_{1}$ and $|G^{\rm D1}-k_{1}|$ modes were extracted for
evaluating $P_{\rm prec,1}$ and $P_{\rm prec,2}$

$P_{\rm prec,1}$ was calculated with $\Delta S11$ spectra measured at
$-30~{}$dBm, where, both the gratings are antiparallel to YIG and the applied
field direction. Whereas, for $P_{\rm prec,2}$, we use the spectra measured at
$-10~{}$dBm. At this microwave power the gratings are completely switched
beneath both the CPWs.

####  6.0.3 S3: Extent of critical field reduction extracted from $\Delta S21$
spectra of devices D1 (consisting 50 nm wide nanostripes) and D2 (consisting
200 nm wide nanostripes) measured at different microwave powers $P_{\rm irr}$

Figure 7: The trend of the difference in critical switching fields extracted
for the devices D1 and D2 consisting of 50 nm wide and 200 nm wide Py
nanostripes respectively. The data points are calculated from the $H_{\rm C1}$
and $H_{\rm C2}$ values obtained from $\Delta S21$ spectra. The trend of this
values is shown in the main text in Fig. 4(d-e).

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