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.
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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.
ii
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
iii
that the non-reciprocity parameter is largely invariant over the range of aspect ratios
studied.
For the stripe with the highest aspect ratio studied
m
nm
2
110, excluding the fundamental
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 spin-
pumping 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.
iv
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.
v
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.
vi
To Eugene Ivanov (Physics, UWA):
Thank you for building the microwave interferometric phase detector, with which high-
sensitivity 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, electron-
beam 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.
vii
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.
viii
Contents
1 Introduction 1
1.1 Thesis outline 2
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 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
3 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
ix
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
4.5 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
x
4.5.5 Non-reciprocity 81
4.6 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
4.7 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
4.8 Chapter conclusion 105
5 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
xi
Appendix C Numerical Simulations 130
Bibliography 132
1
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:
2
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 Damon-
Eshbach 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.
3
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.
4
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 micro-
stripes 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
5
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:
6
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 non-
magnetic 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.
7
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 lock-
in 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
8
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:
)( tiheH
The first two terms of the Taylor expansion (with respect to time) of the susceptibility
are:
dH
dheiH ti
)(
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 f
1noise can be reduced by increasing the modulation frequency. However,
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
9
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
10
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.
11
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:
]cos[)( tAtV LOLOLO
The modulated signal passing through the DUT incident at the RF port is:
)](cos[)()( tttatV RFRF
The mixer mixes the LO and RF signals. The first order output signal at the IF port,
with conversion factor K, is:
)()()( tVtKVtV RFLOIF
)](cos[)()cos()( tttatKAtV RFLOLOIF
)]()cos[()]()cos[()(5.0)( tttttaKAtV LORFLORFLOIF
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):
12
)(5.0)( taKAtV LOIF
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).
13
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 (signal-
ground 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.
14
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.
15
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.
16
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:
17
dt
dMM
MHM
dt
dM
s
eff
)( → Equation 3.1.1a
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 eigen-
frequencies 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
:
)4(22 MHHf → Equation 3.1.2a
18
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 non-
zero 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 d
nk
(Figure
1.2.2a). The Kittel equation is then modified 29
:
)4)((22 MHHHHf exex → Equation 3.1.3a
2DkHex 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 41-
44. 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.
19
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 magneto-
plasmonic 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-beam-
assisted 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
20
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
.
21
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.
22
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.
23
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.
24
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
25
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 nano-
structured 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:
)4)((22
dexdex HMHHHHHf → 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.
26
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 unchanged
49. 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 eigen-
frequencies 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:
MHHfdH
dfex
2
→ Equation 3.5b
27
From Equation 3.5b, one easily sees that the slope is determined by contribution from
the exchange (increasedH
df) and dipolar (decrease
dH
df) energies. One observes that the
low field dominant mode of the MNS array has a smaller dH
dfslope compared to other
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
28
for low aspect ratio
1
width
thicknessstructures. Therefore, one expects inaccuracy in
extracting a small Hex contribution on top of a strongly dominating Hd contribution for
the high aspect ratio
26.0
width
thicknessMNS array studied here.
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).
29
Figure 3.5c: Simulated in-plane dynamic magnetisation 2D profiles across the cross-
section 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.
30
Resonance feature Hex (Oe) Hd (Oe) Mode identification
MNS high-field
(Green diamond)
430 ± 5 0 MNS 1st SSWM
MNS low-field
(Blue triangle)
670 ± 40 1110 ± 70 MNS FM
MNS extra shoulder
(Purple star)
1551 ± 4 0 MNS 3rd
SSWM
Film high-field
(Black circle)
0 0 Film FM
Film low-field
(Red square)
291 ± 4 0 Film 1st SSWM
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-of-
31
plane component of the current density is important and may give rise to significant in-
plane 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.
32
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 r
1decay). As plotted in
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 sub-
skin-depth thin conducting films 42
.
33
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
34
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 anti-
symmetric 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 anti-
symmetric microwave magnetic field efficiently excites the first anti-symmetric SSWM.
35
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 r
1
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).
36
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.
37
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.
38
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.
39
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).
40
Figure 3.6.4c: Field sweep spin wave resonance of the MNS array at 14 GHz.
41
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
One notes a spread of nearx 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
102
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 nearx values are consistent with this, to the accuracy
limits of this experiment. One also notices that 0 < nearx < 3.75 μm for all the cases.
nearx >0 implies that the phase accumulation starts on the side of the excitation antenna
that is closer to the receiving antenna, and nearx < 3.75 μm implies that it starts below the
whole antenna structure (figure 4.7.2b). This is very important to know given the non-
reciprocity of the antenna, because it is not obvious a-priori that for a non-reciprocal
antenna nearx > 0.
One also notes that there seems to be a trend for nearx 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.
103
4.7.3 Effective propagation distance
Experimentally-wise, with knowledge of nearx , one can then determine the effective
propagation distance effx 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,
104
for each of the data set. In figure 4.7.3a -b, all the Hx 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 nearx . In fact, these offsets can be used to extract nearx , which is mathematically
identical to the approach in figure 4.7.2a.
In figure 4.7.3a -c, the antennae gaps were corrected with nearx = 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 effx . Thus, we
demonstrate here that the effective propagation distance effx (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 effx (not the antennae gap x ). Thus, the near-
field 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 effx 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.
105
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 )1010(5.5 24 . 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 non-
uniformity 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
with the highest aspect ratio studied in this work
m
nm
2
110. Remarkably, 6 higher order
width modes (excluding the fundamental MSSM) were found in the excitation spectra.
106
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 backward-
volume-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.
107
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 properties
127-129.
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
108
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
.
109
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:
110
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.
111
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.
112
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.
)4(22 MHHf → Equation 3.1.2a
f
HFWHM
2
→ Equation 4.5.4e
Atmosphere Effective saturation
magnetisation,
4πM (Oe)
Damping coefficient,
α (10-2
)
Nitrogen 12500 ± 200
2.30 ± 0.08
Hydrogen 13300 ± 200 1.73 ± 0.05
Difference 800 ± 400 0.6 ± 0.1
113
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
effectiveM is equal to the difference between the real saturation magnetisation realM and the
114
effective anisotropy field anisotropyH (equation 5.5a). Therefore, we experimentally
observe increase in effective saturation magnetisation in hydrogen atmosphere (down-
field shifts in FMR peaks).
anisotropyrealeffective HMM → 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.
115
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
116
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
117
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.
118
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 cobalt-
palladium 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 spintronic-
based 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
119
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-of-
plane 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
120
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.
121
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 (N-
methyl-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
122
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-beam-
assisted 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.
123
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 non-
vanishing 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 zj ,
z zj E (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) zE 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.
124
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:
0/ /
1( ) /
( ) /
r z
z
r
E z E r i H
rH r Er
rH z E
(2)
Here rE 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 7
0 4 10 /Hn m . Several important equations and
identities can be derived from Eq. (2):
2 2 2 2
0
1/ / / 0z z z zE z E r E r i E
r (3)
2 2 2 2
0 2
1 1/ / / ( ) 0r r z zE z E r E r i E
r r (4)
2 2 2
0/ / ( )r z zE z i E E 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
0 0( , 0) 1, ( , 0) 0z zE r r z E r z (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.)
125
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
0
0
( )z zkE E J kr kdk
(6)
1
0
( )r zkE E J kr kdk
(7)
where 0 ( )J x and
1( )J x are Bessel functions of the zeroth- and the first-order
respectively, and zkE and
rkE are the respective Hankel-components of the fields:
( ) ( ) 0(1)
0
( )z r z r kE E J kr rdr
(8)
On substitution of (6) and (7) in (3) and (4) respectively one obtains:
2 2 2/ 0zk k zkE z E (9)
2 2 2/ 0rk k rkE z E (10)
where
2 2
0k k i (11)
The general solutions to (9) and (10) have the form:
( ) ( ) ( )exp( ) exp( )z r k z r k k z r k kE A z B 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 zkE (12) reduces to
sinh( ( ))zk zk kE A z L (13)
The coefficients zkA are obtained from the boundary condition (5). The Hankel J0
transform of the step-function (5) is 0 1 0( ) /r J kr k . Thus, from (5) and (13) one obtains:
0 1 0( )sinh( ( )) / [ sinh( ( ))]zk k kE r J kr z L k L (14)
126
Similarly, from (10) and (12) we obtain:
0 1 0( )( exp( ) exp( )) /rk rk k rk kE r J kr A z B z k (15)
Finally, using the identity (11) from (14) and (15) taking into account (8) one easily
finds that
2
0 1 0( )cosh( ( )) / [ sinh( ( ))]rk k k kE r J kr z L k L (15)
and then from the first of Eqs.(2) that
2
0 1 0( )sinh( ( )) / [ sinh( ( ))]k k kH r J kr z L k L (16)
Here one has to note that following (8) the Hankel J0 transform should be used to
calculate zE from zkE and the Hankel J1 transform to restore rE and H from rkE and
kH 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:
1 1 2 2( )cos( ) ( )cos( )yH H r H r (17)
where 2 2 1/2
1(2) 1(2)[( ) ]r x x y , 1(2) 1(2) 1(2)cos( ) ( ) /x x r , 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.
127
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 zE 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 yH . Since
sinh( ( )) ( )k kz L z L for 1Lk (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 Hankel-
components 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.
128
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.
129
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:
1 1 2 2( )cos( ) / 2 ( )cos( ) ( )cos( ) / 2y c cH H r H r H r (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 anti-
symmetric linear profile for the magnetic field: ( ) ( 0)y yH z L H z 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
yH 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 zE existing right beneath the contact.
130
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 Landau-
Lifshitz 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
131
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
132
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