Magnetic Dynamics in Antiferromagnetically-Coupled Ferrimagnets: The Role of Angular Momentum [1st ed.] 9789811591754, 9789811591761

Table of contents :
Front Matter ....Pages i-xiii
Introduction (Takaya Okuno)....Pages 1-24
Spin-Transfer Torques for Domain Wall Motion in Antiferromagnetically-Coupled Ferrimagnets (Takaya Okuno)....Pages 25-48
Gilbert Damping Parameter of Ferrimagnets Probed by Domain Wall Motion (Takaya Okuno)....Pages 49-59
Gilbert Damping Parameter of Ferrimagnets Probed by Magnetic Resonance (Takaya Okuno)....Pages 61-73
Conclusion (Takaya Okuno)....Pages 75-75

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Springer Theses Recognizing Outstanding Ph.D. Research

Takaya Okuno

Magnetic Dynamics in AntiferromagneticallyCoupled Ferrimagnets The Role of Angular Momentum

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

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Takaya Okuno

Magnetic Dynamics in AntiferromagneticallyCoupled Ferrimagnets The Role of Angular Momentum Doctoral Thesis accepted by Kyoto University, Kyoto, Japan

123

Author Dr. Takaya Okuno Japan Patent Office Chiyoda, Tokyo, Japan

Supervisor Teruo Ono Institute for Chemical Research Kyoto University Kyoto, Japan

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-15-9175-4 ISBN 978-981-15-9176-1 (eBook) https://doi.org/10.1007/978-981-15-9176-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Supervisor’s Foreword

Conventional electronics utilize mostly the “charge” of electrons, whereas traditional magnetic devices use mainly the “spin” degree of freedom of electrons. The field of spintronics, which aims at controlling both the charge and spin in a single solid-state device, is developing rapidly and reshaping the information technology field. The focus of spintronics has long been ferromagnets because they have significant net magnetization, which enables an easy control of the magnetic state using the magnetic field. Recently, antiferromagnets, a class of magnets where neighboring moments are coupled to align antiparallel so that net magnetization is zero, have been gaining increasing attention for device applications. Robustness against perturbation caused by magnetic fields, absence of the production of stray fields eliminating the unintentional magnetic crosstalk between neighboring devices, and ultrafast magnetic dynamics are advantageous characteristics of such devices. However, the studies on antiferromagnets have been significantly less than those on ferromagnets because of the difficulty in inducing antiferromagnetic dynamics owing to their high stability against the magnetic field, which prevents the manipulation of magnetic moments. 3d transition-metal and rare-earth alloys are types of ferrimagnets, in which the magnetic moments of the transition metal and rare earth, each with a different magnitude of magnetization, couple antiferromagnetically, resulting in nonzero net magnetization. Therefore, ferrimagnets are expected to have both ferromagnetic-like (nonzero net magnetization) and antiferromagnetic-like (neighboring moments are coupled antiparallel) characteristics. In 2017, Kab-Jin Kim et al. reported that the field-driven magnetic domain wall (DW) velocity of a ferrimagnet increased remarkably at the angular momentum compensation temperature (T A Þ, at which the net angular momentum of the ferrimagnet was canceled out. The stark increase in DW velocity is attributed to antiferromagnetic-like dynamics in ferrimagnets. This study has opened up the possibility that the ultrafast antiferromagnetic dynamics in ferrimagnets can be driven by magnetic field, thereby overcoming the difficulty in inducing antiferromagnetic dynamics. It has led to many investigations on ferrimagnets that seek to realize spintronic devices based on ferrimagnets. v

vi

Supervisor’s Foreword

In this thesis, Takaya Okuno has investigated the role of angular momentum in ferrimagnetic dynamics in a gadolinium–ironcobalt (GdFeCo) alloy, in which rare-earth (Gd) and transition-metal (FeCo) moments are coupled antiferromagnetically. First, the current-induced torque on magnetic DW motion, spin-transfer torque (STT), is examined in a wide range of temperatures across T A . It is concluded that the non-adiabatic STT in antiferromagnets acts as a staggered magnetic field, i.e., it exerts effective magnetic fields of equal magnitudes and opposite signs on the antiferromagnetically-coupled moments, which can be used for the efficient control of antiferromagnetic DWs. Moreover, the non-adiabaticity parameter is significantly larger than the Gilbert damping parameter, challenging the conventional understanding of the non-adiabatic STT based on ferromagnets. Next, the Gilbert damping parameter of ferrimagnets, i.e., the rate of energy dissipation from ferrimagnetic dynamics, is investigated using two independent experiments: the field-driven DW velocity and ferrimagnetic resonance. In contrast with the previous reports, in which the effective Gilbert damping parameter of ferrimagnets increases remarkably as the temperature approaches T A and diverges at T A , the theoretical and experimental investigations in this thesis reveal that the Gilbert damping parameter of ferrimagnets is insensitive to temperature even across T A . These studies uncover the nature of ferrimagnetic dynamics in terms of angular momentum and serve as a significant step forward to clarify the dynamics of antiferromagnetic moments. This thesis provides a scientific platform for further investigation into the fundamental physics of ferrimagnetic and antiferromagnetic dynamics as well as realizing the next-generation spintronic devices. Kyoto, Japan August 2020

Teruo Ono

List of Published Articles Parts of this thesis have been published in the following journal articles: [1] Takaya Okuno, Duck-Ho Kim, Se-Hyeok Oh, Se Kwon Kim, Yuushou Hirata, Tomoe Nishimura, Woo Seung Ham, Yasuhiro Futakawa, Hiroki Yoshikawa, Arata Tsukamoto, Yaroslav Tserkovnyak, Yoichi Shiota, Takahiro Moriyama, Kab-Jin Kim, Kyung-Jin Lee, and Teruo Ono, ‘Spin-transfer torques for domain wall motion in antiferromagnetically coupled ferrimagnets’ Nature Electronics 2, 389-393 (2019). [2] Duck-Ho Kim, Takaya Okuno, Se Kwon Kim, Se-Hyeok Oh, Tomoe Nishimura, Yuushou Hirata, Yasuhiro Futakawa, Hiroki Yoshikawa, Arata Tsukamoto, Yaroslav Tserkovnyak, Yoichi Shiota, Takahiro Moriyama, Kab-Jin Kim, Kyung-Jin Lee, and Teruo Ono, ‘Low Magnetic Damping of Ferrimagnetic GdFeCo Alloys’, Physical Review Letters 122, 127203 (2019). [3] Takaya Okuno, Se Kwon Kim, Takahiro Moriyama, Duck-Ho Kim, Hayato Mizuno, Tetsuya Ikebuchi, Yuushou Hirata, Hiroki Yoshikawa, Arata Tsukamoto, Kab-Jin Kim, Yoichi Shiota, Kyung-Jin Lee, and Teruo Ono, ‘Temperature dependence of magnetic resonance in ferrimagnetic GdFeCo alloys’, Applied Physics Express 12, 093001 (2019).

vii

Acknowledgements

Many people have supported me in my research and personal life during my five years with the Ono group, two years as a master’s student and three years as a Ph.D. scholar. First, I would like to express my sincere gratitude to my supervisor Prof. Teruo Ono for his valuable suggestions and advice not only in my research but also in personal life. He inspired me to develop interest in new physical phenomena and develop ideas. I would also like to thank Prof. Takahiro Moriyama, who has taught me how to think logically and how to deal with experimental data. Professor Yoichi Shiota helped me with the simulation program for macrospin simulation and reviewed my abstracts for conferences. I also thank Dr. Duck-Ho Kim, with whom I discussed many experiments. The idea of the spin-transfer torque experiment took shape during my discussions with him. I would also like to express my thanks to my collaborators overseas. Professor Kab-jin Kim at the Korea Advanced Institute of Science and Technology was my supervisor in the Ono group for one and a half years. While he performed his research on the ferrimagnetic DW (published in Nature Materials), he handed down some research on ferrimagnets to some of us at the Ono lab. He is not only a distinguished scientist but also a thoughtful gentleman. He is one of my role models as a scientist. The research in this thesis is owed mostly to him. Furthermore, I would like to thank Prof. Se Kwon Kim at the University of Missouri. I have learned a lot from my discussions with him, most of which took place over e-mails. His valuable advice and suggestions have improved the quality of my research. My interactions with him helped me realize the importance of being prompt and thoughtful in research and business. Next, I would like to thank Prof. Kyung-Jin Lee at Korea University. His suggestions have been path-breaking. Notwithstanding his preoccupations as a leading theoretician in spintronics, I have benefitted from his views whenever I needed guidance in my research. I would like to express my gratitude to the students in the Ono group: Mizuno san, Nishimura san, Ishibashi san, Hirata san, Li san, Ikebuchi san, Hung san, and the master’s course students. In particular, I would like to thank my friends Ando san and Oda san, with whom I have spent five years. Moreover, I have received indirect support from many people. I am grateful to Kusuda san and Ichikawa san, ix

x

Acknowledgements

the managing staff at the Liquid Helium Experiment facility. I also express my sincere gratitude to the secretaries, Yamaguchi san and Tanigawa san, of the Ono group. I really appreciate all the kindness and support from each one of them. Finally, I would like to show my greatest appreciation to my family. I could not have accomplished my Ph.D. degree without their moral support and warm encouragements.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Current-Driven Domain Wall Motion . . . . . . . . . . . . . . . . . . 1.1.1 What is Magnetic Domain Wall? . . . . . . . . . . . . . . . 1.1.2 Spin-Transfer Torque Driven Domain Wall Motion . . 1.1.3 Spin–Orbit Torque Driven Domain Wall Motion . . . . 1.2 Spin-Torque Ferromagnetic Resonance . . . . . . . . . . . . . . . . 1.2.1 What is Ferromagnetic Resonance? . . . . . . . . . . . . . . 1.2.2 Spin–Orbit Torque Induced Ferromagnetic Resonance 1.3 Ferrimagnets in Spintronics . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Antiferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Advantage of Ferrimagnets: Field-Driven Antiferromagnetic . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Experiments on Antiferromagnetic Dynamics in Ferrimagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Research Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

1 1 1 3 9 14 14 16 17 17

....

17

.... .... ....

19 22 23

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. . . . . . . . . .

2 Spin-Transfer Torques for Domain Wall Motion in Antiferromagnetically-Coupled Ferrimagnets . . . . . . . . . . . . . . . . 2.1 Research Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Brief History of Field-Driven Domain Wall Motion . . . . 2.1.2 Mechanism of Field-Driven Domain Wall Motion: Steady and Precessional Motions . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Walker Breakdown in Field-Driven Domain Wall Motion in Ferrimagnets . . . . . . . . . . . . . . . . . . . . 2.1.4 Scope of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Theory of Ferrimagnetic Domain Wall Velocity . . . . . . . . . . . . 2.3 Experimental Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Film Preparation and Device Fabrication . . . . . . . . . . . .

. . . . . . . . . .

.. .. ..

25 25 25

..

26

. . . . .

29 30 31 33 33

. . . . .

xi

xii

Contents

2.3.2 Experimental Setup for Field-Driven Domain Wall Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Domain Wall Detection Technique . . . . . . . . . . . . . . . 2.4 Experimental Results on Field-Driven Current-Assisted Domain Wall Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Qualitative Explanation for Temperature Dependence of Spin-Transfer Torques . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Significantly Large Ratio of j b j=a . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Supplementary Information . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Magnetization Compensation Temperature of GdFeCo Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Estimation of the Temperature Dependence of stotal and ds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

... ...

34 35

... ...

37 40

. . . .

. . . .

40 43 43 44

...

44

... ...

46 47

3 Gilbert Damping Parameter of Ferrimagnets Probed by Domain Wall Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Physical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Theoretical Background for Determining the Gilbert Damping Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Gilbert Damping Parameter Obtained from Field-Driven Domain Wall Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

.. ..

49 49

..

53

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55 58 58

.....

61

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61

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61 62 65 66

..... ..... .....

70 72 73

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

4 Gilbert Damping Parameter of Ferrimagnets Probed by Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Derivation of Equation for Magnetic Resonance in Ferrimagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Landau–Lifshitz–Gilbert Equation for Experimental Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Resonance Frequency . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Spectral Linewidth . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Experimental Methods and Results . . . . . . . . . . . . . . . . . . 4.3 Analysis and Discussion Based on Theory of Ferrimagnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

Abbreviations

DMI DW FiMR FMR IMA MFM PMA RE SHE SOT SQUID ST-FMR STT TM

Dzyaloshinskii–Moriya interaction Domain wall Ferrimagnetic resonance Ferromagnetic resonance In-plane magnetic anisotropy Magnetic force microscopy Perpendicular magnetic anisotropy Rare earth Spin Hall effect Spin–orbit torque Superconducting quantum interference device Spin-torque ferromagnetic resonance Spin-transfer torque Transition metal

xiii

Chapter 1

Introduction

Abstract This thesis addresses the current-driven magnetic domain wall (DW) motion in antiferromagnetically-coupled ferrimagnets. There are two aspects behind this topic: current-driven DW motion in ferromagnetic materials, which has been an area of intense research for several decades, and antiferromagnetic-like dynamics of ferrimagnets, which has been recently attracting increasing attention. Because ferrimagnets and antiferromagnets are expected to exhibit advantageous characteristics, such as the ultrafast spin dynamics, for device applications, it is necessary to clarify the current-driven ferrimagnetic/antiferromagnetic dynamics. This chapter introduces the physics required to understand the main results of this thesis. In particular, the current-driven DW motion is explained in Sect. 1.1, and two mechanisms of the current-driven DW motion, namely, the spin-transfer torque (STT) and spin– orbit torque (SOT), are described in detail. In Sect. 1.2, spin-torque ferromagnetic resonance, which is a type of ferromagnetic resonance excited by SOT, is introduced. This is used as an experimental method in Chap. 4. In Sect. 1.3, recent studies on ferrimagnetic DW motion are summarized. In addition to helping in understanding the physical backgrounds of this thesis. This chapter also introduces readers to one of the recent topics of the spintronic research field. Keywords Current-driven magnetic domain wall motion · Spin-torque ferromagnetic resonance · Antiferromagnetically-coupled ferrimagnets

1.1 Current-Driven Domain Wall Motion 1.1.1 What is Magnetic Domain Wall? The magnetic domain in a ferromagnetic material is a region with nearly uniform magnetization, within which spontaneous magnetizations are aligned almost in one direction. A magnetic DW is a boundary of two magnetic domains that form the transition regime of the magnetization direction. Inside DW, magnetization rotates gradually from one direction to the other. In a simple model of the DW structure,

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 T. Okuno, Magnetic Dynamics in Antiferromagnetically-Coupled Ferrimagnets, Springer Theses, https://doi.org/10.1007/978-981-15-9176-1_1

1

2

1 Introduction

the width of the DW is governed by the balance between the anisotropy energy and exchange energy. A small DW width is desirable in terms of the anisotropy energy. The magnetic moments inside a DW are tilted away from the anisotropic easy axis, and they are energetically unstable; thus, a small DW width, i.e., a small number of magnetic moments inside the DW, is stable. However, a large DW width is energetically stable in terms of the exchange energy. The exchange coupling between two magnetic moments aligns them parallel to each other in ferromagnets; thus, a large DW width, i.e., a small angle between two neighboring moments inside the DW, is stable owing to the exchange coupling. The DW width is determined as the√minimum point of the sum of the anisotropy energy and exchange energy: λ = K /A, where λ is the DW width, K is the anisotropy energy, and A is the exchange stiffness. In ferromagnetic thin films with perpendicular magnetic anisotropy (PMA), two types of the DW structure are known: the Bloch wall and Néel wall [1–6]. Figure 1.1a, b depict the schematics of the Bloch and Néel walls, respectively, in a ferromagnetic thin film with PMA. In the Bloch wall, the direction of magnetic moments rotates gradually from up (+z) to down (−z) within the yz-plane; the magnetization component normal to the wall plane (along the x-axis) remains zero. However, the direction of magnetic moments is tilted gradually within the xz-plane in the Néel wall; the magnetic moment at the DW center is directed normal to the wall plane (parallel to the x-axis). In ferromagnetic thin films with PMA, the Bloch wall is energetically more stable when compared with the Néel wall. In the Néel wall, the magnetic moment inside the DW, aligned normal to the wall plane (in the x-direction), creates

Fig. 1.1 Schematic of magnetic domain wall in a ferromagnetic thin film with perpendicular magnetic anisotropy. a Bloch wall and b Néel wall

1.1 Current-Driven Domain Wall Motion

3

magnetic poles on both sides of the DW. These magnetic poles generate a demagnetizing field in the opposite (–x) direction, destabilizing the Néel wall. However, the Néel wall can be more stable than the Bloch wall in systems such as the magnetic heterostructures with the Dzyaloshinskii–Moriya interaction (DMI) [7–10] and the magnetic wires with extremely small wire widths [11].

1.1.2 Spin-Transfer Torque Driven Domain Wall Motion 1.1.2.1

Outline

In 1984, Berger suggested a new theoretical model of current-driven magnetic DW motion originating from the exchange interaction between conduction electrons spin and magnetic moments inside DW (referred to as spin-transfer torque (STT)) [12]. Soon, Freitas and Berger experimentally observed the current-driven DW motion in an in-plane magnetized ferromagnetic NiFe wire through the Faraday effect [13]. However, this study did not attract much attention at that time because (1) the  threshold current density for DW motion was so large ∼ 1011 A/m2 that the DW motion was successfully observed only in two out of the twenty devices tested (The rest were destroyed during current injection.), and (2) because of the existence of the Ampère field caused by the electric current, the quantitative evaluation of the current effect on DW motion was difficult. However, in recent years, the situation has changed considerably. The development of fine processing technology has enabled the fabrication of nanoscale magnetic wires, making experiments on a single DW possible. Thus, the quantitative comparison between experiment and theory is possible. In addition, it is expected that a smaller current can drive the DW motion in nanoscale magnetic wires than in milliscale wires [13]. Especially, in 2004, the current-driven motion of a single DW was first observed in a NiFe (permalloy, Py) submicron nanowire [14]. Moreover, in 2008, a new type of non-volatile magnetic memory, the DW racetrack memory, was proposed by Parkin et al. [15, 16]. The DW racetrack memory utilizes the currentdriven DW motion to drive the magnetic domains, where the information is stored as the direction of the magnetization. These studies have attracted attention to the DW motion phenomenon in terms of memory application as well as fundamental physics. The current-driven DW motion so far has been intensively investigated both theoretically and experimentally.

1.1.2.2

Theory of Spin-Transfer Torque Driven Domain Wall Motion

In the history of current-driven DW motion research, Berger first suggested in a series of his papers [12, 13, 17] that the exchange interaction between conduction electrons spin and magnetic moments inside the DW can cause the currentdriven DW motion. These theoretical works were based on his deep insight into

4

1 Introduction

physics of magnetization dynamics, but it appears to be phenomenological and lacks transparency as a self-contained theory. On the other hand, in 1996, Slonczewski [18] and Berger [19] proposed a theoretical mechanism of magnetization switching induced by spin-polarized electric current, referred to as the STT. It was then considered that the current-driven DW motion proposed by Berger [12, 13, 17] was attributed to the STT taking place inside the DW. In 2004, the general theory of STT-driven. DW motion starting from a microscopic description was formulated by Tatara et al. [20], who suggested that STT includes two mechanisms; spin-transfer and momentum-transfer. The spin-transfer is dominant for thick DWs (e.g., for the case of ferromagnetic metals). While the conduction electrons flow through the DW adiabatically—the direction of conduction electrons spin changes as it is always parallel to the local magnetic moments inside the DW (see Fig. 1.2)—the spin angular momentum is transferred from the conduction electrons to the local magnetic moments due to the exchange interaction. This gives rise to the torque acting on the local magnetic moments, resulting in DW displacement (Fig. 1.2). On the other hand, the momentum-transfer is dominant for thin DWs (e.g., for the case of magnetic semiconductors). Conduction electrons are reflected when they pass through the DW, and by its reaction, the DW is pushed in the direction of electron flow. Note that the spin-transfer is not equivalent to the magnetic field, while the momentum-transfer is equivalent to the magnetic field.

Fig. 1.2 The schematic of the adiabatic spin-transfer mechanism for the domain wall motion

1.1 Current-Driven Domain Wall Motion

5

The dominant effect in DW motion is determined by the relative comparison between the time during which the conduction electrons precess around the molecular field of the local magnetic moment, τex ≡ (/Jex )(gμB /M), and the time during which the conduction electrons pass through the DW [21]. Here,  is the reduced Planck’s constant, Jex is the exchange integral (for s-d exchange interaction for the case of ferromagnets), g is the Landé g-factor, μB is the Bohr magneton, and M is the magnetization. Spin-transfer occurs while the conduction electrons precess around the local magnetic moments. Therefore, if the time during which the conduction electrons pass through the DW is larger (or smaller) than τex , the spin-transfer (or momentum-transfer) mechanism is dominant. In this thesis, I worked on a ferrimagnetic metal where the DW width was tens to several hundreds of a nanometer. Thus, it is considered that the DW width is wider (larger) than the Larmor precession length of conduction electrons and that the spin-transfer mechanism is dominant over the momentum-transfer mechanism. So far, the adiabatic spin-transfer mechanism has been discussed, where the whole spin angular momentum of conduction electrons passing through the DW is given to the local magnetic moments (Fig. 1.2). However, according to literature, the nonadiabatic spin-transfer also needs to be considered in terms of the inconsistency between theory and experiments [22–26]. The non-adiabatic STT, namely the β term, originates from spin relaxation during when the conduction electrons are passing through the DW. Note that the adiabatic and non-adiabatic STTs are orthogonal to each other. The non-adiabatic spin-transfer, similar to the momentum-transfer, is equivalent to the magnetic field, thus called field-like torque and described by the equivalent magnetic field, Hβ , as follows [27]: Hβ =

βu γ

.

(1.1)

Here, β is the non-adiabaticity which characterizes the magnitude of non-adiabatic STT,  is the DW width, and γ is the gyromagnetic ratio. u = μB p J/eMS is the current-driven DW velocity through full spin-transfer [23, 27, 28], where p is the spin polarization of current, J is the current, −e is the electron charge, and MS is the saturation magnetization. The magnitude of β of ferromagnets is still the issue of debate. The study in Chap. 2 challenges the conventional understanding of β by investigating the value of β in a ferrimagnetic metal. To formulize the aforementioned explanation, the Landau–Lifshitz–Gilbert (LLG) equation (the equation of motion of magnetic moments) including the adiabatic and non-adiabatic STTs is represented as follows [23, 24, 27]: ∂m ∂t

= −γ m × H + αm ×

∂m ∂t

− u ∂m + βm × u ∂m . ∂x ∂x

(1.2)

Here, m is the unit vector along the magnetization direction, H is the effective external magnetic field, and α is the Gilbert damping parameter. The third and fourth terms on the right-hand side are the adiabatic and non-adiabatic STTs, respectively. The resultant DW velocity is derived as [27]

6

1 Introduction

Vsteady =

γ α



H+

 − α H+ V = γ 1+α 2

βu γ

βu γ



 ,

+

u 1+α 2

(1.3) (1.4)

Note that Eqs. (1.3) and (1.4) represent the DW velocity in the steady and precessional motion regimes, respectively (See Sect. 2.1.2 for the details of steady and precessional DW motion.). The DW is primarily driven by magnetic field H ,  u/ 1 + α 2 represents the velocity modification due to the adiabatic STT, and βu/γ (Eq. (1.1)) represents the effective magnetic field caused by the non-adiabatic STT.

1.1.2.3

History of Experiments on Spin-Transfer Torque Driven Domain Wall Motion

Soon after Berger suggested the current-driven DW motion mechanism in 1984 [12], he reported that the spontaneously formed DWs in a ferromagnetic thin film can be driven by an electric current [13]. The DWs in the NiFe thin film was observed through the Faraday effect, and it was found that the DWs were moved in the direction of electron flow by applying electric current in the order of 1011 A/m2 . However, in this experiment, several effects other than STT (e.g., hydromagnetic DW drag [29, 30]) were expected to be included, which makes the quantitative evaluation of DW motion difficult. Nonetheless, the recent developments of fine processing technology have enabled the fabrication of nanoscale magnetic wires, leading to a large number of studies being devoted to the investigation of current-driven DW motion in nanowires. A remarkable breakthrough was achieved by Yamaguchi et al. in 2004 [14]. Figure 1.3a shows the current-driven motion of a single DW in a submicron permalloy wire under zero magnetic field. The device is 240 nm wide and L-shaped, and the position of the DW was observed by magnetic force microscopy (MFM). The current pulse with a density of 1.2 × 1012 A/m2 and a width of 5 µs drove the DW in the electron flow direction. Figure 1.3b shows the DW velocity as a function of the current density. The maximum observed DW velocity was ≈6 m/s. Here, the mechanism of DW motion was considered to be STT despite the observation that the threshold current density (Jth ) for the DW motion was one or two orders smaller than that expected from the intrinsic pinning [20]. This small Jth implied that the non-adiabatic STT was dominant in the submicron permalloy wire, but soon they suggested that the Joule heating effect should be taken into account in the current-driven DW motion [31]. The current-driven DW motion in permalloy nanowires was then observed with pulse current in the order of nanoseconds [15, 32, 33]. Here, the pulse width was short (≤100 ns) enough to suppress the Joule heating effect of the wire, thus only the STT was considered to exist. Hayashi et al. reported that the DW velocity reached up to ~110 m/s with the current density of ~1.5 × 108 A/cm2 [32]. The DW velocity of ~110 m/s was higher than the DW velocity which was estimated from the assumption

1.1 Current-Driven Domain Wall Motion

7

Fig. 1.3 Current-driven domain wall (DW) motion observed by the magnetic force microscopy (MFM). a The MFM images of the magnetic DW in the submicron permalloy wire. Top: after the introduction of a DW. Middle: after an application of a pulsed current from left to right. Bottom: after an application of a pulsed current from right to left. b Average DW velocity as a function of current density. Reprinted figures with permission from A. Yamaguchi et al., Physical Review Letters 92, 077,205 (2004). Copyright (2004) by the American Physical Society. https://doi.org/10. 1103/PhysRevLett.92.077205

that the DW is driven solely by the adiabatic STT, i.e., u = μB P J/eMS [23, 27, 28]. This indicated the existence of some other mechanisms for DW motion, such as momentum-transfer (The non-adiabatic β term includes the contribution from the momentum-transfer.). In addition, Parkin et al. reported that the threshold current density (JC ) for DW motion was proportional to the pinning field as shown in Fig. 1.4 [15]. JC in the low pinning regime appeared to be zero when the pinning field is zero, implying a purely extrinsic origin for DW motion. As the finite intrinsic pinning is Fig. 1.4 Current-driven domain wall motion in a permalloy nanowire with short current pulses. Threshold current density JC for depinning a vortex domain wall near zero field is shown as a function of pinning field. The blue (black) dots represent the results in the wire with the thicknesses of 10 (30) nm and width of 100 (300) nm. From S. S. P. Parkin et al., Science 320, 190 (2008). Reprinted with permission from AAAS

8

1 Introduction

predicted for the case of adiabatic STT [20], this indicates that the non-adiabatic STT was dominant in the DW depinning in permalloy. More recently, current-driven DW motion in ferromagnetic metals with PMA has been attracting attention. The threshold current density (Jth ), in the adiabatic approximation, is expected to be smaller than that of ferromagnetic metals with in-plane magnetic anisotropy (IMA), because the DW width () and the hard-axis anisotropy (K H ) are smaller in PMA films than in IMA films (Jth ∝ K H ) [20, 34]. In 2011, Koyama et al. investigated Jth for DW motion in perpendicularly magnetized Pt/Co/Pt nanowire with structural inversion symmetry [11]. They found that Jth was minimum at the wire width, w ≈ 59 nm, as shown in Fig. 1.5c. This was explained as follows: when a magnetic DW is driven by an electric current through an adiabatic STT, the theory predicts a threshold current due to the intrinsic pinning (see Fig. 1.5a) even for a perfect wire without any extrinsic pinning [20]. Here, the threshold current due to the intrinsic pinning is proportional to K ⊥ /P, where K ⊥ the difference in the energy density between the Bloch wall (red array in Fig. 1.5a) and the Néel wall (green array),  is the DW width, and P is the carrier spin polarization. Considering that the demagnetizing field of the transverse direction of the wire increases as w reduces, there exists a critical w, where the energies of the Bloch and Néel walls are equivalent (ideally K ⊥ = 0). The minimum Jth at w ≈ 59 nm indicated that the energies of the Bloch and Néel walls were equivalent when w ≈ 59 nm. Therefore, they concluded that the DW motion in perpendicularly magnetized Pt/Co/Pt nanowire was attributed to the adiabatic STT [11].

Fig. 1.5 a Schematic of the intrinsic pinning of the DW. b Schematic of the experimental setup. c The threshold current density Jth and the DW pinning field Hpin as a function of the wire width w. Jth becomes the minimum when w ≈ 59 nm, where the energies of the Bloch and Néel walls are equivalent. Reprinted by permission from Springer Nature: Nature, Nature Materials, “Observation of the intrinsic pinning of a magnetic domain wall in a ferromagnetic nanowire” T. Koyama et al., ©2011

1.1 Current-Driven Domain Wall Motion

9

To summarize Sect. 1.1.1, I would like to note that the STT includes two orthogonal components, the adiabatic and non-adiabatic torques, and the dominant mechanism for current-driven DW motion depends on material systems.

1.1.3 Spin–Orbit Torque Driven Domain Wall Motion 1.1.3.1

Origins of Spin–Orbit Torque

Apart from STT, experiments have shown that an in-plane current can influence or directly control the magnetic dynamics in a ferromagnet/heavy metal bilayer for heavy metals, such as Pt or Ta. This is referred to as spin–orbit torque (SOT) and has become one of the main focuses in spintronic research today. Two different origins of SOT have been proposed, spin currents arising from the spin Hall effect (SHE) within heavy metal and the Rashba effect at the magnetic interface with structural inversion asymmetry. At present, it is widely accepted that these two origins, the SHE and Rashba effect, are phenomenologically indistinguishable (see Fig. 1.6). SHE refers to the spin-dependent scattering in metals owing to the spin–orbit interaction that results in the generation of a flow of spin angular momentum, namely, the spin current (see Ref. [35] for the details of SHE). Early theoretical studies predicted a SHE originated from asymmetries in electron scattering for up and down spins [36–39], referred to as an extrinsic SHE. More recently, it has been pointed out that there may exist an intrinsic SHE that arises because of the band structure,

Fig. 1.6 Schematic of spin–orbit torques for the cases of Rashba effect (left) and spin Hall effect (right) [51]. Reprinted figure with permission from A. Manchon et al., Rev. Mod. Phys. 91, 035,004 (2019). Copyright (2019) by the American Physical Society. https://doi.org/10.1103/RevModPhys. 91.035004

10

1 Introduction

even in the absence of scattering, referred to as an intrinsic SHE [40, 41]. SHE was experimentally confirmed in semiconductors using optical techniques [42, 43] and then in normal metals using electrical techniques [44–46]. The attempt to manipulate magnetic moments by use of spin current arising from SHE was successful through the excitation of spin-wave oscillations [47], magnetic resonance [48], magnetization switching [49, 50], and DW motion [7–9]. Figure 1.6 (right) shows the schematic of SHE [51]. SHE results in the spin accumulation at the interfaces of a heavy metal film with opposite spin polarizations at the top and bottom interfaces. The resultant dissipative spin current into the adjacent magnetic layer leads to the SOT on magnetic moments owing to the spin-transfer. As shown in Fig. 1.6 (right), SOT has two orthogonal components, the damping-like torque (T AD ) and field-like torque (TFL ). In general, the damping-like torque is dominant and can alone induce magnetic dynamics. Bulk and structural symmetry-breaking in semiconductors generate electric fields inside themselves. The electric fields act as magnetic fields to moving electrons, as a relativistic effect, and the direction of magnetic fields are determined as the cross product of the electrons momentum and the electric fields. Then, the magnetic fields interact with the electrons spin, aligning them parallel to the direction of magnetic fields. As a result, the electrons spin are coupled with the electrons momentum or wave vector (k), leading to the spin splitting of moving electrons. In zinc-blende type semiconductor crystals, e.g., GaAs, the electric fields caused by bulk inversion asymmetry in crystals give rise to the Dresselhaus term in the Hamiltonian [52]. In heterostructures, further electric fields are introduced due to structural inversion asymmetry, producing the Rashba term [53] (see Fig. 1.6 (left)). Both the Dresselhaus and Rashba fields have long been intensively investigated [54, 55] because the use of electrons spin in devices requires precise and sensitive control of spin’s environment. In 2008, theoretical work suggested that the Dresselhaus and Rashba fields, which had been detected in non-magnetic semiconductors, could be intrinsic to ferromagnets [56, 57], thereby providing a new mechanism for the manipulation of magnetization in ferromagnetic systems. Here we focus on the Rashba field that is produced by the spin–orbit interaction on the conduction electrons of a two-dimensional system characterized by structure inversion asymmetry. The Hamiltonian of the Rashba field HR is given by   HR = αR k × z · σ, 



(1.5)

  with the effective Rashba field, hR = αR k × z . Here, αR is the Rashba coefficient that depends on the strength of the spin–orbit coupling, k is the average electron wavevector, z is a unit vector parallel to the interfacial electric field. σ is the Pauli spin vector that represents the direction of conduction electrons spin. In the absence of a charge current, k = 0, thus hR = 0. However, in the presence of a charge current, k = 0, thus the finite hR induces a non-equilibrium spin density perpendicular to both k (the direction of the charge current) and z (the normal vector of the interface). In Fig. 1.6 (left), je represents the flow of conduction electron, and the red arrows at the 







1.1 Current-Driven Domain Wall Motion

11

interface represent the non-equilibrium spin density, which is aligned perpendicular to both je and z-axis. In a thin film heterostructure consisting of a ferromagnet and a heavy metal with strong spin–orbit interaction, the non-equilibrium spin density at the interface couples with the local magnetic moments via the s-d exchange interaction, generating torques (see Fig. 1.6 (left)) and inducing magnetic dynamics, such as the reversal of local magnetic moments [56, 57]. The first experimental observation of Rashba-induced spin-torque was achieved by Miron et al. in 2010 [58]. They investigated the magnetization reversal in Pt/Co/AlOx wire and found that the asymmetry in the reversal property comes from the Rashba-induced effective field. Soon after this study, they reported that the current-induced Rashba field can induce fast DW motion [59] and magnetization reversal in a ferromagnetic dot [60]. As mentioned above, the SHE and Rashba effect are phenomenologically indistinguishable, so the torques originating from them are currently called SOT, indicating that both effects come from spin–orbit interaction. The LLG equation which includes the SOT terms is represented as follows [49, 61] ∂m ∂t

= −γ m × H + αm ×

∂m ∂t

    + γ cJ m × m × y + γβ J m × y , 



(1.6)

where β is the coefficient that represents the ratio of the field-like torque to damping-like torque, J is the current density in the heavy metal layer, and cJ is (/2e)(θSH J/MS tF ), where e is the elementary charge, θSH is the spin Hall angle, and tF is the thickness of the ferromagnetic layer. The third and fourth terms describe the damping-like and field-like torques, respectively.

1.1.3.2

Spin–Orbit Torque Driven Domain Wall Motion

Experiments have shown that SOT can induce DW motion [7–9, 59]. Figure 1.7 shows the DW velocity as a function of current density in Pt (3 nm)/Co (0.6 nm)/AlOx (2 nm) with structural inversion asymmetry [59]. At that time, this result was not interpreted as the SOT coming from the interfacial spin accumulation but as the non-adiabatic STT that is enhanced by the Rashba effective field. Later, as it was proven that SHE within heavy metal layer also results in DW motion in the adjacent magnetic layer, it was gradually recognized that both the Rashba effect and SHE result in the interfacial spin accumulation, giving rise to the SOT. Note that the SOT acts as an effective magnetic field. The inset of Fig. 1.7 shows a measurement of the DW velocity as a function of the magnetic field. When H > 1.0 kOe, the DW mobility saturates (indicated by the red solid line) and thus, demonstrates a similar behavior to the current-induced DW velocity. The mechanism of SOT-driven DW motion is related to the chirality of the DW structure stabilized by the interfacial Dzyaloshinskii–Moriya interaction (DMI) [8, 9]. Figure 1.8a shows the schematic of the interfacial DMI [9]. The interfacial DMI

12

1 Introduction

Fig. 1.7 Domain wall (DW) velocity as a function of the current density. The solid line is a linear fit to the DW flow regime (occurring for j > 1.8 × 1012 A/m2 ). The inset shows a measurement of the DW velocity as a function of the magnetic field. When H > 1.0 kOe, the DW mobility saturates (indicated by the red solid line) and thus demonstrates a similar behavior to the currentinduced velocity. Reprinted by permission from Springer Nature: Nature, Nature Materials, “Fast current-induced domain-wall motion controlled by the Rashba effect” I. M. Miron et al., © 2011

occurs at the interface between a ferromagnetic metal and a metal with a strong spin– orbit coupling (SOC), where the inversion symmetry is broken. The Hamiltonian of DMI is expressed as HDM = −D12 · (S1 × S2 ),

(1.7)

where D12 is the DMI vector and S1 and S2 are the spin vector of the neighboring atoms in a ferromagnetic metal. Starting from a ferromagnetic state with S1 parallel to S2 , the DMI tilts S1 with respect to S2 by a rotation around D12 . In addition, the chirality of this rotation (either clockwise or counterclockwise with respect to D12 ) is determined by the sign of D12 . When it comes to DW structure, the interfacial DMI stabilizes the Néel DW instead of the Bloch DW, and the chirality of the Néel DW (either left-handed or right-handed) is determined by the sign of D12 . Figure 1.8b shows the Néel type DW with the left-handed chirality of magnetic moments inside DW [8]. When the spin current is injected into the Néel DW from the adjacent heavy metal layer, the magnetic moment inside the DW (directed in the x-axis) is exerted to the damping-like torque to align the magnetic moment parallel to the spin (directed in

1.1 Current-Driven Domain Wall Motion

13

Fig. 1.8 a Schematic of the interfacial Dzyaloshinskii–Moriya interaction (DMI) at the interface between a ferromagnetic metal (grey) and a metal with a strong spin–orbit coupling (SOC) (blue). The DMI vector D12 related to the triangle composed of two magnetic sites and an atom with a large SOC is perpendicular to the plane of the triangle. Reprinted by permission from Springer Nature: Nature, Nature Nanotechnology, “Skyrmions on the track” A. Fert et al., © 2013. b Schematic of the left-handed chiral Néel domain wall (DW). Reprinted by permission from Springer Nature: Nature, Nature Materials, “Current-driven dynamics of chiral ferromagnetic domain walls” S. Emori et al., © 2013

the y-axis). This results in the slight tilting of magnetic moments to the transverse direction (parallel to the y-axis), inducing the transverse magnetic field owing to the demagnetizing field. This transverse field again rotates the magnetic moment upward and downward, moving the DW. In other words, the damping-like torque is considered as the out-of-plane effective magnetic field (denoted as HSL in [8] named after Slonczewski [18]) as shown in Fig. 1.8b This out-of-plane effective field drives the DW. The SOT is considered to lead to an efficient (i.e., with low current density) and high-speed DW motion and magnetization reversal that are thus intensively investigated at present. Note that the conventional STT is still of interest in terms of fundamental physics as well as applications, and in this thesis, the conventional STT in ferrimagnetic DW is studied.

14

1 Introduction

1.2 Spin-Torque Ferromagnetic Resonance 1.2.1 What is Ferromagnetic Resonance? In 1946, ferromagnetic resonance (FMR) was first reported by Griffiths [62], followed by a further confirmation by Yager et al. [63]. In 1947, the theoretical interpretations of the experimental observation were given by Kittel [64, 65]. One of the typical experimental setups for FMR consists of a microwave cavity and an electromagnet. A ferromagnetic thin sheet is attached to the end wall of a microwave cavity so that the microwave is incident perpendicular to the ferromagnetic wall plane. Next, a static magnetic field is applied using an electromagnet in the direction parallel to the ferromagnetic wall plane and orthogonal to the microwave magnetic field. For measurement, the microwave absorption is measured while the magnitude of the static magnetic field is swept with the frequency of the microwave field fixed. When the frequency of magnetization precession equals the resonance frequency of the microwave cavity, microwave absorption increases remarkably. The mechanism of the drastic increase in the microwave absorption can be understood based on the Landau-Lifshitz-Gilbert (LLG) equation: ˙ = −γ m × H + αm × m. ˙ m

(1.8)

The LLG equation describes the two kinds of torques exerted on magnetic moment ˙ The schematic of m: precession torque −γ m × H and damping torque αm × m. the LLG equation is depicted in Fig. 1.9. When m is placed in magnetic field H, it precesses around H owing to the precession torque, decaying gradually to the direction of H owing to the damping torque. When microwave magnetic field is applied orthogonal to H, m experiences two kinds of torques with the microwave frequency: one is parallel to the precession torque (denoted by τp ), and the other is antiparallel to the damping torque (denoted by τd ). In particular, when the frequency of the Fig. 1.9 Schematic of the LLG equation under microwave magnetic field

1.2 Spin-Torque Ferromagnetic Resonance

15

microwave magnetic field coincides with that of the magnetic precession, damping ˙ can be totally canceled out by τd ; thus m continually precesses torque αm × m around H without damping. In other words, when the frequency of the microwave magnetic field is equivalent to that of magnetization precession, the microwave magnetic field and magnetic precession are coupled. Energy is converted from the microwave magnetic field to magnetic precession. Therefore, the energy gain from the microwave magnetic field compensates energy dissipation via magnetic damping, and the magnetic precession around H continues. When H is exactly at the resonance field, the phase of the magnetic precession is exactly the same as that of the microwave magnetic field, where the microwave absorption is the largest. As H is deviated from the resonance field, the phase difference between the magnetic precession and microwave field arises and results in weak microwave absorption. When H is deviated further, the phase difference becomes larger and microwave absorption becomes weaker. Therefore, when microwave absorption is measured as a function of H, it indicates a peak at the resonance field with a linewidth. Figure 1.10 shows the FMR spectrum in a Ni film reported by Griffiths [62]. A FMR peak with a linewidth is evident. The linewidth, or the full width at half maximum, of the FMR peak depends on the Gilbert damping parameter (α), i.e., the rate of energy dissipation from magnetic dynamics. As α is one of the important

Fig. 1.10 Ferromagnetic resonance peak of a ferromagnetic metal (Ni) film. The vertical axis represents the product of the electrical resistivity ρ and the magnetic permeability μ of the conductor, which describes the energy loss of microwave; the horizontal axis represents the external field; λ represnts the wavelength of microwave. Reprinted by permission from Springer Nature: Nature, Nature, “Anomalous High-frequency Resistance of Ferromagnetic Metals” J. H. E. Griffiths, © 1946

16

1 Introduction

parameters to characterize the magnetic dynamics, FMR is most commonly used to obtain α experimentally.

1.2.2 Spin–Orbit Torque Induced Ferromagnetic Resonance FMR can be excited not only by the microwave magnetic field but also by the electric current-induced radio-frequency (RF) spin-torque. In 2011, Liu et al. reported that, when the RF current is applied in the Py/Pt bilayer structure, FMR can be induced in the ferromagnetic Py layer by the combination of the SOT from the heavy metal Pt layer and the Ampère field generated by the current that flows through the Pt layer [48]. As the SOT plays an essential role in exciting FMR, this phenomenon is referred to as the spin-torque FMR (ST-FMR). Figure 1.11a shows the schematic of the experimental system. The static magnetic field, Hext , is applied in-plane and tilted by θ = 45◦ from the direction of the RF current, JC . When the RF current is applied in the bilayer structure, the spin current injection due to the SHE in Pt layer exerts the torque τSTT on the magnetic moments in the Py layer. In addition, the RF current flowing through the Pt layer gives rise to the Ampère field Hrf within Fig. 1.11 a Schematic of a Pt/Py bilayer thin film with the spin-transfer torque τSTT , the torque τH induced by Ampère field, and the damping torque τα . b Spectra of the spin-torque ferromagnetic resonance on a Pt (6 nm)/Py (4 nm) sample measured under frequencies of 5–10 GHz. Reprinted figures with permission from L. Liu et al., Physical Review Letters 106, 036,601 (2011). Copyright (2011) by the American Physical Society. https://doi.org/10.1103/Phy sRevLett.106.036601

1.2 Spin-Torque Ferromagnetic Resonance

17

the adjacent Py layer, exerting the torque τH . Here, τH and τSTT are time-dependent (RF) and orthogonal to each other; τH is parallel to the precession torque caused by Hext whereas τSTT is antiparallel to the damping torque τα . Therefore, similar to the FMR excitation by microwave magnetic field, the FMR is induced in the Py layer. The LLG equation for the magnetic moment in the Py layer is written as [48] ˙ + γ 2eμ0MS tF · JS,rf (m × σ × m) − γ m × Hrf . ˙ = −γ m × Heff + αm × m m (1.9) Here, JS,rf (/2e) = θSH J (/2e) represents the oscillating spin current density injected into Py, σ is the direction of the injected spin moment, and Hrf is the Ampère field generated by the RF current. The third and fourth terms on the right-hand side of Eq. (1.9) represent the spin current-induced torque τSTT and the Ampère field-induced torque τH , respectively. Figure 1.11b shows the consequent FMR spectra at various frequencies. Each of the FMR spectra consists of the symmetric and antisymmetric Lorentzian functions with the same linewidth, which are attributed to τSTT and τH , respectively.

1.3 Ferrimagnets in Spintronics 1.3.1 Antiferromagnets In the spintronic research field, considerable effort has been devoted to reducing device power consumption and scale. Antiferromagnets have been gaining increasing attention for recent years in this viewpoint [66–71]. The advantages of antiferromagnets are (1) the robustness against perturbation owing to magnetic fields, (2) the absence of the production of stray fields, eliminating unintentional magnetic crosstalk between neighboring devices, (3) ultrafast spin dynamics, and (4) the generation of large magneto-transport effects. However, in spite of these advantages, antiferromagnets have been far less studied than ferromagnets because the stability against the external magnetic field results in the difficulty in manipulating magnetic moments of antiferromagnets, thus the experimental investigation of antiferromagnetic dynamics is rather difficult. This is also a big problem for building the virtual computing system.

1.3.2 Advantage of Ferrimagnets: Field-Driven Antiferromagnetic Another class of magnetic materials that have the antiferromagnetically exchangecoupled magnetic moments with finite net magnetization are ferrimagnets. With

18

1 Introduction

Fig. 1.12 Schematics of a ferromagnets, b antiferromagnets, and c ferrimagnets

ferrimagnets, one can avoid the difficulty in experimental investigations of antiferromagnetic dynamics mentioned in Sect. 1.3.1. Figure 1.12 shows the schematics of (a) ferromagnets, (b) antiferromagnets, and (c) ferrimagnets, respectively. In antiferromagnets, the neighboring moments having the same magnitude are aligned antiparallel, thus totally canceling the net magnetization. In ferrimagnets, on the other hand, the neighboring spins having different magnetic moments are aligned antiparallel, resulting in finite net magnetization. The RE-TM alloy is a kind of ferrimagnet where the magnetic moments of RE and TM are coupled antiferromagnetically [72]. Figure 1.13a shows the schematic of gadolinium-iron-cobalt (GdFeCo) alloy, where Gd (RE) and FeCo (TM) moments are coupled antiferromagnetically. Considering that the magnetizations of Gd, MGd , and FeCo, MFeCo , are different, there is finite net magnetization, Mnet . A remarkable characteristic of RE-TM ferrimagnetic alloy is that because the temperature dependences of MGd and MFeCo are different, there exists the magnetization compensation temperature, TM , at which Mnet = MGd + MFeCo = 0 (Fig. 1.13b). Similarly, because the temperature dependences of the angular momentum of Gd (sGd ) and FeCo (sFeCo ) are different, there also exists the angular momentum compensation temperature, TA , at which snet = sGd + sFeCo = 0. Interestingly, because the gyromagnetic ratios (the ratio of magnetic moment to angular momentum) of Gd and FeCo are different (γGd = γFeCo ), the temperature dependences of Mnet and δs (= MGd /γGd + MFeCo /γFeCo ) are different, resulting in TM = TA . Figure 1.13b shows the schematic of Mnet and δs as a function of temperature T . The time evolution of a magnet state is governed by the commutation relation of the angular momentum, not of the magnetization. This means that the magnetic dynamics of ferrimagnets is antiferromagnetic-like at TA , not at TM . In other words, the nature of the ferrimagnetic dynamics is expected to change from ferromagnetic to antiferromagnetic as T approaches TA . Furthermore, Mnet is non-zero at TA (see Fig. 1.13b) and can couple to an external magnetic field, achieving the field-driven antiferromagnetic spin dynamics [73]. Consequently, the existence of TA in ferrimagnets provides a framework to investigate antiferromagnetic spin dynamics by use of the external magnetic field.

1.3 Ferrimagnets in Spintronics

19

Fig. 1.13 a Schematic of ferrimagnetic GdFeCo. In a rare-earth transition-metal ferrimagnet, the magnetizations of the rare-earth (Gd) and the transition-metal (FeCo) are coupled antiferromagnetically, resulting in the net magnetization, Mnet . b The magnetization of the rare-earth, MRE , that of the transition-metal, MTM , the net magnetization Mnet , and the net angular momentum, δs , are illustrated as a function of temperature, T . TM , TA , and TC represent the magnetization compensation temperature, the angular momentum compensation temperature, and the Curie temperature, respectively

1.3.3 Experiments on Antiferromagnetic Dynamics in Ferrimagnets 1.3.3.1

Magnetic Resonance in Ferrimagnets

In 1953, the theory on ferrimagnetic resonance (FiMR) predicted a strong temperature dependence of the dynamic behavior in RE-TM ferrimagnetic compounds [74, 75]. In particular, the frequency of the homogeneous spin precession, as well as the Gilbert damping parameter α, was expected to diverge at TA . However, the experimental confirmation of the FiMR theory had been lacking until 2006. In 2006, Stanciu et al. investigated the laser-induced magnetic dynamics and its decay to equilibrium in the RE-TM ferrimagnetic GdFeCo alloy across the TM and TA by the use of an alloptical pump and probe technique [76]. They demonstrated experimentally that both spin precession frequency and α increase significantly when the temperature of the sample approaches TA . The conventional understanding of FiMR is as follows. There are two resonance modes in FiMR: the ferromagnetic mode and the exchange mode. The ferromagnetic

20

1 Introduction

mode can be described by a single LLG equation with an effective gyromagnetic ratio γeff given by γeff (T ) =

MRE (T )−MTM (T ) MRE (T ) M (T ) − TM γ γ RE

=

TM

Mnet (T ) , δs (T )

(1.10)

and an effective Gilbert damping parameter αeff written as λRE

αeff (T ) =

|γRE |2

+

λTM

|γTM |2

MRE (T ) MTM (T ) γRE − γTM

=

A0 . δs (T )

(1.11)

Here, Mi and γi (i = REorTM) are the sublattice magnetization and the gyromagnetic ratio, respectively, and Mnet and δs are the net magnetic moment and net angular momentum, respectively. Note that Eqs. (1.10) and (1.11) state that both γeff and αeff diverge at TA since δs = 0 [74–76]. For the experiment, the divergence of γeff (αeff ) at TA was observed as the drastic increase in the precession frequency (spectral linewidth) in the vicinity of TA [76]. Since then, it has been believed that the magnetic resonance and the magnetic damping phenomenon in ferrimagnets are clarified. Most studies on ferrimagnetic dynamics have focused on how to excite and extract the ultrafast magnetization reversal for the application of the magnetic recording [72, 77–79]. Our study in Chaps. 3 and 4 deal with the magnetic damping phenomenon in ferrimagnets. The previous theory of FiMR [74–76] basically considers the FiMR phenomenon as two sublattice magnetizations follow the LLG equation for ferromagnets; when one writes the simultaneous LLG equations for the two sublattices as one effective LLG equation, one can obtain γeff and αeff . However, the study by Kim et al. suggested to include the antiferromagnetic dynamics term in the LLG equation, with which the ferrimagnetic dynamics is simply understood [73]. Based on this study, in Chaps. 3 and 4, we try to obtain the correct description of magnetic resonance and magnetic damping phenomenon in ferrimagnets by including the antiferromagnetic term into the LLG equation.

1.3.3.2

Domain Wall Motion in Ferrimagnets

In 2017, Kim et al. reported that the field-driven DW velocity in ferrimagnetic GdFeCo alloy becomes the fastest at TA [73]. They concluded that this is attributed to the antiferromagnetic dynamics at TA as follows. The DW velocity V and the precession frequency under the external field H are theoretically derived as V =

λ(α1 s1 +α2 s2 )(M1 −M2 ) (α1 s1 +α2 s2 )2 +(s1 −s2 )2

H,

(1.12)

=

(s1 −s2 )(M1 −M2 ) (α1 s1 +α2 s2 )2 +(s1 −s2 )2

H.

(1.13)

1.3 Ferrimagnets in Spintronics

21

Here, λ is the DW width, αi , si, and Mi are the Gilbert damping parameter, the spin angular momentum density, and the magnetization of sublattice i (i = 1, 2), respectively. Equations (1.12) and (1.13) indicates that, as the system approaches TA (i.e., |s1 − s2 | → 0), the DW velocity V increases, whereas the precession frequency decreases. At T = TA , V becomes the maximum while becomes zero. This means that the DW position X and angle are completely decoupled, and the pure translational dynamics of the DW is obtained, implying that the ferrimagnetic DW effectively acts as antiferromagnetic DW. In terms of the energy conservation of the system, the energy dissipation due to the precession of the angle vanishes at TA , so that all Zeeman energy gain is used for the translational motion energy of DW (i.e., V becomes the maximum). Kim et al. experimentally investigated the DW velocity in ferrimagnetic GdFeCo wire (30 nm thick) at various temperatures [73]. Figure 1.14 shows the DW velocity v as a function of temperature T . One can find that v remarkably increased as T approached 310 K and became the maximum at 310 K. Based on the above theory on field-driven DW velocity for ferrimagnets, they concluded that the TA of the ferrimagnetic GdFeCo alloy was 310 K. This study clearly shows that field-driven antiferromagnetic dynamics is definitely achievable at T = TA , and furthermore, that the field-driven DW velocity is fast (~2000 m/s) at T = TA , opening a possibility for ultra-high-speed device operation at TA of ferrimagnets. SOT is considered to be the effective magnetic field. Therefore, it is straightforward to expect that SOT-driven DW velocity also becomes the maximum at TA , which is advantageous for a virtual device application. Siddiqui et al. investigated the SOT-driven DW velocity in ferrimagnetic CoTb at room temperature by tuning the composition [80]. They found that the SOT-driven DW velocity became the maximum when the Co composition was 0.74, indicating that the TA of Co0.74 Tb0.26 was near room temperature. Similarly, Caretta et al. investigated the SOT-driven DW velocity in ferrimagnetic GdCo at various temperatures [81]. They experimentally found that, when the current density is large enough, the DW velocity became Fig. 1.14 The domain wall (DW) velocity v as a function of temperature T at several external magnetic field μ0 H in the out-of-plane (z) direction. Reprinted by permission from Springer Nature: Nature, Nature Materials, “Fast domain wall motion in the vicinity of the angular momentum compensation temperature of ferrimagnets” K.-J. Kim et al., © 2017

TM

TA

22

1 Introduction

Fig. 1.15 a The schematic of the measurement setup (top) and the Kerr microscopy images (bottom) showing current-driven domain wall motion near TA , where a train of current pulses with amplitude jHM = 1.5 × 1012 A/m2 was injected. b The DW velocity vDW as a function of temperature for various current densities. Reprinted by permission from Springer Nature: Nature, Nature Nanotechnology, “Fast current-driven domain walls and small skyrmions in a compensated ferrimagnet” L. Caretta et al., © 2018

the maximum at 260 K (Fig. 1.15b). Comparing the experimental results with the theoretical model, they concluded that this indicates that TA = 260 K. Note that, more recently, SOT-driven DW motion has been reported in a ferrimagnetic insulator, thulium-iron-garnet Tm3 Fe5 O12 [82].

1.4 Research Purpose The research purpose of this thesis is to investigate the role of angular momentum in ferrimagnetic dynamics, particularly the STT effects (Chap. 2), magnetic damping (Chaps. 3 and 4), and magnetic resonance (Chap. 4). As for the STT effects, the physical understanding has been inadequate in that, except for the study in [83], experimental investigations on STT in ferrimagnetic DWs have been lacking. As for magnetic damping and magnetic resonance, it should be noted that the conventional understanding needs to be changed in that the antiferromagnetic dynamics term has been missing in the LLG equation. Magnetic damping is investigated via the DW mobility (Chap. 3) and magnetic resonance (Chap. 4). These comprehensive studies on ferrimagnetic dynamics in terms of the angular momentum will provide a scientific platform for further investigating the fundamental physics of ferrimagnetic dynamics as well as realizing the next-generation spintronic devices based on ferrimagnets.

References

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.

Landau L et al (1935) Phys Z Sowjetunion 8:153 Néel L (1956) J Phys Radium 17:250 Middlehoek S (1962) J Appl Phys Suppl 33:1111 Middlehoek S (1963) J Appl Phys 34:1054 Torok EJ et al (1965) J Appl Phys 36:1394 DeJong MD et al (2015) Phys Rev B 92:214420 Haazen PPJ et al (2013) Nat Mater 12:299 Emori S et al (2013) Nat Mater 12:611 Ryu K-S et al (2013) Nat Nanotechnol 8:527 Fert A et al (2013) Nat Nanotechnol 8:152 Koyama T et al (2011) Nat Mater 10:194 Berger L (1984) J Appl Phys 55:1954 Freitas PP et al (1985) J Appl Phys 57:1266 Yamaguchi A et al (2004) Phys Rev Lett 92:077205 Parkin SSP et al (2008) Science 320:190 https://www.ibm.com/ibm/history/ibm100/us/en/icons/racetrack/ Berger L (1992) J Appl Phys 71:2721 Slonczewski JC (1996) J Magn Magn Mater 159:L1 Berger L (1996) Phys Rev B 54:9353 Tatara G et al (2004) Phys Rev Lett 92:086601 Waintal X et al (2004) Europhys Lett 65(3):427 Gun L et al (2000) IEEE Trans Magn 36(5):3047 Zhang S et al (2004) Phys Rev Lett 93:127204 Thiaville A et al (2005) Europhys Lett 69(6):990 Barnes SE et al (2005) Phys Rev Lett 95:107204 Tatara G et al (2006) J Phys Soc Jpn 75:064708 Mougin A et al (2007) EPL 78:57007 Koyama T et al (2012) Nat Nanotechnol 7:635 Carr WJ (1974) J Appl Phys 45:394 Berger L (1974) J Phys Chem Solids 35:947 Yamaguchi A et al (2005) Appl Phys Lett 86:012511 Hayashi M et al (2007) Phys Rev Lett 98:037204 Thomas L et al (2007) Science 315:1553 Fukami S et al (2008) J Appl Phys 103:07E718 Sinova J et al (2015) Rev Mod Phys 87:1213 D’yakonov MI et al (1971) JETP Lett 13:467 D’yakonov MI et al (1971) Phys Lett A 35:459 Hirsch JE (1999) Phys Rev Lett 83:1834 Zhang S (2000) Phys Rev Lett 85:393 Murakami S et al (2003) Science 301:1348 Sinova J et al (2004) Phys Rev Lett 92:126603 Kato YK et al (2004) Science 306:1910 Wunderlich J et al (2005) Phys Rev Lett 94:047204 Valenzuela SO et al (2006) Nature (London) 442:176 Saitoh E et al (2006) Appl Phys Lett 88:182509 Kimura T et al (2007) Phys Rev Lett 98:156601 Kajiwara Y et al (2010) Nature (London) 464:262 Liu L et al (2011) Phys Rev Lett 106:036601 Liu L et al (2012a) Phys Rev Lett 109:096602 Liu L et al (2012b) Science 336:555 Manchon A et al (2019) Rev Mod Phys 91:035004 Dresselhaus G (1955) Phys Rev 100:580

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1 Introduction

53. Bychkov YA et al (1984) J Phys C Solid State Phys 17:6039 54. Winkler R (2003) Spin–orbit coupling effects in two-dimensional electron and hole systems. Springer tracts in modern physics, vol 191. Springer, Berlin 55. Meier L et al (2007) Nat Phys 3:650 56. Manchon A et al (2008) Phys Rev B 78:212405 57. Manchon A et al (2009) Phys Rev B 79:094422 58. Miron IM et al (2010) Nat Mater 9:230 59. Miron IM et al (2011a) Nat Mater 10:419 60. Miron IM et al (2011b) Nature (London) 11:189 61. Lee K-S et al (2013) Appl Phys Lett 102:112410 62. Griffiths JHE (1946) Nature 158:670 63. Yager WA et al (1947) Phys Rev 72:80 64. Kittel C (1947) Phys Rev 71:270 65. Kittel C (1948) Phys Rev 73:155 66. MacDonald AH et al (2011) Phil Trans R Soc A 369:3098 67. Jungwirth T et al (2016) Nat Nanotechnol 11:231 68. Baltz V et al (2018) Rev Mod Phys 90:015005 69. Železný J, Wadley P, Olejník K, Hoffmann A, Ohno H (2018) Nat Phys 14:220 70. Jungfleisch MB, Zhang W, Hoffmann A (2018) Phys Lett A 382:865 71. Gomonay O, Baltz V, Brataas A, Tserkovnyak Y (2018) Nat Phys 14:213 72. Kirilyuk A et al (2013) Rep Prog Phys 76:026501 73. Kim K-J et al (2017) Nat Mater 16:1187 74. Wangsness RK (1953) Phys Rev 91:1085 75. Wangsness RK (1954) Phys Rev 93:68 76. Stanciu CD et al (2006) Phys Rev B 73:220402(R) 77. Stanciu CD et al (2007) Phys Rev Lett 99:217204 78. Kirilyuk A et al (2010) Rev Mod Phys 82:2731 79. Mekonnen A et al (2011) Phys Rev Lett 107:117202 80. Siddiqui SA et al (2018) Phys Rev Lett 121:057701 81. Caretta L et al (2018) Nat Nanotechnol 13:1154 82. Avci CO et al (2019) Nat Nanotechnol 14:561 83. Aoshima K et al (2018) Jpn J Appl Phys 57:09TC03

Chapter 2

Spin-Transfer Torques for Domain Wall Motion in Antiferromagnetically-Coupled Ferrimagnets

Abstract Spin-transfer torque (STT) effects on DW motion in rare earth-transition metal (RE-TM) ferrimagnets are investigated both theoretically and experimentally across the angular momentum compensation temperature, TA . First, theoretical equation for the field-driven STT-assisted DW velocity in ferrimagnets is derived based on Landau–Lifshitz–Gilbert equation. Second, this theory is tested experimentally using the DW motion experiment on a ferrimagnetic GdFeCo alloy, where Gd and FeCo moments are coupled antiferromagnetically. The experimental results are well fitted with the theory, confirming that the adiabatic STT component in DW velocity reverses its sign across TA , whereas the non-adiabatic STT component is the maximum at TA without a sign change. Given that STT effects at TA in ferrimagnets represent those in antiferromagnets, this finding indicates that the non-adiabatic STT in antiferromagnets acts as a staggered magnetic field, which can induce antiferromagnetic DW motion. It is also found that non-adiabaticity parameter, β, which characterizes the magnitude of non-adiabatic STT, is remarkably larger than Gilbert damping parameter, α. This finding challenges the conventional understanding of the non-adiabatic STT based on experiments on ferromagnets. Keywords Spin-transfer torque · Domain wall motion · Angular momentum compensation temperature of ferrimagnets

2.1 Research Background 2.1.1 Brief History of Field-Driven Domain Wall Motion Until the beginning of the 1970s, the experiments on field-driven DW motion were compared with and discussed based on the primitive forms of theory [1–6]: v = G(H − H0 ), i.e., DW velocity v is linear to magnetic field H with DW mobility G and intersection H0 . However, there were several experimental studies that reported some deviation from linearity. Therefore, there was confusion regarding the equation

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 T. Okuno, Magnetic Dynamics in Antiferromagnetically-Coupled Ferrimagnets, Springer Theses, https://doi.org/10.1007/978-981-15-9176-1_2

25

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2 Spin-Transfer Torques for Domain Wall Motion …

of the field-driven DW velocity [7]. In the 1970s, a one-dimensional model of fielddriven DW motion was developed by Walker [8] and Slonczewski [9, 10]. Since then, the phenomenon of the field-driven DW motion has been understood based on this one-dimensional model (explained in Sect. 2.1.2). At that time, the experiments on field-driven DW motion were limited to those performed in magnetic thin films. Therefore, observing the field-driven DW motion in one single submicron-scale magnetic wire was difficult owing to technological limitations. The development of the atomic-layer deposition and nanofabrication technologies made it possible to perform experiments on field-driven DW motion in single submicron ferromagnetic wires. In 1999, Ono et al. were the first to report quantitative investigations on field-driven DW velocity in a single ferromagnetic submicron wire consisting of NiFe/Cu/NiFe [11]. In this paper, DW mobility (∂ V /∂(μ0 H ), where V is DW velocity and H is field) was small (2.6 m/s Oe), which was attributed to a large α value. Later in 2003, Atkinson et al. reported the field-driven DW velocity in a ferromagnetic permalloy nanowire, with the DW mobility exceeding 30 m/s Oe, which was significantly larger than that reported by Ono [12]. This quantitative disagreement in DW mobilities was settled by Beach et al. in 2005 [13]. They measured field-driven DW velocity in a submicron permalloy wire and confirmed the existence of two linear regimes, where DW mobility at high fields was reduced to 1/10th from that at low fields. This finding provided an experimental confirmation of the one-dimensional model [8–10], where the steady motion regime (high mobility) and precessional motion regime (low mobility), as well as the Walker breakdown between these regimes, are derived. Based on this study, the low mobility reported in Ref. [11] is attributed to the DW precessional motion, whereas the large mobility in Ref. [12] is concluded to have originated from the DW steady motion. After the study by Beach et al. [13], field-driven DW motion has been studied in various types of magnetic systems based on the one-dimensional model.

2.1.2 Mechanism of Field-Driven Domain Wall Motion: Steady and Precessional Motions In Chaps. 2 and 3, we focus on the field-driven STT-assisted DW motion. In this section, the one-dimensional model of field-driven DW motion [13, 14] is introduced so that the results presented in Chaps. 2 and 3 can be understood. According to the one-dimensional model, there exist two regimes in the fielddriven DW motion: steady motion regime and precessional motion regime [8–10, 13, 14]. We consider the field-driven motion of a Bloch DW in a ferromagnetic thin film with PMA (see Fig. 1.1a). When low magnetic field, H (H < HW ), is applied along the anisotropy easy axis (+z), m at the DW center position starts precession owing to the Larmor precession. This precession tilts m away from the y-axis. The magnetization component perpendicular to the DW plane (yz-plane) creates magnetic charges on both sides of DW, as depicted in Fig. 2.1. These magnetic charges generate

2.1 Research Background

27

Fig. 2.1 Diagram of the xy-plane of a Bloch domain wall in a ferromagnetic thin film with perpendicular magnetic anisotropy under the application of out-of-plane magnetic field H

demagnetizing field Hd along the +x-direction. Hd gives rise to the Larmor precession of the magnetic moment around itself, leading to the out-of-plane torque that aligns m in the out-of-plane (+z) direction. Furthermore, Hd gives rise to damping torque to align m parallel to Hd . This damping torque halts the magnetic precession around H (+z-direction) by canceling the precession torque induced by H. Therefore, DW moves smoothly with the time-independent Φ (Φ˙ = 0). This type of DW motion is called DW steady motion. Its velocity is given by Vsteady =

γ μ0 H. α

(2.1)

As the external field increases, the time-independent Φ increases. When Φ = π/4, the equilibrium state balanced between the damping torque (due to Hd ) and precession torque caused by the external field is broken. The DW velocity no longer obeys Eq. (2.1) and reaches its maximum at Φ = π/4. This is called the Walker breakdown. The Walker breakdown field is calculated as   HWB = 2π α Ms  N x − N y ,

(2.2)

where N x and N y are the demagnetizing factors along the x- and y-axes, respectively. Above HWB , the azimuthal angle (Φ) precesses around the out-of-plane (+z) external field, i.e., the DW structure continuously changes between the Bloch wall and Néel wall alternatively. Because the torque for the precession of m is periodic with respect to Φ, DW motion is oscillatory. This is called the DW precessional motion. The −

average DW velocity, V , in the precessional motion is given by −

V=

αγ μ0 H. 1 + α2

(2.3)

28

2 Spin-Transfer Torques for Domain Wall Motion …

Note that DW mobility ∂ V /∂(μ0 H ) in the steady motion regime is larger than that in the precessional motion regime. This is attributed to energy dissipation via magnetic precession inside DW. A small portion of the Zeeman energy gain by the magnetic field is converted into energy dissipation via the DW translational motion, resulting in small DW velocity in the precessional motion regime. The DW velocities in the steady and precessional motion regimes as functions of H are depicted in Fig. 2.2a. Before the Walker breakdown, the DW velocity in the steady motion regime increases linearly with H , reaching its maximum at HWB . Thereafter, the DW velocity starts to decrease, before it again starts to increase linearly with H in the precessional motion regime with a smaller DW mobility than that in the steady

Fig. 2.2 Schematic of domain wall velocity as a function of a an external magnetic field H and b a driving force combining field and current effects. These figures illustrate the two linear regimes of velocity below and significantly above the Walker breakdown. Reprinted figures with permission from “Domain wall mobility, stability and Walker breakdown in magnetic nanowires” A. Mougin et al., EPL 78, 57,007 (2007). https://dx.doi.org/10.1209/0295-5075/78/57007. c Experimental verification of the steady and precessional motion regimes, as well as the Walker breakdown between them. Reprinted by permission from Springer Nature: Nature, Nature Materials, “Dynamics of field-driven domain-wall propagation in ferromagnetic nanowires” G. S. D. Beach et al., © 2005. d Simulation results of field-driven domain wall velocity, V , in ferrimagnets as a function of the out-of-plane field (Bz ). The Walker breakdown field, BWB , increases as the net angular momentum, δs , approaches 0 (i.e., the temperature approaches the angular momentum compensation temperature, TA ), and diverges at δs = 0 (at TA , light blue dots labeled 5). Reprinted by permission from Springer Nature: Nature, Nature Materials, “Fast domain wall motion in the vicinity of the angular momentum compensation temperature of ferrimagnets” K.-J. Kim et al., © 2017

2.1 Research Background

29

motion regime. This is the one-dimensional model of field-driven DW motion. It was experimentally confirmed in 2005 by Beach et al. [13] (see Fig. 2.2c). In this thesis, we investigate the STT effects on the field-driven DW velocity. Based on the one-dimensional model [14], the field-driven DW velocity under the effect of STT is modified by including additional terms (redisplaying Eqs. (1.3) and (1.4)):   βu γ H+ , α γ   − βu u α H+ + . V = γ 2 1+α γ 1 + α2 Vsteady =

(2.4) (2.5)

Figure 2.2b depicts the DW velocity driven by the magnetic field and the adiabatic and non-adiabatic STTs. Note that the non-adiabatic STT component (the effective field βu/γ ) exists both in the steady and precessional  regimes, whereas the  motion adiabatic STT component (the additional velocity u/ 1 + α 2 ) is included only in the precessional motion regime. This implication that the STT affects the DW velocity differently depending on the DW motion regime is related to our conclusions in this chapter. To elaborate this, the mechanism of the field-driven DW motion in ferrimagnets is explained in detail in the following section (see Sect. 1.3.3.2 for a brief explanation).

2.1.3 Walker Breakdown in Field-Driven Domain Wall Motion in Ferrimagnets The Walker breakdown field plays an essential role in the fast DW motion at TA of ferrimagnets. According to Ref. [15], the drastic increase in DW velocity at TA of ferrimagents is attributed to the reduction of magnetic precession inside the DW; therefore, energy dissipation due to precession is reduced, resulting in large energy dissipation due to the DW translational motion (i.e., fast DW velocity). In the precessional DW motion regime, the DW velocity (V ) and precession frequency (Ω) under the external field (H ) are theoretically derived as (see Sect. 1.3.3.2) V =

λ(α1 s1 + α2 s2 )(M1 − M2 ) H, (α1 s1 + α2 s2 )2 + (s1 − s2 )2

(2.6)

Ω=

(s1 − s2 )(M1 − M2 ) H, (α1 s1 + α2 s2 )2 + (s1 − s2 )2

(2.7)

with the Walker breakdown field given by

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2 Spin-Transfer Torques for Domain Wall Motion …

HWB =

α1 s1 + α2 s2 κλ . 2(s1 − s2 ) (M1 − M2 )λ

(2.8)

Here, αx , sx , and Mx are the Gilbert damping parameters, spin angular momentum densities, and magnetizations of sublattices, respectively; κ is the small easy axis anisotropy energy along the y-axis. Therefore, as the temperature approaches TA , |s1 − s2 | → 0 and thus V approaches the maximum value, Ω → 0 and HWB → ∞. The DW velocity as a function of magnetic field with the Walker breakdown field is analytically obtained, as shown in Fig. 2.2d. The DW mobility in the steady motion regime is the same for all the temperatures, including TA , because energy dissipation due to the precession of the magnetic moment is absent; therefore, the Zeeman energy gain is converted solely into energy dissipation via the DW translational motion. However, the DW mobility in the precessional motion regime changes depending on |s1 − s2 |, and thus on the temperature. Significantly away from TA , the DW mobility in the precessional regime is small owing to large energy dissipation through magnetic precession; moreover, HWB is also small. In Fig. 2.2d, the smallest DW mobility and HWB corresponding to the dots labeled as 1 and 9 are the numerical solutions under the farthest conditions from TA . As the temperature approaches TA , the DW mobility in the precessional regime increases. Because Ω decreases as the temperature approaches TA , the energy dissipated through it also decreases, resulting in an increase in the DW translational velocity. In Fig. 2.2d, it is evident that DW mobility and HWB increase gradually as the label of the dots shifts from 1 or 9 to 5. Exactly at TA , HWB diverges, leading to the vanishing of magnetic precession. Therefore, DW exhibits only steady motion, and DW mobility for all the field range coincides with that in the DW steady motion regime (the dots labeled 5 in Fig. 2.2d). The focus of the study in this chapter is on the STT effects on DW motion in ferrimagnets. Given that the DW motion regime (steady or precessional motion) changes based on the temperature in ferrimagnets, the adiabatic and non-adiabatic STT effects should also change based on temperature. In particular, the adiabatic STT effect is expected to vanish at TA , consistent with the vanishing of the precessional DW motion regime. In the following sections, this qualitative understanding is formulated and thoroughly tested experimentally.

2.1.4 Scope of Research As mentioned in Sect. 1.3.3.2, recent experiments have shown that field-driven [15] or spin–orbit-torque driven [16, 17] DW motion in antiferromagnetically-coupled ferrimagnets is fastest at TA , where the ferrimagnetic dynamics are antiferromagnetic. However, the STT effects in antiferromagnetic DWs have only been investigated in theoretical studies [18–22]. The theoretical studies suggest that STT for antiferromagnetic DWs consists of adiabatic and non-adiabatic STT components, as is the case for ferromagnetic DWs (see Sect. 2.1.2). In particular, theory suggests that nonadiabatic STT in antiferromagnets exerts a staggered magnetic field [20, 21], that

2.1 Research Background

31

is, it exerts effective magnetic fields of equal magnitude and opposite sign on the two different sublattices. In contrast, non-adiabatic STT for ferromagnetic DWs acts like a uniform magnetic field (the effective field Hβ = βu/γ , see Eq. (2.5)). The origin of non-adiabatic STT for ferromagnetic DWs has been intensively debated not only theoretically but also experimentally over the past decade [23–35]. Nevertheless, the nature of non-adiabatic STT for antiferromagnetic DWs has not been investigated experimentally and remains unresolved. The scope of research in this chapter is on the experimental exploration of the adiabatic and non-adiabatic STTs in antiferromagnetic textures. Because experiments on STT effects in pure antiferromagnetic materials are so difficult, here we have chosen to investigate STT effects on DW motion in a RE-TM ferrimagnet, where RE and TM moments are coupled antiferromagnetically.

2.2 Theory of Ferrimagnetic Domain Wall Velocity In this section, the field-driven STT-assisted DW velocity in ferrimagnets is formulated. The LLG equation for the magnetization of a ferrimagnet including the STT effects [30, 36] is given as ˙ − αsm × m ˙ − ρm × m ¨ = −m × heff + P(J · ∇)m − β Pm × (J · ∇)m, δs m (2.9) where m is the unit vector along the direction of magnetization, δs is the net spin density along −m (i.e., the difference between the spin densities of two sublattices), α > 0, s is the saturated spin density (i.e., the sum of the spin densities of two sublattices), ρ is the inertia associated with the dynamics of m, heff ≡ −δU /δm ˙ is the effective field conjugate to m, and U [m] is potential energy. Parameters m ¨ denote the first- and second-order time derivatives of m, respectively. The and m second and third terms on the right-hand side of Eq. (2.9) are the adiabatic and nonadiabatic STT terms;J is the current density; P is the spin conversion factor and is    given as P = (/2e) σ↑ − σ↓ / σ↑ + σ↓ (with the electron charge e > 0); and β is the non-adiabaticity parameter. Parameter P represents the polarization of the spindependent conductivity, σs (s = ↑ or ↓ with ↑ chosen along −m), and characterizes the adiabatic STT term, whereas β parametrizes the non-adiabatic STT term. It should be noted that Eq. (2.9) is in contrast with Eq. (1.2), the LLG equation for ferromagnets, in some respects. (1) In Eq. (2.9), the precession term includes δs and the damping term includes s. This is based on the model where magnetic precession is governed by δs , whereas the energy dissipation that causes the magnetic damping is originated from s (Chapters 3 and 4 deal with the confirmation of this model.). As for ferromagnets, the net spin density is identical to the saturated spin density, thus δs /s = 1.

32

2 Spin-Transfer Torques for Domain Wall Motion …

¨ which represents the (2) Equation (2.9) includes the additional third term −ρm× m, antiferromagnetic dynamics of the two sublattices [20]. It is absent in Eq. (1.2). Note that the remaining terms (including the adiabatic and non-adiabatic torque terms) in Eq. (2.9) are essentially the same as those in Eq. (1.2). Before the equation of DW motion is derived, we present the coordinates of the magnetic moments of DW in the one-dimensional model. Here, Φ is defined as the azimuthal angle from the x-axis, and X is the position at the DW center. The m inside the DW at the position x is calculated as m(x) = (sech((x − X )/λ)cosΦ, sech((x − X )/λ)sinΦ, tanh((x − X )/λ)). (2.10) The magnetic moment at the DW center (x = X ) is tilted from the x-axis by angle Φ, where Φ = π/2 for the Bloch wall and Φ = 0 for the Néel wall (see Fig. 1.1). When we discuss DW motion, X (t) and Φ(t) are time-dependent variables; X (t) and Φ(t) change continuously with the time, t, as the DW moves. Using Eq. (2.10) in Eq. (2.9), the following coupled equations of motion for X and Φ are obtained [37]: δs Φ˙ +

αs X˙ ρ X¨ βPJ + = μ0 M H − , λ λ λ

PJ δs X˙ − αs Φ˙ − ρ Φ¨ = − , λ λ

(2.11)

(2.12)

where J is the charge current density along the x-axis, M is the magnitude of magnetization along m, and μ0 H is the external magnetic field along the z-direction. Solving Eqs. (2.11) and (2.12) for the steady-state velocity X˙ → V , the DW velocity, V , above the Walker breakdown [13, 14] is obtained as V (μ0 H, J ) =

1 δs2 + (αs)2

(αsλMμ0 H − δs P J − αsβ P J ),

(2.13)

where the first term represents the DW velocity driven by the magnetic field μ0 H , and the second and third terms represent those originated from the adiabatic and non-adiabatic STTs, respectively. DW velocity measurements under the positive and negative currents enable us to separate V (μ0 H, J ) into field component Vfield (μ0 H ) and STT components V (J ). We can extract Vfield (μ0 H ) by adding the DW velocity for positive current V (μ0 H, +J ) to that for negative current V (μ0 H, −J ), canceling out the STT components as follows: Vfield (μ0 H ) =

αsλMμ0 H V (μ0 H, +J ) + V (μ0 H, −J ) = . 2 δs2 + (αs)2

(2.14)

2.2 Theory of Ferrimagnetic Domain Wall Velocity

33

Similarly, we can extract the STT components in DW velocity V (J ) by subtracting V (μ0 H, −J ) from V (μ0 H, +J ): V (J ) =

δs P J V (μ0 H, +J ) − V (μ0 H, −J ) αsβ P J =− − . (2.15a) 2 2 2 2 δs + (αs) δs + (αs)2

Equations (2.13) and (2.15a) give us some insights into the STT-driven DW velocity. First, Eq. (2.15a) indicates that the adiabatic STT contribution to DW velocity is zero at TA (i.e., δs = 0) and reverses its sign across TA . The sign reversal means that the direction of DW motion driven by the adiabatic STT is opposite for T > TA and T < TA . Second, Eq. (2.13) is rewritten as   δs P J βPJ μ0 H − − , V (μ0 H, J ) = 2 2 2 λM δs + (αs) δs + (αs)2 αsλM

(2.16)

where β P J/λM is the effective field caused by the non-adiabatic STT, which is identical to Eq. (1.1). Given that this effective field acts on the saturated spin density, s, Eq. (2.16) suggests that this effective field is staggered, meaning that the nonadiabatic STT induces opposite effective magnetic fields on the two sublattices of antiferromagnets, thereby enabling the antiferromagnetic DW motion driven by the non-adiabatic STT. This finding is consistent with previous theories for antiferromagnetic DWs [18–22]. In contrast, the non-adiabatic STT for ferromagnetic DWs is an effective uniform magnetic field (see Eq. (1.1)), which couples linearly to local magnetization. Third, Eq. (2.15a) also suggests the possibility that the STT components in DW velocity V (J ) can be separated into adiabatic and non-adiabatic STT components. Equation (2.15a) indicates that the dependence of V on δs is different for the adiabatic and non-adiabatic STT components. Provided that δs is significantly dependent on temperature in ferrimagnets, Eq. (2.15a) implies that the adiabatic STT component is antisymmetric with respect to δs = 0 (i.e., T = TA ), whereas the non-adiabatic STT component is symmetric. These insights will be discussed in detail in Sect. 2.5.1.

2.3 Experimental Method 2.3.1 Film Preparation and Device Fabrication The studied sample was an amorphous ferrimagnetic thin film of 5 nm Si3 N4 /30 nm Gd23.5 Fe66.9 Co9.6 /100 nm Si3 N4 on an intrinsic Si substrate deposited by magnetron sputtering. The Si3 N4 buffer and capping layers were used to avoid oxidation of the GdFeCo layer. It was confirmed by polar magneto-optical Kerr effect microscopy that the GdFeCo film exhibits perpendicular magnetic anisotropy (PMA). GdFeCo

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2 Spin-Transfer Torques for Domain Wall Motion …

Fig. 2.3 Schematic of the device and the measurement setup. The direction of the bias current, DW motion, and the applied magnetic field are indicated. Copyright (2019): Springer Nature

microwires 5 μm wide and 500 μm long were fabricated by the use of electron beam lithography and Ar ion milling. A negative-tone electron beam resist (maN-2403) mixed with a thinner for resists (T-1047, at a volume ratio of 1:1) was used for lithography at a fine resolution (~5 nm). After the milling process, 100 nm Au/5 nm Ti electrodes were stacked on the substrate for current injection (see the gold areas in Fig. 2.3).

2.3.2 Experimental Setup for Field-Driven Domain Wall Motion The schematic of the experimental setup is shown in Fig. 2.3. A pulse generator (Picosecond 10,300B) was used to generate the current pulse (Iwrite in Fig. 2.3) and

2.3 Experimental Method

35

create the DW at one end of the wire. A 2 mA, 100 ns current pulse was used to create the DW. For field-driven DW motion, a 2, 2.5, or 3 mA bias D.C. current (which corresponds to a current density of 1.3, 1.7, or 2.0 × 1010 A/m2 , respectively) was injected along the wire to generate an anomalous Hall voltage, VH . The direction of the positive bias D.C. current (+J in Fig. 2.3) is defined as the same as that of the DW motion. A Yokogawa 7651 was used as a D.C. current source. The VH at the Hall cross was measured by an oscilloscope (Tektronix 7354) through a 46 dB differential amplifier. A low-temperature probe station was used to investigate the DW motion for wide ranges of temperature.

2.3.3 Domain Wall Detection Technique We used a time-of-flight measurement of DW propagation to obtain the DW velocity [38]. This measurement scheme is based on the real-time detection of a change in VH by an oscilloscope. The detailed procedure for measuring the DW velocity was as follows. (1) A large perpendicular magnetic field μ0 Hsat = −150 mT was applied to saturate the magnetization (Fig. 2.4a). (2) A drive field μ0 Hd , in the range of |μ0 HP | < |μ0 Hd | < |μ0 HC |, was applied in the opposite direction (Fig. 2.4b). Here, μ0 HP is the pinning field of DW motion, where a DW “pinned” in the wire starts to move, and μ0 HC is the coercive field of the sample, where DWs start to nucleate resulting in the magnetization reversal in the whole wire. Since μ0 Hd is smaller than μ0 HC , μ0 Hd does not reverse the magnetization nor create DWs. (3) A current pulse (2 mA, 100 ns) Iwrite (the red arrow in Fig. 2.3) was injected into the contact line by a pulse generator in order to create a DW through a current-induced Oersted field (Fig. 2.4c). The local Oersted field creates a reversed domain (upward domain) next

Fig. 2.4 Schematic of the flow of real-time detection of domain wall. a The magnetizations of the whole wire are saturated downwards due to a large magnetic field μ0 Hsat . b A drive field μ0 Hd is applied, yet the magnetizations are still saturated downwards because μ0 Hd is smaller than the coercive field. c When a current pulse Iwrite was injected into the contact line, a DW is created through an Oersted field. d As soon as the DW is created, μ0 Hd pushes the DW so that the upward magnetic domain expands. e The DW passes through the Hall cross region, resulting in the sign change of the transverse voltage measured at the Hall cross region due to the anomalous Hall effect

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2 Spin-Transfer Torques for Domain Wall Motion …

Fig. 2.5 An example of the subtraction of “reference trace” from “signal trace” (blue dots) and the signal from writing pulse current (red dots). The steplike signal indicates that the domain wall passes the Hall cross region. The flight time of the domain wall from the contact line to the Hall cross can be obtained

Signal [arb. unit]

to the contact line. (4) As soon as the DW is created, μ0 Hd pushes the DW so that the reversed domain expands because μ0 Hd is larger than μ0 HP (Fig. 2.4d). The DW propagates along the wire from the contact line and passes through the Hall cross region (Fig. 2.4e). The anomalous Hall effect (AHE) creates the transverse voltage in the wire; when the electric current flows through the magnetic materials, the electrons are scattered in the direction orthogonal to both electrons flow and the magnetic moments. This results in the accumulation of electrons in the transverse direction, generating the transverse voltage. Therefore, when the direction of magnetic moments is reversed, the transverse voltage due to AHE is also reversed. The transverse voltage is measured at the Hall cross. Therefore, at the moment the DW passes through the Hall cross region, the transverse voltage changes because the magnetizations of the Hall cross reverse as a result of the DW passage. This transverse voltage change is recorded by the oscilloscope through the differential amplifier. We refer to this as a “signal trace”. Since the detected transverse voltage change includes a large background signal, we subtract the background from the “signal trace” by measuring a “reference trace”. The “reference trace” is obtained in the same manner as the signal trace except that the saturation field direction is reversed (μ0 Hsat = +150 mT). In this reference trace, no DW is nucleated; hence, in the “reference trace”, no DW propagates through the Hall cross region and only electronic noise can be detected in the oscilloscope. Figure 2.5 shows an example of the subtraction of “reference trace” from “signal trace” in our GdFeCo wire. The vertical axis represents the signal measured using an oscilloscope, whereas the horizontal axis corresponds to the elapsed time. The measured transverse signal indicates a steplike change, which corresponds to the passage of DW at the Hall cross. The DW arrival time was estimated from the fitting of the traces with the hyperbolic tangent function:

0.2

Writing pulse current Transverse voltage

0.1

Time of flight

0.0 0

1

2 3 Delay time [μs]

4

5

2.3 Experimental Method

37

 VH = tanh

 x −t , c

(2.17)

where t is the DW arrival time, and c is the parameter that represents how rapidly VH changes.   The DW velocity is given as V = L/ t − t , where L is the DW travel length,  which was 400 m in our measurement, and t is the DW generation time, obtained as  t = (tfinal + tinitial )/2, where tinitial (tfinal ) is the initial (final) time of the pulse current for DW generation. In Fig. 2.5, the signal from the writing pulse current is indicated as red circles. tinitial and tfinal are obtained from the time of the abrupt rise and drop of the signal, respectively. In the measurement in this chapter, we averaged V from five (when 230 < T < 250 K) or three (otherwise) repeated measurements (note that TA = 241 K in our sample). The error bar in V was determined as the standard deviation of V values from multiple measurements.

2.4 Experimental Results on Field-Driven Current-Assisted Domain Wall Motion To verify the theoretical prediction described in Sect. 2.2, STT effects on the fielddriven current-assisted DW motion in a ferrimagnetic GdFeCo compound is investigated, in which Gd and FeCo moments are coupled antiferromagnetically. Because the film (5 nm SiN/30 nm Gd23.5 Fe66.9 Co9.6 /100 nm SiN film deposited on an intrinsic Si substrate) lacks a nonmagnetic heavy metal layer as a spin-current source, the effects of the spin–orbit-torque [16, 17, 39–41] are ignored. In this study, our focus is on the DW motion in the precessional regime, where the DW angle changes continuously (see Sect. 2.1.2 for details). Figure 2.6 shows V as a function of μ0 H under positive (+J , red dots) and negative (−J , blue dots) bias currents. It is confirmed that V increases linearly with an increase in μ0 H and shifts under positive and negative bias currents. The Fig. 2.6 Domain wall velocity V as a function of magnetic field μ0 H at 211.6 K under bias current J = ±1.3 × 1010 A/m2 . The dotted green line represents Vfield (μ0 H ) = [V (μ0 H, +J ) + V (μ0 H, −J )]/2 and the orange arrow represents 2V (J ) = [V (μ0 H, +J ) − V (μ0 H, −J )]. Copyright (2019): Springer Nature

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2 Spin-Transfer Torques for Domain Wall Motion …

results shown in Fig. 2.6 can be understood in terms of Eqs. (2.13)–(2.15a), i.e., V (μ0 H, J ) = Vfield (μ0 H ) + V (J ). The dotted green line in Fig. 2.6 represents Vfield (μ0 H ), which increases linearly with an increase in magnetic field μ0 H in the precessional regime (see Sect. 2.1.2). Under the positive and negative bias currents, V (μ0 H, +J ) and V (μ0 H, −J ) are shifted from Vfield (μ0 H ) by V (J ) in the opposite direction. The double orange arrow in Fig. 2.6 represents the difference between V (μ0 H, +J ) and V (μ0 H, −J ), which is equal to 2V (J ). Note that, as mentioned already, Vfield (μ0 H ) and V (J ) can be extracted from V (μ0 H, +J ) and V (μ0 H, −J ) based on Eqs. (2.14) and (2.15a). Figure 2.7a shows Vfield (μ0 H = 85 mT) as a function of temperature T . It shows that Vfield reaches its maximum at 241 K regardless of the current density |J | = 1.3, 1.7, or 2.0 × 1010 A/m2 , indicating that TA = 241 K [15] for our device (Note that TM = 160 K for our device, see Sect. 2.7.1). Figure 2.7b shows V (J )/J as a

Fig. 2.7 Field and current contributions to domain wall velocity as functions of temperature T . a Vfield (μ0 H = 85 mT) and b V /J as functions of temperature under bias current density |J | = 1.3, 1.7, and 2.0 × 1010 A/m2 . The error bar of V was determined as the standard deviation of V values from a series of measurements under different magnetic fields (e.g., seven V values for the case of |J | = 1.3 × 1010 A/m2 and T = 211.6 K as shown in Fig. 2.6). The fitting result of V /J based on Eq. (2.15b) is indicated by the black line. c The non-adiabatic and adiabatic STT components, V N,STT /J (red line) and V A,STT /J (blue line), respectively, in V /J , which are calculated from the fitting result in Fig. 2.7b. Copyright (2019): Springer Nature

2.4 Experimental Results on Field-Driven Current-Assisted Domain Wall Motion

39

function of T . First, it should be noted that data points collapse onto a single curve in Fig. 2.7b regardless of their current densities. This is consistent with Eq. (2.15a), where V (J ) linearly depends on J . Second, it is found that V /J strongly depends on T ; (1) The sign of V /J is reversed in the vicinity of TA . In terms of Eq. (2.15a), this indicates that the sign of the adiabatic STT component is reversed at TA . (2) V /J is asymmetric with respect to TA . Based on Eq. (2.15a), this asymmetry implies that there is a finite non-adiabatic STT component in the vicinity of T A . To divide V /J quantitatively into the adiabatic and non-adiabatic STT components, V /J as a function of T is fitted by Eq. (2.15a). However, before the fitting is performed, we need to clarify the fitting conditions. The fitting equation is δs P αβs P V =− − , J δs2 + (αs)2 δs2 + (αs)2

(2.15b)

where T varies but is not included explicitly on the right-hand side of Eq. (2.15b). As will be explained in Sect. 2.7.2, s and δs are obtained independently as functions of T . Therefore, V /J can be fitted by Eq. (2.15b) by varying T and fitting parameters α, β, and P. Note that this fitting still requires the assumption that α, β, and P are constant regardless of T for the measured temperature range. This assumption is justified as follows: (1) Parameter α is considered to be insensitive to temperature across TA . Note that this statement is in contrast with a previous experimental report, where the effective Gilbert damping parameter (αeff ) of ferrimagnets, determined using the ferrimagnetic resonance (FiMR) linewidth, is strongly temperature dependent [42]. Parameter αeff increases remarkably as the temperature approaches TA and diverges at TA . However, a recent theoretical study has suggested that the increase in the FiMR linewidth in the vicinity of TA can originate from the change in the nature of the magnetic dynamics from ferromagnetic-like (significantly away from TA ) to antiferromagnetic-like (in the vicinity of TA ) rather from the increase in αeff [43]. In addition to Ref. [43], our group has reported experimental evidences that α of ferrimagnets is insensitive to temperature [44, 45], confirming the results of Ref. [43]. The details of these reports are provided in Chaps. 3 and 4. (2) It can be assumed that the sign of P does not change at TA . Spin polarization is given as   σ↑ − σ↓ , p= σ↑ + σ↓

(2.18)

where σ↑ (σ↓ ) represents the density of states of spin-up (spin-down) electrons at the Fermi level. Given that the 4f electron band, responsible for the magnetic moment of Gd, lies significantly lower (approximately 8 eV) than the Fermi

40

2 Spin-Transfer Torques for Domain Wall Motion …

level, σ↑ and σ↓ (thus p) are determined mostly by the 3d electron band, which contributes largely to the magnetic moment of FeCo. This characteristic of RETM ferrimagnets is justified by an earlier experiment [46] reporting that p for Co is four times larger than that for Gd at temperatures below 1 K, even though the magnetic moment of Gd is larger than that of Co at that temperature. However, TA of ferrimagnets is determined by integrating the net spin density over the whole Fermi sea (spin densities of both FeCo and Gd sublattices below the Fermi level). Therefore, given that a FeCo-rich ferrimagnet is used in this study, it is speculated that TA (determined using the net spin density over the whole Fermi sea) is not special for p (determined using only the spin density near the Fermi level). Therefore, p and P = p/2e cannot change signs at TA . (3) Similarly, it can be also assumed that the sign of β does not change at TA . Parameter β arises from the spin dissipation processes, which can be caused by several distinct microscopic Hamiltonian terms independent of the net spin density; examples include spin–orbit coupling and interactions with random magnetic impurities [28, 29, 32]. This suggests that the sign of β does not change at TA . Finally, it should be noted that even if α is insensitive to the temperature in the vicinity of TA , and β and P do not change signs at TA , they cannot necessarily be constant for a temperature range across TA . However, α, β, and P are assumed to be constant regardless of the temperature for simplicity (only for the measured temperature range). Therefore, they are treated as the fitting parameters in this analysis. Based on the preceding discussions, the data in Fig. 2.7b are fitted using Eq. (2.15b). The best fitting is represented by the black solid line in Fig. 2.7b, yielding the fitting parameters of α = (3.17 ± 0.09) × 10−3 , β = −0.53 ± 0.02, and p = 0.109 ± 0.002. fitting parameters, the adiabatic component   Based on these 2 2 VA,STT /J = −δs P/ δs + (αs) (blue curve) and the non-adiabatic component   2 VN,STT /J = −αβs P/ δs2 + (αs) (red curve) are recalculated as functions of T , as shown in Fig. 2.7c. It is evident that the temperature dependence of nonadiabatic component VN,STT /J is symmetric, whereas that of adiabatic component VA,STT /J is antisymmetric with respect to TA , as expected from Eq. (2.15a).

2.5 Discussion 2.5.1 Qualitative Explanation for Temperature Dependence of Spin-Transfer Torques From the aforementioned experimental results, it is confirmed that the adiabatic STT component of the DW velocity exhibits an antisymmetric temperature dependence,

2.5 Discussion

41

Fig. 2.8 Schematic of adiabatic and non-adiabatic spin-transfer torques in a ferromagnets and b ferrimagnets

whereas the non-adiabatic STT component exhibits a symmetric temperature dependence with respect to TA . The difference in the temperature dependences of the adiabatic and non-adiabatic STTs is discussed in detail below. First, we provide a qualitative understanding of Eq. (2.15a), where the adiabatic STT component is proportional to the net spin density, δs , whereas the non-adiabatic STT component is proportional to the saturated spin density, s. A schematic of STTs in a ferromagnet is depicted in Fig. 2.8a. When a conduction electron comes in the vicinity of a local magnetic moment, it exerts two types of torques orthogonal to each other: adiabatic torque τA and non-adiabatic torque τNA on the angular momentum, s, of the local moment. Parameter τA is simply understood as the angular momentum transfer due to the exchange interaction. It aligns s parallel to the spin of the conduction electron, σ. Therefore, τA has the same symmetry as the damping torque. However, τNA is orthogonal to τA and has the same symmetry as the precession torque. Figure 2.8b depicts the schematic of STTs in a ferrimagnet. The two sublattice angular momenta of the ferrimagnet are denoted s1 and s2 . The non-adiabatic torques, and τNA denoted τNA 1 2 , rotate s1 and s2 , respectively, in the same direction,. This indicates that the non-adiabatic torque is additive for s1 and s2 and depends on the saturated spin density, s = s1 + s2 . However, the adiabatic torques, denoted as A τN 1 and τ2 , align s1 and s2 , respectively, to the same direction of σ. Given that s1 and s2 are coupled antiferromagnetically, this indicates that the adiabatic torque is rather subtractive for s1 and s2 and depends on the net spin density, δs = s1 − s2 . This explanation qualitatively indicates that the non-adiabatic torque depends on s, whereas the adiabatic torque depends on δs . However, this explanation is not enough to understand the symmetric (antisymmetric) temperature dependence of the non-adiabatic (adiabatic) STT; the clear peaks in the magnitudes of STTs in the vicinity of TA are still not understood. As for the non-adiabatic STT, providing a simple explanation for the VN,STT /J peak at TA is

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2 Spin-Transfer Torques for Domain Wall Motion …

rather difficult because the microscopic mechanism of the non-adiabatic STT is not yet fully understood. As for the adiabatic STT, the maximum (the minimum) point of VA,STT /J just below (above) TA can be explained in terms of the conservation of angular momentum. It is well known that the DW motion driven by the adiabatic STT can be understood based on the conservation of angular momentum. The adiabatic STT-driven DW motion is understood as follows: while the conduction electrons pass through the DW, they adiabatically transfer their angular momentum to the local magnetic moments. The local magnetic moments are rotated so that DW is shifted (see Fig. 1.2). The angular momentum lost by the conduction electrons during the passage through the DW is calculated as the difference between the spin angular momenta of conduction electrons in the up and down domains:  J × p× , e 2

(2.19)

where J is the current density, e is the electron charge, p is the spin polarization, and /2 is the angular momentum of an electron. The angular momentum that the up domain acquires per unit time is given by V × S,

(2.20)

where V is the DW velocity and S is the spin density of the ferromagnetic material. In terms of the conservation of angular momentum, these values are identical: J  × p × = V × S, e 2

(2.21)

which yields V =

(/2) p J . eS

(2.22)

Note that this expression is identical to the aforementioned DW velocity with full spin-transfer, u = μB p J/eMS (see Sect. 1.1.2.2) [14, 47, 48]. It can be understood easily that Eq. (2.22) holds for the ferrimagnetic DW motion given that S is replaced by the net spin density, δs . Therefore, as the temperature approaches TA , δs → 0 and V diverges, that is, the DW velocity driven by the adiabatic STT diverges at TA . However, as mentioned already, the adiabatic STT is subtractive for the two sublattices of ferrimagnets and is thus zero at TA . The product of these effects results in an antisymmetric temperature dependence with respect to TA for VA,STT /J .

2.5 Discussion

43

2.5.2 Significantly Large Ratio of |β|/α From the fitting results, |β|/α of the order of 100 is obtained. In most previous studies on ferromagnetic DWs [26, 33, 35], |β|/α ranges from ~ 1 to ~ 3. Thus, |β|/α obtained in this experiment is extremely larger than those values from the previous reports. Extremely large |β|/α makes an impact on device application and fundamental physics; (1) As the DW velocity driven by the non-adiabatic STT is proportional to |β|/α, this result suggests that efficient manipulation of antiferromagnetic DWs by the non-adiabatic STT is possible. (2) In addition to the merits for application, the large ratio |β|/α has an important physical meaning. For ferromagnetic DWs, there has been longstanding debate about the ratio |β|/α [23–35]. This debate is related to the underlying spin relaxation processes for α and β. In this study, it is speculated that the large |β|/α observed in this ferrimagnetic GdFeCo sample originates from the spin mistracking process with the small effective exchange interaction, which has been predicted to yield a large increase in β [27, 29]. In RE-TM ferrimagnets, the effective exchange averaged over two sublattices is smaller than that of typical ferromagnetic DWs because of the antiferromagnetic alignment of RE and TM moments. Another interesting observation is the negative value of β in GdFeCo, which is critically different from most, if not all [49], previous reports so far. To clarify the origin of negative β is beyond the scope of this thesis, but we speculate that this negative β might be related to the electron band structure of GdFeCo because one theory [32] has predicted that β can be negative in systems with both holelike and electronlike carriers.

2.6 Conclusion The STT effects on DW motion in antiferromagnetically-coupled ferrimagnets are clarified theoretically and experimentally. Theoretical investigations suggest that the adiabatic and non-adiabatic STT components of the DW velocity in RE-TM ferrimagnets exhibit different dependences on net spin density. Given that the net spin density in RE-TM ferrimagnets can be tuned by controlling temperature, each component in the DW velocity exhibits different dependence on temperature, enabling us to extract each component separately. The experimental results on the field-driven STTassisted DW motion are well fitted by the theory, which shows that the adiabatic (nonadiabatic) STT component in the DW velocity exhibits an antisymmetric (symmetric) temperature dependence with respect to TA . This confirms the theoretical prediction that the non-adiabatic STT in antiferromagnets acts like a staggered magnetic field, which can be used for the efficient control of antiferromagnetic DWs. In addition, the fitting results indicate that the ratio of |β|/α is significantly large compared with those values reported for ferromagnets, which can lead to a fast current-induced antiferromagnetic DW motion. This study demonstrates that ferrimagnets are useful

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2 Spin-Transfer Torques for Domain Wall Motion …

for investigating the current-driven magnetic dynamics of antiferromagneticallycoupled systems and suggests further theoretical and experimental studies on STT in inhomogeneous antiferromagnetic spin textures [50].

2.7 Supplementary Information 2.7.1 Magnetization Compensation Temperature of GdFeCo Wire The magnetization compensation temperature, TM , of the GdFeCo wire can be determined by measuring the anomalous Hall effect (AHE) resistance, RH [15, 51]. AHE is the electron scattering that depends on the direction of the local magnetic moment. It is attributed to the electrons at the Fermi level. Therefore, for the RE-TM alloy ferrimagnets, the transverse voltage due to AHE is originated solely from TM magnetization, without the contribution from the RE magnetization; 3d electrons responsible for the magnetization of TM elements are located at the Fermi level, whereas 4f electrons that generate the magnetization of RE elements are significantly below the Fermi level. Figure 2.9a depicts the schematic of the AHE measurement for the ferrimagnetic GdFeCo wire. As the GdFeCo wire exhibits PMA, the transverse voltage, Rxy , of the GdFeCo wire shows a hysteresis loop by sweeping the external magnetic field in the out-of-plane direction (see Fig. 2.9b). When the magnetic field is swept from the positive (+z) to the negative (−z) direction, denoted by the orange dots, Rxy changes its sign suddenly at the negative magnetic field (but the absolute value of the voltage remains the same). However, when the magnetic field is swept from the negative (−z) to the positive (+z) direction, denoted by the green dots, Rxy changes its sign suddenly at the positive magnetic field. The magnetic fields where the transverse voltages abruptly change are the coercive field, μ0 HC , at which the whole magnetizations of the GdFeCo wire are reversed simultaneously. Figure 2.9b also shows the hysteresis loops at temperatures ranging from 300 to 10 K. First, it is evident that μ0 HC increases monotonically as the temperature approaches 140 K. Figure 2.9c shows μ0 HC as a function of T . It is evident that μ0 HC increases remarkably at approximately 140 K. This is a well-known behavior of ferrimagnets; a small net magnetization in the vicinity of TM requires a large magnetic field to obtain a large Zeeman energy enough to overcome the magnetic anisotropy energy. Therefore, Fig. 2.9c indicates that the TM of this GdFeCo wire is approximately 140 K. Second, by comparing the hysteresis loops at the temperature ranges below and above 140 K, we can notice that the direction of the hysteresis loops for T < 140 K is reversed from that for T > 140 K. Figure 2.9d shows the transverse resistance difference, RH , as a function of T . Despite the sign change of RH at 140 K, its absolute value (open circles in Fig. 2.9d) does not vary significantly with respect to T across

2.7 Supplementary Information

45

Fig. 2.9 Determination of magnetization compensation temperature. a Schematic of GdFeCo wire. b Hysteresis loops (transverse voltage as a function of out-of-plane magnetic field) for temperatures ranging from 300 to 10 K. Orange points represent the measurements where the magnetic field is swept from the positive (+z) to the negative (−z) direction, whereas the green points represent the cases where the magnetic field is swept from the negative (−z) to the positive (+z) direction. c Coercive field, μ0 HC , as a function of T . d Transverse resistance difference, RH , as a function of T . Reprinted figures with permission from T. Okuno et al., Applied Physics Express 9, 073,001 (2016). Copyright (2016) The Japan Society of Applied Physics

140 K. This trend is understood as follows: the 4f shell, which is responsible for the magnetic properties of the rare-earth metal, is located significantly below the Fermi energy level. Therefore, in the RE-TM ferrimagnet GdFeCo, the magneto-transport properties, including AHE, are governed by the TM (FeCo) moments rather than by both the RE (Gd) and TM moments. In terms of energy stability, the net magnetization is always parallel to the magnetic field. Therefore, the above trend indicates that the FeCo moments are aligned parallel (antiparallel) to the net magnetization for T > 140 K (T < 140 K), providing the evidence that TM ≈ 140 K. Both the divergence of μ0 HC and the sign change of RH can be the probes for determining TM . In this study, the TM of the GdFeCo wire used for the DW motion experiment was determined using the sign change of RH . Figure 2.10 shows the transverse resistance difference, R H ≡ RH (+μ0 H sat ) − RH (−μ0 Hsat ), where μ0 Hsat

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2 Spin-Transfer Torques for Domain Wall Motion …

6

Fig. 2.10 Anomalous Hall resistance as a function of temperature. Copyright (2019): Springer Nature

3

0 RH [Ω]

ΔRH [Ω]

3

-3 -6

0

-3 -0.4

100

150

200 T [K]

ΔRH

0.0

μ0H [T]

250

0.4

300

is the saturation field (see the inset to Fig. 2.10). It is evident that the sign of R H is reversed at 160 K, which suggests that TM = 160 K for the GdFeCo wire.

2.7.2 Estimation of the Temperature Dependence of stotal and δ s In the fitting process of V /J using Eq. (2.15b), the saturated spin density s and the net spin density δs are represented as functions of temperature. This section deals with how to obtain s(T ) and δs (T ). First, it is assumed that the magnetization of each sublattice can be expressed by the power-law criticality [15, 52] as MGd (T ) = MGd (0)(1 − T /TC )βGd ,

(2.23a)

MFeCo (T ) = MFeCo (0)(1 − T /TC )βFeCo ,

(2.23b)

where MGd (0) (MFeCo (0)) is the saturation magnetization of the Gd (FeCo) sublattice at 0 K, βGd (βFeCo ) is the critical exponent of the Gd (FeCo) sublattice, and TC is the Curie temperature of GdFeCo. Then the net magnetization of GdFeCo is expressed as Mnet

    T βFeCo T βGd = MFeCo (T ) − MGd (T ) = MFeCo (0) 1 − − MGd (0) 1 − . TC TC (2.24)

According to the previous papers on GdFeCo, 0.65 < βGd < 0.7 and 0.45 < βFeCo < 0.5 [53, 54]. Therefore, here βFeCo = 0.475 and βGd = 0.675 are assumed for simplicity. In addition, based on the power-law criticality, there is a correlation

(a)

47

(b)

Angular momentum density [×10-6 J⋅s/m3]

2.7 Supplementary Information

s = sFeCo+sGd

8

δs = sFeCo-sGd

6 4 2 0 0

100

200 300 T [K]

400

Fig. 2.11 Estimation of the temperature dependence of s and δs . a The net magnetization of the GdFeCo film Mnet as a function of temperature T measured by the superconducting quantum interference device (SQUID) (black points), and the fitting result by the power-law criticality (red curve). The blue and green curves indicate the magnetizations of the FeCo (MFeCo ) and Gd (MGd ) sublattices, respectively. MFeCo and MGd are calculated from the fitting. b The saturated spin density s and the equilibrium net spin density δs as a function of T , which are calculated from figure (a). Copyright (2019): Springer Nature

between TM , TA , and the Curie temperature TC : TA /TC = TM /TC +η, where η = 0.19 for the case of GdFeCo [52]. Substituting TA and TM of our GdFeCo film, TC = 394 K is obtained. Based on these assumptions, the net magnetization Mnet is fitted as a function of temperature with Eq. (2.24) and obtains MFeCo (0) = 7.09 × 105 and MGd (0) = 8.00 × 105 A/m as the fitting parameters. The fitting result is shown in Fig. 2.11a. Therefore, MFeCo (T ) and MGd (T ) are expressed as the one-variable function of T . s and δs are obtained by s = sFeCo + sGd = MFeCo /γFeCo + MGd /γGd and δs = sFeCo − sGd = MFeCo /γFeCo − MGd /γGd , respectively. Here, γFeCo (γGd ) is the gyromagnetic ratio of FeCo (Gd), which is expressed as γi = gi μB /, where gi is the Landé g-factor of the sublattice i (i = FeCo or Gd), μB is the Bohr magneton, and  is the reduced Planck’s constant. By using gFeCo ∼ 2.2 and gGd ∼ 2.0 [55– 57], we obtain s(T ) and δs (T ) as the one-variable function of T , which are shown in Fig. 2.11b. Then, by substituting s(T ) and δs (T ) into Eq. (2.15b), Eq. (2.15b) can be expressed as the one-variable function of T with the three parameters of the Gilbert damping parameter α, the non-adiabaticity β, and the spin conversion factor P. Therefore, those data in Fig. 2.7b can be fitted by Eq. (2.15b).

References 1. Landau L et al (1935) Physik Z Sowjetunion 8:153 2. Williams HJ et al (1950) Phys Rev 80:1090 3. Bean CP et al (1955) J Appl Phys 26:124

48 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57.

2 Spin-Transfer Torques for Domain Wall Motion … Rodbell DS et al (1956) Phys Rev 103:886 Ford NL (1960) J Appl Phys 31:S300 Patton CE et al (1966) J Appl Phys 37:4269 Konishi S et al (1971) IEEE Trans Magn 7:722 Schryer NL et al (1974) J Appl Phys 45:5406 Slonczewski JC (1972) Int J Magn 2:85 Slonczewski JC (1974) J Appl Phys 45:2705 Ono T et al (1999) Science 284:468 Atkinson D et al (2003) Nat Mater 2:85 Beach GSD et al (2005) Nat Mater 4:741 Mougin A et al (2007) Europhys Lett 78:57007 Kim K-J et al (2017) Nat Mater 16:1187 Siddiqui SA et al (2018) Phys Rev Lett 121:057701 Caretta L et al (2018) Nat Nanotechnol 13:1154 Xu Y et al (2008) Phys Rev Lett 100:226602 Swaving AC et al (2011) Phys Rev B 83:054428 Hals KMD et al (2011) Phys Rev Lett 106:107206 Tveten EG et al (2013) Phys Rev Lett 110:127208 Yamane Y et al (2016) Phys Rev B 94:054409 Tatara G et al (2004) Phys Rev Lett 92:086601 Zhang S et al (2004) Phys Rev Lett 93:127204 Thiaville A et al (2005) Europhys Lett 69:990 Hayashi M et al (2006) Phys Rev Lett 96:197207 Xiao J et al (2006) Phys Rev B 73:054428 Tserkovnyak Y et al (2006) Phys Rev B 74:144405 Tatara G et al (2007) J Phys Soc Jpn 76:54707 Tserkovnyak Y et al (2008) J Magn Magn Mater 320:1282 Boulle O et al (2008) Phys Rev Lett 101:216601 Garate I et al (2009) Phys Rev B 79:104416 Burrowes C et al (2009) Nat Phys 6:17 Gilmore K et al (2011) Phys Rev B 84:224412 Sekiguchi K et al (2012) Phys Rev Lett 108:017203 Kim SK et al (2017) Phys Rev B 95:140404(R) Tretiakov OA et al (2008) Phys Rev Lett 100:127204 Yoshimura Y et al (2015) Nat Phys 12:157 Haazen PPJ et al (2013) Nat Mater 12:299 Emori S et al (2013) Nat Mater 12:611 Ryu K-S et al (2013) Nat Nanotechnol 8:527 Stanciu CD et al (2006) Phys Rev B 73:220402(R) Kamra A et al (2018) Phys Rev B 98:184402 Kim D-H et al (2019) Phys Rev Lett 112:127203 Okuno T et al (2019) Appl Phys Express 12:093001 Kaiser C et al (2005) Phys Rev Lett 95:047202 Zhang S et al (2004) Phys Rev Lett 93:127204 Koyama T et al (2012) Nat Nanotechnol 7:635 Thevenard L et al (2017) Phys Rev B 95:054422 Okuno T et al (2019) Nat Electron 2:389 Okuno T et al (2016) Appl Phys Express 9:073001 Hirata Y et al (2018) Phys Rev B 97:220403(R) Ostler TA et al (2011) Phys Rev B 84:024407 Kirilyuk A et al (2013) Rep Prog Phys 76:026501 Kittel C (1949) Phys Rev 76:743 Scott GG (1962) Rev Mod Phys 34:102 Min BI et al (1991) J Phys Condens Matter 3:5131

Chapter 3

Gilbert Damping Parameter of Ferrimagnets Probed by Domain Wall Motion

Abstract The Gilbert damping parameter α represents the rate of energy dissipation associated with magnetic dynamics, characterizing the relaxation of magnetic dynamics to its equilibrium state. In Chap. 2, it is assumed that α of ferrimagnetic GdFeCo is constant in the temperature range across TA . However, this assumption of constant α is in contrast with earlier experiments reporting that the effective α of ferrimagnets, estimated from the FiMR linewidth, increases remarkably as the temperature approaches TA and diverges at TA . Recently, a theoretical study by Kamra et al. has suggested that the drastic increase in the FiMR linewidth in the vicinity of TA can be attributed to the change in the nature of the magnetic dynamics from ferromagnetic (significantly away from TA ) to antiferromagnetic (at TA ) rather than the increase in the effective α. In Chap. 3, α of ferrimagnetic GdFeCo is investigated using the field-driven DW mobility extracted from the DW motion experiment in Chap. 2. It is confirmed that α of ferrimagnets is almost insensitive to temperature across TA , supporting the validity of the theoretical study by Kamra et al. and the analysis in Chap. 2. Keywords Rayleigh dissipation function · Gilbert damping parameter · Magnetic domain wall mobility

3.1 Physical Background First, the physical background, on which the study explained in Chaps. 3 and 4 is based, is introduced, namely, the meaning of α in magnetic dynamics [1]. As stated in Chap. 2, the dynamics of magnetization is described phenomenologically using the LLG equation. This equation was first derived by Landau and Lifshitz in 1935 [2] and was modified by Gilbert in 1955 [3]. The LLG equation describes the dynamics of M located in H (Fig. 3.1): ˙ ˙ = −γ M × H + α M × M, M M

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 T. Okuno, Magnetic Dynamics in Antiferromagnetically-Coupled Ferrimagnets, Springer Theses, https://doi.org/10.1007/978-981-15-9176-1_3

(3.1)

49

50

3 Gilbert Damping Parameter of Ferrimagnets Probed by Domain Wall Motion

Fig. 3.1 Schematic of the Landau–Lifshitz–Gibert (LLG) equation

˙ = ∂M/∂t is the first-order time derivative of M, γ is the gyromagnetic where M ratio, and M is the magnitude of M. Here, M × H represents the precession torque ˙ represents the damping torque in the direction of H. In the around H and M × M LLG equation, α is considered to be the rate of the parallel alignment of M to H. To obtain an insight into α, the LLG equation is derived within the classical mechanics [1]. Within the Lagrangian mechanics, the equation of motion of an object is represented as follows: ˙ ∂ L[q, q] ˙ d ∂ L[q, q] − = 0, dt ∂ q˙ ∂q

(3.2)

where Lagrangian L is a function of generalized coordinate q and its time derivative q. ˙ Equation (3.2) describes the case when the energy is conserved within a system; thus, the dynamics of the object is determined by the kinetic and potential energies. Next, the energy dissipation from the system is considered. A term representing energy dissipation needs to be added to Eq. (3.2) as follows: ˙ ∂ L[q, q] ˙ δ R[q] ˙ d ∂ L[q, q] − + = 0, dt ∂ q˙ ∂q ∂ q˙

(3.3)

where R[q] ˙ is the Rayleigh dissipation function, which is given as. R[q] ˙ =

η 2 q˙ . 2

(3.4)

η is the viscosity coefficient for the motion of the object. Parameter R is the energy dissipation per unit time from the system. Therefore, η is assumed to be the “damping parameter” for the motion of the object.

3.1 Physical Background

51

For α in magnetic dynamics, we can adopt the analogy of energy dissipation described in Eqs. (3.3) and (3.4). By replacing q in Eq. (3.3) with M, the equation of motion of M is represented as       ˙ ˙ ˙ δR M ∂ L M, M d ∂ L M, M + − = 0, ˙ ˙ dt ∂M ∂M ∂M

(3.5)

where   η ˙ R M(r, t) = 2



˙ ˙ M(r, t) • M(r, t)dr.

(3.6)

As mentioned above, the first and second terms on the left side of Eq. (3.5) represent the dynamics of M under energy conservation, whereas the third term represents energy dissipation from the system. Note that Eq. (3.5) is equivalent to the LLG equation (Eq. 3.1); the first and second terms are equivalent to the precession ˙ = −γ M × H), and the third term is equivalent to the damping torque torque (M ˙ Therefore, by comparing the damping torque (energy dissipation) ((α/M)M × M). term, we can obtain the following correlation between R and α: R=

α/γ M 2



αS ˙ ˙ M(r, t) · M(r, t)dr = 2

 ˙ t) · n(r, ˙ t)dr, n(r,

(3.7)

where n = S/S is the unit vector along S of magnetization M(= −γ S). Because R is energy dissipation per unit time from the system, it can be concluded that α indicates energy dissipation per unit angular momentum. Significantly, R is a physical quantity that represents the energy dissipation from the system; α is a dimensionless coefficient that represents energy dissipation per unit angular momentum. The focus of the discussions in Chaps. 3 and 4 is the definition of R in ferrimagnets. As illustrated in Fig. 3.2a, the conventional understanding of ferrimagnetic dynamics is based on the approximation where the dynamics of antiferromagnetically-coupled ferrimagnetic moments, sTM and sRE , is described using the single moment of the net angular momentum, δs = sTM − sRE . This approximation is based on the fact that the precession of the magnetic moment is governed by δs . In this approximation, energy dissipation is also attributed to δs (the wavy arrow in Fig. 3.2a). Therefore, R=

αeff |δs | 2

 n˙ 2 d V,

(3.8)

where αeff is the effective Gilbert damping parameter [4, 5]. Because R is a physical quantity that represents the energy dissipation rate from the system, it is always positive and finite. Therefore, at TA of ferrimagnets, where δs = 0, αeff diverges. In

52

3 Gilbert Damping Parameter of Ferrimagnets Probed by Domain Wall Motion

Fig. 3.2 a Schematic of the conventional understanding of ferrimagnetic dynamics. The dynamics of antiferromagnetically coupled moments sTM and sRE is approximated as that of the single magnetic moment, δs , and energy is dissipated from δs . b Schematic of the correct understanding of ferrimagnetic dynamics. Energy is dissipated from both sTM and sRE

other words, provided that R is positive and finite for all temperatures, αeff changes remarkably as δs changes with temperature and is ill defined physically at TA . However, the above definition of R ∝ δs αeff does not reflect the actual ferrimagnetic dynamics because energy dissipation in ferrimagnetic dynamics occurs from both sTM and sRE , as illustrated in Fig. 3.2b (the wavy arrows in Fig. 3.2b). Therefore, based on the model illustrated in Fig. 3.2b, R is calculated as R=

αs 2

 n˙ 2 d V,

(3.9)

where s = sTM +sRE is the total angular momentum. In terms of the definition of R in Eq. (3.9), α is always well defined and does not necessarily change with temperature because the temperature dependence of stotal is more moderate than that of δs . Note that α is not necessarily constant with respect to temperature. Yet the temperature dependence of α could be more moderate than that of αeff . In spite of the importance of α in ferrimagnetic dynamics, previous studies have focused only on αeff [4, 5], and the temperature dependence of α in ferrimagents has not been fully clarified. The objective of the discussions in Chaps. 3 and 4 is to verify the temperature dependence of α in ferrimagnets and confirm the validity of the model in Fig. 3.2b and the definition of R in Eq. (3.9). The LLG equation corresponding to the model in Fig. 3.2a is as follows [4] (see the conventional LLG equation Eq. (3.1)): δs n˙ + αδs n × n˙ = n × fn .

(3.10)

However, the LLG equation corresponding to the model in Fig. 3.2b is as follows [6–10] (see Eq. (4.1)): δs n˙ + αsn × n˙ + ρn × n¨ = n × fn ,

(3.11)

3.1 Physical Background

53

where unit vector n represents the order parameter of ferrimagnets directed the same as the spin angular momentum of FeCo lattice atom; ρ > 0 is the moment of inertia for the dynamics (which is inversely proportional to the microscopic exchange field between the two sublattices and describes the antiferromagnetic dynamics of the magnet); fn ≡ −∂U/∂n. Note that Eqs. (2.9) and (3.11) are essentially the same. However, Eq. (3.11) is different from Eq. (2.9) in that (1) the adiabatic and nonadiabatic STTs are ignored (because of the absence of STT effects), and (2) the variable in the LLG equation is n (the unit vector along the angular momentum) instead of m (the unit vector along the angular momentum; n = −m). Next, Eq. (3.10) is compared with Eq. (3.11). The difference between Eqs. (3.10) and (3.11) is that (1) the damping term in Eq. (3.10) includes δs , whereas that in Eq. (3.11) includes s. This difference reflects the difference in the model illustrated ¨ whereas Eq. (3.10) does not. Term in Figs. 3.2a, b. (2) Eq. (3.11) includes term ρn× n, ρn × n¨ originates from the antiferromagnetic dynamics of the antiferromagneticallycoupled ferrimagnetic moments [11]. Therefore, the absence of ρn × n¨ in Eq. (3.10) indicates that Eq. (3.10) does not reflect the antiferromagnetic nature of ferrimagnetic dynamics. In Chaps. 3 and 4, experimental data are analyzed based on Eq. (3.11) to obtain α, so that the validities of Eq. (3.11) and the model in Fig. 3.2b are tested.

3.2 Theoretical Background for Determining the Gilbert Damping Parameter Parameter α can be obtained from the DW velocity experiment in Chap. 2 as follows. As described in Eq. (2.4), the field-driven DW velocity of ferrimagnets in the precessional regime is given by [9, 12] Vfield (μ0 H ) =

αsλMμ0 H δs 2 + (αs)

2

,

(3.12)

where Vfield is the field-driven DW velocity; M and δs are the magnitudes of the net magnetization and net spin density, respectively; s is the saturated spin density (i.e., the sum of the spin densities of two sublattices); and μ0 H is the perpendicular magnetic field. The field-driven DW mobility μ is obtained based on Eq. (3.12) as αsλM ∂ Vfield = , 2 2 ∂(μ0 H ) δs + (αs)

(3.13)

μs 2 α 2 − λs Mα + μδs 2 = 0.

(3.14)

μ=

which is rearranged as

54

3 Gilbert Damping Parameter of Ferrimagnets Probed by Domain Wall Motion

Based on Eq. (3.14), the solutions to α are obtained as α± =

λM ±

 λ2 M 2 − 4μ2 δs 2 . 2μs

(3.15)

Equation (3.15) can be used to determine α from the given μ. Note that α can have two values, α+ and α− , for each value of μ because Eq. (3.13) is a quadratic function of α. Only one of these two solutions is physically valid, which can be determined based on the following discussion on energy dissipation. Parameter R for the DW motion in ferrimagnets is given by [9, 12] R=

αs A 2 V + αs Aλ 2 , λ

(3.16)

where A is the cross section of the DW, V is the DW translational velocity, and is the angular velocity of DW. The first term on the right-hand side of Eq. (3.16), (αs A/λ)V 2 , represents the energy dissipation associated with the translational motion of DW, and the second term on the right-hand side, αs Aλ 2 , represents the energy dissipation associated with the precessional (angular) motion of magnetic moments inside DW. As is explained in Sect. 3.1, energy dissipation of a moving object is given as the square of the velocity multiplied by η. For the case of DW motion, V and are coupled as δs V . αsλ

(3.17)

  δs 2 1 2A 1 αs + V 2 = ηV 2 . 2 λ αs 2

(3.18)

= Therefore, R is obtained as R=

Here, η for the DW motion is   δs 2 2A αs + . η= λ αs

(3.19)

The first and the second terms within the parenthesis in Eq. (3.19) represent the energies dissipated from the translational and angular motions of the DW, respectively. As is evident from Eq. (3.19), the energies dissipated from the translational and angular motions of DW depend on δs and s (thus on temperature) in ferrimagnets. Therefore, the quantitative relationship between these two energy dissipations varies with δs and s (as well as temperature). (1) When the energies dissipated from the translational and angular motions of DW are quantitatively equal, the following

3.2 Theoretical Background for Determining the Gilbert Damping Parameter

55

equation holds: αs =

δs 2 , αs

(3.20)

which yields α = δs /s. Note that, in this case, Eq. (3.20) has repeated root, α+ = α− = δs /s. (2) In the vicinity of TA , the precessional motion of DW is reduced, whereas the DW velocity is enhanced. For this case, it is assumed that energy dissipation from the translational motion of DW is larger than that from the angular motion of DW [9]. Therefore, the following equation holds: αs >

δs 2 , αs

(3.21)

which yields α > δs /s. (3) Away from TA , the precessional motion of DW is enhanced, whereas the DW velocity is reduced. For this case, it is speculated that the energy dissipated from the translational motion of DW is smaller than that from the angular motion of DW [9]. Thus αs
δs /s) is determined to be the correct α. For T < T1 and T > T2 (the blue area in Fig. 3.4a), i.e., significantly away from TA , the energy dissipated from the angular motion of the DW is larger than that from the translational DW motion; thus α− (< δs /s) is determined to be the correct α. Based on this, the correct α is determined out of α+ and α− as shown in Fig. 3.4b. Interestingly, even though the determined α is not perfectly constant with respect to temperature, we can conclude that α is insensitive to temperature without showing the remarkable increase as temperature approaches TA and divergence at TA . In addition, by averaging α over a wide range of temperature (200K < T < 300K ) in Fig. 3.4b, α ∼ 8.0 × 10−3 (see the dotted line in Fig. 3.4b) in ferrimagnetic GdFeCo. This result is in stark contrast with the previous reports [4, 5]. Stanciu et al. investigated the temperature dependence of FiMR and concluded that, based on the ferromagnetic-like description of FiMR, αeff increased remarkably as the temperature

58

3 Gilbert Damping Parameter of Ferrimagnets Probed by Domain Wall Motion

approached TA and diverged at TA . However, our theory for field-driven ferrimagnetic DW motion can describe both the antiferromagnetic-like dynamics in the vicinity of TA and the ferromagnetic-like dynamics away from TA [9]. Therefore, the unphysical singularity, i.e., the divergence of αeff at TA , is absent in our analysis and the welldefined α can be obtained for all the temperatures, including TA . This result is restated as follows: α defined based on R (as illustrated in Fig. 3.2b), which is the rate of energy dissipation in the magnetic dynamics, is insensitive to temperature and δs (which governs the magnetic dynamics) [15]. For ferromagnets, s = δs holds. The angular momentum governing the magnetic precession and the angular momentum contributing to energy dissipation are equal. Thus, α in the LLG equation represents directly the rate of energy dissipation. However, for ferrimagnets, the magnetic dynamics is governed by δs , whereas energy dissipation is attributed to s. Therefore, αeff defined in the ferromagnetic-like LLG equation [4, 5] fails to describe the rate of energy dissipation in ferrimagnets and has the singular point at TA . This result indicates that α defined based on R provides the correct understanding of the ferrimagnetic dynamics and damping phenomenon.

3.4 Conclusion In conclusion, α of a ferrimagnetic GdFeCo alloy, defined based on R, was determined from the field-driven magnetic DW motion experiment. Parameter α was found to be insensitive to temperature (∼8.0 × 10−3 ) without showing divergence at T A . This result is in contrast with the results reported previously, where α eff diverged at T A . This indicates that α defined based on R represents correctly the rate of energy dissipation associated with the magnetic dynamics, confirming the validity of the model where energy dissipation occurs from both the antiferromagnetically-coupled ferrimagnetic moments [16].

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Gilbert TL (2004) IEEE Trans Magn 40:3443 Landau LD et al (1935) Phys Z Sowjetunion 8:153 Gilbert TL (1955) Phys Rev 100:1243 Stanciu CD et al (2006) Phys Rev B 73:220402(R) Binder M et al (2006) Phys Rev B 74:134404 lvanov BA et al (1983) Sov Phys JETP 57, 214 (1983) Chiolero A et al (1997) Phys Rev B 56:738 Kim SK et al (2017a) Phys Rev B 95:140404(R) Kim K-J et al (2017b) Nat Mater 16:1187 Oh S-H et al (2017a) Phys. Rev. B 96:100407(R) Hals KMD et al (2011) Phys Rev Lett 106:107206 Oh S-H et al (2017b) Phys Rev B 96:100407(R) Volkov VV et al (2008) Phys Solid State 50:199

References 14. Ono T et al (1999) Science 284:468 15. Kamra A et al (2018) Phys Rev B 98:184402 16. Kim D-H et al (2019) Phys Rev Lett 122:127203

59

Chapter 4

Gilbert Damping Parameter of Ferrimagnets Probed by Magnetic Resonance

Abstract In Chap. 3, the Gilbert damping parameter α is obtained from the fielddriven domain wall mobility and is found to be temperature-insensitive across the angular momentum compensation temperature, TA . Here, it should be noted that the most common experimental method to determine α is the ferromagnetic resonance. The previous study on the ferrimagnetic resonance (FiMR) concluded that the effective Gilbert damping parameter αeff significantly increases as the temperature approaches TA and diverges at TA . However, a recent theoretical study has provided a new perspective on α of ferrimagnets; the temperature dependence of FiMR is attributed to that of magnetic dynamics, not to that of α. This chapter presents a macroscopic theory and experimental results on FiMR, providing an additional evidence of temperature-insensitive α of ferrimagnets. The theory, which considers both the antiferromagnetic-like and ferromagnetic-like dynamics, can describe FiMR in a wide temperature range across TA without divergence of α. The experiment on FiMR excited by spin–orbit torque is performed, and the spectral analysis based on the theory reveals that α is insensitive to temperature. Keywords Ferrimagnetic resonance · Spin-torque ferromagnetic resonance · Gilbert damping parameter

4.1 Derivation of Equation for Magnetic Resonance in Ferrimagnets 4.1.1 Landau–Lifshitz–Gilbert Equation for Experimental Condition In Sect. 4.1, the equations for FiMR in a ferrimagnet consisting of two antiferromagnetically-coupled sublattices are derived. Throughout this chapter, it is assumed that the temperature range of interest is away from the magnetization compensation temperature, TM , so that the magnetization is finite and well defined.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 T. Okuno, Magnetic Dynamics in Antiferromagnetically-Coupled Ferrimagnets, Springer Theses, https://doi.org/10.1007/978-981-15-9176-1_4

61

62

4 Gilbert Damping Parameter of Ferrimagnets Probed …

The derivation starts with the Landau–Lifshitz–Gilbert (LLG) equation for ferrimagnets [1–5]: δs n˙ + αsn × n˙ + ρn × n¨ = n × fn ,

(4.1)

where the unit vector n represents the order parameter of ferrimagnets which is directed same as the spin angular momentum of FeCo lattice atom; δs = s1 − s2 is the net spin density of two sublattices s1 > 0 and s2 > 0; α > 0 is the Gilbert damping parameter, s = s1 +s2 is the sum of the magnitudes of the two spin densities, ρ > 0 is the moment of inertia for the dynamics (which is inversely proportional to the microscopic exchange field between the two sublattices and describes the antiferromagnetic dynamics of the magnet); fn ≡ −∂U/∂n. The energy U of this system is expressed as U = −K n x 2 − Ml Hext · n,

(4.2)

where Ml is the longitudinal magnetization with respect to the order parameter. The anisotropy coefficient is positive K > 0 so that the ferrimagnet has PMA along the x-axis. Note that Eq. (4.1) is expressed with the variable of the order parameter n, instead of the magnetic moment m as in Eq. (2.9); As n = −m, this change in the variable necessitates no significant modifications in the LLG equation. ˙ represents the magnetic Here, the second term on the left side of Eq. (4.1), αsn× n, damping. As is mentioned in Sect. 3.1, the damping term is based model in  on the 2 ˙ /2 instead Fig. 3.2b; the Rayleigh dissipation function is defined as R = αs d V n  of R = αeff |δs | d V n˙ 2 /2 so that α is well defined even in the vicinity of TA where δs vanishes [6, 7]. The third term governs the antiferromagnetic dynamics in the vicinity of TA [8]. When the temperature varies across TA , the net spin density δs changes its sign, and accordingly, other parameters such as α, ρ, K , and Ml should change with temperature. However, let us assume here that the sign change of δs is the only change of the parameters. Here, we consider the case where Hext is applied along the z (in-plane) direction, Hext = Hext z (Fig. 4.1). Note that this is the case for our experimental setup (see Sect. 4.2). In this case, 





fn = 2K n x x + Ml Hext z.

(4.3)

4.1.2 Resonance Frequency Let us expand Eq. (4.1)along  the positive in-plane z-direction to linear order in the small fluctuations |n x |, n y   1. The resultant equations are given by

4.1 Derivation of Equation for Magnetic Resonance in Ferrimagnets

63

Fig. 4.1 Schematic of the coordinate definitions for the case of out-of-plane external magnetic field Hext

δs n˙ x − αs n˙ y − ρ n¨ y = M Hext n y ,

(4.4)

δs n˙ y + αs n˙ x + ρ n¨ x = −M(Hext + Hani )n x , where Hani = −2K /M is the effective anisotropy field along the x-direction perpendicular to the film (including the effect of the demagnetizing field). We consider small-amplitude elliptical precession along the z-axis as 

nx ny



 =

 δn x −iωt e . δn y

(4.5)

Then Eq. (4.4) can be written as 



   M Hext − ρω2 − iωαs δn x iωδs  = 0. iωδs δn y −M(Hext + Hani ) + ρω2 + iωαs (4.6)

In the presence of an external driving force, Eq. (4.6) becomes 



     iωδs M Hext − ρω2 − iωαs δn x hx  = . iωδs δn y hy −M(Hext + Hani ) + ρω2 + iωαs (4.7)

Therefore,

δn x δn y



⎡

 iωδs

 = ⎣ −M(Hext + Hani ) + ρω2 + iωαs

 ⎤−1  M Hext − ρω2 − iωαs hx ⎦ hy iωδs

64

4 Gilbert Damping Parameter of Ferrimagnets Probed … 1

  M(Hext + Hani ) − ρω2 − (ωδs )2 − i M(2Hext + Hani ) − 2ρω2 αsω   ⎤  iωδs −M Hext + ρω2 + iωαs ⎣   ⎦ hx . × (4.8) hy iωδs M(Hext + Hani ) − ρω2 − iωαs

= 

M Hext − ρω2 ⎡



Here, the second order of α  1 is considered to be negligible. The resonance frequency is obtained from

 M Hext − ρω2 M(Hext + Hani ) − ρω2 − (ωδs )2 = 0.

 ∴ ρ 2 ω4 − δs2 + ρ M(2Hext + Hani ) ω2 + M 2 Hext (Hext + Hani ) = 0.  2 δs2 + ρ M(2Hext + Hani ) ± δs4 + 2ρ Mδs2 (2Hext + Hani ) + ρ 2 M 2 Hani 2 ∴ ω± = . 2ρ 2  2 δs2 + ρ M(2Hext + Hani ) ± δs4 + 2ρ Mδs2 (2Hext + Hani ) + ρ 2 M 2 Hani ∴ f ±2 = . 8π 2 ρ 2 (4.9)



Here, f + and f − are the higher and lower resonance frequencies for the given external field Hext . Significantly away from TA , where the net spin density |δs | is sufficiently large, Eq. (4.1) and the corresponding dynamics are dominated by the first-order time ˙ and thus we can neglect the second-order term (ρn × n) ¨ by derivative term (δs n) setting ρ = 0. Therefore, the expression for the lower frequency f − is reduced to that for the ferromagnetic resonance frequency [9]: f FM =

M 2π|δs |

√

Hext (Hext + Hani ).

(4.10)

Note that M/|δs | is the effective gyromagnetic ratio γeff of the ferrimagnets. As the temperature approaches TA , the net spin density |δs | decreases and thus the resonance frequency is expected to increase. However, Eq. (4.10) cannot be used in the vicinity of TA , where δs vanishes and thus the second-order term ρn × n¨ cannot be neglected. Exactly at TA , the net spin density vanishes δs = 0, which reduces the obtained resonance frequencies (Eq. (4.9)) to f+ =

1 2π



M(Hext +Hani ) , ρ

f− =

1 2π



M Hext . ρ

(4.11)

Inclusion of the second-order time derivative term ∝ ρ in the LLG equation (Eq. (4.1)) is necessary to obtain finite resonance frequencies at TA ; otherwise, the LLG equation lacks the reactive dynamic term ∝ δs at TA and becomes unable to describe the ferrimagnetic dynamics properly.

4.1 Derivation of Equation for Magnetic Resonance in Ferrimagnets

65

4.1.3 Spectral Linewidth Strictly speaking, the spectral linewidth is ill defined from Eq. (4.8) because Eq. (4.8) is not the Lorentzian function. However, once we set the approximation of ρ = 0 (significantly away from TA ), Eq. (4.8) is reduced to 

δn x δn y

 =

1

+ Hani ) − (ωδs )2 + i M(2Hext + Hani )αsω    iωδs −M Hext + ρω2 + iωαs hx  ×

, iωδs hy M(Hext + Hani ) − ρω2 − iωαs (4.12) M 2 Hext (Hext 

which is the Lorentzian function. Therefore, the spectral linewidth significantly away from TA is ωFiM = M(2Hext + Hani )αs.

(4.13)

+Hani ∴ f FiM = 2π ωFiM = α |δss | 2√ H2Hext (H +H ext

ext

ani )

f FiM .

(4.14)

Finally, H can be obtained as follows, where we use γeff = M/|δs |. d Hres 1  f FiM f FiM =  d f FiM d f FiM d Hres √ s 4π |δs | Hres (Hres + Hani ) 2Hres + Hani f FiM ×α = √ |δs | 2 Hres (Hres + Hani ) M(2Hres + Hani ) 2π s = α (4.15) f FiM γeff |δs |

H =

Therefore α in ferrimagnets (denoted as αFiM hereafter) is given by αFiM =

 γeff |δs |  H  . 2π s f FiM

(4.16)

Note that both s and δs appear in the linewidth expression because (1) the energy dissipation rate is proportional to s since two lattices contribute additively whereas (2) the resonance frequency is inversely proportional to δs . On the other hand, in the conventional expressions for ferromagnetic resonance, the spin density that attributes to energy dissipation and the spin density that is related to magnetic resonance are identical, s = δs , and the corresponding expression αFM ≈

 γeff  H  2π

f FiM

(4.17)

66

4 Gilbert Damping Parameter of Ferrimagnets Probed …

was used to analyze the FiMR in the previous reports [10, 11]. Hereafter, these two expressions for the Gilbert damping parameters, αFiM and αFM , are obtained from our experimental results and are compared with each other.

4.2 Experimental Methods and Results The magnetic resonance in the ferrimagnetic GdFeCo was investigated by spin-torque ferromagnetic resonance (ST- FMR, see Sect. 1.2.2). Figure 4.2 depicts the schematic of the experimental setup. In this study, a 5-nm SiN/10-nm Gd25.0 Fe65.6 Co9.4 /5-nm Pt/100-nm SiN/Si substrate film was used. The Pt layer served as a heavy metal layer for injecting the spin current into and generating the Oersted field in the adjacent GdFeCo layer, so that FiMR was induced in it. The insulating SiN layers were the capping and buffer layers to protect oxidization. The film was patterned into a 10µm-wide and 10-µm-long strip structure using optical lithography and the Ar ion milling process. A coplanar waveguide made of 100-nm Au/5-nm Ti was deposited at the ends of the strip (the golden areas in Fig. 4.2). FiMR was induced at a fixed radio frequency current, Irf ( f = 4 − 18 GHz), and an external magnetic field, Hext , swept from the maximum field (300 mT or 800 mT depending on the peak position) to 0 mT. Hext was applied in-plane, 45◦ away from the longitudinal direction of the strip to maximize the signal arising from the anisotropic magnetoresistance. Figure 4.3a shows the FiMR spectra at T values between 220 and 295 K. It shows a clear resonance peak at 295 K. Provided that the spontaneous magnetization lies in-plane at T = 295 K, this peak is originated from the resonance of in-plane magnetization. However, for T ≤ 240 K, another peak is also observed at Hext ≈ 50 mT. This second peak can be attributed to the resonance of the out-of-plane magnetization for the following reasons: (1) The TM of the device was measured to be 110 K (Fig. 4.5a), thus the net magnetization of the GdFeCo decreased as T approaches TM = 110 K. The ferrimagnetic GdFeCo has in itself PMA [12]. Provided that the perpendicular anisotropy energy is insensitive to the temperature, Fig. 4.2 Schematic of the device and measurement setup. Directions of the external magnetic field, Hext , and AC current, Irf , are indicated. Hext was applied in-plane 45◦ away from the long axis of the strip. Reprinted figure with permission from [21]. Copyright (2019) The Japan Society of Applied Physics

4.2 Experimental Methods and Results

(a)

67

200

f = 4 GHz 6 GHz

T = 295 K

150 100

10 GHz

50

14 GHz 18 GHz

0 200

T = 280 K

150 100 50 0 200

T = 260 K

150 100 50 0 200

T = 240 K

150 100 50 0 200

T = 230 K

150 100 50

V [μV]

0 200

T = 220 K

150 100 50 0 0

100

200

300

400

500

600

700

800

Hext [mT]

(b)

fres [GHz]

15

(c) 295 K 260 K 240 K 220 K

60

ΔH [mT]

20

10

40 20

5 0 0

295 K 260 K 240 K 220 K

100

200

300

Hres [mT]

400

500

0 0

5

10 fres [GHz]

15

20

Fig. 4.3 a Ferrimagnetic resonance spectra as a function of external magnetic field Hext at temperatures from 220–295 K. The emerging peak at Hext ≈ 50 mT below 240 K is attributed to the out-of-plane resonance peak and is neglected in this study. b Resonance frequency f res as a function of resonance magnetic field Hres . Solid lines are the fitting results by Eq. (4.18). c Spectral linewidth H as a function of f res . Solid lines are the fitting results by Eq. (4.19). Reprinted figures with permission from [21]. Copyright (2019) The Japan Society of Applied Physics

68

4 Gilbert Damping Parameter of Ferrimagnets Probed …

the perpendicular anisotropy field can increase with respect to the decrease in the net magnetization with temperature [12]. For this device, the perpendicular anisotropy field could be considered to increase as T approaches TM = 110 K. Therefore, for T ≤ 240 K, the spontaneous magnetization directs in the out-of-plane direction, overcoming the demagnetizing field. (2) It is well known that when the demagnetizing field is directed out-of-plane and the external field is applied in-plane, the spectrum shows two peaks: one from the resonance of out-of-plane magnetization and the other from the resonance of in-plane magnetization. In this study, the peak originating from the resonance of in-plane magnetization is focused, so the low-field peak was cut off and the spectra shown in Fig. 4.3a were fit by the combination of symmetric and antisymmetric Lorentzian functions, from which the resonance parameters were obtained [13, 14]. Figure 4.3b shows resonance frequency, f res , as a function of the resonance field, Hres , and Fig. 4.3c shows the spectral linewidth, H (half width at half maximum), as a function of f res . Hereafter, these experimental data are analyzed as follows; first, these data are analyzed based on the conventional equation of FMR [9, 15]. Next, the analysis of the results is reconsidered in terms of the theory of FiMR derived in

(a)

(b)

4.5

0

Hani [mT]

geff

4.0 3.5 3.0 2.5 2.0

220

240

260 T [K]

280

300

-40 -80 -120

220

240

260 T [K]

280

300

220

240

260 T [K]

280

300

(d)

(c)

12

0.20

ΔH0 [mT]

αFM

10

0.15 0.10 0.05

8 6 4

220

240

260 T [K]

280

300

Fig. 4.4 Resonance parameters as functions of temperature extracted by the fitting in Fig. 4.3. a Effective Landé g-factor, geff . b Effective anisotropy field, Hani . c Effective Gilbert damping parameter, αFM . d Frequency-independent linewidth, H0 . Reprinted figures with permission from [21]. Copyright (2019) The Japan Society of Applied Physics

4.2 Experimental Methods and Results

69

Magnetization [×105 A/m]

(a)

4 3

(b) Mnet (by SQUID) MFeCo MGd

2 1 0 -1 0

TM ≈ 110 K

50 100 150 200 250 300 T [K]

(c)

0.20

α

0.15

αFM αFiM

0.10 0.05 0.00 150

TA~160K

200

T [K]

250

300

Fig. 4.5 a Net magnetization, Mnet , and magnetizations of two sublattices, MFeCo and MGd , as functions of temperature. b Net spin density, δs , spin densities of two sublattices, sFeCo and sGd , and the sum of the magnitudes of the two spin densities, s, as functions of temperature. c Effective Gilbert damping parameter, αFM , and properly defined Gilbert damping parameter of ferrimagnets, αFiM , as functions of temperature. Reprinted figure with permission from [21]. Copyright (2019) The Japan Society of Applied Physics

Sect. 4.1. Note that this analysis is based on the assumption that the temperature is significantly away from TA , which is justified later. Based on the conventional FMR equation, f res =

geff μB √ Hres (Hres h

H =

αFM f (geff μB / h) res

+ Hani ),

+ H0 ,

(4.18) (4.19)

where geff is the effective Landé g-factor, μB is the Bohr magneton, h is Planck’s constant, Hani is the effective anisotropy field including the demagnetizing field, αFM is the effective Gilbert damping parameter defined as in Eq. (3.8) [10], and H0 is a frequency-independent linewidth, known as the inhomogeneous broadening, originating from magnetic nonuniformity [16]. Equation (4.18) is the same as Eq. (4.10) if we replace geff μB / with the effective gyromagnetic ratio M/|δs | ( = h/2π is

70

4 Gilbert Damping Parameter of Ferrimagnets Probed …

reduced Planck’s constant) and Hres with Hext . First, f res as a function of Hres in Fig. 4.3b is fitted by Eq. (4.18) (the fitting results are indicated by the solid lines), and geff and Hani are obtained as the fitting parameters. Figure 4.4a and b show the obtained geff and Hani as functions of T , respectively. From Fig. 4.4a, it is observed that geff increases remarkably as T decreases. In this study, TA of the GdFeCo wire is estimated to be ~160 K (see Fig. 4.5b and Sect. 4.3). Therefore, the results shown in Fig. 4.4a are concluded to indicate that geff increases as T approaches TA . This trend in geff is consistent with a previous study [10]. From Fig. 4.4b, it is clear that Hani not only decreases as T decreases but also changes its sign between 295 and 280 K. This observation is, as mentioned above, attributed to the change in magnetic anisotropy, i.e., from the in-plane anisotropy at 295 K to the perpendicular anisotropy at 220 K. Second, H as a function of f res in Fig. 4.3c is fitted by Eq. (4.19) (the fitting results are indicated by solid lines), and αFM and H0 are obtained as the fitting parameters. Figure 4.4c and d show the obtained αFM and H0 as functions of T , respectively. From Fig. 4.4c, it is observed that αFM increases significantly as T decreases, i.e., as T approaches TA . This trend in αFM is also in good agreement with the previous studies [10, 11, 17]. According to previous studies [10, 11], the T dependences of geff and αFM , i.e., the trend that geff and αFM increase as T approaches TA , are understood in terms of the net angular momentum, δs . As for geff , the effective gyromagnetic ratio, geff μB / = Mnet /δs , increases as  T approaches TA and diverges at TA , where δs = 0. As for αFM , R = αFM |δs | d V n˙ 2 /2 as per Ref. [10, 11]. Therefore, provided that R is insensitive to temperature, αFM increases as T approaches TA and diverges at TA , where δs = 0 (see Sect. 3.1). However, the fact that geff and αFM diverge (are ill defined) at TA suggests that the theory of FiMR, on which the previous studied were based, is invalid. By contrast, the FiMR theory explained in Sect. 4.1 can prevent the divergence of the gyromagnetic ratio and α at TA . As for the gyromagnetic ratio, the resonance frequency,  f ± , is well defined at TA (see Eq. (4.9)). Parameter α is defined based on R = αs d V n˙ 2 /2 so that the temperature dependence of the spectral linewidth can be explained by that of δs with constant α (see Eq. (4.15)).

4.3 Analysis and Discussion Based on Theory of Ferrimagnetic Resonance As mentioned in Sect. 4.1, the resonance frequency in FiMR significantly away from TA (where the assumption that ρ = 0 is valid) is approximated to that in FMR, whereas the spectral linewidth in FiMR significantly away from TA is reduced to that in FMR multiplied by δs /s. Hereafter, we focus on α to test the validity of this theory. Based on Eq. (4.15), the Gilbert damping parameter of ferrimagnets significantly away from TA is given by   αFiM = αFM  δss ,

(4.20)

4.3 Analysis and Discussion Based on Theory …

71

where αFM has been already obtained, as shown in Fig. 4.4c. Therefore, δs /s needs to be acquired to obtain αFiM from Eq. (4.20). Although the net spin density, δs , is easy to acquire, obtaining the total spin density, s, is not straightforward. This problem is resolved by the following analysis. The effective gyromagnetic ratio satisfies the following equation [10, 11]: geff μB 

=

Mnet δs

=

MFeCo −MGd MFeCo M − g μGd/ ( Gd B )

(gFeCo μB /)

,

(4.21)

where MFeCo (MGd ) is the magnetization of transition metal FeCo (rare-earth Gd), and gFeCo (gGd ) is the Landé g-factor of FeCo (Gd) sublattice. First, we focus on the left-hand side and the middle of Eq. (4.21), geff μB / = Mnet /δs . geff is obtained as in Fig. 4.4a. In addition, Mnet is independently measured using SQUID, as shown in Fig. 4.5a. Therefore, δs can be obtained from Eq. (4.21); δs = Mnet /geff μB . Second,we focus on the middle and  right-hand side of Eq. (4.21), MFeCo MGd Mnet /δs = (MFeCo − MGd )/ g μ / − g μ / . This equation is considered as ( FeCo B ) ( Gd B ) the simultaneous equations consisting of Mnet = MFeCo − MGd , δs =

MFeCo

(gFeCo μB /)

−

MGd

(gGd μB /)

(4.22a) ,

(4.22b)

with MFeCo and MGd being variable. Note that gFeCo and gGd are obtained from literature (gFeCo ∼ 2.2 and gGd ∼ 2.0) [18–20]. Therefore, by solving the above simultaneous equations, we can obtain the magnetizations of two sublattices, namely, MFeCo and MGd . Figure 4.5a shows Mnet , MFeCo , and MGd as functions of T . Moreover, each of the spin densities of two sublattices, sFeCo and sGd , is obtained based on si =

(gi μB /) (i Mi

= FeCoorGd).

(4.23)

Therefore, we can obtain the total spin density, s = sFeCo +sGd . Figure 4.5b shows δs , sFeCo , sGd , and s as functions of T . By substituting δs and s into Eq. (4.20), αFiM is obtained as shown in Fig. 4.5c. It is evident that αFiM is insensitive to temperature (≈ 0.01). This trend of αFiM is in sharp contrast with that of αFM , which increases significantly as T approaches TA . Note that, from Fig. 4.5b, TA is estimated roughly to be 160 K as a function of T . As TA ∼ 160K is significantly below the lowest T in our measurements (220 K), the assumption that the measured temperature range (220–295 K) is significantly away from TA , i.e., ρ = 0, is valid in the above analysis. Temperature-insensitive αFiM in Fig. 4.5c suggests the validity of Eq. (4.15); the T dependence of the spectral linewidth in FiMR is attributed to that of the net spin density, δs , instead of that of the effective Gilbert damping parameter, αFM . This observation is consistent with a recent theoretical study [6] and the discussion in

72

4 Gilbert Damping Parameter of Ferrimagnets Probed …

Chap. 3 [7]; however, it is in contrast with the conclusions of some previous studies [10, 11], where the T dependence of the spectral linewidth in FiMR was attributed to the change in αeff . Therefore, this result provides an additional evidence that properly defined αFiM of ferrimagnets is insensitive to temperature, supporting the validity of these recent studies [6, 7]. Finally, it should be noted that, even though Fig. 4.5c serves as evidence for the temperature-insensitive αFiM , it lacks the information of αFiM in the vicinity of TA . Simply, it might seem that measuring the FiMR spectrum in our measurement system enables us to obtain αFiM for the entire temperature range, including TA . However, obtaining αFiM in the vicinity of TA based on FiMR experiments is challenging for the following reasons. (1) In a previous paper [10], laser-induced spin dynamics for a certain ferrimagnet was investigated using an alloptical pump and probe technique. The resonance frequency at TA was measured to be larger than 50 GHz. Thus, observing FiMR electrically using the homodyne detection technique (40 GHz at maximum in our measurement system) is expected to be difficult. (2) Without the assumption that ρ = 0, the analysis of the resonance spectrum requires at least four fitting parameters (see Eq. (4.9)), which complicate the fitting process and degrade the quantitative validity of the fitting parameters (Note that these reasons do not eliminate the possibility of obtaining αFiM for the entire temperature range, including TA . By adopting the laser-induced magnetic dynamics using the pump-probe detection technique and by selecting the proper conditions of the experimental setup, obtaining αFiM at TA should be possible.). Therefore, Fig. 4.5c serves as experimental evidence to conclude that αFiM of ferrimagnets is insensitive to temperature.

4.4 Conclusion In conclusion, the discussion in this chapter provides the macroscopic theory of FiMR and the experimental results that support it. The characteristics of our theory are that an antiferromagnetic-like inertial term is added to the equations of motion (LLG equation) and the Gilbert damping parameter is properly defined using the Rayleigh dissipation function. This theory demonstrates that the resonance frequency and spectral linewidth of FiMR are well defined across TA . In addition, the experiments of the spin-torque FiMR are performed in the temperature range away from TA . It is observed that the resonance frequency and spectral linewidth depend on temperature. By analyzing the resonance spectra based on the theory, it is found that the Gilbert damping parameter of ferrimagnets is insensitive to temperature, which has been previously considered to be strongly temperature dependent. This study introduces a new framework for studying FiMR and helps to interpret the ferrimagnetic dynamics for a wide range of temperatures [21].

References

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

lvanov BA et al (1983) Sov Phys JETP 57:214 Chiolero A et al (1997) Phys Rev B 56:738 Kim SK et al (2017a) Phys Rev B 95:140404(R) Kim K-J et al (2017b) Nat Mater 16:1187 Oh S-H et al (2017) Phys Rev B 96:100407(R) Kamra A et al (2018) Phys Rev B 98:184402 Kim D-H et al (2019) Phys Rev Lett 112:127203 Hals KMD et al (2011) Phys Rev Lett 106:107206 Kittel C (2004) Introduction to solid state physics, 8th edn. Wiley, New York Stanciu CD et al (2006) Phys Rev B 73:220402(R) Binder M et al (2006) Phys Rev B 74:134404 Okuno T et al (2016) Appl Phys Express 9:073001 Liu L et al (2011) Phys Rev Lett 106:036601 Fang D et al (2011) Nat Nanotechnol 6:413 Heinrich B et al (1985) J Appl Phys 57:3690 Arias R et al (1999) Phys Rev B 60:7395 Rodrigue GP et al (1960) J Appl Phys 31:S376 Kittel C (1949) Phys Rev 76:743 Scott GG (1962) Rev Mod Phys 34:102 Min BI et al (1991) J Phys Condens Matter 3:5131 Okuno T et al (2019) Appl Phys Express 12:093001

73

Chapter 5

Conclusion

In conclusion, the studies in this thesis have clarified the magnetic dynamics in antiferromagnetically-coupled ferrimagnets in terms of the role of the angular momentum. First, the spin-transfer torque effects on domain wall (DW) motion in ferrimagnets are unraveled. The theoretical and experimental investigations reveal that (1) the adiabatic torque induced DW velocity exhibits an antisymmetric temperature dependence with respect to the angular momentum compensation temperature, TA ; it vanishes exactly at TA and its sign reverses across TA . (2) The non-adiabatic torque induced DW velocity exhibits a symmetric temperature dependence with respect to TA ; it exhibits the maximum at TA without the sign change. This indicates that the non-adiabatic torque in antiferromagnets acts as a staggered magnetic field, which can be useful to drive DWs. In addition, it is found that the nonadiabaticity parameter, β, is significantly larger than the Gilbert damping parameter, α, in our ferrimagentic GdFeCo, challenging the conventional understanding of the non-adiabatic torque, where α and β are comparable. Second, it is confirmed via two independent experiments that α of ferrimagnets is insensitive to temperature, which has been considered to diverge at TA . The DW motion experiment demonstrates that α of ferrimagnets is almost insensitive to the temperature for a wide range of temperatures, including TA , without diverging at TA . Furthermore, the ferrimagnetic resonance experiment demonstrates that α of ferrimagnets is almost constant irrespective of the temperature in the temperature range well above TA . In both experiments, the Landau–Lifshitz–Gilbert equation for ferrimagnets is modified so that α is defined based on the model where the magnetic damping occurs via energy dissipation from both sublattices of ferrimagnets. This modification helps α to be defined well for all the temperature ranges, including TA . In contrast with the previous studies, the analyses of these experiments yield temperature-insensitive α of ferrimagnets, which represents correctly the energy dissipation rate from magnetic dynamics. The two studies in this thesis provide key understanding to clarify fully the magnetic dynamics in antiferromagnetically-coupled ferrimagnets, which would be useful to realize the next-generation spintronic devices with ferrimagnets and antiferromagnets. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 T. Okuno, Magnetic Dynamics in Antiferromagnetically-Coupled Ferrimagnets, Springer Theses, https://doi.org/10.1007/978-981-15-9176-1_5

75