Magnetic Coupling and Relaxation at Interfaces Measured by Ferromagnetic Resonance Spectroscopy and Force Microscopy

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Rohan Adur, M.Sci. Graduate Program in Physics

The Ohio State University 2014

Dissertation Committee: Professor P. Chris Hammel, Advisor

Professor Fengyuan Yang

Professor Mohit Randeria

Professor Richard Furnstahl ⃝c Copyright by

Rohan Adur

2014 Abstract

The emergent field of spintronics, which utilizes the spin of the electron rather thanthe charge for information processing, relies on an understanding of interfaces and surfaces of ferromagnetic thin films. An interface between a ferromagnetic thin film and a neighboring material can be engineered to provide tuneable static and dynamic couplings, which manifest as effective fields on the ferromagnet. Ferromagnetic resonance (FMR) is apowerful spectroscopic technique for studying these effective fields and couplings. In addition, FMR has been used to generate a pure spin current at these interfaces, which allows for the transfer of angular momentum without an accompanying charge current. The technique of magnetic resonance force microscopy (MRFM) has allowed the study of spin dynamics at the nanometer scale and with sensitivity down to single electron spins in paramagnetic materials and it would be illuminating to use this technique to study the spin transport behavior near an interface. MRFM uses the field from a magnetic probe to define a sensitive slice in which the resonance condition is met. The combination of MRFM techniques with

FMR spectroscopy has, until recently, been limited to the measurement of global properties of a sample due to strong spin-spin exchange interactions that lead to collective spin wave modes that are defined by the sample and not sensitive to the probe field. Recently, the negative dipole field from a high coercivity probe magnet has been used to strongly perturb the spin wave spectrum of metallic ferromagnetic films, resulting in the localization of precessing magnetization in the ‘field well’ of the probe magnet into discrete modes, analogous to the discrete modes of a particle in a quantum well. The localized nature of these modes enables their use as a local probe of magnetic properties, and this has been utilized in the demonstration of FMR imaging of a ferromagnetic thin film using ferromagnetic resonance

ii force microscopy (FMRFM).

In this thesis I shall demonstrate the use of FMR spectroscopy and FMRFM to study static and dynamic couplings in ferromagnetic materials with emphasis on interfaces. First,

I introduce the basic concepts of ferromagnetic resonance and spin wave relaxation. Second,

I present the results of using conventional FMR spectroscopy to study the tuneable static effective fields in a ferromagnet, which manifest as anisotropy fields that define theenergy landscape and equilibrium direction of the magnetization. Third, I explore both dipolar and exchange couplings between magnetizations which are dynamic in nature, and only manifest when the magnetizations are precessing. Fourth, I demonstrate the use of FMRFM to observe the modification of localized modes in a ferromagnetic film engineered with alateral interface. Fifth, I describe the design of an FMRFM microscope and management of spurious background effects in an FMRFM experiment. Sixth, I measure relaxation fromthe localized modes in an insulating ferromagnetic film, which reveal a size-dependent damping effect that was unexpected in an insulating system. This suggests that spin transport from the interface of the localized mode can dictate its relaxation, even in the absence of conduction electrons. Finally, I observe a frequency-independent linewidth broadening contribution that also depends on mode size and this may give a measure of the inhomogeneous fields within a ferromagnetic sample.

iii Acknowledgments

I have many people to thank for getting me to the stage in my career where I can write a dissertation that I could be proud of. First and foremost, I’d like to acknowledge the advice and support of my advisor Chris Hammel, without whom I would not have had the resources to accomplish what I have here. He has always encouraged me to get straight to the physics at the heart of all the work I’ve done, and I hope to continue to do this throughout my career.

I wish to thank the senior members of the group who helped me learn about techniques in experimental condensed matter early in my career. In particular, I’d like to thank Inhee

Lee for introducing me to the concept and experimental details of an FMR microscope and conventional FMR, as well as Yuri Obukhov for his role in developing the experimental tools that started my career in this field. I’d also like to thank Michael Boss for getting me started with the basics of cryogenic and vacuum science. I thank Vidya Bhallamudi for his advice and help with scanned probe techniques that have enabled me to design and build my own microscope. I thank Denis Pelekhov for his help with micromagnetic simulations, as well as guidance with experiments and microscope design. I also wish to thank Andrew

Berger for allowing me to bounce ideas off him, but primarily for our numerous discussions on a range of topics from technical details to the philosophy of experimental physics.

In the latter half of my graduate student career I have benefited from the help of several talented individuals to help build and conduct experiments with a room temperature FMR microscope. First, I’d like to thank Chunhui Du for helping me with the FMR microscope and experiment: her tireless efforts to collect a seemingly endless amount of data, aswell as for her input into the physics we’ve managed to obtain from our results were invaluable.

iv Second, I’d like to thank Sergei Manuilov for his role in teaching me a substantial amount from our conversations about spin wave physics and relaxation, and for developing the excellent microwave circuits that have enabled our experiments. I’d like to thank Haidong

Zhang for his contributions towards the microscope design, and Josh Gueth for his excellent machining work to build the parts for the microscope. I’ve also benefited from the help of

Nicholas Scozzaro for cantilever magnetometry experiments. I thank Jeremy Cardellino for discussions on scanned probe techniques, and Chi Zhang and Shane White for discussions on intralayer spin pumping. I’d also like to thank Hailong Wang and Fengyuan Yang for synthesizing excellent YIG films that have made these experiments possible. Finally, I’dlike to thank Kris Dunlap, Stephanie Arend and Rachel Page for their help with administrative matters that enabled me to concentrate my efforts on research.

v Vita

June, 2006 ...... M.Sci. Astrophysics, University College London, London, UK 2007 - 2008 ...... Graduate Teaching Assistant, The Ohio State University, Columbus OH, USA 2008 - 2014 ...... Graduate Research Assistant, The Ohio State University, Columbus OH, USA

Publications

F. Wolny, Y. Obukhov, T. M¨uhl,U. Weißker, S. Philippi, A. Leonhardt, P. Banerjee, A. Reed, G. Xiang, R. Adur, I. Lee, A.J. Hauser, F.Y. Yang, D.V. Pelekhov, B. B¨uchner and P.C. Hammel, Quantitative magnetic force microscopy on permalloy dots using an iron filled carbon nanotube probe, Ultramicroscopy 111 8, 1360 (2011)

R. Adur , S. Lauback , P. Banerjee , I. Lee , V. J. Fratello , and P. C. Hammel, Anisotropy and Field-Sensing Bandwidth in Self-Biased Bismuth-Substituted Rare-Earth Iron Garnet Films: Measurement by Ferromagnetic Resonance Spectroscopy, IEEE Trans- actions on Magnetics 49, Issue 6, 2899 (2013)

C.H. Du, R. Adur, H.L. Wang, A.J. Hauser, F.Y. Yang, and P.C. Hammel, Control of Magnetocrystalline Anisotropy by Epitaxial Strain in Double Perovskite Sr2FeMoO6 Films, Phys. Rev. Lett. 110, 147204 (2013)

H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Large spin pumping from epitaxial Y3Fe5O12 thin films to Pt and W layers, Phys Rev. B. 88 100406(R) (2013)

A. J. Berger, W. Amamou, S. P. White, R. Adur, Y. Pu, R. K. Kawakami, P. C. Hammel, Magnetization dynamics of cobalt grown on graphene, J. Appl. Phys 115, 17C510 (2014)

H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Scaling of spin Hall angle in 3d, 4d and 5d metals from Y3Fe5O12/metal spin pumping, Phys. Rev. Lett. 112, 197201 (2014) vi C. S. Wolfe, V. P. Bhallamudi, H. L. Wang, C. H. Du, S. Manuilov, R. M. Teeling-Smith, A. J. Berger, R. Adur, F. Y. Yang, and P. C. Hammel, Off-resonant manipulation of spins in diamond via precessing magnetization of a proximal ferromagnet, Phys. Rev. B 89 180406(R) (2014)

H. Yu, M. Harberts, R. Adur, Y. Lu, P. C. Hammel, E. Johnston-Halperin, and A. J. Epstein, Ultra-narrow ferromagnetic resonance in organic-based thin films grown via low temperature chemical vapor deposition, Appl. Phys. Lett. 105 012407 (2014)

Fields of Study

Major Field: Physics

Studies in Experimental Condensed Matter Physics: P. Chris Hammel

vii Table of Contents

Page Abstract ...... ii Acknowledgments ...... iv Vita...... vi List of Figures ...... x List of Tables ...... xii

Chapters

1 Introduction to Ferromagnetic resonance and spin-wave relaxation 1 1.1 Introduction ...... 1 1.2 Ferromagnetic Resonance ...... 2 1.3 Spin-wave dispersion and relaxation ...... 4

2 Control of Magnetocrystalline Anisotropy and Self-Biased Ferromag- netic Resonance Frequency in Bismuth-Substituted Rare-Earth Iron Garnet Films by Varying Chemical Composition 8 2.1 Introduction ...... 8 2.2 Ferromagnetic Resonance ...... 9 2.3 Experimental Setup ...... 11 2.4 Results ...... 13

3 Control of Magnetocrystalline Anisotropy by Epitaxial Strain in Double Perovskite Sr2FeMoO6 Films 16 3.1 Introduction ...... 16 3.2 Growth and strain characterization by X-ray diffraction ...... 17 3.3 Ferromagnetic Resonance spectroscopy ...... 21

4 Dynamic Dipolar Coupling Between Exchange Biased Stripes 26 4.1 Introduction ...... 26 4.2 Experiment ...... 27

5 Dual-Frequency Ferromagnetic Resonance to Measure Spin Current Coupling in Multilayers 37 5.1 Introduction ...... 37 5.2 Single frequency FMR of YIG with and without a Py layer ...... 39

viii 5.3 Dual frequency FMR of YIG/Au/Py ...... 42

6 Ferromagnetic Resonance Imaging Across an Exchange Bias Patterned Interface 45 6.1 Introduction ...... 45 6.2 Experiment ...... 46

7 Ferromagnetic Resonance Force Microscope Design and Minimizing Spurious Backgrounds 50 7.1 Introduction ...... 50 7.2 Sub-micron probe magnet shaping and magnetometry ...... 55 7.3 Spurious cantilever coupling to microwaves ...... 55 7.4 Magnetic field control: field steps vs continuous ramp ...... 58 7.5 Measuring vibrations ...... 62

8 Damping of Confined Modes in a Ferromagnetic Thin Insulating Film: Angular Momentum Transfer Across a Nanoscale Field-defined Interface 64 8.1 Introduction ...... 64 8.2 Experiment ...... 65 8.3 Localized Mode Resonance Fields ...... 65 8.4 Size-dependent damping ...... 69 8.5 Discussion ...... 71

9 Numerical, Analytical and Micromagnetic Solutions to the Localized Mode Problem 74 9.1 Introduction ...... 74 9.2 Micromagnetic solution to localized mode precession ...... 74 9.3 Analytical solution to localized precession in a parabolic field well . . . . . 76

10 Inhomogeneity and Vibrational Broadening of Localized Modes 81 10.1 Introduction ...... 81 10.2 Frequency-independent broadening due to vibrations ...... 82 10.3 Size-Dependent Inhomogeneous Broadening ...... 83

References 87

ix List of Figures

Figure Page

1.1 Exchange spin-wave manifold for wavevectors parallel and perpendicular to the external field ...... 5 1.2 Magnon scattering processes relevant for magnon relaxation ...... 6

2.1 Coordinate system used for angular dependence...... 9 2.2 Angular variation of the FMR resonance fields of (BiLu)3(FeGaAl)5O12 .. 12 2.3 Angular variation of the FMR resonance fields of (BiLuGd)3(FeGa)5O12 .. 12 2.4 Angular variation of the FMR resonance fields of (BiTm)3(FeGa)5O12 ... 14

3.1 Semi-log θ − 2θ XRD scans of Sr2FeMoO6 films ...... 19 3.2 Thickness dependence of lattice constants of strained Sr2FeMoO6 films . . . 21 3.3 Angular dependence of FMR spectra and uniaxial anisotropy vs strain . . . 22 3.4 Magnetic hysteresis loops of 100 nm Sr2FeMoO6 on varying substrates showing change from easy-plane to easy-axis due to change in perpendicular uniaxial anisotropy ...... 24

4.1 MOKE images and hysteresis loops of modulated exchange bias sample . . 28 4.2 Hysteresis loops and angular-dependent FMR of as-deposited and ion- bombarded control samples ...... 29 4.3 Angular dependence of resonance field for striped samples showing acoustic and optic mode splitting ...... 31 4.4 Coupling between stripes due to dynamic dipolar interaction ...... 33 4.5 Out-of-plane angular dependence of resonance field for 20 micron wide striped sample ...... 35

5.1 Trilayer sample for dual-frequency FMR detection of spin current coupling . 38 5.2 FMR spectra for bare YIG sample and YIG/Au/Py sample ...... 39 5.3 Angular dependence of the YIG FMR peaks for out-of-plane rotation. . . . 40 5.4 YIG Linewidth before and after permalloy deposition ...... 41 5.5 Dual FMR resonance condition for YIG and Py ...... 42 5.6 Dual FMR result: YIG linewidth as second frequency is swept ...... 43

x 6.1 Field-position FMRFM images showing the localized modes spectra as a function of position across the interface and micromagnetic mode profiles at the interface ...... 48

7.1 CAD image of Ferromagnetic Resonance Force Microscope ...... 51 7.2 CAD zoomed Image of Ferromagnetic Resonance Force Microscope . . . . . 52 7.3 Photograph of Ferromagnetic Resonance Force Microscope ...... 53 7.4 Focused Ion Beam image of Sm1Co5 probe on cantilever and magnetometry data showing its high coercivity ...... 56 7.5 Spurious cantilever coupling to microwaves for silicon and diamond cantilevers with and without magnetic particles ...... 57 7.6 Background and signal forces when using coaxial cable standing wave resonator 59 7.7 Spurious cantilever coupling for amplitude modulation and frequency modulation when using coaxial cable standing wave resonator ...... 60 7.8 FMRFM force signal comparison between field step technique and continuous field ramp ...... 61 7.9 Vibration spectrum measured by touching cantilever to sample measured before and after fixing cantilever mounting issue ...... 63

8.1 Localized mode FMRFM spectra for thin film YIG at several probe-sample separations...... 66 8.2 Resonance field of the first four localized modes as a function of probe-sample separation at 4 GHz ...... 68 8.3 FMRFM spectra at multiple microwave frequencies at a fixed probe-sample separation of 3700 nm ...... 70 8.4 Linewidths for uniform mode and localized modes of radii R = 1600 nm, R = 1860 nm and R = 2230 nm ...... 70 8.5 Gilbert damping parameter α scales inversely with mode radius R equivalent to the surface/volume ratio of the localized mode...... 71

9.1 Micromagnetic solution to localized mode resonance field for thin film YIG 75 9.2 Mode shapes from micromagnetic simulations of localized modes in the out- of-plane geometry ...... 76 9.3 Analytic solution to localized mode resonance field for thin film YIG . 77 9.4 Theoretical localized mode radii obtained by numerical and analytic methods together with experimental mode resolution from imaging permalloy . . . . 79

10.1 Linewidths as a function of probe-sample separation and mode radius . . . 82 10.2 Frequency-independent linewidth as a function of localized mode radius . . 85

xi List of Tables

Table Page

2.1 Anisotropy constants and resulting sensing bandwidths obtained from fits to the angle-dependence of the FMR spectra of three Bi:RIG samples...... 15

3.1 Lattice parameters, lattice mismatch and sign of strain due to buffer layers and substrates used to grow epitaxial Sr2FeMoO6 films ...... 18

xii Chapter 1 Introduction to Ferromagnetic resonance and spin-wave relaxation

1.1 Introduction

The study of magnetization dynamics in ferromagnetic materials is one that has shown itself to be of both practical and academic importance. On the practical side, magnetic resonance and spin-waves have been used in microwave filters, oscillators, isolators and other microwave devices in which the manipulation of electromagnetic signals from very high frequency (VHF) to millimeter-wave is required [1]. On the academic front, magnetization dynamics has been exploited to demonstrate interesting physics such as chaos [2], soliton production [3] and room-temperature Bose-Einstein condensation [4]. Ferromagnetic Reso- nance (FMR) is a spectroscopic technique used to probe the effective fields and relaxation processes that dictate magnetization dynamics in ferromagnets. The field of spintronics has also recently ignited intense research into magnetization dynamics for both practical and academic reasons. Spintronics promises a computer paradigm based on the spin of the electron rather than the charge. A pure spin current involves a number of up-spins moving in an opposite direction to an equal number of down-spins, resulting in only the flow of angular momentum but no net flow of charge. Spintronics promises the benefits ofboth efficiency (as pure spin currents should not suffer from ohmic losses) and non-volatility (the state of the spin system is not destroyed when a device is turned off). As an important

1 step in producing viable spintronic devices, the generation of spin currents by ferromagnetic resonance has been demonstrated. This technique of spin current generation, called spin pumping, has been demonstrated when a ferromagnet is in contact with a normal metal and the ferromagnet is made to precess at its FMR frequency, generating a pure spin current in the normal metal. There are open questions about what happens at the interface between the ferromagnet and normal metal, and how the magnetization in the ferromagnet couples to electrons in the normal metal. This thesis will demonstrate the use of local FMR techniques in understanding the physics of how magnetization dynamics can be controlled and coupled between materials via interfaces, and how these interfaces affect relaxation and spin transport.

1.2 Ferromagnetic Resonance

Since the first ferromagnetic resonance experiment by Griffiths [5] the study of couplings and relaxation in magnetization dynamics of ferromagnetic materials has been intensely investigated. The equation of motion commonly used to study the time-dependence of magnetization M is the Landau-Lifshitz-Gilbert [6] equation

dM α dM = −γM × Heff + M × (1.1) dt Ms dt where γ is the gyromagnetic ratio of the electron, Heff is the total effective field experienced by the magnetization, α is the Gilbert damping parameter, Ms is the magnitude of the magnetization M. The first term on the right hand-side of 1.1 describes precession of M about Heff . The second term describes relaxation of M back towards Heff . The net result is that if the magnetization M is kicked away from equilibrium it will follow a spiral path until it is parallel with the effective field Heff , which is also the equilibrium direction as dM dt = 0 when M ∥ Heff . Equation 1.1 is an inherently non-linear equation, not just because dM it has dt on both sides of the equation, but also because the effective field Heff is strongly dependent on the orientation of M due to dipolar fields. This allows for the interesting nonlinear effects that have been studied in the literature [2], however for the purposes of

2 this thesis we shall mainly consider the linear solutions to this equation.

For a thin film ferromagnet, there are two magnetization directions for which thefer- romagnetic resonance condition is easily calculated: magnetization pointing normal to the film plane and magnetization in the film plane. Ignoring magnetocrystalline anisotropies, the dominant effective field is the dipole field, or demagnetizing field. This fieldisthe energy cost of magnetic moments polarized normal to the film plane having dipolar fields opposite to neighboring moments, equivalent to producing magnetic charges at the film surfaces and so it manifests as a shape anisotropy field that tries to keep the magnetization in the plane of the film. Magnetic flux density must be continuous normal to the surface of the film and in Gaussian units the magnetic flux density for magnetization oriented normal to the film plane is B = H = Heff + 4πMs, where H is the external applied field. Hence the effective field has the form Heff = H − 4πMs. Using Eqn. 1.1 with α = 0, the resonance condition is simple to compute for external field and magnetization out of plane, as

Heff = (H − 4πMs)ˆz and the magnetization with a small circular precessing component is

−iωt −iωt M = Mszˆ + me xˆ + ime yˆ where m ≪ Ms. The resulting resonance condition for out-of-plane magnetization is ω = H − 4πM (1.2) γ s For in-plane magnetization it is important to consider the dipole field from the precessing magnetization, which results in an effective field with a time-dependent component Heff =

−iωt Hzˆ−4πmxe xˆ wherez ˆ is the direction of the external field andx ˆ is the direction normal

−iωt −iωt to the film plane. Together with the magnetization M = Mszˆ + mxe xˆ + imye yˆ this results in the resonance condition for in-plane magnetization

ω 2 = H(H + 4πM ) (1.3) γ s

The resonance condition can also be computed numerically for other angles and anisotropy fields, and this is covered in section 2.2.

The ac susceptibility χxx = mx/hx can be calculated for out-of-plane resonance by

−iωt substituting a time-dependent ac magnetic field hxe into the effective field Heff = (H −

3 −iωt 4πMs)ˆz + hxe xˆ together with Eqn. 1.1, this time with non-zero damping parameter α to obtain the complex susceptibility

γMs(ω0 + iαω) χxx = 2 2 (1.4) (ω0 + iαω) − ω where ω0 = H − 4πMs. In a conventional FMR experiment the imaginary part of the susceptibility is determined by measuring the absorbed power as a function of external applied H at fixed ac frequency ω. The imaginary susceptibility has the form

2 2 ′′ −γMsαω0(ω0 + ω ) χxx = 2 2 2 (1.5) (ω0 − ω ) − 2αωω0 which results in a Lorentzian lineshape with full-width at half maximum of 2αω/γ. There- fore, it is possible to directly obtain the Gilbert damping parameter by measuring the linewidth of the FMR absorption peak at multiple microwave frequencies ω.

1.3 Spin-wave dispersion and relaxation

Thus far we have only considered the case where the entire magnetization of a sample is uniformly precessing in phase, the so-called uniform precession mode. This uniform mode is the expected lowest energy magnetization excitation and is often the strongest resonance absorption line due to the use of uniform microwave fields that couple most strongly tothe uniform mode. In addition, there is an energy cost to neighboring spins having different precession phase due to the exchange energy that tries to keep spins in lock step, as well as dipole fields that can modify the effective field produced by nonuniform precession.

Nonuniform magnetization precession as plane waves has been widely theorized for several cases, such as exchange-dominated spin waves [7], dipole-exchange spin waves [8] and thin films magnetized normally to the plane [9,10].

The simplest spin-wave dispersion is that for exchange-dominated spin waves calculated by Herring and Kittel [7] for an infinite sample

2 2 2 2 ω = ωH + aexωMk ωH + aexωMk + ωM sin θ (1.6)

θDDπ

θDD D D

DkDD Figure 1.1: Exchange spin-wave manifold for wavevectors parallel and perpendicular to the −12 2 external field calculated for a YIG sphere with4πMs=1600 Oe, aex = 3.64 × 10 cm in external field H=3000 Oe. Relaxation of the uniform mode at k = 0 to degenerate magnons with k > 0 is responsible for linewidth broadening by the two-magnon scattering process.

2 where ωH = γH, ωM = γ4πMs, aex is the exchange constant in cm , k is the magnitude of the wavevector and θ is the angle between the wavevector and external field H. The

resulting spin-wave manifold is shown for the extremes of θ = 0 and θ = π/2 in Fig.

1.1. Intermediate values of θ produce magnons in between these curves. Hence, there is

a band of magnons that are degenerate with the uniform mode at k = 0 and relaxation

by scattering from the uniform mode to the degenerate magnons is known as two-magnon

relaxation [11, 12], which has been shown to contribute to the relaxation of the uniform

mode of a thin film when the magnetization is oriented in-plane [13].

The more relevant analysis for our case of a normally magnetized film investigated in

chapters 8-10 is the Kalinikos-Slavin dispersion relation [10]

2 2 2 ω = ωH + aexωMk ωH + aexωMk + f(kt)ωM (1.7)

where f(kt) = 1 − [1 − e−kt]/kt is the effective dipolar field of the spin wave and t is the

thickness of the film. In this case, the spin-wave manifold is independent of wavevector

Figure 1.2: Magnon scattering processes relevant for magnon relaxation. a) Kasuya-Lecraw magnon-phonon scattering: A uniform mode magnon with ku = 0 can relax by scattering with a lattice phonon to produce a final magnon with k > 0. b) Three magnon confluence: A magnon with non-zero wavevector can combine with a thermal magnon to produce a final magnon at high k. c) Four magnon scattering: Initial magnon scatters with a thermal magnon, producing two final magnons.

direction, as expected from azimuthal symmetry, and so there is no band of magnons to scatter into. Hence the two-magnon scattering process would not be expected to contribute to the linewidth when magnetization is oriented normal to the film plane.

Relaxation processes involving three bosons (magnons or phonons) can, however, be important for the geometry of magnetization oriented normal to the film plane and for the size-dependent linewidth broadening investigated in chapter 8. The Kasuya-Lecraw mechanism is a magnon-phonon scattering process that allows the k = 0 uniform mode to relax to a high k magnon with wavevector conservation enabled by a phonon [14, 15], as shown in Fig. 1.2a. This linewidth mechanism is linear in frequency as it depends on the magnon-phonon dispersion crossing, and so agrees with the phenomenological Gilbert damping formalism. In addition, high k magnons can relax by combining with thermal magnons to produce a single final magnon as shown in Fig. 1.2b. This three-magnon confluence process has been measured in parallel-pumping experiments [14] and is found to be linear in k but independent of frequency, which agrees with the theory as the magnon band is unchanged for different fields. Four magnon processes, as shown inFig. 1.2c have also been theorized [16] and measured [17] as a form of magnon relaxation. Four magnon scattering has a k2 or k3 dependence (depending on temperature), and so can be a dominant

6 relaxation mechanism for high k magnons. This contribution is also mentioned in chapter

8, but is ruled out from our experiment due to its nonlinear k-dependence.

7 Chapter 2 Control of Magnetocrystalline Anisotropy and Self-Biased Ferromagnetic Resonance Frequency in Bismuth-Substituted Rare-Earth Iron Garnet Films by Varying Chemical Composition

2.1 Introduction

The magneto-optic sensitivity of iron garnets makes them attractive candidates for magnetic field sensing [1, 18, 19]. This interest has recently stimulated ferromagnetic resonance

(FMR) and Faraday rotation studies of Bi-Fe garnets to explore their potential for high frequency sensing applications [20]. In addition, bismuth-substituted rare-earth iron garnets

(R3−x Bix )Fe5O12 (Bi:RIGs) exhibit giant magneto-optical responses and are transparent in the near infrared wavelengths of telecom interest. While domain wall rotation is often used in magneto-optic applications, the frequency limit in this scheme is due to domain wall resonance which is on the order of MHz. Alternatively, the frequency response of magnetization rotation is limited only by the ferromagnetic resonance frequency and so the upper frequency limit for ac field detection, also known as the bandwidth, can reach GHz frequencies [21]. An external DC biasing field can be used to increase the FMR frequency

8 Figure 2.1: Coordinate system used for angular dependence. In-plane rotation involved varying φH. Out-of-plane rotation involves varying θH

and hence tune the bandwidth [1]. However, Bi:RIG materials with internal anisotropy fields are attractive because they achieve high resonance frequencies without the needfor an applied field [22]. Substrate-induced elastic strain results in anisotropies in Bi:RIG films when grown on gadolinium gallium garnet (GGG) substrates [23], but the effect of this interfacial anisotropy is reduced in thicker films.

Here we report on another approach to increasing the anisotropy fields that involves changing the composition of the Bi:RIG film [1]. We have measured the angular dependence of the room temperature FMR spectra of Bi:RIG films of differing composition to determine the uniaxial and cubic crystal anisotropies, and from these measurements we estimate the upper limit to the frequency response of these films in zero applied field. We report materials that should enable detection bandwidths as large as 4 GHz.

2.2 Ferromagnetic Resonance

Ferromagnetic resonance allows one to measure the magnetic free energy of a material with spectroscopic precision. FMR is typically performed in an applied field sufficient to saturate the sample, thus ensuring a single domain magnetic state. In particular this technique allows precise measurement of the interaction of the sample magnetization with the crystal lattice and how this varies with relative orientation of magnetization and lattice vectors, resulting in effective anisotropy fields. Though these experiments were measured by sweeping an external field, the anisotropy fields and magnetization remanence inthese

9 materials are sufficiently large that they remain in a single domain state even in zero-field.

The zero-field resonance frequency is dictated by the internal anisotropy fields intheso- called self-bias regime [1,24], and henceforth this frequency is referred to as the bandwidth since it is the upper frequency limit for field-sensing applications. In a ferro- or ferrimagnet the strong exchange coupling between spins allows their collective behavior to be considered as a single magnetization vector when assumed to be in this single-domain state. When perturbed from equilibrium by an AC field the magnetization vector precesses about its equilibrium direction. The equilibrium direction and resonance condition for a ferromagnet can be derived from the free energy E of the system; here we include the cubic and uniaxial anisotropy terms, assuming tetragonal crystal structure [25]

M H E = −H · M + 4πM cos2 θ − 4⊥ cos4 θ (2.1) 2 eff 2 H π − 4∥ (3 + cos 4φ) sin4 θ − H sin2 θ sin2 φ − 8 2∥ 4 where we use the coordinate system illustrated in Fig. 2.1: the angle θ denotes the out-of- plane magnetization angle measured from the normal to the sample plane and φ describes

the in-plane magnetization angle measured from the easy axis. The first term in the free en-

ergy is the Zeeman energy of the magnetization M in an external field H. The second term

4πMeff = 4πMs − H2⊥ includes both the demagnetizing energy and perpendicular uniaxial anisotropy grouped together in an effective uniaxial anisotropy term, as they are indistinguishable in FMR due to their identical angular dependencies. H4⊥ and H4∥ describe the perpendicular cubic and parallel cubic anisotropies respectively. H2∥ is the in-plane uniaxial anisotropy constant. The equilibrium angle occurs where the free energy is minimum.

Determining the anisotropy fields from the FMR spectra requires fitting the angular dependence of the resonance frequency. We perform this calculation in the regime of small deviations from equilibrium where the resonance frequency ωres is given by [26]

γ 2 1 ω = (E E − E ) 2 . (2.2) res M sin θ θθ φφ θφ

Here γ = gµB/~ is the gyromagnetic ratio and Eij is the second derivative of the free energy

10 E with respect to the angles i, j. The resonance condition occurs along the equilibrium direction. By observing the changing resonance spectrum as the thin film sample is rotated it is possible to measure the anisotropies present in the film. In practice, an FMR spectrum is obtained at fixed frequency while an external magnetic field is swept through resonance.

When modeling the resonance field an additional complication arises as prior knowledge of the external field is required to calculate the equilibrium direction. This is especially important when the out-of-plane angle θ is varied, as the demagnetizing field can drastically

affect the equilibrium angle. In this study, we have used an iterative numerical procedure to

minimize the free energy to calculate the equilibrium angle and hence the correct resonance

field [27].

2.3 Experimental Setup

Bi:RIG films of composition (BiLu)3(FeGaAl)5O12 (sample 1), (BiLuGd)3(FeGa)5O12 (sam-

ple 2) and (BiTm)3(FeGa)5O12 (sample 3) were grown to a thickness of 3µm on GGG substrates by Liquid Phase Epitaxy. This technique has previously been used to grow magnetic

bubble memory films and thick film (500 micron) Faraday rotators [28]. Samples 1 and 2

were grown on (100) substrates, while sample 3 was grown on a (111) substrate. While

the garnets have cubic anisotropy, the result of growth-induced uniaxial anisotropy renders

them effectively tetragonal, and so the free energy given by(2.1) is applicable. FMR spec-

tra were obtained using a Bruker EMXPlus EPR spectrometer at a frequency of 9.8GHz.

The external field at resonance was determined by fitting Lorentzian peaks to the spectra.

The resonance field was recorded as the sample orientation was varied with respect tothe

external field direction for two configurations. The first configuration, denoted in-plane

rotation, involved fixing θH = π/2 while rotating φH keeping the external field in the plane of the thin film. The second configuration, denoted out-of-plane rotation (OOP), involved

fixing φH = 0 and changing the angle between the film plane and the external field θH. It is useful to rotate the axes for modeling out-of-plane angles to avoid the singularity in (2.2)

at θ = 0.

11 4500 6 1.6x10 3240 (b) Data (a) 1.2 Lorentzian 0.8

0.4 3200 Abs. (a.u) 4000 0.0 2800 3000 3200 3400 External Field (Oe) 3160 3500 3120 Resonance Field (Oe) Field Resonance Resonance Field (Oe)

3080 3000 0 45 90 135 180 225 270 315 360 −90 −60 −30 0 30 60 90 Angle (deg) Angle (deg)

Figure 2.2: Angular variation of the FMR resonance fields of (BiLu)3(FeGaAl)5O12 a) In- plane rotation of φH and b) Out-of-plane rotation of θH, where θH = 0 denotes normal to the film plane. Fits (solid line) to the data (dots) were used to extract the anisotropy parameters given in Table 2.1. Inset: FMR spectrum at φH = 0, θH = 90 with Lorentzian fit

6000 5 4.0x10 Data 2600 (b) Lorentzian (a) 3.0 2.0 5000 Abs (a.u) 1.0 2500 0.0 1800 2000 2200 2400 2600 External Field (Oe)

2400 4000

2300 3000 Resonance Field (Oe) Field Resonance Resonance Field (Oe)

2200 2000 0 45 90 135 180 225 270 315 360 −90 −60 −30 0 30 60 90 Angle (deg) Angle (deg)

Figure 2.3: Angular variation of the FMR resonance fields of (BiLuGd)3(FeGa)5O12 as in Fig.2.2. Fit parameters are given in Table 2.1. Inset: FMR spectrum at φH = 0, θH = 90 with Lorentzian fit

12 2.4 Results

We have measured FMR spectra for three Bi:RIG samples and plotted the position of the most prominent peaks as a function of angle [29], as shown in Figs 2.2-2.4. The angular dependencies of samples 1 and 2 shown Fig. 2.2a and Fig. 2.3a reveal a four-fold symmetry

in the sample plane demonstrating cubic anisotropies. The amplitude of the oscillations as

the applied field rotates between easy and hard axes every 45◦ is determined by the in-plane

cubic anisotropy constant H4∥ while the average resonance field is dictated by the effective

uniaxial anisotropy field 4πMeff .

Sample 3 was a (BiTm)3(FeGa)5O12 film grown on a (111) substrate and was expected to display sixfold symmetry. This was not observed, presumably due to a slight misorien-

tation of the substrate surface with respect to the crystal plane [30], which can destroy the

sixfold symmetry and the effect of second-order anisotropies. Therefore, for our purposes,

it is sufficient to consider sample 3 as a (100) film, where we include an in-plane uniax-

ial anisotropy H2∥ term to obtain good fits. This sample displayed very large linewidths, which caused a large systematic error in the peak positions. Increased linewidth of Tm

iron garnets compared to Lu iron garnets is well known [31]. Nevertheless, for all the data,

good fits were obtained to (2.2) using our numerical fitting method. We obtained values for

the anisotropy constants and g-factors, and these are given in Table 2.1. Vibrating Sample

Magnetometry (VSM) measurements were also used to estimate 4πMeff from the hard axis hysteresis loop saturation fields [32], and the magnetometry measurements are comparable

with those obtained by FMR spectroscopy. It is important to note that the effective uniax-

ial anisotropy field 4πMeff = 4πMs − H2⊥ can be negative if the uniaxial anisotropy H2⊥

is larger than the demagnetizing field 4πMs, causing the easy axis to be normal to the film plane.

The condition for in-plane resonance [29] of a film with cubic anisotropy is found from

(2.2) to be

13 4890 (a) 6000 6 6x10 Data 4870 Lorentzian (b) 5 4 4850 3 5000 2

Abs. (a.u) Abs. 1 4830 0 2000 4000 6000 4810 4000 External Field (Oe) 4790 3000

Resonance Field (Oe) 4770 Resonance Field (Oe) Field Resonance 4750 2000 0 45 90 135 180 225 270 315 360 −90 −60 −30 0 30 60 90 Angle (deg) Angle (deg)

Figure 2.4: Angular variation of the FMR resonance fields of (BiTm)3(FeGa)5O12 as in Fig.2.2. Fit parameters are given in Table 2.1. Inset: FMR spectrum at φH = 0, θH = 90 with Lorentzian fit

ω 2 res = [H + H cos(4φ)] γ 4∥ 3 + cos(4φ) × H + 4πM + H (2.3) eff 4∥ 4

We can use this equation to fit the in-plane data to obtain 4πMeff and H4∥, and the values so-obtained agree well with results of the numerical method described above. This equation is also useful for estimating the bandwidth of these materials for detecting ac fields applied perpendicular to the easy axis if we assume the bandwidth is equal to the FMR frequency when H = 0 [21]. This is the case for samples 1 and 2 where θ = π/2, φ = 0 points along the easy axis. For materials in which the perpendicular uniaxial anisotropy H2⊥ is sufficiently large that the easy axis is normal to the film plane (such as forsample3)we

use the resonance condition for increasing angles normal to the plane for a sample with

perpendicular cubic anisotropy [29] obtained from (2.2) and set θH = θ = 0 to get the out-of-plane resonance condition:

ω 2 res = (H − 4πM )(H + H − 4πM ) (2.4) γ eff 4⊥ eff with H = 0 and the obtained anisotropy values to get the zero-field FMR frequency, which is equivalent to the magneto-optic sensing bandwidth. The resulting bandwidths for all three materials are given in Table 2.1.

In conclusion, we have determined the anisotropy fields of Bi:RIG films from the angular

14 Table 2.1: Anisotropy constants and resulting sensing bandwidths obtained from fits to the angle-dependence of the FMR spectra of three Bi:RIG samples.

4πMeff H4∥ H2∥ H4⊥ g Bandwidth No. Composition (Oe) (Oe) (Oe) (Oe) (unitless) (GHz) 1 (BiLu)3(FeGaAl)5O12 587 ± 12 94 ± 10 - −258 ± 17 2.021 ± 0.002 0.7 2 (BiLuGd)3(FeGa)5O12 2292 ± 30 244 ± 17 - −490 ± 240 2.05 ± 0.03 2.2 3 (BiTm)3(FeGa)5O12 −2100 ± 20 57 ± 15 42 ± 13 −1000 ± 100 1.94 ± 0.02 4.1

variation of their FMR spectra and used these data to estimate the FMR frequency in zero

field, which is equal to the self-biased bandwidth for magnetic field-sensing by magnetiza-

tion rotation. We find that the perpendicular and planar anisotropies are large enoughto

suggest bandwidths on the order of GHz for field-sensing applications. Recently, a high-

frequency magneto-optic sensor using (BiLuGd)3(FeGa)5O12 has been demonstrated [33] with a bandwidth of 2 GHz in zero applied field, equal to our estimated bandwidth. This

confirms that the bandwidth of a magneto-optic sensor in the magnetization rotation regime

is limited by the FMR frequency, and in turn the FMR frequency in zero applied field is

dictated by the crystal anisotropy fields that act to self-bias the film.

15 Chapter 3 Control of Magnetocrystalline Anisotropy by Epitaxial Strain in Double Perovskite Sr2FeMoO6 Films

3.1 Introduction

Magnetization orientation is widely used for information processing and storage applications [34]. There is widespread interest in discovering methods that could grant efficient, local control of magnetization. Ferro- or ferri- magnets (FM) with strong magnetocrystalline anisotropy, in which spin-orbit interactions couple magnetization to crystalline structure, offer a promising route to achieving this capability. High-quality epitaxial FM films arede- sirable for technological applications using strain-tunable magnetocrystalline anisotropy to control magnetization orientation. Half-metallic double perovskite Sr2FeMoO6 (SFMO) is an attractive material [35–43] for the study and application of magnetocrystalline anisotropy due to its high spin polarization, high Curie temperature TC (well above room temperature), and relatively strong spin-orbit coupling provided by the 4d transition metal Mo.

We have succeeded in growing high-quality Sr2FeMoO6 epitaxial films [42] that, in addition to enabling detailed studies, provide a robust platform for technological applications.

The fabrication of Sr2FeMoO6 epitaxial films [39–43] has been extensively studied. However, their chemical complexity and strict reduction/oxidation requirements make it very diffi-

cult to deposit Sr2FeMoO6 films of sufficient quality to reveal intrinsic properties [43–45].

16 Moreover, almost all reported Sr2FeMoO6 film fabrication has employed a single substrate,

SrTiO3 (STO), for which the lattice mismatch η = (as − af )/af with Sr2FeMoO6 is -1.1%,

where as and af are the in-plane lattice constants of the substrate and unstrained films,

respectively. Given the relatively strong spin-orbit coupling in Sr2FeMoO6, one may expect magnetocrystalline anisotropy to be sensitive to structural distortion which can be induced

and controlled by epitaxial strain. In particular, this can enable the large strain-induced

out-of-plane magnetocrystalline anisotropy needed to produce perpendicularly magnetized

FM films that can serve as perpendicular polarizers for a spin-transfer-torque devicewith

low critical current [46] for nonvolatile memory [47] and spin-torque oscillator devices [48].

Shape anisotropy dictates in-plane magnetization for most FM films, while only a hand-

ful of films and heterostructures exhibit out-of-plane anisotropy. Continuous strain con-

trol of magnetocrystalline anisotropy promises to expand the family of such films avail-

able for strain-tuned magnetization as well as perpendicular polarizers. Magnetocrystalline

anisotropy has been investigated in thin films of metals, dilute magnetic semiconductors

and La0.7Sr0.3MnO3 [49–53]. Tunable magnetic anisotropy was observed below 100 K in GaMnAs by epitaxial strain [51] and in GaMnAsP by varying the phosphorous content [52].

However, the correlation between magnetic anisotropy and strain-induced lattice distor-

tion has not been systematically studied. Here we report growth of Sr2FeMoO6 epitaxial films on a selected set of single-crystal substrates and buffer layers to create arangeof

strain-induced tetragonal distortion in Sr2FeMoO6 films and measurements of their out- of-plane magnetocrystalline anisotropy by ferromagnetic resonance (FMR). We find that

sufficiently strained Sr2FeMoO6 films grown on2 Sr GaTaO6 buffer layers can possess a large enough strain-induced anisotropy to overcome demagnetizing fields such that they display

out-of-plane easy-axis behavior.

3.2 Growth and strain characterization by X-ray diffraction

Epitaxial Sr2FeMoO6 films with thicknesses t varying from 50 to 200 nm were grown using a new sputtering technique [42,54,55]; this range of thicknesses spans from fully strained (50

17 Buffer Layer/Substrate a(= b)(A)˚ c(A)˚ Lattice mismatch Sign of strain SrTiO3 3.905 3.905 -1.1% Compressive Sr2GaTaO6/SrTiO3 3.970 3.970 0.58% Tensile Sr2CrNbO6/LSAT 3.950 3.944 0.076% Tensile Sr2CrNbO6/SrTiO3 3.905 3.988 -1.1% Compressive

Table 3.1: Lattice parameters, lattice mismatch and sign of strain due to buffer layers and substrates used to grow epitaxial Sr2FeMoO6 (bulk a = c = 3.947 A)˚ films

nm) to nearly relaxed (200 nm) films. For Sr2FeMoO6 films below 50 nm, the average film quality decreases because it takes several nm to fully establish the Fe/Mo ordering during early stages of film growth, resulting in an Fe/Mo disordered layer several nm thick nearthe interface. Four different (001)-oriented substrates or buffer layers have been used toenable tuning of the lattice mismatch with Sr2FeMoO6 (bulk a = 3.947 A):˚ 1) SrTiO3; 2) a fully strained Sr2CrNbO6 (SCNO) buffer layer on SrTiO3; 3) a Sr2GaTaO6 (SGTO) buffer layer on SrTiO3; and 4) a fully relaxed Sr2CrNbO6 buffer layer on (LaAlO3)0.3(Sr2AlTaO6)0.7 (LSAT). The lattice constants of the four substrates or buffer layers were measured by

high resolution x-ray diffraction (XRD); the lattice mismatch and the sign of thestrain

with respect to Sr2FeMoO6 are shown in Table ??. All buffer layers are around 100 nm thick. The magnetizations of the films were measured using a superconducting quantum

interference device (SQUID) magnetometer and a vibrating sample magnetometer (VSM).

XRD clearly demonstrates the importance of substrate/buffer layer on the crystalline

quality and strain relaxation of the Sr2FeMoO6 films. Figure 3.1 shows representative θ-2θ

scans of the Sr2FeMoO6 films with t = 100 and 200 nm grown on SrTiO3, Sr2CrNbO6/LSAT,

and Sr2GaTaO6/SrTiO3. For Sr2FeMoO6 films grown on SrTiO3 (Fig. 3.1a), the relatively large lattice mismatch (η = -1.1%, compressive) elongates the out-of-plane lattice constant

c, resulting in a tetragonal distortion. The in-plane lattice constants a(= b) were measured

using off-axis XRD scans on the2 Sr FeMoO6 (022) peaks. The tetragonal distortion (tetragonality σ = (c − a)/a) decreases from 1.7% for the 100-nm thick film (a = 3.915 A,˚ c =

3.981 A)˚ to 0.41% for the 200-nm film (a = 3.941 A,˚ c = 3.957 A).˚

The Sr2CrNbO6 buffer layers grown on LSAT (a = 3.868 A)˚ are fully relaxed for all

18 FWHM=0.1285° SFMO/STO (a) 6 100nm 10 200nm

SFMO(004)

4 STO(002) 10 22.6 22.8 23.0 w 2 10

0 10 FWHM=0.0153° SFMO/SCNO/LSAT (b) 6 10

4 SCNO(004) SFMO(004)

10 LSAT(004) 22.8 23.0 23.2 w 2 10 Intensity(c/s) 0 10 SFMO/SGTO (c) FWHM=0.0570° 6 10

SGTO(004) SFMO(004) STO(002) 4 10

22.8 23.0 23.2 w 2 10

0 10 45.0 45.5 46.0 46.5 47.0 2Q (degree)

Figure 3.1: Semi-log θ −2θ XRD scans of Sr2FeMoO6 films with thicknesses of 100 nm (red) and 200 nm (blue) grown on (a) SrTiO3, (b) Sr2CrNbO6/LSAT, and (c) Sr2GaTaO6. The ◦ insets give the FWHM of rocking curves for the 100-nm Sr2FeMoO6 films: 0.129 (SrTiO3), ◦ ◦ 0.015 (Sr2CrNbO6/LSAT), and 0.057 (Sr2GaTaO6).

19 thicknesses (Fig. ) due to the larger lattice mismatch between bulk Sr2CrNbO6 (a =

3.944 A)˚ and LSAT. Because the in-plane lattice constants of Sr2CrNbO6 and Sr2FeMoO6

are nearly identical, the Sr2FeMoO6 films on2 Sr CrNbO6/LSAT are essentially strain-free. These samples serve as the “origin” in the strain tuning of the magnetoelastic coupling. The

Sr2FeMoO6 films grown on2 Sr GaTaO6 buffer layers (a = c = 3.970 A,˚ η = +0.58%) on

SrTiO3 (Fig. 3.1c) show a small tetragonal distortion with an expanded a = 3.963 Aand˚ a compressed c = 3.935 Afor˚ the 200 nm thick film. The satellite peaks in Figs. 3.1b and 3.1c

are XRD Laue oscillations which indicate highly uniform films having smooth surfaces and

sharp interfaces. The Sr2CrNbO6 buffer layers on SrTiO3 (not shown) were fully strained

with a = 3.905 A(same˚ as SrTiO3) and c = 3.988 A.˚ Consequently, the lattice constants of

Sr2FeMoO6 films grown on2 Sr CrNbO6/SrTiO3 are almost identical to those grown directly

on SrTiO3. Strain also plays a dominant role in determining the crystalline quality and uniformity

of the Sr2FeMoO6 films as shown by the full-width-at-half-maximum (FWHM) of theXRD

rocking curves in the insets to Fig. 3.1 for the Sr2FeMoO6 (004) peaks. The FWHM

◦ ◦ ◦ decreases from 0.129 to 0.057 and 0.015 s for Sr2FeMoO6 films on SrTiO3, Sr2GaTaO6,

and Sr2CrNbO6/LSAT with lattice mismatch of -1.1%, 0.58%, and 0.076%, respectively.

Clearly, the nearly perfect lattice matching between Sr2CrNbO6/LSAT and Sr2FeMoO6 enables the highest crystalline quality.

Figure 3.2 shows the lattice constants a and c for all the Sr2FeMoO6 films grown on the four substrates/buffer layers for 50 ≤ t ≤ 200 nm. For Sr2FeMoO6 films on SrTiO3 and fully strained Sr2CrNbO6/SrTiO3, the lattice constants show a clear strain relaxation as t increases with a quasi-linear dependence, while the Sr2FeMoO6 lattice evolves from tetragonal (c > a) to cubic. One notes that a and c change in opposite directions during strain relaxation in order to minimize volume change of the Sr2FeMoO6 lattice. The Sr2FeMoO6 films at 50 nm are almost fully strained with a = 3.908 Aand˚ become fully relaxed at 200 nm with a = 3.944 A.˚ For Sr2FeMoO6 films on2 Sr CrNbO6/LSAT, because of the near perfect lattice match, the Sr2FeMoO6 lattice is essentially cubic and strain free for all thicknesses. The Sr2FeMoO6 films on2 Sr GaTaO6/SrTiO3 show a slight strain relaxation with

20 4.00 4.00

3.98 3.98 c parameter (Å)

3.96 3.96

3.947 Å 3.94 3.94 parameter (Å) a parameter

3.92 SFMO/STO (-1.1%) 3.92 SFMO/SGTO (0.6%) SFMO/SCNO/LSAT (0.08%) SFMO/SCNO/STO (-1.1%) 3.90 3.90 0 50 100 150 200 0 50 100 150 200 Thickness (nm) Figure 3.2: Thickness dependence of lattice constants of (a) the in-plane lattice constant a and (b) out-of-plane lattice constant c of the Sr2FeMoO6 films grown on SrTiO3, Sr2CrNbO6/SrTiO3, Sr2CrNbO6/LSAT and Sr2GaTaO6.

an expanded in-plane lattice constants (c < a). At t = 50 nm, the Sr2FeMoO6 film is fully

strained to Sr2GaTaO6 with a = 3.970 A.˚

3.3 Ferromagnetic Resonance spectroscopy

Ferromagnetic resonance (FMR) spectroscopy was used to measure the magnetic anisotropy of our Sr2FeMoO6 films at 9.60 GHz with 0.2 mW input microwave power at various angles

θH between the dc magnetic field H and the film normal (inset to Fig. 3.3a). Fig. 3.3a shows

four representative FMR spectra from a 50 nm Sr2FeMoO6 film on2 Sr CrNbO6/LSAT at θH = 0◦, 30◦, 50◦, and 90◦. The resonance field is defined as the field where the derivative ofthe

FMR absorption crosses zero. Figure 3.3b shows the angular dependence of the resonance

◦ ◦ field from out-of-plane (θH = 0 ) to in-plane (θH = 90 ) for the 50-nm Sr2FeMoO6 films on four different substrates or buffer layers, which evolves systematically as the lattice

mismatch varies from η = -1.1% to +0.58%. One notes that for Sr2FeMoO6/Sr2GaTaO6,

◦ the resonant field is maximum at θH = 90 (in-plane), indicating that the magnetic easy axis is out-of-plane in spite of magnetic shape anisotropy. The angular dependence of resonance

21 %GGGG % cS cS GGG θOOmGO %G θOOGO

θOOGO GGG OLL θOOGO GGG OL G OLL OL GGG N GGG N%G GGG N% OOcS

OOcTS mGGG NyG yGGG Ny%G %GGG %GGG yGGG mGGG GGG GGG GGG G mG G G OOcS OcS

yGGG OcS OcS OcS y

%GGG E %

G Oc%G S y ⊥ OL N%GGG

G m NyGGG S

OO H NmGGG N%

NGGG Ny N% G % N% G % y N% G % y (c-a)/a η/tOc%GNyO8LS (c-a)/aOc8S Oc8S

Figure 3.3: (a) Room-temperature FMR derivative spectra for a 50 nm Sr2FeMoO6 film on ◦ ◦ ◦ ◦ Sr2CrNbO6/LSAT at θH = 0 , 30 , 50 , and 90 (Inset: coordinate system used for FMR measurement and analysis) (b) Angular dependence (θH ) of the resonance fields for the 50 nm Sr2FeMoO6 films grown on SrTiO3, Sr2CrNbO6/SrTiO3, Sr2CrNbO6/LSAT, and Sr2GaTaO6. The fitting (solid curves) was performed using Eqs. (1) and (2) to obtain 4πMeff , from which H2⊥ was determined for each film. (c) H2⊥ of all the Sr2FeMoO6 films as a function of η/t (mismatch/thickness). The solid line is the least-squares fit to all the data points excluding the two on the very left and the one on the far right (t = 50 nm for all 3 points). (d) H2⊥ vs. tetragonality (c–a)/a of the Sr2FeMoO6 films. All the experimental data fall nicely onto a straight line, indicating strain-induced magnetocrystalline anisotropy. (e) Anisotropy energy Eani as a function of tetragonality.

22 field can be quantitatively characterized once again by fitting Eqns 2.1 and 2.2 to obtain magnetic anisotropy parameters. Fig. 3.3b shows that the fitting curves agree with the experimental data very well and that 4πMeff is obviously different for2 Sr FeMoO6 films

grown on the four substrates/buffer layers, while the other anisotropy terms H4∥, H2∥, and

H4⊥ are substrate-independent and comparatively small. To illustrate the effect of strain

on magnetic anisotropy, Fig. 3.3c shows the linear dependence of H2⊥ (calculated from

4πMeff and Ms) on lattice mismatch η and the inverse of film thickness t, reflecting the

evolution of H2⊥ as strain relaxes (depending on both η and t). Most of the experimental

data fall on a line except for the three points for the 50 nm Sr2FeMoO6 films on SrTiO3,

Sr2CrNbO6/SrTiO3, and Sr2GaTaO6 due to that at 50 nm, the Sr2FeMoO6 films are below the critical thickness and fully strained as shown in Fig. 3.2.

This sensitivity of magnetocrystalline anisotropy to lattice symmetry highlights a key re-

sult of our study: the striking proportionality of H2⊥ to the tetragonality of the Sr2FeMoO6

lattice over a broad range (-0.93% < (c − a)/a < +2.0%), H2⊥ = (32 ± 60) − (162.7 ± 5.9) × 103 × [(c − a)/a] (Oe) as shown in Fig. 3.3d, demonstrating a fundamental relationship between magnetocrystalline anisotropy and lattice symmetry.

The strain-induced anisotropy arises from the magnetoelastic effect [56, 57] in which a change in interatomic distances alters the magnetic properties through spin-orbit coupling.

The magnetoelastic energy density is given by F = −σb cos2 α, where b is the magnetoelastic constant, σ the tetragonality (c − a)/a, and α the angle between M and strain direction.

When M is along the [001] direction, F = −σb. Fig. 3.3e shows the linear dependence of anisotropy energy, 1 E = − MH (3.1) ani 2 2⊥ on tetragonality for all Sr2FeMoO6 films, from which a least squares fit gives

3 5 3 Eani = (−5.0 ± 4.6) × 10 + (92.9 ± 4.5) × 10 × [(c − a)/a] (erg/cm ) (3.2)

The slope of the line in Fig. 3.3e gives –b = (92.9 ± 4.5) × 105erg/cm3. The negative value of b implies that the magnetic easy axis is parallel to the short axis of the tetragonal lattice.

Figure 3.4: In-plane (red) and out-of-plane (blue) magnetic hysteresis loops of 100 nm Sr2FeMoO6 films on (a) SrTiO3, (b) Sr2CrNbO6/LSAT, and (c) Sr2GaTaO6. The evolution of the hysteresis loops reflects the variation of4πMeff from positive in (a), showing easy- plane behavior, to negative in (c), which displays easy-axis behavior.

The ability to tune the magnetization using both thickness and substrates brings in precise

control in fabricating a spintronic device.

The variation of H2⊥ with tetragonal distortion of Sr2FeMoO6 results in evolution of the

total magnetic anisotropy and, most interestingly, perpendicular anisotropy in Sr2FeMoO6

films on Sr2GaTaO6/SrTiO3 by overcoming the demagnetizing field (see the magnetic hys-

teresis loops in Fig. 3.4). The saturation field (Hs) of the out-of-plane hysteresis loops

depends on 4πMeff = 4πMs − H2⊥. For a 100-nm Sr2FeMoO6 film on SrTiO3 (Fig. 3.4a ),

4πMs = 1343 Oe and H2⊥ = -2737 Oe, resulting in a 4πMeff = 4080 Oe which matches well

with the Hs of the out-of-plane hysteresis loop. For Sr2FeMoO6/Sr2CrNbO6/LSAT (Fig.

3.4b), H2⊥ is almost zero and 4πMeff ≈ 4πMs = 1525 Oe which agrees with the observed 24 Hs. For Sr2FeMoO6/Sr2GaTaO6 (Fig. 3.4c), H2⊥ = +1753 Oe which is large enough to compensate the demagnetizing field 4πMs = 1430 Oe, resulting in a negative 4πMeff = -323 Oe and consequently, an out-of-plane easy axis. Thus we have demonstrated the ability to shift the magnetic easy axis from in-plane to out-of-plane via epitaxial strain in Sr2FeMoO6.

The strong magnetocrystalline anisotropy and high TC in Sr2FeMoO6 and other com- pounds (e.g., containing 5d transition metals) with strong spin-orbit coupling provide unique opportunities for studying strain-induced magnetoelastic coupling and point toward potentially important spintronic applications that build on the ability to control magnetism through modification of lattice parameters. FMR measurements reveal a surprisingly straightforward linear dependence of magnetocrystalline anisotropy on the tetragonality of

Sr2FeMoO6, implying the opportunity to gain detailed understanding of the magnetoelastic interaction that will enhance the value of this material for spintronics research and technol- ogy. Our results demonstrate that the magnetocrystalline anisotropy can be tuned through strain engineering over a broad range of values sufficient to switch from in-plane to out-of- plane magnetization states even in zero applied fields. This implies that2 Sr FeMoO6 films grown on piezoelectric substrates or underlayers could be electrically switched for novel room-temperature spintronic applications.

25 Chapter 4 Dynamic Dipolar Coupling Between Exchange Biased Stripes

4.1 Introduction

In previous chapters it has been shown that Ferromagnetic Resonance (FMR) is a useful tool in mapping the energy landscape of a ferromagnet, which manifests as effective fields on the magnetization of the ferromagnet. Magnetic fields are mapped to frequency-space enabling the measurement of field magnitude and direction with spectroscopic precision.

Thus far we have only considered field shifts due to static fields experienced by the single macroscopic spin of the uniform mode precession. In this chapter we consider how precession of the magnetization of an individual element can produce stray oscillating dipolar fields that can manifest as effective fields on neighboring elements. Interacting elements canbe utilized as a controllable medium to transmit information by a collective spin wave (magnon) that traverses the individual elements, and this field of research is known as magnonics.

Patterned structures with interacting elements in one- and two-dimensions have attracted interest as magnonic crystals and have been widely studied using the technique of Brillouin

Light Scattering (BLS) [58–63], with resolution of wavevector (both magnitude and angle) in addition to frequency information. On the other hand, the use of FMR in studying these structures has not been widely used, due to the limitation of uniform microwave fields coupling only to standing wave resonances of uniform modes with zero wavevector. In this

26 chapter we demonstrate the use of FMR to detect the eigenmodes of a one-dimensional array of stripes and show that at certain angles they display characteristics of couplings due to dynamic dipolar fields.

4.2 Experiment

A sample was prepared with spatially modulated magnetic parameters by exploiting the effect of exchange bias (EB), which is an effective bias field that acts on aferromagnet when it is contact with an antiferromagnet [64]. A continuous antiferromagnetic IrMn layer was deposited on a 20 nm thick continuous NiFe (permalloy) layer while an external magnetic field was applied. The external field dictated the preferred orientation ofthe antiferromagnet while it cooled through it’s N´eeltransition temperature, hence controlling the direction of the exchange bias field due to the exchange interaction. The external field was then reversed, and portions of the bilayer were locally bombarded with He-ions, which disturbed the magnetization of the antiferromagnet and interface, allowing the exchange bias direction to be reversed in the bombarded sections. Thus, the continuous ferromagnetic film was transformed into an array of stripe regions with a modulated effective field, without directly modifying the continuity of the film.

Magneto-optic Kerr Effect (MOKE) imaging was used to confirm that the effective bias field changed orientation between neighboring stripes, as seen inFig. 4.1. In the

20 micron wide stripes at zero field it is observed that neighboring stripes have opposite magnetic orientations due to the different exchange bias field direction in each stripe, and this manifests as a double hysteresis loop in a global measurement of the magnetization, as seen in Fig. 4.1b. Magnetic reversal of individual stripes is not observed in the narrow

2 micron stripes, as seen in Fig. 4.1c indicating that individual stripes do not switch without their neighbors also switching, which manifests as a global switching event and single hysteresis loop. Hysteresis loops for a control sample with a completely unbombarded surface and another control sample with a completely bombarded surface were also measured with MOKE to check that the magnitude of exchange bias was the same for both, as seen

EB EB H1 H2

EB -H1 EB -H2

Figure 4.1: MOKE images and hysteresis loops of modulated exchange bias sample. a) Cross-section schematic of the stripe sample. The NiFe film experiences an exchange bias field due to the interaction with an antiferromagnetic IrMn layer. The direction ofthe exchange bias field is reversed by ion-implantation. b) MOKE hysteresis loop of 20micron EB wide stripes show individual stripe reversal at the corresponding exchange bias fields H1 EB and H2 . Inset shows reversed domain magnetizations defined by the stripe pattern. c) MOKE hysteresis loop of 2 micron wide stripe show only a single switching loop. The inset shows the domain walls do not follow the stripe axis.

28 EB EB -H1 -H2

H EB EB 1 H2

EB H1 mmm πmmm H EB mm mm 2 mmm mmm πmm mmm m m Figure 4.2: a) Hysteresis loops of control samples show sign reversal of the exchange bias field shift for the ion-bombarded sample. b,c) Angular-dependent FMR also show the change in direction of the exchange bias field. Circles are experimental data and solid lines are fits to Eqn. 4.1 using the parameters shown.

in Fig. 4.2a.

Angular-dependent FMR was performed on the completely ion-bombarded and as- deposited unbombarded control samples to investigate the effect of the antiferromagnet and ion bombardment on the properties and effective fields of the ferromagnetic film.

The effective fields of interest are exchange bias HEB, effective saturation magnetization

4πMeff = 4πMs − H2⊥, and in-plane uniaxial anisotropy H2∥. The resonance condition as a function of external field H rotated with in-plane angle φ is

29 ω 2 H H = H + 2∥ + 4πM + H cos φ + 2∥ cos(2φ) H + H cos φ + H cos(2φ) γ 2 eff EB 2 EB 2∥ (4.1)

The experimental resonance field for the as-deposited and completely ion-bombarded samples are shown in 4.2b and 4.2c respectively with parameters obtained from fits to Eqn 4.1.

As expected, the exchange bias field was of the same magnitude and reversed sign between as-deposited and ion-bombarded region. In addition, the two types of sample showed slightly different values for in-plane uniaxial anisotropy H2∥ and effective saturation magnetization

4πMeff . The in-plane anisotropy can likely be attributed to the exchange interaction with the antiferromagnet [65]. The change in 4πMeff may also be due to an interfacial uniaxial anisotropy through a change in H2⊥, or a change in saturation magnetization 4πMs, however these two effects are not distinguishable from FMR alone.

Performing the angular-dependent FMR experiment on samples with stripes of reversed exchange bias between neighbors displayed different behavior depending on the width of the stripes. For 20 micron wide stripes there are two resonance peaks of equal amplitude, and the resonance field of the two peaks as a function of in-plane field angle appears astwo

◦ sinusoidal functions of φ with a phase shift of 180 due to the reversed sign of HEB in the two regions, as seen in Fig. 4.3a. The angular-dependence of the resonance field for the

2 micron wide stripes does not display the same behavior. Fig. 4.3b shows that there is a main peak with a weak angular dependence and constant amplitude and another peak that only appears at certain angles and whose resonance field and amplitude are both angle dependent.

To describe the angular dependent resonance we model the system of alternating stripes as two magnetizations M⃗ 1, M⃗ 2 whose total energy per unit volume is given by

⃗ ⃗ ⃗ ⃗ ⃗ EB ⃗ ⃗ EB ⃗ E = − (H · M1) − (H · M2) − (H1 · M1) − (H2 · M2) (4.2) 2 2 2 + 2π(1 − N)(M1⊥ + M2⊥) + 2πN {M1n − M2n}

EB EB where H is the external field, H1 and H2 are the exchange bias fields of the as-deposited 1 2 and ion-bombarded regions respectively, M1⊥ and M1⊥ are the components of the magnetizations perpendicular to the film plane, M1n and M2n are the components of the mag- 30 1060 (a) 20 μm wide stripes 1040

1020 H EB 1 = 50 Oe

4πM1 = 9920 Oe 1000 H EB H 2 = -35 Oe StripeDipole= 13 Oe

4πM2 = 10120 Oe 980 Resonance Field (Oe) Resonance Field 960

940 −90 0 90 180 270 Angle (deg) 1050 (b) 2 μm wide stripes

1000

H 950 StripeDipole= 130 Oe

H EB 900 1 = 25 Oe Resonance Field (Oe) Resonance Field 4πM1 = 10020 Oe H EB 2 = -35 Oe

4πM2 = 10120 Oe 850 −90 0 90 180 270 Angle (deg) Figure 4.3: Angular dependence of resonance field for striped samples showing acoustic and optic mode splitting. Circles are experimental data and solid lines are theoretical resonance field for the acoustic (blue) and optic (red) modes for the system using the parameters shown. a) For 20 micron wide stripes the HStripeDipole field is small due to the small shape anisotropy of the stripe. b) For 2 micron wide stripes the larger shape anisotropy manifests as a large splitting HStripeDipole between optic and acoustic modes.

31 netizations perpendicular to the stripe walls and N is the shape factor for a rectangular stripe. The shape factor N for an infinitely long rectangular prism is given by[66]

2 2 2 4πN = 8 arctan( p ) − 2p ln(1 + ( p ) ) where p is the ratio of the width-to-thickness. A thickness of 20 nm and widths of 2 microns and 20 microns were used in this study.

The last term in Eqn. 4.2 describes the dipolar interaction due to a difference in phase

between transverse magnetization in neighboring regions, and has previously been used

to describe the dipolar interaction between domains in an unsaturated film [26]. This

dynamic interaction is displayed in Fig. 4.4 and describes the behavior of acoustic and

optic coupled modes. When the two magnetizations precess in phase in the acoustic mode,

the dynamic difference term is zero. For out-of-phase precession of an optic mode thereisa

magnetic energy cost to producing magnetic charges at the stripe edges, and this manifests

as an effective field shift, which is largest when the external field is oriented parallelto

the stripe axis. The term is also relevant when the static magnetizations in neighboring

regions are in opposite directions, and this effect is seen in the MOKE hysteresis loops.

For magnetizations in completely opposite directions the effective dipole field is maximum

and has a value of HStripeDipole = 2 × 4πNMs. For 20 micron wide stripes HStripeDipole = 13 Oe, which is less than the exchange bias fields of 40 Oe, and so an energy minimum

can be found with neighboring stripes with opposition magnetizations. For 2 micron wide

stripes HStripeDipole = 130 Oe, which is larger than the exchange bias field, and so an energy minimum with reversed neighboring stripes cannot be found without paying this energy

cost. Hence the 2 micron wide stripes only reverse together, and this causes the single

hysteresis loop as seen in Fig. 4.1c.

To model the behavior of angular-dependent FMR in the stripe samples we once again

consider small deviations from equilibrium with e−iωt time dependence to solve the Landau-

Lifshitz equation of motion and consider only first order terms. In this case we have two

coupled oscillators and we must solve four equations of motion for the four degrees of

freedom θ1,φ1,θ2,φ2. Hence the equation of motion becomes a matrix expression [26, 67]

32 EB EB H EB EB H1 H2 1 H2

M M 1 2 M1 M2

+ H ++ ++ Figure 4.4: Coupling between stripes due to dynamic dipolar interaction. a) In-phase (acoustic) and out-of-phase (optic) modes of a stripe sample. The modes are split in energy due to the cost of producing magnetic poles at the stripe boundaries for the optic mode. b) FMR spectrum showing the acoustic and optic mode. The optic mode amplitude is smaller due to weaker coupling to the uniform magnetic field of a microwave cavity. The splitting between the two modes is given by the HStripeDipole field, which is a function of stripe width.

33     iω Eθ1φ1 Eφ1φ1 Eθ2φ1 Eφ1φ2 + 2 mθ1  γ t1Ms sin θ1 t1Ms sin θ1 t1Ms sin θ1 t1Ms sin θ1 sin θ2     Eθ1θ1 iω Eθ1 φ1 Eθ1θ2 Eθ1φ2     − − − −   mθ   t1Ms γ t1Ms sin θ1 t1Ms t1Ms sin θ2   2      = 0 (4.3) Eθ1φ2 Eφ1φ2 iω Eθ2φ2 Eφ2φ2  + 2   mφ   t2Ms sin θ2 t2Ms sin θ1 sin θ2 γ t2Ms sin θ2 t2Ms sin θ2   1   E E E E    − θ1θ2 − θ2φ1 − θ2θ2 iω − θ2φ2 m t2Ms t2Ms sin θ1 t2Ms γ t2Ms sin θ2 φ2 where Ei,j’s are the derivatives of the energy with respect to the four degrees of freedom

θ1, θ2, φ1, φ2 of the two magnetizations M1,M2; mθi = Msδθi and mφi = Ms sin θiδφi are small deviations of the magnetization along θi and φi respectively. The system will have solutions if the determinant of the 4 × 4 matrix equals zero, and hence the resonance field can be found by solving for H. Four solutions are obtained, two of which will be negative and non-physical. The remaining two solutions correspond to the acoustic and optic modes of the system. The resonance field for these modes is plotted as the thesolid lines in Fig. 4.3 and there is good agreement with the experimental resonance field. The disappearance of the lower field experimental resonance peak when the field is parallel to the stripe axis can be explained by the weak coupling of the uniform microwave field (from the microwave cavity) to the optic mode, due to the transverse component of magnetization in neighboring regions being out-of-phase and hence canceling out when averaged over a macroscopic volume. Nevertheless, at certain angles it is clear that an experimental peak is observed that follows the behavior of the theoretical line, and we suggest that this is due to a mode that has some characteristics of optic mode oscillation but without the transverse magnetizations in neighboring regions canceling completely.

It is also possible to measure the resonance field for external field rotated out-of-plane, and modeling this resonance condition requires no free parameters since all magnetic parameters were obtained from the in-plane rotation. The results are shown in Fig. 4.5 and show that the prediction from the model for the two resonance peaks and the difference between them clearly follows the experiment. The resonance condition for external field

◦ applied normal to the plane at θH = 0 is particularly interesting. Intuitively, one would expect the the resonance condition to be the same for both regions since the exchange bias field is perpendicular to the external field for both. However, there is also theeffectof

− −

− − −

− − Figure 4.5: Angular-dependent resonance conditions of 20 micron wide striped sample for out-of-plane rotation. a) Resonance field for the two peaks. Solid circles are experimental peaks and solid lines are expected resonance field using parameters from the in-plane fit.b) Difference between the two resonance peaks. Empty circles are experiment and solid lines are expected resonance field difference. Inset: zoomed in data shows there is a non-zero difference between the two resonance fields at θH = 0

35 the 200 G difference in saturation magnetization between the two stripes, as noted inFig.

◦ 4.3a), which results in a 200 Oe splitting between the two resonance fields at θH = 0 .A resonance field shift on the order of this splitting is also seen when this same striped sample is imaged using FMRFM, as seen in chapter 6.

In conclusion, we have used angular-dependent FMR spectroscopy to measure the magnitude and direction of the effective fields in a sample engineered with stripes of exchange bias in opposing directions. We find that we are able to measure the magnitude anddi- rection of exchange bias fields even in narrow stripes in which the dipolar fields between neighboring stripes hinder individual magnetization reversal and prevented measurements of individual switching fields by magnetometry. Finally, we show that for narrow stripes at certain angles there is evidence for coupling by dynamic dipolar fields which manifest as an angular-dependent splitting in the resonance field of FMR spectra. This result has consequences for studies of coupled magnetic structures and in particular magnonic crystals, and shows that the transverse uniform microwave field in FMR experiments is capable of coupling to precession that displays optic mode behavior.

36 Chapter 5 Dual-Frequency Ferromagnetic Resonance to Measure Spin Current Coupling in Multilayers

5.1 Introduction

The generation of spin currents by FMR, a field known as spin pumping, is a topic that has received considerable attention in recent years due to its possible use in spintronic devices. The spin pumping effect occurs when a ferromagnet in contact with anormal metal is made to precess at its FMR frequency and is observed as a pure spin current in the normal metal. There are two components to the spin current: a dc component due to the change in the static Mz component of the magnetization when on resonance, and an ac component due to the precessing transverse component of the magnetization on resonance. The dc component has recently been extensively measured in heavy metals with large spin-orbit coupling [68], which allows for the pure spin current to be easily measured as a charge voltage. The ac component of spin pumping, however, was actually measured several years before the dc component, as it manifested as an enhanced damping torque on a ferromagnet when in contact with a normal metal [69] that scales as 1/t where t is

the thickness of the ferromagnet, confirming that it is an interfacial effect [70]. It was

only recently that the theory of spin pumping suggested that this enhanced damping was

due to the generation of a pure spin current in the normal metal [71]. The spin current

generation theory was confirmed in an experiment in a FM1/NM/FM2 trilayer where FM1

Figure 5.1: Trilayer sample for dual-frequency FMR detection of spin current coupling. FM1 precesses on resonance and emits a spin current across the normal metal NM. The spin current acts as a damping-like spin-torque on FM2, modifying the resonance linewidth of FM2.

and FM2 are ferromagnets 1 and 2 respectively and NM is a normal metal. It was found that the linewidth of FM2 was affected by the precession of FM1, and vice versa [72].

This was attributed to a dynamic exchange coupling due to the emission of spin currents from FM1 to FM2 across the normal metal, which acted as a damping or anti-damping torque to modify the linewidth of FM2. This is in contrast to the dynamic dipolar coupling observed in chapter4, which manifested as an effective field-like torque that shifted the resonance frequency of neighboring stripes when they were both close to resonance. The distinction between static and dynamic couplings should now be clear: static couplings are effective static fields experienced by a ferromagnet that affect the equilibrium orientation of the ferromagnet and can be measured by other means e.g. magnetometry, while dynamic couplings only occur when a ferromagnet precesses. In the current chapter we shall describe our attempts to detect the dynamic spin current coupling in a trilayer when the resonance and hence spin current emission from two ferromagnetic layers are controlled independently using two microwave frequencies.

With independent control of two microwave frequencies it should be possible to detect the effect of spin current in a FM1/NM/FM2 trilayer system, as shown inFig. 5.1. When

FM1 (ferromagnet 1) is on resonance, it will emit a spin current across the normal metal

NM. The spin current acts as a damping-like spin-torque [73] on FM2 (ferromagnet 2),

YY Y Y Y

YYh

YYh Figure 5.2: FMR spectra for bare YIG sample and YIG/Au/Py sample

which should manifest as a change in linewidth of the FM2 resonance peak. The reverse effect of spin current emitted by FM2 and absorbed by FM1 can be ignored if the precession cone angle for FM2 is much smaller than FM1, which can be controlled by the microwave power on FM2.

5.2 Single frequency FMR of YIG with and without a Py layer

To measure FMR spectra we use a Bruker EPR spectrometer with a high Q microwave cavity at a fixed frequency of 9.8 GHz. At this frequency we observe two resonance peaks from YIG in Fig. 5.2. One is attributed to the area of YIG covered by Au/Py that is broadened due to spin pumping. A second narrow peak is attributed to uncoupled

YIG regions, and this peak only appeared when some of the Au/Py was removed. Static coupling between two ferromagnets separated by a spacer is well-known and described by a short-ranged RKKY-like exchange coupling that oscillates with spacer thickness [74] on a lengthscale of nanometers. In the present study, the 10nm Au spacer should be sufficient to

II II

I Figure 5.3: Angular dependence of the YIG FMR peaks for out-of-plane rotation. The narrow peak is attributed to uncoupled YIG that has had Au/Py removed and is fit to 4πMs = 1800 Oe and γ = 2π × 2.87. The broad peak is from the regions of YIG covered with Au/Py and is fit toπM 4 s = 2000 Oe and γ = 2π × 2.87

completely suppress the RKKY interaction between Py and YIG, and this was confirmed in the experiment by Heinrich et al. [75] for thicker spacers. Experimentally, however, we observe that the broad peak is shifted in field from the narrow peak. From the angular dependence of the broad peak shown in Fig. 5.3 we can characterize this with a change

in saturation magnetization 4πMs. Static exchange coupling has been ruled out. Dynamic exchange coupling due to the spin current can manifest as a field shift if there is an imaginary

contribution to the spin-mixing conductance. However, this field-like spin-torque would

appear as a shift in the effective gyromagnetic ratio of the sample, which is not reflected in

the angular-dependence of the YIG peak shown in Fig 5.3. An explanation for the change

in saturation magnetization remains outstanding.

To measure the frequency-dependence of the YIG linewidth we use a frequency-tunable

microwave excitation provided by a coplanar stripline (CPS) connected to a broadband

microwave source. The sample was glued onto the CPS, and placed inside the cavity, which

provides the field modulation. A trilayer film of YIG(5nm)/Au(10nm)/Py(5nm) was sputter

deposited [76] on a (111)-oriented Gd3Ga5O12 substrate. The Gilbert damping parameter

40 HH HHαHHHH H HHαHHH

H Figure 5.4: YIG Linewidth before and after permalloy deposition

α of the YIG film was measured from the frequency dependence of the linewidth before and after the Py deposition, and it was found that the Gilbert damping parameter of the

YIG film was enhanced with Py deposition as shown in Fig. 5.4. This indicates that the

YIG was spin pumping across the normal metal Au layer, and the Py absorbed the spin current. The Gilbert damping enhancement defined by α = α0+αsp can be used to calculate the efficiency of spin current emission, characterized by a parameter called the spin-mixing conductance [71] αsp4πMst g↑↓ = (5.1) gµB where 4πMs = 1600 Oe is the saturation magnetization for YIG, g = 2 is the electron g-

factor, µB is the Bohr magneton, and t is the film thickness. From this calculation and with

18 −2 αsp = 0.009 we find the spin-mixing conductance for YIG/Au/Py is g↑↓ = 3.9 × 10 m . This is a factor of 3 larger than that measured in a similar experiment by Heinrich et al. [75], indicating very efficient spin current transport from the YIG interface and efficient spin current absorption at the Py interface in our sample.

πmHHH ω πmHHH ω

HHm

HHm Figure 5.5: Dual FMR resonance condition. The resonance of FM2 at the fixed microwave frequency of the cavity ωcavity occurs at an external field Hres. The resonance of FM1 can be tuned to occur at the same external field using a second microwave source at frequency ωsim. When this condition is met we expect to see the effect of the spin current generated by FM1 on the linewidth of FM2.

5.3 Dual frequency FMR of YIG/Au/Py

To measure the effect of spin current absorption the YIG FMR peak was measured atfixed frequency in the EPR cavity while the Py resonance was excited at multiple frequencies on the broadband CPS. This is in contrast to previous attempts at dual-frequency FMR in which two broadband frequencies were used [77]. In that study, the two signals were separated using lock-in detection and frequency-modulation at two different modulation frequencies. In our study we used field-modulation, but the EPR cavity acts as a narrow- band filter and so the spectra recorded by the EPR spectrometer are not sensitive tothe broadband microwave excitation. From the resonance conditions for the two ferromagnets shown in Fig. 5.5 it is possible to tune the broadband microwave frequency to ωsim, enabling

simultaneous resonance at a field Hres. When the simultaneous resonance condition is met we expect to see a change in linewidth of FM2. The result of measuring the linewidth of

FM2, which is YIG, while sweeping the second frequency through the simultaneous reso-

nance ωsim is shown in Fig. 5.6a. There is a small change in the linewidth at ωsim, which might suggest the presence of a spin current induced linewidth change at the simultaneous

42 ω zl

rrzl

rrrzl

rrzl

rrrωzl

Figure 5.6: Dual FMR result: a) YIG linewidth as second frequency is swept through the simultaneous resonance condition ωsim. b) YIG linewidth as second microwave source is swept in power while the frequency is fixed at the simultaneous resonance condition ωsim

43 resonance condition. However, the change in linewidth is of the same order of magnitude of the spread in the data, and so the result is likely not due to a spin pumping effect. In addition, sweeping the power of the second microwave source while at simultaneous resonance showed no change in linewidth, as seen in Fig. 5.6b.

The absence of a change in linewidth when the dual FMR resonance condition is met can be explained by the distinction between dc and ac spin currents. In electrical measurements, the spin current is usually measured via the inverse spin hall effect in metals [68] but this dc spin current scales quadratically with the precession cone angle, while the ac spin current is linear in the cone angle [78]. It is generally accepted that the ac spin pumping is responsible for the damping modifications in previous experiments when the resonance frequencies are equal [72]. In dual-frequency FMR experiments the resonance frequencies of the two ferromagnetic layers are not equal, and coupling can only occur via the dc spin current [79], which can be two orders of magnitude smaller than the ac spin current [78].

Hence the lack of linewidth change in the present case and in the literature for dual-frequency

FMR [77] confirm this picture of ac and dc spin current emission and their different roles in experiments.

44 Chapter 6 Ferromagnetic Resonance Imaging Across an Exchange Bias Patterned Interface

6.1 Introduction

The microscopic study of magnetization dynamics and couplings at interfaces requires an experimental technique capable of probing spin dynamics at an interface. Magnetic resonance force microscopy (MRFM) is a local probe of spin dynamics that uses the dipole field from a micromagnetic probe to define a sensitive slice in which the resonance condition is met [80]. MRFM has been used to detect single electron spins [81] and for 3D magnetic resonance imaging with nanoscale resolution [82], demonstrating that it is ideal for probing spin dynamics at interfaces. However, the application of MRFM to local probing of ferromagnetic resonance is not trivial due to the exchange and dipolar interactions that couple spins together and so the excitations are spin-wave modes defined by the sample geometry [83]. The geometry-defined modes can be shifted in frequency by the dipolar fieldsfrom a micromagnetic tip if lateral dimensions are on the order of the micromagnet, allowing for local FMRFM imaging [84] but this still requires that the sample be patterened by lithography. A truly non-destructive method for FMRFM imaging is desirable, and this has been demonstrated recently by Lee et al. [85]. They demonstrated that if the dipole field from a micromagnetic tip is sufficiently strong and opposite to the external field then it is sufficient to localize a standing spin wave mode. When the external field and magnetization are nor-

45 mal to the film plane and the magnetic moment of the micromagnet is anti-parallel tothe magnetization the azimuthal symmetry allows for simple numerical analysis by modelling the modes as cylindrical bessel functions, similar to modes observed in perpendicularly magnetized dots [86]. The lateral confinement of this mode makes it suitable for imaging spatial field variations in the lateral dimensions of a thin film sample. In this chapter,we shall demonstrate how this mode is modified as it approaches a lateral interface, and show that it can be used as a sensitive local probe of the interface.

6.2 Experiment

An interface between two different materials is the most common structure examined in the study of interfacial spin transfer. The physics of these interfaces is difficult to directly probe using the technique of localized FMRFM for two reasons. First, the localization occurs only in lateral dimensions, while the mode is uniform across the film thickness and so only lateral interfaces can be directly probed. Second, an interface between a ferromagnet and a normal metal, for example, has a step in the demagnetizing field at the interface that falls from −4πMs to zero on a spatial lengthscale comparable to the film thickness. Hence, a localized mode would be heavily modified and would have more resemblance to an edge mode defined by the sample geometry [87] and not by the dipole field from the tip. To lessen the effect of large changes in demagnetizing field it would be beneficial to investigate a sample with a smaller step in internal field. To achieve this, a continuous ferromagnetic thin film sample of Py was prepared with a modulated internal fieldby covering the film with antiferromagnetic IrMn and exploiting the effect of exchange bias between a ferromagnet-antiferromagnet bilayer. Details of the sample preparation and the dynamic coupling between stripes of different internal field when precessing are discussed in section 4.2 and the same sample with stripes of 20 micron width was used in this chapter.

The important aspect of the sample for the current chapter is the continuous interface

between stripes of varying internal field, which results in a small field step compared to

that of an abrupt change in material [88].

46 A high coercivity Sm1Co5 magnetic particle with dimensions 1.2µm × 1.2µm × 1.5µm

−9 and moment mp = 1.2×10 emu was used with an external field applied normal to the film plane and with the particle moment oriented anti-parallel to the external field to confine the localized modes. With the particle glued to a silicon cantilever, mechanical detection of the resonance was achieved by measuring the force on the cantilever due to dipolar interactions between the sample and magnetic particle. The cantilever was scanned across the field step using a piezotube scanner. Experiments were performed at 10 K with microwaves provided by a stripline resonator at 7.5 GHz and the microwave amplitude modulated at the cantilever frequency. Further details of the microscope and detection are given elsewhere [89].

The resulting images obtained by scanning the cantilever with the magnetic particle across the interface of the sample are shown in Fig. 6.1. The image are presented as field- position scans, in which the y-axis is the external applied field, the x-axis is position relative to the interface and the color scale is the force on the cantilever. When on resonance spins will precess and there is a change in the longitudinal component of their magnetization, which results in a dipolar force on the magnetic particle and hence a force on the cantilever.

Micromagnetic simulations were also performed for this system, and the resulting resonance fields are plotted as solid dots in the figure. While the micromagnetic simulations donot exactly match the resonance fields from the experiment, they show the same qualitative behavior. For the large probe-sample separation of z = 2390 nm the resonance field is distinct for the two different regions and the resonance mode is mostly defined bythe sample geometry and internal field. The splitting of 400 Oe is due to a combination oftwo parameters that change across the interface: the saturation magnetization 4πMs and the exchange bias direction. A splitting with the same order of magnitude was seen in angular- dependent conventional FMR at room temperature and the results were shown previously

4.5. For the smallest probe-sample separation of z = 1340 nm, there is a continuous change in resonance field as the probe is scanned across the interface, which is reflected inthe micromagnetic simulations. For the intermediate probe-sample separation of z = 1765 nm there is a region where two resonance fields are distinct but coexist. This behavior is qualitatively reproduced by the micromagnetic simulations, as the resonance field at higher

47 z z z

NN NN NN Nμ Nμ Nμ

N N N

− − − − Nμ Nμ − − Nμ Figure 6.1: Top row: Field-position FMRFM images showing the localized modes spectra as a function of position across the interface at probe-sample separations of z = 2390 nm, z = 1765 nm and z = 1340 nm. Color scale is force detected by the cantilever. Solid dots are micromagnetic simulations of the resonance field. The position axis is centered at the field interface. Bottom row: Micromagnetic simulations show the mode profile at the interface can be heavily modified due to the change in internal field of the sample. The position axis is centered at the probe moment axis.

field continues past the interface and starts to bend down towards the lower resonance field.

The shape of the localized mode when the probe is positioned directly over the interface is also obtained by micromagnetic simulations and shown in the bottom row of Fig. 6.1. It is clear that the mode profile is heavily modified from a symmetrical bessel mode due tothe internal field step of the interface, and also that the mode has a non-zero amplitude onboth sides of the interface. This is the key result of this study, and demonstrates that localized modes are capable of probing dynamics at an interface. In conclusion, we show that localized modes can be used to image the local fields in a sample with large field variations. Wefind that micromagnetic simulations are required to understand the behavior of the mode in the presence of an internal field step, which can heavily modify the mode shape. Overall, this demonstrates that our technique can be used to probe spin dynamics at an interface

48 or boundary, which is an important step in understanding novel phenomena such as spin pumping across a ferromagnet-metal bilayer interface [71].

49 Chapter 7 Ferromagnetic Resonance Force Microscope Design and Minimizing Spurious Backgrounds

7.1 Introduction

The study of magnetization dynamics in ferromagnetic thin films at room temperature with submicron spatial resolution required the design and implementation of a new magnetic resonance microscope that could meet the following requirements:

• broadband microwave excitation of sample up to 20 GHz

• cantilever interferometry providing piconewton (10−15N) force sensitivity

• three-dimensional scanning elements to enable millimeter-scale alignment and micron-

scale imaging

• high precision (< 0.1G) magnetic field control for resonance spectroscopy

• high coercivity (> 5kG) and micron-sized cantilever probe magnet to enable localized

mode spectroscopy

• high vacuum environment to ensure high quality-factor (Q > 10000) of cantilever

CAD images of the microscope design are shown in Figures 7.1 and 7.2, and a photograph of the implemented microscope inside the magnet is shown in Fig. 7.3.

50 iue71 A mg fFroantcRsnneFreMcocp.Dse ieindi- line Dashed 7.2 Microscope. Force Fig. Resonance of Ferromagnetic view of zoomed Image cates CAD 7.1: Figure

iue72 A omdIaeo ermgei eoac oc Microscope Force Resonance Ferromagnetic of Image zoomed CAD 7.2: Figure

Figure 7.3: Photograph of Ferromagnetic Resonance Force Microscope in magnet.

53 Microwaves were provided by gluing the sample substrate-side down onto a microstrip fabricated from Rogers TMM10 microwave material. The microstrip was connected to a

Gigatronics 12000A microwave source via SMP-connector edge launchers and 0.085” coaxial cable. Cantilever interferometry was provided by an optical fiber connected to a 1550nm laser diode and coupled to a photodetector. The output signal of the photodetector was connected to a digital acquisition card on a computer with LabVIEW, which was then used to extract the frequency and amplitude of the cantilever oscillations. The cantilever was excited by feedback at its natural frequency using oscillations of a piezodisk. The piezodisk was excited by a pulse generator that was triggered by the cantilever signal from the photodetector with a phase delay, completing the feedback loop. Scanning in three dimensions was provided by attocube stages with 3mm of travel in each dimension. The attocube was calibrated for probe-sample separation studies by touching the cantilever to the surface of the sample and by continuing to move the attocube the fiber-cantilever distance goes through interferometry fringes, whose separation are known to be λ/4 where

λ =1550nm. A high coercivity Sm1Co5 magnetic particle was glued to a diamond atomic force microscopy cantilever, and the measurement and imaging of the probe magnet is discussed in 7.2. Modulation of microwave amplitude at the cantilever frequency was used to measure the resonance behavior of the sample, and spurious coupling of this time-dependent force is described in section. Magnetic fields up to 20 kG were provided by a Varian

15” magnet with 1.75” pole spacing and a Walker Scientific magnet power supply. The field was measured using a Lake Shore Transverse Hall Probe and Lake Shore Model475

Gaussmeter, and field feedback control was implemented by connecting the analog output of the gaussmeter to the magnet power supply, which is covered in detail in section 7.4.

Vibration isolation was provided by suspending the whole microscope by rubber O-rings to

prevent high frequency vibrations from reaching the microscope. However, there was still

an issue with vibrations that were indirectly measured as a broadening of resonance lines,

which is covered in section 10.2. These vibrations were measured directly and minimized,

and the results are given in section. The entire microscope was placed inside of a vacuum

can and pumped continuously with a turbo pump to reach 10−5 Torr.

54 7.2 Sub-micron probe magnet shaping and magnetometry

A high coercivity Sm1Co5 magnetic particle of size 3µm was glued to the tip of a diamond atomic force microscopy cantilever whose point had been shaped down using a focused ion beam (FIB). The magnet was then also shaped using the FIB down to a size of 1.75 µm, and the resulting image of the probe magnet is given in Fig. 7.4a. Cantilever magnetometry was performed after installing the cantilever into the microscope and aligning with the optical fiber to allow measurements of the cantilever frequency. Measuring the cantilever frequency as a function of applied external magnetic field Hext allows the determination of the magnetic moment mp and anisotropy HK of the cantilever using the equation

∆ω mpHextHK = 2 (7.1) ω0 2k0Le(Hext + HK ) where ∆ω = ω−ω0 is the change in cantilever frequency from it’s zero-field frequency ω0, k0 is the spring constant of the cantilever, Le = 1.38L is the effective length of the cantilever

(where L is the actual length of the cantilever) and HK is the anisotropy field of the probe magnet. The resulting experimental cantilever frequency is shown in Fig. 7.4b, and fits to

−9 the data show that the probe moment was mp = 3.9 × 10 emu and the anisotropy field

5 was HK = 2 × 10 G.

7.3 Spurious cantilever coupling to microwaves

In FMRFM experiments the magnetization dynamics are usually at GHz frequencies, whereas the detection is by the mechanical oscillation of the cantilever at kHz frequencies.

This discrepancy is avoided by modulating the amplitude of the microwave excitation at the cantilever frequency. When on resonance, the magnetization of the sample being studied will increase and decrease it’s cone angle at the cantilever frequency, resulting in an oscillating dipole field from the sample that is detected by the cantilever asa time-dependent force. The modulation of microwave amplitude at the cantilever frequency also results in a time-dependent background force even when the sample is off-resonance, and this has been attributed to a spurious coupling of microwaves directly to the cantilever.

55 m9

sssss ss bssb s s ss s s

ssm9

sm9

Figure 7.4: a) Focused Ion Beam image of shaped Sm1Co5 probe magnet glued to cantilever. b) Cantilever magnetometry shows the particle magnetization switching around 10kG and the probe’s magnetic moment and anisotropy fields are found from fits toEqn. 7.1

Figure 7.5: Spurious cantilever coupling to microwaves for silicon and diamond cantilevers above open and shorted microstrip and with and without magnetic particles.

To elucidate the nature of this coupling we have measured the background force for cantilevers made of different materials and on transmission lines which were shorted or open (to provide electric field or magnetic field standing wave anti-nodes respectively).

The results of this study are given in Fig. 7.5. First, it is observed that the spurious coupling to the silicon cantilever was smaller for the open microstrip than the shorted microstrip. However, ac magnetic fields are necessary for magnetic resonance experiments, and so shorted microstrip was used for all subsequent experiments. Second, it is clear that the spurious coupling to the diamond cantilever is much smaller than the spurious coupling to the silicon cantilever, probably due to the lower conductivity of diamond. Finally, the spurious coupling increased when a magnetic particle was glued to the diamond cantilever.

We can conclude from this study that the spurious coupling is magnetic in nature and may be due to induced eddy current heating in both the cantilever material and in the probe magnet.

57 The spurious coupling was also investigated in the presence of a resonant circuit. The resonant circuit was demonstrated by inductively coupling the coaxial cable and microstrip inside the microscope to the microwave source. The circuit, which ended in a shorted microstrip, acted like a standing wave resonator whose allowed wavelengths λ = nL/4

(where n=1,3,5...) were dominated by the length of the coaxial cable L=15”. The resulting background coupling to the cantilever, as well as the FMRFM signal are shown in Fig.

7.6. It is clear that the cantilever plays the role of a rectifier, and is a way of directly measuring the ac magnetic field at the position of the sample and cantilever. It isalso clear that both the background force and the signal force are amplified equally when on a standing wave resonance peak, indicating that both have the same dependence on the ac magnetic field. This also shows that the signal-noise is not improved by using the standing wave resonator, but it does allow for lower microwave powers to be used while keeping the signal force and ac magnetic field constant. In addition to amplitude modulation we investigated using frequency modulation for FMRFM experiments. We used the standing wave resonator circuit and found that the frequency modulation background depends only on the slope of the frequency response of the microwave circuit, as can be seen in Fig. 7.7.

This is an advantage over amplitude modulation, as the slope of the frequency response

(and hence the background force) is minimum at a standing wave resonance peak, while the ac magnetic field (and hence the signal force) is maximum. This indicates thatthe signal-noise ratio can be maximized at these frequencies.

7.4 Magnetic field control: field steps vs continuous ramp

The field step measurement protocol in which the current for the magnetic field issetand held constant during data acquisition for one data point and the current is stepped up or down by a set amount between data points has been a popular technique, in the past, for measuring FMRFM spectra [84, 85, 90, 91]. The advantage of this technique is that there

is complete control of both the field interval and the averaging for each data point during

measurement. The main disadvantage is the length of time taken for a single FMRFM

58 w. GG GG GGGG GGw. GGw.

Gw.

w. GG G G GGw.

GGGw.

Gw. Figure 7.6: a) Background and signal cantilever amplitude when using coaxial cable standing wave resonator. b) Zoomed view of amplitudes near 2 GHz show both forces peak near the same frequencies

z z

z Figure 7.7: Spurious cantilever coupling for amplitude modulation and frequency modulation when using coaxial cable standing wave resonator. Amplitude modulation spurious coupling depends on magnitude of ac magnetic field as function of frequency, whereas frequency modulation coupling depends on slope of ac magnetic field as a function of frequency

spectrum can be an hour or more. We have investigated the effect of continuously ramping the field and using feedback to control the field instead of directly the current. Themain result is that continuous ramping of the field enables much faster acquisition times (seconds instead of hours) and spectra comparing the two techniques are given in Fig. 7.8. While it may seem that the signal-noise is reduced for the continuous ramp technique, there are also more data points acquired, and so with interpolation the signal-noise for the continuous ramp is better than for field steps. This can also be attributed to the timescales ofeach measurement: the field step technique is prone to thermal drifts or shifts that can happenin room temperature measurements or due to vibrations that might occur in the lab over a 90 minute period. The faster continuous ramp is not affected by this as it only takes 90 seconds, and can be restarted if a disruptive event is observed. The continuous ramp technique was implemented using a Lake Shore Model 475 Gaussmeter with its analog output connected to the current control input of the magnet power supply. Feedback was used to control the

60 R

RR R R

RR R R Figure 7.8: FMRFM force signal comparison between field step technique and continuous field ramp. a) Field step technique: field is held constant during data acquisition and stepped between fields. Data acquisition takes 90 minutes. b) Field is continuously ramped while data is acquired and data acquisition takes 90 seconds. Data is shown before and after interpolation.

61 field and a field set point above the desired maximum field for the spectrum wassettostart the ramp. The field precision was initially limited to a minimum field stepof1.4G due to the limited precision of the gaussmeter analog output. However, this was soon corrected by adding a resistor in series to the analog output, increasing precision at the cost of decreased range of field.

7.5 Measuring vibrations

Stability is an important issue in force microscope design. In this current design the microscope head, containing the cantilever and force detection components, is mounted on a scanning stage that can vibrate relative to the sample mount if the fixing mechanisms and vibration isolation are not sufficient. Vibration isolation from the outside environment was provided by suspending the entire inner probe of the microscope head and sample mount on rubber o-rings, thereby mechanically isolating the probe from external vibrations. How- ever, there was initially an issue that the cantilever mount was not sufficiently fixed to the scanning stage, and this was directly measured by touching the cantilever to the sample and measuring the frequency response of the photodiode signal. The results before and after fixing this mounting issue are given in Fig. 7.9, and it is clear that the vibrations have been reduced from a peak of 46 nm down to a peak of 15 nm. This is also reflected in the vibrational broadening of the localized mode line, which is discussed in section 10.2. Fits to the vibrationally broadened linewidth also provided a measure of the vibrations before and after the fix to be 39 nm and 11 nm respectively, which are very close to the directly measured values.

ff ff

f Figure 7.9: Vibration spectrum measured by touching cantilever to sample measured before and after fixing cantilever mounting issue. The peak amplitude before fix is 46 nmat218 Hz and the peak after fix is 15 nm at 395 Hz.

63 Chapter 8 Damping of Confined Modes in a Ferromagnetic Thin Insulating Film: Angular Momentum Transfer Across a Nanoscale Field-defined Interface

8.1 Introduction

Spin pumping driven by ferromagnetic resonance (FMR) is a powerful and well-established technique for generating pure spin currents in magnetic multilayers [71,92,93]. Understand- ing the mechanism that couples precessing magnetization to spin transport is an important step in utilizing this phenomenon. In addition, probing the effect of spin pumping on the damping of individual nanostructures is vital for the development of practical spintronic devices, such as spin-torque oscillators [94,95]. Conventional FMR studies at these sub-micron lengthscales becomes difficult due to the sensitivity limitation of power absorption detection. Recent studies have shown that individual nanoscale elements exhibit size-dependent damping effects, such as nonlocal damping from edge modes [96] and wavevector-dependent damping in thickness standing wave modes [97]. These experiments have revealed the effect of intralayer spin pumping due to the transfer of angular momentum from regions of precession to neighboring static regions by a spin current within the same material. A primary challenge in this endeavor is distinguishing intralayer spin pumping from other mechanisms

64 that cause variations in linewidth from sample to sample, such as surface and edge damage [98, 99]. In this paper we measure size-dependent angular momentum transport in a single sample and in the absence of growth-defined interfaces or lithography-induced edge damage. This is achieved by confining the magnetization precession to a mode within an area defined by the controllable dipolar field from a nearby micron-sized magnetic particle [85]. This enables a unique investigation of changes in relaxation due to angular momentum transfer across the field-defined interface between precessing magnetization within a mode to the spin sink of the surrounding quiescent material.

8.2 Experiment

We investigate the size dependence of interfacial damping using the technique of localized mode ferromagnetic resonance force microscopy (FMRFM) [85]. By adjusting the magnitude of the dipolar field from the probe we can control the confinement radius. Localized modes have previously been observed in permalloy when the probe field is out-of-plane [85], in-plane [100] and at intermediate angles [101]. The azimuthal symmetry of the out-of- plane geometry permits simple numerical analysis based on cylindrically symmetric bessel function modes with a well-defined localization radius [85], similar to those seen in perpendicularly magnetized dots [86]. In addition, this geometry eliminates the effect of eigenmode splitting, which can cause additional broadening [102].

8.3 Localized Mode Resonance Fields

We demonstrate the control of confinement radius by the observation of discrete modes in an FMRFM experiment in the out-of-plane geometry in an unpatterned epitaxial yttrium iron garnet (YIG) film of thickness 25 nm grown by off-axis sputtering [76] on a (111)- oriented Gd3Ga5O12 substrate. The probe field is provided by a high coercivity Sm1Co5 particle glued on an uncoated diamond atomic force microscope cantilever and shaped down to a size of 1.75 µm using a focused ion beam. The magnetic moment and coercivity of the particle are measured by cantilever magnetometry to be 3.9 × 10−9 emu and 10 kOe

65 200 Height = 7900 nm n=2

Height = 5900 nm Field n=1 probe field well 150 Height = 4600 nm Lateral position

Height = 3400 nm 100

Height = 2400 nm

Cantileveramplitude (nm) 50

Uniform Mode n = 2 n = 1 3000 3100 3200 3300

External Field Hext (Oe) Figure 8.1: Localized mode FMRFM spectra for thin film YIG at several probe-sample separations. The dashed line indicates the position of the uniform mode peak that does not shift with probe-sample separation. As probe-sample separation is reduced the localized modes shift to higher field relative to the uniform mode peak. Inset: transverse magnetization of the first two spin wave modes confined by the magnetic field well of theprobe magnet. The energy of the confined modes is dictated by the depth of the field well.

66 respectively. When the applied field is anti-parallel to the tip moment, the tip creates a confining field well in the sample that localizes discrete magnetization precession modes immediately beneath it [85, 103], analogous to the discrete modes in a quantum well [104].

The microwave frequency magnetic field that excites the precession is provided by placing the sample substrate side down near a short in a microstrip transmission line. A force- detected ferromagnetic resonance spectrum is obtained by modulating the amplitude of the microwaves at the cantilever frequency (≈ 18 kHz) and measuring the change in cantilever amplitude as a function of swept external magnetic field. Measurements were made ina range of microwave frequencies: 2-6.5 GHz.

Fig. 8.1 shows the evolution of the FMRFM spectra as a function of tip-sample separation obtained at a particular microwave frequency of 4 GHz. At large probe-sample separation we observe a peak at the expected resonance field for the uniform mode in the out-of-plane geometry. As expected [85], several discrete peaks emerge and shift toward higher applied field as the probe-sample separation decreases, thus increasing the (negative) probe field at the sample, while the uniform mode stays at constant resonance field.

The resonance frequency ω of a confined mode with magnetization out-of-plane is[85]

ω 2 = H − 4πM + ⟨H ⟩ + 4πM a k2 H − 4πM + ⟨H ⟩ + 4πM a k2 + 4πM f(k) γ ext s p s ex ext s p s ex s (8.1)

where f(k) = 1 − (1 − e−kt)/kt. The dispersion can be be approximated for wavevectors kt ≪ 1 as ω = H − 4πM + ⟨H ⟩ + πM kt + 4πM a k2 (8.2) γ ext s p s s ex

where γ = 2π×2.8 MHz/Oe is the gyromagnetic ratio, Hext is the external applied magnetic

−12 2 field, 4πMs = 1608 Oe is the saturation magnetization, aex = 3.6×10 cm is the exchange constant of the material, k is the wavevector of the mode, t is the thickness of the film and

⟨Hp⟩ is the dipole field from the probe magnet weighted by the mode m

R 2 2 S Hp(r) m (r) d r ⟨Hp⟩ = R 2 2 (8.3) S m (r) d r

Both the averaged probe field and the wavevector are functions of the mode shape and

67 3300 12 n = 1 n = 1 n = 2 10 n = 2 3250 n = 3 n = 3 8 n = 4 6 3200 4

Mode Radius ( m m) 2 3150 0 2 4 6 8 10 Probe-sample separation (mm) 3100 External Field (Oe) 3050

3000 2 4 6 8 10 Probe-sample separation (mm)

Figure 8.2: Resonance field of the first four localized modes as a function of probe-sample separation at 4 GHz. Filled markers indicate experimental peaks and solid lines indicate expected resonance field obtained numerically. Inset: radius of the first three localized modes obtained during the numerical minimization.

mode radius R, so the frequency is obtained by numerical minimization with variation of radius [85]. Due to cylindrical symmetry a zeroth order Bessel function J0(kr) suitably describes the magnetization profile of the mode, with boundary conditions that define discrete wavevectors kn = χn/R where χn are the zeros of the Bessel function J0(χn) = 0. The minimization of frequency at fixed field is equivalent to the maximization of field atfixed frequency. Hence the deeper field well shifts the modes to higher field when the microwave frequency is fixed, as seen in the spectra ofFig. 8.1. This modeling procedure allows us to obtain both the resonance field and the radius of the mode, and these are given inFig. 8.2.

We see that the resonance field of the experimental peaks are well described by the model, confirming the accuracy of the calculated mode radius.

68 8.4 Size-dependent damping

To measure the damping of the confined modes as a function of mode size we measured

FMRFM spectra for a given mode radius at multiple frequencies, one example of which can be seen in Fig. 8.3. The field shift of the localized modes, relative to the uniform mode

ω Huniform = γ + 4πMs is constant for a given wavevector k, independent of frequency ω, as predicted by Equation (8.2).

By fitting a Lorentzian lineshape to the n = 1 peak we obtain the full-width half-

maximum linewidth of the first localized mode and plot as a function of microwave frequency

to separate intrinsic and extrinsic linewidth broadening mechanisms [105]. Following from

the Landau-Lifshitz-Gilbert equation, the linewidth ∆H is given by

2αω ∆H = ∆H + (8.4) 0 γ where the slope of linewidth vs frequency is a measure of the Gilbert damping parameter

α and the intercept ∆H0 is inhomogeneous broadening due to local variations in magnetic parameters. The frequency dependence is repeated at several probe-sample separations corresponding to several mode radii.

The key result of our study is that compared to the uniform mode the confined modes exhibit an enhanced Gilbert damping and from Fig. 8.4 it can be seen that this enhanced damping shows an unambiguous dependence on the radius of the mode, as seen from the change in slope of linewidth. The Gilbert damping parameter α shows a surprising linear

behavior when plotted against R−1, the reciprocal of the mode radius, as seen in Fig. 8.5.

An enhanced damping is reminiscent of spin pumping observed when a ferromagnet is placed

in contact with a normal metal [71]. If t is the thickness of the film, the enhanced damping

αsp scales like 1/t, which equates to the ratio of the area of the interface with the metal to the volume of the ferromagnet, and hence is given by [75]

γ~g↑↓ 1 αsp = (8.5) 4πMs t where ~ is the reduced Planck constant and g↑↓ is the spin-mixing conductance parameter

69 rmtesoeo h ierfit. linear the parameters of damping slope Gilbert the radii the data. from of to modes fits localized linear and mode and nm uniform 1860 for Linewidths 8.4: Figure probe-sample of fixed radius a mode at a frequencies to microwave equivalent multiple nm, 3700 at of spectra separation FMRFM 8.3: Figure o lrt n h xenlfield external the and clarity for H uniform . R 20n.Fle akr r xeietllnwdh n oi ie are lines solid and linewidths experimental are markers Filled nm. 2230 = Cantilever amplitude (nm) 100 120

OHOl 20 40 60 80 -40 OOO H spotdrltv oteuiommd eoac field resonance mode uniform the to relative plotted is 0

Ol Uniform O OOO

H- Mode O O O R R R 70 H OOOOOαOOOO OOOOOαOOOO OOOOOαOOOO uniform 40 α (Oe) n =2 r eemndfrec oeradius mode each for determined are R n =1 80n.Setaaeoffset are Spectra nm. 1860 = 80 6GHz 2GHz 4GHz R 60nm, 1600 = R = hog h ufc ftemd.Clrsaedntstases antzto precession magnetization transverse denotes scale Color material intralayer surrounding mode. showing to the cross-section mode m of localized Inset: surface the of the mode. volume through the localized from the transfer momentum of angular ratio surface/volume the to hsisltn ytmadta t ffiiny hrceie ythe by characterized efficiency, its that in and transport momentum system angular insulating observe this we that remarkable somewhat and interesting is It Discussion 8.5 from mechanism damping damping interfacial interfacial an of this possibility to volume the the analogy suggest an we As pumping spin pumping. to spin due of efficiency the describes that parameter damping Gilbert 8.5: Figure h nacddmigsol losaea h ufc-ovlm ratio: surface-to-volume the as scale also should damping enhanced the hc so eoac hog h uvdsrae2 surface curved the through resonance off is which eoti pnmxn odcac o hsYGYGsse of system YIG-YIG this for conductance spin-mixing a obtain we Fig and ) (8.6 Equation From t πR 2 t fords-iemd,wihi nrsnne otesronigmaterial surrounding the to resonance, on is which mode, disk-like our of 2.0x10 Gilbert damping a 0.0 0.4 0.8 1.2 1.6 -3 0 hc hw h nacddmigvru oeradius, mode versus damping enhanced the shows which , 8.5 Surface/Volume ratio =Surface/Volume 2/ g ↑↓ 4 = 5.3 x10 α sp α = clsivreywt oeradius mode with inversely scales m 0 8 71 4πM γ t 19 g max max Transfer Momentum Angular m ↑↓ -2 s R πRt 2 12 R rudteeg fteds.Hence, disc. the of edge the around (10 Volume Volume Precession Localized 16 3 cm g -1 ↑↓ ) 20 g (5 = ↑↓ aaee,i larger is parameter, . 3 ± 0. 2) R × equivalent 10 19 m (8.6) −2 . than the spin-mixing conductance measured in YIG-metal bilayers [68, 76, 78]. We suggest that g↑↓ measured here is a parameter that describes a generalization of spin pumping as the transport of energy and angular momentum from an on-resonance spin source to an off- resonance spin sink, even in the absence of a material interface or conduction electrons [106].

The energy and angular momentum from the precessing confined mode can be absorbed by the surrounding ferromagnetic material of the unpatterned film, as depicted in Fig. 8.5(b).

19 −2 The relatively large value of g↑↓ = (5.3 ± 0.2) × 10 m we obtain for YIG-YIG compared

18 −2 to g↑↓ = (6.9±0.6)×10 m previously measured for YIG-Pt [68] can be attributed to the larger exchange coupling in YIG-YIG and that the interface in question is not defined by growth but by a magnetic field. In addition, it might be unexpected for the confined mode to relax via the surrounding material where the lowest energy state, which is the uniform mode, is well above the energy of the confined mode inside the well. However, previous experiments by Heinrich et. al [72, 75] have shown that ferromagnets do act as good spin

sinks when the precession frequency of the spin current is not at a resonance frequency of

the spin sink ferromagnet.

It is possible to rule out other mechanisms for linewidth broadening by analysis of the

phenomenology of our result. The dipolar field from the micromagnetic tip is a potential

source of linewidth broadening as it is produces an inhomogneous field in the sample of

several hundred gauss, and would be a dominant source of inhomogeneous broadening for

a paramagnetic sample. Inhomogeneous broadening from the tip can be ruled out as the

source of increased damping in this study for two reasons. First, any inhomogeneous broad-

ening would be frequency independent, and hence would lead to a change in the intercept

of the frequency-dependence of linewidth shown in Fig. 8.5(a), while the change in slope

alone is a clear indication of a Gilbert damping enhancement. Second, for exchange-coupled

ferromagnetic moments the inhomogeneous field from the tip is cancelled by the dynamic

field from the precession, which allows the effective field to be equal at every position inside

the mode, and hence it can be described as an eigenmode with a single eigenfrequency. The

role of inhomogeneous broadening is further discussed in chapter 10. Other well-established

mechanisms for size- or wavevector-dependent relaxation can also be eliminated due to their

72 smaller magnitude and differing phenomenology; 3-magnon confluence [14,15] manifests as a linewidth broadening that is linear in k but independent of frequency, while 4-magnon scattering [16] scales as k2. These two mechanisms were discussed in section 1.3.

To conclude, we observe robust intralayer spin pumping within an insulating ferromag-

net, which manifests as enhanced damping as the mode radius approaches micron size. This

result has consequences for devices that induce spin precession in confined regions, such as

spin-torque oscillators in the point-contact geometry [107, 108]. In addition, our study

highlights the power of localized mode FMRFM for illuminating local spin dynamics and in

particular for spectroscopic studies of the impact of mode relaxation across a controllable,

field-defined interface.

73 Chapter 9 Numerical, Analytical and Micromagnetic Solutions to the Localized Mode Problem

9.1 Introduction

The analysis of localized mode spectra in the perpendicular geometry has thus far been limited in this thesis and in the literature [85] to numerical solutions due to the nature of the dispersion relation 8.1 and it’s nontrivial dependence on mode radius. In this chapter we compare the results of micromagnetic and analytic methods for solving the problem of localized modes in a ferromagnetic thin film, and compare with the numerical minimization technique to investigate the accuracy of these methods. First, we find that the micromagnetic technique, while computationally more intensive, reveals mode shapes that appear to be more physically permissible than bessel functions that are abruptly set to zero outside of a calculated mode radius. Secondly, an analytic solution is found to be an accurate solution, but only when exchange fields are negligible, which is the case for the first localized mode, and most importantly the analytic solution is found to be computationally much faster than other methods.

9.2 Micromagnetic solution to localized mode precession

Micromagnetics have previously been used to model the more difficult problem of in-plane localized modes [100], for which a simple solution is difficult to obtain due to the non-

μ Figure 9.1: Micromagnetic solution to localized mode resonance field for first four modes in thin film YIG. Solid lines are micromagnetic simulation and markers are experimental data.

symmetric nature of the problem. Here we apply micromagnetics to the modeling of localized modes in an out-of-plane geometry for which the modes are cylindrically symmetric and for which numerical solutions have previously been obtained in chapter 8. The result of micromagnetics in this case shows a close correlation to the experimental data for the first four localized modes, as seen in Fig. 9.1 and the results are almost indistinguishable from the numerical solution shown previously in Fig. 8.2, confirming that micromagnetics accurately models the behavior of these localized modes.

It is illuminating to compare the mode profiles produced by micromagnetic simulations to the bessel functions assumed for the numerical minimization technique. The resulting mode shapes are given in Fig. 9.2. From the figure it is clear that the mode shapes found by micromagnetics are qualitatively similar to the bessel function approximations used by the numerical technique, albeit with some small modifications to the mode shapes. The main difference occurs at the edge of the mode. For the numerical minimization themode is defined by a radius outside which the mode abruptly falls to zero, while the modeshape from micromagnetics has a more gradual mode profile near the mode edge. This gradual

== == − − − − =μ =μ Figure 9.2: Mode shapes for n=1 and n=2 modes from micromagnetic simulations for the out-of-plane geometry compared with bessel functions used for numerical minimization

mode profile is more physically permissible, as the abrupt cutoff of the bessel modewould result in a divergent exchange energy in a continuous film. The computation time for the micromagnetic method, together with the accuracy of the bessel function approximation suggests micromagnetics are unnecessary for the geometry of localized modes in an out-of- plane geometry, but it is reassuring that the two techniques provide similar results for the resonance field of the localized modes.

9.3 Analytical solution to localized precession in a parabolic field well

Magnetization precession localized within an inhomogeneous field was suggested by

Schl¨omannin 1964 [104]. He pointed out the similarity between the Landau-Lifshitz equation to the Schr¨odingerequation for a particle in a potential:

“The internal magnetic field H obviously plays the role of potential energy of this ‘particle.’ It would be interesting to shape the magnetic field in such a way that it has a deep minimum somewhere inside the sample. In that case our quasi-particle would never reach any surface of the sample and its properties might be expected to be particularly simple.”

This simple problem was indeed experimentally verified by Kalinikos et al. in the 1980’s [103] using the dipole field from a magnet pole with a hole along its axis to localize precession

ccH H H H H

ccμ Figure 9.3: Analytic solution to localized mode resonance field for the first four modes in mp thin film YIG and static field (including dipole field) H0 = Hext − 2 z3 . Solid lines are resonance field obtained using Eqn. 9.2 and markers are experimental data.

modes. They calculated an analytic solution for localized precession in a parabolic field well and for sufficiently small wavevectors such that exchange fields are negligible, which is a reasonable assumption for localization radius larger than a micron. Here, we apply

Kalinikos’ solution to localized modes confined by the dipole field from a micron sized probe and find very good agreement with the experimental data for the observed discrete modes in thin film YIG seen in chapter 8.

The analytic solution [103] is calculated for a parabolic magnetic field well of the form

2 H(z) = H0(1 + az ). The boundary condition is

Z zn kz(z)dz = χn (9.1) 0 where zn is defined as the position where kz(z = zn) = 0 and χn is the nth zero of the bessel

function J0(z). Note that the boundary condition used in chapter 8 and in the literature [85]

describes a localized mode with a single wavevector kn = χn/R, where R is the mode radius.

The two boundary conditions are equivalent if it assumed that kn is an average wavevector

77 R zn that satisfies knR = 0 kz(z)dz. However, the boundary condition involving the integral is more general, and physically it suggests that the mode varies its wavevector according to the local field H(z) but is injected at the position zn with wavevector k = 0, which explains how a uniform microwave field (which nominally only couples to k = 0 modes) can couple so easily to these higher order modes.

The resulting analysis [103] yields the resonance field s 3 2 H = − 3 H a χ 4πM t − H + 2H (9.2) n 0 10 n s 0 ext where 4πMs is the effective saturation magnetization field of the film, t is the film thickness

ω and Hext = γ − 4πMs is the applied external field for the uniform mode. For our case, the total field at the center of the field well z = 0 including external applied field and the dipole field from the probe magnet of moment mp at a distance z away is

m H = H − 2 p (9.3) 0 ext z3 and the parameter describing the parabolic field well from this probe magnet is found from a binomial expansion of the dipole field to be

m a = 6 p (9.4) 5 mp z (Hext − 2 z3 )

The expected resonance fields Hn for the first four localized modes n = 1 to 4 are plotted in Fig. 9.3 as solid lines together with the experimental data obtained in chapter8. The

−9 parameters are the same as before, t = 25 nm, 4πMs = 1608 Oe, mp = 3.9 × 10 emu, ω/2π = 4 GHz. This analytic solution shows very good agreement with the experimental data for n = 1, but begins to fail for the higher order modes. Nevertheless, the merit of the analytic solution is in its efficiency: it is at least 2 orders of magnitude faster to compute than the numerical minimization, and is a good representation of the first localized mode resonance field. The discrepancy for higher order modes can be attributed to the factthat exchange fields are not negligible for these higher wavevector modes, and this is confirmedby the fact that similar results are produced when exchange fields are ignored during numerical

78 NN N N

NNN μ

NNN Figure 9.4: Theoretical localized mode radii obtained by numerical and analytic methods together with experimental mode resolution obtained by imaging a permalloy film [85]

minimization.

In addition, the radius of the localized mode can be estimated by the analytic technique

[103] to be r 3 χ 4πM t R = 3 m s . (9.5) 10 aH0 This analytic radius can be compared with the numerical radius obtained by frequency minimization, together with the experimental resolution of the localized mode obtained by imaging a permalloy film in a previous study [85] and the result is shown in Fig. 9.4. The numerical method overestimates the mode radius, while the analytic method underesti- mates it, however both methods at least qualitatively describe the experimentally obtained resolution.

In conclusion it is found that analytic and micromagnetic methods can be used to com- plement the established numerical minimization method to solve the resonance condition for probe localized modes. It is found that while the analytic method is the fastest, it is only accurate for the first localized mode due the omission of exchange energy terms.

79 Adding exchange energy to the calculation would be a useful extension to the technique, but attempts have so far only led to lengthy equations that in the end have to be solved numerically. Micromagnetic simulations were found to accurately model the experimental data for all modes and provide insight into the true shapes of the modes, however this is achieved at the cost of increased computational time.

80 Chapter 10 Inhomogeneity and Vibrational Broadening of Localized Modes

10.1 Introduction

In chapter 8 it was demonstrated that the Gilbert damping parameter α of a localized

mode, measured from the slope of linewidth when plotted against microwave frequency,

is found to depend on the mode radius. There is also frequency-independent linewidth

broadening obtained where the linewidth intercepts the y-axis at ω = 0, which up to this

point has not been considered. This zero-frequency broadening in uniform mode FMR of

macroscopic samples has been attributed to inhomogeneities such as variations in anisotropy,

strain or saturation magnetization, that lead to spatially varying effective fields. These

perturbation fields act to redistribute energy by scattering from the main uniform modeto

nonuniform spin wave modes in the so-called two-magnon model [109,110]. In this chapter,

we shall study the inhomogeneous broadening that arises in localized mode spectra, which

is particularly illuminating in that we are able to independently vary the localization radius

and observe changes in the contribution of inhomogeneous linewidth broadening. This

should allow us to probe the lengthscale and magnitude of the effective field variations

within the sample.

81 9 9

zuuuRuuu uzuuuuRuuu zuuuRuuu uzuuuuRuuu zuuuRuuu uzuuuuRuuu uzuuuuRuuu uuuu Δ H u9 uuuu Δ H u9 u9 u9 Figure 10.1: Linewidths as a function of probe-sample separation z and mode radius R. a) Modes with radii R = 4800 nm to 3070 nm show a decreasing intercept as radius is decreased. b) Modes with radii R = 1420 nm to 1180 nm show an increasing intercept as radius is decreased.

10.2 Frequency-independent broadening due to vibrations

The experimental linewidths of the first localized mode are given in Fig. 10.1 for several

different probe-sample separations z. It can be clearly seen for the large probe-sample

separations in Fig. 10.1a the intercept or frequency-independent linewidth is decreasing

as z is decreased. The trend reverses for the smaller probe-sample separations shown in

Fig. 10.1b: as z decreases the intercept increases. For intermediate heights, the intercept

is almost constant, and this data was previously shown in Fig. 8.4.

The frequency-independent linewidth for small z can be attributed to an experimental

line broadening due to vibrations in the microscope. This broadening occurs due to the

dependence of resonance field H(z) on probe-sample separation z, such as in Fig. 9.3.

At a height z in the presence of vibrations ∆z that occur on a timescale faster than the

measurement timescale there is a smearing effect that results in the measured lineshape

being a sum of the resonance lines between H(z − ∆z) and H(z + ∆z). This leads to a

dHn linewidth broadening that depends on the slope of the resonance field dz , which is largest 82 at small probe-sample separations. From Fig. 9.3 it is clear that the slope of the resonance

2mp field is dominated by the static dipole field from theprobe z3 . Hence, we can determine the broadening of localized modes due to vibrations ∆z of the probe-sample distance:

dH 6m ∆H (∆z) ≈ 2 0 ∆z = 2 p ∆z (10.1) vib,n dz z4

To investigate this broadening as a function of mode radius, we use a quadratic fit to the inset of Fig. 8.2 to obtain a simple expression for mode radius R as a function of probe- sample separation z

R = 2.13 × 10−5 + 771z2 (10.2) substituting into Eqn. 10.1 we get an expression for vibrational broadening as a function of localized mode radius

771 2 ∆H = 12m ∆z (10.3) vib,n p R − 2.13 × 10−5

10.3 Size-Dependent Inhomogeneous Broadening

In addition to the experimental effect of vibrational broadening discussed above there exists a size-dependent frequency-independent linewidth contribution that dominates for larger mode radii such as in Fig. 10.1a. If this frequency-dependent broadening is due to inhomogeneities in the sample we expect it to scale with the mode volume. Hence, we add a phenomenological term to the linewidth that scales as R2, and so the total frequency-

independent linewidth is fit to

771 2 ∆H = 12m ∆z + BR2 (10.4) p R − 2.13 × 10−5

The experimental data for frequency-independent linewidth is shown in Fig. 10.2. There are two sets of data, one of which corresponds to the intercepts from Fig. 10.1 and Fig.

8.4. The second set of data was obtained in the presence of a cantilever mounting issue that amplified vibrations, and clearly the vibrational broadening was enhanced. Fits toEqn.

10.3 show that the magnitude of vibrations inferred from vibrational broadening matches

83 closely with the direct measurement of vibration obtained by touching the cantilever to the sample, which is discussed in section 7.5. The most striking result, however, is that the size-dependent parameter B = 20 Oe/µm2 is the same for both fits, suggesting that this is an intrinsic property of the sample.

More analysis is required to understand the broadening mechanism that scales with the size of the mode. On the one hand, one might expect there to be inhomogeneities in the sample that broaden the linewidth, and it has previously been shown that by patterning a sample down to sub-micron size it is possible to remove this linewidth contribution [111].

This is conceptually easy to understand in the context of a spin system without interactions: each spin is sensitive to the local field which may be spatially varying and hence different regions of the sample have different resonance fields, resulting in a line broadened onthe order of the field variation. A real ferromagnetic system, on the other hand, consists of coupled spins, and while different regions might see different magnetic fields, the exchange and dipole interactions couple these regions resulting in a common resonance frequency. If the inhomogeneous field varies on a lengthscale where dipole fields are dominant(>1 micron) then the inhomogeneously broadened line can be narrowed by dipolar coupling [112]. If the inhomogeneity lengthscale is on the order of interatomic spacing then the exchange interaction can be responsible for narrowing the linewidth [113]. These linewidth narrowing mechanisms are both responsible for uniform mode relaxation, where the volume of the magnetization precession is large and the inhomogeneities act as a pertubation that couple the uniform precession to degenerate magnons. For the case of localized modes, the largest field inhomogeneity comes from the dipole field of the probe itself, which is thesourceof the localization. Any inhomogeneities in the sample field on this lengthscale would bea small pertubation to the probe field, which would cause a frequency shift and not a linewidth broadening. Further analysis is required to explain the experimentally observed broadening.

In conclusion, we have demonstrated that localized mode FMRFM is a useful tool in characterizing frequency-independent broadening in a continuous thin film sample without the need for patterning. We find a linewidth contribution that scales with the size of the mode, suggesting that patterning below this length allows the measurement of in-

84 inise ntemcocp eefxd aapit o fe i r ae fromintercepts taken fix are to Eqn. fits forafter are lines points vibra- Solid Data after 10.1. fixed. and were Fig. before microscope of linewidth Solid the of in radius. frequency-dependence issues mode from tion localized intercepts of the function are a markers as linewidth Frequency-independent 10.2: Figure

88ov 10.4 8ov 888μ Δ888 85 888 888 ihftprmtr ipae nthefigure. on displayed parameters fit with 888μ Δ888 trinsic damping and eliminates extrinsic effects due to inhomogeneity [111]. This is an important result for spin-torque oscillators which can maximize their efficiency if linewidth is minimized. Further studies, such as spatial imaging of the resonance field by localized mode FMRFM, are required to confirm whether this linewidth broadening is due to spatial variations in the internal field of the ferromagnet.

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