# Tailoring crosstalk between localized 1D spin-wave nanochannels using
focused ion beams

Vadym Iurchuk  Institute of Ion Beam Physics and Materials Research,
Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
<EMAIL_ADDRESS>Javier Pablo-Navarro  Institute of Ion Beam Physics and
Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden,
Germany  Tobias Hula  Institute of Ion Beam Physics and Materials Research,
Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  Ryszard
Narkowicz  Institute of Ion Beam Physics and Materials Research, Helmholtz-
Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  Gregor Hlawacek  Institute
of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-
Rossendorf, 01328 Dresden, Germany  Lukas Körber  Institute of Ion Beam
Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, 01328
Dresden, Germany  Fakultät Physik, Technische Universität Dresden, 01062
Dresden, Germany  Attila Kákay  Institute of Ion Beam Physics and Materials
Research, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  Helmut
Schultheiss  Institute of Ion Beam Physics and Materials Research, Helmholtz-
Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  Jürgen Fassbender
Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum
Dresden-Rossendorf, 01328 Dresden, Germany  Institute of Solid State and
Materials Physics, Technische Universität Dresden, 01062 Dresden, Germany
Kilian Lenz  Institute of Ion Beam Physics and Materials Research, Helmholtz-
Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  Jürgen Lindner  Fakultät
Physik, Technische Universität Dresden, 01062 Dresden, Germany

###### Abstract

1D spin-wave conduits are envisioned as nanoscale components of magnonics-
based logic and computing schemes for future generation electronics. À-la-
carte methods of versatile control of the local magnetization dynamics in such
nanochannels are highly desired for efficient steering of the spin waves in
magnonic devices. Here, we present a study of localized dynamical modes in
1-$\mu$m-wide Permalloy conduits probed by microresonator ferromagnetic
resonance technique. We clearly observe the lowest-energy edge mode in the
microstrip after its edges were finely trimmed by means of focused Ne+ ion
irradiation. Furthermore, after milling the microstrip along its long axis by
focused ion beams, creating consecutively $\sim$50 and $\sim$100 nm gaps,
additional resonances emerge and are attributed to modes localized at the
inner edges of the separated strips. To visualize the mode distribution,
spatially resolved Brillouin light scattering microscopy was used showing an
excellent agreement with the ferromagnetic resonance data and confirming the
mode localization at the outer/inner edges of the strips depending on the
magnitude of the applied magnetic field. Micromagnetic simulations confirm
that the lowest-energy modes are localized within $\sim$15-nm-wide regions at
the edges of the strips and their frequencies can be tuned in a wide range (up
to 5 GHz) by changing the magnetostatic coupling (i.e. spatial separation)
between the microstrips.

##  1 Introduction

Magnonic devices—often deemed the candidates for next generation
electronics—take advantage of purely spin-based transport and processing of
information encoded in the amplitude and/or the phase of spin waves, being the
collective excitations of magnetization dynamics in thin films [1]. Increased
endeavors have been taken to reduce the size and energy consumption of the
magnonic blocks and enhance the control over the spin-wave propagation and
their interaction with each other and with other components of the magnonic
circuit [2]. Recent studies have successfully demonstrated possibilities to
scale down magnonic waveguides to $\sim$50 nm width by precise control of the
nanofabrication conditions [3, 4]. Another approach to reduce the lateral size
of spin-wave conduits takes advantage of magnetic domain walls as propagation
channels [5, 6].

The tunable crosstalk between spin waves is a prerequisite of functional
magnonic circuits like, e.g. nanoscale directional couplers and half adders
[7] as well as periodic or quasi-periodic magnonic crystals [8]. In that
respect, on-demand engineering of closely packed magnonic conduits is a key
step towards controlling the magnetostatic interactions between spin waves [9,
10, 11, 12]. One approach, allowing for such manipulation, lies in utilizing
localized edge modes in patterned magnetic micro- and nanostructures [13, 14,
15]. A major weakness of this approach is the fact that the experimental
observation and manipulation of the edge-localized spin-wave modes is not
straightforward due to excessive roughness of the edges of the structures
fabricated by conventional methods. To overcome this drawback, material
modification using focused ion beams (FIB)—in particular using Ne+ ions—was
recently employed for an on-demand precise manipulation of the magnetic
nanostructure and shape to obtain high-quality edges [16, 17]. On the other
hand, conventional methods of magnetization dynamics detection are, in
general, non-local, i.e. they are capable of probing the signal either from
the whole structure (in case of conventional ferromagnetic resonance
technique) or from a macroscopic region of the sample (in case of conventional
Brillouin light scattering technique). When applied to the arrays of closely
spaced magnetic objects, these methods detect an averaged dynamical signal
and, therefore, meet difficulties in resolving the magnetization dynamics of
an individual structure.

Here, we present a compound study of the magnetization dynamics in a single
confined micron-sized Permalloy strip as a function of its shape, modified on-
demand by means of focused Ne+ ion beam milling. More specifically, we use
planar microresonator ferromagnetic resonance ($\mu$FMR) spectroscopy
complemented by Brillouin light scattering (BLS) microscopy and micromagnetic
simulations to investigate the dynamics of a single 5 $\mu$m $\times$ 1 $\mu$m
$\times$ 50 nm strip in an as-prepared state as well as cut along its length
using Ne-FIB, creating consecutively two 5-$\mu$m-long strips separated by 50
and 100-nm-wide gaps. We show that FIB-based modification of the Py microstrip
geometry directly impacts the dynamical spectra, i.e., it leads to the
appearance of additional resonances in the corresponding spectra. Using
micromagnetic modelling we have attributed these resonances to the dynamical
modes localized at the narrow regions of the inner edges of the cut strips. We
demonstrate that the resonance fields of these modes can be effectively tuned
by changing the distance between the two strips. We further confirm the
localization of the emerged modes by BLS microscopy showing good agreement
with the $\mu$FMR measurements. Our analysis shows, that Ne-FIB-assisted
modification of confined magnetic microstructures constitutes a powerful tool
for the on-demand control of the magnetization dynamics in closely packed
magnonic conduits.

Figure 1: (a) Scanning electron microscopy (SEM) image of the $\Omega$-shaped
Cu/Au microresonator loop and the 5 $\mu$m $\times$ 1 $\mu$m $\times$ 50 nm Py
strip sample positioned in the center. During the $\mu$FMR measurements, the
static external magnetic field $H$ is applied in-plane of the microstrip at
angle $\phi_{H}$ with respect to the short axis of the strip. (b-e) SEM images
of the (b) as-prepared Py microstrip, (c) edge-trimmed using Ne-FIB, (d) cut
along by Ne-FIB leaving a nominal gap of 50 nm and (e) 100 nm. (f-g) Atomic
force microscopy (AFM) profiles of the cut strips measured along the red
arrows shown in (d) and (e), respectively. Dashed lines indicates the
substrate level.

##  2 Results and discussion

###  2.1 Sample fabrication and Ne-FIB milling procedure

Figure 1(a) shows a scanning electron microscopy (SEM) image of the fabricated
device comprising a 50 nm thick Py microstrip with 5 $\mu$m $\times$ 1 $\mu$m
nominal planar dimensions, centered inside the $\Omega$-shaped loop of the
microresonator (see section Methods for the fabrication details). Figure 1(b)
shows an SEM image of the as-prepared Py microstrip with clearly visible
imperfections on the edges formed naturally during standard wet lithography
processing. We use focused ion beam-assisted milling by means of a Neon gas
field ion source (GFIS) of a Zeiss Orion NanoFab [18] to obtain a precise
modification of the microstrip shape (see section Methods). Figure 1(c) shows
an SEM image of the Py microstrip after its edges were trimmed by Ne-FIB in
order to obtain smooth sidewalls with a minimum of irregularities. Thereafter,
the central region along the strip was milled in order to obtain two strips
separated by a gap of nominal 50 nm width [see Figure 1(d)]. Finally, the
milled gap was widened to nominal 100 nm width [see Figure 1(e)]. To confirm
the quality of the Ne-FIB-based milling and to visualize the cross-section of
the cut strips, we performed atomic force microscopy (AFM) linescans across
the Py microstrips (red arrows in Figure 1(d,e) indicate the scan directions).

The AFM measurements on the strips with nominal 50 nm and 100 nm gap [see
Figures 1(f) and (g), respectively] confirm the high quality of the Ne-FIB
milling procedure allowing for the smooth shaping of both inner and outer
edges of the strips. However, due to inevitable straggling of the Ne ions and
AFM-tip artifacts in the narrow trench, the inner edges are not straight and
appear to have a rather quasi-Gaussian cross section profile at the bottom.
Nonetheless, the measured full-width-half-maximum of the fabricated gaps is 60
nm and 97 nm for the nominal gap width of 50 nm and 100 nm, respectively [see
Figures 1(f) and (g)].

###  2.2 $\mu$FMR measurements

Figure 2: (a-c) In-plane angular dependences of the field-swept $\mu$FMR
signal measured at 14.059 GHz for (a) the edge-trimmed microstrip, (b) cut
along in half using the Ne-FIB with a nominal gap of 50 nm, and (c) and 100
nm. The red line in (a) shows the analytical fit to the measured data using
the Kittel relation. The edge modes are numbered in the order of their
appearance starting from the high field values (see text for details). (d-f)
Corresponding $\mu$FMR spectra of an infinitely long strip simulated for the
excitation frequency 14.057 GHz. (d) infinitely long 1-µm-wide and 50-nm-thick
strip, (e) two 475-nm-wide infinitely long strips separated by 50 nm, and (f)
two 450-nm-wide strips separated by a 100-nm-wide gap.

After each FIB modification, $\mu$FMR measurements were performed to probe the
dynamical modes associated to the corresponding shape of the sample (as-
prepared (not shown), edge-trimmed, with 50-nm-wide gap and with a 100-nm-wide
gap). The lowest-energy localized edge mode can hardly be detected in the as-
prepared microstrip due to the excessive edge roughness. However, similar to
the work presented by Lenz et al. [16], this mode is clearly observed when the
edges of the microstrip are finely trimmed by focused Ne ion beams. Figure
2(a) shows the $\mu$FMR signal for an in-plane field sweep measured on the
edge-trimmed Py microstrip [see Figure 1(c)] at $f_{\mathrm{res}}$ = 14.059
GHz for different angles of the in-plane bias field $\phi_{H}$ ranging from
90∘ to 202∘. When the bias field is parallel to the strip, i.e., $\phi_{H}$ =
90∘, the $\mu$FMR signal exhibits a strong peak around 180 mT corresponding to
the main dynamical mode excited in the center of the strip. As the applied
magnetic field is rotated towards the direction perpendicular to the strip
($\phi_{H}$ = 180∘), a rich field-dependent mode structure emerges. At
$\phi_{H}$ = 180∘, the mode 1 with the highest resonance field (here
$\mu_{0}H_{\mathrm{res}}$ = 557 mT) corresponds to the true edge mode
localized at the very edge of the long side of the strip. Subsequent modes
appearing at $\mu_{0}H_{\mathrm{res}}$ = 437 mT, 408 mT, and 385 mT are
attributed to the 2nd, 3rd and 4th modes localized in the vicinity of the
strip edge. One should not confuse these modes with the higher order modes of
the true edge mode. These localized modes are essentially confined standing
spin waves. The detailed quantification of the mode character will be
discussed further in the section Dynamical modes definition.

When the field orientation is deviated from the $\phi_{H}$ = 180∘ direction
(across the strip), the effective magnetic field increases due to the dipole-
dipolar energy contribution. Therefore, at a given frequency, the resonance
fields of the corresponding modes downshift towards lower values. At
$\phi_{H}\sim$ 150∘, the effective external field in the direction
perpendicular to the strip becomes lower than the energy threshold required
for the edge mode’s stability, and the edge modes vanish.

Figures 2(b,c) show the $\mu$FMR signal measured on the Py microstrip cut in
the middle by Ne focused ion beam to obtain two 1 $\mu$m long strips separated
by a nominal 50 nm and 100 nm gap, respectively. A qualitative comparison of
the angle-dependent $\mu$FMR spectral maps of the strips reveal additional
modes appearing in the region of the spectral band corresponding to the large
$\phi_{H}$ angles as compared to the edge-trimmed strip. However, we observe
not only the overall shift of the resonance fields of the dynamical modes due
to the modified magnetostatic energy of the system, but a pronounced splitting
of the edge-mode resonances as well. To identify the origin of the measured
modes in detail, we use micromagnetic simulations. The emerged additional
modes shown in Figures 2(b,c) can be attributed to the ones localized at the
_inner edges_ of the fabricated strip pairs as opposed to the ones localized
at the _outer edges_ (already visible in the edge-trimmed strip as well).

###  2.3 Micromagnetic simulations

To reveal the dynamical mode profiles and to define the exact localization of
the modes within the strips, we have performed micromagnetic simulations of
the magnetization dynamics in confined Py microstrips using the two-
dimensional propagating-wave eigensolver of the open source finite-element
micromagnetic package TetraX [19, 20]. First, we simulate the absorption
spectra [21, 22] of the infinitely long Py strip with a 1 $\mu$m $\times$ 50
nm cross section. To mimic the $\mu$FMR measurements on the Ne-FIB-cut strips,
we performed similar simulations of the pairs of infinitely long strips with
475 $\times$ 50 nm and 450 $\times$ 50 nm cross sections spaced laterally by
50 nm and 100 nm, respectively. For all simulations, the static magnetic field
was applied in the sample plane, and the absorption was computed assuming a
homogeneous out-of-plane rf field profile.

Figures 2(d–f) shows the $\mu$FMR absorption spectra for an excitation
frequency of $f$ = 14.057 GHz and for different azimuthal angles $\phi_{H}$ of
the in-plane field. Here, we have simulated the magnetization dynamics in
three different strip geometries: Figure 2(d) shows the result of an
infinitely long, 1 µm wide and 50 nm thick strip. Figure 2(e) shows two
475-nm-wide infinitely long strips separated by a 50-nm-wide gap and Figure
2(f) two 450-nm-wide strips separated by a 100 nm gap, respectively. This data
is in good agreement with the experimentally measured $\mu$FMR spectra of
Figures 2(a–c). More specifically, the simulated $\phi_{H}$ angular dependence
of the FMR signal of the edge-trimmed strip reveals a band of excited
dynamical modes whose resonance fields $\mu_{0}H_{\mathrm{res}}$ decrease with
increasing mode number [see Figure 2(d)]. The first four resonances are
observed at $\mu_{0}H_{\mathrm{res}}$ = 614 mT, 383 mT, 335 mT, and 231 mT,
respectively. The angular dependence directly correlates with the experimental
data exhibiting a resonance field downshift with increased deviation from the
$\phi_{H}$ = 180∘ azimuthal direction of the bias field.

A similar distribution of the modes is observed in the simulated FMR spectra
of the cut strips, i.e., the appearance of the additional modes at bias
fields, which do not match the resonances of the main mode numbers. The origin
of these modes is attributed to the magnetostatic coupling between the closely
spaced strips. For example, for a 50 nm gap, the first additional mode appears
at $\mu_{0}H_{\mathrm{res}}$ = 515 mT, i.e. between the resonance fields of
the 1st and the 2nd modes, and separated by 100 mT from the 1st edge-mode
resonance. Similarly, for the 100-nm-wide gap, the first additional mode is
observed at $\mu_{0}H_{\mathrm{res}}$ = 567 mT, being separated by 50 mT from
the corresponding resonance of the 1st edge mode. This frequency separation
between the first two resonances observed in the cut strips decreases with
increasing gap-width due to a decreasing magnetostatic coupling between the
strips, and eventually vanishes for gaps wider than $\sim$500 nm (not shown
here). For such large spacing, the magnetization dynamics in both strips were
found to be essentially independent, as the strips become magnetically
isolated due to the negligible magnetostatic coupling between them. For the
lower magnetic field values, a distinct splitting of the dynamical modes is
observed too, whose classification is presented in the next section.

###  2.4 Dynamical modes definition

To classify the observed modes and define the distribution of the dynamic
magnetization profile within the cut strips, we examined the $\mu$FMR spectra
for the fixed azimuthal angle of $\phi_{H}$ = 180∘ [Figure 3(a)] and compare
them to the corresponding simulated spectra [Figure 3(b)] and extracted mode
profiles [Figures 3(c–e)].

The experimental $\mu$FMR spectrum of the edge-trimmed strip (top green curve)
in Figure 3(a) shows multiple resonances, and the corresponding simulated
spectrum in Figure 3(b) allows for attributing the observed resonances to
well-defined mode profiles as depicted in Figure 3(c). Note that we plot the
field derivative of the FMR dispersion signal ($d\chi\prime/dH$) giving a
symmetric peak to make the peaks easily identifiable by their local maximum.
While decreasing the field from saturation, we observe the first distinct
$\mu$FMR peak at $\mu_{0}H_{\mathrm{res}}$ = 557 mT. Comparing the position of
this peak with the simulations, and visualizing the mode profile of the
corresponding mode, we attribute this resonance to the 1st edge mode localized
at the long edges of the strip. Detection of this mode by conventional
inductive FMR measurements is often elusive due to its reduced active mode
volume in confined geometries and very sensitive dependence on the quality of
the edges of the sample. Nevertheless, our $\mu$FMR technique with optimized
quality factor of the resonator allows for the detection of such low energy
modes [16]. Notably, the dynamical magnetization profile of this mode is
extremely localized in the narrow edge region of the strip within a width of
$\sim$15 nm.

By further decreasing the bias field, we observe additional localized
resonances, which appear consecutively at $\mu_{0}H_{\mathrm{res}}$ = 437 mT,
408 mT, and 385 mT. These resonances are also present in the simulated data,
allowing for a classification of the observed modes by examining their
dynamical profiles. One can see in Figure 3(c), that the 2nd mode profile is
no longer perfectly localized at the edges of the strip, but moves towards the
center of the strip. The consecutive modes also exhibit periodic magnetization
pattern with the tendency of shifting their maximum precession amplitude
towards the center of the strip. Upon lowering the field the amplitudes of the
observed peaks increases, and the spreading and hybridization of the dynamical
nodes within the strip does not allow for exact classification of the measured
mode. These modes, contrary to the Kittel-like 1st edge mode, correspond to
standing spin-wave resonances, confined in specific regions of the strip.
Indeed, as shown by Pile et al. [23], with decreasing field, these regions of
the mode localization shift from the edges towards the center of the strip,
where two counter-propagating spin waves form quantized nodes.

Figure 3: (a) Field-swept $\mu$FMR spectra measured for $\phi_{H}$ = 180∘ on
the edge-trimmed strip (top green curve), and cut with 50 nm (middle yellow
curve) and 100 nm gaps (bottom red curve) respectively. (b) Corresponding
simulated FMR spectra of an infinitely long strip with 1 $\mu$m $\times$ 50 nm
cross section, two strips with 475 nm $\times$ 50 nm cross section separated
by 50 nm, and two strips with 450 nm $\times$ 50 nm cross section separated by
100 nm gap, respectively. The spectra in (a) and (b) show the field-derivative
of the dispersion signal $d\chi\prime/dH$ and are vertically offset for visual
clarity. Numbers in (a) and (b) denote the mode order (see text). (c-e) Cross
sections of the single strip (c) and FIB-cut strips (d,e) with the mode
profiles labeled corresponding to the resonances observed in b). The color
scale denotes the $x$-components of the dynamical magnetization.

Eventually, when reaching the value of the main FMR resonance, we observe a
large-amplitude signal at $\mu_{0}H_{\mathrm{res}}$ = 249 mT, attributed to
the quasi-uniform resonance mode after visualizing its simulated mode profile
[see Figure 3(c)]. We note, that the low amplitude resonances observed for the
edge-trimmed strip between $\sim$450 and $\sim$550 mT were not reproduced in
the micromagnetic simulations [see top green curve in Figure 3(b)]. Possible
sources of these peaks may be attributed to the nonuniform thickness of the
edge-trimmed strip, asymmetry of the opposite edge shapes and/or edge
inhomogeneities due to the ion-induced modification of the Py crystal
structure or due to the material re-depositions during the milling.

A more complex set of dynamical modes is observed in the cut strips, where the
additional peaks emerge in both experimental and simulated $\mu$FMR spectra.
More specifically, for a 50 nm gap between the strips, a pronounced peak
appears at $\mu_{0}H_{\mathrm{res}}$ = 474 mT [see middle yellow curve in
Figure 3(a)]. A similar peak is also observed in the simulated spectra at
$\mu_{0}H_{\mathrm{res}}$ = 515 mT [see middle yellow curve in Figure 3(b)].
When visualizing the mode profile of this resonance [see Figure 3(d)], we can
clearly attribute it to the so-called mode 1i, i.e. the first edge mode
localized at the _inner_ edges of the two strips after the milling. Upon
further decrease of the bias field, we observe the peak at
$\mu_{0}H_{\mathrm{res}}$ = 436 mT attributed to the mode 2, which is
localized at the _outer_ edges of the cut strip, as confirmed by the
micomagnetic modelling of the corresponding mode profiles. Further reducing
the bias field reveals a resonance at $\mu_{0}H_{\mathrm{res}}$ = 410 mT,
which was expected to be associated with the 2i mode. Surprisingly, the
micromagnetic analysis shows that for the pairs of 475-nm-wide strips
separated by a 50 nm gap, this mode has a lower resonance field, therefore,
higher frequency as compared to the mode 3. Detailed examination of the mode
profile of this particular resonance confirms that the resonant response is
localized in the vicinity of the outer edges, and the alternation of the
precession maxima and minima allows for an unambiguous classification of the
observed mode as mode 3. The 2i mode resonance for this particular geometry
appears between the resonant fields of the modes 3 and 4 [see the resonances
of the yellow curve in Figure 3(a,b) and the corresponding mode profiles in
Figure 3(d)]. The origin of this behavior is attributed to the reduced outer
edges saturation field due to the increased magnetostatic coupling between the
strips. The quasi-uniform mode is detected via $\mu$FMR at
$\mu_{0}H_{\mathrm{res}}$ = 286 mT following the global shift of the resonance
fields due to the modified shape anisotropy as compared to the uncut strip.

When the gap between the two strips increases to 100 nm [see bottom red curve
in Figure 3(a)], the resonances of the corresponding inner edge modes are
shifted towards the outer edge mode resonances, due to the decreased
magnetostatic coupling in the system of two strips. More specifically, the 1i
resonance is now detected at $\mu_{0}H_{\mathrm{res}}$ = 523 mT, i.e. closer
to the mode 1 resonance as compared to the 50 nm gap. Upon decreasing bias
field, we observe the mode 2 resonance at $\mu_{0}H_{\mathrm{res}}$ = 444 mT
followed by the mode 2i resonance at $\mu_{0}H_{\mathrm{res}}$ = 429 mT. The
same is true for the mode 3i, which now is located straight below the mode 3.
Notably, at first glance, we were unable to resolve a distinct peak of the
mode 4 in our $\mu$FMR measurements. However, an analysis of the simulated
mode profiles reveals a superposition of the modes 3i and 4 [see Figure 3(e)],
i.e., the resonant response of both modes occurs at the same field value,
$\mu_{0}H_{\mathrm{res}}$ = 387 mT. The corresponding mode profile shows the
features of the mode 3i localized closer to the inner edges of the strips, and
the mode 4, which is located closer to the outer edges of the strips. This
superposition is the consequence of the reduced width of the strips, where the
spatially distributed dynamical magnetization of the two different but closely
spaced eigen-resonances can be excited at the single resonance frequency in
the confined geometry.

One has to comment on the two additional modes present in the experimental
$\mu$FMR spectrum of the strips with 100 nm gap at $\mu_{0}H$ = $\sim$490 mT
denoted as 1* and 1** in the bottom curve of Figure 3(a). Although these
resonances were not observed in the simulations, the spatially resolved
Brillouin light scattering microscopy measurements presented in the next
section show that these modes are also localized at the inner edges of the cut
strips.

One can see that the difference between the experimental and the simulated
values can reach tens of mT. Here we comment on the possible origins of these
discrepancies. First, the TetraX simulation framework allows working with
exclusively 2D geometries, which automatically sets one of the demagnetizing
factors (here, $N_{x}$) to zero, thus lowering the in-plane shape anisotropy,
which leads to an increased $\mu_{0}H_{\mathrm{res}}$ values when the
transverse bias field is applied to the strip. Second, as can be seen from the
topography of the cut strips [see AFM scans in Figure 1(f,g)], the profile of
the outer and, especially, inner edges is not straight, but significantly
deviates from the vertical. This results in a non-uniform separation between
the strips across the thickness. For example, for the 100-nm-nominal-gap-
witdh, the separation between the strips is $\sim$60 nm at the substrate level
and $\sim$130 nm at the top surface of the Py strip. The canted shape of the
edges leads to considerable redistribution of the edge modes localization
within the inner edge of the cut strips and, therefore, to the shift in the
corresponding resonant magnetic fields. Third, the position of the edge mode
resonances is extremely sensitive to the asymmetries in the system introduced
by the FIB milling. The asymmetric profiles of the gap sides together with the
difference in the widths of the cut strips may contribute to the resonance
field shift due to the relocalization of the modes within the strips. The
described inhomogeneities can also be considered as the origin of the 1* and
1** modes appearance.

###  2.5 Brillouin light scattering microscopy

To further elucidate the mode localization within the cut strips
experimentally, we measured the spin-wave spectra on the strip with the
100-nm-wide gap by means of Brillouin light scattering (BLS) microscopy [24].
The BLS was performed on the identical sample as for the $\mu$FMR using an
excitation frequency of $f_{\mathrm{res}}$ = 14.059 GHz.

Figure 4(a) shows the spatially resolved BLS intensity map as a function of
the laser spot position for different bias magnetic fields applied
perpendicular to the strip’s long axis ($\phi_{H}$ = 180∘). Depending on the
magnetic field magnitude and the position of the beam, one can clearly
distinguish the different resonances, which directly correspond to the modes
defined in the previous section for the strips with 100 nm gap. More
specifically, the large amplitude resonance around $\mu_{0}H$ = 300 mT is
attributed to the quasi-uniform (center) $\mu$FMR mode. The BLS intensity vs.
position across the strip as plotted in Figure 4(b) confirms that this mode is
essentially spread over the whole strip width with the maximum amplitude in
the center of the individual strips. The spatial distribution of the dynamical
response is in agreement with the simulated mode profile of the quasi-uniform
$\mu$FMR mode shown in the bottom-most image of Figure 3(e).

Figure 4: (a) Brillouin light scattering intensity map measured on the FIB-cut
strip with 100 nm gap as a function of the in-plane bias field applied
perpendicular to the strip’s long axis and for different laser beam positions
across the sample (denoted by the red arrow in the top inset). The
measurements were performed at an excitation frequency of 14.059 GHz.
Rectangles on the map are guides to the eye visualizing the strip edges. (b)
BLS intensity as a function of the beam position measured for $\mu_{0}H$ = 295
mT corresponding to the quasi-uniform $\mu$FMR mode. (c-d) BLS intensity vs.
beam position measured for the fields corresponding to (c) the 1st and (d) the
2nd edge mode resonances localized at the outer (black circles) and the inner
(red squares) edges of the cut strips, respectively. Grey rectangles in (b–d)
denote the positions of the strips with respect to the scanned range.

With increasing magnetic field, we can clearly identify the other modes
present in the system. By examining their BLS intensity distributions across
the strip width and comparing the resonance fields with the ones measured by
$\mu$FMR, we can attribute the detected resonances to the corresponding edge
modes of the strips. For example, Figure 4(c) shows the measured BLS intensity
across the structure for $\mu_{0}H$ = 551 mT (black circles) and $\mu_{0}H$ =
508 mT (red squares). The distribution of the BLS intensity across the strip
width confirms that the detected modes correspond to the mode 1 localized at
the outer edge of the strip and mode 1i, localized at the inner edges of the
strips. The same analysis allows for classifying the modes at $\mu_{0}H$ = 435
mT and $\mu_{0}H$ = 420 mT as modes 2 and 2i respectively [see Figure 4(d)].
In addition, here, one can clearly notice the gradual shift of the BLS
intensity of the localized modes towards the central region of the strip, in
accordance with the simulated mode profiles of Figure 3(e).

Notably, the modes 1* and 1**, present in the $\mu$FMR spectra, are also
detected by the BLS microscopy [see Figure 4(a)]. The spatial distribution of
their BLS intensity signal indicates that these modes are localized at the
inner edges of the strips. Therefore the appearance of these modes is
attributed to the inner edges asymmetry of the cut strips (including non-
uniform separation over the thickness and side wall slope) as explained above
in section Dynamical modes definition.

Interestingly, the modes localized at the outer edges are detected only at one
side of the strips. This asymmetry might be explained by a non-homogeneous rf
field distribution across the strips due to the off-centered position of the
strips in the microresonator. Therefore, the opposite strip edges are exited
with different rf field amplitudes leading to the spatial asymmetry of the BLS
intensity map. Additionally, the $\Omega$-shaped antenna has a $\sim$2$\mu$m
opening to allow for the rf current flowing through the resonator [see Figure
1(a)]. Therefore, the strip edge closer to this opening is excited with
different amplitude as compared to the opposite edge, and the corresponding
dynamical signal falls below the detection threshold.

One has to note that the slight discrepancies between the resonance fields
detected by $\mu$FMR and BLS measurements can be attributed to different
scales of the experiments. Whereas in the inductive $\mu$FMR measurements a
volume-averaged magnetic signal is detected, the BLS microscopy is a purely
local technique probing the dynamics in the small area illuminated by the
laser spot. In that respect, laser-induced heating of the sample may also
contribute to the observed resonance fields shift due to locally modified
magnetic parameters. In addition, the BLS signal acquires the whole band of
the excited resonances at different $k$ vectors for a given excitation
frequency, whereas the $\mu$FMR is sensitive to uniform magnetization
dynamics. Nevertheless, a good qualitative agreement between both experiments
allows to designate the observed resonances to the corresponding localized
modes observed in the pair of Py strips.

##  3 Conclusions

This work takes advantage of different techniques, i.e. micro-cavity FMR
spectroscopy, BLS microscopy, and micromagnetic modelling, to study the
magnetization dynamics in confined magnetic microstructures modified by Ne+
GFIS-based FIB. We have demonstrated that ion-induced modification of the
magnetic microstructure geometry directly alters the ferromagnetic resonance
band structure. When the 5 $\mu$m $\times$ 1 $\mu$m $\times$ 50 nm Py
microstrip is cut into two strips along its length, the additional resonances
emerge in the corresponding $\mu$FMR spectra, attributed to the dynamical
modes localized at the narrow ($\sim$15 nm) regions of the inner edges of the
cut strips. Local probing by means of BLS microscopy shows a good agreement
with the $\mu$FMR data and confirms the modes localization at the inner/outer
edges of the cut strips. Micromagnetic modelling using the 2D eigensolver of
the TetraX package helps the identification of the observed modes. Moreover,
preliminary micromagnetic studies show that the inner edge mode resonance
fields can be tuned in a wide range by controlling the spatial separation
between the strips. For example, the 1i mode resonance shifts up by 173 mT (at
14.059 GHz) when going from a 20 nm to a 200 nm gap between the strips, which
corresponds to a $\sim$5.5 GHz shift in the frequency domain at fixed bias
field.

In conclusion, Ne-FIB-assisted modification of the magnetic micro- and
nanostructures is a powerful method of tailoring the nanoscale spin-wave
channels with tunable coupling and dynamical properties. Such combined
experimental approach allows for a complex characterization of the
magnetization dynamics in the closely packed confined magnetic structures
fabricated on-demand using focused ion beams.

##  4 Methods

###  4.1 Sample fabrication

The $\Omega$-shaped microresonator was fabricated on highly resistive Si(001)
substrate by means of conventional UV lithography followed by a deposition of
Cr(5 nm)/Cu(500 nm)/Au(100 nm) by e-beam evaporation and lift-off. The back
side of the substrate was metallized with 5 nm Cr/300 nm Cu/100 nm Au to
create a ground plane for the microresonator. The detailed description of the
microresonator design and fabrication process can be found elsewhere [25, 26].
As a next step, the Py microstrip was prepared using e-beam lithography,
thermal evaporation, and lift-off.

###  4.2 FIB-assisted milling

A Ne gas field ion source (GFIS) based FIB (Zeiss Orion NanoFAB) was used for
the modification of the Py microstrip. For the milling, we used Ne+ ions with
a kinetic energy of 25 keV, a 10 µm2 aperture and spot control 5. This
resulted in an ion beam current of 1.3 pA. A fluence of 5000 ions/nm2 has been
applied for milling the structures.

###  4.3 $\mu$FMR measurements

The FMR was measured using a home-built FMR spectrometer. The field-sweept
$\mu$FMR measurements were performed for different angles $\phi_{H}$ of the
bias magnetic field $H$ applied in the plane of the Py microstrip [see 1(a)].
The out-of-plane rf magnetic field is generated by injecting rf currents into
the $\Omega$-shaped antenna. The geometry of the rf circuit was designed to
yield an out-of-plane microwave excitation field with the resonant frequency
of 14.059 GHz and a maximum rf amplitude of $\mu_{0}h_{\mathrm{rf}}$ = 0.65 mT
at 85 $\mu$W of injected rf power. More details on the $\mu$FMR technique can
be found in Refs. [25, 27, 28].

###  4.4 Micromagnetic simulations

We calculate the eigenvalues and the corresponding eigenstates of the Py
strips for different orientations and magnitudes of the bias magnetic field
using the open source finite element method package TetraX [19, 20]. By
fitting the angle-dependent $\mu$FMR of the edge-trimmed Py using the Kittel
relation [marked by the red curve in Figure 2(a)], the following magnetic
parameters of the Py strip were extracted: saturation magnetization $M_{s}$ =
760 kA/m and the g-factor $g$ = 2.11 corresponding to the reduced gyromagnetic
ratio $\gamma/2\pi$ = 29.547 GHz/T. We also used the exchange stiffness
$A_{ex}$ = 13 pJ/m and the damping constant $\alpha$ = 0.008 for the
simulations. For each geometry, a triangular mesh with cell size of 5 nm was
used. To visualize the angular dependence of the FMR resonance field
$\mu_{0}H_{\mathrm{res}}$, we first calculate the frequency-swept absorption
spectra as a function of the magnitude and the direction of the bias magnetic
field. Then we extract the field-swept absorption spectra and compute their
derivatives for different azimuthal angles $\phi_{H}$ at $f_{\mathrm{res}}$ =
14.057 GHz corresponding to the experimental resonance frequency of the
$\mu$FMR microcavity circuit.

###  4.5 BLS microscopy

For the $\mu$-BLS measurements, a 532 nm continuous-wave laser was focused to
a spot size of approximately 350 nm and scanned across the strips using a high
precision positioning system with a spatial precision of $\sim$10 nm. We have
performed a line scan in the central region of the strips as shown in the top
inset of Figure 4(a), where the red arrow denotes the scanning direction. The
frequency and the intensity of the inelastically scattered light was analyzed
using a high contrast Tandem-Fabry-Pérot interferometer. See Ref. [24] for
more details on BLS microscopy.

##  5 Acknowledgments

We thank the Nanofabrication Facilities Rossendorf team at the HZDR Ion Beam
Center for the support in samples fabrication. LK and AK acknowledge the
financial support by the Deutsche Forschungsgemeinschaft within the grants KA
5069/1-1 and KA 5069/3-1. TH and HS acknowledge the financial support by the
Deutsche Forschungsgemeinschaft within Program No. SCHU 2922/1-1.

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