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

spatial resolution, namely Brillouin light scattering 56

and magnetic force microscopy 57,

58 – the current injection method using a coplanar probe is far simpler and quicker to

utilise.

One drawback of this method is that being a contact method, damage to samples usually

unavoidable. We now consider the physical contact between the coplanar probe and the

material being probed. The probe has a built-in spring which applies a constant force

onto the surface being probed. This ensures good physical contact between the tip and

the probed surface without risking tip breakage or loss of contact from mechanical

vibration. The standard probe tips are available in either tungsten (for long-lasting tips)

or nickel (for better electrical contact and minimal sample damage). In our setup, we use

tungsten tips since we require robustness in our experiments, and the alternative –

nickel – is magnetic and hence undesirable in spin wave experiments. The downside of

probing with a hard tungsten tip is potential physical damage to the surface being

probed, especially if the material is a soft metal.

The probed samples were inspected with an optical microscope for sample damage. No

trace of physical damage was observed on the continuous film probed. Thus, Permalloy

appears to be hard enough to resist the pressure exerted by the probe tips. However,

scratch marks were left by the probe on the surface of the MNS array (figure 3.7a).

Nano-structuring has weakened the material; making it mechanically softer than the

continuous film.

42

Figure 3.7a: Scratch marks left by the tips of the coplanar probe on the MNS array.

The depth profile of the scratch in figure 3.7a is shown in figure 3.7b. The distance

between the three scratch dips is consistent with the pitch of the probe (200 μm); these

are not sample fabrication defects, but that caused by the physical contact of the probe.

The width of the trench left by the central tip is about 10 μm wide. Note that this

dimension cannot be used to estimate the number of contacted stripes as discussed

before because this is the size along the stripe length direction, not across the stripe

width direction. Furthermore, an indentation is typically larger than the size of the

object which causes the indentation. Important from the scratch profile is the depth of

the indentations; as long as the probe exerts minimal force on the MNS array (just

enough to ensure contact), the stripes are not cleaved by the probe tips. The indentations

depths are of the order of nanometres; not enough to cleave the thick 100 nm film in this

case. This result implies that in order to use this method as a non-destructive spin wave

characterisation tool, a probe should be designed with a non-magnetic soft metal tip and

minimal force should be exerted onto the probed surface.

43

Figure 3.7b: Optical profilometer profile taken in the direction perpendicular to the

direction of the scratch mark left by the coplanar probe.

Another drawback of the current injection method is the requirement of current

continuity between the three probe tips. This means that the sample probed must be a

good electrical conductor. The current injection technique was attempted on a

La0.7Sr0.3MnO3 half-metallic film, one of the samples studied in 59

. The typical

resistance between the coplanar tips through the poorly conducting sample was of the

order of 103 Ω, which is consistent with the typical resistivity of unannealed

La0.7Sr0.3MnO3 60

. No spin wave resonances were observed in the current injection

spectra through the half-metal, indicating that the very low microwave current flowing

through the poorly conducting material does not induce sufficient microwave magnetic

field to drive spin wave excitation. Therefore, we conclude that if the material is

continuous but poorly conducting (like most ferrites), the current injection method

cannot be used.

In addition to the requirement of the sample being electrically conducting, there must

also be possibility of current conduction between the signal and ground tips of the

probe. This means that only a subset of patterned films can be probed with the current

injection method. This method cannot be used on a dot array for example 61

, even if the

material is conducting, due to lack of current continuity. The requirement of current

conduction continuity thus limits the method to continuous films, stripes, anti-dots 62, 63

,

or similar nano-structures where continuity of the conducting phase is preserved across

the whole distance between the tips.

44

Finally, impedance mismatch is a potential issue in the current injection technique. In

this method, the probed sample is essentially the load at the terminus of a microwave

transmission line. This means that for efficient transfer of microwave power into the

load, the load should have impedance matching that of the transmission line. However,

designing the sample to impedance-match with the probe defeats the purpose of the

current injection technique as a characterisation tool in the first place. In the work

presented here, the impedance mismatch was quite significant, resulting in most of the

incident power reflected from the probed MNS array. Assuming a purely resistive MNS

array load of 130 Ω and a nominal coplanar probe characteristic impedance of 50 Ω, the

transmission coefficient is calculated to be 56 %. The transmission coefficient of a

purely resistive 3 Ω continuous Permalloy film is even lower at 11 %. However, these

estimates disregard reactance contributions to the impedance and insertion loss. In the

actual experiment, the typical reflected power is nearly at the same level as that of the

incident power for both the MNS array and continuous film.

3.8 Chapter conclusion

In this chapter, the microwave current injection spin wave spectroscopy technique is

demonstrated on a Permalloy nano-stripe array and continuous film. A sub-milimetre

coplanar probe was used to inject microwave current into the samples studied, and the

spin wave excitation response compared with standard macroscopic planar waveguide-

based spin wave spectroscopy. The current injection method is able to efficiently excite

anti-symmetric standing spin wave modes; these modes provide important interfacial

material properties, and the modes are often lacking in planar waveguide-based

methods. The current and radiation field distributions of both the current injection and

planar waveguide techniques were used to explain the mode amplitudes observed in the

spin wave spectra. It is found that the in-plane current flowing through the contacted

nano-stripes induces an anti-symmetric dynamic magnetic field inside the stripes, which

efficiently drives the first anti-symmetric standing spin wave mode. The current

injection technique is quick and allows easy spatial mapping of magnetic properties of

conducting materials with resolution down to 0.1 mm.

45

Chapter 4

Propagating spin wave spectroscopy

The sections in this chapter are organised as follows. A brief theory of propagating

waves in continuous and laterally confined geometry is first presented, followed by case

for work. In the next section, the experiment setup, procedure and data acquisition

methods are highlighted. The presentation and discussion of results in this chapter are

then divided into three main sections, each containing sub-sections of their own. Section

4.5 then deals with the characterisation of the fundamental propagating surface mode in

the series of stripes studied. Section 4.6 focuses on the characterisation of high-order

width modes observed in the stripe with the highest aspect ratio studied in this work.

Section 4.7 elucidates the antenna near-field effect, how it may be quantified, and how

it affects the data. The chapter ends with a summary of major findings of the presented

work.

4.1 Background

If the characteristic wavelength of the excitation source is smaller or comparable to the

mean free path of spin waves in a particular magnetic material, the resultant highly

localised excitation field can excite propagating modes. The wave equation of

propagating spin waves is obtained by solving the Landau-Lifshitz-Gilbert equation

together with Maxwell’s equations in the magnetostatic limit and material constitutive

relations. The resultant wave equation is known as Walker’s equation 64

. Since these are

obtained in the magnetostatic limit, these propagating modes are known as

magnetostatic spin wave modes. The dispersion relations of propagating spin waves in a

specific geometry are obtained by solving Walker’s equation subject to boundary values

of the required geometry.

46

4.1.1 Propagating modes in continuous films

For infinite continuous films, there are three possible field orientation and propagation

direction geometries, each yielding different propagation modes and dispersions:

Figure 1.2.3a: Dispersion relations of magnetostatic modes in a thin film 65

.

Forward volume mode 66

If the film is perpendicularly magnetised, forward volume spin waves can propagate

tangentially in-plane. The characteristics of this mode is indicated by the name; being a

volume mode, it permeates the entire bulk of the solid, and the phase and group

velocities are both positive.

Backward volume mode 67

If the film is tangentially magnetised, two possible wave modes can be excited.

Backward volume waves propagate parallel to the applied field. An interesting property

of this mode is that while the phase velocities are positive, the group velocities are

negative, hence the name “backward” volume waves.

Surface mode 67, 68

In the tangentially magnetised configuration, a second mode of wave propagation is

possible. Surface spin waves can propagate perpendicular to the applied field. This

47

mode is named so because the wave is evanescent; it persists only near the surface of

the film, decaying exponentially into the bulk. Both its phase and group velocities are

positive, so it is a forward wave. The wave is monotonic. This means that every wave

number corresponds to a unique frequency in the dispersion and vice versa. Note that

this is not true for forward and backward volume waves, which are multi-tonic; the

potential function of those volume modes are composed of sinusoids. When the

direction of propagation is reversed (or the field is reversed), the mode fields shift from

one surface to the other 69

. This is known as field displacement non-reciprocity. This

particular magnetostatic mode is also known as Damon-Eshbach (DE) waves, after the

pioneers in the field 68

.

4.1.2 Propagating modes in laterally confined geometry

In the past 15 years, research in the field has focussed on the effects of lateral

confinement on propagating spin waves modes. These have been studied on micro-

stripes and patterned arrays using inductive techniques in the frequency domain 35, 70-75

,

inductive techniques in the time domain 34, 76

, Brillouin light scattering (BLS) in

reciprocal space 77-80

, micro-BLS 81, 82

, and Kerr microscopy in the time domain 83

.

Lateral confinement results in three significant deviations from the infinite continuous

film case. First, lateral confinement results in quantization of magnetostatic modes

across the confined dimensions 36, 50, 77-79, 84, 85

. Due to confinement, the dipolar eigen-

functions of stripes take the form of sinusoids, analogous to the form of spin wave

resonance modes in a perpendicularly magnetised film 50, 79, 84

. Thus, one may observe

higher order confinement modes in the measured spin wave spectra. Second, the static

and dynamic demagnetising fields in confined geometry would increase and become

non-uniform. This in effect, shifts the position and slope of spin wave dispersion 79, 86,

87. Thirdly, the extreme case of increased static demagnetising field is the formation of

potential wells at the edges of tangentially magnetised stripes where demagnetising

fields can be very large 33, 70, 88-90

. This leads to decrease in the effective width of the

stripe where dipolar spin waves propagate (channelling effect) 33

, and the formation of

exchange edge modes in such potential wells 79, 90-92

.

48

Channelling of spin waves in metallic ferromagnetic media confined in stripe geometry

is of technological importance for the future device applications in microwave signal

processing 33, 93, 94

and magnetic logic 2-4

. Of the three propagation modes, the Damon-

Eshbach surface mode has the highest group velocity and consequently, low spatial

attenuation 69, 95

. This makes it a good candidate for spin wave device application 70

, as

evident by the majority of the previous work in the Damon-Eshbach geometry.

The work presented in this chapter is concerned with spin wave propagation through

laterally confined Permalloy stripes in the Damon-Eshbach geometry, where the stripes

are magnetised tangentially and wave propagation perpendicular to the applied field

(along the long axis of the stripes) is considered. The experiment design and technique

is similar to that pioneered by Bailleul et al. 70

, with the exception that the experiment

was performed field-resolved using a lock-in modulation technique, and a much more

sensitive detection method was used.

4.2 Case for work

Thicker stripes

Most of the work done in the past 15 years on spin wave propagation in stripes was on

stripes thinner than 36 nm. There were relatively few studies on thicker stripes 35, 76

. In

this work, thicker stripes were studied (55, 80, and 110 nm). Apart from filling in the

knowledge gap, studying thicker films possess the following advantages. The first

trivial advantage is that the signal reception of spin wave propagation would be better

for thick films simply because there is more material under the transducers. Second, the

group velocity of magnetostatic surface spin waves is directly proportional to the film

thickness. Group velocity, being the speed of energy transfer, directly relates losses in

the time and spatial domains 80

. Since spatial damping is inversely proportional to group

velocity, the greater the thickness, the larger the group velocity and the less is the spatial

attenuation. For signal detection, the consequence of this is that a thicker film would

have better signal per unit thickness compared to that of a thinner film.

49

Wider range of aspect ratios

One way to quantify confinement of a long stripe is by its aspect ratio. One expects the

manifestation of confinement effects as mentioned in section 4.1.2 to scale with aspect

ratio. In particular, demagnetising field 96

is directly proportional to aspect ratio for

stripes magnetised tangentially perpendicular to the long axis 70, 79

. The stripe array with

the highest aspect ratio studied to date was by Huber et al.

110

3.0

30

m

nm

74

, while the

previously done study with the largest range of aspect ratios (10-3

– 10-2

) was by

Vlaminck et al. on forward volume waves 73

. Considering all relevant previous work to

date, there is little work done to systematically investigate the effects of lateral

confinement on spin wave propagation. The series of thick magnetic stripes studied in

this chapter systematically span the largest range of aspect ratios to date, covering two

orders of magnitude (Figure 4.2a). The stripe with the highest aspect ratio studied in this

work falls just slightly short of the aspect ratio of Huber et al.’s array 74

.

Figure 4.2a: Aspect ratio of stripes studied in the field of propagating spin waves.

Improved sensitivity

Lastly, we have built a highly sensitive microwave detector which is able to detect

extremely small microwave signals (see section 2.2.3) specifically for this propagating

spin wave project. Hence, the possibility of detecting higher order width and edge

modes, both of which are relatively elusive by the induction method, warrants further

investigation at vastly improved sensitivity levels.

50

4.3 Experimental setup

The simplest design of a propagating spin wave spectroscopy (PSWS) experiment 70

is

shown in figure 4.3a. Gold coplanar waveguide antennas are overlaid across the

magnetic stripes at both ends with one antenna as the excitation source and the other as

the detector (figure 4.3b). Microwave current passes through one antenna, and the

microwave magnetic field of this current drives spin waves in the underlying Permalloy.

Propagating spin wave modes then travel along the strip, and get detected at the second

antenna via transduction. An insulating spacer (in this case, aluminium oxide) between

the magnetic strips and the coplanar lines ensure no physical electrical contact between

the two.

Figure 4.3a: (a) Diagram illustrating the experiment, showing dimensions and field lines. The

antennae separation gap, stripe width, and stripe thickness, are labelled x, y, and z, respectively.

The applied dc field H is in-plane and parallel to y. Microwave current i(ω) in the left-hand

antenna generates a non-uniform excitation field h(ω,x,z), which in turn drives spin waves,

m(ω,k). 1 denotes the Permalloy stripe, 2 is the 30 nm thick Al2O3 insulating layer, and 3 is the

receiving antenna (the similar structure to the right is the output antenna).

(b) Enlarged view of the antenna cross-section. The origin x=0 of the frame of reference

coincides with the symmetry axis of the input antenna.

51

A series of Permalloy stripes of various widths and thicknesses was fabricated (figure

4.3a) in order to experimentally investigate high aspect ratio confinement effects which

may deviate from the infinite thin film case. Permalloy is well-known for having low

intrinsic magnetic damping 32

, hence it is the ideal metallic ferromagnet for spin wave

propagation. The narrowest and widest stripes are 2 and 100 µm respectively. All the

stripes have the same length 200 µm. Three different Permalloy film thicknesses were

fabricated: 55 nm, 80 nm, and 110 nm. In all, 108 unique stripe geometries were

investigated (figure 4.3a) with aspect ratios ranging [5.5 x (10-4

–10-2

)]. In addition,

large patches of continuous film (3.7 x 3.7 mm2) were patterned on the same wafer

together with the stripes, and all underwent the same fabrication process. These serve as

reference continuous films, and were large enough to enable characterisation using flip-

chip broadband FMR (figure 2.2.1a).

A coplanar waveguide (CPW) design was used in this work to facilitate connection with

a coaxial line via a sub-millimetre coplanar probe. The CPW antenna geometrical

parameters are as follows: the conductor widths and separation gaps are 1.5 µm, and the

thickness is 200 nm. The distances between the excitation and detection antennas were

varied from 12 to 110 µm. Far away from the stripes, the CPW lines gradually become

larger, but still maintain geometrical proportions. This is to ensure matched

transmission line characteristic impedance throughout, which is calculated to be 67 Ω 97

.

Finally, the CPW lines terminate at 100 µm sized contact pads (figure 4.3c). A 30 nm

thick aluminium oxide spacer ensures no direct dc electrical contact between the

overlaying gold antennas and the underlying Permalloy stripes.

52

Figure 4.3b: Optical micrograph of a typical PSWS experiment showing a magnetic

strip and overlaid coplanar lines.

Figure 4.3c: Optical micrograph of spin wave device showing coplanar lines leading to

contact pads.

53

4.4 Experimental procedure

4.4.1 Data acquisition

First, contact with the coplanar probes is made to the DUT using the probe station setup.

Then, microwave coaxial lines were connected to the probes; one coaxial line feeds

microwave power to the excitation antenna on the DUT, while the other coaxial line

relays the transmitted signal from the DUT. The other ends of the coaxial lines were

connected to the microwave receiver detailed in section 2.2.3. 10 GHz microwave

power at 10 dBm was fed into the receiver; this power is split into two channels, with

one driving the mixer’s LO port, and the other exciting spin waves on the DUT. 10 GHz

was used because the microwave receiver is optimised for that frequency. Spin waves

propagate along the magnetic stripe, and get detected at the second antenna (figure

4.3.1a). The detected signal gets fed into the RF port of the mixer. The resultant output

signal at the mixer’s IF port is a product of the reference driving signal, and the spin

wave signal (see section 2.2.3). This signal gets fed into a lock-in amplifier and is

recorded as a function of sweeping field applied tangentially to the magnetic stripes;

this is the Damon-Eshbach geometry. A modulating field of about 9 Oe (rms) at 220 Hz

modulates the sweeping field. The lock-in amplifier refers to this same modulating

field, and the resultant signal acquired is proportional to the field derivative of the

microwave susceptibility. A typical raw trace of the data is shown in figure 4.4.1a. Note

the oscillatory spin wave signal as a function of field due to phase selection at the

detection antenna.

54

Figure 4.4.1a: Raw trace of transmitted spin wave signal propagating through a 55 nm

thick, 2 μm wide Permalloy stripe over a distance of 20 μm, by microwave excitation

frequency of 10 GHz.

4.4.2 Sensitivity

We now evaluate the sensitivity levels of previous works utilising the inductive method,

and compare them to this work. The example from figure 4.4.1a was chosen because the

single stripe geometry is closest to that used in Bailleul et al.’s pioneering work in the

Damon-Eshbach geometry 70

and Vlaminck et al.’s subsequent work in the forward

volume wave geometry 73

. In Vlaminck et al.’s work, the excitation and detection

antenna were meanders (5x), maximum antenna separation was 15 μm, the stripes were

10 – 20 nm thick, and a VNA was used to perform frequency sweeps. Note that even

though the film used in Vlaminck et al.’s work was about 5 times thinner than the

example in figure 4.4.1a, the use of 5 meanders effectively amplifies the signal 5-fold;

hence, both stripe data becomes comparable. At a VNA intermediate frequency

bandwidth of 10 Hz, the transmitted spin wave signal is comparable to the noise floor

(S/N ≈ 1). Averaging of more than 100 scans was needed to bring the spin wave signal

clearly above the noise floor. Assuming that the signal-to-noise ratio (S/N) improves as

the square root of number of trials, then the S/N of Vlaminck’s work for this particular

stripe geometry is about 10.

55

Another comparison may be made with Sekiguchi et al. 34

and Covington et al. 76

’s

time-resolved PSWS work. The former used 35 nm thick, 120 μm wide Permalloy

stripes, while the later used 27 or 104 nm thick 475 μm squares of Permalloy; these are

large-scale structures compared with the narrow 2 μm wide stripe example in figure

4.4.1a. Though there were no explicit mentions of the noise level, the experiments

required averaging of 1024 waveforms to improve S/N.

The other two previous works done to date using the inductive method are Bao et al.’s

work on a 25 nm continuous film 75

, and Huber et al.’s work on an array of nano-stripe

arrays 74

. Evaluation and comparison of sensitivity levels with these two works are not

possible since there were no mentions of the S/N of both these experiments, and also

that these works were not performed on individual stripes.

We now turn attention to our experiment in the frequency domain utilising our sensitive

microwave receiver (section 2.2.3) with the field-modulation method (section 2.2.2).

The typical noise floor of the setup is roughly 0.3 μV. Using the amplitude of the

dominant spin wave band from the example raw trace from figure 4.4.1a, this translates

to a remarkable S/N of about 500! Note that the raw trace (figure 4.4.1a) was obtained

with a single field sweep (without averaging multiple runs), and with a single detection

antenna (without multiple meandering of antennae 70, 73

). As a further testament of the

sensitivity of this setup, we successfully detected multiple higher order width modes on

our stripe with the highest aspect ratio and characterised their dispersion and group

velocities in detail (see section 4.6).

4.4.3 Wave number space

Spin waves are excited by the microwave magnetic field of the microwave current

flowing through the coplanar antennas. The wave number spectrum of spin wave

excitation and absorption depends on the square of the Fourier transform of the current

density in the coplanar antenna 98

. The skin depth of gold at 10 GHz frequency is

calculated to be 0.8 μm 99

. The skin depth is much larger than the thickness of the

coplanar line (0.2 μm) and about half the widths of each conductor (1.5 μm). Hence one

expects the microwave current density in the coplanar lines to be practically uniform

across the thickness of the coplanar lines. Laterally, the current density is described by

56

Dmitriev 100

; the current is concentrated on the outer edges of the central conductor and

the inner edges of the ground planes, as shown in figure 4.4.3a. The negative current

densities in the ground conductors indicate the return current.

Figure 4.4.3a: Spatial current density of the coplanar antennas. The central conductor,

ground conductors, and separation gaps are all 1.5 μm wide.

Figure 4.4.3b: Spatial Fourier transform of the current density of the coplanar lines,

assuming uniform current density distribution.

Although the current density distribution is non-uniform laterally, the Fourier transform

of this non-uniform distribution is quite close to the case where an uniform distribution

is assumed 98

. For ease of computation, Fourier transform for the uniform current

57

density case was taken (figure 4.4.3b). This approximation is reasonable since the

microwave skin depth and the conductor width are comparable. Multiple wave number

bands were found in the spatial Fourier transform of the coplanar line current density, of

which the first three are shown in (figure 4.4.3b). The dominant band is centred on the

most intense peak, at k0 = 10038 cm-1

. This corresponds roughly to the wavelength of

the central conductor width, w, and gap between the conductors, Δ, according to the

expression 98

:

2

20

wk

For simplicity of further data analysis, only the dominant band (0 – 20000 cm-1

) is

considered to effectively contribute to spin wave excitation 43, 101

.

4.4.4 Extracting dispersion

The signal obtained in transmission (wave propagation from one antenna to another), is

the sum of two signals; direct electromagnetic induction through air, and spin wave

transduction. Assuming plane wave propagation, the superposition of the two waves is:

)sin()sin( kxtBtA

The first term in the sum represents direct electromagnetic induction through air, while

the second term represents spin wave transduction after propagating through the

Permalloy stripe a distance x (which is the separation distance between both antennae).

ϕ is an arbitrary initial phase difference between the two waves. The typical group

velocity of magnetostatic surface spin waves (≈ 10 µm/ns for a 100 nm Permalloy film)

is much smaller than the speed of light. Hence from the spin wave’s perspective, the

electromagnetic wave propagating through air has virtually no phase accumulation over

such small propagation distances.

For the case where A = B (where both amplitudes are the same), the expression can be

rewritten as:

2cos

2

2sin2)sin()sin(

kxkxtkxtt

58

The cosine term in the trigonometric identity relation above determines the amplitude

envelope of the superposition of two plane waves. Power is proportional to the square of

the wave amplitude, so power constructive interferences occur when:

12

cos2

kx

Or when (where n = integer):

nkx 2

The phase difference between each successive maxima of the power absorption

interference spectra is thus:

xk

2 → Equation 4.4.4a

This method of using phase accumulation to extract dispersion was also used in

references 35, 73, 85, 102, 103

. From this, one expects that the number of interference patterns

in the spectra increase with propagation distance. This is confirmed experimentally as

shown in figure 4.4.4a. Note that the envelope band over which spin waves occur does

not change with propagation distance; only the number of oscillations within the

interference pattern. One finds a linear relationship between the numbers of amplitude

oscillations with propagation distance (figure 4.4.4b).

59

Figure 4.4.4a: Normalised spin wave raw traces at 10 GHz for a 110 nm thick, 100 µm

wide Permalloy stripe, at various propagation distances, x.

60

Figure 4.4.4b: Number of amplitude oscillations at various propagation distances.

Hence, when field (or frequency) is swept around the resonance, different wave

numbers are selected according to the dispersion relation and within the bandwidth of

the spatial Fourier transform of the current density of the excitation antenna 73

. Note that

the envelope of the interference pattern in the raw trace reflects the wave number

distribution of the Fourier transform of the coplanar antenna. Using this fact, one can

extract the dispersion from a raw trace as follows:

61

Figure 4.4.4c: Extracting dispersion from raw data.

An example raw trace data is shown in figure 4.4.4c. One sees an interference pattern in

the signal as a function of sweeping field for a given frequency. The extrema with the

largest signal amplitude on the raw trace (figure 4.4.4c-a) corresponds to the

fundamental wave number k0 of the coplanar line (figure 4.4.4c-b), and each successive

maxima/minima has a total phase accumulation of 2π (figure 4.4.4c-b). This way, one

can construct a dispersion relation (figure 4.4.4c-c) by mapping each extrema on the

raw trace (figure 4.4.4c-a) to a particular wave number of the Fourier spectra of the

coplanar line (figure 4.4.4c-b). This procedure is repeated for all the stripes studied in

this work, and presented here after.

Note that in contrast to Bao et al.’s proposed absolute phase method using a VNA 75

,

this relative phase difference method is much simpler. Also note that since the

experiment is field-resolved, the plot is technically a pseudo-dispersion relation, rather

than a proper dispersion plot of frequency versus wave number.

62

4.5 Magnetostatic surface mode in confined stripe geometry

4.5.1 Dispersion

For a continuous film, magnetostatic surface modes (MSSM) propagate according to the

dispersion relation:

)1(4)4( 222

2

kdeMMHHf

→ Equation 4.5.1a

Note that by setting k = 0 (wave becomes stationary), one essentially recovers the Kittel

formula for FMR in a tangentially magnetised continuous film (Equation 3.1.2a).

To plot dispersion relation, one has to consider the optimal propagation distance from

which to extract dispersion from. As seen in figure 4.4.4a, the number of oscillations is

proportional to the propagation distance. Thus, more data points can be extracted from

large propagation distances. In addition, one should also minimise antenna near-field

effects by maximising the separation gap between antennae (see section 4.7). However,

signal attenuation at large propagation distances place a practical upper limit on the

antennae gap from which sufficient signals are available. For the 110 nm series reliable

data could be obtained at 60 µm separation gap between the antennae; for the 80 nm and

55 nm series, due to more attenuation for thinner films, the distance is 30 µm. For these

propagation distances, the dispersion relations of the MSSM in the Permalloy stripes are

shown in figure 4.5.1a for the range of thicknesses and widths studied.

Due to sample fabrication defects, some antennae and/or Permalloy stripes were

malformed, resulting in “gaps” in the data. The black dotted curves are the theoretical

dispersion relations for infinite continuous films calculated using equation 4.5.1a based

on the extracted saturation magnetisation values obtained from FMR (figure 4.5.1b).

One immediately observes that the dispersion relation shifts to high field upon

narrowing of stripe width. We claim that this is due to the effect of the static

demagnetising field. As the stripes become narrower, the effective demagnetising field

becomes larger, resulting in decrease in internal field for a given applied external field

70, 79, 96 (see section 4.5.2). This shifts the dispersion up-field.

63

Figure 4.5.1a: Dispersion relation of Permalloy stripes at 10 GHz.

It is found that within the experimental accuracy, for a particular stripe thickness, the

slope of the dispersion is independent of stripe width, at least up to the stripe with the

highest aspect ratio

m

nm

2

110. The dynamic confinement effects are known to modify

64

the slope and curvature of the dispersion curve 87

. This implies that for the available

aspect ratios and wave numbers, the dynamic confinement effects are not important

within the experimental accuracy. Therefore, one can treat the magnetisation dynamics

in the stripes as in continuous films. One only needs to include the static confinement

effect. We argue that this effect can be accounted for by including an effective

demagnetising field term, Hd, into the dispersion relation for continuous film similar to

104. Thus, equation 4.5.1a is modified into:

)1(4)4)(( 222

2

kd

dd eMMHHHHf

→ Equation 4.5.1b

Using the modified dispersion law, one can set Hd as the fitting parameter and extract

the effective demagnetising fields. The saturation magnetisation parameter, M, was

extracted separately from reference film FMR data (figure 4.5.1b). M is assumed to be

constant across all strip widths for particular thicknesses in the dispersion fitting. The

dispersions were fitted with the effective demagnetising field as the only free fitting

parameter; all other parameters were assumed to be constant. In particular, the

saturation magnetisation, gyromagnetic ratio, and thicknesses of the stripes were

assumed to be identical to their respective reference continuous films, since they were

all deposited together on the same wafer in the same deposition process, and underwent

the same lithography fabrication process. The fits are shown with solid curves in figure

4.5.1a.

65

Figure 4.5.1b: FMR data for the reference films showing extracted saturation

magnetisation values. Solid lines are fits with Kittel’s formula (equation 3.1.2a).

As shown, the data fits well to the modified dispersion law across the range of aspect

ratios studied, except for the stripe with the highest aspect ratio

m

nm

2

110. For that

particular stripe, there was noticeable deviation from equation 4.5.1b. This is postulated

to be due to the fit assuming an uniform demagnetising field 86

, but in actual fact, this

particular stripe has the most non-uniform demagnetising field profile across its width

amongst the stripes studied (see section 4.5.2). The thin film model assuming a uniform

effective demagnetising field is insufficient to properly describe a narrow stripe of such

high aspect ratio. Furthermore, for large aspect ratios, the geometrical confinement also

affects the dynamic dipole field. The static demagnetising field shifts dispersion up-

field, while the dynamic dipole field increases the dispersion slope with respect to the

continuous case 79, 86

.

66

Figure 4.5.1c: Experiment and simulation MSSM dispersion for the

m

nm

2

110stripe.

Solid curves are fits with equation 4.5.1b.

To properly model the experiment dispersion of the

m

nm

2

110stripe, numerical

simulations were performed. The demagnetising field profile (see figure 4.5.2a-a in

section 4.5.2) was used as the stripe ground state. From this, the eigen-fields for

particular wave numbers at 10 GHz frequency were numerically simulated. The

simulated dispersion is plotted together with experiment data in figure 4.5.1c. To

compare with the modified continuous film dispersion, both the experiment and

simulation results were fitted with equation 4.5.1b. Note that both the simulation and

equation 4.5.1b did not adequately model the experiment result. The slope of the

experimental dispersion is smaller than predicted by simulation. The failure of the

simulation to model the experiment data might be due to unaccounted peculiarities in

this particular stripe. To further elucidate the matter, the

m

nm

2

110stripe’s dispersion

was fitted with equation 4.5.1b with additional free fitting parameters (figure 4.5.1d).

67

Figure 4.5.1d:

m

nm

2

110stripe dispersion fittings with equation 4.5.1b.

One notes from figure 4.5.1d that better fits could be obtained by setting the saturation

magnetisation, M, or the thickness, d, as fitting parameters in addition to the effective

demagnetising field, Hd. By doing these, M decreased by 10% while d decreased by

15%. From these results, one may speculate that the peculiarity of this stripe may be due

to localised inhomogeneity during fabrication, resulting in reduced saturation

magnetisation, thickness, or both, for this particular stripe.

Interestingly, the simulation data (figure 4.5.1c) fits well to the modified continuous

film dispersion (equation 4.5.1b). This seems to suggest that at least theoretically

(disregarding the peculiarity seen for this particular stripe in experiment), a stripe of

aspect ratio as high as

055.0

2

110

m

nm

can still be modelled by the simple modified

continuous film dispersion by assuming an uniform up-field shift in the dispersion from

an effective static demagnetising field, in the range of wave numbers tested in this

experiment.

68

4.5.2 Static demagnetising field simulations

In order to evaluate the validity of introducing an effective demagnetising field into the

dispersion model, the demagnetising field profiles of the stripes studied were

numerically simulated using LLG Micromagnetics Simulator software 105

(refer to

Appendix C). To simplify computation, the stripes were assumed to be infinitely long,

thus reducing the problem into a two dimensional one.

Figure 4.5.2a: Simulated demagnetising field profiles in infinitely long Permalloy

stripes.

69

The simulated demagnetising field profiles of two extreme cases (smallest and largest

aspect ratios) are shown in figure 4.5.2a. The stripe with the lowest aspect ratio

m

nm

100

55is expected to have the least demagnetising field, and one clearly sees that

this is indeed the case in figure 4.5.2a-b. Demagnetising fields are only significant near

the edges of the stripe, just roughly 2% of the total width. The small demagnetising field

across the bulk of the stripe is practically uniform. For the other extreme case, the stripe

with the highest aspect ratio

m

nm

2

110is expected to have highly non-uniform

demagnetising fields across the width, and indeed this is the case in figure 4.5.2a-a.

Note that in this narrow stripe, significant demagnetised edge regions are present for

low applied fields.

70

Figure 4.5.2b: Comparison between experiment and simulation effective demagnetising

fields.

71

The simulated effective demagnetising field for each strip is approximated by taking the

mean value across the strip width (figure 4.5.2a). The experimentally extracted effective

demagnetising fields are compared against these simulated values in figure 4.5.2b. It is

observed that the simulation tends to overestimate the experimental effective

demagnetising field. Possible explanations for these discrepancies are as follows:

Firstly, taking the arithmetic mean across the demagnetising field profile as an effective

value may not be the appropriate statistical approach. Arithmetic mean tends to be

disproportionately weighted towards large values, and demagnetising fields can be very

large at stripe edge regions while only occupying relatively small volumes 79

.

Secondly, considering the demagnetising field profile alone is insufficient. One also

needs to account for the non-uniformity of the static magnetisation and spin wave mode

profile for a more accurate analysis. The effective demagnetisation factor for a

particular spin wave mode is the proportional to the overlap integral between the mode

profile and the demagnetisation field profile 84

. In addition, dynamic effects due to

confinement affect the dispersion slopes 79, 86

; this is not accounted for by the static

demagnetising field fitting parameter in equation 4.5.1b. Hence, merely taking the mean

value of the simulated demagnetisation field profile alone is insufficient to accurately

model the actual effective demagnetisation field for the particular spin wave mode.

72

4.5.3 Group velocity

The group velocity, Vg, can be calculated from the dispersion data using the

relationship:

k

H

H

f

kVg

2 → Equation 4.5.3a

The k

H

term is simply the slope of the field-resolved dispersion, and the

H

f

term is

obtained by differentiating equation 4.5.1b:

f

MHH

H

f d )2(2

→ Equation 4.5.3b

From equation 4.5.3a, the experimentally calculated group velocities are plotted in

figure 4.5.3a as function of wave number and tabulated in table 4.5.3a for k0 = 10038

rad/cm.

73

Figure 4.5.3a: MSSM group velocities of Permalloy stripes.

74

From figure 4.5.3a, one sees a general trend that the group velocity increases with film

thickness. This is what one expects since the slope k

H

in the dispersion (figure 4.5.1a)

increased with film thickness. Theoretically, this is also what one expects by

differentiating equation 4.5.1b to obtaink

, where the group velocity is proportional to

film thickness:

kd

g ef

dM

kV 2

2238

→ Equation 4.5.3c

For each particular stripe, it is immediately obvious that the group velocity decreases

for increasing wave number. This is consistent with the formulation in equations 4.5.3b

and 4.5.3c; one expects a negative slope for a plot of group velocity versus wave

number.

For particular thicknesses, there seems to be no correlation between stripe width and

group velocity. For the 55 nm and 80 nm thick stripes, the vertical spread range in the

group velocities is roughly 1.5 μm/ns. This is the same for the 110 nm thick stripes, if

the peculiar

m

nm

2

110stripe is excluded. If we attribute the scatter to the accuracy

limitations of the experiment, then we may conclude that for the aspect ratios and wave

number range investigated here, the group velocity is width-invariant for a particular

thickness.

Thickness (nm) Group velocity (μm/ns)

55 5.5 – 7

80 8 – 9.5

110 13 – 14

Table 4.5.3a: Group velocities at 10 GHz and k0 = 10038 rad/cm.

75

4.5.4 Attenuation and relaxation

We now turn attention to the propagation attenuation and relaxation characteristics of

MSSM in the stripes studied. The total signal transmitted, , from the excitation

antenna to the detection antenna is 100

:

)(),()(),( de TkPTk → Equation 4.5.4a

eT and dT are antenna excitation and detection efficiencies, respectively. P is the spin

wave propagation loss. For identical antennae, *ed TT ; the detection efficiency is

simply the complex conjugate of the excitation efficiency. Then, the antennae losses can

be lumped into a pre-factor for the spin wave propagation loss term. In this experiment,

we assume that differences in antennae characteristics are negligible; i.e., the antennae

are similar to each other ( *ed TT assumption holds). Then, the relative drop in signal

transmission between antennae upon propagation will be due to spin wave propagation

losses. Spin wave propagation loss was modelled as an exponential decay (equation

4.5.4b) 34, 71

, where Ld is the attenuation length, defined as the propagation distance

when the signal has dropped to 1/e from its initial value.

dL

x

ePP

0 → Equation 4.5.4b

From equation 4.5.4a, and taking logarithms, equation 4.5.4b becomes:

dL

xPT lnln2ln → Equation 4.5.4c

Thus, a logarithmic plot of the transmitted signal amplitude on the vertical scale versus

a linear plot of the propagation distance on the horizontal scale would enable extraction

of the attenuation length from the linear slope of the plot. Note that the initial spin wave

signal amplitude P and antenna efficiency T both contribute to the vertical intercept.

Real-world variations in the antenna characteristics ΔT would manifest as vertical

spread in the plot. Thus, the accuracy of the linear fit depends on the difference between

ΔT and P. This means that the thicker the film, the larger the initial spin wave signal

amplitude P, and thus the more reliable the fit would be.

76

In order to obtain absolute quantitative values, we measured the transmission scattering

parameter (S21) between the two antennae with field-resolved VNA. Measurements

were performed on wide stripes for each of the three thicknesses in order to obtain the

high signal-to-noise (decrease inaccuracies from ΔT antenna variation contributions).

The amplitude ΔS21 is defined as the range between the central extrema of the spin

wave packet, about the dominant wave number k0 = 10038 rad/cm (figure 4.5.4a).

Figure 4.5.4a: Example VNA raw trace, showing definition of amplitude ΔS21.

The logarithmic amplitudes of the spin wave signals at various antennae gaps are

plotted in figure 4.5.4b. Note the linear trend in the logarithmic plot of the data (figure

4.5.4b), verifying the exponential decay assumption made earlier. It is noteworthy that

the antenna near-field effect (see section 4.7) results reduced effective propagation

distance compared to the physical separation distance between the excitation and

detection antennae. This shifts all the data points horizontally by a constant amount.

However, this would not affect the slope of the plot. The attenuation lengths extracted

from the fittings were tabulated in table 4.5.4a.

77

Figure 4.5.4b: ΔS21 amplitude at various antennae gaps at 10 GHz and k0 = 10038

rad/cm. Solid lines are fits to extract the attenuation lengths.

Comparison can be made with Sekiguichi et al.’s similar work involving spin wave

propagation through a Permalloy stripe with aspect ratio

m

nm

120

35 over distances 5 –

50 μm 34

. In that work, the attenuation length was determined to be 15 μm, which is

similar to the values for the stripes studied here (table 4.5.4a).

The relaxation time τr of MSSM can be calculated from the group velocity, Vg, and

attenuation length, Ld, similar to 95, 106

through the simple relationship:

g

d

MSSMrV

L

)( → Equation 4.5.4b

Similar to attenuation length, relaxation time is defined as the time it takes for a signal

to decay to 1/e from its initial value 71

. The calculated relaxation times are tabulated in

table 4.5.4a. Comparison can be made with Bailleul et al.’s work where similar MSSM

relaxation times were obtained: 2 ns 70

and 1.6 ns 71

. Using equation 2 from reference 107

together with equation 4.5.4b and 4.5.3a, the Gilbert damping coefficient, α, can be

formulated in terms of the MSSM relaxation time, where Hi is the internal field:

)()2(2

1

MSSMri MH

→ Equation 4.5.4c

78

The calculated attenuation lengths, relaxation times, and Gilbert damping coefficients

were tabulated in table 4.5.4a. Note the atypical losses in the 80 nm thick stripes.

Stripe

thickness (nm)

Group

velocity

(μm/ns)

Attenuation

length (μm)

Relaxation

time (ns)

Gilbert

damping

coefficient

(10-3

)

55 6.46 ± 0.04 7.83 ± 0.04 1.21 ± 0.01 7.66 ± 0.06

80 8.17 ± 0.08 7.37 ± 0.04 0.90 ± 0.01 10.8 ± 0.1

110 13.14 ± 0.09 14.7 ± 0.5 1.12 ± 0.04 8.2 ± 0.3

Table 4.5.4a: Stripe MSSM propagation and attenuation parameters at k0 = 10038

rad/cm and 10 GHz.

The MSSM attenuation parameters can also be compared with those obtained from their

respective reference film FMR line widths. The FMR relaxation time in terms of the

FMR line width, FWHMH , reformulated from Stancil’s equation 17c 95

, is:

)2(2)(MHH

f

iFWHM

FMRr

→ Equation 4.5.4d

The Gilbert damping coefficient is given by Stancil’s equation 5 95

:

f

HFWHM

2

→ Equation 4.5.4e

By plotting the FMR line width for various resonance frequencies, the Gilbert damping

coefficient can be easily extracted from the slope using equation 4.5.4e (figure 4.5.4c).

Non-zero intercepts at zero frequency (≈ 10 Oe) were found in the line width plots. This

is attributed to inhomogeneous line width broadening 24, 108, 109

, and is sensitive to the

thermal history of the material 110

. However, most importantly, the slope of the line

width plot is insensitive to sample history, and is a reliable measure of intrinsic damping

109, 110.

79

Figure 4.5.4c: FMR line widths of reference films.

Film thickness

(nm)

Saturation

magnetisation

(emu/cm3)

Relaxation time

at 10 GHz (ns)

Gilbert

damping

coefficient

(10-3

)

55 837 ± 3 1.15 6.6 ± 0.2

80 799 ± 3 1.03 8.2 ± 0.3

110 886 ± 3 1.03 6.8 ± 0.3

Table 4.5.4b: Reference film properties.

The FMR relaxation times and Gilbert damping coefficients are tabulated in table

4.5.4b. The values for the 110 nm and 55 nm films are typical for Permalloy 24, 32, 70, 71

.

Note the ~25% larger than typical intrinsic damping for the 80 nm thick reference film

compared to the other two. This might be due to fabrication defect resulting in a low

quality magnetically “lossy” film. This is noticeably evident in the plot in figure 4.5.4c.

This might explain the reduced attenuation length and relaxation time of MSSM

propagation in the 80 nm stripes (table 4.5.4a).

Note also that the Gilbert damping coefficients calculated from the MSSM propagation

data (table 4.5.4a) were larger (~ 25%) than those extracted from the reference film

80

FMR line widths (table 4.5.4b). Three possible reasons for this discrepancy are as

follows:

First, one expects confinement to contribute additional edge losses (not present in a

laterally infinite continuous film). As covered in section 4.5.2, demagnetised regions

(very low or even negative internal fields) are present at the stripe edges due to large

demagnetising fields at the edges of transversely magnetised narrow stripes 70, 79, 96

. In

fact, this edge effect leads to channelling of dipolar modes at the centre of such narrow

stripes 81

. MSSM propagation into these demagnetised edge regions would be trapped in

the potential wells 79, 90

, contributing to loss. In addition, edge defects from fabrication

imperfections will contribute to scattering of spin waves 19, 20

. All these contribute to

propagation attenuation on top of the intrinsic material damping.

The second possible explanation is that the damping coefficient extracted from FMR

data was at kFMR = 0 rad/cm while the value obtained from MSSM data was at a

different wave number, k0 = 10038 rad/cm. One notes that the inhomogeneous line

width broadening may be slightly different between kFMR = 0 rad/cm and k0 = 10038

rad/cm, thus the damping coefficient may have wave number dependence 111

.

The third explanation is a procedural one. The damping coefficients calculated for

particular wide stripes were obtained indirectly through the group velocities and

attenuation lengths. Both of these (group velocity and attenuation length) themselves

were obtained indirectly; from the dispersion and amplitudes, respectively. Since the

calculation of the damping coefficients from the MSSM data was done through two

levels of indirect methods, inaccuracies would be compounded, and one may question

the reliability of the results. On the other hand, determination of the damping coefficient

from the continuous film FMR line widths is a direct and thus more reliable method. In

fact, this is a standard method to determine the damping coefficient from FMR

experiments 24, 32, 109, 110, 112

.

81

4.5.5 Non-reciprocity

One propagation property of MSSM is its non-reciprocity. For a given tangential

magnetisation orientation, counter-propagating MSSM waves are localised on opposing

surfaces of a ferromagnetic slab 67

. The surface on which the wave propagates is

determined from the vector cross product 113

:

k x H = s n

Where n is the normal vector pointing out of the plane where k propagates along, and s

is a proportional constant. This is illustrated in figure 4.5.5a.

Figure 4.5.5a: MSSM wave propagation on a ferromagnetic slab, showing non-

reciprocity.

For thick insulating ferrites like YIG, the mechanism of non-reciprocity may be

explained by the concentration of surface waves on opposing sides of the film 67, 102, 113

.

However, this mechanism for non-reciprocity is invalid when the wavelength of MSSM

is much larger than the film thickness ( 1kd ) 81

. Note that this is indeed the case for

our stripe thicknesses and the range of available wave numbers ( 1.0kd ).

The antenna-induced mechanism of non-reciprocity for the case of 1kd in metallic

thin films with large magnetic moments is explained by Demidov et al 81

. Consider the

excitation field components of the antenna, hx (in-plane) and hz (out-of-plane),

according to the frame of reference in figure 4.3a. Both components provide the driving

82

torque for magnetisation precession. The torque contribution by hx is always in-phase

with the emitted spin wave, but the torque contribution by hz have opposite phase

relations at the opposite sides of the antenna 34, 74, 81

. This asymmetric excitation field

results in non-reciprocal emission of counter-propagating waves from the antenna.

However, non-reciprocity is weaker for materials with large saturation magnetisation

(like the 3d metallic ferromagnets), owing to large in-plane ellipticity of magnetisation

precession (the asymmetric torque contribution of hz to magnetisation precession is

weaker) 81

.

Experimentally wise, non-reciprocity can be probed by either reversing the antennae

roles (reversing k) or reversing the field. Both were shown to be equivalent by

Sekiguchi et al. 34

; this was verified in our experiments in a wide range of aspect ratios.

However, it is much easier to reverse the applied field than to swap the roles of the

excitation/detection antennae, so we study non-reciprocity by simply sweeping the field

from negative to positive. An example trace raw trace is shown in figure 4.5.5b. The

spin wave amplitude difference upon field reversal is immediately noticeable.

Figure 4.5.5b: Example of MSSM non-reciprocity upon field reversal. Stripe thickness:

110 nm. Stripe width: 50 μm. Propagation distance: 30 μm.

Similar to Demidov et al. 81

and Sekiguchi et al. 34

, we define the non-reciprocity

parameter, η, as the amplitude ratio:

A

A → Equation 4.5.5a

83

A+ is the spin wave amplitude in the favoured propagation direction (larger amplitude)

while A– is the spin wave amplitude in the unfavoured propagation direction (smaller

amplitude). Amplitudes were taken at the extrema where the dominant wave number k0

= 10038 rad/cm occur (figure 4.5.5b).

Figure 4.5.5c: Non-reciprocity parameters at 10 GHz and k0 = 10038 rad/cm.

84

The non-reciprocity parameters were calculated for all the stripes studied in this work

and plotted in figure 4.5.5c. There seems to be a slight trend of non-reciprocity

strengthening as stripes become narrower for the 110 nm series. However, over all, the

non-reciprocity is largely invariant (η ≈ 0.25 ± 0.05) over the range of stripe

thicknesses, widths and propagation distances investigated. The results strongly suggest

an antenna-induced non-reciprocity origin, since the same antenna geometry was used

on all the stripes studied in this work.

Using Demidov et al.’s formula 81

, the theoretical non-reciprocity was calculated to be η

≈ 0.5 for a single antenna. For a PSWS experiment utilising two identical antennae as in

this experiment, the non-reciprocity parameter is twice that for a single antenna. If one

extends Demidov et al.’s formula for our system, we obtain η ≈ 0.25. Thus, the non-

reciprocity of MSSM in our PSWS experiment is as predicted by theory.

4.6 Higher order width modes in confined stripe geometry

Extra modes in addition to the dominant MSSM were observed in the spin wave spectra

for the stripe with the highest aspect ratio studied in this work

m

nm

2

110. These were

identified as higher order width modes appearing due to confinement in such a narrow

stripe. This sub-chapter deals with the identification and characterisation of these

modes.

In a simple qualitative model for confined stripe geometry, the finite width leads to

quantization of backward volume-like spin wave modes across the width in the form 50,

87:

w

nky

→ Equation 4.6.1a

Where n is the mode number and w is the effective width of the stripe. The eigen-modes

take on sinusoidal functions analogous to the case of spin wave resonance modes across

the thickness 38, 50, 79, 84

. The total wave number, kt, for spin wave propagation in a

confined stripe is thus 84

:

85

222

yxt kkk → Equation 4.6.1b

Where kx is propagation in the longitudinal direction (along the long axis) and ky is

propagation in the transverse direction (along the short axis), following the Cartesian

axes of figure 4.3a. One sees from equations 4.6.1a and 4.6.1b that the relative

contributions of the two orthogonal kx and ky modes depends on the stripe’s lateral

dimensions. From this, we define the wave propagation angle, θ:

x

y

k

karctan → Equation 4.6.1c

For θ = 0, the wave is purely longitudinal, and for θ = 90°, the wave is purely

transverse. Thus, from equation 4.6.1c, one sees that the angle θ increases with width

mode number. Thus, for increasing mode number, the total wavevector becomes more

canted away from the longitudinal direction towards the transverse direction.

In our case, longitudinal quantization is not possible since the length of the stripes (200

μm) is much longer than the spin wave attenuation length (table 4.5.4a). The closest

distance between the antenna and the end of the stripe is 50 μm. Due to attenuation,

reflections along the longitudinal direction to form quantized longitudinal modes are not

possible. Thus, for our stripes, wave propagation in the longitudinal direction is

effectively similar to MSSM in an infinite continuous film 67

, without quantization

effects (see preceding section 4.5). Contrast can be made with Mathieu et al 77

and

Roussigne et al 78

’s work in the Damon-Eshbach geometry where their stripes were

magnetised along the long axis and quantization of MSSM were observed across the

stripe width. In their work, dispersion-less quantized MSSM modes were observed for

small wave numbers. Note that in contrast to their work, our stripes were magnetised

transversely and propagation along the long axis was considered (figure 4.3a).

On the other hand, excitation of quantized transverse modes along the stripe width is

plausible since the narrowest stripe width in our stripe series (2 μm) is of spin wave

attenuation length order. In such confinement, waves can bounce back and forth from

the side walls without being significantly attenuated, thus forming standing waves

across the width. Indeed, we observed multiple higher order width modes in the spin

wave spectra of our particular stripe with the highest aspect ratio

m

nm

2

110(see section

86

4.6.1). While this is not the first time these higher order width modes were detected

using an inductive method (observed in Bailleul et al.’s pioneering work 70

), our work

was done so with greatly improved sensitivity to enable further detailed characterisation

of their dispersions and group velocities (see section 4.6.2). Furthermore, we display the

higher order mode signals obtained in reflection and compare them with ones obtained

in transmission. To date, the detailed dispersion characterisation of these higher order

width modes have only been achieved on stripe arrays using Brillouin light scattering 77-

79, 86. In the following sections, the analysis of these width modes was performed similar

to section 4.5.

4.6.1 Mode identification

The raw data traces for the stripe exhibiting multiple higher order width modes are

shown in figure 4.6.1a. In the reflection data, a remarkable 6 higher order width modes

on top of the fundamental MSSM were observed in the spectra. Numerical simulation

was used to generate theoretical eigen-fields and mode profiles in order to identify these

modes. As seen in figure 4.6.1a, the simulated eigen-field positions match up well with

the experimental mode positions. Mode numbers were assigned accordingly upon

evaluation of the mode profiles (figure 4.6.1b), where n is the number of anti-nodes in

the mode profile. The excitation field has a symmetric odd mode profile across the

stripe width. This implies that only modes with odd symmetry can be excited by such a

field 36, 38, 70

, since the mean values of odd mode profiles modes are non-vanishing

(figure 4.6.1b). For the modes n ≥ 3, one sees a pronounced decrease in the amplitude of

the maxima of the standing wave towards the stripe edges. This is due to the non-

uniformity of the magnetisation ground state (see the upper panel in figure 4.5.2a).

The simulated mode profiles are similar to figure 6 in Bayer et al. 79

and figure 4 in

Kostylev et al.’s work 86

, with the exception that our experiment was done field-

resolved. (Refer to Appendix C for the simulation procedure and list of key parameters.)

Note the similarity in the aspect ratio of this particular stripe

055.0

2

110

m

nm

with the

87

stripe array studied by Kostylev et al.

05.0

600

30

nm

nm, where multiple width modes

were detected using Brillouin light scattering 86

.

Figure 4.6.1a: Field-trace at 10 GHz for the

m

nm

2

110aspect ratio stripe, showing

multiple high order width modes. Solid curve: signal received at detection antenna over

12 μm propagation distance. Dashed curve: signal reflected back from excitation

antenna. Vertical lines: simulated mode field positions, where n is the mode number.

88

Figure 4.6.1b: Simulated mode profiles.

89

Figure 4.6.1c: Mode amplitudes at the dominant wave number k0 = 10038 rad/cm

normalised to the fundamental MSSM (n = 1). For the simulation, the mean value of the

mode profile was designated the mode amplitude. For the experimental data, the mode

amplitudes follow the definition in figure 4.5.5b.

Theoretically, the excitation efficiency is proportional to the overlap integral of the

mode profile with the excitation field profile 84

. Assuming a uniform excitation field,

the overlap integral reduces to the mean value of the mode profile. One expects the

mode amplitude to decrease for increasing mode number, since the mean value of

dynamic magnetisation decreases with increasing number of nodes in the mode profile.

Indeed, this was observed in the simulated and experimental reflection data. (figure

4.6.1c). Interestingly, in the experimental reflection data, the mode amplitudes seem to

drop linearly with increasing mode number (for n ≥ 3). In the simulation result, the

reduction in mode amplitude for increasing mode number has a different functional

dependence. At this point, we emphasize that it is often difficult to simulate high order

mode amplitudes to quantitatively match real-world data. It is sufficient for the

simulated and experimental relative mode amplitudes to qualitatively follow a rough

trend for the purpose of mode identification.

Comparing between the experimental reflection and transmission data, one observes

that the transmission data contained progressively less modes for increasing antennae

gap. In the transmission data (figure 4.6.1c), the amplitude of high order modes

decrease even more rapidly than in the reflection data with increasing mode number.

We postulate that this is due to increase in attenuation for increasing mode number upon

90

wave propagation (see section 4.6.3). In fact, for antennae gap of 12 μm, only the modes

up to n = 9 had sufficient transmitted signal to enable further dispersion

characterisation. Even though modes n = 11 and n = 13 were clearly observed in the

reflection data, these modes were attenuated down to below noise levels after

propagating 12 μm in the transmission data. Table 4.6.1a summarises the appearance of

higher order width modes as a function of antennae gap.

Mode

number

(n)

Reflection Transmission separation gap (μm)

12 15 20 30 60

1 ✓ ✓ ✓ ✓ ✓ ✓

3 ✓ ✓ ✓ ✓ ✓

5 ✓ ✓ ✓ ✓

7 ✓ ✓

9 ✓ ✓

11 ✓

13 ✓

Table 4.6.1a: Observation of width modes in experimental data.

4.6.2 Dispersion and group velocity

After having identified the width modes with the aid of numerical simulations, we now

consider their dispersion characteristics. Guslienko et al 84

formulated a non-analytic

dispersion expression for quantized dipole modes of an arbitrary rectangular slab

magnetised tangentially. In this work however, further characterisation of propagation

characteristics were done experimentally. Following the same phase interference

technique of extracting dispersion in section 4.4.4, the dispersions of the width modes

91

were extracted from the 12 μm separation gap transmission data in figure 4.6.1a. Note

that due to significant antennae near-field effects at such a small antennae gap (see

section 4.7), an effective propagation distance of 9.5 μm was used to calculate the phase

difference between the extrema in the raw data. The dispersions are plotted in figure

4.6.2a.

Figure 4.6.2a: Width mode dispersions at 10 GHz.

One immediately notices that the dispersion slope of the higher order modes (n ≥ 3) are

much smaller than the fundamental MSSM (n = 1). Since the slope of the dispersion

curve is proportional to group velocity, this indicates that the group velocities of the

higher order modes (n ≥ 3) are significantly lower than that of the fundamental MSSM.

Unlike the fundamental MSSM, there is no analytic expression to calculate group

velocity for the higher order width modes. Therefore, we will now utilise a Taylor

approximation method to determine the group velocities of these modes about 10 GHz

and the dominant wave number k0 = 10038 rad/cm.

92

The dispersions in figure 4.6.2a were fitted with either a linear or quadratic function

(whichever yielded the least residuals). These are shown as solid lines in figure 4.6.2a.

From these fits, the local gradientk

H

were calculated for k0 = 10038 rad/cm.

The differential conversion factor between frequency and field resolved measurements,

H

f

, can be obtained by performing multiple frequency measurements for k0 = 10038

rad/cm (figure 4.6.2b). Note the excellent linear fits (solid lines), from which

H

f

values were extracted from the slopes for each respective mode.

Figure 4.6.2b: Frequency versus field plots for the width modes for k0 = 10038 rad/cm.

Similar to section 4.5, knowing both k

H

and

H

f

, the group velocity can be calculated

using equation 4.5.3a:

k

H

H

f

kVg

2

Thus, to first degree Taylor approximation, one can calculate the group velocity for the

dominant wave number k0 = 10038 rad/cm and 10 GHz frequency. The results are

plotted in figure 4.6.2c and also tabulated in table 4.6.3a.

93

Figure 4.6.2c: Group velocity of higher order width modes at 10 GHz and k0 = 10038

rad/cm.

As postulated earlier, the group velocities of higher order width modes (n ≥ 3) were

found to be nearly an order of magnitude lower than the fundamental MSSM. (Hence,

for clarity of the higher order mode data, the fundamental MSSM data point was

deliberately left out of figure 4.6.2c.) Furthermore, the group velocity decreased for

increasing mode number (figure 4.6.2c). This results in increased spatial attenuation for

increasing mode number, as evident in the mode amplitudes in the transmission data

(figure 4.6.1a).

As for the reflection data in figure 4.6.1a, the decrease in group velocity for increasing

mode number may explain the discrepancy between the simulated relative mode

amplitudes in figure 4.6.1c. From equation 4 in Dmitriev 100

, one may expect the

efficiency of mode excitation to grow with a decrease in group velocity; this may partly

compensate the decrease in the overlap integral for increasing mode number.

The group velocity of the fundamental MSSM for this stripe obtained using the Taylor

approximation method (12 μm/ns) is consistent with the values obtained in section

4.5.3, within the accuracy of the experiment. The fundamental MSSM group velocity is

directly proportional to the film thickness, and one may assume that this proportionality

also applies to higher order width modes. This may explain why these higher order

width modes were not observed in the thinner stripes studied in this work (very low

94

group velocity and excessive attenuation), and also why their observation using

inductive spectroscopy methods are lacking in the literature.

4.6.3 Attenuation and relaxation

We now evaluate the attenuation characteristics of the width modes. Due to excessive

loss for large mode numbers, only the first 3 modes (n = 1, 3, 5) contain the minimum

of 3 data points for meaningful extraction of attenuation length (figure 4.6.3a). As

mentioned before in section 4.5.4, the scattering of antenna efficiencies from sample to

sample was negligible and thus, 3 points – though not ideal – were sufficient to

determine the attenuation length.

Figure 4.6.3a: Mode amplitude for various antennae gaps. Solid lines are fits to extract

the attenuation lengths.

Mode amplitudes following the definition in section 4.5.4 for k0 = 10038 rad/cm were

extracted from the raw traces. The logarithms of the mode amplitudes were plotted as

function of antennae separation gaps in figure 4.6.3a. The attenuation lengths were

extracted from the slopes following section 4.5.4 and tabulated in table 4.6.3a. As noted

before in section 4.5.4, the antenna near-field effect (see section 4.7) would shift all the

data points horizontally by a constant amount. However, this would not affect the slope

of the plot in figure 4.6.3a.

95

Mode

number H

f

(10-3

GHz/Oe)

k

H

(10-3

Oe

cm)

Group

velocity

(μm/ns)

Attenuation

length (μm)

Relaxation

time (ns)

1 5.25 ± 0.06 37 ± 1 12.2 ± 0.5 14 ± 1 1.2 ± 0.1

3 4.49 ± 0.06 7.6 ± 0.5 2.1 ± 0.2 4.9 ± 0.7 2.3 ± 0.4

5 4.11 ± 0.07 5.0 ± 0.1 1.29 ± 0.05 3.2 ± 0.2 2.5 ± 0.2

7 3.83 ± 0.04 4.1 ± 0.2 0.99 ± 0.06 n/a n/a

9 3.64 ± 0.04 3.3 ± 0.3 0.75 ± 0.08 n/a n/a

Table 4.6.3a: Width mode propagation characteristics about 10 GHz and k0 = 10038

rad/cm.

The attenuation length of the fundamental MSSM in this 2 μm narrow stripe (14 ± 1

μm) was found to be consistent with that of the 50 μm wide stripe of equal thickness

(14.7 ± 0.5 μm) determined in section 4.5.4. The attenuation lengths of the higher order

width modes (3 – 5 μm) were found to be comparable to half the antenna’s physical

width (3.75 μm). Thus, not only do these width modes have very low group velocities

(compared to MSSM); they are very short range, localised in the vicinity of the

excitation source. These come as no surprise, since these higher order width modes are

backward volume wave-like across the stripe width, with weak dispersions and low

group velocities 70, 71

. This explains why the higher order width modes were detected

only in reflection and for closely spaced antennae. In addition, a trend is observed where

the attenuation length decreased for increasing mode number.

Using equation 4.5.4b, the relaxation times were calculated for the first 3 width modes

from their respective group velocities and attenuation lengths. These were tabulated in

table 4.6.3a. The relaxation time for the fundamental MSSM was consistent with the

value calculated in section 4.5.4 for the wide 50 μm stripe. Note the difference in the

extracted relaxation times between the fundamental and higher order modes.

96

4.6.4 Non-reciprocity

Similar to section 4.5.5, the non-reciprocity parameters of the width modes were

evaluated using equation 4.5.5a and plotted in figure 4.6.4a. The n = 3 width mode was

found to have the same non-reciprocity behaviour as that of the fundamental MSSM (η

≈ 0.2). Note that only the first two modes (n = 1, 3) had sufficient transmitted signal

amplitude to evaluate non-reciprocity with reasonable reliability.

Due to tiny signals for the larger mode numbers (barely above noise level), quantitative

evaluation of the non-reciprocity parameter should only be taken with a grain of salt.

However, one can still observe a trend in the data. For larger mode numbers (n ≥ 5), the

non-reciprocity seems to get weaker (η approaching unity) for increasing mode number.

This result is actually consistent with the rotation of the wave vector towards the

transverse direction for increasing mode number; the wave takes on more backward-

volume-like characteristic (a purely backward volume wave is completely reciprocal).

Figure 4.6.4a: Non-reciprocity parameter of width modes at 10 GHz and k0 = 10038

rad/cm.

97

4.7 Antenna near-field effect

In this section, we consider the antenna’s near-field effect on spin wave propagation.

Spin waves are slow electromagnetic waves with a dominating magnetic component.

Therefore, the characteristics of electromagnetic wave radiation and reception by usual

(e.g. radio or TV antennae) should be applicable to the coplanar line antennae. In

particular, it is known that the field of an antenna separates into two regions (with

continuous transition region in between): near-field and far-field 99, 114

. These are

regions of time-varying electromagnetic field around the source for the field.

Far enough from the source, the wave is a purely propagating wave which accumulates

phase on its path between the radiation source and the receiving antenna. In the region

very close to an antenna, the wave’s ac field is dominated by field components

produced directly by currents in the antenna. This field is called the “near-field". An

important consequence which follows from this origin of the field is that the phase of

the field is the same across the whole width of the near-field region (i.e. there is no

effect of retardation in this region). At distances far from the antenna, the propagating

wave becomes dominated by the field components produced by its own ac field. For

instance, in our case of dipole dominated spin waves, the dynamic magnetisation is

produced by the dynamic dipole field and vice versa. Thus the wave’s ac field is

effectively no longer affected by the currents at the source. Due to this origin for the

field, the phase of the field changes (accumulates) with the distance from the source, i.e.

retardation effect is present. This more distant part of the electromagnetic field is the

"radiative field” or "far-field".

4.7.1 Characteristic equations

Analogous to radio waves propagating from a source antenna through free space, we

postulate that a purely propagating wave exist at some characteristic distance from the

antenna. At distances shorter than this characteristic length, we postulate that the phase

of magnetisation precession is dominated by the phase of the current in the antenna and

is the same across the whole near-field region. What this means in terms of propagating

spin wave experiment is that the effective propagation distance, effx , is shorter than the

98

physical separation gap between antennae, x ; the difference between the two gives

twice the antenna characteristic near-field length, nearx :

effnear xxx 2 →Equation 4.7.1a

Note the factor 2 in equation 4.7a originates from accounting for the near-fields of both

the excitation and detection antennae, and that we assume both antennae to be identical

( nearx is defined for a single antenna). In PSWS utilising metallic ferromagnetic films,

the distance between antennae is always comparable to the size of the antennae

themselves. This is due to lossy metallic ferromagnetic films with attenuation lengths of

the order of microns. This means that for closely separated antennae, nearx may no

longer be negligible. In such cases, one needs to determine the proper effective

propagation distance, which may differ significantly from the nominal propagation

distance (between the symmetry axes of antennae). We now derive an expression from

which the antenna characteristic near-field length may be determined from experiment.

From equation 4.5.3a, we have the group velocity:

k

H

H

f

kVg

2 → Equation 4.5.3a

The slope of the field-resolved pseudo-dispersion relation is thus:

H

f

V

k

H g

2

→Equation 4.7.1b

From the dispersion plots in figure 4.5.1a, we see that the data points are nearly linear

over the range of wave numbers available. For a particular stripe, we assume – to first

degree Taylor approximation – that the slope k

H

is constant at a particular frequency

and in the close proximity of the central dominant wave number k0 = 10038 rad/cm. The

phase difference between each maximum in the raw trace is given by equation 4.4.4a:

effxk

2 → Equation 4.4.4a

Substituting the expression for k in equation 4.4.4a into k in equation 4.7.1b, and

rewriting H into H gives:

99

Hx

H

f

Veff

g

→ Equation 4.7.1c

Here, H is the field step corresponding to 2π phase accumulation in the raw trace (see

section 4.4.4). To first degree approximation, the left hand side of equation 4.7.1c is a

constant for a particular stripe. The consequence of equation 4.7c is that H (the field

step over which a phase accumulation of 2π occurs) is inversely proportional to the

effective propagation distance. This is clearly seen in figure 4.4.4b. Essentially,

equation 4.7.1c is a reformulation of equation 4.4.4a in terms of H instead of k .

Let

H

f

VV

g

' , then equation 4.7.1c becomes:

'VHxeff → Equation 4.7.1d

By substituting equation 4.7.1a into equation 4.7.1d for effx , we have:

')2( VHxx near

HxVHx near 2' → Equation 4.7.1e

From equation 4.7.1e, one sees that by plotting Hx (x = antennae gap) versus H for

various antennae gap separations, the slope of the plot gives the antenna characteristic

near-field length.

4.7.2 Antenna characteristic near-field length

We now establish some criteria before using equation 4.7.1e to analyse our data. First,

for a stripe aspect ratio, there must be at least 3 different separation gaps available; this

is the bare minimum for a linear fit. Second, the spread in H should be minimal and

statistically random; there is no significant slope and curvature in the field-resolved

pseudo-dispersion. This allows us to average H data for each raw trace into a single

representative value. This value is multiplied by the respective x ( Hx ) is plotted

against itself for all available stripes which had data meeting the established criteria to

produce plots in figure 4.7.2a.

100

Figure 4.7.2a: Plots of antennae gap times delta H versus delta H, at 10 GHz around the

vicinity of the central dominant wave number k0 = 10038 rad/cm.

101

If there is no antenna near-field effect, then the antennae gaps would equal the

propagation distance, and the plots would be horizontal. However, we see in figure

4.7.2a, one notes that sloping is clearly present. One also notes that for all the cases

when we have at least 3 points, the data are well fitted with straight lines and with

positive slopes. This implies that our approach is physically sound. The antennae

characteristic near-field lengths, nearx can simply be obtained from the slopes of the

plots. These values were tabulated in table 4.7.2a. Note from equation 4.7.1e that the

horizontal intercept is proportional to the group velocity. Thus, the proposed method

here may also be used to extract group velocities in addition to the characteristic

antenna near-field lengths. However, the extraction of group velocities had already been

done in section 4.5.3 and is beyond the focus of this section.

Thickness

(nm)

Width

(μm)

Antenna characteristic near-field length, nearx (μm)

55 5 0.7 ± 0.1

55 100 1.0 ± 0.2

80 2 0.29 ± 0.04

80 20 1.4 ± 0.1

110 2 1.7 ± 0.1

110 20 2.25 ± 0.05

110 100 2.3 ± 0.1

Table 4.7.2a: Extracted antenna characteristic near-field lengths.

One notes a spread of 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

Bibliography

1. J. H. E. Griffiths, Nature 158 (4019), 670-671 (1946). 2. M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven and B. Hillebrands, Applied Physics Letters 87 (15), 3501 (2005). 3. T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps and M. P. Kostylev, Applied Physics letters 92 (2), 2505 (2008). 4. K.-S. Lee and S.-K. Kim, Journal of Applied Physics 104 (5), 3909 (2008). 5. A. Khitun, M. Bao and K. L. Wang, Journal of Physics D: Applied Physics 43 (26), 264005 (2010). 6. The Physics of Ultrahigh-Density Magnetic Recording. (Springer, Berlin, 2001). 7. J. M. Slaughter, R. W. Dave, M. DeHerrera, M. Durlam, B. N. Engel, J. Janesky, N. D. Rizzo and S. Tehrani, Journal of Superconductivity: Incorporating Novel Magnetism 15 (1), 19-25 (2002). 8. S. Tehrani, J. M. Slaughter, M. Deherrera, B. N. Engel, N. D. Rizzo, J. Salter, M. Durlam, R. W. Dave, J. Janesky and B. Butcher, Proceedings of the IEEE 91 (5), 703-714 (2003). 9. A. Ney, C. Pampuch, R. Koch and K. H. Ploog, Nature 425 (6957), 485-487 (2003). 10. B. D. Terris, Journal of Magnetism and Magnetic Materials 321 (6), 512-517 (2009). 11. M. V. Costache, M. Sladkov, S. M. Watts, C. H. Van der Wal and B. J. Van Wees, Physical Review Letters 97 (21), 216603 (2006). 12. K. Hika, L. V. Panina and K. Mohri, Magnetics, IEEE Transactions on 32 (5), 4594-4596 (1996). 13. C. S. Chang, M. Kostylev and E. Ivanov, Applied Physics Letters 102 (14) (2013). 14. F. Klose, C. Rehm, D. Nagengast, H. Maletta and A. Weidinger, Physical Review Letters 78 (6), 1150 (1997). 15. K. Munbodh, F. A. Perez and D. Lederman, Journal of Applied Physics 111 (12), 123919-123919-123916 (2012). 16. J. M. Shaw, H. T. Nembach, T. J. Silva, S. E. Russek, R. Geiss, C. Jones, N. Clark, T. Leo and D. J. Smith, Physical Review B 80 (18), 184419 (2009). 17. R. Law, R. Sbiaa, T. Liew and T. C. Chong, Applied Physics Letters 91, 242504 (2007). 18. H. Gong, D. Litvinov, T. J. Klemmer, D. N. Lambeth and J. K. Howard, Magnetics, IEEE Transactions on 36 (5), 2963-2965 (2000). 19. D. R. Birt, B. O'Gorman, M. Tsoi, X. Li, V. E. Demidov and S. O. Demokritov, Applied Physics Letters 95 (122510) (2009). 20. J. Callaway, Physical Review 132, 2003-2009 (1963). 21. A. O. Adeyeye and N. Singh, Journal of Physics D: Applied Physics 41 (153001), 29 (2008). 22. T. J. Silva, C. S. Lee, T. M. Crawford and C. T. Rogers, Journal of Applied Physics 85 (11), 7849-7862 (1999). 23. G. Counil, J.-V. Kim, K. Shigeto, Y. Otani, T. Devolder, P. Crozat, H. Hurdequint and C. Chappert, Journal of Magnetism and Magnetic Materials 272-276 (1), 290-292 (2004). 24. S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva and J. P. Nibarger, Journal of Applied Physics 99 (9), 093909-093909-093907 (2006). 25. M. Belmeguenai, F. Zighem, Y. Roussigne, S.-M. Cherif, P. Moch, K. Westerholt, G. Woltersdorf and G. Bayreuther, Physical Review B 79 (024419) (2009). 26. A. M. Niknejad, (2005). 27. H. T. Friis, Proceedings of the IRE 32 (7), 419-422 (1944). 28. C. S. Chang, M. Kostylev, A. O. Adeyeye, M. Bailleul and S. Samarin, EuroPhysics Letters 96 (2011). 29. C. Kittel, Introduction to Solid State Physics, 8 ed. (John Wiley & Sons, 2005).

133

30. L. Landau and L. Liftshitz, Phys. Zeit. Sowjetunion 8 (1935). 31. T. A. Gilbert, 1955. 32. C. E. Patton, Z. Frait and C. H. Wilts, Journal of Applied Physics 46 (11), 5002 (1975). 33. V. E. Demidov, S. O. Demokritov, K. Rott, P. Krzysteczko and G. Reiss, Applied Physics Letters 92 (232503) (2008). 34. K. Sekiguchi, K. Yamada, S. M. Seo, K. J. Lee, D. Chiba, K. Kobayashi and T. Ono, Applied Physics Letters 97 (022508) (2010). 35. P. K. Amiri, B. Rejaei, M. Vroubel and Y. Zhuang, Applied Physics Letters 91 (065502) (2007). 36. V. E. Demidov, S. O. Demokritov, K. Rott, P. Krzysteczko and G. Reiss, Applied Physics Letters 91 (252504) (2007). 37. C. Kittel, Physical Review 73 (2), 155- 161 (1948). 38. C. Kittel, Physical Review 110 (6), 1295-1297 (1958). 39. W. Stoecklein, S. S. P. Parkin and J. C. Scott, Physical Review B 38 (10), 6847 (1988). 40. R. D. McMichael, M. D. Stiles, P. J. Chen and W. F. Egelhoff Jr, Physical Review B 58 (13), 8605 (1998). 41. R. Magaraggia, K. Kennewell, M. Kostylev, R. L. Stamps, M. Ali, D. Greig, B. J. Hickey and C. H. Marrows, Physical Review B 83 (5), 054405 (2011). 42. M. Kostylev, Journal of Applied Physics 106 (4), 043903-043903-043914 (2009). 43. K. J. Kennewell, M. Kostylev, N. Ross, R. Magaraggia, R. L. Stamps, M. Ali, A. A. Stashkevich, D. Greig and B. J. Hickey, Journal of Applied Physics 108 (073917), 12 (2010). 44. D. M. Kostylev and P. G. A. Melkov, (Australian Research Council, The University of Western Australia, 2010), pp. 7. 45. Y. V. Khivintsev, L. Reisman, J. Lovejoy, R. Adam, C. M. Schneider, R. E. Camley and Z. J. Celinski, Journal of Applied Physics 108 (2), 023907-023907-023906 (2010). 46. V. V. Kruglyak, S. O. Demokritov and D. Grundler, Journal of Physics D: Applied Physics 43 (26), 264001 (2010). 47. A. A. Grunin, A. G. Zhdanov, A. A. Ezhov, E. A. Ganshina and A. A. Fedyanin, Applied Physics Letters 97 (26), 261908-261908-261903 (2010). 48. N. Kostylev, I. S. Maksymov, A. O. Adeyeye, S. Samarin, M. Kostylev and J. F. Williams, Applied Physics Letters 102 (12), 121907-121907-121905 (2013). 49. G. Gubbiotti, S. Tacchi, M. Madami, G. Carlotti, A. O. Adeyeye and M. Kostylev, Journal of Physics D: Applied Physics 43 (26), 264003 (2010). 50. K. Y. Guslienko, S. O. Demokritov, B. Hillebrands and A. N. Slavin, Physical Review B 66 (13), 132402 (2002). 51. K. Y. Guslienko and A. N. Slavin, Physical Review B 72 (1), 014463 (2005). 52. S. Tacchi, M. Madami, G. Gubbiotti, G. Carlotti, S. Goolaup, A. O. Adeyeye, N. Singh and M. P. Kostylev, Physical Review B 82 (18), 184408 (2010). 53. M. P. Kostylev, A. A. Stashkevich and N. A. Sergeeva, Physical Review B 69 (6), 064408 (2004). 54. M. M. Ney, Electromagnetic Compatibility, IEEE Transactions on 33 (4), 321-327 (1991). 55. J. D. Kraus and D. A. Fleisch, Electromagnetics, 5 ed. (McGraw-Hill Science/Engineering/Math, 1999). 56. S. O. Demokritov and V. E. Demidov, IEEE Transactions on Magnetics 44 (1), 6-12 (2008). 57. Z. Zhang, P. C. Hammel and P. E. Wigen, Applied Physics Letters 68 (14), 2005-2007 (1996). 58. D. Rugar, R. Budakian, H. J. Mamin and B. W. Chui, Nature 430 (6997), 329-332 (2004). 59. R. Magaraggia, M. Hambe, M. Kostylev, V. Nagarajan and R. L. Stamps, Physical Review B 84 (10), 104441 (2011). 60. R. von Helmolt, J. Wecker, B. Holzapfel, L. Schultz and K. Samwer, Physical Review Letters 71 (14), 2331 (1993).

134

61. N. Ross, M. Kostylev and R. L. Stamps, Journal of Applied Physics 109 (1), 013906-013906-013908 (2011). 62. C. L. Hu, R. Magaraggia, H. Y. Yuan, C. S. Chang, M. Kostylev, D. Tripathy, A. O. Adeyeye and R. L. Stamps, Applied Physics Letters 98 (26), 262508-262508-262503 (2011). 63. R. Bali, M. Kostylev, D. Tripathy, A. O. Adeyeye and S. Samarin, Physical Review B 85 (10), 104414 (2012). 64. L. R. Walker, Physical Review 105 (2), 390-399 (1957). 65. S. O. Demokritov, B. Hillebrands and A. N. Slavin, Physics Reports 348 (6), 441-489 (2001). 66. B. A. Kalinikos, Microwaves, Optics and Antennas, IEE Proceedings H 127 (1), 4 (1980). 67. R. W. Damon and J. R. Eshbach, Journal of Physics and Chemistry of Solids 19 (3-4), 308-320 (1961). 68. J. R. Eshbach and R. W. Damon, Physical Review 118 (5), 1208-1210 (1960). 69. D. D. Stancil and A. Prabhakar, Spin Waves Theory and Applications. (Springer, 2009). 70. M. Bailleul, D. Olligs, C. Fermon and S. O. Demokritov, Europhysics Letters 56 (5), 741 (2001). 71. M. Bailleul, D. Olligs and C. Fermon, Applied Physics letters 83 (5), 972-974 (2003). 72. V. Vlaminck and M. Bailleul, Science 322 (5900), 410 (2008). 73. V. Vlaminck and M. Bailleul, Physical Review B 81 (014425) (2010). 74. R. Huber, M. Krawczyk, T. Schwarze, H. Yu, G. Duerr, S. Albert and D. Grundler, Applied Physics Letters 102 (012403) (2013). 75. M. Bao, K. Wong, A. Whitun, J. Lee, Z. Hao, K. L. Wang, D. W. Lee and S. X. Wang, EuroPhysics Letters 84 (27009) (2008). 76. M. Covington, T. M. Crawford and G. J. Parker, Physical Review Letters 89 (23), 7202 (2002). 77. C. Mathieu, J. r. Jorzick, A. Frank, S. O. Demokritov, A. N. Slavin, B. Hillebrands, B. Bartenlian, C. Chappert, D. Decanini and F. Rousseaux, Physical Review Letters 81 (18), 3968 (1998). 78. Y. Roussigne, S. M. Cherif, C. Dugautier and P. Moch, Physical Review B 63 (13), 134429 (2001). 79. C. Bayer, J. Jorzick, B. Hillebrands, S. O. Demokritov, R. Kouba, R. Bozinoski, A. N. Slavin, K. Y. Guslienko, D. V. Berkov, N. L. Gorn and M. P. Kostylev, Physical Review B 72 (064427) (2005). 80. A. A. Stashkevich, P. Djema, Y. K. Fetisov, N. Bizière and C. Fermon, Journal of Applied Physics 102 (10), 3905 (2007). 81. V. E. Demidov, M. P. Kostylev, K. Rott, P. Kryzysteczko, G. Reiss and S. O. Demokritov, Applied Physics letters 95 (11), 2509 (2009). 82. V. E. Demidov, J. Jersch, S. O. Demokritov, K. Rott, P. Krzysteczko and G. Reiss, Physical Review B 79 (054417) (2009). 83. Z. Liu, F. Giesen, X. Zhu, R. D. Sydora and M. R. Freeman, Physical Review Letters 98 (087201) (2007). 84. K. Y. Guslienko, R. W. Chantrell and A. N. Slavin, Physical Review B 68 (2), 024422 (2003). 85. V. E. Demidov, S. O. Demokritov, K. Rott, P. Krzysteczko and G. Reiss, Physical Review B 77 (064406) (2008). 86. M. P. Kostylev, G. Gubbiotti, J. G. Hu, G. Carlotti, T. Ono and R. L. Stamps, Physical Review B 76 (054422) (2007). 87. T. W. O'Keeffe and R. W. Patterson, Journal of Applied Physics 49 (4886) (1978). 88. P. Bryant and H. Suhl, Applied Physics Letters 54 (22), 2224 (1989). 89. R. Matthies, K. Ramstock and J. McCord, IEEE Transactions on Magnetics 33 (5), 3993 (1997).

135

90. J. Jorzick, S. O. Demokritov, B. Hillebrands, M. Bailleul, C. Fermon, K. Y. Guslienko, A. N. Slavin, D. V. Berkov and N. L. Gorn, Physical Review Letters 88 (4) (2002). 91. M. P. Kostylev, A. A. Serga, T. Schneider, T. Neumann, B. Leven, B. Hillebrands and R. L. Stamps, Physical Review B 76 (18), 4419 (2007). 92. B. B. Maranville, R. D. McMichael, S. A. Kim, W. L. Johnson, C. A. Ross and J. Y. Cheng, Journal of Applied Physics 99 (8), 08C703-708C703-703 (2006). 93. C. S. Tsai and D. Young, Microwave Theory and Techniques, IEEE Transactions on 38 (5), 560-570 (1990). 94. R. Hertel, W. Wulfhekel and J. Kirschner, Physical Review Letters 93 (25), 257202 (2004). 95. D. D. Stancil, Journal of Applied Physics 59 (1), 218-224 (1986). 96. W. F. J. Brown, Magnetostatic Principles in Ferromagnetism. (North-Holland Publishing Company, Amsterdam, 1962). 97. R. Simons, Coplanar Waveguide Circuits, Components, and Systems. (Wiley, 2001). 98. K. J. Kennewell, M. Kostylev and R. L. Stamps, Journal of Applied Physics 101 (09D107) (2007). 99. D. K. Cheng, Field and Wave Electromagnetics, 2nd ed. (Addison-Wesley Publishing Company, 1989). 100. V. F. Dmitriev, Soviet Journal of Communications Technology & Electronics 36 (1-3), 34 (1991). 101. V. F. Dmitriev and B. A. Kalinikos, Soviet Physics Journal 31 (11), 875-898 (1988). 102. T. Schneider, A. A. Serga, T. Neumann, B. Hillebrands and M. P. Kostylev, Physical Review B 77 (214411) (2008). 103. V. E. Demidov, S. Urazhdin and S. O. Demokritov, Applied Physics letters 95 (26), 2509 (2009). 104. T. M. Crawford, M. Covington and G. J. Parker, Physical Review B 67 (024411), 1-5 (2003). 105. M. R. Scheinfein, (Mindspring.com, 2007). 106. C. Vittoria and N. D. Wilsey, Journal of Applied Physics 45 (1), 414-420 (1974). 107. K. Sekiguchi, K. Yamada, S.-M. Seo, K.-J. Lee, D. Chiba, K. Kobayashi and T. Ono, Physical Review Letters 108 (1), 017203 (2012). 108. X. Liu, J. O. Rantschler, C. Alexander and G. Zangari, Magnetics, IEEE Transactions on 39 (5), 2362-2364 (2003). 109. Z. Celinski and B. Heinrich, Journal of Applied Physics 70 (10), 5935-5937 (1991). 110. B. Heinrich, J. F. Cochran and R. Hasegawa, Journal of Applied Physics 57 (8), 3690-3692 (1985). 111. A. G. Gurevich and O. A. Chivileva, PISMA V ZHURNAL TEKHNICHESKOI FIZIKI 15 (11), 7-9 (1989). 112. J. M. Shaw, H. T. Nembach and T. J. Silva, Physical Review B 85 (5), 054412 (2012). 113. A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves. (CRC Press, 1996). 114. J. Volakis, Antenna Engineering Handbook, 4 ed. (McGraw-Hill, 2007). 115. M. Bailleul. 116. T. Hubert, L. Boon-Brett, G. Black and U. Banach, Sensors and Actuators B: Chemical 157 (2), 329-352 (2011). 117. S. Okuyama, Y. Mitobe, K. Okuyama and K. Matsushita, Jpn. J. Appl. Phys. 39 (3584) (2000). 118. Y. Imai, Y. Kimura and M. Niwano, Applied Physics Letters 101 (18), 181907-181907-181904 (2012). 119. R. Gupta, A. A. Sagade and G. U. Kulkarni, international journal of hydrogen energy 37 (11), 9443-9449 (2012).

136

120. C. Langhammer, E. M. Larsson, V. P. Zhdanov and I. Zoric̕, The Journal of Physical Chemistry C 116 (40), 21201-21207 (2012). 121. J. H. Jung, Applied Physics Letters 101 (24), 242403-242403-242404 (2012). 122. A. Tittl, C. Kremers, J. Dorfmuller, D. N. Chigrin and H. Giessen, Opt. Mat. Express 2 (111) (2012). 123. N. Liu, M. L. Tang, M. Hentschel, H. Giessen and A. P. Alivisatos, Nature materials 10 (8), 631-636 (2011). 124. R. C. Hughes and W. K. Schubert, Journal of Applied Physics 71 (1), 542-544 (1992). 125. Y.-T. Cheng, Y. Li, D. Lisi and W. M. Wang, Sensors and Actuators B: Chemical 30 (1), 11-16 (1996). 126. M. A. Butler, Applied Physics Letters 45 (10), 1007-1009 (1984). 127. J. Villatoro, D. Luna-Moreno and D. Monzón-Hernández, Sensors and Actuators B: Chemical 110 (1), 23-27 (2005). 128. B. Chadwick, J. Tann, M. Brungs and M. Gal, Sensors and Actuators B: Chemical 17 (3), 215-220 (1994). 129. M. Tabib-Azar, B. Sutapun, R. Petrick and A. Kazemi, Sensors and Actuators B: Chemical 56 (1), 158-163 (1999). 130. W. Gouda and K. Shiiki, Journal of Magnetism and Magnetic Materials 205 (2), 136-142 (1999). 131. K. Ando and E. Saitoh, Journal of Applied Physics 108 (11), 113925-113925-113924 (2010). 132. Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando and K. Takanashi, Nature 464 (7286), 262-266 (2010). 133. J. Foros, G. Woltersdorf, B. Heinrich and A. Brataas, Journal of Applied Physics 97 (10), 10A714-710A714-713 (2005). 134. S. Okamoto, O. Kitakami and Y. Shimada, Journal of Magnetism and Magnetic Materials 239 (1), 313-315 (2002). 135. K. Munbodh, F. A. Perez, C. Keenan, D. Lederman, M. Zhernenkov and M. R. Fitzsimmons, Physical Review B 83 (9), 094432 (2011). 136. M. Kostylev, Journal of Applied Physics 112 (9), 093901-093901-093906 (2012). 137. B. N. Engel, C. D. England, R. A. Van Leeuwen, M. H. Wiedmann and C. M. Falco, Physical Review Letters 67 (14), 1910 (1991). 138. R. L. Stamps, L. Louail, M. Hehn, M. Gester and K. Ounadjela, Journal of Applied Physics 81 (8), 4751-4753 (1997). 139. T. B. Flanagan and W. A. Oates, Annual Review of Materials Science 21 (1), 269-304 (1991). 140. F. A. Lewis, The Palladium Hydrogen System. (Academic Press, London New York, 1967). 141. E. Lee, J. M. Lee, J. H. Koo, W. Lee and T. Lee, international journal of hydrogen energy 35 (13), 6984-6991 (2010). 142. S. Mizukami, E. P. Sajitha, D. Watanabe, F. Wu, T. Miyazaki, H. Naganuma, M. Oogane and Y. Ando, Applied Physics Letters 96 (15), 152502-152502-152503 (2010). 143. U. Gradmann, Journal of Magnetism and Magnetic Materials 54, 733-736 (1986). 144. H. J. G. Draaisma, W. J. M. De Jonge and F. J. A. Den Broeder, Journal of Magnetism and Magnetic Materials 66 (3), 351-355 (1987). 145. Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Physical Review B 66 (22), 224403 (2002). 146. O. Dankert and A. Pundt, Applied Physics Letters 81 (9), 1618-1620 (2002). 147. D. Lederman, Y. Wang, E. H. Morales, R. J. Matelon, G. B. Cabrera, U. G. Volkmann and A. L. Cabrera, Applied Physics Letters 85, 615 (2004). 148. J. Topp, D. Heitmann, M. P. Kostylev and D. Grundler, Physical Review Letters 104 (20), 207205 (2010).

137

149. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics. (John Wiley & Sons, New York, 1978). 150. N. S. Almeida and D. L. .Mills, Physical Review B 53 (18), 12232-12241 (1996). 151. G. Gubbiotti, M. Kostylev, N. Sergeeva, M. Conti, G. Carlotti, T. Ono, A. N. Slavin and A. Stashkevich, Physical Review B 70 (22), 224422 (2004).