Supplementary materials
Towards graded-index magnonics: Steering spin waves
in magnonic networks
C. S. Davies1, A. Francis1, A. V. Sadovnikov2, S. V. Chertopalov3, M. T. Bryan4, S. V.
Grishin2, D. A. Allwood4, Y. P. Sharaevskii2, S. A. Nikitov2,5, and V. V. Kruglyak1
1
School of Physics, University of Exeter, Stocker road, Exeter, EX4 4QL,
United Kingdom
2
Laboratory “Metamaterials,” Saratov State University, Saratov 410012,
Russia
3
Donetsk National University, 24 Universitetskaya Street, Donetsk, 83001,
Ukraine
4
5
1
Department of Materials Science and Engineering, University of
Sheffield, Sheffield, S1 3JD, United Kingdom
Kotel'nikov Institute of Radioengineering and Electronics, Russian
Academy of Science, Moscow 125009, Russia
Note 1. Procedure of the TRSKM imaging experiments
In the main text, we presented images of spin waves propagating in Permalloy
microstructures. The images were acquired using the time resolved scanning Kerr
microscope (TRSKM) [1]. In this supplementary note, we describe in greater detail the
experimental procedure that was adopted during the measurements.
At the first stage, the diffraction-limited optical "probe" spot of the TRSKM was
focused on the "leg" of the T-junction within the sample, as schematically indicated in
Fig. S1 (c). The sample was excited by a global uniform in-plane pulsed magnetic field,
serving as a "pump". The time-resolved magnetic response (Fig. S1 (a)) was measured from
the area under the probe spot by recording the Kerr rotation of the probe's polarisation as a
function of the pump-probe time delay. The fast Fourier transform (FFT) spectrum
calculated from the time-resolved signal was used to identify frequency (-ies) of the dominant
modes in the specific area of the sample (Fig. S1 (b)). Due to the uniformity of the pulsed
field, the spectrum was dominated by the most uniform ("quasi-uniform") precessional mode.
The same procedure was used to identify the frequency of the quasi-uniform mode in the
"arms" of the T-junction.
Fig. S1 The time resolved Kerr signals (a) and their fast Fourier transform (FFT) spectra (b) are
shown for the optical probe spot focused in the two positions on the T-junction's surface
indicated in (c). The blue and red curves (signal frequencies of 5.75 GHz and 8.25 GHz,
respectively) correspond to the signals acquired from the probe positions indicated by disks
of the same colour in the "arms" and the "leg" of the T-junction. (c) Kerr images of the outof-plane component of the magnetization within the T-junction continuously pumped at
2
8.24 GHz are shown for four moments of time 30.4 ps apart within one cycle of the
microwave pump. The bias field of 500 Oe is applied parallel to the leg of the junction.
At the second stage, the pulsed magnetic field was a replaced by a harmonic magnetic
field as the pump. The frequency of the microwave was tuned to the frequency of the quasiuniform mode in the leg of the junction. The images of the excited magnetisation dynamics
were acquired by raster-scanning the optical probe over the sample, while keeping the time of
the probe's arrival fixed relative to the phase of the microwave pump (Fig. S1 (b)). By
combining images acquired at several equidistant moments over the full cycle of the pump
field, the movies of spin wave propagation in the structure were created.
3
Note 2. Angular dependence of the magnetisation dynamics in the T-junction
In the main text, we presented images of the dynamic magnetisation in a Permalloy Tjunction acquired for three angles of the bias magnetic field relative to the junction's
symmetry axis. In this supplementary note, we present additional data that elucidates the
angular dependence of the observed magnetisation dynamics.
Fig. S2 (a) The frequencies of the quasi-uniform modes in the leg (red) and arms (blue) of the Tjunction are plotted as a function of the angle α between the direction of the bias field of
500 Oe and the symmetry axis of the T-junction. (b) and (c) Series of of Kerr images of the
out-of-plane component of the magnetization within the T-junction continuously pumped at
8.24 GHz are shown for four moments of time 30.4 ps apart within one microwave cycle.
The bias field of 500 Oe is applied at -15° (b) and +15° (c) relative to the leg of the
junction.
4
Fig. S2 (a) shows the dependence of the dominant mode frequencies in the leg and arms
of the T-junction on the orientation of the bias magnetic field relative to the symmetry axis of
the sample. The dependence follows the trend expected for the shape anisotropy. In
particular, the frequency is maximised (minimised) when the field is applied parallel to the
length (width) of the relevant part of the sample, as e.g. was observed in Ref. [2]. At the
same time, the frequencies of the dominant modes in the leg and arms of the T-junction
become equal when the field is applied at 45° relative to its symmetry axis.
Fig. S2 (b) and (c) show series of snapshots of the magnetisation dynamics in the sample
acquired for the bias magnetic field was applied at angles of -15° and +15° to the symmetry
axis of the sample, respectively. As in Fig. S1 (c), the different snapshots correspond to
different phases of on cycle of the microwave pump.
5
Note 3. Micromagnetic simulations of spin waves in a Permalloy T-junction
with a narrow leg
In the main text, we presented the measured and numerically simulated images of the
magnetisation dynamics near the Permalloy T-junction. The images clearly show that the
precessional dynamics initiated near the leg-arm boundary then propagate along the arms of
the structure, as prescribed by the direction of the bias magnetic field. However, due to the
same widths of the leg and arms of the structure, the beams of spin waves in terms of which
the observations are interpreted are also quite wide and therefore not as distinct as one could
wish. In this supplementary note, we present results of simulations for a Permalloy Tjunction that has a narrower (1 µm wide) leg, which leads to a better defined spin wave
caustic beam propagating into one of the arms of the structure.
Fig. S3 (a) The space-averaged temporal response of the Permalloy T-junction to excitation by a
uniform pulsed magnetic field is shown for a bias magnetic field of 500 Oe oriented at 15°
to the symmetry axis. (b) The FFT spectrum of the signal in (a) is shown. The peaks at
5.5 GHz and 10.3 GHz correspond to the quasi-uniform resonant modes of the arms and the
leg, respectively. (c) The images of the out-of-plane component of the dynamic
magnetization within the T-junction continuously pumped at 10.3 GHz are shown for four
moments of time 24.3 ps apart within one cycle of the microwave pump.
The simulations were performed using the same methodology as described in the main
text. Fig. S3 (a) shows the sample's time-resolved response to a pulsed excitation, while
Fig. S3 (b) shows the FFT spectrum of the signal from panel (a). The spectral peaks
observed at 5.5 GHz and 10.3 GHz correspond to the quasi-uniform resonant modes of the
arms and the leg, respectively. Fig. S3 (c) shows a series of snap shots of the spin wave beam
excited in the sample by a uniform harmonic magnetic field at the frequency of 10.3 GHz.
The smaller width of the leg leads to the higher frequency of its quasi-uniform mode and also
6
to the observed smaller cross-section of the spin wave beam excited at the frequency in the
right arm of the structure.
Note 4. Depth dependence of the magnetisation dynamics in the Permalloy
structure
The wave vectors of the magnetostatic spin waves observed here to propagate into the
arms of the Permalloy T-junctions are nearly perpendicular to the direction of the static
magnetisation. Such magnetostatic spin waves are known to have surface character [3]. This
means that the precession amplitude decays exponentially from the surface into the depth of
the film. Moreover, the surface near which the maximum amplitude is observed depends on
the direction of the wave propagation. At the same time, the optical skin depth at the probe
wavelength of 400 nm is only about 20 nm, which smaller than the thickness of the
Permalloy film of 100 nm. Hence, there is a possibility that spin waves are actually emitted
into both arms of the T-junction, while one of them is not actually observed due to the
insufficient optical skin depth of the probe. In this supplementary note, we verify and
disprove this hypothesis by numerical micromagnetic simulations.
Fig. S4 The images of the out-of-plane component of the dynamic magnetization are shown for the
top (left) and bottom (right) surfaces within the Permalloy H-shaped sample continuously
pumped at 7.52 GHz.
The simulations were performed for an H-shaped microstructure with dimensions and
magnetic properties identical to those for Fig. 2 of the main text. The bias magnetic field of
500 Oe was applied at 15° to the symmetry axis of the structure parallel to its "leg", as shown
in Fig. S4. The sample was excited by a uniform microwave magnetic field at the frequency
of the quasi-uniform mode of the leg. The results of the simulations are shown in Fig. S4 for
the top and bottom surfaces of the sample. We observe that the colour contrast is slightly
different on the opposite surfaces, confirming the surface character of the spin wave modes
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emitted into the left and right arms. However, the propagation direction of the emitted spin
waves is the same for both surfaces. Thus, the observed uni-directionality of the spin wave
emission cannot be explained by the surface character of the excited spin waves.
However, the difference of the spin wave amplitude near the opposite surfaces does
manifest itself in our data. Indeed, Fig. S2 shows that the spin waves appear to travel further
when the bias magnetic field is applied at α = -15° as compared to the case of α = +15°. This
is because the Damon-Eshbach magnetostatic spin waves are localised at the upper surface at
α = -15°, whereas they are localised at the bottom surface at α = +15°. However, this effect is
quite small here since the spin wave wavelength is more than one order of magnitude greater
than the film thickness.
Note 5. Theoretical formalism used in the analysis
In the main text, we interpreted our observations in terms of the spatial variation of the
isofrequency curves and the directions of the group velocities of magnetostatic spin waves
that were calculated for each point in the samples. In this supplementary note, we state the
equations that were used in the calculations.
We used the theory developed in Ref. [3]. The isofrequencies of magnetostatic spin
waves were plotted using the following equation


f k x , k y     1k x cos   k y sin     2   2  1 k y cos   k x sin   
2
1
2
2
 2   k x2  k y2  k x cos   k y sin    k y cos   k x sin   




2
(1)

1
2
2 
 cot  s  k x cos   k y sin    k y cos   k x sin     0



Here, kx and ky are the projections of the wave vector on the axes of a Cartesian coordinate
system, α is the angle between the local direction of the magnetisation and the Y coordinate
axis, s is the thickness of the magnetic film, while µ and ν are parameters describing the local
response of the film to excitation by alternating magnetic film via the following permeability
tensor


 0
 i
0 i
1
0
0 .
(2)

The tensor is written in local coordinates in which the Y coordinate axis is aligned with the
static magnetisation.
The magnetostatic spin wave dispersion is implicitly defined by the equation (1)
supplemented by the dependence of µ and ν on the frequency ω, the projection of the internal
magnetic field Hi on the direction of the static magnetisation, the saturation magnetisation M
and the gyromagnetic ratio γ given by
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  1
M 
M H
and  
,
2
2
H 2   2
H  
(3)
where ωM = 4πMγ and ωH = γHi.
The group velocity could then be calculated as
xˆ
v g  k  
f
f
 yˆ
k x
k y
(4)
f 
from the equations (1) and (3). However, here, it was determined numerically using the
corresponding tool available in MatLab to find the gradient of an implicitly defined function.
Note 6. Spatial distribution of the directions of the group velocity and wave
vectors
In the qualitative analysis of the spin wave propagation in graded-index media
presented in the main text, we operated with the local directions of the group velocity and
wave vectors, derivation of which was illustrated by plotting the isofrequency curves at each
point of the sample. In this supplementary note, we present an example of the spatial
distributions of the directions of the group velocity and wave vectors over the entire area of
the Permalloy T-junction.
For given values of the spin wave frequency and applied internal magnetic field, the
two in-plane components of the wave vector are related by the equation (1). Hence, in order
to uniquely define the group velocity and wave vector of a spin wave, one has to specify one
of the wave vector components, ideally the one that is conserved (at least approximately) as
the wave propagates. So, we fix the value of kx at 0.94 µm-1, which corresponds to the peak
spectral amplitude of the spin wave launched into the arms of the T-junction as a result of its
pumping with uniform microwave magnetic field at the frequency of 7.52 GHz (see Fig. 3(b)
of the main text). The results of the calculation are shown in Fig. S5 for the incident (a) and
scattered (b) waves. Using such maps of the magnetostatic spin wave group velocity, it is
possible to trace the path that the spin waves can take within inhomogeneously magnetised
structures. We also draw your attention to the small regions at the right of the leg/arms
interface in which there are no solutions in ky (for the specific value of the kx) – this is an
example of how wave-vector-specific ‘forbidden regions’ can be formed within magnetic
media.
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Fig. S5 The vector maps of the spatial variation of the group velocity (top panel) and the wave
vector (bottom panel) in the Permalloy T-junction are shown for the incident (a) and
scattered (b) spin waves with the horizontal projection of the wave vector of kx = 0.94 µm-1.
Each unit vector represents an average over 5 x 5 computational cells.
10
Note 7. Absence of the spin wave beam for negative kx values
From Fig. 3 (b), we could expect that a beam of spin waves with negative kx values
could be excited into the left arm of the T-Junction, and expects its amplitude to be weaker.
In this supplementary note, we elaborate on the reasons why the beam with negative kx values
is not actually observed.
Fig. S6 (a) The static magnetic state of the Permalloy T-junction is shown for the bias magnetic
field applied at +15° to the symmetry axis. The image is a zoomed copy of that presented
in Fig. 3 (a), with a focus on the left side of the junction-arms interface. The arrows
represent the average direction of the magnetisation for 2x2 simulation cells. The colour
scale represents the projection of the internal field on the magnetisation direction. (b)
Three characteristic sets of isofrequency curves are shown for the boxed pixels in panel (a)
in respective order. Indices "i" and "r" correspond to the unit vectors of the wave vectors k̂
and group velocity v̂ the incident and reflected waves, respectively. (c) The spatial
distribution of the group velocity unit vectors is shown for kx = -1 μm-1.
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Fig. S6 (a) shows the static configurations of the magnetisation and internal magnetic
field in the Permalloy T-junction, in particular over its left arm in which the beam of spin
waves with negative kx values could be expected. The distribution of the internal magnetic
field is strongly non-uniform, as expected for this non-ellipsoidal sample. Hence, in order to
understand the propagation of spin waves in this magnetic "landscape", we need to consider
the isofrequency curves in different points of the sample, characterised by different values of
the internal magnetic field and different directions of the static magnetisation. So, Fig. S6 (b)
shows isofrequency curves plotted for the three pixels highlighted in Fig. S6 (a), while
Fig. S6 (c) shows the vector map of the group velocity directions over the entire junction.
Upon excitation at the leg-arms interface, the spin waves propagate crudely parallel to
the interface towards the left arm of the T-junction, as shown in the schematic from the right
panel of Fig. S6 (b). However, as the spin waves then approach the lower left corner of the
T-junction, the variation of the internal field causes the beam to "curve" towards the lower
geometrical edge of the left arm. This results in a strong concentration of the wave amplitude
in the lower left geometrical boundary, as observed e.g. in Fig. 2 (c) of the main text. The
field non-uniformity also causes a distributed reflection of the waves in this region.
However, the reflected waves are steered directly back upon themselves, towards the right
arm, as shown in the left and centre panels of Fig. S6 (b). Hence, even though negative-kx
spin waves are excited at the leg-arms interface, they are prevented from propagating into the
left arm.
Note 8. Methods
Sample fabrication. The Permalloy (Ni80Fe20) T-shaped waveguides of 100 nm
thickness were formed on 0.17 mm thick microscope glass cover slips by a combination of
electron beam lithography, high vacuum magnetron sputtering and lift-off.
Measurements. The Permalloy samples were overlaid "face-up" onto a 0.5 mm wide
signal line of a microwave coplanar waveguide (CPW) etched on a printed circuit board. The
measurements were then performed using a pump-probe scheme implemented within a timeresolved scanning Kerr microscope (TRSKM). A 80 MHz train of 100 fs optical "probe"
pulses of 400 nm wavelength was produced by frequency doubling the output of a
Ti:Sapphire Tsunami laser. The optical beam was focused into a sub-micrometre spot on the
sample's surface. An optical bridge detector in combination with a lock-in amplifier was
used to measure the polar Kerr rotation of the polarisation of the reflected optical beam,
which was proportional to the dynamic out-of-plane component of the precessing
magnetisation. The optical pulse train was synchronised either to a Picosecond Pulse Labs
pulse generator or to a Rohde&Schwarz microwave generator, producing 70 ps long
electrical pulses or 1 - 20 GHz continuous wave (cw) wave forms of electrical current,
respectively. The electrical signals fed into the CPW produced a fast varying magnetic field
that was used as a "pump" to excite the sample's magnetisation. The magnetisation dynamics
were imaged by raster scanning the microscope's objective lens (and therefore also the probe
spot) using a parallel piezo-electric stage. The temporal resolution was achieved by scanning
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the time of arrival of the optical probe pulses using an opto-mechanical delay line. An
external static bias magnetic field was applied along the CPW by means of an electromagnet.
All measurements were performed at room temperature.
Micromagnetic simulations. The simulations were performed using Object Oriented
Micromagnetic Framework OOMMF (http://math.nist.gov/oommf/), closely mimicking the
experimental conditions. The Permalloy microstructure enclosed in a "box" spanning
40 x 40 x 0.1 μm3 was discretised into a mesh of 100 x 100 x 10 nm3 cuboids, assuming a
saturation magnetization of 800 G and a Gilbert damping constant of 0.008. Since the
features studied here occur on length scales much greater than the exchange length of
Permalloy, we completely and deliberately neglected the exchange interaction, which is
justified by the physics underpinning our observations and validated by the agreement
between the experiments and simulations.
The methodology of the simulations was prescriptive. The samples were initially
magnetized to saturation along a nominated direction by a uniform magnetic field. The field
strength was then incrementally reduced to the desired value. The magnetisation dynamics
were induced through the application of an in-plane uniform dynamic field with a maximum
value of 0.175 mT and either a Gaussian (full width at half maximum of 50ps) or harmonic
temporal profile. To provide a fairer comparison with the measurements, the computed
magnetisation distributions were convolved with a Gaussian filter, bringing the resolution
closer to that of the TRSKM imaging, i.e. circa 250 nm.
Note 9. List of supplementary movies and their captions
Movie 1. TRSKM movie of the spin waves propagating in the Permalloy Tjunction. The bias magnetic field of 500 Oe is applied parallel (a) and at angles of α = -15°
(b) and α = +15° (c) relative to the leg of the junction. The frequency of the cw microwave
pump magnetic field was 8.24 GHz. Snapshots from the movie are shown in Fig. 2 of the
main text and also in Fig. S1 and Fig. S2 of this document.
Movie 2. Raw simulated movies of the spin waves propagating in the Permalloy
T-junction. The bias magnetic field of 500 Oe is applied parallel (a) and at angles of α = 15° (b) and α = +15° (c) relative to the leg of the junction. The frequency of the cw
microwave pump magnetic field was 7.62 GHz in panel (a) and 7.52 GHz in panels (b) and
(c).
Movie 3. Smoothed simulated movie of the spin waves propagating in the
Permalloy T-junction. The bias magnetic field of 500 Oe is applied parallel (a) and at
angles of α = -15° (b) and α = +15° (c) relative to the leg of the junction. The frequency of
the cw microwave pump magnetic field was 7.62 GHz in panel (a) and 7.52 GHz in panels
(b) and (c). The movie shows the data from Movie 2 after smoothing to emulate the spatial
resolution of the TRSKM measurements. Snapshots from the movie are shown in Fig. 2 of
the main text.
13
Movie 4. Depth resolved simulations of spin waves in the Permalloy T-junction.
The simulated movies of the out-of-plane component of the dynamic magnetization are
shown for the top (left), middle (centre) and bottom (right) cross-sections of the H-shaped
sample (comprising two T-junctions) continuously pumped at 7.52 GHz. Snapshots of this
movie are shown in Fig. S4 of this document.
14
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15