Magnetisation dynamics
in ferromagnetic continuous and
patterned films:
Microwave current injection ferromagnetic resonance,
propagating spin waves, and
a ferromagnetic resonance-based hydrogen gas sensor
Crosby Soon Chang
Bachelor of Science (Honours)
School of Physics
The University of Western Australia
2013
This thesis is presented for the degree of
Doctor of Philosophy of The University of Western Australia.
Abstract
In recent years, microwave magnetisation dynamics in thin ferromagnetic metallic
films, multi-layers, and nano-structures has attracted a lot of attention due to possible
future applications in microwave signal processing, magnetic logic, and magnetic
sensors. In this work, magnetisation dynamics were studied for ferromagnetic
continuous and patterned films using inductive broadband spin wave spectroscopy
techniques in three projects:
a.) A microwave current injection ferromagnetic resonance (FMR) technique using a
sub-millimetre coplanar probe was demonstrated on a continuous Permalloy film and a
periodic array of Permalloy nano-stripes. It was found that the first standing spin wave
mode (SSWM) with odd symmetry across the material thickness was efficiently excited
in the nano-stripe array. On the contrary, in spin wave resonance spectra measured with
conventional techniques the higher-order SSWMs are often lacking due to symmetry
reasons. However, they are of great importance since they carry important information
about the exchange constant for the material. Calculations of microwave current
distributions by the current injection method were used to explain the spin wave
resonance spectra. The suggested current injection FMR technique is fast and simple.
On top of the efficient excitation of the higher-order SSWMs, it also allows spatial
mapping of magnetisation dynamics with spatial resolution determined by the size of
the coplanar probe tip.
b.) Magnetostatic spin wave modes in the Damon-Eshbach geometry were
systematically studied for a series of Permalloy micro-stripes over a wide range of
aspect ratios using a highly sensitive custom-made microwave detector. The use of the
detector allowed tracking the spin wave dispersion over a wide range of wave numbers
using the simple phase method. It was found that over the range of aspect ratios and
wave numbers studied, the dynamic effects can be neglected and the surface mode
dispersions can be modelled by including an effective static demagnetising field term in
the continuous film dispersion case. The group velocities were found to increase with
thickness and were width invariant over the aspect ratios considered. The attenuation
and relaxation parameters were found to be typical for the material. It was also found
ii
that the non-reciprocity parameter is largely invariant over the range of aspect ratios
studied.
 110nm 
For the stripe with the highest aspect ratio studied 
 , excluding the fundamental
 2m 
mode, up to six higher order width modes with odd symmetry were observed. The
modes were identified from numerical simulations, from which the modal profiles were
obtained. Group velocities, attenuation properties, and non-reciprocity of these higher
order width modes were characterised in detail. It was found that group velocity,
attenuation length, and non-reciprocity decreased for increasing mode number.
Finally, the near-field of the antenna was considered. We propose that spin wave
propagation begins at some finite distance away from the antenna due to the near-field
of the antenna. An expression was derived from which the so-called antenna
characteristic near-field length may be experimentally determined. For our antenna, we
found that this near-field length is non-zero but still lying underneath the total width of
the antenna. This results in the effective wave propagation distance being shorter than
the geometrical antennae separation gap, the difference being twice the antenna
characteristic near-field length.
c.) A cobalt-palladium bi-layer thin film’s functionality as a hydrogen sensor is
demonstrated. Upon hydrogenation of the palladium capping layer, a down-field shift
and line-width narrowing of the ferromagnetic resonance of the underlying cobalt layer
was observed. The resonance shift was attributed to increase in interfacial uniaxial
anisotropy of cobalt due to strain from the expanded hydrogenated palladium capping
layer. We propose that the line-width narrowing is primarily due to reduction in spinpumping into the palladium layer due to reduction of conductivity of the hydrogenated
palladium layer. Finally, the bi-layer film was subjected to repetitive cycling of nitrogen
and hydrogen atmospheres. The ferromagnetic resonance response of the sensor was
consistently reproducible at each cycle with expected palladium hydrogen absorption
and desorption characteristic times. These results open up an exciting new class of
ferromagnetic resonance-based hydrogen sensor.
iii
Acknowledgements
Financial support by the Australian Research Council (ARC), the School of Physics,
The University of Western Australia (UWA), and the Australian-Indian Strategic
Research fund is acknowledged.
This work was performed in part at the University of New South Wales (UNSW) node
of the Australian National Fabrication Facility (ANFF); A company established under
the National Collaborative Research Infrastructure Strategy to provide nano and
microfabrication facilities for Australia’s researchers.
Usage of the facilities of the Sensors & Advanced Instrumentation Laboratory (SAIL),
School of Electrical, Electronics and Computer Engineering, the University of Western
Australia, is acknowledged.
I acknowledge the facilities, and the scientific and technical assistance, of the Australian
Microscopy & Microanalysis Research Facility at the Centre for Microscopy,
Characterisation and Analysis (CMCA), The University of Western Australia.
iv
Thanks
To my main supervisor, Mikhail Kostylev (Physics, UWA):
Throughout the 4 years of this journey, I have learnt so much from your vast
knowledge, experience, and wisdom in the field. I truly appreciate the opportunity given
to work under your guidance at the Spintronics and Magnetisation Dynamics Group.
Thank you for initiating suitable projects for me to work on, and for directing me in the
right direction whenever faced with obstacles. Thank you for helping me to set up the
experimental equipment for the various projects throughout the years. Thank you as
well for training me in the ferromagnetic resonance measurement techniques in the
laboratory, and for the numerical simulation codes. Thank you for always being
available to answer my questions. I have benefited much from our fruitful discussions
and your advices.
To my co-supervisor, Ivan Maksymov (Physics, UWA):
Thank you for your valuable feedback towards the thesis writing and checking up on
my progress.
To my former co-supervisor, Bob Stamps (University of Glasgow):
Thank you for your ideas and input during the early days of the thesis journey.
To Adekunle Adeyeye (National University of Singapore):
Thank you for fabricating samples which made this thesis possible. Your contribution is
greatly appreciated. Thank you for sharing your expertise in discussions regarding
fabrication techniques of patterned magnetic structures.
To Matthieu Bailleul (Institute of Physics and Chemistry of Materials, University of
Strasbourg):
Thank you for your microwave current injection technique suggestion, of which a
publication resulted, and which constituted a significant part of this thesis. Thank you as
well for discussions and your expert advice on propagating spin wave spectroscopy, of
which a major part of this thesis is based on.
v
To Eugene Ivanov (Physics, UWA):
Thank you for building the microwave interferometric phase detector, with which highsensitivity ferromagnetic resonance measurements could be made, especially for the
propagating spin wave and hydrogen sensor experiment. Thank you as well, for useful
discussions on noise and sensitivity of measurements.
To Fay Hudson (ANFF-UNSW):
Thank you for your hospitality in my trips to ANFF-UNSW. Thank you for inducting
me into the facility, training me in clean room techniques, optical lithography, electronbeam lithography, scanning electron microscopy, and thermal evaporative deposition.
Thank you as well for helping me to develop the recipe to fabricate micro-patterned
magnetic structures, without which this thesis would not have been possible.
To the Physics Workshop crew (Physics, UWA):
Thank you for building the probe station and the gas cell; the “hardware” of the thesis!
Thank you also (especially Gary Light and John Moore) for your hard work in fixing
and maintaining the ageing sputtering machine.
To Dave O’Connor (Bandwidth Foundry):
Thank you for your expert advice on design of optical lithographic masks.
To Nils Ross (formerly Physics, UWA):
Thank you for “passing on the baton” to me by training me to use the group’s sputtering
machine.
To Alexandra Suvorova (CMCA-UWA):
Thank you for training me to use the scanning electron microscope at CMCA. Thank
you also for helping us to image particularly challenging samples on a tilted sample
stage.
To Joanna Szymanska (ANFF-UNSW):
Thank you for training and supervising me to use the electron-beam evaporative
deposition equipment at ANFF-UNSW.
vi
To Adrian Keating (Electrical Engineering, UWA):
Thank you for training me to use the optical profilometer in the SAIL laboratory.
To Rhet Magaraggia (Physics, UWA):
Thank you for teaching me the magneto-optical Kerr effect (MOKE) setup in our
laboratory. Thank you also for helping to troubleshoot data acquisition software of our
measurement setups whenever something went wrong.
To Rob Woodward (Physics, UWA):
Thank you for letting me use the Biomagnetics group’s optical microscope to inspect
my samples.
To Nir Zvison (Electrical Engineering, UWA),
Thank you for depositing silicon nitride on my samples for me during the early days of
the thesis.
vii
Contents
1
2
Introduction
1
1.1
2
Experimental setup and techniques
3
2.1
Sample fabrication
3
2.1.1
Film deposition
3
2.1.2
Micro-fabrication
4
2.2
3
Thesis outline
Broadband spin wave spectroscopy
4
2.2.1
Vector network analyser
5
2.2.2
Lock-in with field modulation
7
2.2.3
Interferometric phase detector
9
2.3
Probe station
13
2.4
Gas cell
14
Microwave current injection spin wave spectroscopy
16
3.1
Background
16
3.1.1
Spin waves
16
3.1.2
Ferromagnetic resonance
17
3.1.3
Standing spin wave mode
18
3.2
Case for work
19
3.3
Experiment design
19
3.4
Continuous film mode identification
23
3.5
Nanostripe array mode identification
24
viii
3.6
Microwave electromagnetic field calculations
30
3.6.1 Current injection method on continuous film
30
3.6.2 Current injection method on nanostripes
34
3.6.3 Microstrip method on continuous film and nanostripes
37
3.6.4 Out-of-plane microwave magnetic field contribution
38
3.7
Microwave current injection as a characterisation tool
41
3.8
Chapter conclusion
44
4
Propagating spin wave spectroscopy
45
4.1
Background
45
4.1.1 Propagating modes in continuous films
46
4.1.2 Propagating modes in laterally confined geometry 47
4.2
Case for work
48
4.3
Experimental setup
50
4.4
Experimental procedure
53
4.4.1 Data acquisition
53
4.4.2 Sensitivity
54
4.4.3 Wave number space
55
4.4.4 Extracting dispersion
57
Magnetostatic surface mode in confined stripe geometry
62
4.5.1 Dispersion
62
4.5.2 Static demagnetising field simulations
68
4.5.3 Group velocity
72
4.5.4 Attenuation and relaxation
75
4.5
ix
4.5.5
4.6
4.7
4.8
5
Non-reciprocity
81
Higher order width modes in confined stripe geometry
84
4.6.1
Mode identification
86
4.6.2
Dispersion and group velocity
90
4.6.3
Attenuation and relaxation
94
4.6.4
Non-reciprocity
96
Antenna near-field effect
97
4.7.1
Characteristic equations
97
4.7.2
Antenna characteristic near-field length
99
4.7.3
Effective propagation distance
103
Chapter conclusion
105
Ferromagnetic resonance-based hydrogen gas sensor
107
5.1
Background
107
5.2
Case for work
108
5.3
Experiment design
109
5.4
Experiment results
110
5.5
Discussion of results
113
5.6
Cobalt-palladium film as a hydrogen sensor
115
5.7
Suggestions for further work
118
5.8
Chapter conclusion
120
Appendices
121
Appendix A
Photolithography micro-fabrication recipe
121
Appendix B
Microwave current injection into a continuous film
123
x
Appendix C Numerical Simulations
Bibliography
130
132
xi
Chapter 1
Introduction
The study of magnetisation dynamics in magnetic materials has been around for nearly
seven decades 1. Recently, the focus has been on magnetisation dynamics in thin
ferromagnetic metallic films, multi-layers, and nano-structures. These have attracted a
lot of attention due to potential applications in microwave signal processing [2-12],
magnetic logic 2-5, magnetic memory 6-10, and sensors 11-15. Thus, there is still much
room for research into the characterisation of magnetisation dynamics in such patterned
magnetic media, including the development and improvement of measurement
techniques.
In this thesis, three different magnetic systems were studied using inductive broadband
spectroscopy techniques. The first is the use of a microwave current injection technique
to probe local magnetisation dynamics. This technique – developed as a part of this
thesis – was demonstrated on an array of magnetic nano-stripes and a reference
continuous film. The second – and largest – work in this thesis is the study of
propagating spin waves in confined magnetic stripes. Channelling of spin waves along a
confined stripe is of great technological importance for potential microwave signal
processing and magnetic logic application. The characteristics of magnetostatic surface
waves across a wide range of stripe aspect ratios were systematically studied in that
chapter. Finally, the third work demonstrates the functionality of a metallic magnetic /
palladium bi-layer film as a hydrogen sensor. The state of the hydrogen-absorbing
palladium was probed through the dynamic magnetisation properties of the underlying
magnetic film. This represents a new class of ferromagnetic resonance-based hydrogen
sensor.
Hence, the chapters in this thesis are set out as follows:
1
1.1 Thesis outline
Chapter 2
This chapter details the fabrication techniques, custom-made experimental setups, and
measurement techniques developed for the experiments detailed in this thesis. Many of
these setups and techniques were developed over the course of the thesis work, and
hence deserve a dedicated chapter.
Chapter 3
In this chapter, a microwave current injection ferromagnetic resonance (FMR)
technique was demonstrated on an array of Permalloy nanostripes along with its
reference continuous film. The results were compared with standard microstrip FMR
method. The modes in the ferromagnetic resonance spectra were identified and the
relative amplitudes of the modes explained with the aid of microwave electromagnetic
field calculations. Finally, the merits of the microwave injection technique were
explored.
Chapter 4
Propagating spin wave spectroscopy using our highly sensitive microwave detector was
performed on Permalloy stripes over a wide range of aspect ratios in the DamonEshbach geometry. The dispersion, group velocity, attenuation, and non-reciprocity
properties of the fundamental surface wave propagation through such laterally confined
samples were characterised. Higher order width modes found in the stripe with the
highest aspect ratio studied were also characterised for their dispersion, group velocity,
attenuation, and non-reciprocity. Finally, simple theory for an antenna near-field effect
was proposed and experimentally quantified.
Chapter 5
The functionality of a cobalt-palladium bi-layer thin film as a hydrogen sensor was
demonstrated. Ferromagnetic resonance measurements were performed on the bi-layer
film under nitrogen and hydrogen atmospheres. The results obtained were compared and
explained. Further tests were performed by recording the response of the sensor under
cyclic introduction of hydrogen, and signal detection through a 1 mm barrier.
2
Chapter 2
Experimental setup and techniques
Over the time frame of the work which went into this thesis, many custom-made
experimental setups and measurement techniques were developed at our group. The
experimental setups developed specifically for the projects described in this thesis
include: a probe station, a gas cell, and a highly sensitive microwave detector. The
group gained experience in developing the magnetic microstructure fabrication and
characterisation techniques. All these major milestones warrant a dedicated chapter of
their own.
2.1 Sample fabrication
2.1.1 Film deposition
Most of the metallic continuous thin films used were deposited in-house using the
group’s dc sputter machine. Typically, a 5 nm tantalum seed layer is first deposited onto
silicon substrate, followed by the material of interest (e.g. Permalloy, cobalt,
palladium), and then finally capped with another 5 nm layer of tantalum. The tantalum
seed layer improves adhesion to the silicon substrate and aids in (111) lattice ordering
for the layer above the seed layer 16-18. The tantalum capping layer shields the film of
interest from oxidation. Sputtering is typically done at room temperature with argon
plasma at a pressure of 6 mTorr and regulated power of 60 W.
The group’s sputter machine lacks a monitoring crystal, so deposition rates need to be
pre-determined by calibration. For a particular target material, gun, and sputtering
power, a series of films were sputtered for known exposure times. For calibration, the
silicon substrates were partially covered prior to sputtering, resulting in film depositing
only on the uncovered areas of substrates. The resulting step height at the boundary is
then measured with a white light interferometer profilometer. This step height is the
thickness of the film sputtered. From these, the deposition rates were determined.
3
Calibrations are repeated approximately every 20 hours of target use to check for drifts
in the sputtering rates due to target depletion.
2.1.2 Micro-fabrication
The central part of this PhD thesis involves characterising properties of propagating
spin waves in micro-stripes (detailed in chapter 4). Fabrication was jointly done at the
Australian Nanofabrication Facility node at the University of New South Wales
(UNSW), and by Prof. A.O Adekunle’s group at the Department of Electrical and
Computer Engineering, National University of Singapore (NUS). A series of microstripes of various aspect ratios overlaid with microscopic coplanar waveguides were
fabricated. Lift-off deposition fabrication method was used. The fabrication recipes are
detailed in Appendix A.
It was found that sputter deposition followed by lift-off is unsuitable to fabricate the
magnetic stripes. The non-directional nature of sputtering resulted in side wall coating
of the photoresist pattern, which after lift-off, resulted in rough and steep stripe edges.
This is unacceptable, since irregular submicron-sized physical defects will cause
unwanted scattering of spin waves 19, 20. Following Prof. A.O Adekunle’s group’s
fabrication method at NUS 21, electron beam evaporative deposition was found to be
suitable to form magnetic stripes with straight edges (with defect sizes of the order of
submicrons).
2.2 Broadband spin wave spectroscopy
The inductive method to study excitation of spin wave resonance in a ferromagnetic
film was pioneered by Silva et al. 22. In a typical broadband spin wave experiment,
microwave absorption is measured as a function of the driving microwave frequency
and/or externally applied magnetic field. At resonance, a dip in the spectra indicates
absorption of microwave power into the sample under test (Figure 1.2.2a). The
experiment is usually repeated for a number of frequency and field sweeps, and material
parameters extracted by fitting with the appropriate analytic formula or numerical
simulation. Thus, broadband spin wave spectroscopy is a tool to characterise the
4
magnetisation dynamics of ferromagnetic materials. Various forms of broadband
magnetic resonance techniques were used to characterise the continuous and patterned
magnetic films presented in this thesis. These are detailed in this subchapter.
2.2.1 Vector network analyser
The broadband inductive technique using a network analyser was first developed by
Counil et al.23, and is now widely employed for the measurement of magnetisation
dynamics. Similar to 24, a planar waveguide (Figure 2.2.1a) is placed between the poles
of an electromagnet such that the waveguide is perpendicular to and in-plane to the
direction of the applied field. Out-of-plane configuration is possible as well, but this
geometry is not used in the experiments detailed here. The magnetic sample of interest
to be tested is placed on a top of the waveguide, usually with the film facing the
transducer. The waveguide is connected on both ends to the two ports of a vector
network analyser (VNA).
Figure 2.2.1a: A microstrip waveguide with sample under test across the signal line.
The VNA functions as both the microwave source to excite spin waves in the magnetic
sample, and as a signal receiver. More precisely, it measures the scattering parameters –
S21 (transmission) and S11 (reflection) – of the device-under-test (DUT). There are two
methods to measure the FMR response of the sample:
5
Frequency sweep: The electromagnet field is fixed, and the scattering parameters
measured as a function of frequency. This method is quick, but less sensitive compared
to a field sweep. In addition, frequency sweeps may yield signals which are nonmagnetic in origin, but simply due to variations in the impedance of the DUT as
frequency is swept.
Field sweep: The VNA is set to operate at a single frequency, and the electromagnet
field is swept. The scattering parameters are measured as a function of field for a
particular frequency. This method is slow, but more sensitive than a frequency sweep.
In addition, it only yields signals which vary with magnetic field. This method requires
additional computer codes to enable automation of field sweep and data acquisition. An
example of spectra taken with VNA using field sweep is shown in figure 2.2.1b.
The merit of VNA is that it enables one to measure the absolute value of spin wave
microwave absorption in terms of well-defined scattering parameters. However, the
disadvantage of VNA is that it measures the scattering parameters of the whole DUT;
both the waveguide and the sample of interest. Due to the sheer physical size difference
between the waveguide and the sample, the sample signal is almost always much
smaller than the total DUT signal, appearing as blips on top of the background
waveguide signal. Typically, background subtraction needs to be done to isolate the
sample signal from the total DUT signal.
Figure 2.2.1b: Spin wave absorption spectra of a 100 nm thick Permalloy film at 10
GHz, showing the fundamental mode and the first standing spin wave mode as
microwave absorption dips.
6
2.2.2 Lock-in with field modulation
In light of the disadvantage of VNA pointed out before, the group developed a more
sensitive lock-in and modulation broadband spin wave spectroscopy method. The VNA
is replaced by a dedicated microwave generator, a microwave tunnel diode, and a lockin amplifier. In addition, modulation coils were fixed at the poles of the electromagnet
(Figure 2.2.2a).
Figure 2.2.2a: Lock-in with field modulation broadband method circuitry.
The microwave signal transmitted through the DUT is measured as a function of applied
field for given microwave frequencies. Alternatively, the reflected signal can also be
measured instead by redirecting reflected power from the DUT through a circulator.
Similar to 24, 25, the field is modulated using two small coils attached to the poles of the
electromagnet. Modulation frequency is 220 Hz and the RMS magnetic field produced
by the coils is typically about 9 Oe. The input microwave power is set such that the
rectified bias voltage at the output end of the tunnel diode is between 50 – 100 mV; this
is the most sensitive and linear region of the particular diode’s response. The
transmitted / reflected signal from the DUT is rectified using a tunnel diode and fed into
a lock-in amplifier referenced by the same 220 Hz signal driving the modulation coils.
The signal obtained this way is proportional to the field derivative of the imaginary part
7
of the rf susceptibility as a function of the microwave frequency 25. The mathematical
concept is as follows:
Consider the microwave susceptibility of the DUT as a function of field, H:
 (H )
Modulation produces an ac field on top of the dc field, so the susceptibility becomes:
 ( H  heit )
The first two terms of the Taylor expansion (with respect to time) of the susceptibility
are:
 ( H )  iheit
d
dH
The first term is effectively a dc term, which is removed by the lock-in amplifier. The
second term is an oscillatory signal with the same frequency as the field modulation
frequency. By referencing the lock-in amplifier with the driving frequency of the
modulation coils, the second term gets “locked-in”. Note that the second term is
proportional to the modulation amplitude and the shape of the curve is the first
derivative of the susceptibility curve.
Typically, background signals from the transducer and other potentially magnetic
components between the electromagnet pole gaps are broad while sample spin wave
resonance signals are typically sharp. Hence, the derivative of the background signal is
effectively flat compared to the derivative of the spin wave resonance signal. The
practical absence of background means that the sensitivity of the lock-in amplifier can
be set to the sample signal level.
Note that
1
noise can be reduced by increasing the modulation frequency. However,
f
coil inductance increases with frequency, more so since the modulating coils are
attached to the soft iron poles of the electromagnet. Hence, there is a trade-off between
high frequency (to reduce pink noise) and low frequency (to increase modulation field
amplitude). For our setup, we use 220 Hz as a compromise between these two
limitations. 220 Hz is also not a harmonic of 50 Hz mains. In addition, using the lock-in
8
technique confines the signal to a very narrow bandwidth, there-by eliminating most of
white noise.
All the above considered, the single-run lock-in with field modulation technique yields
much better signal-to-noise ratio compared to single-run VNA without averaging.
Unless otherwise indicated, most of the results presented in the succeeding chapters
were obtained with the lock-in with field modulation method.
2.2.3 Interferometric phase detector
For continuous films thinner than 10 nm and micro-patterned structures, the signals
obtained using the single diode lock-in technique approach the noise levels for the
setup. Thus, a highly sensitive microwave detector with much lower noise threshold is
built to enable broadband measurement of spin wave spectroscopy in such systems.
Prof. Eugene Ivanov (Frequency Standards and Metrology Research Group at UWA
Physics) is credited for building the device for use in our group’s experiments. The
schematic of the detector is shown in figure 2.2.3a:
Figure 2.2.3a: Schematic of microwave receiver circuitry.
In essence, the device is a double Mach-Zehnder type interferometer. The source signal
is split into two paths; one as the reference signal, and the other passing through the
DUT. Both signals are then recombined. In this particular receiver, it has two loops; a
major loop and a minor loop within one path of the major loop. The key component of
this device is the mixer, which is a non-linear device. It is a device that performs
frequency conversion by multiplying two signals 26. A mixer has three ports; the radio
9
frequency (RF) port, the local oscillator (LO) port, and the intermediate frequency (IF)
port. The major loop can be represented in the form of an equivalent circuit containing a
standard interferometer, a diode, and an amplifier whose gain scales as the input power
of the whole double interferometer.
In the schematic diagram (figure 2.2.3a), the microwave source signal is split into two
paths: A and B. Path A is the driving signal at the LO port of the mixer. Path B is
further split again into a minor loop into two paths: C and D. Path D passes through the
DUT, and both signals (C and D) are recombined again into path E. The phase and
attenuation of path C is set such that the carrier signal is completely suppressed by
destructive interference upon recombination at E. The minor loop enables high
microwave power through the DUT, followed by suppression of the carrier wave at E.
This serves a dual purpose. Firstly, it enables only DUT signal to pass through path E,
so that the measurement sensitivity can be set to the DUT signal level, excluding the
carrier wave level. The second purpose of having the minor loop and destructive carrier
wave interference at E is to prevent overload at the RF port. Path E splits into two more
paths: paths F and G. Path F is fed into the RF port of the mixer, and path G is for
monitoring the signal output of the minor loop. The mixer IF port signal H is fed into an
oscilloscope for monitoring, and lock-in amplifier for data acquisition.
The microwave receiver can be tuned to obtain either amplitude or phase sensitivity. For
optimal DUT susceptibility amplitude sensitivity, the phase in path A is set such that the
slope of the IF voltage V, as a function of phase ϕ, is zero (ΔV/Δϕ = 0). Conversely, for
optimal DUT susceptibility phase sensitivity, ΔV/Δϕ is set to maximum. For all the
results presented in succeeding chapters using this microwave receiver, amplitude
sensitivity mode was used.
10
Figure 2.2.3b: Photo of the interferometric phase detector.
This receiver is able to obtain much better signal-to-noise ratio than using a single diode
(as described in section 2.2.2). The mathematical concept of how the mixer does this is
as follows:
The driving signal at the LO port is:
VLO (t )  ALO cos[LOt ]
The modulated signal passing through the DUT incident at the RF port is:
VRF (t )  a(t ) cos[RF t   (t )]
The mixer mixes the LO and RF signals. The first order output signal at the IF port,
with conversion factor K, is:
VIF (t )  KVLO (t )VRF (t )
VIF (t )  KALO cos(LOt )a(t ) cos[RF t   (t )]
VIF (t )  0.5KALOa(t )cos[(RF  LO )t   (t )]  cos[(RF  LO )t   (t )]
Mixing effectively converts the signal into a low and a high frequency component. The
high frequency component is typically filtered out by the lock-in amplifier, leaving only
the low frequency component. Since both the LO and RF signals are at the same
frequency, the IF signal reduces to a dc term with modulation a(t):
11
VIF (t )  0.5KALO a(t )
The resultant IF signal is thus a product of the amplitudes of the large LO signal and the
small RF signal (from the DUT). For our particular mixer, the typical conversion loss is
-6 dB. Note in the schematic (figure 2.2.3a) that an amplifier and a power splitter
precedes the mixer at the RF port (path E to F). The gain of the amplifier is 32 dB and
half the power is used for monitoring (path G). Therefore, the total gain of the DUT
signal at the IF port is:
Mixer conversion loss + amplifier gain + power splitter attenuation = (– 6 + 32 – 3) dB
= 23 dB
This means that the signal obtained using the receiver is boosted by 23 dB compared to
the single diode method (section 2.2). However, a boosted signal on its own is useless if
noise is also amplified by the same amount. What matters is signal-to-noise ratio. Using
Friis’s formula 27 for noise, one can calculate the total noise factor, F, of the cascade of
components in the microwave receiver. Noise factor is defined as the ratio of the input
and output power signal-to-noise ratios. The two critical components which largely
determine the noise level of the receiver are: the mixer and the amplifier (with gain
factor G) preceding it in the signal chain.
Ftotal
= Famp + (Fmixer – 1)/Gamp
= 100.9/10 + (100.5/10 – 1)/1023/10
= 1.23
≈ Famp
The total noise factor is thus dependent only on the noise factor of the amplifier, which
is 0.9 dB. Theoretically, there is a net increase in signal-to-noise of 1 dB, but in practice
a net signal gain of 23 dB more than makes up for it in this microwave receiver. Also,
the carrier signal suppression at junction E largely eliminates non-DUT signals from
passing through. Succeeding chapters will detail results obtained using this receiver to
measure spin wave resonance on thin films with thickness 5 nm (Chapter 5), and
propagating spin waves on stripes as narrow as 2 microns, 55 nm thick (Chapter 4).
12
2.3 Probe station
A probe station was designed and constructed with the help of the Physics Workshop
technicians (figure 2.3a). The function of the probe station is to accommodate the use of
probes (figure 2.3b). A removable and rotatable aluminium sample stage is positioned
between the poles of an electromagnet. An in-plane static field of up to 3500 Oe can be
applied across a DUT placed on the sample stage. Two sub-millimetre-sized
Picoprobe® coplanar probes are positioned over the sample stage facing each other.
Each probe tip has three contacts (ground-signal-ground), with 200 μm pitch (signalground distance) (Figure 2.3b). Commercially, the material used for the probe contacts
are nickel and tungsten. Nickel is ferromagnetic, and therefore unsuitable for use in
magnetic resonance experiments. Thus, we use tungsten probe contacts, which apart
from being non-magnetic, is also more durable than nickel.
Coaxial lines feed microwave power into the DUT through the probes. The probes are
mounted on the arms of two micromanipulators, enabling high-precision movement of
the probes along three translation axes and one rotation axis. The electromagnet, sample
stage, and micromanipulators are bolted together onto an aluminium platform, so that
there is no relative motion between these three core components of the probe station.
The whole assembly is placed on an optical bench for vibration isolation. Auxiliary
equipment typically used together with the core assembly includes a magnetometer, a
Hall probe, an Ohmmeter, and a digital microscope.
Figure 2.3a: The probe station.
13
Figure 2.3b: Coplanar probe.
The probe station is designed specifically for the propagating spin wave spectroscopy
(PSWS) experiments, and is also used for the current-injection ferromagnetic resonance
(CIFMR) method detailed in Chapter 3. In a typical use of the probe station, the DUT is
first placed onto the sample stage. The coaxial line feeding the probe is connected to an
Ohmmeter. A digital microscope is used to monitor the position of a probe as it is
gradually contacted onto the DUT. Electrical contact is established by monitoring the
resistance across the tips of the probe with the Ohmmeter. Once contact is secured,
microwave power is then fed into the DUT through the probe.
2.4 Gas cell
For the hydrogen sensor work detailed in Chapter 5, a custom air-tight cell (4 x 4 x 4
cm3) was made to enable controlled continuous flow of gas at atmospheric pressure
while performing magnetic resonance experiments (Figure 2.4a). The cell houses a
coplanar waveguide on which the samples sit. Coaxial cables feed microwave power
into the waveguide from one end and carry the transmitted power out through the other
end. The cell is fixed between the poles of an electromagnet such that the magnetic field
is applied in-plane and parallel to the waveguide (Figure 2.4b). A modulation coil is
attached onto the outside of the cell such that the ac field is parallel to the dc field of the
electromagnet.
14
Figure 2.4a: Gas cell schematic.
Figure 2.4b: Photo of the gas cell, showing the coplanar waveguide inside the cell, a
sample, coaxial feed lines, modulation coil, poles of the electromagnet, and gas inlets.
15
Chapter 3
Microwave current injection
spin wave spectroscopy
This chapter is based on a published work as first author 28. The sections in this chapter
are organised as follows. First, the theory of ferromagnetic resonance is briefly covered,
followed by case for work and description of the experiment. The broadband
ferromagnetic resonance spectroscopy results on a magnetic nanostripe array taken
using microstrip and current injection techniques are then shown. Next, the modes seen
in the spectra were identified based on simulation and extracted material parameters
from experimental data. Next, the relative amplitudes of the modes observed in the
resonance spectra were explained with aide of microwave electromagnetic field
calculations. Finally, the merits of the presented microwave current injection technique
were evaluated and the findings of this work summarised.
3.1 Background
3.1.1 Spin waves
Figure 3.1.1a 29: A spin wave on a line of spins. (a) The spins viewed in perspective. (b)
Spins viewed from above, showing one wavelength. The wave is drawn through the
ends of the spin vectors.
Spin waves are eigen-excitations in ferromagnetic media, existing in the microwave
frequency range. Classically, spin waves represent the collective motions of individual
spin precessions in a magnetic media (Figure 3.1.1a). The equation of motion of spins is
given by the Landau-Lifshitz30-Gilbert31 equation:
16
dM
 
dM 
  ( M  H eff ) 
M 
 → Equation 3.1.1a
dt
Ms 
dt 
M is the magnetisation vector, γ is the gyromagnetic ratio, Heff is the effective magnetic
field inside the medium, Ms is the saturation magnetisation, and α is the Gilbert
damping coefficient. The first term on the right-hand-side of Equation 1 gives rise to
precession motion of the magnetisation vector about an equilibrium direction
determined by the effective magnetic field, while the second term is the damping term
responsible for the magnetisation vector spiralling back to static equilibrium. Assuming
a plane wave excitation source, Equation 3.1.1a can be solved together with Maxwell’s
equations for particular geometries to yield spin wave eigen-modes. The eigenfrequencies depend on sample shape, external field, material parameters, and
characteristic wavelength of the excitation source.
If the characteristic wavelength of the excitation source is much larger than the
attenuation length of spin waves in a particular magnetic material, then the spin wave
modes excited in the closest vicinity of the source (for example, right above the signal
line of a microstrip) are stationary. For Ni80Fe20 (Permalloy), a low-loss metallic
ferromagnet 32, the attenuation lengths of spins waves are typically of the order of
microns 33-36. Chapters 3 and 5 deal with spin waves of the stationary kind since the
characteristic wavelength of the waveguides used to excite the spin waves are of the
order of millimetres; much larger than the attenuation length of spin waves. Conversely,
if the characteristic wavelength of the excitation source is similar to or smaller than the
attenuation length of spin waves, then the excited spin waves will propagate away from
the excitation source. Such propagating spin waves will be dealt with in Chapter 4.
3.1.2 Ferromagnetic resonance
Ferromagnetic resonance (FMR) – also known as uniform fundamental mode (FM) – is
the case where all the spins precess in phase in the magnetic material. For the thin film
geometry, the eigen-frequencies for field applied in-plane are given by the well-known
Kittel formula 37:
f 2   2 H ( H  4M ) → Equation 3.1.2a
17
f is the resonant frequency, H is the resonant field, and M is the magnetisation. This
mode is efficiently excited if the microwave magnetic field driving source is uniform
across the thickness of the film 38.
3.1.3 Standing spin wave mode
Long wavelength spin waves can be excited in confined geometries if surface spins are
pinned by surface anisotropy or exchange interactions; the magnetisation at the surface
cannot freely precess like in the bulk. These higher order stationary modes with nonzero wave numbers are known as standing spin wave modes (SSWMs). As the name
implies, the dynamic magnetisation profile of SSWMs across the confined geometry
(usually the thickness) d forms stationary waves with wave number k 
n
(Figure
d
1.2.2a). The Kittel equation is then modified 29:
f 2   2 ( H  H ex )( H  H ex  4M ) → Equation 3.1.3a
H ex  Dk 2 is the exchange field, and D is the exchange constant. SSWMs are affected
by inhomogeneous exchange interaction, carrying important information about surfaces
and buried interfaces 38-41. However, SSWMs are only efficiently excited by
inhomogeneous excitation fields which macroscopic-sized planar waveguides cannot
adequately provide for symmetry reasons 42.
In conducting ferromagnetic films, it is possible to increase the excitation efficiency of
higher order SSWMs due to induction of eddy currents in conducting media, but the
fundamental mode remained dominant unless there is significant interfacial pinning 4144
. One way to get around this deficiency is by embedding the magnetic sample into a
microscopic coplanar waveguide 45. The resultant excitation microwave magnetic field
inside the magnetic material is anti-symmetric, thus couples efficiently to the first
SSWM with odd symmetry.
18
3.2 Case for work
In this chapter, the efficient excitation of the first SSWM is achieved in a much simpler
way, without embedding the sample to be characterised. In contrast to Khivintsev et al.
45
’s single stripe, the method is demonstrated on a periodic array of magnetic nano-
stripes (MNS). These nano-structures are promising for magnonic 46 and magnetoplasmonic 47, 48 applications.
The method is based on injection of microwave currents directly into a sample using a
sub-milimetre-sized coplanar probe. Injecting microwave currents into a magnetic
material using such a probe was first tried by Prof. Matthieu Bailleul (Institute of
Physics and Chemistry of Materials, University of Strasbourg). Our group built on this
method to study the spin wave resonance response in this arrangement in detail and
explain the underlying physics 28. This is the goal of this thesis chapter. Furthermore,
we successfully efficiently excited the first SSWM in an MNS array using the current
injection method. The method is quick and conceptually allows easy spatial mapping of
magnetisation dynamics with resolution given by the size of the coplanar probe tip.
3.3 Experiment design
The nano-structure studied is a Permalloy stripe array (Figure 3.3a). The sample was
fabricated using deep ultraviolet lithography by Prof. Adekunle O. Adeyeye’s group at
the Department of Electrical and Computer Engineering (NUS) 21. A reference film of
same thickness was also fabricated. Both films were deposited by electron-beamassisted evaporative deposition. The MNS array geometrical parameters are as follows:
Thickness = 100 nm
Stripe width = 264 nm
Edge-to-edge gap = 150 nm
Macroscopic area of array = 4 x 4 mm2
19
Figure 3.3a: Scanning electron micrograph of the MNS array.
The MNS array is mounted onto the sample stage of the probe station described in
Section 2.3. The stripes are oriented in-plane and parallel to the external dc magnetic
field produced by the electromagnet. The coplanar probe is then carefully lowered until
the tips come into physical contact with the array (Figure 3.3b). Electrical conduction
through the contacted stripes is confirmed by monitoring the electrical resistance across
the probe’s three tips with an Ohmmeter. The dc resistance is typically around 130 Ω.
Based on the conductivity of Permalloy, this suggests 8 stripes being contacted by the
probe with a contact area of 3.3 μm 28.
20
Figure 3.3b: Drawing of the sub-milimetre coplanar probe tips contacting the MNS
array. Note that the size of the stripes has been vastly exaggerated; the probe tips are in
fact contacting 8 stripes. Red arrows represent the direction of injected current flow
along the stripes. The external magnetic field is applied parallel to the stripes.
Microwave current is then injected into the contacted stripes through the coplanar
probe. The reflected microwave power is measured as a function of applied magnetic
field for given microwave frequencies using the lock-in field modulation method
outlined in Section 2.2.2. To investigate the effect of nano-structuring, microwave
current injection was also performed on a reference continuous film.
Broadband spin wave spectroscopy using macroscopic microstrip was also performed
on the MNS array and reference film for comparison between the two methods. The
sample is placed face down, such that the film side faces the microstrip (Figure 2.2.1a).
For the MNS array, the sample is oriented such that the stripes are parallel to the
microstrip (Figure 3.3c). In all cases, the applied magnetic field is always in-plane and
along the stripe.
21
Figure 3.3c: MNS array parallel to the microstrip.
Ferromagnetic resonance of the MNS array and reference film was done in the
frequency range of 4 – 18 GHz, using both the current injection and microstrip method.
Several modes were observed in the FMR spectra of our samples. These are plotted in
Figure 3.3d. Before we consider the efficiency of excitation of the various modes using
various techniques, one needs to first identify these modes. Section 3.4 and 3.5 deal
with the identification of modes in the continuous and patterned film respectively.
22
Figure 3.3d: Spin wave resonance frequency versus field plot for the MNS array and
reference film.
3.4 Continuous film mode identification
Typically for Permalloy film of thickness 30 – 60 nm, the 1st SSWM is located far
down-field and well-separated from the FM. However, our film is unique in that it is
unusually thick. This result in the 1st SSWM located very close to the FM. In our
sample, this is seen as a small feature on the low-field shoulder of the dominant FM
resonance (Figure 3.4a). The modes were fitted with equation 3.1.3a (Figure 3.3d). The
high field dominant mode is trivially identified as the fundamental ferromagnetic
resonance mode (Hex = 0) with saturation magnetisation 4πM = 10150 ± 40 Oe. The
shoulder feature has Hex = 291 ± 4 Oe, and is thus identified as the first anti-symmetric
SSWM. This mode is observed in microstrip measurements due to eddy current
contribution to the microwave driving field 42. Table 3.5a summarises the fitted
parameters.
23
Consider now the amplitude of the modes. Notice that the signal obtained by microstrip
is 13 dB larger than that obtained by current injection. The vertical scale in Figure 3.4a
is set to clarify the mode features obtained by the current injection method, resulting in
clipping of the much larger microstrip signal. The relative amplitudes of these two
modes in the continuous film are the same for both the current injection and microstrip
method. Again, the reasons for this will be explored in Section 3.6.
Figure 3.4a: Field sweep ferromagnetic resonance of the reference film at 14 GHz.
3.5 Nanostripe array mode identification
. For the MNS array, one observes three resolved distinct modes (Figure 3.4b). The
identification of the modes in the MNS array is less straightforward. Nanopatterning
shifts the FM downfield due to dynamic in-plane demagnetization induced by in-plane
confinement49. One then expects the position of the FM peak in the MNS array to lie
between the extreme geometrical cases of a continuous film and a long thin rod. In light
of this, one may expect the dipolar modes and SSWMs to cross-over or even mix in the
MNS array. Thus, the identification of modes in the MNS array is non-trivial.
The problem is compounded by the absence of a well-established theory for thick
stripes, and accuracy limitations of numerical models in the case of strongly mixed
24
modes. Therefore, we employ two independent methods to complementarily and
qualitatively identify the modes observed in the FMR spectra of the MNS array: a.) Fit
the mode positions with an analytical theory for thin stripes, and b.) simulate the mode
profiles and eigen frequencies with our code.
Figure 3.5a: Field sweep ferromagnetic resonance of the MNS array at 14 GHz.
According to the theory from Guslienko et al. 50, 51, the eigen-frequencies of a nanostructured material should obey the approximate dispersion relation for spin waves valid
for continuous films. All peculiarities of confinement due to nano-structuring can be
accounted with a dipolar effective demagnetising field, Hd. For thin patterned films, the
collective fundamental mode is described by equation 11 in reference 49. By including
exchange, the equation is modified into:
f 2   2 ( H  H ex  H d )( H  H ex  4M  H d ) → Equation 3.5a
The MNS modes are plotted and fitted with Equation 3.5a (Figure 3.3d). For each data
set, there is a range of Hd and Hex combinations for which good fits can be obtained.
Therefore, in order to qualitatively identify the modes, we imposed physical constraints
on the fittings (see below). The fitted parameters Hex and Hd are shown in Table 3.4a.
25
Identification of the 1st SSWM
We observe that the high field mode in the MNS spectra lies close to the 1st SSWM of
the continuous reference film. From established theory of magnetization dynamics of
nanostripes and previous Brillouin light scattering studies, nanostructuring strongly
shits the fundamental downfield with respect to the continuous film case, but leaves the
position of the 1st SSWM unchanged49. We expect similar behaviour for our thick MNS
sample. With this foreknowledge, we bias the fittings for this mode by setting Hd = 0 in
order to obtain physically realistic values of Hex. We obtained Hex = 430 ± 5 Oe for this
mode. This value is close to the 1st SSWM of the reference continuous film (Hex = 291 ±
4 Oe). Therefore, we assign this high field mode in the MNS spectra as the 1st SSWM of
the MNS.
To confirm this, we simulated the eigen modes of the MNS array using theory from
Tacchi et al. 52, and found a mode with eigen frequency close to the high field mode in
the experiment. (Refer to Appendix C for simulation details.) A theoretical eigen-mode
with a quasi-uniform distribution of dynamic magnetisation in the array plane but an
anti-symmetric distribution across the stripe thickness matches the experimental eigenfrequencies of this mode (Figure 3.5c-b). The dipole field Hd is vanishing for this mode
due to its anti-symmetric character 53. The main contribution to the mode frequency
originates from the exchange energy; this depends mainly on the smallest dimension of
the structure. In the MNS array studied here, the smallest dimension is given by the
thickness (100 nm). This mode represents the counterpart of the first SSWM for the
continuous film. Since the MNS array thickness is the same as that of the reference
continuous film, one may expect that the eigen-frequencies for the first SSWMs to be
similar.
Identification of the FM
Since the high field mode has been identified as the 1st SSWM, by process of
elimination, it follows that the dominant low field mode could well be the FM. From
Equation 3.4a, the slope of the resonance plot is:
df   
  H  H ex  2M  → Equation 3.5b
dH  f 
26
From Equation 3.5b, one easily sees that the slope is determined by contribution from
the exchange (increase
df
df
) and dipolar (decrease
) energies. One observes that the
dH
dH
low field dominant mode of the MNS array has a smaller
df
slope compared to other
dH
modes (Figure 3.3d). This suggests that this mode may have a significantly larger
contribution of dipolar interactions to the mode eigen-frequency.
From the fit with Equation 3.5a, this is indeed the case. Based on the large value of the
dipolar field Hd (1110 ± 70 Oe), this mode is identified as the fundamental dipolar mode
of the MNS array. This mode’s resonant field is strongly shifted down field due to
strong effective magnetisation pinning at the stripe edges 50 and a large dynamic
demagnetizing dipolar field, both of these due to nano-structuring confinement.
Figure 3.5b: Eigen-frequencies of the MNS array fundamental dipolar mode.
The identification of the MNS FM is further supported by numerical simulation (refer to
Appendix C), where we found a quasi-uniform mode (Figure 3.5c-a) with eigen
frequencies close to the mode of interest (Figure 3.5b).
Noteworthy is the significant exchange field of this mode (670 ± 40). The simulation
mode profile revealed that this mode is hybridized with the third (next order in-plane
symmetric) dipole mode and the third (out of plane symmetric) exchange mode (figure
3.5c-a). The non-uniformity of the modal profile due to hybridization is possibly partly
responsible for the large value of Hex. In addition, the approximate theory 49-51 is valid
27
 thickness

for low aspect ratio 
1 structures. Therefore, one expects inaccuracy in
 width

extracting a small Hex contribution on top of a strongly dominating Hd contribution for
 thickness

the high aspect ratio 
 0.26  MNS array studied here.
 width

Identification of the 3rd SSWM
Finally, one observes a low field feature at the shoulder of the FM of the MNS array.
We suspect this mode could be the 3rd SSWM, hence we set Hd = 0 for the fitting,
similar to what was done for the 1st SSWM. We obtained a value of Hex = 1551 ± 4 Oe
for this mode. The simulated mode profile for this mode is shown in Figure 3.5c-c. The
mode profile is symmetric across the thickness, with two nodes. Thus, this mode is
identified as the third (out-of-plane symmetric) exchange mode of the MNS array. Note
that the close proximity of this mode with the FM is partially responsible for the
distortion of the FM profile from hybridization, as mentioned before (Figure 3.5c-a).
28
Figure 3.5c: Simulated in-plane dynamic magnetisation 2D profiles across the crosssection of a single nanostripe in an array. Numbers on the axes are the mesh indices
across the thickness on the vertical axis and across the width on the horizontal axis.
Colours are proportional to the real part of the in-plane dynamic magnetisation vector.
29
Resonance feature
Hex (Oe)
Hd (Oe)
Mode identification
MNS high-field
430 ± 5
0
MNS 1st SSWM
670 ± 40
1110 ± 70
MNS FM
1551 ± 4
0
MNS 3rd SSWM
0
0
Film FM
291 ± 4
0
Film 1st SSWM
(Green diamond)
MNS low-field
(Blue triangle)
MNS extra shoulder
(Purple star)
Film high-field
(Black circle)
Film low-field
(Red square)
Table 3.5a: Fitted parameters for the MNS array and reference film.
3.6 Microwave electromagnetic field calculations
Once the modes have been identified, we will now explain the differences in relative
mode amplitudes in the spectra. In order to do this, one needs to consider the driving
microwave magnetic field profiles for both the current injection and microstrip method.
The former is done by first calculating the injected microwave current distribution
inside the MNS array and continuous thin film.
3.6.1 Current injection method on continuous film
The 2D microwave current distribution in a finite conducting slab of negligible
thickness was calculated by Ney 54. The important relevant finding from that work is the
strong microwave current repulsion, resulting in highly non-uniform current
distributions in slabs with sizes much larger than the microwave skin depth. Similar to
Ney’s approach, the microwave current density is calculated for our current injection
geometry. In contrast to Ney, the calculation is performed in 3D because the out-of30
plane component of the current density is important and may give rise to significant inplane microwave magnetic field. The full derivation of the theory suggested by Prof.
Mikhail Kostylev is presented in Appendix B. To enable analytical solutions, the
current density is assumed to be out-of-plane and uniform at the probe tip’s point of
contact with the film. Using this theory, we calculate the radial in-plane (figure 3.6.1a),
and in-depth (figure 3.6.1b) microwave current distributions of an infinite continuous
film 100 nm thick.
Figure 3.6.1a: Radial in-plane microwave current density at the film surface.
31
Figure 3.6.1b: In-depth microwave current density underneath the probe.
The radial in-plane component of the microwave current density is given by a modified
Bessel function of the second kind (which approximates as
1
decay). As plotted in
r
Figure 3.6a, the current density is concentrated directly underneath and in the near
proximity of the probe tip due to microwave current repulsion far from the source. The
in-depth component of the microwave current density is given by a hyperbolic sine
function (which approximates as linear decay). As plotted in Figure 3.6.1b, our
calculation shows that the current density is concentrated at the surface at which the
current from the probe is incident on, and is zero at the opposite buried interface. Note
that this distribution is very similar to the perfect microwave shielding effect of subskin-depth thin conducting films 42.
32
Figure 3.6.1c: Magnitude of microwave magnetic field in the vicinity of the probe tip.
White is most intense, while purple is least intense.
Both the in-plane radial and in-depth components of the microwave current induce an
in-plane microwave magnetic field with intensity profile shown in figure 3.6.1c. This
in-plane circulating field (figure 3.6.1d) is concentrated near the probe tip. This in-plane
component of the microwave magnetic field is responsible for the efficient excitation of
the fundamental uniform mode.
The in-plane current between the probes is significantly diffused due to microwave
current repulsion (figure 3.6.1a). The in-plane radial currents from each of the three
probe tips do not perturb each other since the distance between the probe tips (200 μm)
is much larger than the microwave current decay length (a few μm). Without diffusion,
this current would have induced an anti-symmetric field across the thickness of the film,
which would in-turn, efficiently drive the first SSWM. Therefore, this field is not a
33
candidate for the small first SSWM peak observed in the spectra (figure 3.4b). The
origin of this is proposed to be due to the asymmetry of the in-depth microwave
magnetic field (figure 3.6.1c). Similar to the eddy current shielding effect for the
microstrip case 42, the first SSWM is only negligibly excited due to weak interfacial
pinning for the single layer film studied. Hence, as shown in figure 3.4b, the
fundamental mode is much more strongly excited than the first SSWM for thin films, by
both the current injection and microstrip method.
Figure 3.6.1d: Microwave current injection (I) induces an in-plane microwave magnetic
field (h) circulating in the vicinity of the probe tip.
3.6.2 Current injection method on nanostripes
In the MNS array, the absence of medium continuity in the direction of the array
periodicity does not allow current to diffuse in the array plane as in the case of a
continuous film discussed before. The microwave current remains confined in the
contacted stripes between the probe tips (figure 3.3b). This produces a large in-plane
current density over a large length, given by the pitch of the coplanar probe (0.2 mm).
Since the cross section dimensions of the MNS are comparable to the microwave skin
depth (of the order 100 nm), this current flowing through the stripes can be considered
uniform. The resultant microwave magnetic field of this in-plane current is antisymmetric across the MNS depth (figure 3.6.2a); this is essentially similar to the simple
case of the magnetic field generated by a wire carrying a dc current. This antisymmetric microwave magnetic field efficiently excites the first anti-symmetric SSWM.
34
As seen in figure 3.4a, the first SSWM dominates the spectra of the current injection
method on the MNS array.
Figure 3.6.2a: Anti-symmetric microwave magnetic field (h) generated inside the stripes
due to microwave current (I) flowing along the stripes.
Note from figure 3.4a that the fundamental dipole mode is still present in the spectra,
despite being smaller in amplitude compared to the first SSWM. The same microwave
current which generated the anti-symmetric microwave magnetic field as explained
earlier is also responsible for the excitation of the fundamental mode. If we consider the
microwave magnetic field produced outside a single stripe, and how the field interacts
with nearest neighbour stripes, we see that there is an out-of-plane microwave magnetic
field incident on the nearest neighbour stripes (figure 3.6.2b). This field decays as
1
r
away from the source, essentially the same as the simple case of the magnetic field of a
dc current-carrying wire. If we consider only the first nearest neighbour interactions,
then the out-of-plane field contributed by each individual stripe would be cancelled out
by their respective nearest neighbour stripes, except the outer 2 stripes, where there are
unbalanced net out-of-plane field components. This out-of-plane microwave magnetic
field incident near symmetrically on the outer 2 stripes is able to drive the fundamental
dipole mode inside those 2 outer stripes. Thus, the amplitude of the fundamental dipole
mode should be theoretically 25% that of the first SSWM, since the fundamental mode
is excited in only 2 out of 8 of the stripes contacted. This is indeed approximately what
is experimentally observed in the ratio of the amplitude of the first SSWM to the
fundamental mode (figure 3.6.2c).
35
Figure 3.6.2b: Anti-symmetric microwave magnetic field (h) generated outside the
stripes due to microwave current I flowing inside along the stripes.
Figure 3.6.2c: Amplitude ratio of the first SSWM to the fundamental mode for the MNS
array by the current injection method. The missing data points in the vicinity of 16 GHz
are due to the particular microwave generator unable to regulate constant power output
at the power level required for spin wave excitation in that frequency range.
36
3.6.3 Microstrip method on continuous film and nanostripes
Note from figure 3.4a and 3.4b that the fundamental mode is dominantly excited by the
microstrip method for both the MNS array and continuous film. The first SSWM is also
excited, but much less efficiently, especially in the case of the continuous film. To
explain this, consider the radiation field of the microstrip (figure 3.6.3a) 55.
Figure 3.6.3a: Radiation field lines of a microstrip in the parallel orientation.
An in-plane microwave magnetic field is present on top of the microstrip. When the
ferromagnetic continuous film or MNS array is placed on top of the microstrip, this
near-uniform field efficiently drives the fundamental uniform precession mode. This is
why the uniform mode is dominant (figure 3.4a & 3.4b). The first SSWM is also excited
by the microstrip, but much less efficiently than the fundamental mode. This is due to
the eddy current shielding effect of sub-skin-depth thin films resulting in a quasi-linear
profile of the microwave magnetic field across the film thickness 42. The first SSWM is
not strongly excited in both these cases due to lack of interfacial pinning.
37
3.6.4 Out-of-plane microwave magnetic field contribution
Recall earlier that it was proposed that out-of-plane microwave magnetic field is
responsible for excitation of the fundamental mode in the outer two stripes of the MNS
array by current injection method (figure 3.6.2b). To further investigate the contribution
of this field component to excitation of the fundamental mode, additional measurements
with the microstrip were performed in the nominally-called “perpendicular” orientation.
This is where the microstrip is aligned perpendicular to the applied static field, with the
MNS array still parallel to the field (figure 3.6.4a). Note the difference in geometrical
orientation compared to the “parallel” orientation in figure 3.3c.
Figure 3.6.4a: The “perpendicular” orientation of microstrip.
In the “perpendicular” orientation, only the out-of-plane component of the microstrip’s
magnetic radiation field is able to contribute to spin wave excitation; the in-plane
component is parallel to the static magnetic field and hence does not contribute to spin
wave excitation (figure 3.6.4b). In the spin wave spectra for the continuous film, the
signal of in the perpendicular orientation is 30 dB smaller than that of the parallel
orientation. This is due to large ellipticity of magnetisation precession in metallic
ferromagnetic films, where an in-plane microwave magnetic field drives magnetisation
precession much more efficient than an out-of-plane field. In addition, the out-of-plane
component of the microwave magnetic field is present only near the edges of the
microstrip where the associated dynamic electric field curls down to the embedded
ground plane.
38
Figure 3.6.4b: Radiation field lines of a microstrip in the perpendicular orientation.
Considering the out-of-plane microwave magnetic fields in the nanostripe (figure
3.6.4b), one might wonder why this “anti-symmetric” is able to excite the uniform
mode. The out-of-plane component of the excitation field is localised at the edges of the
microstrip. Absorbed electromagnetic energy is proportional to the dot product between
the driving field and the magnetisation vector. While it is true that the direction of this
field is opposite at opposite edges of the microstrip, one must also bear in mind that the
direction of magnetisation precession is also reversed. This means that the total energy
absorbed at resonance has the same sign on either sides of the microstrip. In addition,
since the microstrip is much wider than the typical attenuation length of spin waves,
local magnetisation dynamics at the edges are not able to couple to one another. Thus,
the uniform mode is driven locally at the edges of the microstrip.
We stress that for the MNS array, the fundamental mode is of similar order of absolute
magnitude in both the perpendicular and parallel microstrip orientations (figure 3.6.4c).
This is very different from the case for the continuous film, where the signal obtained in
the perpendicular orientation is much smaller than that in the parallel orientation, as
discussed earlier. This result is in good agreement with evaluation of ellipticity of
precession for MNS from numerical simulations using Tacchi et all’s theory 52. This
confirms that the out-of-plane component of the microwave magnetic field due to the
current-carrying stripes is responsible for driving the fundamental mode observed in the
current injection spectra (figure 3.4a).
39
Figure 3.6.4c: Field sweep spin wave resonance of the MNS array at 14 GHz.
40
3.7 Microwave current injection as a characterisation tool
We now turn attention to evaluate the merits of the current injection method as a
characterisation tool. As demonstrated, the method is able to characterise the
magnetisation dynamics of ferromagnetic materials similar to standard broadband
planar waveguide methods. More than that, the method enables spatial mapping of local
macroscopic magnetisation dynamics with resolution determined by the size of the
coplanar probe (in our case, 400 μm). The resolution can be improved by using the
smallest commercially available probe (100 μm). Even though the resolution is
macroscopic – a far cry from the other two spin wave spectroscopy techniques with
spatial resolution, namely Brillouin light scattering 56 and magnetic force microscopy 57,
58
– the current injection method using a coplanar probe is far simpler and quicker to
utilise.
One drawback of this method is that being a contact method, damage to samples usually
unavoidable. We now consider the physical contact between the coplanar probe and the
material being probed. The probe has a built-in spring which applies a constant force
onto the surface being probed. This ensures good physical contact between the tip and
the probed surface without risking tip breakage or loss of contact from mechanical
vibration. The standard probe tips are available in either tungsten (for long-lasting tips)
or nickel (for better electrical contact and minimal sample damage). In our setup, we use
tungsten tips since we require robustness in our experiments, and the alternative –
nickel – is magnetic and hence undesirable in spin wave experiments. The downside of
probing with a hard tungsten tip is potential physical damage to the surface being
probed, especially if the material is a soft metal.
The probed samples were inspected with an optical microscope for sample damage. No
trace of physical damage was observed on the continuous film probed. Thus, Permalloy
appears to be hard enough to resist the pressure exerted by the probe tips. However,
scratch marks were left by the probe on the surface of the MNS array (figure 3.7a).
Nano-structuring has weakened the material; making it mechanically softer than the
continuous film.
41
Figure 3.7a: Scratch marks left by the tips of the coplanar probe on the MNS array.
The depth profile of the scratch in figure 3.7a is shown in figure 3.7b. The distance
between the three scratch dips is consistent with the pitch of the probe (200 μm); these
are not sample fabrication defects, but that caused by the physical contact of the probe.
The width of the trench left by the central tip is about 10 μm wide. Note that this
dimension cannot be used to estimate the number of contacted stripes as discussed
before because this is the size along the stripe length direction, not across the stripe
width direction. Furthermore, an indentation is typically larger than the size of the
object which causes the indentation. Important from the scratch profile is the depth of
the indentations; as long as the probe exerts minimal force on the MNS array (just
enough to ensure contact), the stripes are not cleaved by the probe tips. The indentations
depths are of the order of nanometres; not enough to cleave the thick 100 nm film in this
case. This result implies that in order to use this method as a non-destructive spin wave
characterisation tool, a probe should be designed with a non-magnetic soft metal tip and
minimal force should be exerted onto the probed surface.
42
Figure 3.7b: Optical profilometer profile taken in the direction perpendicular to the
direction of the scratch mark left by the coplanar probe.
Another drawback of the current injection method is the requirement of current
continuity between the three probe tips. This means that the sample probed must be a
good electrical conductor. The current injection technique was attempted on a
La0.7Sr0.3MnO3 half-metallic film, one of the samples studied in 59. The typical
resistance between the coplanar tips through the poorly conducting sample was of the
order of 103 Ω, which is consistent with the typical resistivity of unannealed
La0.7Sr0.3MnO3 60. No spin wave resonances were observed in the current injection
spectra through the half-metal, indicating that the very low microwave current flowing
through the poorly conducting material does not induce sufficient microwave magnetic
field to drive spin wave excitation. Therefore, we conclude that if the material is
continuous but poorly conducting (like most ferrites), the current injection method
cannot be used.
In addition to the requirement of the sample being electrically conducting, there must
also be possibility of current conduction between the signal and ground tips of the
probe. This means that only a subset of patterned films can be probed with the current
injection method. This method cannot be used on a dot array for example 61, even if the
material is conducting, due to lack of current continuity. The requirement of current
conduction continuity thus limits the method to continuous films, stripes, anti-dots 62, 63,
or similar nano-structures where continuity of the conducting phase is preserved across
the whole distance between the tips.
43
Finally, impedance mismatch is a potential issue in the current injection technique. In
this method, the probed sample is essentially the load at the terminus of a microwave
transmission line. This means that for efficient transfer of microwave power into the
load, the load should have impedance matching that of the transmission line. However,
designing the sample to impedance-match with the probe defeats the purpose of the
current injection technique as a characterisation tool in the first place. In the work
presented here, the impedance mismatch was quite significant, resulting in most of the
incident power reflected from the probed MNS array. Assuming a purely resistive MNS
array load of 130 Ω and a nominal coplanar probe characteristic impedance of 50 Ω, the
transmission coefficient is calculated to be 56 %. The transmission coefficient of a
purely resistive 3 Ω continuous Permalloy film is even lower at 11 %. However, these
estimates disregard reactance contributions to the impedance and insertion loss. In the
actual experiment, the typical reflected power is nearly at the same level as that of the
incident power for both the MNS array and continuous film.
3.8 Chapter conclusion
In this chapter, the microwave current injection spin wave spectroscopy technique is
demonstrated on a Permalloy nano-stripe array and continuous film. A sub-milimetre
coplanar probe was used to inject microwave current into the samples studied, and the
spin wave excitation response compared with standard macroscopic planar waveguidebased spin wave spectroscopy. The current injection method is able to efficiently excite
anti-symmetric standing spin wave modes; these modes provide important interfacial
material properties, and the modes are often lacking in planar waveguide-based
methods. The current and radiation field distributions of both the current injection and
planar waveguide techniques were used to explain the mode amplitudes observed in the
spin wave spectra. It is found that the in-plane current flowing through the contacted
nano-stripes induces an anti-symmetric dynamic magnetic field inside the stripes, which
efficiently drives the first anti-symmetric standing spin wave mode. The current
injection technique is quick and allows easy spatial mapping of magnetic properties of
conducting materials with resolution down to 0.1 mm.
44
Chapter 4
Propagating spin wave spectroscopy
The sections in this chapter are organised as follows. A brief theory of propagating
waves in continuous and laterally confined geometry is first presented, followed by case
for work. In the next section, the experiment setup, procedure and data acquisition
methods are highlighted. The presentation and discussion of results in this chapter are
then divided into three main sections, each containing sub-sections of their own. Section
4.5 then deals with the characterisation of the fundamental propagating surface mode in
the series of stripes studied. Section 4.6 focuses on the characterisation of high-order
width modes observed in the stripe with the highest aspect ratio studied in this work.
Section 4.7 elucidates the antenna near-field effect, how it may be quantified, and how
it affects the data. The chapter ends with a summary of major findings of the presented
work.
4.1 Background
If the characteristic wavelength of the excitation source is smaller or comparable to the
mean free path of spin waves in a particular magnetic material, the resultant highly
localised excitation field can excite propagating modes. The wave equation of
propagating spin waves is obtained by solving the Landau-Lifshitz-Gilbert equation
together with Maxwell’s equations in the magnetostatic limit and material constitutive
relations. The resultant wave equation is known as Walker’s equation 64. Since these are
obtained in the magnetostatic limit, these propagating modes are known as
magnetostatic spin wave modes. The dispersion relations of propagating spin waves in a
specific geometry are obtained by solving Walker’s equation subject to boundary values
of the required geometry.
45
4.1.1 Propagating modes in continuous films
For infinite continuous films, there are three possible field orientation and propagation
direction geometries, each yielding different propagation modes and dispersions:
Figure 1.2.3a: Dispersion relations of magnetostatic modes in a thin film 65.
Forward volume mode 66
If the film is perpendicularly magnetised, forward volume spin waves can propagate
tangentially in-plane. The characteristics of this mode is indicated by the name; being a
volume mode, it permeates the entire bulk of the solid, and the phase and group
velocities are both positive.
Backward volume mode 67
If the film is tangentially magnetised, two possible wave modes can be excited.
Backward volume waves propagate parallel to the applied field. An interesting property
of this mode is that while the phase velocities are positive, the group velocities are
negative, hence the name “backward” volume waves.
Surface mode 67, 68
In the tangentially magnetised configuration, a second mode of wave propagation is
possible. Surface spin waves can propagate perpendicular to the applied field. This
46
mode is named so because the wave is evanescent; it persists only near the surface of
the film, decaying exponentially into the bulk. Both its phase and group velocities are
positive, so it is a forward wave. The wave is monotonic. This means that every wave
number corresponds to a unique frequency in the dispersion and vice versa. Note that
this is not true for forward and backward volume waves, which are multi-tonic; the
potential function of those volume modes are composed of sinusoids. When the
direction of propagation is reversed (or the field is reversed), the mode fields shift from
one surface to the other 69. This is known as field displacement non-reciprocity. This
particular magnetostatic mode is also known as Damon-Eshbach (DE) waves, after the
pioneers in the field 68.
4.1.2 Propagating modes in laterally confined geometry
In the past 15 years, research in the field has focussed on the effects of lateral
confinement on propagating spin waves modes. These have been studied on microstripes and patterned arrays using inductive techniques in the frequency domain 35, 70-75,
inductive techniques in the time domain 34, 76, Brillouin light scattering (BLS) in
reciprocal space 77-80, micro-BLS 81, 82, and Kerr microscopy in the time domain 83.
Lateral confinement results in three significant deviations from the infinite continuous
film case. First, lateral confinement results in quantization of magnetostatic modes
across the confined dimensions 36, 50, 77-79, 84, 85. Due to confinement, the dipolar eigenfunctions of stripes take the form of sinusoids, analogous to the form of spin wave
resonance modes in a perpendicularly magnetised film 50, 79, 84. Thus, one may observe
higher order confinement modes in the measured spin wave spectra. Second, the static
and dynamic demagnetising fields in confined geometry would increase and become
non-uniform. This in effect, shifts the position and slope of spin wave dispersion 79, 86,
87
. Thirdly, the extreme case of increased static demagnetising field is the formation of
potential wells at the edges of tangentially magnetised stripes where demagnetising
fields can be very large 33, 70, 88-90. This leads to decrease in the effective width of the
stripe where dipolar spin waves propagate (channelling effect) 33, and the formation of
exchange edge modes in such potential wells 79, 90-92.
47
Channelling of spin waves in metallic ferromagnetic media confined in stripe geometry
is of technological importance for the future device applications in microwave signal
processing 33, 93, 94 and magnetic logic 2-4. Of the three propagation modes, the DamonEshbach surface mode has the highest group velocity and consequently, low spatial
attenuation 69, 95. This makes it a good candidate for spin wave device application 70, as
evident by the majority of the previous work in the Damon-Eshbach geometry.
The work presented in this chapter is concerned with spin wave propagation through
laterally confined Permalloy stripes in the Damon-Eshbach geometry, where the stripes
are magnetised tangentially and wave propagation perpendicular to the applied field
(along the long axis of the stripes) is considered. The experiment design and technique
is similar to that pioneered by Bailleul et al. 70, with the exception that the experiment
was performed field-resolved using a lock-in modulation technique, and a much more
sensitive detection method was used.
4.2 Case for work
Thicker stripes
Most of the work done in the past 15 years on spin wave propagation in stripes was on
stripes thinner than 36 nm. There were relatively few studies on thicker stripes 35, 76. In
this work, thicker stripes were studied (55, 80, and 110 nm). Apart from filling in the
knowledge gap, studying thicker films possess the following advantages. The first
trivial advantage is that the signal reception of spin wave propagation would be better
for thick films simply because there is more material under the transducers. Second, the
group velocity of magnetostatic surface spin waves is directly proportional to the film
thickness. Group velocity, being the speed of energy transfer, directly relates losses in
the time and spatial domains 80. Since spatial damping is inversely proportional to group
velocity, the greater the thickness, the larger the group velocity and the less is the spatial
attenuation. For signal detection, the consequence of this is that a thicker film would
have better signal per unit thickness compared to that of a thinner film.
48
Wider range of aspect ratios
One way to quantify confinement of a long stripe is by its aspect ratio. One expects the
manifestation of confinement effects as mentioned in section 4.1.2 to scale with aspect
ratio. In particular, demagnetising field 96 is directly proportional to aspect ratio for
stripes magnetised tangentially perpendicular to the long axis 70, 79. The stripe array with
 30nm

the highest aspect ratio studied to date was by Huber et al. 
 101 
 0.3m

74
, while the
previously done study with the largest range of aspect ratios (10-3 – 10-2) was by
Vlaminck et al. on forward volume waves 73. Considering all relevant previous work to
date, there is little work done to systematically investigate the effects of lateral
confinement on spin wave propagation. The series of thick magnetic stripes studied in
this chapter systematically span the largest range of aspect ratios to date, covering two
orders of magnitude (Figure 4.2a). The stripe with the highest aspect ratio studied in this
work falls just slightly short of the aspect ratio of Huber et al.’s array 74.
Figure 4.2a: Aspect ratio of stripes studied in the field of propagating spin waves.
Improved sensitivity
Lastly, we have built a highly sensitive microwave detector which is able to detect
extremely small microwave signals (see section 2.2.3) specifically for this propagating
spin wave project. Hence, the possibility of detecting higher order width and edge
modes, both of which are relatively elusive by the induction method, warrants further
investigation at vastly improved sensitivity levels.
49
4.3 Experimental setup
The simplest design of a propagating spin wave spectroscopy (PSWS) experiment 70 is
shown in figure 4.3a. Gold coplanar waveguide antennas are overlaid across the
magnetic stripes at both ends with one antenna as the excitation source and the other as
the detector (figure 4.3b). Microwave current passes through one antenna, and the
microwave magnetic field of this current drives spin waves in the underlying Permalloy.
Propagating spin wave modes then travel along the strip, and get detected at the second
antenna via transduction. An insulating spacer (in this case, aluminium oxide) between
the magnetic strips and the coplanar lines ensure no physical electrical contact between
the two.
Figure 4.3a: (a) Diagram illustrating the experiment, showing dimensions and field lines. The
antennae separation gap, stripe width, and stripe thickness, are labelled x, y, and z, respectively.
The applied dc field H is in-plane and parallel to y. Microwave current i(ω) in the left-hand
antenna generates a non-uniform excitation field h(ω,x,z), which in turn drives spin waves,
m(ω,k). 1 denotes the Permalloy stripe, 2 is the 30 nm thick Al2O3 insulating layer, and 3 is the
receiving antenna (the similar structure to the right is the output antenna).
(b) Enlarged view of the antenna cross-section. The origin x=0 of the frame of reference
coincides with the symmetry axis of the input antenna.
50
A series of Permalloy stripes of various widths and thicknesses was fabricated (figure
4.3a) in order to experimentally investigate high aspect ratio confinement effects which
may deviate from the infinite thin film case. Permalloy is well-known for having low
intrinsic magnetic damping 32, hence it is the ideal metallic ferromagnet for spin wave
propagation. The narrowest and widest stripes are 2 and 100 µm respectively. All the
stripes have the same length 200 µm. Three different Permalloy film thicknesses were
fabricated: 55 nm, 80 nm, and 110 nm. In all, 108 unique stripe geometries were
investigated (figure 4.3a) with aspect ratios ranging [5.5 x (10-4 –10-2)]. In addition,
large patches of continuous film (3.7 x 3.7 mm2) were patterned on the same wafer
together with the stripes, and all underwent the same fabrication process. These serve as
reference continuous films, and were large enough to enable characterisation using flipchip broadband FMR (figure 2.2.1a).
A coplanar waveguide (CPW) design was used in this work to facilitate connection with
a coaxial line via a sub-millimetre coplanar probe. The CPW antenna geometrical
parameters are as follows: the conductor widths and separation gaps are 1.5 µm, and the
thickness is 200 nm. The distances between the excitation and detection antennas were
varied from 12 to 110 µm. Far away from the stripes, the CPW lines gradually become
larger, but still maintain geometrical proportions. This is to ensure matched
transmission line characteristic impedance throughout, which is calculated to be 67 Ω 97.
Finally, the CPW lines terminate at 100 µm sized contact pads (figure 4.3c). A 30 nm
thick aluminium oxide spacer ensures no direct dc electrical contact between the
overlaying gold antennas and the underlying Permalloy stripes.
51
Figure 4.3b: Optical micrograph of a typical PSWS experiment showing a magnetic
strip and overlaid coplanar lines.
Figure 4.3c: Optical micrograph of spin wave device showing coplanar lines leading to
contact pads.
52
4.4 Experimental procedure
4.4.1 Data acquisition
First, contact with the coplanar probes is made to the DUT using the probe station setup.
Then, microwave coaxial lines were connected to the probes; one coaxial line feeds
microwave power to the excitation antenna on the DUT, while the other coaxial line
relays the transmitted signal from the DUT. The other ends of the coaxial lines were
connected to the microwave receiver detailed in section 2.2.3. 10 GHz microwave
power at 10 dBm was fed into the receiver; this power is split into two channels, with
one driving the mixer’s LO port, and the other exciting spin waves on the DUT. 10 GHz
was used because the microwave receiver is optimised for that frequency. Spin waves
propagate along the magnetic stripe, and get detected at the second antenna (figure
4.3.1a). The detected signal gets fed into the RF port of the mixer. The resultant output
signal at the mixer’s IF port is a product of the reference driving signal, and the spin
wave signal (see section 2.2.3). This signal gets fed into a lock-in amplifier and is
recorded as a function of sweeping field applied tangentially to the magnetic stripes;
this is the Damon-Eshbach geometry. A modulating field of about 9 Oe (rms) at 220 Hz
modulates the sweeping field. The lock-in amplifier refers to this same modulating
field, and the resultant signal acquired is proportional to the field derivative of the
microwave susceptibility. A typical raw trace of the data is shown in figure 4.4.1a. Note
the oscillatory spin wave signal as a function of field due to phase selection at the
detection antenna.
53
Figure 4.4.1a: Raw trace of transmitted spin wave signal propagating through a 55 nm
thick, 2 μm wide Permalloy stripe over a distance of 20 μm, by microwave excitation
frequency of 10 GHz.
4.4.2 Sensitivity
We now evaluate the sensitivity levels of previous works utilising the inductive method,
and compare them to this work. The example from figure 4.4.1a was chosen because the
single stripe geometry is closest to that used in Bailleul et al.’s pioneering work in the
Damon-Eshbach geometry 70 and Vlaminck et al.’s subsequent work in the forward
volume wave geometry 73. In Vlaminck et al.’s work, the excitation and detection
antenna were meanders (5x), maximum antenna separation was 15 μm, the stripes were
10 – 20 nm thick, and a VNA was used to perform frequency sweeps. Note that even
though the film used in Vlaminck et al.’s work was about 5 times thinner than the
example in figure 4.4.1a, the use of 5 meanders effectively amplifies the signal 5-fold;
hence, both stripe data becomes comparable. At a VNA intermediate frequency
bandwidth of 10 Hz, the transmitted spin wave signal is comparable to the noise floor
(S/N ≈ 1). Averaging of more than 100 scans was needed to bring the spin wave signal
clearly above the noise floor. Assuming that the signal-to-noise ratio (S/N) improves as
the square root of number of trials, then the S/N of Vlaminck’s work for this particular
stripe geometry is about 10.
54
Another comparison may be made with Sekiguchi et al. 34 and Covington et al. 76’s
time-resolved PSWS work. The former used 35 nm thick, 120 μm wide Permalloy
stripes, while the later used 27 or 104 nm thick 475 μm squares of Permalloy; these are
large-scale structures compared with the narrow 2 μm wide stripe example in figure
4.4.1a. Though there were no explicit mentions of the noise level, the experiments
required averaging of 1024 waveforms to improve S/N.
The other two previous works done to date using the inductive method are Bao et al.’s
work on a 25 nm continuous film 75, and Huber et al.’s work on an array of nano-stripe
arrays 74. Evaluation and comparison of sensitivity levels with these two works are not
possible since there were no mentions of the S/N of both these experiments, and also
that these works were not performed on individual stripes.
We now turn attention to our experiment in the frequency domain utilising our sensitive
microwave receiver (section 2.2.3) with the field-modulation method (section 2.2.2).
The typical noise floor of the setup is roughly 0.3 μV. Using the amplitude of the
dominant spin wave band from the example raw trace from figure 4.4.1a, this translates
to a remarkable S/N of about 500! Note that the raw trace (figure 4.4.1a) was obtained
with a single field sweep (without averaging multiple runs), and with a single detection
antenna (without multiple meandering of antennae 70, 73). As a further testament of the
sensitivity of this setup, we successfully detected multiple higher order width modes on
our stripe with the highest aspect ratio and characterised their dispersion and group
velocities in detail (see section 4.6).
4.4.3 Wave number space
Spin waves are excited by the microwave magnetic field of the microwave current
flowing through the coplanar antennas. The wave number spectrum of spin wave
excitation and absorption depends on the square of the Fourier transform of the current
density in the coplanar antenna 98. The skin depth of gold at 10 GHz frequency is
calculated to be 0.8 μm 99. The skin depth is much larger than the thickness of the
coplanar line (0.2 μm) and about half the widths of each conductor (1.5 μm). Hence one
expects the microwave current density in the coplanar lines to be practically uniform
across the thickness of the coplanar lines. Laterally, the current density is described by
55
Dmitriev 100; the current is concentrated on the outer edges of the central conductor and
the inner edges of the ground planes, as shown in figure 4.4.3a. The negative current
densities in the ground conductors indicate the return current.
Figure 4.4.3a: Spatial current density of the coplanar antennas. The central conductor,
ground conductors, and separation gaps are all 1.5 μm wide.
Figure 4.4.3b: Spatial Fourier transform of the current density of the coplanar lines,
assuming uniform current density distribution.
Although the current density distribution is non-uniform laterally, the Fourier transform
of this non-uniform distribution is quite close to the case where an uniform distribution
is assumed 98. For ease of computation, Fourier transform for the uniform current
56
density case was taken (figure 4.4.3b). This approximation is reasonable since the
microwave skin depth and the conductor width are comparable. Multiple wave number
bands were found in the spatial Fourier transform of the coplanar line current density, of
which the first three are shown in (figure 4.4.3b). The dominant band is centred on the
most intense peak, at k0 = 10038 cm-1. This corresponds roughly to the wavelength of
the central conductor width, w, and gap between the conductors, Δ, according to the
expression 98:
k0 
2
w  2
For simplicity of further data analysis, only the dominant band (0 – 20000 cm-1) is
considered to effectively contribute to spin wave excitation 43, 101.
4.4.4 Extracting dispersion
The signal obtained in transmission (wave propagation from one antenna to another), is
the sum of two signals; direct electromagnetic induction through air, and spin wave
transduction. Assuming plane wave propagation, the superposition of the two waves is:
Asin(t   )  B sin(t  kx)
The first term in the sum represents direct electromagnetic induction through air, while
the second term represents spin wave transduction after propagating through the
Permalloy stripe a distance x (which is the separation distance between both antennae).
ϕ is an arbitrary initial phase difference between the two waves. The typical group
velocity of magnetostatic surface spin waves (≈ 10 µm/ns for a 100 nm Permalloy film)
is much smaller than the speed of light. Hence from the spin wave’s perspective, the
electromagnetic wave propagating through air has virtually no phase accumulation over
such small propagation distances.
For the case where A = B (where both amplitudes are the same), the expression can be
rewritten as:
 2t    kx   kx   
sin(t   )  sin(t  kx)  2 sin
 cos

2

  2 
57
The cosine term in the trigonometric identity relation above determines the amplitude
envelope of the superposition of two plane waves. Power is proportional to the square of
the wave amplitude, so power constructive interferences occur when:
 kx   
cos 2 
 1
 2 
Or when (where n = integer):
kx  2n  
The phase difference between each successive maxima of the power absorption
interference spectra is thus:
k 
2
→ Equation 4.4.4a
x
This method of using phase accumulation to extract dispersion was also used in
references 35, 73, 85, 102, 103. From this, one expects that the number of interference patterns
in the spectra increase with propagation distance. This is confirmed experimentally as
shown in figure 4.4.4a. Note that the envelope band over which spin waves occur does
not change with propagation distance; only the number of oscillations within the
interference pattern. One finds a linear relationship between the numbers of amplitude
oscillations with propagation distance (figure 4.4.4b).
58
Figure 4.4.4a: Normalised spin wave raw traces at 10 GHz for a 110 nm thick, 100 µm
wide Permalloy stripe, at various propagation distances, x.
59
Figure 4.4.4b: Number of amplitude oscillations at various propagation distances.
Hence, when field (or frequency) is swept around the resonance, different wave
numbers are selected according to the dispersion relation and within the bandwidth of
the spatial Fourier transform of the current density of the excitation antenna 73. Note that
the envelope of the interference pattern in the raw trace reflects the wave number
distribution of the Fourier transform of the coplanar antenna. Using this fact, one can
extract the dispersion from a raw trace as follows:
60
Figure 4.4.4c: Extracting dispersion from raw data.
An example raw trace data is shown in figure 4.4.4c. One sees an interference pattern in
the signal as a function of sweeping field for a given frequency. The extrema with the
largest signal amplitude on the raw trace (figure 4.4.4c-a) corresponds to the
fundamental wave number k0 of the coplanar line (figure 4.4.4c-b), and each successive
maxima/minima has a total phase accumulation of 2π (figure 4.4.4c-b). This way, one
can construct a dispersion relation (figure 4.4.4c-c) by mapping each extrema on the
raw trace (figure 4.4.4c-a) to a particular wave number of the Fourier spectra of the
coplanar line (figure 4.4.4c-b). This procedure is repeated for all the stripes studied in
this work, and presented here after.
Note that in contrast to Bao et al.’s proposed absolute phase method using a VNA 75,
this relative phase difference method is much simpler. Also note that since the
experiment is field-resolved, the plot is technically a pseudo-dispersion relation, rather
than a proper dispersion plot of frequency versus wave number.
61
4.5 Magnetostatic surface mode in confined stripe geometry
4.5.1 Dispersion
For a continuous film, magnetostatic surface modes (MSSM) propagate according to the
dispersion relation:
f


2

  H ( H  4M )  4 2 M 2 (1  e 2 kd ) → Equation 4.5.1a

Note that by setting k = 0 (wave becomes stationary), one essentially recovers the Kittel
formula for FMR in a tangentially magnetised continuous film (Equation 3.1.2a).
To plot dispersion relation, one has to consider the optimal propagation distance from
which to extract dispersion from. As seen in figure 4.4.4a, the number of oscillations is
proportional to the propagation distance. Thus, more data points can be extracted from
large propagation distances. In addition, one should also minimise antenna near-field
effects by maximising the separation gap between antennae (see section 4.7). However,
signal attenuation at large propagation distances place a practical upper limit on the
antennae gap from which sufficient signals are available. For the 110 nm series reliable
data could be obtained at 60 µm separation gap between the antennae; for the 80 nm and
55 nm series, due to more attenuation for thinner films, the distance is 30 µm. For these
propagation distances, the dispersion relations of the MSSM in the Permalloy stripes are
shown in figure 4.5.1a for the range of thicknesses and widths studied.
Due to sample fabrication defects, some antennae and/or Permalloy stripes were
malformed, resulting in “gaps” in the data. The black dotted curves are the theoretical
dispersion relations for infinite continuous films calculated using equation 4.5.1a based
on the extracted saturation magnetisation values obtained from FMR (figure 4.5.1b).
One immediately observes that the dispersion relation shifts to high field upon
narrowing of stripe width. We claim that this is due to the effect of the static
demagnetising field. As the stripes become narrower, the effective demagnetising field
becomes larger, resulting in decrease in internal field for a given applied external field
70, 79, 96
(see section 4.5.2). This shifts the dispersion up-field.
62
Figure 4.5.1a: Dispersion relation of Permalloy stripes at 10 GHz.
It is found that within the experimental accuracy, for a particular stripe thickness, the
slope of the dispersion is independent of stripe width, at least up to the stripe with the
 110nm 
highest aspect ratio 
 . The dynamic confinement effects are known to modify
 2m 
63
the slope and curvature of the dispersion curve 87. This implies that for the available
aspect ratios and wave numbers, the dynamic confinement effects are not important
within the experimental accuracy. Therefore, one can treat the magnetisation dynamics
in the stripes as in continuous films. One only needs to include the static confinement
effect. We argue that this effect can be accounted for by including an effective
demagnetising field term, Hd, into the dispersion relation for continuous film similar to
104
. Thus, equation 4.5.1a is modified into:
f


2

  ( H  H d )( H  H d  4M )  4 2 M 2 (1  e 2 kd ) → Equation 4.5.1b

Using the modified dispersion law, one can set Hd as the fitting parameter and extract
the effective demagnetising fields. The saturation magnetisation parameter, M, was
extracted separately from reference film FMR data (figure 4.5.1b). M is assumed to be
constant across all strip widths for particular thicknesses in the dispersion fitting. The
dispersions were fitted with the effective demagnetising field as the only free fitting
parameter; all other parameters were assumed to be constant. In particular, the
saturation magnetisation, gyromagnetic ratio, and thicknesses of the stripes were
assumed to be identical to their respective reference continuous films, since they were
all deposited together on the same wafer in the same deposition process, and underwent
the same lithography fabrication process. The fits are shown with solid curves in figure
4.5.1a.
64
Figure 4.5.1b: FMR data for the reference films showing extracted saturation
magnetisation values. Solid lines are fits with Kittel’s formula (equation 3.1.2a).
As shown, the data fits well to the modified dispersion law across the range of aspect
 110nm 
ratios studied, except for the stripe with the highest aspect ratio 
 . For that
 2m 
particular stripe, there was noticeable deviation from equation 4.5.1b. This is postulated
to be due to the fit assuming an uniform demagnetising field 86, but in actual fact, this
particular stripe has the most non-uniform demagnetising field profile across its width
amongst the stripes studied (see section 4.5.2). The thin film model assuming a uniform
effective demagnetising field is insufficient to properly describe a narrow stripe of such
high aspect ratio. Furthermore, for large aspect ratios, the geometrical confinement also
affects the dynamic dipole field. The static demagnetising field shifts dispersion upfield, while the dynamic dipole field increases the dispersion slope with respect to the
continuous case 79, 86.
65
 110nm 
Figure 4.5.1c: Experiment and simulation MSSM dispersion for the 
 stripe.
 2m 
Solid curves are fits with equation 4.5.1b.
 110nm 
To properly model the experiment dispersion of the 
 stripe, numerical
 2m 
simulations were performed. The demagnetising field profile (see figure 4.5.2a-a in
section 4.5.2) was used as the stripe ground state. From this, the eigen-fields for
particular wave numbers at 10 GHz frequency were numerically simulated. The
simulated dispersion is plotted together with experiment data in figure 4.5.1c. To
compare with the modified continuous film dispersion, both the experiment and
simulation results were fitted with equation 4.5.1b. Note that both the simulation and
equation 4.5.1b did not adequately model the experiment result. The slope of the
experimental dispersion is smaller than predicted by simulation. The failure of the
simulation to model the experiment data might be due to unaccounted peculiarities in
 110nm 
this particular stripe. To further elucidate the matter, the 
 stripe’s dispersion
 2m 
was fitted with equation 4.5.1b with additional free fitting parameters (figure 4.5.1d).
66
 110nm 
Figure 4.5.1d: 
 stripe dispersion fittings with equation 4.5.1b.
 2m 
One notes from figure 4.5.1d that better fits could be obtained by setting the saturation
magnetisation, M, or the thickness, d, as fitting parameters in addition to the effective
demagnetising field, Hd. By doing these, M decreased by 10% while d decreased by
15%. From these results, one may speculate that the peculiarity of this stripe may be due
to localised inhomogeneity during fabrication, resulting in reduced saturation
magnetisation, thickness, or both, for this particular stripe.
Interestingly, the simulation data (figure 4.5.1c) fits well to the modified continuous
film dispersion (equation 4.5.1b). This seems to suggest that at least theoretically
(disregarding the peculiarity seen for this particular stripe in experiment), a stripe of
 110nm

aspect ratio as high as 
 0.055  can still be modelled by the simple modified
 2m

continuous film dispersion by assuming an uniform up-field shift in the dispersion from
an effective static demagnetising field, in the range of wave numbers tested in this
experiment.
67
4.5.2 Static demagnetising field simulations
In order to evaluate the validity of introducing an effective demagnetising field into the
dispersion model, the demagnetising field profiles of the stripes studied were
numerically simulated using LLG Micromagnetics Simulator software 105 (refer to
Appendix C). To simplify computation, the stripes were assumed to be infinitely long,
thus reducing the problem into a two dimensional one.
Figure 4.5.2a: Simulated demagnetising field profiles in infinitely long Permalloy
stripes.
68
The simulated demagnetising field profiles of two extreme cases (smallest and largest
aspect ratios) are shown in figure 4.5.2a. The stripe with the lowest aspect ratio
 55nm 

 is expected to have the least demagnetising field, and one clearly sees that
 100m 
this is indeed the case in figure 4.5.2a-b. Demagnetising fields are only significant near
the edges of the stripe, just roughly 2% of the total width. The small demagnetising field
across the bulk of the stripe is practically uniform. For the other extreme case, the stripe
 110nm 
with the highest aspect ratio 
 is expected to have highly non-uniform
 2m 
demagnetising fields across the width, and indeed this is the case in figure 4.5.2a-a.
Note that in this narrow stripe, significant demagnetised edge regions are present for
low applied fields.
69
Figure 4.5.2b: Comparison between experiment and simulation effective demagnetising
fields.
70
The simulated effective demagnetising field for each strip is approximated by taking the
mean value across the strip width (figure 4.5.2a). The experimentally extracted effective
demagnetising fields are compared against these simulated values in figure 4.5.2b. It is
observed that the simulation tends to overestimate the experimental effective
demagnetising field. Possible explanations for these discrepancies are as follows:
Firstly, taking the arithmetic mean across the demagnetising field profile as an effective
value may not be the appropriate statistical approach. Arithmetic mean tends to be
disproportionately weighted towards large values, and demagnetising fields can be very
large at stripe edge regions while only occupying relatively small volumes 79.
Secondly, considering the demagnetising field profile alone is insufficient. One also
needs to account for the non-uniformity of the static magnetisation and spin wave mode
profile for a more accurate analysis. The effective demagnetisation factor for a
particular spin wave mode is the proportional to the overlap integral between the mode
profile and the demagnetisation field profile 84. In addition, dynamic effects due to
confinement affect the dispersion slopes 79, 86; this is not accounted for by the static
demagnetising field fitting parameter in equation 4.5.1b. Hence, merely taking the mean
value of the simulated demagnetisation field profile alone is insufficient to accurately
model the actual effective demagnetisation field for the particular spin wave mode.
71
4.5.3 Group velocity
The group velocity, Vg, can be calculated from the dispersion data using the
relationship:
Vg 

f H
→ Equation 4.5.3a
 2
k
H k
The
H
f
term is simply the slope of the field-resolved dispersion, and the
term is
k
H
obtained by differentiating equation 4.5.1b:
f
 2 ( H  H d  2M )
→ Equation 4.5.3b

H
f
From equation 4.5.3a, the experimentally calculated group velocities are plotted in
figure 4.5.3a as function of wave number and tabulated in table 4.5.3a for k0 = 10038
rad/cm.
72
Figure 4.5.3a: MSSM group velocities of Permalloy stripes.
73
From figure 4.5.3a, one sees a general trend that the group velocity increases with film
thickness. This is what one expects since the slope
H
in the dispersion (figure 4.5.1a)
k
increased with film thickness. Theoretically, this is also what one expects by
differentiating equation 4.5.1b to obtain

, where the group velocity is proportional to
k
film thickness:
Vg 
 M 2 d  2 kd

e → Equation 4.5.3c
 8 3 2 
k
 f 
For each particular stripe, it is immediately obvious that the group velocity decreases
for increasing wave number. This is consistent with the formulation in equations 4.5.3b
and 4.5.3c; one expects a negative slope for a plot of group velocity versus wave
number.
For particular thicknesses, there seems to be no correlation between stripe width and
group velocity. For the 55 nm and 80 nm thick stripes, the vertical spread range in the
group velocities is roughly 1.5 μm/ns. This is the same for the 110 nm thick stripes, if
 110nm 
the peculiar 
 stripe is excluded. If we attribute the scatter to the accuracy
 2m 
limitations of the experiment, then we may conclude that for the aspect ratios and wave
number range investigated here, the group velocity is width-invariant for a particular
thickness.
Thickness (nm)
Group velocity (μm/ns)
55
5.5 – 7
80
8 – 9.5
110
13 – 14
Table 4.5.3a: Group velocities at 10 GHz and k0 = 10038 rad/cm.
74
4.5.4 Attenuation and relaxation
We now turn attention to the propagation attenuation and relaxation characteristics of
MSSM in the stripes studied. The total signal transmitted,  , from the excitation
antenna to the detection antenna is 100:
(, k )  Te ( ) P(, k )Td ( ) → Equation 4.5.4a
Te and Td are antenna excitation and detection efficiencies, respectively. P is the spin
wave propagation loss. For identical antennae, Td  Te * ; the detection efficiency is
simply the complex conjugate of the excitation efficiency. Then, the antennae losses can
be lumped into a pre-factor for the spin wave propagation loss term. In this experiment,
we assume that differences in antennae characteristics are negligible; i.e., the antennae
are similar to each other ( Td  Te * assumption holds). Then, the relative drop in signal
transmission between antennae upon propagation will be due to spin wave propagation
losses. Spin wave propagation loss was modelled as an exponential decay (equation
4.5.4b) 34, 71, where Ld is the attenuation length, defined as the propagation distance
when the signal has dropped to 1/e from its initial value.
P  P0e

x
Ld
→ Equation 4.5.4b
From equation 4.5.4a, and taking logarithms, equation 4.5.4b becomes:
ln   2 ln T  ln P 
x
→ Equation 4.5.4c
Ld
Thus, a logarithmic plot of the transmitted signal amplitude on the vertical scale versus
a linear plot of the propagation distance on the horizontal scale would enable extraction
of the attenuation length from the linear slope of the plot. Note that the initial spin wave
signal amplitude P and antenna efficiency T both contribute to the vertical intercept.
Real-world variations in the antenna characteristics ΔT would manifest as vertical
spread in the plot. Thus, the accuracy of the linear fit depends on the difference between
ΔT and P. This means that the thicker the film, the larger the initial spin wave signal
amplitude P, and thus the more reliable the fit would be.
75
In order to obtain absolute quantitative values, we measured the transmission scattering
parameter (S21) between the two antennae with field-resolved VNA. Measurements
were performed on wide stripes for each of the three thicknesses in order to obtain the
high signal-to-noise (decrease inaccuracies from ΔT antenna variation contributions).
The amplitude ΔS21 is defined as the range between the central extrema of the spin
wave packet, about the dominant wave number k0 = 10038 rad/cm (figure 4.5.4a).
Figure 4.5.4a: Example VNA raw trace, showing definition of amplitude ΔS21.
The logarithmic amplitudes of the spin wave signals at various antennae gaps are
plotted in figure 4.5.4b. Note the linear trend in the logarithmic plot of the data (figure
4.5.4b), verifying the exponential decay assumption made earlier. It is noteworthy that
the antenna near-field effect (see section 4.7) results reduced effective propagation
distance compared to the physical separation distance between the excitation and
detection antennae. This shifts all the data points horizontally by a constant amount.
However, this would not affect the slope of the plot. The attenuation lengths extracted
from the fittings were tabulated in table 4.5.4a.
76
Figure 4.5.4b: ΔS21 amplitude at various antennae gaps at 10 GHz and k0 = 10038
rad/cm. Solid lines are fits to extract the attenuation lengths.
Comparison can be made with Sekiguichi et al.’s similar work involving spin wave
 35nm 
propagation through a Permalloy stripe with aspect ratio 
 over distances 5 –
 120m 
50 μm 34. In that work, the attenuation length was determined to be 15 μm, which is
similar to the values for the stripes studied here (table 4.5.4a).
The relaxation time τr of MSSM can be calculated from the group velocity, Vg, and
attenuation length, Ld, similar to 95, 106 through the simple relationship:
 r ( MSSM) 
Ld
→ Equation 4.5.4b
Vg
Similar to attenuation length, relaxation time is defined as the time it takes for a signal
to decay to 1/e from its initial value 71. The calculated relaxation times are tabulated in
table 4.5.4a. Comparison can be made with Bailleul et al.’s work where similar MSSM
relaxation times were obtained: 2 ns 70 and 1.6 ns 71. Using equation 2 from reference 107
together with equation 4.5.4b and 4.5.3a, the Gilbert damping coefficient, α, can be
formulated in terms of the MSSM relaxation time, where Hi is the internal field:

1
→ Equation 4.5.4c
2 ( H i  2M ) r ( MSSM)
77
The calculated attenuation lengths, relaxation times, and Gilbert damping coefficients
were tabulated in table 4.5.4a. Note the atypical losses in the 80 nm thick stripes.
Stripe
Group
Attenuation
Relaxation
Gilbert
thickness (nm)
velocity
length (μm)
time (ns)
damping
(μm/ns)
coefficient
(10-3)
55
6.46 ± 0.04
7.83 ± 0.04
1.21 ± 0.01
7.66 ± 0.06
80
8.17 ± 0.08
7.37 ± 0.04
0.90 ± 0.01
10.8 ± 0.1
110
13.14 ± 0.09
14.7 ± 0.5
1.12 ± 0.04
8.2 ± 0.3
Table 4.5.4a: Stripe MSSM propagation and attenuation parameters at k0 = 10038
rad/cm and 10 GHz.
The MSSM attenuation parameters can also be compared with those obtained from their
respective reference film FMR line widths. The FMR relaxation time in terms of the
FMR line width, H FWHM , reformulated from Stancil’s equation 17c 95, is:
 r ( FMR ) 
f
→ Equation 4.5.4d
 H FWHM ( H i  2M )
2
The Gilbert damping coefficient is given by Stancil’s equation 5 95:

H FWHM
2f
→ Equation 4.5.4e
By plotting the FMR line width for various resonance frequencies, the Gilbert damping
coefficient can be easily extracted from the slope using equation 4.5.4e (figure 4.5.4c).
Non-zero intercepts at zero frequency (≈ 10 Oe) were found in the line width plots. This
is attributed to inhomogeneous line width broadening 24, 108, 109, and is sensitive to the
thermal history of the material 110. However, most importantly, the slope of the line
width plot is insensitive to sample history, and is a reliable measure of intrinsic damping
109, 110
.
78
Figure 4.5.4c: FMR line widths of reference films.
Film thickness
Saturation
Relaxation time
Gilbert
(nm)
magnetisation
at 10 GHz (ns)
damping
(emu/cm3)
coefficient
(10-3)
55
837 ± 3
1.15
6.6 ± 0.2
80
799 ± 3
1.03
8.2 ± 0.3
110
886 ± 3
1.03
6.8 ± 0.3
Table 4.5.4b: Reference film properties.
The FMR relaxation times and Gilbert damping coefficients are tabulated in table
4.5.4b. The values for the 110 nm and 55 nm films are typical for Permalloy 24, 32, 70, 71.
Note the ~25% larger than typical intrinsic damping for the 80 nm thick reference film
compared to the other two. This might be due to fabrication defect resulting in a low
quality magnetically “lossy” film. This is noticeably evident in the plot in figure 4.5.4c.
This might explain the reduced attenuation length and relaxation time of MSSM
propagation in the 80 nm stripes (table 4.5.4a).
Note also that the Gilbert damping coefficients calculated from the MSSM propagation
data (table 4.5.4a) were larger (~ 25%) than those extracted from the reference film
79
FMR line widths (table 4.5.4b). Three possible reasons for this discrepancy are as
follows:
First, one expects confinement to contribute additional edge losses (not present in a
laterally infinite continuous film). As covered in section 4.5.2, demagnetised regions
(very low or even negative internal fields) are present at the stripe edges due to large
demagnetising fields at the edges of transversely magnetised narrow stripes 70, 79, 96. In
fact, this edge effect leads to channelling of dipolar modes at the centre of such narrow
stripes 81. MSSM propagation into these demagnetised edge regions would be trapped in
the potential wells 79, 90, contributing to loss. In addition, edge defects from fabrication
imperfections will contribute to scattering of spin waves 19, 20. All these contribute to
propagation attenuation on top of the intrinsic material damping.
The second possible explanation is that the damping coefficient extracted from FMR
data was at kFMR = 0 rad/cm while the value obtained from MSSM data was at a
different wave number, k0 = 10038 rad/cm. One notes that the inhomogeneous line
width broadening may be slightly different between kFMR = 0 rad/cm and k0 = 10038
rad/cm, thus the damping coefficient may have wave number dependence 111.
The third explanation is a procedural one. The damping coefficients calculated for
particular wide stripes were obtained indirectly through the group velocities and
attenuation lengths. Both of these (group velocity and attenuation length) themselves
were obtained indirectly; from the dispersion and amplitudes, respectively. Since the
calculation of the damping coefficients from the MSSM data was done through two
levels of indirect methods, inaccuracies would be compounded, and one may question
the reliability of the results. On the other hand, determination of the damping coefficient
from the continuous film FMR line widths is a direct and thus more reliable method. In
fact, this is a standard method to determine the damping coefficient from FMR
experiments 24, 32, 109, 110, 112.
80
4.5.5 Non-reciprocity
One propagation property of MSSM is its non-reciprocity. For a given tangential
magnetisation orientation, counter-propagating MSSM waves are localised on opposing
surfaces of a ferromagnetic slab 67. The surface on which the wave propagates is
determined from the vector cross product 113:
kxH=sn
Where n is the normal vector pointing out of the plane where k propagates along, and s
is a proportional constant. This is illustrated in figure 4.5.5a.
Figure 4.5.5a: MSSM wave propagation on a ferromagnetic slab, showing nonreciprocity.
For thick insulating ferrites like YIG, the mechanism of non-reciprocity may be
explained by the concentration of surface waves on opposing sides of the film 67, 102, 113.
However, this mechanism for non-reciprocity is invalid when the wavelength of MSSM
is much larger than the film thickness ( kd  1 ) 81. Note that this is indeed the case for
our stripe thicknesses and the range of available wave numbers ( kd  0.1 ).
The antenna-induced mechanism of non-reciprocity for the case of kd  1 in metallic
thin films with large magnetic moments is explained by Demidov et al 81. Consider the
excitation field components of the antenna, hx (in-plane) and hz (out-of-plane),
according to the frame of reference in figure 4.3a. Both components provide the driving
81
torque for magnetisation precession. The torque contribution by hx is always in-phase
with the emitted spin wave, but the torque contribution by hz have opposite phase
relations at the opposite sides of the antenna 34, 74, 81. This asymmetric excitation field
results in non-reciprocal emission of counter-propagating waves from the antenna.
However, non-reciprocity is weaker for materials with large saturation magnetisation
(like the 3d metallic ferromagnets), owing to large in-plane ellipticity of magnetisation
precession (the asymmetric torque contribution of hz to magnetisation precession is
weaker) 81.
Experimentally wise, non-reciprocity can be probed by either reversing the antennae
roles (reversing k) or reversing the field. Both were shown to be equivalent by
Sekiguchi et al. 34; this was verified in our experiments in a wide range of aspect ratios.
However, it is much easier to reverse the applied field than to swap the roles of the
excitation/detection antennae, so we study non-reciprocity by simply sweeping the field
from negative to positive. An example trace raw trace is shown in figure 4.5.5b. The
spin wave amplitude difference upon field reversal is immediately noticeable.
Figure 4.5.5b: Example of MSSM non-reciprocity upon field reversal. Stripe thickness:
110 nm. Stripe width: 50 μm. Propagation distance: 30 μm.
Similar to Demidov et al. 81 and Sekiguchi et al. 34, we define the non-reciprocity
parameter, η, as the amplitude ratio:

A
→ Equation 4.5.5a
A
82
A+ is the spin wave amplitude in the favoured propagation direction (larger amplitude)
while A– is the spin wave amplitude in the unfavoured propagation direction (smaller
amplitude). Amplitudes were taken at the extrema where the dominant wave number k0
= 10038 rad/cm occur (figure 4.5.5b).
Figure 4.5.5c: Non-reciprocity parameters at 10 GHz and k0 = 10038 rad/cm.
83
The non-reciprocity parameters were calculated for all the stripes studied in this work
and plotted in figure 4.5.5c. There seems to be a slight trend of non-reciprocity
strengthening as stripes become narrower for the 110 nm series. However, over all, the
non-reciprocity is largely invariant (η ≈ 0.25 ± 0.05) over the range of stripe
thicknesses, widths and propagation distances investigated. The results strongly suggest
an antenna-induced non-reciprocity origin, since the same antenna geometry was used
on all the stripes studied in this work.
Using Demidov et al.’s formula 81, the theoretical non-reciprocity was calculated to be η
≈ 0.5 for a single antenna. For a PSWS experiment utilising two identical antennae as in
this experiment, the non-reciprocity parameter is twice that for a single antenna. If one
extends Demidov et al.’s formula for our system, we obtain η ≈ 0.25. Thus, the nonreciprocity of MSSM in our PSWS experiment is as predicted by theory.
4.6 Higher order width modes in confined stripe geometry
Extra modes in addition to the dominant MSSM were observed in the spin wave spectra
 110nm 
for the stripe with the highest aspect ratio studied in this work 
 . These were
 2m 
identified as higher order width modes appearing due to confinement in such a narrow
stripe. This sub-chapter deals with the identification and characterisation of these
modes.
In a simple qualitative model for confined stripe geometry, the finite width leads to
quantization of backward volume-like spin wave modes across the width in the form 50,
87
:
ky 
n
→ Equation 4.6.1a
w
Where n is the mode number and w is the effective width of the stripe. The eigen-modes
take on sinusoidal functions analogous to the case of spin wave resonance modes across
the thickness 38, 50, 79, 84. The total wave number, kt, for spin wave propagation in a
confined stripe is thus 84:
84
kt2  k x2  k y2 → Equation 4.6.1b
Where kx is propagation in the longitudinal direction (along the long axis) and ky is
propagation in the transverse direction (along the short axis), following the Cartesian
axes of figure 4.3a. One sees from equations 4.6.1a and 4.6.1b that the relative
contributions of the two orthogonal kx and ky modes depends on the stripe’s lateral
dimensions. From this, we define the wave propagation angle, θ:
 ky 
 → Equation 4.6.1c
 kx 
  arctan
For θ = 0, the wave is purely longitudinal, and for θ = 90°, the wave is purely
transverse. Thus, from equation 4.6.1c, one sees that the angle θ increases with width
mode number. Thus, for increasing mode number, the total wavevector becomes more
canted away from the longitudinal direction towards the transverse direction.
In our case, longitudinal quantization is not possible since the length of the stripes (200
μm) is much longer than the spin wave attenuation length (table 4.5.4a). The closest
distance between the antenna and the end of the stripe is 50 μm. Due to attenuation,
reflections along the longitudinal direction to form quantized longitudinal modes are not
possible. Thus, for our stripes, wave propagation in the longitudinal direction is
effectively similar to MSSM in an infinite continuous film 67, without quantization
effects (see preceding section 4.5). Contrast can be made with Mathieu et al 77 and
Roussigne et al 78’s work in the Damon-Eshbach geometry where their stripes were
magnetised along the long axis and quantization of MSSM were observed across the
stripe width. In their work, dispersion-less quantized MSSM modes were observed for
small wave numbers. Note that in contrast to their work, our stripes were magnetised
transversely and propagation along the long axis was considered (figure 4.3a).
On the other hand, excitation of quantized transverse modes along the stripe width is
plausible since the narrowest stripe width in our stripe series (2 μm) is of spin wave
attenuation length order. In such confinement, waves can bounce back and forth from
the side walls without being significantly attenuated, thus forming standing waves
across the width. Indeed, we observed multiple higher order width modes in the spin
 110nm 
wave spectra of our particular stripe with the highest aspect ratio 
 (see section
 2m 
85
4.6.1). While this is not the first time these higher order width modes were detected
using an inductive method (observed in Bailleul et al.’s pioneering work 70), our work
was done so with greatly improved sensitivity to enable further detailed characterisation
of their dispersions and group velocities (see section 4.6.2). Furthermore, we display the
higher order mode signals obtained in reflection and compare them with ones obtained
in transmission. To date, the detailed dispersion characterisation of these higher order
width modes have only been achieved on stripe arrays using Brillouin light scattering 7779, 86
. In the following sections, the analysis of these width modes was performed similar
to section 4.5.
4.6.1 Mode identification
The raw data traces for the stripe exhibiting multiple higher order width modes are
shown in figure 4.6.1a. In the reflection data, a remarkable 6 higher order width modes
on top of the fundamental MSSM were observed in the spectra. Numerical simulation
was used to generate theoretical eigen-fields and mode profiles in order to identify these
modes. As seen in figure 4.6.1a, the simulated eigen-field positions match up well with
the experimental mode positions. Mode numbers were assigned accordingly upon
evaluation of the mode profiles (figure 4.6.1b), where n is the number of anti-nodes in
the mode profile. The excitation field has a symmetric odd mode profile across the
stripe width. This implies that only modes with odd symmetry can be excited by such a
field 36, 38, 70, since the mean values of odd mode profiles modes are non-vanishing
(figure 4.6.1b). For the modes n ≥ 3, one sees a pronounced decrease in the amplitude of
the maxima of the standing wave towards the stripe edges. This is due to the nonuniformity of the magnetisation ground state (see the upper panel in figure 4.5.2a).
The simulated mode profiles are similar to figure 6 in Bayer et al. 79 and figure 4 in
Kostylev et al.’s work 86, with the exception that our experiment was done fieldresolved. (Refer to Appendix C for the simulation procedure and list of key parameters.)
 110nm

Note the similarity in the aspect ratio of this particular stripe 
 0.055  with the
 2m

86
 30nm

stripe array studied by Kostylev et al. 
 0.05  , where multiple width modes
 600nm

were detected using Brillouin light scattering 86.
 110nm 
Figure 4.6.1a: Field-trace at 10 GHz for the 
 aspect ratio stripe, showing
 2m 
multiple high order width modes. Solid curve: signal received at detection antenna over
12 μm propagation distance. Dashed curve: signal reflected back from excitation
antenna. Vertical lines: simulated mode field positions, where n is the mode number.
87
Figure 4.6.1b: Simulated mode profiles.
88
Figure 4.6.1c: Mode amplitudes at the dominant wave number k0 = 10038 rad/cm
normalised to the fundamental MSSM (n = 1). For the simulation, the mean value of the
mode profile was designated the mode amplitude. For the experimental data, the mode
amplitudes follow the definition in figure 4.5.5b.
Theoretically, the excitation efficiency is proportional to the overlap integral of the
mode profile with the excitation field profile 84. Assuming a uniform excitation field,
the overlap integral reduces to the mean value of the mode profile. One expects the
mode amplitude to decrease for increasing mode number, since the mean value of
dynamic magnetisation decreases with increasing number of nodes in the mode profile.
Indeed, this was observed in the simulated and experimental reflection data. (figure
4.6.1c). Interestingly, in the experimental reflection data, the mode amplitudes seem to
drop linearly with increasing mode number (for n ≥ 3). In the simulation result, the
reduction in mode amplitude for increasing mode number has a different functional
dependence. At this point, we emphasize that it is often difficult to simulate high order
mode amplitudes to quantitatively match real-world data. It is sufficient for the
simulated and experimental relative mode amplitudes to qualitatively follow a rough
trend for the purpose of mode identification.
Comparing between the experimental reflection and transmission data, one observes
that the transmission data contained progressively less modes for increasing antennae
gap. In the transmission data (figure 4.6.1c), the amplitude of high order modes
decrease even more rapidly than in the reflection data with increasing mode number.
We postulate that this is due to increase in attenuation for increasing mode number upon
89
wave propagation (see section 4.6.3). In fact, for antennae gap of 12 μm, only the modes
up to n = 9 had sufficient transmitted signal to enable further dispersion
characterisation. Even though modes n = 11 and n = 13 were clearly observed in the
reflection data, these modes were attenuated down to below noise levels after
propagating 12 μm in the transmission data. Table 4.6.1a summarises the appearance of
higher order width modes as a function of antennae gap.
Mode
Transmission separation gap (μm)
Reflection
number
(n)
12
15
20
30
60
✓
1
✓
✓
✓
✓
✓
3
✓
✓
✓
✓
✓
5
✓
✓
✓
✓
7
✓
✓
9
✓
✓
11
✓
13
✓
Table 4.6.1a: Observation of width modes in experimental data.
4.6.2 Dispersion and group velocity
After having identified the width modes with the aid of numerical simulations, we now
consider their dispersion characteristics. Guslienko et al 84 formulated a non-analytic
dispersion expression for quantized dipole modes of an arbitrary rectangular slab
magnetised tangentially. In this work however, further characterisation of propagation
characteristics were done experimentally. Following the same phase interference
technique of extracting dispersion in section 4.4.4, the dispersions of the width modes
90
were extracted from the 12 μm separation gap transmission data in figure 4.6.1a. Note
that due to significant antennae near-field effects at such a small antennae gap (see
section 4.7), an effective propagation distance of 9.5 μm was used to calculate the phase
difference between the extrema in the raw data. The dispersions are plotted in figure
4.6.2a.
Figure 4.6.2a: Width mode dispersions at 10 GHz.
One immediately notices that the dispersion slope of the higher order modes (n ≥ 3) are
much smaller than the fundamental MSSM (n = 1). Since the slope of the dispersion
curve is proportional to group velocity, this indicates that the group velocities of the
higher order modes (n ≥ 3) are significantly lower than that of the fundamental MSSM.
Unlike the fundamental MSSM, there is no analytic expression to calculate group
velocity for the higher order width modes. Therefore, we will now utilise a Taylor
approximation method to determine the group velocities of these modes about 10 GHz
and the dominant wave number k0 = 10038 rad/cm.
91
The dispersions in figure 4.6.2a were fitted with either a linear or quadratic function
(whichever yielded the least residuals). These are shown as solid lines in figure 4.6.2a.
From these fits, the local gradient
H
were calculated for k0 = 10038 rad/cm.
k
The differential conversion factor between frequency and field resolved measurements,
f
, can be obtained by performing multiple frequency measurements for k0 = 10038
H
rad/cm (figure 4.6.2b). Note the excellent linear fits (solid lines), from which
f
values were extracted from the slopes for each respective mode.
H
Figure 4.6.2b: Frequency versus field plots for the width modes for k0 = 10038 rad/cm.
Similar to section 4.5, knowing both
H
f
and
, the group velocity can be calculated
k
H
using equation 4.5.3a:
Vg 

f H
 2
k
H k
Thus, to first degree Taylor approximation, one can calculate the group velocity for the
dominant wave number k0 = 10038 rad/cm and 10 GHz frequency. The results are
plotted in figure 4.6.2c and also tabulated in table 4.6.3a.
92
Figure 4.6.2c: Group velocity of higher order width modes at 10 GHz and k0 = 10038
rad/cm.
As postulated earlier, the group velocities of higher order width modes (n ≥ 3) were
found to be nearly an order of magnitude lower than the fundamental MSSM. (Hence,
for clarity of the higher order mode data, the fundamental MSSM data point was
deliberately left out of figure 4.6.2c.) Furthermore, the group velocity decreased for
increasing mode number (figure 4.6.2c). This results in increased spatial attenuation for
increasing mode number, as evident in the mode amplitudes in the transmission data
(figure 4.6.1a).
As for the reflection data in figure 4.6.1a, the decrease in group velocity for increasing
mode number may explain the discrepancy between the simulated relative mode
amplitudes in figure 4.6.1c. From equation 4 in Dmitriev 100, one may expect the
efficiency of mode excitation to grow with a decrease in group velocity; this may partly
compensate the decrease in the overlap integral for increasing mode number.
The group velocity of the fundamental MSSM for this stripe obtained using the Taylor
approximation method (12 μm/ns) is consistent with the values obtained in section
4.5.3, within the accuracy of the experiment. The fundamental MSSM group velocity is
directly proportional to the film thickness, and one may assume that this proportionality
also applies to higher order width modes. This may explain why these higher order
width modes were not observed in the thinner stripes studied in this work (very low
93
group velocity and excessive attenuation), and also why their observation using
inductive spectroscopy methods are lacking in the literature.
4.6.3 Attenuation and relaxation
We now evaluate the attenuation characteristics of the width modes. Due to excessive
loss for large mode numbers, only the first 3 modes (n = 1, 3, 5) contain the minimum
of 3 data points for meaningful extraction of attenuation length (figure 4.6.3a). As
mentioned before in section 4.5.4, the scattering of antenna efficiencies from sample to
sample was negligible and thus, 3 points – though not ideal – were sufficient to
determine the attenuation length.
Figure 4.6.3a: Mode amplitude for various antennae gaps. Solid lines are fits to extract
the attenuation lengths.
Mode amplitudes following the definition in section 4.5.4 for k0 = 10038 rad/cm were
extracted from the raw traces. The logarithms of the mode amplitudes were plotted as
function of antennae separation gaps in figure 4.6.3a. The attenuation lengths were
extracted from the slopes following section 4.5.4 and tabulated in table 4.6.3a. As noted
before in section 4.5.4, the antenna near-field effect (see section 4.7) would shift all the
data points horizontally by a constant amount. However, this would not affect the slope
of the plot in figure 4.6.3a.
94
Mode
number
f
H
-3
H
k
-3
Group
Attenuation
Relaxation
velocity
length (μm)
time (ns)
(10
(10 Oe
GHz/Oe)
cm)
1
5.25 ± 0.06
37 ± 1
12.2 ± 0.5
14 ± 1
1.2 ± 0.1
3
4.49 ± 0.06
7.6 ± 0.5
2.1 ± 0.2
4.9 ± 0.7
2.3 ± 0.4
5
4.11 ± 0.07
5.0 ± 0.1
1.29 ± 0.05
3.2 ± 0.2
2.5 ± 0.2
7
3.83 ± 0.04
4.1 ± 0.2
0.99 ± 0.06
n/a
n/a
9
3.64 ± 0.04
3.3 ± 0.3
0.75 ± 0.08
n/a
n/a
(μm/ns)
Table 4.6.3a: Width mode propagation characteristics about 10 GHz and k0 = 10038
rad/cm.
The attenuation length of the fundamental MSSM in this 2 μm narrow stripe (14 ± 1
μm) was found to be consistent with that of the 50 μm wide stripe of equal thickness
(14.7 ± 0.5 μm) determined in section 4.5.4. The attenuation lengths of the higher order
width modes (3 – 5 μm) were found to be comparable to half the antenna’s physical
width (3.75 μm). Thus, not only do these width modes have very low group velocities
(compared to MSSM); they are very short range, localised in the vicinity of the
excitation source. These come as no surprise, since these higher order width modes are
backward volume wave-like across the stripe width, with weak dispersions and low
group velocities 70, 71. This explains why the higher order width modes were detected
only in reflection and for closely spaced antennae. In addition, a trend is observed where
the attenuation length decreased for increasing mode number.
Using equation 4.5.4b, the relaxation times were calculated for the first 3 width modes
from their respective group velocities and attenuation lengths. These were tabulated in
table 4.6.3a. The relaxation time for the fundamental MSSM was consistent with the
value calculated in section 4.5.4 for the wide 50 μm stripe. Note the difference in the
extracted relaxation times between the fundamental and higher order modes.
95
4.6.4 Non-reciprocity
Similar to section 4.5.5, the non-reciprocity parameters of the width modes were
evaluated using equation 4.5.5a and plotted in figure 4.6.4a. The n = 3 width mode was
found to have the same non-reciprocity behaviour as that of the fundamental MSSM (η
≈ 0.2). Note that only the first two modes (n = 1, 3) had sufficient transmitted signal
amplitude to evaluate non-reciprocity with reasonable reliability.
Due to tiny signals for the larger mode numbers (barely above noise level), quantitative
evaluation of the non-reciprocity parameter should only be taken with a grain of salt.
However, one can still observe a trend in the data. For larger mode numbers (n ≥ 5), the
non-reciprocity seems to get weaker (η approaching unity) for increasing mode number.
This result is actually consistent with the rotation of the wave vector towards the
transverse direction for increasing mode number; the wave takes on more backwardvolume-like characteristic (a purely backward volume wave is completely reciprocal).
Figure 4.6.4a: Non-reciprocity parameter of width modes at 10 GHz and k0 = 10038
rad/cm.
96
4.7 Antenna near-field effect
In this section, we consider the antenna’s near-field effect on spin wave propagation.
Spin waves are slow electromagnetic waves with a dominating magnetic component.
Therefore, the characteristics of electromagnetic wave radiation and reception by usual
(e.g. radio or TV antennae) should be applicable to the coplanar line antennae. In
particular, it is known that the field of an antenna separates into two regions (with
continuous transition region in between): near-field and far-field 99, 114. These are
regions of time-varying electromagnetic field around the source for the field.
Far enough from the source, the wave is a purely propagating wave which accumulates
phase on its path between the radiation source and the receiving antenna. In the region
very close to an antenna, the wave’s ac field is dominated by field components
produced directly by currents in the antenna. This field is called the “near-field". An
important consequence which follows from this origin of the field is that the phase of
the field is the same across the whole width of the near-field region (i.e. there is no
effect of retardation in this region). At distances far from the antenna, the propagating
wave becomes dominated by the field components produced by its own ac field. For
instance, in our case of dipole dominated spin waves, the dynamic magnetisation is
produced by the dynamic dipole field and vice versa. Thus the wave’s ac field is
effectively no longer affected by the currents at the source. Due to this origin for the
field, the phase of the field changes (accumulates) with the distance from the source, i.e.
retardation effect is present. This more distant part of the electromagnetic field is the
"radiative field” or "far-field".
4.7.1 Characteristic equations
Analogous to radio waves propagating from a source antenna through free space, we
postulate that a purely propagating wave exist at some characteristic distance from the
antenna. At distances shorter than this characteristic length, we postulate that the phase
of magnetisation precession is dominated by the phase of the current in the antenna and
is the same across the whole near-field region. What this means in terms of propagating
spin wave experiment is that the effective propagation distance, xeff , is shorter than the
97
physical separation gap between antennae, x ; the difference between the two gives
twice the antenna characteristic near-field length, xnear :
2 xnear  x  xeff →Equation 4.7.1a
Note the factor 2 in equation 4.7a originates from accounting for the near-fields of both
the excitation and detection antennae, and that we assume both antennae to be identical
( xnear is defined for a single antenna). In PSWS utilising metallic ferromagnetic films,
the distance between antennae is always comparable to the size of the antennae
themselves. This is due to lossy metallic ferromagnetic films with attenuation lengths of
the order of microns. This means that for closely separated antennae, xnear may no
longer be negligible. In such cases, one needs to determine the proper effective
propagation distance, which may differ significantly from the nominal propagation
distance (between the symmetry axes of antennae). We now derive an expression from
which the antenna characteristic near-field length may be determined from experiment.
From equation 4.5.3a, we have the group velocity:
Vg 

f H
→ Equation 4.5.3a
 2
k
H k
The slope of the field-resolved pseudo-dispersion relation is thus:
Vg
H

→Equation 4.7.1b
k 2 f
H
From the dispersion plots in figure 4.5.1a, we see that the data points are nearly linear
over the range of wave numbers available. For a particular stripe, we assume – to first
degree Taylor approximation – that the slope
H
is constant at a particular frequency
k
and in the close proximity of the central dominant wave number k0 = 10038 rad/cm. The
phase difference between each maximum in the raw trace is given by equation 4.4.4a:
k 
2
→ Equation 4.4.4a
xeff
Substituting the expression for k in equation 4.4.4a into k in equation 4.7.1b, and
rewriting H into H gives:
98
Vg
 xeff H → Equation 4.7.1c
f
H
Here, H is the field step corresponding to 2π phase accumulation in the raw trace (see
section 4.4.4). To first degree approximation, the left hand side of equation 4.7.1c is a
constant for a particular stripe. The consequence of equation 4.7c is that H (the field
step over which a phase accumulation of 2π occurs) is inversely proportional to the
effective propagation distance. This is clearly seen in figure 4.4.4b. Essentially,
equation 4.7.1c is a reformulation of equation 4.4.4a in terms of H instead of k .
Let V ' 
Vg
, then equation 4.7.1c becomes:
f
H
xeff H  V ' → Equation 4.7.1d
By substituting equation 4.7.1a into equation 4.7.1d for xeff , we have:
( x  2 xnear )H  V '
xH  V '2 xnearH → Equation 4.7.1e
From equation 4.7.1e, one sees that by plotting xH (x = antennae gap) versus H for
various antennae gap separations, the slope of the plot gives the antenna characteristic
near-field length.
4.7.2 Antenna characteristic near-field length
We now establish some criteria before using equation 4.7.1e to analyse our data. First,
for a stripe aspect ratio, there must be at least 3 different separation gaps available; this
is the bare minimum for a linear fit. Second, the spread in H should be minimal and
statistically random; there is no significant slope and curvature in the field-resolved
pseudo-dispersion. This allows us to average H data for each raw trace into a single
representative value. This value is multiplied by the respective x ( xH ) is plotted
against itself for all available stripes which had data meeting the established criteria to
produce plots in figure 4.7.2a.
99
Figure 4.7.2a: Plots of antennae gap times delta H versus delta H, at 10 GHz around the
vicinity of the central dominant wave number k0 = 10038 rad/cm.
100
If there is no antenna near-field effect, then the antennae gaps would equal the
propagation distance, and the plots would be horizontal. However, we see in figure
4.7.2a, one notes that sloping is clearly present. One also notes that for all the cases
when we have at least 3 points, the data are well fitted with straight lines and with
positive slopes. This implies that our approach is physically sound. The antennae
characteristic near-field lengths, xnear can simply be obtained from the slopes of the
plots. These values were tabulated in table 4.7.2a. Note from equation 4.7.1e that the
horizontal intercept is proportional to the group velocity. Thus, the proposed method
here may also be used to extract group velocities in addition to the characteristic
antenna near-field lengths. However, the extraction of group velocities had already been
done in section 4.5.3 and is beyond the focus of this section.
Antenna characteristic near-field length, xnear (μm)
Thickness
Width
(nm)
(μm)
55
5
0.7 ± 0.1
55
100
1.0 ± 0.2
80
2
0.29 ± 0.04
80
20
1.4 ± 0.1
110
2
1.7 ± 0.1
110
20
2.25 ± 0.05
110
100
2.3 ± 0.1
Table 4.7.2a: Extracted antenna characteristic near-field lengths.
One notes a spread of xnear values extracted from the plot, ranging from 0.3 to 2.3 μm.
Recall the dimensions of the coplanar waveguide antennae used in this experiment: the
conductor widths and separation gaps were 1.5 μm, resulting in a total width of 7.5 μm,
and the distance from its symmetry axis to the external edge is 3.75 μm (figure 4.7.2b).
The antennae characteristic near-field length values extracted lie within this 2.25 μm
gap from the central conductor (figure 4.7.2b). The in-plane excitation magnetic fields
101
of a coplanar waveguide are concentrated underneath the signal and ground lines 98, 104.
The in-plane field contributes the most to spin wave excitation, and is maximum
underneath the central signal line. Thus, one may consider the region underneath the
central signal line as the near-field region, and wave propagation begins at some
distance from it. Our extracted xnear values are consistent with this, to the accuracy
limits of this experiment. One also notices that 0 < xnear < 3.75 μm for all the cases.
xnear >0 implies that the phase accumulation starts on the side of the excitation antenna
that is closer to the receiving antenna, and xnear < 3.75 μm implies that it starts below the
whole antenna structure (figure 4.7.2b). This is very important to know given the nonreciprocity of the antenna, because it is not obvious a-priori that for a non-reciprocal
antenna xnear > 0.
One also notes that there seems to be a trend for xnear to increase with stripe thickness
and width. From a fundamental consideration, the electromagnetic fields of a coplanar
waveguide would be perturbed by the close vicinity of magnetic material 115. Thus, one
may expect some dependence of the antennae characteristic near-field length on film
thickness. However, the accuracy limitations of this experiment do not allow us to
definitively quantify this effect. Furthermore, the theoretical framework required to
investigate this effect is beyond the scope of this work.
Figure 4.7.2b: Cross-section of the antenna.
102
4.7.3 Effective propagation distance
Experimentally-wise, with knowledge of xnear , one can then determine the effective
propagation distance xeff by subtracting from the antennae gap x for a particular stripe.
We now demonstrate the effect this has on the raw data of a particular stripe.
Figure 4.7.3a: Evaluation of antennae characteristic near-field length correction on the
data for the 110 nm thick and 2 μm wide stripes.
In figure 4.7.3a -a, the plot of all H points from the raw data were plotted on the
vertical axis with their respective fields on the horizontal axis. For clarity, they were
plotted on the logarithmic scale on the vertical axis. The data can be collapsed onto the
same scale by multiplication with some factor, in this case, the antennae separation gap,
103
for each of the data set. In figure 4.7.3a -b, all the xH points from the raw data were
plotted on the vertical axis with field on the horizontal axis. The horizontal lines are the
mean values for each antennae gap data set. However, note that the mean values do not
coincide due to offset induced from the finite antennae characteristic near-field
length xnear . In fact, these offsets can be used to extract xnear , which is mathematically
identical to the approach in figure 4.7.2a.
In figure 4.7.3a -c, the antennae gaps were corrected with xnear = 3.5 μm to obtain the
effective propagation distances, and the data replotted similar to figure 4.7.3a -a. This
time, the mean values collapsed closer together upon rescaling with xeff . Thus, we
demonstrate here that the effective propagation distance xeff (not the antennae gap x ) is
directly inversely proportional to H .
From this, it follows that the proper length to use to calculate k (phase accumulation
of 2π) is the effective propagation distance xeff (not the antennae gap x ). Thus, the nearfield effect is most significant for small antennae separation gaps, and becomes less
significant for larger gaps. As discussed in section 4.5.1, this is one of the important
factors (the others being number of data points and signal attenuation) to determine the
optimal antennae separation gap from which to reliably plot dispersion. However, in
section 4.5.1, the dispersions were calculated using the antennae gaps x instead of the
effective propagation distances xeff proper. Note that since adequately separated
antennae gaps were used in the dispersion plot in section 4.5.1, the near-field effect
introduced a discrepancy of only approximately 8%. In addition, since this is a
systematic error, it would merely shift calculated quantities uniformly by 8%. We
consider this acceptable within the accuracy and scope of this work.
104
4.8 Chapter conclusion
Spin wave propagation in the Damon-Eshbach geometry was studied in thick Permalloy
stripes (55 – 110 nm) over the aspect ratio range 5.5  (104  102 ) . Micron-sized
antennae were used to excite and detect spin waves with accessible wave numbers
ranging from 2000 to 20000 rad/cm. A highly sensitive phase interferometer detector,
together with a lock-in field-modulation technique, was used in this inductive spin wave
spectroscopy method.
In section 4.5, MSSM propagation across the range of aspect ratios and wave numbers
was studied. It was proposed that the MSSM dispersion can be modelled by introducing
an effective static demagnetising field factor into the continuous film dispersion.
Dynamic effects were negligible in our case. Micro-magnetic simulations were
performed on the stripes to determine the demagnetising field profiles. The nonuniformity of the demagnetising field across the stripe width increased with aspect ratio.
The mean values of the simulated demagnetising fields tend to overestimate the
effective demagnetising fields extracted from experiment. Group velocities calculated
from the dispersions, and these were found to increase with film thickness. There was
no correlation between the group velocity and stripe width for a particular thickness;
thus within the bounds of the experiment, the MSSM group velocity was found to be
width invariant. The attenuation and relaxation characteristics of the stripes were
evaluated. We found that the attenuation lengths increased with stripe thickness.
Relaxation times and Gilbert damping coefficients were calculated from MSSM data
and compared with the reference continuous film FMR data. It was found that the
Gilbert damping coefficients calculated from the stripe data were about 25% larger
those determined from FMR. This discrepancy was proposed to be due to edge losses
due to confinement, wave number dependence on damping coefficient, and/or
compounding of inaccuracies in the indirect methods used to calculate the damping
coefficient from MSSM dispersion data. Non-reciprocity of the MSSM was evaluated
and found to be largely invariant over the aspect ratios studied.
In section 4.6, multiple higher order width modes were found and identified in the stripe
 110nm 
with the highest aspect ratio studied in this work 
 . Remarkably, 6 higher order
 2m 
width modes (excluding the fundamental MSSM) were found in the excitation spectra.
105
Due to symmetry of the excitation field, only modes with odd symmetry were excited
(up to n = 13). Simulation was used to identify the modes in the recorded spectra and
determine the modal profiles. The amplitudes of these modes decrease for increasing
mode number in the excitation spectra, and even more rapidly in the transmission
spectra for increasing propagation distance. The dispersion, group velocity, attenuation,
and non-reciprocal properties of these modes were characterised in detail by an
induction method. It was found that the group velocity and attenuation lengths of the
higher order width modes decrease for increasing mode number. Within the accuracies
of the experiment, we found weakening of non-reciprocity for increasing mode number.
We propose that this is due to the higher order modes taking on more backwardvolume-like character for increasing mode number (a pure backward volume wave is
completely reciprocal).
In section 4.7, we propose that due to the near-field of an antenna, the spin waves
excited only propagate at some distance away from the antenna. We term this as the
“antenna characteristic near-field length”. The geometrical separation gap between the
excitation and detection antennae thus consists of the effective propagation distance
plus the antenna characteristic near-field length. To this end, we derived an expression
from which the antenna characteristic near-field length may be determined from
experiment. We found that the antenna characteristic near-field lengths extracted from
our data were such that wave propagation begins at some finite distance from the central
signal line, but still within the overall width of the coplanar antenna.
106
Chapter 5
Ferromagnetic resonance-based
hydrogen gas sensor
The work presented in this chapter is based on recent published work as first author 13.
The sections in this chapter are organised as follows. The introductory section first
briefly covers some of the proposed hydrogen sensors in the literature, and then moves
on to the unique hydrogen-absorption and spintronic properties of palladium. Following
through, a ferromagnet-palladium bi-layer sensor utilising both hydrogen-absorption
and spintronic properties of palladium is suggested. After description of the experiment
design, FMR experiment results of the bi-layer film are presented, and explained. The
practical functionality of the bi-layer film as a hydrogen sensor is then demonstrated.
Finally, some ideas for further work are suggested and the main findings of the chapter
summarised.
5.1 Background
The development of hydrogen-based energy source is severely limited by many safety
issues stemming from its high permeability, flammability, and explosiveness. The lower
flammability level of hydrogen in air is just 4 vol% while its lower explosive limit is 18
vol% 116. Thus, safety systems for hydrogen environments require the development of
suitable sensors and detection techniques, especially for low concentrations. Many of
these proposed sensors utilise the well-known property of palladium’s large and
selective hydrogen absorption capacity 116-125. Palladium-based hydrogen sensors116
make use of the changes in the physical property of palladium upon hydrogen
absorption, namely: a.) crystal lattice expansion117, 126, b.) change in conductivity119, 124,
125
, or c.) change in optical properties127-129.
In addition to gas absorption properties, palladium is also of great interest to the
magnetic community due to its spintronic effects. Magnetic multi-layered films which
include non-magnetic palladium layers are of great importance for high-density
107
magnetic random access memory utilizing nanoscale magnetic tunnel junctions 121. The
interest stems from the strong perpendicular anisotropy demonstrated for such systems.
Palladium 130 and similarly hydrogen-sensitive niobium 14 non-magnetic metallic
spacers have also been used in magnetic spin valve nanostructures. Charging such
multi-layered structures resulted in variation of exchange coupling between magnetic
layers in these devices. Furthermore, palladium overlaying magnetic layers exhibit large
inverse spin Hall effect 131 which is important for microwave magnonic applications 132.
Ferromagnetic metal / palladium bi-layers also show significant spin-pumping effect 112,
131, 133
.
5.2 Case for work
Considering both the hydrogen absorption capability and spintronic property of
palladium, we aim to use both of these properties to develop a hydrogen sensor based
on the spintronic property of palladium. In this chapter, we demonstrate the
functionality of a cobalt-palladium bi-layer thin film as a hydrogen sensor. The state of
the capping hydrogen-absorbing palladium layer was indirectly probed by measuring
the FMR response of the underlying ferromagnetic layer. Note that although FMR is not
a unique way to characterise magnetic and spintronic properties of a Co/Pd bi-layer, in
terms of hydrogen sensing, our approach has some important advantages over other
works from literature.
Firstly, previous studies of Co/Pd multilayers utilised methods which are extremely
impractical for sensing application: x-ray diffraction, neutron diffraction, and vibrating
sample magnetometry15, 134, 135. Secondly, our proposed method is able to read the state
of the bi-layer through a non-transparent electrically-insulating wall of a vessel
containing hydrogen gas, using microwave radiation. Thirdly, due to the perfect
microwave shielding effect in sub-skin-depth metallic films 42, 136, the microwave
radiation applied to the cobalt side of the bi-layer through an insulating wall will be
practically absent behind the palladium layer i.e. inside the vessel containing the
hydrogen. This eliminates the possibility of arcing, in stark contrast to conductivity
sensing methods requiring generation or application of electrical potentials inside a
flammable environment119, 124, 125.
108
It needs to be stressed at this point that due to time constraint, the work presented in this
chapter is only preliminary. Further comprehensive study of this class of hydrogen
sensor needs to be done in order to understand the fundamental science, refine the
technique, and improve on the sensitivity. Some recommendations of future research in
this area are expounded in section 5.7.
5.3 Experiment design
Four bi-layer films were fabricated in-house using our dc sputtering machine (see
section 2.1.1). The films were sputtered onto silicon wafers with 5 nm of tantalum seed
layers. The films with various different thicknesses of palladium and magnetic layers
were:
Ni80Fe20(5)/Pd(10)
Ni80Fe20(30)/Pd(10)
Co(5)/Pd(10)
Co(40)/Pd(20)
The numbers in brackets indicate the film thickness in nanometres. The magnetic layers
were buried underneath the palladium layer, with later exposed to atmosphere (figure
5.3a).
Figure 5.3a: Bi-layer film cross-section.
In addition, two single layer ferromagnetic films were also sputtered (without palladium
capping layers), functioning as control samples:
109
Ni80Fe20(5)
Co(5)
FMR measurements were made on the films in nitrogen and hydrogen atmospheres
using the custom-made gas cell described in section 2.4. A field-modulation lock-in
method (section 2.2.2) together with a phase interferometry detector (section 2.2.3) was
used for the FMR measurements in order to obtain good signal-to-noise ratios the thin
films. For the thicker films – Ni80Fe20(30)/Pd(10) and Co(40)/Pd(20) – no appreciable
differences in the FMR spectra were observed upon switching between nitrogen and
hydrogen atmospheres. For the thinner films, only Co(5)/Pd(10) exhibited significant
changes in its FMR spectra upon hydrogenation of the palladium layer. Hence, we focus
on this particular film for the remainder of this chapter.
5.4 Experiment results
An example FMR trace of the Co(5)/Pd(10) film at 10 GHz in nitrogen and hydrogen
atmospheres is shown in figure 5.4a. One immediately notices a down-field shift in the
FMR peak, and less obviously, narrowing of the resonance line width in hydrogen
atmosphere. The FMR field positions, resonance shift, and line widths in the frequency
range 4 – 18 GHz were plotted in figures 5.4b-d respectively.
Figure 5.4a: FMR spectra for Co(5)/Pd(10) at 10 GHz.
110
Figure 5.4b: FMR frequency versus field plots for Co(5)/Pd(10). Solid lines are fits with
the Kittel formula (equation 3.1.2a).
Figure 5.4c: FMR down-field shift for Co(5)/Pd(10) when switching from nitrogen to
hydrogen atmosphere.
111
Figure 5.4d: FMR line widths for Co(5)/Pd(10).
The FMR frequencies versus field plots in figure 5.4b were fitted with the Kittel
formula 37 (equation 3.1.2a) to extract the saturation magnetisations of the film under
nitrogen and hydrogen. The damping coefficients were also extracted from the line
width plots in figure 5.4d using Stancil’s formula 95 (equation 4.5.4e). These are
tabulated in table 5.4a.
f 2   2 H ( H  4M ) → Equation 3.1.2a

H FWHM
2f
Atmosphere
→ Equation 4.5.4e
Effective saturation
Damping coefficient,
magnetisation,
α (10-2)
4πM (Oe)
Nitrogen
12500 ± 200
2.30 ± 0.08
Hydrogen
13300 ± 200
1.73 ± 0.05
Difference
800 ± 400
0.6 ± 0.1
112
Table 5.4a: Magnetic properties of Co(5)/Pd(10) extracted from FMR data under
nitrogen and hydrogen atmosphere.
From table 5.4a, one sees that hydrogenation resulted in an increase in the effective
saturation magnetisation of the Co(5)/Pd(10) film by 800 Oe (6%). This is manifested
as resonance down-field shift in the FMR spectra (figure 5.4c). Line width narrowing
upon hydrogenation resulted in decrease in extracted damping coefficient by 0.006
(26%).
Additional FMR measurements were performed on the control Co(5) film without
palladium capping. FMR spectra were identical across the 4 – 18 GHz frequency range
under nitrogen and hydrogen atmospheres. This result shows that hydrogenation did not
affect the magnetic properties of the cobalt film. Consequently, this strongly suggests
that the resonance shift and line width narrowing observed in the Co(5)/Pd(10) film has
origin in the palladium capping layer.
5.5 Discussion of results
We now explain the results presented in section 5.4 based on known properties of cobalt
and palladium. The most noticeable effect caused by hydrogenation of our Co(5)/Pd(10)
film is down-field shift in FMR (figure 5.4c). No resonance shift was observed in the
control Co(5) film, indicating that hydrogen did not affect the saturation magnetisation
of cobalt. We propose then, that the resonance shift is due to change in the strength of
uniaxial anisotropy at the cobalt-palladium interface when palladium expands upon
absorbing hydrogen. Co/Pd is a typical material with perpendicular anisotropy 121, 137. It
is known that the origin of perpendicular anisotropy in Co/Pt-group multilayers is
interfacial strain 138. It is also known that palladium expands on hydrogen absorption
due to phase transformation into either one or both of the hydride phases 118, 139-141.
Hence, the expansion of the palladium layer upon absorbing hydrogen exerts strain at
the cobalt-palladium boundary. This in turn, decreases the interfacial uniaxial
anisotropy field of cobalt. The effective saturation magnetisation measured in FMR
M effective is equal to the difference between the real saturation magnetisation M real and the
113
effective anisotropy field H anisotropy (equation 5.5a). Therefore, we experimentally
observe increase in effective saturation magnetisation in hydrogen atmosphere (downfield shifts in FMR peaks).
M effective  M real  H anisotropy→ Equation 5.5a
This conclusion is consistent with a negligibly small effect observed for the
Ni80Fe20(5)/Pd(10) film since Ni80Fe20 has negligible anisotropy and magnetostriction.
Furthermore, the effect seems to be interfacial in nature due to strong dependence on
film thickness; no significant differences in the FMR spectra were observed in the
thicker films upon hydrogenation. In addition to strain-induced anisotropy, we note that
the strength of the anisotropy is also affected by the d-d hybridization at the layer
interface 142. If hydrogen atoms reach the interface during their diffusion through the
palladium layer, they may potentially affect the strength of the d-d hybridization.
On an important side note, for sufficiently thin films, perpendicular anisotropy in Co/Pd
is strong enough to force the magnetisation vector out-of-plane 137, 143, 144. However, our
Co(5)/Pd(10) film is too thick for perpendicular anisotropy to flip the magnetisation
vector out-of-plane. The ground state magnetisation lies in-plane due to the very large
out-of-plane demagnetizing field (> 1.8T for cobalt films). Thus, the shift in the FMR
upon hydrogenation cannot be attributed to the switching of equilibrium magnetisation
from out-of-plane to in-plane magnetisation. Such a radical change in the magnetisation
ground state would have resulted in significantly larger resonance shifts than observed
in figure 5.4c.
We now turn attention to the FMR line widths. Recall that hydrogenation of the
palladium layer resulted in narrowing of the FMR line width of the underlying cobalt
layer (figure 5.4d). We found no change in the FMR line width of the Co(5) control
sample when switching from nitrogen to hydrogen atmosphere. This means that the
source of FMR line width variation in the Co(5)Pd(10) film has its origin in the
palladium capping layer. We propose three possible contributions to this effect.
First, it is the spintronic effect of spin-pumping 145. This is an effect which occurs in a
bi-layer film consisting of a ferromagnetic layer interfaced with a non-magnetic layer
with large spin-orbit interaction. Magnetisation precession in the ferromagnetic layer
acts as a spin pump which transfers angular momentum into the non-magnetic layer.
114
This loss of angular momentum from the ferromagnetic layer manifests as additional
damping of magnetisation precession, and is experimentally seen as FMR line width
broadening. Palladium is one of the materials in which spin pumping effect is strong 112.
It is also well-known that absorption of hydrogen into palladium reduces its
conductivity 99, 124, 140.Thus, reduction of palladium conductivity upon hydrogenation
reduces spin-pumping from cobalt into palladium, due to reduced spin-mixing
conductance at the interface.
Second, Gilbert damping may vary due to the variation in the d-d hybridization at the
interface 142. The third effect is a trivial effect of reduction of eddy current losses to the
FMR line width upon reduction in the conductivity of the palladium layer. To estimate
this contribution, simulations of the microwave response of a coplanar waveguide
loaded by a Co(5)/Pd(10) film were performed for different conductivity values of the
palladium layer. Reduction in the conductivity from the one typical for bulk palladium
to zero had negligible effect on the FMR line width. Note that in this simulation, only
the eddy current effect was included; the spin pumping and d-d hybridization effects
were excluded. Hence, we conclude that spin pumping into the non-magnetic palladium
layer is the dominant contribution to the FMR line width broadening. Consequently,
reduction in palladium conductivity upon hydrogenation reduces spin-pumping from
cobalt into palladium. This is experimentally observed as FMR line width narrowing of
the cobalt layer upon hydrogenation of the palladium layer.
5.6 Cobalt-palladium film as a hydrogen sensor
The FMR shift in Co(5)/Pd(10) upon hydrogen absorption and desorption is now
exploited to demonstrate functionality as a hydrogen sensor. First, the frequency and
field were set to resonance condition under nitrogen atmosphere. The cell atmosphere
was then repeatedly cycled between nitrogen and hydrogen. Due to shift in the
resonance curve, a net change in the lock-in signal was observed (figure 5.6a). This
signal is recorded as a function of time with a digital oscilloscope over three cycles
(figure 5.6b). Since the frequency and field were fixed to resonance under nitrogen
atmosphere, the change in the signal baseline upon introduction of hydrogen is due to
115
the cobalt layer going out of resonance condition when the palladium layer is
hydrogenated.
Figure 5.6a: FMR spectra for Co(5)/Pd(10) at 10 GHz. The green dashed line represents
the change in the lock-in signal from the nitrogen FMR signal “baseline” upon
switching to hydrogen atmosphere.
Figure 5.6b: Change in the lock-in signal under the cycling of nitrogen and hydrogen
gas through the Co(5)/Pd(10) under resonance conditions at 10 GHz using the nitrogen
FMR as the “baseline”.
Several key features from this cyclic run were noted. First, the signal change due to
sensing of hydrogen is well above noise level. Second, the sensor reliably returns back
to its initial state in each cycle. Long term entropic increase due to film degradation
116
over repeated cycling was not observed in the short time frame of the experiment. Third,
the sensor rise and fall time constants were found to be 5s and 30s respectively. These
values are similar to the response times of a typical electrical resistance-based palladium
film hydrogen sensor 124, 125. This verifies that the cyclic curve obtain in figure 5.6b was
actually due to hydrogen/desorption process, rather than gas flow or hydrogen buoyancy
artefacts.
Finally, we demonstrate the possibility of remote sensing through a physical barrier.
Previously, the sample was placed such that the metal film faced the waveguide in the
hydrogen cell. In this experiment, we flip the sample such that the film faced away from
the waveguide; the film was separated from the waveguide by the 0.9 mm thick
insulating silicon substrate. This mimics a vessel wall between a coplanar waveguide
attached to the external wall and the film on the internal wall of a gas chamber. We
were able to still detect the resonance signal in this configuration (figure 5.6c) even
though the signal dropped by 20 dB. Note that due to the perfect microwave shielding
effect exhibited by metallic films of sub-skin-depth thicknesses, the microwave field in
this configuration is concentrated in the insulator and the metallic film 42, 136. Due to
this effect, the hydrogen is shielded from the externally applied microwave
electromagnetic field. This is advantageous since hydrogen is a serious fire and
explosion hazard.
Figure 5.6c: FMR spectra for Co(5)/Pd(10) at 10 GHz, measured through a 0.9 mm
thick silicon substrate.
117
We now remark on the robustness of our thin film hydrogen sensor. It is well-known
that repeated absorption/desorption of hydrogen on palladium films eventually lead to
hysteric behaviour 141, plastic deformation 146, and eventually mechanical failure 140 due
to repeated expansion/contraction of the crystal lattice 140. This is especially pronounced
for thick films. There are three general approaches to improve mechanical robustness of
palladium film-based hydrogen sensors. The first approach is to limit sensing to low
hydrogen concentrations in order to prevent formation of the highly expanded β phase
of palladium hydride 139. The second approach is to alloy palladium with another metal
to improve its mechanical properties 124, 125, 140.
The third approach is to reduce the thickness of the palladium film in order to reduce
internal strain. Reducing the film thickness is detrimental to sensors which rely on the
bulk property of palladium to function. For example, strain-based sensing requires large
palladium thicknesses to overcome the substrate clamping effect 147. For our cobaltpalladium bi-layer sensor, the substrate clamping effect is actually beneficial, since
perpendicular anisotropy is formed in its presence. Furthermore, modification of the
anisotropy does not require micron-scale deformations of the macro-size of the sensing
body, but just a small change in the crystal lattice size. Therefore, whereas the
sensitivity of electrical-based sensors decrease with palladium thickness, our spintronicbased sensor will operate at palladium thickness of 10 nm (potentially well below 10
nm). Hence, reducing film thickness actually improves our sensor due to the inter-facial
nature of the sensing mechanism (which scales as the inverse of film thickness) 137. The
additional benefit of using a thin film is improved robustness for our sensor.
5.7 Suggestions for further work
There is much room for further work to build on the preliminary FMR-based hydrogen
sensor presented in this chapter. Here are some suggestions:
a.) Optimal thicknesses of magnetic and palladium layers
Due to the interfacial nature of the functionality of the proposed sensor, one may expect
strong dependence of sensor response on the thickness of the magnetic and/or palladium
layer thickness. For both layers, there should be some maximum thickness over which
118
interfacial interactions become insignificant when the bulk property dominates.
Conversely, there should also be some characteristic interfacial thickness at which the
bulk properties of films cease to exist. A systematic study of various samples of
incremental changes in bi-layer thicknesses should enable one to determine the optimal
magnetic and palladium layer thicknesses as a hydrogen sensor.
b.) Flipping between in-plane and out-of-plane magnetisation
As discussed in section 5.5, the films used in this work were too thick to induce out-ofplane magnetisation. For sufficiently thin Co/Pd films (a few angstroms), due to
interfacial anisotropy, films with out-of-plane magnetisation as the ground state may be
obtained 137. Thus, one may be able to fabricate a film of the required thickness such
that the magnetisation flips between out-of-plane and in-plane configuration by
introduction of hydrogen. The direction in which the magnetisation flips in hydrogen
atmosphere would depend on the sign of the induced change in interfacial anisotropy;
this depends on the crystallinity, and crystal axis orientation of the film during growth
(see figure 3 in reference 137). This radical transformation of the magnetisation ground
state would register large signal changes in both static and dynamic magnetisation
measurement techniques.
c.) Multi-layers
One can also investigate the effect of multi-layering on the sensor signal and time
response.
d.) Patterning
Note that for continuous films, external magnetic fields need to be applied in order to
magnetically saturate the sample. The saturated state is an important condition for
observation of FMR. For practical sensor application, the need for application of
magnetic field may be inconvenient. The need for an external magnetic field may be
eliminated by patterning continuous films into nano-sized elements in an array, similar
to optical sensors 120. For example, due to shape anisotropy, nanostripes are naturally
single-domain without the need for application of external magnetic field 148.
e.) Hydrogen partial pressure
119
The preliminary work presented in this chapter was done at atmospheric pressure, with
hydrogen absorption occurring in 100% hydrogen atmosphere. Future work may
investigate the sensor response in various hydrogen partial pressures.
5.8 Chapter conclusion
In this chapter, we demonstrated the functionality of a cobalt-palladium bi-layer film as
a hydrogen sensor. Hydrogenation of the palladium layer resulted in two interfacial
effects: a.) the magneto-crystalline anisotropy of cobalt is modified, and b.) reduction in
microwave magnetic losses in cobalt due to reduction in spin-pumping effect. These
resulted in down-field shift and line width narrowing of the FMR of the underlying
cobalt film, respectively. This means that the hydrogenation state of the upper
palladium layer can be indirectly probed by measuring the FMR response of the
underlying cobalt layer. We utilised the resonance shift property to demonstrate the
functionality of the film as a sensor by repeated cycling of nitrogen and hydrogen
atmosphere. The hydrogen absorption and desorption time constants were found to be
typical for such thin film palladium hydrogen sensors. We also demonstrated remote
sensing capability of our technique through an electrically-insulating non-transparent 1
mm-thick wall.
120
Appendices
Appendix A
Photolithography Micro-Fabrication Recipe
Permalloy strip layer
The silicon substrate is first spin-cleaned with acetone and iso-propyl alcohol (IPA).
Then, the substrate is exposed to HMDS (hexamethyldisilazane) vapour for 2 minutes.
HDMS functionalises the silicon substrate to increase photoresist adhesion. Photoresist
AZ6632 (from AZ Electronic Materials) is then spun-coated onto the substrate at 4000
rpm for 30 seconds. This results in a thick photoresist layer of approximately 3.2 μm.
The photoresist is then soft baked at 95 °C for 5 minutes.
The photolithography mask is then aligned over the photoresist-coated substrate, and
the exposed substrate illuminated with 10 mW/cm2 of ultraviolet for 9 seconds. The
photoresist is then developed with AZ326 (from AZ Electronic Materials) developer for
90 seconds, followed by deionised water rinse for 30 seconds. The patterned photoresist
is then blow-dried with nitrogen, and then ashed for 20 minutes in 340 mTorr of oxygen
at an rf power of 50 W.
Permalloy of required thickness is then deposited onto the patterned photoresist using
electron-beam-assisted thermal evaporative deposition. Lift-off is done in NMP (Nmethyl-2-pyrrolidone) at 80 °C with light ultrasonication. The patterned Permalloy
structures were then rinsed with IPA and then dried with nitrogen.
Aluminium oxide layer
Following through from the process before, the substrate is exposed to HDMS vapour
for 2 minutes. Photoresist AZ6612 is then spun-coated onto the substrate at 4000 rpm
for 30 seconds. This results in a thick photoresist layer of approximately 1.2 μm. The
photoresist is then soft baked at 95 °C for 5 minutes.
The photolithography mask is then aligned over the photoresist-coated substrate, and
the exposed substrate illuminated with 10 mW/cm2 of ultraviolet for 3 seconds. The
photoresist is then developed with AZ326 developer for 1 minute, followed by
121
deionised water rinse for 30 seconds. The patterned photoresist is then blow-dried with
nitrogen, and then ashed for 20 minutes in 340 mTorr of oxygen at an rf power of 50 W.
30 nm of aluminium oxide is first deposited onto the patterned photoresist. This is the
insulating spacer between the underlying Permalloy strips and the overlaying gold
coplanar lines.
Gold coplanar line layer
Next, 10 nm of Ti is deposited over the aluminium oxide. Titanium aids adhesion of
gold onto silicon substrate, without which gold would easily peel off. Finally, 200 nm of
gold is deposited over the titanium. All depositions were done using electron-beamassisted thermal evaporative deposition. Lift-off is done in NMP at 80 °C with light
ultrasonication. The patterned gold coplanar structures were then rinsed with IPA and
then dried with nitrogen.
122
Appendix B
Microwave current injection into a continuous film
We consider a tip of the microscopic coplanar probe in a contact with a
continuous metallic layer. As has been shown by Ney 54 due to strong tendency of
microwave currents to repulse each other a current injected from a quasi-point source
tends to spread over the whole area of the layer plane. The characteristic distance from
the contact, where the whole area of the film is occupied by the current is the
microwave skin depth for the material. Therefore it is appropriate to consider each of
the contacts of the coplanar separately as connected to a ground plane with a nonvanishing resistivity. The ground plane has the shape of the disk of an infinite radius. It
is at zero potential which is applied to the perimeter of the disk. The contact is located
in the centre of the disk and has the radius r0. It is modelled as a current density jz ,
jz   Ez (1)
evenly distributed across the contact circular area and which is injected into the film
perpendicularly to its surface (i.e. along the axis z of the cylindrical coordinate system
with the origin in the centre of the contact (figure A1) In Eq.(1) Ez is the component
perpendicular to the film surface of the microwave electric field E and  is film
conductivity.
Figure A1: Geometry for single contact and the respective cylindrical frame of
reference.
123
The system of the three contacts of the probe with the metallic layer may be then
considered as separate contacts at microwave potentials of the same magnitude but of
the opposite signs each separately loaded to the same ground plane with the zero
potential at infinity.
Consider first one contact with the ground plane at the zero potential. Similar to
Ney’s approach, using the identity (1) we may derive equations for the microwave
electric field in the conducting film. From Maxwell equations in the cylindrical frame of
reference (r ,  , z ) we obtain:
Er / z  Ez / r  i0 H

1
  (rH ) / r   Ez
r
 (rH ) / z   Er
(2)
Here Er is the radial component of the electric field and H  is the azimuthal
component of the microwave magnetic field (both lie in the film plane),  is the
microwave frequency,  is the magnetic permeability of the metal (which we consider
as a scalar quantity here), and 0  4 107 Hn / m . Several important equations and
identities can be derived from Eq. (2):
1
 2 Ez / z 2   2 Ez / r 2  Ez / r  i0 Ez  0 (3)
r
1
1
 2 Er / z 2   2 Er / r 2  Ez / r  (i0 Ez  2 )  0 (4)
r
r
 2 Er / z 2  i0 Ez   2 Ez / (r z )
We also need boundary conditions. Based on (1) the microwave current injected from
the probe through a contact area of radius r0 is modelled as the boundary condition
Ez (r  r0 , z  0)  1, Ez (r 0 , z  0)  0 (5)
(Obviously the real distribution of the current across the injection area is not uniform
for the same reason of the current repulsion; however we use the uniform distribution as
it allows simple analytic treatment.)
124
Solutions to (3) and (4) in the form suitable for application of the boundary conditions
are obtained using Hankel transform 149. Using this transform the solution to (3) and (4)
can be cast in the form

Ez   Ezk J 0 (kr )kdk (6)
0

Er   Ezk J1 (kr )kdk (7)
0
where J 0 ( x) and J1 ( x) are Bessel functions of the zeroth- and the first-order
respectively, and Ezk and Erk are the respective Hankel-components of the fields:

Ez ( r )   Ez ( r ) k J 0(1) (kr )rdr (8)
0
On substitution of (6) and (7) in (3) and (4) respectively one obtains:
 2 Ezk / z 2  k2 Ezk  0 (9)
 2 Erk / z 2  k2 Erk  0 (10)
where
k2  k 2  i0 (11)
The general solutions to (9) and (10) have the form:
Ez ( r ) k  Az ( r ) k exp(k z)  Bz ( r ) k exp(k z) (12)
Obviously, the z-component of the current density should vanish at the film surface
facing away from the contact z=L. This implies that for Ezk (12) reduces to
Ezk  Azk sinh(k ( z  L)) (13)
The coefficients Azk are obtained from the boundary condition (5). The Hankel J0
transform of the step-function (5) is r0 J1 (kr0 ) / k . Thus, from (5) and (13) one obtains:
Ezk  r0 J1 (kr0 )sinh(k ( z  L)) / [k sinh(k ( L))] (14)
125
Similarly, from (10) and (12) we obtain:
Erk  r0 J1 (kr0 )( Ark exp(k z)  Brk exp(k z)) / k (15)
Finally, using the identity (11) from (14) and (15) taking into account (8) one easily
finds that
Erk  r0k J1 (kr0 ) cosh(k ( z  L)) / [k 2 sinh(k ( L))] (15)
and then from the first of Eqs.(2) that
Hk  r0 J1 (kr0 )sinh(k ( z  L)) / [k 2 sinh(k ( L))] (16)
Here one has to note that following (8) the Hankel J0 transform should be used to
calculate Ez from Ezk and the Hankel J1 transform to restore Er and H  from Erk and
H k respectively.
Figure A2: Geometry for two contacts and the respective Cartesian frame of reference.
Consider now two contacts at the distance R along the Cartesian axis x (figure A2). One
of the contacts is located at (x=x1=R/2, y=0) and the second at (x=x2=R/2, y=0). Then the
amplitude of y-component of the total microwave magnetic field is the sum of the fields
of the two contacts:
H y  H (r1 ) cos(1 )  H (r2 ) cos(2 ) (17)
where r1(2)  [( x  x1(2) )2  y 2 ]1/2 , cos(1(2) )  ( x  x1(2) ) / r1(2) , and the negative sign
between the two terms on the right-hand side of eq.(17) accounts for the fact that one of
the contacts is the source for the electric current and the other is the drain.
126
Let us now analyse (16) and (17). First one sees that the y-component of the microwave
field is perpendicular to the static applied field and is in the film plane. Due to large
ellipticity of magnetisation precession in metallic films only the in-plane component of
the microwave field contributes to the excitation of magnetisation precession. Thus,
Eq.(17) gives distribution of the amplitude of the excited magnetisation across the
volume of the film. First from (16) one sees that similar to the excitation with the
microstrip transducer 42 the excitation field is vanishing at the far film surface with
respect to the contacts. Furthermore, from comparison of (14) and (16) one sees that the
in-plane magnetic field is mostly due to the large density of the current  Ez directed
along z right below the contact area. This current induces a circular microwave field
around the contact. A combination of two circular fields of the adjacent contacts gives
rise to H y . Since
sinh(k ( z  L))  k ( z  L) for  k L  1 (18)
this current density linearly decreases with z to zero at z=L. So does the microwave
magnetic field too. This conclusion based on the consideration of the Hankelcomponents is confirmed by numerical calculation of the inverse Hankel-transform of
(16) for our geometry (figure A2).
Figure A3: Magnitude of the in-plane microwave magnetic field (arbitrary units) as a
function of depth into a 100 nm thick film. Red: Centre of contact area. Blue: Edge of
contact area.
127
Figure A4: Magnitude of the in-plane microwave magnetic field (arbitrary units) as a
function of distance along the line connecting the probe tips.
Figure A5: Magnitude of the in-plane microwave magnetic field (arbitrary units),
radially from the edge of the contact (red), and along y at x=z=0 of Fig. 1(b) (blue).
Figure A4 demonstrates the result of the numerical calculation of the microwave
magnetic field using (17) along the line connecting the probe tips (y=z=0) and figure A5
shows the field distribution along y for x=z=0. One sees that the magnetic field is
concentrated in the closest vicinities of the contacts. Thus one may expect that the main
contribution to the magnetic absorption originates from the areas near the probe tips.
128
Similar to (17) the field of the real coplanar probe having one signal contact at x=xc=0
in the middle and two ground contacts at both sides from the signal line (x1=x2=R/2) can
be calculated:
H y  H (r1 ) cos(1 ) / 2  H (rc ) cos(c )  H (r2 ) cos(2 ) / 2 (19)
The coefficients ½ in the first and in the last terms account for the continuity of the
current density and for the proper amplitudes of electric fields induced by application of
a microwave voltage between the signal and the ground plane contacts. Figure A4
demonstrates the microwave field calculated with (19) for y=z=0.
Turn now to the quasi-linear asymmetric profile of the microwave magnetic field across
the film thickness (figure A3). Obviously, this is the consequence of the microwave skin
effect originating from the first of Eqs.(1), since for x=0 one can expect an antisymmetric linear profile for the magnetic field: H y ( z  L)  H y ( z  0) which follows
from Ampere’s law (the last two equations of system (2)). It is clear that the injection of
the current in the z-direction from one of the surface breaks the symmetry of the current
due to the skin effect. This effect looks similar to the asymmetry of the total microwave
magnetic field of a conducting film in a vicinity of a microstrip line 42.
This theory explains well why the fundamental mode is efficiently excited in our
geometry. As a final note we would like to emphasize that the magnetic character of the
material can be taken into account approximately by introducing the effective scalar
microwave permeability for the film (see Eq.(2.7) in 150). In the resonance this
permeability can take rather large values (several hundred). We made calculations of
H y with   500 in (11) and found that the spatial field profile does not vary noticeably
with  which confirms that the magnetic field is largely the microwave magnetic field
of the perpendicular current  Ez existing right beneath the contact.
129
Appendix C
Numerical Simulations
Numerical simulations were performed to obtain the theoretical eigen frequencies and
mode profiles of magnetic slabs studied in this work. First, the static magnetization
ground state of the particular slab geometry is determined using LLG Micromagnetics
Simulator (v2.63d). Mesh sizes are chosen such that each unit cell is smaller than 5 x 5
nm2.
The dynamic response of the slab is then simulated using this magnetization ground
state. The numerical model used is based on Green’s function description of the
dynamic dipole field of the precessing magnetization. See reference 49 for details. Since
the stripes studied in this work have lengths much larger than their cross section
dimensions, the length can be considered infinite, thus reducing the problem into a 2D
one. The cross section is divided up into square unit cells. The stray field at the mesh
point (i,j) induced by the dynamic magnetization at position (i’,j’) can be evaluated
based on the analytical formulas from reference 151. The discretized Green’s function of
the dipole and effective exchange fields are substituted into the linearized LandauLifshitz equation to produce a matrix. The eigen values of this matrix represent the spin
wave eigen frequencies, while its eigen vectors represent the mode profiles. The
problem is coded in Mathcad 15, and the eigen value problem solved using numerical
tools built into the software.
The key simulation parameters used are:
Chapter 3
LLG Micromagnetics Simulator (v2.63d)
Stripe dimensions:
260 nm (w) x 100 nm (h)
Mesh size:
64 (w) x 32 (h)
Applied field:
500 Oe along stripe length
Saturation magnetization:
800 emu/cm3
Mathcad 15
130
Frequency:
14 GHz
Stripe dimensions:
260 nm (w) x 100 nm (h)
Mesh size:
26 (w) x 10 (h)
Gap between stripes:
150 nm
Saturation magnetization:
10150 Oe
Gyromagnetic ratio:
2.82 MHz/Oe
Applied field:
Along stripe length
Chapter 4
Simulations were done only for micro-stripes with smallest and largest aspect ratios
studied in the chapter.
LLG Micromagnetics Simulator (v2.63d)
Stripe dimensions:
100 μm (w) x 55 nm (h)
Mesh size:
32768 (w) x 16 (h)
Applied field:
950 Oe along stripe width
Saturation magnetization:
800 emu/cm3
Stripe dimensions:
2 μm (w) x 110 nm (h)
Mesh size:
512 (w) x 32 (h)
Applied field:
1300 Oe along stripe width
Saturation magnetization:
800 emu/cm3
Since the stripe with the smallest aspect ratio nearly resemble that of an infinite
continuous film, we next consider only the stripe with the highest aspect ratio.
Mathcad 15
Frequency:
10 GHz
Stripe dimensions:
2 μm (w) x 110 nm (h)
Mesh size:
64 (w) x 8 (h)
Saturation magnetization:
10150 Oe
Gyromagnetic ratio:
2.82 MHz/Oe
Applied field:
Along stripe width
131
Bibliography
1.
J. H. E. Griffiths, Nature 158 (4019), 670-671 (1946).
2.
M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven and B. Hillebrands, Applied Physics
Letters 87 (15), 3501 (2005).
3.
T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps and M. P. Kostylev,
Applied Physics letters 92 (2), 2505 (2008).
4.
K.-S. Lee and S.-K. Kim, Journal of Applied Physics 104 (5), 3909 (2008).
5.
A. Khitun, M. Bao and K. L. Wang, Journal of Physics D: Applied Physics 43 (26), 264005
(2010).
6.
The Physics of Ultrahigh-Density Magnetic Recording. (Springer, Berlin, 2001).
7.
J. M. Slaughter, R. W. Dave, M. DeHerrera, M. Durlam, B. N. Engel, J. Janesky, N. D.
Rizzo and S. Tehrani, Journal of Superconductivity: Incorporating Novel Magnetism 15 (1), 1925 (2002).
8.
S. Tehrani, J. M. Slaughter, M. Deherrera, B. N. Engel, N. D. Rizzo, J. Salter, M. Durlam,
R. W. Dave, J. Janesky and B. Butcher, Proceedings of the IEEE 91 (5), 703-714 (2003).
9.
A. Ney, C. Pampuch, R. Koch and K. H. Ploog, Nature 425 (6957), 485-487 (2003).
10.
B. D. Terris, Journal of Magnetism and Magnetic Materials 321 (6), 512-517 (2009).
11.
M. V. Costache, M. Sladkov, S. M. Watts, C. H. Van der Wal and B. J. Van Wees,
Physical Review Letters 97 (21), 216603 (2006).
12.
K. Hika, L. V. Panina and K. Mohri, Magnetics, IEEE Transactions on 32 (5), 4594-4596
(1996).
13.
C. S. Chang, M. Kostylev and E. Ivanov, Applied Physics Letters 102 (14) (2013).
14.
F. Klose, C. Rehm, D. Nagengast, H. Maletta and A. Weidinger, Physical Review Letters
78 (6), 1150 (1997).
15.
K. Munbodh, F. A. Perez and D. Lederman, Journal of Applied Physics 111 (12), 123919123919-123916 (2012).
16.
J. M. Shaw, H. T. Nembach, T. J. Silva, S. E. Russek, R. Geiss, C. Jones, N. Clark, T. Leo
and D. J. Smith, Physical Review B 80 (18), 184419 (2009).
17.
R. Law, R. Sbiaa, T. Liew and T. C. Chong, Applied Physics Letters 91, 242504 (2007).
18.
H. Gong, D. Litvinov, T. J. Klemmer, D. N. Lambeth and J. K. Howard, Magnetics, IEEE
Transactions on 36 (5), 2963-2965 (2000).
19.
D. R. Birt, B. O'Gorman, M. Tsoi, X. Li, V. E. Demidov and S. O. Demokritov, Applied
Physics Letters 95 (122510) (2009).
20.
J. Callaway, Physical Review 132, 2003-2009 (1963).
21.
A. O. Adeyeye and N. Singh, Journal of Physics D: Applied Physics 41 (153001), 29
(2008).
22.
T. J. Silva, C. S. Lee, T. M. Crawford and C. T. Rogers, Journal of Applied Physics 85 (11),
7849-7862 (1999).
23.
G. Counil, J.-V. Kim, K. Shigeto, Y. Otani, T. Devolder, P. Crozat, H. Hurdequint and C.
Chappert, Journal of Magnetism and Magnetic Materials 272-276 (1), 290-292 (2004).
24.
S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva and
J. P. Nibarger, Journal of Applied Physics 99 (9), 093909-093909-093907 (2006).
25.
M. Belmeguenai, F. Zighem, Y. Roussigne, S.-M. Cherif, P. Moch, K. Westerholt, G.
Woltersdorf and G. Bayreuther, Physical Review B 79 (024419) (2009).
26.
A. M. Niknejad, (2005).
27.
H. T. Friis, Proceedings of the IRE 32 (7), 419-422 (1944).
28.
C. S. Chang, M. Kostylev, A. O. Adeyeye, M. Bailleul and S. Samarin, EuroPhysics Letters
96 (2011).
29.
C. Kittel, Introduction to Solid State Physics, 8 ed. (John Wiley & Sons, 2005).
132
30.
L. Landau and L. Liftshitz, Phys. Zeit. Sowjetunion 8 (1935).
31.
T. A. Gilbert, 1955.
32.
C. E. Patton, Z. Frait and C. H. Wilts, Journal of Applied Physics 46 (11), 5002 (1975).
33.
V. E. Demidov, S. O. Demokritov, K. Rott, P. Krzysteczko and G. Reiss, Applied Physics
Letters 92 (232503) (2008).
34.
K. Sekiguchi, K. Yamada, S. M. Seo, K. J. Lee, D. Chiba, K. Kobayashi and T. Ono, Applied
Physics Letters 97 (022508) (2010).
35.
P. K. Amiri, B. Rejaei, M. Vroubel and Y. Zhuang, Applied Physics Letters 91 (065502)
(2007).
36.
V. E. Demidov, S. O. Demokritov, K. Rott, P. Krzysteczko and G. Reiss, Applied Physics
Letters 91 (252504) (2007).
37.
C. Kittel, Physical Review 73 (2), 155- 161 (1948).
38.
C. Kittel, Physical Review 110 (6), 1295-1297 (1958).
39.
W. Stoecklein, S. S. P. Parkin and J. C. Scott, Physical Review B 38 (10), 6847 (1988).
40.
R. D. McMichael, M. D. Stiles, P. J. Chen and W. F. Egelhoff Jr, Physical Review B 58
(13), 8605 (1998).
41.
R. Magaraggia, K. Kennewell, M. Kostylev, R. L. Stamps, M. Ali, D. Greig, B. J. Hickey
and C. H. Marrows, Physical Review B 83 (5), 054405 (2011).
42.
M. Kostylev, Journal of Applied Physics 106 (4), 043903-043903-043914 (2009).
43.
K. J. Kennewell, M. Kostylev, N. Ross, R. Magaraggia, R. L. Stamps, M. Ali, A. A.
Stashkevich, D. Greig and B. J. Hickey, Journal of Applied Physics 108 (073917), 12 (2010).
44.
D. M. Kostylev and P. G. A. Melkov, (Australian Research Council, The University of
Western Australia, 2010), pp. 7.
45.
Y. V. Khivintsev, L. Reisman, J. Lovejoy, R. Adam, C. M. Schneider, R. E. Camley and Z. J.
Celinski, Journal of Applied Physics 108 (2), 023907-023907-023906 (2010).
46.
V. V. Kruglyak, S. O. Demokritov and D. Grundler, Journal of Physics D: Applied Physics
43 (26), 264001 (2010).
47.
A. A. Grunin, A. G. Zhdanov, A. A. Ezhov, E. A. Ganshina and A. A. Fedyanin, Applied
Physics Letters 97 (26), 261908-261908-261903 (2010).
48.
N. Kostylev, I. S. Maksymov, A. O. Adeyeye, S. Samarin, M. Kostylev and J. F. Williams,
Applied Physics Letters 102 (12), 121907-121907-121905 (2013).
49.
G. Gubbiotti, S. Tacchi, M. Madami, G. Carlotti, A. O. Adeyeye and M. Kostylev, Journal
of Physics D: Applied Physics 43 (26), 264003 (2010).
50.
K. Y. Guslienko, S. O. Demokritov, B. Hillebrands and A. N. Slavin, Physical Review B 66
(13), 132402 (2002).
51.
K. Y. Guslienko and A. N. Slavin, Physical Review B 72 (1), 014463 (2005).
52.
S. Tacchi, M. Madami, G. Gubbiotti, G. Carlotti, S. Goolaup, A. O. Adeyeye, N. Singh and
M. P. Kostylev, Physical Review B 82 (18), 184408 (2010).
53.
M. P. Kostylev, A. A. Stashkevich and N. A. Sergeeva, Physical Review B 69 (6), 064408
(2004).
54.
M. M. Ney, Electromagnetic Compatibility, IEEE Transactions on 33 (4), 321-327 (1991).
55.
J. D. Kraus and D. A. Fleisch, Electromagnetics, 5 ed. (McGraw-Hill
Science/Engineering/Math, 1999).
56.
S. O. Demokritov and V. E. Demidov, IEEE Transactions on Magnetics 44 (1), 6-12
(2008).
57.
Z. Zhang, P. C. Hammel and P. E. Wigen, Applied Physics Letters 68 (14), 2005-2007
(1996).
58.
D. Rugar, R. Budakian, H. J. Mamin and B. W. Chui, Nature 430 (6997), 329-332 (2004).
59.
R. Magaraggia, M. Hambe, M. Kostylev, V. Nagarajan and R. L. Stamps, Physical Review
B 84 (10), 104441 (2011).
60.
R. von Helmolt, J. Wecker, B. Holzapfel, L. Schultz and K. Samwer, Physical Review
Letters 71 (14), 2331 (1993).
133
61.
N. Ross, M. Kostylev and R. L. Stamps, Journal of Applied Physics 109 (1), 013906013906-013908 (2011).
62.
C. L. Hu, R. Magaraggia, H. Y. Yuan, C. S. Chang, M. Kostylev, D. Tripathy, A. O. Adeyeye
and R. L. Stamps, Applied Physics Letters 98 (26), 262508-262508-262503 (2011).
63.
R. Bali, M. Kostylev, D. Tripathy, A. O. Adeyeye and S. Samarin, Physical Review B 85
(10), 104414 (2012).
64.
L. R. Walker, Physical Review 105 (2), 390-399 (1957).
65.
S. O. Demokritov, B. Hillebrands and A. N. Slavin, Physics Reports 348 (6), 441-489
(2001).
66.
B. A. Kalinikos, Microwaves, Optics and Antennas, IEE Proceedings H 127 (1), 4 (1980).
67.
R. W. Damon and J. R. Eshbach, Journal of Physics and Chemistry of Solids 19 (3-4),
308-320 (1961).
68.
J. R. Eshbach and R. W. Damon, Physical Review 118 (5), 1208-1210 (1960).
69.
D. D. Stancil and A. Prabhakar, Spin Waves Theory and Applications. (Springer, 2009).
70.
M. Bailleul, D. Olligs, C. Fermon and S. O. Demokritov, Europhysics Letters 56 (5), 741
(2001).
71.
M. Bailleul, D. Olligs and C. Fermon, Applied Physics letters 83 (5), 972-974 (2003).
72.
V. Vlaminck and M. Bailleul, Science 322 (5900), 410 (2008).
73.
V. Vlaminck and M. Bailleul, Physical Review B 81 (014425) (2010).
74.
R. Huber, M. Krawczyk, T. Schwarze, H. Yu, G. Duerr, S. Albert and D. Grundler, Applied
Physics Letters 102 (012403) (2013).
75.
M. Bao, K. Wong, A. Whitun, J. Lee, Z. Hao, K. L. Wang, D. W. Lee and S. X. Wang,
EuroPhysics Letters 84 (27009) (2008).
76.
M. Covington, T. M. Crawford and G. J. Parker, Physical Review Letters 89 (23), 7202
(2002).
77.
C. Mathieu, J. r. Jorzick, A. Frank, S. O. Demokritov, A. N. Slavin, B. Hillebrands, B.
Bartenlian, C. Chappert, D. Decanini and F. Rousseaux, Physical Review Letters 81 (18), 3968
(1998).
78.
Y. Roussigne, S. M. Cherif, C. Dugautier and P. Moch, Physical Review B 63 (13), 134429
(2001).
79.
C. Bayer, J. Jorzick, B. Hillebrands, S. O. Demokritov, R. Kouba, R. Bozinoski, A. N.
Slavin, K. Y. Guslienko, D. V. Berkov, N. L. Gorn and M. P. Kostylev, Physical Review B 72
(064427) (2005).
80.
A. A. Stashkevich, P. Djema, Y. K. Fetisov, N. Bizière and C. Fermon, Journal of Applied
Physics 102 (10), 3905 (2007).
81.
V. E. Demidov, M. P. Kostylev, K. Rott, P. Kryzysteczko, G. Reiss and S. O. Demokritov,
Applied Physics letters 95 (11), 2509 (2009).
82.
V. E. Demidov, J. Jersch, S. O. Demokritov, K. Rott, P. Krzysteczko and G. Reiss, Physical
Review B 79 (054417) (2009).
83.
Z. Liu, F. Giesen, X. Zhu, R. D. Sydora and M. R. Freeman, Physical Review Letters 98
(087201) (2007).
84.
K. Y. Guslienko, R. W. Chantrell and A. N. Slavin, Physical Review B 68 (2), 024422
(2003).
85.
V. E. Demidov, S. O. Demokritov, K. Rott, P. Krzysteczko and G. Reiss, Physical Review B
77 (064406) (2008).
86.
M. P. Kostylev, G. Gubbiotti, J. G. Hu, G. Carlotti, T. Ono and R. L. Stamps, Physical
Review B 76 (054422) (2007).
87.
T. W. O'Keeffe and R. W. Patterson, Journal of Applied Physics 49 (4886) (1978).
88.
P. Bryant and H. Suhl, Applied Physics Letters 54 (22), 2224 (1989).
89.
R. Matthies, K. Ramstock and J. McCord, IEEE Transactions on Magnetics 33 (5), 3993
(1997).
134
90.
J. Jorzick, S. O. Demokritov, B. Hillebrands, M. Bailleul, C. Fermon, K. Y. Guslienko, A. N.
Slavin, D. V. Berkov and N. L. Gorn, Physical Review Letters 88 (4) (2002).
91.
M. P. Kostylev, A. A. Serga, T. Schneider, T. Neumann, B. Leven, B. Hillebrands and R. L.
Stamps, Physical Review B 76 (18), 4419 (2007).
92.
B. B. Maranville, R. D. McMichael, S. A. Kim, W. L. Johnson, C. A. Ross and J. Y. Cheng,
Journal of Applied Physics 99 (8), 08C703-708C703-703 (2006).
93.
C. S. Tsai and D. Young, Microwave Theory and Techniques, IEEE Transactions on 38
(5), 560-570 (1990).
94.
R. Hertel, W. Wulfhekel and J. Kirschner, Physical Review Letters 93 (25), 257202
(2004).
95.
D. D. Stancil, Journal of Applied Physics 59 (1), 218-224 (1986).
96.
W. F. J. Brown, Magnetostatic Principles in Ferromagnetism. (North-Holland Publishing
Company, Amsterdam, 1962).
97.
R. Simons, Coplanar Waveguide Circuits, Components, and Systems. (Wiley, 2001).
98.
K. J. Kennewell, M. Kostylev and R. L. Stamps, Journal of Applied Physics 101 (09D107)
(2007).
99.
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed. (Addison-Wesley Publishing
Company, 1989).
100. V. F. Dmitriev, Soviet Journal of Communications Technology & Electronics 36 (1-3), 34
(1991).
101. V. F. Dmitriev and B. A. Kalinikos, Soviet Physics Journal 31 (11), 875-898 (1988).
102. T. Schneider, A. A. Serga, T. Neumann, B. Hillebrands and M. P. Kostylev, Physical
Review B 77 (214411) (2008).
103. V. E. Demidov, S. Urazhdin and S. O. Demokritov, Applied Physics letters 95 (26), 2509
(2009).
104. T. M. Crawford, M. Covington and G. J. Parker, Physical Review B 67 (024411), 1-5
(2003).
105. M. R. Scheinfein, (Mindspring.com, 2007).
106. C. Vittoria and N. D. Wilsey, Journal of Applied Physics 45 (1), 414-420 (1974).
107. K. Sekiguchi, K. Yamada, S.-M. Seo, K.-J. Lee, D. Chiba, K. Kobayashi and T. Ono,
Physical Review Letters 108 (1), 017203 (2012).
108. X. Liu, J. O. Rantschler, C. Alexander and G. Zangari, Magnetics, IEEE Transactions on 39
(5), 2362-2364 (2003).
109. Z. Celinski and B. Heinrich, Journal of Applied Physics 70 (10), 5935-5937 (1991).
110. B. Heinrich, J. F. Cochran and R. Hasegawa, Journal of Applied Physics 57 (8), 36903692 (1985).
111. A. G. Gurevich and O. A. Chivileva, PISMA V ZHURNAL TEKHNICHESKOI FIZIKI 15 (11), 79 (1989).
112. J. M. Shaw, H. T. Nembach and T. J. Silva, Physical Review B 85 (5), 054412 (2012).
113. A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves. (CRC Press,
1996).
114. J. Volakis, Antenna Engineering Handbook, 4 ed. (McGraw-Hill, 2007).
115. M. Bailleul.
116. T. Hubert, L. Boon-Brett, G. Black and U. Banach, Sensors and Actuators B: Chemical
157 (2), 329-352 (2011).
117. S. Okuyama, Y. Mitobe, K. Okuyama and K. Matsushita, Jpn. J. Appl. Phys. 39 (3584)
(2000).
118. Y. Imai, Y. Kimura and M. Niwano, Applied Physics Letters 101 (18), 181907-181907181904 (2012).
119. R. Gupta, A. A. Sagade and G. U. Kulkarni, international journal of hydrogen energy 37
(11), 9443-9449 (2012).
135
120. C. Langhammer, E. M. Larsson, V. P. Zhdanov and I. Zoric̕, The Journal of Physical
Chemistry C 116 (40), 21201-21207 (2012).
121. J. H. Jung, Applied Physics Letters 101 (24), 242403-242403-242404 (2012).
122. A. Tittl, C. Kremers, J. Dorfmuller, D. N. Chigrin and H. Giessen, Opt. Mat. Express 2
(111) (2012).
123. N. Liu, M. L. Tang, M. Hentschel, H. Giessen and A. P. Alivisatos, Nature materials 10
(8), 631-636 (2011).
124. R. C. Hughes and W. K. Schubert, Journal of Applied Physics 71 (1), 542-544 (1992).
125. Y.-T. Cheng, Y. Li, D. Lisi and W. M. Wang, Sensors and Actuators B: Chemical 30 (1),
11-16 (1996).
126. M. A. Butler, Applied Physics Letters 45 (10), 1007-1009 (1984).
127. J. Villatoro, D. Luna-Moreno and D. Monzón-Hernández, Sensors and Actuators B:
Chemical 110 (1), 23-27 (2005).
128. B. Chadwick, J. Tann, M. Brungs and M. Gal, Sensors and Actuators B: Chemical 17 (3),
215-220 (1994).
129. M. Tabib-Azar, B. Sutapun, R. Petrick and A. Kazemi, Sensors and Actuators B: Chemical
56 (1), 158-163 (1999).
130. W. Gouda and K. Shiiki, Journal of Magnetism and Magnetic Materials 205 (2), 136-142
(1999).
131. K. Ando and E. Saitoh, Journal of Applied Physics 108 (11), 113925-113925-113924
(2010).
132. Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H.
Kawai, K. Ando and K. Takanashi, Nature 464 (7286), 262-266 (2010).
133. J. Foros, G. Woltersdorf, B. Heinrich and A. Brataas, Journal of Applied Physics 97 (10),
10A714-710A714-713 (2005).
134. S. Okamoto, O. Kitakami and Y. Shimada, Journal of Magnetism and Magnetic
Materials 239 (1), 313-315 (2002).
135. K. Munbodh, F. A. Perez, C. Keenan, D. Lederman, M. Zhernenkov and M. R.
Fitzsimmons, Physical Review B 83 (9), 094432 (2011).
136. M. Kostylev, Journal of Applied Physics 112 (9), 093901-093901-093906 (2012).
137. B. N. Engel, C. D. England, R. A. Van Leeuwen, M. H. Wiedmann and C. M. Falco,
Physical Review Letters 67 (14), 1910 (1991).
138. R. L. Stamps, L. Louail, M. Hehn, M. Gester and K. Ounadjela, Journal of Applied Physics
81 (8), 4751-4753 (1997).
139. T. B. Flanagan and W. A. Oates, Annual Review of Materials Science 21 (1), 269-304
(1991).
140. F. A. Lewis, The Palladium Hydrogen System. (Academic Press, London New York,
1967).
141. E. Lee, J. M. Lee, J. H. Koo, W. Lee and T. Lee, international journal of hydrogen energy
35 (13), 6984-6991 (2010).
142. S. Mizukami, E. P. Sajitha, D. Watanabe, F. Wu, T. Miyazaki, H. Naganuma, M. Oogane
and Y. Ando, Applied Physics Letters 96 (15), 152502-152502-152503 (2010).
143. U. Gradmann, Journal of Magnetism and Magnetic Materials 54, 733-736 (1986).
144. H. J. G. Draaisma, W. J. M. De Jonge and F. J. A. Den Broeder, Journal of Magnetism
and Magnetic Materials 66 (3), 351-355 (1987).
145. Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Physical Review B 66 (22), 224403
(2002).
146. O. Dankert and A. Pundt, Applied Physics Letters 81 (9), 1618-1620 (2002).
147. D. Lederman, Y. Wang, E. H. Morales, R. J. Matelon, G. B. Cabrera, U. G. Volkmann and
A. L. Cabrera, Applied Physics Letters 85, 615 (2004).
148. J. Topp, D. Heitmann, M. P. Kostylev and D. Grundler, Physical Review Letters 104 (20),
207205 (2010).
136
149. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics. (John Wiley & Sons, New
York, 1978).
150. N. S. Almeida and D. L. .Mills, Physical Review B 53 (18), 12232-12241 (1996).
151. G. Gubbiotti, M. Kostylev, N. Sergeeva, M. Conti, G. Carlotti, T. Ono, A. N. Slavin and A.
Stashkevich, Physical Review B 70 (22), 224422 (2004).
137