Spin Wave Confinement: Propagating Waves, Second Edition [2 ed.] 9814774359, 978-981-4774-35-2, 9781351617215, 1351617214, 978-1-315-11082-0, 117-118-119-1, 162-164-165-1

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Spin Wave Confinement

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Spin Wave Confinement Second Edition

Propagating Waves

editors

Preben Maegaard Anna Krenz Wolfgang Palz

edited by

Sergej O. Demokritov

The Rise of Modern Wind Energy

Wind Power

for the World

Published by Pan Stanford Publishing Pte. Ltd. Penthouse Level, Suntec Tower 3 8 Temasek Boulevard Singapore 038988

Email: [email protected] Web: www.panstanford.com British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Spin Wave Confinement: Propagating Waves (2nd Edition) Copyright © 2017 Pan Stanford Publishing Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-4774-35-2 (Hardcover) ISBN 978-1-315-11082-0 (eBook)

Printed in the USA

Contents



Introduction

1

1. Graded Magnonic Index and Spin Wave Fano Resonances in Magnetic Structures: Excite, Direct, Capture 11

V. V. Kruglyak, C. S. Davies, Y. Au, F. B. Mushenok, G. Hrkac, N. J. Whitehead, S. A. R. Horsley, T. G. Philbin, V. D. Poimanov, R. Dost, D. A. Allwood, B. J. Inkson, and A. N. Kuchko

1.1 Introduction

11

1.4 Spin Wave Steering

25

1.2 Spin Wave Dispersion 1.3 Spin Wave Excitation 1.5 Spin Wave Output

1.6 Spin Wave Control and Magnonic Devices 1.7 Conclusions and Outlook

2. Coupled Spin Waves in Magnonic Waveguides

15 18 31 32 37

47

Yu. P. Sharaevsky, A. V. Sadovnikov, E. N. Beginin, M. A. Morozova, S. E. Sheshukova, A. Yu. Sharaevskaya, S. V. Grishin, D. V. Romanenko, and S. A. Nikitov

2.1 Introduction

2.2 Theoretical Approach

2.3 Spin Waves in Coupled Magnetic Stripes

2.4 Nonlinear Spin Wave Coupling in Magnonic Crystals 2.5 Multilayer Magnonic Crystals

2.6 Frequency-Selective Tunable Spin Wave Channeling 2.7 Conclusions

47

48 51 56

60

67 72

vi

Contents

3. Tuning of the Spin Wave Band Structure in Nanostructured Iron/Permalloy Nanowire Arrays 77

G. Gubbiotti, S. Tacchi, R. Silvani, M. Madami, G. Carlotti, A. O. Adeyeye, and M. Kostylev

3.1 Introduction 3.2 Sample Fabrication and Brillouin Light Scattering Measurements 3.3 Micromagnetic Simulations 3.4 Results and Discussion 3.4.1 Spin Wave Band Structure and Mode Spatial Profiles for the NWs with Rectangular Cross Section 3.4.2 Spin Wave Band Structure and Mode Spatial Profiles for the NWs with L-Shaped Cross Section 3.5 Conclusions

4. Magnetization Dynamics of Reconfigurable 2D Magnonic Crystals

78 80 83 84

84 88 91

99

G. Shimon, A. Haldar, and A. O. Adeyeye

4.1 Introduction 4.2 Experiments and Simulations 4.2.1 Sample Fabrication 4.2.2 Ferromagnetic Resonance Spectroscopy 4.2.3 Microfocused Brillouin Light Scattering Spectroscopy 4.2.4 Micromagnetic Simulations 4.3 Coupled Nanodisks 4.3.1 Effect of Interdisk Separations 4.3.1.1 BLS spectra 4.3.1.2 Simulated spectra 4.3.1.3 2D mode profiles 4.3.1.4 Dipolar field estimation 4.3.2 Configurational Anisotropy

100 102 102 104

105 107 108 108 109 111 111 113 114

Contents



4.3.2.1 Resonant spectra and 2D mode profiles 4.3.2.2 Dipolar field models and estimation 4.4 Reconfigurable Magnetization Dynamics 4.4.1 Rhomboid Nanomagnets 4.4.2 Two- or Three-Coupled Rhomboid Nanomagnets 4.4.3 2D Magnonic Crystals 4.4.3.1 Synthetic antiferromagnets 4.4.3.2 Synthetic ferrimagnets 4.5 Summary

5. Spin Wave Optics in Patterned Garnet

115

117 118 119 121 124 124 126 129

139

Ryszard Gieniusz, Andrzej Maziewski, Urszula Guzowska, Paweł Gruszecki, Jarosław Kłos, Maciej Krawczyk, and Alexander Stognij

5.1 Introduction 5.2 Spin Waves in Patterned YIG Micrometer Films 5.2.1 Experimental Methods and Samples Details 5.2.2 Spin Waves Interaction with a Single Antidot in YIG Micrometer-Thick Films 5.2.3 Spin Wave Interaction with a Line of Antidots in YIG Micrometer Films 5.2.4 Modeling Spin Waves Total Nonreflection Effect 5.2.5 Application of Total Nonreflection Effect for Spin Wave Beam Switching 5.3 Optics of Spin Waves in Nanometer-Thick YIG Film 5.3.1 Reflection of Spin Waves from the Edge of the YIG Thin Film: Goos–Hänchen Effect 5.3.2 Molding of Spin Wave Refraction in Two-Dimensional YIG Antidots Lattice

140 144 144

145 149 152 156

157

157 161

vii

viii

Contents

5.3.2.1 Angular filtering 5.3.2.2 All-angle collimation 5.4 Summary

6. Spin Waves in Circular and Linear Chains of Discrete Magnetic Elements

162 164 165

171

Yu. N. Barabanenkov, S. A. Osokin, D. V. Kalyabin, and S. A. Nikitov



6.1 Introduction 6.2 Multiple-Scattering Method 6.2.1 Circular Arrays 6.2.2 Linear Chains 6.3 Radiation Losses 6.4 Results 6.5 Conclusion

171 174 179 180 181 187 192

7. Magnonic Grating Coupler Effect and Microwave-to-Magnon Transducers for Exchange-Dominated Spin Waves 197

Haiming Yu and Dirk Grundler

7.1 Introduction 7.2 Mix-and-Match Lithography for Mesas with Magnonic Grating Couplers 7.2.1 Photolithography to Prepare a Film Mesa 7.2.2 Electron-Beam Lithography and Lift-Off Processing for Magnetic Nanostructures 7.2.3 Integrated Coplanar Waveguide 7.3 Antenna Design for Spin Wave Excitation and Detection 7.3.1 Coplanar Waveguide 7.4 All-Electrical Spin Wave Spectroscopy 7.4.1 Scattering Parameters 7.4.2 VNA Calibration 7.4.3 Measurement Configuration and Data Analysis 7.5 Spin Wave Properties Studied by Experiments 7.5.1 Spin Wave Group Velocity

197 200 200 201 202

202 202 204 206 206

207 207 207

Contents



7.5.2

Decay Length and Nonreciprocity Parameter 7.6 Performance of a Spin Wave Grating Coupler 7.6.1 Grating Coupler–Induced Spin Wave Modes 7.6.2 Towards Omnidirectional Spin Wave Emission 7.6.3 Enhanced Magnon Excitation via Resonant Nanodisks 7.6.4 Sub-100 nm-Wavelength Spin Waves 7.6.5 Angular Dependance of Propagating Grating Coupler Modes 7.7 Conclusions and Outlook

8. Spin Waves on Spin Structures: Topology, Localization, and Nonreciprocity

212

212 213

213 215

219

Robert L. Stamps, Joo-Von Kim, Felipe Garcia-Sanchez, Pablo Borys, Gianluca Gubbiotti, Yue Li, and Robert E. Camley

8.1 Introduction 8.2 Chiral Interactions and Spin Waves 8.2.1 Nonreciprocity: Symmetry Breaking through the DMI 8.2.2 Caustics 8.3 Localization and Reconfigurability 8.3.1 Domain Wall Channeling 8.3.2 Edge (Partial Wall) Channeling 8.3.3 Magnetic Configurations in Artificial Spin Ice 8.3.4 Reprogrammable Microwave Response 8.4 Outlook

9. Steering Magnons by Noncollinear Spin Textures

208 209 211

219 222 223 226 230 231 235 241 247 250

261

Katrin Schultheiss, Kai Wagner, Attila Kákay, and Helmut Schultheiss

9.1 Introduction 9.2 Magnon Transport and Dispersion in Magnonic Waveguides

262

265

ix



Contents

9.3 Steering and Multiplexing Magnons by Current-Induced, Local Magnetic Fields 9.4 Channeling Magnons in Magnetic Domain Walls 9.5 Conclusions and Outlook

10. Current-Induced Spin Wave Doppler Shift

270 280 288

295

Matthieu Bailleul and Jean-Yves Chauleau





10.1 Introduction 10.2 A Doppler Shift for Spin Waves 10.2.1 Spin Waves in a Drifting Electron Population 10.2.2 Influence of Spin Transfer Torque on Spin Wave Dynamics 10.3 Experimental Observations 10.3.1 Frequency Domain Inductive Measurements 10.3.2 Time Domain Inductive Measurements 10.3.3 Magneto-Optical Measurements 10.4 Parametrizing the Two-Current Model 10.4.1 Definitions of the Degree of Spin Polarization 10.4.2 Spin-Dependent Electron Scattering 10.4.3 Spin-Polarized Transport in Permalloy Films 10.4.4 Spin-Polarized Transport in Other Materials 10.5 Extraction of the Non-Adiabatic Spin Transfer Torque Parameter 10.6 Other Types of Spin Wave Frequency Shifts 10.6.1 Zero-Current Spin Wave Frequency Non-Reciprocity 10.6.2 Reciprocal Oersted-Field-Induced Frequency Shift 10.6.3 Non-Reciprocal Oersted-Field-Induced Frequency Shift 10.7 Conclusion and Perspectives

296 297 297 299 301

301 304 306 309

310 311 312 314

314 318 319

321

322 324

Contents

11. Excitation and Amplification of Propagating Spin Waves by Spin Currents

Vladislav E. Demidov and Sergej O. Demokritov

11.1 Introduction 11.2 Experimental Technique 11.3 Excitation of Guided Spin Waves by Spin-Polarized Currents 11.4 Control of the Propagation Length of Spin Waves by Pure Spin Currents 11.4.1 SHE Spin-Wave Control in All-Metallic Magnonic Waveguides 11.4.2 SHE Spin-Wave Control in YIG-Based Magnonic Waveguides 11.5 Excitation of Spin Waves by Pure Spin Currents 11.5.1 Excitation of Continuous Propagating Spin Waves 11.5.2 Excitation of Short Spin-Wave Packets 11.6 Conclusions

12. Propagating Spin Waves in Nanocontact Spin Torque Oscillators

329 332

335 339

339 342 346 346 351 355

363

Randy K. Dumas, Afshin Houshang, and Johan Åkerman



12.1 Introduction 12.2 Nanocontact Spin Torque Oscillators 12.3 Magnetodynamical Modes 12.3.1 Role of the Oersted Field 12.4 Asymmetric Spin Wave Propagation 12.5 Spin Wave Beam–Driven Synchronization 12.6 Conclusions and Future Directions

13. Parametric Excitation and Amplification of Spin Waves in Ultrathin Ferromagnetic Nanowires by Microwave Electric Field

329

363 365 368 369 370 373 377

385

Roman Verba, Mario Carpentieri, Giovanni Finocchio, Vasil Tiberkevich, and Andrei Slavin

13.1 Introduction

386

xi

xii

Contents

13.2 Excitation of Spin Waves 13.2.1 Efficiency of the Parametric Interaction and Excitation Threshold 13.2.1.1 Perpendicularly magnetized nanowire 13.2.1.2 Nanowire with in-plane static magnetization 13.2.1.3 Notes on multimode waveguides 13.2.2 Nonlinear Spin Wave Dynamics under Parametric Pumping: Stationary Amplitudes of Excited Spin Waves 13.3 Amplification of Spin Waves by Parametric Pumping 13.3.1 Linear Regime of the Parametric Amplification 13.3.2 Amplification of Large-Amplitude Spin Waves: Stabilization of Spin Wave Amplitudes 13.4 Effect of Interfacial Dzyaloshinskii–Moriya Interaction on Parametric Processes 13.4.1 Spin Wave Nonreciprocity Induced by Interfacial Dzyloshinskii–Moriya Interaction 13.4.2 Parametric Amplification of Nonreciprocal Spin Waves 13.5 Summary

Index

388 388

388 396 399

402

407

407 410

412

412 414 418

427

Introduction The book Spin Wave Confinement: Propagating Spin Waves, which you are about to read, reflects an increasing interest of the magnetic community in dynamic excitations in magnetic systems of reduced dimensions. It flashes the development of the field since 2008, when the first edition of the book, Spin Wave Confinement, appeared. Even at that time it was clear to all contributors that future reduction in the sizes and dimensions of the studied magnetic systems is inevitable. Meanwhile, the successful development of magnonics [1–4] as an emerging subfield of spintronics, which includes not only the traditional branches of spinwave research but also considers spin waves (or their quanta, magnons) as a basis for smaller, faster, more robust, and more power-efficient electronic devices, on the basis of propagating spinwaves, is obvious. The concept of spin waves was introduced by Bloch [5], who theoretically considered quantum states of exchanged-coupled spins slightly deviating from their equilibrium orientations. He found that these disturbances were dynamic: they propagate as waves through the medium. Later, Holstein and Primakoff [6] and Dyson [7] generalized this model by taking into account weaker magnetic dipole interaction between spins and their interaction with the external magnetic field. This development was followed by the introduction of the term for the quanta of spin waves, “magnons,” in the late 1950s. One of the important distinctions of magnons from photons and phonons is their anisotropic dispersion: the energy/frequency of a magnon depends on the orientation of the corresponding wave vector relative to the orientation of the static magnetization due to the contribution of the relativistic anisotropic magnetic dipole interaction. The unique features of magnons, such as the possibility to carry spin information over relatively long distances, the possibility to achieve submicrometer wavelength at microwave frequencies, and controllability by electronic signals via magnetic fields, make



Introduction

magnonic devices uniquely suited for implementation of novel integrated electronic schemes characterized by high speed, low power consumption, and extended functionalities. The utilization of magnons for integrated electronic applications is addressed within the emerging field of magnonics. Recent advances in spintronics and nanomagnetism created essentially new possibilities for magnonics and brought it onto a new development stage. Of particular importance here is the recent discovery of the spintransfer torque [8–10] and the spinHall effect [11–13], both of which have already been demonstrated to enable novel device geometries and functionalities [14–18]. The discovery of the spinwave Doppler effect [19] has opened an alternative way to control spin waves by electric currents. An important recent development in nanomagnetism is the understanding of the fact that the relativistic Dzyaloshinskii–Moriya interaction can exist not only due to broken bulk inverse symmetry [20, 21] but also due to asymmetric interfaces of ultrathin magnetic layers [22]. Coming back to spin wave confinement, the history of spin wave/magnonic studies clearly shows a progressively increasing interest toward confined magnetic systems. Although the famous T3/2 Bloch law is valid for three-dimensional magnets, the focus of the researchers has moved since 1950s to two-dimensional objects, thin films and magnetic multilayers, resulting in the discovery of quantized resonances [23, 24], the surface Damon– Eshbach mode [25, 26], and interlayer coupling [27]. Later in the 1990s quasi-one-dimensional stripes became the most actively studied magnetic systems, enabling the discovery of lateral quantization [28] and edge modes [29, 30]. An important step toward the development of magnonics was the experimental demonstration of collective propagating spin waves in closely packed arrays of stripes [31, 32], followed by studies of collective modes in arrays of single- and bicomponent dots and antidots [33]. Later the investigation of the magnonic effect was extended to two-dimensional arrays of single- and bicomponent magnetic dots and antidots. The theoretical prediction of the spintorque effect [8, 9] and the development of novel techniques for nanofabrication allowed for the investigation of magnetic zerodimensional objects such as the spintorque nano-oscillator [34–40].

Introduction

The book is structured as follows:

In the first chapter, Kruglyak et al. present the basic classification of magnonic effects and emphasize the propagating nature of spin wave excitations. In fact, due to the inevitably non-uniform magnetization and an internal magnetic field in confined magnetic systems, propagation of spin waves takes place in a non-uniform environment, which can be characterized by a graded index of refraction. Surprisingly, such a simple approach brings about a deep understanding of complex spin wave phenomena, such as the spin wave analogue of Fano resonance. Besides the interesting physics, the chapter highlights technical opportunities associated with spin wave propagation in systems with a graded index, important for future magnonic devices and networks. In chapter 2, Sharaevsky et al. address spin wave–/magnoniccoupled waveguides as the most promising candidates for effective channeling of spin waves between the functional units of magnonic networks. They develop a theoretical approach for description of magnonic couplers in both lateral and vertical geometries. The presented theory is compared with experimental studies of spin wave propagation in such structures using both broadband microwave spectroscopy and Brillouin light scattering spectroscopy. Chapter 3, written by Gubbiotti et al., presents results of combined experimental and theoretical investigations of propagating spin waves in bilayered iron/permalloy (Fe/Py) nanowaveguides with both regular cross sections (bottom and upper layers of equal width) and L-shaped cross sections (upper layer of half width). Particular attention is paid to the bands created in the spin wave spectrum because of finite sizes of the waveguides. For the case of nanowires with an L-shaped crosssection, two dispersive modes with a sizeable magnonic band have been observed. These are interpreted as the fundamental modes of either the thick or the thin portion of each nanowaveguide. In chapter 4, Shimon, Haldar, and Adeyeye investigate the spin wave dynamics in reconfigurable two-dimensional magnonic crystals. A simple example of such system is a cluster of two or more magnetically coupled disks. It is shown that the frequencies of the spin wave eigenmodes of one disk depend on the orientations of its magnetization with respect to its neighbors. The second, more complex, studied system is a network of





Introduction

rhomboid nanomagnets, where a reliable reconfiguration between ferromagnetic, antiferromagnetic, and ferrimagnetic ground states can be realized deterministically. Correspondingly, the microwave properties of such system can be efficiently controlled. In chapter 5 Gieniusz et al. present a comprehensive study of spin wave optics in patterned yttrium iron garnet films. A variety of effects, such as reflection, refraction, interference, diffraction, and focusing of spin waves, as well as the Goos–Hänchen effect are demonstrated. Experimentally, Brillouin light scattering spectroscopy was used for imaging of spin wave interaction with a single antidot and a line of antidots. Especially interesting is the demonstration of the total nonreflection of spin waves by a line of antidots. In this case, a highly focused beam of spin waves propagates along and behind the line. Low-pass angular filtering and high-pass angular filtering of spin waves are observed in twodimensional antidot lattices for specific ranges of the spin wave frequencies. Chapter 6, written by Barabanenkov et al., is devoted to a theoretical description of spin waves in circular and linear chains of confined magnetic elements, which provides a spin wave analogue to the optical theorem for the T-scattering operator. In particular, a general theory for multiple scattering of forwardvolume magnetostatic spin waves by a two-dimensional ensemble of cylindrical magnetic inclusions in a ferromagnetic matrix is built. Using this theory, the authors show that such elements arranged periodically along a circle demonstrate spin wave eigenmodes with high values of the mode quality factor. Besides, the authors also demonstrate that a linear chain of such elements can serve as an efficient microwaveguide for spin wave propagation. In chapter 7 Yu and Grundler investigate the magnonic grating coupler effect in magnonic crystals and its application to realization of multidirectional microwave-to-magnon transducers for excitation of high-frequency spin waves with wavelengths below 100 nm. It is also shown that by further optimization, spin waves with wavelengths of about 20 nm may be excited. In chapter 8 Stamps et al. address the propagation of spin waves in nontrivial magnetic structures (e.g., along domain walls) and the consequences of the Dzyaloshinskii–Moriya interaction on their dispersion. In particular, the authors discuss how the

Introduction

Dzyaloshinskii–Moriya interaction affects the gap between the frequencies of freely propagating spin waves and the spin waves channeled along domain walls, as well as consequent nonreciprocities. Moreover, they analyze the possibility of creating a mesoscopic metamaterial analogue of domain wall channeling, based on artificial spin ice, which is an arrangement of interacting single-domain nanomagnets. Through manipulation of the static magnetic configurations in artificial spin ice by application of magnetic fields, many details of the allowed microwave frequency spin wave eigenmodes can be controlled. In chapter 9 Schultheiss et al. discuss propagation of magnons in noncollinear spin textures. They present experimental studies on magnon transport in metallic, ferromagnetic microstructures, with a focus on actively controlling the magnon propagation path by using two inherent characteristics of magnons: the anisotropy of the magnon’s dispersion law and its sensitivity to changes in the internal magnetic field distribution. They show how these features can be utilized toward realizing complex magnonic networks, a magnon multiplexer for switching the magnon propagation path and channeling magnons inside magnetic domain walls being good examples. In chapter 10 Bailleul and Chauleau provide an overview of the current-induced spin wave Doppler effect, which is observed when a spin wave propagates in a metallic ferromagnet, subjected to an electrical current. In this case, the spin wave continuously accumulates an additional phase due to motion of the conducting electrons in the metal. Predicted a long time ago, this phenomenon has recently been experimentally demonstrated. Here this effect is interpreted in terms of adiabatic spintransfer torque, and the experimental demonstration of the effect using both inductive (frequency or time domain) and magneto-optical techniques is presented. Application of the spin wave Doppler effect to extract relevant information on the spin-polarized electrical transport is shown. Moreover, a related effect, namely current-induced modification of spin wave attenuation associated with nonadiabatic spin transfer torque, is discussed. Chapter 11 by Demidov and Demokritov is devoted to excitation and amplification of propagating spin waves by spin current microscopic waveguides by using the spintransfer torque effect. First, the excitation of propagating guided spin waves by utilizing





Introduction

traditional spintransfer torque devices driven by spin-polarized electric currents is discussed. Then experiments on the control of spin wave propagation by using pure spin currents created by the spinHall effect for the cases of all-metallic magnonic waveguides and waveguides based on ultrathin insulating yttrium iron garnet films are presented. Finally, it is shown that pure spin currents created by the nonlocal spininjection mechanism can be utilized for the efficient excitation of propagating spin waves with a large propagation length and short spin wave packets with the duration down to a few nanoseconds. In chapter 12 Dumas, Houshang, and Åkerman focus on nanocontact spintorque oscillators—devices in which highamplitude precession of magnetization is driven by the spintransfer torque due to locally injected spin-polarized electric or pure spin currents. Particular attention is given to spin waves associated with such oscillators and to the synchronization of two or more oscillators mediated by such spin waves. By “daisy chaining” of several oscillators one can extend the spinwave propagation length, forming a type of spin wave repeater capable of transporting information over much larger distances than the intrinsic propagation length of the spin wave. Besides, the oscillators are considered as potential building blocks for next-generation neuromorphic computing architectures, which aim to mimic the neurobiological functionality found in the human brain. Finally, in chapter 13 Verba et al. present an analytic theory of parametric excitation and amplification of spin waves by microwave electric fields. In particular, the authors focus on the application of the voltage-controlled magnetic anisotropy effect, which can be used for spinwave signal processing by electric fields. It is shown that a microwave electric field allows excitation of propagating spin waves or amplification of small-amplitude spin waves if the spinwave frequency is half of the microwave frequency. Besides, the effect of the interfacial Dzyaloshinskii– Moriya interaction on the amplification of spin waves in ultrathin ferromagnetic waveguides is analyzed. This book, together with the first book of this series, provides a rich spectrum of information on spinwave dynamics for scientists

References

working in the area of spintronics and magnonics and will serve as a textbook for graduate students starting in this field. I am grateful to all the contributors to the book. Without their efforts, this project could not have been successful. Last but not least, I would like to thank the Deutsche Forschungsgemeinschaft and the Russian Ministry of Education and Science (Megagrant № 14.Z50.31.0025) for financial support. Sergej O. Demokritov Yekateringburg January 2017

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29. Jorzick, J., Demokritov, S. O., Hillebrands, B., Berkov, D., Gorn, N. L., Guslienko, K., and Slavin, A. N. (2002). Spin wave wells in nonellipsoidal micrometer size magnetic elements, Phys. Rev. Lett., 88, 047204.

30. Park, J. P., Eames, P., Engebretson, D. M., Berezovsky, J., and Crowell, P. A. (2002). Spatially resolved dynamics of localized spin-wave modes in ferromagnetic wires, Phys. Rev. Lett., 89, 277201. 31. Gubbiotti, G., Tacchi, S., Carlotti, G. Singh, N., Goolaup, S., Adeyeye, A. O., and Kostylev, M. (2007). Collective spin modes in monodimensional magnonic crystal consisting of dipolarly coupled nanowires, Appl. Phys. Lett., 90, 092503. 32. Wang, K., Zhang, V. L., Lim, H. S., Ng, S. C., Kuok, M. H., Jain, S., and Adeyeye, A. O. (2009). Observation of frequency band gaps in a onedimensional nanostructured magnonic crystal, Appl. Phys. Lett., 94, 083112.

33. Krawczyk, M., Mamica, S., Mruczkiewicz, M., Klos, J. W., Tacchi, S., Madami, M., Gubbiotti, G., Duerr, G., and Grundler, D. (2013). Magnonic band structures in two-dimensional bi-component magnonic crystals with in-plane magnetization, J. Phys. D: Appl. Phys., 46, 495003. 34. Tsoi, M., Jansen, A. G. M., Bass, J., Chiang, W.-C., Seck, M., Tsoi, V., and Wyder, P. (1998). Excitation of a magnetic multilayer by an electric current, Phys. Rev. Lett., 80, 4281–4284. 35. Tsoi, M., Jansen, A. G. M., Bass, J., Chiang, W.-C., Seck, M., Tsoi, V., and Wyder, P. (2000). Generation and detection of phase-coherent current-driven magnons in magnetic multilayers, Nature, 406, 46–48.

36. Kiselev, S. I., Sankey, J. C., Krivorotov, I. N., Emley, N. C., Schoelkopf, R. J., Buhrman, R. A., and Ralph, D. C. (2003). Microwave oscillations of a nanomagnet driven by a spin-polarized current, Nature, 425, 380–383.

37. Rippard, W. H., Pufall, M. R., Kaka, S., Russek, S. E., and Silva, T. J. (2004). Direct-current induced dynamics in Co90Fe10/Ni80Fe20 point contacts, Phys. Rev. Lett., 92, 027201.



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Introduction

38. Krivorotov, I. N., Emley, N. C., Sankey, J. C., Kiselev, S. I., Ralph, D. C., and Buhrman, R. A. (2005). Time-domain measurements of nanomagnet dynamics driven by spin-transfer torques, Science, 307, 228–232.

39. Demidov, V. E., Urazhdin, S., and Demokritov, S. O. (2010). Direct observation and mapping of spin waves emitted by spin-torque nano-oscillators, Nat. Mater., 9, 984–988. 40. Demidov, V. E., Urazhdin, S., Ulrichs, H., Tiberkevich, V., Slavin, A., Baither, D., Schmitz, G., and Demokritov, S. O. (2012). Magnetic nanooscillator driven by pure spin current, Nat. Mater., 11, 1028–1031.

Chapter 1

Graded Magnonic Index and Spin Wave Fano Resonances in Magnetic Structures: Excite, Direct, Capture V. V. Kruglyak,a C. S. Davies,a Y. Au,a F. B. Mushenok,a G. Hrkac,a N. J. Whitehead,a S. A. R. Horsley,a T. G. Philbin,a V. D. Poimanov,b R. Dost,c D. A. Allwood,c B. J. Inkson,c and A. N. Kuchkod aSchool

of Physics, University of Exeter, Exeter, United Kingdom National University, Donetsk, Ukraine cDepartment of Materials Science and Engineering, University of Sheffield, Sheffield, S1 3JD, United Kingdom dInstitute of Magnetism of NAS of Ukraine, Kiev, Ukraine bDonetsk

[email protected]

Starting from the general topic and fundamentals of magnonics, we discuss and provide demonstrations of exciting new physics and technological opportunities associated with the graded magnonic index and spin wave Fano resonances, highlighting them as the next big thing in magnonics research.

1.1  Introduction

The most general definition of magnonics [1, 2], as the study of spin waves [3, 4], leaves a lot of freedom for interpretation Spin Wave Confinement: Propagating Waves (2nd Edition) Edited by Sergej O. Demokritov Copyright © 2017 Pan Stanford Publishing Pte. Ltd. ISBN  978-981-4774-35-2 (Hardcover), 978-1-315-11082-0 (eBook) www.panstanford.com

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Graded Magnonic Index and Spin Wave Fano Resonances in Magnetic Structures

and scientific discussion of directions of the field’s further development. Thus, we have recently seen a number of excellent review papers with emphasis on different aspects of spin wave research and technology, e.g., magnonic crystals and metamaterials [5–9], photomagnonics [10, 11], spin caloritronics [12], magnon spintronics [13–15], nanoscience [16–19], and applications of spin waves in microwave signal processing and data manipulation [20–22]. There is, however, an aspect of magnonics that has been both ubiquitous and somewhat underrated so far: magnonics is the study not only of spin but also (and most importantly) of waves, which have an extremely rich and peculiar dispersion that is nonlinear, anisotropic, and nonreciprocal. The spin wave dispersion is very sensitive to the sample’s magnetic properties and micromagnetic state, including both the internal magnetic field and magnetization, so that spin waves are rarely observed to propagate in uniform media. Inspired by and feeding from other fields of wave physics, such as quantum mechanics [23], graded-index optics [24], and transformation optics [25], we have recently tried to formulate the concept of graded-index magnonics as a unifying theme focusing on general aspects of spin wave excitation and propagation in media with continuously nonuniform properties [26, 27]. Graded-index magnonics is the main topic of this chapter, and is also highlighted throughout the rest of this book. The term graded-index magnonics implies existence of a single quantity, often dubbed magnonic or spin wave refractive index or more simply magnonic index, that describes fully the spin wave dispersion in a continuous and uniform sample. Yet, it would be extremely difficult (if at all possible) to do this, given the plethora of factors that influence the (already complex) spin wave dispersion, with their sheer diversity necessarily limiting any definition to special cases and approximations [28–35]. So, we treat the term magnonic index here as a tag for the entirety of the spin wave dispersion and its modifications dictated by the variation of the magnetic medium’s properties. With a historical perspective on graded-index magnonics already presented by Davies and Kruglyak [27], we limit coverage of this chapter mostly to our own published and forthcoming results, which deal with the spatial variation of the magnonic index induced by patterning thin magnetic films of the same

Introduction

material (Permalloy). Such patterning has been extensively discussed in the context of spin wave confinement in the previous book of this series [36]. So, given the additional focus of this book on spin wave propagation, this leads to another topic of central importance for this chapter: spin wave Fano resonances [37]. Resulting from the interaction between systems with a discrete spectrum and continuous spectra, Fano resonances have been widely studied for other wave excitations [38] but have been explored and exploited much less in magnonics [39, 40]. Here, we will demonstrate retrospectively their relevance to some of our previous observations. It is conventional to discuss spin wave phenomena in the context of magnonic technology, focusing on data or microwave signal processing. The areal density of transistors in integrated circuits was famously conjectured by Moore [41] in 1965 to double every two years as part of Moore’s law. This growth has been consistently observed, and modern handheld computing devices can easily contain gigabytes of memory and process data at gigahertz rates. However, in the most recent period, the transistor areal density took three years to double, and the clock rate of modern processers has not advanced since 2004 [42, 43]. To safeguard the technological progress, condensed matter researchers are now striving to develop technologies that will enable computing devices to break past the barriers faced by electronics, with magnonics being one of such emerging alternatives. More generally, magnonics is envisioned to become a natural companion to electronic, spintronic, and microwave technologies, which could offer additional functionalities (e.g., nonvolatility and magnetic field tunability/programmability) to the more conventional technologies. The place of magnonics among its sister fields of research and associated technologies is illustrated in Fig.  1.1a. Direct current (dc) electricity and conventional semiconductor electronics use the one-way translational motion of charge to transmit energy and information across a circuit, while the same goal is achieved using charge oscillations in ac electricity and electromagnetics [44]. Each charged particle in an ensemble can experience a local electromagnetic oscillation relative to its individual equilibrium position. Collectively, these phasecoherent oscillations give rise to a net wavelike motion, which

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Graded Magnonic Index and Spin Wave Fano Resonances in Magnetic Structures

transmits energy (signal) without charge or particle transport. An immediate successor of electronics, spintronics, instead exploits the translational motion of spins. However, the use of the wave motion of spins to carry and process information has only just begun to be explored, giving rise to the recent burst in magnonics research. The relationship between spintronics and magnonics is analogous to that between dc electricity and electromagnetics [45], with an important difference that spin waves can transfer not only energy but also angular momentum [12–15].

Figure 1.1  (a) The relationship between magnonics and its sister fields of research and technology. (b) A block diagram of a generic magnonic device and its constituents. Adapted from Davies et al. [46].

This chapter is organized using the scheme introduced earlier by Kruglyak et al. [1]. Specifically, we consider the generic magnonic device shown in Fig.  1.1b. Then, even findings of a fundamental nature can be discussed in terms of their relevance

Spin Wave Dispersion

to one of the four constituents of the device: source, functional medium, output, and control (mechanism). We show that the concepts of graded magnonic index and spin wave Fano resonances are relevant to each of the constituents. In Section 1.2, we remind the reader about the main aspects of the spin wave dispersion. In Sections 1.3, 1.4, and 1.5, we discuss how the graded magnonic index and spin wave Fano resonances can be used to aid spin wave emission, steering, and detection, respectively. In Section 1.6, we discuss spin wave control and show some conceptual device designs, for illustration.

1.2  Spin Wave Dispersion

Generally, magnetic systems contain ordered ensembles of tiny magnetic moments—spins—coupled by the exchange interaction. On a quasi-classical basis, the magnetic moments are individually capable of precessing about their equilibrium orientation. Due to the coupling between spins, it is possible to excite phasecoherent precessional waves of magnetization—the average magnetic moment per unit volume. It is these waves that are called “spin waves,” and their quanta are called “magnons” [3, 4]. Like other waves, spin waves are characterized by their amplitude, phase, frequency, wave vector, and group and phase velocities, each representing a resource for signal manipulation. The key feature that makes spin waves unique is their dispersion (Fig.  1.2a), which can be peculiarly anisotropic depending on the dominant interaction between magnetic moments [3, 4]. There are two main interactions to consider. The quantum-mechanical exchange interaction (responsible for the magnetic ordering) dominates over nanometer wavelengths and gives rise to an isotropic, parabolic dispersion of so-called exchange spin waves. In the wavelength range from hundreds of nanometres and above, spin waves are said to be magnetostatic (or dipolar) in nature, since their dispersion is dominated by the anisotropic magneto-dipole interaction. In the intermediate wavelength range, when both exchange and dipolar energies contribute noticeably to the dispersion, the spin waves are said to be of dipole exchange character. In either case, spin waves travel with typical speeds of several kilometers per second,

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Graded Magnonic Index and Spin Wave Fano Resonances in Magnetic Structures

and so are admittedly rather slow. However, this also means that, at the same frequency, they have a much shorter wavelength compared to, e.g., electromagnetic waves, paving the way toward device miniaturization.

Figure 1.2 (a) The magnetostatic spin wave dispersion plotted in one quadrant of reciprocal space for a film in the yz plane with in-plane magnetization in the z direction. Above and below the FMR frequency fFMR, there is a single and infinite manifold of dispersion branches, referred to as the surface (Damon–Eshbach) mode and backward volume modes, respectively. Only the first branch of the volume spin wave modes is shown, corresponding to the uniform precession across the film thickness. (b) The isofrequency curves characterizing the propagation of magnetostatic spin waves, where f2 > fFMR > f1. (c)  The isofrequency curves characterizing the propagation of dipole exchange spin waves, where f2 > f1 > fFMR.

Spin Wave Dispersion

The frequency of spin waves with infinite wavelength (and therefore also zero wave vector) is called the ferromagnetic resonance (FMR) frequency, fFMR, which corresponds to the magnon energy gap. An incident microwave at this frequency will couple to the magnetization precession in the sample most strongly, exciting its uniform (fundamental) precessional mode. A constant magnetic field applied to the sample (i.e., bias magnetic field) shifts fFMR up or down, along with the rest of the dispersion. The direction and speed of the energy transfer by spin waves is determined by their group velocity, defined as the gradient of the angular frequency in reciprocal space. In addition to the bias magnetic field, the spin wave group velocity depends dramatically on the angle between the spin wave’s wave vector and the magnetization. In particular, the group velocity is negative (roughly antiparallel to the wave vector k) for backward volume spin waves, i.e., spin waves with their frequencies below fFMR and their wave vectors roughly parallel to the magnetization. Yet, it is positive (roughly parallel to k) for Damon–Eshbach spin waves, i.e., spin waves with their frequencies above fFMR and their wave vectors roughly orthogonal to the magnetization. In general, the direction of the group velocity is convenient to predict using isofrequency curves, i.e., curves of constant frequency [47]. The group velocity is always orthogonal to the isofrequency curves in the reciprocal space, in the same way as the electric field is always orthogonal to the curves of constant potential in real space. The dispersion anisotropy is strongest for magnetostatic spin waves, quickly diminishing as the wavelength decreases. The main attraction of the spin wave dispersion for wave physicists is the variety of factors that determine and therefore could be used to tailor its character. The compositional modulation of magnetic media represents the most obvious (albeit technologically challenging) way to achieve this. The dispersion of magnetostatic spin waves in thin-film magnetic samples is also sensitive to the variation of the film thickness. Moreover, the finite thickness of magnetic film structures leads to quantization and thus appearance of several dispersion branches for spin waves (of any sort) propagating within the plane of the film.

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Graded Magnonic Index and Spin Wave Fano Resonances in Magnetic Structures

The same is true for the effect of the lateral quantisation in waveguides with a finite width. The dispersion is different for each branch and depends on the film thickness, which suggests the continuous variation of the thickness and/or width as a means by which to control the spin wave propagation in patterned magnetic structures. The presence of lateral boundaries leads to a non-uniform demagnetizing field and internal magnetic field (and therefore graded magnonic index) in their vicinity [36]. The non-uniform internal magnetic field can lead to a non-uniform configuration of the magnetization. Furthermore, the magnetization is often nonuniform in samples with a significant antisymmetric exchange interaction (Dzyaloshinskii-Moriya interaction) [48, 49] and those with spatially varying directions of the easy and/or hard magnetization axes of the magnetic anisotropy [3, 4]. The field and magnetization non-uniformity can also be delivered by a non-uniform bias magnetic field, which in addition offers an opportunity to study magnonic phenomena in time-varying gradedindex magnonic landscapes [22]. The non-uniform magnetization textures modify the magnonic index not only through the dispersion anisotropy (Fig.  1.2) but also via modification of the demagnetizing field [36] and emergence of topological effects, such as Berry phase [50], geometrical anisotropy [51, 52], and topological protection [53, 54]. The non-uniformity breaks the translational invariance, limiting the use of the wave vector and therefore the notion of wave dispersion. The notable exceptions are given by the periodic and slowly varying (spatially) non-uniformities, the former giving rise to magnonic crystals [1, 2, 5–9] and the latter allowing the use of the geometrical optics (quasi-classical) approximation [26, 28].

1.3  Spin Wave Excitation

To excite propagating spin waves, it is necessary to perturb the magnetization both quickly and locally, so that the frequency and wave vector of the spin wave to be excited are covered by the spectrum of the perturbation. Conventionally, this is done using microwave microstrips [2, 7] or spin-transfer torque techniques [13, 14]. Another promising mechanism of spin

Spin Wave Excitation

wave excitation involves coupling free-space microwaves to spin waves through the use of local magnetic inhomogeneities, which can have the form of either a graded magnonic index [55–60] or Fano resonators [26, 61–63]. The main idea behind the method is that the spatial non-uniformity breaks the translational symmetry in the system, thereby enabling coupling between the microwave magnetic field and spin waves irrespective of their wavelengths. In the most basic case, the FMR frequency of magnetic samples depends on the saturation magnetization of the material, the sample’s dimensions and the applied bias magnetic field. The same is true for spin wave (higher-order) resonances. It is possible to design a system of two neighboring (or connecting) magnetic elements that have different dominant resonance frequencies for a given bias magnetic field value/orientation. When the entire system is pumped by a harmonic microwave magnetic field at the higher resonance frequency, the resonance is excited in one element only (the “transducer”). The coupling between the magnetizations of the two elements then leads to injection of spin waves into the second element (the “waveguide”), with their wavelength dictated by the magnonic dispersion in the waveguide. Such a resonance, in which the energy from one resonantly excited element with a discrete spectrum is “leaked” into wave modes propagating in the element with a continuous spectrum, is an example of a Fano resonance [37, 38]. This Fano resonance–assisted mechanism of spin wave excitation is illustrated in Fig.  1.3 with the help of micromagnetic simulations [64] (see Ref. [62] for associated experimental results and specific simulation parameters). The sample consists of two cuboidal magnetic elements, aligned orthogonally and overlaid with a vertical separation of 10 nm (Fig.  1.3a). Both the Fano transducer and waveguide are made of Permalloy but differ in thickness and width. When no bias magnetic field is applied, the shape anisotropy of the two elements compels the magnetization to align along each element’s long axis. The frequency spectrum of the sample is shown in Fig.  1.3b, from which the resonance frequencies of the transducer and waveguide are identified as ft  =  11.5  GHz and fwg  =  9  GHz, respectively. A global microwave magnetic field of 11.5  GHz frequency applied along the x axis

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Graded Magnonic Index and Spin Wave Fano Resonances in Magnetic Structures

resonates with the transducer only. The precessing magnetization of the transducer generates an oscillatory stray magnetic field that excites propagating spin waves in the waveguide (Fig. 1.3c).

Figure 1.3  (a) The geometry of the transducer–waveguide Fano resonance system. The arrows show the orientation of the static magnetization within each element. (b) The frequency spectrum of the sample. (c, d) Snapshots of the out-of-plane component of the magnetization of the waveguide are shown for time steps of 22  ps, when the entire sample is excited at 11.5  GHz. The Fano transducer is magnetized (white arrows) parallel and antiparallel to the y axis in panels (c) and (d), respectively. Adapted from Davies et al. [46].

The Fano transducer has two roles in the design. Firstly, it localizes the high-frequency magnetic field to the nanoscale, enabling coupling to short-wavelength spin waves. Secondly, it resonantly amplifies the incident field [65], helping achieve a stronger spin wave emission. In addition, the orientation of the transducer’s magnetization has a marked impact on its functionality. When the transducer is excited at resonance, its magnetization precesses with a well-defined chirality. The handedness of this chirality leads to the unidirectional excitation of propagating spin waves, similar in manner to the water flow

Spin Wave Excitation

generated by a rotating water mill. In Fig.  1.3c, the magnetization in the transducer is parallel to the y axis, and spin waves are excited toward the negative x direction. However, if the direction of the transducer’s magnetization is flipped (Fig.  1.3d), the spin waves instead propagate toward the positive x direction. Several periodically spaced Fano transducers create a periodic highfrequency magnetic field [65], leading to the idea of a resonant grating coupler, discussed in detail by Arikan et al. [63] and Yu et al. [66, 67] and in Chapter 7 of this book. In a graded magnonic medium, one could consider the incident microwave magnetic field as tuned in frequency to some of its regions better than to others. One could therefore tune the microwave frequency so as to excite certain (specifically targeted) regions of the medium at resonance. The resonantly driven magnetisation precession could then launch spin waves of finite wave vector into the adjacent regions, if such propagating spin wave modes are at all allowed by the dispersion relation in those regions. In this so-called Schlömann mechanism of spin wave excitation [55, 56], the resonating regions of the sample do not confine spin waves and therefore are not characterized by a discrete spectrum. Hence, the mechanism is free from reliance on the resonance with discrete normal modes of well-defined parts of the sample, inherent to the case with Fano resonances. This means that the frequency tuning range of the emitted spin waves is no longer limited by the resonance linewidth of the Fano transducer. Instead, the tuning is determined by the graded magnonic index, leading to a continuous distribution of the local FMR frequency, fFMR(r), assigned to each and every point of the sample under the assumption of negligible impact of magnetization gradients on the precession frequency [68]. In other words, the local FMR frequency is defined by setting the wave vector to zero in the spin wave dispersion relation, with the latter defined by the loal values of the magnetic parameters, field strength, and static magnetization orientation. The mechanism is illustrated in Fig. 1.4. Specifically, we have studied a 10  μm wide, 100  nm thick stripe of Permalloy, magnetized along its width [107]. The magnetic charges dynamically induced at the stripe’s left end by the magnetization precession create a local increase in fFMR(r), which then gradually decreases to the bulk value (i.e., the value far from the stripe ends) as the

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Graded Magnonic Index and Spin Wave Fano Resonances in Magnetic Structures

distance from the end increases (Fig.  1.4a). The different values of fFMR(r) correspond to different propagating spin wave modes, the dispersion of which is shown in Fig.  1.4b. Hence, when the stripe is excited by the uniform microwave magnetic field with a frequency matching fFMR(r) at a particular distance from the end, the magnetization at the distance gets resonantly excited and becomes a source of spin waves propagating into the stripe. The spin waves are imaged using the time-resolved scanning Kerr microscope (TRSKM) [26, 61], with the acquired images presented in Fig.  1.4c,d for the excitation at frequencies of 5.76 and 7.52 GHz, respectively. Further measurements (not shown here) have demonstrated successful excitation of spin waves across a frequency range of more than 4 GHz, which far surpasses in bandwidth the Fano resonance–assisted mechanism described by Au et al. [61, 62]. A peculiar example of a graded magnonic landscape is given by magnetic domain walls, i.e., boundaries between regions of different magnetization orientation. In particular, it has been recently shown that a domain wall can generate spin waves when excited by an external magnetic field or a spin-polarized current [69–74], while arrays of domain walls were proposed as spin wave grating couplers [75]. The spin wave emission has traditionally been attributed to the effect of the domain wall oscillations. However, at least in two of the studies referenced above, the spin waves were observed to have the frequency of the driving stimuli, rather than twice its value (as one would expect for a nonlinear process of interaction between two oscillatory modes). So, to uncover the mechanism of the emission, we have developed a linear analytical theory [108] in the exchange approximation using the formalism from Ref. [58]. Figure 1.5a shows a domain wall excited by a uniform microwave magnetic field oriented perpendicular to the magnetization in both the domain wall and domains. Our calculations show that propagating spin waves at the frequency of the incident field are emitted from the domain wall (Fig. 1.5b), as a result of a linear process. The underlying mechanism is similar to that proposed by Schlömann [55], except the domain wall creates a natural graded magnonic index landscape with a reduction of

Spin Wave Excitation

the local FMR frequency. This reduction prevents the incident microwave field from matching the local FMR frequency in any point of the sample. Although our theory is developed in the exchange approximation, we believe the described mechanism for spin wave generation is general and therefore applicable to the experimental observations referred to above.

Figure 1.4 (a) A cross section of the calculated fFMR(r) profile along the length of the stripe is shown for the first 5  μm from its left end and y  =  5  μm. (b) The spin wave dispersion calculated from simulations of an infinitely long stripe is shown in grayscale, with the overlaid red circles showing the points deduced from the measurements of the finite length stripe. For the dispersion calculation, the results of the simulations were spatially smoothed so as to mimic the experimental resolution. The horizontal dashed lines extending from panel (a) to (b) illustrate the correspondence between the source region and wave number of propagating spin waves excited at frequencies of 5.76 and 7.52 GHz, the Kerr images of which are shown in panels (c) and (d) respectively. After Mushenok et al. [107].

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Figure 1.5 (a) Geometry of a thin film extending infinitely in the yz plane with blue arrows showing the calculated magnetization of a domain wall (centered at y  =  0) and adjoining antiparallel domains. The driving magnetic field is aligned along the y direction. (b) Calculated magnetization vectors at phase values of 0 (main plot) and p (inset), showing spin waves emanating from the domain wall in (a). After Whitehead et al. [108].

Curiously, we find that the graded magnonic index profile induced by a domain wall (that of the Pöschl–Teller potential well [76]) is naturally sized so that spin wave emission does occur and is locally optimized. For other values of the profile’s aspect ratio, the emission could be suppressed or even completely eliminated—that is, although a graded magnonic index may be able to generate spin waves, their emission is not guaranteed. As a final note of this section, we point out that the distinction between the spin wave emission mechanisms due to Fano resonances

Spin Wave Steering

and graded magnonic index is very subtle. Indeed, depending on their taste, one could also consider the continuously varying local FMR frequency as an array of coupled Fano resonators, each with a slightly different yet discrete spectrum. At the same time, the spin wave emission from a domain wall could still be argued to be of resonant character, albeit with a large detuning from the domain wall resonance. A more detailed discussion of this will be presented elsewhere.

1.4  Spin Wave Steering

The functional medium element of the generic magnonic device shown in Fig.  1.1b has two main purposes: to deliver the signals from the input to the output and, as magnonic devices can have multiple inputs and outputs, to steer the signals between them. A graded magnonic index between the inputs and outputs can be used to channel [77–80] or focus [81] or defocus [82] spin waves, or to “cloak” an object from them [32], in analogy to a similar research topic in electromagnetics [25]. Spin wave steering is the key prerequisite for creation of efficient magnonic interferometers [83–85], Boolean and analog computing primitives [21, 86–89], splitters (demultiplexers and inverse multiplexers) [26, 90–92], and combiners (multiplexers) [93–95]. Figure  1.6 demonstrates the function of a magnonic inverse multiplexer formed by a Permalloy T junction with 5  µm wide features [26]. The spin waves are excited in the central “leg” of the T junction by a uniform microwave magnetic field through the Fano resonance–assisted mechanism, as described in the previous section. The frequency of the incident microwave field is tuned to fFMR of the leg, which acts as an element with a discrete spectrum. The sample was biased by a uniform in-plane static magnetic field, HB, of 500 Oe strength, applied either parallel to or slightly tilted from the long axis of the leg. Due to the magnetic shape anisotropy, the arms of the T junction have a lower fFMR compared to the leg. So, the resonantly excited magnetization of the leg launches propagating magnetostatic spin waves into either one or both of the T junction’s arms, which act as elements with a continuous spectrum. The spin waves were imaged by the TRSKM and modeled using micromagnetic simulations.

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Graded Magnonic Index and Spin Wave Fano Resonances in Magnetic Structures

Figure 1.6 Snapshots of spin waves propagating in the arms of the T junction. The bias magnetic field HB of 500  Oe strength is applied (a) parallel to and at angles of (b) –15° and (c) +15° relative to the leg of the junction. In each case, the top and bottom panels show results of the TRSKM imaging and micromagnetic simulations, respectively. The frequency of the continuous-wave (cw) pump was 8.24 GHz for experiments, while for simulations it was 7.62 GHz in panel (a) and 7.52 GHz in panels (b) and (c). In panel (c), the extracted directional vectors of the incident (index “i,” solid lines) and reflected (index “r,” dashed lines) group velocities v and wave vectors k are shown for kx = 0.94 µm–1. Adapted from Davies et al. [26].

The observed switching of the spin wave propagation in the arms of the T junction via the tilting of the in-plane bias magnetic field is a direct result of the graded magnonic index in the patterned structure and an excellent illustration of the opportunities in graded index magnonics in terms of spin wave steering. Firstly, Fig.  1.6a shows that the phase fronts of the spin waves in the arms are somewhat tilted relative to their symmetry axis, even though the bias field is applied symmetrically. This is because the bias magnetic field does not fully saturate the magnetization in the arms, which therefore tilts from their symmetry axis. At the

Spin Wave Steering

same time, the energy flow along the arms dictates the direction of the group velocity, which is therefore bound to be parallel to their horizontal symmetry axis. Hence, the anisotropic dispersion of the magnetostatic spin waves (Fig.  1.2) leads to a small, non-zero angle between the directions of the group velocity and wave vector (phase velocity), explaining the tilt of the phase fronts. The same explains the tilts of the spin wave phase fronts observed in Fig.  1.4c,d. When the bias magnetic field is rotated from the symmetry axis by just ±15°, we observe only one spin wave beam propagating into one of the arms (Fig.  1.6b,c). The direction of the propagation is “switched” between the two arms by the sign of the tilt angle. In each case, the spin wave beam (emitted from the leg-arm junction) propagates at an oblique angle to the arm’s axis, hits its edge, and is reflected into a much broader beam, propagating approximately along the arm’s length. The incidence and reflection angles are different, again resulting from the anisotropy of the magnetostatic dispersion relation and the tilt of the static magnetisation. We interpret our observations in terms of the graded magnonic index induced by the spatial variations of the orientations of the magnetisation and the value of the internal magnetic field in the sample. Using the convincing agreement between the measured and numerically simulated results, we apply the theory from Vashkovsky and Lock  [96, 97] to the numerically computed static magnetization and field distributions (Fig.  1.7a) to derive the local directions of the wave vectors and group velocities of the propagating spin waves (shown Fig.  1.6c). The confinement of the precessing magnetization to the width of the T junction’s leg results in a broad kx spectrum (Fig. 1.7c). For each kx value, the isofrequency curve corresponding to the frequency of the incident microwave field returns allowed (by the magnetostatic dispersion relation) values of ky, while the normals to the isofrequency curves show the group velocity directions (Fig.  1.7d–h). The field and magnetization distributions in the arms are quite uniform along the x axis starting from about 1  µm from the leg-arm boundary, which ensures conservation of the kx value of the spin wave propagating across the arm’s width. In contrast, the values of ky and the group velocity adjust adiabatically to the variation of the internal field magnitude and direction of the magnetization [98]. The non-uniformity also

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Graded Magnonic Index and Spin Wave Fano Resonances in Magnetic Structures

leads to a distributed partial reflection of the spin wave amplitude, with the group velocity directions of the reflected waves also shown in Fig.  1.7c–h. The overall wave field is given by the superposition of the incident and scattered waves.

Figure 1.7 (a) The calculated distributions of the static magnetization (arrows) and the projection of the internal magnetic field onto the magnetization (color scale) are shown for the bias magnetic field HB of 500 Oe applied at 15° to the vertical symmetry axis. Each arrow represents the average of 5 × 5 mesh cells. (b) The calculated distribution of fFMR(r) across the T junction [109]. (c) kx spectra of the dynamic magnetization distributions across the leg and along the arms (amplified ×5) of the T junction excited at 7.52 GHz are shown by the dotted and solid curves, respectively. (d–h) Constructions of the isofrequency curves and the group velocities of the incident (index “i”) and reflected (index “r”) beams are shown for the pixels boxed in (a), for the transverse wave vector components kx  =  ±0.94 μm–1 (as indicated by the dashed grey lines). Adapted from Davies et al. [26].

Spin Wave Steering

Apart from the small region at small wave vectors, the spin wave isofrequency curves depicted in Fig.  1.7d–h consist of nearly straight lines. This leads to virtually the same direction of group velocity for a wide range of wave vectors, giving rise to the formation of spin wave caustic beams [47, 96, 97]. This explains the strongly directional beam emitted from the leg-arm boundary for the tilted-bias magnetic field, but not the absence of the other beam. Due to the inhomogeneities of the internal field and magnetization (Fig. 1.7a), the beam curves slightly and experiences distributed scattering, with the group velocities of the scattered waves being roughly aligned with the arm’s length (Fig.  1.7d,e). The group velocity of the reflected beam switches direction near the far edge of the arm (Fig.  1.7f), leading to the phenomenon of “back reflection” [96]. The reflected beam is confined by the non-uniform demagnetizing field and magnetization near the arm’s edge. In addition to this, some spin waves with small (negative or positive) kx values are not supported in parts of the magnetic landscape at all. In Fig. 1.7h, for example, there is no intersection between the line kx  =  0.94 μm–1 and the isofrequency curves, giving rise to a “forbidden” path for spin waves of certain wave vector. Finally, and quite surprisingly, we find that the beam formed from spin waves with negative kx values cannot possibly propagate into the left arm of the junction (for the bias field direction in Fig.  1.7). Indeed, the beam is curved into the nearest edge of the left arm from which it is then scattered backwards into the right arm (Fig.  1.7g). The observed complete disappearance of one of the two beams in favor of the other one would be impossible without the graded distribution of the magnonic index. The anisotropic dispersion is also required for the effects to take place, but on its own it could only lead to a tilt of both beams and asymmetry of their intensities [92]. Figure  1.6 clearly shows that the spin wave beam initiated near the leg-arm boundary then propagates along the arms of the structure as prescribed by the direction of the bias magnetic field. However, the beam is also quite wide and not as distinct as one could wish. So, Fig.  1.8 presents results of simulations for a Permalloy T junction that has a narrower (1  µm wide) leg, which leads to a better-defined spin wave caustic beam propagating into one arm of the structure. The smaller width

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of the leg leads to the higher frequency of its quasi-uniform mode and to the observed smaller cross section of the spin wave beam excited at the frequency into the right arm of the structure. Notably, the observations and interpretations developed for the wider leg sample also remain valid here.

Figure 1.8 (a) A numerically calculated snapshot of the dynamic out-ofplane component of the magnetization in the T junction. The bias magnetic field HB of 500 Oe strength is applied at an angle of a = 15° relative to the leg of the T junction driven by a microwave magnetic field of 10.3  GHz frequency. (b) An example isofrequency curve calculated for the frequency of excitation and with the internal field and magnetization corresponding to the region at the center of the arms (above the leg). The arrows represent examples of the wave vector (k) and group velocity (vi) vectors: the mutual collinearity of the group velocity vectors gives rise to the spin wave caustic beam observed in panel (a). Adapted from Davies et al. [26].

The non-uniformity of the magnetisation and internal magnetic field plays a key role in defining the spatial variation of the graded magnonic index and thereby in steering the direction of the spin wave propagation in the T junction. However, the other key ingredient—the anisotropy of the magnetostatic spin wave dispersion—is only (strongly) present at micrometer

Spin Wave Output

to millimeter length scales, impeding miniaturization of any magnonic devices that would exploit the type of spin wave steering discussed here. In contrast, the non-uniformity of the internal magnetic field and the magnetization persists to much shorter length scales and could still lead to useful device concepts. Moreover, on the nanometer length scales, the non-uniform exchange field (completely neglected here) becomes more important and could therefore be exploited, while additional opportunities arise from the use of the highly localized magnetic field due to magnetic domain walls [70–75, 83]. In this context, the main challenge is that the configurations of the internal magnetic field and the static magnetization in magnetic nanoand microstructures are not arbitrary but are determined by the magnetostatic Maxwell equations. This limits the range of magnetic configurations that could be exploited and favors alternative pathways to creation of the graded magnonic index.

1.5  Spin Wave Output

In research labs, the spin wave outputs from magnonic devices are often detected using Kerr microscopy [17, 26, 33, 61–63, 72] and Brillouin light scattering [9, 16, 22, 59, 77, 91–93], the two most popular magneto-optical techniques in magnonics. However, the outputs of realistic magnonic devices will need to be more congenial to logic circuitry of interest, with their nature depending on the envisaged purpose of the magnonic chip. For example, spin waves can be interfaced with high-frequency electrical signals inductively [19] or magnetoresistively [13, 14], or their action could be encoded into the micromagnetic configuration of domain walls [99] or magnetic nanoelements [9, 16, 17], or both [100]. Indeed,propagating spin waves have been demonstrated to be able to drive domain walls across magnetic samples [101], which could be used either to toggle binary logic states or for a memristor-type function [14]. Arguably, the most important missing element in the technological toolbox of modern magnonics is the direct interfacing of two or more magnonic devices in series. The only demonstration of this kind was done using micromagnetic simulations in Ref. [86], with none reported experimentally.

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It is more challenging to realize the fan-out function, whereby the output of one device (e.g., a logic gate) is connected and can drive the input of more than one identical devices. However, the expanding use of ultrathin films of yttrium-iron garnet (YIG) [67], which have naturally low magnetic damping [102], and the recent demonstration of low damping in metallic CoFe alloys [103, 104] should help further experimental progress in this direction as well as in magnonics, in general. In the context of detection of output signals from magnonic devices, the graded magnonic index could be used to fine-tune the wavelengths of propagating spin waves to the range of values optimal for a specific output detection method. Indeed, in a reverse process to the Schlömann excitation mechanism, an increase in the local FMR frequency fFMR(r) in a medium with a positive dispersion results in an increase of the spin wave wavelength [98, 105]. Thus, a scheme can be realized in which the data or signal processing is done by spin waves of shorter wavelength that are then upscaled to a larger wavelength, e.g., about twice the size of the detector, thereby optimizing the spin wave coupling to the external circuitry. In the same spirit, we have shown [106] that the transducer– waveguide Fano resonance system shown in Fig.  1.3 can be used in the reverse direction: to couple an incident propagating spin wave to the uniform precession in the “transducer,” with the latter acting in this case as a receiving antenna. As a result, the spin wave can be fully absorbed, leading to a pronounced uniform precession of the receiving element. This uniform precession could then be more easily outcoupled to the external circuitry, in a function that is similar to that of the case of the graded magnonic index discussed in the previous paragraph. The only experimental demonstration of this kind was reported by Yu et al.  [67], albeit without a detailed discussion of the detection mechanism.

1.6  Spin Wave Control and Magnonic Devices

The magnonic dispersion itself and the spin wave excitation, steering, and detection mechanisms discussed in the preceding sections of this chapter are all determined by the internal magnetic field and magnetization texture in the sample. Hence,

Spin Wave Control and Magnonic Devices

the mechanisms lend themselves readily to external control, which could be realized through application of the bias magnetic field or even the history of its application. The former can enable construction of magnetically tunable magnonic devices, while the latter property describes devices reprogrammable by the applied (and eventually removed) magnetic field. However, the physics of magnetic switching and control of micromagnetic textures (either by a magnetic field or otherwise, e.g., electrically, optically, acoustically, spintronically, etc.) forms a separate and extremely broad research field, which is beyond the scope of this chapter. So, in this section, we discuss only a few examples of magnonic devices that exploit the graded magnonic index and/or Fano resonances and can be controlled either by the applied magnetic field or by switching their micromagnetic configuration. The results of our TRSKM imaging experiments and micromagnetic simulations presented in Fig.  1.6 demonstrate the efficiency by which an external bias magnetic field can be used to steer magnetostatic spin waves across a T junction [26]. When the bias magnetic field is applied symmetrically along the leg, the T junction acts as a magnonic inverse multiplexer, i.e., a spin wave splitter [91]. The demonstrated magnetic field control of the spin wave beam enables the T junction to be also used as an analog time division demultiplexer, i.e., a device that can steer timeseparated signals between different outputs. The switching of the 500 Oe magnetic field from +15° to –15° is equivalent to applying a constant bias field of 486 Oe along the leg and toggling a control magnetic field of ±117  Oe to switch the signal between outputs. A similar device fabricated from YIG could be switched by an orthogonal control field of ±127  Oe for a constant bias magnetic field of 1153 Oe [92]. Magnetically reprogrammable magnonic devices can be built using Fano resonances, the function of which could be “programmed” by switching the magnetization of the Fano resonator (above a magnonic waveguide) discussed earlier in the context of spin wave excitation and detection, i.e., the magnonic transducer from Fig. 1.3. In particular, this Fano resonator can act as a control element performing the function of either a valve or phase shifter, depending on the distance between the element and the waveguide (Fig.  1.9) [106]. For one orientation of the element’s magnetization (the bottom images in panels (b) and (d)), the

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element does not couple to the propagating spin wave at all. However, for the opposite orientation of the magnetization (the middle images in panels (b) and (d)), the stray magnetic field generated by a spin wave passing underneath can resonantly excite its precession. This precession is then partly re-emitted (in the same manner as the process shown in Fig.  1.3) as a spin wave propagating in the same direction as the originally incident wave, but with opposite phase. For the 20  nm elementto-waveguide separation, the partly transmitted original and the re-emitted spin waves have similar amplitudes and opposite phases. Thus, they nearly cancel each other, yielding the observed valve functionality. For the 5  nm element-to-waveguide separation, the incident spin wave is first fully converted into the element’s precession and then fully re-emitted with a 180° phase shift. In this case, no interference occurs, since there is no directly transmitted wave, and only the observed phase shifter functionality is observed.

Figure 1.9 (a) The Fano control element is positioned 5 nm above the waveguide. A spin wave (SW) excited elsewhere propagates along the waveguide in the negative x direction. (b) The first (top) snapshot shows the out-of-plane component of magnetization (mz) when the control element is absent. The second (middle) and third (bottom) snapshots show mz with the control element present and magnetized parallel and antiparallel to the y axis, respectively. (c, d) The same as (a, b) but for the control element positioned 20 nm above the waveguide. Adapted from Davies et al. [46].

Spin Wave Control and Magnonic Devices

The described methods of spin-wave excitation, control, and detection can be used to construct a complete magnonic logic architecture. The simplest of logic gates is the NOT gate, and this can be straightforwardly realized using the scheme discussed in Fig.  1.9d. The binary input, which can take the values 0 or 1, is assigned to the polarity of the valve’s static magnetization to be respectively toward negative and positive y. A spin wave detector, positioned on the waveguide past the valve, records an output of 1 if non-zero (i.e., above a set threshold) spin wave amplitude is detected, or 0 if spin waves are absent. Hence, the valve can act as a simplistic magnonic NOT gate, as an input of 0 generates an output of 1, and vice versa. Of course, this will require a subsequent amplification stage if the output is to be used in another magnonic logic gate. Gates more sophisticated than a NOT gate require two input signals. This can be implemented by considering either two control elements, or two input transducers with a shared waveguide. For example, an XNOR gate can be constructed using two spin wave phase shifters combined to form a magnonic interferometer shown in Fig.  1.10a. Here, the two phase shifter elements are positioned 5  nm above each branch of the interferometer. The magnetization follows the twists of the interferometer, the symmetry of which causes the propagating spin wave to split equally between its two branches. If the magnetizations in both phase shifters are parallel, i.e., input (0,0) or (1,1), the spin waves from the branches will interfere constructively upon recombination. Hence, a spin wave will be observed by the detector, i.e., output of 1. If instead the phase shifters are antiparallel in the magnetization polarity, i.e., input (0,1) or (1,0), the phase of one spin wave is shifted by 180°. So, the spin waves interfere upon recombination destructively, generating an output of 0. This action is shown in Fig.  1.10b, where the results of micromagnetic calculations are presented for the (0,0) and (0,1) inputs. Figure  1.11a shows a magnonic NAND gate constructed using two input transducers. Similar to the encoding used in Fig.  1.10a, the transducers’ static magnetization polarity, at A and B, is encoded by 0 when the magnetization is aligned along the positive and negative y directions, respectively, and encoded by 1 for the reversed case. The entire sample is excited globally by a harmonic microwave field. As shown in the calculated snapshots of spin

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Figure 1.10  (a) Magnonic XNOR gate realised as a spin-wave interferometer. The directions of the static magnetisation and spin wave (SW) propagation in the interferometer are shown. The magnonic phaseshifter elements are positioned above each branch of the interferometer. (b) Snapshots of the out-of-plane component of magnetisation (mz) are shown for two of the four possible XNOR gate input combinations, as calculated using micromagnetic simulations. (After Ref. [46]).

Figure 1.11  (a) Magnonic NAND gate realized as a pair of two transducers positioned on a shared waveguide. (b) Snapshots of mz corresponding to the same moment of time are shown for the four possible NAND gate input combinations. Adapted from Davies et al. [46].

Conclusions and Outlook

wave propagation in Fig.  1.11b, spin waves are only absent from the center of the waveguide when an input of (1,1) is used. The NAND gate functionality is therefore obtained, as demonstrated in the truth table shown in the inset of Fig. 1.11a.

1.7  Conclusions and Outlook

As conventional electronics is beginning to become restricted in its potential for growth, the research and development of technologies that use alternative means of data processing and communication is gathering significant attention. Among others, magnonic technology promises devices that will have a small footprint, moderately high operational frequencies, and intrinsic nonvolatility, with scope for efficient interfacing with electronic and other emerging research devices. However, in order to advance the development of spin wave devices, research should not just focus on the design, experimental construction, and miniaturization of prototype magnonic devices within existing paradigms. While deepening our understanding of known, fundamental spin wave concepts, we should also proactively search for novel spin wave phenomena and use this knowledge to imagine new devices, which are designed to harness the unique properties of spin waves. In this chapter, we attempted to demonstrate the interesting physics and highlight technical opportunities associated with the graded magnonic index and spin wave Fano resonances. The research into these two exciting (yet underexplored in magnonics) wave phenomena is rapidly gaining momentum, as evidenced by the increased attention they receive throughout this book. We foresee three main research avenues for further advances and the ultimately bright future of these fields. First, the science of spin wave propagation in media with a graded magnonic index and/or Fano resonators is unchartered territory in terms of mathematical physics, with lots of challenges and surprises awaiting theoreticians’ attention. Second, similarly vast opportunities will open to experts in numerical micromagnetic simulations that will venture into the world of media and devices, including compositionally modulated structural elements that extend into three dimensions. And last but not least, the continuing progress in nanotechnology and materials science and the most recent

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advances in reducing the magnetic damping hold promise for the latter exciting phenomena to be observed experimentally and implemented within realistic magnonic device architectures.

Acknowledgments

The research leading to these results has received funding from the Engineering and Physical Sciences Research Council of the United Kingdom (Project Nos. EP/L019876/1, EP/L020696, and EP/P505526/1), and from the European Union’s Horizon 2020 research and innovation program under Marie SkłodowskaCurie Grant Agreement No. 644348 (MagIC).

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68. Hermsdoerfer, S. J., Schultheiss, H., Rausch, C., Schafer, S., Leven, B., Kim, S. K., and Hillebrands, B. (2009). A spin-wave frequency doubler by domain wall oscillation, Appl. Phys. Lett., 94, 223510. 69. Marchenko, A. N., and Krivoruchko, V. N. (2012). Magnetic structure and resonance properties of a hexagonal lattice of antidots, Low Temp. Phys., 38, 157. 70. Roy, P. E., Trypiniotis, T., and Barnes, C. H. W. (2010). Micromagnetic simulations of spin-wave normal modes and the resonant field-driven magnetization dynamics of a 360 degrees domain wall in a soft magnetic stripe, Phys. Rev. B, 82, 134411. 71. Boone, C. T., and Krivorotov, I. N. (2010). Magnetic domain wall pumping by spin transfer torque, Phys. Rev. Lett., 104, 167205.

72. Mozooni, B., and McCord, J. (2015). Direct observation of closure domain wall mediated spin waves, Appl. Phys. Lett., 107, 042402.

73. Van de Wiele, B., Hamalainen, S. J., Balaz, P., Montoncello, F., and van Dijken, S. (2016). Tunable short-wavelength spin wave excitation from pinned magnetic domain walls, Sci. Rep., 6, 21330.

74. Sluka, V., Weigand, M., Kakay, A., Erbe, A., Tyberkevych, V., Slavin, A., Deac, A., Lindner, J., Fassbender, J., Raabe, J., and Wintz, S. (2015). Stacked topological spin textures as emitters for multidimensional spin wave modes, Abstract DE-03 in the Book of Abstracts of the 2015 IEEE Intermag Conference (May 11–15, 2015, Beijing, China).

75. Truetzschler, J., Sentosun, K., Mozooni, B., Mattheis, R., and McCord, J. (2016). Magnetic domain wall gratings for magnetization reversal tuning and confined dynamic mode localization, Sci. Rep., 6, 30761. 76. Pöschl, G., and Teller, E. (1933). Bemerkungen zur Quantenmechanik des anharmonischen Oszillators, Z. Phys., 83, 143.

77. Demidov, V. E., Jersch, J., Demokritov, S. O., Rott, K., Krzysteczko, P., and Reiss, G. (2009). Transformation of propagating spin-wave modes in microscopic waveguides with variable width, Phys. Rev. B, 79, 054417. 78. Demidov, V. E., Urazhdin, S., Zholud, A., Sadovnikov, A. V., and Demokritov, S. O. (2015). Dipolar field-induced spin-wave waveguides for spin-torque magnonics, Appl. Phys. Lett., 106, 022403. 79. Lan, J., Yu, W. C., Wu, R. Q., and Xiao, J. (2015). Spin-wave diode, Phys. Rev. X, 5, 041049.

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80. Garcia-Sanchez, F., Borys, P., Soucaille, R., Adam, J. P., Stamps, R. L., and Kim, J. V. (2015). Narrow magnonic waveguides based on domain walls, Phys. Rev. Lett., 114, 247206. 81. Perez, N., and Lopez-Diaz, L. (2015). Magnetic field induced spinwave energy focusing, Phys. Rev. B, 92, 014408.

82. Dzyapko, O., Borisenko, I. V., Demidov, V. E., Pernice, W., and Demokritov, S. O. (2016). Reconfigurable heat-induced spin wave lenses, Appl. Phys. Lett., 109, 232407. 83. Hertel, R., Wulfhekel, W., and Kirschner, J. (2004). Domain-wall induced phase shifts in spin waves, Phys. Rev. Lett., 93, 257202.

84. Vasiliev, S. V., Kruglyak, V. V., Sokolovskii, M. L., and Kuchko, A. N. (2007). Spin wave interferometer employing a local nonuniformity of the effective magnetic field, J. Appl. Phys., 101, 113919.

85. Kanazawa, N., Goto, T., Sekiguchi, K., Granovsky, A. B., Ross, C. A., Takagi, H., Nakamura, Y., and Inoue, M. (2016). Demonstration of a robust magnonic spin wave interferometer, Sci. Rep., 6, 30268. 86. Lee, K. S., and Kim, S. K. (2008). Conceptual design of spin wave logic gates based on a Mach-Zehnder-type spin wave interferometer for universal logic functions, J. Appl. Phys., 104, 053909. 87. Csaba, G., Papp, A., and Porod, W. (2014). Spin-wave based realization of optical computing primitives, J. Appl. Phys., 115, 17C741.

88. Klingler, S., Pirro, P., Bracher, T., Leven, B., Hillebrands, B., and Chumak, A. V. (2015). Spin-wave logic devices based on isotropic forward volume magnetostatic waves, Appl. Phys. Lett., 106, 212406.

89. Gertz, F., Kozhevnikov, A., Khivintsev, Y., Dudko, G., Ranjbar, M., Gutierrez, D., Chiang, H., Filimonov, Y., and Khitun, A. (2016). Parallel read-out and database search with magnonic holographic memory, IEEE Trans. Magn., 52, 3401304. 90. Vogt, K., Fradin, F. Y., Pearson, J. E., Sebastian, T., Bader, S. D., Hillebrands, B., Hoffmann, A., and Schultheiss, H. (2014). Realization of a spin-wave multiplexer, Nat. Commun., 5, 3727.

91. Sadovnikov, A. V., Davies, C. S., Grishin, S. V., Kruglyak, V. V., Romanenko, D. V., Sharaevskii, Y. P., and Nikitov, S. A. (2015). Magnonic beam splitter: the building block of parallel magnonic circuitry, Appl. Phys. Lett., 106, 192406.

92. Davies, C. S., Sadovnikov, A. V., Grishin, S. V., Sharaevskii, Y. P., Nikitov, S. A., and Kruglyak, V. V. (2015). Field-controlled phase-rectified magnonic multiplexer, IEEE Trans. Magn., 51, 3401904. 93. Braecher, T., Pirro, P., Westermann, J., Sebastian, T., Lagel, B., Van de Wiele, B., Vansteenkiste, A., and Hillebrands, B. (2013). Generation

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94. Nanayakkara, K., Anferov, A., Jacob, A. P., Allen, S. J., and Kozhanov, A. (2014). Cross junction spin wave logic architecture, IEEE Trans. Magn., 50, 3402204. 95. Unfortunately, the terminology used in Refs. 26, 90, 92 was different from that conventionally accepted in the field of electronics. So, we correct it here. 96. Vashkovsky, A. V., and Lock, E. H. (2006). Properties of backward electromagnetic waves and negative reflection in ferrite films, Phys. Usp., 49, 389. 97. Lock, E. H. (2008). The properties of isofrequency dependences and the laws of geometrical optics, Phys. Usp., 51, 375.

98. Smith, K. R., Kabatek, M. J., Krivosik, P., and Wu, M. Z. (2008). Spin wave propagation in spatially nonuniform magnetic fields, J. Appl. Phys., 104, 043911. 99. Allwood, D. A., Xiong, G., Faulkner, C. C., Atkinson, D., Petit, D., and Cowburn, R. P. (2005). Magnetic domain-wall logic, Science, 309, 5741. 100. Xing, X. J., Jin, Q. L., and Li, S. W. (2015). Frequency-selective manipulation of spin waves: micromagnetic texture as amplitude valve and mode modulator, New J. Phys., 17, 023020.

101. Han, D. S., Kim, S. K., Lee, J. Y., Hermsdoerfer, S. J., Schultheiss, H., Leven, B., and Hillebrands, B. (2009). Magnetic domain-wall motion by propagating spin waves, Appl. Phys. Lett., 94, 112502. 102. Krivoruchko, V. N. (2015). Spin waves damping in nanometre-scale magnetic materials (Review Article), Low Temp. Phys., 41, 670.

103. Turek, I., Kudrnovsky, J., and Drchal, V. (2015). Nonlocal torque operators in ab initio theory of the Gilbert damping in random ferromagnetic alloys, Phys. Rev. B, 92, 214407. 104. Schoen, M. A. W., Thonig, D., Schneider, M. L., Silva, T. J., Nembach, H. T., Eriksson, O., Karis, O., and Shaw, J. M. (2016). Ultra-low magnetic damping of a metallic ferromagnet, Nat. Phys., 12, 839.

105. Toedt, J.-N., Mansfeld, S., Mellem, D., Hansen, W., Heitmann, D., and Mendach, S. (2016). Interface modes at step edges of media with anisotropic dispersion, Phys. Rev. B, 93, 184416. 106. Au, Y., Dvornik, M., Dmytriiev, O., and Kruglyak, V. V. (2012). Nanoscale spin wave valve and phase shifter, Appl. Phys. Lett., 100, 172408.

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107. Mushenok, F. B., Dost, R., Davies, C. S., Allwood, D. A., Inkson, B. J., Hrkac, G., and Kruglyak, V. V. (2017). Broadband conversion of microwaves into propagating spin waves in patterned magnetic structures, arXiv:1706.04409. 108. Whitehead, N. J., Horsley, S. A. R., Philbin, T. G., Kuchko, A. N., and Kruglyak, V. V. (2017). Theory of linear spin wave emission from a Bloch domain wall, arXiv:1705.01852. 109. Davies, C. S., Poimanov, V. D., and Kruglyak, V. V. (2017). Mapping the magnonic landscape in patterned magnetic structures, arXiv:1706.03212.

Chapter 2

Coupled Spin Waves in Magnonic Waveguides Yu. P. Sharaevsky, A. V. Sadovnikov, E. N. Beginin, M. A. Morozova, S. E. Sheshukova, A. Yu. Sharaevskaya, S. V. Grishin, D. V. Romanenko, and S. A. Nikitov Laboratory “Metamaterials,” Saratov State University, 83 Astrakanskaya Str., Saratov 410012, Russia Kotel’nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow 125009, Russia [email protected]

2.1  Introduction Recent developments in magnetic thin-film technology led to the fabrication of basic functional elements for spin wave (SW) signal transmission and processing in the microwave range in micro- and nanoscales: SW waveguides, delay lines, resonators, and oscillators [1–5]. A magnonic waveguide, formed from a magnetic stripe, is a building block of any complex integral reconfigurable magnonic network [6, 7]. Control over the dispersion of SWs can be achieved, for example, by periodic patterning of thin magnetic films. Periodic variation of the magnetic materials’ parameters allows fabrication of magnonic crystals (MCs) [3, 4, 8], Spin Wave Confinement: Propagating Waves (2nd Edition) Edited by Sergej O. Demokritov Copyright © 2017 Pan Stanford Publishing Pte. Ltd. ISBN  978-981-4774-35-2 (Hardcover),  978-1-315-11082-0 (eBook) www.panstanford.com

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Coupled Spin Waves in Magnonic Waveguides

which can be widely used for spin-wave-based computing applications. It was shown, that MCs can demonstrate a complicated magnonic band structure with strong dispersion and anisotropy. To fabricate the topology of a magnonic network, signal processing devices should be put together to create integrated circuits where magnons carry data. The most promising candidates for effective channeling of SWs between the functional units of a magnonic network are magnonic couplers, which are essential to control the interelement coupling in the horizontal and vertical directions. Horizontal couplings can be realized on the base of laterally coupled magnetic waveguide segments, whereas vertical couplings can be performed by using layered waveguiding stripes. In this chapter, we develop a theoretical approach and propose lateral and vertical topologies of magnonic couplers as candidates to the internode elements in magnonic networks. We consider the features of SW propagation and the formation of forbidden zones for coupled planar and layered SW elements. Theoretical investigation of wave processes in SW waveguides is performed on the basis of a common mathematical approach using the method of coupled waves [9]. This method is well defined under an assumption of single-mode SW propagation and weak interwaveguide coupling. Influence of intermode coupling may be accounted for by numerical simulation of Maxwell’s equations by means of the finite element method (FEM) and micromagnetic simulations of high-frequency magnetization dynamics in thin-film magnonic structures in space and time domains. SW transmission and dispersion in the proposed topology of magnonic couplers were investigated by broadband microwave spectroscopy using a vector network analyzer (VNA). By means of optical imaging Brillouin light scattering (BLS) [10, 11] techniques the dynamics of SWs at different frequencies and input signal power levels in lateral and vertical coupled magnonic stripes and crystals were addressed directly.

2.2  Theoretical Approach

We consider two coupled magnetic waveguiding structures. Using the coupled wave approach [9] the magnetic field in each waveguide can be represented in the following form:

Theoretical Approach



H1,2(t )= H0 + h1,2(t )+ Kh2,1 (t ),

(2.1)



∂ 2m1,2 w2M d ∂ (m + Km2,1)= 0 , 2 + wH ( wH + wM )m1,2 + ∂t 2 ∂y 1,2

(2.2)



m1,2 = A1,2exp[ j(wt – k0 y )]+B1,2exp[ j( wt + k– y )],

(2.3)

where dynamic magnetic microwave field in each layer H1,2(t )= H0 + h1,2(is t )+the Kh2,1 (t ) and K is the coupling coefficient that determines the coupling between microwave magnetic fields of each waveguide. The external magnetic H1,2field (t )= H0 +ish1,2directed (t )+ Kh2,1along (t ) the x axis. We can obtain the following set of wave equations for magnetic waveguides using the equation of motion for high-frequency components of magnetization and the equation for the magnetostatic potential in each film, as well as the appropriate boundary conditions in the long-wave approximation for magnetostatic surface waves (MSSWs) [12] propagating along the y axis [13, 14]:

where m1,2 = my1,2/M0 are dynamic magnetization components in each waveguide, M0 is the saturation magnetization, d is the magnetic film thickness, wM = g4pM0, wH = gH0, and g is the gyromagnetic ratio. This approach can be adopted to a periodic structure using expansion into Fourier series for dynamic magnetization components in each waveguide. We consider further two coupled 1D MCs separated by a dielectric layer with thickness D (for a layered structure) or width D (for lateral magnetic stripes). The surfaces of each MC (MC 1 and MC 2) are periodically corrugated with the period L. In general, it is assumed that the MCs’ periods are shifted relatively to each other in the direction of the y axis by the value of q. The solution of the wave equations in each MC can be represented as a sum of spatial harmonics [15]. We shall consider only the zero-order harmonic of forward waves and the –1 harmonic of backward waves in the first Brillouin zone. In this case the solution of Eq. 2.2 for each of the crystals is the superposition of forward and backward waves: where A1,2 and B1,2 are the amplitudes of the forward and backward waves [14], respectively; k0 is the propagation constant of the zero-order harmonic; k– = k0 – 2p/L refers to the −1

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Coupled Spin Waves in Magnonic Waveguides

harmonic; and w is the frequency. Assuming in Eq. 2.2 that the thickness of each film is a periodic function, it can be represented as [14, 16]:

d1,2 = d01,02[1+ dd1,2cos( py L + q1,2 )],

(2.4)

where dd1,2 = 2Dd1,2sin(p(L – a)/L)/(pd01,02) and d01,02 = d + Dd1,2 (L – a)/L are the parameters that depend only on the geometry; Dd1,2 and a are the grooves’ depths and width, respectively; and q1 = 0, q2 = q. By substituting Eqs. 2.3 and 2.4 in Eq. 2.2 we can obtain the equations for waves in the coupled periodic structures:

  A1,2 A1,2  + – –  j B1,2 q1,2 + 1,2 KB2,1 = 0, + D A + 1,2k0 KA2,1  q1,2 y  1,2 1,2   t  B  –   A + +  j 1 ,2 – 1 ,2 1,2 + D1,2 KA2,1 = 0, B1,2 + 1,2kKB2,1  q1,2 A1 ,2  q1,2 y    t (2.5)

– D1,2 = – w2 + w2H + wM wH + 1 ,2k– , 1,2 = w2Md01,02 /2, where and ± ± i ( y1 ,2 ) ± ± q1,2 = e d1,2; and d1,2 = k± dd1,2/2, y1 = 0, and y2 = 2pq/L. If dd1,2 = 0 and B = 0 Eq. 2.5 describes a structure consisting of two coupled homogeneous film (HFs). If K = 0 then Eq. 2.5 describes SW dynamics in separated MCs. We determine the dispersion relation for waves in a coupled structure by equating the determinant of the set of equations in Eq. 2.5 to zero:



 D1+    2 k0 K  q+ 1   +  q2 K

 1 k0 K D2+ q1+K q2+

q1– q2– K

D1– 2k– K

q1– K   q2–  = 0 . 1 k – K    D2– 

(2.6)

If the diagonal components of the determinant in Eq. 2.6 are equal to zero, one can obtain the dispersion relations for forward and backward MSSWs in HFs. The off-diagonal components, which include the coupling coefficient K, describe the coupling ± between MC 1 and MC 2, d1,2 are the coupling parameters between ± the forward and backward waves in each MC, and q1,2 depend on

Spin Waves in Coupled Magnetic Stripes

the phase shifts y1 and y2 between the MCs. Note that when K ≠ 0 ± and d1,2 = 0, Eq. 2.6 describes the dispersion relation for MMSWs ± in the structure of the two coupled HFs. When K = 0 and d1,2 ≠0 in Eq. 2.6 we obtain the dispersion equations for MC 1 and MC 2 separately [17]. If d2± = 0 in Eq. 2.6 for MC 2, the second layer is an HF. The main problem in the coupled wave approach is the lack of definition of the coupling coefficient K and geometry of the magnetic structures. Moreover, it is necessary to take into account the multimode coupling between the transverse width modes of each finite-width magnetic waveguide [18]. Thus the micromagnetic numerical simulations reveal the properties of modes coupling and can be used for geometry design of the planar and layered directional multimode SW coupler. Using the finite element method (FEM) [19] the spectra of eigenmodes of the coupled magnetic structures and the coupling parameter K can be calculated, whereas both the time- and space-dependent magnetization evolution and SW transmission can be numerically simulated by means of the finite-difference method [20].

2.3  Spin Waves in Coupled Magnetic Stripes

The coupling of MSSWs propagating in a layered sandwich structure [21, 22], where two magnetic films are located in parallel with a small gap between them, can be used to fabricate a directional coupler. In another way, the conventional planar topology should be used for implementation in the magnonic architecture due to tunable frequency filtering and frequency demultiplexing characteristics of the coupled waveguiding structures [22–24]. Therefore side-coupled magnetic waveguides offer a range of further opportunities in planar magnonics [25, 26]. During the last three decades, the BLS technique has been used extensively to study multilayered magnetic structures. Nowadays high-resolution BLS spectroscopy [10, 11] opens up the possibility of dynamic magnetization study in planar sidecoupled magnetic stripes. Thus a single-crystalline ferrimagnetic yttrium iron garnet (Y3Fe5O12 (111)) (YIG) film with saturation magnetization M0 = 139 G was used for fabrication of side-coupled waveguides using ytterbium fiber laser [27]. The sketch of the

51

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Coupled Spin Waves in Magnonic Waveguides

fabricated structure is shown in Fig. 2.1a. The exactly identical stripes with a width w = 200 µm and a thickness of 10 µm were formed on a 500 µm thick gadolinium gallium garnet (Gd3Ga5O12 (111)) substrate to provide the efficient coupling of the propagating SWs. Magnetic stripes have a trapezoidal form in order to minimize the reflection of SWs, propagating along the y axis, and are separated from each other by a gap d = 40 µm. The short base of the first trapezoid (stripe a) was Sa = 8 mm and for the second trapezoid (stripe b) was Sb = 4 mm. Microwave transducers with a width of 30 µm and a length of ~2 mm are used for SW excitation and detection. The input and output transducers are attached to the magnetic stripe a at a distance of 8 mm from each other. A uniform static magnetic field H0 = 600 Oe was applied in the plane of the waveguides along the x direction. In this case the MSSW was effectively excited [12, 28].

Figure 2.1  (a) Schematic of the experiment. (b) Profile of the static internal magnetic field Hint along the x axis.

To understand how the guided power of SWs may be interchanged between side-coupled magnetic stripes, we calculate the dispersion characteristics of the coupled waveguides using FEM [19]. It should be noted that the spectra of eigenmodes of two identical waveguides consist of a symmetric and an antisymmetric transverse mode. The symmetric mode in the x direction corresponds to the case when the amplitudes of the magnetic potentials in two YIG films have the same phase (Fig. 2.2e,g), and in the antisymmetric mode they are out of phase by 180° (Fig. 2.2f,h). The lateral confinement of the each YIG stripe leads to the reduction of the internal magnetic field in the

Spin Waves in Coupled Magnetic Stripes

MSSW configuration. Figure 2.1b shows the calculated distribution of static internal magnetic field across the x direction. The internal field at the center of the waveguide is Hi = 550 Oe. This leads to the approximate shift of the lower cutoff frequency [29] of the MSSW, f0 = g Hi ( Hi + 4 pM0 ), where g = 2.8 MHz/Oe is the electronic gyromagnetic ratio for YIG. The dispersion characteristics, shown in Fig. 2.2a, were calculated using FEM and taking into account the non-uniform internal magnetic field profile. We consider nth-order transverse modes of coupled ) stripes: symmetric (wavenumber is kns) and antisymmetric (knas ). It can be seen that the lower cutoff frequency for the lowestorder symmetric modes is higher than that of the other modes. This cutoff frequency defines the beginning of the frequency range of effective dipolar coupling of SWs. The dispersion for nth-order transverse modes of the single magnetic stripe is depicted with dotted curves, denoted by kn in Fig. 2.2a. Figure 2.2b shows the frequency dependence of the modulus of the transmission coefficient of coupled magnetic stripes, which was measured using a VNA. Three well-pronounced stop-band dips, denoted with numbers and arrows in Fig. 2.2b, correspond to the frequencies at which the power of SWs does not effectively returns to the stripe a. This regime is possible while the output transducer is placed to the end of the stripe a; thus the laterally coupled stripes can demonstrate the frequency filtering regime, like in Ref. [22]. The transmission characteristic in Fig. 2.2b is plotted with the calculated curves. The dashed line shows the results of calculation of the transmission spectrum, taking into account only the first mode, while the dotted line shows the transmission of the superposition of the first, second, and third modes. It is worth noting that the frequencies of the first and second dips (denoted by arrows) in the experimental curve are in good correspondence with single-mode simulation. The frequency value of the third dip can be predicted by numerical simulation with all the first three modes of the coupled structure. This leads to the necessity of the multimode dispersion calculation for the design of coupled magnetic stripes. The distance for which maximum guided power is transferred from one waveguide to other is called the coupling length. In particular, for coupled magnetic stripes, the length necessary to fully transfer the power of the nth mode from one waveguide to the other can be written

53

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Coupled Spin Waves in Magnonic Waveguides

as Ln = p /|kns – knas|. Figure 2.2c demonstrates the frequency dependence of the coupling length for first three width modes. Coupling length increases with the increase of the frequency and the transverse mode order. Variation of the lateral gap d between the stripes leads to the tuning of Ln (Fig. 2.2d). The dotted line denotes the value d/w = 0.2, corresponding to the coupled YIG stripes in our experiments. Thus the coupling lengths are longer when operating in the multimode regime than in a single-mode regime. We note that the coupling length becomes shorter as the separation distance between the waveguides is shortened. To show this, we analyzed the mode profiles of first three modes corresponding to the eigenmodes of the coupled magnetic waveguides (Fig. 2.2e–h). The coupling length is longer for high-order modes because of the reduced overlap area of modes profiles for higher mode orders. In particular, using this configuration of modes profiles, we are able to do mode decomposition from the SW intensity (BLS map). Using BLS [30] we can estimate the effects of the coupling of the transverse width modes of the magnetic stripes that are difficult to estimate with alternative experimental methods. The BLS technique allows us to map SW intensity across the sample with the spatial resolution of 25 µm. The BLS intensity is directly proportional to the dynamic magnetization squared. Figure 2.3a demonstrates the coupling between two waveguides at the excitation frequency of f1 = 3.125 GHz. It can be seen that the guided power is transferred from one guide to the other in a periodic manner. The mode beating effect in the confined magnetic stripes leads to the strong spatial modulation of the SW intensity. To prove the concept of the transverse mode coupling we perform FEM and micromagnetic simulation of a multimode magnonic coupler. The map of SW intensity in coupled stripes can be represented as the superposition of the symmetric and antisymmetric modes. The beating of the transverse symmetric and antisymmetric modes results in guided power exchanges between two magnetic waveguides. Figure 2.3b is introduced to demonstrate the map of the squared SW amplitude calculated using FEM. Using micromagnetic simulation [20] of SW propagation in dipolar coupled waveguides we also can estimate the coupling length. The map of the mz component of magnetization is shown in

Spin Waves in Coupled Magnetic Stripes

Fig. 2.3c after a transient process (after a time of 200 ns since the excitation source is turned on). As it seen from Fig. 2.3a–c, the coupling length is the same in FEM, in micromagnetic simulation, and in BLS data. To prove the validity of the calculated frequency dependence of the coupling length, we show in Fig. 2.2c the value of coupling length calculated from micromagnetic simulation results (diamonds) and from experimental BLS data (open squares). We hope that using nanometer-thickness YIG films will make possible the fabrication of the nanoscale magnetic coupler in the near future. A further study is pointed to SW propagation in an array of coupled magnetic waveguides that can be used as a frequency multiplexer or a multidirectional multimode magnonic coupler. Thus coupled magnetic stripes can act both as a magnonic splitter and a multimode directional coupler, which might be employed in integrated planar magnonic systems as important building blocks.

Figure 2.2  (a) Dispersion characteristics for first three symmetric (solid lines) and antisymmetric (dashed lines) modes. (b) Transmission characteristics measured with a signal network analyzer (solid line) and calculated with only first symmetric and antisymmetric modes (dashed lines): the results of the calculation for first and third modes are plotted with a dotted line. (c) Frequency dependence of the coupling length obtained with FEM (solid lines), micromagnetic simulation (closed circles), and BLS technique (open squares). (d) Coupling length versus dimensionless ratio d/w at a frequency of 3.125 GHz. (e–h) Profiles for first (n = 1) and second (n = 2) transverse symmetric and antisymmetric modes.

55

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Coupled Spin Waves in Magnonic Waveguides

Figure 2.3  (a) Normalized color-coded BLS intensity map. (b) Calculated intensity (FEM) obtained by taking into account first three symmetric and antisymmetric modes. (c) Micromagnetic simulation of SW amplitude.

2.4  Nonlinear Spin Wave Coupling in Magnonic Crystals

The magnon transport from SW sources to magnonic circuits is central to any all-magnonic or integrated magnonic systems [2–5]. Over the last few years, periodic magnetic structures, MCs, have become a subject of striking interest in the scientific community because of the possibility of fabrication of predefined SW transmission properties. The coupling of periodic waveguiding structures opens up a new possibility of dispersion and transmission control, for example, due to multiple bandgap (BG) formation. The coupled stripes could operate in the nonlinear regime, where predominantly the power exchange ratio between the stripes is defined by the intensity of the input signal [31–33]. The idea of using a side-coupled MC in nonlinear application is associated with the intensity-dependent nonlinear shift of the magnonic forbidden (rejection) band [17] in each adjacent MC. The schematic

Nonlinear Spin Wave Coupling in Magnonic Crystals

of the experimental layout is shown in Fig. 2.4a [33]. The coupled magnetic stripes of width w = 720 µm and edge-to-edge spacing d = 40 µm were fabricated from 10 µm a thick monocrystalline ferrimagnetic YIG film. The length of the first magnonic stripe is 9 mm and of the second is 5.0 mm. The periodic sequences of the grooves with period D = 200 µm were fabricated on the surface of both magnonic stripes using precise ion-beam etching. Both MCs had a length of 20 periods in the direction parallel to the long axis of the stripes. Excitation of SWs in the first magnetic stripe is performed with the 50 ohm–matched microstrip transmission line. It is convenient to denote the input of the microwave signal as port C0 and the output of the first MC as port C1, while the output of the second MC is port C2 (Fig. 2.4a). The uniform static magnetic field H0 = 1300 Oe was applied in the plane of the waveguide along the z direction for the effective excitation of the guided MSSW [12, 28]. The BLS technique in the backscattering configuration was used to measure the 2D 5 × 1.5 mm2 spatial maps of the SW intensity and to demonstrate the efficient coupling of MCs. Figure 2.4b,c presents the BLS map of the dynamic magnetization squared at a frequency above and below the frequency of ferromagnetic resonance, respectively. As expected, the SW power transfers from one guide to the other in a periodic manner. Transmission and dispersion of MSSWs were measured using a VNA. The solid blue line in Fig. 2.5a shows the transmission response (absolute value of S21) for MSSWs. A well-pronounced rejection band, where SWs are not allowed to propagate, is clearly observed at a frequency of f2 = 5.608 GHz. The frequency width of the forbidden zone is DfB = 0.03 GHz at the level of 35 dB. To verify that the frequency f2 is the central frequency of the magnonic forbidden gap we acquire the dispersion characteristics by means of the measurement of the phase frequency response of the microstrip line with the coupled MCs (see the red dashed-dotted curve in Fig. 2.5a). The frequency f2 corresponds to the Bragg wavenumber kB = p/D = 157 cm–1. The noticeable dips at the transmission response, denoted by I, II, and III, are typical for the side-coupled magnonic stripes [19] (see Fig. 2.2b). The nature of these dips is revealed from the BLS experiments. Figure 2.5b shows the frequency dependence of the intensity of the BLS signal I(z, x) at the x position, which corresponds to the position of ports C1 and C2.

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At the frequency region in the magnonic forbidden gap signal, attenuation in both ports is visible. The striking difference between the frequency response of MC 1 and MC 2 is the inequality of the forbidden zone edge shifts due to the power increase. The transmission coefficient (power exchange coefficient) T21(P0, f ) = P2(P0, f )/P1(P0, f ) defines the ratio between the power transmitted in port C2 from MC 1 and is plotted in Fig. 2.5c as a function of frequency detuning df = f – fc from the center of the magnonic forbidden zone, fc = 5.615 GHz. The shift of the forbidden gap for MC 1 occurs at a power level lower than that for MC 2. In particular, this leads to a dramatic increase of T21 with an input power increase at frequencies near the lower cutoff frequency of the forbidden gap.

Figure 2.4 (a) Schematic of the experimental layout with side-coupled magnonic crystals, (b) normalized color-coded BLS intensity map of magnetization squared for the excitation frequencies f = 5.58 GHz (b) and f = 5.48 GHz (c).

Nonlinear Spin Wave Coupling in Magnonic Crystals

Figure 2.5 (a) Transmission (blue solid curve) and dispersion (red dash-dotted curve) characteristics, measured with the VNA. Calculated transmission is shown with the dashed green curve; (b) shows the dynamic magnetization along the z-coordinate of ports C1 and C2 as a function of frequency. Yellow area is the guide for the eye to show the frequency and wavenumber region of the magnonic forbidden zone; vertical dasheddotted line shows the central frequency of magnonic forbidden gap. (c) Power exchange coefficient as a function of frequency detuning from the center of magnonic forbidden zone. The dotted curve shows the numerical results. Open circles, squares, and triangles show the experimental data for power levels 26, 10, and−10 dBm, respectively. Vertical dashed lines show the frequency detuning at frequencies f1, f2 and f3.

Numerical modeling of the MSSW coupling in two side-coupled MCs was performed by using a direct numerical integration of

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the four coupled nonlinear Schrodinger equations [17], which were obtained from the Landau–Lifshitz equation for the magnetization dynamics, taking into account Kerr-type nonlinearity at the frequencies in the vicinity of the first Brillouin zone. Therefore, the decrease of the effective saturation magnetization with an increase of the amplitude of the magnetization dynamic part was taken into account. Numerical simulation (dotted curve in Fig. 2.5c) provides a good qualitative description of the observed phenomena. The parameters for numerical simulation were chosen to fit the nonequal shift of the low-frequency edges of both MCs. Due to more pronounced shift of the low-frequency edge of the magnonic forbidden gap, the simulation results are in good accordance for df < 0. The technology of side coupling of MCs described here represents an important step toward experiments with nonlinear magnonics. We emphasize that the coupling length is almost constant with input power increase, even to the highest value of 30 dBm. Only the nonlinear shift of the magnonic forbidden gap for each MC opens up the possibility of a coupled periodic magnonic structure to operate as a nonlinear switching device. The main strategy to utilize the nonlinear effect is to use the coupling of identical periodic structures and tuning the magnetic field for choosing the operating regime near the magnonic forbidden gap frequency. The nonlinear mechanism of the dynamic BG shift can induce intensity-dependent SW transmission.

2.5  Multilayer Magnonic Crystals

The control of the characteristics of the BGs in the spectrum of propagating waves should expand the functionality of MC waveguides. It was shown that coupled waveguides with the control parameter as the coupling between the layers are used in microwave electronics [1, 9] and in optics [34]. In particular, coupling of two waveguides leads to the existence of two normal waves in the structure—fast and slow waves. These waves propagate with different phase and group velocities, which can be controlled by variation of the coupling parameter in linear [1, 13, 14, 35] and nonlinear [16, 36] cases. It is expected that in

Multilayer Magnonic Crystals

the case of coupled periodic structures based on MCs, the coupling will play an important role in the mechanisms of BG formation. Here we present the results of theoretical and experimental studies of layered periodic structures in the form of MCs separated by a dielectric layer. We consider the influence of the coupling between the layers and the asymmetry of the structure on the mechanisms of BG formation. A layered structure consisting of two MCs separated by a dielectric layer with thickness D is shown in Fig. 2.6. The dispersion relation Eq. 2.6 was used to study the features of BG formation of MSSWs in this structure and the MC-HF. The results of the calculation of dispersion characteristics based on Eq. 2.6 for the two MCs with the same groove thicknesses and different shifts of MCs are shown in Fig. 2.7a–c with solid lines. The horizontal axis in Fig. 2.7 shows the wavenumber value k = Re (k0). The inset in Fig. 2.7a shows the dispersion characteristic corresponding to a single MC (D  ∞, К = 0) (black solid lines). In this case, only one BG is formed (G-MC, shaded gray) at the frequency of the first Bragg resonance at k = kB and f = fMC because of the interaction of the incident wave (black dashed line 1) and the reflected wave (black dashed line 1¢).

Figure 2.6  Scheme of the ferromagnetic structure in the form of two one-dimensional magnonic crystals separated by a dielectric layer.

As it is well known [1, 9], the dispersion curve for the MSSW splits into two normal modes, which correspond to fast and slow waves in coupled HFs (dd1,2 =0 and K ≠ 0). Dispersion characteristics for these waves are shown as dashed lines in

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Fig. 2.7a–c: red lines 2 for the fast incident wave and blue lines 3 for the slow incident wave. Corresponding dispersion characteristics of the reflected waves are denoted by lines 2¢ and 3¢. Interaction between the waves of four described types leads to the formation of BGs (shaded region in Fig. 2.7) at frequencies of the phase synchronism. As can be seen from Fig. 2.7а, in the symmetric structure at y = 0 the two BGs G-1 and G-2 at k = kB can be formed. BG G-1 is observed at frequencies higher than the frequency of the Bragg resonance for a single MC, fMC (red region in Fig. 2.7a). BG G-1 is formed because of the interaction of the incident and reflected fast waves (red dashed lines 2 and 2¢ in Fig. 2.7a). BG G-2 is observed at Re(k) = kB and at frequencies lower than the frequency of the Bragg resonance for a single MC, fMC (blue region in Fig. 2.7a). BG G-2 is formed by the interaction of the incident and reflected slow waves (blue dashed lines 3 and 3 in Fig. 2.7a). The presence of only two BGs in Fig. 2.7a is a symmetric degenerate case when the structure consists of two identical MCs placed symmetrically.

Figure 2.7  Dispersion characteristics of MSSWs in the structure MC-MC with the shift: (a) y = 0, (b) y = 0.6 p, and (c) y = p. (d) Dependence of the transmission and reflection coefficients for the fast wave and the slow wave on the frequency of the input signal at y = 0.6 p.

When y = 0.6 p (see. Fig. 2.7c) three BGs (G-1, G-2, G-3) within the dispersion curve are visible. An additional band G-3 (green

Multilayer Magnonic Crystals

region in Fig. 2.7c) is formed by the interaction of incident fast and reflected slow waves and incident slow and reflected fast waves. If y = p (periodic structures are shifted relative to each other by half of the period), only one BG (G-3) is visible (green region in Fig. 2.7d). The dependencies of the transmission coefficients for a fast wave T+ (solid red curve) and a slow wave T– (solid blue curve) and reflection coefficients for a fast wave R+ (dashed red curve) and a slow wave R– (dashed blue curve) on the frequency of the input signal, calculated on the basis of Eq. 2.5 at y = 0.6 p, are shown in Fig. 2.7d. Three minimums of coefficients T± at different frequencies that correspond to the three BGs in such a structure (G-1, G-2, G-3) are observed. In this case two minimums correspond to the each curve T±, and the average minimum G-3 corresponds to the transmission coefficients of both T+ and T–. This is consistent with the mechanism of BG formation (see Fig. 2.7c). Dependence of the widths and positions of the BGs on the parameter y are shown in Fig. 2.8a. It is seen that at y = 0 and y = 2p there are two BGs (G-1 and G-2). When y = p there exists one BG (G-3). When 0 < y < p and p < y < 2p all three BGs can be formed in the system (G-1, G-2, G-3). Thus, by changing the phase shift between the MCs, one can effectively control the characteristics of the BGs (their number and width). In the case of a structure consisting of two MCs with different geometric dimensions the mechanism of BG formation is similar to the case presented in Fig. 2.2b. In this case also three BGs are formed. As it follows from the presented results, the third BG, G-3, is formed due to violation of symmetry in the geometry of the structure relatively to the axis along the direction of wave propagation. The influence of the coupling coefficient K on the characteristics of BGs is shown in Fig. 2.8b in the structure MC-HF. The dashed curves in Fig. 2.8b show the shift of the central frequencies of the BGs, depending on K. It is seen that if K increases, the central frequency of the gap G-1 shifts up (red area G-1 in Fig. 2.8b), the central frequency of the gap G-2 shifts down and the BG G-2 narrows (blue area G-2), and the central frequency of the gap G-3 shifts down and the BG G-3 narrows

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(green area G-3). Note that a similar situation exists for the structure based on MCs with a shift.

Figure 2.8  (a) Dependence of the width and position of bandgaps from the phase shift between MCs. (b) Dependence of the width of bandgaps (G-1, G-2, G-3) (shaded areas) and central frequencies (dashed lines) on values of coupling K for the structure MC-HF.

For an experimental study of wave propagation in periodic structures, a YIG film with thickness d = 12 µm was used. The schematic of the experimental layout is shown in the inset in Fig. 2.9a. On the surface of the film a periodic system of grooves with period L = 200 µm; width a = 100 µm; and depth Dd1 = Dd2 = 1 µm was developed. On the top surface of one of the MCs a dielectric mica plate with a thickness of D = 25 µm was placed. A second MC with the same parameters or a homogeneous YIG film was placed on top of the dielectric plate. The length of the second MC (or HF) was smaller than the length of the first MC and was equal to 4 mm. For excitation and detection of the magnetostatic SW in an MC microstrip transducers were used. The region of overlap of the first MC and the second MC (or HF) was 4 mm, or 20 periods of MCs. A magnetic field H0 = 300 Oe was

Multilayer Magnonic Crystals

applied to the film parallel to transducers; thus the MSSW was excited.

Figure 2.9  Normalized color-coded BLS intensity map of magnetization squared, demonstrating the coupling in a layered structure MC-HF. (b) Transmission characteristics of MSSWs in a single MC (black curve 1) and in a structure MC-MC (blue curve 2).

In the first step, to confirm that the selected thickness of the dielectric plate provides a sufficient value of coupling and the layered structure behaves as a coupled system, the experimental BLS technique was used. Figure 2.9a shows the map of the scattered light intensity at a frequency of the BG for the layered structure MC-HF. The focus of the optical system was tuned to the layer of MCs. The observed picture is due to periodic power exchange between MCs and HFs in a layered structure along the direction of propagation.

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In the second step, we investigated the transmission response for the studied structures (Fig. 2.9b). In the transmission response of a single MC (black curve 1) one BG, G-MC, is clearly observed, which corresponds to the first Bragg resonance. For a symmetric structure MC-MC at y = 0 the formation of two BGs in the region of the first Bragg resonance, G-1 and G-2, can be seen (blue curve 2). Figure 2.10 shows the results of an experimental study of BG formation in the structure MC-HF. From the results in Fig. 2.10a it follows that in such a structure three BGs in the region of the first Bragg resonance are formed. To compare theoretical and experimental results we plot the dispersion characteristics of MSSWs (blue curve 1, experiment; red curve 2, calculation based on the analytical model) in Fig. 2.10b. It can be seen that three BGs in such a structure are formed.

Figure 2.10  (a) Transmission characteristic of MSSWs in a single MC (black curve 1) and in an MC-HF (blue curve 2). (b) Experimental (blue curve 1) and theoretical (red curve 2) dispersion characteristics of MSSWs in an MC-HF.

We consider the layered structure consisting of two coupled MCs. It is shown that in the case of symmetry breaking (in contrast to the symmetric structures in which two BGs can be formed) in an asymmetric structure, three BGs near the first Bragg resonance are formed. Symmetry breaking can be caused by a change of the shift between the MCs, the geometrical parameters of one of the MCs, in the structure MC-HF. The frequency interval between zones is determined by the coupling coefficient between the MCs, which depends on the distance between the MCs and asymmetric properties.

Frequency-Selective Tunable Spin Wave Channeling

2.6  Frequency-Selective Tunable Spin Wave Channeling The control over the dispersion of SWs can be achieved, for example, by the periodic patterning of thin magnetic films. MCs can demonstrate a complicated magnonic band structure with strong dispersion and anisotropy. The control of the BG characteristics of MCs can be performed by the fabrication of defects in the periodic structures. There are two main types of defects, a local defect and a line defect (LD). The first type of defect is constructed within some region of a periodic structure, and its size is comparable with the MC spatial period. The LD is a homogeneous region located within a periodic structure in the direction of SW propagation. The sizes of such type defects are much greater than the MC spatial period. The absence of a BG in SW spectra of 2D periodic structures with the LD [37] leads to the necessity of 1D MC fabrication, which supports both SW propagation and BG formation. Here we demonstrate the realization of BG control in a 1D MC with an LD by variation of the defect’s geometry. Figure 2.11a shows a 1D MC with an LD. The MC is formed on the surface of a 7.7 μm thick and 2.2 mm wide YIG film. The periodic copper stripes with a period of L = 300 μm form the periodic boundary conditions on one surface of YIG. The bias static magnetic field H0 = 370 Oe is applied in the film plane. Figure 2.11b demonstrates that the first BG of the 1D MC does not disappear when an LD is present within the periodic system. Moreover, the dependence of the BG frequency shift on the LD width has a complicated form. To explain experimental results we consider an LD magnonic waveguide as a structure consisting of two identical regions with periodic metal stripes of width d, separated by a region between them with an HF of width D. Such structure can be considered as two periodic regions with the value of coupling K. In Section 2.2 we obtained the dispersion relation for waves in such a structure [35]. Figure 2.12a shows the dispersion characteristics for an MC with an LD (blue curve 1) and without an LD (black dashed curve 2). BGs are formed in SW spectra in MCs with LDs and in MCs without LDs (shaded areas). According to the theoretical model, the variation of the

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value of D leads to the BG of the MC with an LD shift (shaded blue G-LD MC in Fig. 2.12a) relative to the BG position for the MC without an LD (shaded gray G-MC in Fig. 2.12a).

Figure 2.11  (a) Scheme of a 1D MC with a line defect. (b) Transmission responses of an MC measured at various values of the line defect width: D = 0 (solid line), D = 0.2 mm (dotted line), and D = 1 mm (dashed line).

Figure 2.12 (a) Theoretical dispersion characteristics of SWs in an LD magnonic waveguide. (b) Central frequency of a G-LDMC as a function of LD width.

Figure 2.12b shows the dependence of the central frequency of the G-LD MC, f (solid curve), on the value of D at w = const. Circles in Fig. 2.12b denote the experimental values of the central frequency of the BGs for samples with different D values. Here we denote the BG central frequency at D = 0 by fMC (I in Fig. 2.12b). With increasing D the values of f are shifted first up in frequency and are greater than fMC (II), and then frequency is shifted down and is below fMC (III). Thus, the fabrication of LDs in a 1D MC allows control of the position of the BG in the SW spectrum.

Frequency-Selective Tunable Spin Wave Channeling

The spatial and frequency filtering features of MCs have straightforward advantages in magnonic applications. Magnetic materials can be engineered to generate anisotropy and dispersion, which open up the possibility of nondiffractive SW propagation or self-collimation. The nondiffractive propagation of SWs is possible in a magnonic crystal array (MCA) [38] that supports self-collimation in the frequency range of a magnonic forbidden gap. An MCA was fabricated from a single-crystalline ferrimagnetic YIG film (Fig. 2.13a). A magnetic waveguide width wm = 3.5 mm as produced using laser scribing [38]. An array of grooves with period L = 200 µm was fabricated at the YIG surface using precise ion-beam etching. The width of each groove was wg = 500 μm, the groove depth was 1 μm, and the length was lg = L/2 = 100 μm. The distance between grooves in the x direction was wd = 500 μm. This structure forms three channels separated by 1D MCs. The sample has a length of 30 periods in the y direction.

Figure 2.13 (a) Schematic of the experimental setup. (b) Transmission characteristics measured with a VNA for an MC (blue curve) and an MCA (red curve). Inset: Detailed frequency region of the magnonic forbidden zone.

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The only four periods in the x direction allow us to consider this structure as the MCA with linear waveguide channels rather than as a 2D square-lattice MC. As a reference, we fabricated a conventional MC, which is similar to the structure with the array of MCs but with the value of wd = 0. Input and output transducers are attached to the YIG film at a distance of 8 mm from each other. The uniform static magnetic field H0 = 1185 Oe was applied in the plane of the waveguide along the x direction. The solid blue line in Fig. 2.13b shows the measured MSSW intensity for the MC. The frequency f0 = 5.222 GHz corresponds to the ferromagnetic resonance. A well-pronounced stop band where SWs are not allowed to propagate is clearly observed for the frequency f2 = 5.333 GHz. The frequency width of the first BG is DfB = 0.05 GHz at the level of −35 dB. The frequency f2 = 5.333 GHz corresponds to the Bragg wavenumber kB = p/L = 157 rad/cm. The microwave measurements of the fabricated MCA show that the first forbidden frequency band is clearly distinguishable at the transmission response (see red curve in Fig. 2.13b). To demonstrate the distribution of the dynamic magnetization in the MCA, the BLS technique in the backscattering configuration was used. Figure 2.14a–c represents the results of 2D 4 × 3.5 mm2 mapping of the sample area. At the frequency f1 = 5.3 GHz the spatial profile of the BLS map corresponds to the typical result of width modes beating [11, 18, 28] (Fig. 2.14a). The estimated half-period of the beating of the first and third transverse modes is about 3.5 mm in the frequency range of f1 < f < f2. This is in good accordance with the BLS map, where the transverse size of the SW beam [30] is decreased at y = 3 mm. Distribution of the magnetization squared is almost not affected by the array of MCs on the surface of YIG. Distribution of SW intensity changes as the frequency reaches the forbidden band. At frequency f2 the spatial map of magnetization, shown in Fig. 2.14b, demonstrates the channeling of SWs. Confinement of SWs in the x direction in each channel is clearly seen. In the central channel, denoted by CH2 in Fig. 2.14a, the SW beam propagates almost nondivergently over a distance of 3 mm, while in two adjacent channels (along the lines x = 0.7 mm [CH1] and x = 2.8 mm [CH3]) SW collimation allows the beam to propagate over a distance of about 2 mm. Collimation allows for a wave to travel in a medium along one

Frequency-Selective Tunable Spin Wave Channeling

particular dimension without a significant diffraction in any orthogonal dimensions. To show that the frequency-selective collimation of SWs is associated with the frequency range of the first forbidden zone, we performed spatially resolved measurements of SW intensity along the line y = 4.0 mm in the frequency range from 5.15 GHz to 5.5 GHz. The results of these measurements are shown in Fig. 2.14d. In the frequency range f0 < f < (f2 – DfB)/2 the intensity of SWs is allocated in the center of the MCA structure. As the frequency reaches the frequency range of BG (f2 – DfB)/2 < f < (f2 + DfB)/2, three SW beams are observed in three channels inside the MCA. As the frequency increases the confinement of SWs vanishes. Since the selfcollimation in the MCA structure manifests itself in the frequency range corresponding to the frequencies inside the BG, the MCA can exhibit the spatial-frequency-selective regimes for SWs.

Figure 2.14 Normalized color-coded BLS intensity map at f = 0° for frequencies: f1 = 5.3 GHz (a), f2 = 5.33 GHz (b), and at f = 15° for frequency f2 (c). (d) Map of the BLS intensity as a function of frequency. Gray horizontal dotted lines depict the edges of the adjacent waveguides. The vertical dashed line is a guide for the eye to show the central frequency of the first magnonic forbidden zone of the MC. The frequency f0 is depicted with the vertical dashed-dotted line.

By variation of the orientation of the applied direct current (DC) magnetic field (angle f), the control over the SW propagation length in magnonic channels is possible. Figure 2.14c shows the SW distribution when a magnetic field is applied at f = 15°.

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The propagation length of the collimated SW beam in channel CH1 is considerably larger than in CH2 and CH3. We analyze and verify the angular dependencies of SW transmission with the micromagnetic simulation technique [20] by calculation the integral value of SW intensity in the nth channel Cn at different values of f (Fig. 2.15c). Rotating the magnetic field enables controllable manipulation of SW propagation over the magnonic channels.

Figure 2.15 Dependencies of spin wave transmission on the in-plane magnetic field angle orientation. Open symbols denote the simulation results, and closed symbols correspond to the experimental data.

2.7  Conclusions

In this chapter we considered the concept of SW coupling in lateral and vertical magnonic structures. We showed that the functionalities of simple magnetic stripes can be extended by lateral or vertical coupling with adjacent stripes or MCs. We demonstrated nonlinear SW switching in two side-coupled MCs. We showed that in a fabricated 2D MCA the control over a propagation distance in SW channels can be implemented by the orientation of an external magnetic field. Collimated SW beams can be used as signal carriers in the magnonic platform for applications such as signal multiplexing. Thus, coupled magnonic stripes and crystals can act both as a magnonic splitter and a multimode

References

directional coupler, which might be employed in the integrated planar and vertical magnonic networks as important building blocks.

Acknowledgments

This work was supported, in part, by the grant from the Russian Science Foundation (Project No. 16-19-10283, 14-19-00760), the Russian Foundation for Basic Research (Project No. 15-07-05901), and the grant from the president of the Russian Federation (No. MK5837.2016.9).

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24. Annenkov, A. Y., and Gerus, S. V. (1996). Propagation of magnetostatic waves in two coupled channels created by a magnetic field, J. Commun. Technol. Electron., 41, 196. 25. Sadovnikov, A. V., Davies, C. S., Grishin, S. V., Kruglyak, V. V., Romanenko, D. V., Sharaevskii, Yu. P., and Nikitov, S. A. (2015). Magnonic beam splitter: the building block of parallel magnonic circuitry, Appl. Phys. Lett., 106, 192406.

26. Sadovnikov, A. V., Grachev, A. A., Beginin, E. N., Sheshukova, S. E., Sharaevskii, Yu. P., and Nikitov, S. A. (2017). Voltage-controlled spin-wave coupling in adjacent ferromagnetic-ferroelectric heterostructures, Phys. Rev. Appl., 7, 014013. 27. Sheshukova, S., Beginin, E., Sadovnikov, A., Sharaevsky, Y., and Nikitov, S. (2014). Multimode propagation of magnetostatic waves in a width-modulated yttrium-iron-garnet waveguide, IEEE Magn. Lett., 5, 1–4. 28. Bajpai, S. N. (1985). Excitation of magnetostatic surface waves: effect of finite sample width, J. Appl. Phys., 58, 910.

29. Kittel, C. (1951). Ferromagnetic resonance, J. Phys. Radium, 12, 291–302. 30. Demidov, V. E., Demokritov, S. O., Rott, K., Krzysteczko, P., and Reiss, G. (2008). Mode interference and periodic self-focusing of spin waves in permalloymicrostripes, Phys. Rev. B, 77, 064406.

31. Zvezdin, A., and Popkov, A. (1983). Three-and four-magnon decay of nonlinear surface magnetostatic waves in thin ferromagnetic films, Sov. Phys. JETP, 57, 350.

32. Boyle, J. W., Nikitov, S. A., Boardman, A. D., and Xie, K. (1997). Observation of cross-phase induced modulation instability of travelling magnetostatic waves in ferromagnetic films, J. Magn. Magn. Mater., 173, 241. 33. Sadovnikov, A. V., Beginin, E. N., Morozova, M. A., Sharaevskii, Yu. P., Grishin, S. V., Sheshukova, S. E., and Nikitov, S. A. (2016). Nonlinear spin wave coupling in adjacent magnonic crystals, Appl. Phys. Lett., 109, 042407.

34. Marcuse, D. (1972). Light Transmission Optics (Van Nostrand Reinhold Company, New York). 35. Morozova, M. A., Grishin, S. V., Sadovnikov, A. V., Romanenko, D. V., Sharaevskii, Yu. P., and Nikitov, S. A. (2015). Band gap control in a linedefect magnonic crystal waveguide, Appl. Phys. Lett., 107, 242402.

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36. Morozova, M. A., and Matveev, O. V. (2015). Propagation of nonlinear pulses of magnetostatic waves in coupled magnonic crystals, Phys. Wave Phenom., 23(2), 114–121.

37. Schwarze, B. T., Grundler, D. (2013). Magnonic crystal wave guide with large spin-wave propagation velocity in CoFeB, Appl. Phys. Lett., 102, 222412.

38. Sadovnikov, A. V., Beginin, E. N., Odincov, S. A., Sheshukova, S. E., Sharaevskii, Yu. P., Stognij, A. I., and Nikitov, S. A. (2016). Frequency selective tunable spin wave channeling in the magnonic network, Appl. Phys. Lett., 108, 172411.

Chapter 3

Tuning of the Spin Wave Band Structure in Nanostructured Iron/Permalloy Nanowire Arrays G. Gubbiotti,a S. Tacchi,a R. Silvani,a,b M. Madami,b G. Carlotti,b A. O. Adeyeye,c and M. Kostylevd aIstituto

Officina dei Materiali, Consiglio Nazionale delle Ricerche, Sede Secondaria di Perugia, Via A. Pascoli, Perugia 06123, Italy bDipartimento di Fisica e Geologia, Università di Perugia, Via A. Pascoli, Perugia I-06123, Italy cInformation Storage Materials Laboratory, Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore dSchool of Physics, University of Western Australia, Crawley 6009, WA, Australia [email protected]

In the developing field of magnonics, based on the utilization of spin waves in magnetic microstructures for information  processing and computation, it is very important to understand  and tune the spin wave band structure. In this chapter, we  present the results of a combined experimental and theoretical investigation of the spin wave band structure in bilayered iron/ permalloy (Fe/Py) nanowires (NWs) with either “rectangular” cross section (bottom and upper layers of equal width) or “L-shaped” cross section (upper layer of half width). The results of Brillouin  Spin Wave Confinement: Propagating Waves (2nd Edition) Edited by Sergej O. Demokritov Copyright © 2017 Pan Stanford Publishing Pte. Ltd. ISBN  978-981-4774-35-2 (Hardcover),  978-1-315-11082-0 (eBook) www.panstanford.com

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light scattering (BLS) experiments, performed while sweeping  the wave vector perpendicularly to the NW length over three Brillouin zones of the reciprocal space, have been interpreted  thanks to micromagnetic simulations based on a GPU-based software. For the rectangular Fe(10 nm)/Py(10 nm) NWs, the lowest-frequency fundamental mode exhibits a sizable and  periodic frequency dispersion. A similar dispersive mode is also present in L-shaped NWs, but the mode amplitude is concentrated in the Py NW portion not covered by Fe. An overall blue shift in frequency of the entire band structure is observed on increasing  the Fe NW thickness from 2 to 20 nm and corresponds to a localization of the mode in the Py NW portion not covered by the  Fe NW. Moreover, for the L-shaped NWs it is shown that there is at least another dispersive mode, at higher frequency, whose  amplitude is large in the thick side of the NW.

3.1  Introduction

Magnonic crystals (MCs) consist of periodically modulated  magnetic material and make use of fundamental properties of  waves, such as scattering and interference, to create “bandgaps”— ranges of wavelength (or frequency) within which spin waves  (SWs) cannot propagate through the structure. The concept of MCs followed by two decades the analogous concept of photonic crystals for the propagation of electromagnetic waves [1, 2]  where the band gap is caused by a periodic variation in the  refractive index of an artificially structured material. In this  respect, MCs offer opportunities to tailor-made novel properties for SW propagation such as high group velocity [3], magnetic  field tunability, and controllable bandgap opening [4–15].  Interesting phenomena such as anisotropic damping [16], Bragg diffraction [17, 18], and SW mode conversion [19] have also  been demonstrated. Besides, bi-component MCs (BMCs), consisting of two different periodically arranged magnetic materials, have additional degrees of freedom thanks to the contrasting properties of the ferromagnetic (FM) materials [20–22]. The progress in the field of MCs goes in parallel with the capability to realize magnetic nanostructures over a large area by

Introduction

using interference [23, 24] and nanosphere [25, 26] lithographic techniques. However, these techniques are limited in the types of geometry of the nanostructure that can be fabricated and it  is only recently that synthesizing of large area MCs consisting  of one or two different ferromagnetic materials with dimension control, resolution, size, and shape homogeneity became possible [27–30]. In one dimension (1D), BMCs have been experimentally demonstrated, in the form of periodic arrays of alternating contacting cobalt (Co) and permalloy (Py) nanowires (NWs).  The SW dispersion relations were mapped for both the parallel [31, 32] and the antiparallel [33, 34] configurations, studying the dependence of the bandgap position and width on the external magnetic field. Due to the rather different resonance frequency of the two materials, the main dispersive mode is formed on  the basis of standing wave oscillations in uncoupled Py NWs.  For such excitations, the Co NWs act as amplifiers of dipole  coupling between the Py NWs. These studies also suggested that there is a strong direct exchange coupling across the Co–Py  interface, which influences the pinning of the dynamical  magnetization at the interface [10, 35]. Furthermore, experimental studies on 2D BMCs revealed that their band structure is very rich because of the large density of modes and their consequent hybridization [36–38]. Here, additional complexity with respect to the 1D case derives from the pronounced non-uniformity of  the internal static field, which induces the formation of channels  for spin wave propagation in the direction perpendicular to that  of the applied magnetic field [2, 7, 10, 30–41]. In the recent past, a new class of magnonic materials  constituted by closely packed Py NWs with step-modulated thickness has been proposed, showing that this nonplanar array supports the propagation of collective spin waves in the  periodicity direction [42]. These results indicated that vertically layered MCs, where ferromagnetic materials are placed in direct contact or separated by a nonmagnetic material, can be  exploited to add new functionalities to control the in-plane propagation of collective SWs [43]. The advantage of vertically modulated NWs over 2D systems is that they have all the benefits of BMCs without the complexity of the inhomogeneity of the 

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Tuning of the Spin Wave Band Structure in Nanostructured Iron/Permalloy NW Arrays

internal magnetic field, since when the NWs are longitudinally magnetized the internal field is uniform and equal to the magnetic applied field value. The intent of this chapter is to review recent results relative  to the spin wave band structure of longitudinally magnetized bilayered (Fe–Py) NW arrays characterized by either a rectangular  or a L-shaped cross section [43, 44]. Our investigation has been carried out by means of Brillouin light scattering (BLS) from  thermally excited spin waves and GPU-based micromagnetic simulations. All the samples have fixed width (w2) of the Py NWs and differ by the width of the Fe (w1) ones. For NWs with “rectangular” cross section the bottom Py and upper Fe layers  are of equal width w1 = w2 = 340 nm, while for “L-shaped” NWs w1 = 170 nm and w2 = 340 nm. BLS spectra were recorded by sweeping the wave vector perpendicularly to the wire length  over three BZs of the reciprocal space. Remarkably, for all the investigated NWs, the fundamental mode, lying at the lowest frequency, shows the largest frequency dispersion. Moreover, for  the L-shaped cross-section NWs, it is shown that there is also  another, higher-order, mode with a sizable dispersion.  Quantitatively, the experimental results of the spin wave band structure have been successfully reproduced my means of micromagnetic simulations performed with the GPU-based  software MUMAX3. This approach also enabled us to calculate the spatial profile of the dynamical magnetization corresponding  to each of the detected eigenmodes, showing that those modes which exhibit a sizable dispersion are characterized by the  absence of nodes across the NW width.

3.2  Sample Fabrication and Brillouin Light Scattering Measurements

Seven NWs arrays, whose geometrical and magnetic parameters are summarized in Table 3.1, have been studied. NWs arrays were fabricated over a large area (4 × 4 mm2) on silicon substrates by using deep ultraviolet lithography in combination with tilted  shadow deposition technique. Differently from the multilevel electron beam lithographic process, this recently developed  method does not require the alignment between the two FM layers

Sample Fabrication and Brillouin Light Scattering Measurements

and the deposition of the two materials can be performed without breaking the vacuum in the same process step, thus ensuring high quality of the interface between the two FM materials. (a)

(b)

Figure 3.1  (a) Scanning electron microscope images of Samples #3 and  #6 with L1 = L2 = 10 nm, together with the coordinate axes and the  direction of the external applied magnetic field (H) and wave vector k.  (b) Schematic representation of the NW with either rectangular and  L-shaped cross section.

In all the samples, the lower NWs, formed by permalloy (Ni80Fe20, Py), have a fixed width w2 = 340 nm and thickness  L2 = 10 nm. The Py NWs were capped with Fe NWs having a either  a width w1 = 340 nm and a thickness L1 = 10 nm (“rectangular”  cross-section, sample #3) or w1 = 170 nm (“L-shaped” cross section) and L1 = 2, 5, 10, or 20 nm (sample #4 to #7). Single-layer NWs of either Py (w2 = 340 nm, L2 = 10 nm, sample #1) or Fe (w1 = 340 nm, L1 = 10 nm, sample #2) were also fabricated and used as reference samples. For all the arrays, the inter-wire edge-to edge distance  was d = 100 nm (as measured at the level of Py layers) and the  array periodicity a = (w2 + d) = 440 nm, resulting in the boundary 

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Tuning of the Spin Wave Band Structure in Nanostructured Iron/Permalloy NW Arrays

of the first BZ located at p/a = 0.71 × 107 rad × m–1. Scanning  electron micrographs of the “rectangular” (sample #3) and  “L-shaped” (sample #6) cross-section NWs are shown in Fig. 3.1, while the geometric parameters of the investigated NW arrays  are listed in the third and fourth columns of Table 3.1. Table 3.1  Geometric and magnetic parameters for the single- and bilayered NWs Sample Number of Layer width label layers w1, w2 (nm)

Layer thickness L1–L2 (nm) 4pMs

1

1,Py

340

10

7000

3

2,Fe–Py

340–340

10–10

16500–10000

2 4 5 6 7

1,Fe

2,Fe–Py 2,Fe–Py 2,Fe–Py 2,Fe–Py

340

170–340 170–340 170–340 170–340

10

2–10 5–10

10–10 20–10

12500

12500–7500 16500–8500 18500–9000 18000–7500

The same set of NW array has been studied by ferromagnetic resonance (FMR) [45]. However, with the above technique, it  is not possible to sweep the wave vector of measured excitations in order to map the SW frequency dispersion across the whole  first BZ. FMR allows accessing the G point of the first BZ only. In other words, with FMR one probes only spin  waves with the Block wave vector k = 0, and only modes for  which the magnetization dynamics represents a family of  in-plane standing spin waves with a nonvanishing net magnetic moment. For recording the BLS spectra, about 200 mW of  p-polarized monochromatic light from a solid state laser l = 532 nm  was focused onto the sample surface and the s-polarized backscattered light was analyzed with a (3+3) tandem Fabry–Pérot interferometer [46]. A magnetic field of H = 500 Oe was applied along the NWs length ( y direction) and spectra were recorded for different incidence angles of light (qi). The incident plane of light  was orthogonal to the direction of H (Damon–Eshbach  geometry). This corresponds to sweep the in-plane wave vector k in the orthogonal direction (x direction) [47]. Thanks to the

Micromagnetic Simulations

conservation of the in-plane component of the wave vector in  the scattering process, these two quantities are related by the following relation k = (4p/l) × sin q i.

3.3  Micromagnetic Simulations

In order to reproduce the dispersion of collective SW excitations  and the cross-sectional profiles of the modes, we used the MUMAX3 GPU-based micromagnetic simulator [48]. Periodic boundary conditions were used in the array plane to reproduce the periodic nature of an artificial MC. A magnetic field H = 500 Oe was  assumed along the NWs length, while a sinc-shaped field pulse

h(t )= h0

sin[2pf0 (t – t 0 )] 2pf0 (t – t 0 )

directed perpendicularly with respect to the sample plane (i.e.,  along the z axis), with the maximum amplitude h0 = 30 Oe and frequency f0 = 30 GHz, was used to excite the NW placed in the center of the simulated area. The pulse deflects the magnetization vector from its equilibrium direction. The precession is spatially uniform along each individual NW (y direction) but is  non-uniform across the NW cross section (x direction), owing to the effect of confinement in the cross-section plane. The dynamic stray field of individual NWs is strong and long-ranging  enough to dynamically couple the NWs and produce collective magnetization dynamics on the array in the form of a Bloch spin wave propagating along x, i.e., along the array’s periodicity  direction. The dispersion relation was calculated by performing a Fourier transform of the out-of-plane component of the  magnetization both in space and time. The spatial profile of the  spin wave modes has been calculated at the center of the first BZ  (k = 0). The spatial profiles for the two layers were extracted separately, by first Fourier-transforming the out-of-plane component of the magnetization vector in time, for each simulation cell,  and then plotting the spectral amplitude at a given frequency for each cell. Further information on the micromagnetic simulations  can be found in Gubbiotti et al. [44] and Silvani et al. [49].  The material parameters used in the simulations are as follows:

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Tuning of the Spin Wave Band Structure in Nanostructured Iron/Permalloy NW Arrays

gyromagnetic ratio gPy = 2.90 GHz/kOe and gFe = 3.05 GHz/kOe,  exchange stiffness constant AFe = 2 × 10–6 erg/cm and  APy = 1 × 10–6 erg/cm. Regarding the interlayer exchange field, MUMAX code uses a built-in model assuming the interface-  effective exchange field equal to the average of the exchange fields  of the two ferromagnetic materials. The geometric parameters and the saturation magnetization values assumed in the simulations are summarized in Table 3.1.  It can be seen that the saturation magnetization values vary from sample to sample and are significantly smaller  than the corresponding values of bulk Fe and Py. This is because the extracted values correspond to the “effective” saturation magnetizations—i.e., they include the effect of surface  perpendicular uniaxial anisotropy, as well as the presence of  dead magnetic layers.

3.4  Results and Discussion

3.4.1  Spin Wave Band Structure and Mode Spatial Profiles for the NWs with Rectangular Cross Section In Fig. 3.2, we show BLS spectra measured at k = 0 and p/a for  NW arrays with rectangular cross section (samples #1 to #3).  Spectra have a very good signal-to-noise ratio and are  characterized by the presence of several well-resolved peaks.  It can be seen that the lowest-frequency mode has the largest intensity and exhibits a remarkable frequency evolution with  k, while the modes at higher frequency are much less dispersive,  in agreement with previous investigations of SW in arrays of interacting stripes [50]. The evolution of the frequency as a function of the k vector for these modes is plotted in Fig. 3.3 together with the simulated dispersion that is periodic with a number of BZs in agreement  with the artificial periodicity of the stripes array. The periodicity  of the frequency oscillation (width of the Brillouin zone) is independent of the thickness of the layers, since all the investigated NWs arrays have the same lattice period a.

Results and Discussion

Figure 3.2  BLS spectra for sample #1 to #3 measured for H = 500 Oe  applied along NW length ( y axis) at k = 0 and k = p/a. To avoid overlapping, spectra measured at k = p/a are vertically shifted with respect to  those at k = 0. The drawings (not to scale) give a schematic representation  of the NW layering structure where black represents Py and grey is for Fe.

Figure 3.3  Measured (points) and simulated (intensity plot) frequency dispersion of the collective SW modes for the different NWs with  rectangular cross section (sample #1 to #3). The external magnetic field,  H = 500 Oe, is applied along the length of NWs, while the wave vector is  in the direction perpendicular to the NWs length.

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Tuning of the Spin Wave Band Structure in Nanostructured Iron/Permalloy NW Arrays

Concerning the position of the center of the magnonic band, one sees that it shifts up passing from sample #1 to #3, i.e., from the single-layer Py NWs to the single Fe layer and finally to the bilayered NWs. In addition, also the width of the magnonic band (frequency variation between k = 0 and p/a) is more  pronounced for bilayered NWs than for single-layer ones. In all  the cases, an overall good agreement of the experimental data  with the results of the micromagnetic calculations is achieved.  This is particularly true for the lowest frequency mode of each sample, while for the high-order modes the simulated frequencies  are slightly overestimated. Note, however, that such a small quantitative disagreement between experiment and simulations originates from the fact that we did not perform a real best-fit procedure of the experimental data to the calculated curves. From the calculations, one can also obtain the spatial profiles of the dynamic magnetization amplitude for the different modes  of each sample, as shown in Fig. 3.4. The profiles are calculated at k = 0; the dynamic magnetization distributions have been  averaged over the thicknesses of the respective layers. The spatial profiles for the Py and Fe NWs arrays are practically the same: the profile of the fundamental (quasi-uniform) mode without nodal plane corresponds to the lowest-frequency dispersion branch. Because of a large net dynamic magnetic moment associated with this fundamental type of profile, this mode creates a large  dynamic dipolar field which efficiently couples each NW with its neighbors, resulting in a sizable dispersion [51]. The modes at higher frequencies are characterized by a nonvanishing number of nodes, whose number increases with the mode frequency.  Modes having an odd number of nodes are plotted as dashed lines and their net magnetic moment vanishes for symmetry reasons.  This results in a negligible contribution of these modes to the  BLS cross section at k = 0 [52]. Note that for all the modes recorded for the bilayered NWs (sample #3) the precession is in-phase in the two layers  (“acoustic” oscillation), with a similar amplitude in the Fe layer  than in the Py. In addition, the strong exchange interaction pushes the frequency of the optic mode (out-of-phase precession) of the bilayer structure beyond the highest frequency detected in the experiment (24 GHz). This is different from what was previously observed for isolated (non-interacting) Py/Cu/Py NWs where

Results and Discussion

the exchange interaction at the interface is suppressed by the Cu  spacer so that dipolar coupling leads to the appearance of  stationary modes of both acoustic (in-phase) and optic (antiphase) types at relatively low frequencies [53]. Moreover, in the case of different thicknesses or different materials for the top and  bottom NWs, one could achieve reprogrammable functionalities thanks their different coercivity [54].

Figure 3.4  Calculated profiles of the dynamic magnetization through the  NW width for the lowest five collective modes. Black (grey) curves are  for Py (Fe) layers. The frequency values indicated in the plots are the  mode frequency at the center of the BZ (k = 0).

The magnetic field dependence of both the width and the center of the magnonic band corresponding to the lowest  frequency mode has been investigated by Gubbiotti et al. [43], showing that the frequency of the band center (width)  monotonically increases (decreases) with the intensity of the  applied magnetic field. This is because the band center frequency  is larger for larger saturation magnetization and thickness, while  the band width depends on the strength of the inter-NW dipole coupling (that also increases with magnetization and thickness).

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Tuning of the Spin Wave Band Structure in Nanostructured Iron/Permalloy NW Arrays

3.4.2  Spin Wave Band Structure and Mode Spatial Profiles for the NWs with L-Shaped Cross Section Figure 3.5 presents spectra measured for the NWs with L-shaped cross section measured at k = 0 and p/a. We notice that these  spectra are richer in terms of number of peaks with respect to  those plotted in Fig. 3.2 for the NWs with rectangular cross  section. In all the cases, the lowest frequency mode is the most intense one. #4

#6

#5

-10

-5

0

5

#7

k =Sa

k =Sa

k =Sa

k =Sa

k=

k=

k= 0

k= 0

10

Frequency shift (GHz)

-15 -10 -5

0

5

10 15

Frequency shift (GHz)

-15 -10 -5

0

5

10 15

Frequency shift (GHz)

-15 -10 -5

0

5

10 15

Frequency shift (GHz)

Figure 3.5  BLS spectra measured for L-shaped NWs (samples #4 to #7)  at k = 0 and p/a. A magnetic field H = 500 Oe is applied along the NW  length ( y direction).

Let us now consider the frequency dispersion for the  L-shaped cross-section NWs. The comparison between the experimental and calculated frequency dispersion for samples #4 to #7 is shown in Fig. 3.6. One sees that several peaks have been detected by BLS, whose number increases and whose frequencies  are upshifted on increasing the thickness of the Fe NWs.  Interestingly, the lowest-frequency mode exists in a frequency  range which increases on increasing the Fe thickness (see Fig. 3.5).  In addition, there is at least a second mode, at a higher frequency, which exhibits a sizable periodic frequency dispersion as a  function of the Bloch wave vector k. The oscillation amplitude of this dispersive mode increases on increasing the Fe thickness  until it reaches about 0.5 GHz for sample #7. The presence of the two dispersive modes suggests a strong dynamic dipole



89

Results and Discussion

coupling of individual NWs. This is different from the case of the  non-interacting stripes where, as a consequence of lateral   confinement, stationary and dispersionless quantized spin waves  have been observed [55, 56]. 

#1

#4

#5

#6

#7

 

Figure 3.6  Measured (points) and calculated (grey scale) dispersion  for spin waves propagating perpendicularly to the NWs length  (x direction) for H = 500 Oe applied along the NW length for  sample #4 to #7. The dispersion of sample #1 is here repeated for the  sake of comparison. curves

In order to understand the nature of the observed spin  wave spectra and their dependence on the Fe NW thickness, it is instructive to look at the simulated spatial profiles of the modes,  which are shown in Fig. 3.7. The dynamic magnetization  distributions have been averaged over the thicknesses of the respective layers. Similar to the case of rectangular cross-sectional NW, the magnetization precession for both layers is in phase for  0 < x < 170 nm. Secondly, one sees that on increasing the Fe  thickness, the modal profiles become more complex with an increasing number of nodes. To understand the details of the formation of the modal  profiles of Fig. 3.7, it is useful to exploit some similarity of the  L-shaped NW geometry to the case of bi-component ferromagnetic MCs consisting of two stripes made of different materials  alternated to form a periodic array [34]. In this respect, one may represent the L-shaped NW as two effective 170 nm-wide NWs: we refer to the region (0 < x < 170 nm) as the THICK portion of 

  90



Tuning of the Spin Wave Band Structure in Nanostructured Iron/Permalloy NW Arrays



the L-shaped NW and to the region not covered by Fe  (170 < x < 340 nm) as the THIN portion. Of course, the THICK  and THIN portions, being in lateral contact through a “virtual interface” placed at x = 170 nm running across the Py layer, are coupled by the exchange interaction and also by their dipole  fields. One can easily see from the panels of Fig. 3.7 that in  samples #4 to #7 the spatial profiles of the THICK portion are qualitatively the same in the two layers.

   

#4

#5

(e) 11.0 GHz

(e) 9.1 GHz

(e) 14.7 GHz

(e) 16.7 GHz

(e) 16.7 GHz

(d) 10.0 GHz

(d) 12.3 GHz

(d) 13.2 GHz

(d) 14.9 GHz

(d) 15.7 GHz

(c) 8.9 GHz

(c) 11.4 GHz

(c) 11.6 GHz

(c) 13.7 GHz

(c) 13.8 GHz

(d) 7.9 GHz

(b) 10.2 GHz

(b) 10.5 GHz

(b) 11.7 GHz

(b) 11.4 GHz

(a) 6.3 GHz

(a) 7.1 GHz

(a) 7.7 GHz

(a) 8.3 GHz

(a) 8.2 GHz

Amplitude (arb. units)

#1

0

      

50

100 150 200 250 300

Width, x (nm)

0

50

100 150 200 250 300

Width, x (nm)

0

50

#7

#6

100 150 200 250 300

Width, x (nm)

0

50

100 150 200 250 300

Width, x (nm)

0

50

100 150 200 250 300

Width, x (nm)

Figure 3.7  Calculated out-of-plane (mz) dynamic magnetization as a  function of distance along the NW width for the lowest five frequency  modes of the L-shaped samples. Black and grey curves refer to the  spatial profile within the Py (0 < x < 340 nm) and Fe (0 < x < 170 nm) NW,  respectively. The frequency values indicated in the plots are the mode frequencies at the center of the BZ (k = 0). The profiles calculated for  sample #1 are repeated here for the sake of comparison.

Remarkably, one can notice that the dispersive modes, whose calculated profiles are reported in Fig. 3.7 (for instance in panels  (a) and (c) for samples #5 and #6, and panels (a) and (d) for  sample #7), correspond to the fundamental mode of either the  THIN or the THICK portions of the NW. In the latter case the  frequency is considerably larger, due to the significantly larger thickness and mean saturation magnetization of the THICK  portion. This mismatch of the fundamental mode frequencies between the two sides also explains why for the low-frequency fundamental mode of the THIN portion (panels (a) for sample #4 



Conclusions

to #7) there is no counterpart to couple in the THICK part. Consequently, the magnetization precession in the latter region  is not resonant but represents a forced oscillation (decayingexponent-like) driven by the exchange coupling through the virtual interface and by the long-range dipole field of the THIN part. On the contrary, the fundamental mode of the THICK portion couples to a much higher individual mode of the THIN one,  as seen in the panels (c) and (d) of Fig. 3.7. In particular, for the  lowest frequency fundamental mode, (panels (a) in Fig. 3.7),  one can see that the presence of a thin Fe overlayer noticeably  reduces the dynamic magnetization amplitude in the THICK part,  where the dynamic magnetization profile takes a form typical for an evanescent wave especially for larger Fe thicknesses (samples  #6 and #7). In fact, the precession in the THICK region is  mainly driven through exchange interaction via the “virtual  interface” at x = 170 nm by the standing wave resonance taking place in the part of the Py layer not covered by Fe. In addition to  the variation of the spatial profile, there is also a substantial frequency evolution of this mode, as seen in Fig. 3.5: only 2 nm of  Fe on top of Py are sufficient to shift the mode frequency up  from about 6.3 to 7.1 GHz. As the wavelength of the standing  wave is large with respect to the exchange length, the rise in  frequency is due to the above-mentioned reduction in the  effective wavelength across the NW that leads to an increase in the standing-wave dipole energy. We also notice that the amplitude  of dynamic magnetization is larger for the Fe layer that has a  larger saturation magnetization. Regarding the field dependence of both the center and the  width of the magnonic of the lowest frequency mode, it is found that results of the L-shaped samples are very similar to those of  the reference Py NWs (sample #1), reflecting the marked localization of the lowest frequency mode in the THIN (Py) portion of the NW side.

3.5  Conclusions

In this chapter, we analyzed the dispersion of collective spin waves in arrays of iron/permalloy nanowires with either rectangular

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Tuning of the Spin Wave Band Structure in Nanostructured Iron/Permalloy NW Arrays

or L-shaped cross section by using Brillouin light scattering and numerical simulations. The measurements relative to the  frequency dispersion of the spin waves were satisfactorily  reproduced by micromagnetic simulations, which also allow us to calculate the spatial profiles of the modes in the two layers. For the rectangular nanowires, the fundamental mode lying at the lowest frequency, characterized by an in-phase precession of the magnetization in the two layers and maximum amplitude in  the center of the nanowire, exhibits the largest frequency oscillation amplitude. Its frequency in the bilayer Py/Fe nanowires is  up-shifted with respect to that measured for single-layer Py and Fe nanowires due to the increased thickness. For the case of the nanowires with L-shaped cross section, two dispersive modes with sizable magnonic band were observed. These are interpreted as the fundamental modes of either the THICK or the THIN portion of each nanowire. The effect of the Fe nanowire width and thickness on the magnonic band amplitude and frequency position was also explored. We believe that this work can stimulate design, tailoring, and characterization of magnonic crystals where,  thanks to the presence of two contrasting ferromagnetic materials  and of interlayer exchange interaction at their interface, new  tailored functionalities can be achieved.

Acknowledgments

A. O. A. was supported by National Research Foundation, Prime Minister’s Office, Singapore, under its Competitive Research Programme (CRP Award No. NRF-CRP 10-2012-03).

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33. Zhang, V. L., Lim, H. S., Lin, C. S., Wang, Z. K., Ng, S. C., Kuok, M. H., Jain, S., Adeyeye, A. O., and Cottam, M. G. (2011). Ferromagnetic  and antiferromagnetic spin-wave dispersions in a dipole-exchange coupled bi-component magnonic crystal, Appl. Phys. Lett., 99, 143118. 34. Livesey, K. L., Ding, J., Anderson, N., Camley, R. E., Adeyeye, A. O., Kostylev, M. P., and Samarin, S. (2013). Resonant frequencies of a binary magnetic nanowire, Phys. Rev. B, 87, 064424.

35. Lin, C. S., Lim, H. S., Zhang, V. L., Wang, Z. K., Ng, S. C., Kuok, M. H.,  Cottam, M. G., Jain, S., and Adeyeye, A. O. (2012). Interfacial  magnetization dynamics of a bi-component magnonic crystal comprising contacting ferromagnetic nanostripes, J. Appl. Phys., 111, 033920.

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37. Krawczyk, M., Mamica, S., Mruczkiewicz, M., Klos, J. W., Tacchi, S., Madami, M., Gubbiotti, G., Duerr, D., and Grundler, D. (2013). Magnonic band structure in tangentially magnetized thin film of  two-dimensional bi-component magnonic crystals, J. Phys. D: Appl. Phys., 46, 495003. 38. Zivieri, R. (2014). Bandgaps and demagnetizing effects in a Py/Co magnonic crystal, IEEE Trans. Magn., 50, 1100304.

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54. Gubbiotti, G., Nguyen, H. T., Hiramatsu, R., Tacchi, S., Madami, M., Cottam, M. G., and Ono, T. (2014). Field dependence of the magnetic eigenmode frequencies in layered nanowires with ferromagnetic  and antiferromagnetic ground states: experimental and theoretical study, J. Phys. D: Appl. Phys., 47, 365001. 55. Mathieu, C., Jorzick, J., Frank, A., Demokritov, S. O., Slavin, A. N., Hillebrands, B., Bartenlian, B., Chappert, C., Decanini, D., Rousseaux, F., and Cambril, E. (1998). Lateral quantization of spin waves in  micron size magnetic wires, Phys. Rev. Lett., 81, 3968. 56. Jorzick, J., Demokritov, S. O., Hillebrands, B., Bailleul, M., Fermon, C., Guslienko, K. Y., Slavin, A. N., Berkov, D. V., and Gorn, N. L. (2002).  Spin wave wells in nonellipsoidal micrometer size magnetic elements, Phys. Rev. Lett., 88, 047204.

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Chapter 4

Magnetization Dynamics of Reconfigurable 2D Magnonic Crystals G. Shimon,a A. Haldar,a,b and A. O. Adeyeyea aDepartment

of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117576, Singapore bDepartment of Physics, Indian Institute of Technology Hyderabad, Kandi 502285, Telangana, India [email protected]

This chapter discusses our recent works on magnetization dynamics of reconfigurable 2D magnonic crystals. The chapter is broadly divided into two parts. The first part begins by examining the influence of neighboring dipolar interactions on a single Ni80Fe20 (NiFe) disk using microfocused Brillouin light scattering (µ-BLS) spectroscopy. Using pairs of identical NiFe disks of reducing interdisk spacing, marked spectral and spatial shifts of the resonant mode are observed because of increasing dipolar interactions. Beyond a two-disk system, we found that by coupling more disks as a cluster, a unique dynamic response can be realized depending on the number of disks involved and their configuration. The second part shows examples of design and fabrication strategies for creating nanomagnet networks with reconfigurable Spin Wave Confinement: Propagating Waves (2nd Edition) Edited by Sergej O. Demokritov Copyright © 2017 Pan Stanford Publishing Pte. Ltd. ISBN  978-981-4774-35-2 (Hardcover),  978-1-315-11082-0 (eBook) www.panstanford.com

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magnetic ground states. For instance, using rhomboid nanomagnet (RNM) networks, a reliable reconfiguration between ferromagnetic, antiferromagnetic, and ferrimagnetic ground states can be realized. The deterministic magnetic ground-state configuration is achievable owing to the inherent shape anisotropy that stabilizes the RNMs to a specific ground state upon field initialization along their short axis. The reconfiguration is also apparent in the magnetization dynamics, as systematically investigated using broadband ferromagnetic resonance (FMR) and µ-BLS spectroscopy techniques.

4.1  Introduction

Nanomagnet arrays and clusters are considered vital building blocks for many technological applications such as ultrahighdensity media [1–3], magnetic random access memory [4, 5], logic [6–10], microwave signal processing devices [11–13], magnonics [14–21], and spin torque nano-oscillators (STOs) [22–26]. Some of these applications require nanomagnets to be packed more closely together in order to achieve higher-areal-density devices. Beyond the challenges for device fabrication and integration, the strength of dipolar interactions between neighboring nanomagnets is known to be significantly enhanced when interelement spacing becomes much smaller than their lateral dimensions. Such enhanced dipolar interaction has been found to largely modify magnetization reversal [27–29], switching field distribution [30–32], and dynamic magnetization reversal processes [33–36]. Dipolar interaction can also be engineered to realize functional magnetic systems. For example, magnetic quantum cellular automata (MQCA) have been utilized to perform logic operations and propagate magnetic information [6, 9, 37]. Similarly, artificial spin ice has been shown to produce localized spin wave (SW) or selective SW propagation based on defects or change of magnetization states [38–40]. Additionally, flavors of tunable dynamic behaviors were made possible by exploiting their dipolar interaction [11, 33–36]. Magnonic crystals (MCs), magnetic analogues of photonic crystals [41, 42], hold the possibility to tailor SW propagation and frequency on the basis of geometrical confinement and periodic

Introduction

patterns. Such magnetic metamaterials are promising for further miniaturization of microwave electronics and signal processing [43]. As an information carrier, SW propagation promises faster and greener computing than conventional electronics [44–47]. At the focal point of realizing reconfigurable 2D MCs is the fundamental understanding of how dipolar interaction influences the magnetization dynamics. The strength of dipolar interaction between nanomagnets is a collective function of their shape, size, and interelement spacing, lattice configuration, and coupling orientation. To accurately quantify the effect of dipolar interaction and predict the dynamic behavior, careful analysis of each of these contributing factors must be made. Furthermore, to measure the effect of neighboring dipolar interaction, it is important to develop a reliable method to quantify the response from an individual nanomagnet as a function of the changing energy landscape of its surrounding because of dipolar interaction. This is in contrast to measuring a collective behavior from an array or clusters of nanomagnets. Reconfigurable dynamic responses were demonstrated earlier by switching between magnetic ground states that were obtained through precise minor loop hysteresis [41, 42, 48, 49]. However, these methods are slow and involve complex field initialization, which is not suitable for device integration. It is critical that the initialization and reconfiguration of 2D MCs be of low power, be rapid, and be straightforward in its operation [43, 50]. This chapter is written to outline the effects of dipolar interaction on the magnetization dynamics in coupled nanomagnets and demonstrate its usefulness. The first part of this chapter is devoted to systematically studying the effect of dipolar interaction strength on the dynamic behavior of a single nanodisk as a function of the separation distance and cluster configuration of its neighboring nanodisks using the microfocused Brillouin light scattering (µ-BLS) technique. Using a pair of nanodisks with varying separation, we demonstrate that the resonant frequency shift of the fundamental mode is proportional to the increase of dipolar field strength and there are opposing trends of frequency shift, depending on the relative orientation of the interdisk coupling direction and the applied magnetic field (Happ). Imaging the resonant mode profile reveals strong spatial shift of the resonant area because of dipolar interaction, including an intensity ripple

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in its strongest effect. When more disks are introduced in a cluster of various configurations, a unique dynamic response emerges for each configuration, in which the neighboring dipolar interaction plays a key role. In the second part of this chapter, a strategy for a reconfigurable magnetization dynamic response with no standby power [43, 50] is discussed. To date, experimental evidence of this functionality is rare. One main hindrance is that the dipolarcoupling-driven antiferromagnetic (AFM) ground state requires a complex field initialization process and it is susceptible to structural imperfections. As such, robust designs are desired from a fabrication point of view so that they can be scaled into a large-area pattern, particularly important for manufacturability. We will demonstrate an example of reconfigurable MCs in 1D and 2D using an RNM chain, cluster, and arrays. In particular, we will highlight that reliable switching between ferromagnetic (FM) and AFM remanent states can be achieved using RNMs with a simple field initialization process. The resulting remanent state is largely dependent on the direction of field initialization rather than on the stray field from the nearest neighbors in an array. A distinct magnetization dynamic response was obtained because of different dipolar field distributions for different remanent states.

4.2  Experiments and Simulations 4.2.1  Sample Fabrication

To investigate the dynamic behavior of patterned nanomagnets, high-frequency excitation and detection in the microwave regime is required. For this purpose, a conventional radio frequency (rf) excitation through a coplanar waveguide (CPW) with groundsignal-ground (GSG) configurations is used. The CPW is connected to an external microwave source through the use of microwave picoprobe(s). There are two main variations for which the CPW is incorporated, namely by using the flip-chip technique or by fabricating an integrated CPW. In the flip-chip technique, the substrate with the nanomagnets is placed face down on the preexisting CPW on a sample stage. The technique involving the fabrication of an integrated CPW can be done either by integrating

Experiments and Simulations

the CPW on top of already existing patterned nanomagnets (usually chosen for characterizing a continuous film or nanomagnet array) or by first fabricating the integrated CPW and then following by patterned nanomagnets on top of it. The last approach is usually chosen for a number of specific purposes and limitations, for example, when surface characterization of excited nanomagnets is required (such as through optical probing) or when a special sample’s orientation or geometry is required. In our experimental works, the last approach was chosen to accommodate all of the aforementioned factors. The integrated CPW having a nominal signal line width of 20 µm is made up of Cr (5 nm)/Pt (200 nm) fabricated on a SiO2/Si substrate using the UV lithography and liftoff process. The Cr adhesion layer was deposited using electron beam evaporation at a rate of 0.2 Å/s, while the Pt layer was deposited using sputter deposition at a rate of 0.39 Å/s. These were done successively within a single deposition chamber (base pressure of 4 × 10–8 Torr) without breaking its vacuum. Subsequently, various nanomagnets were patterned on top of the existing CPW’s signal line using electron beam lithography, followed by deposition of Cr (5 nm)/NiFe (25 nm) using electron beam evaporation at a rate of 0.2 Å/s and the liftoff process. Figure 4.1 depicts an example of how the integrated CPW and sample are incorporated in the experimental setup.

Figure 4.1  Schematic of the vector network analyzer ferromagnetic resonance (VNA-FMR) setup.

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4.2.2  Ferromagnetic Resonance Spectroscopy Ferromagnetic resonance (FMR) spectroscopy is a common technique for characterizing the dynamic properties of magnetic materials. When an FM sample is placed under an orthogonal Happ and an rf field (hrf ), the magnetization M will precess according to the equation of motion: дM/дt = –g(M × Heff ), where g is the electron gyromagnetic ratio and Heff is the effective field. The FMR event occurs when the natural resonant frequency of uniform mode in the magnetic sample matches that of hrf and the applied rf energy will be strongly dissipated by the magnetic sample. The detailed mechanism responsible for the FMR absorption in FM materials was explained by Kittel [51, 52]. A generic formula for predicting the FMR frequency in arbitrary sample geometry was derived by considering a general ellipsoid magnet with three principal axes (x, y, z) with the Happ applied along z and hrf applied along x. The demagnetization field (Hdi) of an FM body was also considered through the use of a demagnetization tensor Ni for each axis (i) according to Hdi = –Ni Mi where Mi = M0eiwt. The generic formula, also widely known as the Kittel formula, is shown below as Eq. 4.1.

f res =

g (Hz +( N y – N z)4 pMz ) (Hz +( N x – N z )4 pMz ) 2p

(4.1)

FMR spectroscopy is mostly performed either by sweeping the Happ at a fixed frequency or by sweeping the frequency at a fixed Happ. In the former, the resonant condition will be detected as a resonant field at a given frequency. In the latter, the resonant condition will be detected at a resonant frequency (fres) for a given Happ. In our experiments, the frequency sweep approach was used, and in particular we utilized a broadband vector network analyzer (VNA)-based FMR spectroscopy technique. In this method, hrf is generated by an rf microwave current applied using a VNA. A schematic of the experimental setup for FMR measurement is shown in Fig. 4.1. The CPW geometry was designed to produce a 50 W impedance matching to the VNA. The FMR absorption signal is obtained by measuring the S21 scattering parameter, a direct measure of the transmission characteristic (ports 1 and

Experiments and Simulations

2 as the emitting and the receiving port, respectively). The S21 parameter is quoted as the power ratio between the two ends in decibel units. When a resonant event occurs, the plot of the S21 parameter against the excitation frequency will show a dip (i.e., power loss) at fres, signifying a strong resonant absorption of microwave power by the magnetic sample. The evolution of resonant behavior can be traced in terms of fres versus Happ and be correlated to the sample’s magnetization.

4.2.3  Microfocused Brillouin Light Scattering Spectroscopy

Brillouin light scattering (BLS) is the inelastic scattering of a photon because of its interaction with a magnetic sample at the surface. The inelastic scattering event is categorized into Stokes and anti-Stokes scattering. In the former, the incident photon (ħwi, ħqi) loses some energy and momentum by emitting a spin wave (SW) (ħwSW, ħqSW) and a scattered photon (ħwS = ħ(wi – wSW) and ħqs – || = ħ(qi – || – qSW – ||). In the latter, it gains some energy and momentum by absorbing SW energy into the scattered photon ħws = ħ(wi + wSW) and ħqs – || = ħ(qi – || + qSW – ||). The parallel (//) sign denotes that the momentum transfer occurs with the SW traversing parallel to the sample’s surface. During either of the scattering events, the total energy of the system and the total wave momentum (ħq) at the sample’s surface are conserved. Depending on the sample’s magnetization direction and the SW’s wave vector orientation, various types of SW modes in magnetic films can be detected using BLS [53]. BLS spectroscopy is designed on the basis of the observation of frequency shift between the incoming and scattered photons using an interferometry setup to resolve light intensity at different frequencies using piezo-controlled multipass tandem Fabry–Perot mirror construction. Using the interferometer setup allows for low noise and precise detection of scattered photons. A well-known commercial setup of such construction was developed by Sandercock [54]. The typical spot size of the laser beam in the conventional BLS setup is 50 µm, which is suitable for measuring nanomagnetic arrays or magnetic thin films. Using a microfocusing lens, the laser beam can be focused down to ~250 nm spot size, which enables SW probing for

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selective nanomagnets or localized spots. When coupled with a piezo-controlled sample stage, it will facilitate the SW characterization, with an added high-resolution spatial profile [55]. BLS spectroscopy offers a number of flexibilities and advantages over FMR spectroscopy. BLS spectroscopy allows the collection of SW dispersion characteristics, that is, fres against wave vector (k), which is otherwise limited to only uniform dispersion (k = 0) using FMR spectroscopy. BLS spectroscopy also allows the investigation of thermally excited SWs and a time-resolved measurement. Lastly using the µ-BLS setup, spatial resolution of the resonant mode and the traveling SW can be obtained.

Figure 4.2  Schematic of the microfocused Brillouin light scattering setup.

Figure 4.2 shows the schematic diagram of a µ-BLS spectrometer. A monochromatic laser of 532 nm wavelength is passed through a polarizing beam splitter into the 100x objective lens with numerical aperture (NA) = 0.75 and focused onto the sample’s surface. The nanomagnets fabricated on the signal line of a CPW are uniformly excited by hrf generated by the microwave signal generator through a picoprobe. Note that the CPW is shorted on one end to complete the rf current loop,

Experiments and Simulations

unlike in FMR spectroscopy. The direction of hrf and Happ (or M) is set to be perpendicular to each other, similar to that in the VNA-FMR technique. The back-scattered light is projected back through the lens and polarizing beam splitter and analyzed in the interferometer. A typical BLS spectrum shows photon counts as a function of frequency shift, and intensity peaks signify that resonant has taken place. When a particular resonant mode is selectively excited at its fres, its 2D spatial profile can be obtained by performing a raster scan. The relative photon intensity obtained will give information about the location in which the resonant occurs most strongly.

4.2.4  Micromagnetic Simulations

In characterizing the dynamic behavior of nanomagnets, micromagnetic simulations are often carried out to substantiate the experimental observations. For this chapter, micromagnetic simulations were performed using the Object Oriented Micromagnetic Modeling Framework (OOMMF) [56]. The goal of micromagnetic simulations is to minimize the total energy of the system on the basis of micromagnetic models. The energy minimization will drive the magnetic moments to reach the equilibrium state with respect to Heff. In this process, the magnetic moments will precess according to the modified equation of motion, which includes a damping term:

dM dM  a = –| g | M × Heff + M × . Ms  dt dt 

This is more commonly known as the Landau–Lifshitz–Gilbert (LLG) equation [57]. The first and second terms of the LLG equation correspond to the precessional and the damping term, respectively. The parameter g is known as the gyromagnetic  g



ratio  = 2.8GHz/kOe, and a is known as the damping factor.  2p  It is worth noting that the LLG equation is based on an earlier

Landau–Lifshitz (LL) equation. Typical magnetic parameters for NiFe are used in the simulations. The saturation magnetization is taken as Ms,NiFe = 800–860 emu . cm–3, exchange constant ANiFe = 13 × 10–7 erg . cm–1, and magnetocrystalline anisotropy K1,NiFe = 0. The uniaxial anisotropy of NiFe is assumed to be negligible when compared with the shape anisotropy of the patterned structures.

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Magnetization Dynamics of Reconfigurable 2D Magnonic Crystals

For realistic simulations, the sample volume must be discretized into subsets of smaller unit cells. The unit cell size is chosen to be 

A



less than the exchange length   lex = 2pM2   of a given material to  S  yield accurate results, for example, lex of NiFe is ~5.7 nm. Typically, a unit cell size of 5 nm × 5 nm × 5 nm was chosen for our simulations. There are two common simulation routines used in our work. First was the basic quasi-static simulation, which simulates the steady-state response of magnetic samples to Heff. For quasi-static simulation, a damping coefficient of a = 0.5 was chosen to obtain rapid convergence. Second was the dynamic simulation that simulates the dynamic response and quantifies the spatial characteristics of the SW mode. Time-dependent simulations are performed using a = 0.008 for NiFe. A time-varying sinc wave excitation field hsinc = h0

sin(2pft ) is used to yield uniform t

excitation in the frequency domain, where h0 is the initial amplitude of the sinc wave (within the linear excitation regime, typically up to few tens of oersteds). The dynamic simulation results are analyzed in the frequency domain by performing fast Fourier transform (FFT) processing.

4.3  Coupled Nanodisks

In this section, we study the influence of dipolar interaction from nearby nanomagnets on the dynamic behavior of a single nanomagnet using µ-BLS spectroscopy having microwave excitation. We will consider the effect of interdisk separation in a two-disk system. We will then investigate the effect of configurational anisotropy by coupling more disks as a cluster having various lattice arrangements. A simple method to precisely quantify the strength and orientation of dipolar interaction will be presented. For more details, readers are encouraged to study the published works in the literatures [58, 59].

4.3.1  Effect of Interdisk Separations

The effect of interdisk separation has been investigated using a simple two-disk system. NiFe disks were patterned directly on the

Coupled Nanodisks

signal line of a shorted CPW using electron beam lithography. The disks have a nominal diameter D = 500 nm, thickness t = 25 nm, and varying interdisk separation s (s = 200, 150, 100, 50 nm). An isolated disk from a disk array with s = 500 nm was also examined as a reference. Figure 4.3a–c shows representative top-down scanning electron microscopy (SEM) images of the coupled nanodisks with s = 500, 200, and 50 nm. The green arrow in each SEM image labels the disk under the probe. Here, the two disks are coupled along the x direction, parallel to the Happ direction. We also examined the case where the two disks are coupled along the y direction, perpendicular to Happ. (a)

(b)

(c)

Figure 4.3 (a–c) SEM images of disks with s = 500, 200, and 50 nm. (d) Normalized BLS spectra and (e) the corresponding simulated spectra for different s and orientations. Plots with filled (open) symbols represent the BLS spectra in parallel (perpendicular) orientations between the interdisk coupling direction and Happ. Insets in (d): Schematic diagram of stray field models for the disk pair in the parallel (left) and perpendicular (right) orientations. Adapted with permission from Shimon and Adeyeye [58]. Copyright 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

4.3.1.1  BLS spectra

Figure 4.3d shows the normalized BLS spectra measured at Happ = 1 kOe for the different s and coupling orientations. Prior to each measurement, the disks were saturated with Happ = 2 kOe. At Happ = 1 kOe, the disks are in the quasi-saturated single-domain

109

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Magnetization Dynamics of Reconfigurable 2D Magnonic Crystals

state. For each s (except for s = 500 nm), there are two BLS spectra plotted, representing parallel (filled symbols) and perpendicular (open symbols) coupling orientations with respect to the Happ direction. At Happ = 1 kOe, only one prominent mode was observed in the disk for various s. We named the mode observed for the isolated disk as mode C1, while those in the coupled disks were modes C2A and C2B when their coupling orientation with respect to the Happ direction is parallel and perpendicular, respectively. For the isolated disk, the resonant frequency of mode C1 (fC1) was detected at 9.2 GHz. As s is reduced, we observed a systematic shift of fres with respect to that of the isolated disk. In addition, we observed opposing trends of this fres shift, depending on the coupling orientation (two dotted lines in Fig. 4.3d). For parallel orientation, fC2A increases by 300 MHz as s is reduced from 200 nm to 50 nm. In contrast, for perpendicular orientation, fC2B decreases by 300 MHz as s is reduced from 200 nm to 50 nm. To understand the observed fres shift, we consider two stray field models (see insets in Fig. 4.3d). The first model (right), for coupling in parallel orientation, illustrates how a dipolar field along +x (+Hdx, blue arrow) is exerted on the disk under the probe by its neighbor from the left. Conversely, the second model (left), for coupling in perpendicular orientation, shows how the dipolar field along –x (–Hdx, red arrow) is exerted on the disk under the probe by its neighbor from the top. On the basis of this model, we analyze how additional Hd influences the fres of the disk using the modified Kittel resonant formulation as Eq. 4.2:

f res =

g (H x +( N y – N x )4 pMx ) (H x +( N y – N x )4 pMx ) , 2p

(4.2)

where Nx, Ny, and Nz are the demagnetization factors for x, y, and thickness (z) directions, respectively; g is the gyromagnetic ratio of the materials; and 4pMx and Hx are the magnetization of the sample and the effective field along x, respectively. In the parallel orientation, +Hd–x adds to Hx such that Hx = Happ + Hd–x, and as a result fres will increase as compared to the isolated case, where Hx = Happ. As a larger Hd–x is expected with a reduction of s, Dfres will consequently increase. In the perpendicular orientation, –Hd–x reduces Hx following Hx = Happ – Hd–x and as a result fres will decrease with s (opposite to that with parallel orientation). Note that the dipolar field along

Coupled Nanodisks

y (perpendicular orientation) does not contribute to Hx variation with s. It is important to underline that dipolar interaction would typically produce a resonance mode splitting, in-phase (acoustic) and out-of-phase (optical) coupled modes, when measuring a collective response of an array or pairs of magnetic elements and/or using asymmetric excitation [60–63]. This is in contrast to our measurement, in which the probing was done on a single disk and not by measuring the collective response of the disk pair. Furthermore, the uniform excitation field generated by the signal line in our experiment cannot efficiently excite the optical mode for the BLS detection [64].

4.3.1.2  Simulated spectra

To further verify our analyses, time-dependent (dynamic) simulations were performed on the basis of the magnetization state obtained at a given Happ to simulate the BLS response and quantify the spatial characteristics of the SW mode. Figure 4.3e plots the normalized simulated spectra for different s and coupling orientations at Happ = 1 kOe. In agreement with the experimental spectra, we found a single prominent mode in all the simulated spectra. We also observed two opposite fres shift trends versus s for the different coupling orientation. The simulated fC1 for the isolated disk is observed at 9.18 GHz, close to 9.2 GHz from experiments. In the parallel orientation, fC2A increases by ~600 MHz when s is reduced from 200 nm to 50 nm, while in perpendicular orientation, fC2B decreases by ~600 MHz. The fact that |Dfres| in the simulations is about twice as large as |Δfres| in the experiments indicates a stronger influence of dipolar interaction in simulations in changing the dynamic magnetization (partly due to the 0 K condition in simulations). It is important to note that using the uniform sinc wave excitation in the simulation could not efficiently excite the optical mode (hence no mode splitting) [61], virtually mimicking our experimental microwave excitation.

4.3.1.3  2D mode profiles

To provide direct evidence of dipolar interaction, a series of 2D µ-BLS images were acquired as a function of s (Fig. 4.4a–d)

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Magnetization Dynamics of Reconfigurable 2D Magnonic Crystals

s=100 nm s=150 nm s=200 nm s=500 nm

s=500 s=500 nm nm

at their respective fres values. The crosshairs mark the center position of each disk in the scanned image. For the isolated disk, the fundamental resonant mode is located at the center of the disk. As s is reduced, the resonant mode is gradually shifted toward the gap region in the disk pair because of the influence of the neighboring dipolar field. For s = 50 nm, the mode is strongly Simulated 2D ȝ-BLS of profile shifted toward the edge of the disk. The images changing of the Mode Profiles at 1 mode at s1 kOe clearly highlight resonant mode and the shift resonant of fres versus (j) (e) the influence of the neighboring (a) dipolar field in modifying the Simulated Mode Profiles at 1 kOe (norm.) ȝ-BLS images of dynamic behavior in a2D disk.

s = 50 nm

1.7 μm

1 μm

1.7 μm

(i)

s = 50 nm

1.7 μm

(i)

hrf Y 1.5Z μm

(m)

800 nm

s = 50 nm

(l)

(i)

Y

hrf

800 nm

s = 100 nm

(h) 1 μm

(d)

(k)

(m)

s=50 nm

s=100 nm s=150 nm s=200 nm s=500 nm

s=200 nm s=100 nm s=50 nm

(h)

(l)

s = 100 nm

800 nms=50 nm

s = 100 nm

(d)

s=100 nm s=150 nm s=200 nm s=500 nm

s=50 nm

(d)

1 μm

s=100 nm

s = 200 nm

(c)

s = 200 nm

(c) (g)

s=50 nm

s=100 nm

s=200 nm

s=200 nm

s=500 s=500 nm nm

s=500 s=500 nm nm

resonant mode at 1 kOe (j) (e) (f) 2D ȝ-BLS images of (a) Simulated Mode Profiles at 1 kOe (norm.) (b) resonant mode at 1 kOe (j) (e) (k) (a) (f) (g) s = 200 nm (b) (f) (c) (l) (g) (b) (h)

s=50 nm

112

1.5 μm

+Happ X

1.5 μm

Figure 4.4  (a–d) 2D μ-BLS images of fundamental resonant mode at Happ = 1 kOe for interacting disks with varying s. (e) 2D μ-BLS images of mode A at Happ = 1 kOe for an isolated disk. Color scale bar in (e) represents the normalized BLS intensity of the mode profiles in (b–e). (f–m) Corresponding simulated mode profiles with varying s and orientations. Adapted with permission from Shimon and Adeyeye [58]. Copyright (2015) WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

The corresponding simulated mode profiles with varying s and orientations (Fig. 4.4f–m) were extracted at fres values and found to be in agreement with experiments. For s = 50 nm, in addition to the strong shift of the center mode toward the disk’s

Y Z

+Happ X

Coupled Nanodisks

edge, we observed an intensity ripple toward the gap region because of the strong dipolar interaction. The shifted mode profile may have caused a stronger dynamic interaction and produced an intensity ripple toward the gap region. In the experiment, this intensity ripple may be difficult to resolve because of limited resolution of the probing laser. In contrast, in perpendicular orientation, the resonant mode is unaffected by the reduction of s. The effect of reducing s is evident only for s ≤ 100 nm (Fig. 4.4l–m), where the resonant mode is offset along ±x. No edge mode is observed in our experiment and simulation up to Happ = 2 kOe, which may not be favorable for our disks’ dimension. Only by using a much larger Happ or by measuring a smaller disk could we observe the edge mode [65, 66].

4.3.1.4  Dipolar field estimation

In this section, we describe a simple method to estimate the neighboring dipolar field magnitude on a single disk from BLS measurement. In the first step, BLS spectra were measured as a function of Happ of the isolated disk (Fig. 4.5a) and the disk pair with a particular s, for example, s = 50 nm (Fig. 4.5b,c) for the Happ range where the fundamental resonant mode is observed, that is, 2000 Oe ≤ Happ ≤ 500 Oe (quasi-saturated single-domain state). Next, we fitted the peak fres versus Happ of the isolated disk using Eq. 4.2 to obtain the demagnetization factors of the disk. For the case of the isolated disk, we assume that Hx = Happ (negligible external dipolar field). For the fitting purpose, various parameters and constraints were set as follows: g = 2.8 GHz/kOe, (ii) Mx = 860 emu . cm–3, (iii) Nx = Ny (i) 2p (symmetric disk), (iv) Nx + Ny + Nz = 1, and (v) the fitting precision of Nx, Ny, and Nz was set to be three significant figures. The fitted demagnetization factors are Nx = 0.0390, Ny = 0.0390, and Nz = 0.922. In the final step, the dipolar field of the disk pair was determined by fitting the measured BLS spectra within the same Happ range but using a modified Hx = Happ + Hd–x expression (as discussed earlier) into Eq. 4.2. The demagnetization factors obtained earlier were used in the fitting. The fitted dipolar field is Hd–x = 89 ± 2 Oe for s = 50 nm in parallel orientation, while for s = 50 nm in perpendicular orientation, it is Hd–x = –46 ± 2 Oe.

113

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Magnetization Dynamics of Reconfigurable 2D Magnonic Crystals

The opposite sign of Hd–x confirms the accuracy of the model of dipolar field orientation presented earlier. Furthermore, for disks coupled in perpendicular orientation, the |Hd–x| value was found to be smaller than disks coupled in parallel orientation. This indicates a larger effective distance between magnetic poles of neighboring disks in perpendicular orientation. The detection of a subtle difference in the |Hd–x| value highlights the advantage of using this method for quantifying the effect of dipolar field interaction in coupled disks, whereby the whole range of 500 Oe ≤ Happ ≤ 2 kOe was taken into consideration. This is in contrast to comparing D fres for s = 50 nm with different orientations at Happ = 1 kOe, which results in the same D fres = 0.3 GHz (Fig. 4.3d).

Figure 4.5  (a–c) Frequency versus Happ plot for (a) C1, (b) C2A, and (c) C2B. Symbols are experimentally extracted fres values, lines are fitted curves based on Eq. 4.2. Adapted with permission from Shimon and Adeyeye [58]. Copyright (2015) WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

4.3.2  Configurational Anisotropy

In this section, the dynamic response of a single disk as a function of its neighboring cluster configurations in two-, three-, and five-

115

Coupled Nanodisks

disk clusters is presented. We will show how the neighboring coupling configuration could significantly affect the resulting dynamic behavior of the nanodisk under the probe. The first three SEM images (Fig. 4.6a–c) are of the isolated disk (C1) and the two-disk cluster (C2A and C2B), which have been described earlier. Figure 4.6d–f shows the three-disk cluster (C3) and the five-disk cluster (C5A and C5B). (d)

d=500 nm (b)

C2A

s

(e)

C5A

s=50 nm C2B

s

(c)

s C5B

G hrf S G Happ

s

(f) y

x

(g) BLS Spectra

C3

s

C1

500 nm

Normalized Peak Intensity (arb. units)

(a)

C5B

8.9GHz

C5A

9.5GHz

9.38GHz 8.98GHz

9.5GHz

C2A 1

9.57GHz

10.35GHz

8.9GHz

C2B

C1

8.79GHz

9.8GHz 9GHz

C3

9.7GHz

(h) Simulations

9.2GHz

9.77GHz

8.59GHz

9.77GHz 9.18GHz

0

6

8 10 12 6 8 10 12 Frequency (GHz) Frequency (GHz)

Figure 4.6  (a–f) SEM images of disks with various cluster configurations (D = 500 nm, s = 50 nm). Green arrows in (a–f) label the disk under the probe. (g) Measured BLS spectra and corresponding 2D mode profiles of C1, C2A, C2B, C3, C5A, and C5B with Happ = 1 kOe. (h) Corresponding simulated spectra and 2D mode profiles. Color scale bar in (g) represents the normalized intensity of the resonant mode profile. Adapted with permission from Shimon and Adeyeye [59]. Copyright (2016) American Institute of Physics.

4.3.2.1  Resonant spectra and 2D mode profiles

We first recall the results from an isolated disk and from a coupled two-disk cluster. The bottom three panels in Fig. 4.6g,h are the measured and simulated resonant spectra and mode profiles of the disk under the probe for C1, C2A, and C2B at Happ = 1 kOe, as discussed earlier in Section 4.3.1. We now consider the case of a three-disk cluster such as C3 (see Fig. 4.6d). In this configuration, we observed dual fres peaks where the higher frequency is detected at fC3H = 9.5 GHz, while



116

Magnetization Dynamics of Reconfigurable 2D Magnonic Crystals

the lower one is at fC3L = 9 GHz. Most interestingly, the 2D µ-BLS mode profiles at fC3H and fC3L show that they are the resonant modes coupled along the parallel and perpendicular orientations, respectively, with respect to the Happ direction. Our 2D µ-BLS mode profiles validate the earlier simulated profiles of the threedisk cluster by Liu et al. [67]. These observations are further substantiated by the simulated spectra and mode profiles for the C3 configuration in Fig. 4.6h. We do not observe any edge modes in our experiments. In agreement with the existing literatures [66, 68–71], the observation of these modes is only possible when the disks have an aspect ratio 2t/D larger than a certain critical value, typically ≥0.17. Furthermore, the confinement and hence the observation of edge modes become limited as D is decreased Fcrit. The shape of the SW beam moving along the antidots line for Fcrit was also determined by the geometry of the experimental setup (see Fig. 5.6b), where the antenna was located at the right side of the sample. The SWs excited by the lower part of the antenna reached the antidots first and then transformed into a beam and traveled along the antidots line. This process took place later at the other part of the antenna, which is closer to the top side of the sample. SWs generated by different parts of the antenna couple resulted in the nonreflected beam propagated along the antidots line with an increase in both intensity and width. To compare the effect of the total nonreflection of SWs on the antidots line and on the edge of the magnetic sample, the microBLS technique with high spatial and temporal resolution was employed. The transverse amplitudes of the SW intensity profiles were measured by scanning the BLS laser focus perpendicular to the beams. Figure 5.7a shows spatial profiles of normalized microBLS intensity measured at the same distances from the antenna for traveling SW beams along a line of antidots (circles) and along the edge of the YIG sample (squares). The estimated widths of the two moving SWs beams are almost equal. The shape of the spatial profile is determined by the demagnetizing fields that are responsible for the strong decrease in the effective field near the edge [38]. To explain the experimental results, it was assumed that propagation of SWs in a magnetically ordered medium will obey the laws of geometrical optics [39], and that the projections of the incident and reflected wavevectors on the mirror plane are equal. This condition was used with the dispersion equations of the SWs for the incident and reflected waves to find the directions of the reflected waves. The above dependence was applied to the antidots line, which we consider to act as an ideal mirror for SWs. This approach was used for analyzing

151

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Spin Wave Optics in Patterned Garnet

experimental results from Fig. 5.6. At F = 0° the wavevectors kin and kout of the incident wave and the reflected wave exist and have the same amplitudes and opposite components in the direction perpendicular to the antidots line. Different reflected and incident SW wavevector amplitudes are generally expected while F ≠ Fcrit. For the critical angle Fcrit, no reflected SWs beam can be observed—the projecting line of kin wavevector (on the line of antidots) is parallel to the IFDRL drawn for reflected beam (see Fig. 5.6d).

Figure 5.7 (a) Spatial profiles of the normalized microBLS intensity maps taken perpendicular to the nonreflected beams traveling along the line of antidots (gray open dots) and along the edge of the YIG sample (black open diamonds). The edge position is at the distance of 0 μm (with accuracy of ±0.15 µm). (b) Examples of the dependence of the critical angle Fcrit as a function of the normalized frequency f/fFMR for external magnetic field values of H = 600, 1020, and 2600 Oe. On the dashed curve for H = 1020 Oe, the gray circle indicates the experimental value of Fcrit for the results given in Fig. 5.6.

Exemplary Fcrit dependence as a function of the normalized frequency f/fFMR was calculated from the IFDRL for three values of the external magnetic field (see Fig. 5.7b). Both the Fcrit angle and the range of its occurrence increases while the magnetic field H decreases. The gray circle on the dashed curve indicates the experimental point determined for the field H = 1020 Oe and the results presented in Fig. 5.6.

5.2.4  Modeling Spin Waves Total Nonreflection Effect

In the following part, we will examine the influence of the magnetic (demagnetizing) field, neglected in the previous investigations of total non-reflection of SWs [10, 40, 31]. In Fig. 5.8a one can

Spin Waves in Patterned YIG Micrometer Films

see the calculated demagnetizing field appearing near all edges of the ferromagnetic film, including edges of antidots, whenever the magnetization vector crosses their edge [41, 42]. This field locally decreases the internal magnetic field near the edges of antidots, thus (see discussion in Section 5.3) near the edges of antidots the incident SWs shall propagate through areas with gradually increased refractive index. (Relative refractive index between two media is defined by the phase velocities relation [43]. While decreasing magnetic field, wavevector length increases, resulting in a phase velocity decrease.)

Figure 5.8 Modeling of SW interaction with the antidots line. (a) The demagnetizing field in the YIG film (4.5 μm thickness) near the square antidots row (of 70 μm size). The gray scale shows the demagnetizing field component along the direction of the H oriented at angle 45° with respect to the antidots line. (b) Iso-frequency dispersion relation lines (obtained from Eq. 5.3) at 4.7 GHz in the homogeneous YIG film for various internal magnetic field magnitudes in the range 950–980 Oe. The IFDRLs for 980 Oe and 966 Oe are emphasized with the solid and dash-dotted lines, respectively. The direction of antidots line is marked as e||. The direction of the refracted SW group velocity is marked with vg,out.

To elucidate the role of demagnetizing field on the total nonreflection, let us assume that the internal magnetic field is abruptly reduced to a certain value (we put 966 Oe) in a narrow region close to the line of antidots and parallel to it. This value can be considered as the average internal magnetic field value in an area in which the demagnetizing field changes gradually, over a distance comparable to (or smaller than) the wavelength of the incident SWs (lin ≈ 225 μm, determined for H = 980 Oe from Eq. 5.3).

153

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Spin Wave Optics in Patterned Garnet

Figure 5.8b shows IFDRLs plotted over the (kx, ky) plane obtained from Eq. 5.3 for a frequency of 4.7 GHz with H equal 950, 966, and 980 Oe. The incident SWs are refracted at the interface between the regions of lower and higher values of the refractive index (higher and lower H, respectively) before reaching the edge of the antidots line. The condition kin,|| = kout,|| (index || means the component of the wavevector which is tangential to the interface) provides the wavevector kout,|| of the refracted SWs and the direction of the group velocity vg at the terminal point of kout, which is parallel to the direction of antidote line, e||. Indeed, this was observed in experiment in Fig. 5.6b. Moreover, the direction of energy transfer is not very sensitive to the assumed decrease in the internal magnetic field, as indicated by the IFDRLs plotted for various field magnitudes H. Another interesting conclusion from Fig. 5.8b is that |kout| > |kin|, i.e., the wavelength of the refracted SWs is significantly shorter than that of the incident SWs (at 966 Oe lout = 2p/kout = 65 μm). This decrease is fully confirmed in MS in Fig. 5.9a. In Fig. 5.9, the results of MS performed by means of Mumax3 [44] are presented. They show the out-of-plane component of the magnetization mz or averaged in time square mz (T = 1/f) proportional to the BLS intensity in a steady state (see Fig. 5.5a or Fig. 5.5b–d, respectively. Our simulations well describe creation of strong SWs beam experimentally observed (see Fig. 5.6b). Furthermore, we simulated SW scattering with long rectangular hole. The result of these simulations is shown in Fig. 5.9c, and it clearly demonstrates almost the same features as we found in Fig. 5.9b for the antiodots line. Thus, we conclude that the crucial role in the experimentally observed effects is played by the inhomogeneity of the internal magnetic field, rather than the edge of the ferromagnetic film itself. Nevertheless, the edge of the film is the source of the demagnetizing field there. We have also checked numerically the influence of increasing SW frequency on the refraction and the beam formation. Exemplary result of the simulations at frequency 4.98 GHz is shown in Fig. 5.9d. We clearly see the broadening of the SW beam with respect to the beam of lower frequencies (compare with Fig. 5.9c for 4.7 GHz) and the rotation of the beam propagation direction. This agrees well with the IFDRL analysis

Spin Waves in Patterned YIG Micrometer Films

for contours shown in Fig. 5.8b for SWs with increased frequencies. The dependence was also confirmed experimentally by increasing f to 5.2 GHz. This sensitivity of frequency adds additional functionality, giving rise to the possibility of changing the direction of the beam propagation with the change in frequency or alternatively with the variation in the external bias magnetic field magnitude, which shifts up or down dispersion relation of SWs.

Figure 5.9  Results of MSs showing SWs’ interaction with (a, b) antidots of side 70 μm line with period a = 100 μm at 4.7 GHz and (c, d), long rectangular hole (70 μm width) at 4.7 GHz and 4.98 GHz, respectively. Note that in (a) there is SW amplitude and in (b–d) SW intensity presented. The SWs are excited by microstrip antenna located on the right side of the antidots line (rectangular hole) and propagate on the left side of the antenna. The static magnetic field 980 Oe for (a–d) was oriented along the y axis.

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Spin Wave Optics in Patterned Garnet

The phenomenon of SW reflection at the edge of the sample was analyzed in [10, 40]. From the theoretical considerations, it was shown that the total internal reflection of the SWs appears at the critical angle Fcrit, measured between the edge of the reflection and the direction of the magnetic field H. Based on the theory, it should be possible to observe a highly intense beam of SWs propagating along this edge. The aim of this study was to investigate whether, for the critical angle Fcrit, the line of antidots can act as a reflective edge, and as a kind of magnetic “mirror” directing SWs along the line. Our idea of strong focusing of SW beams on only a few antidots in a plane homogeneous YIG film seems to be useful for future applications.

5.2.5  Application of Total Nonreflection Effect for Spin Wave Beam Switching

In this part we demonstrate an example of the application of the ideas discussed above. We propose an SW switcher using total nonreflection on a line of antidots to create a strong SW beam (see discussion above and in Fig. 5.6) and nonreciprocal magnetostatic spin waves excitation by a microwave strip antenna [45]. Possible realization of SWs switcher is presented in Fig. 5.10. The antenna and the magnetic field H are set along the y direction and the wavevector k of the SWs emitted by the antenna is along the x direction with –k and +k for –H and +H, respectively. This field amplitude H = 980 Oe was chosen to obtain a critical angle Fcrit = 45° for total nonreflection. Two-dimensional maps of the SWs intensity I(x, y) distributions were collected with the BLS spectrometer. The excitation frequency of 4.7 GHz was selected for a close to maximum intensity of the amplitude–frequency characteristics. Figure 5.10a,b illustrates BLS maps for the field directions –H and +H, respectively. SWs were emitted on the left side of the antenna and the strong beam of SWs was formed left-up along the line of antidots for –H direction of the magnetic field (see Fig. 5.10a). With reversed magnetic field to +H (see Fig. 5.10b), SWs are emitted on the right side of the antenna and the high intensity beam of SWs propagates right-down along the antidots line. Thus, we have experimentally demonstrated the formation

Optics of Spin Waves in Nanometer-Thick YIG Film

of a narrow SW beam from the SWs generated by the microwave antenna and by changing the direction of the SW beams by 180° with change of the magnetic field orientation. The relatively wide range of the frequencies where the effect exists (approximately 0.3 GHz) suggests wide band functionality of a possible SW switcher device.

Figure 5.10  Experimental demonstration of the strong SW beam formation combined with changing SW propagation direction. Twodimensional mapping of the SW amplitude (detected with the BLS spectrometer) registered for (a) –H and (b) +H. The SWs are generated by the microwave antenna (red vertical bar). Square holes (50 µm size) in YIG film with period a = 100 µm created the antidots line marked by crosshatch.

5.3  Optics of Spin Waves in Nanometer-Thick YIG Film 5.3.1  Reflection of Spin Waves from the Edge of the YIG Thin Film: Goos–Hänchen Effect

Here, we study the properties related to the reflection of the SW beams in a thin YIG film. The reflection can be described by a complex reflection coefficient R = | R |eiy, where R2 is the reflectance and y is the phase shift between the incident and reflected SWs. The physical effects related to the phase shift is a lateral shift between incident and reflected beam spots along the interface, DX (see Fig. 5.11). This effect, called Goos–Hänchen shift (GHS), was firstly observed for electromagnetic waves in the

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1940s [46], and then for acoustic waves [47], electrons [48], and neutron waves [49]. Recently, this phenomenon was investigated theoretically also for the exchange or dipole-exchange SWs [42, 50, 51]. The magnetic properties at the film edge were shown to be crucial for a lateral shift of the SW beam.

Figure 5.11  Schematic plot of the thin YIG film geometry for consideration of the GHS. The film has thickness t, which is much smaller than the film’s lateral sizes, Lx and Ly. The (x,y,z) coordinating system defines the structure with the film edge at y = 0 (hatched area). kin and kout are wavevectors of incident and reflected SWs beams, respectively. DX is a total shift of the SW beam reflected at the edge.

To show the GHS for SWs, let us analyze the dipole-exchange SW beam of frequency f = 35 GHz reflected from the edge of a thin YIG film (Fig. 5.11). The film has the edge at y = 0 and is magnetically saturated by the in-plane static magnetic field H = 7000 Oe, which is applied along the y axis, perpendicular to the film’s edge. Let us focus on the magnetic properties at the film’s edge, in particular surface anisotropy KS and its contribution to the shift of the SW beam. The thickness of the film, t = 5 nm, is much smaller than the lateral sizes of the film, Lx and Ly used in micromagnetic simulations (MS) performed by means of MuMax3 [44]. Propagation of the SWs is limited to the film plane (x, y). Due to translational symmetry of the system along the x axis and conservation of the x components of the incident and reflected wavevector (kin,x = kout,x), the incidence angle is equal to the reflection angle qin = qout. To calculate the GHS we use the stationary phase method developed for electromagnetic waves [52] and already used for

Optics of Spin Waves in Nanometer-Thick YIG Film

the exchange SWs [50]. This method is based on the observation that the reflected beam can exhibit a phase shift relative to the phase of the incident light. If the incident SW is a wave packet of a Gaussian shape with the x component of the wavevector variation Dkx ≪ kx, the reflected beam will show a shift DX along the x axis:

DX =

2p tan q in arg(R ) , = 2 kx p +(k cos q in )2

(5.4)

2 , KS is surface anisotropy at the where p =[t /2– 2K S /( m0 MS2 )]/ lex film’s edge, and k is SW wavevector. The complex reflection coefficient R used to obtain Eq. 5.4 was derived using boundary conditions described in Ref. [42]. The DX dependence on KS (Eq. 5.4) is very sensitive for the surface anisotropy value for the range of the most typical values near KS = 0 (Fig. 5.12a). For KS = 0, GHS takes a negative value, which we attributed to the effective dipolar pinning at the thin film edge. Comparison of the MSs and analytical results obtained for KS = 0 at different angles of incidence shows quantitative disagreement (see Fig. 5.12b). The values coming from the simulations are characterized by much larger negative GHS than those obtained from the model. The disagreement means that an important factor has not been taken into account in the analytical model. Indeed, due to the chosen magnetic configuration with H oriented perpendicular to the film’s edge, in its vicinity a gradual decrease in the static magnetic field Heff,0 is noticeable (Fig. 5.12c). It is caused by the demagnetizing field. If the nonuniformity of the internal magnetic field influences the lateral shift, the total DX should depends on the saturation magnetization of the sample. Additional simulations performed for the permalloy Py film (which has much larger MS than YIG, MS = 0.7 × 103 Gs, A = 1.1 × 10–6 erg/cm) confirms that prediction (see Fig. 5.12b,c). The additional lateral shift is coming from the SWs’ bending and we can estimate this shift using IFDRL analysis for the exchange dominating the SWs (as in Fig. 5.1d). A gradual decrease in the effective magnetic field causes a gradual increase in the refractive index of the SWs, as shown in Fig. 5.13a. The SW beam propagating through the area of increased refractive index bends,

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providing at the edge an additional lateral negative shift of the SW beam, as shown schematically in Fig. 5.13b.

Figure 5.12  (a) The analytical results of the GHS in the reflection of the SW beam from the edge of the thin YIG film calculated using Eq. 5.4 in dependence on the magnetic surface anisotropy constant KS. The gray square corresponds to the value of GHS for KS = 0. (b) The GHS in dependence on the angle of incidence obtained from Eq. 5.4 (solid lines) and MSs (dashed lines with symbols) for KS = 0 for YIG (solid) and Py (dashed line). (c) Internal magnetic field for YIG (solid line) and Py (dashed line).

Figure 5.13  (a) The refraction of the wave on the interface between media with low refractive index (n-th slice, solid black line) and high refractive index (n + 1 slice, gray dashed line) based on IFDRL analysis. The conservation of the kx components in the refraction is required by the translational symmetry along the x axis. (b) Example of two refractions on the interfaces between n-th and n + 1, and between n + 1 and n + 2 slice corresponding to the different values of Heff,0, as it is presented in the inset on the right side of (b). The beam shift DXbending resulting from the bending is marked schematically. (c) Results of the MSs (gray square points) obtained from the analytical models presenting dependence of the SW beam shift on the surface magnetic anisotropy constant in the YIG film in the external magnetic field of 7 kOe. Solid black line corresponds to the basic analytical model for GHS (Eq. 5.4) without including SW bending. Dashed gray line presents results for the analytical model taking into account SW bending (see details in [42]).

Optics of Spin Waves in Nanometer-Thick YIG Film

To validate this approach, we compared results of the GHS in dependence on the surface anisotropy (for the angle of incidence 60°) obtained from the analytical model including bending due to the demagnetizing field (dashed grey line) with the MS results (gray squares) in Fig. 5.13c. The DX(KS) dependence is now recovered; however, in order to obtain such good agreement, the averaging procedure of the demagnetizing field was required. Such approach was crucial due to the significant changes in the demagnetizing field on the distance shorter than the wavelength of the SWs near the edge, because the analytical model of the SW bending is based on adiabatic assumption (see details in Ref. [42]).

5.3.2  Molding of Spin Wave Refraction in Two-Dimensional YIG Antidots Lattice

In this section we will focus our discussion on magnonic crystals (MCs) [53, 54] based on two-dimensional antidots lattice (ADL) in a thin (thickness 12 nm) YIG film. Similarly, as in photonic crystals [55], the periodic modulation of the structural or material parameters results in the dispersion relation folded back into the first Brillouin zone. In MCs the SWs of the same frequency, but propagating in different directions, can differ significantly in wavelength, even if the material itself has an isotropic dispersion relation, like for the exchange SWs (Fig. 5.14). This anisotropic dispersion relation can lead to an unusual (i.e., not present in uniform medium) relation between direction of phase and group velocity that, finally, is responsible for the refractive properties of the system [56]. In thin-film magnonic systems, operating in a long wavelength range, we observe also the intrinsic anisotropy of SWs (see Section 5.1) resulting from dipolar interaction. Figure 5.14 presents the two-dimensional dispersion relation of the planar MC in the form of the square ADL based on a thin YIG film. We can see that the lowest magnonic band has a saddle-like form exhibiting two-fold symmetry being the result of dominance of dipolar interactions. The higher bands, located in the frequency range, where isotropic exchange interaction overshadow the dipolar interactions, start to manifest the four-fold symmetry of the lattice.

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Figure 5.14  (a) The structure of the magnonic ADL based on the YIG film of thickness 12 nm. The cylindrical antidots of the diameter 84.6 nm are arranged in a square lattice (lattice constant 150 nm). (b) Dispersion relation (band structure) in the first Brillouin zone for the structure shown in (a) with the external magnetic field applied along the y direction calculated using plane wave method.

Using the geometrical analysis of IFDRL we will discuss the unusual refraction at the interface between the homogeneous film (HF) of YIG and the patterned YIG film (MC in the form of the square ADL). We will discuss the effects of (i) angular filtering, where the beam of SWs can enter from one medium to the other only if its incidence angle is limited to specific ranges, and (ii) all-angle collimation [57], where the refracted wave always propagates normal to the interface independently of the incidence angle.

5.3.2.1  Angular filtering

The bottom of the lowest magnonic band for ADL presented in Fig. 5.14 has a saddle-like shape with opposite signs of curvature along the directions parallel and perpendicular to the external magnetic field. As a result, the bottom of the dispersion relation has two minima displaced from the center of the Brillouin zone (see the two closed loops of the iso-frequency lines in Fig. 5.15 for ADL, marked in black). Due to the presence of air holes (antidots) in the ADL, the bottom of this minima is shifted up in the frequency scale in reference to the dispersion of HF. Therefore, for selected frequency 7.67 GHz, the dispersion of HF is already exchange dominated and the corresponding IFDRL is almost circular (see grey loop in Fig. 5.15). We investigate the refraction

Optics of Spin Waves in Nanometer-Thick YIG Film

of SWs incoming from HF and entering into ADL for two different orientations of the magnetic field: perpendicular and parallel to the HF/ADL interface. For the first case (see Fig. 5.15a), we found the superprism effect, where a small change in the incidence angle results in a large change in the refraction angle. The system accepts the transmission at the direction normal to the interface because both ADL and HF have magnonic eigenstates with the tangential component of the wavevector equal to zero at frequency 7.67 GHz. As a result, the superprism effect leads to total internal reflection for the incidence angle exceeding a small critical value. The system in this configuration works as a low-pass angular filter transmitting through the HF/ALD interface only those SWs that come in at the incidence angle lower than some threshold value (we marked this range in Fig. 5.15 by a bright grey circular sector). The structure considered can also work in the other operational mode, high-pass angular filtering. We can switch between these modes by rotating the external field (or the structure) by 90°. In the mode presented in Fig. 5.15b, the field is normal to the HF/ALD interface and the system does not transmit, through the interface, the SWs of incidence angles lower than a specific critical value.

Figure 5.15 The IFDRLs for ADL (black contours) and homogeneous YIG film (gray contour) for frequency 7.67 GHz located close to the bottom of the first magnonic band in ADL in Fig. 5.14. The arrows show the directions and the magnitudes (in arbitrary units) of the group velocity (arrows normal to IFDRLs) and phase velocity (arrows attached at the center of Brillouin zone). The gray (black) arrows refer to the group and phase velocities in HF (ADL). The dotted squares mark the edges of the first Brillouin zone. (a) Low-pass angular filtering and (b) highpass angular filtering. ADL and HF are made of YIG and have structural and material parameters as described in Fig. 5.14.

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5.3.2.2  All-angle collimation In films, the SWs propagate with a larger group velocity in the direction perpendicular to the external magnetic field than along the magnetic field direction. This effect is even more noticeable for ADLs at the crossover of the dipolar-exchange regime, where the competition between the dipolar contribution (in backward configuration) and exchange contribution leads to flattening the dispersion relation. In ADL, the additional factors come from the profile of the demagnetizing field, where the trenches of internal fields are induced perpendicularly to the direction of the magnetic field [58]. This effect is also visible in the dispersion relation in Fig. 5.14, where the slope (and group velocity) of the lowest magnonic bands is higher in the direction perpendicular to the external magnetic field. The top of the first band and the bottom of the second band have a peculiar shape, i.e., the dispersion is almost independent of the component of the wavevector parallel to the magnetic field. This means that the group velocity, being the gradient of the dispersion relation, is directed almost exactly perpendicular to the field direction, independently of the direction of the corresponding wavevector (phase velocity). The SWs of different wavevectors will then propagate mostly in the direction perpendicular to magnetic field, enabling collimation, as demonstrated schematically in Fig. 5.16.

Figure 5.16 All-angle collimation effect. The IFDRLs in ADL and HF for frequency 9.1 GHz (the bottom of second band in ADL). Due to the flat IFDRLs in ADL, the divergent beam coming from HF is collimated in ADL into the beam propagating in the direction normal to the ADL–HF interface. The meaning of the lines and arrows is the same as in Fig. 5.14.

Summary

5.4  Summary In summary, we presented different optic-like effects of SWs in patterned thin magnetic film YIG samples. Brillouin light scattering technique was used for imagining SW interaction with the single antidot and the line of antidots in YIG micrometer-thick films. The SW wavelength was driven by the excitation frequency or the external magnetic field amplitude. Discussion of the results of SW interaction with single antidot studies was focused on caustic SW beam diffraction, like SW reflection and interference. Especially interesting was the SW total nonreflection effect connected with the creation of a highly focused beam of SWs propagating along a line of antidots and behind this line. We showed that a physical phenomenon to obtain highly focused beams originates from the refraction of SWs in an inhomogeneous internal magnetic field (connected with the demagnetization field) near the edge of the antidot line. In the theoretical analysis of SWs in nanometer-thick magnetic films, we showed that a Goos–Hänchen shift depends on both the local surface magnetic anisotropy at the film edge and the variation of the internal magnetic field. In patterned 2D antidots lattice nanometer films, in specific ranges of frequency, we reported the effects of low-pass and high-pass angular filtering and of all-angle collimation. The SW total nonreflection effect was used for the creation of strong SW beams at the 180° switcher. The already available technology of fabricating low-damping, very thin, nanometerthickness films and the various optic-like effects of SWs reported have a promising potential for desirable miniaturization of SW devices.

Acknowledgments

This work was supported by the National Science Centre, Poland, through OPUS grant DEC-2013/09/B/ST3/02669 and Sonata-bis grant UMO-2012/07/E/ST3/00538. The authors would like to thank V. D. Bessonov and H. Ulrichs for their contributions to spin wave studies in YIG micrometer-thick films.

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Chapter 6

Spin Waves in Circular and Linear Chains of Discrete Magnetic Elements

Yu. N. Barabanenkov, S. A. Osokin, D. V. Kalyabin, and S. A. Nikitov Kotel’nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow 125009, Russia Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia Laboratory “Metamaterials,” Saratov State University, 83 Astrakanskaya str., Saratov, 410012, Russia [email protected]

6.1 Introduction Investigations of magnetic micro- and nanostructures as potential candidates for applications in spintronics and for magnetic logic devices have become a hot topic recently [1–7]. In particular, intensive studies of various magnetic structures are performed to understand properties of perspective materials for above-mentioned applications. This, in turn, requires investigation of important physical phenomena related to spin wave dynamics in magnetic materials and especially in micro- and nanostructured magnetic Spin Wave Confinement: Propagating Waves (2nd Edition) Edited by Sergej O. Demokritov Copyright © 2017 Pan Stanford Publishing Pte. Ltd. ISBN 978-981-4774-35-2 (Hardcover), 978-1-315-11082-0 (eBook) www.panstanford.com

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films. Spin wave dynamics is very often related to properties of spin waves propagation in confined magnetic structures or in arrays of magnetic dots, stripes, etc. [8–11]. There are various types of magnetic periodic structures, called magnonic crystals (MCs) [12, 13], which are suitable for investigation of spin waves propagating in such structures promising information processing and concepts of logic devices. These MCs can be represented, in particular, by arrays of holes (antidots) etched in yttrium-iron-garnet (YIG) films [14, 15], dynamic MCs [16], and other patterned films [17–19]. The main manifestation of magnetic film patterning is the appearance of a wave band structure within the wave spectrum. The spin wave band structure can be very complicated and can be controlled by variation of external parameters, for example, an external magnetic field, and, particularly, by metallization of the structure [20, 21]. However, in many of the cited works, investigations of interaction of waves with single inhomogeneity in periodic structures are usually left behind consideration. Basically, only the collective influence of the magnetic structure on the propagating wave properties is taken into account. The problem of spin wave scattering by an infinite set of magnetic or nonmagnetic inclusions (cylindrical pillars) embedded in a ferromagnetic thin film (matrix) was considered recently [22]. It was shown that under certain conditions, spin wave edge modes are excited around these inclusions, and moreover these modes a nonreciprocal character of propagation with respect to the external magnetic field saturating the ferromagnetic matrix and inclusions. Furthermore, investigations of spin wave edge modes became very popular because of the prediction of their existence in various magnetic nanostructures, such as ferromagnetic islands and/or circular magnetic thin rings or circular disks, or semi-infinite arrays of dipolar-coupled magnetic nanopillars [23–27]. As mentioned, investigations of spin wave propagation in MCs or other periodic magnetic systems were performed considering these structures as an infinite set of periodic perturbations located along the spin wave propagation path. On the other hand, it is interesting to study and important to understand how spin waves are scattered by a finite array of perturbations located in a ferromagnetic matrix along the propagation path. It is very important to notice that spin wave propagation in magnetic film and scattering by a finite set of magnetic perturbations differ considerably from the similar phenomena in

Introduction

infinite periodic sets of inclusions. Spin wave scattering by infinite sets of inhomogeneities can be described within the theoretical approach of coupled wave equations based on the so-called Bloch theorem. This approach is well known and can be applied for almost any types of the waves, independently of their physical origin. The problem of spin wave edge modes in a finite array of perturbations as circular magnetic thin rings or circular disks is considered in Ref. [28], treating the circular ring as a 1D chain of spins that are coupled between them via long-range dipole-dipole interactions. Thus, the important problem in spin wave theory is related to the scattering of waves by differently arranged magnetic inhomogeneities. In this chapter we describe a theoretical approach directly aimed at solving this problem. First, we develop a general theory of forward-volume magnetostatic spin wave (FVMSW) multiple scattering by a finite 2D ensemble of cylindrical magnetic inclusions in a ferromagnetic matrix [29] (Fig. 6.1). As it turned out, such finite number of magnetic inclusions arranged periodically along a circle can have spin wave eigenmodes and can be treated as a specific micro–spin wave resonator with a high value of the quality factor. This microresonator can be considered as an element of a magnonic circuit device. The second problem that is intended to be solved is to consider spin wave propagation through a linear finite array (chain) of magnetic inclusions embedded in the ferromagnetic matrix [30] (Fig. 6.2). This chain is an element of a magnonic circuit device, which can play the role of a micro–spin wave waveguide. Our basic result is proved by the optical theorem for the T-scattering operator, which describes spin wave multiple scattering on magnetic inclusions, and derives a theoretical formalism for a collective extinction cross section of spin waves by finite circular and linear chains of inclusions.

Figure 6.1 Ferromagnetic film with ferromagnetic inclusions.

173

174

Spin Waves in Circular and Linear Chains of Discrete Magnetic Elements

Figure 6.2 Ferromagnetic film with a linear chain of cylindrical ferromagnetic inclusions.

6.2 Multiple-Scattering Method We consider a ferromagnetic film (the matrix) of thickness d and saturation magnetization M0s with embedded cylindrical inclusions of another ferromagnetic material having the same thickness and the value of the saturation magnetization as M1s . An external uniform magnetic field Hext is applied normally to the matrix surfaces along the z axis, allowing propagation of FVMSWs (see Fig. 6.1). Neglecting crystalline and surface anisotropy and exchange effects, the effective field inside the matrix and inclusion is written as H = Hext – Hdm, where Hdm is a demagnetizing field. Further we use a simple model for the demagnetizing field in the form 4pM0s and 4pM1s and got the effective fields H0 and H1 inside the matrix and inclusion, respectively. The antisymmetric μ tensor (dyadic) of magnetic susceptibility has the form

È 0 Í m = Í-i a ÍÎ 0

i a 0 0

0˘ ˙ 0˙ . (6.1) 1˙˚

The tensor diagonal and off-diagonal components inside the matrix,  00 and i  0a , and inclusions,  10 and i  1a , are found from the solution of the Landau–Lifshitz equation (see Ref. [31]). Here the indices t = 0 and t = 1 indicate the matrix and the inclusion, respectively; the ferromagnetic resonance frequency wH = γH, with γ being the gyromagnetic ratio; and the frequency wM = g4pMs. The frequencies wt are defined by w t2 = w H2 + w Ht w Mt and restrict the propagating spin wave frequency band from above.

Multiple-Scattering Method

We start with Maxwell’s equations in the magnetostatic approximation [31]  ◊ b = 0 and  ¥ h = 0 for magnetic induction b =  ◊ h and magnetic field h vectors in the spin wave. The magnetic field can be written in terms of the magnetostatic potential Y according to h = – ◊ Y. The magnetic induction is detailed with the aid of the diagonal dyadic  =  0 ( xˆ ƒ xˆ + yˆ ƒ yˆ ) + zˆ ƒ zˆ and the gyration vector g = i ma zˆ , where xˆ , ŷ, and ẑ denote the unit vectors along the x, y, and z axis and the symbol ⊗ denotes the tensor product, as



bt = - m0 ◊ —Y t + g ¥ —Y t , bzt = -∂ z Yt . (6.2)

Substituting this representation into the equation for magnetic induction leads to the Walker equation

(

)

m0t ∂2x + ∂2y Y t + ∂2zY t = 0 (6.3)

for the magnetostatic potential inside the ferromagnetic matrix and inclusions. As the next step of consideration we suppose that the matrix surfaces z = 0 and z = d as well as the inclusion end parts z = 0 and z = d (see Fig. 6.1) are metallized, following the known studies of a cylindrical ferromagnetic resonator [31]. In this case one can put Yt (x, y, z) = Yt(x, y) cos(kzz), where kz = np/d (n = 0, 1, 2,…) and

(∂

2 x

)

+ ∂2y Y t + krt 2Y t = 0. (6.4)

In the 2D Walker equation (Eq. 6.4) the components krt of the wave vector along the (x, y) plane inside the matrix and the

inclusion are defined by krt = -1 / m0t k z . The desired form Yt(x, y, z) of the magnetostatic potential satisfies the boundary conditions bzt ( z = 0, d ) = 0 on the matrix surfaces and the end parts of an inclusion. The first equation (Eq. 6.2) gives brt ( x , y , z brt (r , )cosk and t t hj ( x , y , z ) = hj (r ,j )cos k z z , where r and j are the local cylindrical coordinates of a cylindrical inclusion (see Fig. 6.3). Solving the 2D Walker equation (Eq. 6.4) with boundary conditions, which means the continuity of the magnetic induction normal component and the magnetic field tangential component on each inclusion lateral surface, and substituting the result into Yt(x, y, z) gives in magnetostatic approximation the magnetic induction and magnetic field inside the matrix with inclusions.

175

176

Spin Waves in Circular and Linear Chains of Discrete Magnetic Elements

Figure 6.3 Sketch showing two ferromagnetic inclusions with their local coordinate systems.

In considered case the set of cylindrical inclusions is irradiated by a plane spin wave (incident wave) with magnetic field potential Y inc (r ) = exp(ik 0r ◊ r ) . Each j th inclusion is related to its local Cartesian coordinate system xj, yj, zj, with the center Rj that coincides with the cylinder center and is evaluated relative to the laboratory system x0, y0, z0. All local coordinate systems are obtained via parallel translation of the laboratory system along its x0 axis. The wave vector kr0 = k 0r xˆ of the incident spin wave (denoted by kx in Fig. 6.1) is directed along the unit vector x. Writing r = Rj + rj where r = Rj + rj is a point position with respect to the local coordinate system, and bearing in mind the well-known expansion (see, for example, Ref. [32]) of the scalar plane wave through cylindrical functions, we obtain

Y inc (r j ,f j ) =

•

ÂA

m=-•

imf j 0 jm J m ( kr r j )e

. (6.5)

Here rj, fj are the local cylindrical coordinates of an observation point and Jm(u) is the Bessel function, with m being a cylindrical multipole index. The coefficients Ajm = ejim, where the phase factor e j = exp(ik 0r ◊ R j ) is dependent on the position of the j th cylinder center relative to the laboratory system. During the scattering process scattered Yscat and transmitted Ytrans spin wave fields appear around inclusions and inside them, respectively. They are

Multiple-Scattering Method



Y

and Y

scat

(r j , f j ) =

trans

•

ÂB

m=-•

(r j , f j ) =

•

i mf j 0 jm Hm ( kr r j )e

ÂX

m=-•

imf j 0 jm J m ( kr r j )e

, (6.6) , (6.7)

where Bjm and Xjm denote the scattering and transmission coefficients, respectively, and Hm(u) is the Hankel function. To write out the boundary conditions on an inclusion lateral surface one needs to consider all fields outside the inclusion near its surface. For example, the total field around the first inclusion consists of three components

scat Y1scat + inc (r1 ,f1 ) = Y1scat (r1 ,f1 ) + Y1inc (r1 ,f1 ) + Y21 (r2 ,f2 ). (6.8)

The first and the second terms on the right-hand side (RHS) of this equality present the scattered (Eq. 6.6) and incident (Eq. 6.5) fields, respectively. The third term of the RHS of Eq. 6.8 is the field scattered by the second inclusion and irradiating the first inclusion. This problem is solved with the aid of the addition theorem [32] for cylindrical wave functions, which is taken in the form

einq2 Hn (kr0r2 ) = ( -1)n

•

ÂH

l =-•

- ilq1 0 0 n+1 ( kr R12 ) J1 ( kr r1 )e

,

(6.9)

where the angles q1 and q2 are defined in Fig. 6.3. We denote by Rjj¢ = Rj – Rj¢ the vector distance between the j th and the j¢ th inclusion scat centers, with Rjj¢ = |Rjj¢|. Bearing in mind that Y21 (r2 ,j2 ) is given actually by Eq. 6.6, the application of the addition theorem in Eq. 6.9 provides



scat Y21 (r2 ,f2 ) =

B21m =

•

ÂB

l =-•

•

ÂB

m=-•

2l e

imf1 0 21m J m ( kr r1 )e

i arg R12

(

)

,

(6.10)

Hl-m k r0R12 , (6.11)

and the angle b again defined in Fig. 6.3. arg R12 = p + b denotes the angle between the vector R12 and the unit vector x . Equation 6.11 can be called the invariant form of the addition theorem for cylindrical wave functions and enables one to write out the total field around the second inclusion immediately by simply interchanging the indices 1 ´ 2 in Eqs. 6.8, 6.10, and 6.11.

177

178

Spin Waves in Circular and Linear Chains of Discrete Magnetic Elements

Now we are ready to formulate in detail the boundary conditions on the inner surfaces of both inclusions, which are the continuity of the magnetic field tangential component and the magnetic induction normal components on the surfaces. Dividing the two boundary conditions on the inner surface of each inclusion permits one to get some complete algebraic equations for the scattering coefficients Bim. For example, one can obtain an algebraic equation for the scattering (1) coefficient Bm of the first single inclusion by putting B21 Æ 0. Thus we have obtained one equation for the spin wave amplitudes for multiple scattering in the case of two inclusions. The invariance considerations make it not difficult to write out the set of equations for spin wave multiple scattering by an ensemble of N = 2, 3, … inclusions in the obvious form:

(1) B jm = Bm +

(1) Bm A1m

•

 ÂG

j π j' =1l =-•

with the matrix kernel

'

Gmjj-l = e

i( l -m )arg R

•

jj'

(

jj' m - l B j 'l ,

j = 1,2,º , (6.12)

)

Hl-m k r0R jj’ , (6.13)

where arg Rjj¢ denotes the angle between the vector Rjj¢ and the unit vector xˆ . ( j) The ratio Tm( j ) = Bm / A jm that we meet in the RHS of Eq. 6.12 defines the scattering matrix Tm( j ) of the j th single inclusion, which is dependent on the inclusion parameters only. In particular, the scattering matrix of the first single inclusion is obtained from the algebraic Eq. 6.12 and coincides with the expression found in Ref. [22]. The derived set of Eq. 6.12 for multiple scattering of FVMSWs provides a transparent physical interpretation that can be achieved by an iterative solution of the set of equations. In this case the incident spin wave is singly scattered by the jth inclusion and then propagates along the ferromagnetic matrix to the jth inclusion to be singly-scattered by it, and so on. The spin wave propagation along the ferromagnetic matrix between the two inclusions is described via the matrix kernel Eq. 6.12, the first factor of which takes into account the wave phase shift; the second factor is a component of the Green function for the 2D Walker equation (Eq. 6.4). Let us note that Eq. 6.13 defines a difference kernel that makes it favorable to apply the Bloch substitution for Eq. 6.12 in the form

Multiple-Scattering Method

B jm = exp(iR j )B1m , where  is the Bloch wave vector, in the case of 2D infinite MCs. A similar substitution may be done in the case of spin wave propagation along the circular array of the inclusions. Nevertheless, here we prefer to solve Eq. 6.12 iteratively and made the following transformation:

B jm - Tm( j )

•

ÂB '

j π j =1

H j'm 0

(k

r

)

R jj¢ = B jm . (6.14)

0

The left-hand side (LHS) of this equation set contains the m th multipole terms only. The RHS of Eq. 6.14 has the form

( j) B jm = Bm + Tm( j )

+

•

•

  [B

j π j ' =1d m=1

iB sj(m ,d m) sin(d m arg R jj'

c j( m ,d m ) cos(d m arg R jj' )

)]Hd m (kr0 R jj'

)

(6.15)

where the cosine and sine coefficients are defined by

B cj(m ,d m) = B j(m+d m) + ( -1)d m B j(m-d m),

B sj(m ,d m) = B j(m+d m) - ( -1)d m B j(m-d m) .

(6.16)

As one can see, the RHS of Eq. 6.14 defined by Eqs. 6.15 and ( j) 6.16 describes the incident spin wave in the first term Bm and contributions of all other multipoles m ± dm created at scattering.

6.2.1 Circular Arrays

Let us consider now Eq. 6.14 for spin wave multiple-scattering partial coefficients Bjm by N cylindrical inclusions, with centers placed periodically along a circuit of the radius r. We introduce an eigenmode and eigenvalue problem in the one-multipole approximation using Eq. 6.14:

U( j) -

a(jjm' )

N

Âa

j π j ' =1

( m) ( j' ) U jj'

= l (jm)U ( j ) ; a(jjm' ) = Tm( j )H0 (kr0 R jj' ), (6.17)

where is a coupling matrix between inclusions that are assumed to be identical in the following calculations. The eigenmodes U(j) giving the solution to Eq. 6.17 can be written as N-component vectors

179

180

Spin Waves in Circular and Linear Chains of Discrete Magnetic Elements



U ( j ) = (1, p j , p2j ,º , pNj -1 ),

(6.18) Ê 2p j ˆ p j = exp Á i , j = 1, 2,º , N . ˜ Ë N ¯

The eigenvalues l (jm) corresponding to eigenmodes in Eq. 6.18

are written for the case of an even number of inclusions N = 2n as sums:

l (jm) = 1 -

n

Ân '

j =1

a(m) jj' 11+ j'

,

(6.19) p jj ' ' , j = 1,º , n - 1 n The distance R11+j¢ between the centers of the first and (1 + j¢) th inclusions is defined by the expression R11+j¢ = 2r sin(j¢p/n). The eigenvectors in Eq. 6.21 are orthogonal to each other ·U(j) |U(j¢)Ò = Ndjj¢. Thereby Eq. 6.14 can be written in compact vector form as

n jj' = 2cos

B jm =

N U (jk ) 1 B | U ( j ) (m) , N k =1 lk

Â

(6.20)

which gives, in particular, the solution to the problem of a boundmode excitation due to scattering of the propagating spin wave. Note that in Eq. 6.20 the indices j and k denote the numbers of the inclusion and mode, respectively, with U (jk ) being the j th component of the k th eigenmode. Equation Eq. 6.20 can be solved iteratively. Thus one can replace ( j) in the first step B jm with Bm , getting the first approximation B(1) jm to Bjm. The first approximation B(1) jm is substituted in the RHS of Eq. 6.20 instead of B jm , getting the second approximation B(2) jm , and so on. In the following section we use a simplified version of the iteration process combining the first iteration and the two-multipole approach. Actually we take in the inner sum in the RHS of Eq. 6.15 the terms with dm = 1 only and replace in the RHS of Eq. 6.16 the ( j) quantities Bj(m + dm) with Bm ±d m .

6.2.2 Linear Chains

In this part we consider a problem of FVMSW propagation in a ferromagnetic thin film (matrix) containing a finite 2D linear array

Radiation Losses

of magnetic cylindrical inclusions. Centers of inclusions are placed along the xˆ axis with equal distance R12 between each inclusion (Fig. 6.2). To solve the general equation (Eq. 6.15) for scattered amplitude Bjm, we apply normalization B jm = Tm( j )Bˆ jm and obtain the following equation:



Bˆ jm -

N

Âa

j π j ' =1

( m) ˆ B j 'm jj'

' = Aˆ jm , a(jjm' ) = H0 (kr0 R jj' )Tm( j ) . (6.21)

In Eq. 6.21 the quantity a(jjm' ) denotes the coupling parameter of inclusions numbered j and j¢. The scattering matrix Tm( j ) of the single inclusion is independent from the inclusion number j if all inclusions have identical geometrical and material properties. To solve Eq. 6.21 analytically, we apply the closest-neighbor interaction approximation putting ajj¢ ª 0 if |j – j¢| > 1. Here it is considered that the incident spin wave is propagating along the yˆ axis and only the first inclusion j = 1 is irradiated by the incident spin wave A1m π 0. With such approximation, the matrix of Eq. 6.21 set becomes the Jacobi matrix and one can use the Rayleigh-like solution (the index m is omitted)



sin [( N + 1 - j )q ] 1  Bˆ j = 2( -1) j -1 cosq ,cosq = . (6.22) sin [( N + 1)q ] 2a12

6.3 Radiation Losses

The total amount of spin wave radiation scattered by inclusions is considered in the case of a linear chain of inclusions. We solve this problem with an extinction cross section [33] characterizing the incident spin wave energy loss due to scattering and possible absorption by the linear chain of inclusions. The first step is finding the Green function of a spin wave in a homogeneous ferromagnetic film; to do that the Walker equation (Eq. 6.3) is written in the form

LY = ∂ x ( m0∂ x Y ) + ∂ y ( m0∂ y Y )

+i ÎÈ∂ x ( ma ∂ y Y ) - ∂ y ( ma ∂ x Y )˚˘ + ∂2z Y = 0

. (6.23)

The operator L defined by Eq. 6.23 is the Hermitian operator L+ = L on the function Y(x, y, z) inside the matrix with boundary conditions ∂zY(x, y, z)|z=0d = 0 and a scalar product

181

182

Spin Waves in Circular and Linear Chains of Discrete Magnetic Elements



( Y1 , Y 2 ) =

d

ÚÚ

Ú

dx dy dz Y1* ( x , y , z )Y2 ( x , y , z ). (6.24) 0

Next we suppose that the magnetic susceptibility dyadic m(x, y) has values m0 and m1 inside the matrix and inclusions, respectively, denoting dm(x, y) = m(x, y) – m0 the subtraction of ferromagnetic matrix magnetic susceptibility with inclusions by magnetic susceptibility of the homogeneous matrix. We write the Walker operator as sum L = L0 + L1 of the unperturbed operator L0 = L m Æ m 0 and perturbation L1 = L|m Æ dm. The unperturbed operator describes the magnetostatic spin wave propagation inside the homogeneous matrix, and for perturbation one takes into account the spin wave scattering by inclusions. We write (1 / m00 )L1 Y = -U( x , y )Y and call U(x, y) the magnetostatic scattering operator of spin waves. This operator is Hermitian U+ = U. Now the Walker equation (Eq. 6.23) takes the form

(∂2x + ∂2y )Y +

1

m00

∂2z Y - U Y = j( x , y , z ), (6.25)

where j(x, y, z) is a source term. Note that both the scattering operator and the magnetic susceptibility deviation dm(x, y) have zero values outside the inclusion volume. We denote the Green function G0(r, r¢), where the vector r = (x, y, z) is a point in 3D coordinate space. The Green function for the unperturbed Walker equation (Eq. 6.25) satisfies

Ê 2 2 1 2ˆ Á ∂ x + ∂ y + 0 ∂ z ˜ G0( r , r ¢ ) = d (r - r ¢ ). m0 ¯ (6.26) Ë ∂ z G0(r , r ¢ ) z =0,d = 0.

The solution to the differential Walker equation (Eq. 6.25) is reduced to the integral equation

Ú

Y( r ) = Y0( r ) + G0(r , r ¢ )U(r ¢ )Y(r ¢ )dr ¢ , (6.27)

where the inhomogeneous term on the RHS is the magnetic potential of the incident spin wave

Ú

Y0( r ) = G0(r , r ¢ )j(r ¢ )dr ¢. (6.28)

The T-scattering operator T(r¢, r¢¢) for magnetostatic spin waves is introduced by writing the solution to the integral Eq. 6.27 in the form

Radiation Losses



Ú

Y( r ) = Y0( r ) + G0(r , r ¢ )T ( r ¢ , r ¢¢ )Y0(r ¢¢ )dr ¢¢

(6.29)

T (r , r ¢ ) = U( r )d (r - r ¢ ) + U( r ) G0(r , r ¢¢ )T ( r ¢¢ , r ¢ )dr ¢¢.

(6.30)

and satisfies the integral Lippmann–Schwinger (LS) equation

Ú

The operator T(r¢, r¢¢) depends on both of its arguments and has non-zero values only inside the inclusions. On the basis of Eqs. 6.27 and 6.29 it is useful to introduce a quantity

Ú

P( r ) = U( r )Y( r ) = T (r , r ¢ )Y(r ¢ )d r ¢ , (6.31)

which has a physical meaning of dynamic magnetization displacement current excited inside inclusions by spin wave scattering. In terms of this current the magnetic field potential scattered by inclusions Ysc(r¢) is written from Eqs. 6.27 and 6.29 as

Ú

Y sc ( r ) = G0(r , r ¢ )P(r ¢ )dr ¢. (6.32)

Because of the scattering potential U hermiticity one can derive from the LS Eq. 6.30 the optical theorem for the T-scattering operator in the form

T (r , r ¢ ) - T * ( r ¢ , r ) =

Údr ¢¢ Ú ÈÎG (r ¢¢ , r ¢¢¢) - G (r ¢¢¢ , r ¢¢)˘˚ ¥ T (r ¢¢ , r )T(r ¢¢¢ , r ¢)dr ¢¢¢. * 0

0

*

(6.33)

Now we consider vector r in cylindrical coordinates (r, z), by translating 2D Cartesian coordinates (x, y) into polar coordinates (, j). With this transformation we introduce a complete and orthogonal set Gn(z),n = 0, 1, 2,… of homogeneous film (matrix) transversal eigenmodes defined by

z 1 2 G n ( z ) = bn cos p n , b0 = , bn = , n = 1,2,º. (6.34) d d d One can verify that all quantities in Eqs. 6.26–6.33 can be easily expanded along the set of transverse eigenmodes of Eq. 6.34. For example, the 3D unperturbed Green function G0(r¢, r¢¢) satisfying the differential Eq. 6.26 can be expanded as





G0(r , r ¢ ) =

•

ÂG n=0

(0) n ( -  ¢ )G n ( z )G n (z ¢ ), (6.35)

where the 2D unperturbed Green function Gn(0)( r ) is evaluated directly in the form

183

184

Spin Waves in Circular and Linear Chains of Discrete Magnetic Elements



1 (1) 0 H (krn  ). (6.36) 4i 0 (1) On the RHS of this equation Hm (u) denotes the Hankel function Gn(0)(  ) =

of the first kind and order m = 0 and k 0rn = -1 / m00 (p n / d ) is a component of the spin wave wave vector along the (x, y) plane inside the homogeneous matrix for the n th transverse mode. Equation 6.32 for the magnetic field potential Ysc(r) scattered by inclusions in terms of 2D quantities takes the form

Ú

Y scn (  ) = Gn(0)(  -  ¢ )Pn (  ¢ )d ¢ , (6.37)

where a 2D current Pn() is obtained by transformation of the second equation from Eq. 6.31 and is written as

Ú

Pn (  ) = Tn (  ,  )Y0n (  )d ¢. (6.38)

On the RHS of Eq. 6.38 Tn(, ¢¢) denotes the 2D scattering operator that is connected to the 3D scattering operator T(r¢, r¢¢) by transformation similar to one in Eq. 6.35. The quantity Y0n(¢) appears in expansion of the magnetic potential of the incident spin wave in Eq. 6.28 along transverse eigenmodes. The application of this expansion along transverse eigenmodes. The Fourier transformation Tn(k, ¢) and a denotion of the 2D unit vector s to the optical theorem Eq. 6.33 give

1 Im È Y0* n (  )Pn (  )d ˘ = ÎÍ ˚˙ 8p

Ú

Ú

2p

2

0 Pn (krn s ) d s.

(6.39)

The obtained relation is a basic optical theorem for magnetostatic spin waves under consideration. It is useful to rewrite this relation in the form

Cext = Csc



Cext = -

(6.40)

where Cext and Csc are cross sections of extinction and scattering, respectively, for spin wave scattering by inclusions. These values are defined by the relations

and C sc =

1 Im È Y0* n (  )Pn (  )d ˘ , (6.41) 0 ˚˙ krn ÎÍ

1 0 8p krn

Ú

Ú P (k n

0 rn s )

2

d s. (6.42)

Radiation Losses

To clarify the physical meaning of the optical theorem in Eqs. 6.32–6.42, one can consider the spin wave magnetic field potential Yscn()(Eq. 6.37) scattered by single inclusion centered at the 2D point R1 =0 (Fig. 6.3) for the observation point  placed in the far wave zone of an inclusion. Applying the asymptotics for the Hankel function with the big argument’s value, we obtain

Y scn (  )  Æ• ª e

-

T( s) =

ip 4

0 2 ei krnr T ( s ), 0 p krn r (6.43)

1 P (k 0 s ). 4i n rn

The quantity T(S) can be called the scattering amplitude in the direction with unit vector s = /r Equations 6.41 and 6.42 can be rewritten as

Cext = -

4 2 Re ÈÎ T ( s0 )˘˚ , C sc = 0 0 krn p krn

Ú

2p

2

T ( s ) d s. (6.44)

Consider the case of N inclusions centered in points with 2D coordinates Rj (j = 1, 2,…) (Fig. 6.2). The 2D T-scattering operator for an ensemble of N inclusions is evaluated with the help of the Watson composition rule [34] as follows

Tn ( ,  ¢ ) =

N

ÂT j =1

( j) n ( ,  ¢ ), (6.45)

where the self-consistent 2D T-scattering operators Tn( j )( ,  ¢ ) satisfy the set of equations

Ú Ú

Tn( j )( ,  ¢ ) = Tn(0)(  - R j ,  ¢ - R j )

+ d ¢¢ d ¢¢¢ Tn(0)(  - R j ,  ¢¢ - R j ) ¥ Gn(0)(  ¢¢ -  ¢¢¢ ) N

¥

ÂT

j π j' =1

(j ¢ ) n (  ¢¢¢ ,  ¢ ).

(6.46)

On the RHS of this set Tn(0)( - R j ,  ¢ - R j ) denotes the 2D T-scattering operator of a single inclusion centered in Rj . The Eq. 6.46 set shows, in particular, that a self-consistent 2D T-scattering operator Tn( j )( ,  ¢ ) of a j th inclusion is confined in dependence on

185

186

Spin Waves in Circular and Linear Chains of Discrete Magnetic Elements

its first argument  inside the j th inclusion. According to Eq. 6.38 the 2D current Pn() in the case of N inclusions takes the form of the sum Pn (  ) =



Ú

N

ÂP j =1

( j) n (  ),

Pn( j )(  ) = Tn( j )(  ,  ¢ )Y0n (  ¢ )d ¢ ,

(6.47)

where Pn( j )( ) is a self-consistent current excited inside a j th inclusion. The scattered spin wave field in Eq. 6.37 takes the form j) of the sum of self-consistent fields Y(scn (  ) scattered by different inclusions N

Y scn (  ) =



Ú

ÂY j =1

( j) scn (  ),

j) Y(scn (  ) = Gn(0)( -  ¢ )Pn( j )( ¢ )d ¢.

(6.48)

Now the general optical theorem in Eqs. 6.40–6.42 is taken in the case of incident plane spin wave scattering by an ensemble of inclusions in the form similar to a single inclusion case in Eq. 6.44, with

Ts ( s ) =

N

ÂT j =1

( j)

( s ).

(6.49)

T (1)( s0 ).

(6.50)

We consider propagation of the incident spin wave narrow beam that irradiates only the first inclusion and is scattered by the linear chain of inclusions. The geometry of this case is depicted in Fig. 6.2, where the linear chain is located along the xˆ axis and the incident wave is propagating along the direction of the unit vector s0 = yˆ of the y axis irradiating the first inclusion j = 1 located in the R1 = 0 point of the film. Substituting this last evaluation in Eq. 6.41 gives for a general definition of the extinction cross section



Cext ª -

4

0 krn Re

0 Here T(1)(S) is defined with Fourier transform Pˆn( j )(krn s)

1 0 0 s )Pˆn( j )(krn s ), (6.51) exp( -ikrn 4i and defines scattered potential in the far wave zone

T ( j )( s ) =

Results



Y(j) scn ( )



Cext = -

 Æ•

ªe

-

ip 4

0 2 eikrnrT ( j )(s). (6.52) 0 p krn r

With the general expression for scattered magnetostatic potential, Eq. 6.6, we can show the extinction cross section in onemultipole approximation using the partial scattered amplitude from Eq. 6.22:

1  Re(Tm Aˆ 1m Bˆ 1m ). (6.53) 0 krn

Defining the extinction cross section of a single inclusion as 0 C(1)ext = -(4 / krn )ReTm and putting Aˆ 1m ª 1 , we present the ratio Cext / C(1)ext in the form



Cext ImTm = ReFN ImFN , (6.54) C(1)ext ReTm

where the collective extinction factor FN is defined by

 FN = Bˆ 1m . (6.55)

The major property of Eq. 6.50 is that the extinction cross section of the linear chain in the case of an incident narrow spin wave beam irradiating only the first inclusion formally coincides with the extinction cross section in Eq. 6.44 for a single inclusion. Thus only the irradiated inclusion makes a direct contribution in the collective extinction cross section despite the fact that the total number of inclusions in the linear chain that makes the direct summarized contribution of all other inclusions in spin wave scattering almost invisible; we call this dark mode.

6.4 Results

As is known, excitation of edge spin waves in different magnetic structures is connected with reciprocity breaking at the spin wave scattering by non-uniformity in the antisymmetric magnetic susceptibility tensor. We give a simple interpretation of this phenomenon as the effect of spatial phase modulation in the spin wave scattered field, with the appearance of a helical component in the scattered field.

187

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Spin Waves in Circular and Linear Chains of Discrete Magnetic Elements

In Fig. 6.4 we present the spatial distribution of the sum Re Y scat =

4

ÂY j =1

scat j (r j , f j )

for m = 1, –1 to show the spatial phase

modulation. The calculation results show that like in the case of scattering by one inclusion, the helical line behavior is present for the low value of the collective wave parameter kr0R12 = 1.2 < p in Fig. 6.4a. For big values of the collective wave parameter kr0R12 = 4.3 > p in Fig. 6.4b, helical lines are absent and a distribution of the potential with radial nodal lines is explained by Bragg diffraction. Here R12 is the distance between the two closest inclusions in a circular array of four inclusions (Fig. 6.1).

Figure 6.4 Helical lines in the distribution of the real part of the magnetic field sum potentials scattered by two inclusions: the collective wave parameter kr0R12 = 1.2 for case (a) and kr0R12 = 4.3 for case (b).

Further study shows that a given mode may have several resonances, the positions of which in the frequency range and their quality factors can strongly depend on the cluster radius at dense and almost dense packing of inclusions inside the cluster. The revealed resonance peculiarities cause the circular array of four magnetic inclusions inside a ferromagnetic matrix to be a specific Fabry–Perot interferometer of spin waves. We consider first the resonance property of eigenmode U4 (Eq. 6.18) with eigenvalue l4(m) (Eq. 6.19) and the dipole case m = –1. In Fig. 6.5a the dependence (-1) of Re(1 / l4 ) on the normalized frequency W = w / w H0 - 1 at four closely packed cylindrical inclusions is plotted. One sees two

Results

resonance peaks of different resonance frequencies, heights, and widths related to two clusters with slightly different radii.

Figure 6.5 (a) Dependence of Re(1 / l4( -1) ) on the normalized frequency W at four almost closely packed cylindrical inclusions with parameter R12 = 6.2 mm (the deeper and narrower peak) and R12 = 6.6 mm (the less deep and wider peak), and inclusion radius R = 1.8 mm . (b) Dependence of quality factor Q of peaks on the distance R12.

Now we consider the investigation of the distant transfer of spin wave excitation along the linear chain of coupled magnetic inclusions. The most important point appears to be the resonant case when the ¢ ¢¢ imaginary part of the coupling parameter a12 = a12 + ia12 takes a zero value:

¢¢ a12 = 0 (6.56)

As studies show [30], under this condition it is possible to ¢ ¢ determine separate cases of small 2| a12 |< 1 and big 2| a12 |≥ 1 values ¢ of the real part of the coupling parameter a12 in the dispersion (Eq. 6.22). In these two cases, solutions of the dispersion (Eq. 6.22) are



q ¢ = 0,coshq '' =

1 '' ¢ = 0, - 1 < 2a12 < 0, (6.57) , a12 ¢ 2| a12 |

q ¢¢ = 0, coshq ' =

-1 ¢¢ ¢ , a12 = 0, 2a12 < -1. (6.58) ¢ 2a12

According to Eq. 6.57 the imaginary part q¢¢ of the complex variable q changes in a semi-infinite interval 0 £ q¢¢< •, and according to Eq. 6.58 the value q¢ changes in an interval 0 £ q¢< p/2,. If resonant ¢¢ ¢ conditions a12 = 0 and a12 Æ -0.5 are matched, then both values

189

190

Spin Waves in Circular and Linear Chains of Discrete Magnetic Elements

satisfy conditions q¢¢Æ 0 and q¢Æ 0 in both Eqs. 6.57 and 6.58. Therefore, Eq. 6.22 leads to the limiting formula

j ˆ 1 ˆ Ê Ê  Bˆ j Æ ( -1) j -1 2Á 1 , FN Æ 2Á 1 . ˜ Ë ¯ Ë N +1 N + 1 ˜¯

(6.59)

 Bˆ j Æ ( -1) j -1 exp[-( j - 1)q ¢¢ ] , FN Æ 1.

(6.60)

On the other hand, if the condition q¢¢π 0 or even q¢¢Æ • is satisfied, then Eq. 6.22 gives

The limiting formulas of Eq. 6.59 describe the case of the distant resonant transfer of spin wave excitation along the linear chain of coupled inclusions at resonant value of the coupling parameter a12 Æ –0.5, which shows a linear dependence of excitation decrease on the number of particles. The collective extinction factor according to Eq. 6.59 is equal to approximately FN ª 2 for N ? 1; the fact that FN π 1 shows some indirect effect of the particles influence with numbers j > 1 on the collective extinction factor via influence on the self 1 in the first inclusion. Equation consistent scattering amplitude B 6.59 describes a short transfer of spin wave excitation along the chain, with exponential decrease of excitation, and the collective extinction factor has a more physically understandable value FN Æ 1. Another calculation result is that the case of distant transfer of the spin wave is obtainable with resonant conditions 2a¢12 < -1 and q¢¢Æ 0 (Eq. 6.57). Here the distribution of the scattered amplitude  Bˆ j over inclusions will not be described by Eq. 6.59 or 6.60 but will be described by the general expression Eq. 6.22. In Fig. 6.6 we present dependencies of normalized resonance frequency Wres = wres/wH on geometrical parameters R and R12 (Fig. 6.2) of the linear chain when resonance conditions in Eqs. 6.58 and 6.59 for the coupling parameter are satisfied. As usually, wH denotes the ferromagnetic resonance frequency of the ferromagnetic film. All curves in Fig. 6.6 are obtained by numerical solution of the dispersion equation (Eq. 6.22) under the additional condition q¢¢Æ 0 in accordance with the first Eq. 6.58. All calculations were performed for the following material parameters: external magnetic field Hext = 5 kOe, saturation magnetization of the film and inclusions in ferromagnetic materials M0s = 1620 Oe and M1s = 1740 Oe , and film thickness d = 10 mm.

Results

The left boundary linear curve in Fig. 6.6 is related to the case Eq. 6.59. Parameters presented by this curve can be approximately described by the relation (Wres – 1)R12/R = const. Other data points in Fig. 6.6 outside the curve, which represent data for linear decay of scattering amplitudes, depict the case of distant transfer with conditions from Eqs. 6.56 and 6.58. This case is represented by the Fig. 6.7, where scattering amplitudes and collective extinction factors are described by general equations (Eqs. 6.22 and 6.55).

Figure 6.6 Curves presenting the dependence of the resonant frequency on ¢¢ ¢ the inclusion radius R at a fixed ratio R12/R when a12 = 0 and a12 £ -0.5 .

The calculation for the resonant case with small coupling ¢ ¢¢ parameters 2| a12 |< 1 , a12 = 0 shows that the condition q¢¢Æ 0 for the distant transfer cannot be satisfied. This means that partial scattering amplitudes will decrease exponentially (Eq. 6.60), and the signal will not transfer for a big distance. Analyzing Fig. 6.7 we need to note that the collective extinction factor value at N = 23,47 becomes as big as FN ª 103, which is caused by a small value of the sine function in the denominator of the ratio in Eq. 6.22 and consequently leads to big values of scattering amplitudes according to Eq. 6.55. On the other hand, one can see at N = 24, 48 (Fig. 6.7a), that the extinction factor can have values down to FN ª 10–3 due to the same properties of Eqs. 6.22 and 6.55.

191

192

Spin Waves in Circular and Linear Chains of Discrete Magnetic Elements

 Figure 6.7 Illustrations of the behavior of scattering amplitudes Bˆ j of inclusions, depending on the inclusion number j for the case N = 24 (a), N = 23 (b), and collective extinction factor FN (c) of a linear chain under condition of dark-mode filtering from radiation losses.

6.5 Conclusion We presented the quantum-mechanical-type T-scattering operator approach to study FVMSW multiple scattering by a finite ensemble of cylindrical magnetic inclusions in a ferromagnetic film (matrix) metallized from both sides. The substantial result consisted of deriving an optical theorem for the T-scattering operator and as a consequence deriving a new formula for the collective extinction cross section of the inclusion ensemble in terms of the incident spin wave influence on dynamic magnetization displacement currents excited inside inclusions by spin wave scattering. An analogy to the known quantum mechanics scattering theory, the Watson composition rule for the T-scattering operators of particles is formulated in the case of the magnetic inclusion ensemble. This composition rule was used to derive the equation set for partial spin wave multiple-scattering amplitudes. The general results of the T-scattering operator approach were applied to study the Bloch-like spin wave eigenmodes propagation along the inclusion circular array and the distant spin wave excitation transfer along the finite linear array of inclusions. By studying the Bloch-like spin wave eigenmodes we described high-quality-factor-specific micro-spin wave resonator consisting of four magnetic inclusions placed periodically along the circuit. In connection with distant spin wave excitation transfer along a finite linear array of inclusions, we concretized our new formula for the collective extinction cross section of the inclusion ensemble in the

References

case of only the first inclusion of the chain being irradiated by the incident narrow spin wave beam. Such formula showed that directly only the irradiated inclusion makes a contribution to the collective extinction cross section despite the total number of inclusions being big, which makes the direct summing contribution of all another inclusions in spin wave scattering as unviewed (dark mode). We found also a resonant mechanism of filtering the dark mode from radiation losses, which makes the linear chain of magnetic inclusions to be a micro–spin waveguide, which transfers distantly information in the form of the above-mentioned dark mode without radiation losses.

References

1. Stamps, R. L., Breitkreutz, S., Akerman, J., Chumak, A. V., Otani, Y., Bauer, G. E. W., Thiele, J.-U., Bowen, M., Majetich, S. A., Klui, M., et al. (2014). J. Phys. D: Appl. Phys., 47, 333001. 2. Blugel, S., Burgler, D., Morgenstern, M., Schneider, C. M., and Waser, R. (2009). Lecture Notes of the 40th Spring School, Julich, Germany. 3. Zutic, I., and Fuhrer, M. (2005). Nat. Phys., 1, 85.

4. Morris, D., Bromberg, D., Zhu, J., and Pileggi, L. (2012). Int. J. High Speed Electron. Syst., 21, 1250005. 5. Pulizzi, F. (2012). Nat. Mater., 11, 367.

6. Kruglyak, V. V., Keatley, P. S., Neudert, A., Hicken, R. J., Childress, J. R., and Katine, J. A. (2010). Phys. Rev. Lett., 104, 027201. 7. Ding, J., and Adeyeye, A. (2012). Appl. Phys. Lett., 101, 103117.

8. Jorzick, J., Demokritov, S. O., Hillebrands, B., Bailleul, M., Fermon, C., Guslienko, K. Y., Slavin, A. N., Berkov, D. V., and Gorn, N. L. (2002). Phys. Rev. Lett., 88, 047204. 9. Gubbiotti, G., Conti, M., Carlotti, G., Candeloro, P., Fabrizio, E. D., Guslienko, K. Y., Andre, A., Bayer, C., and Slavin, A. N. (2004). J. Phys.: Condens. Matter, 16, 7709. 10. Kruglyak, V. V., Demokritov, S. O., and Grundler, D. (2010). J. Phys. D: Appl. Phys., 43, 264001. 11. Ciubotaru, F., Chumak, A. V., Obry, B., Serga, A. A., and Hillebrands, B. (2013). Phys. Rev. B, 88, 134406. 12. Krawczyk, M., and Puszkarski, H. (1998). Acta Phys. Pol. A, 93, 805.

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13. Nikitov, S. A., Tailhades, P., and Tsai, C. S. (2001). J. Magn. Magn. Mater., 236, 320. 14. Gulyaev, Y. V., Nikitov, S. A., et al. (2003). JETP Lett., 77, 567. 15. Vysotsky, S. L., Nikitov, S. A., et al. (2005). ZhETF, 128, 636.

16. Chumak, A. V., Pirro, P., Serga, A. A., Kostylev, M. P., Stamps, R. L., Schultheiss, H., Vogt, K., Hermsdoerfer, S. J., Laegel, B., Beck, P. A., et al. (2009). Appl. Phys. Lett., 95, 262508.

17. Kostylev, M. P., Serga, A. A., Schneider, T., Neumann, T., Leven, B., Hillebrands, B., and Stamps, R. L. (2007). Phys. Rev. B, 76, 184419.

18. Demidov, V. E., Hansen, U.-H., and Demokritov, S. O. (2007). Phys. Rev. Lett., 98, 157203. 19. Chumak, A. V., Serga, A. A., Hillebrands, B., and Kostylev, M. P. (2008). Appl. Phys. Lett., 93, 022508.

20. Mruczkiewicz, M., Krawczyk, M., Gubbiotti, G., Tacchi, S., Filimonov, Y. A., Kalyabin, D. V., Lisenkov, I. V., and Nikitov, S. A. (2013). New J. Phys., 15, 113023. 21. Mruczkiewicz, M., Pavlov, E. S., Vysotsky, S. L., Krawczyk, M., Filimonov, Y. A., and Nikitov, S. A. (2014). Phys. Rev. B, 90, 174416. 22. Lisenkov, I., Kalyabin, D., and Nikitov, S. (2013). Appl. Phys. Lett., 103, 202402.

23. Shindou, R., Ohe, J.-i., Matsumoto, R., Murakami, S., and Saitoh, E. (2013). Phys. Rev. B, 87, 174402.

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25. Shindou, R., Matsumoto, R., Murakami, S., and Ohe, J.-i. (2013). Phys. Rev. B, 87, 174427.

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31. Gurevich, A. G., and Melkov, G. A. (1996). Magnetization, Oscillations and Waves (CRC Press, New York). 32. Stratton, J. (1941). Electromagnetic Theory (McGraw-Hill, New York).

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Chapter 7

Magnonic Grating Coupler Effect and Microwave-to-Magnon Transducers for Exchange-Dominated Spin Waves Haiming Yua and Dirk Grundlerb aFert Beijing Research Institute, School of Electronic and Information Engineering, BDBC, Beihang University, Beijing 100191, China bLaboratory of Nanoscale Magnetic Materials and Magnonics, Institute of Materials and Institute of Microengineering, School of Engineering, Ecole Polytechnique Fédérale de Lausanne (EPFL), STI-IMX-LMGN, Station 17, Lausanne CH-1015, Switzerland

[email protected]; [email protected]

7.1  Introduction Magnonic crystals (MCs) offer enhanced control of spin waves (SWs) via their artificial SW band structures. Periodic magnetic patterns with lattice constants on the order of the SW wavelength l allow one to tailor eigenfrequencies w = 2pf and group velocities dw/dk for a given wavevector k = 2p/l [1–5]. Bicomponent MCs, composed of two different ferromagnetic materials, have showed backfolding of spin wave (magnon) dispersion relations, formation of minibands, and opening of Spin Wave Confinement: Propagating Waves (2nd Edition) Edited by Sergej O. Demokritov Copyright © 2017 Pan Stanford Publishing Pte. Ltd. ISBN  978-981-4774-35-2 (Hardcover),  978-1-315-11082-0 (eBook) www.panstanford.com

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Microwave-to-Magnon Transducers for Exchange-Dominated Spin Waves

bandgaps at Brillouin zone boundaries (Fig. 7.1). Relevant SW frequencies have been in the microwave regime and up to a few ten gigahertz.



Figure 7.1  Two-dimensional bicomponent magnonic crystal (sketch on the top right) consisting of two materials [5]. The lattice constant is a. Measured (open circles) and simulated (dark gray) mode frequencies as a function of the SW wavevector q in the plane of the MC perpendicular to magnetic field H0 (see inset). There exist forbidden frequency gaps near a Brillouin zone boundary (vertical line) due to the periodic modulation of magnetic properties. On the right side at the bottom two characteristic mode profiles are depicted for the two low-frequency modes: bright color indicates large spin-precession amplitude. Adapted from Tacchi et al. [5].

The control of SWs is at the heart of magnonics [6] and magnon spintronics [7–9]. SWs with small wavelengths are of prime importance when one aims at, e.g., parallel processing in magnetic cellular nonlinear networks (CNNs) [10]. The perspective of novel non-charge-based interconnections with low power consumption and logic devices based on SWs stimulate further developments of spin-based electronics, which has entered partly the international technology roadmap for semiconductors already. However, it still remains a challenge to design on-chip GHz components that act as emitters for short-wavelength plane waves. Here we revisit the design and realization of multidirectional microwave-to-magnon transducers that allow one to emit short-wavelength SWs with relatively large





Introduction

intensity in different spatial directions from conventional GHz antennas [11, 13].

Figure 7.2  Magnonic grating couplers. (Left) One-dimensional device consisting of periodic Py nanowires (green) coated with CoFeB (blue). (Right) 2D lattice consisting of periodic nanotroughs in Py (green). Geometrical parameters such as lateral dimensions a and d as well as thicknesses t1 to t3 are used to tailor the performance. Graphs extracted from figure “Magnonic grating couplers provoking giantly enhanced spin waves,” rearranged from Yu et al. [11], published under CC-BY-NC-SA.

For such microwave-to-magnon transducers, we have fabricated ferromagnetic structures whose magnetic properties are periodically modulated (Fig. 7.2). Short-wavelength SWs are stimulated by an intrinsic magnonic grating coupler effect experimentally evidenced in different periodically patterned ferromagnetic material [11]. The wavelength and propagating direction of the SWs are precisely determined by reciprocal lattice vector G of the periodic structure. Compared to several of the original modes excited by the microwave antenna, the signal strengths of grating coupler–induced modes are largely enhanced. The amplitude enhancement is crucially determined by the magnetic materials, lattice constant of the artificial lattice patterns, and the ferromagnetic resonance of the transducer material [11, 13]. The following sections are organized as follows: in Section 7.2 we outline relevant fabrication steps to realize a magnonic grating coupler that is resonant and operated on yttrium iron garnet (YIG) [13]. In Section 7.3 we highlight parameters for microwave antennas. In Section 7.4 we describe the spectroscopy technique and in Section 7.5 parameters that we evaluated from corresponding spectra. In Section 7.6 we summarize experiments on two different types of grating couplers, i.e., resonant and nonresonant couplers, before, in Section 7.7, we conclude with a summary and outlook.

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 Microwave-to-Magnon Transducers for Exchange-Dominated Spin Waves

7.2  Mix-and-Match Lithography for Mesas with Magnonic Grating Couplers In what follows, we present the processing of a sample starting from thin-film YIG grown by pulsed laser deposition [12]. The substrate consisting of (111) gadolinium gallium garnet was covered by 20 nm thick YIG. To explore spin waves, first mesas were etched from the YIG thin film (Fig. 7.3). For clarity, we focus on the example of CoFeB nanodisks with a diameter of 350 nm arranged as a square lattice with a lattice constant a = 800 nm on a YIG thin film.

Figure 7.3 Process using photolithography and etching: (a) YIG (blue) on the substrate (gray). (b) Photoresist (brown) covering the sample. (c) After exposure. The exposed area (yellow) becomes soluble in the developer. (d) After development. The mesa pattern and markers are transferred to the resist layer. (e) After etching. The ferromagnetic layer has been etched by argon ion milling. (f) After removal of the resist. Markers and mesa remain.

7.2.1  Photolithography to Prepare a Film Mesa

Resist mesas are exposed and the YIG is etched in order to confine spin excitations to a specific region (Fig. 7.3). At the same time, markers are prepared on the sample to align precisely the microwave-to-magnon transducers and coplanar waveguides (CPWs) prepared via electron-beam lithography at a later stage (mix-and-match lithography).

Mix-and-Match Lithography for Mesas with Magnonic Grating Couplers

Before spinning resist on the YIG we heated the sample to about 100°C to dry the surface and enhance the resist’s adhesion. After exposure using a chromium mask and UV contact lithography, we put the sample in the developer. Etching by an argon ion plasma transferred the pattern from the photoresist layer to the ferromagnetic YIG layer. The etching depth amounted to about 40 nm. After removing the remaining resist, we washed the sample with ethyl alcohol and dried it.

7.2.2  Electron-Beam Lithography and Lift-Off Processing for Magnetic Nanostructures

We applied electron-beam lithography (EBL) to create nanostructures with high spatial resolution on the YIG mesa. Here, a focused electron beam locally modified a positive resist that was put on the substrate. We generated a periodic array of holes in the resist that we used for lift-off processing of subsequently sputtered amorphous CoFeB [13]. The detailed process is as follows:

(1) Design of a resist layout with a periodic square lattice of holes and markers. The diameter of the nanoholes is 350 nm and the lattice constant of the square lattice is a = 800 nm. (2) Deposition of a 10 nm thick aluminum layer on the surface. This provides a conductive cover layer for the electronbeam lithography process that follows, thereby preventing charging effects that deteriorate the electron-beam exposure. (3) Spin coating of a two-layer resist and baking. The commercially available electron-beam resist AR-P 639.10 50k is spin-coated with spin speed of 6000 rpm for 120 seconds and baked at 160°C for 3 minutes. Then AR-P 679.04 950k is spin-coated on top and baked with the same condition. The two-layer resist provides undercut for an effective lift-off processing due to the 20% higher sensitivity of the 50k resist compared to the 950k. (4) Electron-beam lithography in that the focused electron beam exposes the lattice of holes. Beforehand, the microscopy mode is used to detect the markers and to align the subsequently exposed pattern accordingly.

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Microwave-to-Magnon Transducers for Exchange-Dominated Spin Waves

(5) Development done by AR-600 56 for 60 seconds. Removing the exposed resist produces a resist mask with a lattice of holes. The sample is put into MF-26 for 45 seconds and water for 30 seconds to remove the aluminum layer. (6) Deposition of 15 nm thick amorphous CoFeB thin film using magnetron sputtering. (7) Lift-off processing done by using remover 1165f. (8) Growth of a thin Al2O3 layer with a thickness of about 4 nm using atomic layer deposition (if isolation between the metallic CPW and the sample is needed).

7.2.3  Integrated Coplanar Waveguide

The integration of CPWs [14] was achieved by electron-beam lithography by making use of the same alignment markers and liftoff processing of about 150 nm thick evaporated gold. To enhance the adhesion, a Cr or Ti layer about 5 nm thin was deposited beforehand in situ. Stitching of different write fields was avoided to obtain straight edges and optimized microwave transmission characteristics.

7.3 Antenna Design for Spin Wave Excitation and Detection

We use CPWs to irradiate the magnetic samples with a radiofrequency magnetic field hrf and detect the spin-precessional motion via induction [15, 16]. The magnetic field accompanying the microwave current that is applied by a microwave source to the CPW’s signal (S) line (Fig. 7.4) excites the spins. At the same time, precessing spins induce a voltage signal in the metallic lead that is guided to a detector by the CPW. This technique is called inductive detection [15, 16]. In the following we highlight parameters that are relevant for the design of a CPW.

7.3.1  Coplanar Waveguide

The coplanar waveguide (CPW) consisting of three metallic lines deposited on a substrate (Fig. 7.4) was first demonstrated by



Antenna Design for Spin Wave Excitation and Detection

C. P. Wen in 1969. The signal (S) line of width ws is surrounded by two ground (G) lines, each with a width wg. The gap width is given by wgap.

Figure 7.4 Cross section of a CPW. The two ground lines with width wg and the signal line with width ws are separated by a gap of width wgap. t is the thickness of the dielectric substrate. Field lines of the oscillating magnetic field surrounding the CPW leads are sketched.



Specifically designed CPWs allow for a small electromagnetic crosstalk between two adjacent CPWs due to the shielding effect of the intermediate ground lines. A small crosstalk is advantageous if spin waves are excited at one CPW and detected at the neighboring CPW (Fig. 7.5). A parasitic direct electromagnetic crosstalk can lead to unwanted interference effects between the electromagnetic wave and the detected spin wave signal.

Figure 7.5  The top view of two neighboring CPWs with ground (G) and signal (S) lines by which a spin wave transmission measurement is performed. The parameter s denotes the center-to-center separation between signal lines.

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Microwave-to-Magnon Transducers for Exchange-Dominated Spin Waves

Following [14] the characteristic impedance Zc of a CPW on a dielectric substrate of finite thickness can be calculated by 30p K (d0 ) , eeff K (d0 )



Zc =



eeff = 1 +

where K is the complete elliptic integral of the first kind and the effective permittivity eeff is given as er –1 K (d0 ) K (d1 ) , 2 K (d0 ) K (d1 )

where er is the relative permittivity and the values dr and dr (r = 0, 1) are given by d0 =



d0 =

ws ; d0 = 1 – d02 ws + wgap

sinh( pws /4t ) ; d  = 1 – d12 sinh{[ p( ws + 2wgap )]/4t } 1

with t as the thickness of the dielectric substrate. According to the formulas the impedance can be optimized by an appropriate ratio of ws and wgap in the case of a given substrate material and thickness of the metal. The width of a CPW can be varied along its length while keeping the characteristic impedance Zc adjusted to the source impedance allowing for optimized microwave transmission characteristics. In our case, ws = 2 μm and wgap = 1.6 μm in the transmission region, and ws = 160 μm and wgap = 100 μm in the pad region. We use wide metal leads in the pad region (Fig. 7.5) to allow microwave probes to land on the CPWs. In the transmission region in which spin waves are launched and detected we use small widths to provide a locally confined field hrf as will be explained below.

7.4  All-Electrical Spin Wave Spectroscopy

For the broadband detection of propagating spin waves, we use an all-electrical spin-wave spectroscopy (AESWS) setup making

All-Electrical Spin Wave Spectroscopy

use of a vector network analyzer (VNA). The VNA can apply and detect microwave signals with different frequencies ranging from 10 MHz to 26.5 GHz. In conventional cavity-based ferromagnetic resonance (FMR) experiments a small sized magnetic sample is placed in the standing wave pattern of a microwave cavity of large volume. Thereby a uniform excitation is possible. For our technique based on integrated CPWs we intentionally excite the magnetic sample only locally. Thereby spin waves of non-zero wavevector k are excited. Depending on their damping (decay) length they can propagate a certain distance and induce a voltage in a remotely positioned detector CPW (Fig. 7.5). Using a two-port VNA we can excite and phase-sensitively detect the propagating spin waves. Due to the phase-sensitive detection it is possible to analyze the real and imaginary part of the scattering parameter S12 as discussed in the following.



Figure 7.6  Sketch of the AESWS setup with a pair of magnet coils applying a field along the CPW. The VNA is connected via microwave cables to tips which have a GSG geometry and connect the microwave source and detector to the CPW. A sample might be mounted in flip-chip configuration to a preexisting CPW or contain an integrated CPW.

The ports of the VNA are connected to semiflexible impedance-matched microwave cables that end in commercial microwave tips (Fig. 7.6). These tips have a ground-signal-ground

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(GSG) geometry which fits to the pad region of the CPW (Fig. 7.5) and make electrical connections accordingly. The microwave tips are held by x,y,z translational stages to ensure precise and rigid positioning. The contacted CPWs plus sample are located between magnet coils which generate an external magnetic field (Fig. 7.6). The whole setup is on a vibration-cushioned table in a climate controlled laboratory.

7.4.1  Scattering Parameters

The scattering parameters Sij are measured by the VNA for all permutations of i and j, where i, j = 1, 2. Here, i stands for the detecting port receiving voltage signal b and j is the excitation port emitting signal a. The VNA data provide us with a 2 × 2 matrix

 b1   S11    b2   S12

S 21 a1    S 22 a2 

that relates emitted microwave signals ai at the two different ports with detected voltage signals bj obtained in either reflection or transmission configuration.

7.4.2  VNA Calibration

The microwave part of the AESWS setup is calibrated before measurements. For this a commercial calibration kit is used consisting of specifically designed microwave elements such as through (T), open (O), short (S), and matching (M) impedance. The TOSM calibration cancels out the mismatch of the used cables and probe tips by shifting the reference plan to the end of the tips. Via calibration the VNA evaluates errors and subtract them from the measured signal. Using a well-designed CPW it is possible to directly display detected spin wave resonances on the VNA monitor. For an uncalibrated setup, the measured spectrum would contain a series of parasitic resonance features that arise from the microwave circuit and obscure the weak spin wave signal.

Spin Wave Properties Studied by Experiments

7.4.3  Measurement Configuration and Data Analysis In the following we discuss spectra obtained for magnetic field H applied in the plane of a thin-film sample. The orientation of the field H is given by the angle q that is measured between H and the normal direction of the signal line of the CPW. Wavevectors kI, kII, … generated by the locally confined excitation field of the CPW are orthogonal with respect to its signal line. Accordingly, for q = 0° the wavevectors kI, kII, … are collinear with H. If, at the same time, the magnetization M is parallel to H, the CPW excites a spin wave in the so-called backward volume magnetostatic spin-wave (BVMSW) configuration. For q = 90° the so-called Damon–Eshbach (DE) mode configuration is realized for which kI, kII, … are perpendicular to M. To increase the signal-to-noise ratio and cancel remaining parasitic signals that do not originate from the magnetic sample we subtract two spectra Sij from each other. The second spectrum is taken for a magnetic field value or angle q at which magnetic resonances are shifted that much that we obtain the magnetic field independent background signal in the frequency region of interest.

7.5 Spin Wave Properties Studied by Experiments 7.5.1  Spin Wave Group Velocity

The group velocity vg of spin waves is derived from dispersion relations w(k) according to

vg =

dw df = 2p dk dk

where f is the frequency in hertz as provided by the VNA. The group velocity is relevant for the propagation speed of a spin wave signal and hence data processing with spin wave–based devices in magnonics. For a transmission signal S12 detected by the VNA between two CPWs, spin waves are excited at the second antenna and propagate to the first antenna through a certain distance s. As

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CPWs can excite spin waves within a distribution of wavevectors around, e.g., kI and kII spin waves with different wavevectors k arrive at the detector antenna when changing f. Due to the phase-sensitive detection of the spin-precession-induced voltage the recorded voltage signal oscillates between positive and negative values as a function of f whenever there is a relative phase change Dj of p [15, 16]. In the analysis of group velocities we therefore focus on oscillatory signals in the real or imaginary parts of S12 spectra. The phase variation j of the individual spin wave along the path depends on its wavevector k and propagation distance s between signal lines according to

j = k . s.

We introduce the frequency variation Df between two neighboring maxima in a spin wave transmission spectrum S12 (compare Fig. 7.8b). From maximum to maximum the phase changes by Dj = 2p. Thereby one can rewrite the group velocity as

vg =

2pdf 2pDf 2pDf Dj = = Df . s  dk Dk Dj Dk

Based on this approach we measured the group velocity in 20 nm thick YIG and obtained about 1.2 km/s at small magnetic field [17]. The group velocity of spin waves in CoFeB was determined to be up to 25 km/s [18] consistent with a Heusler alloy [19]. The large spin wave group velocity makes such materials suitable for magnonic devices even if they exhibit a much larger damping than YIG.

7.5.2  Decay Length and Nonreciprocity Parameter

When analyzing spin wave signal strengths in spectra S12 and S21 taken by a VNA, one can encounter an intensity difference due to nonreciprocity effects. A similar difference can appear when comparing spin wave signal strengths in, e.g., S12 for positive and negative magnetic field values. The nonreciprocity parameter b is defined via signal amplitudes aij as

Performance of a Spin Wave Grating Coupler

a21 a11 b= a21 a12 + a11 a22



where a11, a12, a21, a22 are extracted from VNA spectra taken in transmission and reflection configurations. The nonreciprocity parameter enters when evaluating transmitted amplitudes in terms of the decay length ld [20]:

a21 = ( b )a11e

–

s ld

–

s

, a12 = (1 – b)a22e ld .

Considering this we evaluated the decay length in 20 nm thick YIG according to

ld =

–s .  a21  In   ba11 

Using the extracted decay length ld and group velocity vg, we recalculated spin relaxation times t by

t=

ld . vg

Relaxation times amounted to about 400 to 600 ns in 20 nm thick YIG [17]. The decay length was determined to be up to about 600 μm, which was much longer than in a Heusler alloy (16.7 μm) and CoFeB (23.9 μm) [18, 19].

7.6  Performance of a Spin Wave Grating Coupler

We studied spin wave resonances and spin wave propagation in different 2D bicomponent magnonic grating couplers. We investigated nanodisks of different ferromagnetic materials embedded in an antidot lattice etched into a ferromagnetic thin film (Fig. 7.7), and nanodisks deposited on the surface of a plane film (Fig. 7.8). The nanodisks were arranged on periodic square

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lattices with lattice constants a = 800 nm. In both approaches we detected spin waves propagating with wavevectors kI, kII, … excited directly by the CPW (compare Fig. 7.7a). In addition, spin waves were resolved that corresponded to wavevectors different from kI, kII, …. Analyzing their eigenfrequencies, field dependencies and frequency variations when changing the orientation of the in-plane magnetic field (Fig. 7.7b) we attributed such spin waves to grating coupler modes, i.e., modes with a combination of kI and reciprocal lattice vectors G originating from the periodic nanodisk array (Fig. 7.7c). In Fig. 7.7, one can see a direct comparison between spin waves excited by a CPW on a bare CoFeB thin film and excited via an array of nanodisks positioned between the CPW and the nominally identical CoFeB film [11].



Figure 7.7  Spectra taken on a (a) CoFeB film and (b) a BGC with nanodisks in reflection configuration for different angles q. White dotted lines represent model calculations for the CPW modes kI, kII, and kIII [11]. White squares highlight the resonance frequencies of these three modes at q = 90°. The BGC contained Py nanodisks on a lattice with period a = 800 nm. The red dashed lines in (b) represent calculated angular dependencies for modes with k = kI + G01 (uppermost line) and k = kI – G01 (second-highest line). Modes attributed to the grating coupler effect with reciprocal lattice vectors G that are tilted with respect to G01 are marked with blue arrows. (c) Calculated dispersion relation f (k) of DE modes at 40 mT in a CoFeB plane film for positive and negative k (solid lines). The vertical dashed line indicates kI. The squares highlight frequencies expected for |kI|, |kII|, and |kIII|. Horizontal arrows and dashed curves indicate the back-folding of spin wave branches due to the grating coupler. Graphs extracted from figure “Angular-dependent spin-wave spectroscopy and back-folded dispersion relation,” rearranged from Yu et al. [11], published under CC-BY-NC-SA.

Performance of a Spin Wave Grating Coupler

Figure 7.8 (a) Sketch of the experiment. Large magnetization ferromagnetic nanodisk arrays (green) were positioned between CPWs (yellow) and insulating YIG (violet). The magnetization M of YIG is parallel to the CPWs. (b) Oscillating transmission signal (imaginary (IMG) part) around 7.6 GHz attributed to k = kI + 6G = 48 rad/μm. The corresponding wavelength amounts to 131 ± 3 nm. Graphs extracted from figure “Resonantly driven nanodisks for injection and detection of large-amplitude spin waves in thin YIG,” rearranged from Yu et al. [13], published under CC-BY.

7.6.1  Grating Coupler–Induced Spin Wave Modes

In the spectra Fig. 7.7b, we find numerous additional modes at large frequencies that we interpret as the back-folding of the thin-film dispersion relation f (k) via reciprocal-lattice vectors G (Fig. 7.7c). The high-frequency modes in Fig. 7.7c exhibit a significantly larger signal strength than modes with, e.g., kII and kIII directly excited by the CPW. Depending on the geometry and materials, modes reached a signal enhancement as high as ~20,000% compared to the original CPW excitation at the relevant k. The angular dependence of the grating coupler modes (highlighted by dashed lines in Fig. 7.7b) was similar to the modes excited directly by the CPW at kI, kII, and kIII. The similarity indicated that these two high-frequency modes exhibited wavevectors k being collinear with kI, kII, and kIII. At the same time, the large eigenfrequencies at q = 90° suggested wavevectors k > kIII. Considering the dispersion relation of DE modes, we attributed the two high-frequency modes to two modes that are back-folded to kI in the first Brillouin zone of the periodic lattice using +G01 and –G01 (Fig. 7.7c). The absolute value of a reciprocal lattice vector |G01| = 2p/a amounts to 7.85 rad/μm that is larger than |kI| by about a factor of 8.

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7.6.2  Towards Omnidirectional Spin Wave Emission Not only were magnons excited with reciprocal-lattice vectors that were collinear with kI, but also with other propagating directions. The two modes with wavevectors kI ± G11 and kI ± G1(–1) with |G11|=| G1(–1)| = 2p/a exhibit maximum eigenfrequencies near q = 45° and 135° instead of 90°. Such modes cross at about 15 GHz near 0° and 180° (faintly seen in Fig. 7.7b). Thanks to the 2D grating couplers acting both as emitters and detectors, we perceived even “longitudinal” modes in transmission signals whose wavevectors were nearly parallel to the CPW. Thus, magnonic grating couplers enable one to emit plane-wave SWs in a multitude of in-plane directions by adding (or subtracting) specific reciprocal-lattice vectors from a CPW-induced wavevector k. Using different Bravais lattices, different lattice constants and differently wide CPWs spin waves might be emitted in almost arbitrary directions. This is interesting when aiming at complex integrated magnonic circuits.

7.6.3  Enhanced Magnon Excitation via Resonant Nanodisks

Note that we discussed so far nonresonant grating couplers for which the FMR of the nanodisks was below the eigenfrequencies of excited spin waves. In the following we revisit a nanodisk array that was prepared from a metallic ferromagnet with large saturation magnetization MS such as Py or CoFeB on top of a 20 nm thin YIG film (Fig. 7.8). YIG possesses a saturation magnetization that is a factor of about 5(9) smaller compared to Py (CoFeB) [13]. The FMR eigenfrequency f0 of a thin YIG film is much smaller compared to a Py and CoFeB thin film assuming the same film thickness and in-plane magnetic field H. In a field regime where short-wavelength spin waves in YIG became degenerate with the field-dependent FMR of the nanodisks, we found that the transmitted spin-wave signal was pronounced (Fig. 7.8b) and significantly enhanced (by about factor of 6 in case of Py) compared to the signals measured for the nonresonant grating coupler. Au et al. [21] argue that the dynamic stray field hdip of precessing spins in a nanomagnet-based transducer locally enhances the

Performance of a Spin Wave Grating Coupler

microwave magnetic field hrf and enlarges the torque acting on the magnetic moments in a neighboring film. Following their modeling, the enhancement of hrf at f0 via hdip depends on materials properties according to (2a(1 + 2H/Ms))–1 where a denotes the Gilbert damping parameter of the given transducer material. Slightly larger signal amplitudes in YIG observed for CoFeB nanodisks were attributed to the larger saturation magnetization and smaller damping parameter of CoFeB compared to Py nanodisks [13].

7.6.4  Sub-100 nm-Wavelength Spin Waves

As CoFeB exhibits a large saturation magnetization MS [18], the field-dependent FMR frequency for the saturated CoFeB nanodisks was high and underwent crossings with spin waves in YIG that had large wavevectors k. We observed a pronounced propagation signal at a frequency consistent with k = kI + 9G = 71 rad/μm. The wavelength l corresponded to 88 ± 2 nm [13]. The eigenfrequencies f attributed to different grating coupler modes followed a dependence f  k2 which suggested exchangedominated spin waves. Note that the original wavelength of electromagnetic waves at the same frequency in free space was on the order of cm, i.e., larger than l = 2p/k by more than a factor of 105. The magnonic grating coupler thus allowed to shrink the wavelength of a microwave signal enormously. The relative signal strength at large k amounted to up to 38 % of the signal strength at kI.

7.6.5  Angular Dependance of Propagating Grating Coupler Modes

In addition to field-dependent measurements at fixed angle as shown in Ref. [13], we also conducted measurements, in which we fixed the field strength at 50 mT and rotated the field direction. Figure 7.9 shows the experimental data measured on a YIG plain film (top row) as well as YIG film with Py nanodisks on top (bottom row). Both reflection spectra S11 (left) and transmission spectra S12 (right) are depicted. We assume that H is large enough that the YIG film and nanodisks are saturated for the different

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field directions. DE modes are pronounced at –90°, 90°, and 270°. At these angles, several mode branches can be resolved in Fig. 7.9 (top row) consistent with Fig. 7.7a. At 0° and 180°, mainly one pronounced BVMSW mode branch is detected.

Figure 7.9  Angular dependence of spin wave modes in YIG film (top row) and YIG with a Py nanodisk array (bottom row). The applied field value is 50 mT. Zero degree is defined as field parallel to kI. Horizontal lines are due to subtraction of a reference spectrum. White arrows indicate grating coupler modes of weak signal-to-noise ratio at high frequencies.

The spectra become richer in Fig. 7.9 (bottom row) when the nanodisk array is present. As discussed, spin waves are detected up to higher frequencies compared to the bare YIG film. These high-frequency modes are due to the grating coupler effect. With angular dependence measurements, we follow how such modes transform from the DE mode configuration into BVMSW mode. Some of the modes have “frequency maxima” near 45°

Conclusions and Outlook

or 30°, which can be understood as a result of grating coupler modes such as kI + G11 or kI + G12, respectively. In a specific angle regime we observed weak transmission signals S12 around 12 GHz from which we estimated the so far shortest wavelength l of 68 ± 1 nm. The angular dependence of the main branch related to kI is modified in the grating coupler sample compared to the bare thin film sample (Fig. 7.9, at the bottom on the left). We detect a splitting of modes near, e.g., +/–45°. This observation might indicate that the band structure of the YIG is modified in the presence of the nanodisk array similar to a bicomponent magnonic crystal [5]. We now relate the grating coupler effect to some other schemes that can excite spin waves with short wavelengths. Among these schemes there are parametric pumping [22], wavelength conversion [23], spin-transfer torque nanocontact oscillators (NCO) [24, 25], magnetic vortices [26], nanostructured antennas [27, 28], and edge excitation of a ferromagnetic thin film [29]. NCOs and grating couplers form opposing limiting cases in that NCOs represent point-like emitters, whereas grating couplers allow for plane waves. The research fields of nanomagnonics [9, 30] and magnon spintronics [8] certainly benefit from the recent developments concerning the different spin wave emitters.

7.7 Conclusions and Outlook

Taking advantage of nanofabrication technology and material science, spin waves with wavelengths l below 100 nm were excited in magnetic thin films using conventional coplanar waveguides. This was achieved via the resonant magnonic grating coupler effect. By further optimization (via nanodisks made from ferrites with higher FMR frequency and reduced lattice parameters), spin waves with l of about 20 nm might be excited, i.e., in a wavelength regime known from electromagnetic waves in the X-ray regime. Combined with reconfigurable magnonic crystals one expects advanced wave control in solids that promises a compelling future for nanomagnetic devices using spin waves for computing and information processing.

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Acknowledgments We thank S. Tacchi, G. Duerr, R. Huber, M. Bahr, T. Schwarze, F. Brandl, M. Madami, S. Neusser, G. Gubbiotti, G. Carlotti, O. d’ Allivy Kelly, V. Cros, R. Bernard, P. Bortolotti, A. Anane, F. Heimbach, and Y. Stasinopoulos for their contributions. The experimental work received funding from the Deutsche Forschungsgemeinschaft (DFG) via the German excellence cluster Nanosystems Initiative Munich (NIM), as well as project GR1640/5-1/2 in the framework of the DFG Priority Program Spincaloric Transport Phenomena SPP 1538. We thank the following colleagues for their help in editing the text: Che Ping, Tobias Stueckler, Sa Tu, Chuanpu Liu, Junfeng Hu, Jilei Chen, Jianyu Zhang, Xing Chen, and Yinan Wang.

References

1. Kostylev, M., Gubbiotti, G., et al. (2007). Partial frequency band gap in one-dimensional magnonic crystals, Appl. Phys. Lett., 92, 132504. 2. Wang, Z. K., Adeyeye, A. O., et al. (2009). Observation of frequency band gaps in a one-dimensional nanostructured magnonic crystal, Appl. Phys. Lett., 94, 083112.

3. Chumak, A. V., et al. (2009). A current-controlled, dynamic magnonic crystal, J. Phys. D: Appl. Phys., 42(20), 205005. 4. Kruglyak, V. V., et al. (2010). Imaging collective magnonic modes in 2D arrays of magnetic nanoelements, Phys. Rev. Lett., 104, 027201.

5. Tacchi, S., et al. (2012). Forbidden band gaps in the spin-wave spectrum of a two-dimensional bicomponent magnonic crystal, Phys. Rev. Lett., 109, 137202. 6. Kruglyak, V. V., Demokritov, S. O., and Grundler, D. (2010). Magnonics, J. Phys. D: Appl. Phys., 43, 264001. 7. Nikitov, S. A., et al. (2015). Magnonics: a new research area in spintronics and spin wave electronics, Phys. Usp., 58, 1002–1028.

8. Chumak, A. V., Vasyuchka, V. I., Serga, A. A., and Hillebrands, B. (2015). Magnon spintronics, Nat. Phys., 11, 453–461. 9. Grundler, D. (2016). Nanomagnonics, J. Phys. D: Appl. Phys., 49, 391002.

References

10. Khitun, A., Bao, M., and Wang, K. L. (2010). Magnetic cellular nonlinear network with spin wave bus for image processing. Superlattices Microstruct., 47, 464–483.

11. Yu, H., et al. (2013). Omnidirectional spin-wave nanograting coupler, Nat. Commun., 4, 2702. 12. Kelly, O. A., et al. (2014). Inverse spin Hall effect in nanometerthick yttrium iron garnet/Pt system, Appl. Phys. Lett., 103, 082408.

13. Yu, H., et al. (2016). Approaching soft X-ray wavelengths in nanomagnet-based microwave technology, Nat. Commun., 7, 11255.

14. Simons, R. N. (2001). Coplanar Waveguide Circuits, Components, and Systems (Wiley-Interscience John Wiley & Sons, Inc., New York, NY, USA). 15. Silva, T. J., Lee, C. S., Crawford, T. M., and Rogers, C. T. (1999). Inductive measurement of ultrafast magnetization dynamics in thin-film Permalloy, J. Appl. Phys., 85, 7849–7862.

16. Bailleul, M., Olligs, D., Fermon, C., and Demokritov, S. O. (2001). Spin waves propagation and confinement in conducting films at the micrometer scale, Europhys. Lett., 56, 741. 17. Yu, H., et al. (2012). Magnetic thin-film insulator with ultra-low spin wave damping for coherent nanomagnonics, Sci. Rep., 4, 6848.

18. Yu, H., et al. (2012). High propagating velocity of spin waves and temperature dependent damping in a CoFeB thin film, Appl. Phys. Lett., 100, 262412. 19. Sebastian, T., et al. (2012). Low-damping spin-wave propagation in a micro-structured Co2Mn0.6Fe0.4Si Heusler waveguide, Appl. Phys. Lett., 100, 112402.

20. Huber, R., Krawczyk, M., Schwarze, T., Yu, H., Duerr, G., Albert, S., and Grundler, D. (2013). Reciprocal Damon-Eshbach-type spin wave excitation in a magnonic crystal due to tunable magnetic symmetry, Appl. Phys. Lett., 102, 012403. 21. Au, Y., et al. (2012). Resonant microwave-to-spin-wave transducer, Appl. Phys. Lett., 100, 182404. 22. Sandweg, C. W., et al. (2011). Spin pumping by parametrically excited exchange magnons, Phys. Rev. Lett., 106, 216601. 23. Demidov, V. E., et al. (2011). Excitation of short-wavelength spin waves in magnonic waveguides, Appl. Phys. Lett., 99, 082507. 24. Madami, M., et al. (2011). Direct observation of a propagating spin wave induced by spin-transfer torque, Nat. Nanotechnol., 6, 635–638.

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25. Bonetti, S., et al. (2015). Direct observation and imaging of a spinwave soliton with p-like symmetry, Nat. Commun., 6, 8889.

26. Vlaminck, V., and Bailleul, M. (2010). Spin-wave transduction at the submicrometer scale: experiment and modeling, Phys. Rev. B, 81, 014425.

27. Che, P., Bi, L., Yu, H., et al. (2016). Short-wavelength spin waves in yttrium iron garnet micro-channels on silicon, IEEE. Mag. Lett., 7, 3508404.

28. Wintz, S., et al. (2016). Magnetic vortex cores as tunable spin-wave emitters, Nat. Nanotechnol., 11, 948–953. 29. Davies, C. S., and Kruglyak, V. V. (2016). Generation of propagating spin waves from edges of magnetic nanostructures pumped by uniform microwave magnetic field, IEEE Trans. Magn., 52, 2300504.

30. Grundler, D. (2016). Spintronics: nanomagnonics around the corner, Nat. Nanotechnol., 11, 407–408.

Chapter 8

Spin Waves on Spin Structures: Topology, Localization, and Nonreciprocity Robert L. Stamps,a Joo-Von Kim,b Felipe Garcia-Sanchez,c Pablo Borys,d Gianluca Gubbiotti,e Yue Li,a and Robert E. Camleyf aSUPA

School of Physics and Astronomy, University of Glasgow, UK for Nanoscience and Nanotechnology, Université Paris-Saclay, France cIstituto Nazionale di Ricerca Metrologica, Turin, Italy dRIKEN Center for Emergent Matter Science, Tokyo, Japan eIstituto officina dei Materiali, CNR, Perugia, Italy fDepartment of Physics and Energy Science, University of Colorado at Colorado Springs, Colorado, USA bCentre

[email protected]

8.1  Introduction Spintronics exists because of an extra degree of freedom provided by electron spin that can be used for carrying information. Whereas information can be manipulated and transported using charge, this comes at a cost due to the concurrent and inherent generation of Joule heating. An alternative mechanism for transporting information through the spin variable is available and, in fact, has been studied for over 80 years. Spin waves and their particle-like counterpart, magnons, are the low-lying energy Spin Wave Confinement: Propagating Waves (2nd Edition) Edited by Sergej O. Demokritov Copyright © 2017 Pan Stanford Publishing Pte. Ltd. ISBN  978-981-4774-35-2 (Hardcover),  978-1-315-11082-0 (eBook) www.panstanford.com

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states of spin systems and were first predicted by Bloch [4, 25, 38, 43]. Not only do spin wave excitations exhibit a wide variety of linear and nonlinear properties (which makes them interesting for fundamental research), they also exist in the gigahertz to terahertz region of the frequency spectrum, which is appropriate for telecommunications and information technologies. New technologies that allow the fabrication of devices in the nanoscale together have led to the discovery of phenomena such as spin pumping [88], spin transfer torque [2, 80], and spin Hall effects [24, 42]. The field now called magnonics is concerned with consequences of the fact that the transport and processing of information can be achieved without physical charge transport. Challenges addressed in this field pertain to issues related with spin wave dissipation, device miniaturization [82], and fabrication of artificial magnonic crystals [48–50]. Most recently, consequences of the Dzyaloshinskii–Moriya interaction (DMI) on spin wave properties has been studied extensively, especially in regards to interface-induced DMI. The DMI arises in low-symmetry materials with a strong spin-orbit coupling and is modeled as an antisymmetric form of the exchange interaction. Dzyaloshinskii first postulated this interaction in order to explain weak ferromagnetism in antiferromagnets [26]. A few years later Moriya calculated the second-order energy terms associated with spin-orbit couplings for the exchange interaction, thereby establishing a mechanism for the interaction [68, 69]. In noncentrosymmetric magnetic crystals the DMI is responsible for the spontaneous formation of helicoidal and skyrmionic structures [6, 7]. From the viewpoint of applications, the most exciting recent development has been experimental demonstration that an interface form of the DMI can appear because of inversion symmetry breaking at the surface between magnetic films on heavy metal nonmagnetic substrates that provide the spin-orbit coupling. Experiments have shown that this induced form of the DMI leads to chiral spin structures in manganese monolayers on top of tungsten [5] and skyrmion lattices in iron monolayers on iridium [37]. This interfacial DMI also exists for sputtered multilayer films such as Pt/Co/Ni [23], Pt/Co/AlOx [1], Pt/Co/Ir [66], Pt/Co/MgO [9], and (TaN, Hf, W)/CoFeB/MgO [81], which have strong perpendicular magnetic anisotropy. This form of DMI has helped explain

Introduction

puzzling observations obtained for domain wall dynamics. The DMI stabilizes a Néel-type domain wall with a given handedness [15, 36, 86] with significant consequences on domain wall mobilities. The mobilities are now understood in terms of the interfacial DMI [87]. The interfacial DMI is of particular interest for magnonics. Udvardi and Szunyogh in 2009 suggested the possibility that the spin wave chiral degeneracy (resulting from the isotropic part of the exchange) could be lifted in the presence of the DMI [90]. From a first-principles calculation they found an asymmetric magnon dispersion for a Fe monolayer on tungsten for a certain direction of propagation that was explained by the presence the DMI. Shortly thereafter, Zakeri et al. demonstrated a DMIdriven asymmetry in the spin wave dispersion using spin-polarized electron energy loss spectroscopy on a Fe double layer grown on tungsten [100]. Theoretical studies suggested that DMI-induced nonreciprocity should exist [17, 18, 54, 65], and inelastic light scattering studies provided evidence for the nonreciprocal dispersion phenomenon and have been used to obtain measures of its strength [1, 23, 81, 83]. Localized spin wave modes have been studied for many years and are particularly important for thin-film geometries and inhomogeneous magnetic configurations. In particular, Winter calculated spin wave properties for propagation along a Bloch domain wall in the context of nuclear magnetic resonances in 1961 [98] and outlined the properties of a walllocalized mode that appears in the modified spin wave dispersion. An unpinned wall supports a mode with zero energy for propagation perpendicular to the plane of the wall and a quadratic gapless dispersion for propagation parallel to the plane of the wall in which there is no spatial variation of the static magnetization. We discuss in this chapter the propagation of spin waves along domain walls and the consequences of the DMI on their dispersion. We also discuss how the DMI affects the gap between the energies of freely propagating spin waves and the spin waves channeled along walls, as well as consequent nonreciprocities. These features provide the essential ingredients for a new type of application whereby domain walls are used to guide and control the flow of spin wave information.

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In the second half of this review, we discuss a possibility for creating a mesoscopic metamaterial analogue of domain wall channeling. The idea in this case is very different and relies upon a new emerging technological concept sometimes referred to as artificial spin ice (ASI). Artificial magnetic spin ice is an arrangement of interacting nanomagnets with emergent collective magnetic properties. A straightforward and well-studied example is square ASI, wherein elements are arranged such that the dipolar interactions result in a type of antiferromagnetic alignment. The single-domain magnetic elements in these structures can spontaneously order into two sublattice arrays of alternating magnetic orientations on a two-dimensional square lattice. Relatively simple alterations of the array geometry can be made to produce other types of ordering or create frustration through competing interactions as occurs in spin glasses. Static magnetic configurations in ASI can be manipulated through application of magnetic fields. We review how the magnetic configuration determines many details of the allowed microwave frequency excitation spectra. The individual ferromagnetic single-domain elements from which ASI is constructed have resonances in the microwave region. We discuss aspects of ongoing work aimed at using magnetic configurations in spin ice to manipulate microwave resonances. Ultimately it may be possible to employ ASI as reconfigurable magnonic crystals, a magnetic analogy to photonic crystals, where new and useful dynamic properties emerge by patterning magnetic thin films. We illustrate this idea using results from micromagnetic simulations for a type of configurable microwave resonator in a square ice geometry. The configurability arises from resonances associated a spin ice analogue to a microscopic domain wall.

8.2  Chiral Interactions and Spin Waves

As discussed in the introduction, the presence of the DMI leads to nonreciprocal propagation of spin waves. In this section, we discuss two specific consequences of this interaction, namely the appearance of an underlying drift current in certain geometries and focusing effects such as caustics.

Chiral Interactions and Spin Waves

8.2.1  Nonreciprocity: Symmetry Breaking through the DMI To understand how the nonreciprocal behavior comes about, it is useful to consider the magnetization dynamics in the micromagnetics approximation. Let m(x, t) be a unit vector that describes the magnetization in time and space. The basic equation of motion governing the dynamics is the Landau–Lifshitz–Gilbert equation,

dm dm , = – gm0m × Heff + am × dt dt



Heff = –



d dm = – gm0 (m0 × dHeff + dm × Heff,0 ). dt

(8.1)

where a is a dimensionless damping parameter, g is the gyromagnetic constant, and m0 is the permeability of free space. Heff (x, t) is the local effective field and is given by the variational derivative of the magnetic energy, U, with respect to the magnetization vector, 1 dU , m0 MS dm

(8.2)

where MS is the saturation magnetization. The energy contains contributions from the exchange interaction, dipole-dipole interaction, magnetocrystalline anisotropies, and the DMI. The spin wave dispersion relation can be obtained by linearizing the equation of motion about the equilibrium magnetic configuration. Let m(x, t) = m0(x) + dm(x, t), where m0 represents the static equilibrium state and dm represents the fluctuations. This generates corresponding terms in the effective field, Heff = Heff,0 + dHeff. Neglecting the damping term, the linearized form of the dynamics is obtained by keeping terms that are linear in the fluctuations,

(8.3)

For the interfacial form of the DMI, the largest nonreciprocities occur in the Damon–Eshbach (DE) geometry, which describes the configuration in which the magnetization lies in the film plane and where spin waves propagate in the film plane in a direction

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perpendicular to the magnetization. For an ultrathin film of thickness d, the dispersion relation is given by [53]

w( k )= w||( k )w( k ) – 2gDk x / MS ,

(8.4)



w||( k )= w0 + wex (k )+ gm0 MSdk x2/2k ,

(8.5)



w( k )= w0 + wex (k )– wK – gm0 MSdk/2k.

(8.6)

where and

Here, w0 = gm0H0 is the Zeeman term, where H0 is the applied field in the film plane along the y axis that saturates the magnetization in this direction. wex = 2gAk2/MS is the contribution from the exchange interaction, where A is the exchange constant. wK = gm0HK is the contribution from the effective perpendicular anisotropy, with 2K0/m0MS and K0 = Ku – m0MS2/2 being the anisotropy field and constant, respectively. From Eq. 8.4, one can immediately deduce the nonreciprocal propagation introduced by the linear term in kx, where D is the strength of the DMI. Note that propagation is reciprocal along the y direction, parallel to the magnetization. Another interesting consequence of the shifted dispersion relation can be seen in Fig. 8.1, where results of micromagnetics simulations of the transient response of the dynamical magnetization to a pulsed field excitation are shown. The opensource code MuMax3 [93] was used to perform these simulations on a 40 μm × 40 μm × 1 nm thick film. A magnetic field 5% larger than the effective anisotropy field is applied to saturate the magnetization along the y direction. The simulations were used to compute the transient magnetization response to a 5 GHz sinusoidal field excitation applied for one period [53]. After the application of the field pulse, we observe a ripple structure that represents spin waves radiating outward from the excitation source, a common feature of wave phenomena like the water ripples observed after a pebble is thrown into a pond. An important feature for the DMI case is that the center

Chiral Interactions and Spin Waves

of the ripple is observed to drift along the –x direction as its size grows, which can be seen from the snapshots in Fig. 8.1a taken at different instants after the field pulse. In Fig. 8.1b, the displacement of this ripple is shown as a function of time after the application of the field pulse for different values of the DMI constant. The drift velocity of the ripple depends on D, where the lines indicate the expected displacement given by the drift part of the dispersion relation, that is, vdrift = дwdrift/дkx = wdrift/kx = –2gD/MS, which represents the component for which the phase and group velocities are identical. The simulated displacement of the ripple agrees very well with this equation, which indicates that the DMI leads to an underlying drift in the spin wave propagation in the DE geometry. This is consistent with the recent proposal that the DMI can be interpreted in terms of a Doppler shift by an intrinsic spin current [51].

Figure 8.1  DMI-induced drift of a spin wave ripple. (a) Time evolution of a spin wave ripple at three instants (2, 4, and 8 ns) after the application of a sinusoidal field pulse at the center of the image, with D = 1 mJ/m2. The image dimensions are 10 μm × 10 μm. Dx denotes the displacement of the ripple center. (b) Ripple displacement as a function of time for three different values of D. Symbols correspond to simulation data while solid lines are based on Eq. 8.4. Reprinted with permission from Kim et al. [53]. Copyright 2016 American Physical Society.

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8.2.2  Caustics The DMI-induced drift in spin waves has interesting consequences for power flow. With an interfacial DMI and for propagation in the DE geometry, the dispersion curve is approximately parabolic but with the minimum shifted away from the origin along the wavevector axis. Because of this, dw/dk is negative in some regions, and this indicates the group velocity is opposite to the phase velocity. However, this simple analysis is not sufficient to capture all the important features of the anisotropic power flow created by the DMI. The study of focusing patterns for bulk [85] and surface phonons [13] in crystals is well known. The corresponding investigations in thick-film magnetic systems have begun only recently with both experimental [19, 22, 78, 79] and theoretical results [94]. The focusing results have already shown remarkable behaviors, including focusing effects of energy well below the expected diffraction limit and an interesting reflection behavior for energy where the angle of incidence is not equal to the angle of reflection. In many respects the magnetic system offers richer phenomena because the external magnetic field offers an opportunity to tune the dispersion relations and alter the focusing patterns, something that is not available in phonon focusing. Let us discuss how this drift leads to focusing effects and caustics. In general, the far-field radiation pattern of waves excited by a point source can be predicted from the slowness surface, that is, a constant frequency curve in k-space. The radiation or focusing pattern can then be determined from the power flow, directed along the normal to the slowness surface, and with an amplitude that is inversely proportional to the square root of the curvature of the slowness surface [94]. Caustics appear at points along the slowness surface at which its curvature goes to zero, resulting in a divergence in the power flow. To understand how caustics appear for spin waves in ultrathin films with an interfacial DMI, let us return to the dispersion relation in Eq. 8.4 from which the slowness surfaces can be computed. There are three main contributions to the spin wave energy that are wavevector dependent. First, the exchange term, wex  k2, gives a circular component to the slowness surface that results in a finite and positive curvature for all propagation directions in the film plane. As such, radiation of spin wave

Chiral Interactions and Spin Waves

power from an excitation point source is isotropic with only the exchange term. Second, the DMI results in simple displacement of the slowness surface in wavevector space but does not influence the curvature in any way. This results in the overall drift of excitation patterns, as discussed in the previous section. Third, the dipole-dipole interaction not only leads to elliptical precession (w|| ≠ w) but also anisotropic propagation in the film plane with respect to the magnetization orientation. The combination of these three terms leads to nontrivial spin wave flow in ultrathin films. Some examples are shown in Fig. 8.2, where the slowness surface and focusing patterns are presented at five different frequencies for a 2-nm-thick film. In Fig. 8.2a, the slowness surface for each frequency is shown, where elements of the three interaction terms can be seen. The group velocity vectors are also indicated along each slowness surface. The expected focusing patterns are shown in Fig. 8.2b, computed from the curvature of the slowness surface in Fig. 8.2a. For the lowest frequency considered (4.2 GHz), a caustic can be seen for spin wave propagation in the –x direction, which is a consequence of the flattening on the left part of the slowness surface. As the frequency is increased to 5 and 6 GHz, a dent develops in the slowness surface, leading to two caustics propagating outward in the −x direction. The presence of the dent leads to the curvature vanishing at two points along the slowness surface, resulting in the two focused beams predicted. As the frequency is further increased, the dent vanishes and a single caustic is recovered at 6.5 GHz. For higher frequencies, the exchange terms become dominant in the dispersion relation and the slowness surfaces recover a more circular shape, resulting in smaller focusing effects, as seen for 7.0 GHz. The predicted focusing patterns can be compared with results from micromagnetics simulations, with which the spin wave power flow from a point source excitation can be computed. The same geometry as for Fig. 8.1 is considered, but instead the response to a continuous sinusoidal point source field excitation is computed. In Fig. 8.2c, the spin wave power is presented for five difference excitation frequencies, which is computed by averaging the z component of the dynamic magnetization, dmz(x, t)2, following the application of the field excitation. The excitation frequencies used in the simulations were chosen to match as

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closely as possible the focusing patterns predicted from the dispersion relation (Fig. 8.2b). While the agreement in the frequencies is only semiquantitative, the simulations reproduce well the different focusing patterns predicted, namely the orientation and trends in the different caustics as the excitation frequency is increased. The discrepancy is likely due to the local approximation used for the dipolar interaction in Eq. 8.4. Nevertheless, relatively good agreement between the simple analytical theory and full micromagnetics can be achieved.

Figure 8.2 Spin wave power flow and caustics. (a) Slowness surfaces for different frequencies determined from Eq. 8.4. vg denotes the group velocity vector. (b) Predicted focusing patterns based on the slowness surfaces in (a). (c) Simulated focusing patterns due to a sinusoidal point source excitation at different frequencies. Each image represents an area of 20 μm × 20 μm, and the point source is located at the center. The frequencies are chosen to match the focusing patterns in (b). Reprinted with permission from Kim et al. [53]. Copyright 2016 American Physical Society.

Chiral Interactions and Spin Waves

Figure 8.3 Interference patterns. (a) Slowness surfaces for two frequencies determined from Eq. 8.4. vg denotes the group velocity vector. (b) Wavevector k as a function of the direction of vg for the slowness surfaces in (a). The shaded regions denote propagation directions for which several k are possible. In the top inset, a schematic real space representation of propagation directions along which interference is expected, where the numbers indicate the number of allowed k. (c) Simulated interference patterns due to a point source excitation at different frequencies. Each image represents an area of 5 μm × 5 μm, with the point source located at the center. The frequencies are chosen to match the interference patterns expected from (b). Reprinted with permission from Kim et al. [53]. Copyright 2016 American Physical Society.

Another remarkable feature of Eq. 8.4 is the possibility of generating interference patterns from a single point source. Some evidence of interference can already by seen in Fig. 8.2c for the excitation frequencies of 4.7 and 5.2 GHz in the region bounded by the two focused beams. To illustrate how such effects arise, we consider an example in which the dent in the slowness surface evolves into two distinct surfaces between 5.7 and 5.8 GHz, as shown in Fig. 8.3a. Consider first the response to the excitation at 5.7 GHz, which results in a C-shaped slowness surface. If we examine how the

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group velocity vector, vg, evolves around this surface, we notice that certain orientations of vg appear at multiple points along this surface, which indicates that propagation along these directions can involve partial waves with different wavevectors. To see this more clearly, we plot in Fig. 8.3b the magnitude of the wavevector k as a function of the angle of vg with respect to the kx axis (in the film plane), vg ,f, for the two excitation frequencies considered. For 5.7 GHz, a range of propagation angles can be identified in which three values of the wavevector k are allowed, while only a single k is allowed outside this range. This is illustrated schematically above the graph in Fig. 8.3b, which suggests that three-wave interference should only be seen for propagation near the –x direction, while no interference is expected for propagation along +x. This picture is verified in micromagnetics simulations at a similar excitation frequency of 5.56 GHz, where interference is mostly localized to the x < 0 region. On this basis, the existence of two slowness surfaces for 5.8 GHz (Fig. 8.3a) should result in interference for all propagation directions. By following a similar analysis, four-wave interference is expected within a narrow range of propagation angles about the –x direction, while two-wave interference for all other directions (Fig. 8.3b). This feature is confirmed in simulation at an excitation frequency of 5.66 GHz, where one can distinguish two different interference patterns with the expected angular dependence. Such caustic beams and interference patterns, induced by an interfacial DMI, could be useful for magnon-based computation and memory [49, 50, 59] and for exploring magnetic analogs of wave phenomena seen in other physical systems such as vortices in electron optics [73].

8.3  Localization and Reconfigurability

In this section, we discuss spin wave localization and channeling effects due to spin textures, such as magnetic domain walls and tilted-edge magnetization states in nanostructured films and in ASI. These localized excitations can potentially allow spin waves to be guided efficiency in thin films without any lithography and provide schemes for reconfigurable magnonic

Localization and Reconfigurability

circuits by virtue of using different domain structures and magnetization states.

8.3.1  Domain Wall Channeling

The basic principle of the domain wall magnonic waveguide is illustrated in Fig. 8.4 for a thin rectangular ferromagnetic wire with dimensions of 1000 nm × 250 nm × 1 nm and a perpendicular magnetic anisotropy along the z axis. A domain wall separates two uniformly magnetized up and down states with the wall axis along y, which is perpendicular to the wire axis x. The spin waves considered are associated with localized domain wall eigenmodes that propagate along the x direction, parallel to the domain wall. In the following, we discuss results of micromagnetics simulations in which these modes are driven by a local excitation field hrf. With the isotropic exchange interaction, uniaxial anisotropy, and dipole-dipole interactions, the Bloch-type domain wall minimizes the volume dipolar interaction and it is characterized by moments that rotate in a plane (xz) perpendicular to the wall direction (y). For this wall type, there exists a family of spin wave eigenmodes, known as Winter modes, that are localized in the direction perpendicular to the domain wall (y) on a length scale l but propagate as plane waves parallel to the domain wall (x) [98]. Here l = A / K 0 is the characteristic wall width parameter. A particular feature of these modes is that they are gapless, in contrast to the bulk spin wave modes that are quadratic in wavevector with a gap at zero wavevector defined by the perpendicular anisotropy energy (Fig. 8.4b). For a microwave field excitation in the frequency gap of the bulk modes, only the localized Winter modes are excited and are effectively channeled along the domain wall center (Fig. 8.4c; excitation at 10 GHz), which acts as a local potential well for the spin waves. The wavelength at 10 GHz is approximately 60 nm, which means there is subwavelength confinement in both the film thickness (1 nm) and across the width of the domain wall (~18 nm); such localized modes therefore represent true one-dimensional propagation of spin waves. When the microwave field is applied in the frequency band of the bulk modes, the channeling is preserved whereby the localized modes can be seen to propagate

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with a higher wavevector than the bulk modes (Fig. 8.4c; excitation at 50 GHz). hrf

(a)

DW y z

x

– 0.01

0

0.01

W / 2p (GHz)

50

50 GHz

bulk

40 30

(c)

20

wall

10

10 GHz

(b) 0 –0.2

0

0.2

kx (nm-1) 50

W / 2p (GHz)

232

50 GHz

bulk

40 30

(e)

20

wall 10 GHz

10

(d) 0 –0.2

0

0.2

kx (nm-1)

Figure 8.4  Spin wave channeling in domain walls (DWs). (a) Geometry for channeling along the center of the wall, where a radio-frequency antenna generating an alternating field, hrf, excites spin waves that propagate along the x direction. (b, d) Dispersion relation for channeled Bloch (b) and Néel (d) domain wall spin wave modes in comparison with bulk spin waves. For the Néel wall case, D = 1.5 mJ/m2. (c, e) Simulation results of propagating modes for excitation field frequencies in the bulk (50 GHz) and in the gap (10 GHz) for Bloch (c) and Néel (e) walls. These driving frequencies are shown as dashed lines in (b, d). Reprinted with permission from Garcia-Sanchez et al. [28]. Copyright 2015 American Physical Society.

Above a critical value of the DMI [36, 87], Néel domain walls are favored energetically over Bloch walls. For example, left-handed Néel walls are favored at equilibrium in the asymmetric Pt/Co/AlOx multilayer [86], which possesses a strong interfacial DMI [1]. The moments in this wall type rotate in a plane (yz) parallel to the wall direction (y), which leads to an increase in the volume dipolar interaction but which is subsequently compensated by the DMI above a critical value [87]. In this case, the inclusion

Localization and Reconfigurability

of the DMI leads to a hybridization of the Winter modes and it is possible to obtain an expression for the channeled mode frequencies from perturbation theory by using the Winter modes as a scattering basis [8]. In addition to an ellipticity in the precession, the DMI results in a linear wavevector dependence for the mode frequency, as seen for the case discussed before for propagation in the DE geometry in continuous films. However, this linear dependence does not lead to a simple shift in the quadratic dispersion relation as a result of the ellipticity. Instead, the dispersion relation becomes markedly asymmetric with respect to kx (Fig. 8.4d), where a quasi-linear variation is seen for kx > 0, while a strongly quadratic variation is preserved for kx < 0. This asymmetry leads to pronounced differences in the left- and right-propagating wavevectors, which can be seen for microwave field excitations in the frequency gap and in the frequency band of the bulk spin wave modes (Fig. 8.4e). The channeling properties of the Néel-type wall are preserved, even in cases where the localized and propagating mode frequencies are closely spaced, which can be seen for the kx propagation at around 50 GHz in Fig. 8.4e. It is interesting to note that the energies of the channeled and bulk mode become degenerate for a certain value of kx for finite D. This value of kx represents an inversion of the gap separating the localized from the bulk states. Because of the strong localization of Winter modes, domain walls can act as effective conduits for spin waves even in curved geometries. An example is shown in Fig. 8.5, where results of micromagnetics simulations are presented for channeled wall modes for different excitation frequencies. In the simulations, the spin waves are excited by a microwave antenna located at one end of the domain wall and their propagation along a curved wall structure, with a radius of curvature of 89 nm, is computed. For wavelengths shorter than and comparable to the radius of curvature 40 ≤ l ≥ 120 nm, transmission of the spin waves around the corner is observed to be possible with minimal scattering. For l = 120 nm, one can observe a slight phase shift at the corner where the node of the wave profile is slightly stretched at the bend. For l = 140 nm, some transmission is observed but with amplitudes greatly reduced in comparison to the incident wave. For l ≥ 160 nm, no perceptible transmission of the spin waves around the corner is seen. These results

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suggest that the curved waveguide structure proposed can allow full transmission of spin waves as long as the wavelengths involved are comparable to or smaller than the radius of curvature of any corners encountered. Challenges for applications would therefore involve designing suitably curved conduits, for example, through domain wall pinning and engineering material properties to obtain narrow or wide domain walls.

Figure 8.5 Spin wave channeling along curved domain walls. (a) Geometry of the curved domain wall used in simulation, where a radio-frequency antenna generating an alternating field, hrf, excites spin waves that propagate along the wall. (b) Snapshots of channeled spin wave modes at different wavelengths with the corresponding excitation frequencies. rd denotes the radius of curvature of the domain wall channel. Reprinted with permission from Garcia-Sanchez et al. [28]. Copyright 2015 American Physical Society.

Domain walls also allow for the prospect of constructing magnonic circuits that do not require lithography [28, 55, 95]. In Fig. 8.6, an example is shown of how stripe domains can be used as multiple channels through which spin waves can be propagated without interference. In this example, the domain structure was calculated from micromagnetics simulations of a 1-nm-thick film with lateral dimensions of 1 µm × 1 µm by allowing a stripe domain pattern to relax with different pinning conditions along the left and right edges of the system. For the Bloch walls considered, an excitation in the frequency gap (10 GHz) results in channeled modes that propagate along the domain wall conduits with no perceptible cross-talk between the channels, as shown in Fig. 8.6b.

Localization and Reconfigurability

Figure 8.6 Example of spin wave channeling in a multidomain structure with Bloch domain walls. (a) Magnetization configuration of the domain structure in a 1-nm-thick 1 µm × 1 µm film with a microwave antenna along the middle of film. (b) Spin wave channeling along the domain walls for an excitation frequency of 10 GHz, which lies in the frequency gap of the bulk spin wave modes (cf. Fig. 8.4). Reprinted with permission from Garcia-Sanchez et al. [28]. Copyright 2015 American Physical Society.

This scheme opens up a number of interesting possibilities for reconfigurable magnonic circuits. For example, the orientation of the stripe domain pattern and the spacing between walls can be modified by external applied magnetic fields. The positions of the walls can also be modified in a similar way. Indeed, this feature has been demonstrated in a recent experiment involving an in-plane magnetized film in which the domain wall conduit for spin waves was displaced with a small magnetic field and the propagation was probed using microfocus Brillouin light scattering (BLS) (95; see Chapter 9 of this book).

8.3.2  Edge (Partial Wall) Channeling

Another important consequence of the DMI in finite-size nanostructures is the appearance of twisted spin states at boundary edges. To see how this arises in the micromagnetics description (continuum limit), it is useful to recall that the variational procedure leading the to the torque equation (Eqs. 8.1 and 8.2) also gives rise to a boundary condition of the form n . дU/д(m) = 0, where n is a unit vector normal to the surface of the material considered [29, 76]. With only energy terms due to the exchange interaction and perpendicular

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anisotropy, one obtains the usual free boundary condition, дnm = 0, in the absence of any surface pinning. Crucially, the inclusion of the DMI requires satisfaction of twisted boundary conditions. For example, the boundary surface n = yˆ has the conditions

Dmz + 2A∂ymy = 0, –Dmy + 2A∂mz = 0,

(8.7)

which couples the perpendicular magnetization mz with gradients in the transverse components mx,y and vice versa [29, 76]. Such conditions lead to tilts in the magnetization at the edges even if the system is uniformly magnetized in the bulk. An example of magnetization tilts at edges is shown in Fig. 8.7. The profiles were computed with micromagnetic simulations by first allowing a uniformly magnetized state in a 512 nm × 512 nm × 1 nm square dot to relax under several values of the DMI. Stronger tilts occur when the strength of the DMI is increased, and the sign of the transverse component of the tilts is reversed along with the sign of the DMI (Fig. 8.7a,b). In fact, these profiles are well described by partially expelled Néel walls. This is shown by the solid curves in Fig. 8.7c, which represent the theoretical wall profile mz(y) = tanh [(–y – yc)/l] at the right edge, where yc is the position of the domain wall center outside the film, as illustrated schematically in Fig. 8.7b. On the basis of this wall profile and the boundary conditions in Eq. 8.7, it is possible to derive an analytical expression for the partial wall position, as sketched in Fig. 8.7b (for yc > 0),

yc –

 2A  w = l cosh–1 ,  Dl  2

(8.8)

where w is the wire width. This behavior is reminiscent of the partial twists encountered in exchange-spring systems and ferromagnet-antiferromagnet bilayers where the gradual rotation of the uniformly magnetized hard (ferromagnetic) layer creates torques at the interface that are compensated by formation of a partial wall structure in the soft (antiferromagnetic) layer [52]. Here, the DMI acts to pin a partial wall at the edges through Eq. 8.7, and the strength of the DMI governs the extent to which the partial wall enters the film. The analytical model agrees well with the simulation results (Fig. 8.7d).

Localization and Reconfigurability

Figure 8.7  Twisted spin states at boundary edges due to the DMI. (a) Transverse magnetization component at the boundary edges (located at y = ±256 nm) of a 512-nm-wide rectangular wire. (b) Illustration of the magnetization tilts for D > 0, with the shaded regions representing the tilts shown in panels (a) and (c). The partial wall (dashed curve) is shown schematically, with yc denoting the wall center and w the wire width. (c) The perpendicular component at the boundary edges, where the solid lines correspond to fits to a partial Néel wall profile. (d) Partial wall center yc as a function of D. Points are simulation data and the solid line represents Eq. 8.8. Reprinted with permission from Garcia-Sanchez et al. [29]. Copyright 2014 American Physical Society.

The consequences for propagation along the edges of the spin texture induced by the DMI follow from the previous discussion on channeling in Néel walls. Since the tilted magnetization at the edges are partial Néel walls, the tails that extend into the system possess a specific chirality determined by the DMI, and as a result, the energies of spin wave states propagating along a given edge will depend on their propagation direction relative to the (partial) domain wall orientation. As a consequence, the lowest-energy spin waves propagate only along one direction when localized on one side of the wire and flow in the opposite direction when localized on the other side.

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Figure 8.8 Nonreciprocal propagation in a thin rectangular wire. Spatial profiles of the dynamic magnetization resulting from a microwave field excitation, hrf, at (a) 50 GHz and (b) 16 GHz. The different wave vector components considered are illustrated. In panel (a), the excitation frequency is in the spin wave band and nonreciprocal propagation occurs for ktop and kbot, while kcen propagation is symmetric. In panel (b), the excitation frequency is in the gap of the bulk modes and only edge modes are excited. (c) Dispersion relations computed from simulations for D = 4.5 mJ/m2. Dots represent simulation results. The solid black curve represents the theoretical dispersion relation for exchange modes. (d) Dispersion relation for D = 2.5 mJ/m2. Reprinted with permission from Garcia-Sanchez et al. [29]. Copyright 2014 American Physical Society.

This nonreciprocal propagation can be seen in more detail in Fig. 8.8, where results from micromagnetic simulations of spin wave propagation in a thin rectangular wire are shown. For excitation frequencies in the spin wave band (Fig. 8.8a), three distinct wavevectors can be identified for propagation along one direction, which correspond to the top (ktop), center (kcen), and bottom (kbot) of the wire. For propagation towards the right, +x, we note that |ktop| < |kcen|< |kbot|, while for propagation towards the left, –x, the opposite inequality applies, |ktop| > |kcen| > |kbot|. Moreover, |ktop| = –kbot, which is a clear signature of nonreciprocal propagation. We observe a shifted dispersion

Localization and Reconfigurability

relation for the edge modes, while the central modes remain symmetric about kcen = 0 (Fig. 8.8c). For the central modes, the dispersion relation is well described by exchange-dominated spin waves, where the theoretical curve using our micromagnetic parameters agrees well with the simulated curves. For the edge modes, the shifted dispersion relation for D = 4.5 mJ/m2 is well described by a reduction in the spin wave gap due to the reduced anisotropy field at the edge in addition to a linear wavevector term that describes the nonreciprocity. As Fig. 8.7d shows, the center of the partial wall is located farther outside the wire for smaller values of the DMI, which results in a weaker nonreciprocal channeling effect. This can be seen in the dispersion relation of the edge modes in Fig. 8.8d, where the shifts become less pronounced as D decreases. This phenomenon is reminiscent of edge modes in topological insulators. Channeling for the wire geometry is robust with regard to the curvature of the edge in the same way as for curved domain walls. We now discuss the consequences for finite-size nanostructures. In a circular dot, for example, it is known that clockwise (CW)- and counterclockwise (CCW)-propagating azimuthal spin waves are degenerate in frequency. The inclusion of the DMI, however, lifts this degeneracy by favoring one handedness over the other. To appreciate how this might occur, one can imagine the edge modes in a circular dot constructed by deforming a rectangular wire bent into a ring-shaped structure. The lowest-frequency spin waves traveling along the outer circumference can propagate with only one handedness. Spin waves traveling along the inner circumference travel with the opposite handedness at the same frequency. Figure 8.9 illustrates the spin wave eigenmode spectra for a circular dot 100 nm in diameter and a square dot 100 nm in width. A key feature is the frequency splitting of certain modes as the strength of the DMI is increased. The frequency of other modes, on the other hand, are only slightly affected by the DMI. For a similar dot size, the magnitude of the splitting appears to be larger for the circular dots, which suggests that the azimuthal component of the eigenmodes plays an important role. For the circular dots, the frequency splitting with increasing DMI is associated with lifting in the degeneracy of eigenmodes with a strong azimuthal character, such as Modes 2 and 3 in Fig. 8.9c.

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Figure 8.9 Map of the eigenmode power spectral density (PSD) as a function of D for (a) 100-nm-diameter circular dots and (b) 100-nmwide square dots. Selected profiles of the four lowest modes for different strengths of the DMI for the (c) circular and (d) square dots. Reprinted with permission from Garcia-Sanchez et al. [29]. Copyright 2014 American Physical Society.

While there is no discernible change in the spatial profile of these modes, a frequency splitting of around 1 GHz appears at D = 2.5 mJ. Modes with a strong radial character, such as Modes 1 and 4 in Fig. 8.9c, experience only a slight decrease in their frequency with increasing D and little change in their spatial profile. These differences can be understood in terms of the nonreciprocal wall channeling described earlier, where radial modes are similar modes that traverse a domain wall, while azimuthal modes are similar to modes that are channeled by the domain wall, which are strongly nonreciprocal. Similar features

Localization and Reconfigurability

are also seen in the square dots, but the distinction between radial and azimuthal modes is not as sharp. One difference can be seen in Mode 4 in Fig. 8.9d, which represents a mixed radialazimuthal excitation for which splitting due to the DMI results in an asymmetric profile at higher frequencies.

8.3.3  Magnetic Configurations in Artificial Spin Ice

Artificial spin ice (ASI) is a class of magnetic materials created by patterning single-domain ferromagnetic islands in such a way as to introduce some degree of frustration through competing interactions [40]. The magnetic elements are typically fashioned as elongated islands with nanoscale dimensions in order to ensure that their magnetic state is single domain with a large uniaxial shape anisotropy so as to approximate a rigid block spin [27, 67, 71]. In magnetic ASI composed of discrete elements, interactions are provided by the stray magnetic fields associated with the individual elements. Square ASI is one of the first and best-studied ice geometries [92, 96], along with the fully frustrated kagome lattices [62]. A variety of other structures have also been since studied, including Penrose [3] and Shatki lattices [16, 30] as well as suggestions for many others [31, 75]. In what follows we will discuss exclusively square ASI for which the magnetic elements are aligned in two sets of rows on a square lattice. One set has elements aligned along one axis of the lattice, and the other set of elements is aligned along the other orthogonal axis. An example of the unit cell of a square lattice is shown in Fig. 8.10. Each vertex, defined by the four ends of adjacent magnetic elements, can be characterized by an average magnetization magnitude (called the charge) and its direction. There are 16 unique configurations possible with charges varying from –4 to +4 and 8 possible alignments for non-zero magnetization values. The dipolar coupled square ASI ground state complies with the ice rule, that is, a two-in, two-out configuration, and can be thought of as a mesoscopic antiferromagnet with two well-defined ground states. An appealing aspect of ASI is the ability to modify macroscopic magnetic properties through design of the ASI geometry and generate new functionality [3, 30, 58].

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The soft transition metals commonly used to fabricate ASI exhibit high-frequency electromagnetic resonances in the gigahertz range. The resonances correspond to band center, longwavelength spin waves. We discuss below results obtained using BLS. This technique has been used to obtain experimental values for a number of magnetic parameters: magnetic anisotropy [41, 60], gyromagnetic ratio [84], exchange constants [35], and saturation magnetization. The mode structure of a spin wave in individual elements is well understood [45]. Spin wave modes observed in micronand submicron-size magnetic elements are confined modes with stationary character and their energies are primarily magnetostatic [32]. Most importantly, modes exist which are localized to edges and ends of the magnetic elements. These modes have dipolar stray fields [45], which extend away from the material, although they decay exponentially away into the surrounding medium. In addition, since the elements are in a single-domain state, there is a non-uniform distribution of demagnetizing fields in the elements, which is most pronounced near the edges and corners. These can lead to some curling of the magnetization within an exchange length of the edges and serves to affect the frequency of end-localized modes. The magnetic elements lie along the sides of squares and the frequencies of modes depend upon whether the magnetization of individual elements are aligned by the applied field along easy or hard directions of the elements. Micromagnetic simulations were used to identify the excitations measured in the independent elements. Details of sample fabrication and experimental procedure can be found in Ref. [57]. Results from two BLS configurations are discussed. First, the frequency dependence on the wavevector (qk) was studied with a 3 kOe magnetic field applied along directions at 0° and 45° with respect to the ASI lattice. The angle of incident light, q, upon the sample varied from 0° to 60° corresponding to the in-plane wavenumber qk from 0 to 2 × 1017 m–1. Two scattering geometries were studied: the DE for spin waves with wavevector k perpendicular to the external field H, and the backward volume (BA) mode configuration for spin waves with the wavevector parallel to the applied field. In the second configuration, the angle of incidence of the illuminating laser

Localization and Reconfigurability

was fixed at a = 20°. The external field H was, however, varied from +4 kOe to –4 kOe and applied along the 0° and 45° orientations with respect to the ASI lattice. Spin wave dispersions were measured to indicate the possible inter-island dynamic coupling and propagation of collective spin waves through the array. Parts (a) and (b) of Fig. 8.10 are shown for the two different magnetic field orientations. When the field is applied at 45°, there are two well-defined peaks in the spectra, while at 0° up to seven peaks are visible in two different frequency ranges with the larger in-plane wavevector.

Figure 8.10 Sequence of BLS spectra measured at different incidence angles with the external field of 3 kOe applied at (a) 0° and (b) 45°. The wavevector of the incident light parallel to the applied field in the Damon–Eshbach configuration. Spin wave dispersion curve of ASI at a 3 kOe (c) parallel field and (d) diagonal field. Dots are experimental results and lines are guides for the eye. Reprinted with permission from Li et al. [57]. Copyright 2016 IOP Publishing Ltd.

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Frequencies measured for different wave numbers are shown in Fig. 8.10c,d with the field applied along an array edge and diagonal orientation. The dispersion curve is almost flat, indicating that dynamic coupling between individual island resonances is negligible in the ASI system with this spacing. In fact, careful micromagnetic studies of square ASI with this element design suggest that it is not possible to produce magnonic bands with the DE geometry because of the difficulty in bringing edge mode localizations on adjacent elements sufficiently close in order to produce strong dynamic interactions. Some examples of the localized mode profiles for these element geometries are shown in Fig. 8.11.

Figure 8.11 (a) Frequencies of the spin wave eigenmodes as a function of the applied field parallel to the ASI islands. Dashed lines are the cutoff points between saturated and unsaturated regions of the hard-axis magnetization. (b) Spatial profiles of the eigenmodes at different field strengths for H1 (upper panel) and H89 (lower panel), with frequency increasing from left to right. Reprinted with permission from Li et al. [57]. Copyright 2016 IOP Publishing Ltd.

Localization and Reconfigurability

We note that Iacocca et al. [45] have theoretically calculated the magnonic band structure of a square ASI array and show that the Brillouin zone energy variations of band structure are in the order of 0.1 GHz for spacing similar to our system. Their results, however, do not take into consideration coupling via localized modes. This suggests that with careful element design and spacing, it may be possible, at least in principle, to create magnonic bands in a square ASI lattice geometry. The field dependence of the frequencies obtained in the DE scattering configuration is presented in Fig. 8.11a. The measured frequencies are shown by square symbols, and the magnetic field is oriented along the 0° direction. Several distinct modes are identified from the spectra, and each exhibits different behavior for fields in the region of hysteresis between +4 and –4 kOe. The frequencies were recorded from spectra obtained by decreasing the field from positive to negative saturation, thereby following the upper branch of the magnetization loop. The behavior of the frequencies is roughly linear with field outside this region, as one expects for saturated elements. At the coercive fields, several modes appear to merge with others or disappear entirely. The two lowest-frequency modes have minima near the coercive fields. At a zero field, mode crossings appear in two higher-frequency modes. These modes in the horizontal islands appear to be softening at negative applied fields. Except for the mode crossings, the behavior of the mode frequencies for the vertical islands with the applied field are symmetrical so that minima again appear for the lowestfrequency modes at around a ±1.3 kOe field, marked by two black dashed lines, and there is a linear increase of frequencies for fields outside the hysteresis region. An analysis of the mode structure was performed using micromagnetic simulations with parameters detailed in Ref. [57]. The frequencies were calculated in the following way. At each field step after relaxation to a steady-state configuration a field pulse is applied and oriented along the z axis. This drives oscillations in the components mx, my, and mz in each micromagnetic discretization cell, and their responses are recorded every picosecond. Frequencies and intensities of spin wave modes are then calculated using a discrete Fourier transform (in time and space) of the magnetization component, mz, for each cell [61].

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The frequencies calculated in this way are shown in Fig. 8.11a by solid lines for the horizontal elements (for which the applied field is collinear) and the dotted blue lines for the vertical elements (for which the applied field is perpendicular to the element axes). The simulations describe well the measured frequencies. The small discrepancies may be due in part to the effects of edge roughness [34] and neglect of static interaction fields. These static fields can be quite large, on the order of several hundred oersteds in the saturated state. To identify which of the confined modes are responsible for the spectra, the spatial profile of the magnetization dynamics mz was calculated, and examples are shown Fig. 8.11b. For the analysis which follows, we use the same classification protocol as in Ref. [33]. The modes are classified into four categories: backward (m-BA), Damon–Eshbach (m-DE), edge (m-EM), and fundamental (F). In this classification, the integer m indicates the number of nodal lines. The m-BA mode is a mode with the nodal line perpendicular to the magnetization. Nodal lines parallel to the magnetization are called m-DE. The edge modes, m-EM, are localized at the ends of the islands and normally have a small intensity in the BLS spectrum. The fundamental F is the Kittel uniform resonance (m = 0). This mode typically has the largest intensity. The modes associated with the horizontal elements are labelled in Fig. 8.11 as 1-EM1 and F1, representing, respectively, the EM and fundamental modes. These mode profiles remain unchanged in intensity for magnetic fields between 3 kOe to 1 kOe. F1 appears to soften for fields more negative than −50 kOe, consistent with reversal of the magnetization of the horizontal element. We note that the calculated 1-EM1 mode has two minima in the unsaturated region. This corresponds to curling of the edge magnetization. Note also the difference in amplitude of the F mode for the 1 and 3 kOe fields. Modes for the vertical elements are labelled EM89, DE89, E89M, F89, EM89, EM89, and EM89. The corresponding spatial profiles shown in the bottom of Fig. 8.11b. The EM89 mode possesses the lowest frequency. Hybridization is more apparent in the higherfrequency modes, where a mix of an F mode with EM and DE modes occurs. As for the horizontal elements, there is significant

Localization and Reconfigurability

dependence of the mode amplitudes on field, as seen by comparing the profiles for 2 and 3 kOe. Furthermore, the frequency of the EM mode is smallest at 1.5 kOe as the magnetization begins to saturate perpendicular to the element axis [64].

8.3.4  Reprogrammable Microwave Response

The example mode profiles displayed in Fig. 8.11 illustrate how simple magnetic configurations can be used to generate a rich spectrum of microwave responses for uniformly magnetized particle arrays. When the applied magnetic field is small, in the range between +1.5 kOe and –1.5 kOe, the magnetization of the array is not uniform although the magnetization of individual elements is still single domain. In the spin ice discussed so far, inter-element static interactions are relatively weak, and the ordering of neighboring element magnetizations are not strongly correlated. However, in spin ice with stronger static interactions, obtained, for example, by patterning thicker films with closer inter-element spacing, correlations can be observed. In some systems mesoscopic domains of ordered elements have been observed, with boundaries defined by conjoined lines of mesoscopic analogies to domain walls. If we are to use such complex arrangements of dipolar coupled magnets for device applications [46], then a first important step is to be able to control the configurations resulting from inter-element correlations. One way to do this is to modify individual islands, for example, making them narrower (wider) than the rest of the islands in the array so that the shape anisotropy, and therefore the island switching field, is higher (lower). In this way, one can determine in a controlled manner where Dirac string avalanches in quasi-infinite ASI arrays start and where they stop [44, 62]. In small clusters, one can use this to control the field-induced states and vortex chirality [39]. Another intriguing approach is to ask whether it is possible to create a specified configuration of magnetic moments with a sequence of applied magnetic fields. In particular, a field oriented along certain directions can drive specific element reversals and avalanches, where the reversal details depend upon the magnitude of the field and disorder in the system. An analysis

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of this concept was made in simulations whereby specific configurations were tracked for sets of possible field sequences [11]. It was found that a sequence of such fields can access a small subset of states but not all states. However, it is possible to increase the number of states accessible to a finite sequence of fields by controlling a small number of element orientations. An example is shown in Fig. 8.12. In Fig. 8.12a, the four configurations in a 16-spin square array that can be accessed for a certain applied field strength are shown. The starting point is a fully polarized lattice, and for this particular field strength there are only two configurations available to evolve into. Which configuration appears depends on the direction of the applied field. One additional configuration is possible from each state and arrived at by applying the field in a new direction. The exact configurations are depicted in Fig. 8.12a where the top and bottom pairs of configurations correspond to the two possible evolutions. Controlling one of the corner spins, indicated by the circular frame, allows one to access many more configurations, as shown in Fig. 8.12c, where one is able to access 128 different configurations through alignments of the control spin and different orientations of the applied field. Micromagnetic simulated examples of how control of magnetic array configurations on mesoscopic length scales can be used to configure microwave response is shown in Fig. 8.13. In this figure are two examples of how mesoscopic analogies to domain walls can be used to channel microwave resonances of strongly interacting magnetic elements. Square ASI, coupled by dipolar fields, is antiferromagnetic with two ground-state configurations represented by a two-in, two-out spin ice rule for each vertex of four magnetic elements. These ground states are labelled as “Type I.” The geometry of the square spin ice defines the two ground states as incommensurate arrangements, which cannot coexist without a topological boundary separating the arrangements. The boundary is the equivalent of a microscopic magnetic domain wall and is labelled as “Type II” in the figure. In Fig. 8.13a the Type II configuration defines a straight path through the magnetic array, and in Fig. 8.13b the Type II configuration represents a corner. In both cases, microwave wave frequency excitations exist for the Type IIs that are distinct in frequency and profile from the excitations that can exist in the

Localization and Reconfigurability

Type I regions. The profiles of these resonances are shown in the panels on the right-hand side of Fig. 8.13. The excitations are resonances strongly localized to the magnetic element ends defining the vertices between neighboring elements.

Figure 8.12 Effect of a control spin on allowed configurations of a 16element array. The reference configuration for the array is with all spins pointing to the right, that is, the black arrows. Without a control spin, a field strength of h = 11.5 is large enough to access only four configurations. These configurations are shown in (a), begin with reversals of edge element spins, and are accessed by applying the field along specific directions. The corresponding map of allowed configurations is shown in (b), where each dot represents a unique configuration of element spins and each line represents an orientation of the field (with fixed magnitude h). To illustrate the effect of including a control spin, we specify the orientation of an element spin at the lower-left corner of the array, indicated with a circular frame in (a), in addition to, and independently of, the applied field. This allows, in principle, two different flipping processes for each field orientation corresponding to the two different orientations of the control spin. The cumulative effect of these different flipping processes is that a multitude of configurations can be accessed depending on the orientation of the control spin and the direction the field is applied, even though its magnitude is still h. In (c) the corresponding configurations are shown. Reprinted with permission from Budrikis et al. [11]. Copyright 2012 IOP Publishing Ltd.

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Figure 8.13 Control of microwave resonances in different spin ice configurations. In (a), two domains of ground-state configurations of magnetic elements in square spin ice (designated “Type I”) are separated by a line of differently aligned elements (designated “Type II”). The Type II configurations are the mesoscopic equivalent of a magnetic domain wall. The magnetic resonances associated with the Type II configurations are distinct from the resonances of the Type Is and represent a channeling of resonances along the direction defined by the Type IIs. This channeling is illustrated in (b), where the Type II configuration is directed around a corner separating two Type I domains. The resonances associated with this configuration are localized strongly to the element edges at the vertices of four neighboring elements.

The ability to spatially localize and configure microwave frequency resonances in an ASI array is only one example of how patterned magnetic elements may be used to control microwave properties on micrometer length scales. Reconfigurability, perhaps through some scheme such as that outlined in Fig. 8.12, may enable a new type of microwave device based on artificial magnetic materials created through mesoscopic engineering, as exemplified by the construction of ASI.

8.4  Outlook

We close with a few comments regarding the important topics of charge and spin transport. It may be possible to utilize and control

Outlook

spin wave excitations in new ways, for example, through spin caloritronics in which spin currents can be created by thermal gradients. Spin-orbit coupling, discussed earlier in relation to chiral interactions, can also affect the transport of angular momentum via spin currents across interfaces and give rise to spin Seebeck and spin Hall effects. These are so named due to their analogies to conventional charge transport [20, 21, 89, 91], and spin currents are believed to interact with spin waves. This opens up exciting possibilities for electric field control of spin waves. Further afield are other growing areas of activity. Plasmonic metamaterials have many possible applications and are, in principle, subject to control via applied magnetic fields. It turns out that magnetically polarizable elements in a plasmonic array can enable different properties to be “tuned” using externally applied magnetic fields. An example is negative refraction [12, 56, 63, 70]. Additional developments include laser-induced switching [72] and optically controlled charge and spin currents for information encoding and transfer. These advances may allow the generation and guidance of spin waves using all-optical techniques [77]. There are enormous possibilities for the manipulation of magnetic element structure and composition with nanoscale precision in two and even three dimensions. Because of a rapidly evolving lithographic technology, there exist new, unexplored opportunities for designer metamaterials. It is possible, for example, to examine topics from entirely different fields within the context of spin wave materials. As an example, electrical conduction in topological insulators is a phenomenon that arises from the way electronic states can form near material boundaries. The topological phase is a property of waves that can be realized for wavelike excitations that are distinct from electron waves and electronic band structure. In particular, analogies to topologically protected surface states have been proposed for spin waves in magnetic materials [48]. Essential for realization of these phenomena is the ability to geometrically pattern magnetically functional matter on appropriate length scales. The fascinating interplay between symmetry, interactions, and physical properties is now being explored at the micro- and nanoscale in ways that were not possible a few years ago [10].

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For over one hundred years, wave interference in periodic structures has been studied [74] and has led to our understanding of, for example, localization [47] and the engineering of bandsand bandgaps [99]. In terms of magnonics, we now have available a large number of ways to introduce novel electric, magnetic, thermal, elastic, and electromagnetic responses in artificially designed materials. In this chapter we have described a number of recent examples of how relatively simple material designs can produce spin wave phenomena, such as channeled nonreciprocity, that is at once interesting and potentially useful. The prospects are very encouraging, indeed, for exploiting new capabilities of sample fabrication at mesoscopic length scales in order to create new magnonic metamaterials and devices for broad application across information and communications technologies.

Acknowledgments

The authors acknowledge fruitful discussions with J.-P. Adam, M. Bailleul, V. Cros, T. Devolder, Y. Henry, S. Rohart, J. Sampaio, R. Soucaille, A. Thiaville, and A. Vansteenkiste. This work was partially supported by the Agence Nationale de la Recherche (France) under grant numbers ANR-11-BS10-003 (NanoSWITI), ANR-14CE26-0012 (Ultrasky), and ANR-16-CE24-0027 (Swangate); the Engineering and Physical Sciences Research Council (UK); the University of Glasgow; and the National Council of Science and Technology of Mexico (CONACyT). The work of Y. L. and R. L. S. was supported by the China Scholarship Council and EPSRC (grant number EP/L002922/1). P. B. acknowledges support from a Grant-in-Aid for Scientific Research on Innovative Areas (Grant No.26103006) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. We would like to acknowledge that some of the results of micromagnetics simulations presented here made use of the Emerald High Performance Computing facility made available by the Centre for Innovation. The Centre for Innovation is formed by the universities of Oxford, Southampton, Bristol, and University College London in partnership with the STFC Rutherford-Appleton Laboratory.

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Chapter 9

Steering Magnons by Noncollinear Spin Textures

Katrin Schultheiss, Kai Wagner, Attila Kákay, and Helmut Schultheiss Helmholtz-Zentrum Dresden–Rossendorf, Institute of Ion Beam Physics and Materials Research, Bautzner Landstraße 400, 01328 Dresden, Germany [email protected]

One of the grand challenges in cutting edge quantum and condensed matter physics is to harness the spin degree of electrons for information technologies. While spintronics, based on charge transport by spin polarized electrons, made its leap in data storage by providing extremely sensitive detectors in magnetic hard-drives [1], it turned out to be challenging to transport spin information without great losses [2]. With magnonics, a visionary concept inspired researchers worldwide: Utilize spin waves—the collective excitation quanta of the spin system in magnetically ordered materials—as carriers for information [3–8]. Spin waves, which are also called magnons, are waves of the electrons’ spin precessional motion. They propagate without charge transport and its associated Spin Wave Confinement: Propagating Waves (2nd Edition) Edited by Sergej O. Demokritov Copyright © 2017 Pan Stanford Publishing Pte. Ltd. ISBN 978-981-4774-35-2 (Hardcover), 978-1-315-11082-0 (eBook) www.panstanford.com

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Ohmic losses, paving the way for a substantial reduction of energy consumption in computers. While macroscopic prototypes of magnonic logic gates have been demonstrated [9, 10], the full potential of magnonics lies in the combination of magnons with nano-size spin textures. Both magnons and spin textures share a common ground set by the interplay of dipolar, spin-orbit, and exchange energies, rendering them perfect interaction partners. Magnons are fast, sensitive to the spins’ directions, and easily driven far from equilibrium. Spin textures are robust, nonvolatile, and still reprogrammable on ultrashort timescales. The vast possibilities offered by combining these magnetic phenomena add value to both magnonics and the fundamental understanding of complex spin textures. The scope of this chapter is about experimental studies on magnon transport in metallic ferromagnetic microstructures with focus on actively controlling the magnon propagation. Two inherent characteristics of magnons enable for lateral steering: the anisotropy of the magnon dispersion and its sensitivity to changes in the internal magnetic field distribution. We intend to give an idea of how these magnon features can be utilized toward realizing functionalized magnonic networks.

9.1 Introduction

The main idea behind magnonics is to explore the foundations of a novel type of information processing technology, in which logic operations are performed by the superposition of waves rather than by the motion of electric charges [3–8]. But what are the reasons behind searching for alternatives for state of the art CMOS technologies? Since the information transport by magnons does not rely on charge transport it does not suffer from Ohmic losses and the associated Joule heating, one of the drawbacks and limitations of conventional electronics and their scalability. Moreover, the wavebased phenomena such as interference, diffraction or refraction allow for novel types of logic architectures and a broad frequency

Introduction

operating range from MHz up to THz. In contrast to electromagnetic waves in this frequency range, the magnons’ wavelengths and, therefore, the size of magnonic circuits approaches nanometer dimensions, allowing for high integration densities in future devices. In fact, prototypes of magnonic logic gates based on the ultralow damping material yttrium iron garnet (YIG) demonstrated coherent signal propagation over millimeters and wave-based logic operations [9]. Chumak et al. even succeeded in the realization of an all-magnon transistor for controlling the primary magnon propagation by nonlinear interaction with secondary magnons [10]. Most of the realized prototypes of magnonic logic gates are based on the interference of magnons in macroscopic devices made from YIG. The terminals for controlling the magnons’ phases or amplitudes are either macroscopic current lines for generating magnetic fields or microwave antennas for pumping secondary magnons. However, the route towards highly integrated magnonic circuits requires efficient mechanisms for controlling magnons on length scales ideally given by the magnons’ wavelengths, ultimately only a few nanometers. Thinking about these requirements provides the not-so-obvious answers to the question, why magnons as carriers of information could be a valuable extension to state-of-the-art electronic and spintronic approaches. In short, the answer lies in the interactions between magnons and noncollinear spin textures, present even in the linear regime for small amplitude excitations. The origin of these interactions lies in the fact that magnon transport is strongly determined by the local orientation of the magnetic moments with respect to the magnons’ wavevectors. This allows one to control magnon propagation not only by different source geometries or waveguide designs (as typically the case for linear and isotropic waves), but additionally by tailoring the magnetic material parameters and their magnetic  texture itself. Here, any non-zero derivative of the vector field M given by the magnetization generates additional effective magneticfields due  2to the symmetric, Heisenberg-like exchange energy Hex µ (—M ) and magnetic  volume  dipolar energy Hdip µ div( M ) . Note that these two terms are only accounting for fields generated by magnetic moments within the sample volume. The boundaries of any finite magnetic body  surface  cause an additional dipolar magnetic field Hdip µ M ◊ nˆ once the

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magnetic moments have a component parallel to the surface normal nˆ . This results in a strong anisotropy of the magnon dispersion for thin, in-plane magnetized films [11, 12]. While the exchange field is strongly localized, the bulk and surface contributions of the dipolar magnetic fields are nonlocal. This means that for calculating the dipolar magnetic field acting on a magnetic moment at a particular position, the dipolar magnetic fields generated by all other magnetic moments have to be summed up, which makes the analytical handling of the dipolar contribution to the effective magnetic field to be complicated. A theory for calculating the magnon dispersion in infinite thin films incorporating both exchange and dipolar interaction was given by B. Kalinikos and A. N. Slavin in 1986 [13]. Even though this dispersion relation was not derived for patterned films it could be used to quantitatively describe discrete magnon spectra and magnon propagation in microstructures [14–19]. To sum this up, even in conventional ferromagnets such as Fe, Ni, and Co a noncollinear magnetization texture will create an inhomogeneous internal magnetic field and, therefore, locally affect the magnon dispersion. Hence, spin textures like magnetic domain walls, vortices, and skyrmions, where magnetic moments are not parallel, are ideal candidates to manipulate magnons on a nanometer length scale. A pioneering work giving evidence for the interaction of magnons with topological spin textures by R. Hertel and coworkers [20] demonstrated, that a magnon pulse travelling through a magnetic domain wall experiences a phase shift large enough for realizing magnon based logic gates [21]. The full potential of using spin textures for manipulating magnons is given by the fact that they are topologically protected and, therefore, nonvolatile. But still there are various means for their manipulation for example by magnetic fields, electric currents, or ultrashort laser pulses [22–24]. This chapter is outlined as follows: In the next section, we will give a short introduction to magnon transport in ferromagnetic stripes with a width in the micrometer range. We will explore the magnon dispersion and the impact of its anisotropy, in order to sensitize the reader to challenges that emerge when guiding or steering magnons in two-dimensional structures. In the following section, we will introduce a method based on locally generated magnetic fields for steering magnons and even switching

Magnon Transport and Dispersion in Magnonic Waveguides

their propagation path. In the final section, we discuss active channeling of magnons in magnetic domain walls.

9.2 Magnon Transport and Dispersion in Magnonic Waveguides

The experiments discussed in this chapter are all based on magnon transport in microstructures patterned from Ni81Fe19 (Permalloy, Py) films with thicknesses below 50 nm and lateral dimensions larger than one micrometer. For such geometries the magnetization is oriented within the sample plane and the magnon dispersion can be well approximated following the general formalism for corresponding continuous films derived by Kalinikos and Slavin [13]. A simpler form of the dispersion is discussed in [25], in particular for confined magnons with uniform amplitude profiles over the film thickness. Note that magnons with non-uniform amplitude profiles across the film thickness have much smaller group velocities for propagation within the film plane and, therefore, are typically not considered for magnonic applications. In thin films the magnon dispersion is defined by a continuous surface in the reciprocal space as shown in Fig. 9.1a, where the magnon frequency is plotted as a function of their wavevector components parallel and perpendicular to the equilibrium direction of the magnetization. For the sake of simplicity we omitted quantitative values for frequencies and wavevectors, which are dependent on material parameters, film thickness and the magnitude of the applied magnetic field. The main feature we devote our attention to is the anisotropy of the dispersion relation with respect to the angle between the wavevector k and the magnetization M. For k ^ M the magnon frequency monotonously increases, whereas for k || M it initially decreases with increasing wavevector k until the exchange interaction dominates over the dipolar interaction and causes an increase of the magnon frequency. It is also worth mentioning the significant difference in the absolute value of the slope of the dispersion, which actually defines the group velocity of magnons and, hence, the signal propagation speed in magnonic circuits.

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Figure 9.1 Magnon dispersion relation in the wavelength range in which the magnon energy is mainly determined by dipolar interaction. (a) The dispersion for an infinite thin film exhibits a strong anisotropy. For larger wavevectors k perpendicular to the magnetization M the frequency increases, whereas it decreases for magnons propagating parallel to M. (b) The dispersion of the first magnon mode of a perpendicular magnetized stripe with a width w can be calculated by intersecting the dispersion surface with a plane defined by k^ = p/w.

As a general rule, magnons with a wavevector perpendicular to the magnetization are faster compared to any other propagation direction. With respect to information transport via magnons, this has a strong impact on the design of magnonic waveguides. In the simplest scenario, i.e., magnon transport within a magnetic stripe, the magnetic shape anisotropy forces the magnetization to align parallel to the stripe’s long axis. Thus, in order to realize most efficient signal transport with high group velocities, one needs to overcome the shape anisotropy and actively pull the magnetization perpendicular to the transport direction. The calculation of the exact dispersion relation for such geometries is tedious because of the inhomogeneous internal magnetic field distribution caused by the demagnetizing field. An analytic expression can be derived using a Green’s function formalism as shown by Kostylev and coworkers [26]. However, in good approximation one can get results following the idea sketched in Fig. 9.1b. Confining a wave to a waveguide with finite width results in discretization of the wavevector components perpendicular to the transport direction in analogy to the TEM (transversal

Magnon Transport and Dispersion in Magnonic Waveguides

electromagnetic) modes of a laser resonator or microwave waveguide. The dispersion for these transversal magnon modes of a stripe waveguide with a width w and mode number n can be extracted from the dispersion relation of a continuous film by intersecting the dispersion surface with a plane defined by k^ = np/w, where k^ is the component of the magnon wavevector perpendicular to the magnetization. To account for the finite size of the magnetic structure when using the dispersion of a continuous film a slight adjustment has to be made to the width of the magnonic waveguide. As pointed out by Guslienko and coworkers [27] effective dipolar boundary conditions for the dynamic magnetization in thin magnetic stripes have to be considered. This results in a slight increase of the effective width w of the discretization volume for the waveguide modes used for calculating the dispersion relation. Deriving the dispersion for a finite element from the dispersion relation of a continuous thin film works surprisingly well and has been applied to understand magnon transport in numerous studies [14–19, 28, 29]. One example is shown in Fig. 9.2a that is adopted from the work of Pirro and coworkers [18]. It shows a spatially resolved magnon intensity map of two counter propagating magnon beams excited by microwave antennas in a 4 µm wide and 40 nm thick Permalloy stripe. The magnon intensity was measured with Brillouin light scattering microscopy (µBLS), which is in detail described in [30, 31]. The stationary interference pattern shown in Fig. 9.2a for an excitation frequency of 7.13 GHz and an externally applied magnetic field of 30 mT oriented perpendicularly to the magnon waveguide demonstrates the coherent propagation of magnons over distances larger than the decay length. The periodicity of the interference pattern allows for a direct measurement of the magnon wavelength. Repeating this for different excitation frequencies yields the magnon dispersion relation shown in Fig. 9.2b for wavevectors oriented along the magnon waveguide, i.e., in transport direction. From the periodic oscillations of the width of the magnon beams one can already conclude that the resulting interference pattern is a superposition of many transverse waveguide modes, an effect which was first observed experimentally in micron sized magnonic waveguides by Demidov and coworkers [16, 17].

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Figure 9.2 (a) Intensity profile of two counter propagating magnon beams in a Permalloy stripe with a width of 4 µm revealing a clear stationary interference pattern. (b) Plot of the magnon frequency as a function of the wavevector extracted from the periodicity of the interference pattern. Images adopted from [15].

With the example described above and the listed references, we conclude that coherent magnon transport over distances larger than the magnon decay length is possible in metallic microstructures. Additionally, the parameters that describe the transport properties, in particular the dispersion relation, are well known and understood. This conclusion, however, only holds for configurations, where the magnetic moments are homogeneously aligned along the principle directions, either perfectly perpendicular or parallel to the magnon wavevector. The situation changes drastically once this symmetry is broken and the magnetic moments have an alignment with the magnon wavevector that is neither parallel nor perpendicular. Two examples of magnon transport in magnonic waveguides with a finite width are shown in Fig. 9.3. The most straightforward approach of introducing a skew section in a 2.5 µm wide stripe patterned from a 60 nm thick Permalloy film was realized by Clausen and coworkers [32]. Part of the results are displayed in Fig. 9.3a, where magnon intensities are measured with µBLS for an excitation frequency of 7 GHz and an external magnetic field of 50.7 mT applied perpendicularly to the waveguides. The key result of this study was that the magnon flow is not simply redirected to follow the waveguide but that instead the magnons undergo a mode conversion upon passing a simple skew section. In the straight waveguide a continuous variation of the width of the magnon beam is observed, which Demidov and coworkers [16, 17] identified as the beating

Magnon Transport and Dispersion in Magnonic Waveguides

pattern resulting from the interference of several waveguide modes with an odd mode number. In contrast, the waveguide with the skew section exhibits a snake like pattern as a consequence of the mode conversion in the skew section and the subsequent interference of the first and second waveguide mode. The challenges of magnon transport are even more evident in the second example shown in Fig. 9.3b. Vogt and coworkers [33] launched a magnon beam in a 2 µm wide and 30 nm thick Permalloy waveguide which exhibits a Y junction with an opening angle of 60 degree in 5 µm distance to the microwave antenna. In case all magnetic moments are aligned perpendicularly to the initially straight part of the waveguide by applying a magnetic field, magnons propagate away from the excitation antenna towards the junction point. The magnon beam impinging on the Y junction does neither split into two beams nor is it converted into other magnonic waveguide modes. The experimental results show that magnons do not propagate into the angled waveguide sections, where the magnetization is not aligned perpendicularly to the transport direction.

Figure 9.3 (a) Magnon transport in 2.5 µm wide waveguides. One straight and the other with a skew section, where the waveguide is shifted transversally by 1 µm over a distance of 3 µm. A magnetic field of 50.7 mT was applied perpendicularly to the waveguide and the excitation frequency was 7 GHz. (b) Magnons propagating towards a Y junction, excited by microwaves in the lower part (not shown). Magnon intensities were measured with µBLS. Images adapted from Clausen et al. [32] and Vogt et al. [33].

These two examples should give a taste of the challenges arising in more complex magnonic circuits. New design rules are required for steering magnons in two-dimensional networks. Intriguing approaches are for example graded-index magnonics introduced by Davies and coworkers [34] or the magnonic grating

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coupler effect discovered by Yu and coworkers [35] that are both covered as separate chapters in this book. In the following sections of this chapter we introduce two alternative concepts for steering magnons, one based on locally generated magnetic fields for aligning the magnetization in the magnonic waveguide and one based on channeling magnons inside magnetic domain walls.

9.3 Steering and Multiplexing Magnons by Current-Induced, Local Magnetic Fields

In the previous section, we gave two examples that showed redirecting magnon transport to be increasingly complex in magnonic waveguides when the relative angle between the magnetization and the magnon wavevector component in transport direction was altering from the starting 90° (see Fig. 9.3) as a result of the geometric patterning. But what if one could ensure a transversal alignment between magnetization and propagation direction even in more complex geometries like a bended magnonic waveguide? It would require an effective magnetic field that is oriented transversally to the waveguide’s edges and sufficiently strong to locally compensate the demagnetizing field caused by the shape anisotropy. It turned out that such a locally oriented magnetic field can be generated via an electric current line placed underneath the magnon waveguide [36]. The working principle of this idea is sketched Fig. 9.4a,b for a curved magnon waveguide with a thickness of 30 nm, that is directly deposited on top of a 50 nm thick Au current line. Let us first consider the case without electric current but with a uniform, external magnetic field, strong enough to overcome the shape anisotropy (Fig. 9.4a). Such an external field has two major disadvantages regarding magnon transport in the curved section: First, the magnetization is rotating with respect to the transport direction which affects magnon propagation as discussed in the previous section. Second, the angle of the magnetization with respect to the waveguide’s edges is continuously rotating, resulting in an inhomogeneous demagnetizing field and, thus, also affecting magnon transport. However, both of these disadvantages can be avoided when magnetizing the magnon waveguide via the Oersted field generated

Steering and Multiplexing Magnons by Current-Induced, Local Magnetic Fields

from an electric current running underneath the magnonic waveguide. Already in the simple case of a ferromagnet/metal hybrid conduit with a composition of Py(30 nm)/Au(50 nm) as shown in Fig. 9.4 calculations reveal that 90% of the current flows in the Au conductor because of its lower resistivity. Since the current flow follows the direction of the magnon transport, the resulting magnetic field is aligned perpendicularly to the magnon wavevector at any position of the magnon waveguide. However, the open question is if the generated magnetic field for moderate current densities (without creating too much Joule heating) is sufficient to overcome the shape anisotropy of the magnon waveguide in order to align the magnetization perpendicularly to the waveguide boundaries and, hence, perpendicularly to the magnon wavevector.

Figure 9.4 (a, b) Illustrating the difference in the magnetization texture of a bended magnon waveguide depending on the magnetic field source. External  magnetic fields Hext cause a uniform alignment of the magnetic moments, where the angle between the magnon wavevector and the magnetization changes in the curved section of the waveguide. The alternative of an electric current flowing in a conductor below the magnon waveguide generates a magnetic field (Oersted field) that can force the magnetization to always align perpendicularly to the magnon transport direction. (c) Schematic sample layout for investigating the guidance of spin waves with locally rotating magnetic fields. Images adapted from Vogt et al. [36].

This concept was tested for a 2 µm wide hybrid magnon waveguide with a composition of Py(30 nm)/Au(50 nm) and a microwave antenna for magnon excitation and dc contacts for applying electric currents as shown in Fig. 9.4c. In order to experimentally verify that the magnetic field generated by the electric current creates similar conditions for magnon transport as an externally applied

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magnetic field oriented perpendicularly to the magnon waveguide, Vogt and coworkers proceeded as follows: In 1 µm distance to the exciting antenna (position A in Fig. 9.4c), the magnon intensities were measured as a function of the excitation frequency by means of µBLS. This was recorded as a function of the externally applied field, without dc current (Fig. 9.5a) and as a function of the applied dc current, without externally applied magnetic field (Fig. 9.5b). If only an external magnetic field is applied, magnons can be detected starting from approximately 10 mT and 2 GHz. Starting from this value the magnitude of the magnetic field is sufficient to overcome the shape anisotropy at least in the middle of the magnon waveguide. For fields below 10 mT hardly any magnon intensity is recorded since the magnetization is still aligned parallel to the magnon waveguide. In this geometry, the magnon dispersion is very flat (Fig. 9.1) and, consequently, the group velocity of magnons is so small that any magnons excited at the microwave antenna do not reach the measurement position. The solid black line in Fig. 9.5a indicates the minimum of the magnon band for modes in a perpendicularly magnetized stripe, which is essentially the magnon frequency for  k^ = 0, because the magnon dispersion monotonously increases for this geometry. The white dashed line marks the magnetic field and frequency combination for which the resulting magnon wavevector coincides with the minimum of the microwave antenna’s excitation efficiency. Note that the solid black and white dashed lines are calculated based on the dispersion relation for a 2 µm wide, perpendicularly magnetized waveguide and taking into account the applied magnetic field. The excellent quantitative agreement of both of these curves demonstrates once again the thorough theoretical understanding of magnon transport in micron sized ferromagnetic conduits. The periodic oscillation of the magnon intensity as a function of the excitation frequency is an artifact of the experimental setup. The absolute microwave power that arrives at the exciting antenna depends on the applied frequency due to impedance mismatches of the microwave contacts to the sample. The scenario without externally applied field, in which the Oersted field of an electric current flowing beneath the magnon conduit magnetizes the magnon waveguide, is shown in Fig. 9.5b. For dc currents below 20 mA no magnons are detected in 1 µm

Steering and Multiplexing Magnons by Current-Induced, Local Magnetic Fields

distance to the antenna, indicating that the magnetic field generated by the dc current is not strong enough to rotate the magnetization perpendicularly to the transport direction. Starting from 30 mA the measured magnon intensities increase. For each current value the spectra as a function of the excitation frequency (each column in the intensity plot) are comparable to Fig. 9.5a, for which the magnon waveguide was magnetized with an externally applied magnetic field. In order to avoid heating, we applied current pulses with duration of 150 ns and a repetition rate of 3 µs. Magnons were excited by 100-ns long microwave pulses synchronized to the dc pulses so that magnons are only excited in the 150 ns time window defined by the current pulses.

Figure 9.5 (a) Magnon intensities measured in the vicinity of the microwave antenna used for magnon excitation as a function of the excitation frequency and the externally applied magnetic field or (b) a direct current flowing in the Py(30)Au(50) magnon waveguide. The black solid line indicates the  lower limit of the magnon band ( k^ = 0 ), the white dashed line marks the magnetic field and frequency combination, where the resulting magnon wavevector coincides with the minimum of the microwave antenna’s excitation efficiency.

The magnon intensities in Fig. 9.5 were recorded in 1 µm distance to the antenna. For selected excitation frequencies, the same data is shown in a different representation in panel A of Fig. 9.6a. We directly compare magnon intensities measured with µBLS as a function of the external magnetic field (lower x axis, empty symbols) and the dc pulse amplitude (upper x axis, filled symbols).

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In the investigated current range, the magnon spectra show the same characteristics, independent of whether an external field or a dc current is applied to align the magnetic moments. A direct current of 54 mA is needed to generate a magnetic field of 10 mT. Duty cycle measurements performed with shorter dc pulses and the same repetition rate yield the same excitation spectra ensuring that Joule heating can be neglected. Note that the maximum applied current density of 1.1 · 1011 A/m2 is too small to expect a contribution of spin transfer torque to the measured magnon intensity [37].

Figure 9.6 (a) Empty symbols show the BLS intensity measured as a function of an externally applied magnetic field for excitation frequencies ranging from 2.1 to 3.9 GHz. Filled symbols show the results of the BLS measurement with dc current pulses through the Au/permalloy hybrid waveguide and without applying an external magnetic field. The measurement positions of panels A, B, and C are indicated in (b); the results were obtained 1 µm, 8 µm, and 11 µm away from the antenna. (c) Two-dimensional magnon intensity distribution with an externally applied magnetic field of 12.3 mT at an excitation frequency of 2.1 GHz and (d–g) applied dc pulses with an amplitude of 66.7 mA for excitation frequencies ranging from 2.1 GHz to 3.9 GHz. The intensity scale is logarithmic and color-coded. Images adopted from [36].

Steering and Multiplexing Magnons by Current-Induced, Local Magnetic Fields

To verify that local magnetic fields generated by dc pulses can turn magnons around the corner, the same measurements were performed right before and inside the curved section. The results are plotted in panels B and C of Fig. 9.6a, respectively. At position B, the magnon intensity starts to increase at higher fields and higher currents compared to the measurements at position A. This can be attributed to forced magnon excitations outside the magnon band that can still be detected in the vicinity of the antenna. These magnons do not propagate along the waveguide and, thus, vanish when moving away from the antenna. However, the results for an applied magnetic field (empty symbols) and for a direct current flowing through the waveguide (filled symbols) show similar behavior. Since the magnetization configuration in region B is the same for both the external magnetic field and the applied direct current, the magnon spectra should not differ. In particular, the rise in the magnon intensity coincides for the direct current as well as for the external fields, which shows that the propagation characteristics of magnons in the straight section of the waveguide is the same for both magnetization configurations. A comparison of the magnon intensity at position B for the highest external magnetic fields and dc currents shows that for some frequencies, there is a strong difference in the measured intensities. The reason for this is that in the case of an externally applied magnetic field, the magnons cannot propagate into the curved section and are reflected. As a result, a standing wave pattern is created in front of the curved section, and the intensity measured at point B oscillates as a function of the applied excitation frequency. Moving further away from the antenna into the curved region of the waveguide, significant differences in the magnon spectra are observed. Panel C of Fig. 9.6 compares the results of the measurements made 11 µm away from the antenna, inside the curved section. As the empty symbols show, no magnons are detected inside the bend when an external field is employed to homogeneously magnetize the waveguide. However, with a current flowing through the hybrid magnon waveguide (filled symbols), the intensity of magnons excited at 2.1 GHz strongly increases when the currents exceed 63 mA, corresponding to a field of about 11.7 mT. For excitation frequencies up to 3 GHz, those differences in the magnon intensity remain

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visible. The overall magnon intensity drops for higher frequencies because their group velocity decreases, which generally results in an increased spatial decay. Due to damping the magnon intensity at the center of the S-shaped bend is strongly attenuated and vanishes altogether behind it. These results show that magnon propagation in curved waveguides is not possible as long as all magnetic moments are aligned in one direction by a global external field. Only when a transversal alignment of the magnetization of the waveguide is maintained it is possible for magnons to turn a corner. To better visualize this result, the two-dimensional magnon intensity distribution was mapped in the bent region of the waveguide. In Fig. 9.6c–g, the results of spatially resolved µBLS measurements are displayed together with the schematic layout of the sample and the corresponding magnetization texture. The intensity scale is logarithmic and color-coded, where white (red) represents minimum (maximum) values. When an external field of 13.3 mT is applied, magnons decay as soon as they enter the curved section of the waveguide (Fig. 9.6c). However, if a direct current flows through the hybrid magnon waveguide, magnons are guided within the curved section for various excitation frequencies ranging from 2.1 GHz to 3.9 GHz (Fig. 9.6d–g). With 66.7 mA, the amplitude of the current pulses was chosen to yield the same magnetic field as for the externally applied. From the measurements shown in Fig. 9.6 two key results are obvious: First, if no external magnetic field or dc current is applied, magnons do not propagate far into the waveguide. The shape anisotropy aligns the magnetization parallel to the waveguides boundaries, which results in low group velocities and, hence, magnons quickly decay. Second, a favorable geometry for magnon transport, where the magnetization is always perpendicular to the propagation direction, can be achieved in micro structured waveguides without applying any external magnetic field. The Oersted field generated by a direct current flowing in a conductor underneath the magnon conduit is sufficiently high to attain transverse magnetization of the waveguide. Employing the direct current for locally aligning the magnetization, magnon propagation inside a curved waveguide can be realized. This demonstrates that artificially guiding the

Steering and Multiplexing Magnons by Current-Induced, Local Magnetic Fields

propagation direction of magnons in the sample plane is possible, which is a crucial step towards new magnon based applications.

Figure 9.7 (a) Microscope image showing the magnon multiplexer (Permalloy, Py), the microwave antenna (CPW: coplanar waveguide) and the leads for dc connections (Au). Blue arrows indicate the oscillating Oersted fields around the antenna for magnon excitation. The leads for the dc connection are designed to allow for an electric current flowing either from bottom to the left or from bottom to the right part of the Y junction. (b, c) Schematics of the magnetization configurations with locally generated Oersted fields (b) and with an externally applied magnetic field (c), respectively. Images adapted from Vogt et al. [33].

In the following part of this section this concept for steering magnons is applied in order to switch the magnon propagation path in a Y junction, effectively building a magnon multiplexer [33]. The fundamental question behind this experiment is: Can one use the anisotropy of the magnon dispersion not only for guiding magnons in bended waveguides but also for actively controlling the propagation path in geometries with multiple options for the magnon propagation direction? To address this question a hybrid magnon waveguide forming a Y junction as displayed in Fig. 9.7a was prepared. In order to optimize the electric current flow for generating the locally oriented magnetic field, the hybrid magnon waveguide was improved compared to the structure used for bending magnons in curved geometries as discussed above. First, the electric current lines were patterned in shape of a Y from 50 nm Au with a width of 3 µm by electron-beam lithography. Subsequently a second Y from MgO (50 nm) and permalloy (30 nm) with a width of 2 µm was patterned on top of the current lines, where the MgO layer ensures

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that the electric current is only flowing in the Au lines. Afterwards, an insulation pad was patterned from a hydrogen silsesquioxane (HSQ, 50 nm) layer covering the entire magnon conduit and then the coplanar waveguides (CPW, 100 nm Au) for magnon excitation were processed on top. The schematic configurations of the magnetization with locally generated Oersted fields and with an externally applied magnetic field are shown in Fig. 9.7b and 9.7c, respectively. If an electric current flows in the Y junction from the bottom to the top right lead, the resulting Oersted field aligns the magnetization in the magnon conduit perpendicularly to the magnon transport direction, whereas the magnetization in the top left arm of the Y junction is still aligned parallel to the transport direction due to the shape anisotropy. Hence, magnons propagating from the bottom into the right part of the junction will always sense a magnetization oriented perpendicularly to their wavevector component in the direction of propagation and have maximum group velocity. Thus, since the magnon transport is connected to the electric current, switching the dc configuration is expected to switch the magnon flow.

Figure 9.8 Magnon intensities measured with µBLS in the left (red circles) and right (blue squares) part of the magnon waveguide in 4.5 μm distance to the excitation antenna (effective propagation distance of the magnons). Magnons are excited at frequencies ranging from 2 to 4 GHz and an electric current of 100 mA was flowing in (a) the left and (b) the right arm of the Y structure. Images adapted from Vogt et al. [33].

In Fig. 9.8 the switching capability of this device based on the anisotropy of the magnon dispersion was quantified. For a current

Steering and Multiplexing Magnons by Current-Induced, Local Magnetic Fields

of 100 mA flowing in the left (Fig. 9.8a) and right (Fig. 9.8b) part of the Y-junction magnon intensities were measured with µBLS in both the top-right arm (red squares) and the top left arm (blue circles) for frequencies ranging from 2 GHz to 4 GHz. There is clear evidence for an overall asymmetry in the magnon propagation between the left and right arm of the Y junction and maximum magnon intensity is observed always in the part of the structure in which the magnetization is aligned by the current-induced magnetic field. Ultimate evidence for the functionality of the magnon multiplexer is obtained from the measured two-dimensional magnon intensity distributions in Fig. 9.9. Excitation with a frequency of 2.75 GHz, which showed maximum magnon transmission in Fig. 9.8 for an electric current of 100 mA, demonstrates switching of magnon beams in the direction of the current flow. Figure 9.9c shows that magnon propagation is entirely blocked behind the Y junction if a uniform external magnetic field is applied, as discussed in the previous section. Magnons are known to adiabatically adjust their wavevector in regions of varying internal magnetic fields to compensate for the resulting shifts of the dispersion [38–43]. But, apparently, magnons do not propagate in stripe-shaped magnon waveguides if the relative orientation between the magnetization direction and the magnon wavevector changes.

Figure 9.9 (a, b) Two-dimensional map of the magnon intensity illustrating the switching process: For a direct current of 100 mA applied to the left (a) and right (b) arm of the Y structure, respectively, the spin waves at 2.75 GHz are guided through the Y junction and only propagate in the same direction as the current flow. (c) Saturation with an external magnetic field completely blocks magnon propagation beyond the junction. Image adapted from Vogt et al. [33].

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To summarize this section, magnon transport can be guided in curved magnonic waveguides and even switched in more complex structures such as a Y junction using locally generated magnetic fields. Note that the dimensions of the magnon conduits discussed here were chosen to allow for optical characterization of their operation. However, there are no physical limitations for miniaturizing them into the nanometer regime, because the important parameter for the local magnetic fields is the current density below the magnon conduit. Therefore, smaller devices will also require less absolute values of electric currents. Regarding materials for the magnon conduits, the same approach can be adapted to ferrimagnetic insulators, such as yttrium iron garnets, or metallic compounds with ultralow magnetic damping [28, 44], in which the magnon propagation distance is significantly increased. Another promising approach to reduce the currents required in order to magnetize the magnon conduit perpendicularly to the transport direction is to use ion implantation to imprint magnonic waveguides into paramagnetic [45] or ferromagnetic films [46, 47]. In such magnon waveguides the shape anisotropy is reduced due to the surrounding ferro- or paramagnetic environment. Lastly, while we use Oersted fields to locally rotate the magnetization, it is also conceivable to use programmable, nonvolatile stray fields from adjacent magnetic elements [48] or direct exchange coupling [49] which would enable a low-power operation of the magnon multiplexer in more complex magnonic devices.

9.4 Channeling Magnons in Magnetic Domain Walls

In the previous section we introduced the concept of using the anisotropy of the magnon dispersion for steering magnons by a local, magnetic field induced rotation of the magnetization. In this section we transfer this knowledge to a new concept that also uses a locally rotating magnetization direction but does not require any externally applied magnetic fields or electric currents for operation. We explore the intrinsic magnons that are confined across the width of a magnetic domain wall and possess a well-defined wavevector

Channeling Magnons in Magnetic Domain Walls

along the wall [50]. By targeting this class of magnons in Néel walls, we focus on the potential of using domain walls as nanometer-scaled channels to open new perspectives for energy-efficient control of magnon propagation in two dimensions. The top part of Fig. 9.10a shows a schematic of a 180° Néel wall. The magnetic moments rotate within the sample plane, giving rise to magnetic volume charges, with opposite sign on the two sides of the domain-wall center. The red curve in Fig. 9.10a displays these volume charges as a function of position perpendicular to the domain wall calculated by means of micromagnetic simulations. They create a strong dipolar magnetic field (blue curve in Fig. 9.10a) oriented antiparallel to the magnetization in the center of the domain wall, resulting in a locally decreased effective magnetic field. Since the magnon energy depends on this internal magnetic field, the domain wall effectively forms a potential well for magnons comparable to magnons localized near edges of magnetic elements [38, 51]. The width of this potential well is given by the domain wall width and strongly depends on various parameters of the magnetic material. The Néel wall studied here has a width of about 40 nm. However, note that it can be tuned towards even smaller sizes if other materials are used. In the following, we demonstrate experimentally and numerically that such a potential well hosts magnons that are strongly localized to the domain-wall width but still travel freely along the wall. Figure 9.10b presents a scanning electron microscopy (SEM) image of the structure used in the experiment. A 40 nm thick permalloy film is patterned into a magnon waveguide that is 5 μm wide at one end and gradually broadens until it reaches a constant width of 10 μm. This variable width stabilizes the 180° Néel wall under the microwave antenna. A domain wall parallel to the long axis of the waveguide is initiated by applying a sinusoidal, exponentially decaying magnetic field parallel to its short axis, as depicted in the bottom right inset in Fig. 9.10b. Magneto-optical Kerr microscopy was used to confirm the magnetic remanence state of the magnon waveguide. The overlay in Fig. 9.10c shows a Kerr micrograph, with the black–white color code representing the magnetization component along the y direction. A Landau-like domain pattern is formed with a 180° Néel wall in the center separating two domains with opposite magnetization. Additional micromagnetic simulations

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(discussed in the following) confirm the Néel-type character of the domain wall, with the magnetic moments rotating in the sample plane as is expected for the material parameters and thickness of the Py film [52].

Figure 9.10 (a) Schematic illustration of a 180° Néel wall carrying a  magnon. The divergence of the magnetization div(M) across the width of the wall shows opposite sign on the two sides of the domain wall center and  results in a strong magnetostatic field Hdemag oriented antiparallel to the magnetization direction. (b) Scanning electron microscopy image showing the magnon waveguide and the microwave antenna. Inset (lower right): magnetic field sweep used to initialize the domain configuration. (c) The black–white contrast of the Kerr microscopy image confirms the formation of a Landau-like domain pattern. Arrows indicate the magnetization direction in the domains. Image adapted from Wagner et al. [50].

To analyze the magnon spectra in different areas of the waveguide, the magnon intensity was measured as a function of the excitation frequency at two different positions: inside the domain wall and in the domains. These measurements were performed in the 10 μm wide bottom part of the waveguide at a distance of 1 μm from the antenna. Figure 9.11a summarizes the results and clearly shows two distinct spectra. Inside the domain wall (square symbols), the maximum intensity was observed at the lowest possible detection frequency of 500 MHz, whereas in the domains (triangular symbols) the highest intensities can be found at frequencies around 2.8 GHz. To understand the nature of these different magnons, their intensity profiles across the width of the waveguide were detected and are plotted in Fig. 9.11b. Measurements were carried out at the excitation frequencies where the maximum intensity was observed within the wall and the domains, respectively. Figure

Channeling Magnons in Magnetic Domain Walls

9.11b summarizes the results. The magnons show a clear spatial separation, with magnons at 0.5 GHz strongly confined inside the domain wall (squares). In contrast, magnons at 2.8 GHz (triangles) are spread throughout the domains at both sides of the wall and almost vanish at the domain-wall position. Thus, the potential well formed by the domain wall can be used to confine magnons in a certain frequency range on the nanoscale. The slightly different intensity profiles detected at 2.8 GHz in the two domains are attributed to a misalignment of the domain wall relative to the center position. This asymmetry leads to different lateral confinement conditions in the domains and, consequently, to a relative frequency shift of the excitation spectra. However, quantization due to confinement in micrometer-size waveguides has already been studied extensively [14–19, 25–33]. Here, we focus on magnons inside the nanochannel formed by the domain wall. The strong confinement becomes even more evident from the two-dimensional µBLS measurement presented in Fig. 9.11c. Solely at the position of the domain wall, the signal of the magnons propagating away from the antenna is detected, revealing the channeling character of the domain wall.

Figure 9.11 (a) Magnon intensity as a function of excitation frequency inside the domain wall (squares) and the domains (triangles), respectively. The measurement clearly shows different excitation spectra depending on the probing position. (b) Line scans of the magnon intensity across the width of the Py waveguide for two excitation frequencies that yielded maximum magnon intensity inside the domain wall (squares) and in the domain (triangles). (c) Two-dimensional magnon intensity distribution at 0.5 GHz. Images adapted from Wagner et al. [50].

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To obtain a deeper insight into the propagation characteristics within the wall, micromagnetic simulations were carried out [53]. The modeled system was a 5 μm long, 1 μm wide, and 10 nm thick Py rectangle. This reduction in size compared to the experiment allowed for a sufficiently fine discretization required to simulate the nanosize domain wall without affecting its character, which is mainly determined by the material parameters and not by the different dimensions. Therefore, the comparison between experiment and simulation is still assumed to be valid for magnons within the domain wall. Figure 9.12a shows the equilibrium magnetization configuration. The red–blue color codes the magnetization component along the short axis of the rectangle, and the arrows indicate the magnetization direction. The microstructure exhibits a flux-closure Landau domain pattern with a 180° Néel wall separating two domains with opposite magnetization.

Figure 9.12 (a) Simulated domain configuration of a rectangular Py thinfilm element. The red–blue color represents the in-plane magnetization component parallel to the short axis of the rectangle, and the arrows display the net magnetization direction in the domains. (b) The x component of the effective magnetic field. (c) Amplitude profiles of magnons excited locally at the green dot. The red–blue color represents the out-of-plane component of the magnetization. Green bars indicate the magnon wavelengths that can be extracted from the data. (d) Dispersion relation extracted from simulations. Images adapted from Wagner et al. [50].

The magnitude of the effective magnetic field calculated from this remanence state is presented in Fig. 9.12b with the red–blue

Channeling Magnons in Magnetic Domain Walls

color representing the field component along the short axis of the rectangle. The data show strong effective fields across the domain wall oriented antiparallel to the magnetization direction. An out-of-plane field pulse with a Gaussian spatial profile was used to locally excite spin dynamics at the domain-wall position at a distance of 1.6 µm from the bottom edge (green dot in Fig. 9.12c). The subsequent analysis yields the magnon spectrum and dispersion relation, which is discussed in the next paragraph. To illustrate the mode profiles of the magnons, the response to a continuous microwave excitation at four different frequencies was simulated. In Fig. 9.12c, the normalized z component of the magnetization is plotted for a given time once the system reached a steady state. For the lower frequencies of 0.52, 1.28 and 2.16 GHz, the magnons are strongly localized inside the domain wall. However, a general trend is that the strong localization within the wall softens with increasing frequency. Particularly for modes with higher frequencies, for example 5.68 GHz, the radiation from the domain wall indicates the onset of the magnon band in the domains and, hence, loss of the channeling effect. Although the strong localization for lower frequencies inside the domain wall is in good agreement with the experiment, the simulated domain spectra appear at higher frequencies than in the experiment. In general, magnons shift to higher frequencies when reducing the domain size, which therefore allows the guiding of magnons inside the wall, even for higher frequencies and smaller wavelengths. In addition to the spatial and spectral characteristics, the micromagnetic simulations also shed light on the dispersion relation. The wavelength for a given frequency—illustrated by the green bars in Fig. 9.12c—can be determined by a Fourier analysis of the dynamic magnetization along the domain wall. Figure 9.12d shows the resulting positive dispersion that enables the transport of information via magnons propagating within the domain wall. Even though the magnons are confined transverse to the wall on a length scale given by the domain wall width, the dispersion is mainly dominated by dipolar energy. For the first three modes shown in Fig. 9.12c, the dynamic part of the dipolar energy originating from the magnons is by a factor of three to five larger than the dynamic

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exchange energy. The observation of well-defined wavevectors along the propagation path is a crucial precondition for numerous concepts that rely on the interference of magnons [6, 9, 54–58] and highlights the potential of domain walls in magnonic circuits for data processing. In contrast to the previously mentioned geometrical and, therefore, rigid designs for confining magnon transport, magnon nanochannels based on domain walls allow for intriguing flexibility by actively controlling the magnon propagation path. The domain walls and, hence, the magnon flow can be manipulated by several means, including magnetic fields, charge or spin currents. Such manipulations open a new perspective for the control of magnon transport in two-dimensional nanostructures and towards the realization of reconfigurable magnonic circuits. The position of the domain wall was controlled via external magnetic fields applied along the long axis of the waveguide. Depending on the polarity of the field, the growth of either the left or the right domain was favored, resulting in a shift of the domain wall. Figure 9.13 shows µBLS scans across the width of the waveguide for magnetic fields of −0.15, −0.05, 0.05, and 0.23 mT. The measurements show that the detected magnons are shifted, together with the domain wall, by the applied field. The overall left–right asymmetry is attributed to the initial displacement of the domain wall, even in the absence of external fields. In fact, the magnon propagation path can be moved over a distance of 2 µm within a field range of only ΔH = 0.38 mT. The inset in Fig. 9.13 illustrates this displacement as a function of field with a proportionality constant of 5.57 µm per mT. These data establish a novel mechanism to control magnon transport in two dimensions. Moreover, they pave the way for reconfigurable but nonvolatile magnonic nanocircuitry. In future scenarios, artificial pinning centers—for instance induced by ion implantation [59, 60]—might be used to enable switching between stable remanence states, with different domain-wall configurations acting as nanochannels to guide magnons in logic devices. Furthermore, multiple domain walls with a separation of a few tens of nanometers can form in materials with perpendicular magnetic anisotropy [61–64], which allows for an even higher integration density of these magnonic nanochannels.

Channeling Magnons in Magnetic Domain Walls

Figure 9.13 Magnon intensities measured across the waveguide for different external magnetic fields applied parallel to the long axis of the magnon waveguide for 0.52 GHz. The measurements show that the domain wall can be easily shifted with small applied fields over micrometer distances in both directions, therefore allowing for fine control of the magnon channel position. Inset: displacement of the magnon propagation path as a function of applied magnetic field, yielding a proportionality constant of 5.57 µm per mT. Image adapted from Wagner et al. [50].

In this section, we discussed experimental and numerical results on the channeling of magnons in nanometer-wide magnetic domain walls with a width of about 40 nm. Micromagnetic simulations allowed for further analysis of the propagation characteristics. Magnons propagating inside domain walls exhibit a well-defined wavevector along their propagation path, enabling data transport and processing using wave properties. Finally, we have demonstrated a major advantage of domain-wall-based magnonic waveguides: manipulating the domain configurations with tiny fields below 1 mT allows shifting the propagation path over a distance of several micrometers. These observations pave the way for the realization of reconfigurable, nonvolatile magnonic circuitry by switching between different remanence states and, thus, for the realization of energy-efficient and programmable magnon logic devices on the nanoscale.

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9.5 Conclusions and Outlook In this chapter, we discussed methods for steering magnons laterally in two-dimensional structures. Both presented concepts—active rotation of the magnetization by local, current-induced magnetic fields and channeling magnon flow inside magnetic domain walls— are based on the anisotropy of the magnon dispersion relation and the magnons’ sensitivity to effective magnetic fields. We are excited about the future opportunities for magnon transport arising from novel material compositions and multilayer geometries just being discovered. The experimental evidence of asymmetric exchange interaction opened new perspectives: Due to the chiral nature of magnons the dispersion relation in samples with interfacial Dzyaloshinskii–Moriya interaction (iDMI) is not only anisotropic but it is also nonreciprocal. Magnons with the same frequency have different wavelengths upon inversion of the propagation direction, i.e., reversal of the magnon wavevector [65, 66]. It has been demonstrated in micromagnetic simulations that domain walls in such systems allow for nonreciprocal and even narrower magnon channels operating at even higher frequencies [61, 62] and that these domain walls can be programmed in microstructures by electric currents. In a recent work it was shown [63] that one can combine the magnon switch based on a Y junction and the channeling inside domain walls in an innovative way, where the domain wall path acting as a magnon guide is permanently switched by a short electric current pulse. Furthermore, nonreciprocal effects modifying the magnon dispersion relation are not limited to material compositions with iDMI, but can also be observed in antiferromagnetically coupled multilayers [67] or on curved surfaces [68]. Combining different classes of materials and making use of their richness in topological spin textures sheds new light on magnonics. Spin textures such as skyrmions are on the real nanoscale and nonvolatile but still can be manipulated by electric currents or magnetic fields. The marriage of both topological spin textures and magnonics, as for example the magnon Hall effect [69], could link both high-frequency information carriers and long-time information storage in one unified, nanoscale framework.

References

Acknowledgments The samples studied in the original publications were prepared at the Center for Nanoscale Materials (CNM) and the Materials Science Division (MSD) at the Argonne National Laboratory as well as at the Nanofabrication Facilities (NanoFaRo) at the Institute for Ion Beam Physics and Materials Research at the Helmholtz-Center DresdenRossendorf (HZDR). Without these facilities and the commitment of the people behind them we would not be able to conduct our experimental research. Therefore, special thanks go to L. E. Ocola and R. Divan (CNM), J. E. Pearson (MSD), and T. Schönherr, C. Neisser, B. Scheumann, and A. Erbe (HZDR). Furthermore, we are grateful to A. Henschke and T. Hula for their support in designing, building, and maintaining the experimental setups. The work discussed in this chapter was inspired by ideas, contributions, and discussions from F. Y. Fradin, S. D. Bader, A. Hoffmann, B. Hillebrands, T. Sebastian, J. Lindner, and J. Fassbender. K.S. acknowledges funding from the Helmholtz Postdoc Programme. Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged within programme SCHU2922/1-1.

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Chapter 10

Current-Induced Spin Wave Doppler Shift Matthieu Bailleula and Jean-Yves Chauleaub, aInstitut

de Physique et Chimie des Matériaux de Strasbourg, UMR7504 CNRS-Université de Strasbourg, F-67000 Strasbourg, France bLigne SEXTANTS, Synchrotron SOLEIL, L’orme des Merisiers Saint-Aubin, F-91192 Gif-sur-Yvette, France [email protected]

When subjected to an electric current, the electrons of a metal ferromagnet are put in motion, which translates into a Doppler shift for the spin waves. Predicted a long time ago, this phenomenon has recently been measured by several groups in different experimental situations. In this chapter, we will discuss the interpretation of this effect in terms of adiabatic spin transfer torque and its measurement using both inductive (frequency or time domain) and magneto-optical techniques. We will also discuss how the spin wave Doppler shift can be used to extract relevant information on spin-polarized electrical transport and present a related effect, namely the current-induced modification Formerly at Department of Physics, Regensburg University, 93053 Regensburg, Germany. Spin Wave Confinement: Propagating Waves (2nd Edition) Edited by Sergej O. Demokritov Copyright © 2017 Pan Stanford Publishing Pte. Ltd. ISBN  978-981-4774-35-2 (Hardcover),  978-1-315-11082-0 (eBook) www.panstanford.com

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of spin wave attenuation associated with non-adiabatic spin transfer torque.

10.1  Introduction

Since the discovery of spin transfer torque (STT) in the late 1990s, a number of current-induced effects have been explored experimentally, interpreted theoretically, and, for some of them, even put in application. Let us mention current-induced domain wall motion in nanostrips and current-induced magnetic switching or current-induced magnetic oscillations in magnetoresistive nanopillars or nanocontacts. All these effects involve the flow of spin-polarized electrons (the so-called spin current) in singleor multilayer ferromagnets having a non-uniform magnetic configuration. A spin wave, whose instantaneous magnetization distribution provides such a non-uniform magnetic configuration, is also subjected to spin transfer. In this case, the STT translates into a simple shift of the spin wave frequency. This shift can be identified as a Doppler effect. Indeed, everything happens as if the whole spin system were drifting at an effective velocity u = Q/Ms, where Q is the spin current expressed in Bohr magneton per unit time and MS is the saturation magnetization. This so-called currentinduced spin wave Doppler shift (CISWDS) had been predicted theoretically in 1966. It has been observed experimentally in 2008. Since that time, the CISWDS has proven to be a relevant tool for measuring spin-polarized current in different materials and for different experimental conditions. In 2012, another effect, namely the current-induced modification of spin wave relaxation, has been observed that is related to a non-adiabatic correction of the standard (adiabatic) STT. In this chapter, we review the work done on current-induced modification of spin wave dynamics. For this purpose, we first discuss the general idea of a Doppler effect for spin waves and derive the expression of the frequency shift from the standard equations of the STT (Section 10.2). Then, we present the three experimental approaches that have been used to observe the effect (Section 10.3) and discuss how such measurements can be used to extract spin-dependent resistivities, which are essential ingredients in the description of spin-polarized diffusive electrical

A Doppler Shift for Spin Waves

transport (Section 10.4). Then, we describe the other effects that accompany the CISWDS, namely the frequency shifts induced by magnetic asymmetries and by the Oersted field generated by the current (Section 10.4) and the current-induced modification of spin wave relaxation, which allows one to evaluate the nonadiabatic correction to the STT (Section 10.5). Finally we conclude and give general perspectives (Section 10.6).

10.2  A Doppler Shift for Spin Waves

10.2.1  Spin Waves in a Drifting Electron Population A typical example of the Doppler effect is the firefighter siren, which sounds higher pitched when the vehicle is moving toward the listener than when it is moving away. A similar effect arises when both source and observer are immobile and the medium supporting the propagation of waves is moving. This situation is illustrated in the second panel of Fig. 10.1a, which shows water surface waves generated by the oscillation of a wave maker: in the presence of a global water current, the wavelength is longer for water waves moving downstream than for waves moving upstream [1]. The same situation is expected to occur for a metal ferromagnet subjected to an electric current. Indeed, the conduction electrons that govern the electrical transport also participate in the magnetism of the material. As a consequence, putting these electrons in motion with the help of an external electrical field will induce some linear momentum in the spin system, which will affect the waves carried by it, namely spin waves. In a first approximation, the kinematics of the spin system can be modeled as a global drift with velocity vd. Let us now consider a localized and immobile source inducing a precession of the magnetization; this will generate propagating spin waves whose wavelength will be different for upstream and downstream propagations (bottom panel of Fig. 10.1a). For illustration purposes, the figure shows a situation in which the frequency is fixed and the wavelength adapts to the spin (or water) flow. The same effect can be described in terms of a frequency shift for spin waves generated at a given wave vector: the magnetization precession in the frame R’ moving with the spins is written as M(x¢, t) = Meq +

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m0 exp[i(wt – k x)], where Meq is the equilibrium magnetization, m0 is the complex spin wave amplitude, k is the wave vector, w = w0(k) is the angular frequency, and w0(k) is the spin wave dispersion relation governed by the magnetic interactions acting on the spin system. Then, the magnetization evolution in the rest frame R is obtained from the Galilean transform x = x + vdt, which yields M(x, t) = Meq + m0 exp[i(wt – kx + kvdt)], thus leading to a Doppler-shifted angular frequency w = w + kvd.

Figure 10.1  (a) Sketches illustrating the Doppler effect for water waves and spin waves in a drifting medium. (Top) Water surface waves generated by the oscillation of a wake maker. (Middle) Same in the presence of a water current. (Bottom) Spin waves generated locally in a magnetic medium subjected to a global drift. (b) Calculated spin wave dispersion in the absence of drift (full line) and for large drift velocities (dashed lines). The long dashed line shows a hypothetical situation in which the electric current is strong enough to induce instability of the uniform ground state. After McMichael and Stiles [1] and Fernandez-Rossier et al. [4].

This idea of a Doppler effect for spin waves was first introduced by Lederer and Mills as a possible test of the nature of ferromagnetism in 3D transition metals (itinerant vs. localized) [2]. It was argued that the observation of a sizeable currentinduced Doppler shift for spin waves would be a strong indication that the carriers of magnetism are actually band electrons, because the effect of an electric field on purely localized spins was calculated to be much smaller. At that time, the available macroscopic-scale experimental techniques did not allow to explore this prediction. Indeed, drift velocities in bulk metal do not exceed a few centimeters per second, which, for a wavelength

A Doppler Shift for Spin Waves

of a few millimeters, translates into a frequency shift of a few tens of hertz, which is much smaller than the typical spin wave frequencies, of the order of a few gigahertz. About 30 years later, with the development of nanofabrication and high-frequency measurement techniques, it became possible to inject much larger current densities in nanostructures and also to measure spin waves with a much shorter wavelength. Interestingly, the effect was also revisited theoretically in a different context: Bazaliy et al. derived the  CISWDS within the general framework for spin transfer [3] and Fernandez-Rossier et al. [4] obtained it from different theoretical descriptions of itinerant magnetism, including the spin-polarized Hubbard model [4]. In the following section, we will describe explicitly the derivation of the effect from the basic equation of motion of the magnetization, including the STT.

10.2.2  Influence of Spin Transfer Torque on Spin Wave Dynamics

In the micromagnetic framework, the action of a spin-polarized electric current on the local magnetization can be modeled by augmenting the Landau–Lifshitz–Gilbert (LLG) equation with two STT terms as follows:

dM M dM M = – 0 gM × Heff + a × –(u .  )M + b [(u .  )M)], dt Ms dt Ms

(10.1)

with M(r, t) being the local magnetization, Heff the effective magnetic field, a the Gilbert damping parameter, and u the spin drift velocity. This latter quantity, which has the dimension of velocity, is proportional to the spin-polarized current density in the following way:

u=

– g B JP , 2 Ms | e |

(10.2)

with J being the electrical current density, P its spin polarization, Ms the saturation magnetization, and | e | the absolute value of the electron electrical charge. As electrons responsible for electrical conduction flow through a non-uniform magnetic distribution,

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their spins tend to align with the local magnetization direction because of s-d exchange interaction. This process of realignment results in a torque generated onto the local magnetization by virtue of the principle of conservation of angular momentum. In the approximation where magnetic moment of the conduction electron fully aligns with the local magnetization direction, this torque is given exactly by the first STT term (third term on the right-hand-side of Eq. 10.1), commonly named adiabatic STT [3]. In continuous magnetic distributions, the STT will take place when a spin-polarized current crosses regions with a finite gradient of magnetization (Fig. 10.2a). Such a situation occurs in current-induced domain wall motion experiments (see, for example, Ref. [5] and references wherein). However, it has been shown that the current density threshold calculated for putting a domain wall in motion by the adiabatic STT is at least 1 order of magnitude higher than the one reported in these experimental observations [6]. A second term accounting for deviations to the adiabatic approximation and weighted by the b parameter was, therefore, needed [6, 7]. This second STT term (fourth term on the right-hand-side of Eq. 10.1) is named non-adiabatic STT. Note that this torque is perpendicular to both the local magnetization and the adiabatic STT, [6]. The possible origins of the non-adiabatic STT will be discussed later in this chapter.

Figure 10.2  Schematic representation of the adiabatic spin transfer torque (a) and its consequences on spin wave characteristics (b, c). Reproduced from Vlaminck [8].

Experimental Observations

Spin waves are dynamical non-uniform magnetization textures which have, in the simplest description, a constant gradient characterized by the complex wave vector k. Subjected to a spin-polarized flow of electrons, they are therefore expected to be affected by the STT (Fig. 10.2b) [8]. Considering spin waves as plane waves with small deviations m(r, t) from the equilibrium magnetization (Meq), that is, m(r, t) = m0 exp[i(wt – k . r)] and, the linearized LLG becomes   

i( w – u . k )m = – 0 g(m × Heq + Meq × h )+

iw  bu . k  a – M × m , w  eq MS 

(10.3)

where Heq and h are the equilibrium and dynamic parts of the effective field. Interestingly, one can note that the here-above equation is similar to the spin wave equation in the absence of the STT if one proceeds to the following substitutions [8, 9]:

wres  wres + u . k(adiabatic STT),

(10.4)



a  a–

(10.5)

bu . k (non-adiabatic STT). w

This means that while the adiabatic STT leads to a frequency shift of the spin wave mode, the non-adiabatic STT affects the dissipative part and modifies the apparent damping coefficient (Fig. 10.2b,c). Both changes are proportional to u, which reflects the essence of the STT. Current-induced modifications of spin wave dynamics are therefore a direct approach to probe the STT. Note that the spin drift velocity u plays the role of the global drift velocity vd introduced in Section 10.2.1.

10.3  Experimental Observations

10.3.1  Frequency Domain Inductive Measurements A convenient method for measuring spin wave propagation with a good frequency resolution consists in using a pair of narrow conducting lines (the so-called spin wave antennas) inductively

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coupled to the ferromagnetic film and to measure the microwave transmission between these two antennas. This technique, known as propagating spin wave spectroscopy (PSWS), was introduced in the late 1960s at the millimeter scale to measure spin wave propagation in yttrium-iron-garnet (YIG) slabs [10] in which spin waves could propagate over such long distances. In the early 2000s, this technique was miniaturized to the micrometer scale to adapt to the much shorter attenuation length of metal ferromagnets [11]. Measuring the CISWDS, it has been necessary to miniaturize the technique further to the submicrometer scale [9, 12] so as to increase the wave vector to a few radians per micrometer (see the device pictures in Fig. 10.3a,b and the corresponding spatial Fourier transform in Fig. 10.3c).

Figure 10.3 Inductive frequency domain measurement of the currentinduced spin wave Doppler shift. (a) Optical micrograph of a device showing the ferromagnetic strip, DC pads, coplanar waveguide ports, and two antennas. (b) Scanning electron microscope image showing the central part of the strip and the two antennas. (c) Fourier transform of the spatial distribution of the microwave current flowing in one antenna. (d) Principle of the propagating spin wave spectroscopy measurement. (e) Propagating spin wave waveforms measured for a DC current of +6 mA. The red and blue curves correspond to k > 0 and k < 0, respectively. (f) Same for a current of –6 mA. After Vlaminck and Bailleul [9].

The principle of the technique is illustrated in Fig. 10.3d. Each antenna consists of a set of conducting tracks connected such that the injected microwave current alternates in direction

Experimental Observations

between adjacent tracks. The microwave current generates a microwave field around each track, which, in turn, generates a strongly non-uniform microwave magnetic field. This field forces the precession of the magnetization in the ferromagnetic film located underneath, thus exciting spin waves with a wavelength equal to the pitch of the antenna. The excited spin wave propagates in the ferromagnetic film and reaches a second antenna located at a distance D on which it generates a microwave voltage. The response functions for such measurements are the mutual inductance ΔLij, which is the ratio of the time-varying magnetic flux at the detection antenna i to the microwave current injected in the excitation antenna j (i, j = 1, 2). These mutual inductances are conveniently extracted from the S parameters measured by connecting the two ports of a microwave vector network analyzer to the antennas. Due to the phase delay accumulated by the wave during its propagation between the two antennas, the measured mutual inductances are oscillating signals with a period equal to the inverse of the spin wave group delay time. For a typical group velocity of about 1 km/s and a distance between antennas of a few micrometers, this time amounts to a few nanoseconds, resulting in an oscillation with a period of a few hundreds of megahertz [12]. To measure the CISWDS, a direct electric current is injected along the ferromagnetic metal strip during the microwave measurements, and the mutual inductances DL21 and DL12 are measured as a function of the direct current (DC) [9]. DL21 corresponds to a wave propagating from antenna 1 to antenna 2, that is, k > 0, and DL12 corresponds to a wave propagating from antenna 2 to antenna 1, that is, k < 0. Figure 10.3e,f shows the waveforms measured for a current of +6 mA and –6 mA, respectively. One recognizes clearly a frequency shift of about 18 MHz between the two waveforms, which, as expected for a Doppler effect, changes sign upon reversing the current. This frequency shift was measured as a function of the current, wave vector, external field, and ferromagnetic strip width to make sure it follows the predicted form of Eqs. 10.2 and 10.5. In this first observation, a degree of spin polarization of the electric current of about 0.5 was extracted. We will discuss in detail the meaning of this value in Section 10.4.

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Let us mention possible artifacts likely to affect this measurement (see supplementary materials of Ref. [9]). Because a relatively large current is applied in DC mode, a sizeable Joule heating of several tens of Kelvin is observed. This effect can be ruled out by comparing measurements taken for the same magnitude of the current, that is, at the same temperature. The electric current also generates a sizeable Oersted field (about 1 mT on each film surface), which might perturb the spin dynamics. However, for the measurement mode used here, this does not influence the extracted frequency shift. Indeed, the measurement is performed in the so-called magnetostatic forward-volume wave configuration (equilibrium magnetization saturated out of the film plane by a large external magnetic field H0). This configuration is known to be reciprocal for spin wave propagation (i.e., the –k spin wave has the same amplitude and mode profile as the +k one), so the Oersted field affects in the same way two counterpropagating spin waves, and the frequency shift between +k and –k is expected to be solely due to the Doppler effect. This simplification cannot be invoked for the so-called magnetostatic surface wave (MSSW) configuration (equilibrium magnetization in the film plane, perpendicular to the spin wave vector), which is intrinsically non-reciprocal. This configuration, used in subsequent observations of the CISWDS [13–18], mostly because it is easier to implement, requires therefore specific data analysis procedures, which will be explained in Section 10.6. Before discussing the other techniques used, let us review briefly the pros and cons of PSWS.  Because it works intrinsically in the frequency domain, this technique has the advantage of giving directly access to the frequency shift. Another strong point is the relatively good definition of the wave vector allowed by the meander design of the antennas. Main disadvantages are the difficulties of fabricating such meanders with track widths and interspacings of the order of 100 nm and the need for a careful microwave calibration before network analyzer measurements.

10.3.2  Time Domain Inductive Measurements

Inductive time domain PSWS [19] has proven to be an interesting alternative approach [20]. A burst of magnetic field that is created

Experimental Observations

by a first asymmetric coplanar stripe (Fig. 10.4) triggers a several nanosecond-wide spin wave pulse in a large permalloy element (typically 120 × 120 µm² and 35 nm thick). After propagating through the permalloy layer, the spin wave pulse induces an inductive signal in a second asymmetric coplanar stripe (Fig. 10.4b), which is detected with time resolution.

Figure 10.4 (a) Experimental configuration for time domain inductive investigation of the current-induced spin wave Doppler effect. (b) Induced inductive signals detected after spin wave propagation for different gaps (x) between the two antennae. (c) Dependence of the shift in group velocity on the applied current density. (d) Extracted values of the degree of spin polarization for different samples with different antenna gaps (x). Adapted from Sekiguchi et al. [20].

In this approach, spin waves are characterized by wave packets and the main experimental observables are the time delay and the amplitude of the spin wave packet, as well as their current-induced changes. First of all, by assessing the delay and the amplitude as a function of the distance between the antennae, the spin wave group velocity (here Vg = 13.1 km/s) and the decay length (here l = 15 µm) can be extracted. A key parameter of magnetization dynamics, namely the damping parameter (a), can be deduced from the decay length. Then, in the presence of an electric current, the spin wave group velocity is augmented by the spin drift velocity:

DVG = u.

(10.6)

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Current-Induced Spin Wave Doppler Shift

For nanosecond current density pulses of 0.8 × 1011 A/m² (equivalent to a current intensity of 330 mA for the described geometry), a change of group velocity of about 4 m/s could be measured. As expected, this current-induced shift in group velocity varies linearly with the amplitude of the applied electric current (Fig. 10.4c). By assessing the slope of this variation, the degree of spin polarization P = 0.6 could be extracted. This analysis has been performed for samples with different antenna gaps and shows fairly robust reproducibility (Fig. 10.4d).

10.3.3  Magneto-Optical Measurements

Besides inductive techniques, another experimental approach can be used to probe current-induced spin wave dynamics, namely time-resolved magneto-optical Kerr microscopy (TR-MOKE). As denominated, the heart of this approach is the magneto-optical Kerr effect, which manifests itself as a modification of light polarization characteristics (direction and ellipticity) after being reflected upon a magnetic layer. Although all components of magnetic distributions can, in principle, be probed [21]; for currentinduced spin wave dynamics investigations, solely the out-of-plane component is accessed in a so-called polar MOKE configuration. Characteristic timescales of magnetization dynamics in ferromagnetic systems are typically in the range of hundreds of picoseconds. Therefore, the use of TR-MOKE to probe magnetization dynamics has been made possible, thanks to the emergence of mode-locked lasers with subpicosecond light pulse durations. The system used to investigate spin wave dynamics is an 80 MHz repetition rate mode-locked Ti-sapphire laser, which provides 150–200 fs laser pulses. The access to magnetization dynamics with TR-MOKE is based on an optical stroboscopic pump-probe technique, which provide a snapshot of the dynamical out-of-plane component of the magnetic configuration at a given excitation time (Fig. 10.5). The potential of TR-MOKE to investigate magnetization dynamics processes has been demonstrated by the pioneer work of Hiebert et al. [22], where the temporal evolution of magnetic distributions in micron-size circular shaped permalloy structures is assessed in the time domain after being subjected to short magnetic field pulses. In addition, it was demonstrated that by focusing the laser pulse, local magnetic

Experimental Observations

resonance measurements can be performed with a spatial resolution limited by the diffraction limit of far-field visible optics, that is, a few hundreds of nanometers. Consequently, by measuring locally magnetization dynamics as a function of the distance to a localized excitation, propagating spin wave dynamics can be accessed and even imaged, as demonstrated in nonpatterned permalloy thin layers [23, 24]. TR-MOKE has been used in a large variety of studies of magnetization dynamics, including investigations related to magnonics.

Figure 10.5 Principle of TR-MOKE as used for spin wave Doppler investigations. Adapted from Bauer et al. [41]. (Upper left inset) Schematic representation of the synchronization of short light pulses with out-of-plane oscillations of the magnetic component. (Upper right inset) Typical experimental spin wave line scan.

Regarding current-induced modifications of spin wave dynamics, an 855 nm wide and 15 nm thick permalloy stripe is continuously excited at a fixed frequency by a continuous wave (CW) microwave magnetic field delivered by a unique gold antenna (Fig. 10.5) similar to the ones described for the inductive techniques (see Section 10.3.1). A 400 nm wavelength probing light pulse, phase-locked on the CW microwave excitation (Fig. 10.5), is focused on the sample using an objective lens allowing a spatial resolution of about 300 nm. Note that an additional in-plane static magnetic field is applied and kept constant to adjust the spin wave resonance conditions. By

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Current-Induced Spin Wave Doppler Shift

scanning the laser spot over the permalloy stripe, an image of the excited spin wave mode is thus acquired (Figs. 10.5 and 10.6a–c). Alternatively, the position of the laser spot can be kept at a given position, while the magnitude of the static magnetic field is swept across the different resonance conditions, allowing therefore to probe locally spin wave resonance spectra (not shown).

Figure 10.6 Characterization of spin wave modes under applied electric current: Idc = 1.2 mA. (a–c) Set of spin wave mode images on the left side of the antenna and for different magnitudes of the applied field. The region of the antenna is shaded in red. (d) Real part of the wave vector extracted from fits to the line scans as a function of H. Measurements have been performed at different excitation frequencies: 9.92 GHz (red squares), 10.00 GHz (black circles), and 10.08 GHz (blue triangles). Solid lines are the associated linear fits. (e) Decay length as a function of H, measured at 10 GHz. (f) Deduced group velocity as a function of H. Note that the error bars are smaller than the dot size and are therefore not shown in the graphs. (g) Histograms of the current-induced shift Δk of the real part of the complex wave vector Re(k) = 5.33 µm–1  measured on the left (green) and right (orange) sides of the antenna at a fixed applied field (H = –105 mT). Adapted from Chauleau et al. [25].

The analysis of a spin wave line scan (see, for example, inset of Fig. 10.5) leads to a direct access to its spatial characteristics: the real and imaginary parts of the k-vector. A full characterization of spin wave modes is performed, including spin wave mode imaging, which enables the evaluation of important additional quantities such as group velocity (Fig. 10.6f). Similarly to the inductive studies, the permalloy stripe is subjected to an electric current (±1.2 mA, equivalent to ±0.935 × 1011 A/m²). A TR-MOKE study will thereby aim at addressing the current-induced changes in the real and imaginary parts of the

Parametrizing the Two-Current Model

k-vector [25]. Note that the case of the real part of the k-vector is the exact realization of McMichael and Stiles’s picture of the spin wave Doppler effect (see Fig. 10.1a) [1]. The currentinduced shift in the resonant magnetic field was also measured; however, unlike PSWS, TR-MOKE is not suitable to measure frequency shifts accurately. Indeed, because of the stroboscopic approach, the excitation frequency has to be a multiple of the laser repetition rate, that is, 80 MHz. The frequency resolution is therefore limited to this value. The current-induced shift of the real part of the spin wave vector can be related to the spin drift velocity (u) as follows:

u=

VG  Dk  Re , 2  k 

(10.7)

where VG is the group velocity of the considered spin wave mode. After ruling out the possible additional effects discussed in Section 10.6 by considering both sides of the antenna (i.e., ±k) and of course both current polarities (±I), the degree of spin polarization could be estimated as P = 0.65 ± 0.03.

10.4  Parametrizing the Two-Current Model

As can be seen from Eqs. 10.2 and 10.5, CISWDS measurements give access directly to the degree of spin polarization of the electric current P once the spin wave vector k, saturation magnetization Ms, and charge current density J are known. This degree of spin polarization plays an essential role in many spintronics phenomena, including all manifestations of the adiabatic STT, but also some magnetoresistive effects. Up to now, this quantity could only be measured in quite indirect ways, either by modeling the deviations from Matthiessen’s rule (DMRs), observed when measuring the electrical resistivity of binary or ternary ferromagnetic alloys [26], or by following the ferromagnetic film thickness dependence of the CPP-GMR [27] (giant magnetoresistance in the current-perpendicular-to-theplane configuration). As explained in this section, CISWDS measurements give access to P with good precision in different materials and for different experimental conditions, and these

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measurements provide an unprecedented realistic description of spin-polarized transport in ferromagnetic metal films.

10.4.1  Definitions of the Degree of Spin Polarization

Before discussing the value of this degree of spin polarization, let us first discuss its meaning. Indeed, there are several different definitions of spin polarization, each of them being related to a specific context and/or measurement technique. Let us start with the expression of the electrical conductivity s = e2 < Nv2 > Fτ derived from Boltzmann theory. Here, N is the electron density of states (DOS), v is the electron velocity, F indicates an average over the Fermi surface, and τ is the electron linearmomentum relaxation time. In the two-current model, the electrical transport is assumed to proceed via two independent channels, one for majority spins and one for minority spins. Accounting for the spin dependence of the electron properties, the degree of spin polarization of the electric current is therefore:

P=

N(E F )v 2 t – N(E F )v 2  t

N(E F )v 2 t + N(E F )v 2 t

.

(10.8)

This expression of spin polarization is relevant for a purely diffusive regime of electrical transport in which the Boltzmann approach holds. This is the case for our Doppler measurements in which the spin wavelength is much larger than all characteristic lengths of electron transport, but also for resistivity measurements in bulk alloys [26], for some measurements of CPP-GMR [27] and for some ab initio calculations of spinpolarized transport [28, 29]. In the case of tunnel or ballistic regimes of transport, the expression of the electric current is different, and so is that of its degree of spin polarization. For tunnel transport through amorphous barriers or photoemission measurements, one usually uses the spin polarization of the DOS: PN = (N – N)/ (N ­ + N). For point contact Andreev reflection measurements in the clean limit, one uses the spin polarization of the ballistic current: PNv = (­ – –)/( ­+ –) [30]. Although all these expressions involve the DOS at the Fermi level N, the two other parameters v and t also play a role and, if they are strongly

Parametrizing the Two-Current Model

spin dependent, can lead to large discrepancies between the different definitions of the degree of spin polarization.

10.4.2  Spin-Dependent Electron Scattering

Once we have identified the context, let us review the different microscopic mechanisms that determine the spin-dependent conductivity in the diffusive regime. Figure 10.7c shows a resistor model of the main sources of electron scattering in a ferromagnetic metal film:





• Within the majority and the minority spin channels, electrons undergo spin-conserving momentum scattering due to impurities (random alloy disorder), phonons, grain boundaries (if the film is polycrystalline), and also film surfaces. In the resistor model of Fig. 10.7c, it is assumed that Matthiessen’s rule applies within each channel, that is, the spin-up and spin-down resistivities corresponding to the different sources of scattering add up to form ρ and ρ, respectively. • Electrons can also undergo spin-flip scattering due to thermal magnons or due to processes involving the spin-orbit interaction. The corresponding spin-flip resistivity ρ tends to increase the total resistivity and to depolarize the current: r  r + r( r  + r) r  + r + 4 ( r  – r) . P= r  + r + 4 r r=

(10.9)

Measuring the degree of spin polarization of the current P and the total resistivity ρ simultaneously, one obtains two parameters, which is not enough to determine all spin-dependent resistivities in the model of Fig. 10.7c. However, the relative magnitudes of these contributions can be tuned by varying the experimental parameters, including the temperature, the film thickness, and the film composition. Let us now review the different measurements that have been carried out on permalloy (Py = Ni80Fe20), the prototypical material of nanomagnetism.

311

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Current-Induced Spin Wave Doppler Shift

Figure 10.7  Current-induced spin wave Doppler shift measurements of the degree of spin polarization of the electric current in permalloy films. (a) Temperature dependence for a film thickness of 20 nm. (b) Film thickness dependence at room temperature. (c) Two-current resistor model of electrical transport used to interpret these results. (d) Comparison of these experimental results (crosses and triangles) to an ab initio calculation accounting only for phonons (open circles) and phonons plus thermal magnons (filled circles). After Zhu et al. [13], Haidar and Bailleul [16], and Liu et al. [32].

10.4.3  Spin-Polarized Transport in Permalloy Films

Figure 10.7a shows the temperature dependence of the degree of spin polarization extracted from spin wave Doppler shift measurements on a 20-nm-thick permalloy strip by Zhu et al. [13]. One recognizes a clear decrease of the degree of spin polarization

Parametrizing the Two-Current Model

as the temperature increases, from 0.75 at 80 K to 0.58 at 340 K. This behavior is understood as follows: At low temperature, the spin polarization is determined to a large extent by the random alloy disorder (Fe atoms in the Ni host), and this contribution is known to be strongly spin polarized (P = 0.8 to 0.95 according to resistivity measurements in dilute Ni-based ternary alloys [26] and ab initio calculations [31]). Interestingly, the low temperature value extracted from spin wave Doppler measurements is also in good agreement with the ones extracted from CPP-GMR experiments [27]. When the temperature increases, thermal excitations, that is, phonons and magnons, start to play a role, both of them tending to reduce the spin polarization. Indeed, the (mostly spin-conserving) electron-phonon scattering is expected to be less strongly spin polarized (P ≈ 0.6 in Ni from measurements in dilute alloys [26]). Moreover, the electron-magnon spin-flip scattering, which contributes through the r term in Eq. 10.9 is expected to reduce P significantly [26]. Figure 10.7b shows the film thickness dependence of the degree of spin polarization extracted from spin wave Doppler shift measurements on permalloy strips at room temperature by Haidar et al. [16]. In this case, the degree of spin polarization decreases as the film thickness decreases, from 0.63 at 20 nm to 0.42 at 6 nm. This decrease is attributed to a large extent to diffuse electron scattering by the surfaces, whose relative weight increases as the film thickness decreases. Because of the relatively long mean free path of majority electrons, this effect is particularly strong for the majority channel. In addition, two other thickness-dependent effects are included in the model of Fig. 10.7c: a gradual change of stoichiometry of the film as a function of the film thickness, due to an unwanted selective oxidation at the top Py/Al2O3 interface, and a surface spin-flip scattering that might result from the spin-orbit interaction. The results of Fig. 10.7a,b have been used as an experimental reference in a recent ab inito study of spin-polarized transport at finite temperature by Liu et al. (Fig. 10.7d) [32]. The good agreement between theory and experiment indicates the consistency of this picture of spin-polarized diffusive electrical transport.

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Current-Induced Spin Wave Doppler Shift

10.4.4  Spin-Polarized Transport in Other Materials In addition to these studies on pure permalloy films, three other systems have been explored by the Doppler shift method, in which both smaller and larger values of the degree of spin polarization were extracted. In (Ni80Fe20)1–xGdx, Thomas et al. observed a significant depolarization of the current [15] with P decreasing from 0.71 at x = 0% to 0.30 at x = 5.5% (see Fig. 10.8a). This indicates that the electron scattering on Gd impurities significantly depolarizes the current, which might be associated with negatively polarized spin-conserving electron scattering and/or to spin-flip scattering. On the other hand, in (CoFe)1–xGex, Zhu et al. measured a large spin polarization [14] ranging from 0.84 at x = 0% to 0.95 at x = 35% (see Fig. 10.8b), which has been attributed to the emergence of a pseudo gap in the minority spin band in this pseudo Heusler compound. Note finally the value of about 0.3 reported recently for Co0.9Fe0.1 films [18].

Figure 10.8  Degree of spin polarization of the electric current extracted in (Ni80Fe20)1–xGdx (a) and (CoFe)1–xGex films (b). Measurements are performed at room temperature in 30-nm-thick strips. After Thomas et al. [15] and Zhu et al. [14].

10.5  Extraction of the Non-Adiabatic Spin Transfer Torque Parameter

As mentioned in Section 10.2.2, the so-called non-adiabatic STT has been introduced in order to overcome the poor matching

Extraction of the Non-Adiabatic Spin Transfer Torque Parameter

between current-induced domain-wall experiments and their micromagnetic modeling when including solely the adiabatic STT. In the LLG description, the norm of the magnetization remains constant, so only the component of the torque perpendicular to the local magnetization is active. The only possible direction for a torque in addition to (u  .  )M is therefore M × [(u  .  )M] [6]. Thus, an orthogonal basis is formed, and any other terms should be projected on this basis. Furthermore, the non-adiabatic STT term is weighted by a parameter b, which reflects the different microscopic mechanisms involved in the nonadiabaticity. The first mechanism is the actual deviation from the adiabatic approximation, that is, a real geometrical mistracking of the conduction electron magnetic moment with the local magnetization distribution. The magnitude of this contribution has been discussed in details by several studies [33, 34]. It should be dominant only in magnetic distributions showing very large spatial gradients, that is, a magnetic structure comparable to a characteristic length L = EF/Eex . kF, with EF and kF being the Fermi energy and the wave vector, respectively, and Eex the s-d exchange energy. The geometrical contribution to the nonadiabatic STT is expressed with a parameter b, which depends on the magnetic object width (w) as follows:  b = 1/w exp(–gw/L), with g being a sharpness indicator, as defined by the authors [33]. Note that typically, L is expected to be in the range of a few nanometers for metallic ferromagnets. The second mechanism, addressed by the pioneer work of Zhang et al. [7], involves the spin accumulation generated by an electric current crossing a non-uniform magnetization distribution. This contribution to b is linked to the spin-flip scattering time (tsf ) and the s-d exchange energy interaction (  J) as follows: b = ħ/tsf J. It is interesting to note that unlike the pure mistracking contribution, this expression is solely dependent on materials parameters and not on characteristic sizes of the considered magnetization distributions. Finally the most advanced theoretical work that tackles the microscopic origins of b has been proposed by Garate et al. [35]. In this study, it is pointed out that the obtained expression for b “can be applied for real materials with arbitrary band-structures” and can be considered as a correction to the Gilbert damping where the spin-orbit coupling plays a major role. This theoretical work opens new routes for material-

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Current-Induced Spin Wave Doppler Shift

specific predictions in the field of current-induced magnetization dynamics. Regarding current-induced modifications of spin wave dynamics, the non-adiabatic STT contribution appears as a current-induced change of effective damping (see Section 10.2.2). Straightforwardly, one can see the attenuation length as being the main spin wave characteristic that is affected by the nonadiabatic STT. In other words, to assess the non-adiabatic b parameter from spin wave dynamics, the current-induced changes in the imaginary part of the spin wave vector have to be measured. Indeed, b can be expressed as (supplementary materials of Ref. [25])

 k Dk   Im    w k   b= + a,  k  Dk   Re    gu0H k  

(10.10)

where k is the complex spin-wave wavevector, Dk its complex current-induced changes, w the spin wave frequency, and a the Gilbert damping parameter. (Note: The * denotes the complex conjugate).   One consequence of the change of spin wave attenuation length is the variation of its amplitude at a given location. This first approach has been addressed by Sekiguchi et al. [20]. Using inductive time domain PSWS, the change of amplitude of a spin wave packet is measured as a function of the applied electric current (Fig. 10.9a). For the considered current densities, amplitude changes are typically in the range of 0.1%. Similarly to the spin polarization (Section 10.3.2), b is obtained from the slope of the current-induced change in amplitude as a function of the applied current density (Fig. 10.9a). Yet the changes in spin wave amplitude are actually the sum of several contributions. Substantial effort has been performed in this work in order to disentangle these latter contributions and therefore extract the sole non-adiabatic parameter. In particular, the CISWDS, that corresponds here to the change in group velocities, leads to a

Extraction of the Non-Adiabatic Spin Transfer Torque Parameter

change in amplitude at a given position. Besides, because the sample is relatively thick (35 nm) and because the Damon– Eschbach configuration is used, the effect of the generated Oersted field substantially affects the evaluation of b. Therefore it was essential to carry out micromagnetic simulations in order to extract a b parameter corrected from the spurious Oersted field contribution. Eventually, for a central wave vector of k = 0.5 µm–1 (wavelength = 12.6 µm), a frequency of w/2p = 3.6 GHz, and a spin polarization of P = 0.6 (see supplementary materials of Ref. [20]), the non-adiabatic parameters of four devices with different antenna gaps have been evaluated, ranging from about b ≈ 0.02 to b ≈ 0.05, and leading to an average value of = 0.033 ± 0.012. This latter average value is reduced to about breal = 0.02 after modeling correction.

Figure 10.9  (a) Current-induced changes of normalized spin wave amplitude as a function of the applied current (antenna gap x = 20 µm). Adapted from Sekiguchi et al. [20]. (b) Histograms of the currentinduced shift Dk of the imaginary part of the complex wave vector Im(k) = 0.502 µm–1  measured on the left (green) and right (orange) sides of the antenna at a fixed applied field (µ0H = –105 mT). Adapted from Chauleau et al. [25].

In a second work, TR-MOKE microscopy was used to directly and precisely image [25] current-induced changes in the imaginary part of the wave vector (Fig. 10.9b). Furthermore, one can note that after some approximations (which lead to less than a 2% error), Eq. 10.10 can be slightly simplified and has the interesting advantage to present quantities that are all experimentally accessible on the very same magnetic stripe:

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Current-Induced Spin Wave Doppler Shift



  Dk     Im  –1  k  Im( k )t   – + a, b  k   Dk  1 – Re( k )t    Re VG Re    gì 0H   k 

(10.11)

where VG is the group velocity and t the nanostripe thickness. Typical changes in the wave vector imaginary part of Im(Dk) =0.005 µm–1 with 25% uncertainty are measured (error bars are substantially reduced by the averaging  procedure). Subsequently, for k = 5 µm–1, w/2p = 10 GHz and a = 0.0075, the value b = 0.035 ± 0.011 has been evaluated. Despite relatively large error bars (about 30%), the nonadiabatic parameters have been evaluated from current-induced modification of spin wave dynamics. More importantly in these two studies a, b, and the spin drift velocity have been measured simultaneously, that is, from the same samples and on the same magnetic objects, which is of the utmost relevance in order to optimize current-induced magnetization dynamics.

10.6  Other Types of Spin Wave Frequency Shifts

As already mentioned in the experimental section, the Doppler shift might combine with other sources of frequency shift, and it is important to distinguish them, particularly in the MSSW/ Damon–Eshbach configurations, which have been used in most experimental studies. By nature, the Doppler effect is odd in both the spin wave vector k and the electric current I and even in the magnetic field H. In the following section we discuss three other types of frequency shift: a zero-current frequency nonreciprocity of purely magnetic origin (odd in k and H, even in I), the standard Oersted-field-induced frequency shift (odd in I and H, even in k), and a non-reciprocal Oersted-field-induced frequency shift (NR-OFIFS) (odd in k and I, even in H, that is, the same symmetry as the Doppler effect). Understanding each of these effects will make clear how to rule them out for a given measurement.

Other Types of Spin Wave Frequency Shifts

10.6.1  Zero-Current Spin Wave Frequency Non-Reciprocity The so-called MSSWs, that is, spin waves with a wave vector perpendicular to the equilibrium magnetization, both vectors lying in the film plane, have specific non-reciprocal properties, in that the amplitudes and sometimes also the frequencies of two counterpropagating spin waves are different. This is illustrated in Fig. 10.10c, which shows the two counterpropagating PSWS signals measured on a 10 nm permalloy film at a wave vector of +/–7.8 rad/µm. One recognizes clearly a large difference of amplitude of the two waveforms, the k < 0 one being about four times more intense than the k > 0 one. One recognizes also a clear frequency shift, the k < 0 waveform lying 34 MHz higher in frequency than the k > 0 one.

Figure 10.10  Magnetostatic surface wave (MSSW) non-reciprocities. (a) Sketch of a propagating spin wave spectroscopy (PSWS) experiment in the MSSW configuration. The asymmetries of the modal profiles across the film thickness are sketched for both propagation directions. (b) Sketch of the distribution of the dipole fields generated by the dynamic magnetization. (c) PSWS waveforms measured at zero current for a 10 nm film and k = +/–7.8 rad/µm. The solid line is for k < 0 and the dashed line is for k > 0. After Gladii et al. [37].

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Current-Induced Spin Wave Doppler Shift

The amplitude non-reciprocity is well-known in the spin wave community: it is attributed to the matching of the chiralities of the microwave field generated by the antenna to that of the dynamic magnetization of the spin wave. The less documented frequency non-reciprocity originates from the combination of two asymmetries: the asymmetry of the modal profile (see Fig. 10.10a, where one recognizes that the amplitude of spin precession is maximal close to the bottom surface for k > 0, while it is maximal close to the top surface for k < 0) and the asymmetry of the magnetic properties of the film. In Fig. 10.10a, the magnetic surface anisotropies at the top and bottom surface are different. For each direction, the spin wave frequency is influenced mostly by the value of the surface anisotropy of the surface on which it has the largest amplitude, resulting in a frequency non-reciprocity proportional to the difference of surface anisotropies. For a realistic description of MSSWs in the film thickness range of interest, one requires a dipole exchange theory accounting for the hybridization between the uniform FMR mode and the antisymmetric first perpendicular standing wave mode [36]. In the small-thickness, small-wave-vector limit, the frequency non-reciprocity derived in this description is [37]

fNR 

8 g K Sbot – K Stop MS p3

k . L2 p2 1+ 2 t

(10.12)

K Stop the ksurface magnetic anisotropy K bot – K Stop and 8kg K bot –are 8 gHere . fNR  32 2S . fNR  3 S 2 constants and bottom surfaces, respectively, p top L2 pfilm MS for theL p p M S 1 + 1 + 2 2 and L is the exchange length.t Note that the interfacial t Dzyaloshinskii–Moriya interaction, of importance in ferromagnetic metal/heavy metal bilayers, also results in an MSSW frequency non-reciprocity (see chapter 8 by Stamps et al. for a discussion of the specificities of spin waves in the presence of this interaction). Note also that both amplitude and frequency non-reciprocities reverse when the direction of the equilibrium magnetization is reversed. Consequently, these two effects can be ruled out by comparing (+k, +H) and (–k, –H) spectra, as done by Zhu, Thomas, Sugimoto, and coworkers [13–15, 18].

Other Types of Spin Wave Frequency Shifts

10.6.2  Reciprocal Oersted-Field-Induced Frequency Shift As sketched in Fig. 10.11a, a direct current flowing in a film leads to an Oersted field directed perpendicular to it. Assuming the electric current density J is uniform, this field is perfectly antisymmetric with respect to the film midplane and averages out to zero when integrated over the film thickness. However, any asymmetry of the electrical properties of the film (e.g., a difference of the diffuse electron scattering efficiencies at the top and bottom surfaces) translates into a deviation from this antisymmetry and a non-zero residual Oersted field . In the MSSW configuration, this field combines with the external field H, and the spin wave frequency becomes f (H + ). In this picture, the current-induced frequency shift is expected to be the same for k > 0 and k < 0, that is, it is a reciprocal effect. As a consequence, this reciprocal Oersted-field-induced frequency shift (R-OFIFS) will add to the (non-reciprocal) current-induced Doppler shift for one propagation direction and subtract from it for the opposite one. This is illustrated in Fig. 10.11b, which shows PSWS signals measured for the two propagation directions and for the two current polarities. In this plot, one recognizes first the signature of the magnetic frequency non-reciprocity: the pair of k < 0 waveforms (dashed lines) being shifted about 15 MHz higher in frequency with respect to the pair of k > 0 ones (solid lines). One also recognizes qualitatively the current-induced Doppler effect, with a positive current-induced frequency shift for one direction (df21 = f21(–I) – f21(+I) > 0) and a negative current-induced frequency shift for the opposite one (δf12 = f12(–I) – f12(+I) < 0). However, contrary to the expectation for a pure Doppler effect, these two shifts are of very different amplitudes, which is attributed to the combination of the R-OFIFS with the Doppler effect. The two contributions can be extracted as dfOe,R = (df21 + df12)/4 and dfDop = (δf21 – df12)/4. In the example in Fig. 10.11b the residual Oersted field is estimated to be about 12 A/m, which corresponds to a top/bottom asymmetry of the distribution of electric current of about  3% for this 10 nm film. Note that the R-OFIFS is odd in both H and k [38], so it is ruled out when comparing (+k, +H,I)

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and (–k, –H, I) spectra, as done by Zhu, Thomas, Sugimoto, and coworkers [13–15, 18].

Figure 10.11 Reciprocal Oersted-field-induced frequency shift (R-OFIFS). (a) Sketch of the geometry. (b) Current-induced spin wave frequency shifts measured for k = +/–3.8 rad/µm, I = +/–7.5 mA, and µ0H = 28 mT. Solid lines are for k > 0, and dashed lines are for k < 0. The Al2O3/Py(10nm)/ Al2O3 strip is 8 µm wide. After Haidar and Bailleul [16].

10.6.3  Non-Reciprocal Oersted-Field-Induced Frequency Shift

Let us now combine two effects mentioned before and examine the situation, sketched in Fig. 10.12b, in which a nominally antisymmetric Oersted field acts on a spin wave with a pronounced modal profile asymmetry. In this case, the residual field dHOe can be evaluated by a weighted average in which the Oersted field is multiplied by the normalized amplitude of the spin wave and integrated over the film thickness. Due to the opposite localization of the +k and –k spin waves, this weighted average results in dHOe of different signs for two counterpropagating spin waves (solid and dashed arrow in the top of Fig. 10.12b), that is, the effect is non-reciprocal. It is also clear that this effect is odd in I, and it can be shown that the effect is even in H [17]; when the static magnetic field reverses, the modal profile asymmetry also reverses. This effect is called the NR-OFIFS. It is odd in I, odd in k, and even in H, which is the same symmetry as the CISWDS, so the two effects cannot be distinguished from each other by any symmetry analysis. Hopefully, the two effects

Other Types of Spin Wave Frequency Shifts

have very different dependences on the film thickness and wave vector. Indeed, using the same dipole exchange description as mentioned before, Haidar et al. could derive an explicit expression of the NR-OFIFS [17]:

dW =

P –P 4 2 Jt Q01 00 11 . 2 p Ms W– W 

(10.13)

Figure 10.12  Non-reciprocal Oersted-field induced frequency shift (NR-OFIFS). (a) Wave vector dependence of measured and calculated non-reciprocal current-induced spin wave frequency shifts (NR-OFIFS + Doppler) for two film thicknesses. Open and solid squares are measured data for a 10 nm and a 40 nm permalloy film, respectively. The lines are calculated using values of the degree of spin polarization and exchange constant P = 0.6 and A = 11.5 pJ/m, respectively. (a) Sketch of the effect showing the distribution of the Oersted field (arrows) and of the spin wave amplitude (mode profiles) across the film thickness. After Haidar et al. [17].

Here, J is the electric current density, t is the film thickness, W0 and W1 are the normalized angular frequencies of the fundamental mode and the first perpendicular standing wave mode, respectively, and P00, P11, and Q01 are dipolar matrix elements representing the dipole field generated by the two

323

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modes over themselves and their non-reciprocal mutual coupling, respectively. Figure 10.11a shows the non-reciprocal frequency shift measured for two film thicknesses (open squares: 10 nm; solid squares: 40 nm). Using the symmetry analysis mentioned in the last section, the part of the current-induced frequency shift dfodd, which is odd in k, has been extracted, which allows one to eliminate both the zero-current frequency non-reciprocity and the R-OFIFS. The dfodd extracted for different devices with different values of wave vectors are then normalized by the electric current density J and plotted as a function of k. For the 10 nm film, one obtains a linear dependence on k, as expected for a pure Doppler effect. Indeed, the NR-OFIFS correction calculated in that case is extremely small, because the exchange spitting makes the denominator of Eq. 10.13 very large.  For the 40 nm film, on the other hand, one observes a complicated wave vector dependence (solid squares in Fig. 10.12e), with a saturation around 4 rad/µm and a change of sign at higher values. This behavior is accounted for by adding to the Doppler shift of Eq. 10.5 the NR-OFIFS of Eq. 10.13 (dashed line in Fig. 10.12e). Indeed, the NR-OFIFS is opposite to the Doppler shift and its magnitude increases strongly as the wave vector increases, basically because the frequency splitting between the uniform and first antisymmetric modes decreases, which leads to the sign change of the total frequency shift.  From this analysis, it is clear that the best way to avoid the NR-OFIFS is to deal with thinenough films (for permalloy films thinner than 20 nm, the NR-OFIFS can be safely neglected for the wave vectors commonly used) [17]. For thicker films, one should examine the wave vector dependence of the effect and subtract it from the measurements.

10.7  Conclusion and Perspectives

Together with magnetic domain walls, vortices, and, nowadays, skyrmions, spin waves represent the most commonly studied magnetic objects, and the research carried out on them has already led to tremendous breakthroughs in the understanding

References

of the magnetism of nanostructures. Measuring the currentinduced modification of the spin wave dynamics has proven to be a particularly elegant approach to probe STT mechanisms. Several experimental configurations have been implemented, which have led to the evaluation of the effect of a spin-polarized current on most spin wave characteristics, namely changes in frequency [9, 13–15, 18], changes in group velocity and amplitude [20], and changes in the complex wave vector and resonant field [25]. The access to the CISWDS has allowed us to measure precisely the degree of spin polarization of the electric current in various conditions. Furthermore, it has been demonstrated that the three key parameters that rule currentinduced magnetization dynamics can be extracted self-consistently from spin wave Doppler experiments. This is of the utmost importance in order to improve the experimental understanding regarding the microscopic origins of the non-adiabatic parameter. Indeed, it is now well established that the Gilbert damping coefficient a varies depending on the considered magnetic object [39, 40] and P varies with the sample thicknesses [16]. A precise quantitative determination of b requires a and P to be measured for the same magnetic object in the same nanostructure. Current-induced modification of spin wave dynamics can therefore be envisioned as a relevant approach to investigate the microscopic origins of b by measuring the set (a, b, P) as a function of wave vector, temperature, and material compositions. Finally, it is interesting to note that the spin wave Doppler experiment could be implemented to investigate spin currents of different kinds such as thermal magnonic spin currents.

References

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3. Bazaliy, Y. B., Jones, B., and Zhang, S.-C. (1998). Modification of the landau-lifshitz equation in the presence of a spin-polarized current in colossal- and giant-magnetoresistive materials, Phys. Rev. B, 57, R3213.

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4. Fernandez-Rossier, J., Braun, M., Nunez, A. S., and MacDonald, A. H. (2004). Influence of a uniform current on collective magnetization dynamics in a ferromagnetic metal, Phys. Rev. B, 69, 174412.

5. Malinowski, G., Boulle, O., and Kläui, M. (2011). Current-induced domain wall motion in nanoscale ferromagnetic elements, J. Phys. D: Appl. Phys., 44, 384005. 6. Thiaville, A., Nakatani, Y., Miltat, J., and Suzuki, Y. (2005). Micromagnetic understanding of current-driven domain wall motion in patterned nanowires, Europhys. Lett., 69, 990. 7. Zhang, S., and Li, Z. (2004). Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets, Phys. Rev. Lett., 93, 127204. 8. Vlaminck, V. (2008). Décalage Doppler d’onde de spin induit par un courant électrique, PhD thesis, Université de Strasbourg, http://scdtheses.u-strasbr.fr/1569/

9. Vlaminck, V., and Bailleul, M. (2008). Current-induced spin-wave Doppler shift, Science, 322, 410.

10. Brundle, L. K., and Freedman, N. J. (1968). Magnetoelastic surface waves on YIG slab, Electron. Lett., 4, 132.

11. Bailleul, M., Olligs, D., Fermon, C., and Demokritov, S. O. (2001). Spin waves propagation and confinement in conducting films at the micrometer scale, Europhys. Lett., 56, 741. 12. Vlaminck, V., and Bailleul, M. (2010). Spin-wave transduction at the submicrometer scale: experiment and modeling, Phys. Rev. B, 81, 014425. 13. Zhu, M., Dennis, C. L., and McMichael, R. D. (2010). Temperature dependence of magnetization drift velocity and current polarization in Ni80Fe20 by spin-wave Doppler measurements, Phys. Rev. B, 81, R140407.

14. Zhu, M., et al. (2011). Enhanced magnetization drift velocity and current polarization in (CoFe)1-xGex alloys, Appl. Phys. Lett., 98, 072510.

15. Thomas, R. L., Zhu, M., Dennis, C. L., Misra, V., and McMichael, R. D. (2011). Impact of Gd dopants on current polarization and the resulting effect on spin transfer velocity in Permalloy wires, J. Appl. Phys., 110, 033902. 16. Haidar, M., and Bailleul, M. (2013). Thickness dependence of degree of spin polarization of electrical current in Permalloy thin films, Phys. Rev. B, 88, 054417.

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18. Sugimoto, S., Rosamond, M., Linfield, E. H., and Marrows, C. H. (2016). Observation of spin-wave Doppler shift in Co90Fe10/Ru microstrips for evaluating spin polarization, Appl. Phys. Lett., 109, 122405. 19. Covington, M., Crawford, T. M., and Parker, G. J. (2002). Time-resolved measurement of propagating spin waves in ferromagnetic thin films, Phys. Rev. Lett., 89, 237202.

20. Sekiguchi, K., et al. (2012). Time-domain measurement of currentinduced spin wave dynamics, Phys. Rev. Lett., 108, 017203.

21. Hubert, A., and Schäfer, R. (1998). Magnetic Domains: The Analysis of Magnetic Microstructures (Springer-Verlag, Berlin, Heidelberg). 22. Hiebert, W. K., Stankiewicz, A., and Freeman, M. R. (1997). Direct observation od magnetic relaxation in a small Permalloy disk by time-resolved scanning kerr microscopy, Phys. Rev. Lett., 79, 1134. 23. Liu, Z., Giesen, F., Zhu, X., Sydora, R. D., and Freeman, M. R. (2007). Spin wave dynamics and the determination of intrinsic damping in locally excited Permalloy thin films, Phys. Rev. Lett., 98, 087201.

24. Perzlmaier, K., Woltersdorf, G., and Back, C. H. (2008). Observation of the propagation and interference of spin waves in ferromagnetic thin films, Phys. Rev. B, 77, 054425.

25. Chauleau, J.-Y., et al. (2014). Self-consistent determiantion of the key spin-transfer torque parameters from spin-wave Doppler experiments, Phys. Rev. B, 89, R020403. 26. Fert, A., and Campbell, I. A. (1976). Electrical resistivity of ferromagnetic nickel and iron based alloys, J. Phys. F: Metal Phys., 5, 849.

27. Bass, J., and Pratt Jr., W. P. (1999). Current-perpendicular (CPP) magnetoresistance in magnetic metallic multilayers, J. Magn. Magn. Mater., 200, 274.

28. Mertig, I. (1999). Transport properties of dilute alloys, Rep. Prog. Phys., 62, 237.

29. Starikov, A. A., Kelly, P. J., Brataas, A., Tserkovnyak, Y., and Bauer, G. E. W. (2010). Unified first-principles study of Gilbert damping, spin-flip diffusion, and resistivity in transition metal alloys, Phys. Rev. Lett., 105, 236601.

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30. Zhu, Z. Y., et al. (2008). intrinsic anisotropy of degree of transport spin polarization in typical ferromagnets, J. Phys.: Condens. Matter, 20, 275245. 31. Banhart, J., Ebert, H., and Vernes, A. (1997). Applicability of the two-current model for systems with strongly spin-dependent disorder, Phys. Rev. B, 56, 10165.

32. Liu, Y., et al. (2015). Direct method for calculating temperaturedependent transport properties, Phys. Rev. B, 91, R220405. 33. Xiao, J., Zangwill, A., and Stiles, M. D. (2006). Spin-transfer torque for continously variable magnetization, Phys. Rev. B, 73, 054428. 34. Waintal, X., and Viret, M. (2004). Current-induced distortion of a magnetic domain wall, Europhys. Lett., 65, 427.

35. Garate, I., Gilmore, K., Stiles, M. D., and MacDonald, A. H. (2009). Non-adiabatic spin-transfer torque in real materials, Phys. Rev. B, 79, 104416. 36. Kostylev, M. (2013). Non-reciprocity of dipole-exchange spin waves in thin ferromagnetic films, J. Appl. Phys., 113, 053907.

37. Gladii, O., Haidar, M., Henry, Y., Kostylev, M., and Bailleul, M. (2016). Frequency nonreciprocity of surface spin wave in Permalloy thin films, Phys. Rev. B, 93, 054430.

38. Haidar, M. (2012). Role of surfaces in magnetization dynamics and spin polarized transport: a spin wave study, PhD thesis, Université de Strasbourg, https:tel.archives-ouvertes.frtel-00869643 39. Li, Y., and Bailey, W. E. (2016). Wave-number-dependent Gilbert damping in metallic ferromagnets, Phys. Rev. Lett., 116, 117602.

40. Weindler, T., et al. (2014). Magnetic damping: domain wall dynamics versus local ferromagnetic resonance, Phys. Rev. Lett., 113, 237204.

41. Bauer, H. G., Chauleau, J.-Y., Woltersdorf, G., and Back, C. H. (2014). Coupling of spinwave modes in wire structures, Appl. Phys. Lett., 104, 102404.

Chapter 11

Excitation and Amplification of Propagating Spin Waves by Spin Currents Vladislav E. Demidova and Sergej O. Demokritova,b aInstitute for Applied Physics, University of Muenster, Corrensstrasse 2-4, Muenster 48149, Germany bInstitute of Metal Physics, Ural Division of RAS, Yekaterinburg 620041, Russia

[email protected]

11.1 Introduction The unique features of spin waves, such as the possibility to achieve submicrometer wavelength at microwave frequencies and electronic controllability by static magnetic fields make these waves uniquely suited for implementation of novel integrated electronic devices characterized by high speed, low power consumption, and extended functionalities. The utilization of spin waves for integrated electronic applications is addressed within the emerging field of magnonics [1–8]. Although the application of spin waves for microwave signal processing has been intensively explored for many decades (see, e.g., [9, 10]), recent advances in spintronics and nanomagnetism, as well as the development of novel techniques for nanofabrication and measurements of high-frequency magnetization dynamics,

Spin Wave Confinement: Propagating Waves (2nd Edition) Edited by Sergej O. Demokritov Copyright © 2017 Pan Stanford Publishing Pte. Ltd. ISBN 978-981-4774-35-2 (Hardcover), 978-1-315-11082-0 (eBook) www.panstanford.com

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created essentially new possibilities for magnonics and brought it onto a new development stage. Of particular importance here is the discovery of spin-transfer torque (STT) produced by spin-polarized electric currents [11, 12] or by pure spin currents [13, 14], which has already been demonstrated to enable novel geometries and functionalities of spin-wave devices [5, 8, 15–22]. STT phenomena have been traditionally studied in nanodevices based on the tunneling or giant magnetoresistance spin-valve structures [23–26], where STT is induced by the electric current flowing through a multilayer that consists of a “fixed” magnetic spinpolarizer and the active magnetic layer, separated by a non-magnetic metallic or insulating spacer. In these structures, the electric charges must cross the active magnetic layer to excite its magnetization dynamics. The flow of the electrical current through the active layer results in a significant Joule heating. In addition, the inhomogeneous Oersted fields induced by the localized currents can complicate the dynamical states induced by STT. Moreover, to enable current flow through the active magnetic layers, STT devices operating with spinpolarized electric current require that current-carrying electrodes are placed both on top and on the bottom of the spin valve, which severely reduces the flexibility of the device geometry. An alternative approach to the implementation of STT devices that avoids these shortcomings utilizes pure spin currents—flows of spin not accompanied by directional transfer of electrical charge. This approach does not require the flow of electrical current through the active magnetic layer, resulting in reduced Joule heating and electromigration effects. One can also eliminate the electrical leads attached to the magnetic layer to drain the electrical current, enabling novel geometries of the STT devices. Additionally, it becomes possible to use insulating magnetic materials [27] such as yttrium iron garnet (YIG) [28]. The main advantage of this material is its exceptionally low dynamic magnetic damping. Since the expected density of the driving current necessary for the current-induced auto-oscillations is proportional to damping, YIG-based STT devices can be significantly more efficient than the traditional devices based on the transition-metal ferromagnets. Pure spin currents can be created by using the spin-Hall effect (SHE) [13, 14]. Devices utilizing SHE benefit from many advantages of pure spin currents. However, they are also not free

Introduction

from shortcomings. In particular, efficient spin-Hall materials such as Ta and Pt exhibit a high resistivity. As a consequence, a significant fraction of the driving current is shunted through the active magnetic layer, which must be in electrical contact with the SHE material to enable device operation. This problem is avoided in the SHE devices based on magnetic insulators, but even in this case the high resistivity of the SHE material results in a significant Joule heating of both this material and the active magnetic layer in direct thermal contact with it. The heating adversely affects the dynamical characteristics and can even damage the device. Another shortcoming of the SHE devices is the increase of magnetic damping generally observed in the bilayers of active magnetic layers with the SHE materials. This increase is difficult to avoid because it is caused by the enhanced electron–magnon scattering due to the same spin– orbit interactions that enable the operation of SHE devices. The shortcomings described above can be avoided in devices that utilize spin currents generated by the nonlocal spin-injection (NLSI) mechanism, which does not require electric current flow through highly resistive materials [29–31]. In contrast to the SHE-based devices, current flow through the active magnetic layer is negligible in the NLSI-based oscillators. Their dynamical characteristics are also not compromised by the detrimental effects of layers with strong spin–orbit coupling. Moreover, since the driving current in the NLSI devices flows mostly through low-resistivity layers, the Joule heating effects are minimized. Here we review our recent experimental investigations on the excitation and control of spin waves propagating in microscopic magnonic waveguides by using STT phenomena. We first discuss the excitation of propagating guided spin waves by utilizing traditional STT devices driven by spin-polarized electric current. Then we describe experiments on the control of spin-wave propagation by using pure spin currents created by the SHE mechanism for the cases of all-metallic magnonic waveguides and waveguides based on ultra-thin insulating YIG films. Finally, we show that pure spin currents created by the NLSI mechanism can be utilized for the efficient excitation of propagating spin waves with large propagation length and short spin-wave packets with the duration down to a few nanoseconds.

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11.2 Experimental Technique All the experiments described below were performed by using the micro-focus Brillouin light scattering (BLS) spectroscopy. BLS spectroscopy has been known for many decades as an experimental tool enabling direct visualization of spin-wave propagation with the spatial and temporal resolution and unprecedented sensitivity [32]. Thanks to the recent developments, nowadays the spatial resolution of this technique is in the submicrometer range, which enables imaging of spin waves in individual magnetic structures with microscopic dimensions. Since the first demonstration of the micro-focus BLS technique [33], the technical details about this experimental tool have been discussed in several review papers [6, 34]. Therefore, here we limit ourselves to a short introduction of its main principles considering typical implementation of microfocus BLS measurements of spin-wave propagation in a microscopic magnonic waveguide. Figure 11.1a shows a schematic of a typical experiment. A ferromagnetic film with the thickness of 5–50 nm is patterned into a stripe with the width in the range of 0.1 to 5 µm serving as a magnonic waveguide. To excite spin waves, a stripe microwave antenna oriented perpendicular to the waveguide axis is manufactured on top of it. The antenna is made of highly conductive material, such as Au or Cu, and has typical width in the range of 0.5 to 2 µm and thickness of 100–150 nm. The antenna is electrically isolated from the waveguide by a 20–40 nm thick dielectric layer. The excitation of spin waves [35, 36] is performed by transmitting through the antenna a microwave-frequency current. This current creates a local dynamic magnetic field, which couples to the magnetization in the waveguide underneath the antenna, resulting in a creation of a spin wave propagating away from the excitation area. Although the detection of spin waves in the described system can, in principle, be done electronically by using secondary spin-wave antenna, utilization of spatially resolved magneto-optical methods, such as micro-focus BLS technique, allows one to obtain significantly more detailed information about the spin-wave propagation and essentially improve the experimental sensitivity. To implement BLS measurements, the probing light generated by a single-frequency laser with the wavelength of 532 or 473 nm and the power

Experimental Technique

IC, where IC is the critical current, at which the damping is completely compensated by the spin current resulting in the onset of current-induced magnetization auto-oscillations [21]. We note

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that at I > IC, the magnetic damping should be overcompensated by the STT, which is expected to result in a spin-wave amplification. In contradiction to this naive expectation, the propagation length decreases rapidly at I > IC, so that already at I = 2.65 mA, the BLS signal from the spin wave excited by the antenna completely vanishes. This experimental observation can be attributed to the strong nonlinear scattering of coherent spin waves from large-amplitude currentinduced magnetic auto-oscillations due to the nonlinear magnon– magnon scattering processes, which are known to be highly efficient in low-damping YIG films [58].

Figure 11.7 (a) Normalized spatial intensity map of the propagating spin wave excited by the antenna. The map was recorded for I = 2.55 mA. The mapping was performed by rastering the probing spot over the area 1.6 by 10 µm, which is larger than the waveguide width of 1 µm. Dashed lines show the edges of the waveguide. Inset shows the transverse profile of the spin-wave intensity. (b) Dependences of the spin-wave intensity on the propagation coordinate for different currents, as labeled, in the log-linear scale. Lines show the exponential fit of the experimental data. Reprinted from [21], with the permission of AIP Publishing.

To characterize the variation of the propagation length with current in detail, we plot in Fig. 11.8 its inverse value—the decay constant (down-triangles), which is proportional to effective Gilbert damping constant αeff. In agreement with the simple theoretical model assuming the linear variation of αeff with current, the

Control of the Propagation Length of Spin Waves by Pure Spin Currents

decay constant shows a linear dependence on I. Additionally, one expects the linear dependence in Fig. 11.8 to cross zero at I = IC, which corresponds to an infinitely large propagation length under conditions of the complete damping compensation. The data of Fig. 11.8 show, however, that the linear fit yields the intercept value larger than IC. This disagreement can be attributed to the contribution of the nonlinear scattering of propagating spin waves from spin currentenhanced magnetic fluctuations [59]. Indeed, as shown in [21], even at I < IC the intensity of magnetic fluctuations in YIG can increase by more than one order of magnitude in comparison to the case, when no spin current is applied, resulting in additional propagation losses due to the magnon–magnon scattering.

Figure 11.8 Current dependences of the propagation length and of the decay constant. Vertical dashed line marks the critical current IC, at which the damping is completely compensated by the spin current. Solid line is the linear fit of the experimental data at I < IC. The data were obtained at H0 = 1000 Oe. Reprinted from [21], with the permission of AIP Publishing.

As seen from the data discussed above, pure spin currents do not allow one to completely compensate propagation losses of spin waves. We note, however, that the maximum achieved propagation length of 22.5 µm is nearly by a factor of two larger compared to the value of 12 µm estimated for a waveguide made of a bare YIG film without Pt on top (α = 5 ¥ 10–4). To further characterize the significance of the achieved propagation-length control for the performance characteristics of spin-wave transmission lines, we calculate the intensity of the spin wave at the output of a transmission line with the length 10 µm based on the experimental values of the propagation length x(I) (Fig. 11.8). The results normalized by the value at I = 0 are shown in Fig. 11.9. As one clearly sees from

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Excitation and Amplification of Propagating Spin Waves by Spin Currents

these data, the achieved propagation-length control provides the opportunity to increase the intensity of spin waves at the output of a transmission line with technologically relevant length by more than three orders of magnitude.

Normalized spinwave intensity

346

1000 100 10

1.0

1.5

2.0 I, mA

2.5

Figure 11.9 Current dependence of the intensity of the spin wave at the output of a 10 µm long transmission line calculated based on the experimentally measured propagation length. The intensity is normalized by the value at I = 0. Reprinted from [21], with the permission of AIP Publishing.

11.5 Excitation of Spin Waves by Pure Spin Currents

In spite of the quick developments in the excitation of coherent localized magnetization dynamics by pure spin currents [38, 56, 57, 60–66], excitation of propagating spin waves by this mechanism remained an unsolved problem for a long time. One of the main difficulties was associated with the limitations imposed by the geometry of spin-current devices based on the spin-Hall effect, which place significant constraints on their compatibility with magnonic devices. Only with the demonstration of the novel type of nanooscillators driven by pure spin currents created by the nonlocal spin injection mechanism [67–69], the efficient excitation of propagating spin waves with large propagation length by pure spin currents became practically possible [8, 22].

11.5.1 Excitation of Continuous Propagating Spin Waves

The schematic of the NLSI-based magnonic system is shown in Fig. 11.10. It consists of a 5 nm thick Permalloy (Py) active magnetic

Excitation of Spin Waves by Pure Spin Currents

film separated from the 8 nm thick CoFe spin injector by a 20 nm thick layer of Cu. The electric current is injected into the multilayer through a 60 nm circular nanocontact fabricated on the CoFe side. Because of the large difference in the resistivities of the materials comprising the device, most of the current is drained through the Cu layer, while only 3% of the current is shunted through the active Py layer. The arrow in Fig. 11.10 shows the corresponding flow of electrons. The injected electrons become spin polarized due to the spin-dependent scattering in CoFe and at the Cu/CoFe interface, resulting in spin accumulation in Cu above the nanocontact. Spin diffusion away from this region produces a spin current flowing into the Py layer, exerting STT on its magnetization. The magnetizations of both CoFe and Py layers are aligned with the saturating static inplane magnetic field H0. For positive driving electric currents, as defined by the arrow in Fig. 11.10, the magnetic moment carried by the spin current is antiparallel to the magnetization of the Py layer, resulting in the STT compensating the dynamic magnetic damping. When damping is completely compensated by the spin current, the magnetization of the Py layer exhibits highly coherent autooscillations in the spatial area with the size of about 300 to 400 nm determined by the spin current injection region [67, 68].

Figure 11.10 Schematic of a NLSI-based system enabling excitation of propagating spin waves by pure spin currents. The inset illustrates the dipolar field HD in the stripe waveguide caused by uncompensated magnetic charges at its edges. Reprinted from [76], with the permission from Elsevier.

To convert these localized magnetization oscillations into a propagating spin wave with a sufficiently large propagation length, a 20 nm thick and 500 nm wide Py strip aligned perpendicular to the direction of the static field H0 is fabricated on the surface of

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the extended Py film. The waveguide is terminated at a distance of 150 nm from the center of the nanocontact. This distance is sufficiently small to ensure efficient dynamic coupling between the current-induced magnetic auto-oscillations in the thin film and the magnetization in the strip. Because of this coupling, the localized oscillations are expected to excite a propagating spin wave, provided that there are available spin-wave spectral states at the frequency of the auto-oscillations. Figure 11.11a shows the dispersion spectra of spin waves in an extended Py film with the thickness of 5 and 25 nm (solid curves) and for spin waves propagating in the stripe waveguide (symbols) calculated for H0 = 1000 Oe. As seen from these data, the slope of the dispersion curve, which is proportional to the spin-wave group velocity, is significantly larger for thicker extended film. Since the group velocity determines the propagation length, the latter increases from less than 1 µm for a 5 nm film to several micrometers for a 25 nm thick film. The slope of the dispersion curve calculated for the 20 nm thick stripe waveguide manufactured on top of 5 nm thick extended Py film is close to that of the thick extended Py film. Therefore, one can also expect a large propagation length for the spin waves in the waveguide. A distinct feature of the dispersion spectrum of spin waves in the waveguide, which is of particular importance for the possibility to achieve spin-wave radiation, is its significant frequency downshift in comparison with the curves for an extended film. This downshift originates from the reduction of the internal static magnetic field in the waveguide caused by the dipolar field HD produced by the uncompensated magnetic charges at the waveguide edges (see inset in Fig. 11.10 and Ref. [70]). The downshift of the spin-wave spectrum results in the appearance of spin-wave spectral states at frequencies below the frequency of the uniform ferromagnetic resonance (FMR) (dashed line in Fig. 11.11a), which enables the radiation of spin waves by the localized currentinduced auto-oscillations, even though the frequency of the latter is always smaller than the FMR frequency [22, 67]. The auto-oscillation characteristics of the NLSI oscillator integrated into the hybrid spin-wave device described above are

Excitation of Spin Waves by Pure Spin Currents

summarized in Fig. 11.11b, which shows the current dependence of the frequency and the intensity of the auto-oscillations obtained from BLS measurements with the probing laser spot located directly at the position of the nanocontact. The integrated oscillator transits to the auto-oscillation regime at the critical current IC = 3.6 mA, which is close to that for stand-alone devices [67]. This indicates that the oscillation characteristics are not adversely affected by the integration of the oscillator into the magnonic system. The intensity of the auto-oscillations gradually increases with increasing I > IC, while their frequency decreases due to the nonlinear frequency shift. We emphasize that the auto-oscillation frequency is below the FMR frequency in the extended Py film (horizontal dashed line in Fig. 11.11b) within the entire used range of current. As follows from the data of Fig. 11.11a, there are no propagating spin-wave states available in the Py film at the frequency of auto-oscillation, and therefore the oscillation does not radiate spin waves into the surrounding Py(5 nm) film. In contrast, the auto-oscillation frequency range is well matched with that of the waveguide mode, which should result in an efficient radiation of spin waves into the waveguide.

Figure 11.11 (a) Solid lines: dispersion spectra of spin waves in an extended Py film with the thickness of 5 and 25 nm, as labeled. Symbols: dispersion spectrum of a spin-wave mode in a 500 nm-wide and 20 nm-thick stripe waveguide manufactured on top of 5 nm-thick Py film. Calculations were performed for H0 = 1000 Oe. Dashed horizontal line marks the FMR frequency. (b) Current dependences of the auto-oscillation frequency (point-down triangles) and the intensity (point-up triangles) of the NLSI nano-oscillator. H0 = 1000 Oe. Reprinted from [76], with the permission from Elsevier.

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Figure 11.12a shows the spatial BLS intensity map recorded by rastering the probing laser spot over a 3 µm by 1.2 µm region encompassing the nanooscillator and the adjacent area of the waveguide. It clearly shows two merged but distinct dynamical regions. The first circular high-intensity region is centered on the nanocontact. In this region, the magnetization oscillations are excited by the spin current. It is merged with another arrow-shaped increased-intensity region aligned with the strip waveguide, which is indicated by a dashed contour in Fig. 11.12a. The increased intensity is entirely confined to the waveguide, as shown by the transverse section of the map, inset in Fig. 11.12b. These observations are consistent with the directional propagation of a spin wave excited in the waveguide by the spin current-induced oscillations.

Figure 11.12 (a) Normalized spatial map of the dynamic magnetization recorded by BLS at I = 4 mA. Dashed line indicates the contour of the waveguide. (b) Propagation-coordinate dependence of the spin-wave intensity. Solid curve shows the result of the fit of the experimental data (symbols) by the exponential function. Inset: transverse profile of the spinwave intensity at x = 2 µm. Reprinted from [76], with the permission from Elsevier.

The propagation characteristics of the excited spin wave can be analyzed based on the measured dependence of the BLS signals on the propagation coordinate x, which is defined as the position along the waveguide strip with the origin at the location of the nanocontact. The point-down triangles in Fig. 11.12b show the BLS intensity integrated across the transverse sections of the intensity map. These data plotted on the logarithmic vertical scale show that the excited spin wave exhibits a well-defined exponential decay ~exp(–2x/x)

Excitation of Spin Waves by Pure Spin Currents

along the waveguide. By fitting these data with the exponential function (curve in Fig. 11.12b), one obtains the propagation length x = 3.0 µm, which is significantly larger than that obtained for devices driven by spin-polarized currents [5] and is sufficiently large for the practical implementations of magnonic nanosystems. The BLS data also allow one to determine the efficiency of spinwave excitation in the waveguide due to the dynamical coupling to the nano-oscillator. One can extrapolate the exponential spin-wave decay curve to the position x = 150 nm corresponding to the edge of the waveguide, and find the ratio between this value and the intensity at the position x = 0 of the nanocontact, which characterizes the energy of the localized auto-oscillation mode. From the data of Fig. 11.12b, one obtains the coupling of about 35%, which is superior to the coupling efficiency of 2% achieved for traditional STNO devices [5]. This significant improvement is likely due to the large size of the auto-oscillation area in the NLSI oscillators providing a very efficient coupling of localized oscillations to the propagating spin waves.

11.5.2 Excitation of Short Spin-Wave Packets

For the practical implementations of high-speed integrated magnonic circuits, it is particularly important that the spinwave source is capable of generation of short wave packets. The performance of the traditional inductive excitation technique is very limited in this respect, since the externally generated microwave signal has to be pulse-modulated by semiconductor switches, which are generally characterized by a relatively low power efficiency, and on-off times of at least several nanoseconds. The fastest spin-wave excitation rate demonstrated so far was achieved by utilizing ultrashort laser pulses [71–73]. However, this approach requires a highpower femtosecond optical source, and therefore has significant technological limitations. Recent experimental investigations of the NLSI-based spin-wave generation have shown that this mechanism is sufficiently fast to enable generation of short spin-wave packets with the duration down to a few nanoseconds, close to the best results achieved by using optical-pulse excitation [8]. Figure 11.13a shows the dynamic response of the NLSI oscillator built into the magnonic system (Fig. 11.10) recorded by applying driving electrical current I = 7 mA in the form of pulses with different

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widths wd in the nanosecond range. As seen from these data, the generated pulse of the dynamic magnetization maintains a nearly rectangular shape and a constant peak intensity for wd down to 10 ns. As wd is reduced to 3 ns, the dynamic-magnetization pulse becomes almost Gaussian-shaped, and its peak intensity starts to decrease. This width defines the characteristic time scale, at which the finite response time of the NLSI oscillator starts to affect the excitation efficiency. The dependence of the peak oscillation intensity on the width of the driving-current pulse (Fig. 11.13b) shows that the efficiency of the driving mechanism rapidly diminishes at wd < 3 ns. In particular, at wd < 2.5 the intensity of the dynamic-magnetization pulse falls below 50% of its maximum value achieved with long pulses.

Figure 11.13 (a) Time dependencies of the auto-oscillation intensity recorded by applying pulses of the driving current with the amplitude of 7 mA and different widths, as labeled. The data for the 3-ns wide pulse are fitted by the Gaussian function. (b) Dependence of the peak intensity of the auto-oscillation pulse on the width of the driving-current pulse. Curve is a guide for the eye. The data were obtained at H0 = 1000 Oe. The autooscillation frequency is 8 GHz. Reprinted from [8], with the permission of AIP Publishing.

To estimate the shortest pulse achievable without an appreciable loss of intensity, we note that the actual temporal width of the generated pulse of the dynamic magnetization is smaller than that of the driving-current pulse. For example, Gaussian fitting of the magnetization pulse induced by the pulse of current with the width wd = 3 ns (Fig. 11.13a) yields a half-maximum width of 2.1 ns. Therefore, one can conclude that the NLSI mechanism allows one to excite pulses of the dynamic magnetization with the width down

Excitation of Spin Waves by Pure Spin Currents

to about 2 ns, without significantly compromising the efficiency of the conversion of the dc current pulse into a microwave-frequency signal. Generation of shorter microwave pulses can also be achieved, but at the expense of the reduced power efficiency. Figure 11.14 characterizes the spin waves excited in the nanowaveguide by the pulses of current with the width wd = 3 ns and amplitude 7 mA. The shown time-resolved BLS maps were recorded by rastering the probing laser spot over a 4.5 µm ¥ 0.8 µm region encompassing the NLSI oscillator and the waveguide. To better visualize the spatial characteristics of the propagating spin-wave packet, these maps are compensated for the spatial decay of spin waves by multiplying the experimental data by exp(2x/x).

Figure 11.14 Normalized decay-compensated maps of the spin-wave intensity recorded at delays of 1.6, 2.4, and 3.2 ns with respect to the start of the driving current pulse, as labeled. Dashed lines indicate the contour of the nanowaveguide. The data were obtained at H0 = 1000 Oe. The width of the drivingcurrent pulse is 3 ns and its amplitude is 7 mA. The spin-wave frequency is 8 GHz. Reprinted from [8], with the permission of AIP Publishing.

The maps shown in Fig. 11.14 demonstrate that the pulse of the current-induced magnetization precession of the NLSI oscillator efficiently couples to spin waves in the waveguide, producing a propagating spin-wave packet. At t = 1.6 ns, a dynamic signal emerges in the waveguide near its edge facing the NLSI oscillator. At t = 2.4 ns, the increased-intensity region spreads away from the oscillator, indicating the propagation of the leading front of the spin-wave

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packet. Finally, at t = 3.2 ns the packet occupies almost the entire analyzed length of the waveguide. We note that the studied wave packet has a small temporal width of about 2.1 ns, but a large spatial width of more than 4 µm comparable to the size of the measured maps. The relation between these characteristics is determined by the large group velocity of spin waves in the waveguide vg = 2.5 µm/ ns, which is advantageous for the reduced spatial propagating losses, but complicates the analysis of the propagation characteristics in the spatial domain. Figure 11.15 shows the results of the analysis of the wave packet propagation in the time domain. We fit the temporal profiles of the spin-wave packet recorded at different distances from the NLSI oscillator by a Gaussian function (Fig. 11.15a), and determine the dependence of the temporal width of the packet (Fig. 11.15b) on the propagation coordinate x. The obtained dependence reveals an intriguing behavior. Based on the general theory of waves in dispersive media, one can expect that the short wave packet should experience a temporal broadening due to the different velocities of its spectral components. Contrary to these expectations, the data of Fig. 11.15b demonstrate that the wave packet experiences a noticeable compression from the temporal width of 2.06±0.05 ns at the edge of the guide, to 1.73±0.05 ns at the distance of 1.5 µm from the edge. The initial compression is followed by a monotonic broadening at larger propagation distances. The only known mechanism that can be responsible for the observed temporal compression is the dynamic magnetic nonlinearity, which under certain conditions can counteract the dispersion broadening and lead to the formation of spin-wave solitons [74, 75]. This interpretation is consistent with the observed dependence on the propagation distance: the compression is observed only at the initial stage of the packet propagation, where the amplitude of the dynamic magnetization in the spin wave is sufficiently large. As the amplitude of the propagating wave decreases due to damping, the nonlinear effects disappear, and the wave packet starts to broaden due to the expected effects of dispersion. We emphasize a significant potential of the observed nonlinear phenomena for applications, where they can be utilized to further reduce the width of the generated wave packets, improving the information transmission capacity of magnonic nanocircuits.

Conclusions

Figure 11.15 (a) Temporal profile of the wave packet at x = 0, fitted by the Gaussian function. (b) Propagation-coordinate dependence of the temporal width of the spin-wave packet. The data were obtained at H0 = 1000 Oe. The width of the driving-current pulse is 3 ns and its amplitude is 7 mA. The spin-wave frequency is 8 GHz. Reprinted from [8], with the permission of AIP Publishing.

11.6 Conclusions In conclusion, we would like to emphasize the large importance of the advancements in current-induced excitation and control of spin waves for the development of the research field of magnonics, which until recently has evolved independently from the field of spin-torque phenomena. Although the limitations imposed by the geometry of traditional spin-torque devices have discouraged researchers from using them in magnonic circuits, we believe that recent advances in studies of pure spin currents will dramatically accelerate the integration of spin-torque and magnonic devices. An additional encouraging benefit of pure spin currents for magnonics is the possibility to excite and control spin waves in magnetic insulators such as yttrium iron garnet, which is presently viewed as the most suitable material for future nanomagnonic circuits.

Acknowledgments

We would like to acknowledge A. Anane, J. Ben Youssef, V. Bessonov, P. Bortolotti, M. Collet, V. Cros, B. Divinskiy, M. Evelt, K. GarciaHernandez, T. Kendziorczyk, O. Klein, T. Kuhn, J. Leuthold, R. Liu, G. de Loubens, M. Munoz, V. V. Naletov, J. L. Prieto, G. Reiss, A. B.

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Rinkevich, A. Telegin, H. Ulrichs, S. Urazhdin, and G. Wilde for their contributions to this work. This work was supported in part by the Deutsche Forschungsgemeinschaft and the program Megagrant N° 14.Z50.31.0025 of the Russian Ministry of Education and Science.

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64. Duan, Z., Smith, A., Yang, L., Youngblood, B., Lindner, J., Demidov, V. E., Demokritov, S. O., and Krivorotov, I. N. (2014). Nanowire spin torque oscillator driven by spin orbit torques, Nat. Commun., 5, 5616. 65. Liu, R. H., Lim, W. L., and Urazhdin, S. (2015). Dynamical skyrmion state in a spin current nano-oscillator with perpendicular magnetic anisotropy, Phys. Rev. Lett., 114, 137201.

66. Awad, A., Dürrenfeld, P., Houshang, A., Dvornik, M., Iacocca, E., Dumas, R., and Åkerman, J. (2017). Long-range mutual synchronization of spin Hall nano-oscillators, Nat. Phys., 13, 292–299.

67. Demidov, V. E., Urazhdin, S., Zholud, A., Sadovnikov, A. V., Slavin, A. N., and Demokritov, S. O. (2015). Spin-current nano-oscillator based on nonlocal spin injection, Sci. Rep., 5, 8578.

68. Demidov, V. E., Urazhdin, S., Divinskiy, B., Rinkevich, A. B., and Demokritov, S. O. (2015). Spectral linewidth of spin-current nanooscillators driven by nonlocal spin injection, Appl. Phys. Lett., 107, 202402.

69. Urazhdin, S., Demidov, V. E., Cao, R., Divinskiy, B., Tyberkevych, V., Slavin, A., Rinkevich, A. B., and Demokritov, S. O. (2016). Mutual synchronization of nano-oscillators driven by pure spin current, Appl. Phys. Lett., 109, 162402.

70. Demidov, V. E., Urazhdin, S., Zholud, A., Sadovnikov, A. V., and Demokritov, S. O. (2015). Dipolar field-induced spin-wave waveguides for spin-torque magnonics, Appl. Phys. Lett., 106, 022403.

71. Satoh, T., Terui, Y., Moriya, R., Ivanov, B. A., Ando, K., Saitoh, E., Shimura, T., and Kuroda, K. (2012). Directional control of spin-wave emission by spatially shaped light, Nat. Photonics, 6, 662–666.

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72. Au, Y., Dvornik, M., Davison, T., Ahmad, E., Keatley, P. S., Vansteenkiste, A., Van Waeyenberge, B., and Kruglyak, V. V. (2013). Direct excitation of propagating spin waves by focused ultrashort optical pulses, Phys. Rev. Lett., 110, 097201.

73. Iihama, S., Sasaki, Y., Sugihara, A., Kamimaki, A., Ando, Y., and Mizukami, S. (2016). Quantification of a propagating spin-wave packet created by an ultrashort laser pulse in a thin film of a magnetic metal, Phys. Rev. B, 94, 020401(R). 74. Kalinikos, B. A., Kovshikov, N. G., and Slavin, A. (1988). Envelope solitons and modulational instability of dipole-exchange spin waves in yttrium-iron garnet films, Sov. Phys. JETP, 67, 303–312.

75. Kovshikov, N. G., Kalinikos, B. A., Patton, C. E., Wright, E. S., and Nash, J. M. (1996). Formation, propagation, reflection, and collision of microwave envelope solitons in yttrium iron garnet films, Phys. Rev. B, 54, 15210.

76. Demidov, V. E., Urazhdin, S., de Loubens, G., Klein, O., Cros, V., Anane, A., and Demokritov, S. O. (2017). Magnetization oscillations and waves driven by pure spin currents, Phys. Rep., 673, 1–31.

Chapter 12

Propagating Spin Waves in Nanocontact Spin Torque Oscillators Randy K. Dumas, Afshin Houshang, and Johan Åkerman Department of Physics, University of Gothenburg, Origovägen 6B, Gothenburg 412 96, Sweden [email protected]

12.1  Introduction Spin torque oscillators (STOs) comprise a diverse class of nanomagnetic devices that exhibit ultrawide operating frequencies and modulation rates. Furthermore, their manufacturing processes are compatible with radio frequency (RF) complementary metaloxide semiconductor (CMOS) fabrication standards, which makes them particularly well-suited for easy integration into existing and future technologies. STOs combine several spintronic and nanomagnetic phenomena for their operation, such as giant magnetoresistance (GMR), tunneling magnetoresistance (TMR), spin transfer torque (STT), and, depending on device architecture and the constituent magnetic materials employed, a plethora of possible fundamentally intriguing magnetodynamic modes. Spin Wave Confinement: Propagating Waves (2nd Edition) Edited by Sergej O. Demokritov Copyright © 2017 Pan Stanford Publishing Pte. Ltd. ISBN  978-981-4774-35-2 (Hardcover),  978-1-315-11082-0 (eBook) www.panstanford.com

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Central to the operation of an STO is the phenomenon of STT, which was first theoretically postulated just over 20 years ago in two seminal papers by Slonczewski [1] and Berger [2]. In its most basic manifestation an electron spin current can be thought of as a flow of angular momentum, given each constituent electron carries a quantized unit of spin with it. When this spin-polarized current enters, or reflects from, a nonmagneticferromagnetic interface the constituent spins will begin to precess if the spin polarization is not collinear with the local magnetization. Furthermore, over a very short distance, typically on the scale of a few atomic lattice spacings, the average spin angular momentum in the direction transverse to the local magnetization direction within the ferromagnet is lost. Owing to the conservation of angular momentum in a closed system, such a change in angular momentum must be accompanied by a corresponding torque, which then acts on the local magnetization. Depending on the sign of the current and spin polarization this so-called in-plane torque has a component that is either parallel or antiparallel to the intrinsic damping torque, which acts to align the local and magnetization along the local effective field. When predominantly antiparallel to the damping toque the time-averaged STT acts to reduce the magnetic damping and, if strong enough, can destabilize the magnetization, thus exciting large angle dynamics, or oscillations, of the magnetization, typically in the gigahertz frequency range for most applied field strengths. In the interest of completeness, it is important to note that this in-plane torque is generally accompanied by an out-ofplane torque that has the same symmetry as an effective field. While this out-of-plane torque is small for all-metallic STOs, which will be our primary focus here, it can be substantial in devices with tunnel barriers [3, 4], where it, for example, plays an important role in promoting mutual STO synchronization [5]. A more rigorous treatment of the STT can be found in the seminal work of Ralph and Stiles [6]. How is a spin-polarized current generated? There are several ways in which initially spin-unpolarized electrons can become partially, or in some cases completely, spin polarized. We will briefly discuss two methods. The first, and in many ways the most straightforward, is by a process simply known as spin filtering. Here, the spin current arises directly from the unequal population

Nanocontact Spin Torque Oscillators

of spin-up and spin-down electrons at the Fermi surface of a ferromagnetic metal. In short, when an unpolarized charge current flows through a ferromagnetic layer it will exit partially (~40% for Fe, Ni, and Co) spin polarized and the generated spin current is accompanied by a charge current. Interestingly, and very topical today, is that spin currents do not require any charge current to flow. If one utilizes the spin Hall effect (SHE) [7, 8] by passing a charge current longitudinally through a material with a large spin-orbit coupling (e.g., Pt, Pd, W, etc.) a transverse pure spin current is established, where spins of opposite signs move in opposite directions, thus establishing a net flow of angular momentum without a flow of charge (in that direction). STOs that rely on the SHE are often referred to as spin Hall nano-oscillators (SHNOs) [9–13]. Pure spin currents are especially useful if one wants to make an STO with insulating magnetic materials, such as the ferromagnetic insulator yttrium-iron-garnet (YIG) [14, 15], which possesses an ultralow damping, making it particularly exciting for a variety of future technologies based on the functionalization of spin waves, that is, magnonics [16–20]. A recent review [21] describes both spin torque and spin Hall nano-oscillators, as well as their technological potential, at length. The first experimental demonstration of STT-driven magnetization precession was reported by Tsoi et al. in 1998 [22] using a device based on a mechanical point contact, where a highly sharpened metallic tip was used to make electrical contact, typically with an area of ~102 nm2, to an extended multilayer film (see Fig. 12.1a). Such small contacts are necessary to increase the charge current density and, therefore the resulting spin current density, needed for the STT to stabilize steadystate magnetization dynamics in the nearby ferromagnetic layer. Interestingly, these early point contact devices can be thought of as the predecessors of nanocontact STOs (NC-STOs) [23–27] that are the primary focus of this chapter.

12.2  Nanocontact Spin Torque Oscillators

As advances in nanofabrication techniques evolved, the ability to reliably fabricate STOs with nanoscopic dimensions became a

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reality. Typical fabrication techniques for defining nanocontacts (NCs) with nanodimensions are electron-beam lithography or hole colloidal lithography techniques [28]. These techniques combined with argon ion milling, reactive ion etching, and optical lithography can produce NCs with well-defined sizes, cross sections, and array geometries within an insulating layer, as shown schematically in Fig. 12.1b. In both point contact and NC devices, Figs. 12.1a and 12.1b, respectively, it is only the current injection site that is confined to a nanoscopic region, and the underlying layers can be thought of as being effectively infinite in extent for most practical purposes. The current density is largest close to the NC, and therefore at the top of the multilayer film stack. As the electric current enters the extended multilayer structure it spreads out dramatically decreasing in density and is ultimately collected by a ground contact far from the NC.

Figure 12.1  Schematic representation of (a) point contact and (b) nanocontact spin torque oscillators highlighting the multilayer composition.

The most oft-utilized layer structure relies on a fixed layer/ spacer/free layer trilayer where both fixed and free layers are magnetic. The fixed layer serves two purposes. First, it provides the necessary spin polarization through reflections at the spacer/fixed layer interface. The reflected electrons will be preferentially spin polarized in a direction opposite to the spin polarization of the fixed and free layer. Second, the fixed layer provides the necessary GMR, for a conductive spacer such as Cu,

Nanocontact Spin Torque Oscillators

necessary for electrical detection of the STT-induced oscillations in the free layer. As the device is current biased, the oscillating magnetization will manifest itself as a microwave voltage across the device, which can routinely be measured after amplification in the time domain using an oscilloscope or in the frequency domain using a spectrum analyzer. Note that insulating spacer layers can be employed where the typically much larger TMR effect leads to larger microwave voltages and therefore higher output powers [29–31]. The free layer is typically thin (2–5 nm) to lower the necessary drive currents needed to sustain oscillations. In the interest of completeness, we should also mention that STT-driven oscillations have also been observed in NC-STOs with only a single free magnetic layer [32–34]. Here, the spin polarization is established by unequal spin populations on either side of the free layer facilitated by asymmetric top/bottom interfaces, and the resulting microwave voltage is generated by the anisotropic magnetoresistance (AMR) effect. The materials, and resulting easy magnetization directions of the constituent fixed and free layers found in NC-STOs can vary greatly. For example, in all-perpendicular devices [35, 36] both fixed and free layers have perpendicular magnetic anisotropy (PMA). Typical materials of choice include both Co/Ni and Co/ Pt multilayers. Such all-perpendicular devices show particular promise for STO operation in zero applied fields but are not optimal for maximizing output power. An interesting compromise has been to consider STOs with a tilted free layer [37–41] where the magnetization has both in-plane and out-of-plane components and can therefore work in a zero field, while maintaining relatively high output powers [42]. Yet another material combination establishes an orthogonal equilibrium direction of the magnetizations of the fixed and free layers [43, 44]. For example, a fixed/free layer combination of Co, which naturally has an in-plane anisotropy, and Co/Ni multilayers with PMA has recently generated significant attention as such an arrangement allowed for the first experimental demonstration [45] of dissipative magnetic droplets [46–50]. However, both the original experiments and the devices discussed in this chapter rely on all in-plane devices where both fixed and free layers have a dominant in-plane anisotropy. More specifically, an 8-nm-thick Co layer

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fixed layer will be separated from a 4.5-nm-thick permalloy (Ni80Fe20) free layer by an 8 nm Cu spacer [51–53].

12.3  Magnetodynamical Modes

NC-STOs exhibit a diversity of fundamentally different dynamical modes that depend greatly on the device architecture, constituent magnetic layers, applied field strength, and applied field angle. Here, we focus on the types of modes commonly exhibited by NCSTOs where the fixed and free layers have an intrinsic in-plane anisotropy. This is best described by considering the out-of-plane angular dependence, that is, the angle of the applied field with respect to the film plane, of these excitations. The initial experiments [26, 54, 55] using in-plane fields revealed a strongly nonlinear solitonic bullet mode, as later described by both analytical calculations [56] and micromagnetic simulations [57]. However, when a field of sufficient strength orients the free layer out-of-plane an exchange-dominated propagating spin wave is excited, as originally predicted by Slonczewski in 1999 [58]. The wave vector of the excited propagating spin waves is inversely proportional to the NC radius. The first direct experimental observation of such propagating spin waves was made in 2011 by Madami et al. [59] using microfocused Brillouin light scattering (m-BLS), a technique described in detail in the previous chapter. For intermediate angles the initial calculations [60], simulations [61], and experiments [54] showed that the solitonic bullet mode can only be generated for applied field angles up to a certain critical angle, qC, and the propagating spin wave mode, while technically possible for any applied field angle, should not coexist with the bullet mode. The reasoning behind this “winner takes all” interpretation is that the bullet mode has a substantially lower threshold current since radiative losses do not have to be compensated for. However, these initial considerations for oblique applied field angles did not properly take into account the critically important Oersted field generated in NC-STOs. For example, a simple application of Ampère’s law reveals that for typical drive currents and NC dimensions, the Oersted field is

Magnetodynamical Modes

on the order of 0.1 T at the NC edge. This is a significant fraction of the externally applied field and cannot be considered an inconsequential perturbation. The remainder of this chapter will therefore be devoted to the consequences of the Oersted field on the magnetodynamics.

12.3.1  Role of the Oersted Field

The Oersted field has three important consequences on the fundamental spin wave excitations that were not considered in the initial studies [51]: (i) localization of the propagating mode at low applied field angles, (ii) multimode coexistence, and (iii) asymmetric spin wave propagation. The first two consequences can be easily observed in the angular dependence of the electrically measured microwave frequency spectra, as shown in Fig. 12.2. Three distinct angular regions can be identified. For qex > 60° a single spin wave mode lies well above the ferromagnetic resonance (FMR) frequency (solid white line) of the Ni80Fe20 free layer and corresponds to a propagating spin wave, as originally theorized by Slonczewski [58]. For qex < qC = 60° the threshold current of the Slavin–Tiberkevich solitonic bullet mode [56, 60] becomes finite, and we enter an angular range where both the propagating mode and the bullet mode can be excited by the STT. However, for these intermediate angles it is difficult for these two modes to coexist as they spatially overlap, and we do not observe a “winner takes all” situation. Instead, the primary feature observed for these intermediate angles is a broad low-frequency ( f < 3 GHz) signature, consistent with the relatively slow stochastic switching between modes, consistent with mode hopping [62]. As the applied field angle is lowered further, the Oersted field now acts to localize the propagating spin wave mode. It is clearly shown in Fig. 12.2 that for qex < 45° the propagating mode frequency now becomes smaller than the FMR frequency and can therefore be considered self-localized. Furthermore, micromagnetic simulations reveal that the Oersted field acts to spatially separate the bullet mode and the now-localized propagating mode. This spatial separation then ensures that these two modes can now coexist, and the low-frequency

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mode-hopping signature in the microwave frequency spectra disappears. Some of the most interesting consequences of the Oersted field on the STT-driven dynamics in NC-STOs can be observed for the propagating spin wave mode and are not immediately apparent by just analyzing the electronically measured microwave spectra shown in Fig. 12.2.



Figure 12.2  Experimentally measured angular dependence of the frequency spectra for an NC-STO with an NC diameter of 90 nm and a bias current of –20 mA. Three distinct angular regimes are indicated, each exhibiting unique magnetodynamics. © (2016) IEEE. Reprinted, with permission, from [21].

12.4  Asymmetric Spin Wave Propagation

In Fig. 12.2 the FMR frequency was superimposed on the measured microwave spectra. However, the magnetic field used to calculate this FMR frequency only utilized the large (~1 T) external field applied by an electromagnet and neglects the spatially inhomogeneous current induced Oersted field in the vicinity of the NC. Therefore, the FMR frequency shown in

Asymmetric Spin Wave Propagation

Fig. 12.2 is strictly only valid in the far field, that is, at a considerable distance from the NC. It is important to instead think in terms of a spatially varying FMR frequency, calculated on a cell-by-cell basis, that takes into account all relevant magnetic fields. Such an FMR frequency spatial map near an NC is shown in Fig. 12.3a. For this particular geometry, the in-plane component of the external field points to the right and the flow of electrons is into the plane of the page, establishing the sense of the indicated Oersted field shown in Fig. 12.3b. Therefore, regions near the top (bottom) of the NC experience a smaller (larger) effective field and the FMR frequency varies, as indicated in Fig. 12.3a. Clearly, the current-induced Oersted field has dramatic consequences on the local FMR frequency [63], which shows variations on the order of 6 GHz. The regions with a higher local FMR frequency will tend to block the propagation of spin waves in that direction, downward in this case. The spatial variation of the power of the simulated spin wave propagation is shown in Fig. 12.3b for an applied field angle of qex = 70°. A clear spin wave beam is observed, with the majority of the spin wave energy travelling upward toward the regions with locally smaller FMR frequencies.

Figure 12.3  The calculated FMR frequency landscape for a single NC with a diameter of 90 nm. The inhomogeneous FMR frequency landscape promotes the formation of spin wave beams, as shown in the micromagnetic simulations (b).

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Figure 12.4 (a) Scanning electron microscope (SEM) image of an NC-STO device showing the ground (G) and signal (S) top contacts. The location of the NC and the sense of the Oersted field are also highlighted. The m-BLS beam is scanned over the 2 × 1 µm2 area indicated. The experimentally measured spin wave intensity maps are shown in (b) and (c) for two opposite orientations of the external field (|µoHext| = 0.7 T), where the in-plane component points either to the left or to the right, respectively. Reprinted with permission from Madami et al. [52].

To experimentally observe such asymmetric spin wave propagation directly, a scanning microscopy probe is necessary. As discussed at length in the previous chapter, scanning m-BLS is an ideal probe for spin wave excitations in general and in STOs [64]. However, the NC-STO geometry poses several challenges for m-BLS measurements; most notably the top contact used to supply current to the NC is usually very thick (~micrometers), which is much more than the propagation length of the probing light. It is therefore necessary to alter the device architecture

Spin Wave Beam–Driven Synchronization

slightly to allow for the probing light to more efficiently probe the magnetodynamically active Ni80Fe20 free layer. In the original study [59], which definitely and directly probed propagating spin waves in NC-STOs, this was done by using a focused ion beam to open a window in the top contact. More recently we have slightly modified our top contact geometry to allow for a considerable space above the NC for the laser to probe, as shown in Fig. 12.4a. The NC-STOs studied only generate STT-driven dynamics for a single current polarity corresponding to electrons flowing into the plane of the page. Therefore, in the scanned region above the NC, the Oersted field will always point predominantly to the left, as indicated in Fig. 12.4a–c. However, the in-plane component of the external field (HIP) provided by an electromagnet can easily change sign. When the Oersted field and HIP point in opposite directions there is a local field minimum, and therefore FMR frequency minimum in the scanned region and spin waves are clearly observed (Fig. 12.4b). However, when the sign of HIP is reversed, a local FMR frequency maximum thwarts spin wave propagation upward and little to no BLS signal is observed (Fig. 12.4c). This work [52] was the first to experimentally probe the asymmetric propagation of spin waves in NC-STOs, having only been shown using simulations before.

12.5  Spin Wave Beam–Driven Synchronization

The synchronization of coupled nonlinear oscillators is a common natural phenomenon, and STOs provide an ideal playground to study synchronization at the nanoscale. Synchronization in NC-STOs has been relatively straightforward, and two seminal works were published in 2005 [65, 66], demonstrating the successful synchronization of two oscillators, just one year after the first experimental demonstration of what we now refer to as an NC-STO. The necessary coupling is achieved by pattering multiple NCs on top of a shared free layer. It was then shown that propagating spin waves in the free layer are the dominant coupling mechanism [67]. The necessary coupling needed to enforce synchronization has typically been described as being mutual in nature, with each oscillator playing a more or less equal role. Interestingly, since these early demonstrations of NC-

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STO synchronization, progress in synchronizing more than two oscillators had remained relatively slow. Only in 2013 was the successful synchronization of three high-frequency NCSTOs mediated by spin waves published [28]. The most recent advancements in synchronization of NC-STOs has focused on taking advantage of the highly directional spin wave beams discussed in the prior section [53]. The critical importance of spin wave beams on synchronization can be easily understood by considering Fig. 12.5a–d, where the NCs are fabricated in what is called a horizontal geometry, where HIP points along the direction joining the two oscillators (Fig. 12.5a). Figure 12.5b shows the calculated FMR frequency landscape in the vicinity of the two NCs and Fig. 12.5c the resulting simulated propagating spin wave emission pattern. As can be clearly seen, in the horizontal geometry the majority of the spin wave energy propagates in a direction away from the other NC or oscillator. It is therefore not surprising that synchronization can prove to be difficult in this geometry. Figure 12.5d shows the experimentally measured microwave spectra measured as a function of bias current. Over the entire current range two distinct modes, corresponding to each oscillator, are observed and synchronization is never achieved. It is important to note that in this horizontal geometry, which is in fact the same geometry used in many of the initial experiments, synchronization is sometimes observed but with a relatively low success rate. The solution to robustly synchronizing multiple oscillators also now seems relatively obvious, namely one should fabricate the NCs in a vertical geometry such that the spin waves travel in a direction toward the other NC (Fig. 12.5e–h). We note a dramatically different spin wave emission pattern (Fig. 12.5g) and have successfully observed synchronization at distances up to 1300 nm (Fig. 12.5h), which is not only a record distance for NC-STOs but also consistent with the finite propagation length typically observed for Ni80Fe20 free layers. Yet another interesting consequence of this spin wave beam–mediated synchronization is that the character of the synchronization can no longer be considered mutual in nature but is instead driven. That is, the frequency of the bottommost oscillator sets the frequency of the synchronized pair.

Spin Wave Beam–Driven Synchronization



Figure 12.5  SEM images (a, e), calculated FMR frequency landscapes (b, f), simulated spin wave intensity maps (c, g), and experimentally measured frequency spectra (d, h) of two NC-STOs spatially arranged so the in-plane component of the external field is either parallel or orthogonal to the line joining the NCs, that is, either a horizontal (left panels) or a vertical (right panels) array geometry. Due to the asymmetric spin wave beam propagation, synchronization is much more preferred in the vertical array geometry and can be observed for NC separations up to 1300 nm. Adapted by permission from Macmillan Publishers Ltd: Nature Nanotechnology [53].

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Figure 12.6  (a, top) Experimentally measured frequency spectra of 5 NC-STOs in a vertical array geometry showing robust synchronization over the entire range of bias currents. When the in-plane component of the applied field is rotated by 30°, synchronization is broken (a, bottom). By comparing the integrated power and linewidth (b) of the fully and partially synchronized states shown in (a) it can be concluded that there exists pairwise synchronization of four of the oscillators and a single unlocked oscillator in (a, bottom). Reprinted by permission from Macmillan Publishers Ltd: Nature Nanotechnology [53].

The results shown in Fig. 12.5e–h for a vertical NC geometry can be extended to an even larger number of NCs by simply daisychaining them together, as shown in Fig. 12.6a (upper panel) for

Conclusions and Future Directions

five such NC-STOs. As the synchronization is driven in nature, from the onset of auto-oscillations the system is in a synchronized state and only a single mode is observed over the entire current range. To break the synchronization, the in-plane angle of HIP is rotated by 30° so that the spin waves propagate in a direction away from the other NCs (Fig. 12.6a, lower panel). Note, that complete desynchronization is not observed. However, after analyzing the mode power and linewidths (Fig. 12.6b), we conclude that four of the oscillators are pairwise synchronized and one oscillator is independent.

12.6  Conclusions and Future Directions

In conclusion, this chapter summarizes some of the most recent work since the first experimental observation of propagating spin waves in NC-STOs was published in 2011 [39]. A central theme has been the critical role of the inhomogeneous Oersted field on the magnetodynamics, which has shown to give rise to not only spin wave localization but also asymmetric spin wave propagation. This asymmetric spin wave propagation has farreaching consequences. For example, synchronization mediated by such spin wave beams cannot be considered mutual in nature but is instead driven by the NC from which the spin waves originate. In addition, one can daisy-chain any number of NC-STOs and extend the spin wave propagation length, forming a type of spin wave repeater capable of transporting information over much larger distances than the intrinsic propagation length would ordinarily allow. Taking advantage of the directionality of the spin waves, one can also envision techniques to further manipulate and control the synchronization state in a simple two-dimensional array of NC-STOs by changing the direction of the in-plane applied field, thereby steering the spin wave beam in a particular direction, as was also shown in Fig. 12.6a (lower panel) to break the synchronization of the 5-NC chain. This concept is schematically shown in Fig. 12.7, in which the NCs with the same color are synchronized and the arrows indicate the spin wave propagation direction under different in-plane applied field directions.

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In a different context, STOs can also act as potential building blocks for next-generation neuromorphic computing architectures that aim to mimic the neurobiological functionality found in the human brain [68]. Biological neurons can be modelled as nonlinear oscillators that adjust their rhythms in response to external stimuli. Furthermore, neurons form massively parallel and interconnected networks of coupled oscillators where the coupling is mediated by adaptable and programmable synapses. In analogy to spintronic nano-oscillators, neural networks can self-synchronize and can play a key role in, for example, associative memory tasks. In fact, if properly harnessed, spintronic nano-oscillators, which work on nanometer length scales and at gigahertz frequencies, may even be more scalable than biological neurons, which work on much larger (~micrometers) and slower (~10 Hz) scales. Recently, simulations have shown that pulse-coupled STOs with dynamic synapses can show various cooperative phenomena such as synchronous, clustered, and coherent states based on synaptic interactions [69]. Furthermore, recent experiments showed the realization of the first magnetic tunnel junction (MTJ)-based memristor, which paves the way for bioinspired magnetic neural computing [70]. Finally, non-Boolean computation with artificial oscillatory neural networks, which operate using the frequency domain representation of an analog signal where the frequency, phase, and amplitude play the role of information carriers, is projected to be much more energy efficient than its present, and even future, Boolean counterpart.

Figure 12.7  Schematic representations of synchronization in a simple two-dimensional triangular array of NC-STOs. By controlling the direction of the in-plane component of the applied field (HIP), synchronization (indicated by circles with the same color) can be controlled between the bottommost and (a) upper-left, (b) upper-right, or (c) all three oscillators.

References

Acknowledgments This work was supported, in part, by the Swedish Research Council (VR), the Swedish Foundation for Strategic Research (SSF), and the Knut and Alice Wallenberg Foundation. It was also partially supported by the European Research Council (ERC) grant no. 307144 “MUSTANG” and the European Commission FP7-ICT-2011 contract no. 317950 “MOSAIC.” We would also like to thank M. Madami for the m-BLS measurements presented in this chapter.

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12. Dürrenfeld, P., Awad, A. A., Houshang, A., Dumas, R. K., and Åkerman, J. (2017). A 20 nm spin Hall nano-oscillator, Nanoscale, 1–12.

13. Mazraati, H., et al. (2016). Low operational current spin Hall nano-oscillators based on NiFe/W bilayers, Appl. Phys. Lett., 109(24), 242402. 14. Collet, M., et al. (2016). Generation of coherent spin-wave modes in yttrium iron garnet microdiscs by spin–orbit torque, Nat. Commun., 7, 10377. 15. Haidar, M., et al. (2016). Controlling Gilbert damping in a YIG film using nonlocal spin currents, Phys. Rev. B, 94(18), 180409. 16. Neusser, S., and Grundler, D. (2009). Magnonics: spin waves on the nanoscale, Adv. Mater., 21, 2927–2932.

17. Kruglyak, V. V., Demokritov, S. O., and Grundler, D. (2010). Magnonics, J. Phys. D: Appl. Phys., 43(26), 260301.

18. Serga, A. A., Chumak, A. V., and Hillebrands, B. (2010). YIG magnonics, J. Phys. D: Appl. Phys., 43(26), 264002.

19. Lenk, B., Ulrichs, H., Garbs, F., and Münzenberg, M. (2011). The building blocks of magnonics, Phys. Rep., 507(4–5), 107–136. 20. Dumas, R. K., and Åkerman, J. (2014). Spintronics: channelling spin waves, Nat. Nanotechnol., 9(7), 503–504.

21. Chen, T., et al. (2016). Spin-torque and spin-Hall nano-oscillators, Proc. IEEE, 104(10), 1919–1945. 22. Tsoi, M., et al. (1998). Excitation of a magnetic multilayer by an electric current, Phys. Rev. Lett., 80(19), 4281–4284.

23. Myers, E., Ralph, D., Katine, J., Louie, R., and Buhrman, R. (1999). Current-induced switching of domains in magnetic multilayer devices, Science, 285, 867–870. 24. Rippard, W. H., Pufall, M. R., and Silva, T. J. (2003). Quantitative studies of spin-momentum-transfer-induced excitations in Co/Cu multilayer films using point-contact spectroscopy, Appl. Phys. Lett., 82(8), 1260. 25. Pufall, M. R., Rippard, W. H., and Silva, T. J. (2003). Materials dependence of the spin-momentum transfer efficiency and critical current in ferromagnetic metal/Cu multilayers, Appl. Phys. Lett., 83(2), 323.

26. Rippard, W., Pufall, M., Kaka, S., Russek, S., and Silva, T. (2004). Direct-current induced dynamics in Co90Fe10/Ni80Fe20 point contacts, Phys. Rev. Lett., 92(2), 27201.

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29. Deac, A. M., et al. (2008). Bias-driven high-power microwave emission from MgO-based tunnel magnetoresistance devices, Nat. Phys., 4(10), 803–809. 30. Maehara, H., et al. (2013). Large emission power over 2 µW with high Q factor obtained from nanocontact magnetic-tunnel-junctionbased spin torque oscillator, Appl. Phys. Express, 6(11), 113005.

31. Tsunegi, S., Yakushiji, K., Fukushima, A., Yuasa, S., and Kubota, H. (2016). Microwave emission power exceeding 10 μW in spin torque vortex oscillator, Appl. Phys. Lett., 109(25), 252402. 32. Özyilmaz, B., Kent, A., Sun, J., Rooks, M., and Koch, R. (2004). Currentinduced excitations in single cobalt ferromagnetic layer nanopillars, Phys. Rev. Lett., 93(17), 176604.

33. Sani, S. R., Durrenfeld, P., Mohseni, S. M., Chung, S., and Akerman, J. (2013). Microwave signal generation in single-layer nano-contact spin torque oscillators, IEEE Trans. Magn., 49(7), 4331–4334.

34. Dürrenfeld, P., et al. (2016). Low-current, narrow-linewidth microwave signal generation in NiMnSb based single-layer nanocontact spin-torque oscillators, Appl. Phys. Lett., 109(22), 222403. 35. Mangin, S., Ravelosona, D., Katine, J. A., Carey, M. J., Terris, B. D., and Fullerton, E. E. (2006). Current-induced magnetization reversal in nanopillars with perpendicular anisotropy, Nat. Mater., 5(3), 210–215. 36. Sim, C. H., Moneck, M., Liew, T., and Zhu, J.-G. (2012). Frequency-tunable perpendicular spin torque oscillator, J. Appl. Phys., 111(7), 07C914.

37. Zhou, Y., Zha, C. L., Bonetti, S., Persson, J., and Åkerman, J. (2008). Spin-torque oscillator with tilted fixed layer magnetization, Appl. Phys. Lett., 92(26), 262508.

38. Zhou, Y., Zha, C. L., Bonetti, S., Persson, J., and Åkerman, J. (2009). Microwave generation of tilted-polarizer spin torque oscillator, J. Appl. Phys., 105(7), 07D116. 39. Zhou, Y., Bonetti, S., Zha, C. L., and Åkerman, J. (2009). Zero-field precession and hysteretic threshold currents in a spin torque nano device with tilted polarizer, New J. Phys., 11(10), 103028. 40. Chung, S., et al. (2013). Tunable spin configuration in [Co/Ni]-NiFe spring magnets, J. Phys. D: Appl. Phys., 46(12), 125004.

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41. Nguyen, T. N. A., et al. (2011). [Co/Pd]-NiFe exchange springs with tunable magnetization tilt angle, Appl. Phys. Lett., 98(17), 172502.

42. Skowroński, W., Stobiecki, T., Wrona, J., Reiss, G., and Van Dijken, S. (2012). Zero-field spin torque oscillator based on magnetic tunnel junctions with a tilted CoFeB free layer, Appl. Phys. Express, 5(6), 63005. 43. Rippard, W. H., et al. (2010). Spin-transfer dynamics in spin valves with out-of-plane magnetized CoNi free layers, Phys. Rev. B, 81(1), 14426. 44. Mohseni, S. M., et al. (2011). High frequency operation of a spintorque oscillator at low field, Phys. Status Solidi RRL, 5(12), 432–434. 45. Mohseni, S. M., et al. (2013). Spin torque-generated magnetic droplet solitons, Science, 339(6125), 1295–1298.

46. Hoefer, M., Silva, T., and Keller, M. (2010). Theory for a dissipative droplet soliton excited by a spin torque nanocontact, Phys. Rev. B, 82(5), 54432.

47. Mohseni, S. M., et al. (2014). Magnetic droplet solitons in orthogonal nano-contact spin torque oscillators, Phys. B: Condens. Matter, 435, 84–87. 48. Chung, S., et al. (2016). Magnetic droplet nucleation boundary in orthogonal spin-torque nano-oscillators, Nat. Commun., 7, 11209.

49. Macià, F., Backes, D., and Kent, A. D. (2014). Stable magnetic droplet solitons in spin-transfer nanocontacts, Nat. Nanotechnol., 9(12), 992–996. 50. Xiao, D., et al. (2017). Parametric autoexcitation of magnetic droplet soliton perimeter modes, Phys. Rev. B, 95(2), 24106.

51. Dumas, R. K., et al. (2013). Spin-wave-mode coexistence on the nanoscale: a consequence of the oersted-field-induced asymmetric energy landscape, Phys. Rev. Lett., 110(25), 257202. 52. Madami, M., et al. (2015). Propagating spin waves excited by spin-transfer torque: a combined electrical and optical study, Phys. Rev. B, 92(2), 024403.

53. Houshang, A., Iacocca, E., Dürrenfeld, P., Sani, S. R., Åkerman, J., and Dumas, R. K. (2016). Spin-wave-beam driven synchronization of nanocontact spin-torque oscillators, Nat. Nanotechnol., 11(3), 280–286. 54. Bonetti, S., et al. (2010). Experimental evidence of self-localized and propagating spin wave modes in obliquely magnetized currentdriven nanocontacts, Phys. Rev. Lett., 105(21), 217204.

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55. Bonetti, S., Puliafito, V., Consolo, G., Tiberkevich, V. S., Slavin, A. N., and Åkerman, J. (2012). Power and linewidth of propagating and localized modes in nanocontact spin-torque oscillators, Phys. Rev. B, 85(17), 174427. 56. Slavin, A., and Tiberkevich, V. (2005). Spin wave mode excited by spin-polarized current in a magnetic nanocontact is a standing selflocalized wave bullet, Phys. Rev. Lett., 95(23), 237201.

57. Consolo, G., Lopez-Diaz, L., Torres, L., and Azzerboni, B. (2007). Magnetization dynamics in nanocontact current controlled oscillators, Phys. Rev. B, 75(21), 214428. 58. Slonczewski, J. (1999). Excitation of spin waves by an electric current, J. Magn. Magn. Mater., 195(2), 261–268.

59. Madami, M., et al. (2011). Direct observation of a propagating spin wave induced by spin-transfer torque, Nat. Nanotechnol., 6(10), 635–638.

60. Gerhart, G., Bankowski, E., Melkov, G., Tiberkevich, V., and Slavin, A. (2007). Angular dependence of the microwave-generation threshold in a nanoscale spin-torque oscillator, Phys. Rev. B, 76(2), 24437. 61. Consolo, G., et al. (2008). Micromagnetic study of the abovethreshold generation regime in a spin-torque oscillator based on a magnetic nanocontact magnetized at an arbitrary angle, Phys. Rev. B, 78(1), 1–7.

62. Muduli, P. K., Heinonen, O. G., and Åkerman, J. (2012). Decoherence and mode hopping in a magnetic tunnel junction based spin torque oscillator, Phys. Rev. Lett., 108(20), 207203.

63. Hoefer, M., Silva, T., and Stiles, M. (2008). Model for a collimated spin-wave beam generated by a single-layer spin torque nanocontact, Phys. Rev. B, 77, 144401. 64. Demidov, V. E., Urazhdin, S., and Demokritov, S. O. (2010). Direct observation and mapping of spin waves emitted by spin-torque nano-oscillators, Nat. Mater., 9(12), 984–988.

65. Kaka, S., Pufall, M. R., Rippard, W. H., Silva, T. J., Russek, S. E., and Katine, J. (2005). Mutual phase-locking of microwave spin torque nanooscillators, Nature, 437(7057), 389–392.

66. Mancoff, F. B., Rizzo, N. D., Engel, B. N., and Tehrani, S. (2005). Phase-locking in double-point-contact spin-transfer devices, Nature, 437(7057), 393–395. 67. Pufall, M., Rippard, W., Russek, S., Kaka, S., and Katine, J. (2006). Electrical measurement of spin-wave interactions of proximate spin transfer nanooscillators, Phys. Rev. Lett., 97(8), 87206.

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68. Grollier, J., Querlioz, D., and Stiles, M. D. (2016). Spintronic nanodevices for bioinspired computing, Proc. IEEE, 104(10), 2024–2039.

69. Nakada, K., and Miura, K. (2016). Pulse-coupled spin torque nano oscillators with dynamic synapses for neuromorphic computing, in IEEE 16th International Conference on Nanotechnology (IEEENANO), pp. 397–400. 70. Lequeux, S., et al. (2016). A magnetic synapse: multilevel spin-torque memristor with perpendicular anisotropy, Sci. Rep., 6, 31510.

Chapter 13

Parametric Excitation and Amplification of Spin Waves in Ultrathin Ferromagnetic Nanowires by Microwave Electric Field

Roman Verba,a Mario Carpentieri,b Giovanni Finocchio,c Vasil Tiberkevich,d and Andrei Slavind aInstitute

of Magnetism, 36b Vernadskogo Blvd., Kyiv 03680, Ukraine of Electrical and Information Engineering, Politecnico di Bari, via E. Orabona 4, Bari I-70125, Italy cDepartment of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, V.le F. d’Alcontres 31, Messina I-98166, Italy dDepartment of Physics, Oakland University, 2200 N. Squirrel Road, Rochester, MI 48309, USA bDepartment

[email protected]

Electric field control of magnetization of ferromagnets via magnetoelectric effects attracts a lot of attention as it makes possible the development of novel magnetic devices with ultralow power consumption. In particular, it could allow energy-efficient excitation and processing of spin wave signals in ferromagnetic films and nanowires. In this chapter we focus on the application of the voltageSpin Wave Confinement: Propagating Waves (2nd Edition) Edited by Sergej O. Demokritov Copyright © 2017 Pan Stanford Publishing Pte. Ltd. ISBN 978-981-4774-35-2 (Hardcover), 978-1-315-11082-0 (eBook) www.panstanford.com

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controlled magnetic anisotropy (VCMA) effect to excite, amplify, and control propagating spin waves in magnetic nanostructures by means of an externally applied microwave electric field. It is shown that a microwave electric field signal of a certain frequency applied to a nanoscale VCMA gate can parametrically excite half-frequency spin waves, propagating from the gate. It is also shown that a similar microwave electric field signal applied to a “control” VCMA gate, situated along the propagation path of the excited half-frequency spin wave, can effectively parametrically amplify the propagating spin wave if the initial wave amplitude is sufficiently small, or can stabilize the amplitude of the propagating wave if the initial wave amplitude is sufficiently large. In addition, we discuss the effect of the interfacial Dzyaloshinskii–Moriya interaction (IDMI) on the parametric amplification of spin waves and demonstrate that IDMI can be used for the improvement of the operational characteristics of the spin wave signal processing devices based on the VCMA effect.

13.1 Introduction

The research field of magnonics has attracted growing attention due to the potential applications of propagating spin waves (SW) (or magnons) in the next generation of signal-processing devices as information carriers replacing the electrons that are used for this purpose in traditional CMOS devices [8, 26, 28]. SWs as carriers of information have several important advantages: (i) small wavelengths, , down to tens of nanometers [1], (ii) the possibility to vary SW dispersion both by the patterning of a ferromagnetic film [11, 41] and by dynamic manipulation of the external bias magnetic field [7, 26], (iii) the possibility to achieve nonreciprocal SW propagation [16, 24, 54, 63], and (iv) the possibility to control SWs by various nonlinear and parametric processes [16, 31]. All these features allow one to design nanoscale devices for both digital and analog data processing [12, 21, 46]. One of the important drawbacks of the existing SW technology is that the excitation and control of SW are performed using external magnetic fields, which are, usually, created by electric currents in adjacent conducting lines. The generation of microwave currents

Introduction

leads to substantial Ohmic losses, limits the minimum time constants of the devices due to the finite inductance of the conducting lines, and complicates device miniaturization and compatibility of SW devices with conventional voltage-controlled CMOS microelectronics. Modern methods of magnetization manipulation using spinpolarized dc current [23, 51] or pure spin current produced using the spin-Hall effect [9, 10, 29] can be used at nanoscale, but have insufficient energy efficiency. A possible way to create energy-efficient nanoscale SW devices lies in the use of magnetoelectric effects, which, in general, allow one to manipulate the magnetization of a ferromagnet or the effective field in it directly by the application of an electric field or voltage. Several different magnetoelectric effects are now intensively investigated for this purpose, e.g., stress-mediated effects in piezoelectric-piezomagnetic heterostructures, magnetoelectric effect in multiferroic materials, spin flexo-electric interaction, etc. [15, 33, 36, 49]. In this chapter we focus on the recently discovered effect of voltage-controlled magnetic anisotropy (VCMA) [13, 42, 60]. This effect takes place at the interface between a ferromagnetic metal (e.g., Fe, CoFeB) and a nonmagnetic insulator (e.g., MgO) and originates from the different rates of filling of d-like electron bands in response to electric field applied perpendicularly to the interface [13]. Since electrons in different bands contribute unequally to the uniaxial perpendicular magnetic anisotropy at the interface, the electric field can be used to modulate perpendicular magnetic anisotropy of a ferromagnetic metal. The VCMA effect has many attractive features, including linearity (variation of the anisotropy energy is directly proportional to the applied voltage) [13], absence of a hysteresis [17], possibility of relatively large variations of the anisotropy field [32, 39], and vanishing inertia (at least, in the gigahertz frequency range) [43, 65]. These features make VCMA promising for various practical applications. In particular, the VCMA effect has been already proposed for use in magnetic recording [47, 59], control of motion of domain walls [2, 45] and skyrmions [64], phase control of SWs [40], and linear and parametric excitation of standing SWs in ferromagnetic nanodots [6, 43, 48, 65]. In this chapter we demonstrate several principles, which could be used for SW signal processing using VCMA effect, following, mainly,

387

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Refs. [55–58]. First, in Section 13.2, the excitation of SWs in ultrathin ferromagnetic nanowires by microwave electric field is considered. It is known that ultrathin ferromagnetic films and nanowires can exist in the out-of-plane-magnetized or in-plane-magnetized ground state depending on their thickness [18, 61], and both these cases are discussed below. It is shown that in both above mentioned cases the SW excitation is possible only via the parametric mechanism, when the SW frequency is one-half of the frequency of the applied voltage. Then, in Section 13.3, the amplification and stabilization of SW amplitude by microwave voltage is demonstrated. Finally, in Section 13.4, we discuss the effect of the SW nonreciprocity induced by the interfacial Dzyaloshinskii–Moriya interaction (IDMI) [14, 38] on the amplification of SWs, and show how it can improve the characteristics of SW signal-processing devices.

13.2 Excitation of Spin Waves

As it has been already pointed out and will be explained in more detail below, the SW excitation by VCMA microwave signal in the absence of an external magnetic field is possible only via the parametric excitation mechanism. The parametric excitation is the threshold process—SWs are excited only when the amplitude of the driving signal (microwave voltage in our case) at the double SW frequency exceeds a certain threshold level [16]. This excitation threshold depends on the geometry of the ferromagnetic nanowire and on the wavenumber and structure of the excited SW, as discussed in Section 13.2.1. In the supercritical regime, when the amplitude of the driving voltage exceeds the threshold value, the nonlinear interactions between the excited SWs become important and determine the amplitudes of the excited SWs and stability of the parametric excitation. This regime is briefly discussed in Section 13.2.2.

13.2.1 Efficiency of the Parametric Interaction and Excitation Threshold 13.2.1.1 Perpendicularly magnetized nanowire

A layout of a device for the excitation of SWs by electric field via the VCMA effect is presented in Fig. 13.1. A ferromagnetic metal

Excitation of Spin Waves

(Fe) strip of the width wy and thickness h grown, typically, on the GaAs(100) substrate is covered by a thin dielectric layer (MgO). The microwave-frequency voltage applied between the top gate electrode of the length Lg and the Fe strip causes the variation of the magnetic anisotropy at the Fe/MgO interface. The variation of the surface anisotropy constant Ks is linearly proportional to the electric field E at the interface: ΔKs = βE, where β is the magnetoelectric coefficient [13]. Since VCMA is a purely interfacial effect, it is more pronounced in ultrathin ferromagnetic metal films (h ≈ 1 nm), in which the perpendicular surface magnetic anisotropy significantly affects the magnetic ground state [18, 61]. In the absence of a bias magnetic field, which is very desirable for applications, the direction of the static magnetization of a magnetic film depends on its thickness—below some critical value (~ 0.78 nm for Fe film) the static magnetization is out-of-plane, while above this value, it is inplane. We shall start our discussion with the first case.

Figure 13.1 A layout of a device for the VCMA-induced parametric excitation of propagating SWs.

Only the perpendicular anisotropy can be modulated via the VCMA effect. In the considered case of a perpendicular static magnetization this leads to the modulation of the perpendicular (z) component of the effective field inside the ferromagnet at the frequency of the applied voltage, i.e., the microwave component of the effective field is parallel to the static magnetization. It is known that in such a case the linear excitation of SWs (when the frequency of the excited wave is equal to the frequency of the driving voltage) is impossible, since the linear excitation requires the non-zero component of the microwave field to be perpendicular to the static magnetization [16]. At the same time, the geometry of an unbiased magnetic strip (Fig. 13.1) is ideally suitable for the parametric excitation of SWs, when the frequency of the excited wave is two times smaller than

389

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the frequency of the applied microwave electric field. This is a socalled geometry of a parametric “parallel pumping” [16, 31], when the external microwave field (or the effective field, as in our case) is parallel to the static magnetization. The mechanism of the parametric excitation can be described as follows. Owing to the thermal fluctuations, the magnetization of the film performs stochastic precession around the equilibrium direction M0 = Msez (Ms is the saturation magnetization of the ferromagnetic film). If the precession of magnetization with the frequency ωk (Fourier component of the stochastic precession) goes over an elliptic trajectory (e.g., due to the shape or crystallographic anisotropy), it creates a modulation of the longitudinal (mz) magnetization component with double frequency 2ωk. The externally driven time-varying effective anisotropy field ΔBan(t) = ΔBan(t)exp[–iωpt] couples to these out-of plane oscillations of the film magnetization and pumps energy into the magnetic system of the film. This pumping process is efficient (resonant) if the pumping frequency is close to ωp ≈ 2ωk. When the pumping amplitude exceeds a certain threshold ΔBan,th, determined by the magnetic damping in the film material and the radiation losses, it excites a propagating SW of the frequency ωp/2. The parametric excitation of SWs by both spatially uniform and localized parallel pumping has been extensively discussed in literature [16, 31, 34]. In general, the magnetization dynamics in a ferromagnetic nanowire is described by the well-known Landau–Lifshitz equation [16], in which the effect of VCMA can be taken into account by time-dependent effective anisotropy tensor Nan (t ) = (e z ƒ e z )b E (t )/( m0Ms2h) with the anisotropy axis ez. Representing the nanowire magnetization as a sum of static magnetization and a series of propagating SWs having wavevector k = kex, frequency ωk, damping rate Γk and vector structure mk(y) = mk,xex + mk,yey,

È M (r , t ) = M s Íe z + ÍÎ

 (c m e ( k

k

k

i kx -w k t )

)

˘ + c.c. ˙ , ˙˚

(13.1)

one can obtain the following equation for the SW amplitude ck under the action of a parallel parametric pumping [34]:

dck + iw k ck + G k ck = dt

Lg

Âi l k¢

x

Vkk ¢ bk + k ¢ e

- iw pt * ck '

.

(13.2)

Excitation of Spin Waves

Here bk is the Fourier image of the spatial profile of the effective pumping magnetic field bp(x), which, in our case, is equal to the voltage-induced anisotropy field bp(x) = ΔBan(x) = 2βE(x)/(Msh), Lg and lx are the lengths of the gate electrode and the magnetic nanowire in the x direction, respectively, and the superscript * denotes the complex conjugation. For a spatially uniform gate of the length Lg the Fourier profile bk = sinc[kLg /2] , and the parametric pumping most efficiently couples to the SWs having opposite wavevectors k and –k, which is a consequence of the momentum conservation law. Note that, in general, the SW spectrum of a nanowire consists of infinitely many SW modes, which differ by the mode profile along the nanowire width (i.e., mk = mk,n(y)). However, since the pumping is uniform along the nanowire width, it does not couple the SW modes having different width profiles, and the dynamics of each SW mode is described by the same Eq. 13.2) (with replacement ωk Æ ωk,n, etc.). Below we shall restrict our attention to the case of the lowest SW mode n = 0 of the nanowire, which has a uniform profile along the nanowire width. Notes on the excitation of higher-order SW modes are given in Section 13.2.1.3. The parameter Vkk’ describes the efficiency of the parametric coupling of SWs with the pumping. In this case, when the condition of exact parametric resonance ωk = ωp/2 is satisfied for a certain SW wavevector k, the coupling parameter is equal to

Vk( - k ) = Vkk = -g

m*k ◊ m*k 2 A k

,

(13.3)

where γ is the gyromagnetic ratio of the ferromagnetic material, angular brackets mean averaging over the nanowire width, and A k is the norm of the SW mode (see Appendix A, which provides details on the calculation of the SW spectrum and the mode vector structure mk). Noting that for a perpendicular static magnetization the SW mode structure can be represented as mk = |mk,x|ex + i|mk,y|ey, one can easily find that the parametric interaction efficiency is proportional to the difference of the squares of the dynamic magnetization components, Vkk ~ (|mk,x|2 – |mk,y|2). In other words, |Vkk| depends on the ellipticity of magnetization precession. So, for circular precession, when |mk,x| = |mk,y|, the parametric coupling is equal to zero, and this coupling increases with the increase in the precession ellipticity.

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The ellipticity of SW depends on the nanowire geometry and on the SW wavenumber k, resulting in the dependence of Vkk on these parameters. The dependence of Vkk on the SW wavenumber k for the lowest mode (n = 0) is shown in Fig. 13.2b. At small wavenumbers k, the precession ellipticity is determined mainly by the in-plane shape anisotropy of the nanowire and the long axis of the precession ellipse is directed along the nanowire length. Naturally, this shape anisotropy is higher for narrower nanowires, which results in larger parametric interaction efficiency of long-wavelength SWs in narrower nanowires. With an increase in k (decrease in the SW wavelength) the dynamic demagnetization along the x axis increases and, at a certain wavenumber k0, the precession becomes circular, leading to zero parametric interaction efficiency. This compensation of ellipticity occurs at smaller k0 for wider nanowires. Above the compensation point k > k0 the precession is also elliptical, but with the long axis of the precession ellipse directed along the y axis, which is a consequence of the increased dynamic demagnetization in the x direction. Finally, at even higher k the properties of the SW mode are mainly determined by the exchange interaction, which is isotropic, leading to vanishing of the SW ellipticity and parametric interaction coefficient Vkk.

Figure 13.2 Eigenfrequency fk = ωk/2π (a), parametric interaction efficiency Vkk/γ (b), and minimum threshold electric field for the parametric excitation of SW Eth (c) as functions of SW wave number for the lowest mode of a ferromagnetic nanowire. Calculations were made for a Fe nanowire of thickness h = 0.75 nm and widths wx = 20 nm (solid lines) and wx = 100 nm (dashed lines). The material parameters are μ0Ms = 2.1 T, exchange length λex = 3.4 nm, constant of surface perpendicular anisotropy Ks = 1.36 mJ/m2, Gilbert damping αG = 0.004, non-uniform line broadening Δωnu = 2π×230 MHz, magnetoelectric coefficient b = 100 fJ/Vm [42, 52, 61].

Excitation of Spin Waves

As it was mentioned above, the parametric excitation starts when the energy transfer from the pumping to SWs overcomes the total losses. The total losses consist of the losses associated with the intrinsic SW damping and the radiation losses due to the SW propagation outside of the pumping region. The minimum possible threshold is achieved when the pumping region is significantly larger than the SW propagation length. In that case, the threshold is determined only by the intrinsic damping rate Γk = αGωk + Δωnu, which consists of the homogeneous damping αGωk and the nonhomogeneous line broadening characterized by the parameter Δωnu. As one can show from Eq. 13.2, taking into account only the SWs having opposite wavevectors k and –k (since bk Æ d (k ) for an infinitely large pumping region), the pumping threshold amplitude is given by the simple expression bp,th = Γk / Vkk. Above this level, bp > bp,th, the SW amplitudes start to grow exponentially, which means that the parametric excitation of SWs takes place [16, 31]. The exponential growth of SWs is limited by nonlinear interactions between the excited SWs, and this nonlinear regime is considered below in Section 13.2.2. The threshold electric field can be found as Eth = bp,thMsh/2β, and Fig. 13.2c demonstrates this threshold electric field as a function of SW wavenumber k. Due to the existence of the point of circular polarization k = k0, in which the threshold becomes infinite, the dependence Eth(k) has two local minima, one at k = 0 and another one at a certain large k > k0 (for wx = 20 nm the second minimum is out of the plot region). As one can see from Fig. 13.2c, in narrow nanowires the minimum parametric excitation threshold is achieved for relatively long SWs (small k). In contrast, for a sufficiently wide nanowire, the second local minimum at k > k0 becomes deeper than the minimum at k = 0, meaning that the excitation of relatively short SWs becomes more efficient. However, one should be aware of the possibility of excitation of higher-order modes in wide nanowires (see Section 13.2.1.3). Typical magnitudes of the threshold electric field are about 1 V/nm, which is readily achievable in experiment. Note that here we used the parameters of a common VCMA material: Fe/MgO multilayer (β ≈ 100 fJ/Vm). Recent studies have found more effective VCMA structures like Cr/Fe/MgO multilayer with β ≈ 300 fJ/Vm [44].

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Another possibility to decrease the threshold is to use materials with ultralow damping, e.g., CoFeB/FeZr (αG = 0.0027) [22]. For real applications, of course, the gate size Lg should be of the order of or, better, much smaller than the SW propagation length, so that the excited SWs can propagate away from the gate to be processed and received by other gates. In this case the excitation threshold becomes higher due to increased radiation losses. To find the threshold theoretically it is convenient to introduce envelope amplitudes of SWs a1,2 (t , x ) = S k 'ck ' (t )exp[iw k 't + i(k ' ∓ k )x ] , which can be defined as the complex amplitudes of SW packets having the mean wavevectors k (a1) and –k (a2), respectively. Using this representation in Eq. 13.2, one can derive the following equation, which describes temporal and spatial evolution of the SW envelope amplitudes under the resonant (ωp = 2ωk) parametric pumping:



∂ Ê ∂ ˆ * * ÁË ∂t + v ∂x + G ˜¯ a1 = -iVkk bp ( x )a2 - iVkk a bp ( x )a1 ,

(13.4)

and the second equation is obtained by the replacement a1 ´ a2 and v Æ –v. Here v = ∂ω/∂k is the SW group velocity, bp(x) describes spatial profile of the pumping and a ∫ b2k /b0 is the so-called pumping non-adiabiticity parameter. These equations are often called the Bloembergen equations [20, 34]. As one can see, the nonadiabatic term (last term on the righthand side of Eq. 13.4) describes the interaction of pumping with two SWs having the same wavevector, while the adiabatic term (first term on the right-hand side) describes the interaction of pumping with counter-propagating SWs. The non-adiabatic term is proportional to the Fourier component of the spatial pumping profile b2k . As a result of the momentum conservation law, the pumping can split into 2 SWs with wavevector k only if it has the wavevector equal to 2k (i.e., has non-zero Fourier component at 2k). In the considered case of a piecewise uniform pumping (bp(x) = bp under the gate and bp(x) = 0 otherwise) the coefficient |α| = sinc[kLg], i.e., the non-adiabatic term is significant only if the pumping length is comparable to the SW wavelength. To find the excitation threshold, one should calculate the amplification rate for one of the SWs (which is done below in Section 13.3.1), and find the pumping amplitude at which the amplification rate becomes infinite, which means that any small fluctuation will

Excitation of Spin Waves

grow from the thermal level up to stationary level determined by nonlinear interactions [20]. The threshold can be calculated from the following implicit equation: 

| Vkk bth |2 - (G- | aVkk bth |) G- | aVkk bth |

2

È 2 Lg ˘ = - tan Í | Vkk bth |2 - (G- | aVkk bth |) ˙. v ˚ Î (13.5)

In the limiting case of a small gate length (ΓLg/v >1 the threshold approaches the value bth = Γ/(1+|α|). Note, however, that the level of pumping nonadiabaticity α also depends on the gate length (which is not accounted for in Fig. 13.3), so in a real situation of a uniform pumping created by a long gate, the threshold amplitude is bth Æ Γ/Vkk. Of course, if the pumping profile under the gate in nonuniform, the trend will be different, but this situation lies out of the scope of this chapter.

Figure 13.3 The dependence of the parametric excitation threshold on the length of the pumping localization at different levels of the pumping nonadiabaticity.

It should be also noted that there is no direct relation between the SW wavenumber and the gate length Lg in Eq. 13.5. The parametric

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Parametric Excitation and Amplification of Spin Waves

pumping can excite SWs having the wavelength of the order of Lg or larger, as well as SWs with the wavelength that is much smaller than the gate size. This is a substantial advantage of the parametric excitation in comparison to the more common linear excitation of SWs by microwave antennas. In the linear case the excitation of SW with wavelength λSW = 2π/k < Lg by a single strip-line antenna of width Lg is almost impossible.

13.2.1.2 Nanowire with in-plane static magnetization

Here we shall consider the case of nanowires having the thickness that is above the critical one, in which the static magnetization of a nanowire in the absence of an external magnetic field is directed along the nanowire (x direction in Fig. 13.1). As it was mentioned above, VCMA is purely interfacial effect, so its overall efficiency decreases with the nanowire thickness. At the same time, the thicker nanowires are usually easier to fabricate, and they have better uniformity. Therefore thicker wires have a lower inhomogeneous broadening of the SW linewidth [52]. Also, due to the vanishing static demagnetization fields in the in-plane case, it is possible to achieve in that case higher SW frequency at zero bias field. At a first glance, it may seem that in this case it is possible to excite SWs in the linear regime, since the microwave electric field creates the variation of the perpendicular anisotropy, while the static magnetization lies in the nanowire plane. However, a microwave variation of the anisotropy is not equivalent to the appearance of a microwave magnetic field. Microwave electric field E(t) ≈ cos[ωpt] at the frequency ωp leads to the variation of the perpendicular surface anisotropy ΔKs = βE. The effective magnetic field associated with the surface anisotropy can be calculated as Beff = –δW/δM, where W = –∫ Ks Mz2/(hM2s ) dxdy is the anisotropy energy, and is equal to

DBeff (t , r ) =

2b E (t , r ) M z (t , r ) hMs2

ez .

(13.6)

Note that Mz(t) is the dynamic magnetization component having zero static value and oscillating at the SW frequency ωk if SW is excited. Thus, there are no terms proportional solely to the external force E(t), varying at the pumping frequency ωp and directed perpendicularly to the static magnetization. Consequently, the linear SW excitation in the case of the in-plane static magnetization (for any

Excitation of Spin Waves

in-plane direction, not only along the nanowire) is also impossible. It could be shown that the linear excitation becomes possible if a nanowire is magnetized at a finite angle to the surface θ ≠ 0, π/2, as it was experimentally realized in Ref. [65] (indeed, in this case Mz contains both static and dynamic component and Beff has term at ωp, which is not orthogonal to dynamic magnetization). One can also see from Eq. 13.6 that if the frequency of the electric field is ωp ≈ 2ωk, the effective magnetic field DBeff contains a resonant term at the SW frequency ωk, which indicates the possibility of the parametric excitation of SWs. Using the standard expansion of magnetization in a series of SW modes Eq. 13.1 (with a proper direction M0 = Msex of the static magnetization) in the Landau–Lifshitz equation, one can obtain the same Eq. 13.2 describing the dynamics of SW amplitudes. The only change is in the efficiency of the parametric interaction, which in the in-plane case becomes

Vkk = g

(m ) * k ,z

2

/2 A k .

(13.7)

For the lowest (uniform) SW mode, this expression is simplified to Vkk = γ|mk,z|/(4|mk,y|). As one can see, the parametric interaction efficiency is proportional to the out-of-plane magnetization component, but not to the ellipticity of magnetization precession, as in the common case of a parallel pumping discussed above. This is a consequence of different coupling mechanisms—the coupling of field (or effective field) to the longitudinal magnetization component in the out-of-plane case and the coupling of dynamic anisotropy field with dynamic magnetization component in the in-plane case. The wavenumber dependence of the parametric interaction efficiency for the lowest SW mode is shown in Fig. 13.4b. Vkk increases as the SW becomes shorter, and saturates at the value γ/4 in the high-k range, in which, due to the dominant role of the isotropic exchange interaction, the SW polarization becomes circular, |mk,y| = |mk,z|. Thus, in the case of an in-plane magnetization there are no principal limits on the wavenumber of the excited SW, in contrast to the case of an out-of-plane magnetization, for which Vkk could be equal to zero at a certain wavenumber, and vanishes at high k. At a small k the interaction efficiency is higher for narrower nanowires, as they have stronger in-plane (y) dynamic demagnetization

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fields resulting in a larger out-of-plane component of the dynamic magnetization.

Figure 13.4 Eigenfrequency (a), parametric interaction efficiency (b), and minimum threshold electric field for parametric excitation (c) as functions of the SW wavenumber for the lowest SW mode of a ferromagnetic nanowire with in-plane static magnetization. Parameters: Fe nanowire of thickness h = 1 nm and widths wx = 20 nm (solid lines) and wx = 100 nm (dashed lines), material parameters being the same as in Fig. 13.2.

Figure 13.4c shows the wavenumber dependence of the minimum threshold of a parametric SW excitation Eth = ΓkMsh/(2βVkk). It is clear that in a wide nanowire it is easier to excite relatively short SWs, while the excitation of SWs with k Æ 0 requires a higher electric field pumping. In a narrow nanowire this difference becomes less and less pronounced. Consequently, the use of narrow nanowires with width of the order of 10–20 nm gives advantages only when the excitation of SWs with a small k, (i.e., close to the ferromagnetic resonance frequency) is required. Away from the ferromagnetic resonance (at a distance of several gigahertz) the difference in the nanowire width is practically insignificant. In the case of an outof-plane static magnetization, in contrast, the dependence on the nanowire width is more complex (see Fig. 13.2). It also should be noted that the minimum threshold values in the in-plane case are similar to that in the out-of-plane case and are of the order of 0.5 V/nm. However, in the in-plane case the range of the excited SW frequencies is much larger. For example, for a nanowire of the 20 nm width the threshold electric field varies insignificantly, and is below 0.5 V/nm in the whole considered frequency range fk = ωk/2π = 6 – 45 GHz (see Fig. 13.4ac). For the case of an out-of-plane static magnetization, the excitation of SWs even at 20 GHz requires the electric field amplitude of about 2 V/nm (see Fig. 13.2; note that

Excitation of Spin Waves

the material parameters are the same as in Figs. 13.2 and 13.4). Thus, one can conclude that in the case of the in-plane static magnetization the parametric excitation of SWs by microwave VCMA is more effective than in the case of the out-of-plane magnetization. This is related to the different mechanisms of coupling of the pumping with SWs—via precession ellipticity in the case of the out-of-plane magnetization, and via perpendicular dynamic magnetization component in the case of the in-plane magnetization.

13.2.1.3 Notes on multimode waveguides

Above, we considered only the lowest SW mode, uniform across the nanowire width. This is a natural assumption for a narrow nanowire, since higher-order SW modes have much higher frequencies and do not satisfy the condition of a parametric resonance. However, when the nanowire width increases, the SW spectrum becomes denser, and the condition ωp = 2ωk,n can be satisfied for several SW modes simultaneously. An example of such a situation is shown in Fig. 13.5a,c: one can see that only in a narrow frequency range there are no degenerate SW modes (3.6–4.8 GHz for 100 nm wide nanowire with the out-of-plane static magnetization, Fig. 13.5a, and 3.8–6.1 GHz for the case of in-plane magnetization, Fig. 13.5c). Within this range the parametric pumping excites only the lowest SW mode. However, above this range the resonance condition is satisfied for two or more modes with different width profiles simultaneously. It is important to find out which mode will be excited in this case. When the pumping amplitude gradually increases from zero, the first excited mode is the mode with the lowest threshold. Moreover, with a further increase in the pumping amplitude, the subsequent mode with a higher threshold (calculated in a linear approximation as above) will not be excited at this slightly higher threshold. This happens due to the energy transfer from the pumping to the first excited mode and nonlinear interaction between the modes, which results in an increase in the excitation threshold for the next mode. Often, the second mode is excited only when the first one is close to its saturation due to nonlinear processes [31]. Thus, in most cases the only mode that can be excited by parametric pumping is the mode with the lowest excitation threshold.

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Figure 13.5 SW spectrum (a, c) and parametric interaction efficiency as functions of the SW frequency (b, d) for different width modes of a Fe nanowire of the width 100 nm. (a, b) Out-of-plane static magnetization (h = 0.75 nm); (c, d) in-plane magnetization (h = 1 nm). Material parameters are the same as in Figs. 13.2 and 13.4, and free boundary conditions are used for the calculation of the SW profile.

As already mentioned, the excitation threshold depends on the SW damping rate, radiation loses, and the efficiency of the parametric interaction. The damping rate for all the degenerate modes with the same frequency is almost the same. For ultrathin nanowires the group velocity of SWs is mainly determined by the exchange interaction, and is approximately given by v ≈ 2ωMλex2k. Thus, the group velocity is smaller for higher-order SW modes and, consequently, the radiation losses Γrad = v/Lg at a given length of the pumping localization also decrease with an increase in the mode number n. The behavior of the parametric excitation efficiency depends on the direction of static magnetization. For the in-plane static magnetization the coupling coefficient Vkk is higher for higher-order SW modes (see Fig. 13.4d), which is explained by the increased out-

Excitation of Spin Waves

of-plane dynamic magnetization component due to the additional dynamical stiffness in the y direction. Consequently, both factors— the highest Vkk and the lowest group velocity—are achieved for the highest SW mode at a given frequency. Thus, in the in-plane magnetized case, the parametric pumping excites the highest SW mode that satisfies the condition of parametric resonance. For the out-of-plane static magnetization the behavior of Vkk is much more complicated (see Fig. 13.5b). In a certain frequency range, the maximum parametric interaction efficiency is achieved for the highest mode (for f < 9.75 GHz in Fig. 13.5b). In contrast, in other regions, it is achieved for the lowest SW mode (see the interval 9.75– 10.75 GHz in Fig. 13.5b). Depending on the material and geometrical parameters, the modes with intermediate mode numbers may have the higher interaction efficiency than the lowest and the highest SW modes. Thus, in the out-of-plane magnetization case, the largest Vkk and the lowest group velocity may be achieved for two different SW modes. Consequently, depending on the pumping localization length (gate length), which determines the influence of radiation loses on the excitation threshold, different SW modes can be excited by the pumping of the same frequency. Finally, it is important to note, that the use of ferromagnetic nanowires or waveguides is crucially important. Indeed, if a continuous two-dimensional film or a strip having a large width (larger than SW propagation length) is used, the SW spectrum becomes two-dimensional, containing a continuum of degenerate SW modes having the same frequency and different propagation direction. The lowest threshold of the parametric excitation corresponds to the SW mode, which propagates along the gate (in the y direction in Fig. 13.1 if wy is extended to infinity), since this SW mode never leaves the pumping region, and has, consequently, zero radiation losses. Such a situation is not suitable for applications, since the parametrically excited SWs should leave the excitation gate region to be processed and received by another gate. Thus, in order to guarantee a definite propagation direction of the excited SWs, one should use ferromagnetic nanowires or strips with a sufficiently small width for the formation of discrete SW modes propagating along the nanowire.

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Parametric Excitation and Amplification of Spin Waves

13.2.2 Nonlinear Spin Wave Dynamics under Parametric Pumping: Stationary Amplitudes of Excited Spin Waves It has been shown above that using VCMA microwave pumping, it is possible to overcome the threshold of the parametric excitation of propagating SWs. It should be noted, however, that overcoming the parametric excitation threshold does not guarantee a stable excitation of the propagating SWs. In an externally driven parametric process, the excitation of monochromatic half-frequency propagating SWs could be hindered by many parasitic nonlinear phenomena, such as modulational instability, auto-oscillations, and even developed turbulence of parametrically excited SWs [31]. The exact manifestation of these nonlinear phenomena depends on the strength of the microwave driving signal, SW spectrum of a magnetic sample, and peculiarities of the parametric interaction between the driving signal and parametric SWs in a given sample geometry. The conditions necessary for a stable parametric excitation of SWs were studied in detail for two- and three-dimensional geometry in the case of spatially uniform parametric pumping in the framework of the so-called L’vov–Zakharov–Starobinets “S-theory” [31, 62]. The S-theory shows that the amplitudes of the excited SWs are limited by the “phase mechanism,” which manifests itself as a reduction of the phase correlation between the pumping signal and the excited SWs due to the nonlinear interaction between the SWs. The S-theory allows one to calculate the stationary SW amplitudes in the abovethreshold regime, and, thus, can be used for theoretical estimation of the parametric excitation efficiency. Unfortunately, the conclusions of the conventional S-theory [31] cannot be directly applied to our case of VCMA-induced parametric excitation of SWs, where the ferromagnetic sample is quasi-one-dimensional (nanowire) and the microwave parametric pumping is localized in a relatively small region under the VCMA gate electrode. In order to verify if and when the stable parametric excitation of SWs is possible, the micromagnetic simulations using GPMagnet solver [30] were performed. Below, we present a micromagnetic study for the case of Fe nanowire of 20 nm width, 1 nm thickness, which has an in-plane static magnetization in zero magnetic field.

Excitation of Spin Waves

A similar study for the case of the out-of-plane magnetization is presented in [57], and shows the same features. In our simulation we used the same material parameters as in Fig. 13.4, except the effective Gilbert constant, which was equal to αG = 0.033. This value was chosen to give the magnitude of the total SW damping rate (Gilbert damping plus nonuniform damping) in the studied frequency range similar to the one used in the Fig.13.4. The gate length was Lg = 100 nm, and the thermal fluctuations corresponding to the temperature of 1 K were taken into account. For these parameters the FMR frequency of the nanowire is equal to f0 = ω0/2π = 6 GHz, which is the lowest SW frequency in the spectrum (see Fig. 13.4). When the pumping frequency is above the double FMR frequency, fp > 2f0, the micromagnetic simulations showed that propagating SWs were excited at a certain threshold pumping amplitude. The frequency of the excited SW is exactly half of the pumping frequency. Above the threshold, the amplitude of the excited SW monotonically increases from the thermal level, as one can see from Fig. 13.6, where the amplitude of the dynamic magnetization at the center of the pumping gate is shown. The simulations demonstrated that the parametric SW excitation remains stable even for pumping amplitudes significantly exceeding the threshold value. This is confirmed by the presence of a single narrow peak in the frequency spectrum of the excited magnetization dynamics, linewidth of which is several times smaller than the linewidth of linear SW excitation. Thus, the micromagnetic simulations confirmed that a microwave VCMA pumping can be used for stable parametric excitation of monochromatic SWs in ferromagnetic nanowires. The SW wavenumber k is determined by the pumping frequency (through the relation ωk = ωp/2), and increases with an increase in ωp, as one can see from Fig. 13.7. The SW wavenumber also depends on the excitation amplitude due to the nonlinear frequency shift. One can also see from Fig. 13.7, the excited SWs propagates away from the gate, symmetrically in both directions, and can be detected at distances several times larger than the gate length. Further increase in the pumping frequency results in the excitation of shorter SWs having larger propagation lengths. For the parameters of our simulations, the maximum SW propagation distance is about 1 µm for an SW having the wavenumber of k ≈ 0.2 nm–1.

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Parametric Excitation and Amplification of Spin Waves

Figure 13.6 Amplitudes of the excited SWs (normalized magnetization my = My/Ms at the center of the excitation gate) as functions of the pumping amplitude at different pumping frequencies fp (see legend). Symbols: results of the micromagnetic simulations. Lines: theoretical fit by Eq. 13.8.

Figure 13.7 Profiles of parametrically excited SWs in a nanowire with in-plane static magnetization (micromagnetic simulations). Gate region is shown by a black rectangle. Excitation frequency and amplitude are fp = 11.6 GHz, bp = 80 mT (a), fp = 12.2 GHz, bp = 94 mT (b), fp = 12.6 GHz, bp = 110 mT (c), and fp = 12.8 GHz, bp = 120 mT (d).

The simulations have also shown parametric excitation of SWs in the case when the pumping frequency was slightly below the double FMR frequency (see curve for fp = 11.6 GHz in Fig. 13.6), but in this case the excitation is of a subcritical type: the amplitude of the excited SW has a large finite value at the threshold of excitation. The excitation in this frequency range becomes possible due to a negative nonlinear frequency shift T < 0: an increase in the SW amplitude leads to a decrease in its frequency ωk(|ck|2) = ωk,0 + T|ck|2, and finite-amplitude SWs can satisfy the parametric resonance condition ωk(|ck|2) = ωp/2 even when half of the pumping frequency lies outside the linear SW spectrum. The corresponding SW mode has an evanescent character and does not propagate away from the gate (see Fig. 13.7a). This nonlinear SW mode has a frequency which lies below the spectrum of linear propagating SWs and, therefore, is nonlinearly localized in the region close to the excitation gate.

Excitation of Spin Waves

In Ref. [57] we developed an approximate analytical theory describing the evolution of SW amplitudes in a nonlinear supercritical regime, and allowing one to calculate stationary amplitudes of the excited SWs. Detailed consideration of the supercritical regime lies out of the scope of this chapter, and we present below only the main qualitative features of the theory developed in [57] and the final result. The two main nonlinear processes which affect supercritical regime of the parametric excitation in nanoscale quasi-1D samples are (i) nonlinear shift of SW frequency, described by the coefficient Tk: ωk(|ck|2) = ωk,0 + Tk|ck|2, and (ii) 4-wave (4-magnon) interaction between the parametrically excited pairs of SWs having wavevectors k and –k, which is described by the Hamiltonian H = ΣkSk|ck|2|c–k|2. The magnitudes of the nonlinear coefficients T and S can be evaluated using the formalism developed in Ref. [27]. For the lowest, SW mode, uniform across the nanowire width, these coefficients are equal to Tk = –Ak + Bk2ωM(4λex2k2 – F0xx + F2kxx)/(2ωk2), Sk = –Ak + Ak2ωM(4λex2k2 – F0xx + F2kxx)/(2ωk2), where Ak = γBint + ωM(2λex2k2 + Fkyy+Fkzz)/2, Bk = ωM(Fkzz+Fkyy)/2, Bint is the static internal field in the nanowire,  and Fk is the dynamic demagnetization tensor of the nanowire (see Appendix A). These expressions are derived for a nanowire with inplane static magnetization (in x direction) at zero external magnetic field. For the case of the out-of-plane static magnetization, one should make a cyclic replacement of indices (x, y, z) Æ (z, x, y). In bulk samples the 4-magnon interaction of SW pairs, described by the coefficient Sk, is the only important mechanism limiting (or determining) the amplitudes of the parametrically excited SWs. Due to this interaction phases of SWs with opposite wavevectors deviate from the optimal condition φk + φ–k = φp – π/2, which reduces the efficiency of energy transfer from pumping to the excited SWs and limits the growth of the SW amplitudes at a certain stationary level. This mechanism is still important in nanoscale samples. However, we have found an additional limiting mechanism, associated with nonlinear frequency shift, which does not play a significant role in bulk samples, and becomes essential only for localized pumping at a submicrometer length scale. As it was pointed out above, the nonlinear frequency shift leads to the change of the SW wavenumber with an increase in the SW amplitude (through the

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Parametric Excitation and Amplification of Spin Waves

relation ωk,0 + Tk|ck|2 = ωp/2). Variation in SW wavevector may lead to a change of all the SW characteristics, e.g., damping, parametric interaction efficiency, group velocity. Among these characteristics only the SW group velocity v changes significantly with the wavenumber variation. In our case the nonlinear frequency shift coefficient is negative, Tk < 0 (both for in-plane and out-of-plane static magnetization). Thus, with an increase in the SW amplitude the SW spectrum shifts down, and half of the pumping frequency will now correspond to a higher SW wavenumber and, therefore, to a higher value of the SW group velocity. Thus, with an increase in the SW amplitude the radiation losses also increase, which leads to the additional limitation of the amplitude of the excited SWs. Accounting for both these nonlinear limiting mechanisms, one can obtain following expression for the stationary amplitudes of excited SWs ai (defined as in Eq. 13.4) [57]: 2 È 2 2  ˘ = Í Vkk bp - Vkk bth + T - T ˙ 2| S k | Î ˚

C



ai

max



ai

max

(

)(

)

1/2

,

(13.8)

2 where T = w 2 lex Cv C 2 | Tk | / L2g | S k | , the coefficient Cv decreases from Cv = π for extended to infinity pumping to Cv = 1 for strongly localized nonadiabatic pumping, and C ≈ 1–2 is the fitting coefficient of the approximate model which describes effect of non-uniform SW profiles within the pumping region (C = 1 for extended pumping and increases to C ≈ 2 for strongly localized pumping). Close to the threshold Eq. 13.8 can be simplified to

=

Lg

2lex 2w 2Cv | T |

2

Vkk bp - Vkk bth

2

.

(13.9)

It is important to note that nonlinear variation of SW group velocity leads to different dependence of SW amplitude on the pumping amplitude (square root of (b2 – bth2), while in bulk samples it is fourth-order root). Using Eq. 13.8, one can describe the dependence of the excited SW amplitudes on the pumping amplitude, obtained from the micromagnetic simulations, making common conversion from SW envelope amplitude ai to dynamic magnetization component (see details in [57]). As one can see from Fig. 13.6, Eq. 13.8 gives a good quantitative description of the results of our micromagnetic

Amplification of Spin Waves by Parametric Pumping

simulations and, thus, can be used for the estimation of parametrically excited SW amplitudes (except in the case fp = 11.6 GHz < 2f0, for which our model is not applicable). Finally, let us discuss the issue of stability of the SW parametric excitation. As was shown, the only important nonlinear interactions between the excited SWs are the 4-wave processes responsible for “self-action” (described by the nonlinear coefficient T) and the processes of the interaction between the SW “pairs” (described by the nonlinear coefficient S). All the other 4-wave scattering processes, which could lead to SW instability, are weak due to quasione-dimensional geometry of the nanowire. The 3-wave processes in this geometry are prohibited as long as the static magnetization is aligned with one of the symmetry axes of the nanowire [27]. The 2-magnon scattering processes, which could take place due to defects presented in a nanowire, in the studied case of ultrathin magnetic nanowires should be weak, since the characteristic size of the possible defects in the nanowire material (nm) is substantially smaller than the characteristic SW wavelength (~100 nm). Finally, the non-adiabatic character of the applied parametric pumping fixes not only the sum of phases of the excited SWs, but also the difference of these phases [34]. All these peculiarities of parametric SW excitation in nanowire geometry make the excitation process stable with respect to the appearance of secondary instabilities, such as, e.g., magnetization auto-oscillations [31]. This means that parametric VCMA cells can serve as reliable and tunable sources of high-amplitude propagating SWs in ultrathin magnetic nanowires.

13.3 Amplification of Spin Waves by Parametric Pumping 13.3.1 Linear Regime of the Parametric Amplification

It has been shown above, that parametric VCMA pumping can excite propagating SWs if the pumping amplitude exceeds a certain threshold. However, the parametric pumping with the amplitude smaller than the excitation threshold also interacts with SWs. In particular, if SW propagates through a region of parametric pumping, the interaction with the pumping leads to a partial compensation

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Parametric Excitation and Amplification of Spin Waves

References

of SW propagation losses or an amplification of a propagating SW. Another consequence of such a sub-threshold parametric process is the formation of an idler SW with an opposite wavevector –k. In the case of a reciprocal SW spectrum this idler SW is counter-propagating to the incident signal SW (the case of a nonreciprocal spectrum is considered in Section 13.4). As in the case of the parametric excitation, the parametric amplification is the most efficient if the resonance condition is satisfied: ωk + ω–k = 2ωk = ωp. The parametric pumping is a well-known method for the amplification of SWs. Its main advantage is the frequency selectivity, since the amplification rate decreases fast with the frequency mismatch (ωk – ωp/2). The parametric amplification in the linear regime, when the amplitudes of both incident and amplified SWs are small enough to neglect nonlinear SW interaction, is well studied [20, 31, 34]. The only specific feature of the VCMA parametric pumping in ferromagnetic nanowires is the value of the parametric interaction efficiency (Eqs. 13.3, 13.7), so here we shall briefly describe the main results from the existing literature. The amplification rate can be calculated from Eq. 13.4, setting the boundary conditions as a1(x = 0) = a0, which determines the incident SW amplitude, and a2(x = Lg) = 0, which corresponds to the absence of the incident idler SW (recall that it propagates in the opposite, –x, direction). In the case of exact parametric resonance and adiabatic pumping (α = 0) the amplification rate is [20]

G È ˘ K = Ícosk Lg + k sink Lg ˙ vk Î ˚

-1

,k=

1 2 Vkk bb - G 2k . v

(13.10)

The dependence of the amplification rate K on the pumping amplitude bp is shown in Fig. 13.8. At zero pumping bp = 0 the rate (Eq. 13.10) equals to exp[–ΓkLg/v] < 1, which corresponds to a partial decay of the SW due to the propagation losses across the pumping region. With an increase in bp, the parametric pumping partially compensates the SW propagation losses. At a certain pumping amplitude the losses are fully compensated (K = 1) and, with further increase in bp, amplification of the signal SW is achieved. Finally, at bp = bth the amplification rate becomes infinite, which corresponds to the threshold of the parametric excitation—any small-amplitude fluctuation will grow into a finite-amplitude excited SW. To use the VCMA gate as the parametric amplifier one should, of course, operate

Amplification of Spin Waves by Parametric Pumping

in the subthreshold regime bp < bp,th; otherwise the excited SWs will significantly distort the incoming signal SW. As one can see from Fig. 13.8, when the ratio ΓLg/v is large (wide pumping gate), the region of proper amplification • > K > 1 is rather narrow, and it might be difficult to operate in this regime given the uncertainties of the experimental parameters of nanoscale VCMA gates. Thus, the use of narrow VCMA gates ΓLg/v < 1 is preferable in practical applications. If the incident SW does not exactly satisfy the parametric resonance condition, the amplification rate decreases. This frequency mismatch Δω = ωk – ωp/2 can be accounted for by the replacement Γk Æ Γk –i Δω, in Eq. 13.10 [20].

Figure 13.8 The dependence of the amplification rate on the pumping amplitude (Eq. 13.10) for different lengths of the pumping localization region. Vertical dotted lines show the parametric excitation thresholds.

In the case of a non-adiabatic pumping (α ≠ 0) the solution becomes more complex. The amplification rate becomes dependent on the phase of the incident SW with respect to the pumping phase [34]. The minimum and maximum (with respect to the variation of the incident SW phase) amplification rates are given by Eq. 13.10 with the replacement Γk Æ Γk ± |αVkkbp| [34]. In conventional amplifiers the phase dependence of the amplification gain is undesirable, and should be excluded by a proper choice of the gate length and SW wavenumber so that α = sinc[kLg] h(ωH + ωM)/ (4ω0), which is derived in the approximation that the waveguide width in substantially larger that the magnetic film thickness wy >> h. An example of a SW spectrum having minimum at kmin ≠ 0 is shown in Fig. 13.10. This spectrum was calculated for a permalloy nanowire deposited on a platinum substrate. The parameters for

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Parametric Excitation and Amplification of Spin Waves

this calculation (nanowire thickness h = 0.7 nm, width wy = 100 nm, μ0Ms = 1 T, external field Be = 0.2 T, λex = 4 nm, D = 3 mJ/m2, di = 0.248 nm, damping constant αG = 0.01) were taken from Refs. [25]. We use these parameters for all the numerical calculations described below, but all the qualitative features of the presented results are applicable to any other materials, as long as the minimum of SW spectrum is located at kmin ≠ 0.

13.4.2 Parametric Amplification of Nonreciprocal Spin Waves

Similarly to Section 13.3.1, we consider evolution of a spectrally narrow SW packet having the central frequency ωs, wavevector ks > 0, and describe it by the envelope amplitude as(x). Naturally, in the absence of the parametric pumping the amplitude decreases in the propagation direction as as(x) = as(0) exp[–Γsx/vs], where Γs and vs are the damping rate and the group velocity of the signal SW, respectively. The parametric pumping leads to the coupling of the signal SW with idler SWs. If the pumping region is sufficiently large (see condition below) and the pumping field is almost uniform within this region, the signal SW is efficiently coupled with only one idler SW having the wavevector –ks. Due to the IDMI-induced SW non-reciprocity the frequencies of the signal and idler SWs are different, ωks ≠ ω–ks, and the parametric interaction is the most efficient (resonant) if the pumping frequency satisfies the condition ωp = ωks + ω–ks. Also, as one can see from Fig. 13.10, in a large range of the wavevectors of the signal SW, k s Œ(0, -kmin ) , the group velocities of both the signal and the idler SWs have the same sign, vsvi > 0. This means that both SWs propagate in the same direction, and the counterpropagating idler SW vanishes. Moreover, if the wavevector of the signal SW is opposite to the wavevector position of the spectral minimum, ks = –kmin, the group velocity of the idler SW vanishes, and this idler wave becomes evanescent outside the pumping region (see Fig. 13.11). It is clear that this is the best situation for the effective use of the SW parametric amplifier, as energy losses of the idler SW are minimized and possible parasitic influence of the idler SW on the other SW signal processing devices is negligible.

Effect of Interfacial Dzyaloshinskii–Moriya Interaction on Parametric Processes

Figure 13.10 SW spectrum of a permalloy nanowire on a platinum substrate. Red arrows show the positions of the signal SW at ks = 0.01 nm–1, main idler SW at –ks, and 2 nonadiabatic idler SWs at k̃ 1, k̃ 2. Green arrows show the minimum of the SW spectrum kmin = –0.05 nm–1 and the corresponding optimum position of the signal SW at –kmin = 0.05 nm–1.

For the quantitative description of the amplification rate and profiles of the interacting SWs we use the same system of Bloembergen equations (13.4) with a1 = as and a2 = ai being the amplitudes of signal and idler SWs, respectively. We take into account different group velocities vs and vi of the signal and idler SWs and neglect the non-adiabatic term (α = 0). For the proper description of the case of vanishing group velocity of the idler SW, vi Æ 0, one has to take into account the dispersion of the group velocity by adding the term +i(s i /2)∂2ai* /∂x 2 to the left-hand side of Eq. 13.4

for idler SW amplitude ai* , where s i = ∂2w k /∂k 2 . The dispersion of group velocity σ is not important for the signal SW, since its group velocity is not vanishing. For simplicity, we use identical damping rates for the signal and idler waves Γs = Γi = Γ = αGωs. The parametric interaction efficiency is given by Eqs. 13.3 or 13.7, depending on how the pumping is created—microwave magnetic field with polarization in the y direction or VCMA-induced microwave variation of the perpendicular anisotropy. The calculated amplification rate K = as(Lg)/as(0) in a stationary regime is shown in Fig. 13.11a. The pumping intensity is characterized by the supercriticality parameter ξ = (Vkkbp/Γ – 1). The value ξ = 0 corresponds to the damping compensation for an

415

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Parametric Excitation and Amplification of Spin Waves

infinitely extended pumping. For the parameters of our calculation, this is achieved for the effective pumping field bp ≈ 10–15 mT, depending on the value of the signal SW wavenumber. For a finite gate length Lg the threshold value of ξ, required to compensate the damping of the signal SW, increases with the reduction in the length Lg of the pumping region, similarly to the reciprocal case (Section 13.3.1). The amplification rate is, naturally, proportional to ξ and depends inversely on the group velocity vi of the idler SW, since the smaller values of vi correspond to lower radiation losses for the idler SW. The highest amplification rate at a given pumping amplitude can be achieved for the case when vi = 0, when the idler SW is not propagating at all.

Figure 13.11 (a) Amplification rate of the signal SW as a function of the pumping localization length at different supercriticalities ξ. Dashed lines correspond to the wavevector of the signal SW ks = 0.01 nm–1, and green solid lines to the optimum case ks = –kmin = 0.05 nm–1. (b, c) Profiles of the signal (solid lines) and idler (dashed lines) SWs for the wavenumber of the signal SW ks = 0.01 nm–1 (b) and for the optimum case ks = 0.05 nm–1 (c). Shaded area shows the pumping region; the pumping supercriticality ξ = 1.5.

It should be noted that for a finite pumping length Lg and any finite pumping amplitude, the amplification rate K remains finite if the group velocities of the signal and idler SWs satisfy the condition vsvi > 0. In this case the SW parametric instability could develop in space, but not in time [3]. The analysis of initial equations shows that in the case vi = 0 the amplification rate is also always finite. Thus, the resonant parametric excitation of SWs from the thermal level is impossible in this case, in contrast to the case when the signal and

Effect of Interfacial Dzyaloshinskii–Moriya Interaction on Parametric Processes

idler SW are counterpropagating. The condition vsvi < 0 is satisfied for a certain pair of SWs only away from the parametric resonance (at a given pumping frequency ωp < ωkmin + ω–kmin), meaning that the parametric excitation of these SWs requires much stronger pumping. The absence of the parametric excitation is an important feature of the parametric interaction of nonreciprocal SWs, because one can achieve practically arbitrary SW amplification gain without risk of starting spurious SW generation. The profiles of the signal and idler SWs in a nonreciprocal parametric amplifier are shown in Fig. 13.11b,c. In the case (c), when vi = 0, the profile of the idler SW is almost symmetric with respect to the center of the pumping gate and, at the center of the gate, can even exceed the signal amplitude as. Outside of the pumping region, the amplitude of the idler wave exponentially decays with the localization length li, determined by the dispersion of the group velocity: li ≈ (σi/Γ)1/2. For the parameters of our calculations, li ≈ 80 nm, which is rather small compared to the mean free path of signal SW (in order of 1 µm). Now we consider the limits of applicability of the presented results. Above, we used the adiabatic approximation in the description of the parametric interaction, assuming that only the SWs with the wavevector ks and –ks are coupled parametrically. In a general case, the resonant parametric pumping leads, also, to the coupling with the other 2 SWs having frequencies ωks, ω–ks and wavevectors k̃ 1, k̃ 2 (see Fig. 13.10). Note that these SWs have negative group velocities, i.e., they are counterpropagating to the signal SW. Thus, they may lead to the formation of a nonvanishing reversed idler SW, in particular the one having the frequency equal to the frequency of the signal wave, which is undesirable. The wavevectors of these idler SWs can be estimated as k̃ 1,2 ≈ 2kmin ± ks. The interaction efficiency with these idler SWs is proportional to Vkkb̃ ks+k̃ 1 and Vkkb̃ –ks+k̃ 2, respectively [34], where b̃ k is the Fourier image of the pumping profile. It is clear that interaction with these idler SWs can be neglected if the length of the pumping gate Lg is much larger than 1/kmin (20 nm in our case). The non-adiabatic parametric processes, which create counterpropagating spin waves, are undesirable for the parametric amplification of SWs. At the same time, these processes allow one to achieve parametric excitation of nonreciprocal SWs, since counterpropagating waves create a feedback necessary for the

417

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Parametric Excitation and Amplification of Spin Waves

operation of any auto-oscillatory system. Thus, nonreciprocal SWs in thin ferromagnetic films with IDMI can be both excited and amplified by parametric VCMA pumping. The choice of operation is dictated by the length Lg of the pumping gate: narrow “non-adiabatic” gates Lg ≈ 1/kmin can be used for the SW excitation, while wide “adiabatic” gates Lg >> 1/kmin are ideal for the parametric amplification.

13.5 Summary

In this chapter we described several phenomena, which can be used for the excitation and processing of SWs in ultrathin ferromagnetic nanowires using microwave electric field via the VCMA effect. First, it has been shown that microwave pumping using the VCMA effect can excite propagating SWs. In the absence of an external magnetic field for both in-plane and out-of-plane ground states only the parametric excitation is possible, when SWs are excited at half the driving frequency. The in-plane magnetized geometry is more effective for parametric excitation, since in this geometry the excitation efficiency is proportional to the out-of-plane magnetization component, rather than the SW precession ellipticity, which determines the threshold in the case of the out-of-plane magnetization. Consequently, in the in-plane magnetized case it is possible to excite SW in a significantly wider frequency range with the magnitudes of the threshold electric fields below 1 V/nm (for a typical Fe/MgO structure). We have also considered a nonlinear stage of parametric excitation and have shown that excited SW amplitudes are determined by two mechanisms: (i) 4-magnon “pair” interaction and (ii) amplitude dependence of SW group velocity, resulting from non-zero nonlinear frequency shift. The last mechanism becomes important only at the nanoscale. Second, it has been shown that the application of a parametric pumping at a certain location along the path of a propagating SW leads to the compensation of the propagation losses and, if the pumping is sufficiently large, to the amplification of a propagating SW. If the amplitude of the incident SW becomes sufficiently large, the nonlinear SW interaction leads to a decrease in the amplification rate. It is also shown that in a certain range of the SW amplitudes the parametric amplifier can be used for the stabilization of SW

Calculation of Spin Wave Dispersion and Vector Structure

amplitude. The output SWs can have the same mean value, but significantly smaller amplitude spread compared to the input SWs. Finally, it has been shown how the IDMI-induced nonreciprocity can improve the characteristics of a VCMA SW parametric amplifier. Under certain conditions, the signal and idler SWs in a parametric amplifier are co-propagating, which removes the disruptive influence of the idler SW on the operation of all the preceding SW processing gates. At the optimal conditions, the idler SW has a nonpropagating evanescent character, which minimizes both the energy losses of the idler SW and its parasitic influence on other gates in a SW processing device. Also, it has been found that the parametric excitation of copropagating SWs is impossible in the case of spatially extended (“adiabatic”) pumping, which prevents spurious noise generation in the parametric amplifier. All the described features make the VCMA-based processing of SWs in ferromagnetic nanowires (especially in the ones with IDMI) attractive for applications in the SW-based microwave signal processing devices at the nanoscale.

Appendix A. Calculation of Spin Wave Dispersion and Vector Structure

The vector structure of SWs in ferromagnetic nanowires and SW dispersion relation can be calculated using the general formalism of linear collective SW excitation [53]. The SW frequency ωkx and vector structure mkx (which describes ellipticity) can be determined from the following equation:

-

kx mkx

=

¥ Wkx ◊ mkx ,

(A.1)

where μ is the unit vector in the direction of static magnetization and    2 Wkx = g B + w M lex k x2 + k 2y - w M Nan + w M Fkx (A.2)   is the Hamiltonian tensor. Here B =  ◊ (Be - w M F0 + w M Nan ) is the static internal magnetic field, Be is the external field, ky is the effective wavenumber describing the SW profile across the nanowire width, ωM = γμ0Ms. The uniaxial anisotropy tensor is defined as  Nan = (e z ¢ ƒ e z ¢ )K an /( m0 M s ) , where Kan is the anisotropy constant,

(

)

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Parametric Excitation and Amplification of Spin Waves

which, in our case, is related to the surface anisotropy constant Ks by the expression Kan = Ks/h, and z’ is the anisotropy axis (z axis in our  case). The dynamic demagnetization tensor Fkx , which depends on the nanowire height and width, is defined by the expression:  

 1 Fkx = 2p

Ú

sky

wy

2

Ê f (kh)k x2/k 2 Á Á f (kh)k x k y /k 2 Á 0 ÁË

f (kh)k x k y /k 2 f (kh)k 2y /k 2 0

ˆ ˜ ˜ dk y , 0 ˜ 1 - f (kh)˜¯ 0

(A.3)

where σky = ∫g(y)exp[–ikyy]dy is the Fourier transform of the normalized SW profile g(y) across the nanowire width, f(kh) = 1– (1– exp[–kh])/(kh) and k2 = kx2+ky2 (note that in the main text we used brief notation k = kx, which is not applicable here). For the lowest SW mode with uniform profile g(y) = 1 the function σky = sinc[kywy/2]. In the approximation of free boundary conditions normalized profiles of higher order modes are given by g(y) = 21/2cos[πny/wy], n = 2, 4, 6 … for even modes and by g(y) = 21/2sin[πny/wy], n = 1, 3, 5 … for odd modes. In more complex cases, one should use theoretical formalism from [19] for calculation of the mode profiles. The norm of a SW mode, used in Eqs. 13.3 and 13.7, is equal to Ãk = imk*·μ×mk.

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425

Index addition theorem 177 adiabatic STT 300–301, 309, 315 AESWS, see all-electrical spin-wave spectroscopy all-electrical spin-wave spectroscopy (AESWS) 204 angular momentum 14, 251, 364–365 antenna gaps 305–306, 317 antennae 305 antidot lattices 209 antidots 2, 4, 139, 144–153, 155–156, 161–162, 165, 172 antiparallel, oriented 281–282, 285 artificial spin ice (ASI) 5, 100, 222, 230, 241–243, 250 ASI, see artificial spin ice auto-oscillations 348–349, 377, 402 backward volume magnetostatic spin-wave (BVMSW) 207 BLS, see Brillouin light scattering BLS maps 54, 57, 70, 147, 156 BLS spectra 80, 82, 84–85, 88, 109–110, 113, 115, 123–124, 243 BLS spectroscopy 105–106, 123–124, 332

BLS technique 51, 54–55, 57, 70, 140 micro-focus 332, 334 Brillouin light scattering (BLS) 31, 48, 78, 80, 86, 88, 105, 140, 145, 150, 235, 332, 334, 350, 368, 372 bulk spin wave modes 231, 233, 235 bullet mode 334, 368–369 BVMSW, see backward volume magnetostatic spin-wave CISWDS, see current-induced spin wave Doppler shift configurational anisotropy 108, 114 coplanar waveguide (CPW) 102–103, 106, 200, 202–208, 210–212, 277–278 coupled magnetic stripes 51, 53, 55, 57 spin waves in 51, 53, 55 coupled spin waves in magnonic waveguides 47–48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72 coupling 15, 19, 49–51, 54, 56, 60–61, 64–65, 89, 99, 108, 110–111, 245, 335, 340, 348, 351, 378, 391, 397, 399, 414, 417

428

Index

coupling coefficient 49–51, 63, 66, 400 coupling configuration 119, 129–130 coupling direction, interdisk 101, 109 coupling orientations 101, 109–111 CPW see coplanar waveguide integrated 102–103, 205 current densities 271, 280, 309, 316, 365–366 current-induced spin wave Doppler shift (CISWDS) 295–300, 302–304, 306, 308, 310, 312, 314, 316, 318, 320, 322, 324–325 cylindrical inclusions 175–176, 179, 188–189 cylindrical wave functions 177 demagnetizing field 18, 141, 151, 153–154, 159, 161, 164, 174, 242, 266, 270 dipolar field 110, 113, 117, 125, 248, 347–348 dipolar interaction 100–101, 108, 111, 117–119, 127, 130, 142, 161, 228, 264–266 neighboring 99, 101–102 dipolar interaction strength 100–101, 130 dispersion 4, 12, 15, 17–18, 22, 48, 53, 56–57, 59, 61, 69, 83, 89, 91, 142–144, 164, 189, 207, 221, 265–267, 269, 279, 285, 354, 415, 417

dispersion relation magnetostatic 27 magnon 266–267, 288 dispersive modes 3, 78, 88, 90, 92 domain walls 4–5, 22, 24–25, 31, 221, 231–235, 240, 247–248, 281–288, 300, 387 Doppler effect 296–298, 303–304, 318, 321 Doppler shift for spin waves 297, 299 dynamic magnetization 54, 57, 59, 70, 87, 90–91, 111, 227, 238, 267, 285, 319–320, 340, 350, 352, 354, 397–398, 403 dynamic magnetization amplitude 86, 91, 145 dynamic magnetization components 49, 391, 397, 399, 401, 406 Dzyaloshinskii-Moriya interaction 18 Dzyaloshinskii–Moriya interaction, interfacial 412–413, 415, 417 EBL, see electron-beam lithography edge modes 2, 113, 116, 238–239, 246 eigenfrequencies 210, 212–213 eigenmodes 51–52, 54, 179–180, 188, 239, 244 electron-beam lithography (EBL) 200–202, 277, 366 element spins 249

Index

excitation frequencies 30, 54, 58, 105, 146, 149, 156, 165, 227–230, 233, 235, 238, 267–269, 272–275, 282–283, 308–309, 342, 404 excitation threshold 388, 394, 399–401, 404, 407 parametric 393, 395, 402, 409, 411 excited spin waves 80, 212, 303, 340, 350, 402 Fano transducer 19–21 FEM, see finite element method ferromagnetic films 141, 153–154, 173–174, 190, 192, 280, 302–303, 332, 385–386, 390, 412–413 ferromagnetic ground states 122, 124–125, 128, 130 ferromagnetic matrix 4, 172–173, 175, 178, 188 ferromagnetic metal 365, 387–388 ferromagnets 264, 364, 385, 387, 389 field initialization 100, 102, 120–121, 126 field pulse 224–225, 245 finite element method (FEM) 48, 51, 54–56 FMR frequency 104, 348–349, 369–371, 403 local 21, 23, 25, 32, 371 FMR spectroscopy 104, 106–107

forward-volume magnetostatic spin wave (FVMSWs) 4, 173–174, 178 frequencies angular 17, 298 auto-oscillation 335, 337, 349, 352 radio 102, 363 spin wave 4, 17, 296, 299, 316, 320–321 frequency-selective tunable spin wave channeling 67, 69, 71 frequency shift, nonlinear 335, 349, 403, 405, 410 frequency splitting 239–240, 324 FVMSWs, see forward-volume magnetostatic spin wave Gaussian function 352, 354–355 graded magnonic index 11, 15, 18–19, 21, 24–27, 30–33, 37 Green function 178, 181–182 unperturbed 183 ground-signal-ground (GSG) 102, 205–206 group velocity 17, 26–30, 60, 147–149, 154, 161, 163–164, 207–209, 225–226, 276, 303, 306, 308–309, 316, 318, 325, 348, 400, 406, 414–416 group velocity directions 27–28, 148 group velocity vectors 30, 149, 227–230

429

430

Index

GSG, see ground-signal-ground guided spin waves 5, 331, 335, 337 Happ direction 109–110, 116, 130 Happ range 113 homogeneous film 51, 64, 162–164, 183, 220 IDMI, see interfacial Dzyaloshinskii–Moriya interaction IFDRLs, see iso-frequency dispersion relation lines interfacial Dzyaloshinskii–Moriya interaction (IDMI) 220–221, 226, 230, 288, 320, 386, 388, 412–413, 415, 417–419 interference 4, 34, 78, 139, 147, 165, 229–230, 234, 262, 269, 334 interference patterns 229–230, 267–268 interferometer 35–36, 107 iso-frequency dispersion relation lines (IFDRLs) 142, 144, 147–148, 152–154, 162–163 isofrequency curves 16–17, 27–30 L-shaped NWs 78, 80, 88–90 LD, see line defect

LD magnonic waveguide 67–68 line defect (LD) 67–68, 209 local magnetic fields 270–271, 273, 275, 277, 279–280 magnet coils 205–206 magnetic field 141–142, 159, 264, 270, 284, 288, 396–397, 413 bias 17, 19, 26–27, 29, 33, 129, 389 dipolar 264, 336 static internal 52–53, 419 uniform microwave 22, 25 magnetic force microscopy (MFM) 120–122, 125 magnetic ground states 100, 119, 123, 125, 128–130, 389 magnetic inclusions 173, 188, 192–193 magnetic induction 175, 178 magnetic insulators 331, 355 magnetic moments 15, 107, 126, 213, 247, 263–264, 268–269, 271, 274, 276, 282, 340 magnetic multilayers 2 magnetic nanowires, ultrathin 407 magnetic properties 129, 158, 198–199, 320 magnetic quantum cellular automata (MQCA) 100 magnetic structures, patterned 18 magnetic tunnel junction (MTJ) 378 magnetic waveguides, coupled 54–55

Index

magnetization equilibrium 298, 304, 319–320 in-plane 397, 399–400 local 299–300, 315, 364 out-of-plane 397, 399, 403, 418 magnetization configuration 235, 275, 277 magnetization dynamics 60, 82, 99–102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 223, 305–307, 330, 390 reconfigurable 118–119, 121, 123, 125, 127 magnetization oscillations 333–334, 338, 350 magnetization reversal 100 magnetization vector 83, 141, 223 magneto-optical Kerr effect (MOKE) 123, 126–127, 306 magnetodynamical modes 368–369 magnetoelectric effects 385, 387 magnetostatic interaction 143–144 magnetostatic spin waves 16–17, 27, 33, 182, 184 magnetostatic surface spin wave (MSSW) 49, 52–53, 57, 62, 65–66, 143, 304, 318–320 magnon band 272–273, 275, 285 magnon beams 267–269, 279 counter propagating 267–268 magnon conduits 276, 278, 280 magnon dispersion 262, 264–265, 272, 277–278, 280 magnon intensities 267–269, 272–273, 275–276, 278–279, 282–283, 287

magnon multiplexer 5, 277, 279–280 magnon propagation 5, 262, 264, 270, 276, 279, 281 magnon spintronics 12, 198, 215 magnon transport 5, 56, 262–263, 265–272, 276, 278, 280, 286, 288 magnon waveguide 267, 270–273, 278, 280–282, 287 hybrid 275–277 magnon wavevector 268, 271–273, 288 magnonic applications 69, 265, 335 magnonic band structures 245 magnonic bands 86–87, 129, 244–245 lowest 161–162, 164 magnonic channels 71–72 magnonic couplers 3, 48 magnonic crystal array (MCA) 69–72 magnonic crystals (MCs) 4, 12, 18, 47–49, 51, 56–57, 59–61, 63–67, 69–70, 72, 78, 92, 99–102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 161, 172, 197 magnonic devices 2–3, 25, 31–33, 35, 208, 335, 339, 346, 355 magnonic grating couplers 199–201, 212–213 magnonic index 12, 18, 29 magnonic logic gates 35, 262–263 magnonic NAND gate 35–36 magnonic networks 3, 48

431

432

Index

magnonic waveguides 33, 47–48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 265–271, 332–333, 343 microscopic 331–332 magnonics 2, 7, 11–14, 31–32, 35, 37, 56–60, 69, 77, 91, 100, 161, 197–198, 207, 220–221, 252, 261–262, 288, 307, 329–330, 355, 365 graded-index 12, 269 magnons secondary 263 steering 261–262, 264, 266, 268–270, 272, 274, 276–278, 280, 282, 284, 286, 288 steering and multiplexing 270–271, 273, 275, 277, 279 MCA, see magnonic crystal array MCs, see magnonic crystals MFM, see magnetic force microscopy micromagnetic simulations 19, 26, 33, 48, 54–55, 78, 80, 83, 92, 107, 141, 158, 222, 224, 227, 230–231, 233–234, 236, 238, 242, 252, 281, 284–285, 287–288, 317, 368–369, 371, 402–404, 406, 411 microwave 6, 140, 202, 269, 302–303, 334–335, 386, 388, 396, 402, 410 microwave antenna 144, 149, 157, 199, 233, 235, 263, 267, 269, 271–273, 277, 281–282, 396 excitation efficiency 272–273

microwave frequencies 1, 6, 21, 329 MOKE, see magneto-optical Kerr effect MQCA, see magnetic quantum cellular automata MSSW, see magnetostatic surface spin wave MSSW interactions 145, 149 MTJ, see magnetic tunnel junction multilayer magnonic crystals 60–61, 63, 65 multiple-scattering method 174–175, 177, 179 nanocontact 296, 347–351 nanocontact spin torque oscillators 363–368, 370, 372, 374, 376, 378 propagating spin waves in 363–364, 366, 368, 370, 372, 374, 376, 378 nanodisk array 210, 212, 214–215 nanodisks 101, 115, 209–213, 215 coupled 108–109, 111, 113, 115, 117 Py 210, 213 nanomagnets 100–102, 106–108, 120–121, 123, 127–128 nanometer-thick YIG film, optics of spin waves in 157, 159, 161, 163 nanowaveguide 3, 337–338, 353 nanowires 3, 77, 79, 92, 336–337, 385, 388, 391–392, 396–398, 401–405, 407, 412–413

Index

ferromagnetic 388, 390, 392, 398, 401, 403, 408, 413, 419 magnetic 336, 391 narrow 393, 398–399 Py 79–81 wide 393, 398–399 Néel walls 236–237, 281–282, 284 non-adiabatic spin transfer torque parameter 315, 317 noncollinear spin textures 5, 262–264, 266, 268, 270, 272, 274, 276, 278, 280, 282, 284, 286, 288 nonlinear interactions 263, 388, 393, 395, 399, 402, 410 nonlinear spin wave coupling in magnonic crystals 56–57, 59 nonreciprocity parameter 208–209 Oersted field 270–272, 276, 297, 304, 321–323, 341, 368–370, 372–373 parametric amplification 407–408, 410–412, 417–418 parametric amplifier 408, 411, 414, 418–419 parametric excitation 6, 387–388, 390, 392–393, 396, 398, 401–402, 405, 408, 410, 412, 417–419 parametric processes 386, 402, 408, 412–413, 415, 417

parametric pumping 215, 390–391, 394, 399, 401–402, 407–409, 411–412, 414, 417–418 patterned YIG micrometer films 145, 147, 149, 151, 153, 155 Permalloy 13, 19, 21, 25, 29, 77, 79, 81, 265, 277, 311–312 perpendicular magnetic anisotropy (PMA) 231, 286, 367 PMA, see perpendicular magnetic anisotropy propagating mode frequencies 233, 369 propagating spin wave modes 21–22, 368–370 propagating spin wave packet 353 propagating spin wave spectroscopy (PSWS) 302, 304, 309, 319 propagating spin waves 1, 3, 5–6, 18, 20, 22–23, 26–27, 31–32, 34–35, 89, 172, 180, 204–205, 221, 297, 329–332, 334–336, 338–340, 342, 344–348, 350–352, 354, 356, 368–369, 373, 377, 386 propagation, nonreciprocal 222, 224, 238 propagation-length control 345–346 PSWS, see propagating spin wave spectroscopy pure spin currents 6, 330–331, 339, 341–343, 345–347, 349, 351, 353, 355, 365 Py film 282, 336–337, 349

433

434

Index

rhomboid nano-magnet (RNMs) 100, 119–128, 130 RNMs, see rhomboid nano-magnet SHNOs, see spin Hall nano-oscillators spin currents 251, 286, 325, 329–332, 334, 336, 338, 340, 342, 344, 346, 348, 350, 352, 354, 356, 365 spin Hall effect 220, 251, 365 spin Hall nano-oscillators (SHNOs) 365 spin ice 222, 247 spin-orbit coupling 220, 251, 315 spin-orbit interaction 311, 313 spin polarization 299, 303, 305–306, 309–314, 316–317, 323, 325, 364, 366–367 degree of 303, 305–306, 309–314, 323, 325 spin-polarized currents 335, 337, 351 spin structures 219–220, 222, 224, 226, 228, 230, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252 spin waves on 219–220, 222, 224, 226, 228, 230, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252 spin textures 230, 237, 262, 264, 288 spin-torque devices 342 spin-torque nano-oscillators (STNO) 335–338

spin torque oscillators (STOs) 100, 363–368, 370, 372–374, 376, 378 spin transfer torque (STT) 220, 274, 296–297, 299–301, 330, 339–340, 342, 344, 347, 363–365, 369 spin wave band structure 77, 80, 172 spin wave beams 4, 27, 29–30, 33, 371, 374, 377 spin wave channeling 232, 234–235 spin wave control 15, 355 spin wave dispersion 12, 15, 17, 23, 221, 243, 419 spin wave Doppler experiments 325 spin wave Doppler shift measurements 312–313 spin wave edgemodes 172–173 spin wave eigenmodes 3–4, 173, 192, 231, 244 spin wave emission 20, 22, 24–25, 335 spin wave excitation 3, 12, 18–19, 21–23, 32–33, 189–190, 202–203, 220, 251, 332, 339–341, 349, 351, 353, 372, 388–389, 391, 393, 395, 397, 399, 401, 403, 405 spin wave excitation transfer, distant 192 spin wave Fano resonances 11, 13, 15, 37 spin wave Fano resonances in magnetic structures 11–12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38 spin wave frequency 337, 341, 353, 355

Index

spin wave frequency shifts 319, 321, 323 spin wave grating coupler 22, 209, 211, 213 spin wave group velocity 17, 207–208, 305 spin-wave intensity 333, 340–341, 343–344, 350, 353 spin wave modes 83, 214, 242, 245, 301, 308–309, 349 degenerate 399, 401 higher-order 391, 399–400 localized 221 spin wave nonreciprocity 388, 412–413 spin wave optics in patterned garnet 139, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166 spin wave power flow 227–228 spin wave propagation 3–4, 6, 13, 18, 26, 30, 37, 48, 54–55, 67, 69, 72, 78–79, 151, 172–173, 178–179, 209, 225, 227, 238, 304–305, 331–333, 393, 413 spin wave relaxation, current-induced modification of 296–297 spin wave ripple 225 spin wave scattering 172–173, 182–184, 187, 192–193 spin wave steering 25, 27, 29 spin wave vector 304, 309, 316, 318 spin wave wavelength 334 spin waves dispersion spectra of 348–349 excitation of 52, 57, 388–389, 396–398, 418

excitation of propagating 6, 331, 335, 346 forward-volume magnetostatic 4, 173 nonreciprocal 417–418 propagation characteristics of 342–343 propagation length of 339, 341, 343, 345, 377 propagation of 3–4, 172, 221, 339, 343, 371 radiation of 181, 226, 348 short-wavelength 20, 198–199, 212 spintronic nano-oscillators 378 static magnetization 1, 20, 28, 31, 35, 221, 340, 389–391, 396–402, 404–407, 413, 419 STNO, see spin-torque nano-oscillators STOs, see spin torque oscillators stripe waveguide 267, 347–348 STT, see spin transfer torque STT devices 330 T-scattering operator 4, 173, 182–183, 185, 192 two-current model 309–311, 313 ultrathin films 32, 224, 226–227 VCMA, see voltage-controlled magnetic anisotropy

435

436

Index

vector network analyzer (VNA) 48, 53, 57, 59, 69, 104, 205–208 VNA, see vector network analyzer voltage-controlled magnetic anisotropy (VCMA) 386–387, 389–390, 396, 408 waveguide coplanar 102, 200, 202, 215, 277–278 curved 276 waveguide modes 267, 269, 336–337, 349 wavevectors 141–142, 148–149, 152, 154, 156, 158–159, 163–164, 207–208, 210,

212, 226, 229–231, 242–243, 263, 265, 267–268, 279, 390–391, 394, 405, 414, 416–417 opposite 391, 393, 405, 408, 412 YIG, see yttrium-iron garnet YIG film 52, 64, 140–144, 153, 157, 160–161, 213 thin 143, 157–158, 160–161, 212 yttrium-iron garnet (YIG) 32–33, 51, 53, 67, 69–70, 139–140, 144–145, 149, 157, 159–163, 165, 172, 199–201, 208–209, 211–215, 263, 302, 330, 342, 345, 365