Surface and interface anisotropies measured using inductive magnetometry

Transcription

1 Surface and interface anisotropies measured using inductive magnetometry Kim Kennewell Bachelor of Science (Hons) A dissertation submitted to the University of Western Australia for the degree of Doctor of Philosophy School of Physics (2008)

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3 iii Abstract In this thesis, an inductive ferromagnetic resonance (FMR) technique is developed to measure the magnetisation dynamics in thin films across a wide range of frequencies and fields. In particular, this project concentrates on measuring higher order exchange dominated modes to observe surface and interface effects in bilayer films. The experimental technique was first developed as a time domain technique, utilising a fast rise time ( 50 ps) step pulse to disturb the equilibrium position of the magnetisation. The subsequent precessional damped decay was measured at different applied fields to observe the resonant modes. The data is Fourier transformed to extract a frequency dependent susceptiblity, and results are presented for the frequency and linewidth dependence of excitations of a permalloy film as a function of applied field. This technique is limited to a frequency range dictated by the rise time of the pulse. The technique was then extended so as to use a continuous wave perturbation, utilising a network analyser as both the excitation source and the measurement device. The scattered wave parameters of both the transmission and reflection from the sample were measured, and a magnetic susceptibility is extracted. This method has a frequency range which is dictated by the bandwidth of the network analyser and the microwave circuit. In this project, results are presented for frequencies up to 15 GHz. The signal to noise ratio was also found to be lower than the pulsed technique. Fundamental resonant mode studies are presented for a Fe/MnPd exchange bias bilayer film. Crystalline and exchange anisotropies are extracted from angular measurements, and the behaviour of the magnetisation is investigated during its reorientation to a hard axis direction. Information about the distribution of the local exchange field strength and direction is predicted. Fundamental mode studies are also presented for a Py/Co exchange spring bilayer film. Two modes are observed, approximating an optical and acoustical excitation.

4 iv Film systems were also designed with suitable thicknesses to observe in the experimentally available frequency range non-uniform exchange dominated excitations through the thickness of the film. The broadband nature of the experiment allowed the frequency of the modes to be measured as a function of field. Results from a single permalloy layer showed two observable modes, the fundamental and the first exchange mode. Measurements were also taken of bilayer films where permalloy is coupled to cobalt. In this system the effect of the cobalt is seen to shift the single layer Py mode frequencies, as well as introduce new modes. The relative intensities of the modes also change with the addition of cobalt. Results are shown for a Pt/Co multilayer coupled to a permalloy layer through a Cu spacer of varying thickness. The observation of excitations through the thickness of the film motivated the development of a suitable theory. A system of integro-differential equations were derived which account for dipole and exchange coupling in the film as well as the field screening by the metal of the coplanar line. The conductivity of the sample and the finite wavevector excitation of the stripline are also included. Numerical solution of the equations results in a spectrum of acoustical, optical and higher-order modes. Fitting of the model to the experimental results allowed extraction of the film parameters including; the exchange constants in the film; the surface pinning from any surface layer anisotropy; as well as the interlayer exchange coupling across the interface.

5 v Acknowledgements I would firstly like to thank my principal supervisor, Professor Robert Stamps. Bob has an enthusiasm for physics that is very contagious, and Bob is never short of a new idea. He was a great guide for my postgraduate studies, teaching me a more critical approach to problem solving and always offering great advice. I would also like to thank my co-supervisor, Dr Mikhail Kostylev. Mikhail taught me to think like a theorist and remember my well forgotten undergraduate mathematics. I would also like to thank Dr Robert Woodward and Dr David Crew. Both Rob and David were always happy to provide technical support in the laboratory. They made the basement a vibrant and fruitful working environment. In particular, Rob went well above and beyond to continue to support me even after he changed research groups. A big thank you to fellow students in the condensed matter group for the laughs and special moments that broke up the working day. A quick thanks to Pete Metaxas for the theorist jokes, Becky Fuller for the one too many conference beers and Karen Livesey for the scrabble games. Thank you also to Karen for critically reading this dissertation. And a thank you to the next generation of students; Zoe Budrikis, Rhet Magaraggia and Vincensius Gunawan for providing an instant uplift to the group. It is a pleasure to acknowledge the School of Physics and the administration staff that make being a student here so enjoyable. In particular Professor Ian McArthur, Ede Lappel, Lydia Brazzale and Amanda Atkinson have always had problems fixed before you even ask. Also a thank you to the workshop staff, who are a great asset to the school. Without the help of Gary Light, Dave McPhee, Craig Grimm, Marcus Frankenberger and Peter Hay this project would not be possible. Thanks also for the lunch time beers. Also a thank you to Lance Maschmedt and Joe Coletti for their help in the teaching laboratories. I must also thank Eric May from Oil and Gas, and Tony Preston from Measurement

6 vi Innovations for the loan of their network analysers. This project relied on a borrowed network analyser to perform the microwave measurements, and both Eric and Tony were very patient whilst I was using their equipment. I was fortunate enough to spend a couple of months at the University of Leeds. My time at Leeds was a fantastic experience, and the condensed matter group were incredibly welcoming. I would like to thank Professor Denis Greig, Professor Bryan Hickey, Dr Chris Marrows and Dr Mannan Ali for their support and incredible hospitality during my stay. A personal thank you to Mannan for his help in the lab. A thank you to the postgraduate crew at Leeds, who have made Leeds seem like a second home. I have also been lucky enough to collaborate with some other incredible physicists during my studies. I would like to thank Professor Kannan Krishan from the University of Washington, Professor Shiva Prasad from the Indian Institute of Technology and Dr Jing-Guo Hu from the University of Yangzhou for working with me. I would also like to thank Professor Kevin Coffey from the University of Central Florida for his hospitality in Orlando and for giving me a tour of the universities growth facilities. Also a thank you to Dr Tom Silva and Dr Pavel Kabos for their hospitality and tour of the facilities at NIST, as well as a thank you to Professor Zbigniew Celinski for showing me around the University of Colorado. A huge thank you to my family; Mum, Dad, and my brother Robin. They have always been there to support me. My dog, Inu, also deserves a thank you for the daily stress relieving walks. Finally, the one person who has made all of this possible through their endless support and love is my newlywed wife, Catherine. I dedicate this thesis to you.

7 Contents 1 Introduction Preliminary: Pulsed Inductive Microwave Magnetometer (PIMM) PIMM setup Susceptibility Field dependence PIMM conclusions Thesis Outline RF susceptibility RF setup Scattering parameters Waveguide Susceptibility Data manipulation Propagation Coplanar line description Discontinuity reflection Loaded line Chapter summary vii

8 viii Contents 3 Fundamental mode studies Fe/MnPd Angular dependence model Equilibrium direction Resonance as a function of angle Experimental results Hard axis reorientation Transition behaviour Py/Co exchange spring Effective medium theory Chapter Summary Dynamical field - exchange spin wave model Magnetostatic waves Maxwell s equations derivation Exchange regime Dissipation Single layer solution Pinning conditions Current profile Dispersion Conductivity of sample Conductivity of ground planes Bi-layer Interface exchange conditions

9 Contents ix Bilayer solution Example: Py/Co exchange spring revisited Mode intensities Chapter summary Higher order SSWM results Sample growth Magnetron sputter deposition Sputtering system Growth procedure X-ray reflectivity Experimental Results Py reference film Bilayer Py/Co films Py/[Pt/Co] x Chapter summary Future work and conclusions Conclusions Future work Cryo-system Other films Model A Magnetisation dynamics 127 A.1 Isolated moment resonance

10 x Contents A.2 Equation of motion A.3 Linearisation A.4 Uniform excitation A.4.1 General ellipsoid A.4.2 Thin film geometry B Experimental techniques 135 B.1 Cavity resonance B.2 Brillouin light scattering B.3 Coplanar line technique C RF theory 139 C.1 Guided waves C.2 Telegrapher s equations C.3 Characteristic Impedance C.3.1 Coaxial line C.3.2 Wave velocity C.3.3 Coplanar D Table of films 149 D.1 Leeds films D.1.1 Py/Co films D.1.2 Py/Gd films D.1.3 Py/[Co/Pt] films D.1.4 Ultra-thin films (repeated) D.1.5 Spinvalve trilayer structures

11 Contents xi D.1.6 Exchange bias multilayers D.2 Other films Bibliography 163 List of Figures 179 List of Tables 183

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13 We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about and. Eddington, Arthur ( ) xiii

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15 Chapter 1 Introduction Magnetic materials can behave differently as their dimensions are reduced. The ratio of surface to bulk atoms increases significantly, and at a surface or interface site the local symmetry is generally lower than in the interior of the material. The electronic structure may also be modified at a surface or interface. For example, the first few surface atomic layers in a thin film structure often show a perpendicular surface anisotropy [130, 133]. This was first discussed many years ago by Néel to explain the unique magnetic properties of small particles [114]. The surface anisotropy effect is easily visible if the film s thickness is reduced to less than a few monolayers as then its spontaneous magnetisation is observed to be preferentially orientated out of plane [64]. Similarly, the few atomic layers surrounding a magnetic interface may also show an additional anisotropy, in this case dictated by the coupling between the layers [32,74]. In metallic ferromagnets the coupling may be due to: long range dipolar interactions; direct ferromagnetic exchange coupling; or an indirect exchange interaction of the Ruderman- Kittel-Kasuya-Yosida (RKKY) type. The dipolar interaction only arises in the case of locally non-uniform magnetisation, and is considerably weaker than the exchange interaction. Studying these interactions across an interface is difficult due to its buried nature in the film. One method is to measure magnetic films of different thicknesses. The interface effects will show a 1/t relationship where t is the thickness of the film. Quasi-static measurements can be performed, where the measurement time scale is much greater than the dynamics in the magnetic system. These measurements have the disadvantage in that they only measure the average magnetic moment at a local equilibrium from 1

16 2 Introduction across the entire film. It is difficult to pin-point any one influence on the magnetisation properties of the film. A different approach involves measuring the magnetic properties through dynamic perturbation techniques. In coupled ferromagnet/ferromagnet bi-layer films, magnetisation dynamics of the fundamental mode have been widely studied for several decades [23,26,41,42,40,67,70,73,172,177]. Since the discovery of the Giant Magneto-Resistance effect (GMR) (for a review of GMR see references [58,121]) much work has been done using FMR to characterise the exchange coupling over ferromagnetic, non-magnetic, ferromagnetic tri-layer and multilayer structures [85,135,136]. Typically these experiments measure a resonance localised in each ferromagnetic layer, with a frequency shift in the resonances due to the indirect coupling across the non-magnetic interlayer. The disadvantage in only measuring the fundamental mode is that its behaviour has only a very small dependence on the bulk exchange parameters of the materials, and the influence from the surface and interface anisotropies is small. An exciting proposition is to use higher order exchange dominated modes to observe surface and interface effects. These modes are excitations which are spatially nonuniform in either amplitude or phase and were first discussed in the context of FMR by Rado and Ament [10, 131]. Standing modes were first observed in ferromagnetic resonance experiments performed in spheroids with highly non-uniform applied fields [109, 168, 171]. In a thin film structure, these modes form standing spin waves (SSW) through the thickness of the film. The lowest energy modes present in a single film with no surface pinning are illustrated in Figure 1.1. Kittel first explained that it would be possible to see these higher order standing modes in a uniform field if a surface inhomogeneity was present [89]. He noted that the thin film geometry was perfect for standing wave excitations through the thickness of the film. Seavey and Tannenwald [139] were the first to measure these standing spin wave excitations in Since then, considerable work has been concentrated on observing the standing spin wave response in single films, both insulating (for examples see references [48, 76, 151, 159, 176]), and conducting (for examples see references [56, 57, 100, 101, 148]) films. Finding exchange constants, saturation magnetisations and surface pinning anisotropies in the measured films were the main motivation for these studies. Numerous theoretical treatments have risen to understand and fit the experimental measurements [14, 29, 79, 140, 147, 148, 151, 152, 153, 154,155].

17 Introduction 3 Fund SSWM 1 SSWM 2 Py Figure 1.1: The uniform and first 2 non-uniform Standing Spin Wave Modes (SSWM) in a single layer Permalloy film. What is lacking in the literature is a systematic approach to using these higher order SSW modes to observe both the surface and interface effects in a thin film geometry. It was pointed out by Puszkarski et al. [129] that it is theoretically possible to measure these higher order modes in more complicated bi-layer film structures, however little experimental work has followed. Experimentally, the frequency of these modes depends on the wavelength of the oscillation. If there is pinning at either surface, or an interface exists between different ferromagnetic layers, the mode profile through the thickness of the film will change. For example, in contrast to the unpinned modes illustrated in Figure 1.1, a completely pinned surface will result in an amplitude node at the surface. This change in mode profile results in a mode with a different wavelength, and consequently a different frequency. A key point in this dissertation is to carefully measure these frequency shifts in higher order modes and relate these shifts to changes in surface and interface anisotropies. Sample H A H RF I M ~300 µm RF field Stripline Figure 1.2: Coplanar line illustration. RF/Pulsed current creates microwave field perpendicular to coplanar line. A short overview of possible techniques to measure ferromagnetic resonance is pro-

18 4 Introduction vided in Appendix B. To measure the resonance modes in this project a coplanar line technique was developed. The geometry of the coplanar line technique is illustrated in Figure 1.2. The coplanar line technique gives a precise measure of the resonant behaviour of a magnetic film. It also has the ability to measure over a broad band of frequencies at a relatively low cost. The remainder of this introductory chapter is devoted to explaining time resolved magnetometry using the coplanar geometry. This was the first technique used in this project. In the following section the experimental setup is described, along with the extraction of the precessional data as a function of time. The relevant background magnetisation dynamics theory is presented in Appendix A. A Fourier transform of the time domain is used to extract the magnetic susceptibility. The field dependence of both the frequency and damping of the resonance are presented. In future chapters, the coplanar technique is extended to use a continuous wave excitation. This proved to have a greater frequency bandwidth and a better signal to noise ratio than the time resolved excitation. Also, a detailed theory is developed to include the inhomogeneities present in the coplanar technique. These include the ground planes of the waveguide, the conductivity of the sample and the non-zero wavevector excitation of the coplanar antenna. In combination with the refined experiment, the theory is used to accurately model the mode behaviour in bilayer films. The dissertation outline can be found at the end of the introduction. 1.1 Preliminary: Pulsed Inductive Microwave Magnetometer (PIMM) This section will introduce in detail a temporally resolved technique using an excitation pulse through the coplanar line. This was the first technique developed in this project and was based on a similar setup first described by Silva et al. [143], denoted the Pulsed Inductive Microwave Magnetometer (PIMM) PIMM setup The setup is illustrated in Figure 1.3. The sample is placed upon the coplanar line. The sample is electrically isolated from the waveguide by a thin separating layer of teflon.

19 Introduction 5 The sample can then be placed face-side down to maximise the inductive coupling with the field to the coplanar line. The coplanar line is connected to the system via SMA coaxial cables. PULSE GENERATOR 10V Pulse Microwave Probe OSCILLOSCOPE Trigger signal Static Field WAVEGUIDE SAMPLE z y x Pulse Field Attenuator Figure 1.3: Schematic of PIMM. Excitation field is created by the pulse generator, and the transmitted pulse is measured with a sampling oscilloscope. The magnitude of the field produced by the coplanar line can be estimated from an approximation of the field around a current carrying wire. In the quasi-static approach, H = J, where J is the current density. Integrating over the area of the coplanar center conductor gives the total current I: I = H da = H dl. (1.1) A A Assuming that the coplanar center conductor width is much greater than its thickness, an approximate field amplitude is given by: H x I/(2w), (1.2) where w is the width of the central conductor. This gives wider central conductors a smaller excitation field for a given transmission current, and this was seen experimentally as a smaller inductance from the sample. For this reason, the narrowest central conductor was chosen, with w = 0.35 mm (see Table 2.1.2). Given that the characteristic impedance of the coplanar line is close to 50 Ω, and the pulse voltage V P = 10 V, this gives a current

20 6 Introduction of 0.2 A through the coplanar line. The approximate excitation field acting on the sample is thus 4 Oe. An excitation field is launched onto the waveguide from a commercial step pulse generator (Picosecond 4050). The pulses have a nominal rise time of 50 ps, a pulse duration of 10 ns, and are outputted with a repetition rate of 10 khz. This excitation field disturbs the equilibrium position of the magnetisation by altering the effective field direction. The magnetisation precesses around this new equilibrium position. The precessing magnetisation induces a voltage back into the coplanar line. Using the reciprocity between the sample and the coplanar line, the flux created around the coplanar line from a small magnetic moment in the film dm takes the form [102]: dφ = µ o H I dm, (1.3) where H is the field at the magnetic moment created by current I through the coplanar line. Substituting for the H from the approximate field given in equation (1.2), and integrating over the sample assuming it covers the complete center conductor width w, and has thickness t and length l, gives: φ µ 0tl 2 M x. (1.4) This is only an approximation because the actual field at the small magnetic moment dm may vary on its location over the width of the center conductor, and the distance from the coplanar line. From Faraday s Law the induced voltage becomes: V = dφ dt = µ otl dm x 2 dt. (1.5) For small oscillations, M can be considered harmonic, giving dmx = iωm dt x, with the maximum oscillation magnitude M x calculated from the geometry of the applied fields; M x M s (H x /H eff ), (1.6) where M s is the saturating magnetisation, and H eff is the effective field parallel to the coplanar line acting on the magnetisation. Typical values for permalloy give µ o M s = 1 Tesla, and the minimum of H eff is given by H K, the anisotropy field. Assuming H eff to be on the order of 10 Oe, the sample to be 10 mm 2 and 10 nm thick, an expected voltage of the order of a hundred millivolts is to be expected.

21 Introduction 7 This harmonic signal is exponentially damped due to energy dissipation within the sample, and is super-imposed onto the original much larger 10 V excitation pulse. This is measured by a 20 GHz sampling oscilloscope (86100 Infiniium DCA), measuring 512 pulses to form one trace. As the period between the pulses (0.1 ms) is much larger than the relaxation time of the magnetisation, the measurement is completely repeatable. Noise is reduced by averaging the measurement a few hundred times. To recover the magnetic signal it is necessary to remove the excitation pulse contribution. This is done by measuring a reference pulse with zero magnetic excitation. This can be done in one of two ways. The first involves a smaller bias field coil creating a field perpendicular to a larger field from an electromagnet. This setup is used when small bias fields are required, with the advantage that the bias field coil has no remanence at zero field. In this setup, a measurement of the magnetic signal is taken with the bias coil active. Then a reference measurement is taken with the larger electromagnet saturating the magnetisation in the direction of the pulse field, prohibiting the magnetisation from being excited by the pulse field. In the second setup, a single electromagnet is used for both the excitation and reference measurement. To create the reference measurement, the electromagnet is set to maximum field ( 1 Tesla). This moves any resonances above the measurement frequency, and no magnetic signal is recorded. Using the single magnet setup allows for larger bias fields than would be possible with the smaller coil in a two magnet arrangement. The reference signal is now subtracted from the excitation signal, leaving only the magnetic response. However, due to the steep gradient of the step pulse any small jitter between the pulse generator and the oscilloscope produces spurious signals (see Figure 1.4). Even a 100 fs difference between the pulse leading edges results in tens of millivolts of error when subtracting the two pulses. To minimise this error, both the leading edge and the end of the pulse plateau (after the magnetisation signal has dissipated) are matched in post-processing of the data in a similar way as described by Kos et al. [91]. A measurement for different applied bias fields is shown in Figure 1.5 for a 50 nm Py film grown at NIST. Note that as predicted in equation (1.6) the larger the applied field on the sample, the smaller the amplitude of the measured signal. However, for very small applied fields as seen in Figure 1.5b that the amplitude also seems to decrease. This can be explained by domain formation in the ferromagnet below the coercivity field leading to smaller nett signal measured. As V dm/dt, the integration of the measured signal gives the magnetisation in the x-direction as a function of time. Shown in Figure 1.6 are plots of the evolution

22 8 Introduction a) stripline b) INITIAL STATE Height difference creates DC offset z FINAL STATE Time difference introduces artifacts y x c) - = DATA REFERENCE SIGNAL Figure 1.4: a. Illustration of the magnetisation disturbed by the PIMM pulse. The magnetisation vector precesses about the new equilibrium position. b. Demonstration of pulse matching before subtraction. The rising edge must be matched in time, and the pulse height matched in amplitude for a successful subtraction. c. Illustration of data retrieval. The reference pulse is subtracted from the data pulse to obtain the measured signal. of the magnetisation for the applied fields shown in Figure 1.5 a. The magnetisation reorientates itself to the new equilibrium position created by the pulse field, described by the in-plane angle φ 0. As expected, the magnetisation obtains a larger x-component for the smaller applied fields, indicating φ 0 is larger. The amplitude relationship is not perfect as the measured amplitude also depends on the frequency response of the coplanar line. A simple model for the small amplitude oscillations assumes a sinusoidal oscillation with exponential damping: m x (t) = m 0 x e t τ sin(ω0 t + ψ). (1.7)

23 Introduction 9 a) 65 Oe Amp(V)[offset] Oe 25 Oe 5 Oe Time(ns) b) Amp (V) Time (ns) Field (Oe) Figure 1.5: a. Subtracted and corrected PIMM response for 50 nm Py at different applied bias fields. b. Complete time response for low applied fields of same sample.

24 10 Introduction Oe Integrated signal (V) Oe 45 Oe Time (ns) Figure 1.6: Time integrated data of a 50 nm Py film measured by the PIMM for various applied fields. Substituting this into equation (1.5), it is seen that the voltage is of the form: V e t τ (cos(ω0 t + ψ) 1 ω 0 τ sin(ω 0t + ψ)). (1.8) For the Py results in Figure 1.5, τ is 1ns, and ω = 2πf, where f is > 1 GHz in general. This gives ωτ > 6, and in general for under-damped oscillators the second term can be ignored. However, a more accurate description of the magnetisation includes the reorientation of the magnetisation due to the pulse field of the step pulse, including the equilibrium reorientation angle φ 0. In equation (1.7) the direction x is expressed by x = x cos φ 0. m 0 x is the maximum amplitude of the oscillations. The magnetisation in-plane precessional angle as measured from the coplanar line including the offset φ 0 is: φ(t) = φ 0 + φ m e t τ sin(ω0 t + ψ), (1.9) where φ m is the oscillation magnitude to be fitted from the data. This gives m x (t): m x (t) = M s sin(φ(t)). (1.10) An expansion of equation (1.9) substituted into equation (1.10) is derived by Silva et

25 Introduction 11 al. [143], which includes contributions from second order harmonics. Using the underdamped approximation derived before, the induced voltage in this case is: V cos(φ 0 )φ m e t τ cos(ω0 t + ψ) + sin(φ 0 ) φ2 m 2 e 2t τ sin(2(ω0 t + φ)). (1.11) 0.06 Signal (mv) Data Fit Fit inc. SHD 5 Oe Signal (mv) Oe Time (ns) Figure 1.7: 50 nm Py data fitted using the relation shown in equation (1.8) and the relation including second harmonic distortion shown in equation (1.11). It is clear that these second harmonics are only large when the equilibrium angle set by the pulse field is large from the sin φ 0 term; this occurs when the effective field in

26 12 Introduction the coplanar line direction is small. Fits using both relations (1.8) and (1.11) are shown in Figure 1.7. Here the difference in fits from the inclusion of second order harmonics (as in relation 1.11) is insignificant, even in the low field measurements where φ 0 is the largest. It could be postulated that the difference between the data and fits is from the linear approximation applied to the oscillating magnetisation, however even removing the initial larger oscillations from the data does not improve the fit. An interesting observation is that the experimentally measured oscillation period seems to vary slightly in time compared with the fits. This shows the approximation that the magnetisation is oscillating at a single harmonic frequency as assumed in the simple expressions presented above in equation (1.11) is good, but it is not perfect Susceptibility Rather than fitting the temporal data to a function with many assumptions, it is more convenient to Fourier transform the data and obtain a measured susceptibility of the magnetic film. This will show any dispersion of frequencies, or harmonics of the resonance without the need to include them a priori. However, first consider the form expected by the simple time evolution presented in equation (1.7): χ(ω) = 0 e (iωt+ t τ ) sin(ω 0 t)dt (1.12) ω iω + ω 2 τ 2 τ 0 ω 2. Here the resonant behaviour can be seen in the denominator as ω ω 0. A singularity would be present if the decay time τ. Note that this is the transform of the measured signal, proportional to d m/dt, where m is the dynamical magnetisation. For a single harmonic frequency as given in equation (A.10), d m/dt = iω m. However, the step pulse used to excite the magnetisation has a non-uniform spectral density. If assumed to be a perfect step, the spectral density is derived from a Laplace transform and is seen to be 1/ω. Thus the spectrum of the measured data (d m/dt) is very close to the spectrum of the actual magnetisation. To transform the measured data, consideration must be given to the starting point of the discrete transform; including data before the pulse begins will lead to spurious signals and a change in phase of the transformed data. The resonant frequency occurs at the maximum absorption of energy, which corresponds to a peak in the imaginary spectrum.

27 Introduction Oe Fourier Amplitude (arb. units) 25 Oe 45 Oe 65 Oe 0 3 Frequency (GHz) Figure 1.8: Py suceptibility spectra for different applied fields calculated from Fourier transformed PIMM data. Both the real and imaginary parts are shown (red open circles), the real data exhibits a crossing and the imaginary data exhibits a peak at the samples resonant frequency. The solid blue lines represent fits from equation (1.13).

28 14 Introduction Figure 1.9: Resonant frequency 2 vs field relationship of 50 nm Py film measured along an in-plane easy axis. The solid line is a linear fit to Kittel s equation suitable for when H b + H K << M s. The correlation coefficient for the fit is R = The subtracted data is first truncated at the initial rise in the measured voltage. The data is then padded with zeros after the resonance has dissipated, with a padding factor of 20 to achieve the required frequency resolution in the transform, and to minimise convolution errors from having such a small measured time domain. A discrete transform of the truncated and padded data is then multiplied by a calculated phase factor to obtain the correct susceptibility. This phase factor is calculated such that the imaginary part of the susceptibility fits the expected Lorentizian profile around resonance. The validity of this phase factor can be checked by comparing the resonant frequency as read from the real data crossing compared with the imaginary data maximum. The susceptibility for the Py film is shown in Figure Field dependence In the results from the PIMM shown in Figure 1.8 the resonant frequencies are seen to be field dependent, with the frequency increasing with field. It was shown in Section A.4.2 that the relationship along the easy axis is: ω 2 = γ 2 (H K + H b )(H K + H b + M s ) γ 2 (H K + H b )M s, (1.13) where H K and H b are the anisotropy and applied bias fields respectively, and the approximation is taken in the small field range so that M s >> (H K + H b ). A plot of small

29 Introduction 15 bias fields vs frequency with a linear fit for 50 nm Permalloy is shown in Figure 1.9. The sample has its in-plane easy axis aligned with the bias field. Knowing µ 0 M s = 1.03 T (NIST Py as seen in Table 5.3) it is easy to measure H K = 4.58 ± 0.05 Oe from the x- intercept, and g = ± from the gradient. Note that if larger field regimes are measured it is possible to fit for all of the samples properties simultaneously, including M s. Larger field results will be discussed in Chapter 5. Damping as a function of field Plotted in Figure 1.10a (open circles) is f/f, a measure of the Py linewidth, where f is defined as the full width half maximum (FWHM) of the resonance line. This is directly proportional to the Gilbert damping constant α since f = 2αf [71]. It is interesting how f/f increases for low fields. This can be explained by considering the geometry of the coplanar line. As the center conductor has a small width ( 350µm), a non-homogeneous field is created across the sample. The inhomogeneity is enough to excite non-uniform travelling wave modes with non-zero wavevectors. These wavevectors give an apparent increased broadening of the measured absorption line in addition to other dissipative processes such as Gilbert damping [63]. There are two observable consequences of the spin wave excitation. As the measured response is a convolution of the intrinsic damping and the spin waves excited, the measured resonance frequency will be shifted up and the measured linewidth increased [37, 138]. Here the increase in linewidth will be illustrated with an intuitive argument [83], with a more complete inclusion of wavevector effects described in the model derived in Chapter 4. The simplest model of the coplanar line is that of a uniform current distribution across the line in the x-direction, assuming the ground planes have zero potential (A more accurate current distribution is also included in Chapter 4). This geometry generates wavevectors with a sinc distribution in k x (See inset of Figure 1.10b). The majority of wavenumbers with appreciable magnitude are contained before the first pole of sinc(k x ) at k x = 2π/d. This distribution can be approximated with a discrete set of wavenumbers of constant amplitude with values in the range 2π/d < k x < 2π/d. The frequency at which these wavenumbers precess is given by the dispersion equation for a thin film [104]: f(k x ) = γ 2π [H(H + 4πM s) + (2πM s ) 2 (1 + e 2kxt )] 1/2. (1.14)

30 16 Introduction f/f 1 FWHM by PIMM FWHM by FMR First approx FMR + approx Applied Bias field (Oe) (a) Experimental linewidth from PIMM data (circles) and extrapolated FMR data (solid line). Also shown is the predicted contribution from the k x 0 wavenumbers. When added to the FMR extrapolation the similarity to the PIMM data is clearly seen FT of stripline Frequency (GHz) fo f 0 Amp 2 /d Range of k excites range of frequencies k /d 4 /d Wavevector k x k x (b) The dispersion equation for low k x. As shown in the inset, the range of excited wavevectors below 2π/d have significant amplitude. These correspond to the frequencies in the range shown by f. Figure 1.10: Convolution of excited wavevectors with linewidth to obtain measured linewidth.

31 Introduction 17 The wavevectors are radiated perpendicular to the coplanar line (x-direction), and consequently assumed to travel perpendicular to M. This is valid if M is initially aligned along the coplanar line and as long as the amplitude of the pulse field is less than the amplitude of the bias field. The range of excited wavenumbers produces a range of excited frequencies which are dependent on the thickness of the film. This has been experimentally investigated by Nibarger et al. [116]. To a first approximation, the frequency range f is related to the wavenumber range k by the gradient of the dispersion equation (see Figure 1.10b.): This gives a change in f as: f k df dk. (1.15) k 0 f = k(2πm 2 s t)[h(h + 4πM s )] 1/2. (1.16) For comparison, the same film was measured in a conventional FMR at 9.5 GHz aligned along the same axis. The FMR technique measures the derivative of the resonance curve as a function of field. The linewidth is extracted from the observed maxima and minima (corresponding to the FWHM of a peak) as a derivative linewidth H der. In the NIST Permalloy sample H der is measured to be 24 Oe. It is easy to calculate that the true FWHM field linewidth H = ( 3) H der. This can be compared to a frequency linewidth f using the relation f = ω H. For the fundamental FMR H mode, in general α/f is constant at all fields; the FMR point (as calculated in terms of f above) extrapolates as a constant on the plot of f/f. This is shown in Figure 1.10a as a solid line. The most obvious feature of the PIMM data is the increase in damping at small applied fields. The dashed line in Figure 1.10a illustrates the additional contribution to the intrinsic linewidth expected from any excited spin wave modes. Note from the approximations made in the intuitive approach (uniform field over the central conductor; limited range of modes considered) this is taken only as a first approximation. Also plotted in Figure 1.10a (dotted line) is the sum of the extrapolated FMR value with this approximate increase in modes. Even with such a simple approach the trend towards larger damping at low bias fields is seen in this model. Also, consideration of an inhomogeneous resonant line broadening due to anisotropy dispersion would need to be considered at these low bias fields. This is because the film

32 18 Introduction was sputtered, resulting in a polycrystalline texture. Each grain has a slightly different orientation, so when the applied field is small, the resonance conditions in each grain are notably different. The measured response is a summation over all of the grains, resulting in a measured increase in linewidth. In the NIST Py sample with an average hard direction anisotropy of 5 Oe, this effect would be limited to very small bias fields (< 20 Oe). A combination of this and the calculated spin wave modes would account for the experimentally observed increase at low bias fields PIMM conclusions It was illustrated that an effective susceptibility can be found from a Fourier transform of the temporal data. However, the pulsed technique has several limitations. The data processing is difficult, requiring data manipulation for the subtraction step, and before Fourier transforming. Also, the maximum resonant frequency clearly observed is dictated by the pulse rise time, which with the additional impedance from the sample could be as slow as 200 ps, or 5 GHz. The technique was adapted to measure directly in frequency space, in a similar way to reference [37]. The RF field has the advantage of only being frequency limited by the microwave components bandwidth and sensitivity of the detectors. Also, the samples magnetisation equilibrium position can be set parallel to the coplanar line with an applied field, without the deviation caused by the step pulse. This technique will be presented in the next chapter. 1.2 Thesis Outline So far, this introduction has given an overview of dynamic magnetometry measurements measured with a broadband time domain measurement technique. Chapter 2 introduces a new technique developed in this thesis using a continuous wave excitation through the same coplanar line introduced in Section 1.1. This technique uses a network analyser to both provide the excitation signal and to measure the response of the sample. This chapter describes the setup, as well as a detailed analysis of the microwave propagation in the coplanar line geometry. Background microwave theory is provided in Appendix C. Chapter 3 presents results from fundamental mode studies on both an exchange biased and an exchange spring system. Two modes are seen in the exchange spring system, motivating a detailed theory to explain non-uniform excitations through the

33 Introduction 19 thickness of the film. This theory is detailed in Chapter 4. The theory includes effects from the conductivity of the sample, the conductivity of the shield and the non-zero wavevector excitation of the coplanar line. Both the dynamical magnetic field and exchange contributions are included in single and bilayer thin film configurations, with the appropriate boundary conditions. At the end of this chapter, the results from the exchange spring system explored in Chapter 3 are fitted with this model. In Chapter 5 experimental results are shown for higher order SSW modes. The films used in this chapter were grown at the University of Leeds, and the first section of the chapter details the growth procedure. Finally, Chapter 6 presents future directions for new work, as well as concluding remarks on the dissertation.

34 20

35 Chapter 2 RF susceptibility This chapter extends on the PIMM setup introduced in the Chapter 1 to a continuous RF setup using the same coplanar line. In the microwave range considered, it is necessary to give special attention to the transmission lines as at high frequencies the currents demonstrate wave-like behaviour. This chapter will illustrate the experimental technique as well as detail the necessary data manipulation to extract an effective magnetic susceptibility. Background RF theory can be found in Appendix C. 2.1 RF setup In this setup both the excitation field and resulting measurement are performed by a network analyser. As seen in Figure 2.1, the network analyser replaces the pulse generator and the oscilloscope and is used to complete the circuit with the waveguide. The network analyser sends an AC signal down the microwave coplanar line with amplitude 10 dbm, which when considering a characteristic impedance in the circuit of 50 Ω, is approximately 0.7 V. This is a smaller signal than the DC pulse used to excite the sample in the coplanar line, however the complete power is at a single frequency, resulting in a better signal to noise ratio. The AC signal also does not change the equilibrium position of the sample, removing any problems with second harmonic distortion. The AC field created by the coplanar line will be absorbed by the sample when the resonant properties of the film are reached. The absorption at the input frequency is measured by the network analyser. 21

36 22 RF susceptibility Microwave Probe NETWORK ANALYSER PORT 1 PORT 2 Static Field WAVEGUIDE SAMPLE z y x RF Field Figure 2.1: Experimental continuous wave setup. The coplanar waveguide is connected in series to the network analyser, which provides the excitation signal whilst simultaneously measuring the microwave absorption by the sample. Network analyser A network analyser measures the linear response of an electrical network in both reflection and transmission geometries. Network analysers rely on either diode (broadband) detection, or tuned-receiver (narrow band) detection [4]. Scalar network analysers rely on diode detectors to convert the RF input signal into a DC signal. This loses any phase information, and the noise floor is dictated by the measurement bandwidth, resulting in poor dynamic range. To accurately measure susceptibility it is necessary to have phase information, as then the real (dispersion) and imaginary (dissipation) parts can be separated. The measurements in this project were taken on a vector network analyser. Vector network analysers use a tuned-receiver architecture, which gives a much better sensitivity and dynamic range compared to the diode detector. This is because the receiver samples only a narrow bandwidth around the excitation frequency, resulting in rejection of spurious and harmonic signals. The tuned receiver translates the high frequency RF signals to lower intermediate frequencies (IF) to be measured by sampling receivers. Noise signals are also translated, but the IF signals are filtered through a narrow band filter, with the majority of the noise falling outside of the measurement range. This gives a low noise floor and a large dynamic range, whilst the IF processing can preserve the phase information. A schematic of a vector network analyser is illustrated in Figure 2.2.

37 RF susceptibility 23 RF source Harmonic generator Filters Detectors REF Directional coupler REV PORT 1 PORT 2 IF convertors FWD A/D conv, processing & OUTPUT Figure 2.2: Schematic of a vector network analyser. The incoming signals are lowered by IF converters and passed through narrow band filters before the signal is detected Scattering parameters When working with electronic measurements in the microwave region, the travelling waves amplitude and phase are more easily quantified than conventional voltages and currents. This is particularly the case when dealing with single frequency CW measurements. However, even when working with components sensitive to waves (directional couplers for example), the circuits can still be described in terms of voltages and currents as seen in Appendix C, or as reflected and transmitted waves. Here an equivalence between the two descriptions is shown. From the description first shown by Penfield [124] and later explained in detail by Kurokawa [93], the incident power wave and reflected power wave for each port i is described by a i and b i respectively:

38 24 RF susceptibility V 1 + S 21 V 2 + S 11 S 22 PORT 1 PORT 2 V 1 - S 12 V 2 - Figure 2.3: Illustration of the relationship between the voltage waves and the scattering parameters. a i = V i + Z 0 I i 2 Re[Z 0 ], (2.1a) b i = V i Z 0 I i 2 Re[Z 0 ], (2.1b) where V i and I i are the voltage across and the current flowing into the ith port, and Z 0 is the characteristic impedance across all ports. This is illustrated in Figure 2.3. Simple algebra returns V i and I i from a i and b i. In this way, the use of reflected and transmitted waves is a suitable substitute in place of voltage and current for any circuit analysis. Using these definitions of incident and reflected waves it is possible to introduce scattering parameters denoted as a tensor ( S) to describe the relationship between the different directions. For a general two port circuit the scattering parameters expressed in matrix form are: b 1 b 2 = S 11 S 12 S 21 S 22 a 2. a 1, (2.2) where the reflected waves (b i ) depend on the independent incident waves (a i ). Hence the scattering parameters S nm describes the scattering of the incident wave m to the reflected or transmitted wave n. The scattering parameters describe the behaviour of

39 RF susceptibility 25 a linear electronic network when excited by small continuous wave stimuli. The term scattering is analogous to the scattering of an optical plane wave in optical physics, with the distinction in RF electronics being the reflection and transmission of an RF wave along a transmission line when it meets a discontinuity. The discontinuity can simply be an impedance change in the transmission line. Each scattering parameter is a complex unitless number, representing amplitude and phase. The results measured experimentally from the network analyser are in terms of scattering parameters, so a thorough interpretation is required. If the system is reciprocal, measurements are the same regardless whether they are measured in the forward direction (from port 1 to port 2) or the reverse direction (from port 2 to port 1). If this is the case, S 21 = S 12 and S 11 = S 22. Normally ferromagnetic films are not reciprocal due to the surface on which the propagation of magnetostatic surface waves occurs, hence the large range of ferrite microwave devices used such as directional couplers, isolators and circulators. However, in the particular case of the thin films and small wavevectors in this project it is possible to disregard the non-reciprocal nature of the scattering. Representing the voltage and current in a two port system in matrix form is commonly represented using ABCD parameters in the following way [128]: V 1 I 2 = A B C D. V 2. (2.3) I 2 If for example, the sample causes a change in the line impedance of Z s in series with the waveguide system, the ABCD parameters become: A B C D = 1 Z s 0 1. (2.4) A relationship between the ABCD matrix and the scattering parameters can be found using the wave definitions (equation (2.1)) and equation (2.2) [128]: S 11 = A + B/Z 0 CZ 0 D A + B/Z 0 + CZ 0 D, 2 S 21 = A + B/Z 0 + CZ 0 + D. (2.5a) (2.5b) These relationships will be used to derive a magnetic susceptibility from the experimentally measured scattering parameters, however first consideration needs to be given to

40 26 RF susceptibility the complete microwave circuit containing the waveguide and sample. Calibration and errors Network analysis is plagued with calibration errors and care must be taken to reduce these problems. The errors can be grouped into three categories: Systematic errors Random errors Drift errors FREQ TRACKING REV/REF & FWD/REF REF REV FWD LOAD MISMATCH DIRECTIVITY WAVEGUIDE RF SOURCE CROSSTALK SOURCE MISMATCH Figure 2.4: The six error terms for the forward direction. Directivity, crosstalk, source mismatch, load mismatch, and tracking in both detectors. If the reverse direction is considered, twelve terms are necessary. The largest errors in the equipment are the systematic errors. These include: directivity and crosstalk errors relating to signal leakage; source and load impedance mismatches relating to reflections at the inserted waveguide; and frequency response errors caused

41 RF susceptibility 27 by reflection and transmission tracking errors. This means that there are six error terms to compensate for in each direction, so for a complete two port setup there are twelve error terms to minimise. Figure 2.4 demonstrates where each of these errors enter. In the coplanar line case these errors are minimised by measuring the properties of the coplanar line. This is best done using a through, reflect, line (TRL) calibration [3, 52]. This requires different length transmission lines to correctly measure the wave propagation in the coplanar line. As the setup used in this work relied on commercially available waveguides (see Section 2.1.2), different transmission lengths were not available. It was possible to do a short, open, load, through (SOLT) calibration ignoring the waveguide. This minimises errors up to the insertion of the waveguide including the cables and connectors and is performed using coaxial standards. The insertion of the waveguide brings back error into the system. It should be noted here that all of these errors are frequency dependent, making a true frequency susceptibility difficult to obtain. To this note, the easiest way to avoid these calibration errors is to work with single frequency, field swept measurements. This does not remove the errors, but the response of the system is measured both at and away from resonance, and a normalised value of the susceptibility as a function of field is possible with none of the forementioned errors playing a role. Keeping the frequency constant and only sweeping the field also has the benefit that the frequencies can be chosen away from any spurious waveguide and connector resonances. This means accurate results can be obtained independent of the frequency response of the waveguide. However, the random errors cannot be removed in this fashion, with the largest random error being noise. This comes in the form of sampling noise, as well as the instrument noise floor. The sample also introduces random noise through thermal fluctuations. Noise is reduced by using the largest stable power on the network analyser (10 dbm) and by averaging each measurement at any one field by a factor of 4. The noise can also be reduced by decreasing the IF filter bandwidth of the network analyser. Also, the drift errors occur when the test systems performance changes after calibration. This is reduced by waiting for the system to reach operating temperature (a couple of hours), but as already mentioned, the calibration becomes arbitrary if using field sweeps. As long as the drift is on length scales longer than it takes to do a single field sweep, it can be neglected. The drift can be checked by comparing the zero field measurement before and after a field sweep. It was found not to be a problem.

42 28 RF susceptibility Waveguide Connection between the SMA cables and the coplanar line involved adaptation of two different geometries with the least impedance disruption. Commercial microwave end launch connectors were adapted from Southwest Microwave, as illustrated in Figure 2.5. These were constructed from 303 stainless, specified to be non-magnetic, but proved to give an unacceptable signal in large DC fields. Working with Southwest Microwave new end launch connectors were developed which were constructed of brass, with an alballoy plating (a copper, tin, zinc composite plating). The center conductors are BeCu also with an alballoy plating. These proved to be completely non-ferromagnetic. Unfortunately there is still a small field dependent background because the non-magnetic SMA connectors on the cables proved to be slightly magnetic (most likely a nickel coating on the center pins). This background becomes important when measuring small signals and is subtracted as outlined in section a) b) Overhang allows groundconnection close to launch point Pin centered on trace Ground stitching close to ground edge Figure 2.5: a. The Southwest Microwave end launch connectors mounted on a test board to give a coplanar waveguide. b. Illustration of the microwave launch pin to the coplanar geometry. The end launch connectors are matched to commercial coplanar boards also produced by Southwest Microwave. The boards are 25 mm long by 12.5 mm wide. This gives a suitable surface area for a 10 mm 2 sample. The boards use a glass reinforced hydrocarbon/ceramic laminate, with board specifications given in Table The boards are fragile, so they are reinforced with a 1 mm thick copper backing plate. Referring to

43 RF susceptibility 29 substrate substrate thickness (mm) dielectric constant conductor width (mm) Rogers R Rogers R Rogers R Rogers R gnd-gnd spacing (mm) Table 2.1: Southwest Microwave coplanar test board specifications the impedance derivation in Section C.3 it can be seen that the Southwest Microwave test boards used in this project have a nominal 50 Ω characteristic impedance. This provides a good match between the cables and coplanar line. Coplanar line was chosen as the preferential stripline geometry as it is a lower loss transmission geometry than microstrip line, and it is easily adapted to different size centre conductors. Also, as seen in Figure C.3 the coplanar line exhibits a larger and more uniform field density above the center conductor than the microstrip geometry. The complete field profile for the coplanar line can be found in reference [144]. 2.2 Susceptibility At low frequencies it is possible to measure the susceptibility considering the sample as a discrete element [49]. To do this we use the simple transmission theory. From equation (1.5) and considering that d m dt x = iωm x, the voltage due to the sample is: V (H) = µ 0 idlωm x, (2.6) where the voltage is a function of applied field if the microwave field h = hˆx is applied at a constant frequency. The constant frequency is chosen to avoid any spurious waveguide resonances and to minimise any drift in the calibration. The magnitude of h can be approximated in the same way as in equation (1.2), h = h I/(2w). The

44 30 RF susceptibility magnetisation can be represented by a susceptibility m x = χ xx (H) h x. On substitution: V (H) = µ 0idlωI 2w χ xx(h). (2.7) The change in voltage acts like an extra inductance due to the susceptibility. In fact, using the telegraphers equation (equation (C.2)), equation (2.7) can be expressed as a inductance: L(H) = µ 0dl 2w χ xx(h). (2.8) This inductance is complex, with the imaginary part representing the radiation resistance caused by absorption of energy in the magnetic sample. This gives a complex susceptibility in a similar form as the Fourier transformed PIMM data Data manipulation The magnetic susceptibility has been related to a change in inductance of the line, but it is necessary to relate the measured S parameters to this inductance change. Solving for an impedance change as shown in equation (2.4) with the scattering parameter relationships in equation (2.5) gives: Z s = 2Z 0 + 2Z 0 S 21. (2.9) The effect of loading the coplanar line with a sample is dependent on the magnetic susceptibility of the sample. To characterise the non-magnetic influence of the sample, the sample is subject to a saturating field. This is done using one of the two ways introduced in the PIMM description in Chapter 1. Either a saturating field is applied perpendicular to the coplanar line so the excitation field has no influence on the magnetisation, or a large saturating field is applied parallel to the coplanar line such that the resonant frequencies in the sample are much larger than the measured frequency. This reference measurement can be subtracted from the data measurement, giving: 1 Z(H) = 2Z 0 ( S 21 (H) 1 ), (2.10) S 21 (0) where Z(H) is the complex impedance change due to the magnetisation, S 21 (H) is the field swept measurement, and S 21 (0) is the reference measurement. The susceptibility is

45 RF susceptibility 31 realised by Z(H) = iω L(H): χ xx(h) = 4Z 0wC iωµ 0 dl ( 1 S 21 (H) 1 ). (2.11) S 21 (0) The star on χ xx denotes that this is an effective susceptibility, not the true susceptibility due to the approximation of a lumped impedance change. Also the absolute magnitude will vary with constant C which describes the field to waveguide separation. Magnetic influence Unfortunately even though considerable effort was spent minimising the magnetic contributions due to the waveguide itself, the connectors in the cables had a small Nickel coating which superimposes a broad resonance on the measurements. Following a similar method to above, a sample with similar thickness and dimensions to the rest of the project, but which is non-magnetic was measured over the field range. This mimics any reflections from the sample placement on the waveguide, but the influence from the sample ( Z) is zero. An effective susceptibility of the connectors is measured. The simple subtraction of this broad resonance returns the susceptibilities expected from the measured samples (see Figure 2.6). Unfortunately the subtraction introduces more noise into the measurement. Future improvement could be had by obtaining completely non-magnetic connectors, or fabricating the coplanar line so the end launch adaptors and connectors are outside of the magnet. As the unwanted signal is very broad, it does not shift the position of the maximas of the signal from the sample. Thus it is best to read the resonant frequencies from the nonsubtracted data to have a lower noise floor. It is also possible to gain the relationships of the resonances in field by simply measuring the maximum absorption in transmitted signal (S 21 ) without any calculation. However this does not give any phase information and is not appropriate for reading true resonant linewidths. 2.3 Propagation In the previous section the effective susceptibility of the system was found assuming that the sample acted as a lumped impedance change on the coplanar line. This assumption is correct so long as the sample dimensions are small compared to 1/4λ where λ is the

46 32 RF susceptibility a) b) 3.4 GHz 5 GHz 3.4 GHz 5 GHz χ xx (arb. units) SUBTRACT χ xx (arb. units) Field (Oe) Field (Oe) Figure 2.6: a. Effective susceptibility as a function of field from a 60nm Py film calculated from a lumped element method with the Real (crossing) and Imaginary (peak) parts shown for 2.2GHz (red) and 3GHz (blue). b. Susceptibility from the sample after subtraction of susceptibility measured from connectors. wavelength of the microwave as it travels in the coplanar line. Already seen in Figure 2.6, the two resonant frequencies show a different profile. Using equation (C.18) for the wave velocity in the coplanar line, and assuming that ǫ r = 3.5, gives v m.s 1. The largest frequency we observe, 15 GHz, gives a wavelength of λ = v/f 6.5 mm. The presence of the sample and substrate conductivities only slow this velocity down, giving even shorter wavelengths. With the sample dimensions in the z-direction being of the order of 1 cm, more than one half-wavelength extends over the sample at these high frequencies, giving rise to propagation (and even lateral in-plane standing mode) effects in the sample Coplanar line description As the sample does not transverse the entire coplanar line, the line should be modelled as a distributed impedance, with three discrete sections as shown by Barry [13]. The three sections consist of free space regions I and III, and the sample loaded region II as illustrated in Figure 2.7. The free space regions have an impedance Z a, close to the characteristic impedance of the system Z 0 = 50 Ω. The loaded region of the waveguide will have a different impedance than Z s even far away from resonance. The metallic films conductivity dramatically alters the resulting microwave field lines and changes the wave propagation in the coplanar line. Even with non-conducting ferrites, the influence from

47 RF susceptibility 33 the substrate on which the sample is grown having a dielectric different than air alters the propagation properties. These effects are included by defining an impedance Z s for the loaded region II. Z 0 Z a Z s Z a Z 0 l a l s l a Figure 2.7: Illustration of the different regions of impedance in the coplanar line setup. Three regions are noteable, the characteristic impedance Z 0 is assumed in the cables and connectors, Z a is the impedance in the unloaded coplanar line and Z s is the impedance of the coplanar line loaded with the sample. The characteristic impedance is assumed to hold for the rest of the system, including the cables and connectors. Even if this is not entirely true, a SOLT calibration (see Section 2.1.1) was performed to the waveguide end launch adapters ensuring that the cable and connector properties are not measured. Consider now the ABCD matrix (equation (2.3)) for the free space regions I and III [166]: Ã I,III = coshγ al a 1 Z a sinhγ a l a Z a sinhγ a l a coshγ a l a, (2.12) where γ a is the propagation constant in the regions I and III, γ a = iβ a + α a. Similarly, the ABCD matrix in the loaded region II: Ã II = coshγ sl s 1 Z s sinhγ s l s Z s sinhγ s l s, (2.13) coshγ s l s

48 34 RF susceptibility and γ s = iβ s + α s. Here the real part α s denotes the loss due to the sample Discontinuity reflection Figure 2.8 shows the magnitude and phase of S 21 measured with both an unloaded and loaded coplanar line. Following is a description of characterising the transmission properties of the coplanar line and the effect on measurements. a) b) S 21 (db) -1-2 Stripline Stripline + Sample Freq (GHz) Phase (deg) Stripline Stripline + Sample Freq (GHz) Figure 2.8: a. Plot of the scattering parameter S 21 with and without a sample on the coplanar line. Oscillations are seen from standing waves. b. The phase of the scattering parameter S 21 with and without a sample on the coplanar line. At the boundary of the connector and region I, the complex reflection coefficient, Γ, is the amplitude of the reflected voltage wave normalised to the amplitude of the incident wave: Γ = V 0 V 0 + = Z a Z 0 Z a + Z 0. (2.14) A similar but negative coefficient exists at the exit of the waveguide to the cables, namely Γ. Consider the unloaded coplanar line where equation (2.12) now applies to the complete coplanar line length l. Applying the ABCD to S parameter transformation as shown in equation (2.5) to equation (2.12) gives the scattering parameters as:

49 RF susceptibility 35 S 11 = S 21 = (Za 2 Z0)sinhγ 2 a l Z 0 Z a (cosγ a l + coshγ a l) + (Z0 2 + Za)sinhγ 2 a l, (2.15) 1 coshγ a l (Za Z 0 + Z 0 Z a )sinhγ a l. (2.16) In this simple case where Z a = Z 0 the scattering parameters reduce to: S 11 = 0, S 21 = e γal. (2.17) Transmission propagation coefficient Consider a transmission line with resistance R, conductance G, capacitance C and inductance R as explained in Section C.2. To an initial approximation it is possible to assume the coplanar line is lossless, and in this case, γ a is purely negative. From equation (C.6) with R and G zero, γ a = iω L a C a = iω/v a, where v a is the phase velocity in the coplanar line. This means with zero losses there is simply a phase shift as a function of frequency as measured on the unloaded waveguide (Figure 2.8b). Including losses, γ a can be rearranged from equation (C.6) to read: γ a = iω L a C a 1 i( R a + G a ) R ag a. (2.18) ωl a ωc a ω 2 L a C a Assuming the losses in the line are small, the second order term RG/(ω 2 LC) can be ignored. Expanding the root containing the rest of the loss terms in a Taylor series to the first higher order real term gives: γ a = α + iβ, (2.19) with: α = 1 2 (R a Z a + GZ a ), (2.20) β = ω L a C a, (2.21)

50 36 RF susceptibility where by the same approximation Z a = L a /C a, the lossless characteristic impedance case of equation (C.10). Considering Z a Z 0, the transmitted wave S 21 given by equation (2.17) now includes small losses in the following way: S 21 = e iωl va +αal, (2.22) where α increases slightly with frequency due to the reduction of the skin depth with frequency. This effectively reduces the propagation cross-section of the wave, and increases the resistance of the line, as seen in the magnitude of S 21 in Figure Loaded line The results from such a measurement are compared to the measurement of the unloaded waveguide and shown in Figure 2.8. It is clearly seen that there is a difference in the phase and amplitude of the transmitted signal S 21. The oscillations seen in the magnitude of S 21 without the sample result from standing waves setup on the length of the coplanar line, and with the sample result from a combination of standing waves on the coplanar line and sample. To correctly represent the line, consideration must be made for the reflections at each side of the sample, the waveguide connectors, and the transmission through both the loaded and unloaded line sections. Here the waveguide connector reflections are ignored, and in terms of the so-called cascading scattering parameters, it is a simple matter of multiplying the sections together: T = e iωla va +αala 0 e iωls vs 0 e +iωla va +αala +α sl s 0 0 e +iωls +α vs sl s e iωla va +αala 0 0 e +iωla va +αala (1 Γ) 1 Γ(1 Γ) 1 Γ(1 Γ) 1 (1 Γ) 1 (1 + Γ) 1 Γ(1 + Γ) 1 Γ(1 + Γ) 1 (1 + Γ) 1. (2.23)

51 RF susceptibility 37 The cascading scattering parameters (T) are related to the S-parameters by: S 11 S 12 S 21 S 22 = T 12 T 22 T 11 T 22 T 12 T 21 T 22 1 T 22 T 21 T 22. (2.24) The interesting part of the equation (2.23) is in the velocity of the wave through the system. This will introduce an arbitrary phase to the measurement and this should be accounted for. The problem is greatly simplified if the reflections from the sample edges (Γ) are ignored. This is a valid approximation if Γ does not change significantly between the reference and magnetic measurement. From equation (2.23) and equation (2.24), the transmission S 21 is found to be: S 21 = e iωls +α vs sl s e 2 iωla va +αala. (2.25) If the loss terms α i are ignored this reduces to a phase factor. The loss terms are frequency dependent, but will only affect the amplitude of the calculated susceptibility. As the measurements are performed as a function of field, this does not affect the shape of the measured resonance, so is not necessary to consider. The phase factor is found from the total propagation time for the wave to traverse the waveguide: τ = l s v s + 2l a v a (2.26) If the measured scattering parameter is multiplied by an appropriate phase factor e iωτ, then the lumped element method (equation (2.11)) for extracting the susceptibility gives an approximately correct phase. This method relies on an accurate measure of τ, which can be found from the linear relationship between phase and frequency as shown in Figure 2.8. Note, that if the measurements would be taken as a function of frequency, the loss parameter α should really be calculated as it would change the resonance shape. A more accurate method to eliminate the need to calculate τ or measure α is used in this project (also demonstrated by Counil et al. [36]), and it involves dividing by the reference signal, rather than a subtraction as used in the lumped element method. Consider the propagation change in the sample caused by the magnetisation: 1 1 = (L + L)C LC 1 L, (2.27) v s v s0 2 Z s where L is a change in impedance of the coplanar line due to the magnetisation as

52 38 RF susceptibility shown in equation (2.8); L(H) = µ 0dl 2w χ xx(h). v s0 is the velocity in region II away from resonance, and v s is the velocity in the region II as a function of applied field. The approximation is an expansion to first order. Adding this field dependent term to equation (2.25) gives the field dependent scattering parameter: S 21 (H) = e iωls vs +α sl s e 2 iωla va +αala e iωl2 s µ 0 dχxx(h) 4wZs. (2.28) Dividing this by the reference scattering parameter measurement (S 21 (0)) cancels out the first two exponentials, and only leaves this last field dependent term. From this it is easy to extract the susceptibility: χ xx (H) = log[ S 21(H) S 21 (0) ] 4wZ s C iωl 2 sµ 0 d. (2.29) Shown in Figure 2.9 are plots of the real and imaginary components of a permalloy sample at high (λ/4 < l s ) frequencies calculated by both the lumped element method, and the distributed method shown here. Note, the magnetic influence of the connectors as discussed in Section has been subtracted. a) b) 3.4 GHz Im[χ xx ] (arb units) 5 GHz 8.6GHz 11GHz 10GHz 12.6GHz Im[χ xx ] (arb units) 3.4 GHz 5 GHz 8.6GHz 10GHz 11GHz 12.6GHz Field (Oe) Field (Oe) Figure 2.9: a. Imaginary part of effective susceptibility of Py (60nm) over wide range of frequencies as calculated by lumped element method, showing an inconsistancy in phase with frequency resulting from not considering the propagation of the wave along the sample. b. Same results as (a) calculated with division method (equation (2.29)). Note that an arbitrary phase is not present and the Im[χ xx ] is as expected from Chapter 1. It must be remembered this method is only suitable when the reflection parameter Γ is not significant in the measurement. This is the case for the continuous coplanar line

53 RF susceptibility 39 where the sample only introduces a perturbation on the otherwise impedance matched circuit. If however, shorted lines were used [51] or another geometry where the impedance change from the sample is significant, the more accurate but overly complex calculation of the complete cascading scattering matrix (equation (2.23)) would need to be considered. 2.4 Chapter summary Shown in this chapter is a continuous wave technique using the same coplanar line as the pulsed technique introduced in Chapter 1. This technique exhibits a larger signal to noise ratio and a larger useable bandwidth than the PIMM setup. In particular the RF technique is superior at applied fields greater than 500 Oe, where the PIMM signal reduces dramatically due to a reduced transient response. Spurious waveguide reflections can be ignored by choosing well behaved frequencies and sweeping the field to find the resonances. Also, as the samples measured are kept large to increase magnetisation response, they introduce a phase into the measured scattering parameters. The simpliest way to recover an effective susceptibility is to divide the magnetic response by a reference measurement taken away from resonance. This gives an effective susceptibility with the correct phase.

54 40

55 Chapter 3 Fundamental mode studies In this chapter results which focus on measuring the fundamental resonance mode are presented. This chapter will detail what parameters can be extracted from measuring the fundamental mode in coupled systems. Firstly a Fe/MnPd exchange biased bilayer is explored, followed by a Py/Co exchange spring bilayer. In both cases a simple theory is presented to fit the measured data. 3.1 Fe/MnPd In this section angular measurements are described for a strongly crystalline Fe film with definite anisotropies. The Fe film is coupled to a MnPd antiferromagnet to demonstrate how a unidirectional anisotropy in the Fe film created by the antiferromagnetic/ferromagnetic coupling can be observed. These interfacial coupling effects between the ferromagnet (FM) with the antiferromagnet (AFM) were first observed by Meiklejohn and Bean [107, 108] and are referred to as an exchange bias. Commonly, the exchange bias effects are characterised by quasi-static hysteresis measurements of M(H). Two effects are generally seen, an enhancement of the coercivity field, as well as a shift in the hysteresis loop along the applied field axis (see exchange bias review articles [17, 90, 119, 156]). This shift is from a uni-directional anisotropy caused by the FM/AFM coupling. There is a great technological interest in characterising exchange bias phenomena. One technological application uses the biasing effect as a pinning mechanism in spin valve structures in order to utilise the giant magnetoresistive effect [38, 46, 47, 103]. 41

56 42 Fundamental mode studies Measuring the shift from the hysteresis loop requires the interpretation of complicated magnetisation processes involved with coercivity. This may lead to ambiguities in determining the exchange bias unidirectional anisotropy, particularly if the sample has different reversal mechanisms for different applied field directions. On the other hand, ferromagnetic resonance measurements characterise the curvature of the free energy, allowing a direct measurement of the exchange field. Local fields such as anisotropies and shape demagnetising fields can be studied [104, 106, 111, 119, 123, 170, 173, 174]. The film was grown by Professor Kannan Krishnan s group at the University of Washington, and is a Fe(22 nm)/mnpd(100 nm) bilayer with the anti-ferromagnet having composition Mn 56 Pd 44. It was grown on a single crystalline MgO substrate oriented along the [001] direction using ion-beam sputtering in a vacuum at less than 10 8 Torr. More growth details can be found in reference [31]. The ferromagnet was biased by sputtering the sample in a magnetic field of 300 Oe applied along the [100] easy axis of the Fe film. The structure of the film was determined using x-ray diffraction measurements (Figure 3.1b). From these results it was determined that the Fe layer is single crystal BCC and the MnPd layer is a single crystal, chemically ordered, L10 structure with its c-axis normal to the film plane. The epitaxial relationships are Fe[001] MnPd[001] MgO[001] (out-of-plane directions) and MnPd [110] Fe[100] MgO[110] (in-plane directions). Loops measured on a VSM show a bias of 35 Oe (Figure 3.1a). Similar Fe/MnPd films have been explored using quasi-static measurements before [19,20], largely from the point of view of reversal mechanisms H C1 a) b) H C2 Moment (emu) Field (G) Intensity 2x105 1x105 0 MnPd(001) MgO(002) MnPd(002) Fe(002) Figure 3.1: a. Hysteresis loop showing the exchange bias shift, measured along the cooling field direction. H C1 and H C2 are the two coercivities. b. High angle XRD showing crystalline structure of the sample. The peaks are labeled with the appropriate lattice type and direction. 2

57 Fundamental mode studies Angular dependence model The high crystalline quality of the sample, supporting well defined high order anisotropies, motivates the use of a simple model for the equilibrium magnetisation as a function of applied field. The large in-plane anisotropies involved in the single crystal iron structure allow the magnetisation direction to differ from the applied field direction at small fields. A theory is required which is capable of modelling the resonant frequency when the applied field and magnetisation are not coincident. The resonant frequencies of the SSWMs will be above the measurement range because the sample is very thin and they can be ignored. The conductivity is negligible with a sample this thin, and the in-plane wavevector k x is assumed to be equal to zero for simplicity. The magnetisation is assumed to be uniform throughout the Fe film. The orientation of the magnetisation is expressed by the angles φ FM and θ FM, measured from the film normal (y direction), and the in plane z direction (crystal axis [001]) respectively. All crystal directions are given relative to the Fe crystal. The orientation of the film axes can be seen in Figure 3.2. The free energy F T for the Fe film including unidirectional and four fold anisotropies is given by: F T =(K 1 2 M2 S) sin 2 (φ FM ) + K 4 (sin2 (2φ FM ) + sin 4 (φ FM )cos 2 (2θ FM )) + K U sin 2 (φ FM )sin 2 (θ FM ) J int M s sin(φ FM )cos(θ FM ) HM s sin(φ FM )cos(β θ FM ). (3.1) The first term is the effective demagnetizing energy, where 1/2Ms 2 is the out-of-plane demagnetization and K is a uniaxial anisotropy directed normal to the film plane. The second and third terms are the four-fold and uniaxial anisotropy energies, with anisotropy constants K and K U respectively. The symmetry axis of the uniaxial anisotropy is assumed to lie in plane along a [100] direction. The fourth term represents an exchange anisotropy due to exchange coupling at the interface between the FM and the AFM films. The energy J int corresponds to an effective volume unidirectional exchange anisotropy J int = H E M S aligned along a four fold axis where H E is the interface exchange anisotropy field. Note that H E should depend upon the Fe film thickness according to 1/t Fe where

58 44 Fundamental mode studies t Fe is the Fe film thickness. The last term is the Zeeman energy associated with the applied field H at an angle β from the z direction. Hard Axis x Hard Axis H eff y x M Min FM z [001] Bias Axis M in z [001] Figure 3.2: Illustration of the axes and angles involved in the Fe/MnPd bilayer Equilibrium direction Assuming the demagnetising energy is sufficient to keep the magnetisation in the plane of the film (ie φ = π/2), the magnetisation angle θ FM is determined by minimising the free energy (equation (3.1)): Hsin(β θ FM ) H E sin(θ FM ) 1 2 H Asin(2θ FM ) 1 4 H K1sin(4θ FM ) = 0. (3.2) Consider the case of the field applied along one of the four-fold anisotropy hard directions (β = π/4) if there is no uniaxial anisotropy. At zero field, the magnetisation lies along an easy direction in the positive direction defined by the four fold anisotropy. The magnetisation will align with the applied field for a sufficiently large applied field. The angle between the hard axis and the magnetisation as a function of applied field will be the same regardless of which easy axis the magnetisation is initially along at zero field. This is illustrated in Figure 3.3a. The situation is different if an unidirectional exchange anisotropy exists. Since H E is defined to be along a four fold anisotropy easy direction, there will be a discontinuous change possible in the magnetisation angle θ FM as the applied field is increased from

59 Fundamental mode studies 45 (a) /2 (b) /2 (rad) /4 (rad) /4 no bias Applied Field (Oe) bias Applied Field (Oe) Figure 3.3: Calculated magnetisation angle as a function of applied field along a fourfold hard axis with anisotropy 560 Oe. The blue line shows the rotation with the magnetisation initially along one easy direction, whereas the red line shows the magnetisation initially along a perpendicular easy direction. a. without exchange bias. b. with 10 Oe exchange bias (θ = 0). zero. If the magnetisation in zero applied field lies along the in plane axis which is perpendicular to the unidirectional axis, then the system is in a meta-stable state. At a critical applied field strength, it is more energetically favourable for the magnetisation to follow the same path as if it had originated from along the uni-directional exchange anisotropy axis. At this field the magnetisation will abruptly rotate to lie along this direction. This is shown in Figure 3.3b Resonance as a function of angle Following the theory of Smit and Beljers [146], ferromagnetic resonance occurs at a frequency given by: ( ) 2 ω = γ [ ] 1 2 F 2 F M 2 sin 2 (θ) θ 2 φ ( 2 F 2 θ. φ )2, (3.3) where the derivatives are taken with respect to the free energy of equation (3.1) evaluated with the magnetisation at the equilibrium defined by equation (3.2). In equation (3.3), ω is the microwave driving frequency, and γ the gyromagnetic ratio. This expression is valid for spin wave excitations with very long wavelengths and for resonance in very thin films,

60 46 Fundamental mode studies and it incorporates pinning effects as effective volume anisotropy terms. The frequency is a measure of the curvature of the energy at a local minimum. The significance of this is that contributions from the unidirectional anisotropy (and all other effective fields, for that matter) are evaluated with the magnetisation near equilibrium. The frequency therefore contains information about the local unidirectional exchange anisotropy. The resonance frequency can be found from the substitution of equation (3.1) into equation (3.3): ( ) 2 ω = [Hcos(β θ FM) + H E cos(θ FM) + H A cos(2θ FM) + H K1cos(4θ FM)] γ [ Hcos(β θ FM ) H ueff + H E cos(θ FM ) H A sin 2 (θ FM ) + H K1 (1 1 ] 2 sin2 (2θ FM )), (3.4) where H E = J int M s, H ueff = (2K Ms 2 ) M s, H K1 = 2K M s each applied field from Equation 3.2. and H A = 2Ku M s. θ FM is determined for Experimental results The microwave coplanar line technique can determine resonance conditions at any field applied in-plane. Experimental results for a complete rotation of the sample at a fixed frequency are shown in Figure 3.4, with the frequency fixed at 10.8 GHz. A fit to the data is shown using equation (3.4). θ = 0 is the field cooled direction, aligned along one of the four-fold easy axes. The applied field on the sample was created by a small coil setup to eliminate remnant fields that would be present in the larger electromagnet pole pieces. This allowed setting an accurate field (within 2 Oe) over the sample however, it limited the applied field to a maximum of 900 Oe. Unfortunately this field range was not sufficient to observe the complete resonance profile as a function of angle for any one frequency. However, at 10.8 GHz, a four-fold symmetry is clearly seen, as four peaks in the resonant field profile over a complete revolution of the sample. This agrees with the four fold energy term seen in Equation 3.1. The unidirectional exchange coupling can be seen as a difference in resonance field between θ = π and 2π. The unidirectional anisotropy is measured as 10 ± 2 Oe.

61 Fundamental mode studies 47 Field[Oe] Four 0 /2 3 /2 2 Angle[Rad] Figure 3.4: Resonant field as a function of angle at a constant frequency of Fe/MnPd sample. Error in angle ± 2 degrees ( 0.01π rad), error in field ± 2 Oe. The evidence for four-fold anisotropy is clearly seen. The solid line is a theoretical fit derived from the free energy, H E = 10Oe, H A = 0Oe, H K1 = 560Oe Hard axis reorientation Measurements of resonance frequency were made with the field aligned along one of the four-fold hard axes and the frequency was recorded as a function of applied field strength. Examples are shown in Figure 3.5, where results for the field applied along both directions of one hard axis are shown. Notice within the available applied field range, the measured frequencies along the hard direction are all below 10 GHz. The results are shown for the field along the π/4 direction by squares, and for the field along the 5π/4 direction by stars. Comparison with equation (3.4) is shown in both cases. The solid lines represent the solutions for the π/4 direction whilst the dashed lines represent the solutions for the 5π/4 direction. There are two solutions for each measurement. The solutions correspond to the magnetisation starting at zero applied field along either direction of the Fe easy axis. However, as the sample has a uni-directional exchange field along one of the directions, it is seen that only one of the solutions correctly fits the experimental data. Several general features appear in both alignments. As seen in Figure 3.5 the fre-

62 48 Fundamental mode studies quency lowers as the magnetisation is pulled into the hard direction, reaching a minimum at the same field where the magnetisation aligns with the applied field. Note that in the fitted results two resonant branches are included. One branch corresponds to the magnetisation at zero field aligned along one easy axis, and the other branch corresponds to the zero field alignment along the other easy in-plane axis. The unidirectional field is associated with a discontinuous change in the magnetisation orientation. A softening of one branch of the resonance appears, and indicates an orientational instability for the alignment of the magnetisation preceding an abrupt reorientation transition. This branch corresponds to the magnetisation being aligned along the easy four-fold axis perpendicular to the exchange field. There are no experimental data points with frequencies that lie on this branch as the exchange field dictates the initial magnetisation orientation. 10 Freq [GHz] Field[Oe] Figure 3.5: FMR response as a function of field in the Fe/MnPd sample with the applied field aligned along the hard axis at π/4 (results shown from field applied along both directions). A best fit overlayed from the model with parameters mentioned in the text. The red and blue colours represent solutions shown in Figure 3.3, whilst the solid lines is a fit to the π/4 direction, and the dashed line to the 3π/4 direction. Increasing the applied field to a value greater than the effective field of the four fold anisotropy causes the resonant frequency to increase, and accounts for the high field behaviour above the frequency cusp where the magnetisation is aligned with the applied field. The thin film approximation provides a good fit to the data in this high frequency

63 Fundamental mode studies 49 region, and also for low frequencies. The best fit parameters for the experimental data shown in Figure 3.5 are H E = 10 Oe and H K1 = 560 Oe. The same parameters are used for both orientations of the applied field. There is a large qualitative difference between the fitted data for the π/4 and 5π/4 orientations. The fit for the 5π/4 case is not as good in general than for the π/4 case, mainly in the lower frequency region Transition behaviour Interestingly, there appears to be a jump in frequency for the transition to the hard direction for fields at the frequency cusp. This jump was confirmed not to display hysteretic behaviour - it was identical whether the field was swept from positive to negative or vice-versa. To explain the frequency behaviour for these field values, and for field values near the frequency cusp, it is proposed that there may be a distribution of unidirectional H E fields. Small variations in the unidirectional magnitude and orientation have effects similar to changing slightly the orientation angle of the applied field. Examples are shown in Figure 3.6a where frequencies are calculated for small variations of the applied field angle from the π/4 and 5π/4 values. a) b) Oe Freq [GHz] Freq [GHz] 4-10 Oe Field [Oe] Field [Oe] Figure 3.6: Experimental data from Fe/MnPd with the applied field direction π/4 overlayed with fits from a distribution of: a. small deviations in bias angle from π/4. b. large deviations in bias strength from h E = +10 to 10 Oe corresponding to complete reversal of bias direction for some regions of the interface. In both cases only the experimentally observed branch is shown, and the solid line shows best fit parameters. A similar behaviour is seen when altering the strength of H E in Figure 3.6b. The

64 50 Fundamental mode studies result in either case is that a spread of frequencies appears, and mimics what might be expected for resonance frequencies with a distribution of local exchange anisotropy fields. A distribution of exchange anisotropy fields can thus account for the observed continuity of resonance frequencies as the magnetisation orients into the applied field direction. As noted above, a discontinuity is expected in the frequency for fields where the magnetisation rotates abruptly into the field direction. A distribution of exchange anisotropy fields would wash out this discontinuity, and provide measurable resonances for all field values in the region of the reorientation. One physical argument for varying H E properties would come from the existence of a interface pinning responsible for the exchange bias which varies spatially according to some distribution of strengths. It is not possible to say whether the interface pinning is a results of pinned spins directly near the interface, or spins positioned further away from the interface. This means that these results cannot make any statements as to the mechanism of exchange bias due to either domain state or pinned domain wall models. In either case, this measurement demonstrates that it is possible to obtain a measure of unidirectional exchange anisotropy fields, including information concerning the magnitude and distribution of exchange bias fields at the interface. A possible continuation from this work would involve measuring samples with different growth qualities. It would be interesting to see if the growth quality at the interface would alter the distribution of frequencies around the low frequency region. 3.2 Py/Co exchange spring In this section, results from the fundamental mode in an exchange spring permalloy/cobalt bilayer are presented. An estimation of the films properties will be shown by a very simple effective medium method. These measurements were taken early in the project on the PIMM [40,83], so only low field data is available. This also means temporal data was taken at constant field, and a Fast Fourier Transform (FFT) as described in Chapter 1 was used to obtain the susceptibility as a function of frequency. The film is a Co(100nm)/Py(50nm) bilayer grown at UWA. The substrate is glass, sputtered with a 5 nm Ta buffer layer before deposition of the cobalt and permalloy layers and the ferromagnetic layers are protected from oxidation by a 10 nm Ta capping

65 Fundamental mode studies 51 layer. All films were RF sputtered at room temperature at a power of 60W and an argon pressure of 5 mtorr in a system with a base pressure of 10 7 Torr. Even though Co is magnetically harder than Py, it still has resonances in the measurement range. When both films are thick, observation of modes localised in either of the films is possible. An exchange spring [24,41,42,60,59,78,96,158] can be realised for the bilayer of Py and Co. Exchange spring structures consist of two different anisotropy ferromagnetic films exchange coupled across an interface. The denotation spring refers to the possibility of forming a noncollinear configuration of spins through the thickness of the film. With the correct application of field, it is possible to realign the soft Py layer, whilst leaving the Co magnetisation direction unchanged. This allows a non-uniform magnetisation to exist at the interface of the films, where the magnetisation in one film is orientated anti-parallel to the magnetisation in the other. In these measurements it is necessary to know the orientation of the magnetisation, so the film was saturated in the measurement direction before each point was taken, ensuring the magnetisation in both films was parallel. Measurements were taken from Oe in 1 Oe steps. The frequency vs field results are shown as solid circles in Figure 3.7. It is interesting that two modes are seen, with a change in relative intensity of the modes with frequency. At low fields, only the lower frequency mode is observed. There is a small field window in which both modes are seen, and as the field is increased further, only the higher frequency mode is observed. In the following section a model is devised to understand what the modes represent Effective medium theory In effective medium theory the two layers are approximated by a single layer with properties which are an average of the two individual layers. Using the method outlined in reference [157] the average permeability in the film is given by: µ xx = f 1 µ (Co) xx + f 2 µ (Py) xx, (3.5) where f 1 = 2/3, f 2 = 1/3 describe the filling factors of the two layers, defined by the ratio of the films relative thicknesses. µ (Co) zz and µ zz (Py) are the permeabilities for Co and

66 52 Fundamental mode studies 70 Oe 137 Oe 190 Oe 6 Frequency (GHz) Bias Field (Oe) Figure 3.7: Frequency vs field plot of Py(50nm)/Co(100nm) bilayer sample. Solid circles are measured data points. Dashed line is fit from effective medium method, parameters in Section Solid lines are fits from model constructed in Chapter 4, parameters also given in Chapter 4. Inset shows FFT data from PIMM measurements at various bias fields. Py respectively which can be simplified in the long wavelength limit by: µ xx = 1 + 4πM sh i Hi 2 ω2 /γ2, (3.6) where H i = H 0 + H K ; H 0 is the applied field and H K is the effective anisotropy field of the material. The fundamental mode calculated from this effective permeability is fitted to the experimental data as seen by the dotted line in Figure 3.7. Using this theory and measured saturation magnetisations of M s(py) = 650 G ( A/m) and M s(co) = 1200 G ( A/m), our best fit was found with the anisotropies H K(Py) = 5 Oe and H K(Co) = 1000 Oe. The single dashed line fits an average of the two modes, however this simple approximation does not resolve the detail of the two modes seen experimentally. To model these modes more accurately a theory which accounts for the magnetisation oscillations as a function of thickness needs to be developed. In the following chapter,

67 Fundamental mode studies 53 a continuum model accounting for both dipolar and exchange contributions in the film is developed. The solid line fits shown in Figure 3.7 were modelled using this more complete model. This data is revisited with an application of the continuum model in Section Chapter Summary In this chapter it has been demonstrated that using angular measurements it is possible to accurately measure anisotropy values for a ferromagnetic film. In the Fe/MnPd system observed, it was possible to also measure the coupling between the FM and AFM as an effective unidirectional exchange anisotropy field. A spread in measured frequencies for the transition field magnitude where the magnetisation aligns with the applied field suggests a distribution of the exchange coupling strength and direction over the interface. In the Py/Co bilayer, two modes were seen. A simple effective medium theory was not sufficient to distinguish the two modes, and it is necessary to model how the resonance behaves in each film individually. This motivates a more detailed theory to be developed in the following chapter.

68 54

69 Chapter 4 Dynamical field - exchange spin wave model This chapter describes a theory used to understand the effective susceptibility measured from coplanar line experiments. The nature of the coplanar line breaks the spatial field symmetry that exists in conventional cavity FMR measurements, creating a challenge in understanding the results. This theory will account for the conducting nature of the ground planes and magnetic film, as well as the non-zero wavevector excitation of the waveguide. The theory is suitable for single layer or ferromagnet/ferromagnet bilayer films. 4.1 Magnetostatic waves The geometry of the coplanar line as seen in Figure C.3 shows a highly non-uniform field spatially, so the excited magnetisation will also be non-uniform. In the film, MagnetoStatic Waves (MSW) will be excited propagating perpendicular to the coplanar line. MSW are long wavelength spin waves in the magnetic medium created by the non-uniform dipole-dipole interactions. The wavelength of the wave is large enough that exchange interactions can be neglected. The first paper dealing with propagating magnetostatic waves was by Eshbach and Damon [53]. MSW in a tangentially magnetised ferrite plate were first observed in 1961 [44] and they were later observed with the magnetisation normal to the plate [45]. In this section the excitation of propagating magnetostatic waves within the sample is introduced into the theory. 55

70 56 Dynamical field - exchange spin wave model z a) b) t y x t k k Figure 4.1: a. A magnetostatic wave travelling parallel to the magnetisation direction (MagnetoStatic Backward Volume Wave). b. A magnetostatic wave travelling perpendicular to the magnetisation direction (MagnetoStatic Surface Wave). Two particular cases for propagating waves within the film are illustrated in Figure 4.1. The dynamical magnetisation is considered as propagating as a plane wave: m(x, t) = me i(ωt kx). (4.1) Illustrated is propagation of a wavevector k in the two semi-infinite directions x and z of the film in relation to the in-plane equilibrium magnetisation direction z. In the thin film case where the magnetisation is uniform in the thickness of the film, it is observed that free magnetic charges accumulate from the propagating wave. These charges affect the demagnetising fields experienced by the magnetisation and consequently alter the resonant frequency of the film. In the case where the wavevector is propagating along the direction of M (Figure 4.1a, the free charges accumulate on the surfaces of the film. These free charges are localised on opposite sides of the film at spatial distances λ/2 along the wavevector which

71 Dynamical field - exchange spin wave model 57 consequently lowers the effective demagnetising field in the y direction. The lowering of the effective demagnetising field depends inversely on the wavelength of the propagating wave, and consequently the frequency of the magnetisation reduces as the wavevector k z increases. This results in a negative group velocity (v g = dω ) and consequently this dk mode is refered to as a MagnetoStatic Backward Volume Wave (MSBVW). In the case where the wavevector is propagating perpendicular to the direction of M (Figure 4.1b), free charges accumulate on the surfaces of the film, but also inside the film. The repulsive nature of these charges introduces an effective demagnetisating field in the x direction as well as reducing the demagnetising field in the y direction as seen before. The nett result is an increase in observed resonant frequency with wavevector k x [44]. As k x increases the oscillation tends to be localised on one side of the film, where the interaction of the surface charges and interlayer charges attract. For this reason this mode is referred to as a MagnetoStatic Surface Wave (MSSW) Maxwell s equations derivation Spin waves have a small phase velocity compared to the speed of light so Maxwell s equations can be used in the following approximation: h = J, b = 0, e = b t, (4.2a) (4.2b) (4.2c) where J is the current density in the film and is given by J = σe z. Retardation effects from the displacement term are ignored for simplicity in this calculation. This allows for the film to have a finite conductivity σ, which influences the resonance response measured in the metallic films [126]. The time derivative b t is given by b t = iωµ 0(m + h). The coplanar line acts as an antenna, radiating excited waves perpendicular to its current flow. It is assumed that the coplanar line length is much greater than its width so it acts like a line source. Consequently we expect the radiated waves to take the form of plane waves propagating in the x direction (y direction is the film normal, the magnetisation lies along the z direction). The solutions of Equations 4.2 should then

72 58 Dynamical field - exchange spin wave model take the form: h(x, y) = m(x, y) = e(x, y) = h k (y)e ikx dk, m k (y)e ikx dk, e k (y)e ikx dk. (4.3a) (4.3b) (4.3c) Note that the Fourier transform dictates k to be real. However the damping of k as the wavevector propagates into the material (due to the conductivity in the sample) must be taken into consideration. This damping is included by considering the frequency ω to be complex. As we consider the film to have infinite in-plane dimensions, both the dynamic field and it s derivative are zero in the fixed z-direction (perpendicular to the plane wave). In the limiting case of σ 0, the form of Maxwell s equations equation (4.2) reduces to a somewhat simple expression. If a magnetostatic potential is introduced that satisfies h = φ, equation (4.2) reduces to the Walker equation [168,169]: (1 + χ)( 2 x y2)φ(x, y) = 0. (4.4) However, including the conductivity, it is not such a simple matter to reduce the equations. Rearranging Maxwell s equations we can find two non-zero components of the dynamical field, h k x and h k y, as second order differential equations: 2 y 2hk x b 2 h k x = ρ 1 (ω,k, y), 2 y 2hk y b 2 h k y = ρ 2 (ω,k, y). (4.5a) (4.5b) Here b 2 = k 2 +iσωµ 0, ρ 1 (ω,k, y) = b 2 m k x ik y mk y and ρ 2 (ω,k, y) = ik y mk x+( 2 y 2 + Iσωµ 0 )m k y. Green s functions At the simplest level, a Green s function describes the response of a linear system to a point source of unit strength represented by dirac delta δ(x x ) [54]. It is essentially a solution of the Sturm-Liouville equation with the delta function as its source term.

73 Dynamical field - exchange spin wave model 59 In this thesis Equations 4.5a,b will be expressed as a Green s function, representing the field in space due to the magnetisation at one point. The total dynamic field at any one location can be calculated by the integration of the magnetisation over the sample. In the geometry of the coplanar line, a travelling wave vector is assumed in the x-direction, so integration is only required over the y-direction. In calculating the dynamic field response, the equation of motion can be solved simultaneously with the Green s functions, allowing exchange and anisotropy terms to be included as effective fields with no further complications. Several previous studies of magnetic systems have been done using a Green s function method, considering purely exchange interactions [110,112,113], purely dipolar interactions, [8,33], or dipolar-exchange interactions [80,81,145]. Most of these works deal with Brillouin light scattering (for a full review see reference [35]). The work presented here departs from these works in that it is a calculation incorporating the inhomogeneities of the stipline, the conductivity of the sample and the inhomogeneous driving term directly incorporated from the coplanar line current [84]. It deviates from the work of Almeida et al. [8] by calculating the magnetic field distinct from the equation of motion, allowing easy inclusion of exchange interactions. It deviates from Dmitriev [50] by including both conductivity in the sample and the waveguide, as well as including multiple film layers. To calculate the Green s function an appropriate solution to h x (y) in equation (4.5)a is found. Initially a Fourier transform of the solution h x (y) is taken: Substituting this solution into equation (4.5)a gives: h k x(κ) = 1 h k 2π x(y)e iκy dy. (4.6) ( κ 2 )h k x(κ)e iκy dκ b 2 h k x(κ)e iκy dκ = ρ 1 (ω,k, y). (4.7) To simplify equation (4.7), a Fourier transform is taken with respect to κ. Using the relation 1 )y 2π ei(κ κ dy = δ(κ κ ), equation (4.7) now becomes: κ 2 h k x(κ) b 2 h k x(κ) = ρ(ω,k, κ) h k x(κ) = ρ(ω,k, κ) κ 2 b 2. (4.8)

74 60 Dynamical field - exchange spin wave model The inverse Fourier transform of h x again gives h x as a function of y: h k x(y) = h k x(κ)e iκy ρ(ω,k, κ) dκ = κ 2 b 2 eiκy dκ. (4.9) The non-homogeneous term is then transformed out of κ-space: h k x(y) = 1 2π [ρ 1 (ω,k, y )]dy ) [ eiκ(y y ]dκ. (4.10) κ 2 + b2 The indefinite integral over κ only has non-zero even parts and is simplified by: ) [ eiκ(y y κ 2 + b 2 ]dκ = 2 0 cos(κ(s)) κ 2 + b 2 dκ = π b e s b, (4.11) where s = y y. This gives the solution of h x (y) in integral form: and consequently for h y (y): h k x(y) = 1 2b h k y(y) = 1 2b e b s ρ 1 (ω,k, y )dy, (4.12) e b s ρ 2 (ω,k, y )dy. (4.13) From these integral equations of h it is possible to calculate the Greens function coefficients in the form h k (y) = t 0 Ĝk (s)m k (y )dy, where the superscript k denotes the x direction in reciprocal space, t is the film thickness, and Ĝk (s) is given by: Ĝ k (s) = Gk xx G k yx G k xy G k yy. (4.14) Considering equation (4.12), G k xx is easily found from the m k x component of ρ 1 (ω,k, y ): G k xx = b exp[ b s ]. (4.15) 2 The component G k xy component is a little more involved considering the m k y dependence of ρ 1 (ω,k, y ) includes a partial derivative in y: t 0 G k xy(y, y )m k y(y )dy = ik e y y b 2b y (mk y[h(y) H(y t)])dy, (4.16)

75 Dynamical field - exchange spin wave model 61 where [H(y) H(y t)] are Heaviside step functions bounding the sample in the y direction. Using the chain rule on the y derivative results in two integrals: LHS = ik ( e y y b [H(y) H(y t)] ) 2b y mk ydy e y y b m k y[δ(y) δ(y t)]dy. (4.17) Integration by parts gives only one term that doesn t cancel, and with application of the step functions on the bounds of the integration gives: This clearly gives: LHS = ik t 2 Sgn[s] e s b m k y(y )dy, (4.18) 0 G k xy = ik 2 Sgn[s]e s b. (4.19) Using similar procedures both G k yx and G k yy can be found. The complete set of Green s functions is: b 2 e b s Ĝ k (s) = ik 2 Sgn[s]e s b ik 2 Sgn[s]e s b δ(s) + k2 2b e s b. (4.20) Notice that from the thin film geometry used in deriving these components, a demagnetising field appears in G k yy described by δ(s). When the conductivity is set to zero these result in: k 2 e s k Ĝ k (s) = ik 2 Sgn[s]e s k which is in agreement with Dmitriev [50]. ik 2 Sgn[s]e s k δ(s) + k 2 e s k, (4.21) Electromagnetic boundary conditions To complete the boundary conditions of the dynamic field h k d, it is necessary to also consider the homogeneous solution of Maxwell s equations both inside and outside of the film and match the coefficients using conventional electrodynamic boundary conditions. The homogeneous Maxwell s equations relate to 4.2 with the magnetisation set to zero:

76 62 Dynamical field - exchange spin wave model h = J, h = 0, e = h t, (4.22a) (4.22b) (4.22c) The time derivative is still equal to iω. Inside the film J = σe, inside the shield J = σ s e, and above the film J = 0. The area above the film is approximated to be air and the substrate is ignored. This is appropriate for insulating substrates. No gap is assumed between the waveguide and the film which is a good approximation if the sample is placed film side down. If a gap is required the following method can still be followed with the consideration of extra boundary conditions either side of the gap. The second order differential equations (4.5a,b) describing the dynamic field within the film in the homogeneous solution become: 2 y 2hk x b 2 h k x = 0, 2 y 2hk y b 2 h k y = 0, (4.23a) (4.23b) with corresponding solutions: h k x = α x e by + β x e by, h k y = α y e by + β y e by. (4.24a) (4.24b) From equation (4.22b) we have ikh k x + y hk y = 0 and on substitution of the solutions (equation (4.24)) we can deduce a relationship between the x and y coefficients: α y = ik b α x, β y = ik b β x. (4.25) Outside the sample we have similar solutions, but with the fields approaching zero as y ±. For the shield this results in a solution of: h k,sh x = η x e b shy, (4.26a) h k,sh y = η y e b shy. (4.26b)

77 Dynamical field - exchange spin wave model 63 And above the sample in the air: h k,air x = γ x e ky, (4.27a) h k,air y = γ y e ky. (4.27b) To solve for the coefficients we equate fields at the boundaries, normal b must be continuous and in-plane h must be continuous (for zero surface currents). These fields must include both the homogeneous and Greens function solution within the film. To be able to gauge the intensities of the magnetisation oscillations it is necessary to add a driving term describing the excitation field from the waveguide. We introduce the excitation term as a free surface current in the electrodynamic boundary conditions expressed in equation (4.28). It is assumed that this surface current exists at the boundary of the film and the conducting waveguide, and has an infinitely thin propagation skin depth. The current j(k) is dependent on the wave vector k as will be shown in Section This gives four simultaneous equations, written in matrix form: ik b ik b 0 ik b s e bt e bt e kt 0 ik b ebt ik b e bt ie kt 0 α x β x γ x η x h k xg (t) j(k) h = k yg (t) mk y(t) h k xg (0) h k yg (0) mk y(0), (4.28) where h k xg and hk yg are the inhomogeneous fields as derived from the Green s functions. The first two rows relate to the upper boundary between the film and the air, whilst the bottom rows relate to the lower boundary between the film and the shield. If the system of equations in equation (4.28) is represented by Ĉ a = b, then a is solved in two parts, a homogeneous term a h and a driving term a d, separating the dependence of m:

78 64 Dynamical field - exchange spin wave model a = a h + a d = Ĉ 1 h k xg (t) h k yg (t) mk y(t) h k xg (0) h k yg (0) mk y(0) + Ĉ 1 j(k) (4.29) Solving this system gives values for the coefficients. The two important terms are α h and β h as these describe the film in the sample: α h = 1 D [b(b k)(khk xg(0) ib s (h k yg(0) + m k y(0))) b(b + b s )e bt k(h k xg(t) + i(h k yg(t) + m k y(t)))], (4.30) β h = 1 D [bebt ( e bt (b + k)(kh k xg(0) ib s (h k yg(0) + m k y(0))) +(b b s )k(h k xg(t) + i(h k yg(l) + m k y(t))))], (4.31) where: D = (b b s )(b k)k + (b + b s )e 2bt k(b + k). (4.32) The dynamic fields can be given by a sum of the inhomogeneous and homogeneous parts combined; h d : h k d(y) = t 0 (Ĝk (s)m k (y )dy + α h e by + β h e by ). (4.33) An exchange-free calculation could be realised by including this dynamic field with an effective anisotropy field and an external field into a total effective field. This is then used to solve the equation of motion. Before this is done, consider the effective field from the exchange term Exchange regime The films considered in this project have a thickness small enough that to correctly model any standing excitations exchange interactions must be included. The exchange interaction is a quantum mechanical effect between two or more electrons when their wavefunctions overlap. As the wavefunctions must overlap, this is a short range interac-

79 Dynamical field - exchange spin wave model 65 tion and only nearest neighbours need be considered. In a ferromagnetic material with positive exchange energy, this interaction tends to align neighbouring spins. The energy is represented in the semi-classical limit by the Heisenberg Hamiltonian: W = 2J S i S j, (4.34) where the dash denotes summation over nearest neighbours only, S i,j are the spin angular momenta and J is the exchange constant. If the sum is expanded in an even power series, and only the first angular dependent term is considered, the free energy of the system can be represented in the continuum approach: F ex = A sample Ms 2 [( M x ) 2 + ( M y ) 2 + ( M z ) 2 ]dv, (4.35) where A depends on the lattice geometry and for a simple cubic with lattice spacing r, A is related to the exchange value J by: A = 1 6 JS2 r 2, (4.36) where the sum is over nearest neighbours. The relationship between magnetic energies and effective fields can be derived from variational calculations with [25]: F M = µ oh eff. (4.37) This gives in the continuum approximation for the coplanar line geometry: h k ex = 2A M s 2 (m k x + m k y) = 2A µ 0 M 2 s ( k 2 m k x + 2 y 2mk y). (4.38) Surface boundary conditions The effective field introduced into the equation of motion from the non-uniform exchange interactions shown in equation (4.38) introduces a derivative of the magnetisation in the y-direction. This derivative results in the solution of the dynamical equation (A.5) in the thin film geometry having extra arbitrary constants to be solved. These arise because of the neighbouring spin symmetry breaking at the surface of the magnetic film. Consequently, the local effective fields felt by the surface layer moments of the magnetic

80 66 Dynamical field - exchange spin wave model film are different than felt by the interior moments. To compensate for these different fields, it is necessary to explicitly introduce exchange boundary conditions into the equation of motion. These can be introduced as an effective field denoted H s and the equation of motion (equation (A.5)) is written with this surface field term in the following way: 1 γ M t = M H tot + 2A M 2M 2 M + M H s, (4.39) where H tot is the total effective field minus the exchange and surface field contributions. The surface field is considered to be a result of a surface anisotropy, with the anisotropy easy axis normal to the film with an effective field given by [149]: H s = 2K s M M0 2 y ŷ. (4.40) Within the thin surface layers of the film it is assumed that only the last two terms of equation (4.39) give non-negligible contributions. Integrating over the volume of the surface layer for these two terms gives the resulting boundary condition [71]: 2A M 2 0 M M y + 2K s M M0 2 y M ŷ = 0. (4.41) Linearising this equation using equation (A.8) gives the exchange boundary conditions [149]: m x y + dm xcos2φ = 0, (4.42) m y y + dm ycos 2 φ = 0, (4.43) where φ is the angle of the magnetisation from the film normal. Here d is an exchange pinning parameter, d = K s /A. Consider the case of in-plane magnetisation, φ = π/2: m x y dm x = 0; m y y = 0. (4.44) Here, for zero pinning (d = 0) the gradient of the magnetisation in the y-direction at the surface is zero. However, for the limiting case of perfect pinning (d ) the gradient is infinite at the interface, dramatically altering the nature of the oscillating magnetisation distribution within the film.

81 Dynamical field - exchange spin wave model Dissipation To calculate a finite absorption of energy by the magnetisation precession it is necessary to include some kind of dissipation term, otherwise a singularity arises. The microscopic physical processes leading to the dissipation are complex and quantum mechanical in nature. The microscopic processes have been studied extensively (for reviews see [5, 11, 150]) and will not be described here. The following subsections discuss phenomenological approaches for the allowance of magnetisation relaxation, and show the description assumed in this work. Landau-Liftshitz equation The phenomenological approach proposed by Landau and Liftshitz introduces an additional torque term that pushes the magnetisation in the direction of the effective field. equation (A.5) becomes: M t = γm H γλ M (M H), (4.45) Ms 2 where λ is a phenomenological damping constant of the material. This equation ensures the conservation of the length of M, ie. M = M s Landau-Liftshitz-Gilbert equation Gilbert proposed the most natural way to phenomenologically explain the damping was not via the restoring torque derived by Landau and Liftshitz, but rather by a viscous force which is proportional to the time derivative of the magnetisation. He altered the LL equation (A.5) to what is now known as the Landau-Liftshitz-Gilbert (LLG) equation: M t = γm H + α M s M M t, (4.46) where α is known as the Gilbert damping parameter. Mathematically however, this equation is very similar to equation (4.45). For example, one can quickly retrieve equation (4.45) from equation (4.46). Vector multiplying both sides of equation (4.46) by

82 68 Dynamical field - exchange spin wave model M: M M t = γm (M H) + M ( α M s M M t ). (4.47) Using the vector identity a (b c) = b(a c) c(a b) and M M t = 0 results in: M M t = γm (M H) αm s M t. (4.48) Substituting this for the torque on the RHS of equation (4.46) and rearranging, we obtain: M t = γ 1 + α 2M H γα (1 + α 2 )M s M (M H). (4.49) This is often referred to as the LL equation in Gilbert form. It is easy to see that the substitution: γ γ αm 1 + α2, λ 1 + α2, (4.50) transforms the LL-equation to the LLG equation exactly. An advantage of the LLG equation is that it keeps the dissipation term separate from calculating the equilibrium condition. Also, as noted by Kikuchi [87], in the limit of infinite damping (λ in equation (4.45) and α in equation (4.46)) the LL equation shows M, whilst t the LLG equation shows M 0. It is sensible for a large damping that the change in t magnetisation would be small, showing the LLG equation to be consistent. Linear regime Here it will be demonstrated that when working with linear regime of magnetisation dynamics with small damping these different phenomenological damping terms are degenerate and provide the same solution. Following the approach in Section A.3 we linearise equation (4.45): iωm + γm H 0 + γλ H 0 M 0 m = γm 0 h + γλh (4.51)

83 Dynamical field - exchange spin wave model 69 and equation (4.46): iωm + γm H 0 + iαω M 0 m M 0 = γm 0 h (4.52) Remembering that ω = γh 0 and m y and m x are out of phase by π/2 (m y = im x ) we can simplify equation (4.52) to: iωm + γm H 0 + αγh 0 m = γm 0 h (4.53) The Bloch Bloembergen equation 1 also takes the same form as equation (4.51) with ω r = γλh 0 /M 0. So in the linear approximation with small damping, these phenomenological damping conditions are all the same with α = λ/m 0 = ω r /ω H. Moreover, it can be seen from a simple substitution into equation (4.53) of ω = ω(1 + αi) we retrive equation (4.52). As mathematical packages such as Mathematica are easily capable of manipulation of complex variables we shall introduce damping into the problem with a complex ω. 4.2 Single layer solution With terms for exchange and dynamical magnetic field energies, both with boundary conditions, and the introduction of a complex frequency to account for damping everything exists to solve for the fields simultaneously with the linearised equation of motion. The solution is found numerically, with the sample being divided into N layers. The complete effective field at layer i is given by Heff i with contributions from dynamical magnetic field interactions, non-uniform exchange interactions, an effective anisotropy field and an applied field as shown in equation (A.6). This field is included into the linearised equation of motion (equation (A.11)) where it is assumed that the magnetisation 1 Both the LL and LLG descriptions of dissipation ensure conservation of the vector length M = M s. However in some dissipation processes this vector length is not conserved. An example of a dissipative equation which allows for change in the magnetisation vector length is the Bloch-Bloembergen (BB) equation, commonly referred to as the modified Bloch equation: M t = γm H ω r (M M 0 H 0 H), (4.54) where ω r is the relaxation frequency. The dissipative term here is proportional to the difference between the instantaneous magnetisation M and the magnetisation which would result if the field H was constant.

84 70 Dynamical field - exchange spin wave model is aligned along the applied field and consequently also the coplanar line. This leads to two dynamical components given by: iωm k x(y i ) = ω H m k y(y i )δ ij [ N ω M (G k yx(y i, y j )m k x(y j ) + G k yy(y i, y j )m k y(y j )) j=1 2A + ( k 2 δ µ 0 Ms 2 ij m k y(y i ) + mk y(y i+1 ) + m k y(y i 1 ) 2m k y(y i ) ) 2 + α hy e by + β hy e by + α dy e by + β dy e by], (4.55) iωm k y(y i ) = ω H m k x(y i )δ ij [ N + ω M (G k xx(y i, y j )m k x(y j ) + G k xy(y i, y j )m k y(y j )) j=1 2A + ( k 2 δ µ 0 Ms 2 ij m k x(y i ) + mk x(y i+1 ) + m k x(y i 1 ) 2m k x(y i ) ) 2 + α hx e by + β hx e by + α dx e by + β dx e by]. (4.56) Here ω H = γµ 0 H eff includes applied field and anisotropy terms, and ω M = γm s. The second derivative in the exchange term is represented by a numerical three point derivative. is the spacing between layers, and δ ij represents the dirac delta function, effectively including multiplied terms only when i = j. The only terms not dependent on the dynamical magnetisation are the driving terms α d e by and β d e by. The system of equations over all the layers is represented in matrix form: m k x(y 1 ) m k x(y 1 ) α dx e by 1 + β dx e by 1 m k y(y 1 ) m k y(y 1 ) α dy e by 1 + β dy e by 1... m k x(y i ) iω = m k y(y i ) Ĉ m k x(y j ) α dx e by i + β dx e by i +, (4.57) m k y(y j ) α dy e by i + β dy e by i... m k x(y n ) m k x(y n ) α dx e byn + β dx e byn m k y(y n ) m k y(y n ) α dy e byn + β dy e byn

85 Dynamical field - exchange spin wave model 71 where Ĉ is a coefficient matrix including all of the magnetisation dependent terms in the equation of motion. This is solved from matrix inversion: m k = ( i(ω αiω)i Ĉ) 1 a d, (4.58) where I is the identity matrix and a d is the driving vector. The α term represents the complex component of the resonant frequency, acting as a damping as discussed in Section The magnetisation profile is now solved across the film for a particular intensity current through the coplanar line. The measured net magnetisation can be found from the summation of the magnetisation across the film. The susceptibility can be easily found as either a function of field or frequency by repeating the solution whilst varying the components of Ĉ in the correct manner Pinning conditions i=-1 i=1 i=2 i=n-1 i=n i=n+1 Figure 4.2: Illustration of layers considered outside of the film To correctly account for pinning, the boundary conditions on the surface must be taken into account. Expressing equation (4.44) in discrete form: m k x(y i+1 ) m k x(y i 1 ) 2 d 1,n m k x = 0, (4.59) where d 1,n is the pinning parameter at either surface, i = 1 and i = n. This equation needs to hold at the surfaces, so a virtual layer (outside) of the film must be considered. This is illustrated in Figure 4.2. Consider the layer i = 1. From equation (4.59), the imaginary layer has value: m k x(y i 1 ) = m k x(y i+1 ) + 2 d 1 m k x. (4.60)

86 72 Dynamical field - exchange spin wave model This is substituted into the discrete second derivative term in the exchange effective field: 2 m k x(y 1 ) = mk x(y 1 ) + m k x(y 2 ) 2m k x(y 1 ) 2 = 2mk x(y 2 ) 2(1 + d i )m k x(y 1 ) 2. (4.61) Similar reasoning deduces the correct effective field for the m y component at the i = 1 layer. The process is repeated for the i = n layer. It can be seen that when d 1 = 0, m k x(y 1 ) = m k x(y 2 ) and consequently the gradient over the surface is zero; resulting in an unpinned surface. And when d 1, m k x(y 1 ) = and the gradient over the surface must be infinite; resulting in a perfectly pinned surface. Figure 4.3 shows results of the measured response in a Py film for different pinning levels at one surface. The inset shows the solution of the magnetisation as a function of distance through the film at resonance, as given by equation (4.60). It should be noted that as pinning increases at only one surface, the symmetry of the magnetisation through the film is destroyed, so higher order modes gain a nett magnetisation. This allows these higher order modes to be measured Current profile The driving vector a d is solved from equation (4.29). However, the current profile is not uniform across the coplanar line. As mentioned before, the coplanar line acts as an antenna and excites wavevectors in the x direction. An expression for the distribution of current over the width of the waveguide is given in reference [50]: j(z) = a w K 1 e ( w ) [ [ (1 z 2 ) ( a ]] 1/2 a w )2 z 2. (4.62) Here w is the width of the central conductor, a is the ground to ground spacing, and K 1 e is the complete elliptical integral. This distribution assumes the ground planes are an ideal metal - ie, finite conductivity in the ground planes is not taken into account. It is well known that if lower frequencies were passed through the waveguide, the current would spread over the centre conductor, whilst higher frequencies would result in the current being largely localised to the edges of the center conductor. It must be noted however, that this will not change the maximum positions of the excited wavevectors, justifying this ideal metal approximation for the calculation. Figure 4.4 shows the profile of the waveguide (center conductor width 200µm field both in real space, and transformed into k-space.

87 Dynamical field - exchange spin wave model 73 Absorption(arb units) Increasing pinning at one surface Amp (arb. units) film thickness Freq(GHz) Figure 4.3: Absorption as a function of frequency (imaginary part of susceptibility) calculated for 100nm Py film at a constant applied field of 1000 Oe at a wavevector of k 0. Stacked plot shows results for different surface pinnings, from zero pinning to complete pinning at one surface in the direction of the arrow. The inset shows the magnetisation amplitude through the thickness of the film, with zero pinning only exciting the fundamental mode shown dashed, and strong pinning exciting the fundamental and higher order modes shown as solid lines. The measured response requires calculation of the magnetisation at the complete range of wavevectors, and then summing them with a weighting as shown in Figure 4.4b. In practice, this is too computationally time consuming and the response must be found from a discrete sum of the wavevectors exhibiting the largest amplitude Dispersion Although the measured results involve a summation over the relatively small wavevectors excited by the coplanar line as shown in Figure 4.4b, the calculation is valid at the complete range of wavevectors. As technological devices are striving to be excited by smaller central conductors of the coplanar lines it is advantageous to explore the complete range of wavevectors. In this section, consider that the conductivity in both the film and the ground planes are zero.

88 74 Dynamical field - exchange spin wave model a) b) Jx [arb units] Distance [cm] Jk 2 [arb units] Wavevector [rad/cm] Figure 4.4: a. Current distribution across the coplanar line in real space with center width 1.1 mm and slot width 0.24 mm. b. Current distribution in k-space. In Figure 4.5a, the frequency vs wavevector is shown for different pinning conditions. The negative frequency reflects a wave travelling with wavevector k x, emitted in the opposite direction. Because of the surface nature of the fundamental Damon-Eshbach mode, this will show a non-reciprocal behaviour as k x increases. This can be easily seen in Figure 4.5b for the case kt = 1, where the wavevectors travelling in one direction are clearly preferred. The complete absorption as measured by the experiment is a summation of the waves excited in both directions. In thicker films a non-reciprocity also is created due to the excitation field of the coplanar line being localised to one side of the film. This is largest in conducting films as eddy currents in the film limit the field penetration. Figure 4.5c shows the mode profiles through the thickness of the film for the modes visible below 15 GHz for different pinning conditions at a low wavevector k x. Only in the case of asymmetric pinning are all of the SSWMs visible. For symmetric pinning the odd modes are visible, and for zero pinning at both surfaces only the fundamental mode shows any nett magnetisation. It is interesting to note that the more asymmetric the pinning is, the greater the amplitude of any standing spin wave modes. Figure 4.5d shows the same mode profiles, but now for a higher wavevector (k x t = 1). The surface wave nature of the film not only shows a preferential propagation direction, but it also provides non-uniform excitation of the magnetisation through the thickness of the film. This makes it possible for higher order modes to be seen even in the case of zero surface pinning.

89 Dynamical field - exchange spin wave model 75 a) 15 Freq[GHz] NO PINNING PINNED AT ONE SURFACE PINNED AT BOTH SURFACES -15 b) k x k x t k x t = 1 Im[ xx ] Im[ xx ] Freq[GHz] Freq[GHz] c) d) k x t = 1 k x NO PINNING PINNED AT ONE SURFACE PINNED AT BOTH SURFACES Amp arb.units Amp arb.units Amp arb.units DASHED = m y SOLID = m x FILM THICKNESS Amp arb.units Amp arb.units Amp arb.units Figure 4.5: a. Resonant dispersion of a single layer film modelled with t = nm, M s = A/m, γ = 28GHz/Tesla, A = J/m, α = b. Susceptibility shown for low ( 60000m 1 ) and high (kt = 1) wavevector at different pinnings. The maximum in Im[χ xx ] were recorded to plot (a). c. The resonant mode profiles through the thickness of the film for m x (solid lines) and m y dashed lines for different pinning conditions at low k. d. Same as (c) but for high k.

90 76 Dynamical field - exchange spin wave model Conductivity of sample It is useful to explore the influence of the finite conductivity over the complete range of spin waves in the calculation. As was seen in Maxwell s equations, the precession induces a magnetic induction which by Faraday s Law is coupled to the electric field. If conductivity is finite, this leads to eddy current damping of the precession in the magnetic layer, and consequently an influence on the damping of the magnetisation by an additional ohmic dispersion. This also has the opportunity to change the dispersion of the modes. The calculated results shown in Figure 4.6 are similar to Almeida et al. [8], however, it must be noted that Almeida assumed the skin depth of the material to be constant, whereas it is highly dependent on the frequency (see equation (4.63)). The classical microwave skin depth for a good conductor is given by: δ = 1 πfµσ. (4.63) The conductivity will thus have the greatest effect when the skin depth comparable to the thickness of the film. Also, careful observation of equation (4.63) shows a dependence on µ, so the skin depth can decrease sharply at resonance making it necessary to consider conductivity even for thin films. The resonant frequency of the fundamental mode is the same as an insulating film for k = 0, and is lower than in an insulating film for a positive k. As k becomes very large, the frequency converges to the insulator case again (around kt = 3). A similar phenomena occurs for the damping, with the damping increasing with k from an insulating film, and converging again as k becomes very large. Figure 4.6 shows the largest change in the resonant behaviour close to k 1. This behaviour relates to the competition between the skin depth and the surface wave depth. If the electrical skin depth is smaller than the surface wave penetration, the mode is constricted and decays like the skin depth into the film. This alters the behaviour of the dispersion. However, if the k value is large, and the surface wave penetration is less than the skin depth, then the mode propagates relatively unimpeded by the conductivity. This is why the dispersion returns to that of an insulating film at large k values.

91 Dynamical field - exchange spin wave model 77 a) Freq GHz DISPERSION CURVE kt b) Freq GHz CONDUCTIVE LINEWIDTH kt c) Im xx Arb. kt Freq GHz Im xx Arb. Im xx Arb. kt Freq GHz CONDUCTIVE xx kt Freq GHz Im xx Arb. Im xx Arb. kt Freq GHz kt Freq GHz Figure 4.6: a. The dispersion of thick films with and without conductivity. The film thickness is large (2µm) to illustrate the effect clearly. As the film is so thick, the exchange modes would be effectively degenerate, so the exchange term has been omitted. Conductivity in the film is S/m. b. Linewidth of the resonant line as a function of k. c. Calculated susceptibilities (imaginary component) for different k values.

92 78 Dynamical field - exchange spin wave model Conductivity of ground planes In this section the results of introducing a positive σ s into the boundary conditions expressed for Maxwell s equations are shown. As seen in Section 4.1.1, the ground planes are included as a continuous semi-infinite shield. Seshadri [141] pointed out that in an isolated ferrite slab, the dispersion characteristic for a wave propagating in the +x direction was degenerate with a wave in the x direction, however, a conducting ground plane removes this degeneracy. The wavevector that travels along the shield side of the film is heavily influenced by the reflections the shield presents, so its dispersion is shifted quite significantly. The wavevector on the opposite side of the film is less effected so a non-reciprocity exists in the film. Freq GHz s = Sm -1 x s = Sm -1 x kt Figure 4.7: The change in dispersion by considering the ground planes form a conductive sheet. The dispersion for both directions is shown, illustrating the non-reciprocal nature induced by the ground planes. Shown in Figure 4.7 is the dispersive behaviour altered by inclusion of the conductive sheet, given conductivity of copper S/m. The dispersion for wavevectors travelling in both directions is illustrated, clearly showing the non-reciprocity of the two wavevector directions. At zero wavevector there is no dynamical field. As the wavevector increases, the dynamical field increases, and the reflection from the shield increases the dynamical field in the film. This reflection is concentrated nearest the shield, resulting in the negative wavevector travelling at a faster speed than the positive wavevector. As the wavevector value increases further, the wavevector becomes more localised to the surface, and the dynamical field reflection also becomes more localised to the shield s

93 Dynamical field - exchange spin wave model 79 surface. As the shield has a positive conductivity, this results in a smaller reflected field, and consequently for very large wavevectors the dispersion converges with the unshielded case. This is in agreement with Bongianni [21]. 4.3 Bi-layer This section will expand the calculation to a bilayer film. This is essentially a straight forward extension, with the exception that care must be taken with the exchange conditions at the interface of the film Interface exchange conditions The effective exchange fields at the interface layers are calculated in a similar way to the surface spins. First consider interlayer boundary conditions valid at the film interface as proposed by Rado and Weertman [132] in a linearised manner (in a similar form to reference [165]): Film 1: m 1y y m 1x y + A 12 A 1 m 1x A 12 A 1 M 1 M 2 m 2x = 0, (4.64) + (d i1 + A 12 A 1 )m 1y A 12 A 1 M 1 M 2 m 2y = 0. (4.65) Film 2: m 2y y m 2x y A 12 A 2 m 2x + A 12 A 2 M 2 M 1 m 1x = 0, (4.66) (d i2 + A 12 A 2 )m 2y + A 12 A 2 M 2 M 1 m 2y = 0, (4.67) where A 1 and A 2 are the exchange constants in films 1 and 2 respectively, A 12 is the interlayer exchange constant between films 1 and 2, and d i1 and d i2 are interlayer surface pinning values. Consider now the three point second derivative of the y-component around the last layer m in film 1 (see Figure 4.3.1): 2 m y 2 = m m+1 + m m 1 2m m 2 = m m+1 mm mm m m 1. (4.68)

94 80 Dynamical field - exchange spin wave model i=-1 i=1 i=2 i=m+2 i=m-1 i=m i=m+1 i=n-1 i=n i=n+1 Figure 4.8: Figure illustrating imaginary layers outside the bilayer film for surface pinning, as well as the virtual layers either side of the interface. This is simply splitting the second derivative into two successive single derivatives. The star on layer m m+1 denotes it as a virtual value in film 1 due to the discontinuity in magnetisation parameters at the film interface. This point must be related to the corresponding real value as described in layer 2 using the boundary conditions. Assuming the first order derivative at layer m can be approximated numerically with: m m y m m+1 m m, (4.69) the boundary condition given equation (4.65) can be used to give: m m y m m+1 m m = A 12 A 1 (m m M 1 M 2 m m+1 ). (4.70) This is valid for the x-component of the magnetisation, and is valid also for the y- component of the magnetisation with the addition of the interlayer pinning constants. Substituting this result into equation (4.68) the effective exchange field in layer m is given by: h ex m = 2A 1 A12 A 1 (m m M 1 M 2 m m+1 ) mm m m 1 M 1. (4.71) Again the y-component needs the addition of the interlayer pinning constants. In a

95 Dynamical field - exchange spin wave model 81 similar way, the effective exchange field for the first layer in film 2 can be found: h ex m+1 = 2A m m+2 m m+1 2 A 12 A 2 (m m+1 M 2 M 1 m m ). (4.72) M 2 If there are subsequent films (ie. trilayer structure) the procedure must be repeated at each discontinuity of magnetisation parameters Bilayer solution Now that the exchange interlayer boundary conditions are expressed as effective fields, it is simple to extend equation (4.57) to a bilayer film: iω m k x1(y 1 ) m k y1(y 1 ). m k x1(y m ) m k y1(y m ) m k x2(y m+1 ) m k y2(y m+1 ). m k x2(y n ) m k y2(y n ) = Ĉ m k x1(y 1 ) m k y1(y 1 ). m k x1(y m ) m k y1(y m ) m k x2(y m+1 ) m k y2(y m+1 ). m k 2x(y n ) m k 2y(y n ) + α dx e by 1 + β dx e by 1 α dy e by 1 + β dy e by 1. α dx e bym + β dx e bym α dy e bym + β dy e bym α dx e by m+1 + β dx e by m+1 α dy e by m+1 + β dy e by m+1. α dx e byn + β dx e byn α dy e byn + β dy e byn. (4.73) Care needs to be taken to include the right elements in Ĉ for the exchange boundary conditions at both surfaces and the interface between the films. Once the matrix is setup correctly, the calculation is solved in exactly the same way as the single layer case. With the inclusion of the correct boundary conditions, it is possible to extend this calculation to as many magnetic films as required.

96 82 Dynamical field - exchange spin wave model 4.4 Example: Py/Co exchange spring revisited In the previous chapter the results from the Py/Co bilayer film grown at UWA were modelled using a simple effective medium theory. In this section the continuum model developed in this chapter will be used to model the measured data. Fitting the data with this model gives the solid line fits shown in Figure 3.7. The best fit for this sample was obtained with anisotropy parameters H K(Co) = 20 Oe and H K(Py) = 5 Oe. The main difference in this model compared to the effective medium model is the allowance for the dynamic magnetisation to change through the thickness of the bilayer. The largest influence of the thickness dependence of the magnetisation is the interlayer coupling. Fitting parameters are g = 2.1, with exchange parameters A (Co) = Jm 1, A (Py) = Jm 1, with an interlayer coupling of A (Py/Co) = Jm 1. is the separation between layers in the model. Surface boundary conditions were not assumed known, and a best fit was found with zero pinning. The low interlayer coupling could be an indication of the low quality interface growth. The sputtering system this film was grown in has only one power supply, so to sputter a bilayer the connection to the targets must be physically swapped. It requires a few minutes for this swap and ignition of the second gun. Within this time it is expected the interface quality has been compromised from contaminants in the sputtering chamber. From this model, two modes approximating non-uniform acoustic and optic exchange modes are found to contribute to the response. The acoustic mode corresponds to the magnetisation in the permalloy and cobalt layers precessing in phase, whilst the optic mode corresponds to the two layers precessing out of phase. The magnetisation profiles for the two modes through the thickness of the film as a function of field are shown in Figure 4.9. The discontinuity seen at the interface is due to the change in saturation magnetisation values. The largest influence on the modelled frequencies is the interlayer coupling constant. This constant changes the coupling between the films and consequently the inhomogeneous profiles of the magnetisation through the thickness of the films as seen in Figure 4.9. These profiles are also dependent on the values of A (Co) and A (Py), but as the magnetisation profile within each film is only slowly varying, altering the magnitude of these exchange constants does not have as significant impact as altering the interlayer coupling.

97 Dynamical field - exchange spin wave model 83 ACOUSTIC MODE OPTIC MODE M Amp arb. units Py 20 Oe 80 Oe Co 160 Oe 200 Oe M Amp arb. units Py 20 Oe 80 Oe 160 Oe Co 200 Oe Distance into film nm Distance into film nm Figure 4.9: Cross-section of Py/Co bilayer showing calculated magnetisation intensities for the two observed modes - optic and acoustic - at various measurement fields Mode intensities Experimentally, at some fields it is possible to observe only one mode, and other fields both modes appear. Examples are shown in the inset of Figure 3.7. To explain this, the calculated intensities of the two modes as a function of bias field are shown in Figure The acoustic mode is dominant at low fields, and rapidly loses intensity as the bias field is increased. The optic mode is pinned at the Co/Py interface, and its intensity is only weakly dependent of field. At a field of about 130 Oe, the acoustic mode intensity drops below the optic mode intensity. This explains the observed intensities experimentally in the cross-over region. The intensities of these modes is controlled by the exchange coupling between the films. The Co film has the largest saturation magnetisation combined with the largest thickness, which means the measured signal is dominated by the intensity of the Co mode. The acoustic and optic mode frequencies have largest amplitudes in the Py and Co films respectively. At low fields, these two modes are relatively close together in frequency, and it is possible for the Py film to drive the Co film off resonance with significant amplitude. As the applied field increases, the frequencies of the Py and Co mode diverge, so the driven amplitude of the Co diminishes. At these higher frequencies the Co acts to pin the Py at the interface. This explains the drop off in amplitude in the acoustic mode with frequency. Experimentally in the PIMM setup both modes are only

98 84 Dynamical field - exchange spin wave model Intensity (arb units) Optic Acoustic Field (Oe) Figure 4.10: Calculated intensities of the acoustic and optic modes in the Py/Co bilayer showing the acoustic mode decay in intensity with field. This explains the crossover of modes seen in the experimental data. seen for a very short field range as seen in Figure 3.7. The crossover region corresponds nicely to that seen in Figure 4.10, and it is proposed that either side of this small field range the lower amplitude mode is masked by the linewidth of the other mode. In a similar situation, at low fields the precession at the Co resonance drives an optic mode in the film where the Py is excited away from its fundamental resonance with an amplitude that is out of phase with the Py. As the field is increased, the Py fundamental resonance frequency diverges from the Co. Instead, the Co resonance is now close to the first exchange mode in the Py and drives this mode significantly. This can be seen in optic moment cross-sections shown in Figure 4.9. Using the exchange coupling at the interface of the materials, the amplitudes (and frequencies) of different mode excitations are no longer only affected by the applied fields and intrinsic properties of one material. If the materials parameters are correct, it is possible to tune the modes of a ferromagnetic layer using a second exchanged coupled layer. If the second layer has resonances near the frequency and field values of the first layers resonance, coupled modes are excited and substantial amplitude is observed. On the contrary, if the second layer does not exhibit resonances near the same frequency and field as the first layer, it will act as a pinning potential, effectively damping any resonances that could occur.

99 Dynamical field - exchange spin wave model Chapter summary In this chapter a detailed calculation was introduced to calculate an effective susceptibility to model the experimentally measured results. In particular, the calculation included effects from the film conductivity, from the shielding nature of the waveguide and from the finite wavevectors excited by the coplanar line. The coplanar line used in this project excites quite low wavevectors, and in combination with the relatively thin samples studied in this project, the film conductivity and waveguide shielding have minimal impact on the results. It should be noted however, that if the coplanar center conductor was made considerably narrower, the wavevector would increase and it would be necessary to include these effects to extract an appropriate susceptibility. Finally, the theory was applied to the Py/Co bilayer film studied in Chapter 3. The theory accurately modelled both the frequencies and amplitudes of the two observed modes and an interlayer coupling was extracted.

100 86

101 Chapter 5 Higher order SSWM results Results from measuring the Py/Co bilayer in Chapter 3 would have provided more accurate information about the exchange parameters if higher order modes across the films thickness were able to be measured, rather than just the fundamental modes. These exchange modes were not measured in the film using the PIMM technique as it is not sensitive enough at the higher frequencies at which the exchange mode was excited. Instead the RF technique is needed to be employed. With the sensitive measurement technique and the continuum model, it is possible to measure and model non-uniform excitations through the thickness of the thin film. As illustrated in Chapter 4, nonuniform SSW modes are highly sensitive to surface and interface pinning, and in this chapter results from these modes will be presented. The sample thicknesses are required to be large enough to give SSW mode frequencies in the measurable range of this project. In contrast, the interface influence has a 1/t relationship, with t being the thickness of the film. Consequently thinner films would be better to characterise the interface, and a reference Py film is grown with sufficient thickness so as to see at least one SSWM. A second magnetic film is grown on top of the Py film to create an interface. Two systems are shown here; firstly a Co film grown from a sub-monolayer to several nm on the Py film; and secondly a Pt/Co multilayer film coupled to the Py film. As these modes are highly dependent on the thickness of the film, high film quality and spatially consistent surfaces and interfaces are required to measure a nett signal from these modes. The Py/Co film shown in Chapter 3 was grown in a less than adequate sputtering machine at UWA, with a high base pressure and inconsistent sputtering pres- 87

102 88 Higher order SSWM results sure. Also, only one sputtering power supply was available at UWA, so interface quality suffered as the film surface was contaminated in the time taken to physically change magnetron guns. To observe higher order exchange modes the surfaces and interfaces of the film must be as uniform as possible, because inhomogeneities changes the apparent film thickness, consequently shifting the resonant SSWM frequency locally. The coplanar line technique measures an average of the magnetisation, and if the sample is not spatially resonating at the same frequency due to interface inconsistencies, any higher order modes smear out and are not observable. To improve on sample quality, the film systems shown in this chapter were grown during a trip to the University of Leeds, and the first part of this chapter will outline the sample growth. 5.1 Sample growth This section will illustrate the growth process of the films used in this chapter, as well as film characterisation. The films were grown and characterised under the guidance of Professor Bryan Hickey, Professor Denis Gregg and Dr Mannan Ali. Firstly the principles used in growing films through sputtering will be introduced; followed by an introduction to the sputtering system used at Leeds University to grow the films; and a description of the films grown, including x-ray characterisation Magnetron sputter deposition Magnetron sputtering is a low pressure technique. It has the advantage of being able to sputter many different materials with little contamination and a high deposition rate. It was first observed by Penning in 1935 [125] when he superimposed a transverse magnetic field onto a dc glow discharge tube. He found that the application of even a small field (300G) allowed him to reduce the sputtering gas pressure by a factor of 10 and simultaneously achieve a greater deposition rate. The ionisation process in the gas discharge begins when free electrons collide with inert gas molecules. If the electrons have sufficient energy, the molecules are ionised and result in positive ions. These positive ions are attracted to and bombard the cathode surface, sputtering the target and generating secondary electrons (shown in Figure 5.1a.).

103 Higher order SSWM results 89 Ar + Ion created by electron impact + Sputtered target atom Target surface at negative V Electron released from target Ar + Ion + (a) Sputtering process. Ion bombardment of target results in ejection of target atoms and electrons. The electrons create more ions and the process repeats. Substrate (anode) + DC PSU Magnetic field lines _ Target Cooled Cu Gun N S S N Permanent magnets Earthed shield Water in Water out Negative (cathode) (b) Magnetron gun in DC setup. A voltage is applied between the substrate and target to initiate the sputtering process. Figure 5.1: Sputtering process

104 90 Higher order SSWM results These secondary increase the ionisation of the gas molecules generating a self-sustained discharge. With small currents ( ma) a luminous glow (plasma) begins to form around the cathode. This is known as the glow discharge region and as the current increases the plasma covers more of the cathode with a constant current density. To ensure complete coverage of the cathode the negative potential to the cathode is increased, entering the abnormal glow discharge region. This is the appropriate region for film growth as the plasma smothers the cathode whilst its energy, and hence deposition rate, can be controlled by regulating either the cathode voltage or the total current (resistive plus discharge). In the abnormal glow discharge region V ln(i), greater control of the sputtering rate is possible by regulating the current. Magnetron sputtering generally uses an Argon (Ar) atmosphere with planar targets, and a cathode configuration in the magnetron gun as illustrated in Figure 5.1(b). Negative voltage is applied to the target, attracting any Ar ions to bombard the surface and sputter a target atom. The target atoms are neutrally charged so will be unaffected by the electric field. The permanent magnets embedded within the cathode create a strong magnetic field where the Lorentz force traps any ejected electrons in helical paths close to the surface of the magnetron. This results in more ionising collisions with Ar atoms near the target, and consequently the resulting ions increase the deposition rate. The confined nature of the plasma allows a lower operating pressure, and lower heating of surrounding objects - most importantly the substrate. With the low operating pressure, collisions of the ejected atoms with the gas molecules are so few that they can be neglected. This ensures a large deposition rate. However, the plasma is greatest where magnetic field lines are concentrated, creating a circular cathode glow resulting in non-uniform eroding of the target. This considerably shortens the targets useable life. If ferromagnetic materials are used as a target in the magnetron gun the magnetic field lines from the magnetron are distorted and largely contained in the target. To successfully sputter ferromagnetic materials the magnetron field must be large, and very thin targets must be used so that the field can saturate the magnetic target Sputtering system The growth apparatus used at Leeds university is illustrated in Figure 5.2. The vacuum chamber reaches base pressure in two stages. Firstly, a positive displacement rotary pump is used to rough the chamber from atmospheric pressure to roughly 20 mtorr. After this stage, the chamber is shut off from the rotary pump and switched to a cryop-

105 Higher order SSWM results 91 Residual Gas Analyser LN 2 Meissner cold-trap Samples Viewport Sample wheel Growth magnets Shutter wheel FM growth gun Magnetron guns Rubber vacuum seal Roughing pump line DC power and cooling water Gate valve Cryo-pump Figure 5.2: Illustration of the sputtering machine at the University of Leeds. When operating the magnetron guns confine an Argon plasma above the target which deposits materials on the substrates (upside down). The shutter and the sample wheel are independently rotated so the substrate to be grown is open to the appropriate target. umping system. The cryopump captures vapours and gases by condensing them onto a 10-20K cold stage. This stage is kept at constant temperature by a closed-cycle 4 He refrigerator. Gases with very low condensing temperatures like He and H 2 do not condense on the cooling stage, but are trapped through cryosorption in charcoal panels bonded to the cold elements. The chamber reaches its ultimate pressure with the cryopump overnight ( 10 hours). Most of the residual gas at this time is from water vapour, which is further reduced using a Meissner cold-trap (see Figure 5.2). The final base pressure before growth is better than

106 92 Higher order SSWM results Gas Partial pressure Partial pressure species before cooling(torr) after cooling (Torr) H N H 2 O O Table 5.1: Typical partial pressures before and after cooling the Meissner trap Torr. Chamber pressures above 0.1 mtorr are measured with a MKS Baratron capacitance pressure gauge, with lower pressures being measured using a Spectramass Dataquad quadrupole residual gas analyser (RGA). Average partial pressures in the chamber as measured by the RGA before and after cooling the cold-trap are shown in Table 5.1. Inside the chamber are six planar magnetron sputtering guns, above which is a shutter wheel and a sample wheel. The guns are powered by DC MDX-500 power supplies, operating in current regulated mode. Four of the guns take standard 2 inch diameter targets, whilst the other two guns are smaller in diameter and have stronger magnetic fields for growing ferromagnetic materials. The sample wheel holds 16 substrates around its circumference, and can be rotated in combination with the shutter wheel allowing each sample to be grown independently. One significant advantage to the system is all 16 samples can be grown in one vacuum cycle, minimising sample variation in a series of samples. The sample wheel is made from a significant mass of copper, and is in direct thermal contact with the substrates, reducing heating whilst growing. The shutter wheel shields all samples except for the sample being grown from any deposition. This has the added advantage that the guns may be lit before the sample is rotated over them for deposition, and in the case of multilayer growth, multiple guns can be on simultaneously. These respectively minimise debri from the ignition phase of the plasma creating nonuniform growth, and minimise time between subsequent layer growth reducing the level of impurities collecting at the interface. The distance between the target and the film to be sputtered is 40 mm. The substrates are mounted upside down to minimise accumulated particles whilst the film is in the chamber. Even so, a few larger particles have little undesireable effect to the resulting film as it covers a relatively much larger area.

107 Higher order SSWM results Growth procedure The substrates are cut to a size of approximately 10mm 2 from polished silicon [100] oriented wafers using a diamond tipped scribe. The variation in thickness of the sputtered film over this size substrate is negligible. Each sample is numbered on the rear with the scribe. Silicon is an ideal choice for a substrate as it is relatively strong, the surface is close to atomically flat, and the metals deposited have good adhesion to the surface. The substrates are cleaned thoroughly in an ultrasonic bath of acetone, and then rinsed in isopropyl alcohol. This ensures that the films are free from oils and residue, as well as any dust particles. The samples are mounted onto copper sample mounts using small pieces of vacuum tape. The mounts fit into the sample ring shown in Figure 5.2. The growth procedure including the guns, shutter and sample wheel control are automated and computer controlled to ensure consistency between samples. 5.2 X-ray reflectivity To accurately be able to extract the wavelength of the SSWMs it is necessary to know the exact thickness of the films. At Leeds, the films are characterised with a Siemens twocircle diffractometer, with the setup shown in Figure 5.3. The setup consists of a fixed x-ray beam at which the sample is tilted to the beam at an angle θ, whilst the detector rotates at twice this angle, always giving equal incident and reflected beam angles to the films surface. This is consequently referred to as a θ 2θ measurement, and the setup is known as the Bragg-Brentano configuration. The x-ray radiation is produced from high energy electrons ( 10 4 ev) bombarding a Cu target. Two wavelengths of x-rays are excited from the Cu K-shell, denoted K α and K β. K β is excited with smaller intensity, and is filtered from the beam with a thin Ni foil. This foil is highly attenuating to K β whilst almost transparent to K α. Any Bremstrahlung radiation produced in the bombardment is filtered out using a monochromator, leaving only a monochromatic K α beam with a wavelength of λ = 1.54 Å. From to the symmetry of the Bragg-Brentano geometry, the only momentum transfer measured in elastic scattering is from the y direction, normal the films surface. The wavelength of this normal wavevector incident on the sample can be calculated from

108 94 Higher order SSWM results x -ray detector x-ray beam 2 sample Figure 5.3: X-ray reflectometry setup. geometry to be [9]: Q y = 4π λ sinθ. (5.1) Consequently the wavevector depends on the incident angle of the x-ray. To be able to use these wavevectors to characterise the thickness of samples, or a multilayer repeat thickness, it is necessary to work in the low angle region with θ approximately Figure 5.4 shows the reflectivity intensity from a single layer of cobalt. An interference pattern is seen, with the modulation resulting from the interference between x-rays reflected from the top and bottom of the film. These are referred to as Kiessig fringes [86], and the peaks occur when there is constructive interference: mλ = 2t sin 2 θ m sin 2 θ c, (5.2) where m is an integer, t is the thickness of the film, θ m is the angle of the mth maximum, and θ c is the critical angle for total external reflection. As the incident angles θ m are small, this can be simplified to: θ 2 m θ 2 c = m 2 ( λ 2t )2. (5.3) So it is possible to calculate the thickness from the slope of θ 2 m vs m 2.

109 Higher order SSWM results Intensity (counts) θ (degrees) Figure 5.4: X-ray intensity as a function of angle for a single Co film grown for 200 seconds at 50mA current. Calculated thickness from the fringes is 182 Å ± 6 Å Kiessig fringes would be difficult if not impossible to observe and characterise the thickness of the bilayer structures, including the cap and seed layers. For this reason, calibration films are grown as single layer films, and deposition rates are calculated for each target at the corresponding growth current. The initial growth rate is uncertain as it is not known how the first few monolayers form, but nonetheless, this is the most accurate method available to characterise the total amount of material deposited on the substrate. For characterising the growth rates in the Pt/Co multilayer films it can be more accurate to characterise a bi-layer repeat structure. In a multilayer structure, Kiessig fringes exist from the complete film thickness, however, these are superimposed with extra modulations from the Bragg reflections off the bi-layer repeat surfaces. The position of the n th maximum of these Bragg reflections is given by Bragg s law (nλ = 2dsinθ). A more accurate measure of the bi-layer thicknesses can be gained by keeping one materials thickness constant and altering the second materials thickness over a few samples. The relationship of d versus growth time of the second layer allows calculation of both layer growth rates.

110 96 Higher order SSWM results 5.3 Experimental Results Py reference film Firstly measurements were made on Py films of different thicknesses. Hysteresis loops for the single layer film are illustrated in Figure 5.5. The saturation magnetisation value as measured by SQUID (Superconducting QUantum Interference Device) magnetometry can be found in Table 5.3. This saturation magnetisation compares well with the field required to pull the magnetisation out of plane (suggesting the surface anisotropy is small). From the ferromagnetic resonance measurements the fundamental mode of the different thicknesses measures with almost identical frequency vs field profiles. In the single permalloy layers, one higher order mode was observed. The raw absorption as measured from the transmitted wave (S 21 ) is shown in Figure 5.6a for the different film thicknesses. The resonant frequencies of the two modes are shown for three film thicknesses in Figure 5.6b In plane Out of plane Moment (emu) Field (Oe) Figure 5.5: Loops for a single Py film 60.5nm thick both in and out of plane. To model the single layer system, approximate parameters are found from fitting the fundamental mode with the expression for a thin film resonance given in equation (1.13). These are used as starting values for fitting the complete model shown in Chapter 4 to the data. Note that these permalloy films were grown at Leeds, whilst the permalloy films shown in Chapter 1 were grown at NIST. The difference in the field vs frequency

111 Higher order SSWM results 97 a) response at 8GHz: 605Å 740Å 910Å b) 12 SSWM 10 Freq[GHz] Fundamental modes 605Å 740Å 910Å Field[Oe] Figure 5.6: a. Measured absorption for three permalloy film thicknesses at 8 GHz. b. Resonant frequency vs field for the three Py film thicknesses, with two modes seen from each film. The fundamental mode of all three films is almost identical, and the first SSWM depends on the thickness of the film. The orange lines are fits to the modes with parameters explained in the text. profile between the two films is shown in Figure 5.7. The frequencies for the Leeds film are lower than the NIST film, meaning the saturation magnetisation and/or the g value are lower for the Leeds Py. The static measurements showed a lower saturation magnetisation in the Leeds film (seen in Table 5.3), and a complete fit including the g value is found from measuring the resonance frequency vs field profile over a large ( 3000 Oe) field range. These higher field results (up to 8 koe) were taken using a

112 98 Higher order SSWM results single electro-magnet, with the reference measurement taken at high field ( 1T) using the same electromagnet. The Leeds permalloy films fundamental mode does not change significantly with film thickness (as seen in Figure 5.6), and an approximate fit using Kittel s thin film formula (from equation (1.13)) results in µ 0 M s = 0.8 T, g = 2.05 and H K = 5 Oe. The error in the measured anisotropy value is quite large ( 5 Oe) compared to results taken in the coil setup as the pole pieces in the electromagnet hold a remnant field. If the difference in M s between the Leeds and NIST Py films were to be from a surface anisotropy effect, this difference would be dependent on the thickness of the film. As a thickness dependence is not seen, it must be assumed that the bulk saturation magnetisation value is indeed lower. Freq GHz NIST 50nm Py Leeds 60.5nm Py Field Oe Figure 5.7: Comparision of the frequency vs field profiles for the observed fundamental mode in permalloy grown at Leeds University and at NIST, showing a difference in gradients. Fits are from the thin film Kittel equation using parameters in the text. The complete model is fitted using values for the conductivity in the sample σ Py = Sm 1 [105] and in the shield σ Cu = Sm 1, and g = As a calculation of the convolution of the complete k vector range is time consuming, the calculation was considered for the k vector corresponding to the maximum J k as shown in Figure 4.4b, corresponding to k = 3000 m 1. The complete convolution of the resonance linewidth with the k-range excited would shift the resonant frequency slightly, and increase the linewidth. Using only a single k value in the model will not predict the linewidth increase, but will approximate this shift in resonant frequency. As resonance

113 Higher order SSWM results 99 linewidths are not being scrutinised, this approximation will introduce little error. If the sample was truly excited by a uniform field resulting in only odd modes excited with zero surface effects, no SSWMs would be visible. This is because these modes show zero nett M [139]. For the film thicknesses under consideration, the model does not show significant asymmetry in the higher order modes due to the film conductivity and the shield effects for these modes to be visible. To observe a SSWM in the model (as is seen experimentally) it is necessary to add some pinning at either one or both sides of the film. The pinning is introduced as described in Chapter 4, as an effective surface anisotropy. The pinning at the surface of the permalloy film could be the result of the Ta/Py interface. As the film structure is symmetric, it would seem likely to assume an identical pinning at both surfaces. If pinning is added to both sides of the film, the observed SSWM with the highest amplitude is n = 3, where n = 1 represents the fundamental mode. This is the first non-uniform even mode. Using the model, the exchange value required to fit the data (Figure 5.6) to a film with very small pinning at each surface was A Jm 1. If the pinning is increased, the exchange value must be made smaller to match the same frequency. However, the exchange constant generally accepted for Py is A Jm 1 [72,100,118]. It is clearly incorrect to fit this even mode to the data. If pinning is added to one side of the film, then the observed SSWM with the highest amplitude would relate to mode n = 2. To match this mode to the observed mode, the exchange parameter would have to be A Jm 1. Analytical check It seems sensible here to check the values of the modes calculated by the model to the well known Kittel analytical form for standing spin waves [27]: ( ) 2 ω = (B + Dq 2 )(B + µ 0 M + Dq 2 ) (5.4) γ Where D is the spin wave stiffness constant D = 2A, and q is the wavelength of the M excitation; for unpinned spins, q = nπ where t is the thickness of the film, and n is the t mode number. As the model does not give amplitudes for SSWM s that are unpinned (as they excite no nett M), values are compared from the eigenvalues of the matrix Ĉ given in equation (4.57). Table 5.2 shows the comparison of the first five modes from the model to the analytic expression given in equation (5.4). Values are shown from the model for

114 100 Higher order SSWM results the case where conductivities in the shield and sample are included at k = 3000 m 1, as well as the case where the conductivities are zero, and k = 0. It is clearly seen that for the case where conductivities and wavevector are zero, the model calculates values very close to the analytic expression. Small discrepancies arise as the mode number increases, due to the finite size of the matrix. If the matrix size is increased, these values converge. It is also noted that the shift considering a finite wavevector and conductivities is quite small and only effects the fundamental mode. These effects certainly cannot be used to explain the large difference between the exchange constant found from experiment and the accepted value. mode number Analytical Calc (σ = 0; k = 0) Calc (complete) Table 5.2: Calculated frequencies (GHz) for unpinned spins using Kittel s formula (equation (5.4)) with values t = 60.5 nm, γ = Hz/T, B = 0.1 T, A = J/m and M = A/m. This is compared to calculation with same values (σ, σ s = 0 and k = 0), And calculation with conductivities (σ = S/m, σ s = S/m) and k = 3000 m 1. Asymmetry It is surprising that the first exchange mode is seen (see Figure 5.6). This means an asymmetry exists through the thickness of the film, as suggested by pinning only on one side of the film. It is possible that lattice spacing variations between the Si substrate and the Ta buffer introduces a strain in the film. If this is the case, the strain would only be present at the Py/Ta interface closest to the substrate, creating an asymmetric pinning. It is also possible that the effects of the sample conductivity and waveguide shield are larger than the model predicts for the thickness of the film measured. As seen in Chapter 4, an asymmetry is introduced from both the sample conductivity and the shielding of the waveguide in thicker films. If these effects are larger than the model predicts, an asymmetry in thinner films would be possible.

115 Higher order SSWM results 101 Also seen in Chapter 4 is that any change in the pinning at one or more surfaces increases all of the frequencies of the resonant modes. Taking this into consideration, when fitting the data with pinning M s must be reduced. As the magnetisation value is already small for a Py film, the results were fitted with a small asymmetric pinning (d = 0.01) at one surface, such that the first exchange mode was just visible. Experimentally it is the only odd mode observable because the intensity of higher order modes falls rapidly with increasing n. The final best fit parameters were µ 0 M s = T, H a = 5 Oe and A = Jm 1. Exchange value The exchange value is much lower than would be expected for bulk Permalloy (A J/m), and this should be investigated. Firstly note that the frequency of the exchange mode relative to the fundamental mode is dependent not on A, but on the ratio 2A. If the magnetisation value used in the model is lower than it should be M then A must be artificially lowered to keep the ratio constant. Using Kittel s thin film formula (equation (1.13)) over the complete field range measured, it is seen that the gradient of the frequency vs field curve for the fundamental mode depends on both M and γ. It is only possible to increase M by decreasing γ. The Kittel thin film formula shows a relationship with frequency squared equal to γ 2 (H K + H b )(H K + H b + M s ) (equation (1.13)), where H b is the applied bias field. However, performing a non-linear regression fit to the fundamental mode gives 95% confidence intervals in µ 0 M of [0.69, 0.81]T and in g of [2.01, 2.17]. Even if the maximum value in the confidence range is chosen for M, keeping the ratio 2A/M constant gives a maximum value for A of A = Jm 1. Fitting the exchange modes also is highly dependent on the thickness of the film. The films thicknesses are determined as outlined in Section 5.1, with low angle x- ray reflection measurements taken on calibration films. This measures the structural thickness of the films with an error of less than 5%. It is possible that the growth rate between films changes slightly, particularly if they were grown in separate vacuum cycles. Unfortunately this error is difficult to predict, as all the films except the calibration films are grown with a seed and cap layer, making x-ray measurements on these of the films difficult. The position of the SSWMs of the same film structure grown in different runs is within the FMR measurement error ( 10 Oe), giving evidence that the growth rate between films is relatively constant. It is however possible that the magnetic thickness

116 102 Higher order SSWM results is different than the structual thickness. To increase the value of A, it is necessary to increase the magnetic thickness t. This seems unlikely, because any oxidation at the interfaces would likely decrease the magnetic thickness. One possible explanation for the reduced exchange value is the structure of the film. Due to the sputtering process it is assumed that the Py grows as a collection of nano-crystalline grains. Depending on the distribution of the elements through the grain, and the alloying nature of each grain, it is possible that the exchange interaction is reduced between each grain. This would dominate the standing wave response, as the film is many grains thick. This would give a perceived lower value of A for the permalloy. It also must be remembered that adding pinning in the model increases the frequency of the expected exchange modes. To successfully fit the data with pinning requires a smaller exchange constant. It is possible the pinning parameter is of incorrect magnitude in the model. An increase in the pinning at either surface would reduce the value of the exchange constant further Bilayer Py/Co films The idea to measure the pinning at the interface of Py and Co came with successful measurements using the coplanar line dynamics with a Py (50nm) and Co (100nm) bilayer studied earlier [40]. The results for this bilayer film are found in Section 3.2. To explore the interface dynamics it was appropriate to begin with a reference Py film, with different thickness Co layers grown on the surface of the Py. The saturation magnetisation values for the materials used were found from measuring the total moment by SQUID when the sample was saturated. The values measured for the materials are shown in Table 5.3. Co Py Figure 5.8: Illustration of Co addition to increase interface effects. Figure 5.9 shows the easy and hard direction of the film grown with the largest Co to Py ratio. In the easy direction of the film a two phase loop is clearly visible. The

117 Higher order SSWM results Long Moment (emu) HARD DIRN EASY DIRN Long Moment (emu) Field (Oe) Field (Oe) Figure 5.9: Loops for a Py(60.5nm)/Co(10nm) film in the hard and easy directions. anisotropy for the Co was measured using a single layer of Co 10nm thick from the saturation point on the hard axis of approximately 50 Oe. Material µ 0 M s (T) H Kin (Oe) H K (T) UWA Py (Ni 80 Fe 20 ) UWA Cobalt NIST Py (Ni 81 Fe 19 ) Leeds Py (Ni 80 Fe 20 ) Leeds Cobalt Leeds [Pt(25 Å)/Co(3 Å)] Leeds [Pt(74 Å)/Co(7 Å)] Table 5.3: Saturation magnetisations and anisotropies measured from hysteresis loops. Saturation magnetisation calculated from nett measured moment and sample volume, anisotropies calculated from hard axis saturation field. In the case of the H K, demagnetising fields were added to the saturation field. Ferromagnetic resonance measurements were taken on permalloy films with a dusting of cobalt, less than a complete monolayer. No significant change was seen in the resonance profiles for films with cobalt thicknesses less than 1nm (see Appendix D for films py py001 12, py py and py py002 15). Shown in Figure 5.10 are colour plots of the imaginary component of the susceptibility as a function of frequency and field for different thicknesses of cobalt above 1nm on 60nm of permal-

118 104 Higher order SSWM results loy. Several features can be noted with the addition of the cobalt with thicknesses greater than 1nm. Firstly, the fundamental mode gradient changes very slightly, but the first SSWM gradient changes more significantly. Secondly, a third mode appears for Co thicknesses between 2 and 10 nm. Thirdly, the intensity of the first SSWM increases with frequency. In fact, with 10nm of cobalt on the 60nm Py sample, the intensity of this first SSWM becomes larger than the intensity of the fundamental mode at frequencies above 10 GHz. This is particularly clear if cross-sections of the imaginary part of the susceptibility are observed for different frequencies in this sample, shown in Figure As shown in the single Py film case, the SSWM resonances depend strongly on the film thickness, with thicker films showing lower resonance frequencies. Thus, when the Co (10nm) is grown on a thicker (74nm) Py film, four modes are seen in the measurement range. Shown in Figure 5.12a are the resonant frequencies of these modes compared to the single Py layer film of the same thickness. Also in Figure 5.12b the intensities of the four observed modes are plotted as a function of resonant frequency. These intensities are normalised to the first fundamental mode, to remove any intensity variation due to the waveguide itself. It is clear that the standing spin wave modes increase with intensity relative to the fundamental mode as the resonant frequency (and field) increases. This is not seen in the single permalloy layer films. Pinning model The simplest way to model the Py/Co sample is to assume that the resonances measured come from the Py layer, and the Co affects pinning at one of the surfaces of the permalloy due to its larger saturation magnetisation. The results from assuming a pinning (d = 0.05 at the Py/Co interface are shown in Figure This does not account for the change in gradient in the SSWM as a function of field, as the pinning simply translates the complete SSWM profile upwards in frequency. It does however mimic the slight increase in frequency seen in the fundamental mode, along with the larger increase in frequency seen in the SSWM mode. More importantly, the pinning parameter at one surface of a single layer film does not change the intensity as a function of frequency of the SSWM significantly. The amplitude of the SSWM can be increased by an increase in the pinning parameter, but it is not field dependent, and the calculation predicts the SSWM intensities will never exceed 15% of the fundamental amplitude in a single film calculation. The third mode seen experimentally in the Py(60.5nm)/Co(10nm) film is not seen in the calculation

119 Higher order SSWM results 105 Py(60.5nm)/Co(1nm) Py(60.5nm)/Co(5nm) Py(60.5nm)/Co(2nm) Py(60.5nm)/Co(10nm) Py(60.5nm)/Co(4nm) Figure 5.10: Colour plots of the resonant behaviour (Im[χ xx ]) of 60.5nm Py with 1-10nm Co grown on top. The darker colour represents a larger amplitude of the susceptibility. As the Co thickness is increased a third mode is observed, and the amplitude of the 1st SSWM increases with frequency. It can be seen that for 10nm of Co, the SSWM has larger amplitude than the fundamental for large frequencies.

120 106 Higher order SSWM results Py(60.5nm)/Co(10nm) 4 GHz 7 GHz Im[χ xx ] 10 GHz 13 GHz Field [Oe] Figure 5.11: Imaginary part of the calculated susceptibility from experimental measurements for different fixed frequency sweeps of Py(60.5nm)/Co(10nm). The intensity change of the peaks is clearly visible. until the pinning introduced from the Co layer is quite large. This gives significant asymmetry to the magnetisation through the thickness of the film so this mode can be observed. However, such a large pinning shifts the calculated fundamental and first SSWM frequencies above the measured values. If the pinning is to be this large, it would be necessary to reduce the exchange constant further. All of these observations mean the experimental results seen in the thicker Co films can not be explained by a simple pinning. This is to be expected, particularly for the case of a 10nm cobalt layer, as it makes up approximately 15% of the volume of the sample. The saturation magnetisation and anisotropies of the Co give the Co moments their own resonant behaviour in the frequency and field range observed. The complete response of the sample can be considered to be dependent on the properties of both films as well as the interlayer coupling.

121 Higher order SSWM results 107 a) 14 Freq[GHz] Py (74nm) Py(74nm) / Co(5nm) b) Field[Oe] Intensity fund 1st SSWM 2nd SSWM 3rd SSWM MODE INTENSITIES Frequency[GHz] Figure 5.12: a. Resonant frequency vs field profiles shown for the Py(74nm)/Co(10nm) and the single layer Py(74nm) films. At this thickness the bilayer shows two extra modes. b. The intensity of the modes measured in the bilayer as a function of frequency, normalised to the fundamental mode amplitude.

122 108 Higher order SSWM results Figure 5.13: A colour density plot of Py (60.5nm)/Co(10nm) as seen in Figure Overlaid in red (solid circles) are the modes expected for a single Py layer with pinning at one surface (substrate induced) as shown in Figure Overlaid in green (solid squares) are the expected modes seen in a single Py layer with small pinning at one surface (substrate) and a larger pinning (assumed to be from the Co layer) on the other surface. A shift in the mode frequencies are seen from the large pinning, but the slope remain constant. Bilayer model The Ms of the cobalt layer as grown on the Py is difficult to measure as the Py dominates any quasi-static measurements. A single Co layer (10 nm) was measured using SQUID magnetometry with the saturation magnetisation calculated from a volume measurement to be µ0 M = 1.78 T and an anisotropy calculated from the in-plane hard axis saturation field of 55 Oe. Measuring the single layer Co film with the coplanar line FMR, gives a fit of µ0 M = 1.76 T and an anisotropy of 20 Oe. The anisotropy value from the FMR is much more accurate as it is difficult to determine the saturation point from the SQUID hysteresis loop. The anisotropy in the cobalt is expected to be low, as the sputtered growth results in a nanocrystalline structure. A texture is present in the sample from the growth field, however the anisotropy as averaged over the grains is rather small. The

123 Higher order SSWM results 109 gyromagnetic ratio was very close to the Py value with g = However, the measurements on single cobalt layers may also show different effective saturation magnetisations as the surface anisotropies will strongly depend on the Py/Co interface. This would affect the value of the demagnetising field in the film. Also, the texture in the film is likely to be thickness dependent. For this reason both the anisotropy and saturation magnetisation will be left as fitting parameters in the bilayer model and the values measured in the 10 nm single Co layer will be taken as starting values. Freq GHz Field Oe Figure 5.14: Calculated modes for a single Py layer (blue circles), and calculated Co layer fundamental frequency (red squares) overlaid onto Py(60.5nm)/Co(10nm) data. Parameters in text. The increase in frequency seen in the first SSWM as a function of field with the addition of cobalt can be explained because the fundamental mode in the cobalt overlaps the mode in this region. The two modes measured for the single layer are shown in Figure 5.14 over-layed with a resonant mode for cobalt showing this trend. For this to be the case, the cobalt must have a significantly reduced effective saturation magnetisation, with M s A/m and an in-plane uniaxial anisotropy of 50 Oe. This means that in the case of the coupled films the cobalt mode will resonate in conjunction with the first SSWM of the permalloy where they overlap. There will be an increase in the nett magnetisation oscillation and an increase in intensity will be observed. However, in practice, the interlayer coupling between the Py and the Co reduces the frequencies in the Co, and a higher saturation magnetisation for the Co layer is required in the bilayer model. Best fits to the modes from the model is shown in Figure 5.15a.

124 110 Higher order SSWM results a) b) 3GHz 6GHz CalculatedIm[ xx ] 9GHz 12GHz Field(Oe) Figure 5.15: a. Colour plot of Py(60.5nm)/Co(10nm) as seen in Figure 5.10, overlaid with modelled fits for a bilayer with an interlayer coupling. b. Calculated susceptibility of the bilayer for various frequencies. The increase in the SSWM amplitude with frequency is seen in the calculation, however it is larger in the measured results. The fit to the second mode in the Py(60.5nm)/Co(10nm) was found using the same Py parameters required to fit the single layer film, with an unpinned Co/Ta interface, M s(co) = A/m, A (Co) = Jm 1, H k = 50 Oe and an interlayer coupling of A (Co/Py) = Jm 1. is the separation between sublayers in the model. It should be noted that other cobalt anisotropy and saturation magnetisation parameter combinations can also fit the frequencies. For example, a saturation magnetisation of M s(co) = A/m with an anisotropy of 100 Oe also gives similar frequencies. It is necessary in this case to consider the values measured in the single Co layer to verify the model parameters. The susceptibilities as a function of field for various fixed frequencies are shown in Figure 5.15b. The model shows an increase in the intensities of the SSWMs with field, but not as significant as observed experimentally. It is necessary to model the Co layer with a thickness >30nm before the first SSWM mode will gain greater intensity than the fundamental mode. It is possible that a larger pinning at the Py/Ta interface would be more appropriate, or a surface anisotropy at the Py/Co interface exists. Either of these

125 Higher order SSWM results 111 options would pin the Py oscillations, decreasing the intensity of the first mode which is localised in the permalloy. If the amplitude of this mode was reduced, the relative amplitudes of this mode and the first SSWM mode would model the observed results better. Also of importance is the third observed mode in the Py(60.5nm)/Co(10nm). The parameters used for the bilayer model fit both the frequencies and amplitudes of this mode well Py/[Pt/Co] x In this section the resonant behaviour of permalloy is observed when coupled to a Pt/Co multilayer. In the previous section, where the permalloy was coupled to cobalt, there were two problems. Firstly, the saturation magnetisation and anisotropies of the Co layer as grown on the Py are not known. Secondly, the cobalt exhibits resonant frequencies in close proximity to the frequencies of the permalloy resonances. This leads to strong interactions between modes in both films. To avoid this hybridisation of modes while still pinning the Py surface, a new film was needed with resonances well seperated from the Py film. A [Pt/Co] x multilayer was chosen as its large anisotropy ensures its resonance frequencies will be much larger than the Py. The Permalloy / Cobalt bilayer system was also seen (at least in the instance of the thicker Cobalt layers) to have both layers magnetisations in the plane of the film. The Co/Pt multilayer is chosen to have a definite out of plane anisotropy and thus complements the Py/Co system by showing pinning across an interface with different magnetisation geometries. To alter the interfacial coupling strength between the Py and Pt/Co films it was decided to keep the Pt/Co structure constant, and instead alter the coupling using a separating layer. Py Cu [Pt/Co]x Figure 5.16: Illustration of Cu spacer between Py and [Pt/Co] x layers to alter interface exchange. Increasing exchange coupling as the Cu spacer decreases. The first challenge was to find a well behaving Pt/Co multilayer with a strong out of plane anisotropy. Different out of plane hysteresis loops as measured by MOKE are seen

126 112 Higher order SSWM results M (normalised) Field (Oe) Saturation Field (Oe) Remnant Moment (normalised) (a) Out of plane loops with varying Pt thickness with a 10Å Co thickness (b) Saturation field and remanence with varying Pt thickness with a 10Å Co thickness M (normalised) Field (Oe) Saturation Field (Oe) Remnant Moment (normalised) (c) Out of plane loops with varying Co thickness with a 24Å Pt thickness (d) Saturation field and remanence with varying Co thickness with a 24Å Pt thickness Figure 5.17: Hysteresis loops for differing Pt/Co thicknesses as measured by MOKE

127 Higher order SSWM results 113 in Fig. 5.17a,b for various Pt/Co structures. Two sets of films were grown, one set with a varying Co thickness, and one with a varying Pt thickness. From observing the out of plane remnant moment and required saturation field as a function of both Co and Pt growth times, the coercivity and anisotropy could be engineered to provide a film with a strong out of plane moment. It can be seen in Figure 5.17c,d that the best films have thin Co layers, and thicker Pt layers. These give a squarer loop and a larger coercivity. Two final configurations were chosen to grow on top of the Py; one with 20 repeats of 25Å Pt and 3Å of Co, and a second with 10 repeats of 74Å Pt and 7Å of Co. The repeats were kept high to ensure the films magnetisation was not highly influenced due to interactions with the Py layer. Out of plane anisotropies were found for these films from the field required to saturate the magnetisation in-plane. A large field was required to saturate the film in-plane, so the 7 T magnet of the SQUID magnetometer was required. The out of plane anisotropy is given by a sum of the field required to saturate the magnetisation out of plane, as well as the demagnetising field (M s ). The parameters for the Pt/Co films chosen for coupling to Py can be found in Table 5.3. The Pt/Co repeats were grown on the Py layer with varying Cu spacer distances of 0-5nm. In the Pt/Co multilayer, the anisotropies can be modified by varying the thicknesses of the platinum and cobalt layers, thereby creating an out of plane anisotropy. The anisotropy was measured by SQUID magnetometry to be above 2 Tesla, so the Pt/Co layer magnetisation is assumed to be normal to the plane in the FMR measurement fields (<0.3T). This creates a difficulty in that the model for a bilayer developed in Chapter 4 assumes that the magnetisation in both layers points in the same direction as the applied field. It is necessary to extend the model to arbitrary magnetisation directions in each sublayer (similar to reference [98]). This will not be considered here, because although the anisotropy is large, the coercivity is only a couple of hundred oersteds (Figure 5.17), so domain formation in the Pt/Co layer is likely. This makes the model derived in Chapter 4 inappropriate, because it assumes a uniformly saturated magnetisation in each layer. The results for two different Pt/Co configurations coupled to Py are shown in Figure The Pt/Co is grown straight on to the Py, as well as in a tri-layer structure with a small Cu spacer layer. When the Pt/Co layer is directly in contact with the Py film two modes are seen. In Figure 5.18 the experimental frequency vs field profiles are shown in colour plots, with the modes as measured in a single Py film overlaid as red lines. A Pt/Co film was measured with no Py, and no nett signal was observed. This suggests the two modes seen in the case of zero Cu spacer layer are indeed the funda-

128 114 Higher order SSWM results 13.4 GHz 10GHz Im[ xx ] 6GHz [Pt(25Å)/Co(3Å)] x20 /Py(60.5nm) Field[Oe] 13.4 GHz 10GHz Im[ xx ] 6GHz [Pt(25Å)/Co(3Å)] x20 / Cu(0.68nm)/Py(60.5nm) Field[Oe] 13.4 GHz 10GHz Im[ xx ] 6GHz [Pt(25Å)/Co(3Å)] x20 / Cu(1.33nm)/Py(60.5nm) Field[Oe] Figure 5.18: Colour plots of the frequency vs field profile for [Pt(25Å)/Co(3Å)] x20 /Py(60.5nm) with various Cu separations. The red lines represent fits to the single Py(60.5nm) film showing the first two SSWMs. To the right are measured susceptibilities at various frequencies showing the change in resonant shape.

129 Higher order SSWM results 115 mental and SSWM localised in the Py. However, it is observed that these two modes are lower in frequency than for the single Py film. It is proposed that the coupling between the Pt/Co and the Py pulls the Py magnetisation near the interface out of plane. As domains are likely to exist in the Pt/Co film, Py moments near the surface will lie at some oblique angle with respect to the film normal. This would reduce the effective magnetisation significantly in this area, causing a shift downwards in the resonant frequency. The non-uniform moments also change the effective thickness of the film. As the SSWM frequency is seen to reduce, the effective thickness of the film must increase. The first SSWM also proves interesting as it has an amplitude significantly larger than measured in the single layer Py film. This reinforces the idea that the moments in the Py are more non-uniform through the thickness of the film, because this nonuniformity would result in a more asymmetric mode. If the mode is more asymmetric, a greater nett magnetisation will be measured. In the case of a spacer Cu layer, the copper is introduced to reduce the exchange coupling by removing any direct coupling, and relying purely on indirect RKKY coupling [137,164,178]. The effect of introducing a Cu layer is very clear, except it seems that even the thinnest Cu layer of 0.3nm is enough to remove any strong coupling effects. This is observed by the fundamental mode returning to the measured position for the single layer film (see Figure 5.18). However, an increase in the linewidth in comparison to the single layer film is still visible. This may be an additional damping from a spinpumping effect in the tri-layer structure [95,160] as the geometry is similar to a Giant Magneto-Resistance (GMR) spin valve [18, 68, 122]. This additional damping has been observed in similar systems with different materials using FMR [15,16,75,99,162,161]. Also the first SSWM becomes unmeasurable with the addition of the Cu layer. This is an indication that the Cu does not provide the same pinning effect that the Ta layer did, reducing the intensity of the SSWMs so they are no longer observable. Similar results are found for the Permalloy film grown on the [Pt(74 Å)/Co(7 Å)] x10 multilayer. Again, two modes with reduced frequency are seen, where the amplitude of both modes are equal at large frequencies. Also, the addition of any Cu once again prohibits observation of higher order modes and only the fundamental mode is seen. The fact that a different multilayer structure (with different anisotropy and saturation magnetisation) provides almost identical results is consistent with the idea that the resonance is not influenced directly by any excitation in the Pt/Co layer, but is influenced by the non-uniform Py magnetisation created by the coupling between the films.

130 116 Higher order SSWM results 5.4 Chapter summary In this chapter resonance studies have been shown to provide information from the fundamental resonance mode as well as higher order SSWMs. From the fundamental mode, it has been possible to extract information about the films anisotropies, saturation magnetisation and the gyromagnetic ratio. With the broadband frequency range available in the coplanar line setup, an accurate measure of the interlayer coupling is possible. This is because the frequencies can be modelled at a large range of fields, fitting the gradients of the modes. From the higher order SSWMs, it is possible to extract exchange constants for the magnetic films. These constants were shown to be highly dependent on surface pinning (and film thickness), so it is difficult to extract an accurate value of the exchange constant without knowing the pinning conditions. However, it is possible to see small relative changes in the mode frequencies from altering one interface with another magnetic material. This is useful in understanding the coupling across a ferromagnetic interface.

131 Chapter 6 Future work and conclusions This thesis has demonstrated how a broadband FMR system can be used to measure higher order excitations in magnetic thin films. In particular, the frequencies and amplitudes of these higher order excitations prove to be very sensitive to the coupling between different films. With a correct understanding of the measurement, coupled with a detailed theory, interactions between the different films can be characterised and magnetisation parameters can be extracted. Understanding these interactions will prove vital in future technological applications involving multi-layer magnetic structures. In this chapter, firstly a concluding summary of the results presented in the first five chapters will be presented. This is followed by an overview of the future directions of the project. These include sample systems grown in this project but not measured due to limited equipment access, further applications of the constructed theory, as well as appropriate future extensions to the theory. 6.1 Conclusions Several key results are shown in the thesis: Angular measurements on the Fe/MnPd sample demonstrated a strong four-fold anisotropy, with a weak anisotropy from the exchange bias. The measurement of the reorientation of the magnetisation to the hard axis showed a distribution of frequencies for the lowest energies (where the applied field and the magnetisation become collinear). This distribution of frequencies offers a unique measure of the coupling, and can be 117

132 118 Future work and conclusions explained by either a variation in local bias field strength, or direction. The Py/Co exchange spring bilayer grown at UWA exhibited two fundamental resonances. These related to a lower frequency acoustic mode where the magnetisation in both films oscillated in phase, and a higher frequency optic mode where the oscillation in the films was out of phase. One benefit of the broadband FMR was the possibility to measure the response of these modes over a range of fields and observe the change in frequency and amplitude. The intensities of the modes changed as the spectrum of the Py and Co overlap. The coupling between the films acts like a pinning where the Py and Co modes are far apart, and as a driving force where the modes converge. The theory developed demonstrated that both the film conductivity and the ground plane shielding introduce an asymmetry in the excitation through the thickness of the film. The dispersion with wavevector k was seen to be altered in both cases. In the case of a highly conducting film, the group velocity reduces for k values around kt = 1. The shield increases the group velocity on the wave propagating on the shield surface for k values around kt = 1, however the wave travelling on the other surface (in the other direction) is largely unaffected. Both the shielding and the conductivity of the surface have little effect for very small or very large k values, with the dispersion converging to the unshielded, insulating film case for both of these limits. One of the main motivations to include these effects into the theory was to see if they provided a large enough asymmetry to explain the observation of higher order SSW modes in the single layer Py film. The asymmetry is required to provide the nett magnetisation measured by the stripline for these modes. Although, with the low wavevector and relatively thin samples used in this project, the asymmetry caused by both the shield and the conductivity is simply not predicted to be significant enough to observe the higher order modes. This asymmetry must be explained by some other means, and in this thesis it was presented as a resulting pinning from a surface anisotropy. With an estimate of the pinning magnitude from the SSWM amplitude, it was possible to accurately determine the exchange constant in the Py layer. The exchange constant was remarkably lower (A = J/m) than the accepted value for Py, but this was explained in terms of the nano-crystalline grain structure. A comparison of the growth procedure between the Py/Co bilayers grown at Leeds and UWA predict that interface quality dictates the coupling quite strongly. The Py/Co bilayer grown at UWA required a rather small interlayer coupling to match the optic and acoustic modes. It is possible this is due to a poor quality interface due to a long delay between the deposition of different films. On the other hand, the Py/Co film grown at Leeds university showed

133 Future work and conclusions 119 an interlayer coupling A 12 of the same order as the coupling in the bulk of the Py and Co layers. These films were grown in better controlled conditions, with an almost zero delay between subsequent layer deposition. In the Py/[Pt/Co] x bilayer system, the addition of a Cu interlayer is seen to quickly influence the interlayer coupling. 6.2 Future work The films grown at Leeds University were kept as consistent as possible, with each sample set grown in a single vacuum cycle. This provided accurate comparisons between films within each sample set, for example, different Co thicknesses on the Py. Unfortunately, work presented in this thesis involved measuring samples that were available and not necessarily from the same sample set, leading to inconsistencies in parameters. For example, the parameters fitted to the Py/Co bilayers with a relatively thin Co layer cannot be directly compared to the Py/Co bilayer sample with a thick layer of Co as they were grown with different targets and techniques. The best fit parameters in the model gave remarkably different interlayer exchange coupling. Without consistency between the samples it is difficult to draw any strong conclusions about the results; for example, between the interlayer coupling and the Co layer thickness. One conclusion is that the interface quality is much better in the Leeds sample leading to a higher interlayer coupling. The most important consideration in future work would be to have a reliable source of samples to be able to draw conclusions which are strongly correlated between sample systems. To further understand the results in relation to the interface behaviour, it would be useful to compare the results with other techniques. For example characterising the interface quality via structural characterisation such as cross-sectional Transmission Electron Microscopy (TEM), or observing the magnetic coupling with another technique such as polarised neutron reflectometry would be interesting. To increase the influence of the surface effects such that a stronger conclusion about the interlayer coupling could be derived, it would be useful to decrease the sample thickness significantly. To decrease the thickness whilst still measuring higher order modes would require measurement of much higher frequencies, but this is becoming possible with modern network analysers having bandwidths greater than 100 GHz. Working at such high frequencies would need special care in the continuity of the waveguide and sample structure such that the attenuation is not too large from impedance mismatch.

134 120 Future work and conclusions In the following sections some future work already partly underway is presented Cryo-system Some of the film systems grown at Leeds University were designed to employ a pinning mechanism which could be switched on and off with temperature. This was achieved by choosing materials with a paramagnetic phase transition at a temperature slightly below room temperature. To measure these samples with the FMR coplanar line technique, it was necessary to design a low temperature system to enclose the sample. The equipment was constructed at a near zero cost and is designed to cool using liquid nitrogen (LN 2 ). A temperature of 80 K is easily obtainable. a) b) N 2 in N2 out N 2 out N 2 in Thermocouple LN 2 Figure 6.1: a. Housing designed around waveguide. Cold N 2 gas is circulated inside the housing before being expelled out vent holes. b. Heat tranfer system for cooling N 2 gas with LN 2 stored in a dewar. The system is designed with an enclosure around the waveguide into which cold N 2 gas is pumped (Figure 6.1a). Nitrogen gas is cooled via a heat exchanger to almost liquid temperatures using a LN 2 dewar as pictured in Figure 6.1b. The temperature is determined by regulating the gas flow, and the temperature inside the enclosure is measured with a T-type thermocouple. Using cold gas rather than liquid to cool the sample is preferred as fluctuations in temperature are not as severe. The gas is kept at a sufficient rate to keep the enclosure flushed with pure N 2 gas, ensuring condensation