Magnetisation dynamics in ferromagnetic continuous and patterned films:

Transcription

1 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.

2 Abstract In recent years, microwave magnetisation dynamics in thin ferromagnetic metallic films, multi-layers, and nano-structures has attracted a lot of attention due to possible future applications in microwave signal processing, magnetic logic, and magnetic sensors. In this work, magnetisation dynamics were studied for ferromagnetic continuous and patterned films using inductive broadband spin wave spectroscopy techniques in three projects: a.) A microwave current injection ferromagnetic resonance (FMR) technique using a sub-millimetre coplanar probe was demonstrated on a continuous Permalloy film and a periodic array of Permalloy nano-stripes. It was found that the first standing spin wave mode (SSWM) with odd symmetry across the material thickness was efficiently excited in the nano-stripe array. On the contrary, in spin wave resonance spectra measured with conventional techniques the higher-order SSWMs are often lacking due to symmetry reasons. However, they are of great importance since they carry important information about the exchange constant for the material. Calculations of microwave current distributions by the current injection method were used to explain the spin wave resonance spectra. The suggested current injection FMR technique is fast and simple. On top of the efficient excitation of the higher-order SSWMs, it also allows spatial mapping of magnetisation dynamics with spatial resolution determined by the size of the coplanar probe tip. b.) Magnetostatic spin wave modes in the Damon-Eshbach geometry were systematically studied for a series of Permalloy micro-stripes over a wide range of aspect ratios using a highly sensitive custom-made microwave detector. The use of the detector allowed tracking the spin wave dispersion over a wide range of wave numbers using the simple phase method. It was found that over the range of aspect ratios and wave numbers studied, the dynamic effects can be neglected and the surface mode dispersions can be modelled by including an effective static demagnetising field term in the continuous film dispersion case. The group velocities were found to increase with thickness and were width invariant over the aspect ratios considered. The attenuation and relaxation parameters were found to be typical for the material. It was also found ii

3 that the non-reciprocity parameter is largely invariant over the range of aspect ratios studied. 110nm For the stripe with the highest aspect ratio studied, excluding the fundamental 2 m mode, up to six higher order width modes with odd symmetry were observed. The modes were identified from numerical simulations, from which the modal profiles were obtained. Group velocities, attenuation properties, and non-reciprocity of these higher order width modes were characterised in detail. It was found that group velocity, attenuation length, and non-reciprocity decreased for increasing mode number. Finally, the near-field of the antenna was considered. We propose that spin wave propagation begins at some finite distance away from the antenna due to the near-field of the antenna. An expression was derived from which the so-called antenna characteristic near-field length may be experimentally determined. For our antenna, we found that this near-field length is non-zero but still lying underneath the total width of the antenna. This results in the effective wave propagation distance being shorter than the geometrical antennae separation gap, the difference being twice the antenna characteristic near-field length. c.) A cobalt-palladium bi-layer thin film s functionality as a hydrogen sensor is demonstrated. Upon hydrogenation of the palladium capping layer, a down-field shift and line-width narrowing of the ferromagnetic resonance of the underlying cobalt layer was observed. The resonance shift was attributed to increase in interfacial uniaxial anisotropy of cobalt due to strain from the expanded hydrogenated palladium capping layer. We propose that the line-width narrowing is primarily due to reduction in spinpumping into the palladium layer due to reduction of conductivity of the hydrogenated palladium layer. Finally, the bi-layer film was subjected to repetitive cycling of nitrogen and hydrogen atmospheres. The ferromagnetic resonance response of the sensor was consistently reproducible at each cycle with expected palladium hydrogen absorption and desorption characteristic times. These results open up an exciting new class of ferromagnetic resonance-based hydrogen sensor. iii

4 Acknowledgements Financial support by the Australian Research Council (ARC), the School of Physics, The University of Western Australia (UWA), and the Australian-Indian Strategic Research fund is acknowledged. This work was performed in part at the University of New South Wales (UNSW) node of the Australian National Fabrication Facility (ANFF); A company established under the National Collaborative Research Infrastructure Strategy to provide nano and microfabrication facilities for Australia s researchers. Usage of the facilities of the Sensors & Advanced Instrumentation Laboratory (SAIL), School of Electrical, Electronics and Computer Engineering, the University of Western Australia, is acknowledged. I acknowledge the facilities, and the scientific and technical assistance, of the Australian Microscopy & Microanalysis Research Facility at the Centre for Microscopy, Characterisation and Analysis (CMCA), The University of Western Australia. iv

5 Thanks To my main supervisor, Mikhail Kostylev (Physics, UWA): Throughout the 4 years of this journey, I have learnt so much from your vast knowledge, experience, and wisdom in the field. I truly appreciate the opportunity given to work under your guidance at the Spintronics and Magnetisation Dynamics Group. Thank you for initiating suitable projects for me to work on, and for directing me in the right direction whenever faced with obstacles. Thank you for helping me to set up the experimental equipment for the various projects throughout the years. Thank you as well for training me in the ferromagnetic resonance measurement techniques in the laboratory, and for the numerical simulation codes. Thank you for always being available to answer my questions. I have benefited much from our fruitful discussions and your advices. To my co-supervisor, Ivan Maksymov (Physics, UWA): Thank you for your valuable feedback towards the thesis writing and checking up on my progress. To my former co-supervisor, Bob Stamps (University of Glasgow): Thank you for your ideas and input during the early days of the thesis journey. To Adekunle Adeyeye (National University of Singapore): Thank you for fabricating samples which made this thesis possible. Your contribution is greatly appreciated. Thank you for sharing your expertise in discussions regarding fabrication techniques of patterned magnetic structures. To Matthieu Bailleul (Institute of Physics and Chemistry of Materials, University of Strasbourg): Thank you for your microwave current injection technique suggestion, of which a publication resulted, and which constituted a significant part of this thesis. Thank you as well for discussions and your expert advice on propagating spin wave spectroscopy, of which a major part of this thesis is based on. v

6 To Eugene Ivanov (Physics, UWA): Thank you for building the microwave interferometric phase detector, with which highsensitivity ferromagnetic resonance measurements could be made, especially for the propagating spin wave and hydrogen sensor experiment. Thank you as well, for useful discussions on noise and sensitivity of measurements. To Fay Hudson (ANFF-UNSW): Thank you for your hospitality in my trips to ANFF-UNSW. Thank you for inducting me into the facility, training me in clean room techniques, optical lithography, electronbeam lithography, scanning electron microscopy, and thermal evaporative deposition. Thank you as well for helping me to develop the recipe to fabricate micro-patterned magnetic structures, without which this thesis would not have been possible. To the Physics Workshop crew (Physics, UWA): Thank you for building the probe station and the gas cell; the hardware of the thesis! Thank you also (especially Gary Light and John Moore) for your hard work in fixing and maintaining the ageing sputtering machine. To Dave O Connor (Bandwidth Foundry): Thank you for your expert advice on design of optical lithographic masks. To Nils Ross (formerly Physics, UWA): Thank you for passing on the baton to me by training me to use the group s sputtering machine. To Alexandra Suvorova (CMCA-UWA): Thank you for training me to use the scanning electron microscope at CMCA. Thank you also for helping us to image particularly challenging samples on a tilted sample stage. To Joanna Szymanska (ANFF-UNSW): Thank you for training and supervising me to use the electron-beam evaporative deposition equipment at ANFF-UNSW. vi

7 To Adrian Keating (Electrical Engineering, UWA): Thank you for training me to use the optical profilometer in the SAIL laboratory. To Rhet Magaraggia (Physics, UWA): Thank you for teaching me the magneto-optical Kerr effect (MOKE) setup in our laboratory. Thank you also for helping to troubleshoot data acquisition software of our measurement setups whenever something went wrong. To Rob Woodward (Physics, UWA): Thank you for letting me use the Biomagnetics group s optical microscope to inspect my samples. To Nir Zvison (Electrical Engineering, UWA), Thank you for depositing silicon nitride on my samples for me during the early days of the thesis. vii

8 Contents 1 Introduction Thesis outline 2 2 Experimental setup and techniques Sample fabrication Film deposition Micro-fabrication Broadband spin wave spectroscopy Vector network analyser Lock-in with field modulation Interferometric phase detector Probe station Gas cell 14 3 Microwave current injection spin wave spectroscopy Background Spin waves Ferromagnetic resonance Standing spin wave mode Case for work Experiment design Continuous film mode identification Nanostripe array mode identification 24 viii

9 3.6 Microwave electromagnetic field calculations Current injection method on continuous film Current injection method on nanostripes Microstrip method on continuous film and nanostripes Out-of-plane microwave magnetic field contribution Microwave current injection as a characterisation tool Chapter conclusion 44 4 Propagating spin wave spectroscopy Background Propagating modes in continuous films Propagating modes in laterally confined geometry Case for work Experimental setup Experimental procedure Data acquisition Sensitivity Wave number space Extracting dispersion Magnetostatic surface mode in confined stripe geometry Dispersion Static demagnetising field simulations Group velocity Attenuation and relaxation 75 ix

10 4.5.5 Non-reciprocity Higher order width modes in confined stripe geometry Mode identification Dispersion and group velocity Attenuation and relaxation Non-reciprocity Antenna near-field effect Characteristic equations Antenna characteristic near-field length Effective propagation distance Chapter conclusion Ferromagnetic resonance-based hydrogen gas sensor Background Case for work Experiment design Experiment results Discussion of results Cobalt-palladium film as a hydrogen sensor Suggestions for further work Chapter conclusion 120 Appendices 121 Appendix A Photolithography micro-fabrication recipe 121 Appendix B Microwave current injection into a continuous film 123 x

11 Appendix C Numerical Simulations 130 Bibliography 132 xi

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13 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 Thus, there is still much room for research into the characterisation of magnetisation dynamics in such patterned magnetic media, including the development and improvement of measurement techniques. In this thesis, three different magnetic systems were studied using inductive broadband spectroscopy techniques. The first is the use of a microwave current injection technique to probe local magnetisation dynamics. This technique developed as a part of this thesis was demonstrated on an array of magnetic nano-stripes and a reference continuous film. The second and largest work in this thesis is the study of propagating spin waves in confined magnetic stripes. Channelling of spin waves along a confined stripe is of great technological importance for potential microwave signal processing and magnetic logic application. The characteristics of magnetostatic surface waves across a wide range of stripe aspect ratios were systematically studied in that chapter. Finally, the third work demonstrates the functionality of a metallic magnetic / palladium bi-layer film as a hydrogen sensor. The state of the hydrogen-absorbing palladium was probed through the dynamic magnetisation properties of the underlying magnetic film. This represents a new class of ferromagnetic resonance-based hydrogen sensor. Hence, the chapters in this thesis are set out as follows: 1

14 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. 2

15 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 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 The tantalum capping layer shields the film of interest from oxidation. Sputtering is typically done at room temperature with argon plasma at a pressure of 6 mtorr and regulated power of 60 W. The group s sputter machine lacks a monitoring crystal, so deposition rates need to be pre-determined by calibration. For a particular target material, gun, and sputtering power, a series of films were sputtered for known exposure times. For calibration, the silicon substrates were partially covered prior to sputtering, resulting in film depositing only on the uncovered areas of substrates. The resulting step height at the boundary is then measured with a white light interferometer profilometer. This step height is the thickness of the film sputtered. From these, the deposition rates were determined. 3

16 Calibrations are repeated approximately every 20 hours of target use to check for drifts in the sputtering rates due to target depletion Micro-fabrication The central part of this PhD thesis involves characterising properties of propagating spin waves in micro-stripes (detailed in chapter 4). Fabrication was jointly done at the Australian Nanofabrication Facility node at the University of New South Wales (UNSW), and by Prof. A.O Adekunle s group at the Department of Electrical and Computer Engineering, National University of Singapore (NUS). A series of microstripes of various aspect ratios overlaid with microscopic coplanar waveguides were fabricated. Lift-off deposition fabrication method was used. The fabrication recipes are detailed in Appendix A. It was found that sputter deposition followed by lift-off is unsuitable to fabricate the magnetic stripes. The non-directional nature of sputtering resulted in side wall coating of the photoresist pattern, which after lift-off, resulted in rough and steep stripe edges. This is unacceptable, since irregular submicron-sized physical defects will cause unwanted scattering of spin waves 19, 20. Following Prof. A.O Adekunle s group s fabrication method at NUS 21, electron beam evaporative deposition was found to be suitable to form magnetic stripes with straight edges (with defect sizes of the order of submicrons). 2.2 Broadband spin wave spectroscopy The inductive method to study excitation of spin wave resonance in a ferromagnetic film was pioneered by Silva et al. 22. In a typical broadband spin wave experiment, microwave absorption is measured as a function of the driving microwave frequency and/or externally applied magnetic field. At resonance, a dip in the spectra indicates absorption of microwave power into the sample under test (Figure 1.2.2a). The experiment is usually repeated for a number of frequency and field sweeps, and material parameters extracted by fitting with the appropriate analytic formula or numerical simulation. Thus, broadband spin wave spectroscopy is a tool to characterise the 4

17 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 Vector network analyser The broadband inductive technique using a network analyser was first developed by Counil et al. 23, and is now widely employed for the measurement of magnetisation dynamics. Similar to 24, a planar waveguide (Figure 2.2.1a) is placed between the poles of an electromagnet such that the waveguide is perpendicular to and in-plane to the direction of the applied field. Out-of-plane configuration is possible as well, but this geometry is not used in the experiments detailed here. The magnetic sample of interest to be tested is placed on a top of the waveguide, usually with the film facing the transducer. The waveguide is connected on both ends to the two ports of a vector network analyser (VNA). Figure 2.2.1a: A microstrip waveguide with sample under test across the signal line. The VNA functions as both the microwave source to excite spin waves in the magnetic sample, and as a signal receiver. More precisely, it measures the scattering parameters S21 (transmission) and S11 (reflection) of the device-under-test (DUT). There are two methods to measure the FMR response of the sample: 5

18 Frequency sweep: The electromagnet field is fixed, and the scattering parameters measured as a function of frequency. This method is quick, but less sensitive compared to a field sweep. In addition, frequency sweeps may yield signals which are nonmagnetic in origin, but simply due to variations in the impedance of the DUT as frequency is swept. Field sweep: The VNA is set to operate at a single frequency, and the electromagnet field is swept. The scattering parameters are measured as a function of field for a particular frequency. This method is slow, but more sensitive than a frequency sweep. In addition, it only yields signals which vary with magnetic field. This method requires additional computer codes to enable automation of field sweep and data acquisition. An example of spectra taken with VNA using field sweep is shown in figure 2.2.1b. The merit of VNA is that it enables one to measure the absolute value of spin wave microwave absorption in terms of well-defined scattering parameters. However, the disadvantage of VNA is that it measures the scattering parameters of the whole DUT; both the waveguide and the sample of interest. Due to the sheer physical size difference between the waveguide and the sample, the sample signal is almost always much smaller than the total DUT signal, appearing as blips on top of the background waveguide signal. Typically, background subtraction needs to be done to isolate the sample signal from the total DUT signal. Figure 2.2.1b: Spin wave absorption spectra of a 100 nm thick Permalloy film at 10 GHz, showing the fundamental mode and the first standing spin wave mode as microwave absorption dips. 6

19 2.2.2 Lock-in with field modulation In light of the disadvantage of VNA pointed out before, the group developed a more sensitive lock-in and modulation broadband spin wave spectroscopy method. The VNA is replaced by a dedicated microwave generator, a microwave tunnel diode, and a lockin amplifier. In addition, modulation coils were fixed at the poles of the electromagnet (Figure 2.2.2a). Figure 2.2.2a: Lock-in with field modulation broadband method circuitry. The microwave signal transmitted through the DUT is measured as a function of applied field for given microwave frequencies. Alternatively, the reflected signal can also be measured instead by redirecting reflected power from the DUT through a circulator. Similar to 24, 25, the field is modulated using two small coils attached to the poles of the electromagnet. Modulation frequency is 220 Hz and the RMS magnetic field produced by the coils is typically about 9 Oe. The input microwave power is set such that the rectified bias voltage at the output end of the tunnel diode is between mv; this is the most sensitive and linear region of the particular diode s response. The transmitted / reflected signal from the DUT is rectified using a tunnel diode and fed into a lock-in amplifier referenced by the same 220 Hz signal driving the modulation coils. The signal obtained this way is proportional to the field derivative of the imaginary part 7

20 of the rf susceptibility as a function of the microwave frequency 25. The mathematical concept is as follows: Consider the microwave susceptibility of the DUT as a function of field, H: (H) Modulation produces an ac field on top of the dc field, so the susceptibility becomes: ( H he i t ) The first two terms of the Taylor expansion (with respect to time) of the susceptibility are: ( H) i he i t d dh The first term is effectively a dc term, which is removed by the lock-in amplifier. The second term is an oscillatory signal with the same frequency as the field modulation frequency. By referencing the lock-in amplifier with the driving frequency of the modulation coils, the second term gets locked-in. Note that the second term is proportional to the modulation amplitude and the shape of the curve is the first derivative of the susceptibility curve. Typically, background signals from the transducer and other potentially magnetic components between the electromagnet pole gaps are broad while sample spin wave resonance signals are typically sharp. Hence, the derivative of the background signal is effectively flat compared to the derivative of the spin wave resonance signal. The practical absence of background means that the sensitivity of the lock-in amplifier can be set to the sample signal level. 1 Note that noise can be reduced by increasing the modulation frequency. However, f coil inductance increases with frequency, more so since the modulating coils are attached to the soft iron poles of the electromagnet. Hence, there is a trade-off between high frequency (to reduce pink noise) and low frequency (to increase modulation field amplitude). For our setup, we use 220 Hz as a compromise between these two limitations. 220 Hz is also not a harmonic of 50 Hz mains. In addition, using the lock-in 8

21 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 Interferometric phase detector For continuous films thinner than 10 nm and micro-patterned structures, the signals obtained using the single diode lock-in technique approach the noise levels for the setup. Thus, a highly sensitive microwave detector with much lower noise threshold is built to enable broadband measurement of spin wave spectroscopy in such systems. Prof. Eugene Ivanov (Frequency Standards and Metrology Research Group at UWA Physics) is credited for building the device for use in our group s experiments. The schematic of the detector is shown in figure 2.2.3a: Figure 2.2.3a: Schematic of microwave receiver circuitry. In essence, the device is a double Mach-Zehnder type interferometer. The source signal is split into two paths; one as the reference signal, and the other passing through the DUT. Both signals are then recombined. In this particular receiver, it has two loops; a major loop and a minor loop within one path of the major loop. The key component of this device is the mixer, which is a non-linear device. It is a device that performs frequency conversion by multiplying two signals 26. A mixer has three ports; the radio 9

22 frequency (RF) port, the local oscillator (LO) port, and the intermediate frequency (IF) port. The major loop can be represented in the form of an equivalent circuit containing a standard interferometer, a diode, and an amplifier whose gain scales as the input power of the whole double interferometer. In the schematic diagram (figure 2.2.3a), the microwave source signal is split into two paths: A and B. Path A is the driving signal at the LO port of the mixer. Path B is further split again into a minor loop into two paths: C and D. Path D passes through the DUT, and both signals (C and D) are recombined again into path E. The phase and attenuation of path C is set such that the carrier signal is completely suppressed by destructive interference upon recombination at E. The minor loop enables high microwave power through the DUT, followed by suppression of the carrier wave at E. This serves a dual purpose. Firstly, it enables only DUT signal to pass through path E, so that the measurement sensitivity can be set to the DUT signal level, excluding the carrier wave level. The second purpose of having the minor loop and destructive carrier wave interference at E is to prevent overload at the RF port. Path E splits into two more paths: paths F and G. Path F is fed into the RF port of the mixer, and path G is for monitoring the signal output of the minor loop. The mixer IF port signal H is fed into an oscilloscope for monitoring, and lock-in amplifier for data acquisition. The microwave receiver can be tuned to obtain either amplitude or phase sensitivity. For optimal DUT susceptibility amplitude sensitivity, the phase in path A is set such that the slope of the IF voltage V, as a function of phase ϕ, is zero (ΔV/Δϕ = 0). Conversely, for optimal DUT susceptibility phase sensitivity, ΔV/Δϕ is set to maximum. For all the results presented in succeeding chapters using this microwave receiver, amplitude sensitivity mode was used. 10

23 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: V LO ( t) A cos[ t] LO LO The modulated signal passing through the DUT incident at the RF port is: V RF ( t) a( t)cos[ t ( t)] RF The mixer mixes the LO and RF signals. The first order output signal at the IF port, with conversion factor K, is: V V V ( t) KV ( t) V ( t) IF IF IF LO RF ( t) KA cos( t) a( t)cos[ t ( t)] LO LO RF ( t) 0.5KA a( t) cos[( ) t ( t)] cos[( ) t ( t)] LO RF LO 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): RF LO 11

24 V ( t) 0.5KA a( t) IF LO 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 = ( ) 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. F total = F amp + (F mixer 1)/G amp = /10 + (10 0.5/10 1)/10 23/10 = 1.23 F amp The total noise factor is thus dependent only on the noise factor of the amplifier, which is 0.9 db. Theoretically, there is a net increase in signal-to-noise of 1 db, but in practice a net signal gain of 23 db more than makes up for it in this microwave receiver. Also, the carrier signal suppression at junction E largely eliminates non-dut signals from passing through. Succeeding chapters will detail results obtained using this receiver to measure spin wave resonance on thin films with thickness 5 nm (Chapter 5), and propagating spin waves on stripes as narrow as 2 microns, 55 nm thick (Chapter 4). 12

25 2.3 Probe station A probe station was designed and constructed with the help of the Physics Workshop technicians (figure 2.3a). The function of the probe station is to accommodate the use of probes (figure 2.3b). A removable and rotatable aluminium sample stage is positioned between the poles of an electromagnet. An in-plane static field of up to 3500 Oe can be applied across a DUT placed on the sample stage. Two sub-millimetre-sized Picoprobe coplanar probes are positioned over the sample stage facing each other. Each probe tip has three contacts (ground-signal-ground), with 200 μm pitch (signalground distance) (Figure 2.3b). Commercially, the material used for the probe contacts are nickel and tungsten. Nickel is ferromagnetic, and therefore unsuitable for use in magnetic resonance experiments. Thus, we use tungsten probe contacts, which apart from being non-magnetic, is also more durable than nickel. Coaxial lines feed microwave power into the DUT through the probes. The probes are mounted on the arms of two micromanipulators, enabling high-precision movement of the probes along three translation axes and one rotation axis. The electromagnet, sample stage, and micromanipulators are bolted together onto an aluminium platform, so that there is no relative motion between these three core components of the probe station. The whole assembly is placed on an optical bench for vibration isolation. Auxiliary equipment typically used together with the core assembly includes a magnetometer, a Hall probe, an Ohmmeter, and a digital microscope. Figure 2.3a: The probe station. 13

26 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 cm 3 ) was made to enable controlled continuous flow of gas at atmospheric pressure while performing magnetic resonance experiments (Figure 2.4a). The cell houses a coplanar waveguide on which the samples sit. Coaxial cables feed microwave power into the waveguide from one end and carry the transmitted power out through the other end. The cell is fixed between the poles of an electromagnet such that the magnetic field is applied in-plane and parallel to the waveguide (Figure 2.4b). A modulation coil is attached onto the outside of the cell such that the ac field is parallel to the dc field of the electromagnet. 14

27 Figure 2.4a: Gas cell schematic. Figure 2.4b: Photo of the gas cell, showing the coplanar waveguide inside the cell, a sample, coaxial feed lines, modulation coil, poles of the electromagnet, and gas inlets. 15

28 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 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-Lifshitz 30 -Gilbert 31 equation: 16

29 dm dt dm ( M Heff ) M Equation 3.1.1a M dt s M is the magnetisation vector, γ is the gyromagnetic ratio, H eff is the effective magnetic field inside the medium, M s is the saturation magnetisation, and α is the Gilbert damping coefficient. The first term on the right-hand-side of Equation 1 gives rise to precession motion of the magnetisation vector about an equilibrium direction determined by the effective magnetic field, while the second term is the damping term responsible for the magnetisation vector spiralling back to static equilibrium. Assuming a plane wave excitation source, Equation 3.1.1a can be solved together with Maxwell s equations for particular geometries to yield spin wave eigen-modes. The eigenfrequencies depend on sample shape, external field, material parameters, and characteristic wavelength of the excitation source. If the characteristic wavelength of the excitation source is much larger than the attenuation length of spin waves in a particular magnetic material, then the spin wave modes excited in the closest vicinity of the source (for example, right above the signal line of a microstrip) are stationary. For Ni 80 Fe 20 (Permalloy), a low-loss metallic ferromagnet 32, the attenuation lengths of spins waves are typically of the order of microns 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 Ferromagnetic resonance Ferromagnetic resonance (FMR) also known as uniform fundamental mode (FM) is the case where all the spins precess in phase in the magnetic material. For the thin film geometry, the eigen-frequencies for field applied in-plane are given by the well-known Kittel formula 37 : f 2 2 H( H 4 M ) Equation 3.1.2a 17

30 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 Standing spin wave mode Long wavelength spin waves can be excited in confined geometries if surface spins are pinned by surface anisotropy or exchange interactions; the magnetisation at the surface cannot freely precess like in the bulk. These higher order stationary modes with nonzero wave numbers are known as standing spin wave modes (SSWMs). As the name implies, the dynamic magnetisation profile of SSWMs across the confined geometry n (usually the thickness) d forms stationary waves with wave number k (Figure d 1.2.2a). The Kittel equation is then modified 29 : f 2 2 ( H H ex )( H H ex 4 M ) Equation 3.1.3a 2 H ex Dk 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 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 One way to get around this deficiency is by embedding the magnetic sample into a microscopic coplanar waveguide 45. The resultant excitation microwave magnetic field inside the magnetic material is anti-symmetric, thus couples efficiently to the first SSWM with odd symmetry. 18

31 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 nanostripes (MNS). These nano-structures are promising for magnonic 46 and magnetoplasmonic 47, 48 applications. The method is based on injection of microwave currents directly into a sample using a sub-milimetre-sized coplanar probe. Injecting microwave currents into a magnetic material using such a probe was first tried by Prof. Matthieu Bailleul (Institute of Physics and Chemistry of Materials, University of Strasbourg). Our group built on this method to study the spin wave resonance response in this arrangement in detail and explain the underlying physics 28. This is the goal of this thesis chapter. Furthermore, we successfully efficiently excited the first SSWM in an MNS array using the current injection method. The method is quick and conceptually allows easy spatial mapping of magnetisation dynamics with resolution given by the size of the coplanar probe tip. 3.3 Experiment design The nano-structure studied is a Permalloy stripe array (Figure 3.3a). The sample was fabricated using deep ultraviolet lithography by Prof. Adekunle O. Adeyeye s group at the Department of Electrical and Computer Engineering (NUS) 21. A reference film of same thickness was also fabricated. Both films were deposited by electron-beamassisted evaporative deposition. The MNS array geometrical parameters are as follows: Thickness = 100 nm Stripe width = 264 nm Edge-to-edge gap = 150 nm Macroscopic area of array = 4 x 4 mm 2 19

32 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

33 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 To investigate the effect of nano-structuring, microwave current injection was also performed on a reference continuous film. Broadband spin wave spectroscopy using macroscopic microstrip was also performed on the MNS array and reference film for comparison between the two methods. The sample is placed face down, such that the film side faces the microstrip (Figure 2.2.1a). For the MNS array, the sample is oriented such that the stripes are parallel to the microstrip (Figure 3.3c). In all cases, the applied magnetic field is always in-plane and along the stripe. 21

34 Figure 3.3c: MNS array parallel to the microstrip. Ferromagnetic resonance of the MNS array and reference film was done in the frequency range of 4 18 GHz, using both the current injection and microstrip method. Several modes were observed in the FMR spectra of our samples. These are plotted in Figure 3.3d. Before we consider the efficiency of excitation of the various modes using various techniques, one needs to first identify these modes. Section 3.4 and 3.5 deal with the identification of modes in the continuous and patterned film respectively. 22

35 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 nm, the 1 st 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 1 st 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 (H ex = 0) with saturation magnetisation 4πM = ± 40 Oe. The shoulder feature has H ex = 291 ± 4 Oe, and is thus identified as the first anti-symmetric SSWM. This mode is observed in microstrip measurements due to eddy current contribution to the microwave driving field 42. Table 3.5a summarises the fitted parameters. 23

36 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 confinement 49. One then expects the position of the FM peak in the MNS array to lie between the extreme geometrical cases of a continuous film and a long thin rod. In light of this, one may expect the dipolar modes and SSWMs to cross-over or even mix in the MNS array. Thus, the identification of modes in the MNS array is non-trivial. The problem is compounded by the absence of a well-established theory for thick stripes, and accuracy limitations of numerical models in the case of strongly mixed 24

37 modes. Therefore, we employ two independent methods to complementarily and qualitatively identify the modes observed in the FMR spectra of the MNS array: a.) Fit the mode positions with an analytical theory for thin stripes, and b.) simulate the mode profiles and eigen frequencies with our code. Figure 3.5a: Field sweep ferromagnetic resonance of the MNS array at 14 GHz. According to the theory from Guslienko et al. 50, 51, the eigen-frequencies of a nanostructured material should obey the approximate dispersion relation for spin waves valid for continuous films. All peculiarities of confinement due to nano-structuring can be accounted with a dipolar effective demagnetising field, H d. For thin patterned films, the collective fundamental mode is described by equation 11 in reference 49. By including exchange, the equation is modified into: f 2 2 ( H H H )( H H 4 M H ) Equation 3.5a ex d ex The MNS modes are plotted and fitted with Equation 3.5a (Figure 3.3d). For each data set, there is a range of H d and H ex 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 H ex and H d are shown in Table 3.4a. d 25

38 Identification of the 1 st SSWM We observe that the high field mode in the MNS spectra lies close to the 1 st 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 1 st SSWM unchanged 49. We expect similar behaviour for our thick MNS sample. With this foreknowledge, we bias the fittings for this mode by setting H d = 0 in order to obtain physically realistic values of H ex. We obtained H ex = 430 ± 5 Oe for this mode. This value is close to the 1 st SSWM of the reference continuous film (H ex = 291 ± 4 Oe). Therefore, we assign this high field mode in the MNS spectra as the 1 st SSWM of the MNS. To confirm this, we simulated the eigen modes of the MNS array using theory from Tacchi et al. 52, and found a mode with eigen frequency close to the high field mode in the experiment. (Refer to Appendix C for simulation details.) A theoretical eigen-mode with a quasi-uniform distribution of dynamic magnetisation in the array plane but an anti-symmetric distribution across the stripe thickness matches the experimental eigenfrequencies of this mode (Figure 3.5c-b). The dipole field H d 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 1 st SSWM, by process of elimination, it follows that the dominant low field mode could well be the FM. From Equation 3.4a, the slope of the resonance plot is: df dh H Hex 2 M Equation 3.5b f 26

39 From Equation 3.5b, one easily sees that the slope is determined by contribution from df the exchange (increase dh df ) and dipolar (decrease dh ) energies. One observes that the df low field dominant mode of the MNS array has a smaller dh slope 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 H d (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 H ex. In addition, the approximate theory is valid 27

40 thickness for low aspect ratio 1 structures. Therefore, one expects inaccuracy in width extracting a small H ex contribution on top of a strongly dominating H d contribution for the high aspect ratio thickness width MNS array studied here. Identification of the 3 rd 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 3 rd SSWM, hence we set H d = 0 for the fitting, similar to what was done for the 1 st SSWM. We obtained a value of H ex = 1551 ± 4 Oe for this mode. The simulated mode profile for this mode is shown in Figure 3.5c-c. The mode profile is symmetric across the thickness, with two nodes. Thus, this mode is identified as the third (out-of-plane symmetric) exchange mode of the MNS array. Note that the close proximity of this mode with the FM is partially responsible for the distortion of the FM profile from hybridization, as mentioned before (Figure 3.5c-a). 28

41 Figure 3.5c: Simulated in-plane dynamic magnetisation 2D profiles across the crosssection of a single nanostripe in an array. Numbers on the axes are the mesh indices across the thickness on the vertical axis and across the width on the horizontal axis. Colours are proportional to the real part of the in-plane dynamic magnetisation vector. 29

42 Resonance feature H ex (Oe) H d (Oe) Mode identification MNS high-field (Green diamond) MNS low-field (Blue triangle) MNS extra shoulder (Purple star) Film high-field (Black circle) Film low-field (Red square) 430 ± 5 0 MNS 1 st SSWM 670 ± ± 70 MNS FM 1551 ± 4 0 MNS 3 rd SSWM 0 0 Film FM 291 ± 4 0 Film 1 st 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 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- 30

43 plane component of the current density is important and may give rise to significant inplane microwave magnetic field. The full derivation of the theory suggested by Prof. Mikhail Kostylev is presented in Appendix B. To enable analytical solutions, the current density is assumed to be out-of-plane and uniform at the probe tip s point of contact with the film. Using this theory, we calculate the radial in-plane (figure 3.6.1a), and in-depth (figure 3.6.1b) microwave current distributions of an infinite continuous film 100 nm thick. Figure 3.6.1a: Radial in-plane microwave current density at the film surface. 31

44 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 1 Bessel function of the second kind (which approximates as decay). As plotted in r Figure 3.6a, the current density is concentrated directly underneath and in the near proximity of the probe tip due to microwave current repulsion far from the source. The in-depth component of the microwave current density is given by a hyperbolic sine function (which approximates as linear decay). As plotted in Figure 3.6.1b, our calculation shows that the current density is concentrated at the surface at which the current from the probe is incident on, and is zero at the opposite buried interface. Note that this distribution is very similar to the perfect microwave shielding effect of subskin-depth thin conducting films

45 Figure 3.6.1c: Magnitude of microwave magnetic field in the vicinity of the probe tip. White is most intense, while purple is least intense. Both the in-plane radial and in-depth components of the microwave current induce an in-plane microwave magnetic field with intensity profile shown in figure 3.6.1c. This in-plane circulating field (figure 3.6.1d) is concentrated near the probe tip. This in-plane component of the microwave magnetic field is responsible for the efficient excitation of the fundamental uniform mode. The in-plane current between the probes is significantly diffused due to microwave current repulsion (figure 3.6.1a). The in-plane radial currents from each of the three probe tips do not perturb each other since the distance between the probe tips (200 μm) is much larger than the microwave current decay length (a few μm). Without diffusion, this current would have induced an anti-symmetric field across the thickness of the film, which would in-turn, efficiently drive the first SSWM. Therefore, this field is not a 33

46 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 Current injection method on nanostripes In the MNS array, the absence of medium continuity in the direction of the array periodicity does not allow current to diffuse in the array plane as in the case of a continuous film discussed before. The microwave current remains confined in the contacted stripes between the probe tips (figure 3.3b). This produces a large in-plane current density over a large length, given by the pitch of the coplanar probe (0.2 mm). Since the cross section dimensions of the MNS are comparable to the microwave skin depth (of the order 100 nm), this current flowing through the stripes can be considered uniform. The resultant microwave magnetic field of this in-plane current is antisymmetric across the MNS depth (figure 3.6.2a); this is essentially similar to the simple case of the magnetic field generated by a wire carrying a dc current. This antisymmetric microwave magnetic field efficiently excites the first anti-symmetric SSWM. 34

47 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 1 field incident on the nearest neighbour stripes (figure 3.6.2b). This field decays as r away from the source, essentially the same as the simple case of the magnetic field of a dc current-carrying wire. If we consider only the first nearest neighbour interactions, then the out-of-plane field contributed by each individual stripe would be cancelled out by their respective nearest neighbour stripes, except the outer 2 stripes, where there are unbalanced net out-of-plane field components. This out-of-plane microwave magnetic field incident near symmetrically on the outer 2 stripes is able to drive the fundamental dipole mode inside those 2 outer stripes. Thus, the amplitude of the fundamental dipole mode should be theoretically 25% that of the first SSWM, since the fundamental mode is excited in only 2 out of 8 of the stripes contacted. This is indeed approximately what is experimentally observed in the ratio of the amplitude of the first SSWM to the fundamental mode (figure 3.6.2c). 35

48 Figure 3.6.2b: Anti-symmetric microwave magnetic field (h) generated outside the stripes due to microwave current I flowing inside along the stripes. Figure 3.6.2c: Amplitude ratio of the first SSWM to the fundamental mode for the MNS array by the current injection method. The missing data points in the vicinity of 16 GHz are due to the particular microwave generator unable to regulate constant power output at the power level required for spin wave excitation in that frequency range. 36

49 3.6.3 Microstrip method on continuous film and nanostripes Note from figure 3.4a and 3.4b that the fundamental mode is dominantly excited by the microstrip method for both the MNS array and continuous film. The first SSWM is also excited, but much less efficiently, especially in the case of the continuous film. To explain this, consider the radiation field of the microstrip (figure 3.6.3a) 55. Figure 3.6.3a: Radiation field lines of a microstrip in the parallel orientation. An in-plane microwave magnetic field is present on top of the microstrip. When the ferromagnetic continuous film or MNS array is placed on top of the microstrip, this near-uniform field efficiently drives the fundamental uniform precession mode. This is why the uniform mode is dominant (figure 3.4a & 3.4b). The first SSWM is also excited by the microstrip, but much less efficiently than the fundamental mode. This is due to the eddy current shielding effect of sub-skin-depth thin films resulting in a quasi-linear profile of the microwave magnetic field across the film thickness 42. The first SSWM is not strongly excited in both these cases due to lack of interfacial pinning. 37

50 3.6.4 Out-of-plane microwave magnetic field contribution Recall earlier that it was proposed that out-of-plane microwave magnetic field is responsible for excitation of the fundamental mode in the outer two stripes of the MNS array by current injection method (figure 3.6.2b). To further investigate the contribution of this field component to excitation of the fundamental mode, additional measurements with the microstrip were performed in the nominally-called perpendicular orientation. This is where the microstrip is aligned perpendicular to the applied static field, with the MNS array still parallel to the field (figure 3.6.4a). Note the difference in geometrical orientation compared to the parallel orientation in figure 3.3c. Figure 3.6.4a: The perpendicular orientation of microstrip. In the perpendicular orientation, only the out-of-plane component of the microstrip s magnetic radiation field is able to contribute to spin wave excitation; the in-plane component is parallel to the static magnetic field and hence does not contribute to spin wave excitation (figure 3.6.4b). In the spin wave spectra for the continuous film, the signal of in the perpendicular orientation is 30 db smaller than that of the parallel orientation. This is due to large ellipticity of magnetisation precession in metallic ferromagnetic films, where an in-plane microwave magnetic field drives magnetisation precession much more efficient than an out-of-plane field. In addition, the out-of-plane component of the microwave magnetic field is present only near the edges of the microstrip where the associated dynamic electric field curls down to the embedded ground plane. 38

51 Figure 3.6.4b: Radiation field lines of a microstrip in the perpendicular orientation. Considering the out-of-plane microwave magnetic fields in the nanostripe (figure 3.6.4b), one might wonder why this anti-symmetric is able to excite the uniform mode. The out-of-plane component of the excitation field is localised at the edges of the microstrip. Absorbed electromagnetic energy is proportional to the dot product between the driving field and the magnetisation vector. While it is true that the direction of this field is opposite at opposite edges of the microstrip, one must also bear in mind that the direction of magnetisation precession is also reversed. This means that the total energy absorbed at resonance has the same sign on either sides of the microstrip. In addition, since the microstrip is much wider than the typical attenuation length of spin waves, local magnetisation dynamics at the edges are not able to couple to one another. Thus, the uniform mode is driven locally at the edges of the microstrip. We stress that for the MNS array, the fundamental mode is of similar order of absolute magnitude in both the perpendicular and parallel microstrip orientations (figure 3.6.4c). This is very different from the case for the continuous film, where the signal obtained in the perpendicular orientation is much smaller than that in the parallel orientation, as discussed earlier. This result is in good agreement with evaluation of ellipticity of precession for MNS from numerical simulations using Tacchi et all s theory 52. This confirms that the out-of-plane component of the microwave magnetic field due to the current-carrying stripes is responsible for driving the fundamental mode observed in the current injection spectra (figure 3.4a). 39

52 Figure 3.6.4c: Field sweep spin wave resonance of the MNS array at 14 GHz. 40

53 3.7 Microwave current injection as a characterisation tool We now turn attention to evaluate the merits of the current injection method as a characterisation tool. As demonstrated, the method is able to characterise the magnetisation dynamics of ferromagnetic materials similar to standard broadband planar waveguide methods. More than that, the method enables spatial mapping of local macroscopic magnetisation dynamics with resolution determined by the size of the coplanar probe (in our case, 400 μm). The resolution can be improved by using the smallest commercially available probe (100 μm). Even though the resolution is macroscopic a far cry from the other two spin wave spectroscopy techniques with spatial resolution, namely Brillouin light scattering 56 and magnetic force microscopy 57, 58 the current injection method using a coplanar probe is far simpler and quicker to utilise. One drawback of this method is that being a contact method, damage to samples usually unavoidable. We now consider the physical contact between the coplanar probe and the material being probed. The probe has a built-in spring which applies a constant force onto the surface being probed. This ensures good physical contact between the tip and the probed surface without risking tip breakage or loss of contact from mechanical vibration. The standard probe tips are available in either tungsten (for long-lasting tips) or nickel (for better electrical contact and minimal sample damage). In our setup, we use tungsten tips since we require robustness in our experiments, and the alternative nickel is magnetic and hence undesirable in spin wave experiments. The downside of probing with a hard tungsten tip is potential physical damage to the surface being probed, especially if the material is a soft metal. The probed samples were inspected with an optical microscope for sample damage. No trace of physical damage was observed on the continuous film probed. Thus, Permalloy appears to be hard enough to resist the pressure exerted by the probe tips. However, scratch marks were left by the probe on the surface of the MNS array (figure 3.7a). Nano-structuring has weakened the material; making it mechanically softer than the continuous film. 41

54 Figure 3.7a: Scratch marks left by the tips of the coplanar probe on the MNS array. The depth profile of the scratch in figure 3.7a is shown in figure 3.7b. The distance between the three scratch dips is consistent with the pitch of the probe (200 μm); these are not sample fabrication defects, but that caused by the physical contact of the probe. The width of the trench left by the central tip is about 10 μm wide. Note that this dimension cannot be used to estimate the number of contacted stripes as discussed before because this is the size along the stripe length direction, not across the stripe width direction. Furthermore, an indentation is typically larger than the size of the object which causes the indentation. Important from the scratch profile is the depth of the indentations; as long as the probe exerts minimal force on the MNS array (just enough to ensure contact), the stripes are not cleaved by the probe tips. The indentations depths are of the order of nanometres; not enough to cleave the thick 100 nm film in this case. This result implies that in order to use this method as a non-destructive spin wave characterisation tool, a probe should be designed with a non-magnetic soft metal tip and minimal force should be exerted onto the probed surface. 42

55 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 La 0.7 Sr 0.3 MnO 3 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 10 3 Ω, which is consistent with the typical resistivity of unannealed La 0.7 Sr 0.3 MnO No spin wave resonances were observed in the current injection spectra through the half-metal, indicating that the very low microwave current flowing through the poorly conducting material does not induce sufficient microwave magnetic field to drive spin wave excitation. Therefore, we conclude that if the material is continuous but poorly conducting (like most ferrites), the current injection method cannot be used. In addition to the requirement of the sample being electrically conducting, there must also be possibility of current conduction between the signal and ground tips of the probe. This means that only a subset of patterned films can be probed with the current injection method. This method cannot be used on a dot array for example 61, even if the material is conducting, due to lack of current continuity. The requirement of current conduction continuity thus limits the method to continuous films, stripes, anti-dots 62, 63, or similar nano-structures where continuity of the conducting phase is preserved across the whole distance between the tips. 43

56 Finally, impedance mismatch is a potential issue in the current injection technique. In this method, the probed sample is essentially the load at the terminus of a microwave transmission line. This means that for efficient transfer of microwave power into the load, the load should have impedance matching that of the transmission line. However, designing the sample to impedance-match with the probe defeats the purpose of the current injection technique as a characterisation tool in the first place. In the work presented here, the impedance mismatch was quite significant, resulting in most of the incident power reflected from the probed MNS array. Assuming a purely resistive MNS array load of 130 Ω and a nominal coplanar probe characteristic impedance of 50 Ω, the transmission coefficient is calculated to be 56 %. The transmission coefficient of a purely resistive 3 Ω continuous Permalloy film is even lower at 11 %. However, these estimates disregard reactance contributions to the impedance and insertion loss. In the actual experiment, the typical reflected power is nearly at the same level as that of the incident power for both the MNS array and continuous film. 3.8 Chapter conclusion In this chapter, the microwave current injection spin wave spectroscopy technique is demonstrated on a Permalloy nano-stripe array and continuous film. A sub-milimetre coplanar probe was used to inject microwave current into the samples studied, and the spin wave excitation response compared with standard macroscopic planar waveguidebased spin wave spectroscopy. The current injection method is able to efficiently excite anti-symmetric standing spin wave modes; these modes provide important interfacial material properties, and the modes are often lacking in planar waveguide-based methods. The current and radiation field distributions of both the current injection and planar waveguide techniques were used to explain the mode amplitudes observed in the spin wave spectra. It is found that the in-plane current flowing through the contacted nano-stripes induces an anti-symmetric dynamic magnetic field inside the stripes, which efficiently drives the first anti-symmetric standing spin wave mode. The current injection technique is quick and allows easy spatial mapping of magnetic properties of conducting materials with resolution down to 0.1 mm. 44

57 Chapter 4 Propagating spin wave spectroscopy The sections in this chapter are organised as follows. A brief theory of propagating waves in continuous and laterally confined geometry is first presented, followed by case for work. In the next section, the experiment setup, procedure and data acquisition methods are highlighted. The presentation and discussion of results in this chapter are then divided into three main sections, each containing sub-sections of their own. Section 4.5 then deals with the characterisation of the fundamental propagating surface mode in the series of stripes studied. Section 4.6 focuses on the characterisation of high-order width modes observed in the stripe with the highest aspect ratio studied in this work. Section 4.7 elucidates the antenna near-field effect, how it may be quantified, and how it affects the data. The chapter ends with a summary of major findings of the presented work. 4.1 Background If the characteristic wavelength of the excitation source is smaller or comparable to the mean free path of spin waves in a particular magnetic material, the resultant highly localised excitation field can excite propagating modes. The wave equation of propagating spin waves is obtained by solving the Landau-Lifshitz-Gilbert equation together with Maxwell s equations in the magnetostatic limit and material constitutive relations. The resultant wave equation is known as Walker s equation 64. Since these are obtained in the magnetostatic limit, these propagating modes are known as magnetostatic spin wave modes. The dispersion relations of propagating spin waves in a specific geometry are obtained by solving Walker s equation subject to boundary values of the required geometry. 45

58 4.1.1 Propagating modes in continuous films For infinite continuous films, there are three possible field orientation and propagation direction geometries, each yielding different propagation modes and dispersions: Figure 1.2.3a: Dispersion relations of magnetostatic modes in a thin film 65. Forward volume mode 66 If the film is perpendicularly magnetised, forward volume spin waves can propagate tangentially in-plane. The characteristics of this mode is indicated by the name; being a volume mode, it permeates the entire bulk of the solid, and the phase and group velocities are both positive. Backward volume mode 67 If the film is tangentially magnetised, two possible wave modes can be excited. Backward volume waves propagate parallel to the applied field. An interesting property of this mode is that while the phase velocities are positive, the group velocities are negative, hence the name backward volume waves. Surface mode 67, 68 In the tangentially magnetised configuration, a second mode of wave propagation is possible. Surface spin waves can propagate perpendicular to the applied field. This 46

59 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 Propagating modes in laterally confined geometry In the past 15 years, research in the field has focussed on the effects of lateral confinement on propagating spin waves modes. These have been studied on microstripes and patterned arrays using inductive techniques in the frequency domain 35, 70-75, inductive techniques in the time domain 34, 76, Brillouin light scattering (BLS) in reciprocal space 77-80, micro-bls 81, 82, and Kerr microscopy in the time domain 83. Lateral confinement results in three significant deviations from the infinite continuous film case. First, lateral confinement results in quantization of magnetostatic modes across the confined dimensions 36, 50, 77-79, 84, 85. Due to confinement, the dipolar eigenfunctions of stripes take the form of sinusoids, analogous to the form of spin wave resonance modes in a perpendicularly magnetised film 50, 79, 84. Thus, one may observe higher order confinement modes in the measured spin wave spectra. Second, the static and dynamic demagnetising fields in confined geometry would increase and become non-uniform. This in effect, shifts the position and slope of spin wave dispersion 79, 86, 87. Thirdly, the extreme case of increased static demagnetising field is the formation of potential wells at the edges of tangentially magnetised stripes where demagnetising fields can be very large 33, 70, 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,

60 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. 48

61 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 to scale with aspect ratio. In particular, demagnetising field 96 is directly proportional to aspect ratio for stripes magnetised tangentially perpendicular to the long axis 70, 79. The stripe array with 30nm the highest aspect ratio studied to date was by Huber et al m 1 74, while the previously done study with the largest range of aspect ratios ( ) was by Vlaminck et al. on forward volume waves 73. Considering all relevant previous work to date, there is little work done to systematically investigate the effects of lateral confinement on spin wave propagation. The series of thick magnetic stripes studied in this chapter systematically span the largest range of aspect ratios to date, covering two orders of magnitude (Figure 4.2a). The stripe with the highest aspect ratio studied in this work falls just slightly short of the aspect ratio of Huber et al. s array 74. Figure 4.2a: Aspect ratio of stripes studied in the field of propagating spin waves. Improved sensitivity Lastly, we have built a highly sensitive microwave detector which is able to detect extremely small microwave signals (see section 2.2.3) specifically for this propagating spin wave project. Hence, the possibility of detecting higher order width and edge modes, both of which are relatively elusive by the induction method, warrants further investigation at vastly improved sensitivity levels. 49

62 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 Al 2 O 3 insulating layer, and 3 is the receiving antenna (the similar structure to the right is the output antenna). (b) Enlarged view of the antenna cross-section. The origin x=0 of the frame of reference coincides with the symmetry axis of the input antenna. 50

63 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 ( )]. In addition, large patches of continuous film (3.7 x 3.7 mm 2 ) were patterned on the same wafer together with the stripes, and all underwent the same fabrication process. These serve as reference continuous films, and were large enough to enable characterisation using flipchip broadband FMR (figure 2.2.1a). A coplanar waveguide (CPW) design was used in this work to facilitate connection with a coaxial line via a sub-millimetre coplanar probe. The CPW antenna geometrical parameters are as follows: the conductor widths and separation gaps are 1.5 µm, and the thickness is 200 nm. The distances between the excitation and detection antennas were varied from 12 to 110 µm. Far away from the stripes, the CPW lines gradually become larger, but still maintain geometrical proportions. This is to ensure matched transmission line characteristic impedance throughout, which is calculated to be 67 Ω 97. Finally, the CPW lines terminate at 100 µm sized contact pads (figure 4.3c). A 30 nm thick aluminium oxide spacer ensures no direct dc electrical contact between the overlaying gold antennas and the underlying Permalloy stripes. 51

64 Figure 4.3b: Optical micrograph of a typical PSWS experiment showing a magnetic strip and overlaid coplanar lines. Figure 4.3c: Optical micrograph of spin wave device showing coplanar lines leading to contact pads. 52

65 4.4 Experimental procedure 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 GHz microwave power at 10 dbm was fed into the receiver; this power is split into two channels, with one driving the mixer s LO port, and the other exciting spin waves on the DUT. 10 GHz was used because the microwave receiver is optimised for that frequency. Spin waves propagate along the magnetic stripe, and get detected at the second antenna (figure 4.3.1a). The detected signal gets fed into the RF port of the mixer. The resultant output signal at the mixer s IF port is a product of the reference driving signal, and the spin wave signal (see section 2.2.3). This signal gets fed into a lock-in amplifier and is recorded as a function of sweeping field applied tangentially to the magnetic stripes; this is the Damon-Eshbach geometry. A modulating field of about 9 Oe (rms) at 220 Hz modulates the sweeping field. The lock-in amplifier refers to this same modulating field, and the resultant signal acquired is proportional to the field derivative of the microwave susceptibility. A typical raw trace of the data is shown in figure 4.4.1a. Note the oscillatory spin wave signal as a function of field due to phase selection at the detection antenna. 53

66 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 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 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

67 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) Wave number space Spin waves are excited by the microwave magnetic field of the microwave current flowing through the coplanar antennas. The wave number spectrum of spin wave excitation and absorption depends on the square of the Fourier transform of the current density in the coplanar antenna 98. The skin depth of gold at 10 GHz frequency is calculated to be 0.8 μm 99. The skin depth is much larger than the thickness of the coplanar line (0.2 μm) and about half the widths of each conductor (1.5 μm). Hence one expects the microwave current density in the coplanar lines to be practically uniform across the thickness of the coplanar lines. Laterally, the current density is described by 55

68 Dmitriev 100 ; the current is concentrated on the outer edges of the central conductor and the inner edges of the ground planes, as shown in figure 4.4.3a. The negative current densities in the ground conductors indicate the return current. Figure 4.4.3a: Spatial current density of the coplanar antennas. The central conductor, ground conductors, and separation gaps are all 1.5 μm wide. Figure 4.4.3b: Spatial Fourier transform of the current density of the coplanar lines, assuming uniform current density distribution. Although the current density distribution is non-uniform laterally, the Fourier transform of this non-uniform distribution is quite close to the case where an uniform distribution is assumed 98. For ease of computation, Fourier transform for the uniform current 56

69 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 k 0 = 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 : k 0 2 w 2 For simplicity of further data analysis, only the dominant band ( cm -1 ) is considered to effectively contribute to spin wave excitation 43, Extracting dispersion The signal obtained in transmission (wave propagation from one antenna to another), is the sum of two signals; direct electromagnetic induction through air, and spin wave transduction. Assuming plane wave propagation, the superposition of the two waves is: Asin( t ) Bsin( t kx) The first term in the sum represents direct electromagnetic induction through air, while the second term represents spin wave transduction after propagating through the Permalloy stripe a distance x (which is the separation distance between both antennae). ϕ is an arbitrary initial phase difference between the two waves. The typical group velocity of magnetostatic surface spin waves ( 10 µm/ns for a 100 nm Permalloy film) is much smaller than the speed of light. Hence from the spin wave s perspective, the electromagnetic wave propagating through air has virtually no phase accumulation over such small propagation distances. For the case where A = B (where both amplitudes are the same), the expression can be rewritten as: 2 t kx sin( t ) sin( t kx) 2sin cos 2 kx 2 57

70 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: cos 2 kx 1 2 Or when (where n = integer): kx 2 n The phase difference between each successive maxima of the power absorption interference spectra is thus: 2 k Equation 4.4.4a x This method of using phase accumulation to extract dispersion was also used in references 35, 73, 85, 102, 103. From this, one expects that the number of interference patterns in the spectra increase with propagation distance. This is confirmed experimentally as shown in figure 4.4.4a. Note that the envelope band over which spin waves occur does not change with propagation distance; only the number of oscillations within the interference pattern. One finds a linear relationship between the numbers of amplitude oscillations with propagation distance (figure 4.4.4b). 58

71 Figure 4.4.4a: Normalised spin wave raw traces at 10 GHz for a 110 nm thick, 100 µm wide Permalloy stripe, at various propagation distances, x. 59

72 Figure 4.4.4b: Number of amplitude oscillations at various propagation distances. Hence, when field (or frequency) is swept around the resonance, different wave numbers are selected according to the dispersion relation and within the bandwidth of the spatial Fourier transform of the current density of the excitation antenna 73. Note that the envelope of the interference pattern in the raw trace reflects the wave number distribution of the Fourier transform of the coplanar antenna. Using this fact, one can extract the dispersion from a raw trace as follows: 60

73 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 k 0 of the coplanar line (figure 4.4.4c-b), and each successive maxima/minima has a total phase accumulation of 2π (figure 4.4.4c-b). This way, one can construct a dispersion relation (figure 4.4.4c-c) by mapping each extrema on the raw trace (figure 4.4.4c-a) to a particular wave number of the Fourier spectra of the coplanar line (figure 4.4.4c-b). This procedure is repeated for all the stripes studied in this work, and presented here after. Note that in contrast to Bao et al. s proposed absolute phase method using a VNA 75, this relative phase difference method is much simpler. Also note that since the experiment is field-resolved, the plot is technically a pseudo-dispersion relation, rather than a proper dispersion plot of frequency versus wave number. 61

74 4.5 Magnetostatic surface mode in confined stripe geometry Dispersion For a continuous film, magnetostatic surface modes (MSSM) propagate according to the dispersion relation: f 2 2 H( H 4 M ) 4 M 2 (1 e 2kd ) Equation 4.5.1a Note that by setting k = 0 (wave becomes stationary), one essentially recovers the Kittel formula for FMR in a tangentially magnetised continuous film (Equation 3.1.2a). To plot dispersion relation, one has to consider the optimal propagation distance from which to extract dispersion from. As seen in figure 4.4.4a, the number of oscillations is proportional to the propagation distance. Thus, more data points can be extracted from large propagation distances. In addition, one should also minimise antenna near-field effects by maximising the separation gap between antennae (see section 4.7). However, signal attenuation at large propagation distances place a practical upper limit on the antennae gap from which sufficient signals are available. For the 110 nm series reliable data could be obtained at 60 µm separation gap between the antennae; for the 80 nm and 55 nm series, due to more attenuation for thinner films, the distance is 30 µm. For these propagation distances, the dispersion relations of the MSSM in the Permalloy stripes are shown in figure 4.5.1a for the range of thicknesses and widths studied. Due to sample fabrication defects, some antennae and/or Permalloy stripes were malformed, resulting in gaps in the data. The black dotted curves are the theoretical dispersion relations for infinite continuous films calculated using equation 4.5.1a based on the extracted saturation magnetisation values obtained from FMR (figure 4.5.1b). One immediately observes that the dispersion relation shifts to high field upon narrowing of stripe width. We claim that this is due to the effect of the static demagnetising field. As the stripes become narrower, the effective demagnetising field becomes larger, resulting in decrease in internal field for a given applied external field 70, 79, 96 (see section 4.5.2). This shifts the dispersion up-field. 62

75 Figure 4.5.1a: Dispersion relation of Permalloy stripes at 10 GHz. It is found that within the experimental accuracy, for a particular stripe thickness, the slope of the dispersion is independent of stripe width, at least up to the stripe with the 110nm highest aspect ratio. The dynamic confinement effects are known to modify 2 m 63

76 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, H d, into the dispersion relation for continuous film similar to 104. Thus, equation 4.5.1a is modified into: f 2 ( H H d )( H H d 2 4 M ) 4 M 2 (1 e 2kd ) Equation 4.5.1b Using the modified dispersion law, one can set H d as the fitting parameter and extract the effective demagnetising fields. The saturation magnetisation parameter, M, was extracted separately from reference film FMR data (figure 4.5.1b). M is assumed to be constant across all strip widths for particular thicknesses in the dispersion fitting. The dispersions were fitted with the effective demagnetising field as the only free fitting parameter; all other parameters were assumed to be constant. In particular, the saturation magnetisation, gyromagnetic ratio, and thicknesses of the stripes were assumed to be identical to their respective reference continuous films, since they were all deposited together on the same wafer in the same deposition process, and underwent the same lithography fabrication process. The fits are shown with solid curves in figure 4.5.1a. 64

77 Figure 4.5.1b: FMR data for the reference films showing extracted saturation magnetisation values. Solid lines are fits with Kittel s formula (equation 3.1.2a). As shown, the data fits well to the modified dispersion law across the range of aspect 110nm ratios studied, except for the stripe with the highest aspect ratio. For that 2 m particular stripe, there was noticeable deviation from equation 4.5.1b. This is postulated to be due to the fit assuming an uniform demagnetising field 86, but in actual fact, this particular stripe has the most non-uniform demagnetising field profile across its width amongst the stripes studied (see section 4.5.2). The thin film model assuming a uniform effective demagnetising field is insufficient to properly describe a narrow stripe of such high aspect ratio. Furthermore, for large aspect ratios, the geometrical confinement also affects the dynamic dipole field. The static demagnetising field shifts dispersion upfield, while the dynamic dipole field increases the dispersion slope with respect to the continuous case 79,

78 110nm Figure 4.5.1c: Experiment and simulation MSSM dispersion for the stripe. 2 m Solid curves are fits with equation 4.5.1b. 110nm To properly model the experiment dispersion of the stripe, numerical 2 m simulations were performed. The demagnetising field profile (see figure 4.5.2a-a in section 4.5.2) was used as the stripe ground state. From this, the eigen-fields for particular wave numbers at 10 GHz frequency were numerically simulated. The simulated dispersion is plotted together with experiment data in figure 4.5.1c. To compare with the modified continuous film dispersion, both the experiment and simulation results were fitted with equation 4.5.1b. Note that both the simulation and equation 4.5.1b did not adequately model the experiment result. The slope of the experimental dispersion is smaller than predicted by simulation. The failure of the simulation to model the experiment data might be due to unaccounted peculiarities in 110nm this particular stripe. To further elucidate the matter, the stripe s dispersion 2 m was fitted with equation 4.5.1b with additional free fitting parameters (figure 4.5.1d). 66

79 110nm Figure 4.5.1d: stripe dispersion fittings with equation 4.5.1b. 2 m One notes from figure 4.5.1d that better fits could be obtained by setting the saturation magnetisation, M, or the thickness, d, as fitting parameters in addition to the effective demagnetising field, H d. 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 110nm 2 m 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. 67

80 4.5.2 Static demagnetising field simulations In order to evaluate the validity of introducing an effective demagnetising field into the dispersion model, the demagnetising field profiles of the stripes studied were numerically simulated using LLG Micromagnetics Simulator software 105 (refer to Appendix C). To simplify computation, the stripes were assumed to be infinitely long, thus reducing the problem into a two dimensional one. Figure 4.5.2a: Simulated demagnetising field profiles in infinitely long Permalloy stripes. 68

81 The simulated demagnetising field profiles of two extreme cases (smallest and largest aspect ratios) are shown in figure 4.5.2a. The stripe with the lowest aspect ratio 55nm is expected to have the least demagnetising field, and one clearly sees that 100 m this is indeed the case in figure 4.5.2a-b. Demagnetising fields are only significant near the edges of the stripe, just roughly 2% of the total width. The small demagnetising field across the bulk of the stripe is practically uniform. For the other extreme case, the stripe 110nm with the highest aspect ratio is expected to have highly non-uniform 2 m demagnetising fields across the width, and indeed this is the case in figure 4.5.2a-a. Note that in this narrow stripe, significant demagnetised edge regions are present for low applied fields. 69

82 Figure 4.5.2b: Comparison between experiment and simulation effective demagnetising fields. 70

83 The simulated effective demagnetising field for each strip is approximated by taking the mean value across the strip width (figure 4.5.2a). The experimentally extracted effective demagnetising fields are compared against these simulated values in figure 4.5.2b. It is observed that the simulation tends to overestimate the experimental effective demagnetising field. Possible explanations for these discrepancies are as follows: Firstly, taking the arithmetic mean across the demagnetising field profile as an effective value may not be the appropriate statistical approach. Arithmetic mean tends to be disproportionately weighted towards large values, and demagnetising fields can be very large at stripe edge regions while only occupying relatively small volumes 79. Secondly, considering the demagnetising field profile alone is insufficient. One also needs to account for the non-uniformity of the static magnetisation and spin wave mode profile for a more accurate analysis. The effective demagnetisation factor for a particular spin wave mode is the proportional to the overlap integral between the mode profile and the demagnetisation field profile 84. In addition, dynamic effects due to confinement affect the dispersion slopes 79, 86 ; this is not accounted for by the static demagnetising field fitting parameter in equation 4.5.1b. Hence, merely taking the mean value of the simulated demagnetisation field profile alone is insufficient to accurately model the actual effective demagnetisation field for the particular spin wave mode. 71

84 4.5.3 Group velocity The group velocity, V g, can be calculated from the dispersion data using the relationship: V g f H 2 Equation 4.5.3a k H k H The term is simply the slope of the field-resolved dispersion, and the k obtained by differentiating equation 4.5.1b: f H term is f 2 ( H Hd 2 M ) Equation 4.5.3b H f From equation 4.5.3a, the experimentally calculated group velocities are plotted in figure 4.5.3a as function of wave number and tabulated in table 4.5.3a for k 0 = rad/cm. 72

85 Figure 4.5.3a: MSSM group velocities of Permalloy stripes. 73

86 From figure 4.5.3a, one sees a general trend that the group velocity increases with film thickness. This is what one expects since the slope H k 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 obtain film thickness:, where the group velocity is proportional to k V g M d 8 e k f 2kd 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 110nm the peculiar stripe is excluded. If we attribute the scatter to the accuracy 2 m limitations of the experiment, then we may conclude that for the aspect ratios and wave number range investigated here, the group velocity is width-invariant for a particular thickness. Thickness (nm) Group velocity (μm/ns) Table 4.5.3a: Group velocities at 10 GHz and k 0 = rad/cm. 74

87 4.5.4 Attenuation and relaxation We now turn attention to the propagation attenuation and relaxation characteristics of MSSM in the stripes studied. The total signal transmitted,, from the excitation antenna to the detection antenna is 100 : (, k) T ( ) P(, k) T ( ) Equation 4.5.4a e d Te and Td are antenna excitation and detection efficiencies, respectively. P is the spin wave propagation loss. For identical antennae, T d T e *; 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 ( T d T e * 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 L d is the attenuation length, defined as the propagation distance when the signal has dropped to 1/e from its initial value. P 0 P e x L d Equation 4.5.4b From equation 4.5.4a, and taking logarithms, equation 4.5.4b becomes: ln 2lnT ln P x L d 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. 75

88 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 k 0 = rad/cm (figure 4.5.4a). Figure 4.5.4a: Example VNA raw trace, showing definition of amplitude ΔS21. The logarithmic amplitudes of the spin wave signals at various antennae gaps are plotted in figure 4.5.4b. Note the linear trend in the logarithmic plot of the data (figure 4.5.4b), verifying the exponential decay assumption made earlier. It is noteworthy that the antenna near-field effect (see section 4.7) results reduced effective propagation distance compared to the physical separation distance between the excitation and detection antennae. This shifts all the data points horizontally by a constant amount. However, this would not affect the slope of the plot. The attenuation lengths extracted from the fittings were tabulated in table 4.5.4a. 76

89 Figure 4.5.4b: ΔS21 amplitude at various antennae gaps at 10 GHz and k 0 = rad/cm. Solid lines are fits to extract the attenuation lengths. Comparison can be made with Sekiguichi et al. s similar work involving spin wave 35nm propagation through a Permalloy stripe with aspect ratio over distances m 50 μm 34. In that work, the attenuation length was determined to be 15 μm, which is similar to the values for the stripes studied here (table 4.5.4a). The relaxation time τ r of MSSM can be calculated from the group velocity, V g, and attenuation length, L d, similar to 95, 106 through the simple relationship: r( MSSM) L V d g 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 H i is the internal field: 2 ( H i 1 2 M ) r( MSSM) Equation 4.5.4c 77

90 The calculated attenuation lengths, relaxation times, and Gilbert damping coefficients were tabulated in table 4.5.4a. Note the atypical losses in the 80 nm thick stripes. Stripe Group Attenuation Relaxation Gilbert thickness (nm) velocity length (μm) time (ns) damping (μm/ns) coefficient (10-3 ) ± ± ± ± ± ± ± ± ± ± ± ± 0.3 Table 4.5.4a: Stripe MSSM propagation and attenuation parameters at k 0 = 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, r( FMR) 2 H H FWHM FWHM f ( H i, reformulated from Stancil s equation 17c 95, is: Equation 4.5.4d 2 M ) The Gilbert damping coefficient is given by Stancil s equation 5 95 : H FWHM 2 f 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,

91 Figure 4.5.4c: FMR line widths of reference films. Film thickness Saturation Relaxation time Gilbert (nm) magnetisation at 10 GHz (ns) damping (emu/cm 3 ) coefficient (10-3 ) ± ± ± ± ± ± 0.3 Table 4.5.4b: Reference film properties. The FMR relaxation times and Gilbert damping coefficients are tabulated in table 4.5.4b. The values for the 110 nm and 55 nm films are typical for Permalloy 24, 32, 70, 71. Note the ~25% larger than typical intrinsic damping for the 80 nm thick reference film compared to the other two. This might be due to fabrication defect resulting in a low quality magnetically lossy film. This is noticeably evident in the plot in figure 4.5.4c. This might explain the reduced attenuation length and relaxation time of MSSM propagation in the 80 nm stripes (table 4.5.4a). Note also that the Gilbert damping coefficients calculated from the MSSM propagation data (table 4.5.4a) were larger (~ 25%) than those extracted from the reference film 79

92 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 k FMR = 0 rad/cm while the value obtained from MSSM data was at a different wave number, k 0 = rad/cm. One notes that the inhomogeneous line width broadening may be slightly different between k FMR = 0 rad/cm and k 0 = 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,

93 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 nonreciprocity. For thick insulating ferrites like YIG, the mechanism of non-reciprocity may be explained by the concentration of surface waves on opposing sides of the film 67, 102, 113. However, this mechanism for non-reciprocity is invalid when the wavelength of MSSM is much larger than the film thickness ( kd 1) 81. Note that this is indeed the case for our stripe thicknesses and the range of available wave numbers ( kd 0. 1). The antenna-induced mechanism of non-reciprocity for the case of kd 1in metallic thin films with large magnetic moments is explained by Demidov et al 81. Consider the excitation field components of the antenna, h x (in-plane) and h z (out-of-plane), according to the frame of reference in figure 4.3a. Both components provide the driving 81

94 torque for magnetisation precession. The torque contribution by h x is always in-phase with the emitted spin wave, but the torque contribution by h z 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 h z 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 82

95 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 k 0 = rad/cm occur (figure 4.5.5b). Figure 4.5.5c: Non-reciprocity parameters at 10 GHz and k 0 = rad/cm. 83

96 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 Thus, the nonreciprocity of MSSM in our PSWS experiment is as predicted by theory. 4.6 Higher order width modes in confined stripe geometry Extra modes in addition to the dominant MSSM were observed in the spin wave spectra 110nm for the stripe with the highest aspect ratio studied in this work. These were 2 m identified as higher order width modes appearing due to confinement in such a narrow stripe. This sub-chapter deals with the identification and characterisation of these modes. In a simple qualitative model for confined stripe geometry, the finite width leads to quantization of backward volume-like spin wave modes across the width in the form 50, 87 : k y n Equation 4.6.1a w Where n is the mode number and w is the effective width of the stripe. The eigen-modes take on sinusoidal functions analogous to the case of spin wave resonance modes across the thickness 38, 50, 79, 84. The total wave number, k t, for spin wave propagation in a confined stripe is thus 84 : 84

97 k 2 t k 2 x k 2 y Equation 4.6.1b Where k x is propagation in the longitudinal direction (along the long axis) and k y 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 k x and k y modes depends on the stripe s lateral dimensions. From this, we define the wave propagation angle, θ: k arctan k y x 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 110nm wave spectra of our particular stripe with the highest aspect ratio (see section 2 m 85

98 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 Mode identification The raw data traces for the stripe exhibiting multiple higher order width modes are shown in figure 4.6.1a. In the reflection data, a remarkable 6 higher order width modes on top of the fundamental MSSM were observed in the spectra. Numerical simulation was used to generate theoretical eigen-fields and mode profiles in order to identify these modes. As seen in figure 4.6.1a, the simulated eigen-field positions match up well with the experimental mode positions. Mode numbers were assigned accordingly upon evaluation of the mode profiles (figure 4.6.1b), where n is the number of anti-nodes in the mode profile. The excitation field has a symmetric odd mode profile across the stripe width. This implies that only modes with odd symmetry can be excited by such a field 36, 38, 70, since the mean values of odd mode profiles modes are non-vanishing (figure 4.6.1b). For the modes n 3, one sees a pronounced decrease in the amplitude of the maxima of the standing wave towards the stripe edges. This is due to the nonuniformity of the magnetisation ground state (see the upper panel in figure 4.5.2a). The simulated mode profiles are similar to figure 6 in Bayer et al. 79 and figure 4 in Kostylev et al. s work 86, with the exception that our experiment was done fieldresolved. (Refer to Appendix C for the simulation procedure and list of key parameters.) Note the similarity in the aspect ratio of this particular stripe 110nm 2 m with the 86

99 stripe array studied by Kostylev et al. 30nm 600nm were detected using Brillouin light scattering , where multiple width modes 110nm Figure 4.6.1a: Field-trace at 10 GHz for the aspect ratio stripe, showing 2 m multiple high order width modes. Solid curve: signal received at detection antenna over 12 μm propagation distance. Dashed curve: signal reflected back from excitation antenna. Vertical lines: simulated mode field positions, where n is the mode number. 87

100 Figure 4.6.1b: Simulated mode profiles. 88

101 Figure 4.6.1c: Mode amplitudes at the dominant wave number k 0 = rad/cm normalised to the fundamental MSSM (n = 1). For the simulation, the mean value of the mode profile was designated the mode amplitude. For the experimental data, the mode amplitudes follow the definition in figure 4.5.5b. Theoretically, the excitation efficiency is proportional to the overlap integral of the mode profile with the excitation field profile 84. Assuming a uniform excitation field, the overlap integral reduces to the mean value of the mode profile. One expects the mode amplitude to decrease for increasing mode number, since the mean value of dynamic magnetisation decreases with increasing number of nodes in the mode profile. Indeed, this was observed in the simulated and experimental reflection data. (figure 4.6.1c). Interestingly, in the experimental reflection data, the mode amplitudes seem to drop linearly with increasing mode number (for n 3). In the simulation result, the reduction in mode amplitude for increasing mode number has a different functional dependence. At this point, we emphasize that it is often difficult to simulate high order mode amplitudes to quantitatively match real-world data. It is sufficient for the simulated and experimental relative mode amplitudes to qualitatively follow a rough trend for the purpose of mode identification. Comparing between the experimental reflection and transmission data, one observes that the transmission data contained progressively less modes for increasing antennae gap. In the transmission data (figure 4.6.1c), the amplitude of high order modes decrease even more rapidly than in the reflection data with increasing mode number. We postulate that this is due to increase in attenuation for increasing mode number upon 89

102 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) Table 4.6.1a: Observation of width modes in experimental data Dispersion and group velocity After having identified the width modes with the aid of numerical simulations, we now consider their dispersion characteristics. Guslienko et al 84 formulated a non-analytic dispersion expression for quantized dipole modes of an arbitrary rectangular slab magnetised tangentially. In this work however, further characterisation of propagation characteristics were done experimentally. Following the same phase interference technique of extracting dispersion in section 4.4.4, the dispersions of the width modes 90

103 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 k 0 = rad/cm. 91

104 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. H From these fits, the local gradient were calculated for k 0 = rad/cm. k The differential conversion factor between frequency and field resolved measurements, f, can be obtained by performing multiple frequency measurements for k0 = H rad/cm (figure 4.6.2b). Note the excellent linear fits (solid lines), from which f values were extracted from the slopes for each respective mode. H Figure 4.6.2b: Frequency versus field plots for the width modes for k 0 = rad/cm. Similar to section 4.5, knowing both H k f and H, the group velocity can be calculated using equation 4.5.3a: V g f 2 k H H k Thus, to first degree Taylor approximation, one can calculate the group velocity for the dominant wave number k 0 = rad/cm and 10 GHz frequency. The results are plotted in figure 4.6.2c and also tabulated in table 4.6.3a. 92

105 Figure 4.6.2c: Group velocity of higher order width modes at 10 GHz and k 0 = rad/cm. As postulated earlier, the group velocities of higher order width modes (n 3) were found to be nearly an order of magnitude lower than the fundamental MSSM. (Hence, for clarity of the higher order mode data, the fundamental MSSM data point was deliberately left out of figure 4.6.2c.) Furthermore, the group velocity decreased for increasing mode number (figure 4.6.2c). This results in increased spatial attenuation for increasing mode number, as evident in the mode amplitudes in the transmission data (figure 4.6.1a). As for the reflection data in figure 4.6.1a, the decrease in group velocity for increasing mode number may explain the discrepancy between the simulated relative mode amplitudes in figure 4.6.1c. From equation 4 in Dmitriev 100, one may expect the efficiency of mode excitation to grow with a decrease in group velocity; this may partly compensate the decrease in the overlap integral for increasing mode number. The group velocity of the fundamental MSSM for this stripe obtained using the Taylor approximation method (12 μm/ns) is consistent with the values obtained in section 4.5.3, within the accuracy of the experiment. The fundamental MSSM group velocity is directly proportional to the film thickness, and one may assume that this proportionality also applies to higher order width modes. This may explain why these higher order width modes were not observed in the thinner stripes studied in this work (very low 93

106 group velocity and excessive attenuation), and also why their observation using inductive spectroscopy methods are lacking in the literature 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 for k 0 = 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 and tabulated in table 4.6.3a. As noted before in section 4.5.4, the antenna near-field effect (see section 4.7) would shift all the data points horizontally by a constant amount. However, this would not affect the slope of the plot in figure 4.6.3a. 94

107 Mode number f H (10-3 GHz/Oe) H k (10-3 Oe cm) Group velocity (μm/ns) Attenuation length (μm) Relaxation time (ns) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.06 n/a n/a ± ± ± 0.08 n/a n/a Table 4.6.3a: Width mode propagation characteristics about 10 GHz and k 0 = 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 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 for the wide 50 μm stripe. Note the difference in the extracted relaxation times between the fundamental and higher order modes. 95

108 4.6.4 Non-reciprocity Similar to section 4.5.5, the non-reciprocity parameters of the width modes were evaluated using equation 4.5.5a and plotted in figure 4.6.4a. The n = 3 width mode was found to have the same non-reciprocity behaviour as that of the fundamental MSSM (η 0.2). Note that only the first two modes (n = 1, 3) had sufficient transmitted signal amplitude to evaluate non-reciprocity with reasonable reliability. Due to tiny signals for the larger mode numbers (barely above noise level), quantitative evaluation of the non-reciprocity parameter should only be taken with a grain of salt. However, one can still observe a trend in the data. For larger mode numbers (n 5), the non-reciprocity seems to get weaker (η approaching unity) for increasing mode number. This result is actually consistent with the rotation of the wave vector towards the transverse direction for increasing mode number; the wave takes on more backwardvolume-like characteristic (a purely backward volume wave is completely reciprocal). Figure 4.6.4a: Non-reciprocity parameter of width modes at 10 GHz and k 0 = rad/cm. 96

109 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" 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, x eff, is shorter than the 97

110 physical separation gap between antennae, x ; the difference between the two gives twice the antenna characteristic near-field length, x near : 2x x Equation 4.7.1a near x eff 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 ( x near is defined for a single antenna). In PSWS utilising metallic ferromagnetic films, the distance between antennae is always comparable to the size of the antennae themselves. This is due to lossy metallic ferromagnetic films with attenuation lengths of the order of microns. This means that for closely separated antennae, xnear may no longer be negligible. In such cases, one needs to determine the proper effective propagation distance, which may differ significantly from the nominal propagation distance (between the symmetry axes of antennae). We now derive an expression from which the antenna characteristic near-field length may be determined from experiment. From equation 4.5.3a, we have the group velocity: V g f H 2 Equation 4.5.3a k H k The slope of the field-resolved pseudo-dispersion relation is thus: H k Vg f 2 H 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 H degree Taylor approximation that the slope is constant at a particular frequency k and in the close proximity of the central dominant wave number k 0 = rad/cm. The phase difference between each maximum in the raw trace is given by equation 4.4.4a: 2 k Equation 4.4.4a x eff Substituting the expression for rewriting H into H gives: k in equation 4.4.4a into k in equation 4.7.1b, and 98

111 Vg f H x eff H 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 LetV ' Vg, then equation 4.7.1c becomes: f H H instead of k. x eff H V ' Equation 4.7.1d By substituting equation 4.7.1a into equation 4.7.1d for x eff, we have: ( x 2xnear) H V' x H V ' 2 x near H Equation 4.7.1e From equation 4.7.1e, one sees that by plotting x H (x = antennae gap) versus H for various antennae gap separations, the slope of the plot gives the antenna characteristic near-field length Antenna characteristic near-field length We now establish some criteria before using equation 4.7.1e to analyse our data. First, for a stripe aspect ratio, there must be at least 3 different separation gaps available; this is the bare minimum for a linear fit. Second, the spread in H should be minimal and statistically random; there is no significant slope and curvature in the field-resolved pseudo-dispersion. This allows us to average H data for each raw trace into a single representative value. This value is multiplied by the respective x ( x H ) is plotted against itself for all available stripes which had data meeting the established criteria to produce plots in figure 4.7.2a. 99

112 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 k 0 = rad/cm. 100

113 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, x near 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 and is beyond the focus of this section. Thickness (nm) Width (μm) Antenna characteristic near-field length, x near (μm) ± ± ± ± ± ± ± 0.1 Table 4.7.2a: Extracted antenna characteristic near-field lengths. One notes a spread of xnear values extracted from the plot, ranging from 0.3 to 2.3 μm. Recall the dimensions of the coplanar waveguide antennae used in this experiment: the conductor widths and separation gaps were 1.5 μm, resulting in a total width of 7.5 μm, and the distance from its symmetry axis to the external edge is 3.75 μm (figure 4.7.2b). The antennae characteristic near-field length values extracted lie within this 2.25 μm gap from the central conductor (figure 4.7.2b). The in-plane excitation magnetic fields 101

114 of a coplanar waveguide are concentrated underneath the signal and ground lines 98, 104. The in-plane field contributes the most to spin wave excitation, and is maximum underneath the central signal line. Thus, one may consider the region underneath the central signal line as the near-field region, and wave propagation begins at some distance from it. Our extracted xnear values are consistent with this, to the accuracy limits of this experiment. One also notices that 0 < x near < 3.75 μm for all the cases. x near >0 implies that the phase accumulation starts on the side of the excitation antenna that is closer to the receiving antenna, and x near < 3.75 μm implies that it starts below the whole antenna structure (figure 4.7.2b). This is very important to know given the nonreciprocity of the antenna, because it is not obvious a-priori that for a non-reciprocal antenna x near > 0. One also notes that there seems to be a trend for xnear to increase with stripe thickness and width. From a fundamental consideration, the electromagnetic fields of a coplanar waveguide would be perturbed by the close vicinity of magnetic material 115. Thus, one may expect some dependence of the antennae characteristic near-field length on film thickness. However, the accuracy limitations of this experiment do not allow us to definitively quantify this effect. Furthermore, the theoretical framework required to investigate this effect is beyond the scope of this work. Figure 4.7.2b: Cross-section of the antenna. 102

115 4.7.3 Effective propagation distance Experimentally-wise, with knowledge of x near, one can then determine the effective propagation distance x eff by subtracting from the antennae gap x for a particular stripe. We now demonstrate the effect this has on the raw data of a particular stripe. Figure 4.7.3a: Evaluation of antennae characteristic near-field length correction on the data for the 110 nm thick and 2 μm wide stripes. In figure 4.7.3a -a, the plot of all H points from the raw data were plotted on the vertical axis with their respective fields on the horizontal axis. For clarity, they were plotted on the logarithmic scale on the vertical axis. The data can be collapsed onto the same scale by multiplication with some factor, in this case, the antennae separation gap, 103

116 for each of the data set. In figure 4.7.3a -b, all the x H points from the raw data were plotted on the vertical axis with field on the horizontal axis. The horizontal lines are the mean values for each antennae gap data set. However, note that the mean values do not coincide due to offset induced from the finite antennae characteristic near-field length x near. In fact, these offsets can be used to extract x near, which is mathematically identical to the approach in figure 4.7.2a. In figure 4.7.3a -c, the antennae gaps were corrected with x near = 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 x eff demonstrate here that the effective propagation distance x eff directly inversely proportional to H. From this, it follows that the proper length to use to calculate of 2π) is the effective propagation distance x eff. Thus, we (not the antennae gap x ) is k (phase accumulation (not the antennae gap x ). Thus, the nearfield effect is most significant for small antennae separation gaps, and becomes less significant for larger gaps. As discussed in section 4.5.1, this is one of the important factors (the others being number of data points and signal attenuation) to determine the optimal antennae separation gap from which to reliably plot dispersion. However, in section 4.5.1, the dispersions were calculated using the antennae gaps x instead of the effective propagation distances x eff proper. Note that since adequately separated antennae gaps were used in the dispersion plot in section 4.5.1, the near-field effect introduced a discrepancy of only approximately 8%. In addition, since this is a systematic error, it would merely shift calculated quantities uniformly by 8%. We consider this acceptable within the accuracy and scope of this work. 104

117 4.8 Chapter conclusion Spin wave propagation in the Damon-Eshbach geometry was studied in thick Permalloy 4 2 stripes ( nm) over the aspect ratio range 5.5 (10 10 ). Micron-sized antennae were used to excite and detect spin waves with accessible wave numbers ranging from 2000 to rad/cm. A highly sensitive phase interferometer detector, together with a lock-in field-modulation technique, was used in this inductive spin wave spectroscopy method. In section 4.5, MSSM propagation across the range of aspect ratios and wave numbers was studied. It was proposed that the MSSM dispersion can be modelled by introducing an effective static demagnetising field factor into the continuous film dispersion. Dynamic effects were negligible in our case. Micro-magnetic simulations were performed on the stripes to determine the demagnetising field profiles. The nonuniformity of the demagnetising field across the stripe width increased with aspect ratio. The mean values of the simulated demagnetising fields tend to overestimate the effective demagnetising fields extracted from experiment. Group velocities calculated from the dispersions, and these were found to increase with film thickness. There was no correlation between the group velocity and stripe width for a particular thickness; thus within the bounds of the experiment, the MSSM group velocity was found to be width invariant. The attenuation and relaxation characteristics of the stripes were evaluated. We found that the attenuation lengths increased with stripe thickness. Relaxation times and Gilbert damping coefficients were calculated from MSSM data and compared with the reference continuous film FMR data. It was found that the Gilbert damping coefficients calculated from the stripe data were about 25% larger those determined from FMR. This discrepancy was proposed to be due to edge losses due to confinement, wave number dependence on damping coefficient, and/or compounding of inaccuracies in the indirect methods used to calculate the damping coefficient from MSSM dispersion data. Non-reciprocity of the MSSM was evaluated and found to be largely invariant over the aspect ratios studied. In section 4.6, multiple higher order width modes were found and identified in the stripe 110nm with the highest aspect ratio studied in this work. Remarkably, 6 higher order 2 m width modes (excluding the fundamental MSSM) were found in the excitation spectra. 105

118 Due to symmetry of the excitation field, only modes with odd symmetry were excited (up to n = 13). Simulation was used to identify the modes in the recorded spectra and determine the modal profiles. The amplitudes of these modes decrease for increasing mode number in the excitation spectra, and even more rapidly in the transmission spectra for increasing propagation distance. The dispersion, group velocity, attenuation, and non-reciprocal properties of these modes were characterised in detail by an induction method. It was found that the group velocity and attenuation lengths of the higher order width modes decrease for increasing mode number. Within the accuracies of the experiment, we found weakening of non-reciprocity for increasing mode number. We propose that this is due to the higher order modes taking on more backwardvolume-like character for increasing mode number (a pure backward volume wave is completely reciprocal). In section 4.7, we propose that due to the near-field of an antenna, the spin waves excited only propagate at some distance away from the antenna. We term this as the antenna characteristic near-field length. The geometrical separation gap between the excitation and detection antennae thus consists of the effective propagation distance plus the antenna characteristic near-field length. To this end, we derived an expression from which the antenna characteristic near-field length may be determined from experiment. We found that the antenna characteristic near-field lengths extracted from our data were such that wave propagation begins at some finite distance from the central signal line, but still within the overall width of the coplanar antenna. 106

119 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 Palladium-based hydrogen sensors 116 make use of the changes in the physical property of palladium upon hydrogen absorption, namely: a.) crystal lattice expansion 117, , 124,, b.) change in conductivity 125, or c.) change in optical properties In addition to gas absorption properties, palladium is also of great interest to the magnetic community due to its spintronic effects. Magnetic multi-layered films which include non-magnetic palladium layers are of great importance for high-density 107

120 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, 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 magnetometry 15, 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 environment 119, 124,

121 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 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: Ni 80 Fe 20 (5)/Pd(10) Ni 80 Fe 20 (30)/Pd(10) Co(5)/Pd(10) Co(40)/Pd(20) The numbers in brackets indicate the film thickness in nanometres. The magnetic layers were buried underneath the palladium layer, with later exposed to atmosphere (figure 5.3a). Figure 5.3a: Bi-layer film cross-section. In addition, two single layer ferromagnetic films were also sputtered (without palladium capping layers), functioning as control samples: 109