Collective magnetization dynamics in nano-arrays of thin FePd discs
Agne Ciuciulkaite,1, ∗ Erik Östman,1 Rimantas Brucas,2, 3 Ankit Kumar,2 Marc A.
Verschuuren,4 Peter Svedlindh,2 Björgvin Hjörvarsson,1 and Vassilios Kapaklis1, †
1
Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden
Department of Engineering Sciences, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden
3
Ångström Microstructure Laboratory, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden
4
Philips Research Laboratories, High Tech Campus 4, Eindhoven, The Netherlands
(Dated: August 14, 2019)
arXiv:1902.00403v1 [cond-mat.mes-hall] 1 Feb 2019
2
We report on the magnetization dynamics of a square array of mesoscopic discs, fabricated from
an iron palladium alloy film. The dynamics properties were explored using ferromagnetic resonance
measurements and micromagnetic simulations. The obtained spectra exhibit features resulting from
the interactions between the discs, with a clear dependence on both temperature and the direction
of the externally applied field. We demonstrate a qualitative agreement between the measured and
calculated spectra. Furthermore, we calculated the mode profiles of the standing spin waves excited
during a time-dependent magnetic field excitations. The resulting maps confirm that the features
appearing in the ferromagnetic resonance absorption spectra originate from the temperature and
directional dependent inter-disc interactions.
I.
INTRODUCTION
Arrays of closely packed mesoscopic magnets provide
a rich playground for investigations of collective magnetization dynamics. The (stray field induced) interaction
between the discs forms a link between the internal magnetization dynamics of the elements and the global response of the system1–4 . Magnetic discs are interesting
in this context, due to the richness of internal magnetic
textures and the absence of shape induced anisotropy
in their plane. Discs of certain radius and height ratio
exhibit a ground state referred to as a vortex 5,6 characterized by an in-plane magnetic flux closure. Since the
magnetic moments are curling in-plane of the discs, the
stray field from the discs is negligible when vortices are
formed, in stark contrast to the collinear state 7–10 . The
application of an external magnetic field drives the vortex
core out of the center, towards the edge of the disc. At
a given field, the vortex is annihilated and the magnetic
moment is aligned parallel to the direction of applied field
(collinear state). When the discs are in a collinear state,
their stray fields result in inter-disc interactions. Previous investigations of iron-palladium (Fe20 Pd80 ) discs,
arranged in a square array (See Figure 1) showed that
a change of temperature is sufficient to alter the magnetization dynamics10 . Above a given temperature the
energy barrier for switching from a vortex to the collinear
state and vice versa was even found to be free from hysteresis. As a consequence the magnetization state of the
system under the certain applied field becomes bi-stable:
the vortex and the collinear state have the same energy.
Here we explore the effect of changes in the inter-disc
interactions on the magnetization dynamics of soft magnetic iron-palladium alloy discs, using ferromagnetic resonance (FMR) and micro-magnetic simulations11 . The
changes in the inter-disc interactions are obtained by rotating the magnetisation of the discs as well as altering
the stray field from the discs by changing the sample
FIG. 1. Atomic force microscopy image of a square array
of Fe20 Pd80 alloy discs with the high symmetry directions
indicated as [10] and [11] by arrows.
temperature.
II.
A.
METHODS
Sample
The investigated array consists of iron-palladium alloy (Fe20 Pd80 ) discs arranged in a square lattice as illustrated in Figure 1. Each disc has a radius of 225 nm and
a thickness of 10 nm, with a center-to-center distance of
the discs equal to 513 nm in the [10] direction. This results in an inter-disk distance of 53 nm and 275 nm along
the [10] and along the [11] direction, respectively. A detailed description of the sample preparation is provided
by Östman et al. 10 .
2
B.
Ferromagnetic resonance measurements
Magnetization dynamics of the Fe20 Pd80 disc arrays
were measured using X-band cavity FMR equipped with
with variable-temperature sample holder. A static magnetic field was applied in-plane, while a time-dependent
spatially uniform (wave vector ~k=0) magnetic field excitation, with a frequency of 9.8 GHz was applied perpendicular to the plane of the sample7,12 . The static
magnetic field was swept from 0 to 300 mT and the measurements were carried out at temperatures ranging from
80 K to 293 K. The strength of the applied static field
ensured that discs were measured in the collinear state.
A second set of FMR measurements were performed
using a vector-network-analyzer (VNA) at a room temperature, by an in-plane field sweep, utilizing a coplanar waveguide. The detailed description of this measurement setup is described by Wei et al. 13 . The linewidth
∆Ht versus frequency f data obtained from these measurements were fitted to a linear function, extracting the
Gilbert damping coefficient α. The extracted value for
α was 1.8×10−2 and was later employed as a parameter
in the micromagnetic simulations of the magnetization
dynamics in the Fe20 Pd80 disc array. It was adjusted
in order to match the calculated FMR absorption peak
amplitude to one measured experimentally.
C.
Micromagnetic simulations and standing spin
wave map calculations
Micromagnetic simulations were performed using
MUMAX314 . The exchange stiffness constant, Aex , defining the inter-spin coupling in the magnetic material was
adjusted to qualitatively reproduce the experimental observations. It is known that Aex is temperature dependent and decreases with increasing temperature15 .
Therefore a higher temperature implies softer standing
spin wave modes, excited within the discs. After the
initial simulations, a value of Aex =3.36 pJ/m was used
for the subsequent micromagnetic simulations, in order
to qualitatively reproduce the experimentally observed
FMR spectra features. A broader discussion and motivation regarding this choice is provided by Ciuciulkaite 16 .
Finally, the Gilbert damping parameter, α, describing
the losses in the system and proportional to the FMR
absorption linewidth was chosen to be 1.9×10−2 , after
initially taking the value determined from the VNA-FMR
measurements.
In the micromagnetic simulations, four discs with 225
nm radius and 10 nm thickness were placed in 2-by-2
square lattice with a lattice parameter of 513 nm. The
in-plane cell size was defined as 0.513(18)·lex , where lex is
the exchange length, a material parameter determined by
the Aex and Msat 14 . The cell size along the z-direction,
i.e. the thickness of the structure, was set to 5 nm for
all simulations. Periodic boundary conditions (PBC)14
were applied in both lateral directions (PBCx =PBCy =3,
PBCz =0). The FMR simulations were performed applying a static magnetic field in-plane of the lattice, relaxing the system, and then applying a time-dependent
field excitation out-of-plane, with an analytical expression of A·sin(2πf t) or A·sinc(2πf t). The static magnetic field is applied at an angular offset of 2◦ from the
principal in-plane directions, in order to lift any degeneracy in the simulations, related to the high symmetry directions we will be investigating, being parallel to
the [10] and [11] direction of the disc lattice. The amplitude of the time-dependent excitation was A=5 mT,
the frequency was f =9.8 GHz, the duration of the sinusoidal time-dependent magnetic field excitation was
10 ns, and the sampling period was 1 ps. The frequency bandwidth for the sinc function excitation was 20
GHz. The temperature-dependent saturation magnetization, Msat (T ), was used to account for the temperature
dependence of the FMR response of the soft magnetic
discs. The Fe20 Pd80 alloy has a Curie temperature, TC ,
of 463 K10 . To a first approximation the temperature dependence of the magnetization can be described by the
modified Bloch behaviour:
β
T
Msat (T ) = Msat (0) 1 −
,
TC
(1)
where Msat (0) is the saturation magnetization at 0 K
and is 5.9×105 A/m for the Fe20 Pd80 alloy and β=1.6910 .
Here we would like to note that we do not account for
any other temperature induced effects in our simulations
than the temperature dependence of the saturation magnetization.
A complementary method for describing the observed
features in the FMR spectra are standing spin wave
(SSW) mode maps. The maps were calculated from the
spatial micromagnetic evolution of the magnetization as
excited by the time-dependent magnetic field. A fast
Fourier transform (FFT) for each of the discrete spatial
elements was performed, resulting in the spatial maps of
the amplitude of the magnetization dynamics.
III.
RESULTS AND DISCUSSION
A.
FMR
The FMR spectra measured at room temperature,
with the static field applied along [10] and [11] in-plane
directions of the array, as indicated in Figure 1, are
shown in Figure 2. The ferromagnetic resonance shifts
to slightly higher applied magnetic fields, when the field
is applied along [10] direction as compared to the [11]
direction.
This shift reflects the effect of the inter-disc interaction
strength, due to the change in the applied static magnetic
field and hence the stray field direction. Furthermore, the
measured FMR absorption spectra exhibits a split in the
3
FIG. 2. The measured FMR spectra of square arrays at room
temperature with the field applied along [10] and [11] directions. The shaded grey regions indicate characteristic FMR
absorption features of the investigated structure: a shoulder
feature, appearing before the main absorption peak, and a
main FMR field, µ0 Hres0 .
main absorption peak in the [10] direction. We will call
this feature before the main absorption peak a shoulder
feature (See Figure 2 in the following). The change in
interaction strength is due to a modification of inter-disc
coupling by changing the magnetisation direction of the
discs. Effectively, along the [10] direction a disc has two
nearest neighbours while along the [11] direction number
of interacting nearest neighbours is doubled as described
below.
B.
Micromagnetic simulations on a single disc
Micromagnetic simulations of a single disc were carried out to identify the basic features of the elements,
i .e. the response of the discs in absence of interactions.
A FMR frequency versus applied static field map was
calculated for a wide range of frequencies, using the expression A·sinc(2πf t) for the time-dependent magnetic
field excitation and is provided in Appendix A, Figure
A.1. Since the FMR measurements were carried out at
the frequency of 9.8 GHz, a line cut along this frequency
was taken and is shown as an FMR absorption spectrum
in Figure 3 (a). The shaded grey regions indicate the
strength of the applied static magnetic field, for which
the SSW modes excited in the disc were simulated and
are presented as maps of normalized magnetization precession amplitude and phase in Figure 3 (b) and in Appendix B Figure B.1.
Figure 3 (b) presents the SSW maps for an amplitude
and phase for applied magnetic field corresponding to the
region where the shoulder feature in Figure 2 is observed.
It is indicated on the left-hand-side (LHS) of the main
absorption peak as µ0 HLHS in Figure 3 (a). This mode
is mainly constrained near the middle of the disc and
along a line parallel to the direction of the applied mag-
FIG. 3. (a) FMR absorption spectrum for a single disc, taken
as a linecut from the map in Figure A.1 (see Appendix A) at
9.8 GHz frequency. The shaded grey regions indicate external
magnetic fields at which the SSW profiles (mz component,
amplitude of precession) were calculated; (b) FFT amplitude
and phase maps calculated at a shoulder feature at 197 mT
applied magnetic field.
netic field. In the remainder of this communication we
will refer to the area in the middle of a disc as the central
area. The largest precession takes place at two symmetric points outside of the disc center, along the direction
of static magnetic field. In addition to these symmetry
points, the magnetic moments also resonate at the edges
of the disc along a line perpendicular to the magnetic field
direction. In the following, we will refer to this mode as
a perpendicular edge mode. To SSW modes appearing
at the edges along a line parallel to the direction of the
applied magnetic field, we will be referring to as parallel
edge modes. The SSW spatial mode maps calculated at
higher magnetic fields indicate the following spatial mode
profiles: at 210 mT field - uniform precession, at 226 and
255 mT - edge modes (see Appendix B Figure B.1).
C.
Micromagnetic simulations of square arrays
Subsequent simulations were carried out on square arrays containing interacting discs, but otherwise identical
to the building block described in the previous section.
In order to explain the shoulder feature in the FMR spectra for the [10] direction (Figure 2) and to further investigate which SSW modes are excited in the discs, spatial
amplitude and phase maps (see Figure 4 (b)-(c)) were
4
ified, due to effective increase of nearest neighbors along
the [11] direction and the magnetic moment precession
amplitude significantly increases in the center and perpendicular edge areas. Simultaneously, the phase angles
in the center and at the parallel edges of a disc decrease,
while the overall phase angle distribution over each disc
becomes more uniform as compared to the case when external field is along the [10] direction. Furthermore, the
amplitude of the perpendicular edge mode and the extension over the edges is smaller for the [10] external field
direction as compared to the [11] direction. This again
is a hallmark of the effect the proximity to the nearest
neighboring discs has on the magnetization dynamics.
Comparison of the SSW mode amplitude and phase spatial maps, to those of an isolated single disc reveals that
the amplitude of magnetic moments in the center of an
isolated disc is smaller as compared to the same area in
the arrays.
These results clearly demonstrate the effect of the stray
field coupling between discs in an array on the internal
magnetization dynamics of the elements. Modifying the
inter-disc interaction strength alters the magnetic moment fluctuations at the center of a disc which results
in appearance (or disappearance) of the shoulder feature
in an FMR absorption spectrum. A further important
observation arising from the simulations, is the absorption modes appearing at higher fields after the main absorption peak. These features, indicated as µ0 Hres1 and
µ0 Hres2 at around 229 mT and 251 mT, respectively in
Figure 4 (a)), are not observed in the experimental FMR
spectra (Figure 2). They arise from the idealised circular
shape of the discs in the simulations. In real samples the
lithographically fabricated elements are imperfect, leading to the absence of these modes17 . The origin of spin
wave maps for these simulated modes is discussed in more
detail in the Appendix C.
FIG. 4.
(a) Computed FMR absorption spectra with
Msat (T = 300K) and static magnetic field applied along the
[10] and [11] directions. Grey shaded regions represent external magnetic fields at which SSW modes were calculated; (b)
spatial amplitude and (c) phase maps of SSW modes calculated for the two applied field directions. Color bars indicate
normalized amplitude and phase angles in degrees; the bottom
panels in (b) and (c) represent amplitude and phase linecuts
in the respective maps along the [10] and [11] directions.
calculated at external magnetic fields of 191 mT and 186
mT along the [10] and [11] directions, respectively.
The shoulder feature on the left-hand-side of the FMR
absorption peak, observed in the experimentally measured spectra (Figure 2), is also present in the calculated
results, when the external magnetic field is applied along
the [10] direction (Figure 4 (a)). In this case, calculated
SSW mode spatial maps show that the magnetic moments are strongly out of phase at the parallel edge and
center areas of the discs. When the direction is changed
to [11], the interaction strength between the discs is mod-
D. Ferromagnetic resonance response of
iron-palladium arrays: comparison of experiments
and simulations
A comparison of the measured and calculated spectra
is provided in Figure 5, where we display the change in
the ferromagnetic resonance field with increasing temperature, for applied fields in the [10] and [11] directions.
The strength of the interaction alters the resonance field
as observed in the measured FMR spectra. Same qualitative changes are observed in the micromagnetic simulations. The resonance field increases as the temperature is increased (see Figure 5). Thermal fluctuations
become more prominent with increasing temperature and
a stronger magnetic field is needed to align the moments,
shifting the resonance field to higher values. This trend is
seen in both calculated and measured FMR spectra. At
low temperatures the experimental and calculated resonance fields are comparable. At elevated temperatures
the resonance fields calculated from micromagnetic sim-
5
FIG. 5. Comparison of the resonance fields obtained via micromagnetic simulations (red squares) and experimental values (black squares) extracted from FMR measurements of the
Fe20 Pd80 alloy disc array. The external static magnetic field
was applied along [10] and [11] directions (empty and filled
symbols, respectively). Lines are guides to the eye. Error
bars and resonance fields were obtained from fitting of FMR
amplitude to a Lorentzian peak profile.
ulations increase faster than the measured values. The
micromagnetic simulations offer a qualitative agreement
with the experimental data which is more clearly seen
from a comparison of measured (Figure 2) and calculated FMR absorption spectra (Figure 4(a)) at elevated
temperatures.
In Figure 5 we can further observe that the splitting
between the simulated resonance field values along [10]
and [11] directions, is always present in the investigated
temperature range. The curves for the [10] direction
lie higher in field, compared to those for the [11] direction in the investigated temperature range. In related
studies, where neighboring elongated micromagnetic elements were antiferromagnetically coupled by stray fields,
resonance field shifts depending on the strength of the
interaction, were also reported18,19 . However, the difference in resonance field for [10] and [11] directions
(Hres[10] − Hres[11] ) calculated from micromagnetic FMR
results (by fitting amplitude to a Lorentzian profile peak
and extracting its position), follows a different trend as
compared to the resonance field extracted from experimentally obtained data (Figure 5). This can be attributed to the fact that while exchange stiffness and
Gilbert damping parameters are temperature dependent,
in micromagnetic simulations we kept these parameters
constant for all temperatures. Furthermore, no thermal
fluctuations were accounted for in micromagnetic simulations.
Finally, we would like to comment upon the scaling
of the experimental resonance field difference Hres[10] −
Hres[11] versus temperature. The difference decreases
monotonically up to some temperature, where it effec-
tively becomes zero, while beyond that it seems to increase again. This behaviour potentially relates to the
internal magnetization dynamics within the discs. As
already reported previously by Östman et al. 10 , a temperature range exists where bi-stability is attainable for
the collinear and vortex states, which should also be accompanied by a strong modification in the magnetization
dynamics of the individual discs. This range was found
to be independent of applied field direction, for temperatures above ≈ 220 K from measurement protocols involving very low frequencies (hysteresis curves, recorded
employing the magneto-optical Kerr effect and field cycling of 0.4 Hz). In the present study, Hres[10] − Hres[11]
becomes zero at ≈ 270 K, considerably higher than that
in Östman et al. 10 . The time scale of dynamics, probed
in this work, is much shorter (sub-ns, FMR measured
at 9.8 GHz), hinting for this strong apparent temperature shift. A more thorough survey of this effect would
greatly benefit from more detailed micromagnetic simulations, incorporating also the magnetization dynamics
of the material at all relevant length-scales (inter- and
intra-disc).
The shift in the resonance field is attributed to a difference in the stray field induced inter-disc coupling,
along the [10] and [11] directions. In Figure 6, we show
the spatial maps of the demagnetizing field amplitude,
computed for arrays with initial collinear magnetization
states, along [10] and [11] directions. Having the magnetisation of the elements along the [11] direction results
in an increased stray field coupling between neighboring discs. Clearly, a simple point-dipole-like approximation is not sufficient to describe the effective magnetostatic coupling. Even though the spacing between neighboring discs in the [10] direction is smaller - and thus
one would expect a stronger coupling - the demagnetizing field strength is distributed broadly along the disc
perimeter for the [11] case (see the bottom panels in Figure 6). This leads to an stronger coupling for the applied
fields along the [11] direction. The root of these effects
are clearly seen in Figure 7, where we provide spatial
maps of static magnetization components perpendicular
to the direction of applied magnetic field. A map of the
my component with Hres ||[10], shows a dominant positive direction for these, in accordance to the 2◦ offset of
Hres from the [10] direction in our simulations. When
Hres ||[11], the magnetization components mxy perpendicular to applied fields, obtain non-zero values only at
the discs’ rim and with maxima at the positions where
the gap to neighboring disc is minimum. In contrast to
the case where Hres ||[10], now all of the four gaps related to neighboring discs are active. The maps show
that mxy ⊥ [11] is of the same sign at the disc edges
along the [10] and [01] directions which further confirming inter-disc coupling via stray fields.
6
nance field as well as the fine structures of the resonance.
The inter-island interaction is found to be larger when
FIG. 7. Spatial maps of magnetization components distribution perpendicular to the direction of the external magnetic
field. Maps were obtained from micromagnetic simulations
of arrays with initial magnetization along [10] and [11] directions relaxed in a static magnetic field corresponding to Hres
in magnitude for each of the directions.
FIG. 6.
Spatial distribution maps of demagnetizing field
strengths, obtained from micromagnetic simulations of arrays with initial magnetization along [10] and [11] directions
(first and second row, respectively) relaxed in a 0 mT static
magnetic field and a field corresponding to the ferromagnetic resonance field (left and right columns, respectively).
Bottom panels represent polar plots of the demagnetizing
field, along the rim of a disc, as indicated by a circle in the
“m[11]µ0 H= 0 mT” map.
the magnetisation of the islands are in the [11] as compared to [10] principal directions. The origin of this effect
arises from the transverse component of the magnetisation, which enhances the inter-island interactions in the
transverse directions, illustrating the need of including
inner magnetic structure of the magnetisation to explain
both dynamic and static results.
The results presented in this study, show that micromagnetic simulations are suitable for investigations
of spatial profiles of SSW modes, excited in arrays of
stray field coupled nanomagnets. These can be utilized for analyzing and designing the microwave response
of extended arrays of thermally active and interacting
nanomagnets20–22 , having interesting topological and interaction schemes, such as artificial spin ices1,3,4,23–26 .
The latter could enable the design and fabrication of reconfigurable magnonic devices3,24 .
ACKNOWLEDGMENTS
IV.
CONCLUSIONS
We have investigated the magnetization dynamics in a
square array of Fe20 Pd80 alloy discs by means of FMR
measurements and micromagnetic simulations. The ferromagnetic resonance is found to increase in field with increasing temperature which we attribute to the decrease
of the magnetic moment. The results were qualitatively
reproduced and confirmed by micromagnetic simulations.
The effects of stray field induced interaction of the islands is seen in the orientation dependence of the reso-
The authors would like to acknowledge financial support from the Swedish Research Council (VR), the
Swedish Foundation for International Cooperation in Research and Higher Education (STINT) and the Knut
and Alice Wallenberg Foundation project “Harnessing light and spins through plasmons at the nanoscale”
(2015.0060). This work is part of a project which has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement no. 737093.
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agne.ciuciulkaite@physics.uu.se
vassilios.kapaklis@physics.uu.se
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8
Appendix A: Dispersion relation of a single FePd
disc
The complete map of available resonance modes for a
single nanodisc resonator is shown in Figure A.1 as a frequency vs external magnetic field map. The color bar
represents the log FFT amplitude of spin precession. Below the annihilation field, the spectrum is dominated by
low frequency modes, corresponding mostly to the gyrotropic motion of the vortex core27 . The highest intensity peak corresponding to a vortex mode shows similar
dispersion relation as that for isolated cylinders exhibiting single domain state18 . At stronger external magnetic
fields above the saturation field, when the disc is in a
collinear magnetic state, a significant increase of the resonance frequency is observed and highest intensity peak
follows Kittel-like behaviour of resonance frequency for
continuous ferromagnetic films. Below the main resonance peak are two smaller intensity features corresponding to the edge modes.
FIG. A.1.
The calculated map of the log and normalized FMR amplitude response to a 20 GHz bandwidth sinc
function form magnetic field excitation, for different external
static fields.
Appendix B: Magnetization maps of single disc SSW
modes
SSW maps calculated at the resonance field, which is
210 mT for a single disc, reveal that the area of constrained magnetic moments becomes more concentrated
at the edges along the direction of an applied magnetic
field. Meanwhile, magnetic moment fluctuations become
more intense at the perpendicular edges. In contrast, the
SSW modes calculated at 226 mT and 255 mT external
magnetic fields appear at the parallel edges. The spatial log normalized FFT amplitude and phase maps calculated at fields just below the ferromagnetic resonance
field and above it are shown in Figure B.1. The evolution of parallel edge modes can be observed. Just before
and after the ferromagnetic resonance (at 205 mT and
FIG. B.1. The calculated spatial log and normalized FFT
amplitude and phase maps of an isolated single disc at different fields indicated in Figure 3 (a).
216 mT fields, respectively), spin magnetic moment fluctuation amplitude is uniform throughout the whole area
of a disc except for the parallel edges. At the additional
absorption features observed after the FMR at 232 mT
and 256 mT fields, respectively, fluctuation amplitude
becomes larger at the edges while throughout the rest of
a disc, it goes to zero, in other words, the rest of the spins
“freeze in”.
Appendix C: Magnetization maps of edge modes in
an array
In this section we discuss the resonance and edge
modes observed in the calculated FMR spectra in Figure 4 (a). At the resonance field, indicated by an arrow
9
at 216 mT field in Figure 4 (a), the log FFT magnetisation amplitude of spin magnetic moments is highest
at the perpendicular edge area as in the single disc case
but with the constrained magnetic moment area reduced
(compare Figure B.1 amplitude maps of modes at 205
and 216 mT fields with a top panel in Figure C.1. When
the external magnetic fields are approximately 229 mT
and 250 mT, the parallel edge modes are excited. Mode
maps reveal that in this case the magnetic moments fluctuate the most at the edges along disc coupling direction
C.1. This hints that disc interaction occurs through the
fluctuating magnetic moments. As a result, along [10]
direction the system becomes less stiff and resonance occurs at lower fields when the field is applied along [10]
than along [11] direction as can be seen in the absorption spectra in Fig. 4 (a) at around 229 and 251 mT
fields.
FIG. C.1. Spatial maps of SSW modes calculated for the
Fe20 Pd80 alloy disc array at the field values indicated by
shaded grey regions in Figure 4 (a).
In the real arrays such modes are absent, due to shape
imperfections and edge roughness of the discs arising
from a lithographic fabrication method. Furthermore,
due to computational limitations we calculated significantly smaller amount of nanodiscs than the real sample actually contains17 . This is also further supported
by micromagnetic simulations performed accounting for
shape and edge imperfections which do not reproduce
these modes16 .