Magnetic nano- and microwires design, synthesis, properties and applications [2nd ed] 9780081028339, 0081028334

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Magnetic nano- and microwires design, synthesis, properties and applications [2nd ed]
 9780081028339, 0081028334

Table of contents :
Front Cover......Page 1
Magnetic Nano- and Microwires: Design, Synthesis, Properties and Applications......Page 4
Copyright......Page 5
Contents......Page 6
Contributors......Page 16
Preface......Page 26
Part One: Design, synthesis, and properties......Page 28
1.1. Introduction......Page 30
1.2.1. Magnetic nanowires with modulated diameter......Page 33
1.2.2. Composition modulation: Multilayered nanowires......Page 36
1.2.3. Relative orientation modulation: Radial nanowire array......Page 38
1.3.1. Self-sustaining interconnected nanowire networks from ion-irradiated 3D polymeric templates......Page 40
1.3.3. Self-sustaining interconnected nanowire network 3D structures from 3D AAO templates......Page 42
1.3.3.1. Fabrication process and morphological description of the structures......Page 43
References......Page 45
2.1. Introduction......Page 48
2.2. Atomic layer deposition technique......Page 49
2.3. ALD on nanoporous templates: Tailored nanowire arrays......Page 54
2.4. Magnetic nanotubes......Page 58
2.5. Core-shell magnetic nanostructures......Page 62
2.6. Diameter modulated nanowires......Page 64
2.6.1. Magnetic properties of diameter modulated NiFe nanowire array......Page 68
2.6.2. Micromagnetic simulations of single Ni and NiFe bisegmented diameter modulated nanowires......Page 70
2.6.3. Magnetic properties of diameter modulated FeCo nanowires array......Page 73
2.7. Conclusions......Page 78
References......Page 79
Further reading......Page 87
3.1. Introduction......Page 88
3.2. Nanoporous alumina template synthesis: Hard anodization......Page 90
3.3.1. Metallic flux method......Page 93
3.3.2. Metallic flux nanonucleation (MFNN) method......Page 96
3.4.1. Fe3G4 nanowires: An itinerant metamagnetic compound......Page 97
3.4.2. GdIn3: An intermetallic correlated electron system......Page 101
3.4.3. Ga: A type I like superconductor at low dimension......Page 104
References......Page 109
4.1. Introduction......Page 112
4.2. 3D-nanoprinted scaffolds for permalloy evaporation......Page 113
4.3. Magnetic functionalization of scaffolds through permalloy evaporation......Page 117
4.4. Dark-field magneto-optical Kerr effect (MOKE) magnetometry......Page 118
4.5. Characterization of 3D magnetic domain-wall motion using dark-field MOKE magnetometry......Page 119
4.6. Controlling switching mechanism and domain-wall injection in a suspended 3D nanowire......Page 124
4.7. Summary and outlook......Page 127
References......Page 128
Further reading......Page 129
5.1. Introduction......Page 130
5.2. Synthesis......Page 131
5.2.1. Axial heterostructures......Page 133
5.2.2. Radial heterostructures......Page 136
5.2.3. Nanowire arrays......Page 139
5.2.4. Branched heterostructures......Page 140
5.3. Potential applications......Page 142
5.3.2.1. Unidirectional scattering......Page 144
5.3.2.2. Structural colors......Page 146
5.3.3.1. LEDs......Page 148
5.3.3.2. Solar cells......Page 150
5.3.3.3. Hybrid NW heterostructures......Page 151
5.3.4. Spintronic applications......Page 152
5.4. Future issues and outlook......Page 153
5.5. Summary......Page 154
References......Page 155
6.1. Introduction......Page 162
6.2.1. Polycarbonate membranes......Page 163
6.2.2. Nanoporous alumina membranes......Page 165
6.3.1. Electrochemical cell......Page 166
6.3.2. Annular-shaped working electrodes......Page 167
6.3.4. Tuning of the deposition time......Page 168
6.3.6. Formation of hydrogen bubbles......Page 169
6.3.8. Annular nanochannels......Page 170
6.4.1. Atomic layer deposition......Page 172
6.4.3. Sol-gel method......Page 174
6.4.4. Wetting process......Page 175
6.4.6. Thermal oxidation of nanowires (Kirkendall effect)......Page 176
6.4.8. Hydrothermal process......Page 178
6.5. Magnetism of cylindrical nanotubes......Page 180
6.5.1. Magnetic interactions......Page 181
6.5.2. Magnetization reversal......Page 183
6.5.3. Magnetic domain wall dynamics......Page 185
6.5.4. Exchange bias and thermal effects......Page 186
6.6. Applications of magnetic nanotubes......Page 189
6.7. Conclusions......Page 190
References......Page 191
7.1. Introduction......Page 212
7.2.1.1. General principles......Page 213
7.2.1.2. Morphology and structure......Page 214
7.2.2.2. Structure and formation mechanism of cobalt nanorods and nanowires......Page 216
7.2.2.3. Spontaneous formation of 3D arrays of Co nanorods in solution......Page 217
7.3. Optimization of the magnetic properties of individual wires......Page 219
7.3.1.1. Effect of the shape of elongated magnetic particles on the coercive field......Page 220
7.3.1.2. Mean diameter and aspect ratio......Page 223
7.3.2. Comparison with experimental results......Page 225
7.4. 2D arrays of Co nanorods combining epitaxy and organometallic chemistry......Page 227
7.4.2. Magnetic properties of 2D arrays of perpendicular nanorods......Page 228
7.5.1.1. Influence of the degree of alignment......Page 231
7.5.1.2. Effect of dipolar interactions......Page 233
7.5.1.3. Influence of the packing density......Page 234
7.5.2. Dense arrays of parallel nanorods......Page 235
7.5.3. Consolidation of nanowires......Page 239
7.6. Conclusion......Page 242
References......Page 243
Further reading......Page 246
8.1. Introduction......Page 248
8.2.1. Characterization methods......Page 251
8.2.2. Main aspects of the magnetic behavior......Page 254
8.2.3. Magnetostatic and magnetoelastic contributions......Page 257
8.3. Domain wall propagation......Page 260
8.3.1. Experimental techniques......Page 261
8.3.2. Domain wall velocity and mobility......Page 262
8.3.3. Shape of the propagating domain walls......Page 268
8.3.4. Controlled motion of domain walls......Page 270
8.4. Effects of structural transformations......Page 272
8.5. Final remarks and future work......Page 274
References......Page 277
9.1. Introduction......Page 282
9.2. Evaluation of applicability of glass-coated microwires for use in magnetocalorics......Page 284
9.3. Preparation of glass-coated microwires......Page 286
9.4. Experimental techniques used for characterization of Heusler-type microwires......Page 287
9.5. Magnetic, magnetotransport, and structural properties microwires from Heusler alloys......Page 288
9.5.1. Effect of annealing on magnetic properties of microwires from Heusler alloys......Page 291
9.5.2. Magnetoresistance......Page 302
9.5.3. Structure of Heusler-type microwires......Page 304
9.6. Magnetocaloric effect of Heusler-type microwires......Page 309
9.7. Magnetic hardening and exchange bias effect in Heusler-type microwires......Page 312
Acknowledgments......Page 315
References......Page 316
Further reading......Page 321
Part Two: Magnetization mechanisms, domains and domain walls......Page 322
10.1. Introduction......Page 324
10.2.1. Equation of motion......Page 326
10.2.2. Domain wall velocity......Page 327
10.2.3. Depinning threshold current......Page 328
10.2.4. Domain wall inertia......Page 329
10.3.1. Sample preparation and experimental methods......Page 330
10.3.2. Spin-orbit torque......Page 331
10.3.3. Measurements of domain wall velocity......Page 333
10.3.4. Determination of the magnetic chirality......Page 336
10.3.5. Effective mass of chiral domain walls......Page 339
10.3.6. Synchronous motion of highly packed coupled chiral domain walls......Page 343
10.4. Conclusion......Page 347
References......Page 348
Further reading......Page 351
11.1. Introduction......Page 352
11.1.1.1. The thermal conductivity......Page 354
11.1.1.2. Is the injected current pulse short enough?......Page 355
11.2. Thermal behavior of a ferromagnetic nanostrip......Page 357
11.3. Fabrication of the nanostrips: Some considerations......Page 363
11.4.1. Micromagnetic model at zero or finite uniform temperature......Page 364
11.4.2. Micromagnetic model and heat transport to account for the Joule effects......Page 366
11.5.1.1. Field-driven DW motion......Page 370
11.5.1.2. Current-driven DW motion......Page 373
11.5.2. DW nucleation by a current pulse along a bit line......Page 375
11.5.3.1. Deterministic analysis in the absence of the Joule heating effect......Page 377
11.5.3.2. Current-assisted DW in the presence of Joule heating effect......Page 379
11.6. Conclusions......Page 383
References......Page 384
Further reading......Page 387
12.1.1. Perpendicular magnetic anisotropy......Page 388
12.1.2. The Dzyaloshinskii-Moriya interaction......Page 389
12.1.3. Magnetic domain walls......Page 390
12.2.1. Physical mechanism and device geometries......Page 391
12.2.2. E-field control of PMA......Page 394
12.2.3. E-field control of magnetic DW motion......Page 395
12.2.4. E-field control of DMI......Page 397
12.3. He+ ion irradiation......Page 398
12.3.1. Modification of PMA and DMI by ion irradiation......Page 399
12.3.2. Magnetic domain wall motion and irradiation-induced disorder......Page 400
12.4. Conclusion and outlook......Page 402
References......Page 403
13.1. Introduction......Page 408
13.2. Skyrmion lines and monopoles in nanowires......Page 412
13.3. Emergent electromagnetic fields......Page 418
13.4. Other sample geometries and thermal fluctuations......Page 422
13.5. Conclusions......Page 424
References......Page 425
14.1. Introduction......Page 430
14.2. Hysteresis loops of magnetic nanowires......Page 431
14.3. Magnetic domains and domain walls in straight magnetic nanowires......Page 435
14.4. Domain wall velocity in cylindrical magnetic nanowires......Page 438
14.5. Domains and domain walls in nanowires with geometrical modulations......Page 441
14.6. Domain walls and domains in multisegmented Co/Ni nanowires......Page 446
14.7. Conclusions......Page 448
References......Page 449
15.1.1. Fundamental and technological motivations for domain wall pinning......Page 454
15.1.2. Types of pinning for nanowires......Page 455
15.1.3. Existing theories and experiments......Page 456
15.2.1. Domain walls in cylindrical nanowires......Page 457
15.2.2. Geometry of modulation and potential barrier......Page 458
15.2.3. Magnetic charges......Page 461
15.2.4. Magnetic field generated by the modulation......Page 462
15.2.5. Energy of interaction......Page 466
15.3.1. Abrupt modulation......Page 467
15.3.2. Smooth modulation......Page 469
15.3.3. Protrusion: Double abrupt modulation......Page 471
15.4. Modulation under applied current......Page 474
References......Page 477
16.1. Introduction to the imaging techniques......Page 482
16.2. Synthesis and fabrication of modulated nanowires into the anodic aluminum oxide templates......Page 489
16.2.2. Diameter-modulated nanowires obtained by pulsed anodization......Page 491
16.3. Magnetic characterization of nanowire arrays......Page 494
16.4. Magnetic characterization of individual nanowires with uniform diameter......Page 495
16.5. Study of geometrically modulated nanowires......Page 500
16.6. Multisegmented nanowires......Page 503
16.7. Remarks and conclusion......Page 508
References......Page 509
17.1.1. Magnetic nanostructures......Page 518
17.1.2. Ferromagnetic nanotubes......Page 519
17.1.3. Measuring assemblies versus individual magnetic nanotubes......Page 520
17.2. Magnetoresistance......Page 521
17.3. Torque magnetometry......Page 524
17.4. Magnetic imaging with X-rays......Page 528
17.5. Scanning SQUID microscopy......Page 533
17.6. Magnetic force microscopy......Page 536
17.7. Conclusions and outlook......Page 538
References......Page 539
18.1. Introduction......Page 546
18.2.2. Visualization of surface magnetic domain structures......Page 547
18.2.3. Imaging of the magnetization reversal by MOKE microscopy......Page 551
18.3. Elliptic domain structures......Page 552
18.4. Spiral domain structures......Page 555
References......Page 560
Further reading......Page 561
19.1. Introduction......Page 562
19.2. Sixtus and Tonks measurements......Page 564
19.3. Time-resolved measurement of the DW velocity......Page 567
19.4. Braking and trapping a single-domain wall......Page 574
19.5. Injection of domain walls......Page 580
References......Page 582
20.1. Introduction......Page 586
20.2.1. Fixed frequency (cavity-based) FMR spectroscopy......Page 587
20.2.2. Vector network analyzer-FMR spectroscopy......Page 588
20.2.4. Time-resolved magneto-optical Kerr effect microscopy......Page 590
20.3.1.1. Fabrication of ferromagnetic nanowire arrays in polymeric etched ion-track templates......Page 593
20.3.1.2. Fabrication of ferromagnetic nanowire arrays in alumina templates......Page 594
20.3.2. Static properties of ferromagnetic nanowire arrays......Page 595
20.4.1. Dynamical behavior of saturated isolated ferromagnetic nanowires......Page 598
20.4.2. Dynamical behavior of saturated ferromagnetic nanowire arrays......Page 600
20.4.3. Dynamical behavior of unsaturated ferromagnetic nanowire arrays......Page 607
20.4.4. Dynamical behavior of ferromagnetic nanowire arrays considering the magnetocrystalline anisotropy term......Page 612
20.4.5. Dynamical behavior of multilayer ferromagnetic nanowire arrays......Page 616
20.4.6. Dynamical behavior of gradient ferromagnetic nanowire arrays......Page 621
20.5.1. Fabrication of 3-D ferromagnetic nanowire arrays......Page 622
20.5.2. Static properties of 3-D ferromagnetic nanowire arrays......Page 625
20.5.3. High-frequency behavior of 3-D ferromagnetic nanowire arrays......Page 626
20.6. High-frequency applications and future perspectives of ferromagnetic nanowire arrays......Page 629
References......Page 630
21.1. Introduction......Page 640
21.2.1.1. Static magnetic properties of nanowires......Page 642
21.2.1.2. The role of shape anisotropy: Demagnetizing tensor of nonellipsoidal magnetic elements......Page 643
21.2.1.3. Static dipolar interaction effects in nanowire arrays......Page 645
21.2.1.4. Configurational phase transitions in arrays of nanowires......Page 646
21.3. Magnetic nanowires in electromagnetic fields......Page 649
21.3.1. SWs in magnetic nanowires......Page 652
21.3.1.1. Dipolar-exchange spin-wave modes of individual cylindrical nanowires......Page 653
21.3.1.2. Collective spin-wave modes in arrays of interacting nanowires......Page 659
21.3.1.3. Spin waves in ferromagnetic nanowires with noncircular cross section......Page 661
21.3.1.4. Magnetic structure and dynamics of multilayered nanowires and magnetic nanotubes......Page 663
21.3.1.5. Effects of curvature and torsion on the magnetic dynamics in nanowires and nanotubes......Page 669
21.4.1.1. EMW interactions with nanowire structures......Page 671
21.4.1.2. Shape and size effects in the EMW propagation in nanoarrays......Page 676
21.4.1.3. EMW scattering in nanowires at THz frequencies......Page 680
21.4.2.1. Microwave devices based on magnetic nanowires......Page 684
21.4.2.2. Magnetic nanowire metamaterials: Photonics and plasmonics......Page 687
21.5. Conclusions and future trends......Page 689
References......Page 690
Part Three: Sensing, thermoelectric, robotics, biomedical and microwave applications......Page 700
22.1. Introduction......Page 702
22.2. Fe-Ga alloy nanowires used in tactile sensors......Page 703
22.3. Co/Cu multilayered nanowires for CPP-GMR structures......Page 707
22.4. Nanowires used for biomedical applications......Page 710
22.5. Long-range ordered porous AAO fabricated by double imprinting with line-patterned stamps......Page 716
22.6. Conclusions......Page 717
References......Page 720
23.1. Introduction......Page 724
23.2. Biocompatible magnetic nanowires......Page 725
23.3. Magnetic nanowires for drug delivery......Page 728
23.4. Magnetic nanowires for cancer treatment......Page 730
23.5. Magnetic nanowires as MRI contrast agents......Page 732
23.6. Magnetic nanowire cell scaffolds......Page 733
23.7. Conclusion......Page 735
References......Page 736
24.1. Introduction......Page 742
24.2. Short history......Page 743
24.3. Thermopower......Page 744
24.3.2. Thermopower sources in parallel......Page 747
24.4. Temperature-dependent thermopower......Page 748
24.4.1. Phonon-drag thermopower......Page 749
24.4.2. Influence of nanostructuring......Page 750
24.5. Origin of the magnetic field dependence......Page 752
24.5.1. Magneto-thermopower......Page 754
References......Page 757
25.1. Introduction to magnetostrictive Fe-Ga alloys......Page 764
25.2. Modeling and micromagnetics simulations of Fe-Ga nanowires......Page 771
25.3. Fabrication of Fe-Ga nanowires......Page 776
25.4. Structural and magnetic characterization of Fe-Ga and Fe-Ga/Cu nanowires......Page 778
25.5. Actuation using Fe-Ga/Cu nanowires......Page 786
25.6. Sensing using Fe-Ga/Cu nanowires......Page 794
25.7. Closing remarks......Page 799
References......Page 800
26.1. Introduction......Page 804
26.2. Fabrication techniques for nanowire-based swimmers......Page 807
26.3. Magnetically driven nanowire-based swimmers......Page 809
26.3.1. Corkscrew locomotion......Page 811
26.3.2. Surface-walking locomotion......Page 813
26.3.3. Undulatory (S-like) locomotion......Page 815
26.4. Chemically propelled nanowire-based swimmers......Page 816
References......Page 822
27.1. Introduction......Page 828
27.2. Template-assisted electrodeposition of 3D magnetic NW and NT networks......Page 829
27.3.1. Magnetic and magneto-transport properties of interconnected homogenenous NW networks......Page 832
27.3.2. Interconnected Ni NT networks with controlled structural and magnetic properties......Page 839
27.4.1. Spin-dependent thermoelectric transport in multilayered NW networks......Page 843
27.4.2. Magnetic control in heat management......Page 849
27.5. Conclusion and future perspectives......Page 852
References......Page 853
28.1. Introduction......Page 860
28.2.1. Production......Page 861
28.2.2.1. Magnetic field sensing......Page 864
28.2.2.2. Stress sensing using bistable microwires......Page 867
28.2.2.3. Stress in 3-D printed materials......Page 868
28.2.2.5. Sensing intracranial temperature......Page 870
28.2.2.6. Monitoring intracranial temperature in titanium implants......Page 873
28.2.2.7. Biocompatibility and technical compatibility of glass-coated microwires......Page 874
28.2.2.8. Chemical sensors based on bistable microwires......Page 876
28.2.3. Glass-coated Heusler-based SMART actuators......Page 877
28.2.3.1. Magnetocaloric applications......Page 879
28.2.3.2. Shape memory actuators: SMART actuators......Page 885
28.3. Conclusions......Page 888
Acknowledgments......Page 889
References......Page 890
29.2. Working principle......Page 896
29.2.1. Second harmonic mode......Page 897
29.2.2. Fundamental mode......Page 898
29.3. Circumferential excitation field......Page 899
29.4. Bimetallic wires......Page 902
29.5. Noise and thermal treatment of the wire......Page 903
29.6. Offset......Page 907
29.7. Effect of wire geometry......Page 911
29.8. Applications......Page 913
References......Page 914
Further reading......Page 915
30.1. Introduction......Page 916
30.2. Tunable magnetic configuration in amorphous wire......Page 917
30.2.1. Tuning the anisotropy with applied stress and current annealing......Page 919
30.2.2. Temperature effects......Page 921
30.3. Dynamic permeability in microwires......Page 922
30.3.1. Permeability spectra in wires with near-circumferential anisotropy......Page 923
30.3.2. Permeability spectra in wires with the axial anisotropy near Tc......Page 924
30.4. High-frequency impedance: Effects of external dc field, stress, and temperature......Page 925
30.4.1. Magnetoimpedance plots vs. magnetic field......Page 926
30.4.2. Magnetoimpedance vs. external stress......Page 927
30.4.3. Impedance behavior near the Curie temperature......Page 929
30.5.1. Electric polarization of a ferromagnetic wire......Page 931
30.5.2. Tuning the current distribution and polarization......Page 933
30.5.3. Application to wireless sensors......Page 935
30.6. Microwire composites as artificial dielectrics with tunable permittivity and permeability......Page 936
30.6.1. Effective permeability of wire composites......Page 937
30.6.2. Effective permittivity of composites with finite-length wires (wire dipoles)......Page 938
30.6.3. Effective permittivity of composites with continuous wires......Page 939
References......Page 942
31.1. Introduction......Page 946
31.2. Microwave absorption theory......Page 947
31.3.1. CNT-based polymer composites......Page 950
31.3.3. Graphene-based polymer composites......Page 953
31.3.4. Graphene-based magnetic composites and hybrids......Page 954
31.4. Microwave absorption and metamaterial properties of amorphous wire composites......Page 955
31.5. Tunable microwave absorption of polymer composites incorporating nano-carbon/amorphous wire hybrid fibers......Page 957
31.5.1. Microwave absorption of polymer composites incorporating CNT/amorphous wire hybrid fibers......Page 963
31.5.2. Microwave absorption of polymer composites incorporating rGO/amorphous wire hybrid fibers......Page 969
31.6. Tunable negative permittivity of metacomposites incorporating nano-carbon/amorphous wire hybrid fibers......Page 973
31.6.1. Tunable negative permittivity of metacomposites incorporating CNT/amorphous wire hybrid fibers......Page 974
31.6.2. Tunable negative permittivity of metacomposites incorporating rGO/AW hybrid fibers......Page 978
31.7.1. Summary......Page 982
Acknowledgments......Page 983
References......Page 984
Index......Page 992
Back Cover......Page 1012

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Magnetic Nano- and Microwires

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Woodhead Publishing Series in Electronic and Optical Materials

Magnetic Nano- and Microwires Design, Synthesis, Properties and Applications Second Edition

Edited by

Manuel Va´zquez

An imprint of Elsevier

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright © 2020 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/ permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-08-102832-2

For information on all Woodhead publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Kayla Dos Santos Editorial Project Manager: Joshua Mearns Production Project Manager: Debasish Ghosh Cover Designer: Miles Hitchen Typeset by SPi Global, India

Contents

Contributors Preface

Part One 1

2

3

xv xxv

Design, synthesis, and properties

3D porous alumina: A controllable platform to tailor magnetic nanowires Olga Caballero-Calero and Marisol Martı´n-Gonza´lez 1.1 Introduction 1.2 Modulated magnetic nanowires 1.3 Magnetic structures with a 3D geometry 1.4 Summary and outlook Acknowledgments References Electrochemical methods assisted with ALD for the synthesis of nanowires Javier Garcı´a, Miguel M endez, Silvia Gonza´lez, Victor Vega, Rafael Caballero, and Victor M. Prida 2.1 Introduction 2.2 Atomic layer deposition technique 2.3 ALD on nanoporous templates: Tailored nanowire arrays 2.4 Magnetic nanotubes 2.5 Core-shell magnetic nanostructures 2.6 Diameter modulated nanowires 2.7 Conclusions References Further reading Intermetallic nanowires fabricated by metallic flux nanonucleation method (MFNN) K.R. Pirota, K.O. Moura, A.S.E. da Cruz, R.B. Campanelli, P.J.G. Pagliuso, and F. B eron 3.1 Introduction 3.2 Nanoporous alumina template synthesis: Hard anodization 3.3 Crystal growth: Metallic flux and metallic flux nanonucleation (MFNN)

1 3 3 6 13 18 18 18

21

21 22 27 31 35 37 51 52 60

61

61 63 66

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Contents

3.4 Examples of intermetallic nanowires fabricated by MNFM 3.5 Conclusions Acknowledgments References 4

5

6

Fabrication and magneto-optical characterization of 3D-printed permalloy nanowires D edalo Sanz-Herna´ndez and Amalio Ferna´ndez-Pacheco 4.1 Introduction 4.2 3D-nanoprinted scaffolds for permalloy evaporation 4.3 Magnetic functionalization of scaffolds through permalloy evaporation 4.4 Dark-field magneto-optical Kerr effect (MOKE) magnetometry 4.5 Characterization of 3D magnetic domain-wall motion using dark-field MOKE magnetometry 4.6 Controlling switching mechanism and domain-wall injection in a suspended 3D nanowire 4.7 Summary and outlook Acknowledgments References Further reading Growth of nanowire heterostructures and their optoelectronic and spintronic applications Jerome K. Hyun and Shixiong Zhang 5.1 Introduction 5.2 Synthesis 5.3 Potential applications 5.4 Future issues and outlook 5.5 Summary References Cylindrical magnetic nanotubes: Synthesis, magnetism and applications Mariana P. Proenca, C elia T. Sousa, Joa˜o Ventura, and Joa˜o P. Arau´jo 6.1 Introduction 6.2 Porous templates 6.3 Electrochemical deposition of cylindrical nanotubes 6.4 Other template-filling growth methods 6.5 Magnetism of cylindrical nanotubes 6.6 Applications of magnetic nanotubes 6.7 Conclusions Acknowledgments References

70 82 82 82

85 85 86 90 91 92 97 100 101 101 102

103 103 104 115 126 127 128

135 135 136 139 145 153 162 163 164 164

Contents

7

8

9

From soft chemistry to 2D and 3D nanowire arrays with hard magnetic properties and permanent magnet applications Katerina Soulantica, Thomas Blon, Bruno Chaudret, Lise-Marie Lacroix, Guillaume Viau, and Fr ed eric Ott 7.1 Introduction 7.2 Chemical synthesis of magnetic nanowires 7.3 Optimization of the magnetic properties of individual wires 7.4 2D arrays of Co nanorods combining epitaxy and organometallic chemistry 7.5 From nanowires to 3D bulk permanent magnets 7.6 Conclusion References Further reading Recent trends in magnetic nanowires and submicron wires prepared by the quenching and drawing technique Tibor-Adrian O´va´ri, Nicoleta Lupu, and Horia Chiriac 8.1 Introduction 8.2 Magnetic behavior 8.3 Domain wall propagation 8.4 Effects of structural transformations 8.5 Final remarks and future work Acknowledgments References Heusler-type glass-coated microwires: Fabrication, characterization, and properties Arcady Zhukov, Mihail Ipatov, Juan Maria Blanco, Paula Cort e-Leon, and Valentina Zhukova 9.1 Introduction 9.2 Evaluation of applicability of glass-coated microwires for use in magnetocalorics 9.3 Preparation of glass-coated microwires 9.4 Experimental techniques used for characterization of Heusler-type microwires 9.5 Magnetic, magnetotransport, and structural properties microwires from Heusler alloys 9.6 Magnetocaloric effect of Heusler-type microwires 9.7 Magnetic hardening and exchange bias effect in Heusler-type microwires 9.8 Conclusions Acknowledgments References Further reading

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185 186 192 200 204 215 216 219

221 221 224 233 245 247 250 250

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255 257 259 260 261 282 285 288 288 289 294

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Contents

Part Two Magnetization mechanisms, domains and domain walls 10

11

12

13

Current-induced dynamics of chiral domain walls in magnetic heterostructures Jacob Torrejon, Victor Raposo, Eduardo Martinez, Rafael P. del Real, and Masamitsu Hayashi 10.1 Introduction 10.2 The 1D model of domain walls 10.3 Experimental results 10.4 Conclusion Acknowledgments References Further reading Joule heating and its role in current-assisted domain wall depinning in nanostrips Jos e L. Prieto, Manuel Mun˜oz, Vı´ctor Raposo, and Eduardo Martı´nez 11.1 Introduction 11.2 Thermal behavior of a ferromagnetic nanostrip 11.3 Fabrication of the nanostrips: Some considerations 11.4 Micromagnetic model and modeling details 11.5 Micromagnetic results 11.6 Conclusions Acknowledgments References Further reading Controlling magnetism by interface engineering L. Herrera Diez and D. Ravelosona 12.1 Introduction 12.2 E-field control of magnetism 12.3 He+ ion irradiation 12.4 Conclusion and outlook References Skyrmion lines, monopoles, and emergent electromagnetism in nanowires € F. Loffler € Hans-Benjamin Braun, Michalis Charilaou, and Jorg 13.1 Introduction 13.2 Skyrmion lines and monopoles in nanowires 13.3 Emergent electromagnetic fields 13.4 Other sample geometries and thermal fluctuations 13.5 Conclusions References

295 297

297 299 303 320 321 321 324

325 325 330 336 337 343 356 357 357 360 361 361 364 371 375 376

381 381 385 391 395 397 398

Contents

14

15

16

Micromagnetic modeling of magnetic domain walls and domains in cylindrical nanowires J.A. Fernandez-Roldan, Yu.P. Ivanov, and O. Chubykalo-Fesenko 14.1 Introduction 14.2 Hysteresis loops of magnetic nanowires 14.3 Magnetic domains and domain walls in straight magnetic nanowires 14.4 Domain wall velocity in cylindrical magnetic nanowires 14.5 Domains and domain walls in nanowires with geometrical modulations 14.6 Domain walls and domains in multisegmented Co/Ni nanowires 14.7 Conclusions References Domain wall pinning in a circular cross-section wire with modulated diameter A. De Riz, B. Trapp, J.A. Fernandez-Roldan, Ch. Thirion, J.-Ch. Toussaint, O. Fruchart, and D. Gusakova 15.1 Introduction 15.2 Theoretical background 15.3 Modulation under applied magnetic field 15.4 Modulation under applied current 15.5 Conclusion and perspective Acknowledgments References Magnetic imaging of individual modulated cylindrical nanowires C. Bran, M. Va´zquez, E. Berganza, M. Jaafar, E. Snoeck, and A. Asenjo 16.1 Introduction to the imaging techniques 16.2 Synthesis and fabrication of modulated nanowires into the anodic aluminum oxide templates 16.3 Magnetic characterization of nanowire arrays 16.4 Magnetic characterization of individual nanowires with uniform diameter 16.5 Study of geometrically modulated nanowires 16.6 Multisegmented nanowires 16.7 Remarks and conclusion Acknowledgments References

ix

403 403 404 408 411 414 419 421 422

427

427 430 440 447 450 450 450

455

455 462 467 468 473 476 481 482 482

x

17

18

19

20

Contents

Determining magnetization configurations and reversal of individual magnetic nanotubes M. Poggio 17.1 Introduction 17.2 Magnetoresistance 17.3 Torque magnetometry 17.4 Magnetic imaging with X-rays 17.5 Scanning SQUID microscopy 17.6 Magnetic force microscopy 17.7 Conclusions and outlook References Helical magnetic structures in amorphous microwires: Magneto-optical study and micromagnetic simulations A. Chizhik, J. Gonzalez, P. Gawronski, and A. Stupakiewicz 18.1 Introduction 18.2 Magnetization reversal and domain structures 18.3 Elliptic domain structures 18.4 Spiral domain structures 18.5 Conclusions Acknowledgments References Further reading On-time characterization of the dynamics of a single-domain wall in an amorphous microwire E. Calle and R.P. del Real 19.1 Introduction 19.2 Sixtus and Tonks measurements 19.3 Time-resolved measurement of the DW velocity 19.4 Braking and trapping a single-domain wall 19.5 Injection of domain walls 19.6 Conclusions Acknowledgment References Dynamical behavior of ferromagnetic nanowire arrays: From 1-D to 3-D David Navas, Celia Sousa, Sergey Bunyaev, and Gleb Kakazei 20.1 Introduction 20.2 High-frequency characterization techniques 20.3 Nanowire arrays 20.4 High-frequency behavior of ferromagnetic nanowire arrays

491 491 494 497 501 506 509 511 512

519 519 520 525 528 533 533 533 534

535 535 537 540 547 553 555 555 555

559 559 560 566 571

Contents

20.5 20.6

21

3-D nanowire arrays High-frequency applications and future perspectives of ferromagnetic nanowire arrays References

595

Spin waves and electromagnetic waves in magnetic nanowires Martha Pardavi-Horvath and Elena V. Tartakovskaya 21.1 Introduction 21.2 From single magnetic nanowire to 2-D nanowire arrays 21.3 Magnetic nanowires in electromagnetic fields 21.4 Interactions of EMWs with nanowires 21.5 Conclusions and future trends References

613

Part Three Sensing, thermoelectric, robotics, biomedical and microwave applications 22

23

xi

Template-assisted electrodeposited magnetic nanowires and their properties for applications Joseph Um, Jung Jin Park, Alison Flatau, Wen Zhou, Yali Zhang, Rhonda Franklin, Kotha Sai Madhukar Reddy, Liwen Tan, Anirudh Sharma, Sang-Yeob Sung, Jia Zou, and Bethanie Stadler 22.1 Introduction 22.2 Fe-Ga alloy nanowires used in tactile sensors 22.3 Co/Cu multilayered nanowires for CPP-GMR structures 22.4 Nanowires used for biomedical applications 22.5 Long-range ordered porous AAO fabricated by double imprinting with line-patterned stamps 22.6 Conclusions References Nanowire transducers for biomedical applications € Jose E. Perez and Jurgen Kosel 23.1 Introduction 23.2 Biocompatible magnetic nanowires 23.3 Magnetic nanowires for drug delivery 23.4 Magnetic nanowires for cancer treatment 23.5 Magnetic nanowires as MRI contrast agents 23.6 Magnetic nanowire cell scaffolds 23.7 Conclusion Acknowledgments References

602 603

613 615 622 644 662 663

673 675

675 676 680 683 689 690 693 697 697 698 701 703 705 706 708 709 709

xii

24

25

26

27

Contents

Thermopower measurements in magnetic nanowires € Tim Bohnert 24.1 Introduction 24.2 Short history 24.3 Thermopower 24.4 Temperature-dependent thermopower 24.5 Origin of the magnetic field dependence 24.6 Conclusions References Magnetostrictive Fe-Ga Nanowires for actuation and sensing applications Alison B. Flatau, Bethanie J.H. Stadler, Jungjin Park, Kotha Sai Madhukar Reddy, Patrick R. Downey, Chaitanya Mudivarthi, and Michael Van Order 25.1 Introduction to magnetostrictive Fe-Ga alloys 25.2 Modeling and micromagnetics simulations of Fe-Ga nanowires 25.3 Fabrication of Fe-Ga nanowires 25.4 Structural and magnetic characterization of Fe-Ga and Fe-Ga/Cu nanowires 25.5 Actuation using Fe-Ga/Cu nanowires 25.6 Sensing using Fe-Ga/Cu nanowires 25.7 Closing remarks References Magnetically and chemically propelled nanowire-based swimmers Josep Puigmartı´-Luis, Eva Pellicer, Bumjin Jang, George Chatzipirpiridis, Semih Sevim, Xiang-Zhong Chen, Bradley J. Nelson, and Salvador Pan e 26.1 Introduction 26.2 Fabrication techniques for nanowire-based swimmers 26.3 Magnetically driven nanowire-based swimmers 26.4 Chemically propelled nanowire-based swimmers References 3D magnetic nanowire networks ^ Luc Piraux, Tristan da Camara Santa Clara Gomes, Flavio Abreu Araujo, and Joaquı´n de la Torre Medina 27.1 Introduction 27.2 Template-assisted electrodeposition of 3D magnetic NW and NT networks 27.3 Interplay between the magnetic and magneto-transport properties

715 715 716 717 721 725 730 730

737

737 744 749 751 759 767 772 773

777

777 780 782 789 795 801

801 802 805

Contents

28

29

30

xiii

27.4 NW network-based spin caloritronics 27.5 Conclusion and future perspectives Acknowledgments References

816 825 826 826

Sensoric application of glass-coated magnetic microwires R. Jurc, L. Frolova, D. Kozejova, L. Fecova, M. Hennel, L. Galdun, K. Richter, J. Gamcova, P. Ibarra, R. Hudak, I. Sulla, D. Mudronova, J. Galik, R. Sabol, T. Ryba, L. Hvizdos, P. Klein, O. Milkovic, Z. Vargova, and R. Varga 28.1 Introduction 28.2 Glass-coated microwires 28.3 Conclusions Acknowledgments References

833

Orthogonal fluxgates based on magnetic microwires Mattia Butta 29.1 Introduction 29.2 Working principle 29.3 Circumferential excitation field 29.4 Bimetallic wires 29.5 Noise and thermal treatment of the wire 29.6 Offset 29.7 Effect of wire geometry 29.8 Applications References Further reading

869

Magnetic properties of amorphous microwires at microwaves and applications Larissa V. Panina 30.1 Introduction 30.2 Tunable magnetic configuration in amorphous wire 30.3 Dynamic permeability in microwires 30.4 High-frequency impedance: Effects of external dc field, stress, and temperature 30.5 Magnetopolarization effect and application to wireless sensors operating at GHz frequencies 30.6 Microwire composites as artificial dielectrics with tunable permittivity and permeability References

833 834 861 862 863

869 869 872 875 876 880 884 886 887 888

889 889 890 895 898 904 909 915

xiv

31

Contents

Nano-carbon/magnetic microwire hybrid fibers for tunable microwave functionalities D. Estevez and F.X. Qin 31.1 Introduction 31.2 Microwave absorption theory 31.3 Microwave absorption properties of nano-carbon composites and hybrids 31.4 Microwave absorption and metamaterial properties of amorphous wire composites 31.5 Tunable microwave absorption of polymer composites incorporating nano-carbon/amorphous wire hybrid fibers 31.6 Tunable negative permittivity of metacomposites incorporating nano-carbon/amorphous wire hybrid fibers 31.7 Summary and future directions Acknowledgments References

Index

919 919 920 923 928 930 946 955 956 957

965

Contributors

Flavio Abreu Araujo Institute of Condensed Matter and Nanosciences, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium Joa˜o P. Arau´jo IFIMUP and IN—Institute of Nanoscience and Nanotechnology and Departamento de Fı´sica e Astronomia, Universidade do Porto, Porto, Portugal A. Asenjo Institute of Materials Science of Madrid, CSIC, Madrid, Spain E. Berganza Institute of Materials Science of Madrid, CSIC, Madrid, Spain F. Beron Gleb Wataghin Physics Institute, State University of Campinas (UNICAMP), Campinas, Brazil Juan Maria Blanco Departamento de Fı´sica Aplicada, Escuela de Ingenierı´a de Gipuzkoa, Universidad del Paı´s Vasco/Euskal Herriko Universitatea, San Sebastian, Spain Thomas Blon Universite de Toulouse, INSA CNRS UPS, Toulouse Cedex, France Tim B€ ohnert INL—International Iberian Nanotechnology Laboratory, Braga, Portugal C. Bran Institute of Materials Science of Madrid, CSIC, Madrid, Spain Hans-Benjamin Braun Department of Materials, Laboratory of Metal Physics and Technology, ETH Zurich, Zurich, Switzerland; School of Theoretical Physics, Dublin Institute of Advanced Studies, Dublin, Ireland Sergey Bunyaev Institute of Physics for Advanced Materials, Nanotechnology and Photonics (IFIMUP)/Departamento de Fı´sica e Astronomia, Universidade do Porto, Porto, Portugal Mattia Butta Czech Technical University in Prague, Faculty of Electrical Engineering, Prague, Czech Republic Rafael Caballero Physics Department, University of Oviedo, Oviedo, Spain

xvi

Contributors

Olga Caballero-Calero Micro and Nanotechnology Institute, INM-CNM, CSIC (CEI-UAM + CSIC), Tres Cantos, Spain E. Calle Institute of Materials Science of Madrid, CSIC, Madrid, Spain Tristan da C^ amara Santa Clara Gomes Institute of Condensed Matter and Nanosciences, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium R.B. Campanelli Gleb Wataghin Physics Institute, State University of Campinas (UNICAMP), Campinas, Brazil Michalis Charilaou Department of Materials, Laboratory of Metal Physics and Technology, ETH Zurich, Zurich, Switzerland; Department of Physics, University of Louisiana at Lafayette, Lafayette, Louisiana George Chatzipirpiridis Multi-Scale Robotics Lab (MSRL), Institute of Robotics and Intelligent Systems (IRIS), ETH Zurich, Zurich, Switzerland Bruno Chaudret Universite de Toulouse, INSA CNRS UPS, Toulouse Cedex, France Xiang-Zhong Chen Multi-Scale Robotics Lab (MSRL), Institute of Robotics and Intelligent Systems (IRIS), ETH Zurich, Zurich, Switzerland Horia Chiriac National Institute of Research and Development for Technical Physics, Iasi, Romania A. Chizhik Department of Materials Physics, Basque Country University UPV/EHU, San Sebastian, Spain O. Chubykalo-Fesenko Materials Science Institute of Madrid, ICMM-CSIC, Madrid, Spain Paula Corte-Leon Departamento de Fı´sica de Materiales, Facultad de Quı´micas, Universidad del Paı´s Vasco/Euskal Herriko Unibersitatea, San Sebastian, Spain; Departamento de Fı´sica Aplicada, Escuela de Ingenierı´a de Gipuzkoa, Universidad del Paı´s Vasco/Euskal Herriko Universitatea, San Sebastian, Spain A.S.E. da Cruz Gleb Wataghin Physics Institute, State University of Campinas (UNICAMP), Campinas, Brazil Rafael P. del Real Institute of Materials Science of Madrid, CSIC, Madrid, Spain L. Herrera Diez Centre for Nanoscience and Nanotechnology, CNRS, Univ. Paris-Sud, Universite Paris-Saclay, Palaiseau, France

Contributors

xvii

Patrick R. Downey Aerospace Engineering/Materials Science and Engineering, University of Maryland, College Park, MD, United States D. Estevez Institute for Composites Science Innovation (InCSI), School of Materials Science and Engineering, Zhejiang University, Hangzhou, People’s Republic of China L. Fecova Centre of Progressive Materials, TIP, P.J. Safarik University, Kosˇice, Slovakia; Institute of Physics, Faculty of Science, P.J. Safarik University, Kosice, Slovakia Amalio Ferna´ndez-Pacheco Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom; SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom J.A. Fernandez-Roldan Materials Science Institute of Madrid, ICMM-CSIC, Madrid, Spain; Department of Physics, University of Oviedo, Oviedo, Spain; Institute of Materials Science of Madrid, CSIC, Madrid, Spain Alison B. Flatau Aerospace Engineering/Materials Science and Engineering, University of Maryland, College Park, MD, United States Rhonda Franklin University of Minnesota, Electrical and Computer Engineering, Minneapolis, MN, United States L. Frolova Centre of Progressive Materials, TIP, P.J. Safarik University, Kosˇice, Slovakia; Institute of Physics, Faculty of Science, P.J. Safarik University, Kosice, Slovakia O. Fruchart University of Grenoble Alpes, CNRS, CEA, Spintec, Grenoble, France L. Galdun RVmagnetics, Kosˇice, Slovakia; Centre of Progressive Materials, TIP, P.J. Safarik University, Kosˇice, Slovakia J. Galik Institute of Neurobiology, SAS, Kosice, Slovakia J. Gamcova RVmagnetics, Kosˇice, Slovakia Javier Garcı´a Physics Department, University of Oviedo, Oviedo, Spain P. Gawron´ ski AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Krakow, Poland J. Gonzalez Department of Materials Physics, Basque Country University UPV/EHU, San Sebastian, Spain

xviii

Contributors

Silvia Gonza´lez Physics Department, University of Oviedo, Oviedo, Spain D. Gusakova University of Grenoble Alpes, CNRS, CEA, Spintec, Grenoble, France Masamitsu Hayashi Deptartment of Physics, The University of Tokyo, Bunkyo, Japan; National Institute for Materials Science, Tsukuba, Japan M. Hennel Centre of Progressive Materials, TIP, P.J. Safarik University, Kosˇice, Slovakia; Institute of Physics, Faculty of Science, P.J. Safarik University, Kosice, Slovakia R. Hudak Faculty of Mechanical Engineering, TU Kosˇice, Kosˇice, Slovakia L. Hvizdos RVmagnetics, Kosˇice, Slovakia Jerome K. Hyun Department of Chemistry and Nanoscience, Ewha Womans University, Seoul, Republic of Korea P. Ibarra Centre of Progressive Materials, TIP, P.J. Safarik University, Kosˇice, Slovakia Mihail Ipatov Departamento de Fı´sica de Materiales, Facultad de Quı´micas, Universidad del Paı´s Vasco/Euskal Herriko Unibersitatea, San Sebastian, Spain; Departamento de Fı´sica Aplicada, Escuela de Ingenierı´a de Gipuzkoa, Universidad del Paı´s Vasco/Euskal Herriko Universitatea, San Sebastian, Spain Yu.P. Ivanov School of Natural Sciences, Far Eastern Federal University, Vladivostok, Russia M. Jaafar Institute of Materials Science of Madrid, CSIC, Madrid, Spain Bumjin Jang Multi-Scale Robotics Lab (MSRL), Institute of Robotics and Intelligent Systems (IRIS), ETH Zurich, Zurich, Switzerland R. Jurc RVmagnetics, Kosˇice, Slovakia; Faculty of Aeronautics, TU Kosˇice, Kosˇice, Slovakia Gleb Kakazei Institute of Physics for Advanced Materials, Nanotechnology and Photonics (IFIMUP)/Departamento de Fı´sica e Astronomia, Universidade do Porto, Porto, Portugal P. Klein RVmagnetics, Kosˇice, Slovakia

Contributors

xix

J€ urgen Kosel Division of Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia D. Kozejova Centre of Progressive Materials, TIP, P.J. Safarik University, Kosˇice, Slovakia; Institute of Physics, Faculty of Science, P.J. Safarik University, Kosice, Slovakia Joaquı´n de la Torre Medina Morelia Unit of the Materials Research Institute of the National Autonomous University of Mexico, Morelia, Mexico Lise-Marie Lacroix Universite de Toulouse, INSA CNRS UPS, Toulouse Cedex, France J€org F. L€ offler Department of Materials, Laboratory of Metal Physics and Technology, ETH Zurich, Zurich, Switzerland Nicoleta Lupu National Institute of Research and Development for Technical Physics, Iasi, Romania Eduardo Martı´nez Department of Applied Physics, University of Salamanca, Salamanca, Spain Marisol Martı´n-Gonza´lez Micro and Nanotechnology Institute, INM-CNM, CSIC (CEI-UAM + CSIC), Tres Cantos, Spain Miguel Mendez Physics Department, University of Oviedo, Oviedo, Spain O. Milkovic Institute of Materials Research, Slovak Academy of Sciences, Kosice, Slovakia; Institute of Experimental Physics, Slovak Academy of Sciences, Kosice, Slovakia K.O. Moura Gleb Wataghin Physics Institute, State University of Campinas (UNICAMP), Campinas, Brazil Chaitanya Mudivarthi Aerospace Engineering/Materials Science and Engineering, University of Maryland, College Park, MD, United States D. Mudronova Dept. Microbiol. Immun., UVMP, Kosice, Slovakia Manuel Mun˜oz IMN—Institute of Micro and Nanotechnology (CNM-CSIC), Madrid, Spain David Navas Institute of Physics for Advanced Materials, Nanotechnology and Photonics (IFIMUP)/Departamento de Fı´sica e Astronomia, Universidade do Porto, Porto, Portugal

xx

Contributors

Bradley J. Nelson Multi-Scale Robotics Lab (MSRL), Institute of Robotics and Intelligent Systems (IRIS), ETH Zurich, Zurich, Switzerland Michael Van Order Aerospace Engineering/Materials Science and Engineering, University of Maryland, College Park, MD, United States Frederic Ott Laboratoire Leon Brillouin CEA/CNRS UMR12, Centre d’Etudes de Saclay, Gif sur Yvette, France ´ va´ri National Institute of Research and Development for Technical Tibor-Adrian O Physics, Iasi, Romania P.J.G. Pagliuso Gleb Wataghin Physics Institute, State University of Campinas (UNICAMP), Campinas, Brazil Salvador Pane Multi-Scale Robotics Lab (MSRL), Institute of Robotics and Intelligent Systems (IRIS), ETH Zurich, Zurich, Switzerland Larissa V. Panina Institute of Novel Materials and Nanotechnology, National University of Science and Technology (NUST MISIS), Moscow, Russia; Institute of Physics, Mathematics & IT, Immanuel Kant Baltic Federal University, Kaliningrad, Russia; Institute for Design Problems in Microelectronics, RAN, Moscow, Russia Martha Pardavi-Horvath The George Washington University, Washington, DC, United States Jung Jin Park Aerospace Engineering/Materials Science and Engineering, University of Maryland, College Park, MD, United States Eva Pellicer Departament de Fı´sica, Universitat Auto`noma de Barcelona, Cerdanyola del Valle`s (Bellaterra), Spain Jose E. Perez Division of Biological and Environmental Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia; Division of Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia Luc Piraux Institute of Condensed Matter and Nanosciences, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium K.R. Pirota Gleb Wataghin Physics Institute, State University of Campinas (UNICAMP), Campinas, Brazil M. Poggio Department of Physics, University of Basel, Basel, Switzerland

Contributors

xxi

Victor M. Prida Physics Department, University of Oviedo, Oviedo, Spain Jose L. Prieto Institute of Optoelectronic Systems and Microtechnology (ISOM), Universidad Politecnica de Madrid, Madrid, Spain Mariana P. Proenca IFIMUP and IN—Institute of Nanoscience and Nanotechnology and Departamento de Fı´sica e Astronomia, Universidade do Porto, Porto, Portugal; Instituto de Sistemas Optoelectro´nicos y Microtecnologı´a (ISOM), Universidad Politecnica de Madrid, Madrid, Spain Josep Puigmartı´-Luis Institute of Chemical and Bioengineering, ETH Zurich, Zurich, Switzerland F.X. Qin Institute for Composites Science Innovation (InCSI), School of Materials Science and Engineering, Zhejiang University, Hangzhou, People’s Republic of China; Centre for Advanced Materials Technology (CAMT), School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW, Australia Victor Raposo Department of Applied Physics, University of Salamanca, Salamanca, Spain D. Ravelosona Centre for Nanoscience and Nanotechnology, CNRS, Univ. Paris-Sud, Universite Paris-Saclay, Palaiseau, France Kotha Sai Madhukar Reddy University of Minnesota, Electrical and Computer Engineering, Minneapolis, MN, United States; Electrical Engineering, University of Minnesota, Minneapolis, MN, United States K. Richter RVmagnetics, Kosˇice, Slovakia; Institute of Physics, Faculty of Science, P.J. Safarik University, Kosice, Slovakia A.De Riz University of Grenoble Alpes, CNRS, CEA, Spintec, Grenoble, France T. Ryba RVmagnetics, Kosˇice, Slovakia R. Sabol RVmagnetics, Kosˇice, Slovakia Dedalo Sanz-Herna´ndez Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom Semih Sevim Institute of Chemical and Bioengineering, ETH Zurich, Zurich, Switzerland Anirudh Sharma University of Minnesota, Electrical and Computer Engineering, Minneapolis, MN, United States

xxii

Contributors

E. Snoeck CEMES-CNRS, Toulouse, France Katerina Soulantica Universite de Toulouse, INSA CNRS UPS, Toulouse Cedex, France Celia T. Sousa IFIMUP and IN—Institute of Nanoscience and Nanotechnology and Departamento de Fı´sica e Astronomia, Universidade do Porto, Porto, Portugal Bethanie J.H. Stadler Electrical Engineering, University of Minnesota, Minneapolis, MN, United States A. Stupakiewicz Faculty of Physics, University of Bialystok, Bialystok, Poland I. Sulla Dept. Anat. Hist. Phys., UVMP, Kosice, Slovakia Sang-Yeob Sung University of Minnesota, Electrical and Computer Engineering, Minneapolis, MN, United States Liwen Tan University of Minnesota, Electrical and Computer Engineering, Minneapolis, MN, United States Elena V. Tartakovskaya Institute of Magnetism NAS of Ukraine, Kiev, Ukraine Ch. Thirion University of Grenoble Alpes, CNRS, Institut NEEL, Grenoble, France Jacob Torrejon Unite Mixte de Physique CNRS/Thales, Palaiseau, France J.-Ch. Toussaint University of Grenoble Alpes, CNRS, Institut NEEL, Grenoble, France B. Trapp University of Grenoble Alpes, CNRS, Institut NEEL, Grenoble, France Joseph Um University of Minnesota, Electrical and Computer Engineering, Minneapolis, MN, United States R. Varga RVmagnetics, Kosˇice, Slovakia; Centre of Progressive Materials, TIP, P.J. Safarik University, Kosˇice, Slovakia Z. Vargova Institute of Chemistry, Faculty of Science, P.J. Safarik University, Kosice, Slovakia M. Va´zquez Institute of Materials Science of Madrid, CSIC, Madrid, Spain Victor Vega Physics Department, University of Oviedo, Oviedo, Spain

Contributors

xxiii

Joa˜o Ventura IFIMUP and IN—Institute of Nanoscience and Nanotechnology and Departamento de Fı´sica e Astronomia, Universidade do Porto, Porto, Portugal Guillaume Viau Universite de Toulouse, INSA CNRS UPS, Toulouse Cedex, France Shixiong Zhang Department of Physics, Indiana University, Bloomington, IN, United States Yali Zhang University of Minnesota, Electrical and Computer Engineering, Minneapolis, MN, United States Wen Zhou University of Minnesota, Electrical and Computer Engineering, Minneapolis, MN, United States Arcady Zhukov Departamento de Fı´sica de Materiales, Facultad de Quı´micas, Universidad del Paı´s Vasco/Euskal Herriko Unibersitatea, San Sebastian, Spain; IKERBASQUE, Basque Foundation for Science, Bilbao, Spain; Departamento de Fı´sica Aplicada, Escuela de Ingenierı´a de Gipuzkoa, Universidad del Paı´s Vasco/ Euskal Herriko Universitatea, San Sebastian, Spain Valentina Zhukova Departamento de Fı´sica de Materiales, Facultad de Quı´micas, Universidad del Paı´s Vasco/Euskal Herriko Unibersitatea, San Sebastian, Spain; Departamento de Fı´sica Aplicada, Escuela de Ingenierı´a de Gipuzkoa, Universidad del Paı´s Vasco/Euskal Herriko Universitatea, San Sebastian, Spain Jia Zou University of Minnesota, Electrical and Computer Engineering, Minneapolis, MN, United States

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Preface

This book seeks to provide an up-to-date version, in a reasonable number of chapters, of a summary of the current state on the magnetism of wires from the nano- to the micrometric scale. In fact, this is the second edition of the book on Magnetic Nano- and Microwires after the first one, which was published in 2015 and was successfully accepted by magneticians working on that topic. In this edition, special effort has been made to collect information from different points of view and to discuss each topic as developed in the very recent years. As shown in the Table of Contents, there is a broad scope of topics where magnetic wires are relevant. Magnetic nanowires with a rectangular cross section, sometimes called nanostrips, have been considered in detail in many books and series of review articles. A number of magnetic phenomena have been discovered and investigated in these nanostrips and in ultrathin multilayer films, including the dynamics of domain walls with relevance to spintronics. In this edition, we have kept several representative chapters devoted to nanostrips, but it is not our objective to consider them in full depth. On the other hand, increasing interest is emerging on the effects of curvature on the magnetism at the nano- and micrometer scales. Curvature, a defining feature of magnetic wires, induces novel fundamental aspects including magnetochiral behavior or topological spin textures, while it also launches novel functionalities. This second edition pays particular attention to magnetic wires having cylindrical symmetry whose intrinsic curved nature facilitates the appearance of those and related effects. Cylindrical nanowires with diameter in the range of tens to a few hundred nanometers are specially considered. While these are electrochemically synthesized in most cases, alternative routes to synthesize nanotubes or heterostructured wires are also considered. The magnetic domain structure and the magnetization reversal mechanism of these materials are dominated by their cylindrical shape; current important challenges include determination of the effects of the magnetic singularity existing at the nanowire axis, imaging the internal magnetization distribution, and understanding singular effects at the microwave frequency range. In addition, magnetic nanowires offer a wide spectrum of quite interesting properties to enable many technological applications ranging from arrangements for 3-D architectures to sensors and actuators, thermopower, or robotics. Finally a number of chapters are devoted to magnetic microwires fabricated by the quenching and drawing technique, possessing a ferromagnetic core with a diameter typically in the range from less than one to tens of micrometers. Microwires with large magnetostriction reverse their magnetization in a unique way by the motion of a single domain wall. In this area, new alloy compositions have been recently achieved that offer opportunities for advanced sensing applications or for microwave-sensitive functionalities.

xxvi

Preface

This book contains 31 chapters and is organized into three main parts devoted to the following: (I) Design, Synthesis, and Properties; (II) Magnetization Mechanisms, Domains, and Domain Walls; and (III) Technological Applications. In Part I, chapters are mainly devoted to various technologies to fabricate families of nanowires and nanotubes. Additionally, Part I also describes very important properties concerning interconnected nanowires, modulations in nanowire diameter, optoelectronics and spintronics, permanent magnets, and Heusler alloys. In Part II the domain wall dynamics and interface engineering in nanostrips are first considered. Following this, investigations on emergent curvature effects and micromagnetic modeling are introduced. Chapters on relevant experimental aspects are later considered, including imaging of the magnetization configurations in cylindrical nanowires and nanotubes, ferromagnetic resonance, and spin waves. Toward the end of Part II, quite recent characteristics derived from magneto-optical properties and from the dynamics of the single domain wall propagation in magnetic microwires are discussed. Finally, Part III presents a number of technologies where the used of magnetic nanowires is proposed and, in many cases, is already incorporated. Phenomena include magnetostrictive effects for sensing devices; thermopower and thermoelectric effects for spin caloritronics; magnetic nanoswimmers for robotics; and transducers for biomedical applications. Further, magnetic microwires are being considered as sensing elements in advanced orthogonal fluxgates and in a broad spectrum of sensor applications, as well as in microwave absorption and tunable microwave functionalities. To properly cover the very wide spectrum of recent developments concerning magnetic nano- and microwires, the most recognized experts in their fields have been invited to write the chapters of this second edition. I very sincerely appreciate their outstanding contributions that have enabled this compilation. The book is dedicated to all of you, physicists, chemists, engineers, and biomedical professionals, with an interest in the exciting field of magnetism of nano- and microwires. I hope you will enjoy reading it at least as much as I have. Madrid, July 2019 Manuel Va´zquez, Editor Instituto de Ciencia de Materiales de Madrid, CSIC, Madrid, Spain

Part One Design, synthesis, and properties

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3D porous alumina: A controllable platform to tailor magnetic nanowires

1

Olga Caballero-Calero, Marisol Martı´n-Gonza´lez Micro and Nanotechnology Institute, INM-CNM, CSIC (CEI-UAM + CSIC), Tres Cantos, Spain

1.1

Introduction

Understanding the magnetic properties of nanostructures is a field under continuous evolution. Not only the low-dimensional characteristics of the materials confer unique properties to these structures, but also the high degree of freedom when fabricating them increases the possibilities of tailoring their magnetic properties. The recent development of novel technologies that provide ways of both fabrication and characterization at the nanoscale have opened the door to the obtainment of nanostructures with specifically designed magnetic properties. In this chapter, we reviewed some of these tailored nanostructures that are sometimes referred as three-dimensional (3D) nanostructures, such as diameter-modulated nanowires and composition-modulated nanowires. Then, we compared these structures to actual 3D interconnected nanostructures, those that take advantage of their design in three spatial dimensions. Finally, we have discussed in depth a fascinating 3D self-sustained magnetic nanostructure which has been developed within our research group, based on template growth inside 3D tailored alumina, which is obtained using only industrially scalable and highly tunable fabrication methods. The importance of fabricating nano-sized structured magnetic materials, discussed in different sections of this book, has attracted wide attention because they not only present different properties than bulk materials due to their low dimensions, but also have a great potential for miniaturizing technological applications. Moreover, the fabrication of low-dimensional structures presents extra degrees of freedom, which affect their characteristics, such as the nanometer diameter in the case of nanowires, the aspect ratio of the structures, the different behavior of an isolated nano-sized entity compared to their collective behavior (a single nanowire and an array of nanowires, for instance), multilayer arrangements combining different materials, etc. The control over nano-magnetism has a wide range of applications [1] and therefore has received a great deal of attention in recent years. Small-sized magnetoresistive devices or permanent magnets, miniaturized sensors, advanced information storage means, such as perpendicular recording media, are quite relevant nowadays. Controlling their magnetic properties such as anisotropy, coercive field, remanent, and saturation magnetization is fundamental to understanding the basis of magnetism and also to Magnetic Nano- and Microwires. https://doi.org/10.1016/B978-0-08-102832-2.00001-3 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Magnetic Nano- and Microwires

proceed with their applications. For instance, the control of the movements of the magnetic domain walls has been proposed as a way to store information. Moreover, magnetic nanowires fabricated via template-assisted techniques can be directly implemented into conventional devices, while embedded in the matrices, making their usage much more straightforward than using isolated nanowires. Nevertheless, the ultimate aim of using arrays of magnetic nanowires in most cases is to achieve ultrahigh density magnetic recording media by using each of the nanowires as a storage entity or by controlling individually the wall motion of the magnetic domains. Therefore, state-of-the-art nano-fabrication tools and nano-electronics are needed in order to implement these next-generation storage devices and much research is being devoted to the development of such technology. It is important to say that these cutting-edge technological developments will not be the subject of this chapter. Here we will focus on different advances made in the fabrication of 3D magnetic nanowires themselves. Most of these developments have been envisaged as a way of controlling and understanding the magnetic domain wall dynamics, which are affected by the shape anisotropy, dipolar interaction, wire diameter, inhomogeneities, etc. To date, there are still many open questions and these implementations are in most cases looking for the perfect candidates to gain control over the creation and movement of the walls. In order to produce nanowires of magnetic materials, different approaches can be used. For instance, methods such as chemical routes (e.g., solvothermal synthesis) and physical methods (molecular beam epitaxy, for instance) have been used to obtain metallic nanowires [2, 3]. In this chapter, we have focused mainly on the fabrication of magnetic nanowires inside the templates, thanks to techniques such as electrochemical deposition [4, 5] or atomic layer deposition (ALD) [6]. One of the main advantages of these techniques is that the aspect ratio of the nanowires is higher than that in chemical methods. Historically, the first templates used for this purpose were ion track etched polymers [7], or mica [8, 9], where nanowires in the range of the nanometers were fabricated. In 1995, a new family of templates was developed by Masuda and Fukuda [10], based on anodic aluminum oxide (AAO). Their fabrication consists of a rather straightforward process of two anodization steps, which produces highly ordered porous templates. Depending on the anodization conditions, a wide range of pore diameters can be obtained, and this has been deeply studied in order to decrease the diameter, achieving pores of diameter 12 nm [11]. Currently, it is possible to fabricate these templates with controlled pore diameters in the 8 to 350 nm range using different electrolytes [12], as described in Ref. [13]. The most commonly used electrolytes are sulfuric acid for nanopores of diameter around 20 nm, oxalic acid for diameter around 40 nm, and phosphoric acid for diameter 120 nm and over [14]. Another important parameter is the inter-pore distance, because once a certain ordered alumina template has been fabricated, the pore diameter can be increased by chemical etching, and then the maximum achievable pore diameter will be determined by the inter-pore distance. In this way, alumina templates with pore diameter up to 400 nm can be obtained. From the two anodization steps, the first one gives rise to a hexagonal ordering of the pores, and the second one, which maintains this ordering, defines the

3D porous alumina: A controllable platform to tailor magnetic nanowires

5

length of the final pores. In principle, if the anodization conditions are mild, pores with tens or even a hundred micron in length and homogeneous in diameter can be easily achieved. This straightforward self-ordering process to fabricate large areas of nanoporous templates has been the subject of many studies and as a result, new template architectures have been developed [15]. On the one hand, a higher ordering in the templates has been achieved by pretexturing the surface (avoiding the first anodization step) with a master and then performing the anodization [16]. In this way, different patterns can be obtained, with nonhexagonal pore ordering [17], noncircular pore geometries (square or triangular, for instance) via nano-indentation by stamps [18–20], or holographical prepatterning of the aluminum [21], etc. On the other hand, mixing mild anodization conditions with more aggressive conditions (for instance, increasing the applied anodization voltage to reach the so-called hard anodization regime) while growing the pores in the second step, gives rise to diameter-modulated pores [22]. In this way, a whole new family of the so-called 3D templates at the nanoscale has been created from this technique, which is explained in detail later in the chapter. By replicating the 3D interconnected nanohollow template with magnetic materials, one can produce novel nanostructures with tailored magnetic properties. In this chapter, we have reviewed the fabrication of 3D structures of magnetic materials at the nanoscale. We have not discussed the effects such as 3D nano-magnets in the wide description of the concept, as it was already discussed in a thorough review article of 2017 by Ferna´ndez-Pacheco et al. [1], the types of domain walls or effects at the nanoscale such as 3D arrays of magnetic vortices [23] or 3D spin textures [24]. We have focused on the ways to produce a controlled nanometric structure of magnetic materials, mainly via template-assisted fabrication using 3D templates, and also on other methods. Also, we will review the effects that this nanostructuration has in the magnetic properties, which may greatly differ from the expected ones of the non-nanostructured material. For instance, the magnetization and the magnetic easy axis can be controlled with these structures due to the shape anisotropy. This can be used in applications such as magnetic recording or spintronics. The control of the domain wall is crucial in the development of high-density data storage devices, which are also defined as 3D magnetic data storage. Nevertheless, it is still not fully understood how the dynamics of the process work. What is the influence of the shape anisotropy, the interaction between the adjacent nanowires, etc. Most of the developments in this field were triggered by controlling the domain wall motion when the magnetization is reversed, trying to obtain efficient and well-defined pinning sites for these walls. This has been studied in the different nanowire geometries, which will be presented along with the work. To begin with, different nanostructures that are sometimes referred to as “3D-magnetic nanostructures” such as diameter-modulated nanowires, or multilayer nanowires (composed of different materials) will be discussed. Special detail will be devoted to their fabrication and how the magnetic properties are modified in each case. Then, we will focus on what we define as self-sustaining 3D magnetic nanonetworks, that is, artificial arrangements of magnetic nanostructures that can be self-sustained given that they are interconnected in the three dimensions of space.

6

Magnetic Nano- and Microwires

These structures can be envisaged as the natural evolution of the previous ones, but their properties can differ greatly from those found in modulated nanowire arrays.

1.2

Modulated magnetic nanowires

After studying the behavior of 1D magnetic nanowires of materials such as cobalt [25], nickel [26], Ni-Fe [27], etc., the community started to study the magnetic nanowires with modulated diameter or multilayers. In these cases, the magnetic properties differ from those of conventional 1D nanowires. A deep study of their behavior has opened the door to further understanding the domain wall movement and interaction not only between adjacent nanowires, but also between different sections of a single nanowire.

1.2.1 Magnetic nanowires with modulated diameter As described in the introduction, templates with a certain modulation in the diameter can be produced, in the case of AAO, by using two different anodization processes, such as hard anodization and mild anodization (or even changing the anodization electrolytes) while growing the pores. In this way, the pore diameter, which depends on the anodization conditions, changes from a small diameter in the regions where mild anodization has been used to wider diameters in those where the hard anodization was implemented [22]. In the following years, these templates were used for growing inside magnetic materials via electrochemical deposition, reproducing the modulation in the diameter and giving rise to diameter-modulated nanowires. In this case, different magnetic properties associated with different diameters within the same material would be present in the resulting magnetic material. This modulation in the diameter is crucial for the understanding, for instance, the magnetization reversal dynamics, which is driven by the nucleation of the domain walls. How a magnetic domain wall nucleates and propagates depends on the nanowire diameter, and thus, having a nanowire with different diameters is an excellent test bench to explore the transitions from one kind of domain wall to another. In the case of nickel, the magnetic reversal takes place via transverse wall propagation for diameters under 55 nm, while for diameters over 55 nm it occurs via vortex formation and propagation. When a nanowire with diameters both over and under 55 nm is studied, both processes compete in the magnetization reversal, starting with a vortex wall formation in the larger diameter region which propagates and then evolves to a transverse reversal when the diameter decreases [28]. Then, it is necessary to apply a higher nucleation field to continue the propagation of the wall at the interface, and thus the domain wall is pinned at this point. Not only can electrochemistry be used to replicate the morphology of modulated AAOs, but ALD can also be used to obtain arrays of modulated magnetic iron oxide nanotubes [6]. In this work, the strong shape anisotropy of these modulated nanotubes (with diameters varying between 70 and 150 nm) affects their coercive field, which

3D porous alumina: A controllable platform to tailor magnetic nanowires

7

has higher values for tubes with smaller diameters. If only one wide segment is introduced in the middle of a long narrow tube, the coercive field drops over 30% of its original value, which is a proof of the influence of these diameter modulations on magnetic reversal. In the case of magnetic modulated nanowires grown by electrochemical deposition, there has been a number of works that have explored how these variations in diameter can be used, through domain wall pinning, to control the magnetization reversal. Different measurements, both in single nanowires (such as spatially resolved magnetooptical Kerr effect (MOKE), magnetometry, X-ray magnetic circular dichroism (XMCD) [29], or magnetic force microscopy (MFM) [30]) and in arrays of nanowires (superconducting quantum interference device (SQUID) measurements or first-order reversal curves (FORC) [31]) have been performed on a variety of modulated nanowires. In the case of arrays of nanowires, the interactions between neighboring nanowires play an important role that has to be taken into account. Nickel nanowires with different geometries have been extensively studied [32], showing that the interaction between nanowires is higher when there is a modulation in their diameter compared to nanowires with the same spacing between them and constant diameter [33]. Nevertheless, this effect could also be due to the bigger diameters of the modulated waveguides in the regions where the diameter is increased, given that with increasing nanowire diameter, without varying the distance between their centers, the interaction is also higher. Actually, these modulated nanowires show properties which are a combination of those that would be obtained in arrays of nanowires with the diameters present in the modulation [33], opening the door to tune the magnetic properties. In a more recent work, from 2017, Arzuza et al. studied the magnetic properties of nickel nanowires with their diameter single modulated between 35 and 55 nm, which are under and above the threshold for vortex propagation domain walls, respectively [31]. In order to study the behavior of the nanowires as an array, the first-order reversal curve (FORC) was magnetization reversal in the case of diameter modulated nanowires was found to occur through two different processes, a helicoidal vortex wall (which is more complicated than a vortex domain wall) and a transverse mode. Moreover, FORC measurements showed higher interaction fields in the case of modulated nanowires, along with a nonuniform stray field, a consequence of the shape modulation. This stray field is the main difference between these modulated nanowires and homogeneous ones. In 2014, a study on different geometries of the modulated Co nanowires (square and exponential connection between the low-diameter and high-diameter sections of the nanowire) showed that the shape also has an influence on the magnetic properties [34]. In general, the main differences between nonmodulated and modulated nanowires were the same as that was reported before, with higher magnetostatic interactions in the case of the modulated ones, induced by local stray fields. But within the modulated nanowires, there are subtle differences, given that in this case, the ends of the wider segments of the modulated nanowires, where the magnetic charges appear and induce the stray fields, are different for each studied morphology. That reverts on coercivity and remanence angular evolutions particular of each case, which modifies the interactions between neighboring nanowires.

8

Magnetic Nano- and Microwires

Other magnetic materials have been also fabricated as modulated nanowires, showing differences with their single-diameter nanowire counterparts. For example, in CoFe alloy [35], an increase in the squareness (SQ, defined as the ratio of remanence to saturation) and a reduction in the coercive field was found, which, according to the authors, suggest a higher magnetostatic interaction due to the stray field present in the modulated nanowires. Isolated Fe28Co67Cu5 were also investigated as isolated entities with MFM to get an insight into the effect that the diameter modulation has on the spin orientation, suggesting curling of the magnetization at the junction region [30]. In 2016, a study on the behavior of FeCoCu modulated nanowires with two different geometries (one with the big diameter sections longer than the smaller diameter sections, and the second just the other way round) was performed, obtaining from MOKE measurements from single nanowires that the domain walls were somehow pinned at the diameter modulations. In that way, metastable magnetic states are produced during the propagation of the vortex domain walls along them [36]. The spin configuration of nanowires of the same material, which has the magnetization easy axis along the nanowire length, was also studied in the so-called bamboo-like structure (with diameter modulations each 400 nm along the nanowire’s length, see Fig. 1.1). In other work by Bran et al., similar structured nanowires were studied, comparing FeCoCu and Co, which was chosen because it presents the easy axis perpendicular to the nanowire

(A)

(B)

(C)

Fig. 1.1 (A) SEM (scanning electron microscope) image of bamboo-like nanowires with diameter modulation embedded in the AAO membrane. (B) TEM (transmission electron microscope) image of a Ni nanowire with one modulation in diameter, and (C) diagram of the stray field lines (dashed arrows) outside the nanowire and the magnetization (gray arrows) within the nanowire. (A) From E.M. Palmero, C. Bran, R.P. del Real, M. Va´zquez, J. Phys. Conf. Ser., 2016, pp. 012001, under creative commons license. (B, C) reproduced with permission from K. Pitzschel, J. Bachmann, S. Martens, J.M. Montero-Moreno, J. Kimling, G. Meier, J. Escrig, K. Nielsch, D. G€orlitz, J. Appl. Phys., 109 (2011) 033907.

3D porous alumina: A controllable platform to tailor magnetic nanowires

9

length [29]. The combination of photoemission and XMCD allowed an insight into the magnetic structure and spin configuration of both the surface and the bulk of the nanowires. On the one hand, in the case of FeCoCu bamboo-like nanowires, the shape anisotropy dominates, with vortex structures at the end and at the diameter modulations. This confirms the previous results of magnetization reversal through vortex domain wall propagation with certain pinning at the modulations. On the other hand, for the cobalt nanowires, the structure was different, with vortices of alternating chirality that do not seem to appear at determined positions of the nanowire, that is, which are not connected with the diameter modulations. Studies on single-modulated Ni80Fe20 permalloy showed that the magnetic interaction is higher in the case of having two different diameter sections in the nanowire of the same length. Then, the reversal of the magnetization takes place through the propagation of vortex domain walls that nucleate at both ends and in the junction of these two sections [37]. This hypothesis was simulated and nanowires with these characteristics were fabricated and measured, showing a good agreement between predictions and experiments. All these results and studies show that these kinds of modulated nanowires are ideal to provide a configurable test bench for gaining a further understanding of domain wall dynamics and nucleation phenomena within a single nanowire and in an array of nanowires. As it has been shown, the properties of the nanowire arrays of modulated nanowires present a combination of the expected properties of nanowires with diameters corresponding to the diameters found at the wide and narrow parts of the modulated structure. Moreover, the domain wall pinning has been demonstrated in certain cases, mainly when studying single nanowires with modulated diameters, and different ways of magnetic domain reversal have been studied from both the theoretical and experimental points of view.

1.2.2 Composition modulation: Multilayered nanowires Another so-called 3D nanowires are the multilayered ones. They are fabricated by varying the composition of the layers, as in Ref. [38], where these structures are defined as “3D arrays of magnetostatically coupled nanopillars.” These structures can also be done with electrochemical methods, as it was the case for the diameter-modulated nanowires. They are presented as a way to control the magnetization reversal of the nanowires and, in the best case, a way to control the wall movement and introduce pinning states in well-defined places. Different compositional configurations have been proposed and studied, such as layers that alternate ferromagnetic and nonferromagnetic materials, for instance, nickel and copper [39], iron and copper [40], FeCoCu and copper [36], FeGa and copper [41], cobalt and gold [38], etc. Other configuration consists of combining two or more magnetic materials with different properties, such as Co and Ni [23]. In all cases, when they are fabricated via electrochemical composition inside alumina templates, the parameters that determine the behavior of the nanowires (the length of the different segments, the nanowire diameter, inter-pore distance, and materials properties), can

10

Magnetic Nano- and Microwires

be easily controlled, either by the electrochemical process, or by choosing the appropriate alumina template (pore diameter, inter-pore distance, etc.). In case of ferromagnetic and nonferromagnetic segmented nanowires, one obtains ferromagnetic disks embedded along the nanowires. Their interaction is different from 1D conventional nanowires, as it can be seen through FORC measurements [39]. Moreover, certain properties, such as the total magnetic anisotropy, change with the thicknesses of these ferromagnetic disks. It has been shown that maintaining the ferromagnetic layer constant (FeCoCu) if one increases the nonferromagnetic (copper) layer thickness, the coercivity, remanence, and susceptibility increase. On the other hand, for constant copper thickness, the susceptibility decreases with increasing FeCoCu thickness [36]. In this way, the characteristics of the nanowire array can be tailored, obtaining a decrease in the magnetic interactions for thicker nonferromagnetic layers. The main cause is that the stray fields between neighboring magnetic disks within the same nanowire or from adjacent wires depend on the distance between such disks and their dimensions. Similar results were reported in the case of Co/Au multisegmented nanowires, the coercivity and squareness of the measured hysteresis loops are larger than that in the case of pure Co nanowires [38], that is, cobalt nano-disks coupled via magnetostatic interaction. A deeper insight into the interactions between the ferromagnetic nano-disks (FeGa in this case) and nonferromagnetic sections (Cu) within a nanowire was also discussed in Ref. [41]. There, the complex magnetic interactions are described in order to further tailor the properties of such structures. In this work, dipolar interactions are also described, as well as the 3D spin structure displayed by these nanowires, which is quite complex and could be tailored by the geometrical characteristics of the nanowires. A more recent work, where a similar approach was taken, was presented in 2018 [40]. In this case, multisegmented nanowires in which the ferromagnetic section (made of iron) was kept constant in length and samples with different lengths of nonferromagnetic sections (copper) were studied. It was shown that for small copper segments (around 15 nm in length), the magnetic behavior is driven by the dipolar coupling between the ferromagnetic iron disks (these structures are shown in Fig. 1.2). On the contrary, when the nonmagnetic sections are longer, the interactions within a single nanowire are less intense and the magnetic behavior is dominated by the dipolar interactions between nano-disks from neighbor nanowires. When two different magnetic materials are used for these multilayered nanowire structures, such as nickel and cobalt, the interfaces between adjacent nano-disks provide well-defined pinning sites, as shown in Fig. 1.2B and C [23], where again the simulations and the experiments agree and help us to understand the domain wall dynamics in these structures. Further studies on these isolated nanowires have also given an insight into the dynamics of the domain walls in these cases, which propagate via 3D vortexes. The characterization of these nano-magnetic structures has triggered the development of novel measurement systems, able to follow in time the real-time motion of the domain walls (which cannot be made by more conventional methods such as MFM). Therefore, developments in ultrafast methods such as scanning transmission X-ray spectroscopy (STXM) or X-ray magnetic circular dichroism photoemission electron microscopy (XMCD-PEEM) are emerging in the field.

3D porous alumina: A controllable platform to tailor magnetic nanowires

11

(B)

(A)

(C)

Fig. 1.2 (A) SEM image of multilayered nanowire arrays, in this case, formed by alternative layers of Cu an Fe, (B) measured image of a pinned domain wall in the interface of Co and Ni layers of a multilayered nanowire obtained via differential phase contrast virtual bright field detection (VBF-DPC, developed in Ref. [23]), and (C) the micromagnetic simulations (color cones for the magnetization direction, black arrows for the stray field) of the domain wall structure. (A) Taken from S. Moraes, D. Navas, F. Beron, M. Proenca, K. Pirota, C. Sousa, J. Arau´jo, Nano, 8 (2018) 490, under creative commons license. (B, C) Reprinted with permission from Y.P. Ivanov, A. Chuvilin, S. Lopatin, J. Kosel, ACS Nano, 10 (2016) 5326–5332.

As it has been shown, the fabrication of multilayered nanowires via templateassisted electrochemical growth is quite configurable, and a wide variety of structures can be obtained with high quality in the interfaces. Moreover, the method provides the possibility of growing high aspect-ratio multilayered nanowires, which in return gives more precise measurements than that in other kinds of segmented nanowires due to the larger signals produced. These kinds of nanowires have shown to provide pinning sites that are well defined at the interfaces of the different materials that compose the nanowire, and have also been really useful to study the stray fields within the different segments of a single nanowire, providing a deeper understanding of the domain wall motion and the interactions between magnetic domains.

1.2.3 Relative orientation modulation: Radial nanowire array Different works have dealt with radially distributed nanowire arrays, such as those from R. Sanz et al. [42, 43] in 2007. In a more recent work dealing with these kinds of structures from J. Garcı´a et al., this geometry was defined as a 3D ordered array of magnetic nanowires [44], although it does not enter in what we define a 3D geometry which will be exposed in Section 1.3. These structures consist of a single-magnetic material nanowire array formed using an AAO fabricated in a wire as a template (see Table 1.1 for a schematic view on the geometry). Once this template is electrochemically filled with a magnetic material, the nanowires have the conventional hexagonal ordering with their neighbors. But in a long range, they have a cylindrical

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Magnetic Nano- and Microwires

Table 1.1 Summary of the different performances of different magnetic nanostructures. Type of structure

Main features

Advantages

Disadvantages

Modulated

Domain wall pinning.

Easy fabrication One material

Multilayered

Domain wall pinning.

Easy fabrication of the template.

Cylindrical

Radial properties modified with respect to NW-arrays.

Easy fabrication of the template.

Interconnected 3D network in ion-trackmembranes

Properties modified with respect to NW-arrays. Selfsustained structure.

Self-standing structure. Tunability of the diameter, and relative angle of the NW.

Interconnected 3D by two photon lithography

Virtually any geometry can be produced.

Highly tunable structures obtained.

Moderate control over the magnetization parameters. At least two materials grown. Higher control of the magnetic anisotropy. Only cylindrical properties are modified. Other geometries are not accessed. Need of an accelerator to fabricate the template. No control over the ordering of the nanowires. Time-consuming fabrication method. Limited resolution at the nanoscale.

Interconnected 3D network in anodic alumina (3D-AAOs)

Highly tunable geometry should give rise to tunable magnetic properties. Selfsustained structure.

Self-standing structure. High tunability of the interconnection distance. Easily scalable fabrication methods.

To be analyzed.

3D porous alumina: A controllable platform to tailor magnetic nanowires

13

disposition due to the wire where it was fabricated, obtaining nanowires oriented in all the radial directions with respect to the axis of the original aluminum wire. In a study carried out with this kind of cylindrical nickel and cobalt nanowire arrays in the aforementioned works, they showed similar behavior to the conventional nanowire arrays, although the geometry of the cylindrical nanowire array produces a hysteresis loop which lies between the parallel and perpendicular hysteresis loops of the planar nanowire array. Therefore, this geometry shows a dependence on the angle that the nanowires form with the applied magnetic field, as well as on the angle that the nanowires present among them.

1.3

Magnetic structures with a 3D geometry

As described in the introduction, we will define 3D magnetic nanostructures as those that are interconnected in the three dimensions of space. In this section, we introduce different ways of producing 3D geometries, such as those fabricated in ion-irradiated polymeric templates, in lithographically patterned resist, or in modified alumina templates. These structures have the common features of having nanowires along different spatial directions, along with an interconnected structure that allows them to be self-sustained structures once their templates have been eliminated.

1.3.1 Self-sustaining interconnected nanowire networks from ion-irradiated 3D polymeric templates A true 3D arrangement of nanowires is that resulting from the filling of a polymer template which has been ion-irradiated in two different orientations, producing nanopores in both directions that are interconnected [45]. The most common way of producing such templates is by performing two subsequent irradiations, both with a certain angle of the incident beam with respect to the surface of the polymer, which can be tailored at will. Such a template, when filled by ALD or electrochemical methods, produces an array of nanowires which form a certain angle among them and cross at certain points. Thus, once the ion-damage polymer has been removed, a self-standing structure of crossing nanowires is obtained. Moreover, the diameter of the nanowires can also be tailored by controlling the etching of the membranes once the irradiation process has been developed. Nevertheless, what cannot be controlled is the separation or ordering of such nanowires, given that the template is created by the ionic tracks left by the irradiation, and only the density of the beam can be controlled, but not the individual tracks. These kinds of structures (as shown in Fig. 1.3), sometimes referred to as 3D interconnected nanowire networks, have been studied for different applications, such as thermoelectricity [46], microbatteries [47], and magnetism. For instance, in 2015 a work by Araujo et al. was presented, where Ni and Ni/ NiFe were grown inside a template of this kind, with the nanopores forming 30° with each other [48]. The measurements on these structures resulted in magnetic properties quite different from the ones obtained in parallel-aligned nanowire arrays, as far as their anisotropy and demagnetization fields are concerned (see Fig. 1.3C). On the

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Magnetic Nano- and Microwires

(A)

(B) Normalized magnetization

1

(C)

0.5

0

–0.5

Ni –1 –10

–5

0

H(kOe)

5

10

Fig. 1.3 (A) and (B) SEM images of 3D Ni interconnected networks after the dissolution of the template. In the inset of (A), an optical image of the macroscopic network can be seen. (C) Hysteresis loop when the external field is applied out of plane (OOP, red) and in-plane (IP, blue) for a 3D Ni interconnected network. Reproduced with permission from T. da C^amara Santa Clara Gomes, J. De La Torre Medina, M. Lemaitre, L. Piraux, Nanoscale Res. Lett., 11 (2016), protected under a creative commons license.

one hand, the properties depend strongly on the relative angle of the nanowires among them, which is one of the tunable parameters in their fabrication. This, combined with the possibility of changing the material forming the nanowires, gives this configuration a high tunability. In a more recent work, of 2016, NiCo was used to produce a magnetic 3D interconnected network [49] and the magneto-transport of such nanostructure was studied, obtaining different responses at different angles between the magnetic field and the plane of the 3D interconnected network, which can be tuned by varying the composition of the NiCo alloy. Nevertheless, the tunability in these structures is limited to changing the material, given that modifications in the geometry of the 3D interconnections are not so straightforward, assuming that only the relative

3D porous alumina: A controllable platform to tailor magnetic nanowires

15

angle and the porosity can be changed when producing the template, but other parameters, such as the exact position of the connections cannot be tuned. Also the maximum angle between the nanowires is also limited by the preparation technique of the irradiation of the polymer.

1.3.2 3D nanostructures fabricated inside lithography patterned resists Another way of obtaining 3D nanostructures is via lithography. Recent developments in this field have provided the tools to fabricate nanometric structures, such as the two-photon lithography technique. With this technology, different geometries can be produced in a photoresist by illuminating it with a focused laser and producing 3D structures within it. Then, the illuminated areas can be removed and the patterned photoresist can be used as a template to grow inside magnetic materials with techniques that reproduce the obtained structure, such as electrochemical deposition. In 2018, a work where cobalt 3D structures were fabricated following this route was presented by Williams et al. [50]. One of the main advantages of this fabrication method is the versatility to realize different 3D structures that are not achievable by anodic alumina or track-etched polymer membranes, being able to access to complex 3D geometries and 3D networks. This allows the fabrication of structures which have been previously only imagined in theoretical studies, and which would give rise to nanostructured metamaterials with modified properties. Nevertheless, the twophoton lithography technique has also some limitations. For instance, the sizes that can be achieved cannot go under some hundreds of nanometers in size (approximately 430 nm is the smallest lateral feature that Williams et al. could fabricate), given that it depends on the wavelength of the writing laser. Also, the fabrication of such structures is time consuming, given that it is a serial fabrication method. This technique was used to fabricate tetrapods of cobalt (see Fig. 1.4), which were then studied to see the effect of this angled nanowires on the magnetic properties of the structures, which showed a complex domain structure.

1.3.3 Self-sustaining interconnected nanowire network 3D structures from 3D AAO templates In this section we present a technique, developed by our group, to create a 3D nanostructure that can be tailored as desired, presented in 2014 [51]. This method is based on aluminum anodization, obtaining an array of parallel nanowires connected at a certain length between them. This scaffold structure can be used as a template and filled via electrochemistry methods with different materials [52, 53]. In the case of filling it with a magnetic material, one can fine-tune the properties of the 3D magnetic nanonetwork generated. This control over the magnetic properties, such as the direction of the magnetic easy axis, coercivity and remanence values, can be mainly driven by playing with both the nanowires geometry and the separation among them. These affect the shape anisotropy and magnetostatic interaction terms, respectivelly. It is worth mentioning that this can be achieved on virtually any magnetic material, alloy,

16

Magnetic Nano- and Microwires

(A)

(B)

1.0

1.0

0.5

H

M (a.u.)

M (a.u.)

0.5 0.0 –0.5 –1.0

–0.6 –0.4 –0.2

(C)

0.0

H

–0.5 –1.0

0.0

m0H (T)

0.2

0.4

0.6

–0.6 –0.4 –0.2

(D)

0.0

0.2

0.4

0.6

m0H (T)

Fig. 1.4 (A) Outline of the illumination process within a positive resist of the tetrapod structure. Center, the scheme of the resulting structure once the hollow photoresist is filled with cobalt via electrodeposition and the resist is dissolved. (B) An SEM image of an actual cobalt tetrapods. (C) and (D) The magneto-optical Kerr effect (MOKE) loops obtained in a tetrapod array with the fields applied along different directions. Reproduced with permission from G. Williams, M. Hunt, B. Boehm, A. May, M. Taverne, D. Ho, S. Giblin, D. Read, J. Rarity, R. Allenspach, Nano Res., 11 (2018) 845–854.

or multilayers that can be prepared inside this 3D-porous alumina template, as we have shown in a recent publication [54]. We will enter in more detail into these kinds of 3D magnetic nanostructures, their performances, and behavior.

1.3.3.1 Fabrication process and morphological description of the structures The first step is the fabrication of the 3D template, which is based on a two-step anodization of aluminum, as described in Ref. [51]. The first anodization is used to produce ordered pores on the aluminum, and the second one consists of pulsed anodization alternating between mild and hard anodization, similarly as it was done to produce the modulated templates. The difference is that, after the anodization process, the templates are etched in phosphoric acid. Given that the etching rates of the mild and hard regions are different, by controlling the time of reaction one can etch away the hard regions up to a point that adjacent nanopores are connected, and thus a 3D-AAO interconnected nanostructure is produced. This structure can be described as an array of parallel nanotubes of diameter around 50 nm interconnected with a network of perpendicular nanotubes of diameter around 40 nm. The distance between the connecting nanotubes can be fine-tuned given that they are formed in the areas of hard

3D porous alumina: A controllable platform to tailor magnetic nanowires

17

anodization, and they are defined by the high-voltage pulses applied in the second step, and thus by changing the pulses, the interconnections can be made closer to each other or more separated. Then, after dissolving the remaining aluminum and evaporating a layer of chromium and gold on the one side of the 3D template, they can be filled via electrochemical deposition. It is worth noting that all the fabrication methods used here (anodization and electrochemical growth) are cost-effective and easily scalable to the industry. Once the matrix is filled, the alumina template can be dissolved and the self-standing 3D-magnetic structure can be studied. A scheme of this process, described in Ref. [52] for 3D cobalt nanostructures fabrication, is shown in Fig. 1.5. Studies on the magnetic behavior of such structures showed that these magnetic 3D nanowire interconnected networks can be designed to obtain a particular magnetic response, given that the tuning of the spacing of the nanowire transversal connections gives us an extra degree of freedom on controlling the magnetic properties of this kind of nanostructures. This provides a novel tool for obtaining tunable interconnected self-standing 3D interconnected nanowire networks that can be easily integrated into different devices for a variety of different applications, as shown in Ref. [54]. Finally, it is worth saying that the cylindrical nanowire geometry discussed in Section 1.2.3, from Refs. [43, 44], can be also combined with this 3D nano-network structure, as it has been demonstrated in Ref. [55], where a 3D nano-network structure in a cylinder was processed [55], although they have not yet been filled with magnetic material and studied.

Transversal Channels

Transversal Nanowires

Transversal Nanowires Longitudinal Nanowires

Longitudinal Nanowires

Longitudinal Channels

z (OOP)

(A)

y (IP)

x (IP)

(B)

(C)

Fig. 1.5 Images of the different steps of the 3D magnetic nanonetwork fabrication: (A–C) SEM images with an inset of the details of the structure and an schematic view of (A) empty 3D alumina matrix, (B) 3D alumina matrix filled with Ni, (C) 3D Ni nanonetwork once the alumina matrix is dissolved. Reproduced with permission from reference A. Ruiz-Clavijo, S. Ruiz-Go´mez, O. CaballeroCalero, L. Perez, M. Martı´n-Gonza´lez, Phys. Status Solidi Rapid Res. Lett., 13 (2019) 1900263.

18

1.4

Magnetic Nano- and Microwires

Summary and outlook

The possibility that technology gives us of realizing nanometric structures in a controlled way, with high tunability, has opened door to new possibilities in nanomagnetism. The possibility of creating 3D nano-networks of magnetic materials with modified properties could lead to novel applications in a wide range of fields. In the recent years, different nano-magnetic geometries have provided a way to study the different parameters that play a role in controlling the nano-magnetism independently: diameter-modulated nanowires for studying the domain wall movement, multilayer nanowires to see the interactions between different sections of the same nanowire, 3D magnetic nanostructures that show the importance of the interaction between the isolated nanowires, etc. Thanks to these structures, advancements in wall pinning and domain wall movement have been achieved, along with a further understanding of the models that help us to recreate those effects. Nevertheless, along with these novel structures, novel measurement techniques are also being developed, in order to see with enough resolution the magnetic domains or to follow in real-time the domain wall movement. These developments are crucial to finally understand and control the nanomagnetic structures and be able to design actual devices for using their possibilities for magnetic information storage or random access memories, and if the field stays as active as in recent years, it seems that such applications will be a reality in the near future.

Acknowledgments This work was supported by the MAT2017-86450-C4-3-R and Infante project. The authors acknowledge discussions with Dr. Ruy Sanz.

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[42] R. Sanz, M. Vazquez, K. Pirota, M. Hernandez-Velez, J. Appl. Phys. 101 (2007) 114325. [43] R. Sanz, M. Herna´ndez-, K.R. Velez, J.L. Pirota, M.V. Baldonedo, Small 3 (2007) 434–437. [44] J. Garcı´a, V. Prida, V. Vega, W.d.O.d. Rosa, R. Caballero-Flores, L. Iglesias, B. Hernando, J. Magn. Magn. Mater. 383 (2015) 88–93. [45] M. Rauber, I. Alber, S. M€uller, R. Neumann, O. Picht, C. Roth, A. Sch€ okel, M.E. ToimilMolares, W. Ensinger, Nano Lett. 11 (2011) 2304–2310. [46] M.F. Wagner, F. V€olklein, H. Reith, C. Trautmann, M.E. Toimil-Molares, Phys. Status Solidi A 213 (2016) 610–619. [47] A. Vlad, V.-A. Antohe, J.M. Martinez-Huerta, E. Ferain, J.-F. Gohy, L. Piraux, J. Mater. Chem. A 4 (2016) 1603–1607. [48] E. Araujo, A. Encinas, Y. Vela´zquez-Galva´n, J.M. Martı´nez-Huerta, G. Hamoir, E. Ferain, L. Piraux, Nanoscale 7 (2015) 1485–1490. [49] T. da C^amara Santa, J. Clara Gomes, M. De La Torre Medina, L.P. Lemaitre, Nanoscale Res. Lett. 11 (2016). [50] G. Williams, M. Hunt, B. Boehm, A. May, M. Taverne, D. Ho, S. Giblin, D. Read, J. Rarity, R. Allenspach, Nano Res. 11 (2018) 845–854. [51] J. Martı´n, M. Martı´n-Gonza´lez, J. Francisco Ferna´ndez, O. Caballero-Calero, Nat. Commun. 5 (2014) 5130. [52] B. Abad, J. Maiz, A. Ruiz-Clavijo, O. Caballero-Calero, M. Martin-Gonzalez, Sci. Rep. 6 (2016) 38595. [53] A. Ruiz-Clavijo, O. Caballero-Calero, M. Martı´n-Gonza´lez, Nanomaterials 8 (2018) 345. [54] A. Ruiz-Clavijo, S. Ruiz-Go´mez, O. Caballero-Calero, L. Perez, M. Martı´n-Gonza´lez, Phys. Status Solidi Rapid Res. Lett. 13 (2019) 1900263. [55] P.M. Resende, R. Sanz, A. Ruiz-de Clavijo, O. Caballero-Calero, M. Martin-Gonzalez, Coatings 6 (2016) 59.

Electrochemical methods assisted with ALD for the synthesis of nanowires

2

Javier Garcı´a, Miguel M endez, Silvia Gonza´lez, Victor Vega, Rafael Caballero, Victor M. Prida Physics Department, University of Oviedo, Oviedo, Spain

2.1

Introduction

Electrochemical anodization consists in an electrolytic passivation process usually employed for increasing the natural oxide layer thickness on the surface of valve metals. Two-step mild anodization or single-step hard anodizing processes are typical examples of bottom-up strategies for the synthesis of nanoporous anodic aluminum oxide (AAO) membranes or titania nanotube templates, commonly employed nowadays in many research areas and industrial applications. On the other hand, atomic layer deposition, ALD, technique is a thin layer deposition method based on the sequential use of a gas-phase chemical process, which reacts with the surface of a material in a self-limiting manner. Through the repeated consecutive exposure to reactive gas precursors, a very thin cover layer can be slowly deposited on the material surface. ALD can become a key process in the novel fabrication method of conformal coatings protective against corrosion for many devices; very thin films with accurate control of their thickness and composition can be achieved at the atomic level and being part of innovative tools for the synthesis of new nanomaterials, such as nanotubes or nanowires, for magnetic, semiconducting, or photocatalytic applications. Multisegmented magnetic nanowires produced by varying the chemical composition of each segment or by properly tuning the geometrical modulation in the diameter of each nanowire segment have been recently proposed as novel 3D systems of magnetic multibit memories and logical devices. This peculiar assembling of building blocks made of consecutive segments with modulated composition and/or diameter for each nanowire allows for the magnetization confinement in each nanowire segment, giving rise to arrays of nanowires with a magnetic multidomain structure along the wire length, where the interface layer at the modulation can act as pinning center for magnetic domain wall displacement. In this chapter the different approaches based on ALD to tune both surface chemistry and geometry of different types of nanoporous templates will be reviewed and discussed. Furthermore, the interest of such tuning regarding the impact on the magnetic properties of the developed nanomaterials will also be reviewed. Magnetic Nano- and Microwires. https://doi.org/10.1016/B978-0-08-102832-2.00002-5 Copyright © 2020 Elsevier Ltd. All rights reserved.

22

Magnetic Nano- and Microwires

In addition, bisegmented diameter modulated ferromagnetic nanowires are reported paying special effort to the features of the sharp transition region between two segments of different diameters, ranging from 450 nm of the thicker segment down to 30 nm for the thinner one, thus enabling a precise control over the magnetic behavior of the sample. The magnetic properties exhibited by these ferromagnetic nanowires made of pure Ni, nickel-iron, or cobalt-iron alloys were studied by vibrating sample magnetometry (VSM) technique for the bulk nanowires array or well in the single isolated nanowires by magneto-optical Kerr effect, after releasing them from the patterned alumina substrate. The influence of the sharp-diameter modulation on the magnetization reversal process of these ferromagnetic nanowires has been also examined by micromagnetic simulations performed with Mumax3 package, to determine the main magnetization reversal mechanism for each segment of the bisegmented diameter modulated ferromagnetic nanowire. Micromagnetic simulations performed on single bisegmented nanowires predict a double-step magnetization reversal, where the magnetization of the wide nanowire segment switches through a vortex domain wall, while the magnetization of the narrow one can be reversed either through a Bloch domain wall displacement or well by a corkscrew-like mechanism, depending on the diameter dimension of the narrower segment of the nanowire. In the Ni, NiFe, and FeCo bisegmented ferromagnetic nanowires here presented and discussed, synthesized by properly engineering a sharp and well-defined geometrical modulation in the nanowire diameter, each nanowire segment behaves with independent magnetization reversal process. The latter constitutes a fundamental key point for the development of novel ultrahigh-density data storage devices based on magnetic nanowire arrays with controlled reversal modes by tailored movement of domain wall displacement along nanowires. These geometrically diameter modulated ferromagnetic nanowires here studied can be considered as novel magnetic multidomain systems for ultrahighdensity data storage applications.

2.2

Atomic layer deposition technique

Atomic layer deposition (ALD) is a vapor-phase technique capable of producing a wide variety of materials. It consists of the alternation of separate self-limiting surface reactions, which enables accurate control of film thickness at the angstrom level [1]. The origin of ALD technique can be attributed to the research carried out by scientists from the former Soviet Union in the 1960s [2, 3], although it was really developed and popularly introduced by Suntola and Antson as atomic layer epitaxy (ALE), in 1977 [4]. Nearly 23 years after the technique was renamed as atomic layer deposition (ALD) because it was probed that many materials were able to be deposited nonepitaxially [5]. Since the mid-1990s, a rapidly increasing interest toward ALD conformal deposition method has taken place, as a consequence of the ever decreasing device dimensions and increasing aspect ratios needed in integrated circuits (IC) [6].

ALD for the synthesis of nanowires

23

The first industrial applications of ALD technique were mainly based on binary metal oxide ALD processes. The precursors employed in these applications were typically metal-organic compounds that contained the metal(s) or semimetal(s) of interest, such as trimethylaluminum, tetrakis(dimethylamino)tin, or tetraethylorthosilicate. In these cases, H2O, O3, or O2 plasma were employed as coreactants for obtaining metal oxides such as Al2O3 and TiO2, NH3 or N2/H2 plasmas for metal nitrides such as TiN and SiNx, and H2S for metal sulfides such as ZnS. Over time the technique has been gradually improved, allowing for the introduction of more complex materials including doped, ternary, and quaternary compounds [7]. To date a wide variety of technologically important materials has been deposited using ALD. Moreover, the selection of elements that have been successfully incorporated into ALD deposited films is ever increasing. Those reported to date are summarized in Fig. 2.1, where the periodic table of chemical elements shows that a significant number of main-group elements, transition metals, and lanthanides have been reported in films deposited by ALD, either as the principal metal (blue squares) or as the nonmetal component of binary compounds (gray squares) [8]. The ALD technique is considered by some scientists as a variation of chemical vapor deposition (CVD), but in contrast to their CVD analogs, the ALD procedures are based on a process where the chemical precursors are alternated to react and form He

H Be

B

C

N

O

F

Ne

Na Mg

Al

Si

P

S

CI

Ar

Li

Cr

Mn

Fe

Co

Ni

Cu

Zn

Ga

Ge

As

Se

Br

Kr

Zr

Nb Mo

Tc

Ru

Rh

Pd

Ag

Cd

In

Sn

Sb

Te

I

Xe

*

Hf

Ta

W

Re

Os

Ir

Pt

Au

Hg

Tl

Pb

Bi

Po

At

Rn

**

Rf

Db

Sg

Bh

Hs

Mt

Ds

Rg

Cn Uut

FI Uup Lv Uus Uuo

*

La

Ca

Pr

Nd

Pm Sm

Eu Gd

Tb

Dy

Ho

Tm

Yb

Lu

**

Ac

Th

Pa

U

Pu Am Cm

Bk

Cf

Es Fm Md

No

Lr

K

Ca

Sc

Ti

Rb

Sr

Y

Cs

Ba

Fr

Ra

V

Np

Er

Fig. 2.1 Periodic table denoting the materials deposited by ALD (December 2013). Metallic (and metalloid) elements with a blue background are those that have been incorporated into ALD films of compounds such as oxides, nitrides, and carbides. The gray background indicates elements forming the nonmetallic component of the films. Underlined symbols indicate that ALD thin films of the pure element have been reported. Reproduced with permission from H.C.M. Knoops, S.E. Potts, A.A. Bol, W.M.M. Kessels, Atomic layer deposition, In: T. Kuech (Ed.), Handb. Cryst. Growth, second ed., Elsevier, 2015, p. 1101–1134, https://doi.org/10.1016/C2013-0-09792-7.

24

Magnetic Nano- and Microwires

the desired material, often at significantly lower temperatures ( > > f ψ eq > > > π 2MS ∂2 σ

> > > > aJ 2

+Γ γcosψ eq  : ; γ 2 ∂ψ eq (10.12c)

Current-induced dynamics of chiral domain walls in magnetic heterostructures

Here

#

 ∂2 σ

2MS π Δ f ψ eq ¼ + Γ 2 α γ γaJ sin ψ eq ∂ψ 2 eq 



303

"

and ψ eq is the equilibrium (steady-state)

magnetization angle of the wall. The domain wall energy density (σ ) is defined as σ ¼ σ 0 + MS HK Δ cos 2 ψ  πΔMS HDM cos ψ

(10.13)

density that is a constant (i.e., not a function of q where σ 0 is the domain wall  energy

or ψ). Note that mτ ¼ αwtΔFM

2MS γ

gives the friction against the wall motion.

The spin-orbit torque tends to rotate the wall magnetization away from the Neel configuration (ψ eq  0 or π) to that of the Bloch configuration (ψ eq  π2 or  π2). When current is applied, one can substitute ψ eq  π2 or  π2 into Eq. (10.12b) to obtain the acceleration time τA. The deceleration time τD can be evaluated by substituting ψ eq  0 or π in Eq. (10.12b). The acceleration and deceleration times read 1 + α2 τA ¼

π

γ αH K + aJ

2 τD ¼

1 + α2



π



γα HK + HDM

2

(10.14a)

(10.14b)

The relaxation times τA and τD can be associated with the effective mass m using the relation that derives from Eqs. (10.12a), (10.12b), that is, mAðDÞ ¼

2MS αwtFM τAðDÞ γΔ

(10.15)

mA represents the effective mass when the domain wall is driven by current whereas mD corresponds to the effective mass when the wall is at rest (i.e., when the current is turned off ). Since the proportionality factor that relates m and τ is a constant, these equations indicate that the effective mass is different when the domain wall is driven by current and when it is at rest. Note that τA(D) and mA(D) evolve during the transient process and therefore are not constant.

10.3

Experimental results

10.3.1 Sample preparation and experimental methods We study magnetic heterostructures consisting of Substratejd Xj1 Co20Fe60B20 j2 MgOj1 Ta (units in nanometer) using different HM underlayers (X: Hf, Ta, TaN, and W). Films are deposited on thermally oxidized Si(001) substrates (SiO2: 100 nm thick) using magnetron sputtering. The TaN underlayer is formed by reactively sputtering Ta in Ar gas atmosphere mixed with small amount of N2 gas. The atomic composition of TaN is determined by Rutherford backscattering spectroscopy and

304

Magnetic Nano- and Microwires

contains 52  5 at% of N. Films are annealed at 300°C for 1 h in vacuum. The magnetic properties of the heterostructures (the saturation magnetization and the effective magnetic anisotropy energy) are studied in vibrating sample magnetometry (VSM). Hall bars and wires are patterned from the films using optical lithography and Ar ion etching. Subsequent lift-off process is used to form the electrical contact 10 Ta j100 Au (units in nanometer). The Hall bars used to study the spin-orbit torque are 10 μm wide and 25–60 μm long whereas the wires for domain wall motion experiments are 5 μm wide and 30 μm long. Details of the device preparation, magnetic, transport, and structural properties of the heterostructures can be found in Refs. [23, 48, 49].

10.3.2 Spin-orbit torque Adiabatic harmonic Hall measurements [48, 50, 51] are used to estimate the dampinglike and field-like components of the spin-orbit torque [51–53]. A low-frequency (500 Hz) sinusoidal voltage (amplitude: VIN) is applied to the Hall bar and the in-phase first (Vω) and the out of phase second (V2ω) harmonic Hall voltages are measured using lock-in amplifiers [48]. An in-plane field is applied along (Hx) or transverse to (Hy) the current flow. The Hx and Hy dependence of the harmonic Hall voltages provides information of the damping-like (ΔHx) and field-like (ΔHy) components of the spin-orbit torque, respectively, [54]: ΔHX ¼ 2

BX  2ξBY 1  4ξ2

(10.16a)

ΔHY ¼ 2

BY  2ξBX 1  4ξ2

(10.16b)

∂ Vω ∂V2ω ∂ Vω ΔRP 2ω We define BX ¼ ∂V ∂Hx = ∂H 2 , BY ¼ ∂Hy = ∂H 2 and ξ ¼ ΔRA , where ΔRP and ΔRA are the pla2

2

x

y

nar Hall and the anomalous Hall contributions to the Hall resistance, respectively. Both components ΔHX and ΔHY scale linearly with the amplitude of the sinusoidal voltage (VIN) at low excitation. VIN can be converted to the current density that flows through the underlayer (JN) using the resistance of the wire, the resistivity and the thickness of the underlayer X and the CoFeB layer. We fit ΔHX(Y ) versus JN with a linear function to obtain the effective field per unit current density, ΔHX(Y )/JN. Results of ΔHX(Y )/JN measured for the TaN underlayer films (Sub.jd TaN j1 CoFeB j2 MgO j1 Ta) are shown in Fig. 10.1B and C as a function of the TaN underlayer thickness (d). Solid and open symbols correspond to ΔHX(Y )/JN when the equilibrium magnetization direction is pointing along +z and z, respectively. As evident, both components increase with increasing underlayer thickness. This is consistent with the picture of spin-orbit torque that originates from the SHE of the HM layer: the magnitude of the torque scales with the underlayer thickness up to its spin diffusion length [25, 55–57], above which the torque saturates since spin current generated

Current-induced dynamics of chiral domain walls in magnetic heterostructures

(A)

Current

ΔHX

305

M

ΔHY z

y x

Sub.|d nm TaN|1 CoFeB|2 MgO|1 Ta 600 (B) ΔH /J M || +z

ΔH X(Y)/JN [Oe/(108 A/cm2)]

Y

N

M || –z

0

–600 600

(C) ΔHX /JN

0 M || +z M ||–z

–600 0

3 6 TaN thickness (nm)

Fig. 10.1 HM layer thickness dependence of the spin-orbit effective field. (A) Schematic image of the system. The motion of electrons when the SHE takes place in the underlayer (e.g., Ta). The effective field arising from the spin-orbit torque is shown by the large blue and red arrows, representing their size and direction. (B and C) The field-like (B), and damping-like (C) components of the spin-orbit effective field plotted against the TaN underlayer thickness. Solid and open symbols correspond to magnetization directed along +z and –z, respectively. The effective field is normalized by the current density JN that flows through the underlayer (TaN). Source: From reference J. Torrejon, J. Kim, J. Sinha, S. Mitani, M. Hayashi, M. Yamanouchi, H. Ohno, Nat. Commun. 5 (2014) 4655.

far from the HM j FM interface (distance larger than the spin diffusion length) loose its spin information before reaching the interface. Fig. 10.1B and C shows that ΔHY/JN is the same regardless of the magnetization direction whereas ΔHX/JN changes its direction when the magnetization direction is reversed. These results illustrate the correspondence between ΔHX (ΔHY) and aJ (bJ) in Eq. (10.1), that is: ΔHX  aJ ðm  pÞx

(10.17a)

ΔHY  bJ ðpÞy

(10.17b)

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Magnetic Nano- and Microwires

Note that p is the spin direction of the electrons entering the FM layer (see Fig. 10.1A). Due to the form of Eqs. (10.17a), (10.17b), ΔHX and ΔHY are referred to as the damping-like and field-like components of the spin-orbit “effective field,” respectively. The relative size of the damping-like and field-like components depends on the materials and thickness of the underlayer and the magnetic layer. For Ta and TaN underlayer films, we find that the field-like component (ΔHY/JN) is 2–3 times larger than the damping-like component (ΔHX/JN) [23, 48]. The direction of the damping like component is consistent with that originally predicted by Slonczewski if one assumes the electron spin that impinges on the magnetic layer points along the direction dictated by the sign of the spin Hall angle [25, 58]. The direction of the field-like component is opposite to the incoming spin direction, which is rather counterintuitive. The size and the direction of the field-like component may depend on how the electrons reflect and/or transmit the HM/FM interface [59–61]. The domain wall velocity is proportional to the size of the damping like component ΔHX and therefore depends on aJ [see Eq. (10.5)].

10.3.3 Measurements of domain wall velocity An optical microscope image of a typical wire used to evaluate domain wall velocity is shown in Fig. 10.2A. The wire width is 5 μm and we study the propagation of domain wall(s) along a distance of 30 μm using Kerr microscopy imaging. A pulse generator, which can apply constant amplitude voltage pulse (length: 1 to 100 ns), is connected to the wire. Definition of the coordinate axes is the same with that shown in Fig. 10.1A. Positive current corresponds to current flow along the +x direction. Exemplary hysteresis loops of the wire are shown in Fig. 10.2B for two films with different underlayer material (Hf and W). Hysteresis loops are obtained by the Kerr microscopy images of the wire captured during an out of plane magnetic field sweep (Hz). For all films analyzed here, the switching field is governed by the field needed to nucleate reverse domains and ranges between 100 and 500 Oe. Preparation of a domain wall (or domain walls) in the wire is carried out by applying large-amplitude voltage pulses. First, the CoFeB layer is uniformly magnetized by applying large enough Hz. The field is then removed and a large amplitude voltage pulse is applied to the wire, which can trigger magnetization reversal [62, 63]. This typically results in nucleation of one or a few domain walls within the wire. In some cases, additional field is applied after the voltage pulse application in order to form appropriate domain structure. The pulse amplitude needed to trigger magnetization reversal is in general, larger than that needed to move domain walls: thus, this sets the upper limit of the applicable pulse amplitude for studying current-driven domain wall motion. Kerr images are captured before and after the application of the voltage pulse(s) to estimate the distance the wall traveled. Fig. 10.2C and D shows such Kerr images for the two films shown in Fig. 10.2B. The upper image shows the initial state with two domain walls in the wire; the lower image illustrates the magnetic state after the application of the voltage pulse. As evident, the two domain walls travel similar distance under the application of the voltage pulse. The direction to which the walls move depends on the film structure, as will be discussed below.

Kerr intensity (a.u.)

Current-induced dynamics of chiral domain walls in magnetic heterostructures

30 μm Ta|Au y X|CoFeB|MgO

z

20 10 0

-20

(B) -600 -300 4.7 Hf|1 CoFeB|2 MgO|1 Ta

0

0 300 600 HZ (Oe) 3.1 W|1 CoFeB|2 MgO|1 Ta

Current

Current

10

30 μm

20

(C)

0

10

20

30 μm

(D)

20 15

↑↓

10 14 V

↓↑

0 1 2 3 Cumulated pulse length (μs)

Wall position (μm)

4.7 Hf|1 CoFeB|2 MgO|1 Ta

Wall position (μm)

3.5 nm Hf 3.1 nm W

-10

x

(A)

(E)

307

30 20 10

(F)

0

3.1 W|1 CoFeB|2 MgO|1 Ta

↑↓ ↓↑

16 V 0.0 0.2 0.4 0.6 Cumulated lenght pulse (μs)

Fig. 10.2 Current-driven domain wall motion of left- and right-handed chiral domain walls. (A) Optical microscopy image of the wire used to study domain wall velocity. The coordinate axes are shown together. (B) Hysteresis loops obtained by Kerr microscopy imaging. The region of interest of the captured image is converted to numbers to quantify the Kerr intensity. The intensity is plotted as a function of out of plane field (Hz) for two different film structures: W and Hf underlayer films. (C and D) Kerr images of two domain walls before (upper image) and after (lower image) the application of current pulses. The underlayer is Hf and W for the (C) and (D), respectively. (E and F) Successive position of the domain walls, shown in (C) and (D), upon application of voltage pulses plotted as a function of the cumulated pulse length. The symbols represent the position of the domain wall after application of the following voltage pulse train: (E) 14 V, 50 ns long pulses applied five times and (F) 16 V, 50 ns long pulses applied two times, each pulse separated by 10 ms.

Fig. 10.2E and F shows the positions of the two domain walls shown in Fig. 10.2C and D, respectively, as voltage pulses are applied to the wire. The cumulated pulse length is shown in the horizontal axis to extract domain wall velocity from this plot. When the driving force is large enough to overcome the local pinning, domain walls can be driven along the wire with constant velocity. However, in some circumstances when the pinning is strong or when the driving force is weak, domain walls can get

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Magnetic Nano- and Microwires

locally pinned. An example of such pinning is shown in Fig. 10.2E where the position of one of the domain walls does not change with the application of voltage pulses (the corresponding cumulated pulse length is 1–2 μs). In this case, after application of a few voltage pulses, the wall depins and restarts its motion along the wire. The average domain wall velocity is estimated by fitting the wall position as a function of cumulated pulse length with a linear function. We exclude cases when the domain wall is locally pinned. The average value of the slope of the solid lines in Fig. 10.2E and F gives the domain wall velocity. Fig. 10.3 shows the dependence of domain wall velocity on the current density JN flowing through the HM underlayer for four different film structures. Positive velocity means the domain wall moves along the + x direction. The domain walls move against the current flow for the Hf and Ta underlayer films, whereas the walls move along the X|1 CoFeB|2 MgO|1 Ta

6

(A) HP

0

~4 Oe

Domain wall velocity (m/s)

-6 4.7 nm Hf 8

(B)

0

~3 Oe

-8 1.6 nm Ta 2

(C)

0

~30 Oe

-2 25

6.6 nm TaN

(D)

0 -25 -0.8

~15 Oe

-0.4

0.0

3.6 nm W 0.4 0.8

JN (A/cm2) x108

Fig. 10.3 Pulse amplitude dependence of domain wall velocity. (A–D) Domain wall velocity plotted as a function of the current density that flows through the underlayer (JN). The thickness and material of the underlayer are shown in each panel. The domain wall pinning field (HP) for the corresponding device is displayed next to the panel. Source: From reference J. Torrejon, J. Kim, J. Sinha, S. Mitani, M. Hayashi, M. Yamanouchi, H. Ohno, Nat. Commun. 5 (2014) 4655.

Current-induced dynamics of chiral domain walls in magnetic heterostructures

309

current for the TaN and W underlayer films. The threshold current needed to trigger the wall motion depends on the film structure (we do not consider creep motion here). According to Eqs. (10.10a), (10.10b), domain walls can be moved if the spin Hall effective field aJ exceeds aCJ , which depends on the wall structure (Neel- or Bloch-like) and the domain wall propagation field HP. HP is studied using Kerr images and is defined as the average minimum out of plane field needed to move a domain wall along the wire. HP is listed together in each panel in Fig. 10.3. The threshold effective field aCJ can be calculated using the threshold current density JCN obtained from Fig. 10.3 and the effective field per unit JN from Fig. 10.1. For example, for 6.6-nm thick TaN underlayer film, JCN  0.3  108 A/cm2 from Fig. 10.3C and |Δ HX/JN |  235 Oe/(1  108 A/cm2) from Fig. 10.1C; thus aCJ ¼ JCN(Δ HX/JN)70 Oe (we assume Eqs. 10.12a, 10.12b holds here). This is larger than the propagation field (HP  30 Oe), indicating that the domain wall is  not a perfect Neel wall. Using Eq. (10.10a), these values give HDM = π2 HK  0:27 that corresponds to the direction cosine of the magnetization angle with respect to the wire’s long axis (here, along the x-axis).

10.3.4 Determination of the magnetic chirality Fig. 10.3 shows that the domain wall moves along with the current flow for the TaN (d larger than 1.6 nm) and W underlayer films whereas it moves against it for the Hf and Ta underlayer films. The direction to which a domain wall moves is determined by the sign of the spin Hall angle and the wall chirality [19, 20, 22–24], that is, the DM exchange parameter. The effective field measurements (e.g., Fig. 10.1) can, in general, provide the sign of the spin Hall angle. We find that the sign of the spin Hall angle is the same for the underlayer materials used here (Hf, Ta, TaN, and W) [23, 61]. As shown below, here it is the DM exchange parameter that differs depending on the underlayer material and consequently changes the direction to which a domain wall moves. The in-plane field dependence of the domain wall velocity is shown in Fig. 10.4 for the TaN underlayer film. The velocity scales linearly with the in-plane field in this field range. For the longitudinal field (Hx) sweep (Fig. 10.4A and B) the slope of this linear relationship changes its sign when the current direction is reversed or if the wall type is changed between "# and #" walls. In contrast, the slope is the same for "# and #" walls as well as for positive and negative currents for the transverse field (Hy) sweep (Fig. 10.4C and D). This trend agrees with the 1D model: according to Eqs. (10.5), (10.7), the sign of the slope of v versus Hx and v versus Hy is given by ΓaJ and ΓH∗X (Γ is 1 for "# wall and 1 for #" walls), respectively. As aJ depends on the current direction, the sign of the slope for v versus Hx depends on the wall type and the current flow direction. The slope for v versus Hy does not change its sign with the wall type since H∗x also depends on Γ via HDM (see Eq. 10.6). The longitudinal compensation field H∗x , illustrated in Fig. 10.4A and B, is positive (negative) for #" wall ("# wall), indicating that the domain walls are right-handed for this film (TaN underlayer). This is consistent with the negative slope of v versus Hy in Fig. 10.4C and D.

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Magnetic Nano- and Microwires

3.6 TaN|1 CoFeB|2 MgO|1 Ta

Velocity (m/s)

5

(A)

↓↑ wall

(B)

0

+I -I -80

5

-40

(C)

0 40 Hx (Oe)

↓↑ wall

+I -I

-5

80

0

-5

↑↓ wall

Hx*

0

-5

Velocity (m/s)

5

Hx*

-80 5

-40

(D)

0 40 Hx (Oe)

80

↑↓ wall

0

+I -I -80

-5 -40

0 40 Hy (Oe)

80

+I -I -80

-40

0 40 Hy (Oe)

80

Fig. 10.4 Longitudinal and transverse in-plane field dependence of domain wall velocity. (A–D) In-plane field dependence of the domain wall velocity for the TaN underlayer film. The in-plane field is directed along the x-axis (A, B) and the y-axis (C, D). Solid and open symbols represent results for positive and negative currents. (A, C) show results for #" walls and (B, D) are for the "# walls. The current density that flows through the underlayer (JN) is   0.4  108 A/cm2 (the corresponding pulse amplitude is 28 V). The solid and dashed lines are linear fit to the data.

Fig. 10.5 shows the longitudinal field dependence of the wall velocity for three different underlayer films. The sign of the compensation field H∗x is opposite for Hf, Ta, and TaN, W underlayer films (see Fig. 10.4A and B for H∗x of the TaN underlayer films), which is in agreement with the direction to which domain walls move with current, as shown in Fig. 10.3. Eq. (10.6) shows that H∗x is constant regardless of the current amplitude (aJ) if contribution from the STT term is small (u  0). We find that H∗x changes appreciably when current amplitude is varied for the Ta underlayer films. To extract the DM exchange parameter properly using Eq. (10.6), the adiabatic spin torque term u and the domain wall width parameter Δ needs to be determined. First, the current density (J) that flows into the CoFeB layer needs to be substituted in the expression of u. J is calculated using the thickness and resistivity of each layer: ρCoFeB ¼ 160, ρHf ¼ 199, ρTa ¼ 189, ρTaN ¼ 375, and ρW ¼ 124 μΩcm [23]. The saturation magnetization depends on the material and thickness of the underlayer [49]: here for simplicity we assume MS  1500 emu/cm3, which is close to that of bulk Co20Fe60B20. The current spin polarization of the CoFeB layer has been reported to be 0.7 in a similar system [46]: we use this value as a median and use the error bars to show the range of the DM exchange parameter when P is varied from 0 to 1. The domain wall width parameter is inversely proportional to the effective magnetic

Current-induced dynamics of chiral domain walls in magnetic heterostructures

311

Velocity (m/s)

3.5 nm Hf|1 CoFeB|2 MgO|1 Ta JN (A/cm2) ´ 108 10

(A)

0.36 –0.36 0.24 –0.24

0

–10 ¯ –200

Velocity (m/s)

(B)

10

¯ 0

200

–200

0

(C)

0.94 –0.94 0.82 –0.82

(D)

0

–10

200

0.5 nm Ta|1 CoFeB|2 MgO|1 Ta

¯ –200

0

200

¯ –200

0

200

Velocity (m/s)

3.1 nm W|1 CoFeB|2 MgO|1 Ta 70

(E) (F) 0.49

–0.49

0

0.35

–0.35

–70 –200

0 Hx (Oe)

¯



200

–200

0 Hx (Oe)

200

Fig. 10.5 Longitudinal in-plane field dependence of the velocity of chiral and nonchiral domain walls. (A–F) Domain wall velocity as a function of in-plane field directed along the x-axis (Hx) is shown for three different films. (A, B) Hf, (C, D) Ta, and (E, F) W underlayer films. Solid and open symbols represent the results for positive and negative currents. Results from two different values of current density flowing through the underlayer are shown using different symbols. The corresponding pulse amplitude is (A, B) 12 and 16 V, (C, D) 35 and 40 V, and (E, F) 10 and 14 V. (A, C, E) show results for #" walls and (b, d, f ) are for the "# walls. The solid and dashed lines are linear fit to the data to extract H∗x .

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi anisotropy energy KEFF of the film, that is, Δ ¼ A=KEFF , where A is the exchange stiffness constant. KEFF is determined from the magnetization hysteresis loops from VSM measurements [23] and we use A  3.1 erg/cm3 estimated from a different study reported previously [64]. In Fig. 10.6, we show the compensation field H∗x as a function of the underlayer thickness for four film structures. The background color indicates the direction to which a domain wall moves when current is applied. We fit the underlayer thickness dependence of H∗x using Eq. (10.6): the change in KEFF with the underlayer thickness is taken into account for the fitting. This fitting assumes that the DM exchange parameter does not depend on the thickness of the underlayer. The fitted values of the DM exchange parameter are summarized in Fig. 10.6E for all film structures studied.

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Magnetic Nano- and Microwires

Hx* (Oe)

200

(A)

Hf

(B)

Ta

TaN(Q:0.7%)

(C)

(D)

W

0 -200 0

3

60

D (erg/cm2)

0.4

1 20 3 6 0 Underlayer thickness (nm)

3

6

(E)

0.2 0.0 -0.2 -0.4 Hf

0 Ta

N (at%) TaN

1

W

Fig. 10.6 Underlayer material-dependent DM interaction. (A–D) The compensation field H∗x , that is, the longitudinal field (Hx) at which the velocity becomes zero, plotted as a function of underlayer thickness for (A) Hf, (B) Ta, (C) TaN, and (D) W underlayer films. Solid and open symbols represent "# and #" domain walls, respectively. H∗x is evaluated when the wall is driven either by positive or negative voltage pulses: here, both results are shown together. The background color of each panel indicates the direction to which a corresponding domain wall moves; red: along the current flow, blue: against the current flow. Solid and dashed lines represent fitting using Eq. (10.6) to estimate the DM exchange parameter D. (E) D as a function of underlayer material. The center panel shows D against the atomic concentration of N in TaN. The error bars show the range of D when contribution from STT is changed: lower (higher) bound of the error bars corresponds to P ¼ 0 (P ¼ 1) and the symbols assume P ¼ 0.7. Source: Panels A–D: From reference J. Torrejon, J. Kim, J. Sinha, S. Mitani, M. Hayashi, M. Yamanouchi, H. Ohno, Nat. Commun. 5 (2014) 4655. Panel E: From reference J. Torrejon, J. Kim, J. Sinha, M. Hayashi, SPIN 06 (2016) 1640002.

As evident, the DM exchange parameter D changes as the underlayer material is varied. D is negative for Hf underlayer and is nearly zero for Ta. It increases when nitrogen is added to Ta to form TaN, and D takes the largest value here for W underlayer. These results show that the DM exchange parameter can be controlled by the HM/FM interface [23].

10.3.5 Effective mass of chiral domain walls Eq. (10.11) shows that a domain wall can be considered a topological object with an effective mass [41, 47, 65–67]. The effective mass for a classical object is typically a constant, which results in identical acceleration and deceleration times. The acceleration and deceleration times (see Eq. 10.14a and b) of a domain wall have been found to be the same when the domain wall is driven by current [11] via the STT or by

Current-induced dynamics of chiral domain walls in magnetic heterostructures

313

magnetic field [68, 69]. Under such circumstances, the time it takes for a domain wall to accelerate is identical to time it spends to decelerate. Experimentally, when a domain wall is driven by current pulse, identical acceleration and deceleration times manifest itself as a pulse length-independent velocity [8, 11]. In this chapter, we use “velocity” as a measure of speed which is obtained by dividing the total distance the domain wall traveled during and after the pulse application with the pulse length. It turns out that the velocity of chiral domain walls in W/CoFeB-based heterostructures increases as the current pulse length is reduced, indicating that the acceleration and deceleration times are not the same. The film stack studied is Si-sub/W (d)/Co20Fe60B20 (1)/MgO (2)/Ta (1) (units in nanometers). An optical microscopy image of a representative 50-μm wide wire is shown in Fig. 10.7I inset together with the definition of the coordinate axis. Fig. 10.7A–F shows the wall velocity as a function of pulse amplitude for films with different d [70]. The pulse length (tP) is fixed to 10 ns. The velocity increases with increasing pulse amplitude until it saturates. The saturation of the velocity is one of the characteristics of chiral domain walls driven by current [24, 43]. The functional form of the velocity is assumed as

Velocity (m/s)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 JD v ¼ vD 1+ JN  JC

100 50

(B)

(A)

(C)

(D)

(E)

(F)

0 -50

-100

Velocity (m/s)

(10.18)

80

-30

0

30

-30

0

-30

30

(H)

(G)

y z 0

40

80

0

40

30

-30

0

(I)

40

0

0

80

0

30

Pulse amplitude (V)

-30

0

(J)

30

-30

0

(K)

30

(L)

50 μm x 40

80

0

40

Pulse length (ns)

80

0

40

80

0

40

80

Fig. 10.7 Pulse amplitude and pulse length dependence of domain wall velocity. (A–F) Domain wall velocity v plotted against pulse amplitude for a fixed pulse length (tP ¼ 10 ns). The red solid line represents fitting with the 1D model (Eq. (10.18)). (G–L) Pulse length dependence of v for a fixed pulse amplitude (16 V). Symbols represent the average | v| for both positive (16 V) and negative (16 V) pulse amplitudes. The W layer thickness d varies for (A)–(F) and (G)– (L) as 2.3, 2.6, 3.0, 3.3, 3.6, and 4.0 nm. Inset of (i): representative optical (Kerr) microscopy image of the device and the definition of the coordinate axis. The error bars represent the standard deviation of the velocity estimated in three sections of the wire. Film structure is sub./d W/1 CoFeB/2 MgO/1 Ta. Source: From reference J. Torrejon, E. Martinez, M. Hayashi, Nat. Commun. 7 (2016) 13533.

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Magnetic Nano- and Microwires

where vD ¼ γΔHDM is the saturation velocity and JD ¼ αJNHDM/aJ is the current denΓD sity at which the velocity saturates. HDM ¼ ΔM is the DM exchange field and s

ħθSH aJ ¼  2eM JN is the damping-like effective field due to the spin-orbit torque. Here S tFM γ is the gyromagnetic ratio, e(>0) is the electric charge, ħ is the reduced Planck constant. α is the Gilbert damping constant, Ms is the saturation magnetization, Δ is the domain wall width, and tFM is the thickness of the magnetic layer. θSH is the spin Hall angle of the HM (W) layer and D is the DM exchange parameter. We have added an empirical threshold current density JC to Eq. (10.18) in order to account for the pinning. The red solid lines in Fig. 10.7A–F show fitting of the experimental data [70] using Eq. (10.18). Except for the thinnest W layer device, we find that the saturation velocity decreases when the W layer thickness (d) is increased. The corresponding tP dependence of v for each device is plotted in Fig. 10.7G–L. For the thick W underlayer films, v increases with decreasing pulse length. This is evident when tP is shorter than 10–20 ns. These results show that the distance a domain wall travels does not linearly scale with the pulse length, which is in striking difference with the STT-driven domain walls [8, 11, 69] or current-driven narrow domain walls in large magnetic damping system [71, 72]. In contrast, v drops for shorter pulses when the thickness of W is reduced below 3 nm. According to Eqs. (10.14a), (10.14b), the acceleration and deceleration times of chiral domain walls driven by current (via the SHE) depends on HK, aJ, and HDM. HK is estimated using the relation [43] HK ¼ 4tFM MΔs log ð2Þ and the measured magnetic properties (Ms and KEFF) of the film. For the exchange constant, we use A  3.1 erg/cm3 (Ref. [64]). As the harmonic Hall measurements are not able to estimate the effective field for the W/CoFeB-based heterostructures, we use the spin Hall magnetoresistance (SMR) ΔRXX/RZXX to estimate aJ. ΔRXX is the resistance difference of the device when the magnetization of the CoFeB layer points along the film plane perpendicular to the current flow (RYXX) and along the film normal (RZXX), that is, ΔRXX ¼ RYXX  RZXX. The W thickness dependence of SMR can be fitted using the folh i 2 tanh ðd=λN Þ 1 XX lowing equation [73–75]: ΔR ¼ θ 1  SH ðd=λN Þð1 + ζ Þ RZXX cosh ðd=λN Þ . λN is the spin diffusion length of the HM (W) layer. ζ ¼ ρNtFM/ρFMd describes the current shunting effect into the magnetic layer (ρFM is the resistivity of the magnetic layer: we use ρFM ¼ ρCoFeB  160 μΩ cm). From the fitting, we obtain jθSH j  0.24 and λN  1.1 nm, similar to what has been reported previously [74, 75]. Assuming a transparent interface, the spin Hall effective h field (aJ) cani be estimated from the following equation

[25, 61]: aJ ¼ θSH JN 2eMħS tF 1  cosh1ðd=λ



. To calculate the current density JN that

flows into the W layer, we assume two parallel conducting channels (W and CoFeB layers). aJ is plotted in Fig. 10.8A when the pulse amplitude is set to 16 V. aJ decreases when d is larger than 3 nm. This is primarily due to the increase in MS for larger d (results not shown here, see Ref. [70]). The DM exchange field HDM is obtained by the saturation velocity vD via the relation vD ¼ γΔHDM. vD obtained from the results shown in Fig. 10.7A–F is plotted against d in Fig. 10.8B. The resulting DM exchange field HDM is shown in

Current-induced dynamics of chiral domain walls in magnetic heterostructures

-300

(C)

600 400 200

0 0

0.6 0.3 0.0 0

2 4 d (nm)

2 4 W thickness (nm)

(B)

50

0 0

2 4 W thickness (nm)

D (erg/cm2)

HDM (Oe)

-600 0

100 vD (m/s)

(A)

Relaxation time (ns)

aJ (Oe)

0

315

2 4 W thickness (nm)

6 4

(D) tD

2 0 0

tA 2 4 W thickness (nm)

Fig. 10.8 (A–D) Estimated acceleration and deceleration times of current-driven chiral domain walls. (A) Calculated spin Hall effective field aJ, estimated using the measurement results of the SMR, when a pulse with amplitude of 16 V is applied to the wire. (B) The saturation DW velocity (vD) estimated from fitting results of v versus pulse amplitude, shown in Fig. 10.7(A–F) with Eq. (10.18). (C) Calculated DM exchange field HDM plotted as a function of W thickness d. The inset shows the d dependence of the DM exchange parameter D. (D) W thickness dependence of the acceleration time (τA) and the deceleration time (τD) estimated using Eqs. (10.14a), (10.14b), respectively. Film structure is sub./1.5 Ta/d W/1 CoFeB/2 MgO/1 Ta. Source: From reference J. Torrejon, E. Martinez, M. Hayashi, Nat. Commun. 7 (2016) 13533.

Fig. 10.8C. The inset of Fig. 10.8C shows the d dependence of D: we find D of 0.3 erg/cm2 that is nearly thickness independent. With HK, aJ, and HDM in hand, we may calculate τA and τD using Eqs. (10.14a), (10.14b). The calculated values are plotted against d in Fig. 10.8D. We find that τD is significantly larger than τA, giving rise to a remarkable inertia effect. The difference of the two relaxation times, τD  τA, provides a good guide for the degree of inertia. These results can be compared to the pulse length dependence of the wall velocity shown in Fig. 10.7G–L For the thinner W films, we find that v for shorter pulses do not increase from its long pulse limit, indicating that the inertia effect is small. This is in agreement with the d-dependence of the difference in τD and τA which decreases with decreasing d (except for the device with the thinnest W layer). Note that for even thinner W films, the domains consist of small grain-like structures and they no longer form a uniform pattern across the device. For such films, domain walls cannot be driven by current. These results [70] demonstrate that one can tune the inertia by material design, wire dimensions and, in some cases, the size of the driving force (e.g., current pulses). Large inertia can possibly lead to lower drive current for moving domain walls from pinning sites if one makes use of resonant excitation of domain walls [47]. It is possible to tune the DM interaction in such a way that inertia becomes extremely large or small. Note that in order to observe the pulse length-dependent velocity, Eqs. (10.14a),

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(10.14b) indicate that a system with low magnetic damping constant (α) and a moderate DM exchange parameter (i.e., HDM being close to HK) are required.

10.3.6 Synchronous motion of highly packed coupled chiral domain walls In developing domain wall-based devices [7, 76, 77], one of the main challenges that need to be assessed is its scalability. In particular, it remains to be seen how dense domain walls can be packed within a quasi-1D wire. It is known that a topological repulsive force exists among vortex (and in some cases transverse) domain walls in in-plane magnetized systems [78]. Short-range repulsive interaction between domain walls may be advantageous for placing domain walls close to each other, allowing dense packing of domain walls [79, 80]. However, the repulsive interaction among the vortex (and transverse) domain walls depends on the successive alignment of magnetic charges of neighboring domain walls. For chiral domain walls, we find a universal repulsive interaction exists. The film structure used here is Sub.j1.5 Taj d W j1 CoFeB j2 MgO j1 Ta (units in nanometer). The thickness of the W insertion layer (d) is varied to control the DM exchange interaction [23]. The Ta seed layer is used to promote smooth growth of W. With the Ta seed layer, the domain structures do not break up into small grains when the W layer is thin, as was the case when W was grown directly on the Si/ SiO2 substrate. The saturation magnetization MS, effective magnetic anisotropy energy KEFF, the saturation domain wall velocity vD, and the resulting DM exchange parameter D are shown in Fig. 10.9A–D respectively. As the W insertion layer thickness is increased, D increases until it saturates at d  0.6 nm. We start from the multidomain state created by the voltage pulse application. Although the underlying mechanism remains unidentified, a large number of domain walls can be created with the application of a single voltage pulse in certain systems [81] (it is not clear what material parameters control the number of domain walls created). To study the interaction, Kerr images are taken at near-zero field before and after an out of plane field (Hz) is applied. The number of domain walls existing in the wire after application of Hz is plotted in Fig. 10.9E. The corresponding magnetic states of the wire, obtained by Kerr imaging at near-zero field, are shown in Fig. 10.9F. Here the field is applied such that it compresses the width of the bright domains (magnetization pointing along +z). As evident from the Kerr images, the width of bright domains decreases with increasing | Hz | for small fields (| Hz | <  10 Oe). The mean | Hz | at which the domain walls move is equal to the pinning field (HP). When | Hz | exceeds HP, the domain walls will annihilate if repulsive force between them is absent. Fig. 10.9E and F shows that the domain walls exist well beyond HP, indicating the presence of a repulsive force. Note that the images shown in Fig. 10.9F are taken at zero field after applying Hz: the width of the domains further reduces when Kerr image is taken at larger | Hz |. Pairs of domain walls start to annihilate one another when | Hz | >  20 Oe. The last two walls collapse at | Hz |  37 Oe. The mean value of the annihilation field HAN is

(A)

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HAN, HP (Oe)

MS (emu/cm3)

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KEFF (erg/cm3) x10 6

Current-induced dynamics of chiral domain walls in magnetic heterostructures

-33

0.5 1.0 W thickness (nm)

-35

(G)

40

-37

20 0 0.0

HAN HP

0.5 1.0 W thickness (nm)

Fig. 10.9 Repulsive interaction and formation of coupled chiral domain walls. (A–D) Saturation magnetization MS (A), the effective magnetic anisotropy energy KEFF (B), the saturation domain wall velocity vD (C) and the DM exchange parameter D (D) plotted as a function of the W insertion layer thickness d. (E) Variation of the number of domain walls as a function of out of plane magnetic field (Hz) for a film with d  0.7 nm. (F) Snapshots of the magnetic state after application of Hz: Hz is indicated in the right of each image. The images are captured at near-zero field. (G) W thickness dependence of the average Hz needed to annihilate domain walls (HAN) (black squares). The blue diamonds show the mean pinning field (HP) of domain walls. The error bars show standard deviation of HAN and HP from four independent measurements. Film structure is sub./1.5 Ta/d W/1 CoFeB/2 MgO/1 Ta. Source: From reference R.P. del Real, V. Raposo, E. Martinez, M. Hayashi, Nano Lett. 17 (2017) 1814.

defined as the | Hz | at which the number of domain walls is reduced to half of the initial state. The W layer thickness dependence of HP and the mean HAN are shown in Fig. 10.9G. Clearly, the thickness dependence of HP and HAN is different: whereas HP tends to decrease with increasing d, HAN scales with d (the reason behind the abrupt increase of HP and HAN when d approaches 1 nm is not clear). HP and HAN are nearly the same for nonchiral domain walls (D  0) when d  0 nm and the difference increases with increasing d. As studied extensively [82], part of the change in HP with pffiffiffiffiffiffiffiffiffiffi d is related to the variation of the domain wall width, which scales with 1= KEFF . In contrast, the thickness dependence of HAN may be related to the variation of D: both

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Velocity (m/s)

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i Single wall ↑↓ ↓↑ Coupled walls ↑↓ ↓↑

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ii

1x

iii

2x

iv

3x

v

4x

vi

5x

vii

11x

viii

17x

(B)

Fig. 10.10 Current-driven motion of coupled chiral walls under magnetic field. (A) Domain wall velocity versus out of plane field (Hz) for isolated (open symbols) and coupled (solid symbols) walls. Squares and up triangles (circles and down triangles) represent the velocity for "# (#") walls. The line shows linear fit to the velocity of the isolated walls around zero field. (B) Snapshots of the magnetic state of the wire after application of voltage pulses. The number of voltage pulses applied after the initial state is indicated in the right of each image. An out of plane field of 6.5 Oe is applied during the pulse application. (A, B) The pulse amplitude and length are 32 V and 10 ns, respectively. Film structure is sub./1.5 Ta/0.6 W/1 CoFeB/2 MgO/1 Ta. Source: From reference R.P. del Real, V. Raposo, E. Martinez, M. Hayashi, Nano Lett. 17 (2017) 1814.

HAN and D scale with d in a similar way (except when d is close to 1 nm). We find nearly a 60% increase in HAN when comparing wires with D of 0 (d  0.1 nm) and 0.25 erg/cm2 (d  0.6 nm). This is in accordance with micromagnetic simulations [81], which predict a similar increase in HAN with D. These results show that there is a strong repulsive force among the domain walls with large D. We next study the motion of two neighboring domain walls under application of an out of plane field. The open symbols of Fig. 10.10A display the current-driven velocity of an isolated (single) domain wall as a function of Hz applied during the voltage pulse application [81]. The velocity (v) of a single domain wall shows a significant dependence on Hz. Close to zero Hz, we fit the results with a linear function, which are plotted by the solid lines in Fig. 10.10A. As evident from Eq. (10.4), the slope of v versus Hz is inversely proportional to the damping-like effective field aJ. Thus variation of v with Hz is inevitable for chiral domain walls driven by current via the SHE of the HM layer, which is not preferable for technological applications. Fig. 10.10B shows successive Kerr images of the current-driven domain wall motion under the application of Hz (6.5 Oe). As the velocity for "# and #" walls are different due to the out-of-plane field, the two walls initially approach each other (panels i–iv). Interestingly, once the two walls merge to form a coupled state, the walls move together along the current flow (panels v–viii). This indicates that the velocities

Current-induced dynamics of chiral domain walls in magnetic heterostructures

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of the "# and #" walls are the same once they form the coupled state. The velocities of the coupled domain walls are plotted in Fig. 10.10A with solid symbols. We find that velocity of the coupled state is approximately the same with that of a single wall with Hz ¼ 0 and that it shows little dependence on Hz. Note that the walls no longer form a coupled state when Hz is positive for the configuration shown in Fig. 10.10A and B: positive Hz here promotes separation of the two walls. Micromagnetic simulations [81] show that Hz is compensated by the repulsive dipolar field from the neighboring domain wall when the walls form a coupled state: Hz sets the distance between the two domain walls such that the two walls experience net zero field. Thus the current-driven velocity of the coupled walls under nonzero negative HZ remains the same with that of an isolated wall at zero field. The distance between the two walls that form the coupled state decreases with increasing | Hz |: simulations show that the distance decreases by nearly one decade when | Hz |  20 Oe is applied compared to that at | Hz | ¼ 0 Oe. The simulations suggest that it is possible to move coupled domain walls with current pulses, placed as close as 50 nm, owing to the dipolar field between the magnetic moments of chiral Neel walls. We note that the repulsive force between the chiral Neel walls increases with increasing D, allowing further reduction of the separation distance [80]. Note that the interlayer exchange coupling can be used to form coupled domain walls, which can be driven close to the speed of sound via the spin orbit torque [83, 84]. Finally, we show current-induced synchronous motion of highly packed coupled domain walls [81] (number of domain walls is 18–24) in Fig. 10.11. A large number of domain walls are created by an application of a voltage pulse (typically, much longer and larger pulse than what is needed to move a domain wall). At zero field, all domain walls move in sync along the current flow direction (Fig. 10.11B). When a negative out of plane field that compresses the bright domains is applied (Fig. 10.11A) neighboring walls will first move in opposite direction (see the open symbols in Fig. 10.10A). However, owing to the repulsive interaction and the formation of the coupled states we find that the domain walls do not annihilate one another and move synchronously, even under the application of Hz. When a positive Hz is applied, the bright domains are initially expanded and the dark domains become compressed. The coupled walls at Hz  10 Oe also move synchronously with voltage pulses. During this process, however, we find that one coupled state has been annihilated (compare the top and the next image of Fig. 10.11C). The annihilation of the coupled domain walls during their motion needs to be addressed in particular for device applications. The results shown in Fig. 10.9 suggest that the distribution of the annihilation field HAN among the 20 domain walls is quite large. Such distribution may arise either from the spatial variation of pinning or the DM interaction. Recent reports on similar systems have shown that the DM exchange strength may vary locally in a significant way [85]. Such large variation of D along the wire can explain the annihilation [86] of coupled domain walls while they are moving: once the walls enter a region in which D is locally small, the repulsive interaction will reduce and thus may allow easier annihilation.

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~−8 Oe +35 V, 10 ns

~0 Oe +35 V, 10 ns

~10 Oe +35 V, 10 ns

−35 V, 10 ns

−35 V, 10 ns

−35 V, 10 ns

(A)

(B)

(C)

Fig. 10.11 Current-controlled motion of highly packed chiral domain walls. (A–C) Snapshots of the magnetic state of the wire after application of voltage pulses. Between each image, five voltage pulses are applied. The pulse amplitude and length are described in the legend. The out of plane field (Hz) is 8 Oe (A), 0 Oe (B), and 10 Oe (C). Film structure is sub./1.5 Ta/0.6 W/1 CoFeB/2 MgO/1 Ta. Source: From reference R.P. del Real, V. Raposo, E. Martinez, M. Hayashi, Nano Lett. 17 (2017) 1814.

Controlling the uniformity of D may become essential for moving a large number of domain walls with current.

10.4

Conclusion

We have described the underlying physics of current-driven domain wall motion in ultrathin magnetic heterostructures. With the introduction of the spin-orbit torque and chiral magnetic structure, domain walls can be moved along the wire either along or against the current flow depending on the material and stacking order of the magnetic heterostructure. In order to fully utilize spin-orbit torque and chiral magnetic structure to move domain walls formed in ultrathin magnetic heterostructures, it is essential to find a film structure in which the spin-orbit torque and the DM interaction are greatly enhanced. According to the 1D model, Neel walls can be moved with current only when the spin Hall effective field exceeds the wall pinning field. Thus to simultaneously achieve thermally stable domain walls and low threshold current, one needs to find a system in which the spin-orbit torque becomes sufficiently large to overcome the large pinning field needed for high thermal stability. This is in contrast to adiabatic STT-driven domain walls, where the threshold current is not related to the pinning field. With the engineering of the film stack and materials innovation, however, we hope that this field will further grow and develop viable technologies in the near future [76].

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Acknowledgments We thank J. Sinha for sample preparation and film characterization, J. Kim for the measurements of the spin Hall effective field, and M. Yamanouchi, S. Fukami, S. Takahashi, S. Mitani, S. Maekawa, H. Ohno for helpful discussions. The work by V.R. and E.M. was supported by Project No. MAT2017-87072-C4-1-P from the (Ministerio de Economia y Competitividad) Spanish Government and Project No. SA299P18 from Consejeria de Educacion Junta de Castilla y Leon. R.P.d.R. thanks Subprograma de Movilidad from the Spanish Ministry of Education, Culture and Sports (PRX14/00311). This work was partly supported by the Grantin-Aid (25706017, 15H05702, 16H03853) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST) from Japan Society for the Promotion of Science (JSPS) and the MEXT R&D Next Generation Information Technology.

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[82] R.C. O’handely, Modern Magnetic Materials: Principles and Applications, John Wiley & Sons, 2000. [83] S.-H. Yang, K.-S. Ryu, S. Parkin, Nat. Nanotechnol. 10 (2015) 221. [84] A. Hrabec, V. Krizˇa´kova´, S. Pizzini, J. Sampaio, A. Thiaville, S. Rohart, J. Vogel, Phys. Rev. Lett. 120 (2018) 227204. [85] I. Gross, L.J. Martı´nez, J.P. Tetienne, T. Hingant, J.F. Roch, K. Garcia, R. Soucaille, J.P. Adam, J.V. Kim, S. Rohart, A. Thiaville, J. Torrejon, M. Hayashi, V. Jacques, Phys. Rev. B 94 (2016) 064413. [86] S. Woo, K. Litzius, B. Kruger, M.-Y. Im, L. Caretta, K. Richter, M. Mann, A. Krone, R. M. Reeve, M. Weigand, P. Agrawal, I. Lemesh, M.-A. Mawass, P. Fischer, M. Klaui, G.S. D. Beach, Nat. Mater. 15 (2016) 501.

Further reading J. Torrejon, J. Kim, J. Sinha, M. Hayashi, SPIN 06 (2016) 1640002.

Joule heating and its role in current-assisted domain wall depinning in nanostrips

11

Jos e L. Prietoa, Manuel Mun˜ozb, Vı´ctor Raposoc, Eduardo Martı´nezc a Institute of Optoelectronic Systems and Microtechnology (ISOM), Universidad Politecnica de Madrid, Madrid, Spain, bIMN—Institute of Micro and Nanotechnology (CNM-CSIC), Madrid, Spain, cDepartment of Applied Physics, University of Salamanca, Salamanca, Spain

11.1

Introduction

Spin transfer torque (STT) refers to the transfer of magnetic moment from the conduction electrons to the local magnetization [1]. This transfer mechanism allows to manipulate magnetization with a spin-polarized current without the need, in principle, of an external magnetic field. The introduction of some very popular applications such as the racetrack memory [2] led to a very intense research on the effect of spinpolarized currents in magnetic domain walls. A racetrack memory relies on the precise movement of a sequence of magnetic bits (domains separated by magnetic domain walls) in a memory element. The movement of this sequence of magnetic bits should be done exclusively by an electric current, via STT. The experimental studies in this field are usually performed on ferromagnetic nanostrips, so the current required to see STT effects is not prohibitive, and the associated Joule heating is manageable. Nevertheless the minimum current required to move a magnetic domain wall (DW) in a nanostrip (critical current) is quite large, particularly in soft materials with in-plane anisotropy (such as permalloy, Py) or in nanostrips with engineered constrictions (notch), often used to stop the DW at a particular position [3–5]. Current densities of the order of 1012 A/m2 are commonly required to achieve the movement of a DW in a magnetic nanostrip. Understandably, using such a large current density has the by-product of a sizable Joule heating, even when the current is applied in short nanosecond pulses. It is therefore very important to be meticulous while estimating the local temperature generated by Joule heating. A high temperature can take the ferromagnetic material close to or even above its Curie temperature that can easily lead to misinterpretation of the experimental results. Other effects such as the buildup of lateral thermal gradients may play a role in some experiments [6, 7]. The electric current generates important thermal gradients close to the contact pads [8, 9] and also close to the constrictions (notches) along the nanostrip, as we will show later. In this chapter, we will discuss general aspects of the thermal characterization of magnetic nanostrips deposited on a substrate and the effect of the temperature on Magnetic Nano- and Microwires. https://doi.org/10.1016/B978-0-08-102832-2.00011-6 Copyright © 2020 Elsevier Ltd. All rights reserved.

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magnetic transport measurements. The first three sections are dedicated to the thermal problem and some considerations on the fabrication of the nanostrips. The second part of the chapter is dedicated to the micromagnetic model, including the thermal effect. In Section 11.4, we present the micromagnetic model needed to provide a full description of the current-assisted DW depinning process, where the magnetization dynamic excited by current pulses is coupled to the heat transport to account for the relevant Joule heating effects discussed in the previous sections. The micromagnetic results are presented in Section 11.5, and the main conclusions are discussed in Section 11.6. The temperature in a metallic nanostrip attached to a substrate follows heat diffusion equation: r2 T +

Q CV ∂T ¼ k k ∂t

(11.1)

with T the temperature, k the thermal conductivity, CV the heat capacity per unit of volume, and Q the heating term, which for the case of Joule heating is given by Q ¼ ρJ2, with ρ the electric resistivity of the ferromagnetic stripe and J the current density. This equation describes the transport of heat in a diffusive regime, and it is usually applicable in all the problems related to STT because the mean free path and the lifetime of phonons in the materials under study are considerably smaller than the sizes and timescales relevant in the works discussed in this chapter. It is important to note that the Joule heating term Q ¼ ρJ2 is given in units of energy per unit of volume. Therefore, for a given current density, the heat generated is proportional to the volume of the nanostrip (ultimately its cross section), while the heat dissipated to the substrate is proportional to the contact area with the substrate. For instance, a nanostrip with cross section w  t ¼ 300 nm  10 nm would generate four times more heat than another with cross section 150 nm  5 nm, while the dissipation area is only double. Therefore the smaller the thickness to width ratio in the nanostrip, the better to avoid the unwanted Joule heating. This fact is also reflected in some of the few experimental attempts to estimate the temperature in ferromagnetic nanostrips. Yamaguchi et al. [10, 11] estimated a temperature increase of about 500 K in a 240 nm  10 nm permalloy nanostrip for a current density of 7.5  1011 A/m2. On the other hand, Vernier et al. [12] estimated a mild temperature increase of 100 K in a 120 nm  5 nm permalloy stripe (a fourth of the cross section used for Yamaguchi) for a similar current density of 6  1011 A/m2. Finally, Togawa et al. [13] in a 500 nm  30 nm permalloy nanostrip reached the Curie temperature of the permalloy with only 2  1011 A/m2. Minding the particularities of each experiment, these examples reinforce the idea that larger nanostrips would build up a higher temperature for a given current density. Additionally, the temperature in a nanostrip could be estimated using one of the available theoretical expressions for an ideal 2-D geometry [14] or for a more realistic 3-D geometry [15], although this expression needs an experimental input to provide a value of the temperature. These expressions do not contemplate rapid changes of temperature when the current is delivered in nanosecond pulses, and they cannot provide

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the vital information of the local temperature at every point of the device when the nanostrip is not uniform (for instance, in constrictions) or when there are contact pads dissipating heat. In experiments related to STT-assisted DW movement, knowing how the temperature is building up at the nanosecond scale can be vital for the correct interpretation of the results. A magnetic DW on a magnetic nanostrip often travels very fast (tens to hundreds of meters per second). Therefore, in a nanostrip few microns long, the magnetic switching usually happens in the range of nanoseconds. A precise knowledge of the local temperature around the DW and how this temperature evolves as the DW moves or transforms along the nanostrip is only possible with the help of a computer simulation. Numerical methods have been applied recently to characterize the Joule heating in magnetic nanostrips [9, 16]. Despite the useful results reported in these works, there are aspects that may need to be addressed in more detail. For instance, the material properties are usually considered constant with temperature, and some of them change substantially as the nanostrip gets hotter. Another example is the current density, which is often considered constant, but it changes quickly in a matter of nanoseconds when the stripe heats up. Therefore, before entering the particularities of the thermal simulations in a magnetic nanostrip, it is important to summarize some general notions on the parameters that can be of particular importance to obtain a precise result.

11.1.1 General aspects for the thermal characterization of magnetic nanostrips As we have seen in the introduction, Joule heating is a cause of concern when interpreting transport measurements in magnetic nanostructures. The use of computer programs such as COMSOL has a great potential characterizing a particular problem. On the other hand, feeding the right parameters to these programs is not necessarily straightforward, and one can easily reach an incorrect result by making wrong assumptions or by selecting material parameters that do not correspond to the experimental reality. Here, we list some points that we consider quite important when facing a thermal characterization.

11.1.1.1 The thermal conductivity This is obviously a pivotal parameter to determine the temperature that a material will reach in steady or unsteady state. The thermal conductivity is usually assumed constant with temperature, but this is the case only for metals in a range of temperatures close to room temperature. For insulators (the substrate) the thermal conductivity is usually inversely proportional to the temperature. A commercial software should account for this thermal dependency, at least with bulk values. For the metallic nanostrip or for the metallic contacts, which are patterned thin films deposited with a given technique (sputtering, MBE, PLD, e-beam evaporation,

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etc.), the thermal and conductive properties are often far from bulk values. For the particular case of ferromagnetic materials, it is well known that the electric resistivity of a thin film increases as the thickness decreases and its value may change substantially depending on the deposition conditions. For instance, values between 25 and 65 μΩ cm have been reported [17] for permalloy films deposited at different substrate temperatures. As the electric conductivity of the thin film (or a nanostrip) can be experimentally determined with ease, one can use its value to determine the thermal conductivity. Thermal (k) and electric (σ) conductivity in a metal is related by the well-known Wiedemann-Franz (WF) law, k/σ ¼ LT, with L the Lorenz number, which is usually assumed constant, with the value given by the Sommerfeld theory for free electron, 2.45 WΩK2. Assuming a constant value for the Lorenz number is a good first approximation, but for a fine analysis, it is important to keep in mind that it is a material-dependent parameter, and its value can change in thin films [18]. In the literature, it is not rare to find studies where the bulk value of the thermal conductivity is used for a metal deposited in a thin film and patterned as a nanostrip. This may result in a coarse approximation. For instance, copper is often used as a contact with the ferromagnetic nanostrip, but a sputtered thin film of Cu can easily have an electric conductivity less than half of the bulk value, which would lead to a thermal conductivity less than half of the bulk value. Obviously, in these circumstances, taking the bulk value of the thermal conductivity would result in a relevant underestimation of the temperature that the nanostrip is reaching for a given current density.

11.1.1.2 Is the injected current pulse short enough? To minimize the associated Joule heating, a common approach is to use a pulsed current, rather than a DC current. If the current pulse is too long, the nanostrip would reach the steady-state temperature and having pulsed or DC current would make no difference. For a quick estimation of the time required to reach the final temperature (steady state) in a heating process, we may assume that the temperature increases exponentially as   T ðtÞ ¼ Tfinal  Tfinal  TRT et=τ

(11.2)

where the characteristic heating (or cooling) time τ ¼ RthCth is expressed in terms of the capacity of the nanostrip to accumulate thermal energy Cth and the equivalent thermal resistance provided by the substrate to dissipate the heat Rth. As a first approximation, we can calculate the thermal capacity as Cth ¼ CVAz, where CV is the heat capacity per unit of volume of the nanostrip (for instance, permalloy), A the area, and z the thickness of the nanostrip. Also the dissipation thermal resistance can be expressed as Rth ¼ s/(ksub A), with s the thickness of the substrate, ksub the thermal

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conductivity of the substrate, and A the area. Therefore the characteristic heating (or cooling) time is given by τ¼

CV zs ksub

(11.3)

In the vast majority of the experiments involving magnetic nanostrips, the substrate of choice is Si with a thermal SiO2 layer on top. It is the thermal SiO2 layer the one with strongest contribution to the thermal resistance Rth. Using kSiO2 ¼ 1 Wm1K1 and CV ¼ 4.4  106 JK1m3 for permalloy and assuming a typical thickness of the nanostrip z ¼ 20 nm, we obtain a characteristic heating time of 2.2 ns for a thickness of the SiO2 layer of s ¼ 25 nm, that is, the nanostrip would reach 95% of its final temperature in 3τ ¼ 6.6 ns. If we use 100 nm thickness for the SiO2 layer, the permalloy nanostrip would reach 95% of its final temperature in about 26 ns. These values obtained with such a simple calculation are amazingly close to the results obtained with a simulation (see, for instance, fig. S3 in the supplementary information of Ref. [9]). The reason behind obtaining a precise heating time despite the simplicity of the model is the fact that most of the heat dissipates to the substrate beneath the nanostrip. If the nanostrip has metallic contacts, then thermal conduction through the contacts is very relevant for the part of the nanostrip adjacent to this contact. With the earlier approximation, it is easy to realize that, even if the magnetic nanostrip is deposited on a good thermal insulator, it would reach its steady-state temperature in less than a microsecond. To obtain 3τ ¼ 1 μs for a 20 nm-thick permalloy nanostrip, the thermally insulating SiO2 layer should be approximately 4 μm thick. This is an unlikely experimental scenario, as the researchers working in this field want good thermal conduction in the substrate to dissipate the unwanted heat as effectively as possible. Therefore, in experiments with magnetic nanostrips, when the electric current is delivered in microsecond-long pulses, the maximum temperature of the nanostrip is likely as high as it would be working with a DC current. For most practical purposes, only current pulses in the range of a few nanoseconds would reduce the maximum temperature that the nanostrip reaches for a given current density. This implies the use of a nanosecond pulse generator that is considerably more expensive than a microsecond-long pulse generator. Also the pulse has to be delivered through a transmission line with good (50 Ω) impedance matching; otherwise the current transmitted to the ferromagnetic nanostrip will have nothing to do with a square-shaped pulse. If the setup is designed correctly, the current transmitted from the pulse generator to the nanostrip is given by the expression I¼

2Vp Rns ðT Þ + 2Z0

(11.4)

where Vp is the voltage requested to the pulse generator; Z0 is usually 50 Ω; and Rns(T) is the resistance of the nanostrip, which is temperature dependent. As we will see in the

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following sections, as the temperature in the nanostrip increases, the current decreases following the formula (11.4). This happens in nanoseconds, and the current density can decrease appreciably. Therefore one has to be cautious when assuming constant current density if the experiment delivers the current in nanosecond-long pulses.

11.1.1.3 Thermal contact resistance The interface between a ferromagnetic nanostrip and the substrate separates two different materials and often very different crystalline structures. This translates in a nonperfect heat transfer between both materials, which results in an equivalent thermal contact resistance (TCR). TCR is frequently used when dealing heat transfer problems [19], but it has been invariably neglected in studies involving transport measurements with magnetic nanostrips. A typical value for the TCR in a metal/SiO2 interface around room temperature should be in the range of 2–3  108 W1m2K [20, 21]. To have an idea of how relevant this value is, we take the definition of the thermal resistance of a plane wall Δ x/kA, with Δ x the thickness of the plane. For a given heat flow, a TCR of 3  108 W1m2K is therefore equivalent to a 30 nm-thick layer of SiO2 (thermal conductivity kSiO2 ¼ 1 Wm1K1). This means that, even if one chooses a particularly good thermal conductor as a substrate such a diamond, TCR may deteriorate considerably the thermal dissipation of the nanostrip. For instance, some of us have successfully fabricated nanostrips on sapphire, electric insulator with a considerably higher thermal conductivity than SiO2. The resulting devices (data not published) would get irreversibly damaged with a tenth of the current density required to destroy a device deposited over 25 nm SiO2. This outcome is likely the cause of a large TCR. Therefore selecting a good thermal conductor as a substrate without the appropriate choice of buffer material may not solve the Joule heating problem. Admittedly an experimental determination of the TCR is not straightforward [21]. Therefore, although it may not be practical to estimate TCR in most experiments involving magnetic nanostrips, it is important to keep its existence in mind and to be aware that the temperature in the nanostrip could be higher than what the simulation may show. When the nanostrips blow up for a seemingly low current density, a high TCR may well be the reason behind.

11.2

Thermal behavior of a ferromagnetic nanostrip

One method to calibrate the increase of temperature in the nanostrip is with the help of a high-frequency oscilloscope. As shown in Fig. 11.1A, the nanostrip is connected between a nanosecond pulse generator and a high-frequency oscilloscope. Typically the oscilloscope has 50 Ω input impedance, so the voltage that it measures is proportional to the current transmitted through the nanostrip. If the nanostrip keeps the temperature constant during the current pulse, the oscilloscope would record a perfectly square pulse (red dash line in Fig. 11.1B). On the other hand, if the nanostrip experiences a sizable increase of temperature and the associated increase in resistivity, the

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High-frequency oscilloscope

(A) 600

Temperature (K)

Voltage (mV)

160 120 80 40 0 –20 0

(B)

20 40 60 80 100 120

Time (ns)

500

TCR

400 300

(C)

0

20

40

60

80

100

Time (ns)

Fig. 11.1 Schematics of the setup required to calibrate the temperature in a nanostrip for nanosecond-long pulses (A). Red dash line indicates the ideal pulse that the high-frequency oscilloscope would record if there was no heating. The blue experimental curve is what is obtained if the heat increases the resistivity of the nanostrip (B). Experimental curve of temperature vs time obtained with the curve in (B) and with a previous calibration (not shown) of resistivity vs time. The continuous lines are fittings with and without TCR (C). Adapted from E. Ramos, C. Lo´pez, J. Akerman, M. Mun˜oz, J.L. Prieto, Joule heating in ferromagnetic nanostrips with a notch, Phys. Rev. B 91 (2015) 214404.

total resistive load increases, and the current delivered to the oscilloscope would decrease in real time, as shown in the blue pulse in Fig. 11.1B. To convert the time-dependent voltage of Fig. 11.1B to temperature vs time, a previous calibration of the resistivity vs temperature is required. This was done by Hayashi [22] and later by Ramos et al. calibrating the resistivity on nanostrips of the same dimensions as the ones used in the experiment [8]. The result can be seen in the blue experimental curve of Fig. 11.1C. The temperature rises following, roughly, expression (11.2). The rise time depends, as explained previously, on the dissipation thermal resistance, which is proportional to the thermal resistivity of the substrate and, crucially, to the TCR. As shown in Fig. 11.1C, the right rise time is obtained when the correct value of TCR is pumped into the simulation. On the other hand, there may be other parameters that are uncertain in the thermal simulation. Therefore it is a good approach to repeat the calibration for several amplitudes of pulses, as shown in Fig. 11.2, and/or for several widths of the nanostrip. Matching all the experimental curves to the simulations should only be possible if all the fitting parameters are correct. Once the thermal properties of the different materials involved in the experiment are characterized, one can safely simulate what is happening at the nanoscale level of a notch. Fig. 11.3A shows the typical geometry of a nanostrip used repeatedly in experiments exploring spin transfer-assisted DW depinning. The contact lines (or bit lines)

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Temperature (K)

900

750

600

450

300

0

25

50 Time (ns)

75

100

Fig. 11.2 Temperature vs time calibration curves for pulses of different amplitudes, from Vp ¼ 1 V (bottom curve) to Vp ¼ 4 V. The fit is not perfect at the beginning of the pulse likely due to an imperfect impedance matching of the device. Adapted from E. Ramos, C. Lo´pez, J. Akerman, M. Mun˜oz, J.L. Prieto, Joule heating in ferromagnetic nanostrips with a notch, Phys. Rev. B 91 (2015) 214404.

are 500  50 nm2 made out of Au, and they are in contact with the permalloy nanostrip and the substrate SiO2(400 nm)/Si. The ferromagnetic nanostrip is 5 μm long, 300 nm wide, and 12 nm thick, which includes the buffer and the capping layers. The data shown in Fig. 11.3, adapted from the data shown in Ramos et al. [8], include a fairly large TCR of 5.5  108 m2K/W. We will discuss the influence of the value of TCR later. The ferromagnetic nanostrip has a typical triangular notch 300 nm wide and 100 nm deep. Fig. 11.3A shows the temperature profile along the nanostrip at the end of a 100 ns and Vp ¼ 2.5 V current pulse. There are three important aspects to highlight. The first one is the large temperature reached in the nanostrip, which in the notch is well over the Curie temperature. In the particular case of Fig. 11.3A, there is a large thermal insulation with the substrate (400 nm of SiO2 plus 5.5  108 m2K/W of TCR), but as we will see later, the situation in terms of temperature around the notch is rarely good. A second important aspect is the lack of uniformity of the current density around the notch. As shown in Fig. 11.3B, the current density is larger in the notch area, and it is particularly relevant in the area next to the vertex of the notch. This has two main experimental implications. On one hand, such a large concentration of current density around the vertex can trigger irreversible damage to the nanostrip, via electromigration or simply by an excessive local temperature. Additionally, when studying current-assisted DW depinning, one should bear in mind the fact that the current density is considerably larger around the vertex. Therefore, even in the unlikely case that the thermal contribution could be ignored, the STT is going to be considerably more intense close to the vertex than in the rest of the constriction. Finally, we should draw the attention to Fig. 11.3C. The current density in the nanostrip decreases rapidly in the first nanoseconds of the pulse, up to a 25% decrease

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V = 2.

Temp. (K) 1500 1200 1000 800 600 400

(A) (B)

V = 2.

J ( 1012 A/m2)

(D) 1600 Thinner SiO2 1400 layer 1200 1000 800 600 V = 2. 400 0

1

Jave(1012 A/m2)

Temperature (K)

0.1 1 2

2 3 4 Distance (µm)

1.0

3 4

V = 2.

0.8 0.6 1

(C) 10

100

Time (ns)

5

Fig. 11.3 Color-coded temperature of a nanostrip at the end of a 100 ns, 2.5 V pulse (A). Colorcoded map of the current density distribution around the tip of the notch (B). Behavior of the average current density flowing through the nanostrip with time, as the temperature and the resistance of the nanostrip increases (C). Temperature profile along the length of the nanostrip, following the black dashed line in (B), for different thickness of SiO2 layer: square 400 nm, circle 300 nm, top triangle 200 nm, bottom triangle 100 nm, rhomboid 50 nm, right triangle for pure Si, and left triangle for a sapphire substrate (D). Data adapted from E. Ramos, C. Lo´pez, J. Akerman, M. Mun˜oz, J.L. Prieto, Joule heating in ferromagnetic nanostrips with a notch, Phys. Rev. B 91 (2015) 214404.

in only 5–6 ns. This is due to the rapid increase of the resistivity of the nanostrip with the temperature, following formula (11.4). This may not have relevant thermal consequences as the pulse generator delivers a constant power. On the other hand, having a rapidly decreasing current density may be of importance when evaluating STT. Fig. 11.3D shows the temperature profile along the nanostrip following a line that touches the vertex of the notch (dotted line in Fig. 11.3B), for different thicknesses of the SiO2 substrate, including zero thickness (i.e., pure Si substrate). There is obviously a significant reduction of the average temperature as the thickness of the SiO2 layer decreases. On the other hand the difference between the temperature of the notch and the rest of the nanostrip remains more or less constant. This is intuitively unexpected. Being the permalloy a metal (therefore a good thermal conductor), if the substrate is a

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good thermal conductor, such as Si (purple curve, right triangle in Fig. 11.3D), one should not expect a significant lateral buildup of temperature around the notch. This was the case in the results presented by Fangohr et al. [16]. In their simulation of a permalloy nanostrip 5 μm  150 nm  30 nm, on Si and SiN substrate and with a triangular notch 45 nm deep in the middle, they found that for a constant current density of 1012 A/m2 the temperature in the notch was only few degrees hotter than in the rest of the strip. Ramos et al. measured about 100 K difference for the same substrate. The main reason behind this difference is that Fangohr et al. assumed perfect thermal contact between the nanostrip and the substrate, while Ramos et al. included a TCR. The results of Ramos et al. [8] highlighted the potential importance of TCR and the uniformity of the current density to perform a correct quantitative determination of the STT. On the other hand the experiments and simulations were done on a substrate with a 400-nm-thick SiO2 layer, which is not the best choice in these experiments. The layer of SiO2 should be as thin as possible from the thermal point of view, but if the current is delivered in nanosecond-long pulses, it cannot be too thin, or some current would leak to the substrate through the electric capacitor established between the stripe and the substrate. A 25-nm thermal SiO2 layer is possibly as thin as you can go if the substrate is electrically conductive doped Si (the common choice for e-beam lithography). For this minimum electric insulation of 25 nm oxide layer, Lo´pez et al. [23] studied the effect of the TCR, the electric resistivity of the ferromagnetic nanostrip, and the length of the electric pulse. Fig. 11.4A shows the average temperature in the entire nanostrip (same dimensions as shown in Fig. 11.3A) for different values of TCR, vs the current density at the end of a 100 ns current pulse. The dotted black line indicates the Curie temperature of the permalloy. As can be seen, for the current density of 1012 A/m2, which is the typical figure when dealing with STT experiments, unless the TCR is below 2  108 m2K/W (which in most experiments may be unrealistic), the temperature in the nanostrip is going to be close or above the Curie temperature. Of course, in a notch, the temperature would be higher. The resistivity of the ferromagnetic nanostrip is less influential as it can be seen in Fig. 11.4B. Of course, Fig. 11.4 was obtained considering a “long” 100 ns pulse. Intuitively, one may think that using shorter pulses will reduce the temperature. This is true but only for very short pulses. As formulas (11.2) and (11.3) indicate, if the conductive thermal resistance is low (which is desirable), the rise time to the steady-state temperature is very fast. Fig. 11.5 shows how the temperature in the nanostrip rises with time for a Vp ¼ 2 V pulse, for different values of TCR. Even for a 5-ns pulse, the temperature is quite close to its final value. Only for extremely short pulses (2 ns or less), the temperature in the stripe may be manageable. As we have seen, the temperature of a ferromagnetic nanostrip is going to be very relevant, especially in the notch, unless the TCR is (possibly) unrealistically low and the current is delivered in extremely short pulses of the order of 1 ns. Lo´pez et al. [23] also showed that the temperature in the notch could only be reduced to a manageable value for very shallow notches (50 nm deep or less), which would compromise its pinning potential.

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TCR (10–8 m2K/W)

(µW·cm)

335

5

Temp (K) 3000

4 3 2 1

2300

(A)

1600

70 TCR =2·10–8 m2K/W 60

900

50

350

40 30

(B)

0.4

0.6

0.8

1.0

1.2

1.4

Current density (108 A/cm2) Fig. 11.4 Average temperature in the entire nanostrip vs the current density at the end of a 100ns current pulse, for different values of TCR (A). For a fixed low value of TCR, the impact of the resistivity of the ferromagnetic nanostrip on the average temperature of the nanostrip is represented in (B). The black dotted line indicates the Curie temperature of the permalloy. The resistivity of the ferromagnetic nanostrip is not very relevant for the control of the temperature, but the TCR is. Only low values of TCR, below 2  108 m2K/W, may allow an experiment with large current density. Data adapted from C. Lo´pez, E. Ramos, M. Mun˜oz, S. Kar-Narayan, N.D. Mathur, J.L. Prieto, Influence of the thermal contact resistance in current-induced domain wall depinning, J. Phys. D Appl. Phys. 50 (2017) 325001. j (×108 A/cm 2 (

Average temp (K)

800 700

0.71 0.73 0.76 0.8

600

0.85 0.9

500 400 300

6 5 4 3 2 1

(×10–8 m2K/W) R int

0

2

4

6

8

10

Time (ns) Fig. 11.5 Average temperature in the entire nanostrip after a 2 V current pulse is delivered to the stripe, for different values of TCR (in red font on the right-hand side of the plot). The current density after 10 ns of pulse is displayed over each curve in dark green font. Data adapted from C. Lo´pez, E. Ramos, M. Mun˜oz, S. Kar-Narayan, N.D. Mathur, J.L. Prieto, Influence of the thermal contact resistance in current-induced domain wall depinning, J. Phys. D Appl. Phys. 50 (2017) 325001.

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Magnetic Nano- and Microwires

Fabrication of the nanostrips: Some considerations

In the previous sections, we have shown that, when a large current density is delivered to a ferromagnetic nanostrip, the temperature is going to be quite high in most cases. Only if the TCR is unusually (possibly unrealistically) low, if the current density is delivered in extremely short 1–2 ns pulses and if the substrate is a good thermal conductor, the temperature may be manageable (although possibly not negligible). The first main conclusion is that it is better to work with no notches in the nanostrip or with almost negligible constrictions, in the domain wall low pinning regime. Additionally, the considerations that we have made on the Joule heating in ferromagnetic nanostrips bring to play some particularities of the fabrication method. The fabrication of magnetic nanostrips is usually done with e-beam lithography. This implies that the substrate should be electrically conductive so the charge from the electron beam is dissipated. On top of the conductive substrate, it is mandatory to have a thin layer of electric insulation, to avoid an electric short between the nanostrip and the substrate. This is why a common choice of substrate is doped Si with a thin thermal SiO2 layer on top. If the current delivered to the device is DC, the thinner the oxide layer, the better. On the other hand, if the current is delivered in nanosecond-long pulses, the current can still go to the substrate through the oxide layer because there is a parallel capacitive path for the current. As we mentioned earlier, when the current is going to be delivered in nanosecond-long pulses, the SiO2 should not be thinner than 25 nm; otherwise, it may be difficult to know how much current is flowing through the nanostrip and how much through the substrate. An alternative would be to use an insulating substrate but very good thermal conductor, such as sapphire or diamond. In these cases the e-beam lithography may require the use of a conductive polymer, and the choice of buffer layer would be pivotal to avoid an excessively high TCR. Although the best material or materials to use as buffer layer may depend on the deposition technique, on the deposition conditions, or even on the particular chamber used, Cr and Ta tend to work quite well and others such as Ti or Al not always so well. Dedicating some time to optimize the buffer layers may be a good idea in the long run, as minimizing the TCR would reduce the buildup of temperature. For instance, Hayashi [4] used Fe/AlOx as a buffer layer that, according to our estimations, resulted in an unusually low TCR, 1  108 m2K/W or less. When the nanostrip is subjected to very high temperatures, any defect can trigger an irreversible damage. This includes small (undetectable) residues of the resist or the developer. Therefore, independently of the resist used, after developing and before deposition of metallic layers, cleaning the sample using a low power O2 plasma may be a good strategy to get a good yield and reproducibility in your devices. Finally the ferromagnetic nanostrip should be capped to avoid oxidation, and the capping layer should be a good conductor to get a proper electric contact with the contact pads or bit lines. One option is to use Au, a very good electric and thermal conductor. It works very well, and it helps the lateral dissipation of heat. However, a good part of the current delivered to the nanostrip is shorted through the Au capping layer, and it is very important to be very precise with the thickness of this layer; otherwise, it

Joule heating in nanostrips and its role in current-assisted domain wall depinning

337

may be hard to reproduce the results from one batch of devices to the next. A good alternative is to use Pt. It provides a good contact with the current line above, and its electric resistivity is five times bigger than the one of Au. Consequently, more current can be deployed to the ferromagnetic materials than in the case of using a gold capping layer.

11.4

Micromagnetic model and modeling details

The thermal characterization presented in previous sections showed that the injection of current pulses along a Py strip on top of a Si substrate can generate a significant amount of heat. This Joule heating is particularly important when the current is forced to pass through a constriction (notch) in the Py strip. The local temperature in the notch under typical current pulses (J  1012 A/m2, 5 ns) could easily be above the Curie temperature TC of the sample, and therefore the ferromagnetic order would be destroyed. Therefore, before reaching any definitive conclusion on the role of the STT in these systems, any theoretical or numerical description of the DW depinning process needs to take into account Joule heating. We have developed a micromagnetic framework that describes the magnetization dynamics as coupled to the current and heat transport self-consistently. Within this formalism the magnetization dynamics is described by the Landau-Lifshitz-Bloch equation (LLB), which allows a proper description of the current-driven DW depinning for temperatures close or even above TC.

11.4.1 Micromagnetic model at zero or finite uniform temperature Firstly, we assume zero temperature (T ¼ 0), and consequently, we are neglecting any heating effect. To describe magnetization dynamics in this zero-temperature framework, we numerically solve the Gilbert equation: ! !  dm dm ! ! ! ¼ γ 0 m  H eff + H th + α m  + τ STT dt dt !

!

(11.5) ! !

where γ 0 is the gyromagnetic ratio and α is the Gilbert damping parameter. m ð r ,tÞ ¼ ! ! M ð r , tÞ is the normalized local magnetization to the saturation magnetization (M ). Ms ! H eff

s

δE ¼  μ 1Ms δm ! is the effective field, where E is the total energy density of the system 0 ! δE and δm! represents the functional derivative of E with respect to m . The resulting total ! ! ! effective field H eff includes the exchange (H exc ), the magnetostatic (H dmg ), and the ! Zeeman (H dc ) interactions. Further details of these contributions can be consulted

elsewhere [24, 25]. Additionally, when current pulses are injected through an orthogonal conducting line (current line) placed above the Py strip, the effective field also

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Magnetic Nano- and Microwires !

includes the Oersted contribution H Oe , which results from a numerical solution of Ampere’s law and was computed in all points of the Py nanostrip [26]. We will come back to the details of this Oersted field when we discuss the DW nucleation processes to introduce a DW in the Py nanostrip. The last term in Eq. (11.5) is the STT [1, 27–32]. Current consensus is that there are at least two mechanisms that cause DW motion by current. The first one is due to the transfer of angular momentum from the conduction electrons to the local magnetic moments of the ferromagnetic sample. In this case, it is assumed that, when the electrons move across the DW, they  adjusttheir spin orientation adiabatically to the direc! !

tion of the local moments m ð r , tÞ while transferring angular momentum to the !

DW. This is the adiabatic form of the STT ( τ a ). The second mechanism originates from the spatial mistracking of the spin of the conduction electrons to the local magnetization. When the spin of the conduction electrons does not follow the local magnetization, the electron can be reflected and change its direction of motion. This in turn can transfer linear momentum from the conduction electron to the local magnetization, which can also result in DW motion. This second mechanism is generally known ! as the nonadiabatic process, and it introduces an additional nonadiabatic STT ( τ na ). With these contributions the form of the STT in Eq. (11.5) is ! τ STT

!

!

¼ τ a + τ na ¼ 

  i μB PJ ! μ PJ ! h! ! ! uJ  r m + β B m  uJ  r m eMs eMs ! !

(11.6) !

!

where μB is the Bohr magneton, e is the electric charge, and J ð r , tÞ ¼ Jð r ,tÞu J is the ! current density flowing through the ferromagnetic strip, with u J the unit vector along ! the local direction of the current and J ¼ Jð r , tÞ its local magnitude. Notice that, when the electric charge is forced to pass through constrained geometries, such a notch in the ferromagnetic nanostrip, the current density is nonuniform. In such a case the spa!  ! tial distribution of the current J ð r Þ is previously computed by commercial software [33] and taken into account when solving the magnetization dynamics given by Eq. (11.5). P is the polarization factor, and β is the nonadiabatic parameter, which represents the relative contribution of the nonadiabatic STT to the adiabatic STT term. B PJ In what follows, we use the notation u  μeM . s Eq. (11.5) with Eq. (11.6) is the Gilbert equation including the adiabatic and the nonadiabatic STTs. This equation can be transformed to its Landau-Lifshitz-Gilbert version, which reads ! ! h ! !  ! i dm ð1 + αβÞ ! ! ! ! m ¼  γ 00 m  H eff + H th  γ 00 α m  m  H eff + H th  u  dt 1 + β2  ii  i h h ðβ  αÞ ! h! ! ! ! !  m  uJ  r m  m  uJ  r m  u 2 1+β

(11.7)

Joule heating in nanostrips and its role in current-assisted domain wall depinning

339

where γ 00 ¼ γ 0/(1 + α2). We will use the notation P0  P and M0s to denote the polarization factor and the saturation magnetization at zero temperature, respectively. Finite temperature effects can be also analyzed within the presented formalism, under certain conditions. Both Eqs. (11.5) and (11.7) can also include the effect of   !

random thermal fluctuations at finite and uniform temperature T 6¼ f ð r , tÞ . This !

is done by adding an additional stochastic thermal field (H th ) to the deterministic !  effective field H eff in both Eqs. (11.5) and (11.7). This stochastic thermal field has white noise properties with correlator [34–37] D

E 2kB Tα ! ! ! ! δij δð r  r 0 Þδðt  t0 Þ Hth, i ð r , tÞHth, j ð r 0 , t0 Þ ¼ γ 0 μ0 Ms0 V

(11.8)

where kB is the Boltzmann constant, μ0 is the permeability of the free space, and V is the volume of the computational cell. Therefore the deterministic equation (11.7) becomes a stochastic Langevin equation at finite temperature T. Note however, that this Langevin equation (11.8) is only valid for uniform and low temperatures below the Curie temperature (TC) of the ferromagnetic sample, that is, for T ≪ TC. Typical permalloy (Py) parameters were adopted for this zero-temperature analysis: saturation magnetization M0s ¼ 8.6  105 A/m, exchange constant A0ex ¼ 13 pJ/m (A0ex denotes the value of the exchange constant parameter at zero temperature), and a Gilbert damping parameter α ¼ 0.02. Except when otherwise indicated the polarization factor is P ¼ 1, and different values of the nonadiabatic parameter β will be evaluated. Eq. (11.5) was numerically solved using a conventional micromagnetic solver, MuMax3 [25]. The Py nanostrip has a cross section of w  t ¼ 300 nm 10 nm. They were discretized in cubic cells of Δx ¼ 5 nm. At zero temperature, MuMax3 adapts the time step to ensure proper numerical results. Typical time steps are in the order of Δt ¼ 0.1 ps, although larger time steps are also appropriate. At finite temperature the time step is fixed to Δt ¼ 0.1 ps, and it was checked in several tested cases that a reduced time of Δt ¼ 0.05 ps did not modify the results. We shall use J to indicate the module of the current density !

at points distant from the constrained geometry (i.e., far from the notch, if any), where J ! ! is uniform across the strip width and it points along the longitudinal direction (u J ¼ u x ). Positive current (J > 0) will flow along the positive x-axis. Note that the electron flow for J > 0 is along the negative x-axis. As it will be shown later, the STTs push the DWs along the electron flow direction.

11.4.2 Micromagnetic model and heat transport to account for the Joule effects Here, we describe the details of the micromagnetic model developed to describe the DW dynamics under current pulses. COMSOL simulations presented in the previous sections (Figs. 11.1–11.5) clearly indicate that the injection of current along the Py nanostrip generates significant variations of the temperature in the sample, in

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Magnetic Nano- and Microwires

particular, when the Py strip has a notch that serves to trap the DW. Indeed the high current density in this constrained part of the strip promotes high local temperatures that, for typical values of the current pulses, can reach values close or above the Curie temperature (TC  850 K). Note again that the micromagnetic equation (11.5) or (11.7) described in the previous section is only valid for zero or fixed and uniform temperature well below the Curie threshold. Therefore it is needed to develop a full micromagnetic model that allows a description of the current-induced DW depinning when the temperature evolves both spatially and temporally across the sample. The present section introduces the details of the micromagnetic model, which describes the magnetization dynamics coupled to the heat equation. As it was already shown, the current-induced Joule heating of the Py strip evolves in time during and after the current pulse, and the temperature becomes space- and time-dependent along the Py nanostrip. Therefore any realistic description of the experimental depinning processes requires to take into account the magnetization dynamics coupled to the heat transport under both current pulses and the static magnetic field. Since the nanowire reaches temperatures near or even above the Curie temperature of the Py (TC), we cannot use the LLG equation (11.5), which is only valid for ! uniform temperature well below the Curie threshold (T 6¼ f ð r , tÞ and T ≪ TC). Therefore the magnetization dynamics is studied using the LLB equation including the adiabatic and the nonadiabatic STTs [38]: ! ! ! i d m ð r , tÞ α? h !  !  ! ! ! ¼  γ 00 m H eff  γ 00 2 m  m  H eff + H? th dt m      ! !k u βα? ! ! ! 0 αk !  + γ 0 2 m  H eff m + Hth +  1 +  r m u J 2 m m 1+β  αk   i uβα h  ii h u β  m ! h! ! ! ! ! ? !  m m    u  r m  u  r m m  J J m 1 + β2 m

(11.9) where γ 00 ¼ γ 0/(1 + λ2), with λ  α the conventional Gilbert damping. Apart from the conventional precessional and damping terms (first and second terms on the RHS of Eq. 11.9), the LLB equation (11.9) also includes an additional longitudinal relax! ation term (third term in Eq. 11.9) that describes the relaxation of the module of m toward its equilibrium value me(T). me(T) represents the normalized equilibrium magnetization at each local temperature, which we calculate by using the Brillouin function for 1/2 spins, namely [39], me ðT Þ ¼ tanh

 μ0 μNi TC HB + me kB T T

(11.10)

where μNi represents the Ni atomic magnetic moment and HB  Hdc the external field. We assume μNi ¼ μB according to previous calculations [40]. A characteristic feature

Joule heating in nanostrips and its role in current-assisted domain wall depinning

341

of the LLB equation (11.9) is precisely that, contrary to the LLG equation (11.7), ! ! m ð r , tÞ is not unitary but its module varies depending on the local temperature ! Tð r ,tÞ. Note that this longitudinal relaxation is neglected in the LLG formalism since, at T ¼ 300 K, it is much faster than the transverse relaxation. However, it becomes particularly relevant at T close to TC, where the longitudinal and the transverse relaxation times are comparable. α? and αk are the transverse and the longitudinal damping parameters, respectively. They depend on the temperature       [41–43] as α? ¼ α 1  3TTC

and αk ¼ α

2T 3TC

for T < TC, while α? ¼ αk ¼ α

2T 3TC

for T > TC.

Note that, in the limiting case of zero temperature, α? reduces to the conventional !

Gilbert damping α? ¼ α, and αk ¼ α. The effective field H eff in Eq. (11.7) is given by [41–43] !

!

!

!

!

!

H eff ¼ H exch + H dmg + H dc + H Oe + H m !

!

(11.11)

!

where H dmg , H dc , and H Oe are the magnetostatic field, the external field, and the Oer!

sted field, respectively. The Oersted field H Oe describes the classical Ampere field due to the current pulse along an orthogonal bit line placed above the Py strip to generate or nucleate DWs in the Py nanostrips (see Ref. [26] for further details). All these fields are the same as in the conventional LLG equation (11.7). The exchange field now reads as [43] !

H exch ¼

2AðT Þ 2 ! r m m2e Ms0

(11.12)

where A(T) is the temperature-dependent exchange constant. Here, we follow the same assumption adopted in Atxitia et al. [44] and Ramsay et al. [45], where A(T) scales with T as A(T) ¼ A0m2e (T). M0s and A0 represent the saturation magnetization !

and the exchange constant at T ¼ 0. The last term in Eq. (11.11), H m , which is not present in the LLG equation (11.7), is [41–43] 8   1 m2 ! > > T < TC > < 2χ 1  m2 m , ! k e   Hm ¼ > 1 3 TC m2 ! > > 1+ m , T > TC : χk 5 ðT  TC Þ

(11.13)

where χ k is the so-called longitudinal susceptibility, defined as

χk ¼

∂me ∂Hdc



μ0 μB 0 3 B ðxÞ k T 7 ¼ B 5 T C Hdc !0 1  B0 ðxÞ TC T x¼ me T

(11.14)

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Magnetic Nano- and Microwires

where x ¼ μk0BμTB Hdc + TTC me and B0 ðxÞ ¼ ∂tanhx ∂x (see Ref. [38] and references therein for further details). !?,k

The stochastic fields Hth take into account longitudinal and transverse fluctuations of the magnetization. They are considered to have white noise properties with the following correlators [46]: D! E 2k T α  α  ! B ? k ! ! ? ! ? !0 0 Hth,i ð r , tÞHth, j ð r , t Þ ¼ δij δð r  r 0 Þδðt  t0 Þ 2 0 γ 0 μ0 Ms Vα?

(11.15)

D! E 2k Tα !k B ! ! ! ! k k Hth,i ð r , tÞHth, j ð r 0 , t0 Þ ¼ δij δð r  r 0 Þδðt  t0 Þ μ0 Ms0 V

(11.16)

where i, j : {x, y, z} represents the Cartesian coordinates indexes and V is the volume of the computational cell. Again, further details of the LLB equation (11.6) can be consulted in Moretti et al. [38] and references therein. The last three terms in Eq. (11.9) describe the STT as introduced by Schieback et al.i h !

!

!

[47]. Note that, since the module of m is not restricted to 1, the STT term ðu J  rÞ m

has also a longitudinal component. The last term in Eq. (11.9), which originates when transforming the Gilbert version of the LLB into the Landau-Lifshitz version, h i cancels !

!

out the longitudinal component of the STT proportional to βαm? ðu J  rÞ m . Both of these terms do not appear in Schieback et al. [47], since they neglected terms proportional to βα?. To account for the Joule heating, the LLB equation (11.9) has to be numerically solved coupled to the heat transport equation (11.1), which was already introduced in the COMSOL modeling part in Section 11.1 of the present chapter. For convenience, we rewrite the equation as !

!

∂Tð r , tÞ k 2 ! Qð r ,tÞ ¼ r Tð r ,tÞ + ∂t ρC ρC

(11.17)

where here k is the thermal conductivity (in W/(K m)), ρ is the density (in kg/m3), and ! C (in J/(g K)) is the specific heat capacity of the material. T ¼ Tð r ,tÞ represents the ! temperature, and Q ¼ Qð r ,tÞ is the heat source, which in this case comes from the Joule heating effect due to the current injected through the bit line (IBL), that is, !

!

!

Qð r , tÞ ¼ J 2 =σ, with J ¼ Jð r , tÞ and σ is the electric conductivity. Including the complete system (the nanostrip, the electric contacts, the Si substrate, and the surrounding media) in the micromagnetic code would be prohibitive from a computational point of view. A possible solution is to adopt a phenomenological model of the heat transport. As we saw in Section 11.1.1, the characteristic heating (and cooling) time τ depends, in a first approximation, on the heat capacity of the nanostrip, the thermal conductivity of the substrate, and their relevant thicknesses

Joule heating in nanostrips and its role in current-assisted domain wall depinning

343

(expression 11.3) and on the TCR. Therefore, once the special distribution of temper! atures Tð r Þ is determined with COMSOL for a given time, the time evolution of the spatial distribution can be described by adding an additional term in Eq. (11.17) as follows [38]: !

!

!

∂Tð r , tÞ k 2 ! Qð r , tÞ Tð r ,tÞ  T0 ¼ r Tð r , tÞ +  ∂t ρC ρC τ

(11.18)

where the last term in Eq. (11.18) takes into account the heating (and the cooling) of the Py nanostrip in contact with the conducting lines and the Si substrate. T0 represents the temperature of the substrate, and the parameter τ represents the characteristic heating (or cooling) time. The resulting space and time evolution of the Py strip temperature computed by COMSOL simulations of the full systems is described by this phenomenological model (Eq. 11.18) (see Ref. [38] for further details). Note that unlike Eq. (11.17) the phenomenological heat equation (11.18) is only solved for the Py nanowire, and therefore it can be and is implemented to evaluate the heat transport and the magnetization dynamics (given by Eq. 11.9) simultaneously. The following parameters were considered for the Py nanowire: k(Py) ¼ 46.4 W/(K  m), C(Py) ¼ 0.43 J/(g  K), ρ(Py) ¼ 8.7  103 kg/m3, and σ(Py) ¼ 4  106 (Ωm)1. The ambient temperature and the phenomenological characteristic time are T0 ¼ 300 K and τ ¼ 0.9 ns, respectively (see Ref. [38] for further details). To evaluate the DW dynamics along the Py nanowire as coupled to the heat transport, we solved numerically the LLB equation (11.9) and the heat equation (11.18) simultaneously using a Heun’s method. Note that within this magnetothermal framework the magnetization dynamics, Eq. (11.9), is coupled to the temperature dynamics, !?,k

Eq. (11.18), both through the stochastic thermal fields (Hth ) and the temperature dependence of the magnetic parameters (Ms ¼ Ms(T), A ¼ A(T), α? ¼ α?(T) αk ¼ αk(T)).

11.5

Micromagnetic results

11.5.1 Field- and current-driven DW dynamics at zero temperature 11.5.1.1 Field-driven DW motion Here, we firstly review the micromagnetic results for the field-driven case, that is, in the absence of current. We consider a Py strip with a cross section of w  t ¼ 300 nm  10 nm and 20 μm long along the strip x-axis. The initial state consists on a trans!

verse head-to-head DW. Starting from this initial state, a static magnetic field B ext ¼ ! ! Bx u x ¼ μ0 Hx u x is applied along the x-axis, and the LLG equation (11.5) is numerically solved to evaluate the field-driven DW dynamics. Zero-temperature conditions are

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Magnetic Nano- and Microwires

1.0

w = 300 nm; t = 10 nm

0.5 0.0 0

5

(C)

10

15

20

Time (ns)

25

3 2 Bx = 1.5 mT Bx = 5 mT

1

1.5 mT 5 mT 30

0 0

5

(D)

10

15

k

720

4

1.5

x (mm)

x (mm)

2.0

(B)

k

5

m

20

Time (ns)

25

DW angle (deg)

(A)

Bx= 1.5 mT Bx = 5 mT

540 360 180

30

(E)

0 0

5

10

15

20

25

30

Time (ns)

Fig. 11.6 Field-driven DW dynamics along perfect strips. (A) and (B) show the micromagnetic (μM) snapshots for Bx ¼ 1.5 mT and Bx ¼ 5 mT, respectively. The micromagnetic temporal evolution of the DW position is shown in (C), whereas the (D) and (E) depict the temporal evolution of the DW position and the DW as predicted by the 1DM. The assumed 1DM inputs are given in the text.

assumed here. Fig. 11.6A shows the micromagnetic snapshots at different instants of time for a field of Bx ¼ 1.5 mT, which indicates that for this low field the DW propagates along the Py strip adopting a slightly modified transverse configuration with respect to its static state. If the applied field overcomes the so-called Walker breakdown field (BWB), the DW propagates along the strip, but its internal magnetic structure changes between transverse and vortex/antivortex configurations. This micromagnetic temporal evolution is shown in Fig. 11.6B, which corresponds to a field of Bx ¼ 5 mT. In Fig. 11.6C, we plot the temporal evolution of the DW position for the same driving fields as presented in Fig. 11.6A and B. Whereas the DW reaches a steady-state regime with constant velocity for Bx ¼ 1.5 mT, the displacement becomes turbulent for Bx ¼ 5 mT due to the mentioned DW transformations above BWB. The micromagnetic field-driven DW dynamics along this perfect strip (no imperfections) can be described in terms of a one-dimensional model (1DM), which assumes the DW as a rigid object. Within this 1DM the DW dynamics is described by the internal DW angle ψ(t) with respect to the longitudinal x-axis and its position q(t) along the same x-axis. The corresponding 1DM equations are obtained from the LLG equation (11.5) by assuming the Bloch profile of the DW [29, 31, 32] for details and can be expressed as 

1 + α2

 dq dt

¼ ΔQ½ΩA + αΩB  + αβu

(11.19)

Joule heating in nanostrips and its role in current-assisted domain wall depinning



1 + α2

345

 dψ Qðα  βÞu ¼ Δ½αΩA + αΩB   Qðα  βÞ dt Δ

(11.20)

(A)

800 600 400 200 0 –200 –400 –600 –800 –20

mM (tp = 5 ns) 1DM (tp = 100 ns)

v (m/s)

v (m/s)

where ΩA ¼ γ 0 12 Hk cos ½2ψ  and ΩB ¼ γ 0Hx, with Hk being the shape anisotropy field B PJ and β (nonadiabatic parameter) and Δ the DW width. The parameters u  μeM s correspond to the STTs as already introduced in the previous section. For the pure field-driven case, u ¼ 0. The parameter Q ¼ 1 for head-to-head and tail-to-tail DW configurations, respectively. The temporal evolution of the DW angle (ψ(t)) and the DW position (q(t)  x(t)) predicted by the 1DM are shown in Fig. 11.6D and E for the same driving fields, where the following 1DM inputs were considered: Hk ¼ 1.45  105 A/m and Δ ¼ 36 nm. For Bx ¼ 1.5 mT, the DW angle reaches a terminal value, and the DW position increases monotonously as the time elapses. On the contrary, for Bx ¼ 5 mT, the DW angle precesses periodically around the z-axis, and the DW position depicts a nonmonotonous evolution. Although the rigid 1DM cannot account for the internal DW transformations above the Walker breakdown predicted by the micromagnetic simulations, it can describe the main features of the field-driven DW dynamics along perfect Py nanostrips. Fig. 11.7A shows the DW velocity as function of Bx along a perfect strip as computed by the micromagnetic model (μM) and the 1DM. Note that the micromagnetic values of the DW velocity were computed from the DW displacement after a temporal window of tW ¼ 5 ns (i.e., v ¼ ΔX/tw), whereas the 1DM results were obtained for a longer temporal window of tW ¼ 100 ns. Both models are in good quantitative agreement for fields below the Walker breakdown (j Bx j < BWB), where the DW velocity increases linearly with Bx. The DW reaches its maximum velocity at the Walker breakdown field, and it drops above the Walker field (jBx j > BWB). Note that the μM results indicate an increase of the DW velocity in the high-field regime

–10

0

Bx (mT)

10

20

(B)

800 600 400 200 0 –200 –400 –600 –800 –20 –15 –10 –5

0

5

10 15 20

Bx (mT)

Fig. 11.7 DW velocity (v) as function of the driving field (Bx). (A) corresponds to a perfect strip, without pinning sites. Micromagnetic results are shown by open symbols, whereas the predictions of the 1DM are shown by solid line. (B) Micromagnetic results along a strip with edge roughness, which introduces pinning, and consequently a minimum field to promote the DW displacement along the strip. The edge roughness considered here has a characteristic size of 10 nm.

346

(A)

Magnetic Nano- and Microwires

(B)

Fig. 11.8 Micromagnetic snapshots of the field-driven DW dynamics along Py strips with edge roughness. (A) and (B) show the micromagnetic (μM) snapshots for Bx ¼ 1.5 mT and Bx ¼ 5 mT, respectively. The inset in (A) is a zoom view to show the edge roughness.

(jBx j ≫ BWB). The same feature is also observed for the 1DM case but for much higher fields (not shown). In Fig. 11.7B the micromagnetic results of v vs Bx are shown for a realistic Py strip, which depicts some edge roughness. This roughness, which is inevitable in the fabrication process, generates a random local pinning acting against the free DW motion along the Py strip. In our modeling the roughness has a random profile with a characteristic size of 10 nm (see Ref. [35] for details). As it is shown in Fig. 11.7B, the minimum field needed to promote the DW dynamics along the strip with edge roughness is jBp j2 mT. For field below this pinning or propagation threshold, j Bx j < Bp, the DW does not move, whereas above this threshold (jBx j > jBp j) the DW can travel along the strip. Typical micromagnetic snapshots of these pinned and depinned regimes are shown in Fig. 11.8A and B for jBx j < jBp j and jBx j > jBp j, respectively.

11.5.1.2 Current-driven DW motion Once the main features of the field-driven DW dynamics have been presented, here, we introduce and briefly describe the current-driven DW dynamics due to the STTs. Now the applied magnetic field is set off, and 5 ns current pulses are injected along the Py strip. The micromagnetic results of the DW velocity as function of the current density amplitude (J) along a perfect strip are shown in Fig. 11.9A. The predictions of the !

!

1DM are also included for comparison. Note that positive current (J ¼ Ju x , with J > 0) means that the electron flow is along the negative x-axis. Therefore the current-driven DW motion as due to the STTs is along the electron flow. Both models provide similar results of the v vs J curves for all values of the nonadiabatic parameter β. In the perfect adiabatic limit (β ¼ 0), there is an intrinsic DW propagation current threshold below which the DW is initially displaced, but after a short transient, the DW stops. Above this threshold the DW propagates with periodic transformations of its internal structure similarly as the field-driven case above the Walker breakdown. For finite β the DW propagates maintaining a transverse configuration for J smaller than a Walker breakdown-like threshold (JWB). In this linear regime the DW velocity (jvj) increases with J, and the mobility, defined as the slope of jv j vs jJj or Δ j vj/Δ j Jj also increases

Joule heating in nanostrips and its role in current-assisted domain wall depinning 1500

1500

500 0 –500

–1000 Dots: mM (tp = 5 ns) –1500 Lines: 1DM (tp = 100 ns) –20 –15 –10 –5

0

5

J (TA/m2)

0 0.01 0.02 0.04 0.06

10 15 20

1000

v (m/s)

v (m/s)

1000

(A)

347

0 0.01 0.02 0.04 0.06

500 0 –500

–1000 –1500

(B)

Roughness

–20 –15 –10 –5

0

5

J (TA/m2)

10 15 20

Fig. 11.9 DW velocity (v) as function of the driving current (J). (A) corresponds to a perfect strip, without pinning sites. Micromagnetic results are shown by open symbols, whereas the predictions of the 1DM are shown by solid lines. (B) Micromagnetic results along a strip with edge roughness, which introduces pinning, and consequently a minimum current density to promote the DW displacement along the strip. The edge roughness considered here has a characteristic size of 10 nm.

with β. Except for the case of β ¼ α ¼ 0.02, there is a maximum jJ j, which limits the linear regime, and above which the DW moves turbulently, with periodic transformations of its internal structure. This Walker breakdown-like threshold (JWB) increases with β from β < α and decrease again for β > α. Fig. 11.9B presents the micromagnetic results of the current-driven DW dynamics along a strip with edge roughness. Similarly, to the field-driven case, the edge roughness introduces pinning, and it is the responsibility of the minimum current density (Jp) to promote the self-sustained DW motion along the Py strip. Notice that the pinning current density required to promote this motion is Jp  4 TA/m2 for the larger nonadiabatic parameter studied here (β ¼ 3α). Additionally, note that in our modeling we are assuming the largest value of the polarization factor (P ¼ 1), and therefore the pinning threshold current is even larger for more realistic values of the polarization factor (P  0.4). In any case the large value of Jp  4 TA/m2 may be the reason behind the lack of experimental results showing the pure current-driven DW motion along these Py strips. As it will be shown later on, experimental currents injected in these samples are limited below 3 TA/m2. The direct comparison of the experimental observations of pure current-driven !

(B ext  0) DW velocity as function of the injected current to the micromagnetic results and/or to the 1DM predictions would be a natural procedure to extract information about nonadiabaticity. However, we want to recall here the lack of experimental results of pure current-driven DW motion along these soft Py strips, which could be directly ascribed to the STTs. Indeed, there are experimental studies that indicate that a vortex DW can be driven by current pulses along a Py strip in the absence of external magnetic field (see, for instance, figs. 12 and 13 in Refs. [22, 28, 48, 49]). However, these results are not conclusive about the role of the STT. In particular, when a current pulse is injected in a curved strip, the current is not uniform over the strip cross section. Therefore the STT would be not uniform on the DW texture, and this inhomogeneity should promote modifications of the DW structure. Moreover,

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the Joule heating in these curved strips becomes nonuniform also as a consequence of the nonuniform distribution of the current along the curved strip. These two effects may contribute to the modification of the vortex structure of the DW, and this modification can result on DW motion due to the gyrotropic motion of the vortex [50]. Note also that for these vortex DWs the experiments indicate very low velocities of v  1–10 m/s for J 1TA/m2 [51, 52]. On the other hand, there are not measurements of the DW velocity v vs J for transverse DWs in the absence of field to which we can be compared with the prediction of the micromagnetic and 1DM predictions. For these reasons, much of the efforts to estimate the nonadiabatic parameter β have been focused on the experimental analysis of the current-induced DW depinning from a notch in the presence of magnetic fields, a topic that is reviewed in the following sections from a realistic micromagnetic point of view.

11.5.2 DW nucleation by a current pulse along a bit line Before evaluating the current-driven assisted DW depinning from a notch, here, we review the DW nucleation process by the Oersted field generated by a current pulse injected along an orthogonal conducting line placed on top of the Py strip (see Fig. 11.10A and B). We will refer to this conducting line as the bit line. As in previous analysis, here, we consider again a Py strip with w  t ¼ 300 nm  10 nm cross section, but now, it has a triangular notch at the top edge placed in the middle of the strip. Fig. 11.10A and B shows all the geometrical parameters. The notch dimensions, defined in the inset of Fig. 11.10B, are ‘N ¼ 200 nm and pN ¼ 100 nm. Same edge roughness as in the previous section is also considered here. The initial state of the magnetization in the Py nanostrip is essentially along the negative x-axis, that is, ! ! m ð r , 0Þ  1, and at t ¼ 0, current pulses are injected along the negative direction !

!

!

of the y-axis in the presence of static fields along the x-axis (B ext ¼ Bx u x ¼ μ0 Hx u x ). The dimensions and situation of the bit line are defined in Fig. 11.10A and B and are wL  tL ¼ 500 nm  50 nm, s ¼ 3 nm, and xL ¼ 1536 nm, which is the distance between the bit line center and the notch (Fig. 11.10A and B). The amplitude (IL) and the duration (tp) of the current pulses are IL ¼ 100 mA and tp ¼ 5 ns. This current !

pulse generates a transient Oersted field B Oe ðx, tÞ, which acts on the Py strip. The Oersted field was numerically obtained using the Biot-Savart’s law (see Ref. [26] for details on the calculation of the Oersted field). In particular the longitudinal component of BOe, x points along the positive +x-axis, and therefore it supports the local switching of the magnetization below the bit line. The reversed domain is flanked by two DWs, one tailto-tail at the left side and other head-to-head at the right side, and these DWs are driven !

!

!

by the static magnetic field B ext ¼ Bx u x ¼ μ0 Hx u x . The tail-to-tail DW, which is close to the left end of the Py strip, is rapidly expelled from the strip. On the contrary the head-tohead DW is pushed by the field along the +x direction. Depending on the magnitude of Bx, three different regimes are observed. For low field (Bx ¼ 1 mT), the nucleated headto-head DW is pinned before reaching the notch due to the edge roughness. The corresponding final magnetic state is shown in the top snapshot of Fig. 11.10C. For

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349

(A) (B)

(C)

Fig. 11.10 Nucleation of a DW by the injection of a current pulse along a conducting bit line. (A) and (B) represent the geometry and location of the bit line and the Py strip, which has a notch in the middle. The geometrical parameters considered in the present study are given in the text. The initial state prior pulse injection along the bit line is also shown in (B). The amplitude (IL) and the duration (tp) of the current pulses injected along this are IL ¼ 100 mA and tp ¼ 5 ns. The cross section of Py strip far from the notch is w  t ¼ 300 nm  10 nm. It has a triangular notch at the top edge placed in the middle of the strip. (A) and (B) show all the geometrical parameters. The notch dimensions defined in the inset of Fig. 11.10B are ‘N ¼ 200 nm and pN ¼ 100 nm. Same edge roughness as the previous section is also considered here. The initial state of the magnetization in the Py nanostrip is essentially along the negative x-axis, ! ! that is, m ð r ,0Þ  1, and at t ¼ 0, current pulses are injected along the negative direction of !

!

!

the y-axis in the presence of static fields along the x-axis (B ext ¼ Bx u x ¼ μ0 Hx u x ). The dimensions and situation of the bit line defined in Fig. 11.10A and B are wL  tL ¼ 500 nm  50 nm, s ¼ 3 nm, and xL ¼ 1536 nm. The snapshots in (C) represent the final state of the DW under different static longitudinal fields Bx.

intermediate values of Bx, that is larger than the induced edge roughness pinning but smaller than the local pining imposed by the notch, the DW reaches the notch location, and it gets pinned there. Different pinned DW configurations at the notch are observed depending on the magnitude of the Bx. For instance, a transverse DW pinned at the left side of the notch is reached for Bx ¼ 2 mT. For Bx ¼ 3 mT, the pinned DW has its internal magnetization pointing along  y direction and is placed close to the center of the notch. In the case of Bx ¼ 4 mT, the pinned DW is placed at the right side of the notch, as it is shown in the snapshots of Fig. 11.10C. Note that these final DW states are also influenced by the distance between the nucleation bit line and the notch (xL), and it

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was verified that for other distances the resting pinned state can also adopt a vortex configuration. For larger fields, Bx 7 mT, the DW is depinned from notch and driven to the right end of the strip. In the following section, we will study the DW depinning from the notch, starting with an initial state of the pinned trapped at notch. The reader should take into account that the dynamical depinning field (Bd  7 mT) discussed here is different from the static depinning field that will be studied there.

11.5.3 Current-assisted DW depinning from a notch 11.5.3.1 Deterministic analysis in the absence of the Joule heating effect After the analysis of the field-driven and the current-driven DW dynamics along realistic Py strips, here we focus on the description of the DW depinning under static fields Bx and current pulses along the Py strip. The initial state consists on a transverse headto-head initially pinned at the left side of the triangular notch. This initial state was obtained by nucleation of the DW using the current pulse along the bit line under a static magnetic field of Bx ¼ 2 mT (see Fig. 11.10C). To evaluate the DW depinning, the pinned DW was relaxed to Bx ¼ 0 mT. The resulting magnetic state is shown in Fig. 11.11A, where also the electrical contacts used to inject the current pulses along the Py strip are shown by gray rectangles. The sign criteria defining positive and negative current pulses along the Py strip are also included in Fig. 11.11A. We should

(A)

(B)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

Fig. 11.11 (A) Scheme of the Py strip with the transverse DW pinned at the notch. The contacts used to inject the current pulses along the Py are represented by gray rectangles. The sign criteria for the direction of the current pulses and the applied static field (Bx) are also indicated. A positive current goes from the left to the right electrical contact, and therefore the electron flow goes from right to left along the longitudinal x-axis. In this situation the STT pushes the DW to the left, that is, toward the  x direction. On the contrary, negative current pulses (from right to left) push the DW toward + x. (B) Spatial distribution of the current along the Py strip as computed by COMSOL and taken into account in the micromagnetic modeling of the micromagnetic study of the DW depinning.

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remember again that positive current pulses (J > 0) mean that the electron flow is along the negative x-axis. The notch is placed in the middle of the Py strip, and the center of the electrical contacts is at a distance of xL ¼ 1.5 μm from the notch center. Due to the notch the current density is not uniform, and its spatial distribution ! !

!

!

(J ð r ,tÞ ¼ Jð r , tÞu J ) was computed by a preliminary COMSOL simulation. It is shown in Fig. 11.11B. Note that the magnitude of the local current density within the notch is larger than in the unconstrained parts (between the contacts but far from the notch), where it is almost uniform and points along the longitudinal axis. The following results on the DW depinning are indicated with respect to the magnitude of the current far from the notch. Similar edge roughness as in the previous sections is also taken into account here for the DW depinning analysis. The length of the injected current pulses is tp ¼ 5 ns. The polarization factor of the STT and the nonadiabatic parameter are P ¼ 1 and β ¼ 0.02, respectively. In the absence of thermal fluctuations (T ¼ 0) and ignoring Joule heating effects, the phase diagram of the threshold DW depinning current (Jd) as a function of the static magnetic field Bx is shown in Fig. 11.12. Note that these micromagnetic results were obtained by solving the LLG equation (11.7) augmented by the STTs, as it was already discussed in Section 11.4.2. Several interesting features are observed in this diagram. In the absence of current (J ¼ 0), the depinning field (Bd(J ¼ 0)) is different for positive (Bx > 0) and negative (Bx < 0) values. As the DW is initially pinned at the left side of the notch, its field-driven depinning is smaller for negative fields than for positive fields: Bd(J ¼ 0) ¼  6 mT and Bd(J ¼ 0) ¼ + 15 mT. When current pulses are injected (J 6¼ 0), the depinning becomes also asymmetric with respect to the current pulse polarity: J > 0 and J < 0. This is also expected, as the STT pushes the DW

16

J>0

Without JH

J 0 and toward +x for J < 0. Therefore the depinning current jJd j is larger for J < 0 than for J > 0 under negative fields (Bx < 0): see, for example, the large difference in these depinning current values for Bx ¼  5 mT, where Jd  14 TA/m2 and Jd  + 8 TA/m2. The asymmetry also occurs for positive fields (Bx > 0). In this case jJd j is larger for J > 0 than for J < 0: Jd   5 TA/m2 and Jd  + 9 TA/m2 for Bx ¼ + 5 mT. It is also interesting to notice that the DW depinning can be achieved in the absence of field (Bx ¼ 0) within this deterministic description, which neglects Joule heating effects. In this case the depinning currents Jd are very close for both current polarities: Jd  6 TA/m2. Although in our micromagnetic analysis (i) the dimensions of the notch, (ii) the distance between electrical contacts, and (iii) the length of the injected pulses are slightly different to the ones considered in the equivalent experimental work [53], the deterministic phase diagram of Fig. 11.12 shows significant differences with respect to the experimental results [53]. If we compare the depinning field in the absence of current to the experimental one (see fig. 3c in Ref. [53]), the micromagnetic values are very similar to the experimental ones. However, the experimental results presented there indicate that the DW cannot be depinned with only current, in the absence of field (note the missing points in the experimental depinning diagram for Bx ¼ 0 in fig. 3 in Ref. [53]). The extrapolated depinning current in the experiments 2 is Jexp d (Bx ¼ 0)  3 TA/m , which is a factor of 2 smaller than the micromagnetic one (see Fig. 11.12) even taking into account the large value of P considered here. The discrepancy would be larger for more realistic values of the polarization factor. On the other hand the large asymmetry between positive and negative depinning current for a given field observed in the micromagnetic results is not as marked as in the corresponding experimental diagram. Finally and most importantly the depinning currents of the deterministic depinning diagram are in general much larger than the experimental ones. For all these reasons the deterministic model must be missing at least one relevant ingredient to properly describe the experiments. We anticipate here that the key missing ingredient in the modeling is the Joule heating. In what follows, we will review the depinning taking into account Joule heating effects, and we will show that indeed it is playing a significant role in these experiments.

11.5.3.2 Current-assisted DW in the presence of Joule heating effect The former zero-temperature analysis of the DW depinning presented in the previous section is therefore not consistent with experimental observations [53]. On the other hand, it was already shown that Joule heating of the sample is significant [8, 38], in particular in a Py strip with a notch, which not only makes the current distribution nonuniform but also generates remarkable local heating, precisely due to the high current density at the notch. Here, we solve the LLB equation (11.9) coupled to the heat equation (11.18) simultaneously. Within this formalism the magnetization dynamics ! ! (m ð r , tÞ) is evaluated taking into account the nonuniform current distribution in the

Joule heating in nanostrips and its role in current-assisted domain wall depinning ! !

!

353

!

notch (J ð r , tÞ ¼ Jð r , tÞu J ) and the corresponding Joule heating, which makes the ! temperature time and space dependent, T ¼ Tð r ,tÞ, along the notched Py strip. Note again that within this magnetothermal framework the magnetization dynamics, Eq. (11.9), is coupled to the temperature dynamics, Eq. (11.18), both through the sto!?,k

chastic thermal fields (Hth ) and the temperature dependence of the magnetic parameters (Ms ¼ Ms(T), A ¼ A(T), α? ¼ α?(T), and αk ¼ αk(T)). Fig. 11.13 shows the space and time dependence of the temperature along the ! notched Py strip, T ¼ Tð r ,tÞ. These results were obtained by solving the phenomenological heat equation (11.18) in the Py strip, which are in good agreement with the COMSOL simulations of the full system (including the Py nanostrip, the electrical contacts to inject the current pulses, the Si substrate, and the surrounding media). The inputs considered were given in the previous sections. We should note that, to reproduce the temperature values of the experiment [53], we have assumed zero thermal contact resistance, which is, in general, an unlikely experimental scenario. On the other hand, as explained in Section 11.3, the thermal data in this particular experiment [53] could only be explained with an almost negligible TCR.

T (K)

900 600 300

(A) 1200

T (K)

900 600 300

(C)

0 1 2 5 6 8 14

–2

t (ns)

900

–1

0

x (nm)

1

2

|J| = 3 TA/m2

t (ns) 0 1 2 5 6 8 14

–2

1200

|J| = 1.5 TA/m2

T (K)

t (ns)

600 300

(B)

–1

0

x (nm)

1

2

–2

|J| = 2.5 TA/m2

0 1 2 5 6 8 14

–1

0

x (nm)

1200

TCentral (K)

1200

(D)

1

2

|J| = 1.5 TA/m2 |J| = 2.5 TA/m2 |J| =3 TA/m2

900 600 300 0

5

Time (ns)

10

Fig. 11.13 Space and temporal evolution of the temperature along the notched Py strip. ! (A)–(C) show T ¼ Tð r , tÞ for three 5 ns-long current pulses with different current densities: 2 (A) jJ j ¼ 1.5 TA/m , (B) j J j ¼ 2.5 TA/m2, and (C) jJ j ¼ 3.0 TA/m2. (D) represents the temporal evolution of the temperature at the center of notch (TCentral(t)) for the same current pulses.

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Coming back to Fig. 11.13, the maximum temperature is reached at the center of the notch (x ¼ 0), where the current density is larger. This maximum temperature is reached just at the end of the current pulse at t ¼ tp ¼ 5 ns (Fig. 11.13D). For a current density of jJj ¼ 1.5 TA/m2 (Fig. 11.13A), the local temperature remains below the Curie temperature (TC ¼ 850 K) over the full Py strip. For intermediate currents, j Jj ¼ 2.5 TA/m2 (Fig. 11.13B), the local temperature at the notch overcomes TC, whereas it remains below TC in the rest of the strip. For currents of jJ j ¼ 3.0 TA/m2 (Fig. 11.13C), the full region between contacts is temporally above TC. Note that the amplitude of the current pulses in Hayashi’s experiments [53] of the DW depinning in a similar Py strip is limited below jJ j ¼ 3.0 TA/m2. Therefore their current-assisted DW experiments should be significantly influenced by Joule heating effects for current pulses with jJ j ¼ 2 TA/m2, as it will be shown hereafter. After the accurate description of the space and temporal evolution of the local temperature, we now focus on the analysis of the current-driven DW depinning under current pulses and static magnetic fields. The micromagnetic results presented hereafter were obtained by solving the LLB equation (11.9) coupled to the heat equation (11.18) simultaneously, and consequently, they take into account Joule heating effects. Some representative micromagnetic snapshots are shown in Fig. 11.14. For instance, Fig. 11.14A corresponds to the case Bx ¼ +10 mT and a current pulse with J ¼ +2.5 TA/m2, whereas Fig. 11.14B corresponds to the same field (Bx ¼ +10 mT) but a current pulse with opposite polarity (J ¼ 2.5 TA/m2). The electrical contacts are plotted in the snapshots during the current pulse. We observe a clear gray-colored region around the center of the notch and at the end of the current pulse (t ¼ tp ¼ 5 ns), which a clear sign of the local destruction of the ferromagnetic order at that point, as it was

(A)

(B)

(C)

(D)

Fig. 11.14 Some representative transient micromagnetic snapshots showing the evolution of the magnetization dynamics in the presence of Joule heating. The magnetic field and the amplitude of the current pulse are given above each set of snapshots. Except for the notch the gray color indicates the loss of ferromagnetic order as due to Joule heating effects.

Joule heating in nanostrips and its role in current-assisted domain wall depinning

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anticipated from Fig. 11.13B and D. For these intermediate current pulses (j Jj ¼ 2.5 TA/m2), the Joule heating effect becomes dominant over the STT, and the internal structure of the DW is largely altered. Once the current pulse is turned off, the temperature begins to decrease toward room temperature (see Fig. 11.13D), and the ferromagnetic order is recovered. Note that the ferromagnetic order is preserved outside the electrical contacts, and therefore the magnetostatic interaction promotes the appearance of a new DW in the Py strip. As there is a static magnetic field of Bx ¼ + 10 mT, which is larger than the depinning threshold imposed by the edge roughness, the DW depinning is finally achieved, and the DW is expelled from strip at the right extreme (see the last snapshot of Fig. 11.14A and B). These micromagnetic results indicate the key role of the Joule heating in these DW depinning events. The Joule heating effects are even more pronounced for larger current pulses than the ones considered in Fig. 11.14C and D, where a negative field (Bx ¼ + 10 mT) is applied along with current pulses of j Jj ¼ 3.0 TA/m2. Now the ferromagnetic order is temporally destroyed between the electrical contacts during the injection of the current pulse (see gray areas in the snapshots at t ¼ 2 ns and t ¼ tp ¼ 5 ns in Fig. 11.14C and D). Therefore, for these current pulses, the role of the STT is completely irrelevant with respect to the Joule heating. When the current pulse is switched off, the temperature of the Py decreases, and the ferromagnetic order is again recovered by adopting a new vortex DW configuration, which is pushed to the left by the static negative field. Depending on the location of the new DW when the pulse is turned off, the static field can or cannot promote the DW depinning from the notch, as it is clearly shown in the final snapshots of Fig. 11.14C and D. As a conclusion, it is not possible to infer any role of the STT for these current pulses, where the dynamics is dramatically perturbed by Joule heating effects. We have statistically studied the DW processes as function of Bx and J, taking into account Joule heating effects. The resulting phase diagram of the DW depinning is presented in Fig. 11.15B. The deterministic phase diagram at zero temperature presented in Fig. 11.12 is shown again in Fig. 11.15A to facilitate the direct comparison. The direct observation of these graphs clearly shows the key role of the Joule heating, which is dominant for current pulses with amplitudes j Jj 2 TA/m2 over any STT effect. It is also remarkable that the results including Joule heating, shown in Fig. 11.15B, are quantitatively close to the experimental ones described by Hayashi et al. [53]. The micromagnetic results presented here consider a transverse DW initially trapped on the left side of the notch. The magnitude of the polarization factor (P) was intentionally assumed to be very high (P ¼ 1) to maximize the effect of the STTs, although this value is surely exaggerated. Additionally the nonadiabatic parameter was fixed to β ¼ 0.02 ¼ α. Other micromagnetic results starting from a DW initially trapped at the center of the notch and considering other values of P and β can be seen in our previous works [38]. Similar conclusions were obtained, which also indicated that under these circumstances the critical depinning current a function of the applied field was only sensitive to the current polarity, and therefore to the STT, for low current densities (J ≲ 2 TA/m2). Also there the depinning diagram was only in good

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J>0 J0 J Rc a linear instability of curling type is favorable. Here, A is the exchange stiffness, μ0 is the vacuum permeability, and Ms is the saturation magnetization, while q1 ¼ 1.8412 is the smallest root of J10 ðqÞ ¼ 0 with J1 the first Bessel function of the first kind [13]. In the case R < Rc, an effectively one-dimensional (1D) behavior results which has been predicted [15, 22]. Such 1D behavior is now regularly observed (e.g., [23, 24]), and it is possible to gain significant insight into the nonlinear aspects of the magnetization reversal mechanism relevant for finite temperature by means of analytical solutions as indicated in Fig. 13.1C. The solutions reveal that for sufficiently long wires with L > Lc, the linear instability modes are not suitable to describe the switching behavior at finite temperature.a While for L < Lc magnetization reversal will still proceed via a uniform magnetization state as described by the celebrated Neel-Brown theory [25, 26], for longer wires with L > Lc it is sufficient to overcome the anisotropy energy barrier in a localized region along the nanowire [6, 15] as shown in Fig. 13.1C. This configuration indeed resembles a “nucleus” in the thermodynamic sense and consists of a soliton-antisoliton pair of domain walls of opposite chirality. The corresponding analytical solution reveals that this configuration corresponds to the lowest lying saddle point in energy as it exhibits exactly one unstable spin-wave mode [22], and thus replaces the Neel-Brown energy barrier for longer nanowires. This energy barrier can be computed exactly for arbitrary lengths [6, 27], and for long pffiffiffiffiffiffiffiffiffiffiffi wires it takes the following form [15], Es ¼ 8 AKeff A ½tanh Rs  Rs sech2 Rs . Here, A pffiffiffiffiffiffiffiffiffiffiffi is the cross-sectional area of the nanowire, sech2Rs ¼ h, and 8 AKeff A is the energy of two π domain walls across the nanowire (with parameters as defined in footnote “a”). pffiffiffiffiffiffiffiffiffiffiffi Close to Hn, the barrier energy scales as Es ¼ ð16=3Þ AKeff A ð1  hÞ3=2 . In contrast to the Neel-Brown case where the energy barrier scales with the wire volume, the nonuniform nucleation via soliton-antisoliton pairs yields a considerably smaller barrier only proportional to the cross-sectional area of the wire.b Remarkably, the field and temperature-dependent Arrhenius prefactor can be computed exactly for the

a

b

pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi Here the critical length scale is given by Lc ¼ 2πδ= 1  h2 , where δ ¼ A=Keff , Ke is a crystalline easyaxis along the wire, and taking into account the local part of the dipolar interaction, Keff ¼ Ke +ðμ0 =4ÞMs2 is the resulting effective easy-axis anisotropy along the wire [15, 22], and h ¼ μ0MsH/2Keff is the reduced magnetic field. It should be emphasized that the criticism of this mechanism put forward in Refs. [13, 28] is invalid as it ignores that the soliton-antisoliton mechanism includes the local part of the dipolar interaction via the definition of Keff [15, 22] which is valid for small diameters. This local approximation has been put on mathematically firm ground for the even more restrictive case of thin films [29].

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soliton-antisoliton mechanism [6]. In the limit of atomically thin spin chains, this mechanism has been used to explain the relaxation in molecular spin chains [30]. We wish to note that thermal nucleation of soliton-antisoliton pairs in the bulk of nanowires delimits the scalability of racetrack memory. So far we considered nucleation occurring in the bulk of an ideal nanowire, but nucleation may also start from the sample boundary as shown in Fig. 13.1C bottom. Again in the limit of narrow wires, we find that the energy barrier is given by half the bulk value, that is, Es/2, and may occur in samples with length L > Lc/2. Note that this reversal mechanism is by no means restricted to long nanowires: One of the candidates of next-generation data storage media, FePt, has an fcc phase with L10 superstructure [31] exhibiting a large anisotropy of Ke  6.6  106J/m3 which results in a crossover length as small as Lc(h ¼ 0)/2  4 nm. What may come as a little surprise at this point, this length scale is also of the order of the grain size used in FePt media. This is an empirical consequence of the fact that in order to maximize thermal stability, nonuniform reversal via soliton-antisoliton pairs should be avoided. Note that a crossover of magnetic properties as a function of grain size also occurs at zero temperature for nanostructured materials consisting of interacting grains. In this case, nonuniformity is enforced by frustrating boundary conditions due to intergrain interactions [32, 33]. Before turning to the nonlinear aspect of the curling instability treated in the remainder of this chapter, we comment on the inclusion of “superparamagnetism” in Fig. 13.1B as is often done in textbooks (see, e.g., Ref. [14]). This is obviously potentially confusing as (i) finite temperature transitions also extend well into the regime of transversally uniform magnetization as we have seen earlier and (ii) in the case of finite temperature, the instability field Hn loses its significance, since switching already occurs at fields H < Hn. One instead then has to consider a “time-dependent coercivity” Hc(t) where half to the initial states have transgressed the barrier, that is, exp fΓðHc ðtÞÞtg ¼ 1=2. This latter case (with suitably given Γ) also includes the soliton-antisoliton mechanism discussed earlier and would apply to the purple region of Fig. 13.1B. Finally, it should be mentioned that the crossover between the 1D regime described earlier and curling-type behavior at finite temperatures has been verified numerically via Monte Carlo simulations in Ref. [34]. We now turn to the main topic of this chapter, namely the role of nonlinear effects in the regime, where the initial instability in an infinite nanowire is given by the “curling mode” (cf. Fig. 13.1B), and thus we focus on nanowires with radius R > Rc. It has been pointed out that such a reversal cannot progress without the formation of topological defects as a skyrmion line will be formed at intermediate stages [35]. Such a skyrmion line arises through smooth deformation of the initial linear curling mode and is thus homotopically equivalent to the initial state [6, 36]. This implies complete reversibility of the magnetization process up to this stage once the field is removed. Thus, in order to complete irreversible magnetization reversal, the magnetization must change in a discontinuous way, thus violating the premise of analytical micromagnetics [37], where the magnetization is described by a unit vector field m(r, t) defined in a real space continuum r. This may be accomplished via the generation of point defects in the form of Bloch points or so-called hedgehogs, which are also known to be responsible for processes where the skyrmion number changes [6, 36], for

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example, the annihilation or the merging of skyrmion lines [38] in bulk chiral magnets with DMI [7, 8] or in thin films with interfacial DMI [39]. As we shall discuss here, such point defects are characterized by a nonzero flux of the magnetization field through a sphere surrounding them, a method that can easily be extended to the discrete situation of a computational grid or the physical lattice. It is the purpose of this chapter to show that such topologically nontrivial dynamic magnetization textures, namely skyrmions and hedgehogs, which have been discussed in the context of chiral magnets with DMI, also play a significant role in the reversal behavior in traditional magnetic nanowires which are approximately in a singledomain state at remanence [4, 17, 18]. Topological arguments imply that a curling instability must be followed by the creation of a skyrmion line. However, again by virtue of topology, the latter cannot be continuously transformed into the reversed state [35]. The only way to complete reversal is to break these skyrmion lines via topological point defects such as hedgehogs, Bloch points, or monopoles, where the micromagnetic tenet [37] of a continuous magnetization field is violated [6, 38, 40–43]. Our results are based on micromagnetic simulations with material parameters inspired by simple ferromagnets such as permalloy and have been checked with those of cobalt and nickel. We find that in the first reversible state of the reversal process the initial curling instability develops into the formation of a skyrmion line [35]. Upon removal of the applied field, the magnetization returns to its initial state. The second stage is irreversible and involves the breaking of the skyrmion line with concomitant creation of Bloch points in the form of hedgehog-antihedgehog pairs. After creation, these hedgehog-antihedgehog pairs rapidly separate with velocities that significantly exceed that of domain walls [23]. Being topologically nontrivial objects [6], these rapidly moving hedgehogs give rise to significant forces on the conduction electrons which are described by emergent electric fields with strength of the order of MV/ m. In order to put these result into context, we briefly sketch the derivation of the emergent electromagnetic fields that arise from the adiabatic coupling of conduction electrons to the topologically nontrivial background of a spin texture [44–46]. We shall see that a moving hedgehog generates a solenoidal emergent electric field, characterizing the hedgehogs or Bloch points as emergent magnetic monopoles. As the final stages of magnetization reversal involve periodic magnetization oscillations in the picosecond range, the process is expected to generate terahertz (THz) radiation. Our results demonstrate that topologically nontrivial spin textures are not restricted to materials with DMI, but exist in nanowires of widely studied ferromagnets that are accessible to advanced sample preparation techniques. We thus hope that our results motivate a series of novel experiments in these fascinating systems.

13.2

Skyrmion lines and monopoles in nanowires

The nanowires that we study in the following have a radius somewhat larger than the critical radius Rc (13.1) allowing for curling instabilities. We mainly focus on cylindrical nanowires as discussed elsewhere in this book, but our results are also shown to be valid for samples of prismatic or ellipsoidal shape. We build our results on

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micromagnetic simulations based on the GPU-accelerated software package MuMax3 [48] which is based on an energy density with exchange, long-range dipolar interactions as well as single-ion anisotropy and Zeeman interactions. Note that we do not consider asymmetric exchange of the Dzhyaloshinskii-Moriya type. The instantaneous magnetization state is characterized by the magnetization unit vector m(ri, t) M(ri, t)/Ms, which is defined on discrete lattice sites ri and is obtained by solving a discretized version of the Landau-Lifshitz-Gilbert equation ∂tm ¼ γm Heff + αm  ∂tm. Here, α is the damping constant, γ is the gyromagnetic ration, and the effective field is given by the functional derivative μ0Heff(r) ¼ (1/Ms) δE/δ m(r). In our simulations, we consider a wire with diameter of 60 nm and length 300 nm (aspect ratio 5:1). We use material parameters for permalloy (Py ¼ Fe20Ni80), namely exchange stiffness A ¼ 13pJ/m and saturation magnetization Ms ¼ 800 kA/m implying Rc ¼ 10.5 nm. In the simulations, the computational cell size is approx. 1.9nm  1.9nm  2.4nm. Note that Py-nanowires can be grown via standard techniques as discussed elsewhere in this book. For most simulations, we considered a damping constant of α ¼ 0.1, although additional checks with α ¼ 0.01, 0.001 were made, but no significant deviations from the former case were found. We note that our predictions hold for any magnetically soft nanowires with the results of this section also pertaining to insulators, but of course the discussion of emergent electromagnetic fields is restricted to metallic magnetic nanowires. In addition to permalloy, we also considered materials with nonzero values of the magnetocrystalline anisotropy Ku and we found that the phenomena presented here persist as long as Q  2Ku =μ0 Ms2 < 1, and R > Rc. In Fig. 13.2, the time evolution of the magnetization configuration in a nanowire is shown after a field of μ0H ¼ 125 mT has been applied opposite to the remanent magnetization H > Hn as indicated in Fig. 13.2A. The top row shows the magnetization field, while the bottom row displays a contour plot of the z-component of the magnetization mz, together with the z-dependent quantity, QD ¼

1 4π

Z Z dx1 dx2 m  ð∂x1 m  ∂x2 mÞ,

(13.2)

D

where D denotes a disk-like surface across the nanowire at position z and ∂xi  ∂=∂xi . Since the disk has finite radius, and the magnetization in general attains different values on the boundary, this quantity can take noninteger values as is seen from the bottom row in Fig. 13.2. For a disk D of infinite radius, QD takes the form of a usual skyrmion number, topological charge (or winding number) to be discussed next. This can be seen explicitly if Eq. (13.2) is simplified for cylindrically symmetric Blochlike skyrmion-like spin textures, that is, θ(ρ), with θ(0) ¼ 0, and ϕ ¼ φ  π/2, where (ρ, φ) are cylindrical coordinates in real space and the magnetization unit vector is parameterized by m ¼ ðsin θ cos ϕ, sin θ sin ϕ, cos θÞ. In this case, we obtain QD ¼ (1  mz(ρ ¼ a))/2 with a the cylinder radius [36]. Thus the deviation of QD from an integer value is seen to be caused by the deviation of the magnetization from a complete “down” state at the sample boundary, an effect that can be seen in Fig. 13.2. The initial state before applying a magnetic field is shown in Fig. 13.2B and is approximately uniform, except for dipolar induced “curling” toward the sample ends

Fig. 13.2 Formation of skyrmion lines during the magnetization process and snapshots for fields larger than Hn (cf. (A)). (B–D) Snapshots of a micromagnetic simulation showing the initial, reversible, stages of the switching process in a cylindrical permalloy nanowire with diameter 60 nm and length 300 nm at a supercritical value (A) of the applied magnetic field (H > Hn) at various times, (B) at 0 ps, (C) at 100ps, and (D) at 160 ps, after having applied a constant field of 125mT along z. The top panels show vector-field plots and the bottom panels show contour plots of the z-component of the magnetization together with QD, the topological charge density integrated across the disk-like cross-section of the cylinder. The initial curling-mode instability develops into the formation of two skyrmion lines of opposite handedness. (E) The profile across the wire is well described by the variational ansatz (13.3) [22, 47]. From M. Charilaou, H.B. Braun, J.F. L€offler, Monopole-induced emergent electric fields in ferromagnetic nanowires, Phys. Rev. Lett. 121 (2018) 097202.

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with opposite relative circulation. Note that the absolute rotation sense at one sample end is arbitrary and is a consequence of spontaneous symmetry breaking. In the presence of a reversed (“down”) field as shown in panels C and D, the magnetization reverses on the mantle of the cylinder while it stays in the remanent upstate on the cylinder axis as postulated in Braun [35]. The domain wall separating these states is of Bloch type, resulting in a skyrmion-like spin texture as indicated in panel D. The helicity of these skyrmions in the upper and lower half of the cylinder is opposite. As is clear from the lower row of Fig. 13.2, skyrmion lines are created that extend from the middle of the cylinder to its ends, reflected by the fact that QD  1 throughout most of the cylinder. With increasing time, the skyrmion lines shrink as can be seen from Fig. 13.2D. As shown in Fig. 13.2E, the magnetization profile of mz across the cylinder is approximately given by the analytical expression θ ¼ π  θsk with [22, 47] θsk ¼ θ0 ðρ=δ +RÞ +θ0 ðρ=δ  RÞ,

(13.3)

and θ0 ðξÞ ¼ 2arctan eξ being the profile of a π-domain wall. Here, ρ is the radial coordinate, R characterizes the skyrmion radius, and δ is the intrinsic length scale of the skyrmion texture. This expression is inspired by the analytical result for a 2π-domain wall, where both R and δ exhibit an analytical dependence on the applied field [22]. The behavior of the magnetization depends in a very sensitive fashion on the duration of the applied field as illustrated in Fig. 13.3. If the field is applied during a time Δt1 ¼ 0.16 ns as shown in Fig. 13.3A, the magnetization reversibly relaxes to the initial state, similar to a radial exchange spring [49]. Note that in this case, the skyrmion lines remain intact. In contrast, as shown in Fig. 13.3B, if the field is applied just 10 ps longer, then the skyrmion lines break, and hedgehog-antihedgehog pairs are created with a (anti)hedgehog terminating a skyrmion line. Only in this case does full magnetization reversal occur with the volume averaged magnetization attaining the reversed state with m z  1. Thus we see that irreversibility is directly linked to the creation of topological point defects, also known as Bloch points or hedgehogs. They are examples of a topological point defect, where the magnetization is forced to vanish at its center [6]. Hedgehogs can be characterized by evaluating the winding number on a sphere surrounding the point defect. This defines a mapping between two spheres: S2 ! S2, and the corresponding integer winding number can be expressed as [6, 16, 36] 1 QS2 ¼ 4π

Z S2

dϑ dφ sin θ ð∂ϑ θ ∂φ ϕ  ∂φ θ ∂ϑ ϕÞ,

(13.4)

where (ϑ, φ) are the polar and azimuthal angles parameterizing the real space sphere S2. As this winding number is an invariant of a given homotopy class, it is invariant under smooth deformations of the magnetization field, and it is conveniently evaluated for the most symmetric representative of a given homotopy class (cf. Fig. 13.6): R For a rotationally symmetric spin texture, we have ∂φθ ¼ 0 and 1 QS2 ¼ 4π dϑ dφ sin θ ∂ϑ θ ∂φ ϕ, and thus QS2 ¼ 1 for a symmetric, radially outward pointing hedgehog with θ ¼ ϑ, ϕ ¼ φ.

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Fig. 13.3 Pair creation of topological point defects and the onset of irreversibility. Time evolution of the spatially averaged magnetization m z during a field pulse, and the contour plot of mz at the time when the applied magnetic field is switched-off. (A) The magnetization process is reversible if the applied field is switched-off before the skyrmion lines break (Δt1 ¼ 0.16 ns). (B) Irreversibility sets in only when the skyrmion line is destroyed via the creation of a hedgehog-antihedgehog pair before switching-off the applied field (Δt2 ¼ 0.17 ns). Only in the latter case full magnetization reversal occurs (m z  1). From M. Charilaou, H.B. Braun, J.F. L€offler, Monopole-induced emergent electric fields in ferromagnetic nanowires, Phys. Rev. Lett. 121 (2018) 097202.

Armed with this insight, we see that QS2 ¼ 0 everywhere for the reversible situation shown in Fig. 13.3A, while for spheres surrounding the ends of the skyrmion lines in Fig. 13.3B we have QS2 ¼ 1 with the positive sign referring to the hedgehog situated on the top. We thus find that irreversible switching is associated with the emergence of points in the sample such that there exists a surrounding sphere with QS2 6¼ 0. Conversely, if QS2 ¼ 0, the process remains reversible. Note also that these point defects are of purely dynamical origin [16] and exist in a DMI-free nanowire, in contrast to the point defects at the end of skyrmion lines in chiral magnets [38] or chiral “bobbers” in DMI materials [50–52].

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Note that so far we considered applied magnetic fields that are considerably larger than the zero-temperature coercivity Hn shown in Fig. 13.1A. In order to determine the zero-temperature coercivity for our realistic nanowires (which include sample ends in contrast to the idealized considerations summarized in Fig. 13.1B), we study the magnetization dynamics as a function of decreasing fields as shown in Fig. 13.4. As is evident from Fig. 13.4A, the precession frequency is slowing down as a function of applied field and tends to zero upon approaching μ0Hn ¼ (3  0.1)  102 T.

Fig. 13.4 Critical slowing down close to the instability field Hn. The magnetization dynamics exhibits a critical slowing down as the instability field Hn is approached where the barrier vanishes (cf. Fig. 13.1). For a given value of the applied magnetic field, we analyzed the magnetization components as a function of time. Panel (A) shows the reduction of precession frequency upon approaching Hn as determined from the dynamics of the xy-component. Panel (B) shows the switching delay of the mz component along the wire upon approaching Hn from above. From (A) we obtain a value for the instability field of μ0Hn ¼ (3  0.1)  102 T as indicated by the vertical dashed line. The insets show examples for μ0H ¼ 0.032T which is close to the instability field Hn. From M. Charilaou, H.B. Braun, J.F. L€offler, Monopole-induced emergent electric fields in ferromagnetic nanowires, Phys. Rev. Lett. 121 (2018) 097202.

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We identify this value as the zero-temperature instability field Hn, and the critical slowing down is a consequence of the flat energy surface (cf. Fig. 13.1A) which implies vanishing effective fields that enter the rhs of the LLG equations. Note that the value of Hn as shown here is significantly lower than the textbook estimates following from the curling instability, given by μ0 Hncurl ¼ ðμ0 =2ÞMs ðRc =RÞ2  0:06T for our permalloy parameters and vanishing uniaxial anisotropy [13]. This is approximately twice as large as the numerical value for Hn identified earlier. This implies that the sample ends may play a significant role in reducing the instability field from the infinite wire estimate. We now return to the details of the motion of the hedgehog pairs. As shown in Fig. 13.5, once nucleated, the hedgehogs (highlighted by white circles) rapidly move to the sample end or the nanowire center, respectively. It is clear from Fig. 13.5A–C that the breaking of the skyrmion line generates a topologically trivial segment with QD ¼ 0 between the hedgehogs, and we know that a transition from regions with QD ¼ 1 to regions with QD ¼ 0 along the cylinder can only happen via hedgehog-type point defects [6, 36]. Alternatively, we can evaluate Eq. (13.4) in the vicinity of these points as shown in detail in Fig. 13.5D, and obtain QS2 ¼ 1 (for schematics cf. Fig. 13.6). From Fig. 13.5E, we see that the hedgehogs slow down with increasing time, and toward the end of the process the hedgehogs close to the cylinder end leave the sample, and a domain with reversed magnetization remains in the middle of the nanowire forming a 2π-domain wall. This region finally disappears through unwrapping of the 2π-domain wall via “escape through the third dimension” [22].

13.3

Emergent electromagnetic fields

The change in topology due to the created hedgehogs generates a new dynamical state of magnetic matter in the nanowire. In conducting nanowires, the moving spin texture will give rise to forces on the conduction electrons, very similar to the topological Hall effect induced by the topologically nontrivial spin texture of skyrmions. Note that here, however, the hedgehogs are rapidly moving, and they also induce considerable emergent electric fields which are equivalent to the forces exerted by a real electric field of magnitude 0.5–2 MV/m. Quite generally, emergent electromagnetic fields result if the spin of a conduction electron adiabatically follows a topologically nontrivial spin texture. The Schr€odinger equation for its spinor wave function ψ is then as follows 

 p2 iħ ∂t ψðr,tÞ ¼ 1 + J σ  mðr, tÞ ψðr, tÞ, 2m

(13.5)

where J is the sd-exchange coupling between conduction and localized electrons, σ is the vector of Pauli spin matrices with 1 the identity in spinor space, and m is the unit vector of the space-dependent magnetization texture. This Hamiltonian can easily be diagonalized by using a spinor basis jim with quantization axis along the texture, that is,

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Fig. 13.5 Dynamics of separating hedgehog-antihedgehog pairs. Contour plots of mz and QD as a function of z at (A) t ¼ 10ps, (B) t ¼ 30ps, and (C) t ¼ 50ps after the field has been switchedoff. The circles in the contour plots indicate the position of the highlighted QS2 ¼ 1 hedgehog (top) and the QS2 ¼ 1 (bottom) antihedgehog as a function of time. (D) Detailed view of the moment distribution of a hedgehog at the end of the skyrmion line. (E) Hedgehog separation (circles) and location of the top hedgehog (diamonds) as a function of time yielding a maximal relative velocity of  3  103 m/s. (F) Hedgehog speed as a function of damping constant. Note that speeds of up to 7 km/s are observed for applied fields of μ0H ¼ 0.175T and damping constants of α ¼ 0.1. From M. Charilaou, H.B. Braun, J.F. L€offler, Monopole-induced emergent electric fields in ferromagnetic nanowires, Phys. Rev. Lett. 121 (2018) 097202.

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Fig. 13.6 Hedgehog point defect at the end of a skyrmion line. (A) Schematic view of the end of a skyrmion line with the location of the hedgehog highlighted. (B) Magnetization texture at the end of the skyrmion line can be continuously deformed (i.e., is homotopically equivalent) to a spherically symmetric hedgehog with charge QS2 ¼ 1, cf. Eq. (13.4). From M. Charilaou, H.B. Braun, J.F. L€offler, Monopole-induced emergent electric fields in ferromagnetic nanowires, Phys. Rev. Lett. 121 (2018) 097202.

σ  m j  im ¼  j  im :

(13.6)

This spinor is related to the standard z-spinor via a unitary transformation U such that j  im ¼ Uj  i^z , which is unique up to a U(1) transformation. We may now transform Eq. (13.6) into a Schr€ odinger equation with the z-axis as the global quantization axis via ψ ¼ Uχ, where χ is a spinor wave function expressed in the z-basis. The two global spin directions j  i^z then refer to the majority and minority spins in the formerly rotated spin frame. Under the assumption of adiabatic motion, that is that there are no spin flips between the two channels, the Schr€ odinger equation for the two spin channels χ  takes the following form [45] 

 1 2 ðp  q A + Þ +q V +  J χ  ðr, tÞ, iħ ∂t χ  ðr, tÞ ¼ 2m

(13.7)

where the exchange interaction now has become diagonal, at the expense of coupling the electron in a spin-dependent fashion to the gauge fields, A + ¼ 2iħ ^z h +jU + rUj +i^z ¼ 2iħ m h +jrj +im ,

(13.8)

V + ¼ 2iħ ^z h +jU + ∂t Uj +i^z ¼ 2iħ m h +j∂t j +im :

(13.9)

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Thus electrons with spin directions  couple to this gauge field with opposite charges q ¼ 1/2 (we use here the convention of Ref. [44]). Evaluated in the opposite spin direction, both Eqs. (13.8), (13.9) change sign, and from the second form in these expressions it is clear that these gauge potentials have the form of a Berry connection. Using the explicit form for the unitary transformation U (or equivalently the explicit form of the states j+im), these emergent gauge fields can be shown to give rise to the following emergent electromagnetic fields [19, 44, 45]: ħ Bei ¼ Eijk m  ð∂j m  ∂k mÞ, 2 Eei ¼ ħ m  ð∂i m  ∂t mÞ,

(13.10)

which have units of force/velocity and force, respectively. The total force on a conduction electron of momentum k, spin direction σ ¼ , and emergent charge qσ ¼ σ/2 is given by Schulz et al. [44], Fσ ¼ qσEe + qσ vσ, k Be, where vσ, k is the velocity of quasiparticles. In the following, we focus on the emergent electric field that results from the rapid motion of the hedgehog defects. In leading approximation, the end of the skyrmion line does not change its shape upon motion and retains its cylindrical Bloch-type symmetry, and thus we assume m ¼ m(x, y, z  vt) with v positive for a texture moving in positive z direction. The emergent electric field has then only components in x, y direction and takes the form Eej ¼ ħv m  ð∂j m  ∂z mÞ, with j ¼ x, y. Top and bottom hedgehog are of Bloch type, ϕ ¼ φ  π/2, and the rotational invariance of mz implies that θ ¼ θ(ρ, z), with (ρ, φ, z) being cylindrical coordinates. The resulting emergent electric field has then solenoidal form with Eeφ ¼ ðħv=ρÞ ∂z mz with v positive (negative) for the upward (downward) moving hedgehog. This sign is compensated by the opposite sign of ∂zmz for the upward moving hedgehog and the downward moving antihedgehog, and thus the emergent electric field has the same orientation for the two textures as shown in Fig. 13.7. Thus we see that the moving hedgehog gives rise to a solenoidal emergent electric field, in very much the same way as a moving electric charge gives rise to a solenoidal magnetic field. Hence, we conclude that in emergent electrodynamics, moving hedgehogs behave like (emergent) magnetic monopoles. In order to obtain quantitative estimates, we observe that for the majority electrons with q ¼ 1/2 (as their spin orientation is antiparallel to m), the resulting real net ^ =2 with φ ^ the unit vector in φ direction. force is then also solenoidal with Fφ ¼ Eeφ φ Its magnitude is ħv=2λ2 , where λ is the characteristic length scale of the hedgehog, Bloch point, or monopole. For the values obtained in our simulations, we typically have v  1.5km/s (we also observed values up to 7km/s), and for λ  δ  1nm we obtain F  8  1014 J/m which corresponds to the force exerted on an electron by a real electric field of E  0.5MV/m or up to E  2MV/m. These are strikingly large values and are of the order of the dielectric breakthrough in air (3MV/m). Note that even for larger spin textures with dimensions of a few nanometers we still expect appreciable forces onto the conduction electrons. So far, moving topological spin textures have been thought to give rise to only weak emergent electric fields, but this conclusion was reached considering materials with DMI [38, 40, 41, 53].

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Fig. 13.7 Emergent electric field of separating hedgehogs or monopoles. A pair of separating hedgehogs of opposite topological charge (QS2 ¼ 1) and opposite velocity gives rise to an emergent solenoidal electric field (purple), in analogy to the magnetic field created by a pair of separating electric charges of opposite sign. For the process shown in Fig. 13.5, this emergent electric field of such a magnetic monopole is of the order of 0.5 MV/m. From M. Charilaou, H.B. Braun, J.F. L€offler, Monopole-induced emergent electric fields in ferromagnetic nanowires, Phys. Rev. Lett. 121 (2018) 097202.

In the final stages of magnetization reversal, as can be seen from Fig. 13.5, two of the four monopoles move toward the surface of the sample and are absorbed, whereas their counterparts of opposite charge move toward the middle of the wire, where they perform axial oscillations before they disappear. Such oscillations have time constants of the order of 10 ps, and hence generate oscillating forces on the conduction electrons in the THz range. This suggests that simple ferromagnetic nanowires or nanoparticles may emit THz radiation due to the accelerated electrons, and this could potentially be detected in pump probe experiments [16]. The phenomena discussed earlier for an ideal cylindrical nanowire are quite generic and as we shall see in the following section, they also occur in prismatic wires and ellipsoids. Finally, we also verified their existence in sample with uniaxial anisotropies. Generally, these phenomena persist as long as reversal proceeds in a curlingtype fashion. Even inclusion of thermal activation, which acts like disorder in this regime of supercritical fields, does not change the basic aspects of the mechanism. Thus the physics described here could well be used for potential devices that exploit the emergent electromagnetic fields, and its realization in DMI-free materials substantially widens the range of materials in which such phenomena can be detected.

13.4

Other sample geometries and thermal fluctuations

So far we considered the ideal case of a cylindrical nanowire. For supercritical fields, we identified the formation of skyrmion lines and hedgehogs to be the consequence of general topological arguments which are related to the continuity of the magnetization

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field. Given the generality of these arguments we expect similar effects also to occur in samples with geometries different from that of an ideal cylinder. In order to address this issue, we considered two different and commonly used elongated shapes, namely a rectangular prism and an ellipsoidal particle with the same material parameters of Py as earlier. Fig. 13.8 shows that the generic features of the reversal process of an ideal cylinder remain intact in more general geometries: Reversible behavior with the formation of skyrmion lines in Fig. 13.8B, C, F, and G, and subsequent creation of monopole pairs that break the skyrmion line [16]. Finally, we address the issue of thermal fluctuations. As we consider the regime of fields considerably larger than the zero-temperature coercivity (cf. Fig. 13.4), we

Fig. 13.8 Reversal in elongated prisms and ellipsoids: Switching process in nanowires of (A)–(D) prismatic and (E)–(H) ellipsoidal shape showing the same characteristics of the reversal process as in cylindrical nanowires (cf. Fig. 13.2): Initially two propagation fronts of curling character are generated (B, F), leading to two skyrmion lines with opposite handedness (C, G), which eventually break via pair creation of hedgehogs that rapidly separate and complete the reversal (D, H). From M. Charilaou, H.B. Braun, J.F. L€offler, Monopole-induced emergent electric fields in ferromagnetic nanowires, Phys. Rev. Lett. 121 (2018) 097202 (Suppl. Inf.).

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Fig. 13.9 Supercritical switching process at finite temperature. In a cylindrical permalloy nanowire with diameter 60 nm and length 300 nm, the switching process at finite temperature is nearly identical to the one occurring at zero temperature (cf. Fig. 13.1). The application of a supercritical external field generates two propagating fronts with curling-type spin configurations that lead to the formation of two skyrmion lines with opposite handedness. Irreversibility sets in as soon as the skyrmion lines break, at which point hedgehogantihedgehog pairs are created, and the hedgehogs move rapidly along the wire, generating strong emergent electric fields with solenoidal character giving them the character of monopoles. The monopoles are absorbed by the surfaces, while the central collinear region forms a 2π domain wall that escapes through the third dimension, and the wire enters a singledomain state in the direction of the external field. From M. Charilaou, H.B. Braun, J.F. L€offler, Monopole-induced emergent electric fields in ferromagnetic nanowires, Phys. Rev. Lett. 121 (2018) 097202 (Suppl. Inf.).

expect the effect of thermal fluctuations to be considerably less dramatic than in the case of the overbarrier regime shown in Fig. 13.1. Indeed, we expect thermal fluctuations to have a similar effect that structural disorder in this case. Fig. 13.9 shows the reversal with the addition of thermal fluctuations. Indeed, the behavior resembles a disordered version of Fig. 13.2. Nonetheless, clearly visible is the formation of skyrmion lines, and the creation of monopole-antimonopole pairs in the irreversible regime which are both unaffected by thermal fluctuations.

13.5

Conclusions

In this chapter, we discussed the emergence of topological spin textures in nanowires. In the introduction, we recalled that in thin-long nanowires, thermal fluctuations play an important role, and at subcritical fields, thermal magnetization reversal occurs via a localized structure, a soliton-antisoliton pair [15]. This mechanism that involves a texture of zero total winding number is a direct generalization [6, 27] of the widely used

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Neel-Brown model [25, 26] which is valid for wires shorter than the domain wall width. In wider nanowires and for supercritical fields, temperature plays a less important role, but topological constraints turn out to be of crucial importance. Topological spin textures thus not only play a role in materials with asymmetric exchange, so-called DMI, but also are important in standard ferromagnets. In soft magnetic nanowires of infinite length and moderate diameter, it has long been known that magnetization reversal is initiated via divergence-free fluctuations, the so-called curling mode [13, 14]. However, maintaining cylindrical symmetry also during the nonlinear regime of magnetization reversal must lead to the formation of topologically stable skyrmion lines [35]. In this chapter, we have shown how the breaking of these skyrmion lines via creation of topological point defects in the form of hedgehog or Bloch points leads to irreversibility in the magnetization process [16]: As long as the skyrmion lines remain intact and no point defects are nucleated, the magnetization process remains reversible. Irreversibility occurs once hedgehogantihedgehog pairs of opposite topological charge are created which lead to a breaking of the skyrmion line. The skyrmion lines terminated by the (anti)hedgehog rapidly shrink leading to a rapid motion of the hedgehogs. This very generic behavior is not restricted to ideal cylindrical nanowires but also occurs in elongated particles of prismatic or ellipsoidal shape and in the presence of thermal fluctuations. The hedgehogs/Bloch points move very fast with speeds between 1 and 7 km/s in standard materials such as permalloy, and as topologically nontrivial spin textures they give rise to emergent electromagnetic fields that exert physical forces on electrons with their spin adiabatically following the spin texture. Computing the emergent electric field for our hedgehog we find that it has solenoidal character. It thus resembles the magnetic field created by an electric point charge, and thus shows that the hedgehog/Bloch point behaves like a magnetic monopole in emergent electrodynamics [16]. Multiple monopoles could be generated in arrays of nanowires [54], thus offering the possibility of enhancing the previous effect. Finally, we wish to note that at the time of writing of this chapter there are various reports about the observation of static Bloch points in permalloy nanowires [55], in pillars [56], and of slowly moving Bloch points [57]. Thus it appears that the observation of the scenario first proposed in Ref. [16] is within reach.

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Micromagnetic modeling of magnetic domain walls and domains in cylindrical nanowires

14

J.A. Fernandez-Roldana,b, Yu.P. Ivanovc, O. Chubykalo-Fesenkoa a Materials Science Institute of Madrid, ICMM-CSIC, Madrid, Spain, bDepartment of Physics, University of Oviedo, Oviedo, Spain, cSchool of Natural Sciences, Far Eastern Federal University, Vladivostok, Russia

14.1

Introduction

Cylindrical magnetic nanowires constitute the most promising nanostructures for three-dimensional nanoarchitectures in future applications for information technologies, nanoelectronics, sensing, etc. From a more fundamental point of view, they create a fascinating possibility for studying nanomagnetism in curved geometries that offers a plethora of novel static and dynamical properties [1, 2]. Nanowires can be grown by chemical routes or more frequently by inexpensive electrodeposition in porous membranes with pore diameters ranging from several tens to hundreds of nanometers [3]. The lengths of nanowires are typically in the interval between 100 nm and 100 μm. This geometry presents a high aspect ratio of length/diameter. Among the properties of cylindrical nanowires that make them very appealing for future applications, we can summarize the most relevant: l

l

l

l

l

The elongated shape of cylinders induces a natural axial anisotropy via the magnetostatic energy effect, which favors high stability of axial magnetic states. The curved geometry (via the exchange interaction) leads to an effective Dzyaloshinskii-Moriya interaction [4] with no need of the intrinsic one. The overall effect of the intrinsic interactions and confinement in a curved geometry is to promote the formation of nontrivial topological structures in materials with no need of intrinsic Dzyaloshinskii-Moriya interaction or intrinsic anisotropies [4]. Multiple novel magnetization textures have been reported for this geometry: toroidal, helicoidal, and vortex (Bloch point) domain walls [5] and, recently, skyrmion tubes [6, 7]. The dynamical vortex and skyrmion tubes may carry the Bloch point (a “hedgehog” skyrmion) [5, 6, 8, 9]. The Bloch point is a magnetic singularity with vanishing magnetization [10], which possesses some unique properties. The fascinating property of the Bloch point domain wall is that it is highly mobile [8], with velocities higher than the transverse domain wall. Recently, it has been shown that its motion leads to the emergent electromagnetic fields [6]. Cylindrical nanowires are naturally magnetochiral materials. For example, the magnetization mobility and stability of the vortex domain wall depend on its chirality [5, 11]. Magnetic domain walls in cylindrical nanowires lack the so-called Walker breakdown, which consists of a sudden drop of the domain wall velocity under the action of applied

Magnetic Nano- and Microwires. https://doi.org/10.1016/B978-0-08-102832-2.00014-1 Copyright © 2020 Elsevier Ltd. All rights reserved.

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magnetic field or current, observed in nanostrips [5, 11]. However, the domain wall dynamics in cylindrical nanowires is limited by the spin-Cherenkov emission (spontaneous emission of spin waves by the domain wall for velocities higher than the minimum spin-wave group velocity) [5]. This, however, happens at much higher velocities (up to 10 km/s) than the ones achieved in magnetic stripes. Therefore ultrafast domain wall velocities can be expected.

Therefore the cylindrical geometry offers multiple possibilities for observing highly nontrivial chiral magnetization structures in simple geometries. The main problem for nanowire applications is the control of domain wall dynamics: nucleation, mobility, pinning, etc. Only recently the development of the X-ray magnetic circular dichroism (XMCD) technique, highly sensitive magnetic force microscopy, and electron holography has allowed to make first steps in observation of some of these structures such as circular magnetic domains [12–14] (extended vortex structure with a core parallel to the nanowire axis). Micromagnetic simulations play a decisive role in unveiling the observed magnetic contrast since they provide a complementary 3-D information. The proper dynamical measurements on domain wall in cylindrical magnetic nanowires are a completely open field of research doing its steps [15]. Most of studies report pinned domain walls, such as the Bloch point domain wall [8]. The strategies to pin domain walls involve the use of nanowires with different compositions [16–19] or geometries [20–25] such as modulated nanowires or nanowires with notches or antinotches. Here, micromagnetic simulations help to design nanostructures with desired properties and understand the pinning mechanism.

14.2

Hysteresis loops of magnetic nanowires

The hysteresis loops of magnetic nanostructures are strongly affected by their geometry through the shape anisotropy and, particularly, the aspect ratio length/diameter. Fig. 14.1A illustrates the effect of the shape anisotropy in an ideal individual nanowire of permalloy. Permalloy (Fe20Ni80) has a vanishing magnetocrystalline anisotropy; therefore the behavior is dominated by the shape anisotropy effect. A preferential axial orientation of the magnetization resulting in a squared loop and a high remanence for the field applied along the nanowire axis is observed. The state before the switching (see Fig. 14.2A in the succeeding text) is the two so-called open vortex structure, characterized by the predominant magnetization component along the nanowire direction but already developing a circular structure for the perpendicular components at the nanowire ends. The magnetization switching takes place in a unique Barkhausen jump. On the other hand, for a perpendicular magnetic field, the hysteresis loop of a single nanowire exhibits a completely anhysteretic (rotational) behavior, that is, the area enclosed between both branches of the hysteresis loop vanishes. Therefore, for a single nanowire of permalloy, the magnetic properties and the magnetization reversal process are fully determined by the shape anisotropy induced by the cylindrical geometry (magnetostatic energy). Ivanov et al. [26] showed that the magnetization reversal in other nanowires with fcc magnetocrystalline anisotropy in either polycrystalline or single-crystalline nanowires is similar to permalloy ones because it is also

(A)

1.0 0.5 0.0

// ^

0.5 1.0 –10 –8 –6 –4 –2 0 2 H (kOe)

4

6

8 10

Normalized magnetization

Normalized magnetization

Micromagnetic modeling of magnetic domain walls and domains

(B)

405

1.0 0.5 0.0

Easy axis

// ^

0.5 1.0 –10 –8 –6 –4 –2 0 2 H (kOe)

4

6

8 10

Fig. 14.1 Simulated hysteresis loops for individual nanowires of (A) permalloy with zero magnetocrystalline anisotropy and (B) Co hcp with perpendicular uniaxial magnetocrystalline anisotropy having strength 4.5  105 J/m3. Both nanowires have 40 nm diameter and 2 μm length. Material parameters are presented in Table 14.1. Adapted from Y.P. Ivanov, M. Va´zquez, O. Chubykalo-Fesenko, Magnetic reversal modes in cylindrical nanowires, J. Phys. D Appl. Phys. 46 (2013) 485001. doi:10.1088/0022-3727/46/ 48/485001.

Fig. 14.2 Magnetic configurations in cylindrical nanowire (A) precursor of the vortex domain wall (also known as open vortex structure) in shape anisotropy-dominated nanowires. The image corresponds to the permalloy nanowire at fields close to the switching. (B) The remanent state of hcp Co nanowire with easy axis perpendicular to the nanowire, showing two-vortex domains with opposite chiralities. The field was applied parallel to the nanowire axis. The same geometrical and magnetic parameters as in Fig. 14.1 are used. Images taken from Y.P. Ivanov, M. Va´zquez, O. Chubykalo-Fesenko, Magnetic reversal modes in cylindrical nanowires, J. Phys. D Appl. Phys. 46 (2013) 485001. doi:10.1088/0022-3727/46/ 48/485001.

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predominantly determined by the shape anisotropy owing to its large value compared with the magnetocrystalline anisotropy strength [26]. In the array of nanowire, complex features may arise from the interwire dipolar interactions that decrease the effect of the shape anisotropy and introduce a “thin film” demagnetizing effect (see Refs. [2, 33, 34]). The introduction of a strong magnetocrystalline anisotropy, noncollinear with the nanowire axis, changes the hysteresis cycle due to the interplay between the involved energies. The uniaxial transverse anisotropy reported in cobalt hcp nanowires exhibits a strong competition with the shape anisotropy [26, 35] almost completely suppressing it and producing a very soft behavior. For illustration, Fig. 14.1B presents the simulated hysteresis loop of an individual Co nanowire with anisotropy easy axis perpendicular to the nanowire axis. Note also that the saturation magnetization of cobalt is 75% higher than that of permalloy, indicating a very high shape anisotropy. The inclined hysteresis loops for the single Co hcp nanowire in both parallel and perpendicular applied fields confirm that the magnetocrystalline anisotropy is strong enough to compensate the magnetostatic energy and to promote a drastic decrease of the coercive field and a small remanence. The remanent state is characterized by the formation of a vortex state along the whole length of the nanowire [26, 35] (see Fig. 14.2B). The perpendicular field hysteresis cycle resembles that of the vortex in soft magnetic dots characterized by the nucleation and annihilation of the vortex structures. The magnetostatic interactions between nanowires also significantly modify the hysteresis cycle due to the “thin film” shape anisotropy effect [26, 33]. Experimental evidences and micromagnetic simulations confirm that the properties of the hysteresis loops of nanowires generally become independent of the nanowire length for aspect ratio values larger than 8–10 [36–38]. As an illustration, Fig. 14.3 displays the calculated hysteresis loops of permalloy and Co fcc nanowires with different lengths. For short nanowires a lower coercive field is observed, and the hysteresis loops are strongly affected by small variations of the length. The anisotropy of the fcc phase of Co has also a remarkable effect in the inclined hysteresis loops for short nanowires. In fact, 30-nm-length Co nanowire is a pillar that presents the characteristic vortex hysteresis cycle. On the contrary, for nanowires with large aspect ratios, the coercive field and the remanence become practically independent on the length. Note that in this ideal case in nanowires with lengths smaller than ca. 70 nm (estimated for fcc Co) the open vortices with the same chiralities are formed at the opposite ends due to the minimization of the magnetostatic interaction between them. When the length is increasing, the interaction of structures at opposite ends becomes negligible, and the vortices at the opposite ends are formed with the opposite chiralities as in Fig. 14.2B. The latter happens due to the influence of the nanowire internal demagnetizing field (with an inward and outward component at opposite nanowire ends), resulting in opposite magnetic torques. The latter behavior is common in simulations for all nanowires in the absence of defects and granular structure. The nanowire diameter has a more critical impact on its hysteresis properties than the length. As it is presented in Fig. 14.4, the coercive field strongly decreases with the

Micromagnetic modeling of magnetic domain walls and domains

407

Fig. 14.3 Simulated hysteresis loops for individual nanowires with several lengths without and with magnetocrystalline anisotropy. Figures correspond to individual nanowires of (A) permalloy and (B) Co fcc with a 30 nm diameter. Parameters used for simulations can be found in Table 14.1.

1.0

4000

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Mr/Ms

Hc (Oe)

5000

2000 1000 0

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0.4 0.2

20

40

60 d (nm)

80

100

0.0

(B)

20

40

60 d (nm)

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Fig. 14.4 (A) Coercive field and (B) remanence dependence on the nanowire diameters for individual permalloy nanowires.

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increase of the NW diameter that reflects the change of the reversal mode from that of the transverse domain wall to the Bloch point (vortex) domain wall [26] (see next section). At the same time the remanence has a small dependence on the diameter, defined by the formation of the open vortex structures for shape anisotropy-dominated nanowires (see Fig. 14.2A). This dependence becomes more pronounced for nanowires with high saturation magnetization value such as Fe-based nanowires (e.g., Fe bcc or FeCo nanowires) [26].

14.3

Magnetic domains and domain walls in straight magnetic nanowires

As it is indicated in the previous section, shape anisotropy-dominated magnetic nanowires are in the (almost) single domain state in the remanence and demagnetize via the domain wall propagation [26, 39–41]. The type of domain wall by which the nanowire demagnetizes is determined by the nanowire geometry and material [26]. More concretely, only nanowires with very small diameters demagnetize via the transverse domain wall propagation (see Fig. 14.5A). Depending on the relative orientation

Fig. 14.5 Domain walls in cylindrical nanowires (permalloy). (A) Transverse domain wall (TDW). (B) Vortex (Bloch point) domain wall. (C) Hedgehog Bloch point separating two magnetic domains. (D) Skyrmion tube from Ref. [6] formed during the magnetization reversal. (E) Asymmetric transverse (vortex) domain wall from Ref. [42]. Cross sections indicate the inner magnetization of the domain walls at the marked positions.

Micromagnetic modeling of magnetic domain walls and domains

409

between domains, the wall is called tail-to-tail or head-to-head transverse domain wall [9]. The transverse domain wall carries a magnetic charge in volume due to the divergence of magnetization [9]. Larger-diameter (or saturation magnetization) nanowires demagnetize via the Bloch point (vortex) domain wall propagation (see Fig. 14.5B). The vortex domain wall presents an azimuthal magnetization component with the flux-closure structure. In vortex structures the relative orientation between its azimuthal component and the core direction defines the chirality of the vortex. The direction of the core indicates the polarity. In the case of Co hcp nanowires with perpendicular diameter, for diameters equal and larger than 60 nm, the nucleation of vortex structures along the whole nanowire length takes place. The vortex domain wall typically carries a mathematical singularity called Bloch point [39, 43, 44], which is a unique topological defect in ferromagnetic materials [45], where the magnetization vanishes as displayed in Fig. 14.5B. Therefore it eventually received the name of Bloch point wall to differentiate it from the (surface) vortex wall observed in ultrathin nanostrips [9]. This singularity has been already experimentally observed in nanowires by XMCDPEEM technique [8]. The transition diameter between the two reversal modes decreases as the nanowire saturation magnetization value increases, namely, for permalloy this happens for diameters ca. 40 nm and for Co fcc nanowires for diameters ca. 20 nm, and the Fe-based nanowires generally demagnetize via the vortex domain wall propagation [26]. The Bloch point domain wall in reality carries a 3-D hedgehog skyrmion (see Fig. 14.5B). Note that the vortex core is not necessary located in the nanowire center, but may form a spiral [7]. Finally the proper Neel-type hedgehog 3-D skyrmion structure can be also formed as a stable configuration in small-diameter nanowires with very large saturation magnetization such as FeCo (see Fig. 14.5C). There is another name for a frequently metastable domain wall for nanowire diameters close to the transition value between the vortex and the transverse domain walls, depicted in Fig. 14.5E. This domain wall type presents characteristics of both previous wall types and is called asymmetric transverse domain wall [9, 42] (see Fig. 14.5E). In reality, it corresponds to the situation when the vortex core goes out from the nanowire center to its surface [46]. Together with the vortex on the surface of nanowire, there is an antivortex structure opposite to it to preserve the topological charge. Recently the topological transformation from asymmetric transverse domain wall to a vortex domain wall in cylindrical nanowires, mediated by Bloch point injection, has been reported [45]. Other topological transformations between magnetization textures are also theoretically possible by Bloch point ejection [45]. For nanowires with larger diameters ca. 100 nm, another reversal mode called recently a skyrmion tube is possible, especially when the saturation magnetization is high. Differently to the nanowires with smaller diameter, it does not carry a Bloch point but consists of a core-shell structure. The outer shell first rotates toward the field direction, while the nanowire core remains magnetized in the opposite direction (see Fig. 14.5D). The skyrmion tubes were noticed [6] in permalloy during the dynamical demagnetization process. The magnetization in nanowire cross section indeed has

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the same topological structure as the classical Bloch skyrmions but here induced by the curvature of the nanowire in the absence of the Dzyaloshinskii-Moriya interaction [47, 48]. During further demagnetization the inner skyrmion tube core breaks, and a hedgehog skyrmion (Bloch point) is formed. Its propagation completes the demagnetization process. We should stress that these structures in permalloy are dynamical. In FeCo nanowires (high saturation magnetization), they are shown to be stable at negative applied fields [7]. Most domain walls are dynamical objects; however, they can separate different domains and can be pinned at the defects, thus being stable structures. As many promising applications rely on the nucleation and control of the magnetization processes and magnetic domain walls, there is a strong effort in the nanomagnetism community to understand and control their pinning mechanisms [16–25]. Conventionally a magnetic domain has been considered as a uniformly magnetized region in the mesoscale in thermodynamic equilibrium. Then a magnetic state was defined as a collection of magnetic domains. However, the 3-D nature of nanowires and the emerging properties at the nanoscale make us to redefine the concept of magnetic domains as stable configurations of the magnetization, which may lack homogeneity due to the nanowire curvature. For example, we define the part of the nanowire in the vortex state as a vortex domain. Therefore some materials with large saturation magnetization and high transverse magnetocrystalline anisotropy (as Co hcp nanowires) may show a vortex configuration along the nanowire length as the ground state [26, 35]. In multisegmented nanowires, depending on the segment size and material, single-vortex, two-vortex, or multivortex domain states (see Fig. 14.6 in the succeeding text) have been reported [12, 14, 35]. For example, Fig. 14.2B shows a two-vortex domain, with two opposite vortex chiralities. Depending on the micromagnetic parameters and crystalline structure of the nanowire, either vortex domains (see Fig. 14.7), or transverse domains (see Fig. 14.7A), or their combination is possible (see Fig. 14.7B). The latter combination has been observed in Co85Ni15

Fig. 14.6 A multivortex domain state with opposite vortex chiralities along the nanowire. The calculated image corresponds to cobalt nanowire with perpendicular anisotropy. Parameters used in simulations are listed in Table 14.1. Extracted from Y.P. Ivanov, A. Chuvilin, L.G. Vivas, J. Kosel, O. Chubykalo-Fesenko, M. Va´zquez, Single crystalline cylindrical nanowires—toward dense 3D arrays of magnetic vortices, Sci. Rep. 6 (2016) 23844. doi:10.1038/srep23844.

Micromagnetic modeling of magnetic domain walls and domains

411

Fig. 14.7 Calculated magnetic domain pattern in Co85Ni15 nanowires: (A) alternating transverse domains and (B) coexistence of transverse and vortex domains. Micromagnetic parameters used in modeling are found in Table 14.1.

nanowires by XMCD and confirmed by micromagnetic simulations [29]. This shows that the domain structure is not unique and for some parameters set transverse and vortex domains are metastable states. Another possible domain configuration, discussed in Ref. [49], is alternating vortex structures with core on the nanowire surfaces formed between the transverse domains.

14.4

Domain wall velocity in cylindrical magnetic nanowires

Some of the most fascinating characteristics of cylindrical nanowires are the dynamical properties induced by their cylindrical symmetry and magnetochiral effects, already mentioned in the previous sections. The propagation of the domain walls driven by applied magnetic fields and currents in nanostrips is limited in velocities of the order of 100 m/s (in the absence of the Dzyaloshinskii-Moriya interactions). The steady velocity achieved with the driving force is initially increasing (with the increase of the field or current) and then drops at the so-called Walker breakdown field. This phenomenon limits the speed of future information technologies. Theoretical considerations demonstrate that in cylindrical nanowires the Walker breakdown [50] does not exist due to the cylindrical symmetry [51, 52] or at least it is displaced to higher velocity values. Fig. 14.8 presents the calculated domain wall velocity in a cylindrical permalloy nanowire as a function of the applied field showing velocities almost up 10 km/s, saturating at high field values. This saturation occurs due to the so-called spin-Cherenkov effect [5, 52] when the domain wall loses energy with the spin-wave emission. As for the influence of the current, calculations indicate that the direct motion of the vortex domain wall with current requires too high current and the assistance with applied field will be probably necessary. The Oersted field being circular in nature can change the chirality of the vortex domain wall [53]. In Ref. [15] the current-driven VDW dynamics in cylindrical nanowires were studied by in situ Lorentz TEM in the presence of the external magnetic field to

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Fig. 14.8 Calculated velocity of a vortex (Bloch point) domain wall in a Co fcc nanowire with diameter of 40 nm showing saturation due to the spin-wave emission. Micromagnetic parameters used in modeling are summarized in Table 14.1.

promote the DW dynamics for the lower value of the applied current densities. The cylindrical nanowire of 80 nm diameter and 20 μm length and consisting of segments of Co and Ni of 700 nm each was used. Additionally, the experiments have been performed only for the case of the constant current that ruled out the possibility to measure the DW velocity. The interfaces between segments of different magnetic materials were used as an effective site for the domain wall pinning (more details in Section 14.6). Consequently the value of the external magnetic field needed to nucleate DW in nanowire, the field needed to depin the DW from the interface between the segments, and the critical current density for the DW propagation were determined. Those parameters were compared with the ones obtained by the micromagnetic simulations of an 80-nm-diameter nanowire with two segments of 700 nm each (see model in Fig. 14.9). The direction of the magnetocrystalline anisotropy was considered in agreement with TEM data (at 85 degrees with respect to the NW axis for the Co segment and {220} texture for the Ni segments). The spin polarization of 0.42 reported for planar structures [54] was used. The simulations show that these nanowires demagnetize via the vortex domain wall propagation. The simulations have been done for 0 K and for finite temperatures (extracted from the experiments). Fig. 14.10 presents the simulated velocity data for the domain wall propagation from Co to Ni segment as a function of the current density J for several applied field values. It shows the existence of a critical value of Jc above which the current-driven DW motion occurs, and this value depends on the applied external magnetic field. The maximum value for the domain wall speed was found to be as high as 600 m/s in agreement with theoretical predictions. Simulations show that the critical density of the spin-polarized current at 0 K decreases almost linearly with the increase of the external magnetic field (Fig. 14.10B). Note that the simulated value of Jc at 0 K is more than one order of magnitude larger than the experimental data. The Jc value calculated for 327 K (the final state is seen in the right nanowire in Fig. 14.9 as a small triangle visible in the lower end) is very close to the one obtained

Micromagnetic modeling of magnetic domain walls and domains

Mx

327 K

Co

1.0

413

327 K

Hext = 0.9 Hc

–1.0

Je–X = 0.7x1011 A/m2

0

--DW:17.85 ns

x y

z

--DW:19.20 ns

Ni

Fig. 14.9 Micromagnetic model for domain wall propagation for segmented Co/Ni nanowire under applied magnetic field and current at two different times. Dashed lines indicate the interphase between Co and Ni segments. Modeling parameters are summarized in Table 14.1.

60 0.75 Hdep 0.86 Hdep 0.94 Hdep

40

400

J, 1011 A/m

Velocity (m/s)

600

200

20

0.5 0 0

10

20

J, 1011 A/m

30

40

0.0 0.0

0.2

0.4 0.6 H/Hdep

0.8

1.0

Fig. 14.10 (A) Domain wall velocity under different current densities J for various external magnetic field values defined as a fraction of the depinning field Hdep, simulated at 0 K temperature (arrows show the critical current density for DW depinning). (B) Dependence of the critical current density on the external magnetic field values at 0 K (open circles) and at 327 K (filled circle). The line shows linear approximation toward zero-field DW motion. Images from Y.P. Ivanov, A. Chuvilin, S. Lopatin, H. Mohammed, J. Kosel, Direct observation of current-induced motion of a 3D vortex domain wall in cylindrical nanowires, ACS Appl. Mater. Interfaces 9 (2017) 16741–16744. doi:10.1021/acsami.7b03404.

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experimentally. Most probably, there are additional contributions to the currentdriven 3-D DW motion that were not taken into account in the micromagnetic model. The complete understanding requires more experiments with nanosecond DW dynamics and advanced ultrafast imaging technique.

14.5

Domains and domain walls in nanowires with geometrical modulations

Domain wall-based technologies as the racetrack memory [55] require a precise control over the magnetization and domain wall location. To manage the positioning of domain walls along the nanowire length, multiple strategies have been proposed to achieve the local pinning of the domain wall. The fundamental idea is the creation of potential wells and barriers where the domain wall may eventually get pinned. Geometrical strategies consist of creating constrictions along the nanowire length, artificial notches, antinotches, defects, or diameter modulations that act as pinning sites for the domain wall [20–22, 24, 25, 56]. In addition, variations of the material along the nanowire length or at specific locations [16–19] also offer the possibility of promoting domain wall pinning. Micromagnetic simulations play here a decisive role since they allow to rapidly vary geometrical and material parameters with the aim to design and understand pinning. Here, we discuss two possibilities of geometrical constrictions based on modulations in diameter (with alternating diameters) (see Fig. 14.11A) and geometrical notches (see Fig. 14.11B) in cobalt-based nanowires. The micromagnetic parameters correspond to Co and FeCo (see Table 14.1). The modulated nanowires were considered made of FeCo material with high saturation magnetization value. To represent realistic experimental situations, the granular structure was modeled by Voronoi construction, and several disorder distributions Fig. 14.11 Geometry of simulated (A) nanowire with geometrical modulations and (B) “bamboo” nanowires.

Table 14.1 Material parameters used in micromagnetic modeling, microstructure, and derived quantities: saturation magnetization μoMs, exchange stiffness Aex, exchange length lex ¼ (2Aex/μoMs2)1/2, crystal structure, magnetocrystalline anisotropy type, easy axis direction, first magnetocrystalline constant K1, and shape anisotropy Ksh ¼ μoMs2/4. Material

μoMs (T)

Aex (pJ/m)

lex (nm)

Crystal structure

Anisotropy type

Easy axis

K1 (kJ m23)

Ksh (kJ m23)

Permalloy (Fe80Ni20) [26] Co(111) [26]

1.0

10.8

5.2

Polycrystalline





0

199

1.76

13.0

3.3

fcc

Cubic

75

616

Co(100) [26, 57]

1.76

13.0

3.3

hcp

Uniaxial

450

616

Co hcp [26, 27]

1.76

30.0

4.9

hcp

Uniaxial

450

616

Fe30Co70 [28]

2.0

10.7

2.6

bcc polycrystalline

Cubic

10

796

Ni(111) [26] Co85Ni15 [27, 29–32] Co65Ni35 [27, 29–32] Co35Ni65 [27, 29, 30]

0.61 1.60

3.4 26.0

4.8 5.1

fcc hcp

Cubic Uniaxial

4.8 350

74 509

1.35

15.0

4.5

Uniaxial

260

362

1.01

10.0

5.0

hcp polycrystalline fcc

[1,1,1] jj nanowire axis At 75 degrees with nanowire axis At 75 degrees with nanowire axis A random in-plane easy axis component considered in each grain [111] jj nanowire axis At 88 degrees with nanowire axis At 65 degrees with nanowire axis [111] jj nanowire axis

2

203

Cubic

Saturation magnetization and exchange stiffness from CoNi alloys have been obtained by linear interpolation with the Co content of the alloy in Ref. [29].

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Magnetic Nano- and Microwires

Fig. 14.12 Example of simulated hysteresis cycles for individual modulated FeCo nanowires with three different minor diameters for some particular disordered distribution. Modeling parameters are collected in Table 14.1.

were considered. Hysteresis loops for modulated nanowires under a magnetic field, applied parallel to the symmetry axis of the nanowire, depend considerably on the particular polycrystalline nanostructure of the nanowire. As a main characteristic, the hysteresis loops become less squared, the remanence increases, and the switching field decreases as the minor diameter of the nanowire becomes larger (see Fig. 14.12). The demagnetization process always starts in the segments of the larger diameter by means of the nucleation of the open vortex structures at the constriction. These structures depin from the constriction and propagate first inside the wide-diameter segments and later inside the small-diameter segment. When the difference between diameters is small the propagation in both segments, although consequent and not simultaneous, takes place at the same field value (“weak” pinning). In the opposite case of large diameter difference (“strong pinning”), the hysteresis loop consists of a propagation stage inside the large-diameter segment and depinning (switching) of the magnetization from the constriction to propagate inside a small-diameter segment occurring at a different field. Importantly the open vortex structures (similar to those presented in Fig. 14.2A) at the constrictions between large and small segments are formed with arbitrary chirality. Different realizations of granular structures promote different chirality patterns; therefore, in one particular segment, vortices at the opposite ends may be formed with the same or different chirality. These vortices propagate toward the center of the large diameter and span almost the whole segment. In the case of opposite chirality, the propagation leads to a topologically protected structure that requires a large field to annihilate. As the field progresses, the spins in the outer shell rotate toward its direction forming a skyrmion tube (see Fig. 14.13B). Indeed the magnetization structure in the nanowire cross section is formed by a magnetic skyrmion with the core pointing against magnetic field direction and the shell pointing parallel to it. To prove that the structure is indeed a skyrmion, we evaluate in Fig. 14.15 the topological charge as a surface integral evaluated in the nanowire cross section along its length 1 Q¼ 4π

Z

! ! ! ∂m ∂m  m∙ dS: ∂x ∂y S !

Micromagnetic modeling of magnetic domain walls and domains

417

Fig. 14.13 (A) The surface magnetization configuration of the multisegmented FeCo nanowire showing the helicoidal (“corkscrew”) structure. (B–C) The magnetization distribution in the cross sections of the larger segment at indicated positions, showing the skyrmions with displaced core.

Fig. 14.14 Topological charge for magnetization structure in the corkscrew regime for multisegmented FeCo nanowires, evaluated along the nanowire. The letter C stands for the clockwise chirality of the vortex/skyrmion formed at the constriction or the end of the nanowire, while the letter A denotes an anticlockwise chirality. The colored inset shows the surface magnetization distribution with a chiral structure.

Fig. 14.14 presents the topological charge calculated along the nanowire length for two disorder realizations and large diameter difference. In the case of Fig. 14.14A, the vortices at the ends of the large-diameter segments are formed with the same chirality; as the field is progressing, they are transformed into the skyrmion tubes with topological charge almost one in the center of the nanowire. In the case of Fig. 14.14B, the vortices in the left segment were formed with the opposite chiralities; when the field is increasing in the opposite direction, the two tubes meet, and the topological charge in the middle of this segment is almost zero, reflecting the topological protection.

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Magnetic Nano- and Microwires

This situation needs additional field to annihilate magnetic structures and to complete the magnetization reversal. Once the demagnetization is completed in the large-diameter segments, the skyrmion tube remains pinned at the constrictions. Importantly the skyrmion center along the segment is not positioned in the center of nanowires but forms a spiral (see Fig. 14.13A) and insets in Fig. 14.14 called in Ref. [7] a corkscrew. The occurrence of spiral for vortex/skyrmion tube center is a consequence of the minimization of magnetostatic charges and is a precursor of the magnetic domains. These charges are typically created at the constrictions and are redistributed along the nanowire length to minimize the energy. Ref. [58] argues that these charges are proper to all magnetic nanowires. But they should be especially visible in nanowires with large saturation magnetization where the minimization of magnetostatic energy may create structures following this pattern. In our particular case the presence of the constrictions increases the effect since the vortex/tube core is additionally displaced from the center at the constriction to minimize the energy. In simulations the spiral (“corkscrew”) may also appear in straight magnetic nanowires with diameters ca. 100 nm and larger, especially for nanowires with large magnetization saturation value. Similar topologically protected structures and the corkscrew surface structure have been recently reported in FeNi nanowires with chemical barriers [14]. Recently, they were also reported in simulations of magnetization distribution and observed by Kerr microscopy in microwires [59]. However, experimental verification of the “corkscrew” is needed and requires 3D imaging techniques. Fig. 14.15 presents simulated magnetization configurations in a “bamboo” (with antinotches) nanowire with FeCo (A) and hcp Co (B) compositions in the remanent state. In the case of FeCo nanowire we observe an overall axial orientation of the magnetic moments (with the exception of the regions close to the antinotches) that is principally determined by the shape anisotropy being larger than the magnetocrystalline anisotropy of FeCo. The magnetic structure corresponds to a nearly uniform axial domain state with open vortices at each nanowire end. At the positions of the antinotch (pointed by green arrows), we observe a smooth curling of magnetization at the surface of the nanowire that occurs to minimize the formation of magnetic poles at the modulations. This specific behavior where the magnetization is deviated from its axial orientation is in agreement with the periodic contrasts observed in MFM and XMCD measurements [12, 60] for the FeCo-based “bamboo” nanowires. On the other hand the magnetic configuration of the individual Co bamboo nanowire at remanence in Fig. 14.15B displays a more complex structure. Vortex structures with alternating chiralities are observed along the nanowire length as in the case of straight Co hcp nanowires (see Fig. 14.6). The vortices show an elongated core as a result of the distortion of its shape along the easy axis of the high transverse magnetocrystalline anisotropy and displacement in the direction normal to the easy axis. In addition, the vortex cores are not at the nanowire center but again form a spiral structure, similar to Fig. 14.13A. Finally, micromagnetic calculations inform about the lack of correlation between the multivortex structure and the geometric modulations. These nanowires were studied by MFM and XMCD in Ref. [12], and micromagnetic simulations here provided complimentary information helping to interpret the images.

Micromagnetic modeling of magnetic domain walls and domains

419

Fig. 14.15 Simulated magnetic configurations of (A) FeCo and (B) Co bamboo nanowires at remanence. The figures show a half section from the total length of the wires, containing an edge and two modulations marked by green arrows. Below each nanowire, transverse cross sections at the marked positions along the length are presented. The black arrows denote the projection of the magnetization in the corresponding cross section, while the red-white-blue color gradient of the background corresponds to the longitudinal magnetization. Modeling parameters are collected in Table 14.1. Readapted from C. Bran, E. Berganza, E.M. Palmero, J.A. Fernandez-Roldan, R.P. Del Real, L. Aballe, M. Foerster, A. Asenjo, A. Fraile Rodrı´guez, M. Vazquez, Spin configuration of cylindrical bamboo-like magnetic nanowires, J. Mater. Chem. C 4 (2016) 978–984. doi:10.1039/C5TC04194E.

14.6

Domain walls and domains in multisegmented Co/Ni nanowires

Another approach to create the energy landscape needed to control the DW along the length of the cylindrical nanowire was reported in Refs. [18, 61] with alternative materials aimed at creating periodically modulated anisotropy. Co and Ni were chosen due to difference in their magnetization saturation and magnetocrystalline anisotropy values [26]. Additionally, it has been shown the nanowires of pure Co or Ni are grown perfectly by the electrodeposition in alumina templates with large grain sizes for Ni and being monocrystalline for Co [13, 35, 62]. Short-segmented nanowires (with lengths ca. 100 nm) may be used to realize the concept of the “bar code” (see Fig. 14.16).

420

Magnetic Nano- and Microwires

Fig. 14.16 (A) Magnetic force microscopy images of a multisegmented 80-nm-diameter Co/Ni NW showing pinned magnetic structures. (B) Magnified virtual bright-field differential phase contrast transmission electron microscopy image of the vortex domain wall pinned at the Co/Ni interface. The arrows show the direction of the magnetic field, which is constructed from the raw data of two orthogonal components of the magnetic field. The dashed lines show the position of the Co/Ni interface (purple) and domain wall (red). (C) Micromagnetic simulation of the pinned domain wall structure at the Co/Ni interface (color cones show magnetization direction; black arrows are calculated stray fields). Modeling parameters are collected in Table 14.1.

Micromagnetic simulation (Fig. 14.16C) of the nanowire consisting of two segments of Co and Ni show that the Co/Ni interface is able to effectively pin the domain wall during magnetization reversal process [63]. The simulation proceeds from a saturated state of the multisegmented NW. Upon decreasing the field, open vortex structures are formed at the ends of the segments at the remanence. During further increase of the external field in the opposite direction of the initial magnetization, a subsequent reversal of magnetization of the Ni segment was observed. At this point of reversal the magnetization in Co segments remains relatively unchanged. This differential switching of the segments leads to the pinning of domain walls at the Co/Ni interfaces. The MFM and TEM experiments shown in

Micromagnetic modeling of magnetic domain walls and domains

421

Fig. 14.16A and B confirmed the pinning efficiency of the Co/Ni interfaces. The multisegmented Co/Ni nanowires demonstrated periodic pattern of the strong stray fields emanating from interfaces between Co and Ni segments. Such field acts as a pinning potential for the DW. Another key point is the nature of the domain wall studied in the multisegmented Co/Ni nanowires. The structure of the domain walls inside 80-nm Co or Ni nanowire should correspond to the Bloch point vortex domain wall [26]. Fig. 14.16C shows the 3-D spin configuration of the domain wall in the Co/Ni nanowire extracted from the micromagnetic simulations together with the calculated stray fields. The very good agreement with the experimentally observed magnetic induction map (Fig. 14.16B) is clear. A different possibility is offered by longer segments of Co and Ni nanowires. In Ref. [61], CoNi/Ni multisegmented nanowires of ca. 1-μm segment lengths were studied by means of the MFM and interpreted by means of micromagnetic modeling. Importantly, Co65Ni35 segments were grown in the fcc structure on fcc Ni. However, the difference in the saturation magnetization promotes vortex structure in CoNi segments, while Ni is always in the axial domain state. Importantly the transition between single-vortex and multivortex states in CoNi can be controlled by fields perpendicular to nanowire axis.

14.7

Conclusions

Micromagnetic simulations offer important possibilities to design magnetic cylindrical nanowire with the aim to control the type of magnetic domains, domain walls, and their pinning mechanisms that may serve as a basis for future nanotechnological applications of these objects. They also offer a complementary tool for interpretation of magnetic imaging techniques and are typically in agreement with them provided that the crystallographic structure is known in sufficient details. Domain wall types in nanowires are very rich, and probably not all of them are reported and studied yet. Their type is magnetic and geometrical parameters dependent. Cylindrical geometry produces a plethora of nontrivial topologies for magnetic structures that will be the subject of future studies. The variations of magnetic parameters and geometries give rise to additional possibilities to control domain wall structures and their pinning. The topological effects should play an important role in the magnetic response of nanowires. Here, we have presented an example of the magnetization pinning in high saturation FeCo multisegmented nanowire where skyrmion tubes with a curling core structure are formed. Other nontrivial structures will probably be discovered in the future. Importantly, domain walls in cylindrical nanowires are expected to have very high velocities, comparable with those expected from antiferromagnetic spintronics [64]. Magnetic structures in cylindrical nanowires also may have interesting dynamical properties coming from their topology, and emerging electrodynamics effect may play role [6]. We anticipate that some of these effects may found important nanotechnological applications in the future.

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Magnetic Nano- and Microwires

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[30] H. Kronm€uller, R. Fischer, R. Hertel, T. Leineweber, Micromagnetism and the microstructure in nanocrystalline materials, J. Magn. Magn. Mater. 175 (1997) 177–192, https://doi. org/10.1016/S0304-8853(97)00225-4. [31] A. Moskaltsova, M.P. Proenca, S.V. Nedukh, C.T. Sousa, A. Vakula, G.N. Kakazei, S.I. Tarapov, J.P. Araujo, Study of magnetoelastic and magnetocrystalline anisotropies in Co Ni1  nanowire arrays, J. Magn. Magn. Mater. 374 (2015) 663–668, https://doi.org/ 10.1016/j.jmmm.2014.09.036. [32] V. Vega, T. B€ohnert, S. Martens, M. Waleczek, J.M. Montero-Moreno, D. G€ orlitz, V.M. Prida, K. Nielsch, Tuning the magnetic anisotropy of Co–Ni nanowires: comparison between single nanowires and nanowire arrays in hard-anodic aluminum oxide membranes, Nanotechnology 23 (2012) 465709. https://doi.org/, 10.1088/09574484/23/46/465709. [33] J. Fernandez-Roldan, D. Chrischon, L. Dorneles, O. Chubykalo-Fesenko, M. Vazquez, C. Bran, A comparative study of magnetic properties of large diameter Co nanowires and nanotubes, Nanomaterials 8 (2018) 692, https://doi.org/10.3390/nano8090692. [34] L.G. Vivas, Y.P. Ivanov, D.G. Trabada, M.P. Proenca, O. Chubykalo-Fesenko, M. Vazquez, Magnetic properties of Co nanopillar arrays prepared from alumina templates, Nanotechnology 24 (2013) 105703. [35] Y.P. Ivanov, L.G. Vivas, A. Asenjo, A. Chuvilin, O. Chubykalo-fesenko, M. Va´zquez, Magnetic structure of a single-crystal hcp electrodeposited cobalt nanowire, Europhys. Lett. 102 (2013) 17009, https://doi.org/10.1209/0295-5075/102/17009. [36] S. Bance, J. Fischbacher, T. Schrefl, I. Zins, G. Rieger, C. Cassignol, Micromagnetics of shape anisotropy based permanent magnets, J. Magn. Magn. Mater. 363 (2014) 121–124, https://doi.org/10.1016/j.jmmm.2014.03.070. [37] L.G. Vivas, R. Yanes, O. Chubykalo-Fesenko, M. Vazquez, Coercivity of ordered arrays of magnetic Co nanowires with controlled variable lengths, Appl. Phys. Lett. 98 (2011) 232507, https://doi.org/10.1063/1.3597227. [38] J.A. Fernandez-Roldan, D. Serantes, R.P. del Real, M. Vazquez, O. Chubykalo-Fesenko, Micromagnetic evaluation of the dissipated heat in cylindrical magnetic nanowires, Appl. Phys. Lett. 112 (2018) 212402, https://doi.org/10.1063/1.5025922. [39] H. Forster, T. Schrefl, D. Suess, W. Scholz, V. Tsiantos, R. Dittrich, J. Fidler, Domain wall motion in nanowires using moving grids (invited), J. Appl. Phys. 91 (2002) 6914, https:// doi.org/10.1063/1.1452189. [40] H. Forster, T. Schrefl, W. Scholz, D. Suess, V. Tsiantos, J. Fidler, Micromagnetic simulation of domain wall motion in magnetic nano-wires, J. Magn. Magn. Mater. 249 (2002) 181–186, https://doi.org/10.1016/S0304-8853(02)00528-0. [41] R. Hertel, Computational micromagnetism of magnetization processes in nickel nanowires, J. Magn. Magn. Mater. 249 (2002) 251–256, https://doi.org/10.1016/S0304-8853 (02)00539-5. [42] C.A. Ferguson, D.A. MacLaren, S. McVitie, Metastable magnetic domain walls in cylindrical nanowires, J. Magn. Magn. Mater. 381 (2015) 457–462, https://doi.org/10.1016/j. jmmm.2015.01.027. [43] R. Hertel, J. Kirschner, Magnetization reversal dynamics in nickel nanowires, Phys. B Condens. Matter 343 (2004) 206–210, https://doi.org/10.1016/j.physb. 2003.08.095. [44] S.K. Kim, O. Tchernyshyov, Pinning of a Bloch point by an atomic lattice, Phys. Rev. B 88 (2013) 174402, https://doi.org/10.1103/PhysRevB.88.174402. [45] A. Wartelle, B. Trapp, M. Stanˇo, C. Thirion, S. Bochmann, J. Bachmann, M. Foerster, L. Aballe, T.O. Mentes¸ , A. Locatelli, A. Sala, L. Cagnon, J.-C. Toussaint, O. Fruchart, Bloch-point-mediated topological transformations of magnetic domain walls in

Micromagnetic modeling of magnetic domain walls and domains

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Domain wall pinning in a circular cross-section wire with modulated diameter

15

A. De Riza, B. Trappb, J.A. Fernandez-Roldanc, Ch. Thirionb, J.-Ch. Toussaintb, O. Frucharta, D. Gusakovaa a University of Grenoble Alpes, CNRS, CEA, Spintec, Grenoble, France, bUniversity of Grenoble Alpes, CNRS, Institut NEEL, Grenoble, France, cInstitute of Materials Science of Madrid, CSIC, Madrid, Spain

15.1

Introduction

15.1.1 Fundamental and technological motivations for domain wall pinning The interest for domain walls in one-dimensional (1D) conduits is both for the sake of physics and for technological concepts. As regards physics, considering domain walls in nearly 1D systems allows one to reduce the number of internal degrees of freedom to a minimum. In the limit of cylindrical wires with a diameter typically below seven pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi times the dipolar exchange length lex ¼ 2Aex =μ0 MS2 , with Aex the exchange stiffness and MS spontaneous magnetization, one can neglect variations of magnetization across the wire section, boiling down the description of the domain wall to a 1D problem [1]. In any case, compared with extended thin films this reduces the possible complexity of the wall, obviously easing the understanding of any phenomena related with domain wall motion, for example, precessional dynamics and spin-torques. As regards technology, domain walls have been proposed as means to store [2–4], transport, and process information [5, 6]. It may be desirable to modulate the energy landscape of a domain wall in such a 1D conduit. This may include potential barriers or potential wells. On the fundamental side, such modulations can allow to repeatedly initialize the system with a domain wall at a precise location. This is especially useful to implement timeresolved measurements in a pump-probe scheme, which requires the averaging of reproducible events, including the preparation of a given type of domain wall [7]. Also, energy barriers may be used to confine a domain wall in a segment of finite length to ease its investigation [8] (Fig. 15.1). On the applied side, a digital memory device requires that bits of information are allocated specific physical locations. Thus, domain walls may be forced to remain in potential wells, or conversely, be separated by energy barriers. Among others, this prevents that successive walls in a conduit merge together, which would induce the loss of information. Also, similar to the argument given earlier for fundamental devices, defining a precise starting Magnetic Nano- and Microwires. https://doi.org/10.1016/B978-0-08-102832-2.00015-3 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Fig. 15.1 (A, B) Scanning electron micrographs illustrating the existence of two different diameter transition geometries in the multisegmented aluminum oxide membranes from Bochmann et al. [8] and Trapp [39]. (C) Topography of isolated multisegmented nanowire and magnetic force microscopy image showing the domain wall displacement after application of dc field [39].

position can be helpful to clock circuits, for instance in the case of logic functions involving several domain walls. The modulation of potential along the longitudinal direction has been largely developed and exploited in planar strips based on thin film and lithography technologies. Most are based on the modulation of geometry, which is easily achievable with lithography. This includes notches [7, 9], protrusions [10] or more complex designs such as connection to other magnetic pads [11]. Other means have been demonstrated, such as stray field from neighboring magnetic pads [12] or domain walls [13], ion irradiation [14, 15], or reprogrammable electric-field gating [16].

15.1.2 Types of pinning for nanowires In this chapter, we focus on cylindrical conduits, which we will call nanowires. Magnetic nanowires have been synthesized routinely for several decades, mostly by, for example, electroplating in polymer or anodized aluminum templates [17–19]. This synthesis method present constraints to design modulations of the potential for domain walls, however, also offers opportunities, with respect to flat strips. There exist essentially two modulation designs, which have been developed experimentally and considered theoretically in the past 10 years. The first route for creating a potential landscape is through the geometry of the wire, involving the longitudinal modulation of the diameter. Indeed, the energy of a domain wall sensitively depends on the wire (local) diameter, involving changes in both exchange and dipolar energy. The most common means to achieve such a modulation are multistep anodization [19, 20] or pulsed anodization [21] of aluminum.

Domain wall pinning in a circular cross-section wire with modulated diameter

429

While the versatility is lower than with lithography for strips, a large variety of designs has been demonstrated. More exotic routes exist, such as pulsed plating followed by etching [22], or the alternation of wire and tube [23, 24]. The focus of the present work is restricted to the diameter modulation of a plain wire. The second route for creating a potential landscape is through the longitudinal modulation of the material. In nanowires it may be achieved by changing the growth conditions during synthesis either by multibath anodization for more versatility, or by pulsing the plating potential in a bath with several metal salts, for a higher throughput [25,26]. Both methods are more straightforward and versatile in comparison to local irradiation or gating in flat strips. Note that one may use various magnetic materials, especially varying the composition of compounds [27], or nonmagnetic materials such as Cu [26, 28].

15.1.3 Existing theories and experiments The 1D landscape model for domain walls is probably one of the earliest problems tackled in magnetism to explain the physics of coercivity, as described by the Becker-Kondorski model [29–32]. A key conclusion is that while domain walls are found at the bottom of energy wells at rest, the depinning field is associated with local maxima of slope of the potential, themselves coinciding with inflexion points of the potential curve. We will see that this concept is still applicable for the more specific theories developed in our contribution. Later on, the 1D landscape model was used again in specific cases by A. Aharoni and followers, again in the context of the physic of coercivity. Potential wells and steps [33], slopes [34], and others, were introduced and described. These effective models have been made more specific to the geometry of a nanowire, highlighting the local slope. A number of micromagnetic simulations have been made, considering linear modulations [35], sharp single modulations [36], sharp constrictions [22], and smooth modulations of various length [27]. However, often the processes of domain wall nucleation at a wire’s end and the process of going through the modulation are not studied separately, thus not well describing the latter. Besides, some detailed models of walls at modulations have been proposed [37]; however, their complexity does not allow to shed light a general picture on the phenomenon of pinning. Overall, the existing literature shows interesting features, however, does not provide a comprehensive view. This lack has been driven the present work, to deriving simple analytical scaling laws, and compare the field- and current-driven cases. Finally, note that experimental reports of the interaction of domain walls at modulations of diameter are still scarce and incomplete. Letting aside reports of magnetometry of large assemblies of wires still in a matrix, or experiments on single wires, however, not separating the physics of nucleation from the one of going through the modulation, only a handful of reports exist of domain walls in diameter-modulated single wires [38]. These do not provide a comprehensive quantitative picture at present.

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15.2

Theoretical background

The scope of the present section is to recall the basics of nanomagnetism in a circular cross-section nanowire comprising the domain wall, which are relevant for the concepts discussed in the following sections. We start with the energy terms corresponding to ferromagnetic cylindrical nanowires with no modulation, which we will call straight. Then, we introduce the consequences of the diameter modulations, which imply the existence of an extra magnetic field related to the existence of the magnetic charges.

15.2.1 Domain walls in cylindrical nanowires Domain walls and domains in a ferromagnetic material are usually described within the framework of the micromagnetic theory. First introduced by Brown [40], it is based on a continuous description of magnetization M and of all other quantities. The norm of the magnetization vector is assumed to be constant and uniform, so that the local magnetization density can be written as a function of the lateral position r and of time t as M(r, t) ¼ MS m(r, t) with MS being the spontaneous magnetization. A magnetization distribution can thus be described considering solely the unit vector m(r, t), which indicates the local orientation of the magnetization vector. Its time evolution is governed by the Landau-Lifshitz-Gilbert (LLG) equation ∂m ∂m ¼ γ 0 Heff  m + αm  , ∂t ∂t

(15.1)

where γ 0 ¼ μ0γ > 0 is the gyromagnetic ratio and α is the phenomenological Gilbert damping factor. The effective magnetic field Heff is defined as Heff ¼ 

1 δE : μ0 MS V δm

(15.2)

It is related to the system’s energy E, volume V, and magnetic permeability μ0. The first term of the LLG equation reflects the precession of the magnetization vector around the effective field, which may include the internal field contributions as well as the external applied field Happ. Damping of this motion is described in the second term. This equation may be completed by the torque T produced by the spin-polarized current. However, in this section, we do not consider the phenomena related to the spin-polarized current, which are discussed in Section 15.4. In most cases of domain wall motion, it is desirable to minimize extrinsic pinning such as due to spatial fluctuations of magnetic anisotropy, grain boundaries, etc. Consequently, here we consider only magnetically soft materials (such as Fe20Ni80), in particular with no magnetocrystalline anisotropy. Thus, for a straight cylindrical wire [41] under applied magnetic field, the energy E of the system reads E ¼ E 0 + EZ ,

(15.3)

Domain wall pinning in a circular cross-section wire with modulated diameter

where μ E0 ¼ 0  2

431

Z

Z

½rmðrÞ2 dV,

M  Hd ðrÞdV + Aex V

(15.4)

V

Z EZ ¼  μ0 MS

mðrÞ  HðrÞdV:

(15.5)

V

E0 is the internal energy. The first term in Eq. (15.4) corresponds to the magnetostatic contribution. The magnitude and orientation of the dipolar field Hd depend sensitively on the aspect ratio of the ferromagnet. The second term corresponds to the exchange contribution in its continuum form, with the exchange stiffness Aex. When an external field Happ is applied, its contribution is described as the Zeeman energy term (Eq. 15.5). For complex magnetic textures or nontrivial geometries, when handling of the LLG equation could not be done analytically, the evolution of the magnetic system in time is solved numerically using appropriate micromagnetic codes. In this chapter, all numerical simulations have been done using our home-built finite element freeware feeLLGood (Finite Element Landau Lifshitz Gilbert equation Oriented Object Development) [42, 43].a The nonregular finite element mesh of feeLLGood accurately describes the cylindrical geometry without creating an artificial numerical roughness at the cylinder surface. Moreover, feeLLGood’s parallelized SCALFMM libraryb based on so-called Fast Multipole Method for dipolar field calculation makes it competitive with usually less time-consuming finite difference micromagnetic codes. As we are considering soft magnetic material, the characteristic length scale of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi choice is the dipolar exchange length lex ¼ 2Aex =μ0 MS2 , resulting from the competition between exchange and magnetostatic energy contributions. Magnetization tends to be rather uniform over distances smaller than lex, while it may rotate at larger scales under the influence of boundary conditions or dipolar energy. In cylindrical nanowires, depending on the wire diameter, so far two different domain wall types have been theoretically predicted [41, 44] and experimentally observed [45–48]. Moderate wire diameters (D < 7lex) considered in this chapter favor the formation of the so-called transverse-like domain wall [49, 50]. In one dimension, its profile along the z-axis is well described by mz ¼ tanh ðz=ΔÞ and my ¼ 1=cosh ðz=ΔÞ, where Δ is the wall-width parameter. Fully three-dimensional (3D) transverse-like domain wall distributions obtained by solving numerically Eq. (15.1) and corresponding energies are depicted in Fig. 15.2.

15.2.2 Geometry of modulation and potential barrier A modulation of diameter induces a variation of the internal energy of the system, which depends on the longitudinal position of the domain wall. The center of the wall will be named zDW, which does not mean that we are assuming a wall with zero a

See http://feellgood.neel.cnrs.fr.

b

See http://scalfmm-public.gforge.inria.fr/doc/.

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DW energy (10–19 J)

Head-to-head DW

R1

R2 z

(A)

–1

mz

1

80 60 40 20

(B)

0 5

10

15

R (nm)

Fig. 15.2 (A) Micromagnetic distribution of longitudinal magnetization for the transverse-like head-to-head domain wall for radius R1 ¼ 5 nm and R2 ¼ 10 nm in permalloy, obtained numerically using Eq. (15.1). (B) Simulated domain wall energy versus diameter, ins straight wire diameter. The dashed curve corresponds is a third-order polynomial fit serving as a guide to the eye.

thickness. In the following, we study four types of modulation profiles, which connect a smaller cross-section with radius R1 to a larger cross-section with radius R2: abrupt modulation, straight modulation, and circular- and tanh-based modulations (see Fig. 15.3A). The abrupt modulation is described by the simple step-function,  RðzÞ ¼

R1 , z < 0, R2 , z > 0:

(15.6)

The straight modulation with length λ corresponds to the linear function 8 z < λ=2, < R1 , RðzÞ ¼ kz + s, λ=2 < z < λ=2, : R2 , z > λ=2,

(15.7)

with k ¼ (R2  R1)/λ and s ¼ (R2 + R1)/2. The circle-based profile allows for a smooth transition between smaller and larger cross-section parts 8 R1 , > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > < y1  R2mod  ðz + λ=2Þ2 , RðzÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > y2 + R2mod  ðz  λ=2Þ2 , > > : R2 ,

z < λ=2, λ=2 < z < 0,

(15.8)

0 < z < λ=2, z > λ=2,

with Rmod ¼ [(R2R1)2 + λ2]/[4(R2  R1)], y1 ¼ ðR22 + 2R1 R2  3R21 + λ2 Þ=½4ðR2  R1 Þ, and y2 ¼ ð3R22  2R1 R2  R21  λ2 Þ=½4ðR2  R1 Þ. It has been used in Sections 15.3.2

R2

R1

Straight

R2

l

R1

1.0 0.8 0.6

R1

l

R2 O″

0

(A)

0 ZDW (nm)

50

–50

0 ZDW (nm)

50

–50

0 ZDW (nm)

50

1.0 0.8 0.6

1.2 E0 (10–18 J)

Rmod

–50

1.2

O′

Circle-based

433

1.2

E0 (10–18 J)

Abrupt

E0 (10–18 J)

Domain wall pinning in a circular cross-section wire with modulated diameter

1.0 0.8 0.6

z

(B)

Fig. 15.3 (A) Type of modulation geometry considered and (B) corresponding energy of the domain wall E0 versus its position zDW, from micromagnetic simulations. Parameters used for the energy plots are R1 ¼ 5 nm, R2 ¼ 7.5 nm, λ ¼ 100 nm, and μ0MS ¼ 1 T. Gray horizontal lines correspond to the energy of a straight wire with R ¼ 5 nm and R ¼ 7.5 nm.

and 15.4 for the micromagnetic simulations. For the analytic calculations, the circlebased wire profile was approximately replaced by the tanh-based profile RðzÞ ¼ ½R1 + R2 + ðR2  R1 Þtanh ð4z=λÞ=2:

(15.9)

This is an analytic differentiable function, which approximates well the circle-based profile in the case of the gently sloping modulations studied in Sections 15.3.2 and 15.4. For the gently sloping modulation with (R2  R1) ≪ λ, the relative error made by tanh-based shape approximation instead of circular-based profile is less than 10%. To illustrate the energy situation, in Fig. 15.3B we plotted the internal energy E0 as a function of the position of the domain wall. These curves were obtained by solving

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Magnetic Nano- and Microwires

the LLG Eq. (15.1) numerically for domain walls drifting freely from the broader part toward the thinner part of the wire in the absence of any driving force. In that case, we used α ¼ 1, to approach a quasistatic situation. The energy of the domain wall is smaller in the thinner part of the wire. Handwavingly, this makes sense as the area and thus the volume of the domain wall, as well as its total magnetic charge and hence the dipolar energy, scale with the wire cross-section. Far from the modulation the value of E0 recovers the value of energy of a straight wire, as depicted by horizontal gray lines. The width of the transition between the lower and upper values of E0 corresponds approximately to the modulations length λ.

15.2.3 Magnetic charges By analogy with electrostatics based on Maxwell’s equations, the magnetic volume and surface charges ρm ¼ MSrm and σ m ¼ MS ðn  mÞ may be introduced as the source of the dipolar field Hd [51, 52], where n is the outward-pointing unit vector normal to the system surface. The expression for the dipolar field reads Z Hd ðrÞ ¼

0

0

ρm ðr Þðr  r Þ 4πjr  r0 j3

3

0

d r +

I

0

0

σ m ðr Þðr  r Þ 4πjr  r0 j3

dS:

(15.10)

In the case of the uniformly magnetized straight cylindrical wire (Fig. 15.4), each of the wire’s end possess the total magnetic charge q1 ¼  πMS R21 . The total charge is zero and is an invariant, whose conservation imposes that the head-to-head domain wall in a cylindrical nanowire bears a total magnetic charge qDW ¼ 2πMS R21 . In the case of a modulation of diameter, the two end charges are different: q1 ¼  πMS R21

Fig. 15.4 Schematics of magnetic charges distribution in (A) uniformly magnetized cylindrical wire, (B) cylindrical wire with head-to-head domain wall, and (C) modulated diameter nanowire with head-to-head domain wall placed in the thinner part. Red color corresponds to the positive magnetic charge and blue one to the negative magnetic charge.

Domain wall pinning in a circular cross-section wire with modulated diameter

435

Fig. 15.5 Magnetic potential ϕm distribution for different positions of the head-to-head domain wall. Red (resp., blue) color corresponds to positive (resp. negative) values of ϕm.

and q2 ¼  πMS R22 . Considering for instance the case where the domain wall is clearly in the smaller-diameter part, qDW ¼ 2πMS R21 , and a charge is associated with the modulation: q mod ¼ πMS ðR22  R21 Þ (Fig. 15.4C). At this stage, we did not discuss which type of charge (volume or surface) contributes to the total charge. In all cases (end, modulation, domain wall), the total charge is distributed both over volume charge density as well as the surface charge density, whose distributions are nontrivial. The micromagnetic distribution of the magnetic potential ϕm related to the charge distribution (Hd ¼ rϕm) illustrates this fact in Fig. 15.5. Most notably, the modulation charge qmod gives rise to a magnetic dipolar field Hmod, which we calculate in Section 15.2.4. We show that it tends to move the domain wall toward the part with thinner section.

15.2.4 Magnetic field generated by the modulation In this section, we determine the magnitude and the direction of the magnetic field generated by the modulation of diameter. As regards its contribution to the energy of the system through its interaction with the domain wall, for simplicity we consider its value on the wire axis and at the center of the domain wall zDW. Following Eq. (15.10), the elementary magnetic field dH generated by an element of magnetic charge dq at the distance r reads dH ¼

dq r : 4πr 2 r

(15.11)

For the axisymmetrical charge distribution, which is the case here, the resulting magnetic field generated on the axis by the whole modulation is aligned with the z-axis. While the total charge of the modulation is fixed, its distribution over surface and volume contributions is not straightforward. Thus, some approximation that conserves the total charge of the modulation should be made. We assume that the magnetization vector in the modulation is strictly aligned with the z-axis at each point. This simplification limits the magnetic charge of the modulation to the surface charge σ m only, while volume charges are zero. The surface charge approximation allows us to

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Magnetic Nano- and Microwires

Fig. 15.6 (A) Sketch of the magnetic field generated by the elements of the magnetically charged axisymmetric surface in the presence of the head-to-head domain wall at the position zDW for abrupt modulation. Red color corresponds to the positive surface charge and blue color to the negative one. (B) Magnetic field generated by the abrupt modulation μ0Hmod versus domain wall position zDW for several values of R2. Parameters used for this plot are R1 ¼ 5 nm and μ0MS ¼ 1T.

estimate the amplitude of the magnetic field generated by the modulation analytically in some specific cases. Abrupt modulation. Besides being close to applicable in some experimental cases, the abrupt modulation is a text-book case, from which the general features of the impact of a modulation on domain wall motion can be easily illustrated. With the previous assumptions, the abrupt modulation is described by Eq. (15.6). It is charged positively when the head-to-head domain wall is to the left of the modulation, and negatively when it is to the right of the modulation (Fig. 15.6A). The elementary magnetic field generated by the element dS of the charged surface at the domain wall position zDW, and projected on the z-axis, reads dHz ðzDW Þ ¼ 

jzDW j MS ζdζ  : r 2r 2

(15.12)

Here, dq ¼ σdS, σ ¼  MS ðn  mÞ, dS ¼ 2πζdζ, r/r ¼ 1, dHz ¼ dHjzDWj/r, and r2 ¼ z2DW +ζ 2. The summation of all contributions from element charges over the entire charged surface gives MS Hz ðzDW Þ ¼  2

Z

R2

R1



jzDW jζdζ 3=2 : ζ 2 + z2DW

(15.13)

Upon integration, we obtain the magnetic field Hmod  Hz(zDW) generated by the modulation at the center of the head-to-head domain wall ! MS jzDW j 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , H mod ðzDW Þ ¼  2 R22 + z2DW R21 + z2DW

(15.14)

Domain wall pinning in a circular cross-section wire with modulated diameter

437

which is plotted in Fig. 15.6B. The Hmod always opposes the head-to-head domain wall movement to the right, being negative for all zDW. In other words, the charges at the modulation tend to favor motion toward the part with smaller radius. Charged surface with arbitrary profile. While the main physics is captured by the abrupt modulation, it is associated with an unphysical cusp of Hmod at the very center of the modulation. Besides, it may not be realistic for slowly varying modulations such as found in some experimental cases. The present paragraph intends to describe such situations. Following the same method and assuming only surface charges, let us calculate the magnetic field generated by the modulation with an arbitrary profile given by the continuous function R(z). As shown in Fig. 15.7A, the corresponding modulation surface is charged positively to the right of the head-to-head domain wall and negatively to the left of it. We may assume a stepwise jump of surface charges across the domain wall, in the case of gentle modulations. The summation of all contributions from the entire charged surface reads MS H mod ðzDW Þ ¼  2

Z



0

jzDW  zjRðzÞR ðzÞdz h i3=2 : ∞ ðzDW  zÞ2 + R2 ðzÞ

(15.15) 0

This is derived from σ m ¼ MS sin ðαÞ, dS ¼ 2πR(z)dl, dl ¼ dz= cos ðαÞ, tan ðαÞ ¼ R ðzÞ the derivative of R(z), dHz ¼ dHjzDW  zj/r, and r2 ¼ (zDWz)2 + R(z)2. Fig. 15.7B depicts Hmod computed using Eq. (15.15) for a tanh-based profile given by the formula (15.9). Similar to the case of abrupt modulation, Hmod opposes the head-to-head domain wall movement to the right. However, there is now no more cusp at zDW ¼ 0, and the maximum magnitude of Hmod is now found at the center of the modulation. Note that this maximum decreases sharply with increasing modulation length λ.

Fig. 15.7 (A) Sketch of the magnetic field generated by the elements of the magnetically charged axisymmetric surface in the presence of the head-to-head domain wall at the position zDW for the modulation of arbitrary profile given by the continuous function R(z). Red color corresponds to the positive surface charge and blue color to the negative one. (B) Magnetic field generated by the tanh-based profile modulation of the length λ versus domain wall position zDW for several values of λ. Parameters used for this plot are R1 ¼ 5 nm, R2 ¼ 10 nm, and μ0MS ¼ 1 T.

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Fig. 15.8 (A) Sketch of the tanh-based and straight profiles with R1 ¼ 5 nm, R2 ¼ 10 nm, and λ ¼ 100 nm. (B) Magnetic field generated by tanh-based and straight modulation profiles illustrated in (A). (C) Magnetic field generated by tanh-based modulation for several values of R2. All curves are plotted for μ0MS ¼ 1 T.

Straight modulation. In the case of the straight modulation of three segments (Fig. 15.8A) given by the formula (15.7), the integral in Eq. (15.15) should be calculated separately for each segment: 8Z λ=2 > > > FðzDW , zÞdz, > > > λ=2 > > Z Z λ=2 < zDW H mod ðzDW Þ ¼  FðzDW , zÞdz + FðzDW , zÞdz, > zDW λ=2 > > Z > λ=2 > > > > FðzDW , zÞdz, :

(15.16)

λ=2

where FðzDW , zÞ ¼

ðzDW  zÞðkz + sÞk MS h i3=2 : 2 ðzDW  zÞ2 + ðkz + sÞ2

(15.17)

Domain wall pinning in a circular cross-section wire with modulated diameter

439

For very large λ, we may roughly estimate Hmod considering kz negligible in comparison to s. This gives the analytic expression for the field generated by the straight modulation in the following form: 0 1 8 > > > ksMS B 1 1 C > >  @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA, > > 2 > > ðλ=2 + zDW Þ2 + s2 ðλ=2  zDW Þ2 + s2 > > > > 0 1 > > > < ksM 1 1 SB C H mod ðzDW Þ ¼ + @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2A, > 2 2 2 2 2 > ðλ=2  zDW Þ + s ðλ=2 + zDW Þ + s > > > > 0 1 > > > > > ksMS B 1 1 > C > >  @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA: > > 2 2 2 : ðλ=2  zDW Þ + s2 ðλ=2 + zDW Þ + s2 (15.18) Fig. 15.8 compares Hmod calculated for a straight modulation and a tanh-based profile.

15.2.5 Energy of interaction In addition to the local terms (Eqs. 15.4, 15.5) describing the domain wall behavior within the straight cylindrical wire, the energy of interaction of the domain wall with modulation charges Emod must be considered. This is an extra contribution to the internal energy, while the domain wall moves through the modulation, however, also at longer range. The total energy now reads Etot ¼ E0 + EZ + E mod :

(15.19)

The derivative of energy with respect to the wall position can be written under the form of an effective field. The one associated with the supplementary energy term Emod reads, for an axisymmetrical wire: ∂E mod ¼ μ0 qDW H mod ðzDW Þ: ∂zDW

(15.20)

It is unlikely that Emod has an analytic expression in the case of an arbitrary modulation profile and arbitrary domain wall profile. In contrast, the field distribution Hmod(zDW) can be derived analytically by making some assumptions, as shown in Section 15.2.4. Besides, the z-derivative of energy may be sufficient, for example, to calculate the domain wall depinning field. In this case, we do not need the energy Emod expression but only its derivative, as the minimization of the total energy gives the domain wall pinned position.

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15.3

Magnetic Nano- and Microwires

Modulation under applied magnetic field

In this section, we focus on the case of domain wall behavior under a magnetic field applied along the wires axis (Fig. 15.9). In particular, we aim to calculate the critical field needed to depin the domain wall. As both the internal and the Zeeman energies are conservative, one may derive the critical field Hcrit and corresponding critical domain wall position zcrit on the basis of the position-dependent domain wall energy. In the majority of cases, the purely analytical treatment of this problem is tricky or even impossible. For this reason, here we propose an analytical estimation of the Hcrit in particular limit cases, which implies a number of simplifying hypothesis. Despite the limitations of the simplified approach, our analytical analysis focuses on the key ingredients and gives a very reasonable estimation of the behavior of the critical depinning field in response to the modulation parameters. The cases for which the assumptions used below are too drastic should be covered by micromagnetic simulations.

15.3.1 Abrupt modulation In this section, we estimate the critical applied field Hcrit needed to depin the domain wall in a wire with an abrupt modulation of diameter, described by Eq. (15.6) and visualized in Fig. 15.9. The wire axis was taken as the z direction. The modulation was centered at z ¼ 0 and L is the total length of the wire. The head-to-head domain wall was prepared in the narrow section of the wire, and driven toward the larger section by applying a positive magnetic field. Micromagnetic simulations suggest that for such modulation the transition between the two energy levels (or potential barrier) is relatively sharp (Fig. 15.3B). Moreover,

Fig. 15.9 Head-to-head domain wall displacement under the applied magnetic field Happ. The color scale bar indicates the longitudinal magnetization mz.

Domain wall pinning in a circular cross-section wire with modulated diameter

441

magnetization in the domains is mostly perpendicular to the modulation surface, which gives the maximum surface charge σ m ¼ MS(mn) and thus generates the large magnetic field of the modulation (Eq. 15.14, Fig. 15.6B). In this case, it is reasonable to assume that the key ingredient in domain wall pinning is the competition between applied magnetic field Happ and the magnetic field generated by the modulation Hmod. Besides, the abrupt jump of diameter, and thus domain wall energy when crossing the modulation, makes that an abrupt jump may not describe all features of the total depinning process. Rather, this model is suitable to describe the long-range competition between the applied field contribution EZ ¼ 2μ0 MS Happ πR21 zDW + Cste and the energy of interaction between domain wall and modulation R 0 0 2 E mod ¼ 2μ0 MS πR1 H mod ðzDW , z Þdz . This explains the nonmonotonic energy profile with domain wall position zDW, as shown in Fig. 15.10A. Note also that we neglected the inner structure of the domain wall to derive the Zeeman energy, instead we considered the Zeeman energy of two adjacent uniformly magnetized domains on either sides of the domain walls center position, zDW. The energy derivative ∂(EZ + Emod)/∂zDW ¼ 0 have extrema for zmax and zmin. The latter corresponds to the domain wall pinned position, while the former highlights the top of the energy barrier preventing further motion. Using Eqs. (15.14), (15.20) and applied field contribution, we obtain the expression which relates the applied magnetic field to the energy extrema: 0 Happ ¼

1

zmin ,max MS B 1 1 C @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA: 2 2 2 2 2 R +z R +z 1

min ,max

2

(15.21)

min ,max

Fig. 15.10 (A) Total energy EZ + Emod versus domain wall position zDW for several values of the applied field and R2 ¼ 10 nm. Vertical arrows show the pinned domain wall positions. (B) Critical field value Hcrit as a function of the larger radius R2. Solid line corresponds to the analytic formula, and solid circles and open squares correspond to micromagnetic simulations with Aex ¼ 1  1011 J/m and Aex ¼ 0.25  1011 J/m. All curves are plotted for μ0MS ¼ 1 T and R1 ¼ 5 nm.

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Magnetic Nano- and Microwires

At some critical value of applied field Hcrit both extrema zmin and zmax converge into the same point (an inflection point). zcrit may be found using ∂2(EZ + Emod)/∂z2DW ¼ 0. zcrit corresponds to the final pinned position of the domain wall: 2=3 2=3

R1 R 2 zcrit ¼  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2=3 2=3 R1 + R 2

(15.22)

and the corresponding Hcrit needed to depin the domain wall reads ! MS zcrit zcrit pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Hcrit ¼ 2 R22 + z2crit R21 + z2crit

(15.23)

Fig. 15.10B compares formula (15.23) with micromagnetic simulation. This comparison reveals qualitatively and quantitatively similar tendencies. Note that simulations conducted with a value of Aex reduced in comparison to that of the permalloy-like material, fits slightly better the analytic results. It may be explained by the more compact domain wall which probably better suits the model assumptions.

15.3.2 Smooth modulation In this section, we estimate the critical applied field Hcrit needed to depin the domain wall in a smooth diameter modulation described by Eq. (15.9) and schematized in Fig. 15.11A. In practice, the modulation with length λ was centered at z ¼ 0, and L is the total length of the wire. The head-to-head domain wall was prepared in the narrow section of the wire and was driven toward the larger section by applying a magnetic field. To determine the qualitative expression for Hcrit, we considered the domain wall and Zeeman energies E0 and EZ, based on the magnetostatic, exchange, and applied field contributions (Eqs. 15.4, 15.5). For simplicity, here we omit the energy of interaction Emod between the domain wall and the charges of the modulation. It has been shown in Fernandez-Roldan et al. [53] for smooth modulations that the extension of the present model by including the Emod does not have any qualitative impact, and results only in a slight shift in the total energy minima and maxima. Here, we introduce the approximations that can be used to estimate each energy term. The details of calculation may be found in Fernandez-Roldan et al. [53]. For the dipolar energy, we considered that the magnetic charge qDW ¼ 2MSπR2 [54] carried by the head-to-head wall was uniformly distributed within the plain sphere of radius R, thus with a magnetic charge density ρm ¼ 3qDW/4πR3. The real distribution of the magnetic charge is more complex [55, 56]. Nevertheless, our approximation leads to a compact analytical expression for the different energy terms and gives a reasonable order of magnitude. Note that this magnetic charge depends on the domain wall position zDW, through R(zDW). By analogy with electrostatics, a dipolar field Hd is generated by the charged plain sphere, with a total magnetostatic contribution 3πμ0M2SR3/5. This contribution grows with the wire radius like R3 which is

Domain wall pinning in a circular cross-section wire with modulated diameter

443

Fig. 15.11 (A) Schematic illustration of the uniformly charged sphere corresponding to the domain wall. (B) Domain wall energy E0 + EZ as a function of the domain wall position, for several values of applied magnetic field. (C) Critical field Hcrit as a function of modulation length λ for R2 ¼ 6 nm. (D) Critical field Hcrit as a function of larger radius R2 for λ ¼ 100 nm. All curves are plotted for μ0MS ¼ 1 T and R1 ¼ 5 nm.

consistent with the micromagnetic simulations of the domain wall energy plotted in Fig. 15.2B as a function of R. The exchange energy contribution can be estimated by applying the 1D spin chain model [51] with slowly varying magnetization. In this case [rm(r)]2 (π/2R)2, so that the total exchange energy contribution equals Aexπ 3R/3. To estimate the Zeeman energy contribution, we neglected the inner structure of the domain wall and considered the Zeeman energy of two adjacent uniformly magnetized domains located at the domain walls center position, zDW. The domain wall energy excluding the integration constant then becomes EðzDW Þ ¼

3π Aex 3 μ0 MS2 R3 ðzDW Þ + π RðzDW Þ  2μ0 MS Happ π 5 3

Z

z

L=2

R2 ðzÞdz (15.24)

and is depicted in Fig. 15.11B. Note that it is compulsory analytically to consider the finite length of the wire, so that the Zeeman energy is finite. Both local minima and local maxima are found using energy minimization ∂E(zDW)/∂zDW ¼ 0, which gives   4Happ ∂RðzDW Þ 18 l2ex π 2 + 2 ¼0  ∂zDW 5 3R ðzDW Þ MS

(15.25)

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Magnetic Nano- and Microwires

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with lex ¼ 2Aex =μ0 MS 2 . For a tanh-based profile and smooth modulation with (R2  R1)/(R2 + R1) ≪ 1, the coordinates of minimum and maximum of energy reads zmax ,min ¼ 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ arctanh 1  aHapp , 4

(15.26)

" #1 5λ 10l2ex π 2 where a ¼ 1+ . The coordinate of the energy mini9MS ðR2  R1 Þ 27ðR1 + R2 Þ2 mum zmin corresponds to the domain wall pinned position. It corresponds to an internal effective field Heff experienced at this point by the center of the domain wall:  Heff ¼ Happ ¼  1  tanh ð4zmin =λÞ2 =a:

(15.27)

The domain wall depinning condition, at a given critical applied field value Hcrit, can be defined as the convergence of two energy extrema at the same point, zmin ¼ zmax (red curve in Fig. 15.11B). Here, we derive zcrit ¼ 0 for (R2  R1)/(R2 + R1) ≪ 1 (the numerical solution of Eq. 15.25 without this assumption gives slightly different result [53]). The corresponding critical field Hcrit reads 9MS ðR2  R1 Þ 10l2ex π 2 1+ Hcrit ¼ 5λ 27ðR1 + R2 Þ2

! (15.28)

and is depicted in Fig. 15.11C and D as a function of the modulation parameters. The domain wall repulsion from a modulation due to Hmod, when not negligible like for rather abrupt modulations, shifts Hcrit toward higher values. Nevertheless, the analytical formula (15.28) provides a good estimation of Hcrit and the relation between Hcrit and geometric parameters. A key finding is that the critical field is proportional to the slope of the modulation (R2  R1)/λ, with a negligibly small exchange correction for small diameters. The comparison between analytical formula (15.28) and micromagnetic simulations reveals qualitatively similar tendencies. Moreover, small R2/R1 ratios and long λ, corresponding to gently sloping modulations, ensure the best fit between the simulations and the analytical expression. The cases for which the assumptions used in this model are too drastic should be covered by micromagnetic simulations.

15.3.3 Protrusion: Double abrupt modulation In this section, we estimate the critical applied field Hcrit needed to depin the domain wall in a wire with a protrusion with length Λ, as schematized in Fig. 15.12A and given by the formula: 8 < R1 , RðzÞ ¼ R2 , : R1 ,

z < 0, 0 < z < Λ, z > Λ:

(15.29)

Domain wall pinning in a circular cross-section wire with modulated diameter

445

Fig. 15.12 (A) Schematic of the protrusion geometry of the length Λ and double abrupt radius modulation between smaller one R1 and larger one R2. (B, C, and D) Analytical curve for the magnetic field generated by a double modulation, for three values of the protrusion length Λ. Red and blue lines correspond to the contribution of each modulation and black line to the total resulting magnetic field Hmod. All curves are plotted for μ0MS ¼ 1T, R1 ¼ 5nm, and R2 ¼ 10nm.

Similar to Section 15.3.1, the head-to-head domain wall was prepared in the narrow left section of the wire and driven toward the larger section by applying a magnetic field. To calculate Hcrit, the assumptions which may be done as well as the procedure to follow are exactly the same. In order to calculate Hcrit for any length Λ, we should examine the field Hmod created by the charges from both sides of the protrusion. Its expression reads H mod

! MS jzDW j 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 R2 + z 2 R21 + z2DW 0 2 DW

1

MS jzDW  Λj B 1 1 C + @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA: 2 2 2 2 2 R2 + ðzDW  ΛÞ R1 + ðzDW  ΛÞ

(15.30)

The magnitude of Hmod is plotted in Fig. 15.12B–D for different values of the protrusion length Λ. Blue and red lines correspond to the contribution of each side of protrusion separately. The black solid line corresponds to the total Hmod. Almost for all

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values of Λ (except for very small Λ, i.e., for short modulation), Hmod has its deepest minimum to the left of the protrusion (for zDW < 0). Similar to single abrupt modulation (Section 15.3.1), the deepness of this absolute minimum indicates the minimum amplitude of driving force to be applied to overcome the pinning potential. This means, the domain wall should be blocked to the left of the protrusion if the strength of the driving force is not enough to unpin it. For very small Λ, the situation may be reversed (Fig. 15.12B) and the domain wall may be pinned also to the right of the protrusion. In order to quantify this phenomenon, we followed the same procedure for the energy minimization similar to single abrupt modulation case and solved numerically ∂2(EZ + Emod)/∂z2DW ¼ 0 condition. This numerical solution gives Hcrit as a function of protrusion length Λ, and is plotted in Fig. 15.13A. This gives rises to two regimes. As expected, for large Λ (wide protrusion), the critical field Hcrit tends to coincide with the single abrupt modulation value. The pinning position is ahead of the protrusion, like for a single increase of diameter. Once the domain wall enters the protrusion, propagation proceeds beyond the center of the protrusion as the geometry favors expulsion, adding its effect to the Zeeman energy.

Fig. 15.13 (A) Critical field Hcrit as a function of the protrusion length Λ for several values of R2. Red horizontal dashed line corresponds to the critical field value for Λ ¼ ∞ (single abrupt modulation) and R2 ¼ 12.5nm. (B) Micromagnetic simulation of the Hcrit as a function of the protrusion length Λ for R2 ¼ 10nm. The branch with solid squares corresponds to the domain wall pinned to the left of the protrusion. The branch with open circles corresponds to the domain wall pinned to the right of the protrusion. Black horizontal dashed line corresponds to the critical field value for Λ ¼ ∞ (single abrupt modulation). (C) Domain wall pinned to the left of the protrusion for Λ ¼ 30nm and domain wall pinned to the right of the protrusion for Λ ¼ 10nm. The color scale bar indicates the longitudinal magnetization mz. All graphs are plotted for μ0MS ¼ 1T and R1 ¼ 5nm.

Domain wall pinning in a circular cross-section wire with modulated diameter

447

For decreasing protrusion lengths, the pinning field also decreases. This is related to the partial balance of the repulsive charge on the left side of the protrusion and the attractive charge on the right side of the modulation. For very small values of Λ, the critical field Hcrit has nonmonotonic behavior, which corresponds to the case with the domain wall pinned to the right of the modulation. However, the model considers a domain wall with zero width, a hypothesis that may be put at stake for a short modulation. Besides, we stressed in Section 15.3.1 that the model with abrupt modulation may not describe accurately the situation when the domain wall enters the modulation. For these reasons, it is important to check the situation with micromagnetic simulations. Surprisingly, these reproduce rather well the tendencies obtained in the simplified model, as shown in Fig. 15.12B and C. Among others, the existence of two regimes is confirmed. This stresses that care needs to be taken in the analysis of experimental data, for which multiple pinning sites may not results from extrinsic imperfections.

15.4

Modulation under applied current

In this section, we describe the domain wall behavior under applied current. For the sake of providing a realistic picture even when crossing the modulation, we immediately jump to the case of the smooth diameter modulation given by the tanh-based profile formula (15.9) and schematized in Fig. 15.14A. The wire axis is again taken as the z direction. The modulation of the length λ is centered at z ¼ 0 and L is the total length of the wire. The head-to-head domain wall was prepared in the narrow section of the wire and driven toward the larger section by applying a spin-polarized current, with the electrons flowing from the narrow to the broad section. Similar to Section 15.3.2, we make several simplifying assumptions and focus on the key ingredients to estimate the critical current needed to depin the domain wall. We then use micromagnetic simulations to refine the analytic picture. Under the applied spin-polarized current, the domain wall motion obeys the LLG Eq. (15.1) generalized with the so-called adiabatic and nonadiabatic spin-torques [57] T¼

PμB ½ðJ  rÞm  βm  ðJ  rÞm, eMS

(15.31)

with P is the spin-polarization ratio of the current, μB is the Bohr magneton, e is the (positive) elementary charge, β is the nonadiabatic coefficient, and J is the electron current density. It is convenient to express the magnetization vector and the effective field in the spherical coordinates basis {er, eθ, eϕ} (Fig. 15.14B) with m ¼er, Heff ¼ _ θ + sin θϕe _ ϕ . The circular symmetry of the nanowire _ ¼ θe Hrer + Hθeθ + Hϕeϕ, and m leads to the energy rotational invariant, which implies ∂/∂ϕ ¼ 0 and thus Hϕ ¼ 0. Moreover, for simplicity, we neglect any azimuth distortion of the domain wall, which corresponds to the 1D spin chain and implies rϕ ¼ 0. We name Japp the current density in the narrow part of the wire, and assume that the electron current is parallel to the z-axis Japp ¼ Jappez (Fig. 15.14A). This approximation is suitable for a smooth cross-section. Similarly, the electron current density is considered uniform in

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Magnetic Nano- and Microwires

Fig. 15.14 (A) Illustration of the domain wall under applied current in a modulated diameter wire. (B) Spherical coordinate basis {er, eθ, eϕ} for magnetization vector m and Cartesian spatial coordinates x, y, and z. The magnetization vector is drawn in the particular position corresponding to θ ¼ π/2, so that eθ ¼ ez. (C) Critical current density value Jcrit versus the larger radius R2. The solid curve corresponds to Eq. (15.35). Points correspond to the micromagnetic simulations. (D) Domain wall rotation frequency f in a pinned state versus applied current density Japp for R2 ¼ 10nm. The solid curve corresponds to Eq. (15.36). Points correspond to the micromagnetic simulations. Curves (C) and (D) are plotted for μ0MS ¼ 1T, R1 ¼ 5nm, λ ¼ 100nm, α ¼ 0.02, β ¼ 0.04, and P ¼ 0.7.

a cross-section and is related to the applied current density through Jz ðzÞ ¼ R21 Japp =R2 ðzÞ. After some algebra the LLG Eq. (15.1) augmented by the spin-torque (Eq. 15.31) takes the form γ 1 + αβ sin θ PμB θ_ ¼ 0 2 αHθ  Jz ðzÞ, 1 + α2 Δ eMS 1+α

(15.32)

γ β  α sin θ PμB Jz ðzÞ, sin θϕ_ ¼  0 2 Hθ + 1 + α2 Δ eMS 1+α

(15.33)

where Hθ ¼ (μ0MSV)1δE/δθ, θ_ ¼ vsin θ=Δ, v is the forward domain wall velocity, and ϕ_ ¼ 2πf is the angular domain wall velocity. Here, we applied the useful property of a 1D domain wall profile ∂θ=∂z ¼ sinθ=Δ, where Δ is the wall-width parameter [41, 58].

Domain wall pinning in a circular cross-section wire with modulated diameter

449

For simplicity in the following, we omit the internal domain wall structure and study the behavior of the magnetization vector in its center of the domain wall, where θ ¼ π/2. Let us assume that the domain wall settles at a given position zDW, for a given value of applied current Japp. This corresponds to θ_ ¼ 0, which from Eq. (15.32) implies: Hθ ¼

PμB ð1 + αβÞR21 Japp : eMS αγ 0 R2 ðzÞΔ

(15.34)

In the center of the domain wall, where eθ ¼ ez, Hθ is parallel to the z-axis and is pointed to the negative z direction, so that Hθ ¼ Heff. From Eq. (15.3), Hθ ¼ H0 is the internal field, reflecting the z-dependence of the domain wall energy, which we calculated at any position in the field-driven model. From this, for a given Japp we can solve Eq. (15.34) to search for a z-value allowing an equilibrium position. Thus, by combining Eqs. (15.27), (15.28), (15.34), we obtain a relation linking the applied current (Japp) and the resulting steady-state position of the domain wall zeq. Such a position exists for moderate current, however, not for very large current. The crossover determines the depinning current Jcrit in a smoothly varying modulation: ! 9αγ 0 eMS2 ðR1 + R2 Þ3 ðR2  R1 Þ 10l2ex π 2 Jcrit ¼ 1+ : 20PμB R21 λ 27ðR1 + R2 Þ2

(15.35)

Here, similar to Section 15.3.2, we have assumed zcrit ffi 0, and Δ ffi 2R(zcrit) ¼ R1 + R2. This law is plotted in Fig. 15.14. If we compare the domain wall behavior under applied field (Fig. 15.11D) and under applied current (Fig. 15.14C) in a smooth modulation, there is a major difference in the efficiency of the driving force in both cases. The critical field Hcrit is almost linear with growing larger diameter, whereas the growth of Jcrit follows a power law of R2. The reason is the decrease of local current density when the section broadens: not only does the domain wall energy increase, but also the efficiency of spin-torque decreases. Besides, Jcrit is proportional to the domain wall-width parameter which grows in the larger cross-section. Fig. 15.14C compares the analytical solution with micromagnetic simulations. The tendencies are similar, with even an excellent quantitative agreement in the limit of gentle modulation. This validates the model, and the earlier conclusion. The model also predicts the frequency of precession of the transverse component of magnetization of the wall, at the pinned position: f¼

PμB R21 Japp : 2πeMS αR2 ðzÞΔ

(15.36)

The dominant effect is that of the internal field and not of the nonadiabatic spintorque, resulting in the 1/(αΔ) coefficient in this equation. This frequency is plotted in Fig. 15.14C, for which similar to Section 15.3.2, we estimated the wall-width parameter Δ as Δ ffi 2R(z). Again, an excellent agreement is found with numerical simulation.

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Magnetic Nano- and Microwires

Conclusion and perspective

We have derived analytical models to describe how a magnetic domain wall may go through a modulation of diameter in a cylindrical nanowire, under the stimulus of either a magnetic field or a spin-polarized current. Scaling laws are derived, which may be used to quickly design a modulation to reach specific properties. While the depinning field scales with MS and the slope of the modulation (R2  R1)/λ, the depinning current increases much faster with the geometrical strength of the modulation, due to the decrease of local current density, and the increase of wall width. The relevance of these laws are confirmed by micromagnetic simulations, which reveal an excellent quantitative agreement for smoothly varying modulations. The quantitative investigation of experimental domain wall pinning in modulations of diameter is still in its infancy. It has been carried-out under magnetic field so far, with scattered results, however, pointing at the moderate strength of pinning compared with extrinsic pinning on material defects, when considering smoothly varying modulations. The drastically higher efficiency of pinning under current raises hope that modulations of diameter can be designed efficiently for spin-torque fundamental or applied devices.

Acknowledgments The authors acknowledge financial support from the French National Research Agency (ANR) (Grant No. JCJC MATEMAC-3D) and from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 309589 (M3d). J.A. Fernandez-Roldan is grateful for financial support from the Spanish MINECO under MAT2016-76824-C3-1-R and co-support from the ESF though BES-2014-068789 and EEBB-I-16-10934.

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C. Brana, M. Va´zqueza, E. Berganzaa, M. Jaafara, E. Snoeckb, A. Asenjoa a Institute of Materials Science of Madrid, CSIC, Madrid, Spain, bCEMES-CNRS, Toulouse, France

16.1

Introduction to the imaging techniques

The magnetization distribution within ferromagnetic domains where magnetic moments are essentially parallel to themselves has been a classical topic of research in micromagnetism. It reflects the minimization of total free energy, and when in remanence, it is determined by the following: (i) exchange that determines the parallelism of neighboring moments; (ii) anisotropy, of spin-orbit origin, that results in certain preferred magnetization directions; and (iii) magnetostatic energy that is associated to the presence of free magnetic charges. The latter plays a decisive role to determine the local distribution of magnetization and derives not only from the macroscopic (shape anisotropy) or local (i.e., at the corners and surface irregularities and local defects) geometry but also from magnetic charges accumulated at the domain walls separating magnetic domains (i.e., internal and surface charge distribution). For many years, micromagnetic analytical models have been developed by prominent magneticiens [1–3]. However, at the nanoscale, the complexity of mathematical calculations together with the development of the computer capability has led to new approaches and computational codes that are currently widely employed. The experimental observations of magnetic domains and domain walls at the macro- and mesoscopic scale have been achieved by classical techniques as Bitter technique or magneto-optical Kerr effect [4]. However, the reduction of dimensions to the nanoscale level induces such technical difficulties that the magnetization distribution imaging is only possible with the help of sophisticated advanced techniques. The cylindrical geometry of nanowires (NWs) introduces unique characteristics derived from the topology together with serious experimental difficulties. The large length-to-diameter ratio makes the longitudinal direction of magnetization the easy axis. However, when an additional anisotropy is present, their intrinsic geometry curvature fosters the circumferential path leading to the lack of parallelism of the magnetic moments to reduce the otherwise strong stray fields at the surface. Then a mathematical singularity appears at the nanowire axis due to the exchange interaction. Experimentally the curvature of the cylindrical NWs makes the focusing harder for all imaging techniques. Magnetic Nano- and Microwires. https://doi.org/10.1016/B978-0-08-102832-2.00016-5 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Magnetic imaging techniques are very useful not only to visualize the distributions in domains at remanence but also to follow the magnetization reversal processes under applied magnetic field. In this introductory section, we present a short overview on modern techniques for imaging magnetic domains and walls based on light, electron microscopy, near-field, nitrogen-vacancy center in diamond, neutron diffusion, and X-ray scattering with emphasis on those techniques that will be used for magnetic imaging of cylindrical nanowires [5]. Optical microscopy with polarized light is an excellent tool for domain imaging successfully applied to ferromagnetic, ferroelastic, and ferroelectric materials. As a contrast-enhancing technique for optically anisotropic materials, it is a method well suited for the investigation of a wide variety of effects in solid-state physics, as for example, the magneto-optical Kerr (MOKE) and Faraday effects in reflection and transmission, respectively, in ferromagnetic materials or birefringence in ferroelectric crystals. A modern optical technique is the wide-field polarizing microscope that combines a wide-field microscope and a confocal scanning laser microscope with polarization-sensitive detectors for imaging of magnetic flux structure at temperatures from 4 to 300 K [6]. Two different transmission electron microscopy (TEM) techniques can be used to study magnetic configurations: the Lorentz Fresnel and Foucault imaging in defocused and focused modes, respectively [7], and electron holography (EH) [8]. Lorentz TEM and electron holography methods are only sensitive to the in-plane components of the magnetic field (i.e., perpendicular to electron beam), and both allow retrieving the phase of the fast electron beam that has interacted with the electrostatic and magnetic fields within and around the magnetic object. Compared with defocused Lorentz TEM, EH is an in-focused interferometric technique (see Fig. 16.1) allowing the measurement of the phase of the e beam with nanometer spatial resolution and phase sensitivity up to 2π/1000 (few 103 rad) from which quantitative information about magnetic and electric fields in materials can be extracted with the same spatial resolution [9]. A further advantage of these two TEM-based techniques is that in situ stimuli (magnetic fields, electrical current, temperature, etc.) can be applied, which enables, in particular cases, to study the local magnetization reversal [10] or magnetic phase transitions [11] of a sample in real time. EH has then been successfully used to study the magnetization distribution in cylindrical nanowires, the configurations of various domain walls [12], the saturation magnetization of individual nanowires [13], and the magnetic induction in arrays of nanowires [14]. In scanning electron microscopy (SEM), the SEMPA and SPLEEM imaging techniques are also EM techniques carried out for magnetic imaging [15]. In SEM, electrons with reduced energy of 10–100 kV are backscattered or reemitted (secondary electrons). The analysis of their polarization (SEMPA) combined to the usual SEM topographic image provides a high spatial resolution direct image of the magnetization [16, 17]. As it is a surface technique, it is mostly employed in very thin multilayers, and nanostructures applied mostly focused on spintronic devices. Low-energy electron microscopy (LEEM) lies within the electron microscopy techniques having specific differences from conventional ones. Spin-polarized low-energy

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Source

Reference beam

Sample

Lorentz lens Biprism

Hologram

Fig. 16.1 Sketch of the electron holography setup: an electrostatic biprism permits the superimposition of a reference e beam that has passed through the vacuum and the phaseshifted e beam that has interacted with electromagnetic fields in and around the sample to create an hologram from which the phase of the phase-shifted e beam can be retrieved.

electron microscopy (SPLEEM) uses illumination electrons to image the magnetic structure of a surface since the number of electrons backscattered elastically depends on the relative orientation of the spin polarization of the incident electrons and the surface magnetization. SPLEEM has been used to image magnetic domain configurations in low-dimensional structures [18] apart from investigating spin reorientation transition phenomena or to explore magnetic couplings in layered systems [19]. Scanning probe microscopy techniques are also used to characterize nanomagnetic materials. Spin-polarized scanning tunneling microscopy (SP-STM) is a specialized application of scanning tunneling microscopy (STM) that can provide detailed information of magnetic phenomena on the single-atom scale additional to the atomic topography gained with STM. SP-STM opens a novel approach to static and dynamic magnetic processes as precise investigations of domain walls in ferromagnetic and antiferromagnetic systems and thermal and current-induced switching of nanomagnetic particles. An extremely sharp tip coated with a thin layer of magnetic material is scanned systematically over a sample. In the absence of magnetic phenomena, the strength of the current is indicative for local electronic properties. If the tip is magnetized, electrons with spins matching the tip’s magnetization will have a higher chance of tunneling (essentially the effect of tunnel magnetoresistance). In this case a crosstalk between topography, density of states, and magnetic properties could occur [20, 21].

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One the SPM techniques most widely used to study the microscopic magnetic configuration is the magnetic force microscope (MFM) [22, 23]. The MFM signal originates in the magnetostatic interaction between the tip stray field, fixed to a cantilever, and the magnetization of the sample. Due to the mechanical design, the MFM is mainly sensitive to the out-of-plane force gradients, and the contrast is proportional to the magnetic pole density on the surface. Advantages of MFM include relatively high spatial resolution (down to 10 nm) [24, 25], simplicity in operation and sample preparation, and the capability to apply external magnetic fields [26] or current pulses [27, 28] to study magnetization processes [29] and the possibility of operating in different environments [24, 30]. The drawback of the technique is the difficulty to obtain quantitative information about the magnetic moment of the sample [31–33] and the influence of the tip stray field on the magnetic configuration of the sample [34]. In addition, useful information can be obtained by deep analyzing the tip-sample interaction and in particular the magnetic dissipative and nondissipative signals [35]. The MFM works in different dynamic modes (the oscillation amplitude of the cantilever varies between 5 and 100 nm) regarding the ambient conditions. In MFM in air, the two-pass method is used. In the course of the first pass, the feedback keeps the oscillation amplitude constant to obtain the topography, while during the second scan (Fig. 16.2A), performed at a distances between 10 and 100 nm to avoid the van der Waals forces, the feedback loop is opened, and it is possible to record the changes in amplitude and phase. In addition, if the system incorporates a phase-locked loop facility, the frequency of the external oscillator changes to keep the phase constant, that is, the magnetic images have hertz units. The study of cylindrical nanowires by MFM has been a challenge task in the last years. The high aspect ratio, the contamination of the surfaces and its oxidation, and the complex spin configuration are some of the handicaps of the MFM imaging of magnetic nanowires. However, it has been demonstrated that the MFM allows obtaining detailed information about the spin configuration in elongated nanoparticles thanks to the high spatial resolution, the imaging under magnetic fields, and the combination with micromagnetic simulations. Interesting developments of the MFM technique have been implemented recently: advanced MFM modes [29, 36], tip engineering [37–39], MFM probe calibration [40], and improvement of the interpretation of the MFM images [41]. In the case of the so-called 3-D modes, the MFM signal is recorded along a profile, while the magnetic field is sweeping to obtain information about the reversal magnetization processes (see Fig. 16.2B). The isolated electronic spin system of the nitrogen-vacancy (NV) center in diamond offers unique possibilities to be employed as a nanoscale sensor for the detection and imaging of weak magnetic fields. Magnetic imaging has nanometric resolution and field detection capabilities in the nanotesla range enabled by the atomic size and exceptionally long spin coherence times of this naturally occurring defect [42, 43]. This novel method of imaging and sensing magnetic fields has been successfully applied to a single ferromagnetic nanowire with an atomic-scale sensor in diamond, that is, diamond nitrogen-vacancy (NV) defect center. The distribution of static magnetic fields around a single Co nanowire was mapped out by spatially distributed NV centers, and the obtained image is further compared with numerical simulation for quantitative analysis [44].

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Fig. 16.2 (A) Sketch of an MFM. (B) Sketch of the advanced MFM-based 3-D mode operation.

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On the other hand the use of techniques based on large facilities as neutron reactors and synchrotron radiation made the inner magnetic configurations become accessible. Neutron scattering is well suited to obtaining atom-scale magnetic structure to magnetic domains. It can be achieved by neutron diffraction, depolarization, or refraction. Polarization-analyzed small-angle neutron scattering has been successfully employed in cylindrical nanowire arrays to validate 3-D ordering in nanowire arrays [45, 46], discriminating from all competing energetics, such as segment shape anisotropy, intersegment dipole interactions, and interwire dipolar interactions. Photoemission electron microscopy (PEEM) carried out on synchrotron beam lines has matured into a versatile analytical technique on mesoscopic length scales. Applications range from surface physics to chemistry or biology. When operated with circularly polarized light in the soft X-ray regime, however, photoemission microscopy offers a unique access to many aspects in surface and thin film magnetism by combining magnetic and chemical information. Exploiting the high brilliance and circular polarization available at a helical undulator beam line, the lateral resolution in the imaging of magnetic domain structures may be pushed well into the submicrometer range. Moreover, this technique can be employed to imagine the magnetization reversal of nanostructures [47]. From all these techniques, element-sensitive X-ray magnetic circular dichroism combined with photoemission electron microscopy (XMCD-PEEM) has proved to be a unique technique to map the magnetization configuration in cylindrical nanowires due to its capability to record the magnetic information originating not only at the surface but also from the core of the nanostructures. In this technique the samples are illuminated with circularly polarized X-rays at a grazing angle of 16 degrees with respect to the surface, at the resonant L3 absorption edges of the material investigated (Fig. 16.3A). The emitted photoelectrons (lowenergy secondary electron with ca. 1 eV kinetic energy) used to form the surface image are proportional to the X-ray absorption coefficient, and thus the elementspecific magnetic domain configuration is given by the pixel-wise asymmetry of two PEEM images sequentially recorded with left- and right-handed circular polarization [48–50]. In this way the projection of the local magnetization on the photon propagation vector is imaged. The ferromagnetic domains with magnetic moments parallel or antiparallel to the X-ray polarization vector appear bright or dark in the XMCD image, while domains with magnetic moments at a different angle have an intermediate gray contrast. The cylindrical shape of the nanowires allows for a fractional amount of X-rays to be transmitted through the nanostructure, generating photoemission from the Si substrate. Since the transmitted intensity depends on the relative alignment of the nanowire magnetization and the X-ray helicity, the photoemission from the substrate in the area shadowed by the nanowire does as well. Therefore, by analyzing the circular dichroic or pseudomagnetic contrast formed in transmission in the shadow area, information about the magnetic configuration in the bulk of the wire can be obtained. Notice that dark contrast in transmission is equivalent to bright in direct photoemission, since the absorbed and transmitted X-rays are complementary. XMCD-PEEM thus offers the possibility to overcome the technical difficulty imposed by the cylindrical geometry of the nanowires and to gain double information of their surface and

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internal magnetic structure [47, 50, 51]. A sketch of the double sensitivity to direct photoemission and transmission is shown in Fig. 16.2A. It is also possible to work only in transmission mode by using transmission soft X-ray microscopy (MTXM), which is one of the most advanced and powerful techniques to experimentally visualize and quantitatively determine in situ magnetization processes [52, 53]. XMCD-PEEM has successfully been employed for the first time on cylindrical nanowires by Kimling et al. [47]. The investigated core-shell nanowires consist of a nickel (Ni) core and a shell of iron (Fe) and silicon (Si) oxide layers (sketch in Fig. 16.3A). Fig. 16.3B shows PEEM images taken at the Fe L3 edge (shell) and

Fig. 16.3 (A) Schematic view of the XMCD-PEEM principle and (B) PEEM images of core-shell wire scanned at Fe L3 and Ni L3 edge. (C) XMCD-PEEM images of Fe oxide tube (left) and Ni core (right). Adapted from J. Kimling, F. Kronast, S. Martens, T. Boehnert, M. Martens, J. Herrero-Albillos, L. Tati-Bismaths, U. Merkt, K. Nielsch, G. Meier, Photoemission electron microscopy of three-dimensional magnetization configurations in core-shell nanostructures, Phys. Rev. B 84 (2011) 174406.

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Ni L3 edge (core). The large absorption of Ni core is observed in the shadow at the Ni edge. Fig. 16.3C shows the differential XMCD images recorded at the respective absorption edges revealing the remanent magnetization configurations of the Fe oxide shell (left) and the Ni core (right). The blue and red XMCD contrasts correspond to the magnetization parallel and antiparallel to the direction of the in-plane projection of the incident X-ray beam, respectively, indicating the magnetization components along the wire axis. Three-dimensional tomography techniques [54] have recently been extended toward tomographic imaging of three-dimensional patterned, rolled, or curved structures. The introduction of three-dimensionality provides additional degrees of freedom and can result in novel properties such as magnetochirality or curvatureinduced anisotropy. Hard X-ray vector nanotomography has been successfully employed to determine the three-dimensional magnetic configuration at the nanoscale within micrometer-sized samples [55]. The imaging method enables the nanoscale study of topological magnetic structures in systems with sizes of the order of tens of micrometers. The structure of the magnetization within a soft magnetic pillar of 5-μm diameter was imaged with a spatial resolution of 100 nm, and within the bulk, it was observed as a complex magnetic configuration consisting of vortices and antivortices to form crosstie walls and vortex walls along intersecting planes. At the intersections of these structures, magnetic singularities—Bloch points—have been evidenced. A summary of some of the main features of particular techniques to image magnetic domains is enclosed in Table 16.1.

16.2

Synthesis and fabrication of modulated nanowires into the anodic aluminum oxide templates

The cylindrical nanowires with periodical geometrical modulations along their length are prepared by filling the pores of alumina templates [56–59] or by selective etching of uniform (constant diameter) nanowires with modulated composition [60]. Another method (described in Chapter 2) is combining the conventional alumina anodization with atomic layer deposition (ALD) [61, 62]. The aluminum anodization under controlled parameters produces highly ordered, cylindrical nanopores distributed into hexagonal cells (Fig. 16.4A). The work of Masuda and Fukuda allowed the fabrication of various nanoporous anodic aluminum oxide (AAO) templates with various diameters and interpore distances by the so-called mild anodization (MA) process [65, 66]. In typical MA processes, selfordered arrays of alumina nanopores can be obtained within three well-known growth regimes [59]: (1) sulfuric acid (H2SO4) at 20–25 V for an interpore distance (Dint) ¼ 55–63 nm [67–69], (2) oxalic acid (H2C2O4) at 40 V for Dint ¼ 105 nm [66, 70, 71], and (3) phosphoric acid (H3PO4) at 195 V for Dint ¼ 500 nm [66, 72]. Later on, Lee et al. [59] showed a new, faster method, called hard anodization, which uses low temperature and high voltages to produce AAO templates with much larger pore diameters. The growth rates of about 40–80 μm/h are much larger compared with

Table 16.1 Main features of different magnetic imaging techniques. Technique

Ultimate spatial resolution

Qualitative/quantitative measured data

MOKE

200 nm

Lorentz TEM

5 nm

Quantitative Magnetization: M In-plane induction: BII

Electron holography

1 nm

SEMPA

10 nm

Spin-polarized STM

0.1 nm

MFM

20 nm

NV centers in diamond tip X-PEEM

Few nm 50 nm

Quantitative In-plane induction: BII Quantitative In-plane magnetization: MII Qualitative Magnetization: M Qualitative Magnetic stray field: H Quantitative vector field mapping Quantitative Magnetization: M

Time resolution Few seconds Few seconds Few seconds Scanning technique few minutes Scanning technique few minutes Scanning technique few minutes From minutes to seconds Few milliseconds

Environment, in situ limitation In situ experiments possible Surface technique In situ experiments possible Thin samples (electron transparent) In situ experiments possible Thin samples (electron transparent) Surface technique UHV Conducting simple only Surface technique, UHV requested In situ experiments possible Environmental possible Surface technique Different environments Proximity-induced artifacts In situ experiments possible Chemical selectivity Synchrotron required

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the 1–3 μm/h obtained for conventional mild AAO grown in oxalic acid electrolytes under similar temperature conditions [59, 73]. In the recent years, by using these two methods, or variations of them, AAO templates with pores modulated in diameter were prepared. Various diameter-modulated metallic nanowire arrays were fabricated by filling these pores by an electroplating technique. The modulated nanowires produced in the modulated pores of alumina templates offer more possibilities to tailor the geometry of the wire and consequently their magnetic properties. The most used methods to fabricate AAO membranes with modulated pores are as follows.

16.2.1 Diameter-modulated nanowires obtained by varying the anodization parameters Geometrical modulated nanowires were obtained through the variation of the pores in the templates by using the so-called three-step anodization method [74–76]. The first two steps of this process are identical with those used in mild anodization: after the high-purity Al foils are cleaned and electropolished, the first anodization takes place in oxalic acid solution at a constant voltage (40–45 V) and constant temperature. After the irregular alumina layer is etched away to get a well-ordered concave structure on the aluminum substrate, the second anodization takes place in the same conditions, resulting in a regular array of nanopores. The ordered pores with diameters D ¼ 35–40 nm are enlarged in phosphoric acid (D ¼ 50–80 nm). This process is followed by the third anodization in the same conditions as the first two, producing once again nanopores with D ¼ 35–40 nm (Fig. 16.4B). As the same anodization parameters are used throughout the entire process, the third anodization pores grow in the extension of the second anodization, while the interpore distance is kept constant (around 105 nm). Similar pore architecture with much larger diameters (150 and 250 nm, respectively) was obtained by similar anodization method in phosphoric acid electrolyte (Fig. 16.4C) [64]. Krishnan and Thompson [57] studied the effect of changing the electrolyte during anodization. The Al films were prepatterned, prior to anodization, with 200-nm period pit arrays with hexagonal symmetry, fabricated by lithography. The first anodization was done in 5% phosphoric acid solution, at 86 V for a period of time. After the anodization was stopped, the electrolyte was replaced with 0.015 M oxalic acid, and the anodization was continued with the same applied potential. Fig. 16.4D confirms the change in pore diameter with the change of electrolyte. By changing the phosphoric acid electrolyte with the oxalic one, the diameter decreases from about 85 to 55 nm.

16.2.2 Diameter-modulated nanowires obtained by pulsed anodization Another approach for fabricating nanowires with variations in diameter in the modulated pores of AAO templates is by pulsed anodization. Lee et al. [58] showed that combining the advantages of mild and hard anodization (HA) is possible to control the pore structure of AAO. The pulse anodization of aluminum was done in potentiostatic

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Fig. 16.4 (A) Top-view scanning electron microscopy (SEM) picture of a nanowire array embedded in the alumina template [63], (B) schematic view of the three-step anodization process, (C) SEM image of a AAO membrane with pore diameter of 150 nm [64], and (D) nanofunnels with modulation in diameter obtained by two different anodization parameters [57]. Panel (B): adapted from W.S. Im, Y.S. Cho, G.S. Choi, F.C. Yu, D.J. Kim, Stepped carbon nanotubes synthesized in anodic aluminum oxide templates, Diam. Relat. Mater. 13 (2004) 1214–1217. Panel (C) adapted from M.S. Salem, P. Sergelius, R.M. Corona, J. Escrig, D. Goerlitz, N. Nielsch, Magnetic properties of cylindrical diameter modulated Ni80Fe20 nanowires: interaction and coercive fields, Nanoscale 5 (2013) 3941. Panel (D) adapted from C. Bran, Y.P. Ivanov, J. Garcı´a, R.P. del Real, V.M. Prida, O. Chubykalo-Fesenko, M. Vazquez, Tuning the magnetization reversal process of FeCoCu nanowire arrays by thermal annealing, J. Appl. Phys. 114 (2013) 043908.

conditions in H2SO4 electrolyte. Periodic pulses, alternating a low and high potential, were applied, where the duration of each pulse determines the length of anodized segments at the given applied potential. Nanochannel modulations have been archived by Sulka et al. [77, 78], prepared by two-step anodization of aluminum discs in a 0.3 M H2SO4 at 0 °C temperature. The first anodizing step was performed at constant

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potential of 25 V for 16 h. After the anodized alumina was etched, the second anodization step was performed by applying a series of potential pulses comprising a MA pulse of 25 V (for 3 min) and a HA pulse of 35 V (for 0.5–1.5 s) (Fig. 16.5B). Minguez-Bacho et al. [79, 80] have shown that the voltage pulses, applied during anodization, control not only the length and diameter but also the shape: square and exponential signals have been applied (Fig. 16.5A). Modulated nanowires with larger diameters (>100 nm) and defined geometry (Fig. 16.5C and D) were fabricated [59, 81–85] inside the modulated pores of AAO templates obtained by pulsed hard anodization in oxalic acid solution (0.3 M) containing 5 vol% ethanol at a constant temperature of 0 °C. During the anodization a lower voltage was first applied to produce a protective aluminum

Fig. 16.5 (A) HRTEM image of nanowire’s geometrical features using modulated diameter within squared pulse template [79]; (B) cross-sectional SEM image of AAO template with modulated pore diameter formed by pulse anodization in H2SO4 electrolyte [77]; (C) voltagetime transients of pulsed hard anodization, including the first anodization step (a), ramping of voltage (b), and the applied pulses (c). The inset shows a close look of both applied voltage and current transients, during the pulsed anodization [50]; and (D) SEM cross section of modulated nanowires embedded into AAO template. The insets show top-view SEM pictures of individual nanowires released from AAO templates [50].

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oxide layer at the surface of the disc, which avoids breaking or burning effects (Fig. 16.5C—step A) during subsequent pulsed hard anodization. After that the voltage was slowly increased (0.05 V/s) up to 100 V and kept constant for 400 s, which ensures the alignment of the nanochannels (Fig. 16.5C—step B). Nanopores with tailored periodical diameter modulations were produced by applying voltage pulses for various periods of time [50, 84]. The pulses were repeated 30–40 times to obtain a total length of the modulated pores of about 50–60 μm. The resulting cylindrical pores are formed by segments with diameters of about 100 and 140 nm, respectively, while the center-to-center interpore distance is kept constant at 320 nm. The magnetic nanowires are electrodeposited into the modulated pores (Fig. 16.5D). The NWs are released from the alumina template and characterized individually (Fig. 16.5D—insets).

16.3

Magnetic characterization of nanowire arrays

Arrays of cylindrical magnetic nanowires were fabricated by electrochemical deposition into a template of nanometer-sized pores created by different methods as mentioned in the previous section. The morphology of these arrays of nanowires is commonly studied by using SEM or by TEM after dissolving the template. One of the interest of the NWs embedded in alumina membrane is its use in permanent magnet applications. For this purpose, arrays of NWs with single-domain configuration are developed. Magnetic materials with low anisotropy are used to guarantee the axial easy axis induced by its high aspect ratio. The magnetic properties of the arrays of NWs are analyzed by the combination of different techniques. Magnetometry measurements, with the external field applied perpendicular or parallel to the film plane (hence parallel or perpendicular to the nanowires), were used to obtain bulk hysteresis loops. Arrays of Ni NWs were one of the most studied systems in the first works in this topic. The measurements can be performed at different temperatures— with a vibrating sample magnetometer [86] or by SQUID magnetometer measurements [87]—and also by changing the angle between the applied field and the long axis of the NW [88]. Despite its high aspect ratio and even if the reversal magnetization of the individual nanowires occurs through a Barkhausen jump, not always a squareness of nearly 100% is obtained in the bulk hysteresis loops specially due to the influence of the nearest neighbors. For that reason, it is very common to combine the bulk measurements with other techniques, specially MFM [89], which is a very useful tool to analyze the local magnetic behavior of these arrays. This technique can supply information about the behavior of individual entities with very high resolution and provide simultaneous information about the topography and the magnetic configuration. MFM imaging under in situ applied magnetic field is of particular interest [91] since it is possible to study (i) the switching field distribution [92], (ii) the influence of the long- [93] or short-range magnetostatic interactions including the roll of the six nearest neighbors in a hexagonal array [90], or (iii) the impact of the tip stray field in the switching field of each NW [94]. In that sense, several groups have

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Fig. 16.6 MFM image of a labyrinth [90] magnetic structure observed in an hexagonal array of Ni nanowires.

modeled the magnetic imaging data by different statistical methods to achieve quantitative results [93, 95] about the magnetostatic interaction between NWs. The MFM image in Fig. 16.6 shows the labyrinth magnetic structure of an array of Ni nanowires. The magnetostatic interaction is relevant in the magnetic configuration in the case of Co [96] or FeCo [97] NWs where a closure magnetic structure such as vortex appears at the end of the nanowire as deduced from the quantitative MFM measurements. These edge configurations have a relevant role in the magnetization process. In addition to the MFM, other techniques have been used successfully to characterize the arrays of NWs and antidots. For instance magnetic transmission soft X-ray microscopy (MTXM) imaging, with a spatial resolution of 25 nm, allows to visualize the local reversal magnetization process and to obtain a kind of local hysteresis loops [98]. More recently, precession electron diffraction has been used to compare the magnetic behavior observed by electron holography (EH) with the crystallinity of the NWs [99]. This technique has the advantages of the direct visualization of the ferromagnetic ordering within the Ni NW and its evolution under in situ magnetic fields or increasing the temperature. As shown in Fig. 16.7, Ortega et al. demonstrated the antiferromagnetic ordering of the nanowires within the array due to the long-range magnetostatic interaction [99].

16.4

Magnetic characterization of individual nanowires with uniform diameter

Apart from the single-domain configuration NWs shown in the previous section, the design of multidomain nanowires has been attracting great interest both from fundamental and applied points of view. The magnetic imaging of the multidomain NWs

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Fig. 16.7 (A) Electron hologram of a single row of Ni nanowires. (B) Amplification of the magnetic/electrostatic contour lines representing phase variations through the plane of the sample. (C) Same as (B) after an external H ( 18 kOe) was applied perpendicular to the nanowires using the microscope objective lens coils, note the lines of the magnetic field outside the sample. From E. Ortega, U. Santiago, J.G. Giuliani, C. Monton, A. Ponce, In-situ magnetization/heating electron holography to study the magnetic ordering in arrays of nickel metallic nanowires, AIP Adv. 8 (2018) 056813, doi:10.1063/1.5007671, licensed under a Creative Commons Attribution (CC BY) license.

should be done individually, and thus the NWs must be released from the membrane previously. In the last years the development of nanoscale characterization techniques facilitate that many research groups focus on the study and imaging of individual magnetic nanowires [100–102]. Imaging techniques have provided the opportunity to visualize magnetic domains and further understand the magnetic domain configurations and their dynamics, for instance, by applying in situ magnetic field during the measurements. In 2001, Henry et al. [103] carried out a very thorough study of Co nanowires of different diameters by MFM. Notice that, in opposition to the NWs shown in the previous section, the Co NWs present high magnetocrystalline anisotropy. As a result of the energy balance between magnetostatic energy (due to the NW large aspect ratio) that favors axial magnetization and the magnetocrystalline anisotropy, different types of magnetic configurations are developed. With magnetic torque measurements, they were able to calculate the effective first-order anisotropy constant as a function of the NW diameter and to determine (by using TEM measurements) that the crystalline structure changes with the diameter size. Thus Co nanowires with diameter smaller than 50 nm present single-domain configuration, while, for bigger diameters, magnetic domains are formed. Moreover, MFM characterization [103] shows that, in small-diameter NWs, the application of a transverse saturating field makes the single-domain configuration breaks into domains: a succession of longitudinally magnetized antiparallel domains is formed (see Fig. 16.8). The number of domains depends on the aspect ratio of the NWs. However, in thicker Co-based NWs (80–90 nm in diameter), Belliard et al. [104] found more complex configuration as the coexistence of transversal and axially magnetized domains within the same NW. In this work the influence of the topography on the MFM measurement is also analyzed by fitting the experimental signal to a tip-sample interaction model.

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Fig. 16.8 MFM images corresponding to a 10-μm long and 35-nm thick Co nanowire after applying a transverse field of 1.3 T (transverse remanent state). A and B are two subsequent experiments. Adapted from Y. Henry, K. Ounadjela, L. Piraux, S. Dubois, J.-M. George, J.-L. Duvail, Magnetic anisotropy and domain patterns in electrodeposited cobalt nanowires, Eur. Phys. J. B 20 (2001) 35–54.

Recently, Da Col et al. [105] explored the possibility of creating controlled domain walls in straight NWs by bending them and creating spots with higher stress. They show some evidence of the creation of tail-to-tail and head-to-head domain walls and Bloch walls using MFM. The authors combine the MFM imaging with micromagnetic simulations to reinforce this hypothesis. In a previous publication [51], Da Col et al. successfully employed XMCD-PEEM techniques to show the existence of a Bloch point domain wall in cylindrical permalloy nanowires. Bloch point domain walls (DWs) are topologically protected 3-D structures with potential to reach high velocities [106–108]. The XMCD-PEEM imaging of those Py NWs, with diameter of about 100 nm, gives two types of domain walls: orthoradial curling of magnetization (Fig. 16.9A, shadow) and its symmetry with respect to the perpendicular plane to the nanowire axis (Bloch point DW) and monopolar contrast at the center of the DWs, close to the nanowire axis (transversal DW) [51, 110, 111]. Cobalt [112, 113] or cobalt-rich alloy nanowires [109] have been a frequent subject of study due to the different types of domains that can appear as a result of the anisotropy competition. By using XMCD-PEEM, Bran et al. [109] presented the experimental evidence of transverse magnetic domains, previously observed only in nanostrips, in cylindrical nanowires with designed crystal symmetry and tailored magnetic anisotropy. The combination of Co and Ni can be used to tune the magnetic anisotropy and imprint transverse domains in a cylindrical structure. These domains coexist with more conventional vortex domains along the same cylindrical nanowire, denoting a bistable system with similar energies. Fig. 16.9B presents the magnetic configuration observed by XMCD-PEEM of CoNi nanowire alloys. Nanowires

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Fig. 16.9 (A) XMCD-PEEM image (center) of Bloch point domain wall observed in permalloy nanowires together with X-PEEM image (left) and simulated image (right) [51]. (B) XMCDPEEM images of Co65Ni35 nanowire. Inset figures at the bottom are enlarged images of the transverse and vortex domains in the corresponding regions of Co65Ni35 [109]. The arrows (in A–B) indicate the incident X-ray beam.

with two different compositions, Co65Ni35 and Co85Ni15 and 120 nm in diameter, were fabricated by electrodeposition into the pores of anodic alumina templates prepared by hard anodization. In Co85Ni15 nanowire, it was observed that the surface consists of alternating segments of bright/dark contrast, which stand for vortex domains with different chirality along the nanowire surface. Co65Ni35 nanowires show richer magnetic configurations (Fig. 16.9B). The right side of the nanowire shows a sequence of vortices. The gradual contrast observed in the shadow, with increasing intensity as the edge of the wire, is reached (see the inset) indicating that the vortex penetrates nearly fully into the nanowire. On the left side of the nanowire, a regular sequence of segments with much shorter periodicity and alternating contrast (i.e., bright/dark at the surface and dark/bright in the core) is observed. The contrast in the shadow is opposite to that at the surface and remains constant in the transversal direction (see the inset), showing clearly that the magnetization state is homogeneous along the complete circular cross section of the wire. This is interpreted to correspond to transverse domains (defined as regions where all magnetic moments are parallel to themselves and point in the plane perpendicular to the axis of the nanowire) [109]. In addition to the magnetic domain characterization, recent advances in electron holography (EH) have paved the way to the experimental determination of the domain wall type in cylindrical NWs. Biziere et al. [12] address the characterization of the DW type of Ni NWs with EH and micromagnetic simulations. The main difficulty of this work lies on obtaining an adequate resolution to image the 3-D structure of a DW. By applying a saturating field in the direction perpendicular to the nanowire axis, a DW is created. They find that there is a transition from transverse domain to a hybrid domain as a function of the diameter of the NW.

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Another asset of imaging techniques is the possibility to study induction maps. Beeli et al. [114] published a pioneer work on the study of cylindrical nanowires using off-axis EH. They succeeded to image the magnetization flux of Co nanowires of 40 nm of radius with axial magnetization. They were able to quantify the flux of several NWs, and they found out that their flux is quantified, as it is always proportional to h/e, a flux quantum. In this regard, remarkable progress was achieved by Wolf et al. [115], who proposed the combination of EH and electron tomography to reconstruct threedimensional induction maps of a Co nanowire grown by focus electron beam-induced deposition. Generally, 2-D images obtained by electron holography (projections of the magnetic induction) need to be complemented with simulations to picture a 3-D map. In their work, Wolf et al. acquire a sequence of tilted electron holograms of the cobalt nanowire that allows them to perform a tomographic reconstruction, achieving a resolution of 0 was applied in the opposite direction to the direction of the magnetic field that was used for the processing of the reference image. The image of the saturation state of the wire is shown in Fig. 18.3D. The same image processing procedure was used when the direction of the magnetic field was reversed to produce the saturation state. The image of the saturation state with the opposite helicity of the

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Fig. 18.3 The images of the microwire obtained in Co-rich microwire (sample #1) using the L-MOKE geometry: (A) initial optical image and surface magnetic domains for (B) background state; (C) Hax ¼ 0; (D) Hax ¼ 0.2 Oe saturation; (E) Hax ¼ 0.2 Oe  saturation; (F) Hax ¼ 0.

magnetization is presented in Fig. 18.3E. Using the procedure of the image processing [16], we obtained the “black and white” contrast of magnetic domain image with in-plane magnetization components (see Fig. 18.3F). There are two main orientations of the polarization in the L-MOKE geometry with the s- and p-components of the electric field. In the case where the direction of incidence was parallel to axial orientation of the wire, we observed the maximum magnetic contrast in the domains with the magnetization directed along the wire axis. In the second case, when the direction of incidence was perpendicular to the axial

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orientation of the wire, the contrast in axially magnetized domains decreased, while the contrast of the circularly magnetized domains increased [16].

18.2.3 Imaging of the magnetization reversal by MOKE microscopy An external axial magnetic field was applied during the observation of the evolution of the magnetic domain structure and the investigation of the magnetization reversal process. Generally, when MOKE microscopy is used, the light intensity measured at each pixel of the CCD camera is proportional to the local Kerr rotation. The spatial dependence of the Kerr effect and the resolution is defined by the optical system. The magneto-optical Kerr effect caused a rotation of the plane of polarization of the light. The rotation depended on the direction of the magnetization in the domain and can be transformed into contrast using an analyzer. The Kerr contrast was low enough. Therefore a background image processing technique was used for the measurement of hysteresis loops. These loops were obtained by plotting the averaged image contrast as a function of the external magnetic field using the L-MOKE geometry. Two averaging procedures were used: the light intensity was averaged by repeating the data accumulation for each pixel and by taking an average of pixels in the region of interest (ROI) [16]. The hysteresis loops at any pixel point could be visualized when the magneto-optical images were acquired for the sweep of magnetic field between the negative and positive magnetic saturation. Fig. 18.4 shows the dependence of the normalized intensity, which is proportional to the Kerr rotation,

Fig. 18.4 The hysteresis loop obtained in Co-rich microwire (sample #1) using the analysis of the images of the domain structure. The images of the domain structures show the selected points on the hysteresis loop.

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on the amplitude of the axial magnetic field. A clear hysteresis loop was observed for the microwire area, while no signal was obtained for the region outside the ROI. The program recorded the hysteresis loop in the real time, and the images of domain structures were obtained under the external magnetic field using image processing. The results of the visualization of domain structures permit the determination of the orientation of the magnetization on the surface of the studied microwire. [16].

18.3

Elliptic domain structures

The elliptic domain structure, along with the spiral structure, is one of the basic inclined structures that we have discovered in microwires last years. The search of the conditions of the existence of this structure was the basic task of our investigations. Also the methods of the creation and the control of the predicted structure were in the area of our continuing scientific interest. Here, we present the results of the studies Fe-rich microwire (sample #2). We demonstrate that the mechanical torsion stress τ induced a helical magnetic anisotropy in the microwire. Also the torsion stress induced the approach of different magnetic states and facilitated the transitions between them. The response of the surface magnetic structure on the torsion stress in Fe-rich microwire is presented in Fig. 18.5. The effect is realized in the stress-induced inclination of the magnetization. The magnetic domains were separated by an inclined domain wall (DW) (see Fig. 18.5A). It should be mentioned that the rotation of the magnetization occurs across the axial direction in microwire. Also the saturation of the effect takes place: when a certain amount of stress is reached, the increase in deviation does not occur. Fig. 18.5 (right panel) shows schematically the orientations of the stress-induced magnetization. In such a way, we learned how to set a predetermined angle of domain wall inclination that turned out to be important for far experiments. To investigate the influence of the DW inclination on the dynamic

Fig. 18.5 (Left panel) MOKE images of surface domain structure obtained in Fe-rich microwire (sample #2) in the presence of torsion stress: (A) τ ¼ 34 π rad m1; (B) τ ¼ 27 π rad m1; (C) τ ¼ 7 π rad m1. (Right panel) Schematic pictures with the magnetization orientation (arrows) corresponding to images of domain structures.

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properties, we also used a spatially oriented external magnetic field as a combination of circular and axial magnetic fields Hdc (see Fig. 18.1) to establish the angle of DW inclination (see Fig. 18.6). The next step to control the elliptic domain structure was the experiment with axial magnetic field Hax. It seemed that the use of the axial field is the relatively trivial, and it should not cause anything special, but the experiment showed something unexpected. First, we should to select the value of the initial torsion stress and as a result the starting angle of the surface magnetic structure. Earlier, we recognized that the magnetization with the angle of inclination near the axial direction is very sensitive to the different external factors. Therefore we have selected the torsion stress with τ ¼ 3.5 π rad m1. Applying the axial magnetic field, we followed the transformation of surface domain structure. The hysteresis loops also have been plotted. Three types of the experiments have been performed depending on the area of the wire surface from which the MOKE signal has been obtained (see Fig. 18.7). Fig. 18.7 presents the hysteresis loop (black line) obtained using the “large” region (marked by dashed line). Also the images of the domain structure are presented in Fig. 18.7. The magnetization reversal contains the jumps of the inclination of the DW. These jumps occurs on two half branches of the loop. The jumps could be observed on the images of the surface domain structure. The hysteresis loop clearly consists of two regions as “top” (ROI-1) and “bottom” (ROI-2) corresponding to the

Fig. 18.6 MOKE images of surface domain structure obtained in Co-rich microwire (sample #3) in the presence of external magnetic field. The angle of the DW inclination ϕ was measured from the circular direction. Images show domain structure for ϕ ¼ 19° (A), 38° (B), 52°, and 74° (C) under variation of Hdc [17]. Schematic picture represents the elliptical DW (top sketch).

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Fig. 18.7 L-MOKE hysteresis reflecting the magnetization reversal process in Fe-rich microwire (sample #2) obtained within (A) whole observation area of the microwire and (B) different ROI marked by dashed areas. The images of the domain structures show the selected points on the hysteresis loop.

magnetization reversal in two parts of the microwire. To clear the details of the magnetization reversal, we have performed the MOKE experiments with two mentioned regions. The top region is marked in blue in Fig. 18.7. The almost rectangular MOKE hysteresis correspond the quick motion of the inclined DW without the jump. The bottom region is marked in green in Fig. 18.7. Here, we also observe the motion of the inclined DW, but the direction of the inclination is opposite in comparison with Fig. 18.5. The jump of the inclination also is fixed in the series of the images of the domain structure. There are two peculiar properties. The first is the difference in the coercive field and the second enough plane part of the black hysteresis around zero value of the MOKE signal. As it was demonstrated in our paper [5], there are four structures with helical states that could be observed generally in the microwires. These structures are characterized by the different projections of the surface magnetization. We consider that the jumps between, namely, these structures are observed in the present experiment. The existence of the original inclination of magnetization without external stress and magnetic field is the main reason of the observed effects. Application of the torsion stress and/or magnetic fields causes the consequent formation of the structure with inclined magnetization states, but the angle of the inclination is different in different structures. Different angle of the inclination is the reason of the difference in the coercive field. One of the essential parameters affecting the coercivity is the length of the DW. The length of the inclined elliptic DW has direct correlation with the angle of the inclination. This correlation is the main cause of the observed difference in the surface coercive field (see Fig. 18.6) [17].

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The second effect is related to the pronounced deceleration of the DW (right series of the images in Fig. 18.7). This deceleration is determined by the overcoming of the energetic barrier related to the jump between the states with opposite inclination. The performed experiments and the analysis permit to realize the following conclusions. The combination of the external stress and the magnetic field is the power method of the creation and control of the predicted domain structures in the microwire. At the first step the application of the small stress permits to select the direction of the inclination. The subsequent variation of the stress establishes the angle of the magnetization and DW inclination. The high enough torsion causes the large inclination blocking in this way the possibility of the further jumps between the inclined states. In the same time the small torsion lowers the energetic barriers between the inclined states and in such a way opens the possibility of the multiple transitions between metastable states. Namely, these jumps are realized visibly in the jumps of the DW inclination. The other important points are the repeatability and reversibility of the observed transitions. The observed repeatability of the MOKE hysteresis is the reflection of the reversibility of the field-induced transitions between the inclined states. Raising and lowering the energetic barriers, the external magnetic field forms the conditions of the stable and metastable existence of the state with predicted configuration. It is important to note that the axial field inverts not only the direction of the axial magnetization but also the direction of circular magnetization. This is the evidence of the strong correlation between two projections of the inclined states.

18.4

Spiral domain structures

Spiral domain structure is the second type of the helical structure. While the elliptic domain wall could be considered as an inclined circular domain wall with the limited length, the spiral is characterized by the unlimited length. Or more precisely, it is limited only by the length of the real sample of the microwire. This property determines the specific features of the magnetization reversal and DW motion. Figs. 18.8 and 18.9 demonstrate the hysteresis loops and the corresponding domain images related to the nucleation and transformation of the spiral structure. Fig. 18.8 presents the axial field-induced magnetization reversal in the presence of the torsion stress (τ ¼ 2 π rad m1) and the DC electric current (10 mA) studied in the sample #3. Analyzing the obtained MOKE results, we have put our attention on the dip in the hysteresis loop marked by red. The magnetization reversal begins from the saturated state (presented in bright color in the domain image). Appearance of the black inclined wedge-like domain is the first step of the spiral structure formation. Generally the formation of the wedged domains is the one of the main characteristic features of the spiral domain. The second step is the directed migration of the solitary domain along the microwire surface. Appearance of the second domain in the sight of the microscope marks itself the formation of the long infinite domain encircled the wire surface. Fig. 18.9 presents the results of the MOKE experiment obtained only in the presence of the torsion stress (τ ¼ 2 π rad m1). In this configuration, we observe the formation and the transformation of the multidomain spiral structure. A number of the

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Fig. 18.8 Experimental dependence of L-MOKE hysteresis loop and the images of the domain structure. The images of the domain structures show the selected points on the hysteresis loop.

Fig. 18.9 Experimental dependence of L-MOKE hysteresis loop and the images of the domain structure. The images of the domain structures show the selected points on the hysteresis loop.

black inclined wedged domains form in the surface of the microwire. At the second stage, these domains move in parallel along the surface. In such a way, the multidomain spiral structure forms. As we can, the similarity of the structure (spiral helicity) is determined by the torsion stress. Dissimilarity (monodomain structure or multidomain structure) is related to the DC circular field. The DC circular magnetic field suppresses the nucleation of the spiral domains allowing the formed solitary domain to move along the surface.

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A theoretical analysis of the obtained results was performed on the basis of a model assuming that, for a magnetostrictive microwire, the cylindrical symmetry limits the stress components to radial, circumferential, and axial components. Further, the energy of the magnetic anisotropy is equivalent to the barrier energy necessary to rotate a local magnetic moment. The magnetic anisotropy energy possesses a uniaxial character that is equivalent to the cylindrical symmetry of the angular dependence of the magnetic moment. That is to say, for a local easy axis along the axial direction, the energies needed to rotate the magnetic moment within the ZX plane and within the ZY plane are the same (see Fig. 18.10a). The concept of an effective uniaxial anisotropy, Ku, considers that the magnetic moment is rotated within the plane where the energy cost is lower. Thus, if the circumferential, radial, and axial magnetic moment components value as σ qq < σ rr < σ zz, then Ku ¼ (3/2)λ(σ zz  σ rr), where λ is the magnetostriction constant. Thus the energy to rotate the magnetic moment in the ZX plane is less than that necessary to rotate it in the ZY plane. Fig. 18.10 demonstrates the simulated spiral domain structure and the calculated hysteresis loop obtained using the MuMax program [18]. Conceptually the conditions of the calculation coincide with the conditions of the MOKE experiment: the calculated hysteresis corresponds to the small area on the surface of the microwire [2]. During the MOKE experiments the signal was reflected also from the small part of the microwire surface.

Fig. 18.10 Simulated spiral domain structures in a wire (A). Calculated magnetization reversal process obtained in axial magnetic field applied along Z direction in a wire (B).

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It is necessary to note that the calculated hysteresis has the shape similar to the experimental MOKE hysteresis presented in Fig. 18.8. It also has a dip, which is associated with the formation, transformation, and motion of spiral domains in the surface. The main preliminary conclusion from the performed simulation is the following: The classic “inner core/outer shell” model of magnetic structure of microwire should be reconsidered for the case of spiral structure. That is why we have focused on the analysis of the cross sections of the microwire demonstrating the transformation of magnetic structure at various points along the length of the microwires. The observed effect we have named as “surface and volume migration” of the spiral domain. Fig. 18.11 shows the inner migration, while Fig. 18.12 shows the surface migration. Fig. 18.11A demonstrates the image of the magnetic structure at the surface of the microwire, while Fig. 18.11B shows the calculated magnetic structure inside the microwire. The images of cross section Fig. 18.11C–F show the migration of the inner spiral domain along the microwire. The red arrow marks the volume migration of the inner domain. The main result that we have to accent on is the effect of the migration of the inner spiral domain. The center position of the spiral domain changes in sporadic way (cross sections; see Fig. 18.11A–F) up to the position when the domain comes out to the surface. In these conditions the spiral domain is very similar to the inner core in the “core–shell” model. The main essential difference is the excentric position of spiral domain, while the classic inner core is always centrally located.

Fig. 18.11 3-D pictures of the simulated magnetic structure (A) and visualization as a half-tube (B). Calculated cross sections of the inner structure of the microwire at Hz ¼ 3.25 kA/m for varying distance Z from the edge of the microwire of (C) 2.3, (D) 3.5, (E) 5.3, and (F) 6.5 μm. The red arrow in (C) shows the spiral domain migration direction inside the microwire. The colors correspond to a magnetization with + MZ (red) and  MZ (blue) orientations.

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Fig. 18.12 3-D pictures of the simulated magnetic structure (A) and visualization as a half-tube (B). Calculated cross sections of the inner structure of the microwire at Hz ¼ 1.75 kA/m for varying distance Z from the edge of the microwire of (C) 2.3, (D) 3.5, (E) 5.3, and (F) 6.5 μm. The red arrow in (C) shows the spiral domain migration direction inside the microwire. The colors correspond to a magnetization with +MZ (red) and  MZ (blue) orientations.

The interesting conclusion could be extracted from Fig. 18.11. The inner spiral domain shifts gradually to the surface transforming in such a way to the part of the outer shell. The center of the domain (bright red area) is not exactly at the surface, while the edge of the domain reaches the surface that could be clearly observed in Fig. 18.12A. The migration of the surface spiral domain is totally different form the case of the inner domain. The well-ordered migration could be observed in all the images presented in Fig. 18.12. The cross sections (Fig. 18.12C–F) show the structure of the moved spiral domain. The half-tube picture (see Fig. 18.12B) gives the unusual view of the spiral domain migration. Here, we can see the additional confirmation of the “unlimited” character of spiral domain. The periodic concentration of the red or blue colors along the microwire length indicates a periodic rotation of the magnetization between two axial directions. The comparative analysis of the experimental and theoretical study of the spiral structure permits us to extract the main properties of this recently discovered structure. Unlimited length of domain walls is the main difference of this structure from the elliptic domain structure. The spiral structure forms as a directed motion of the wedged domain. This motion is confirmed as at the surface and inside the microwire. The ordered migration of the magnetic center of the spiral domain is demonstrated. Uniaxial anisotropy sets the direction of this migration, while the combination of the magnetic fields determines the amplitude of the migration.

Helical magnetic structures in amorphous microwires

18.5

533

Conclusions

Developing the Kerr effect technique for the unplane surface of the microwires, we directed our efforts to the new sector related to the helical magnetic structures. We have recognized two basic helical structures: elliptic and spiral. Having the differing magnetic properties, these structures in the same time are very energetically close. That permits us to determine the main experimental tools to control and transition between them. This possibility to manage the helical structure is the key ability for the selection of the magnetic structure in microwires elected as an active element in magnetic microwire. External torsion stress and DC electric current are the main tools that we use to establish helical structure with predicted properties. The main difference of these structures is related to the domain wall structure. While the elliptic domain wall is limited by the shape of the ellipse, the spiral domain wall is limited only by the length of the real sample. Another important property is the surface and volume migration of spiral domain that is the consequence of the unlimited length of this domain structure. As a consequence of these effects, the features of the magnetization reversal in elliptic and spiral structures are radically different. The essential contribution to the understanding of the main properties of the magnetic helical structures brings the micromagnetic simulations. Confirming in detail the experimentally obtained results, the simulation helps us direct the experiment to the optimal course with the purpose of the search of the magnetic structure optimal for the magnetic sensor application.

Acknowledgments We acknowledge support from the National Science Centre Poland under Grant No. DEC2016/22/M/ST3/00471. This work was also supported by Spanish MINECO under MAT2013-47231-C2-1-P and MAT2013-48054-C2-2-R, by the University of Basque Country UPV/EHU (Ref.: PPG17/35). The research of P.G. was supported in part by PL-Grid Infrastructure.

References [1] M. Vazquez (Ed.), Magnetic Nano- and Microwires: Design, Synthesis, Properties and Applications, Woodhead Publishing Series in Electronic and Optical Materials, Woodhead Publishing, Cambridge, UK, 2015. [2] A. Chizhik, A. Zhukov, J. Gonzalez, P. Gawronski, K. Kułakowski, A. Stupakiewicz, Spiral magnetic domain structure in cylindrically-shaped microwires, Sci. Rep. 8 (2018) 15090. [3] A. Chizhik, J. Gonzalez, Magnetic Microwires: A Magneto-Optical Study, Pan Stanford Publishing Pte. Ltd, Singapore, 2014. [4] A. Zhukov, High Performance Soft Magnetic Materials (Springer Series in Materials Science), Springer, Cham, Switzerland, 2017.

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[5] A. Chizhik, V. Zablotskii, A. Stupakiewicz, C. Go´mez-Polo, A. Maziewski, A. Zhukov, J. Gonzalez, J.M. Blanco, Magnetization switching in ferromagnetic microwires, Phys. Rev. B 82 (2010) 212401. [6] A. Hubert, R. Sch€afer, Magnetic Domains, Springer, Berlin, Germany, 1998. [7] A.K. Zvezdin, V.A. Kotov, Modern Magnetooptics and Magnetooptical Materials, Institute of Physics Publishing, Bristol, Philadelphia, 1997. [8] A. Stupakiewicz, A. Maziewski, K. Matlak, N. Spiridis, M. S´lęzak, T. S´lęzak, M. Zaja˛c, J. Korecki, Tailoring of the perpendicular magnetization component in ferromagnetic films on a vicinal substrate, Phys. Rev. Lett. 101 (2008) 217202. [9] A. Stupakiewicz, A. Kirilyuk, A. Fleurence, R. Gieniusz, T. Maroutian, P. Beauvillain, A. Maziewski, T. Rasing, Interface magnetic and optical anisotropy of ultrathin Co films grown on a vicinal Si substrate, Phys. Rev. B 80 (2009) 094423. [10] J. Kerr, On reflection of polarized light from the equatorial surface of a magnet, Philos. Mag. 3 (1887) 321. [11] H.F. Ding, S. P€utter, H.P. Oepen, J. Kirschner, Experimental method for separating longitudinal and polar Kerr signals, J. Magn. Magn. Mater. 212 (2000) 5–11. [12] P. Vavassori, Polarization modulation technique for magneto-optical quantitative vector magnetometry, Appl. Phys. Lett. 77 (2000) 1605. [13] A.V. Ulitovsky, I.M. Maianski, A. Avramenco, Method of Continuous Casting of Glass Coated Microwire, 1960. USSR Patent, No. 128427, Bulletin No. 10 14. [14] A. Chizhik, V. Zablotskii, A. Stupakiewicz, A. Dejneka, T. Polyakova, M. Tekielak, A. Maziewski, A. Zhukov, J. Gonzalez, Circular domains nucleation in magnetic microwires, Appl. Phys. Lett. 102 (2013) 202406. [15] R. Sch€afer, Investigation of domains and dynamics of domain walls by the magnetooptical kerr-effect, in: Handbook of Magnetism and Advanced Magnetic Materials, John Wiley & Sons, Ltd, USA, 2007. [16] A. Stupakiewicz, A. Chizhik, M. Tekielak, A. Zhukov, J. Gonzalez, A. Maziewski, Direct imaging of the magnetization reversal in microwires using all-MOKE microscopy, Rev. Sci. Instrum. 85 (2014) 103702. [17] A. Chizhik, A. Zhukov, J. Gonzalez, A. Stupakiewicz, Control of the domain wall motion in cylindrical magnetic wires, Appl. Phys. Lett. 109 (2016) 052405. [18] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, B. Van Waeyenberge, The design and verification of mumax3, AIP Adv. 4 (2014) 107133.

Further reading A. Zhukov, V. Zhukova, Magnetic Properties and Applications of Ferromagnetic Microwires with Amorphous and Nanocrystalline Structure, Nova Science Publishers, NY, USA, 2009.

On-time characterization of the dynamics of a single-domain wall in an amorphous microwire

19

E. Calle, Rafael P. del Real Institute of Materials Science of Madrid, CSIC, Madrid, Spain

19.1

Introduction

The study and control of the domain wall (DW) velocity is very relevant from the scientific point of view. Several key concepts have emerged from the motion of the DW, such as the DW mass, due to the inertia or excess of energy when moving with respect to the DW at rest [1–4], or the Walker breakdown due to the change of the internal structure of the DW as the velocity reaches a limit value [5]. Besides, when a constant driving magnetic field is applied, the DW reaches a constant velocity due to the viscous nature of the movement; there is a linear relationship between the applied magnetic field and the velocity far from the creep regime [6]. We can analyze the variations of the mobility with respect to temperature, stress, or frequency, obtaining information about the origin of the damping mechanism (eddy currents, existence of mobile defects, or magnetic relaxation due to the rotation of the spins that form the DW). Nonetheless, from the technological and economic point of view, the control of the DW velocity is very crucial. For soft magnetic materials, it is critical to reduce the DW velocity because the eddy current losses are proportional to the square of that velocity. On the other side, microwave applications of ferrites imply the resonance motion of the DWs. Recently the new paradigm of information storage and logic magnetic devices [7, 8] is based on the dynamic storage of the information in the sense that the DW moves along the magnetic material until it reaches the read/write head, avoiding the movement of this one and therefore the usual mechanical problems for the standard magnetic hard disks. For this purpose the highest DW velocity is needed to get the largest information speed. The first measurement of the DW velocity was made by Sixtus and Tonks in 1931 in FeNi cylinders [9]. They applied a local magnetic field close to one end to nucleate the DW and a constant magnetic field to move the DW along the wire. Around the wire, they placed two pickup coils separated by a known distance. The velocity of the DW can be calculated measuring the time interval between the e.m.f. generated when the DW crosses the pickup coils. This is a suitable method to measure the DW velocity in bulky samples but only gives us information about the average velocity. Another common method to measure Magnetic Nano- and Microwires. https://doi.org/10.1016/B978-0-08-102832-2.00019-0 Copyright © 2020 Elsevier Ltd. All rights reserved.

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the average velocity is the use of magneto-optical kerr effect (MOKE). This is an appropriate method for thin films [10, 11] or surface magnetization. A setup that combines both methods, Sixtus-Tonks and MOKE, was specially designed for amorphous microwires [12] obtaining information not only about the velocity of the DW but also about its shape. We can obtain the same information about the DW motion [13] using some other techniques like the magnetic force microscopy (MFM). To measure the time-resolved velocity of a DW, other methods must be utilized although they are only useful for specific kinds of samples or geometries. The usual method to measure the time-resolved DW velocity in magnetic nanostripes is based on the anisotropic magnetoresistance (AMR). When a DW enters the region between two measuring contacts, the nanostripe resistance will change compared with the resistance of the magnetically saturated state. The resistance difference depends on the number of DWs in the sample and the value of the transverse component of magnetization of the DWs integrated over its volume. As the difference in resistance is very small for a single DW, the process usually must be averaged thousands of times to be clearly detected [14]. The giant magnetoresistance method (GMR) is used in some multilayered samples. When the magnetization of the magnetic layers is parallel, the resistance takes the smallest value. As we apply a magnetic field in the opposite direction, a DW is nucleated in one of the magnetic layers, and the resistance starts increasing until the DW reaches the other end of the layer, the magnetization of the layers is antiparallel, and the resistance takes the maximum value [15, 16]. Besides AMR or GMR for the electrical detection of the position of the DW, inductive detection can be used as well [17]. Concerning the optical detection, stroboscopic and pulsed-laser MOKE shows picosecond temporal resolution [18, 19] although the spatial resolution is not so high as for time-resolved photoemission electron microscopy combined with X-ray magnetic circular dichroism (XMCD-PEEM) [20, 21]. The disadvantage of those techniques is their high cost. In this chapter, we will make use of the Matteucci effect to measure the timeresolved velocity of a single DW in an amorphous microwire. When a ferromagnetic sample with a nonzero magnetostriction, placed in a time-dependent longitudinal magnetic field, is twisted or shows a large enough helical anisotropy, a change in the azimuthal magnetization arises: consequently, an induced e.m.f, Γ, is generated between the ends of the sample. This is the so-called Matteucci effect [22–25]. This effect is related to the nondiagonal terms of the susceptibility tensor. The opposite effect (the sample is twisted when a helicoidal magnetic field is applied) is called the Wiedemann effect [26]. If the sample shows a uniform helical magnetization (Fig. 19.1A), due to the large torsion applied, when an ac axial magnetic field is applied, the value of the Matteucci voltage amplitude Γ 0 will be proportional to the radius of the wire r, its length l, the frequency ω, and amplitude H0 of the applied magnetic field and the susceptibility χ ϕz: Γ 0 ¼ μ0 rlωH0

dMϕ dHZ

(19.1)

On-time characterization of the dynamics of a single-domain wall

537

Fig. 19.1 (A) Sketch of a wire with uniform helicoidal magnetization. (B) Sketch of a wire with a DW separating two regions with opposite helicoidal magnetization.

If the magnetization occurs due to the motion of a single DW instead of magnetization rotation (as in Fig. 19.1A), Γ(t) is proportional to the azimuthal component of the magnetization Mϕ, the radius of the wire, and the velocity of the DW, v(t): Γ ðtÞ ¼ 2μ0 Mϕ rvðtÞ

(19.2)

For samples without perceptible changes both in Mϕ and/or r, the e.m.f. Γ will be proportional to the instantaneous velocity. Eq. (19.2) will be used to measure the timeresolved DW velocity for specific glass-coated amorphous microwires. Magnetic glass-coated amorphous microwires (MGCAM) are materials composed by an amorphous ferromagnetic metallic core with a radius from hundreds of nanometers to around 30 μm [27–30] and surrounded by a Pyrex shell with a thickness of several micrometers. They show a defined magnetic behavior thanks to the absence of magnetocrystalline anisotropy (the magnetic core is amorphous), a large shape anisotropy (the fabrication method can produce wires with hundreds of meters long), and a large or very small magnetoelastic anisotropy, depending on the composition and therefore on the value and sign of magnetostriction. For instance, CoFe-based alloys with nearly zero magnetostriction (and very small magnetoelastic anisotropy) are very suitable for applications in sensing devices as they show a giant magnetoimpedance effect and a nonhysteretic behavior [31]. Fe-based alloys show a high positive magnetostriction that, related to the distribution of stresses that appear during the fabrication, results in a bistable magnetization process consisting in the nucleation of a single DW in one end and its propagation along the entire wire length [32]. So far, Sixtus-Tonks experiment has been the technique used for MGCAM to characterize the DW dynamics through the average value of the DW velocity.

19.2

Sixtus and Tonks measurements

Fig. 19.2 shows the sketch of the Sixtus-Tonks setup that we have used to measure the average velocity of the DW. It consists of a long solenoid (40.8 cm) with a field constant of 2227.9 m1. Inside the solenoid, we place four pickup coils, named s1, s2, s3, and s4 with 2000 turns each and a length of 3 mm and separated by a known distance

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Magnetic Nano- and Microwires

Fig. 19.2 Sixtus-Tonks setup to measure the average velocity of a single DW.

(ds1–s2 ¼ 5 cm, ds3–s4 ¼ 5 cm, ds2–s3 ¼ 8.4 cm, and ds1–s4 ¼ 18.4 cm). With those four pickup coils, we can check the average velocity in three different intervals of the microwire. The pickup coils s1–s4 and s2–s3 are connected in series opposition in such a way that, when the domain wall crosses the coil in the same direction, the signal will be inverted in every pair of coils, that is, positive (negative if the DW moves in the opposite direction) in s1 and negative (positive) in s4. Besides, analyzing the order of the signals, we can confirm the area where the domain wall has been created or where it moves from. The microwire is placed into the primary coil in such a way that one of the ends is outside the solenoid and therefore the magnetic field is not large enough to nucleate the DW; we are forcing the DW to be created always in the same end of the microwire. A noninductive resistance R is connected in series with the primary solenoid to know the magnetic field applied to the microwire. A signal generator Tektronix AFG3252 was used to apply a 31-Hz squared wave signal. Positive or negative values of s1, s2, s3, and s4 are arbitrary (it depends on the way to make the electrical connections), and the sign only changes with the DW movement direction or the character (tail-to-tail or head-to-head) of the DW. If the domain wall moves from one end of the wire to the other, s1 and s4 and s2 and s3 will be opposite. This is a way to check if we have created several domain walls due to defects [33, 34] in the wire or if it is a domain wall coming back from the end outside the solenoid when the magnetic field is inverted [35]. The electric signals from the noninductive resistance from coils s1–s4 and from coils s2–s3 are analyzed in a digital oscilloscope Tektronix model TDS3034B. Fig. 19.3 shows the signals obtained from the pickup coils when a positive (Fig. 19.3A) and a subsequent negative (Fig. 19.3B) magnetic field of 143 A/m are applied. The pickup coils connected in series opposition are graphed in the same color

On-time characterization of the dynamics of a single-domain wall 0.06

s3

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s1 –0.06 32,300

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32,500

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0.06

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s4 0.04

s2-s3 (V)

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539

s3 32,500

32,400

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Fig. 19.3 Pickup coil signals for (A) positive applied magnetic field and (B) negative applied magnetic field.

(s1 and s4 black and s2 and s3 red). We have used a microwire with composition Fe75Si10B15 (the magnetostriction of the sample is 35.106 p.p.m. [36], and its saturation polarization is 1.43 T), with a length of 43 cm, metallic diameter of 18.6 μm, and a Pyrex shell of 1.5 μm. When a positive field is applied (Fig. 19.3A), a DW is nucleated in the left end and moves through the sample until it stops out of the solenoid. This is confirmed taking into account the order of the pickup coil signals. The sketch in Fig. 19.3.a shows the magnetization reversal in the microwire; pale and dark gray indicate negative and positive magnetization, respectively, and the red arrow on

540

Magnetic Nano- and Microwires

the DW shows the direction of the DW movement. From the time between peaks of the s signals (Δt) and the distances between the coils (Δd), we can obtain the average velocity with the simple equation v ¼ Δd/Δt. The obtained velocities are vs1–s2 ¼ 1000 m/s, vs2–s3 ¼ 1047 m/s, and vs3–s4 ¼ 1120 m/s; vs1–s4 ¼ 1054 m/s, which obviously coincides with the average value of the other three calculated velocities. Therefore, from these figures, we can obtain the velocity, but we do not get any clue about the movement characteristics of the domain wall between coils. Another information that we could obtained from Sixtus-Tonks experiment is the length of the domain wall, analyzing the time duration of the peaks. From Fig. 19.3A, we got a DW length of around 4 cm, which is clearly overestimated due to the electromagnetic influence of the DW dynamics when approaching or moving away the coils [37–39]. In any case, if we take the DW length from the literature [38], the angle between the domain wall and the surface of the microwire is less than 0.5°. After the positive field, we applied a negative one (Fig. 19.3B). In that case, there exist two DWs in the microwire: the domain wall DW2 previously generated with the positive field (stopped out the solenoid) and a new one nucleated in the left end DW1. They move in opposite directions as it is indicated by the red arrows in the sketch of Fig. 19.3B. This fact is deduced from the temporal order of the s signals (s1, s4, s2, s3). However, it is not possible to know if DW1 is generated before DW2 starts moving. We only know with certainty that DW1 crosses s1 some microseconds before DW2 crosses s4.

19.3

Time-resolved measurement of the DW velocity

To get more information of the domain wall dynamics, we improved the setup shown in Fig. 19.2 to measure the Matteucci effect as well [40]. For this purpose, besides the previous Sixtus-Tonks setup, we have connected both ends of the microwire to a preamplifier with a gain factor of 500 (the electrical contacts were made using silver paint after mechanical removing the Pyrex coating), and we have visualized this signal in the oscilloscope together with the signals from the four pickup coils and the signal from the noninductive resistance connected in series with the primary coil. Fig. 19.4 shows this improved setup. Fig. 19.5 shows the voltage from Γ and s1–s4 and s2–s3 when a positive field of 143 A/m is applied to the microwire. The domain wall is generated and starts moving when the magnetic field created by the primary coil has reached a constant value (that field is not shown in the figure), that is, the nucleation of the domain wall occurs after the relaxation time of the primary coil and the variations of the velocity cannot be associated with temporal changes of the applied field. As it was mentioned earlier, the applied field is a square wave signal. The change from positive (negative) to negative (positive) field generates an abrupt peak in Γ; this peak is not shown in Fig. 19.5 because it is previous to the movement or generation of the DWs and it does not provide any magnetic information. Although it is not observed in Fig. 19.5, previous measurements using this setup have shown [40] that the pickup coils could affect the movement of the DW across

On-time characterization of the dynamics of a single-domain wall

541

Fig. 19.4 Sketch of the setup to measure Matteucci effect and Sixtus-Tonks experiment.

Fig. 19.5 Electric signals from pickup coils s1, s2, s3, and s4 and Matteucci e.m.f. for an applied magnetic field H ¼ 143 A/m.

the microwire. The pickup coils could create a magnetic field opposite to the driving one, and they could brake the DW. The reduction of the DW velocity while crossing the pickup coil can reach 20% of the average velocity, and therefore the use of the Sixtus-Tonks setup can underestimate, in some cases, the DW velocity. From the value of the average domain wall velocity obtained using the temporal distance between s1 and s4 (the more distant pick up coils), vs1–s4 ¼ 1054 m/s, and the average value of the voltage from the preamplifier when the domain wall is between s1 and s4, < Γ >, we can calculate the ratio between the voltage from the preamplifier and the domain wall velocity. This value is 2426 m/(Vs). From here,

542

Magnetic Nano- and Microwires

using this constant (notice that the gain factor of the amplifier is 500) and Eq. (19.2), we have obtained the average value of the azimuthal magnetization due to the fabrication process, μ0Mϕ ¼ 2.2  102 T. As μ0Ms ¼ 1.43 T, the twisting angle is around 0.9°. Fig. 19.6 shows the time-resolved velocity of the domain wall. The domain wall dynamics can be separated in three different regimes: (I) The domain wall is generated, and it starts moving at increasing velocity until the equilibrium in the viscous regime is reached. This process takes around 10 μs. (II) The domain wall moves through the microwire with a velocity closed to the equilibrium value. The variations can come from different origins that will be analyzed later, such as the influence of the Sixtus-Tonks pickup coils, variations on the diameter of the metallic part or the Pyrex shell, local variations of the magnetization of the amorphous wire, or local defects. (III) The domain wall moves out of the solenoid, the magnetic field decreases dramatically, and due to the viscosity of the movement related with eddy currents or spin relaxation, finally, the domain wall stops. This process takes around 80 μs.

To study the dynamics of this single-domain wall when a magnetic field H is applied, we are going to use the well-known differential equation where x indicates the position of the domain wall in the axis of a microwire with cross section S: m

d2 x dx + β + kx ¼ 2μ0 MS HS 2 dt dt

(19.3)

In this equation, we suppose that the domain wall behaves like an object with a punctual mass m, a damping coefficient β, and a restoring constant k. The domain wall is

Fig. 19.6 Three regions in the time-resolved DW velocity that can be associated to the acceleration (I), propagation (II), and final braking of the DW when it leaves the primary coil.

On-time characterization of the dynamics of a single-domain wall

543

not moving around an equilibrium position, but it is forced, due to the applied magnetic field, to move from one end through the microwire. Therefore, in practice, this restoring force can be considered a kind of threshold field: a magnetic field below that value it is not strong enough to move the domain wall due to the magnetic friction with any kind of defects m

d2 x dx + β ¼ 2μ0 MS ðH  Hfr ÞS dt2 dt

(19.4)

where Hfr is the friction field, defined as the minimum value of applied magnetic field capable to move completely the domain wall already generated from one end to the other. We can simplify the equation as m

dv + βv ¼ 2μ0 MS ðH  Hfr ÞS dt

(19.5)

where v is the domain wall velocity. When the domain wall reaches the equilibrium, the velocity will be v ¼ ð2μ0 MS =βÞðH  Hfr ÞS

(19.6)

The magnetic field generated with the primary coil has the following theoretical expression: 2 H¼

3

7 NI 6 x + w=2 x  w=2 6rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ffi  r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5  2 2l 4  w2 w + a2 + a2 x+ x 2 2

(19.7)

where N is the number of turns, I is the current intensity, w the length of the solenoid, and a its radius. It is worth noticing that we use the expression earlier (instead of the simple H ¼ NI/l) because the domain moves even in a region of the microwire out of the solenoid (region III), therefore under the influence of a decreasing magnetic field. The center of the x coordinate is at the center of the solenoid. As we know the equilibrium velocity (the average velocity obtained from region II in Fig. 19.6), we can obtain the value of β from Eq. (19.6) if we measure Hfr. The value of the friction field Hfr can be obtained experimentally in the following way [40]. We first apply a pulsed magnetic field with time duration enough to create the domain wall. Then the magnetic field goes to zero, and afterward, we apply a smaller constant magnetic field to move the DW once it has been created. It is shown in Fig. 19.7 for nucleation field of 115 A/m and the subsequent field of 51.2 A/m. It is worth noticing that, for this field, the DW crosses s1, s2, and s3 but it stops before crossing s4. We obtain that the minimum value to start moving the domain wall was 18 A/m and the value necessary to move the domain wall through the whole solenoid was 60.6 A/m. As friction field, we have used the average of those two values < Hfr >, that is, 39.3 A/m.

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Magnetic Nano- and Microwires

0.06 0.04 100 0.02 0.00

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Hprop = 51.2 A/m

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0

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50

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Fig. 19.7 Applied magnetic field (blue), Γ (orange), s1–s4 (green), and s2–s3 (pink). The fields applied to nucleate and move the domain wall were 115 and 51.2 A/m, respectively.

Once we know the equilibrium velocity of a domain wall in the viscous regime and under an applied magnetic field of 143 A/m, as well as the value of the friction field Hfr, we can calculate the value of β from Eq. (19.6), being β ¼ 8.35  1011 Ns/m. Once we know β and Hfr we can solve the differential Eq. (19.5) by means of MATLAB R2010b, using the ode45 algorithm for the Runge-Kutta method. The DW mass, m ¼ 2.52  1016 kg, was obtained matching the theoretical and experimental DW velocity across the microwire. Fig. 19.8 shows both instantaneous velocities, calculated and experimental, for an applied field of 143 A/m. The theoretical fit 1500

Dw velocity (m/s)

1200 900 600 Dw velocity Hfr = 39.3 A/m; m = 0.252e-15 kg

300 0

0

100

200

300

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Time (ms)

Fig. 19.8 Experimental and theoretical curves of the time-resolved velocity.

500

On-time characterization of the dynamics of a single-domain wall

545

reproduces the experimental curve quite well. The nucleation and acceleration times show the dramatic influence of the mass on the DW dynamics. As a comparison, it is worth keeping in mind that the mass of a DW in a nanostripe is around seven orders of magnitude smaller [2] and the time needed to reach the steady velocity is only few nanoseconds [41] in comparison with the tens of microseconds necessary for the DW in the microwires. This could be a handicap of those magnetic materials (besides the big size) to be used for the information transmission. On the other hand, they show very large steady velocities (even thousands of m/s) compared with the DW velocities in nanodevices [42, 43]. Besides, from the theoretical fit, we can approximately know the distance from the primary coil end where the DW stops due to the decrease of the applied field intensity. It is 0.94 cm, and the applied field is 22.9 A/m, which could fit with the friction field of the DW in that point of the microwire. The DW starts decreasing its velocity before the end of the primary coil due to the reduction of the magnetic field in this area far from the center. The total distance with decreasing velocity is 3.3 cm. The theoretical curve in region II of Fig. 19.8 shows a constant velocity because it is assumed that the magnetic and geometric properties are constant all along the microwire. However, the experimental values present spatial variations. In contrast to regions I and III (where Γ varies mainly due to DW velocity), the variations in Γ (Eq. 19.2) in region II (Fig. 19.6) can be due to r, Mϕ, or the DW velocity. To check the influence of the diameter, the metallic core of the microwire was measured along the wire with an optical microscope. Fig. 19.8 shows that the maximum radius should be twice the minimum. Some punctual inhomogeneities were found, but they did not match the large variations in Γ. It permits us to discard the inhomogeneity in diameter as the main cause of the fluctuations in Γ. The variations in the DW velocity are related to β through Eq. (19.6). The damping experienced by the DW during propagation is the sum of some different contributions [44]. One is due to eddy currents, which depend on geometrical factors and the resistivity of the material. Therefore variations in the eddy current damping could be related again to a nonhomogeneous diameter of the microwire and to spatial variations in resistivity [45]. However, a simple calculus gives us the theoretical DW velocities one order of magnitude larger than the experimental ones [46] when using the eddy currents as the source of the DW damping, meaning that its contribution can be neglected in microwires due to its high resistivity and small diameter. Another contribution to β is associated with mobile defects existing in the amorphous matrix that can interact, hindering the DW during its propagation if its relaxation frequency is in the range of frequency of the applied magnetic field. Several studies have been done about the dependence of the damping mechanism on temperature, frequency, and tensile stress [44, 47, 48]. In our measurements, the frequency of the applied magnetic field was 31 Hz, which is a low frequency, so could be expected that the defects had enough time to relax and therefore to contribute to the damping of the DW via the structural relaxation. Some spatial variations in the damping coefficient related to structural relaxation could then occur as a consequence of a different concentration of mobile defects along the length of the wire. We have not, however,

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measured differences in Γ as a function of the frequency of the applied magnetic field in the 1 Hz to 6 kHz range, which means the DW damping cannot be due to structural relaxation. Another term in the damping coefficient is related to the magnetic relaxation due to the rotation of the spins that form the DW. In this case, β ∝ (K)1/2 where K is the magnetic anisotropy of the microwire [49]. Due to the fabrication process, small local variations of the glass coating’s thickness could produce variations in the stress applied and therefore in the magnetoelastic anisotropy. From Eq. (19.2), it is difficult to obtain detailed information because K somehow affects both Mϕ and v (through β and the pinning field Hfr). As a first approach, we can assume that the DW velocity is constant in region II. Fig. 19.9 shows the variation of the azimuthal magnetic polarization of the microwire in this region under this assumption. From these values, we obtain that the twisting angle could vary from 1.1° to 2.2°. On the other hand, some studies on the distribution and role of defects in amorphous bistable microwires have been performed, in which the authors have studied the distribution of the local nucleation fields, obtaining a shape similar to that of Fig. 19.9 [50]. The advantage of the Matteucci effect is that it is possible to instantaneously obtain the information of the whole wire. Fig. 19.10 shows Γ, s1–s4, and s2–s3 when a subsequent negative magnetic field is applied (H ¼  143 A/m). It also shows a sketch of the DWs into the sample. Comparing with a positive applied field (Fig. 19.6), the domain wall dynamics can be separated in four different regimes: (I) The domain wall DW2, which was stopped out of the solenoid starts moving leftward, toward pickup coil s4. (II) A new domain wall DW1 nucleates at the left end of the microwire. The nucleation and acceleration processes until it reaches the steady velocity takes around 12 μs.

Fig. 19.9 Longitudinal variations of Mϕ assuming a constant DW velocity during the propagation of the DW.

On-time characterization of the dynamics of a single-domain wall

547

Fig. 19.10 Signals from pickup coils s1, s2, s3, and s4 and Matteucci e.m.f. for an applied magnetic field H ¼  143 A/m.

(III) Both DWs move in opposite directions. DW1 crosses s1 and s2 and DW2 s4 and s3. (IV) Just after DW2 crosses s3, the DWs collide and annihilate. This process takes around 15 μs, being quite similar to the nucleation.

19.4

Braking and trapping a single-domain wall

In the previous section, we have studied the time-resolved dynamics of a singledomain wall moving through the wire when a constant magnetic field is applied along the wire. We have measured the local variations of the velocity due to the interaction of the domain wall with the local defects, a characteristic feature of the amorphous magnetic materials. In this section, we are going to analyze the domain wall dynamics related to the application of a local antiparallel magnetic field (with respect to the primary solenoid HPrim) with a maximum intensity HLocal at the center of the local coil. For this purpose a small coil with 90 turns and a length of 4 mm was wounded around a small capillary in which the microwire is inserted. This local coil was placed at the center of the primary coil. A Tektronix AFG3252 arbitrary function generator with two output channels was used to apply 31-Hz square waveforms with tunable amplitudes and a phase difference of 180° between the primary and the local coil. The sum of those fields will produce the nucleation and propagation of the DW, followed by the braking and eventually trapping of the DW due to the decrease of the field intensity

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around the local coil. Noninductive resistors (3 Ω) were connected in series with the primary and the local coil, respectively, to measure both magnetic fields. The magnetic field applied with the primary solenoid was 117.34 A/m, large enough to nucleate and move the domain wall from the left end of the microwire. The profile of the total magnetic field, HTotal, through the microwire is sketched in Fig. 19.11 for a local maximum magnetic field of 86.64 A/m and a field at the center of the local coil of 30.70 A/m. The x coordinate is centered at the local coil. It is worth noticing that the spatial influence of this coil covers around 10 mm. From now on, we call total local field, HTL, to the value of the magnetic field at the center of the local coil. This value coincides with the minimum field created by the system (primary plus local coils). As in the previous section, electrical contacts were made with silver paint at the ends of the microwire after mechanical removal of the glass coat, and Γ was amplified with a Stanford Research System Model SR560 amplifier using a gain factor of 500. The voltages in both resistors and Γ were measured with a Tektronix TDS3034B four-channel oscilloscope, ensuring that the depinning and propagation of the DW occurred when the applied magnetic field had reached a constant value as in the previous section. The setup is sketched in Fig. 19.12. For this section, we have used a microwire slightly different to the previous one but with the same composition (Fe75Si10B15), around 18.7 μm of average diameter of the metallic nucleus, 1.15 μm of glass cover (total diameter of 21 μm), and length of 43.3 cm. Fig. 19.13 shows the time-resolved velocity of the DW for a maximum local field of 69.44 A/m. Before and after the local coil influence, DW velocity is the same irrespective of the local magnetic field, that is, the domain wall is created and moves

Hprim = 117.34 A/m

Hprim – Hlocal = 30.7 A/m

120

Magnetic field (A/m)

100 80 60 40 20 0 –20

–10

0

10

20

x (mm)

Fig. 19.11 Profile of the total applied magnetic field. HP ¼ 117.34 A/m and HL ¼ 86.64 A/m.

On-time characterization of the dynamics of a single-domain wall

549

Fig. 19.12 Setup to measure the stopping dynamics of a DW.

Fig. 19.13 Time-resolved velocity of a DW for a total local applied field of 47.9 A/m.

from the left end showing variations in the velocity due to the local defects, and as the DW interacts with the local field, its velocity decreases until it reaches a minimum value [3] and moves again at quasi constant velocity until it reaches the end of the primary coil where, due to the magnetic viscosity and the absence of magnetic field, it stops somewhere outside the solenoid. Fig. 19.14 shows the time-resolved velocity of the domain wall around the local coil when it is braked with different local magnetic fields. It is worth noticing that the deceleration branch does not depend on the local fields—only the minimum velocity does it; as the total magnetic field is always positive, there is a decrease of the force on the domain wall, and it moves slower due to the viscosity term related with β of

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Fig. 19.14 Time-resolved velocity of a DW under the influence of different stopping local magnetic fields.

Eq. (19.5), but during this deceleration process, the DW does not reach the equilibrium velocity related to the viscous regime. Fig. 19.15 shows the distance traversed by the DW under the influence of the local coil. It was calculated by integration of the curves for different fields in Fig. 19.14. The integrated interval was from t ¼ 119 μs (where all the DW starts sensing the effect of the local coil) until the time the DW recovers the quasi static velocity (v ¼ 1200 m/s) under the primary field of 117.34 A/m. There is a reduction of the DW traverse

Total displacement (m)

0.020

0.018

0.016

0.014

0.012

30

40

50

60

70

80

90

100

Total local field (A/m)

Fig. 19.15 Traversed distance by the DW under the influence of the local coil.

On-time characterization of the dynamics of a single-domain wall

551

distance probably related to the spatial influence of the magnetic field created by the local coil: the larger the magnetic field created by the local coil (smaller total local field), the farer the DW will sense this magnetic field. A shrinkage of the DW for small total local field could appear according to some works regarding Fe-rich water quenched microwires [51], but it has not been observed. When the total local applied field (Happ  Hloc) coincides with the local friction field (the minimum field necessary to move the domain wall in the microwire in that point), the domain wall stops. This local friction field is around 30.7 A/m. However, it is not so simple. In the range 25–32 A/m, a stochastic process appears [52] as it is shown in Fig. 19.16. That figure shows the velocity of three different domain walls for the same total local applied magnetic field, Happ ¼ 117.34 A/m, Hloc ¼ 88.61 A/m, and therefore HTL ¼ 28.73 A/m; one domain wall crosses the local coil without stopping (red curve) whereas the other two are stopped during 30.8 μs (the green one) and during 218 μs (the blue curve). This is a proof of the stochastic behavior of the local friction or pinning field. To get more information, using the features of the oscilloscope, we made a sampling of 500 DWs for several applied local fields. Fig. 19.17 shows the average value of Γ, for different total local fields, and normalized to the value for HTL ¼ 33.3 A/m (the DW always crosses the local coil) at t ¼ 258 μs. This is somehow a measurement of the probability that the DW overcomes the pinning barrier after a certain time t. In this figure, we can observe that all the average signals before the influence of the local field are exactly the same for all the applied fields, and the DW braking starts at t ¼ 115 μs. As the total local field decreases, the average Γ amplitude decreases as well. That figure shows that, in all the cases, the maximum time that the domain wall is trapped by the defects is around 350 μs as no graph shows a detectable average for times above 500 μs: after this time the DW is considered definitively trapped.

Fig. 19.16 Time-resolved velocity for three different DWs under the same total local field (28.73 A/m).

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Magnetic Nano- and Microwires

/ (t = 258 ms)

Hprim = 117.34 A/m

Htotal local (A/m)

1.0

33.3 32.16 31.59 30.44 29.87 28.73 28.16 26.44 25.87 25.3

0.5

0.0

0

200

400

600

Time (ms)

Fig. 19.17 < Γ >/< Γ t¼258 μs > for different applied magnetic fields. Every curve is an average of 500 measurements. Red dot line shows t ¼ 258 μs.

Probability for DW escape (t = 258 ms)

Fig. 19.18 shows the probability of the domain wall to escape the pinning field for t ¼ 258 μs. The normalized amplitude at this time shown in Fig. 19.17 is proportional to the probability of the domain wall to cross the local field. We are aware that we are ignoring domain walls that are stopped for longer times. Above a total local field of 31.6 A/m, all the domain walls show the same behavior, and none of them are completely trapped, only braked. From 30.44 to 25.87 A/m the probability to be stopped during several microseconds increases until the applied field is 25.3 A/m when all the domain walls are trapped for at least 110 μs.

1.0 0.8 0.6 0.4 0.2 0.0 24

26

28

30

Htotal local (A/m)

Fig. 19.18 Escape probability of a DW versus HTL for t ¼ 258 μs.

32

34

On-time characterization of the dynamics of a single-domain wall

553

This pinning field could come from different kinds of defects associated to residual stresses, nonmagnetic or less magnetic inclusions [53, 54]. We can obtain a rough value of the activation energy of this pinning by means of an Arrhenius-like law [55]: Ep 1 ¼ f0 e kT τ

(19.8)

where τ is the waiting time before the barrier is overcome, f0 is a frequency constant ( 1012 Hz), Ep is the pinning energy barrier, and kT is the Boltzmann energy. If we assume a waiting time of 100 μs (from Fig. 19.18), then Ep  1019 J. From this figure, we can induce that the concept of local friction field is not static but dynamic. Again, for very low applied magnetic fields, the Sixtus-Tonks experiment can give us noncomplete information about the real velocity of the DW; when the applied field is close to the friction one, the Sixtus-Tonks method will give us a wide dispersion of velocities but no information about the real physical phenomena related with that dispersion.

19.5

Injection of domain walls

In the previous section, we have used a local coil to brake or even trap a domain wall that was generated at the end of the microwire by the primary or driving field. But when this local coil generates a magnetic field parallel to the driving one, it can nucleate a pair of domain walls moving at opposite directions [3]. For a driving field larger than the nucleation field, we will inject three DWs, one created at the end of the microwire and two DW twins (head-to-head and tail-to-tail) created at the center of the local coil. The sketch of the magnetic configuration is shown on top of Fig. 19.19. Black arrows indicate the direction of the domain magnetization and the green arrows the DW velocity direction. DW1 is the domain wall nucleated by the driving field at the end of the microwire, and DWL and DWR are the twin domain walls moving leftward and rightward, respectively. DWL will collapse with DW1 at a certain time, and DWR will move until it is out of the solenoid, and it stops due to the reduction of magnetic field. Fig. 19.19 shows the driving and local fields and Γ for a microwire with a metallic diameter of 20 μm, total diameter of 23 μm, and a length of 44.2 cm. The driving and local fields are 80 A/m and 872 A/m, respectively. DW1 reaches the equilibrium velocity for a time t  120 μs. The local field turns to positive (parallel) when t ¼ 215 μs and DWR and DWL are created, in such a way that the DWL hardly moves due to the collision with DW1. The twin DWs’ nucleation process takes 5 μs, and the time to collapse is 23 μs. Green color means a single DW in the microwire (either DWR or DW1) and the orange three DWs. It is worth noticing that the DW velocity of DWR1 and DWR is the same, that is, the DW generated at the end and the DWs generated by the local coil are identical and they show the same dynamic behavior. We can control the movement of the three DWs just shifting the application of the local field with respect to the driving field, as it is shown in Fig. 19.20. DW1 reaches the equilibrium velocity for a time t ¼ 120 μs. The local field is applied when

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Magnetic Nano- and Microwires

DW1

DWL

DWR

1000

0.6

100 Primary field (A/m) Local field (A/m) e.m.f (´500) (V)

800 50 600 400

0.4 0.3

0

DW1

0.5

0.2

DWR

0.1

–50

0.0

200 –100

–0.1 0

–150

200

–0.2 600

400 Time (ms)

Fig. 19.19 Up: Sketch of the magnetic configuration. Down: Γ for a primary field of 80 A/m and a parallel local field of 872 A/m applied at t ¼ 215 μs.

DW1

DWL

DWR

1000

0.6

100

0.5

800 50

0.4 Primary field (A/m) Local field (A/m) e.m.f (´500) (V)

600 0

DW1

DWR

400

0.2 0.1

–50

0.0

200 –100

–0.1 0

–150

0.3

200

–0.2 400

600

Time (ms)

Fig. 19.20 Up: Sketch of the magnetic configuration. Down: Γ for a primary field of 80 A/m and a parallel local field of 872 A/m applied at t ¼ 135 μs.

On-time characterization of the dynamics of a single-domain wall

555

t ¼ 135 μs. For 32 μs the three domain walls are moving in the microwire, two rightward and one leftward. After this time the DW1 and DWL collide and collapse. Again, DWR keeps moving rightward until it reaches the end of the driving coil, and it stops due to the absence of magnetic field.

19.6

Conclusions

Fe-rich glass-coated microwires are soft magnetic materials characterized by a bistable hysteresis loop in which magnetization reversal takes place by the depinning and propagation of a single-domain wall. This property makes them very convenient systems to study the processes of nucleation and propagation of domain walls. Furthermore, Fe-rich glass-coated microwires exhibit helical anisotropy as a consequence of the coupling between its high magnetostriction and the stress induced during its fabrication process. This helical anisotropy is responsible for the appearance of the Matteucci effect in the absence of torsional stress applied to these type of microwires. So far the study of the DW motion in amorphous glass-coated microwires had been based on the Sixtus-Tonks technique, and only information about the average velocity could be acquired. We could obtain the time-resolved velocity of the domain wall measuring simultaneously the Matteucci voltage induced at the ends of the wire and the average Sixtus-Tonks velocity. This has permitted us to distinguish the different steps involved in the magnetization reversal process: nucleation and acceleration of the domain wall from one end of the wire, propagation at an average velocity (with some deviations due to variations in the local anisotropy and defects), and braking of the DW when it leaves the primary solenoid. To control the DW motion, we have applied a local field antiparallel to the driving field, producing the braking of the DW around the local antiparallel field and even trapping it when the value of the total local applied field is close to the pinning field. It has been proved that this is a stochastic process. Applying a high local field parallel to the driving field, we can inject a pair of domain walls that move in opposite directions. In this case the measurement of the Matteucci effect has permitted us to determine the time needed for the nucleation and annihilation of pairs of domain walls. We have also found that every wall moving through the wire shows the same velocity irrespective of the origin of that domain wall.

Acknowledgment This work has been supported by the CSIC P.I.E. Project 201760E040.

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[42] J. Vogel, M. Bonfim, N. Rougemaille, O. Boulle, I.M. Miron, S. Auffret, B. Rodmacq, G. Gaudin, J.C. Cezar, F. Sirotti, S. Pizzini, Direct observation of massless domain wall dynamics in nanostripes with perpendicular magnetic anisotropy, Phys. Rev. Lett. 247202 (2012) 1–5. [43] J. Onufer, J. Ziman, M. Kladivova´, Unidirectional effect in domain wall propagation observed in bistable glass-coated microwire, J. Magn. Magn. Mater. 396 (2015) 313. [44] K. Richter, R. Varga, A. Zhukov, Influence of the magnetoelastic anisotropy on the domain wall dynamics in bistable amorphous wires, J. Phys. Condens. Matter 24 (2012) 296003. [45] D.-X. Chen, N.M. Dempsey, M. Va´zquez, A. Hernando, Propagating domain wall shape and dynamics in iron-rich amorphous wires, IEEE Trans. Magn. 31 (1) (1995) 781. [46] R.P. del Real, C. Prados, D.-X. Chen, A. Hernando, M. Va´zquez, Eddy current damping of planar domain wall in bistable amorphous wires, Appl. Phys. Lett. 63 (25) (1993) 3518. [47] G. Infante, R. Varga, G.A. Badini-Confalonieri, M. Va´zquez, Locally induced domain wall damping in a thin magnetic wire, Appl. Phys. Lett. 95 (2009) 012503. [48] R. Varga, G. Infante, G.A. Badini-Confalonieri, M. Va´zquez, Diffusion-damped domain wall dynamics, J. Phys. Conf. Ser. 200 (2010) 042026. [49] S. Chikazumi, Physics of Ferromagnetism, Claredon Press, Oxford, 1997. [50] A. Zhukov, J.M. Blanco, M. Ipatov, V. Rodionova, V. Zhukova, Magnetoelastic effects and distribution of defects in micrometric amorphous wires, IEEE Trans. Magn. 48 (4) (2012) 1324. [51] H. Garcı´a-Miquel, D.-X. Chen, M. Va´zquez, Domain wall propagation in bistable amorphous wires, J. Magn. Magn. Mater. 212 (2000) 101–106. [52] J. Akerman, M.l. Mun˜oz, M. Maicas, J.L. Prieto, Stochastic nature of the domain wall depinning in permalloy magnetic nanowires, Phys. Rev. B 82 (2010) 064426. [53] P. Gawronski, A. Zhukov, V. Zhukova, J. Gonza´lez, J.M. Blanco, K. Kulakowski, Distribution of switching field fluctuations in Fe-rich wires under tensile stress, Appl. Phys. Lett. 88 (2006) 152507. [54] R. Varga, K.L. Garcia, A. Zhukov, M. Vazquez, P. Vojtanik, Temperature dependence of the switching field and its distribution function in Fe-based bistable microwires, Appl. Phys. Lett. 83 (2003) 2620–2622. [55] P. Gaunt, The frequency constant for thermal activation of a ferromagnetic domain wall, J. Appl. Phys. 48 (1977) 3470.

Dynamical behavior of ferromagnetic nanowire arrays: From 1-D to 3-D

20

David Navas, Celia Sousa, Sergey Bunyaev, Gleb Kakazei Institute of Physics for Advanced Materials, Nanotechnology and Photonics (IFIMUP)/Departamento de Fı´sica e Astronomia, Universidade do Porto, Porto, Portugal

20.1

Introduction

The great advances in nanoscience and nanotechnology in the last few decades have led to the development of new systems where the physical properties of elements as well as their geometry can be controlled at the nanoscale. Patterned films are a wellknown example of such kind of systems [1]. They are mostly prepared by e-beam lithography technique that uses an electron beam to expose an electron-sensitive resist. The e-beam is controlled by a computer, allowing the preparation of any predefined patterns on the resist, which is subsequently developed to form the desired structure. One of the main advantages of this technique is its versatility for the fabrication of well-defined element and with arbitrary shapes. Moreover, it allows the fabrication of small devices, such as nonvolatile magnetoresistive magnetic random access memories or magnetic disks for storage application. However, the preparation of large-area nanoelement arrays is a time-consuming process. Therefore another approach that relays heavily on the self-organized growth of the desired nanostructures, particularly high-aspect ratio arrays of inorganic nanoparticles, have aroused great interest and shown many potentialities in several areas, ranging from spintronics to nanomedicine. Among the different approaches to the fabrication of these high-aspect ratio nanoparticles, porous anodic aluminum template-based synthetic methods have received considerable attention as they present controllable pore diameter and extremely narrow pore size distribution, together with a highly selfordered honeycomb lattice of nanopores. Especially, template electrodeposition has been proved to be an effective way to fabricate arrays of metallic magnetic nanowires (NWs). Although the static magnetic properties and the magnetization switching of such systems have been widely studied over the last 20 years, their related dynamic properties received much less attention until recently (see, for example, Chapters 15 and 23 in Ref. [2]). Therefore this chapter focuses on the dynamical properties of nanowire arrays and is organized as follows: Section 20.2 provides a brief description of the different experimental techniques, used to probe dynamic properties—ferromagnetic resonance (FMR) and Brillouin light scattering (BLS) spectroscopies, as well as time-resolved magneto-optical Kerr Magnetic Nano- and Microwires. https://doi.org/10.1016/B978-0-08-102832-2.00020-7 Copyright © 2020 Elsevier Ltd. All rights reserved.

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effect (TR-MOKE) microscopy. Section 20.3 is devoted to the fabrication process and static properties of different ferromagnetic nanowire arrays, while Section 20.4 focusses on their related high-frequency properties. Section 20.5 describes both the fabrication methodologies and static and dynamical properties of 3-D nanowire arrays. Finally, in Section 20.6 applications and future perspectives of ferromagnetic nanowire arrays are discussed.

20.2

High-frequency characterization techniques

One of the most powerful techniques to characterize magnetic nanostructures is FMR spectroscopy that provides information on the magnetization, magnetic anisotropy, dipolar interactions, relaxation of magnetization dynamics [3, 4], and on the structural quality and (in)homogeneity of the objects under study [5]. FMR spectrometers can be divided into two types: fixed frequency and broadband. On the other hand, BLS spectroscopy allows the local probing of magnetic dynamics in magnetic materials. Among its main advantages are the possibilities to determine the frequency and the wave vector of magnetic excitations, to measure two-dimensional spatial distributions of the intensity of magnetization dynamic and to analyze the temporal evolution of magnetic processes with a typical time resolution below 1 ns. Although all-optical TR-MOKE microscope have been mainly applied to ferri- and ferromagnetic thin films as well as in antiferromagnetic materials, this technique has been recently used to study the spin-wave modes in high-aspect ratio single-crystal Ni ferromagnetic nanowire arrays [6]. Therefore this section provides a brief description of the mentioned techniques.

20.2.1 Fixed frequency (cavity-based) FMR spectroscopy The FMR spectrometer is used for the detection of the microwave (MW) absorption in magnetic samples. In the simplest configuration a fixed frequency spectrometer consists of electromagnet, MW generator, resonant cavity, and MW detector, connected between each other using a transmission line with circulator (see Fig. 20.1). Usually the sample is fixed at the end of the sample holder and placed in the center of the cavity because in this location the magnetic component of the MW field has the maximum intensity. Static bias field, produced by the electromagnet, is needed to change the resonance conditions during the experiment and should be applied perpendicularly to the direction of MW field to maximize the intensity of the resonance absorption [7]. Orientation of the sample inside the cavity can vary from in-plane to perpendicular, in relation to the external magnetic field, to provide angular resonance measurements. The generator provides periodic microwave signal with a fixed frequency that corresponds to the operation frequency of the cavity (varies from 3 to 100 GHz, the most popular frequency is 10 GHz). The detector registers the microwave power absorption in the sample as a function of swept bias field. Then, FMR can be observed when the angular frequency of the exciting MW field is equal to the frequency of the magnetization precession in the ferromagnetic material under test.

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561

Microwave bridge Circulator Microwave generator

Microwave detector Hall probe

Modulation coils

Power supply

Lock-in

Cavity

Gauss meter

Computer

Fig. 20.1 Basic configuration of FMR spectrometer. The microwave bridge mainly consists of the microwave generator, circulator, and detector.

More sophisticated design, used in most of commercial FMR spectrometers, includes an additional pair of modulating coils supplying modulation field (typically few hundreds of Hz) inside the magnet and connected to a lock-in amplifier (see Fig. 20.1). This allows measuring directly the 1st derivative of the MW absorption spectrum, which increase drastically the sensitivity. It is worth to mention that the use of modern vector network analyzer (VNA) with wide dynamic range as the microwave generator together with a high-Q cavity can provide a good signal-to-noise ratio even without an external amplification [8]. Whereas the combination of a high-Q cavity and a lock-in-amplifier provides stateof-the-art sensitivity for fixed frequency measurements, the use of cavity-based FMR spectrometers for characterization of the material’s damping parameter is rather complicated and requires a set of several cavities with different operational frequencies. Moreover, measurement of bulky or large-area samples (e.g., thick nanowire matrixes) is also complicated due to dramatic decrease of the cavity Q-factor and the geometrical limitations of the maximum sample size.

20.2.2 Vector network analyzer-FMR spectroscopy As it follows from its title, this type of broadband FMR spectrometers uses as a VNA and benefits from its ability to record independently real and imaginary part of the standard scattering parameters of the network under test. Thus the VNA-FMR

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technique yields FMR parameters (most importantly the internal damping constant) from standard microwave S-parameter measurements versus frequency and field. In contrast with cavity-based spectrometers, commercial ready-to-use solutions based on the VNA-FMR technology are still absent on the market, and each research group realizes their own vision of this tool depending on their purposes and financial possibilities. In spite of a relatively simple basic design (see left panel of Fig. 20.2), the VNAFMR requires an accurate calibration. The proper subtraction of reference spectrum is crucially important to obtain accurate results. One of the best descriptions of this method, along with those formulas for the extraction of FMR data from measured complex S21 parameter, was given in Ref. [9]. More complete deembedding approach, which takes advantage of all four measured complex S-parameters of the background to construct a nearly symmetric model of the waveguide, is described in Ref. [10]. Typically a section of 50 Ω coplanar waveguide (CPW) or microstrip line plays the role of measurement cell. However, other types of transmission lines can be used in special cases. The CPW loaded with the sample is placed between the coils of the electromagnet or the Helmholtz rings and connected by means of coaxial connectors or microprobes to coaxial cables fed by the VNA. Two-port scheme is a standard methodology for such setup, but one-port reflection design is also possible [11]. It is crucial to have all components phase stable and absolutely nonmagnetic to avoid field-dependent artifacts and background in recorded spectra. Sample should be placed on the CPW in orthogonal superposition of the MW, the static magnetic fields generated by CPW, and the electromagnet consequently (see right panel of Fig. 20.2).

Sample Vector network analyzer

hrf Sample

Hext Ground

Power supply

metal

Signal

Ground Substrate

w b

h Helmholtz coils

hrf

CPW ground plane CPW center conductor

G

h

S

erf G

er

Fig. 20.2 Left panel: the schematic diagram of a vector network analyzer ferromagnetic resonance spectrometer. The sample is placed on the coplanar waveguide (CPW) structure, as indicated. The mutually perpendicular static applied field Hext and the microwave field h are in the plane of the film [9]. Right panel: the schematic diagram of a coplanar waveguide. The lower diagram shows the magnetic (full lines) and electric (dashed lines) field lines winding around the signal conductor.

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20.2.3 Brillouin light scattering spectroscopy The BLS technique is based on the spectral analysis of the laser light scattered by a magnetic sample. Due to the photon-magnon interaction, in addition to the photons at the laser frequency, the scattered light also contains photons at frequencies shifted by the frequencies of the magnons. Consequently, analyzing these additional spectral components, one can make a conclusion about the frequencies and intensities of the spin waves existing in the point of the sample where the probing light is focused. The spectral analysis is performed with a tandem Fabry–Perot interferometer operating in a multipass configuration. The interferometer is characterized by a very high contrast of 1010 and a frequency resolution better than 100 MHz. To achieve such performance the interferometer is actively stabilized by an external computer, which continuously aligns the Fabry-Perot etalons over 6 degrees of freedom. Typical BLS setup allows to measure stationary and nonstationary spin-wave phenomena in both transparent and nontransparent magnetic films and nanostructures. The propagation of spin waves can be followed at distances up to several tens of millimeters. The simplified layout of the measurement setup for transparent samples is shown in Fig. 20.3. Analyzing the intensity of light inelastically scattered by the magnetic excitations and in different spatial points of the magnetic film, two-dimensional maps showing the spatial distributions of the magnetization dynamics in the sample can be reconstructed. Also, using a pulsed excitation and a stroboscopic time-of-flight detection technique, the setup allows the recording of two-dimensional maps with temporal resolution better than 1 ns.

20.2.4 Time-resolved magneto-optical Kerr effect microscopy Laser-induced femtosecond magnetism or femtomagnetism ([13].) is a new research field in modern magnetism, whose origin is usually dated to 1996, when Beaurepaire et al. [14] showed that ultrafast 60 fs laser pulses could be used to demagnetize a Ni thin film and to study the evolution of the magnetization dynamics within the first few picoseconds after excitation. Since this important discovery, ultrafast pulsed lasers have provided access to magnetic dynamics in nanostructured systems and on unprecedented timescales [15, 16]. Recently an optical pump-probe system coupled with a microscope and a magnetooptical detection method was used to study the spin-wave modes in high-aspect ratio single-crystal ferromagnetic nanowire arrays [6] (see diagram of the experimental setup in Fig. 20.4A). While the sample magnetization dynamics was excited using a femtosecond laser pulse with high fluence (pump beam), their related behavior was determining through the magneto-optical Kerr effect (Kerr rotation or ellipticity) using a femtosecond laser pulse with low fluence (probe beam) and as a function of the time delay between both beams. The detection of the sample magnetic state was performed via the measurement of the MOKE signal obtained from the probe beam after reflection from the sample surface and using a balanced photodiode detector and a

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Electromagnet

S Sample

Single-frequency laser

L1

6-pass tandem Fabry-Perot Interferometer

Y

Single-photon detector

L2 Transmission analyzer

X

Motion controller

Oscilloscope

PC Control unit Data acquisition PC

N

Synchronization signal

Digital bus

Microwave generator Amplifier

Modulation signal

Pulse generator

Fig. 20.3 Schematic diagram of the Brillouin light scattering spectroscopy setup used by Prof. Demokritov’s group at University of Munster [12]. The sample, a transparent magnetic film, is mounted onto the holder, which allows positioning of the film in two dimensions (XY) with the accuracy of 1 μm. The holder is placed into an electromagnet. Excitation of waves of magnetization is realized by means of a microstrip antenna fed from the amplified microwave generator. Similar antenna is used to receive the waves after they propagated in the film for several millimeters. For space- and time-resolved detection of spin waves, the probing laser light is focused onto the film by the objective L1. The objective L2 collects the light passing through the film and sends it to the interferometer for analysis.

lock-in amplifier. For this particular case the Kerr ellipticity was analyzed as this effect is significant greater than the Kerr rotation for a material such as Ni. At zero delay between pump and probe beams, the Kerr ellipticity signal shows a sudden drop within the first 400 fs corresponding to an ultrafast demagnetization process followed by a fast recovery (remagnetization) within a few picoseconds (8 ps) (top panel of Fig. 20.4B). Afterward the magnetization precession appears as an oscillatory behavior of the Kerr ellipticity signal (middle panel of Fig. 20.4B). Finally a fast Fourier transform (FFT) was performed on the time-resolved Kerr ellipticity data to obtain the spin-wave spectra of the sample (bottom panel of Fig. 20.4B). This technique allows the observation of a discrete spin-wave spectrum formed by the presence of a uniform and few quantized spin-wave modes that can be tuned by varying the external magnetic field.

Quarter waveplate

B

Spectral filter MO

H

A–B A+B

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Pump+probe beams

MO

–10

Nanowires in templates

Time-resolve Kerr ellipticity data over short time scale showing ultrafast demagnetization and remagnetization

–2

0

2 4 Time (ps)

0.8

6

8

H = 1.0 kOe

0.4

Time-resolve Kerr ellipticity data over long time scale showing precession of magnetization

0.0 –0.4 –0.8

0

0.9

300

1

0.6

600 900 1200 1500 Time (ps) 2

4

0

10 Frequency (GHz)

FFT

Experimentally observed spin wave spectra in the nanowire at H = 1.0 kOe

3

0.3 0.0

(A)

Remagnetization

–5

A

Kerr ellipt. (mdeg)

Dt

0 Ultrafast demagnetization

Beam combiner

Kerr ellipt. (mdeg)

Probe beam

Power (a.u.)

Pump beam

20

(B)

Fig. 20.4 (A) Schematic showing the measurement geometry with the applied bias field H. The (A – B) signal corresponds to the Kerr ellipticity, and the (A + B) signal corresponds to the total reflectivity. Bottom panel: a magnified view of the overlap of pump and probe beams through the microscope objective onto the sample surface. (B) Top panel: typical time-resolved Kerr ellipticity data from a Ni nanowire array showing ultrafast demagnetization and fast remagnetization. Middle panel: time-resolved Kerr ellipticity data recorded over a long time. Bottom panel: spin-wave spectrum obtained using a fast Fourier transform of the time-resolved data presented in the middle panel [6].

566

20.3

Magnetic Nano- and Microwires

Nanowire arrays

20.3.1 Fabrication of ferromagnetic nanowire arrays Arrays of ferromagnetic nanowires and nanotubes have gained much importance in the last years based on the development of nanometric devices with potential industrial and technological applications. Nanowires have been successfully fabricated using both top-down and bottom-up approaches. The former approach requires the definition of patterns that can be prepared using nanolithographic techniques such as e-beam, X-ray, or focused ion beam lithographies. However, these techniques are expensive and have a low throughput so that new assembling methods are required. One of the most used procedures to control the shape and size of the nanoelements is the template-assisted method. In this approach a membrane, with micro- or manometer pores, acts as template. The most commonly used templates are diblock copolymers [17], track etched [18], and porous alumina membranes [19]. The desired material is growth in the channels or pores of the selected template allowing us to control both the shape and size of the nanostructures. Depending on the used method and experimental conditions, different geometries, such as nanowires or nanotubes, can be achieved. Electrodeposition, electroless plating, atomic layer deposition, chemical vapor deposition, sol-gel methods, wetting process, and high-pressure injection of a melted material are examples of available techniques to fill the template pores. Among these methods the electrodeposition technique presents several advantages, namely, the high deposition rates, its suitability for filling low- and high-aspect ratio pores, the geometry and material (composition) control, and a low fabrication cost [20]. In particular, nanowires are grown from the bottom to the top of the template, yielding a homogeneous replication of template architecture. The growth of metallic nanowires by electrodeposition using a porous template, such a mica template, was already demonstrated by Possin [21]. However, it was only in 1993 when Whitney et al. [22] prepared and studied ferromagnetic nanowires electrodeposited in polycarbonate membranes. Since these pioneering works, arrays of electrodeposited nanoelements have been prepared in different kinds of templates, mainly in polymeric etched ion-track and nanoporous anodic alumina templates.

20.3.1.1 Fabrication of ferromagnetic nanowire arrays in polymeric etched ion-track templates One of the most used templates for the fabrication of nanowires is ion-track etched polymers. The precursor template is usually polycarbonate, polyethylene terephthalate, or polyamide. The fabrication of etched ion-track membranes involves two intendent steps: (i) first the irradiation of the polymeric material with heavy ions that leads to the formation of latent tracks and (ii) then a selective etching is performed to dissolve the ion-track channels and forming the porous template. The control of both steps enables the fabrication of porous templates with the desired geometries, sizes, and aspect ratios.

Dynamical behavior of ferromagnetic nanowire arrays

567

Fig. 20.5 SEM images displaying the cross sections of the following membranes: (A) cylindrical channels in a polyimide template, (B) conical channels in a polyimide template, (C) cigar-shaped channels in a polyethylene terephthalate template, and (D) cylindrical channels with rhombohedral cross section in mica [23].

Compared with other available templates, such as diblock copolymer membranes or porous alumina, etched ion-track membranes (Fig. 20.5) offer the possibility to control the density and orientation of the nanochannels and choose the polymer material that together with the etching conditions determines the geometry of the channels (i.e., cylindrical, conical, and biconical). However, this type of membranes does not show any kind of organization, pores can be tilted with respect to the surface normal, and the pore diameter varies along the template depth. The electrodeposition of Sn, In, and Zn metallic nanowires in ion-track etched mica templates was reported in 1970 [21]. Since then and in particularly in the last two decades, a large variety of materials have been deposited, mainly in polymeric etched ion-track membranes [23–25]. In particular, ferromagnetic nanostructures, such as Ni [26], Co [27], and Fe [28] wires, were template assisted fabricated in polycarbonate etched ion-track membranes.

20.3.1.2 Fabrication of ferromagnetic nanowire arrays in alumina templates Porous anodic alumina templates are produced by a two-step anodic oxidation of Al substrates in an aqueous solution of acidic electrolytes, such as oxalic, sulfuric, and phosphoric acids [29–32]. The resulting membrane consists of closely packed hexagonal arrays of self-organized nanoporous. The pore diameter depends on the type of electrolyte used and the applied voltage; typical pore densities are of the order of 1011 pores/cm2. Highly ordered hexagonal lattices of porous anodic alumina templates have been successfully prepared by this two-step anodization process and structures with pore diameters in the 25–200 nm range, and interpore distances between 65 and 500 nm were obtained depending on the experimental conditions (Fig. 20.6). An important characteristic feature of these templates is its ability to tune the aspect ratio (length divided by diameter) of the resulting nanoelements, which is not so easily accessible with conventional lithographic techniques. In this respect, applications of the organized nanoporous anodic films as templates or replication masters might

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Magnetic Nano- and Microwires

Fig. 20.6 SEM images of nanoporous anodic alumina layers. Anodization was conducted in 0.3 M sulfuric acid at 10°C and under an applied voltage of 25 V (A), 0.3 M oxalic acid at 1°C and with an applied voltage of 40 V (B), and 10 wt% phosphoric acid at 3°C and with an applied voltage of 160 V (C). Pore opening was carried out in 5 wt% phosphoric acid at 30°C for 30 min (A), 35°C for 30 min (B), and 45°C for 30 min (C). The thickness of the oxide films was approximately 120 μm [33].

provide more realistic opportunities for creating nanostructured materials with tailored properties. Since 2000 the template-assisted fabrication of ferromagnetic nanowires in anodic alumina membranes has been reported by several groups [31, 34–36]. Fig. 20.7 shows an example of a Ni nanowire array electrodeposited in an anodic alumina template.

20.3.2 Static properties of ferromagnetic nanowire arrays The magnetic behavior of nanowire arrays has been widely investigated by several groups during the last two decades. As a number of reviews and key references focusing on this topic already exist in literature [2, 35, 37–42], the main aim of this section is to briefly describe the behavior of few classical examples. From a general point of view, the effective magnetic anisotropy of a nanowire array depends on the following three main contributions: -

The magnetocrystalline anisotropy term depends on the wires composition and the crystalline structure or texture. The shape anisotropy term depends on the wire geometry. The dipolar interaction between nanowires depends on the wires separation.

First, we consider the case of an isolated nanowire with high aspect ratio and without any magnetocrystalline contribution. This wire should present a behavior close to a cylinder with a square hysteresis loop and controlled by the wire geometry (shape anisotropy term). The magnetization is a bistable process with two equilibrium configurations (m+ and m) with respect to the wire longitudinal axis [43]. This behavior was observed in nanowire arrays with low diameter, high aspect ratio, and growth in low-porosity nanoporous templates. Fig. 20.8 shows the hysteresis loops of a Ni nanowire array with a diameter of 56 nm and porosity of 4%. As the magnetocrystalline anisotropy and the dipolar interaction terms are negligible for this situation, the hysteresis loops reproduce the behavior of a highly anisotropic system with the easy magnetization axis along the wire axis. The hysteresis loop shows a low saturation field and high remanence when the external field was applied parallel to the wire axis,

Dynamical behavior of ferromagnetic nanowire arrays

Al2O3

569

Nickel

nm 0 10

S S

S

N

N

S

N

N

S

S

N

30 nm

(A)

S

N

Si

500 nm

(B)

1

Fig. 20.8 Normalized hysteresis loops measured at room temperature with the field applied parallel (filled circles) and perpendicular to the wire longitudinal axis (open circles), for a Ni nanowires array with a diameter of 56 nm and porosity of 4% [44].

0.5 M/Ms

»700 nm

N

Fig. 20.7 (A) Diagram of a ferromagnetic nanowire array (Ni) embedded in a nanoporous anodic alumina template and with hexagonal arrangement. (B) Top-view SEM image of a nickel-filled alumina template with an interpore distance of 100 nm. The Ni wires have a diameter 30 nm and a length of 700 nm [37].

0

–0.5 –1 –10

–5

0 H (kOe)

5

10

whereas when the field was applied perpendicular to the wires, the hysteresis loop presents a high saturation field and very low remanence. As soon as nanowire arrays were grown in nanoporous templates with higher porosities, the dipolar interaction term should be considered. For example, the outof-plane hysteresis loops of Ni wire arrays (Fig. 20.9) show an increase of the

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DP = 55 nm

20 0 –20

B

–40

B

–5000

–2500

0 2500 Appied field (Oe)

Magnetization (memu/cm2)

(A)

10

5000

Magnetization (memu/cm2)

Magnetization (memu/cm2)

20 40

DP = 40 nm 10 0

B

–10

B –20 –5000

(B)

–2500

0 2500 Appied field (Oe)

5000

DP = 30 nm

5 0 –5

B

–10

B

(C)

–5000

–2500

2500 0 Appied field (Oe)

5000

Fig. 20.9 Out-of-plane and in-plane normalized hysteresis loops measured at room temperature for Ni nanowires arrays with 100 nm of interwire distance, 1 μm of wire length, and diameters of 55 (A), 40 (B), and 30 nm (C) [45].

saturation field and a reduction of the remanence values when the wire diameter was increased (higher porosity) [45]. Even in the sample with the higher porosity (Dp ¼ 55 nm), an almost isotropic behavior was observed (Fig. 20.9A). Therefore it was demonstrated that the dipolar coupling effects favor an easy magnetization axis perpendicular to the wires. Finally, we have considered the magnetocrystalline anisotropy contribution. For this purpose, we have focused our attention on a material such as Co because it has a strong magnetocrystalline anisotropy term when it is grown in the hcp crystal structure. Co nanowire arrays with a diameter of 30 nm and 3% porosity were fabricated with a preferential c-axis oriented perpendicular (parallel) to the wire longitudinal axis when they were electrodeposited using an electrolyte solution with pH 4.0 (pH 6.4) [46]. Hysteresis loops showed an increase in the coercive field for the nanowire growth from the higher pH solution, which favors a c-axis parallel to the wires. However, remanence kept close to one in both cases, in agreement with the properties expected for low diameter nanowires. Nanowires with larger diameters (70 nm) showed changes in both coercivity and remanence as a function of the pH. Depending on the pH value of the electrochemical solution, magnetocrystalline contribution favors an easy magnetization axis perpendicular (pH 4.0) or parallel to the wires (pH 6.4), as well as without any magnetocrystalline contribution (pH 2.0). The hysteresis loop squareness increased from its lowest value (pH 4.0) to its highest (pH 6.4).

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571

An intermediate behavior was observed for the array in which no magnetocrystalline contribution is present (pH 2.0). Then, the role of the magnetocrystalline contribution, which can be eliminated or set to reinforce or compete with the shape effects, was confirmed. Moreover, we should also remember that any assembly of nanowires shows a coercive field distribution, which arises from inhomogeneities in features such as wire length, interwire distance, misalignment of the wires, and diameter dispersion [47]. In this brief summary, the possibility of controlling the effective magnetic anisotropy in nanowire arrays by combining the magnetocrystalline anisotropy, the shape anisotropy, and the dipolar interaction effects was demonstrated. This tuning opens interesting options in the control and engineering of the magnetic properties of the nanowire arrays and their potential applicability in several technological and research fields.

20.4

High-frequency behavior of ferromagnetic nanowire arrays

From a technological point of view, ferromagnetic nanowires are potential candidates for the development of microwave devices with lower eddy currents than in bulk ferromagnetic metals. Moreover, as the nanowire array periodicity ( p) is shorter than the guided wavelength (λ) at the microwave frequencies ((d/λ) ≪ 103), this kind of systems behaves like an effective homogeneous medium with well-defined macroscopic constitutive parameters. FMR has been successfully used as a characterization technique, providing information on magnetization, dipolar interactions, and damping parameter, among other properties. Since 1999, when Goglio et al. [48] reported on the microwave properties of arrays of Co, Ni, and Py ferromagnetic nanowires embedded in nanoporous tracketched polymer membranes, several studies of the dynamical behavior of ferromagnetic nanowires have been published for different materials and as a function of the wire diameter, length, and interwire distance. Main published results have been summarized along the following sections.

20.4.1 Dynamical behavior of saturated isolated ferromagnetic nanowires From a general point of view, the resonance condition of an infinite cylinder (see Fig. 20.10) can be derived from the total free-energy density showing an effective uniaxial anisotropy field Heff parallel to the wire axis: E ¼ MSat Heff sin 2 ðθÞ  MSat H ½ sin ðθÞsin ðθH Þcos ðϕ  ϕH Þ + cos ðϕÞ cos ðϕH Þ

(20.1)

with (θ, ϕ) and (θH, ϕH) the polar and azimuthal angles of the sample magnetization and the external applied magnetic field (H), respectively. Heff is the effective anisotropy field that can be formed by different anisotropy contributions such as shape, magnetocrystalline, and/or a magnetoelastic anisotropy terms. In most of the experimental

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Fig. 20.10 The coordinate system used throughout this chapter in which the wire axis is placed parallel to the z-axis.

cases, electrodeposited nanowires show a polycrystalline structure, and the magnetocrystalline term can be neglected. Therefore the effective anisotropy field should be dominated by the shape anisotropy term, and it can be defined as Heff ¼ 2πMSat. A mathematically simpler way to determine the resonance frequency was derived from the energy method introduced by Smit and Beljers [49], and it is given by the double derivatives of the total free-energy density:  2 1=2  γ ∂ E ∂2 E ∂2 E  ω¼    MSat sin ðθ0 Þ ∂θ∂θ ∂ϕ∂ϕ ∂θ∂ϕ

(20.2) θ¼θ0 , ϕ¼ϕ0

This equation must be evaluated at the equilibrium angles, θ0 and ϕ0, of the magnetization that should be obtained from the free-energy setting the first derivatives to zero:  

 ∂E ¼0 ∂θ

(20.2a)

 ∂E ¼0 ∂ϕ

(20.2b)

If we focus our attention on the polar configuration (ϕ ¼ ϕH), the external magnetic field should be applied from θH ¼ 0° (parallel to the wire axis) to θH ¼ 90° (perpendicular to the wire axis), and the polar angular dependence of the resonance frequency (ω) can be simply described by  2   ω ¼ H cos ðθ0  θH Þ + Heff cos 2 ðθ0 Þ ½H cos ðθ0  θH Þ + Heff cos ð2θ0 Þ γ

(20.3)

Dynamical behavior of ferromagnetic nanowire arrays

573

where γ ¼ gμb/h is the gyromagnetic ratio in (MHz/Oe) with g the spectroscopy splitting factor, μb the Bohr magneton, and h the Planck’s constant. In the particular case when the external magnetic field (H) is applied along the wire axis (θH ¼ θ0 ¼ 0), Eq. (20.3) is reduced to ω ¼ ½H + Heff  γ

(20.4)

On the other hand, if the magnetic field (H) is applied perpendicular to the wire axis (θH ¼ θ0 ¼ 90), Eq. (20.3) transforms to  2 ω ¼ H ½H + Heff  γ

(20.5a)

for H larger than the effective anisotropy field (H > Heff > 0), and  2  2  ω ¼ Heff  H2 γ

(20.5b)

for H smaller than the effective anisotropy field (H < Heff). Although FMR analyses have been already performed to individual microwires ([2], Chapter 15), it was not experimentally applied to single nanowires yet. Instead the magnetic behavior of ferromagnetic nanowire array growth in low-porosity nanoporous templates can be considered as isolated nanowires. In particular, Ni nanowire arrays with 56 and 180 nm of diameter, over 20 μm in length, and 4%–5% porosity were fabricated and studied by Encinas-Oropesa et al. [44]. Fig. 20.11A shows that while the external magnetic field was large enough to saturate the sample in this lowporosity case, the experimental data are well reproduced by the theoretical uniform resonance mode of a single wire (Eqs. 20.4, 20.5a, 20.5b). However, this approach is not valid for arrays with higher porosities (Fig. 20.11B and C).

20.4.2 Dynamical behavior of saturated ferromagnetic nanowire arrays When nanowires were grown in templates with higher porosities, wires are strongly coupled by dipolar magnetostatic interactions, and the previous description is not valid anymore. Fig. 20.11B and C shows the field dependence resonance frequency spectra of Ni nanowire arrays with 95 and 250 nm of diameter and porosities of 25%–27% and 35%–38%, respectively. It was observed that when the porosity was increased, the parallel field dependence of the resonance frequency decreases, while the perpendicular resonance frequency increases. Encinas-Oropesa et al. [44] suggested that this behavior, related with the reduction of the effective uniaxial anisotropy field (Heff) with increasing porosities, is originated by the magnetostatic dipolar coupling between wires.

574

Magnetic Nano- and Microwires

F (GHz)

40 30

F (GHz)

(A)

20 10 0 40 30

(B)

20 10 0 40

F (GHz)

Fig. 20.11 Continuous lines are the theoretical resonance-field dispersion of isolated wires (according to Eqs. 20.4, 20.5a, 20.5b) with the field applied parallel (//) and perpendicular (?). Resonance frequencies measured in arrays of Ni nanowires with (A) a porosity of 4%–5% and diameters of 56 (circles) and 180 nm (diamonds), (B) a porosity of 25%–27% and diameters of 95 nm (circles), (C) a porosity of 35%–38% and diameters of 250 nm (circles), and when the external magnetic field was applied parallel (filled symbols) and perpendicular (open symbols) to the wire axis [44].

(C)

30 20 10 0 0

2

4

6

8

H (kOe)

Using a phenomenological mean-field approach [50], it was suggested [44, 51] that two extra terms should be added to the self-demagnetizing field of a single wire (2πMSat) when wires are placed closer or porosity (P) is increased. While the first contribution is originated by the magnetic charges on the cylindrical wire surfaces and produce a dipolar field, which opposes the self-demagnetization field (2πMSatP), the second one came from the charges at the top and bottom extremities (4πMSatP). Therefore the effective uniaxial anisotropy field can be described by Heff ¼ 2πMSat  2πMSat P  4πMSat P ¼ 2πMSat ð1  3PÞ

(20.6)

 pffiffiffi  Assuming a hexagonal array, P is defined as P ¼ πd 2 = 2 3D2 where d is the wire diameter and D is the distance between centers of neighbor nanowires. This equation satisfies the two extreme situations such as an isolated wire (P ! 0) and the continuous thin film (P ! 1). Encinas-Oropesa et al. [44] confirmed a good agreement between the experimental fit values of the dipolar interaction field (Hu ¼ 6πMSatP) as a function of porosity (P) and the dipolar interaction field according to Eq. (20.6) (Fig. 20.12). Since then, this approach has been successfully used for understanding the dynamical response of nanowire arrays with different geometries and several materials in the saturated state. Ramos et al. [52, 53] confirmed the dipolar interaction effects in Ni nanowires arrays prepared in alumina templates with 35 nm of diameter, 105 nm of interwire distance (porosity 10%), and lengths ranging between 0.75 and 2.8 μm.

Dynamical behavior of ferromagnetic nanowire arrays

5

575

Fig. 20.12 Dipolar interaction field as a function of the porosity (P). (A) Experimental fit data (dots) compared with the mean-field model Hu ¼ 6πMSatP (dashed line). (B) Calculated total dipolar interaction (open circles) for an array of square wires (inset), compared with the mean-field expression (dashed line) [44].

(A)

HU (kOe)

4 3 2 1

HDIP (kOe)

0 5

z

(B)

y

4

x

1

3

3

2

s

1 0 0

0.1

0.2

0.3

Porosity

0.4

0.5

The angular dependence of the FMR measurements (Fig. 20.13) was fitted according to Eq. (20.3) and taking into account the porosity effect (Eq. 20.6). In 2012, Sousa et al. [5] demonstrated that FMR measurements can be also used to determine the homogeneity, or the filling factor (fp), of the nanowire arrays. Ni nanowire arrays with diameter of 35 and 105 nm of interwire distance were electrodeposited inside nanoporous alumina templates ranging the Ni pore filling percentage (fp), or the percentage of pores that have been filled by the electrodeposition process, from 1% to 93% (see Fig. 20.14). The nanowire array with the largest fp (93%) shows a single resonance peak in the whole range of orientations of the applied magnetic field. This peak was in agreement with the theory that takes into account both the nanowire size and the distance between neighbor wires (Eqs. 20.3 and 20.6). However, arrays with lower fp show a significant broadening of the resonance linewidth for the field orientation along the sample normal and a quite intense second peak in lower fields for the in-plane orientation. Then, Eq. (20.6) is only valid for nanowire arrays where pores were perfectly filled (P ¼ 100%). Otherwise the pore filling fraction should be taken into account, and the effective uniaxial anisotropy field should be defined as [54]   Heff ¼ 2πMSat 1  3Pfp

(20.7)

Instead of the single resonance peak, two peaks were also reported in 2001 by Ebels et al. [55]. In this publication [55], low density Ni nanowire arrays with wire diameters

576

Magnetic Nano- and Microwires

5 2.8 mm 1.1 mm

Resonance field (kOe)

4

0.75 mm 3

2

1

0

0

30

60

90

120

150

180

Applied field angle (deg.)

Fig. 20.13 Symbols are the experimental data of the angular variation of the FMR signal in Ni nanowire arrays with different lengths. Continuous lines are the fits using Eqs. (20.3), (20.6) [53].

Fig. 20.14 SEM images show surface views of Ni nanowire arrays with fp of 1% (A) and 93% (B). Note that bright spots correspond to filled nanopores and dark spots to nonfilled ones.

ranging from 35 to 500 nm were studied. Although the FMR responses with a single resonance peak (Fig. 20.15) were quite similar for wires with the smaller (35 nm) and larger diameters (270 and 500 nm), an unexpected double-peak structure was observed for intermediate diameters (80 nm). This strange behavior could not be explained in the frame of the magnetostatic theory. A theoretical approach to calculate the dynamical response of isolated nanowires with circular cross section and including both exchange and dipolar contributions was suggested in 2001 by Arias and Mills [56]. Exchange dipole modes were introduced using a small pinning of the magnetic modes on the surfaces of nanowires by adding a

Dynamical behavior of ferromagnetic nanowire arrays

577

Absorption derivative

34.4 GHz 500 nm 270 nm 80 nm 35 nm

4

6

8 Ho (kOe)

10

12

Fig. 20.15 Q band (34.4 GHz) absorption derivative spectra at θ ¼ 0° for Ni nanowire arrays of different diameter (from 35 to 500 nm). Arrows indicate the approximate position of the resonance fields for the different absorption peaks [55].

surface anisotropy term, which can be originated if, for example, an oxide layer is presented. Fig. 20.16 shows the calculated FMR spectra of Ni nanowire arrays with different diameters and with a surface anisotropy value of Ks ¼ 0.85 erg/cm2. It was concluded that two peaks could be observed only when the exchange/dipole mode and the FMR frequencies are far away from each other. Otherwise, both peaks are mixed, and only one resonance peak is observed. This approach successfully reproduces the experimental data shown in Fig. 20.15. BLS measurements of Ni nanowire arrays [57], electrodeposited into hexagonally arranged alumina membranes, quantitatively validated the Arias’s and Mills’s theory [56]. Three spin-wave modes, whose frequencies increase with decreasing diameter, were revealed in zero applied magnetic fields (Fig. 20.17). The analysis, based on the dipole-exchange theory and under the assumption of low pinning on the wire surfaces, indicates that the observed discrete modes are a consequence of the quantization of the bulk spin waves due to confinement by the small cross section of the nanowires. In the absence of an external magnetic field, the frequency of the bulk standing modes can be described using the simple analytical form:  1 am 2 a m 2 ω¼ D + 4πMSat R R

 2

(20.8)

where D is the parameter to fit, R is the wire radius, and am values were defined as a1 ¼ 1.84, a2 ¼ 3.05 and a3 ¼ 4.20 for modes m ¼ 1, 2, and 3, respectively, in the case of small pinning. The corresponding numbers for large pinning are a1 ¼ 3.83, a2 ¼ 5.14, and a3 ¼ 6.38. A good agreement between the experimental data and calculations, using Eq. 20.8, was achieved under the assumption of low pinning on the wire surfaces (Fig. 20.17).

578

Fig. 20.16 Calculated ferromagnetic resonance spectra at θ ¼ 0° for Ni nanowires with (A) 35, (B) 80, and (C) 270 nm of diameter and using Ks ¼ 0.85 erg/cm2 [56].

Magnetic Nano- and Microwires

R = 35 nm 0.05 dG dH

0.00

Gauss–1 –0.05 Ho (kg)

(A)

6

7

8

0.04 dG dH

9

10

R = 80 nm

0.00

Gauss–1 –0.04

(B)

–0.08

Ho (kg) 6

7

8

9

10

0.3 R = 270 nm

0.2 dG dH Gauss–1

0.1 0.0 –0.1 –0.2

(C)

–0.3

Ho (kg) 6

7

8

9

10

Collective spin waves in high-density two-dimensional hexagonally ordered Fe48Co52 nanowire arrays was also observed using BLS measurements as a function of the interwire distance and the longitudinal magnetic applied field [58]. The FeCo wire arrays had a diameter of 20 nm, a length of 1.5 μm, and interwire distances ranging from 30 to 55 nm. Successful interpretation of the experimental data was achieved by application of the Arias-Mills theory [56]. They claimed that the origin of the collective spin waves arises from the dipolar coupling, and it is manifested as a reduction in the spin-wave frequency with decreasing the interwire spacing (see Fig. 20.18). Regarding the observation of multiresonance peaks, we should also remember the work of Li et al. [59] in which FMR angular dependence measurements at the frequency of 9.7 GHz (X band) in Ni nanowire arrays were performed. In this configuration, it was assumed that the wire array is not fully saturated and results showed two

Dynamical behavior of ferromagnetic nanowire arrays

579

Fig. 20.17 Variation of the bulk spin-wave frequencies as a function of the nanowire radius in zero applied magnetic field. The experimental data for each radius correspond to the three values of the azimuthal quantum number m. The solid curves represent the fits of the experimental data by the theory (Eq. 20.8). ♦ denotes the frequency of the bulk SW for bulk Ni [57].

Fig. 20.18 Magnetic field dependence of the frequencies of the spin wave with the lowest energy in Fe48Co52 nanowire arrays. Experimental data are denoted by symbols for the arrays with interwire separations of 30 nm (square), 40 nm (circle), 50 nm (triangle), and 55 nm (star). Theoretical collective spin-wave mode frequencies are represented by lines: for the arrays with interwire separations of 30 nm (dashed-dotted line), 40 nm (dashed line), 50 nm (dotted line), and 55 nm (solid line) [58].

580

Magnetic Nano- and Microwires

resonance peaks as soon as the external magnetic field orientation diverged far from the nanowire axis (0°). While one main peak (HFR), associated with the uniform precession mode, appeared at high applied magnetic fields, the second one (LFR) was observed at low applied magnetic fields, and it depends on the direction of microwave pumping field. Then, it was suggested that the low-field resonance peak could be originated by the complex ferromagnetic domain configuration observed in unsaturated nanowires.

20.4.3 Dynamical behavior of unsaturated ferromagnetic nanowire arrays As it was described in previous sections, several experimental and theoretical studies, focused on the dynamical behavior of ferromagnetic nanowire arrays near and above magnetization saturation, have been performed, and their related behavior are mainly understood as soon as the wires are uniformly magnetized. However, the magnetic response of nanowire arrays is less obvious when they are not fully saturated and populations of wires with up or down magnetization coexist. Nanowires magnetized in two different directions have different resonance frequencies. Then complex FMR spectra, showing two peaks related to the presence of these two populations of wires, should be observed. The investigation of unsaturated nanowire arrays are very attractive since it could allow us the development of microwave devices operating in low magnetic fields. As FMR spectra depends on the magnetic configuration through the net magnetization of the system and the effective dipolar coupling, FMR properties can be tuned by controlling the magnetic state. In 2007, Encinas et al. [60] confirmed that different stable magnetic states, controlling the relative fraction of wires with up or down magnetization orientation, could be achieved by applying different demagnetizing cycles to Co bistable nanowire arrays with 40 nm of diameter and low porosity (6%). This configuration could be useful for the development of field programmable microwave devices. Similar results were obtained by Kou et al. [61]. They reported that the FMR frequency can be tuned from 8.2 to 11.8 GHz by choosing different remanent states of a high-density NiFe nanowire array with 35 nm of diameter and 60 nm of interwire distance (Fig. 20.19). To interpret these experimental results, Kou et al. [61] developed a model suggesting that the frequency shift is due to the magnetostatic interaction among nanowires. In this model, two assumptions should be considered: Nanowires are single domain with uniform magnetization along the wire axis (up or down), and the resonance frequency (ω) can be estimated by the general Eq. (20.3) and considering a magnetostatic interaction. The second one is related with the fact that the number of nanowires with up (N ") and down (N #) magnetizations can be determined from the total magnetization according to ðN " N #Þ MðHÞ ¼ ðN " + N #Þ MSat

(20.9)

Dynamical behavior of ferromagnetic nanowire arrays

581

Fig. 20.19 Measured FMR frequency as a function of remanent magnetization normalized to the saturation magnetization (points). The solid line is the theoretical calculation using Eq. (20.12) [61].

Therefore the total magnetostatic field Hstatic along the wire axis at saturation and for an array with hexagonal arrangement can be described by Hstatic ¼ MSat  β ¼ MSat  6πr 2 L

n X n X i¼0 j¼1

1 ½ ði 2

+ ij + j2 ÞR2

+ L2 =43=2

(20.10)

where r, R, and L are the radious, interwire distance, and wire length, respectively. i and j are the number of nanowires and β is defined as a geometric factor. In particular, when the applied magnetic field is zero, Hstatic is Hstatic ð0Þ ¼ Mrem  β

(20.11)

where Mrem is the sample magnetization at remanence. Then the resonance frequency in zero external applied field can be simplified as

  ω 1 1 ¼ Hstatic ð0Þ + MSat ¼ Mrem β + MSat γ 2 2

(20.12)

The signs  are obtained from the fact that the sample is composed of two groups of wires with up and down magnetizations. The linear behavior of Eq. (20.12) properly fits the experimental data (Fig. 20.19) when β ¼ 3.79 was assumed. Moreover, the positive sign in Eq. (20.12) predicts a second resonance peak at higher frequencies. Double FMR resonance peaks were experimentally observed two years later by Carignan et al. [62] in 40-nm diameter CoFeB ferromagnetic nanowire dense arrays

582

Magnetic Nano- and Microwires

Fig. 20.20 Contour plot of the S21 parameter showing the frequency dependence of the resonance peaks when sweeping the external applied field H from 5.5 to 5.5 kOe along the wire axis. (A) Experiment. (B) Model described in reference Carignan et al. [62].

and when the external magnetic field was applied parallel to the wire axis. The frequency-dependent microwave response, S21 transmission parameter (Fig. 20.20), was experimentally and theoretically determined (1–40 GHz). At large negative fields, sample is saturated (majority of nanowires with down magnetization), and the dipolar interaction field is constant and antiparallel to the applied field. Then, one resonance peak is observed, which decreases linearly with the magnetic field. Between 2.5 and +2.5 kOe the antiparallel interaction field is gradually reduced, and an upward curvature of the resonance condition was observed. Around H ¼ 0 a second peak at higher frequencies (around 30 GHz) shows up and corresponding with the minority of nanowires with up magnetization. Above 2.5 kOe the sample moves to the saturation state, dominated by the population with up magnetization, and only one resonance peak is observed again. Assuming populations of wires with up and down magnetizations, the resonance condition can be obtained from solving two coupled Landau-Lifshitz-Gilbert equations [62]: 8"
0, favoring an easy axis parallel to the wires, the CoCu/Cu multilayered sample with l ¼ 36 nm (magnetic layer thickness) and g ¼ 20 nm (nonmagnetic layer thickness) has Heff  0, showing a nearly isotropic behavior. Therefore it was demonstrated that Heff can be reduced (increased) by using multilayers with thicker (thinnest) nonmagnetic spacer. Fig. 20.26B shows the effective

592

Magnetic Nano- and Microwires

6

Heff (kOe)

4 Experiment: CoCu nw-array g = 4 nm g = 20 nm

2

Theory: g = 4 nm g = 20 nm Cylinder

0

–2

(A)

0

1

2

3

rl = l/d

1.0 Heff > 0 0.8 Heff < 0 rl = l/d

0.6

0.4

0.2

0.0

(B)

Heff = 0

0

1

2

3

4

5

6

7

rg = g/d

Fig. 20.26 (A) Effective anisotropy field as a function of the magnetic layer aspect ratio rl for arrays of multilayered CoCu/Cu nanowires. In both sets of multilayered samples, nonmagnetic layer thickness was g ¼ 4 nm and g ¼ 20 nm. Experimental data (symbols) were fitted using Eq. (20.20). (B) Effective anisotropy diagram as a function of the aspect ratios of the magnetic rl ¼ l/d and the nonmagnetic rg ¼ g/d layers. The continuous line shows the values of rl and rg for which the effective anisotropy is zero. Above this curve the effective field is positive while below it is negative. The magnetization easy axis is represented by the white arrows [96].

anisotropy diagram for a single multilayer nanowire. Continuous line corresponds to those aspect ratios of both magnetic and nonmagnetic layers for which Heff ¼ 0. While above this line (Heff > 0), it is favored that the easy magnetization axis would be parallel to the wire axis and it would be perpendicular in the region below the line (Heff < 0). Moreover, it was observed that the magnetic layers are decoupled in wires with thick nonmagnetic elements and the easy magnetization axis is parallel or perpendicular to the cylinder axis when the aspect ratio is above (cylinder) or below

Dynamical behavior of ferromagnetic nanowire arrays

593

(disk) a critical value rl ¼ 0.906. This value corresponds to the one predicted for a single magnetic cylinder. The increase of the nonmagnetic aspect ratio introduces interlayer coupling, and the easy magnetization axis transition takes place at lower magnetic aspect ratio values. More recently the research community has focused on the development of multilayer ferromagnetic nanowire arrays for the development of spin-torque nanooscillators. Although a brief introduction is described here, a detailed description can be found in reference [85]. For this porpoise, it was considered a multilayer nanowire composed of a thicker magnetic layer with fixed magnetization (Mfixed), a nonmagnetic layer and a thinner magnetic layer (Mfree) that can oscillate freely (Fig. 20.27). A dc density, J, is spin polarized when it is passing through the fixed magnetic layer. Based on the conservation of the spin angular momentum, the component of the spin current transverse to Mfree is transferred to the free layer and generates a precession of the magnetization in the microwave frequency range. Considering the spin-torque effect, the Landau-Lifshitz-Gilbert equation of motion is given by ∂M free ¼ Teff + Tdamp + TST ∂t

(20.21)

where Teff ¼  μ0 jγ g j Mfree  H is the torque term from the dc and ac fields, Tdamp ¼  (αG/Mfree)(Mfree  ∂ Mfree/∂ t) is the damping term, and TST is the new spin-torque term: TST ¼

ζJ b fixed  M free M free  M Mfree

(20.22)

with J the current density; ζ ¼ (jγ j ħη)/(2 jej dfree), where e is the electron charge, dfree b fixed is a is the thickness of the free layer, and η is the spin-torque efficiency term. M unit vector in the direction of Mfixed. Solving Eq. (20.21), for arbitrary angles θH of the external dc magnetic field and θfixed of the fixed layer magnetization, leads to the resonance frequency (not valid for high current values):

Fig. 20.27 Diagram of the spintorque phenomenon in multilayer structures of alternating magnetic (FM) and nonmagnetic (No FM) metal layers. (A) Single spin-torque oscillator, constituted of a trilayer FM/No FM/FM structure. Mfree processes under the action of the input dc (J) due to spin angular moment conservation and around an axis, which is determined by Mfixed and Hext.

594

Magnetic Nano- and Microwires

ω2 ¼

i 1 h 2 ω ω + ð ζJM Þ 0x 0y fixed,z 1 + α2G

(20.23)

and a critical current Jc ¼

  αG ω0x + ω0y 2ζMfixed,z

(20.24)

with   ω0x ¼ ωH,Z + ωM Nyy  Nzz  Hdip, z

(20.25a)

  ω0y ¼ ωH,Z + ωM Nxx  Nzz  Hdip, z

(20.25b)

Mfixed,z ¼ sin ðθfixed Þ cos ðφfixed Þsin ðθM Þ + cos ðθfixed Þcos ðθM Þ

(20.25c)

where ωH, Z ¼ μ0 j γ g jH0 cos (θH) cos (θM); ωM ¼ μ0 jγ g jMfree; Hdip,z is the z-component of the dc dipolar field from the fixed layer; and Nxx, Nyy, and Nzz are the shape demagnetization factors of the free layer [97]. Eq. (20.23) transforms to the FMR frequency of the free layer in the absence of the spin-torque contribution when J ¼ 0, and it shows that the resonance frequency of the spin-torque oscillator can be controlled by the dc.

20.4.6 Dynamical behavior of gradient ferromagnetic nanowire arrays Carreon-Gonzalez et al. [98] introduced in 2011 an interesting approach, based on a simple and nonexpensive dip-coating technique, for the fabrication of nanowire arrays with a linear varying length profile (hf) and a lateral span (Δy) of the wire array (Fig. 20.28). A gradient nanowire array with a lateral span of Δy ¼ 500 μm and length L ¼ 1 cm was used to obtain the insertion losses S+ and S, respectively, for the forward (+) and backward () propagation directions at 4 and 8 kOe (Fig. 20.29A). A nonreciprocal behavior was observed, and the difference in absorption depths in both propagation directions led to an isolation per unit length ΔS/L ¼ (S+  S)/L that increased with the applied field (see Fig. 20.29B). The differential phase shift was defined as the difference between the phase shifts of the forward and backward propagation directions (Δφ ¼ φ+  φ). Fig. 20.29C shows the differential phase shift recorded at the same field values and after saturating the sample at 9 kOe. This figure shows that Δφ 6¼ 0 for frequencies in the vicinity and higher than the FMR frequency. Therefore these results suggest that the nanowire arrays with continuous variations of the wire length are interesting for the development of nonreciprocal microwave applications. In particular the fabricated nonreciprocal microstrip lines are demonstrated with the improvement of both isolation and differential phase.

Dynamical behavior of ferromagnetic nanowire arrays

595

Fig. 20.28 (A) Schematic view of the geometrical parameters of a nanowire array with length gradient profile. SEM images of (B) 820 μm wide, (C) 390 μm wide, and (D) 700 μm wide NW arrays with length gradient profile. (E) Nanowire array with a 110 μm wide sawlike wire length gradient profile, where the dotted line is a guide for the eye. In all figures, the scale bars represent 100 μm [98].

20.5

3-D nanowire arrays

Based on the development of new methodologies for the fabrication of threedimensional nanostructured templates, using different materials such as block copolymers [99, 100], nanoporous particle track-etched polycarbonate (PC) [23, 101], or anodic alumina (AAO) membranes [102, 103], 3-D interconnected nanowire [104–106] and nanotube [24] networks have been successfully fabricated. Three-dimensional interconnected nanowire networks have been already suggested for the development of novel and functional devices with application in multiple fields such as TiO2 in solar cells [107, 108], Ni/TiO2 in energy conversion/storage systems [103], polypyrrole nanotubes in selective chemiresistive sensors [109], polystyrene in photonic crystals [102], Au in optical metamaterials [100], Pt in electrocatalyst materials [101], or NiO [99] and V2O5 [110] in electrochromic supercapacitor devices.

20.5.1 Fabrication of 3-D ferromagnetic nanowire arrays Regarding this topic, we should remark that since 2015, the group of Prof. Piraux has performed a great effort in the development of 3-D multifunctional devices with controlled magnetic anisotropy and microwave absorption properties [24, 104–106]. In particular, polycarbonate (PC) templates with parallel and crossed pores, with

596

Magnetic Nano- and Microwires

S± /L (dB cm–1)

0 –10 –20

S–

–30

S+

4

–40

S+

(A)

–50 0 DS/L (dB cm–1)

S–

8

0 2

–5

4 6

–10 –15

(B)

8

Dj/L (degrees cm–1)

80

8

60

6

40

4

20

2

0

0 (+sat)

–20

(C) 0

10

20 f (GHz)

30

40

Fig. 20.29 Measured (A) S/L at 4 and 8 kOe, (B) ΔS/L at 0, 2, 4, 6, and 8 kOe, and (C) the corresponding Δφ/L for the nonreciprocal microstrip line device. The field values denoted as numbers in (A), (B), and (C) are given in kOe [98].

230 nm of pore diameter and 20% of surface porosity, were used for the fabrication of Ni and Ni/NiFe interconnected nanowire arrays [104]. The crossed nanoporous PC templates were prepared by performing a sequential two-step exposure of energetic heavy ions at different angles [101], such as 30° and +30° with respect to the normal of the PC film surface, respectively. The exposed areas were chemically etched in a 0.5M NaOH aqueous solution at 70°C to form 230-nm diameter nanopores and a volumetric porosity of about 20% [104]. This methodology allows achieving templates in which the angle between the porous and the template normal, the volumetric porosity, and the mean pore diameter (from few tens to a few hundreds of nanometer) can be well controlled. After the sputtering of a metallic cathode on one face of the template, the interconnected nanowire networks were grown by an electrodeposition process allowing the control of the material composition and nanostructuration, since

Dynamical behavior of ferromagnetic nanowire arrays

597

Fig. 20.30 SEM images of (A) the surface of a double-exposed polycarbonate porous membrane, (B–D) Ni interconnected nanowires at different magnifications, and (E) crosssectional image of the bimagnetic Ni/NiFe interconnected nanowire array, which was used for the X-ray compositional analysis shown in (F). It is possible to distinguish the gold electrode (purple), the Ni (yellow), and the NiFe (green) segments. The dashed line is just for reference and gives the upper part of the nanowire network [104].

nanowires and nanotubes, to core-shell or multilayered nanowires. Fig. 20.30 shows Ni and Ni/NiFe nanowire networks after dissolving the PC membrane using dichloromethane. Moreover, the bilayered Ni/NiFe interconnected nanowire array was confirmed by using X-ray compositional image where Ni is shown in yellow and NiFe alloy in green (Fig. 20.30F). Using the same methodology, 3-D alloyed and multilayered interconnected nanowire networks have been recently fabricated [106]. In particular, NixCo(1 x) alloys and Co (10 nm)/Cu (150 nm) multilayer interconnected nanowire arrays with 40 nm of diameter and 20% of volumetric porosity were growth by electrodeposition.

598

Magnetic Nano- and Microwires

20.5.2 Static properties of 3-D ferromagnetic nanowire arrays Fig. 20.31A and B shows the hysteresis loops of parallel and interconnected Ni nanowire arrays when the external magnetic field was applied out- and in-plane. As it was described along this chapter, the shape anisotropy term governs the magnetic response of the nanowires, and an easy magnetization axis along the wire axis (OOP) is observed for the parallel Ni nanowire array. However, hysteresis loops of interconnected Ni nanowires show a more complex behavior, and the direction of the easy magnetization axis is not well defined anymore. The same behavior was observed for parallel and interconnected Ni/NiFe nanowire arrays (Fig. 20.31C and D). It was claimed that the changes in the hysteresis loops and in the magnetic anisotropy are related with the effective orientation of the nanowires with respect to the external applied field and affecting both the shape anisotropy and the dipolar interaction terms. On the other hand, reference [106] showed the out-of-plane hysteresis loops of NixCo(1 x) interconnected nanowire networks (0  x  1). It was demonstrated that the magnetic anisotropy can be accurately controlled by changing the alloy composition. For nanowires with high Ni content, the magnetic behavior of the nanowire networks is controlled by the shape anisotropy. However, when the Ni content was decreased, nanowires grow in a hexagonal close-packed lattice with the crystallographic c-axis oriented perpendicular to the wire axis. Therefore a competition between magnetocrystalline anisotropy and shape anisotropy terms was suggested. Authors claimed that the effective magnetic anisotropy of Ni-rich wire networks is higher than that of Co-rich ones.

Fig. 20.31 Hysteresis loops of the (A) parallel Ni nanowire array, (B) interconnected Ni nanowire array, (C) parallel Ni/NiFe wires, and (D) interconnected Ni/NiFe nanowire array with the external field applied out of the plane (OOP) and in the plane (IP) [104].

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20.5.3 High-frequency behavior of 3-D ferromagnetic nanowire arrays To achieve a deeper understanding of their related magnetic responses, FMR measurements were performed at 21 GHz on parallel (PNWA) and interconnected (CNWA) Ni and Ni/NiFe nanowires arrays with the magnetic field applied along the out-of-plane direction (Fig. 20.32). While the parallel Ni nanowire array showed a typical FMR spectrum with a minimum corresponding to the resonance field, this minimum was shifted to higher fields for interconnected Ni nanowires. For parallel Ni/NiFe nanowires, two peaks were observed. While the higher field peak was related with Ni, NiFe produced the lower field peak. In contrast, interconnected Ni/NiFe nanowires showed a single peak with the resonance field close to the peak associated with the Ni section. This fact was associated with the shift to higher fields of both peaks and the smaller contribution of the NiFe section. For interconnected nanowire arrays and applying the external magnetic field in the out-of-plane direction, nanowires show a 30-degree tilt. Then, θ0 ¼ θH ¼ π/6, and Eq. (20.3) can be reduced to



 2 ω 1 3 Heff + H ¼ Heff + H γ 2 4

(20.26)

Using Eq. (20.3) the resonance field dispersion was fitted (Fig. 20.33). An effective field of 1.2 kOe was extracted from the fit of the Ni parallel nanowire array. This value

Fig. 20.32 FMR spectra of parallel (PNWA) and interconnected (CNWA) nanowire arrays of (A) Ni and (B) Ni/NiFe and measured at 21 GHz [104].

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Fig. 20.33 Dispersion relation measured in parallel (PNWA) and interconnected (CNWA) arrays of (A) Ni and (B) Ni/NiFe nanowires. The dashed lines correspond to the fits to Eqs. (20.3), (20.26) for parallel and interconnected wires, respectively. The continuous line corresponds to the dispersion relation calculated using Eq. (20.26) and with the effective field of the parallel Ni nanowire array. It is noted that NiFe (Ni) parallel wire array corresponds to the low (high) field FMR peak shown in Fig. 20.32B [104].

is in agreement with the theoretical value of 1.22 kOe for a Ni nanowire array with 20% of porosity. On the other hand the dispersion relation of the Ni interconnected nanowire array was fitted using Eq. (20.26), and an effective field of 0.9 kOe was obtained. The observed changes of the effective fields for parallel and interconnected wires were associated with the different values of the dipolar interactions. Then, it was suggested that the contribution of the exchange and dipolar interactions is stronger in interconnected nanowire arrays. Regarding the Ni/NiFe nanowire arrays, the two FMR peaks, observed in the parallel array (Fig. 20.32B) and associated with the Ni and Py segments, were also fitted using Eq. (20.3) (Fig. 20.33B). While the Ni segment showed an effective field of 1.2 kOe, in agreement with the value obtained for the Ni parallel nanowire array, NiFe segment showed 4.4 kOe. On the other hand, Eq. (20.26) was used to fit the dispersion relation of the Ni/NiFe interconnected nanowire array, and an effective

Dynamical behavior of ferromagnetic nanowire arrays

601

field of 1.34 kOe was found. This value is slightly larger than the one found for the Ni interconnected nanowire array (0.9 kOe) and resulting from the presence of NiFe segment. FMR measurements were also performed on NiCo alloy interconnected nanowires [106]. In agreement with the hysteresis loops, the Co-rich nanowire networks with magnetocrystalline anisotropy contribution showed lower resonance frequencies than the Ni-rich ones. Experimental data were analyzed quantitatively by determining their corresponding effective anisotropy field, Heff, and using the approach suggested by C^amara Santa Clara Gomes et al. [105]. As the orientation of the wires were distributed along both the x- and y-axes (in-plane axis) and interconnections between wires took place randomly, the FMR condition for an array of wires tilted to an angle θ0 with respect to the out-of-plane direction was considered (Eq. 20.3). The effective field (Heff) can be defined as Heff ¼ Hsh for wires with only shape anisotropy contribution, while Heff ¼ Hsh  Hcry for wires with both shape and magnetocrystalline anisotropy contributions. The sign + () is related with the fact that the c-axis would be oriented in the direction parallel (perpendicular) to the wire axis. Since measurements were performed at saturation, θ0 ¼ θH, and this angle lies in the range  θ  θ0  θ with θ being the maximum angle of all possible orientations of the wires. Due to the sample geometry, it was expected that negative angles, in the range  θ  θ0 < 0, provided FMR contributions that were equivalent to those for positive ones. Under this assumption, Eq. (20.3) over the range 0  θ 0  θ can be average by 0 1  2 Zθ ω 1 3 ¼ H 2 + Heff H @1 + cos ð2θ0 Þdθ0 A γ 2 θ 0 2 3 Zθ 1 2 4 1 + Heff 1 + ð2 cos 2θ0 + cos 4θ0 Þdθ0 5 4 θ 0

2 ¼ H 2 + C1 Heff H + C2 Heff

(20.27)

where constants C1 and C2 are   1 3 1 + sin 2θ C1 ¼ 2 2θ

(20.28a)

  1 1 1 1 + sin 2θ + sin 4θ C2 ¼ 4 θ 4θ

(20.28b)

Using Eq. (20.27) to fit the experimental frequency dispersion relation and since the fitting curve was a straight line, Heff can be determined from the extrapolated zero-field resonance frequency (ω (H ¼ 0) ¼ ω0) and Heff ¼ ω0 =γ

pffiffiffiffiffiffi C2

(20.29)

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with the geometric factor C2 ¼ 0.57 for a maximum angle θ ¼ π/4 of the nanowire orientation [105]. For the particular case of interconnected wires with only the shape magnetic anisotropy contribution, Heff was the product of MSat and the demagnetizing factor (N). As N factor was unknown, a demagnetizing factor originated by only the shape anisotropy term (NM) could be defined by the ratio between Heff ¼ Hsh and the demagnetizing field of an infinite thin film (4πMSat). Then, NM was independent on MSat and reproduced the main features of interconnected wires with only the shape anisotropy contribution: NM ¼ ωNiFe =

pffiffiffiffiffiffi C2

(20.30)

where ωNiFe ¼ ω0/(4πMSatγ) was the normalized resonance frequency at zero applied magnetic field for interconnected nanowires with only the shape anisotropy contribution such as in materials like NiFe. For nanowire networks with magnetocrystalline anisotropy contribution (Heff ¼ Hsh  Hcry), the combinations of Eqs. (20.29), (20.30) led to   ωNiCo Hcry ¼ 4πMSat pffiffiffiffiffiffi  NM C2

(20.31)

where ωNiCo is the normalized zero-field resonance frequency for the NixCo(1 x) (0  x  1) interconnected wires. Therefore using the NM factor (NM ¼ 0.326) obtained from the dispersion relation of the NiFe network, the magnetocrystalline field of the NiCo alloy could be determined [105]. It was claimed that as soon as the Ni content was reduced, the magnetocrystalline field was negative and increased toward the value observed for Co wires with perpendicular c-axis.

20.6

High-frequency applications and future perspectives of ferromagnetic nanowire arrays

Along this chapter, the flexibility to tune the magnetic behavior and dynamical response of ferromagnetic nanowire arrays by controlling the wire length, diameter, aspect ratio, separation, composition, and even the electrodeposition conditions was demonstrated. Based on this flexibility, ferromagnetic nanowires arrays have been already suggested for the development of magnonic applications [111] and in different microwave devices [85, 112]. In particular, inductors with a Q-factor improvement [113], 20 Gigahertz noise suppressors [114], filters [115], nonreciprocal phase shifters [116], isolators [117], and circulators [118] have been already suggested as microwave devices based on ferromagnetic nanowire arrays composed of a single material or alloy. The spin-torque effect has been demonstrated as a suitable approach for the generation of microwave signals though the injection of a dc in magnetic multilayer nanostructures. Although several approached have been studied, the development of

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tunable microwave generators still requires the improvement of both the spectral purity and the power of the generated microwave signal. To overcome these limitations, multilayer ferromagnetic nanowire arrays, or spin valve nanowire arrays, have been suggested as potential candidates for the development of synchronized spintransfer nano-oscillators [119–122]. These systems show several advantages such as its low-cost production, excellent thermal and mechanical properties, wellcontrolled geometry (diameter, length, and separation between nanoelements), and the high density of nanostructures since each element can be formed by several spin valves connected in series. Considering a more complex configuration, nanowire arrays with height gradient profiles were suggested for the development of microwave devices such as phase shifters with improved nonreciprocal behavior [98]. In Sections 20.5 and 20.6, a recent strategy to modify the dynamical response of the nanowire arrays and based on the modification of the template morphology was described. This approach has allowed the preparation of more exotic nanostructures such as 3-D ferromagnetic nanowire arrays, which can be used to fabricate magnetic devices with controlled anisotropy and microwave absorption properties [104, 106]. Although a huge effort has been done during the last two decades, there are still open questions from both the theoretical and technological point of views: -

-

Regarding the fabrication process a deeper control of the nanostructure sizes, shapes, and separations is required. Magnonic devices for transferring information has been already fabricated using a lowdamping material such as YIG [123, 124], but the size of these systems is still too large, on the order of few hundreds of micrometers, for miniaturized devices. Then, ferromagnetic nanolements, such as wires and tubes, could be interesting candidates. For this purpose the development of new manipulation and characterization techniques, which could allow us to study the dynamical response of individual elements, should be required, and it will be one of the main achievements during the following years. Moreover, it is expected that a deeper analysis of the dynamical behavior of nanowire arrays with more exotic geometries, such diameter-modulated nanowires [125], radial multilayer nanowires [126–128], and 3-D nanowires and multilayer nanowires, could provide materials showing new physical properties and with applicability into the THz regimen.

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Spin waves and electromagnetic waves in magnetic nanowires

21

Martha Pardavi-Horvatha, Elena V. Tartakovskayab a The George Washington University, Washington, DC, United States, bInstitute of Magnetism NAS of Ukraine, Kiev, Ukraine

21.1

Introduction

Magnetic and electromagnetic phenomena in nanoscale magnetic materials are particularly sensitive to changes in size and shape of materials and to the frequency and power of electromagnetic fields. During the past decade, many new “nanoelectromagnetic” phenomena were predicted, observed, and understood, and this understanding brought us new knowledge and new devices. Three-dimensional (3-D) magnetic nanocomposites, nanosize magnetic particles embedded into a nonmagnetic matrix, are remarkable systems for both theoretical and experimental investigations. Promising applications in microwave devices, as photonic and plasmonic metamaterials, are being developed. However, there are still challenging tasks in understanding high-frequency phenomena or taking into account the statistical distribution of separation, shape, and size of real particle systems quantitatively. Periodic two-dimensional (2-D) arrays of magnetic nanowires, besides their wide-ranging applications, are ideal objects to test theoretical predictions, related to fundamental properties and basic physics of small magnetic particles, and to investigate magnetic and electromagnetic properties of low-dimensional systems in general. By changing the size and shape of the elements, fundamental magnetization processes can be studied. Changing the separation one can study the transition from single noninteracting nanoelements through an assembly of weakly interacting elements and finally examine the length scales for strong interactions and watch how thin-film properties develop from individual nanoparticles. Examples of the questions raised at the early stages of research are as follows: Is there a phase transition like behavior (percolation) between weak and strong interaction regimes? How does the statistical distribution of properties of the individual elements influence the dynamic properties? What is the relationship between static characteristics, for example, coercivity and switching field distribution (SFD), to high-frequency damping? How these effects depend on arrays’ geometry? Today, measurements of ferromagnetic resonance (FMR), Brillouin light scattering (BLS), and broadband microwave behavior together with static magnetic properties and modeling can answer these questions. While the static magnetic properties and the magnetization switching of such systems are widely studied, surprisingly, the dynamic properties of nanoscale systems until the very recent time received relatively little attention. The measurement of Magnetic Nano- and Microwires. https://doi.org/10.1016/B978-0-08-102832-2.00021-9 Copyright © 2020 Elsevier Ltd. All rights reserved.

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dynamic magnetic properties at microwave frequencies can answer important questions about fundamental properties of arrays of magnetic nanoelements. Selforganized 2-D arrays of very high-aspect-ratio magnetic nanowires are a particularly attractive geometry, due to its excellent orientation uniformity, simplicity, reproducibility, planarity, and cost advantages. The field is vibrant with many groups preparing and studying such materials. The annual INTERMAG, MMM, and PIERS conferences provide ample material to follow the development of the field. Besides the previously mentioned advantages of nanowires in analyzing fundamental properties of 2-D systems, it is an important fact that the matrix is insulating and nonmagnetic; therefore the dominating interactions are simply dipolar magnetostatic. Such systems exhibit a strong uniaxial shape anisotropy. By electrodeposition a great variety of magnetic metals, multilayers, magnetic-nonmagnetic structures can be deposited into the pores of the porous anodic alumina substrate membrane (see Section 21.4). Due to the flexibility of the preparation, several extra degrees of freedom, like wire length, diameter, aspect ratio, separation, and composition, make them ideal for studying interaction and dynamic effects. Fig. 21.1 illustrates the AFM and self-correlation image of a 35-nm-diameter, 65-nm-periodicity permalloy nanowire sample used to study interaction effects. The road to preparation of nanowire arrays was opened in 1970 by Possin, who used mica films to deposit nanowire arrays [1]. The application of the technique, based on alumina substrates, early on attracted the interest as a potential tool for high-density magnetic recording, and the systematic study of magnetic nanowires started [2, 3]. Today, most of the published results are obtained on nanowire arrays deposited in anodized alumina substrates. Hexagonally ordered, close-packed pores can be formed on the surface of anodized alumina, with the pore size and spacing controlled by the anodization conditions such as voltage, current density, and solution pH. Pores below 10 nm have been reported with packing densities corresponding to more than 1010 cm2. Details of preparation technology and characterization of magnetic nanowire arrays are given in Part I of this book.

Fig. 21.1 AFM and self-correlation image of a permalloy nanowire sample. Wire diameter: 35 nm, periodicity: 65 nm [6].

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This chapter focuses on the dynamic properties of nanowires. To understand the electrodynamic and magnetodynamic behavior of magnetic nanowires, first, in Section 21.2, it is shown how the properties of a nanowire system change from a single nanowire (or a large ensemble of noninteracting nanowires) to a strongly interacting nanowire system. The dominating role of the long-range magnetostatic dipolar interactions is demonstrated in establishing the demagnetizing factors (DMF) of a single wire and for a wire array. One of the striking consequences of the dipolar interactions in dense nanowire arrays, the configurational reorientation transition (CRT), is described. Section 21.3 deals with the interaction of magnetic nanowires with electromagnetic waves (EMWs). When a magnetic system is excited by a high-frequency EMW, the excited spin system responds with a spin-wave (SW) excitation spectrum, strongly depending on the shape and size of the particles. In nanosize particles of nonellipsoidal geometry, the SW spectra are peculiar, as it is described in Section 21.3.1. If the shape of the very-high-aspect-ratio nanowire is modified, the resulting hollow nanotubes and multilayered wires offer intriguing objects of theoretical and experimental research, with potential applications. Section 21.4 deals with the interactions of nanowires with EMW. Magnetic nanowires are serious contenders for novel microwave devices, based on the propagation of EMW in the nanostructures. The results of modeling the electrodynamic diffraction problem show how the properties of this flexible system can be tuned up to the terahertz frequency range. Section 21.5 is a brief survey of magnetic nanowire applications in microwave, photonic, and plasmonic devices.

21.2

From single magnetic nanowire to 2-D nanowire arrays

21.2.1 Interactions in nanowire arrays 21.2.1.1 Static magnetic properties of nanowires The important static magnetic features of magnetic nanowires, to be used in the following discussion, are single-domain remanent state; well-defined easy axis, for example, uniaxiality; controllable coercivity and anisotropy; sufficiently narrow SFD; thermally stable magnetization state; and tunable saturation magnetization (by the choice of the magnetic material and by porosity). The single-domain state assumes that at least one dimension of the particle is smaller than the characteristic length for domain formation. The characteristic length has several definitions, all leading to the same order of magnitude: it can be defined as the exchange length λex ¼ [2A/M2s ]1/2, the domain wall width λDW ¼ [A/K]1/2, or the magnetostatic length λD ¼ [2(AK)1/2]/M2s , where A is the exchange interaction energy, K is the anisotropy energy, and Ms is the saturation magnetization at temperature T. For the existence of a thermally stable state at T, the size of the single-domain particle should be just around these limits; otherwise

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the particle is superparamagnetic or multidomain. For traditional ferromagnetic metals (Fe, Co, and Ni), the single-domain size is a few tens of nanometers, ideally fitting the nanowire technology. For a single-domain particle, the anisotropy has three contributions: for polycrystalline wires the magnetocrystalline anisotropy (HA) is usually negligible, unless there is a preferential growth-induced ordering. There might be a magnetoelastic contribution, due to the nonzero magnetostriction of the wire material, that can be optimized by proper choice of the materials. Finally the dominating contribution for nanowires is the shape anisotropy. For single-domain-diameter wire particles, the easy axis is preferred to coincide with the wire axis, thus minimizing the stray fields. For a large enough aspect ratio a, that is, wire length/diameter, a ¼ ‘/d »10, the DMF along the wire can be taken as Nz ffi 0, while Nx ¼ Ny ¼ 0.5, as for an infinitely long ellipsoid of rotation. As a result the nanowires are bistable, having only two magnetic states, “up” and “down” along the easy axis with a remanent magnetization Mr ¼ Ms. However, for nanowire systems, consisting of a large number of wires, an interesting situation takes place: the overall magnetic anisotropy of the array is determined not only by the shape anisotropy of the individual wires, which will induce a magnetic easy axis parallel to the wire axis, but also by the magnetostatic coupling among the wires, attempting to develop a magnetic easy axis perpendicular to the wire axis. Thus the uniaxial anisotropy is expected to be reduced for an interacting system of nanowires. Reducing the anisotropy field corresponds to a reduction in coercivity, measurable experimentally in hysteresis measurements. The nonellipsoidal shape and the relatively large surface area of a small size particle produce a nonuniform internal field in nanoparticles. Close to saturation the magnetic moments at the wire ends are still canted to minimize the stray fields, and due to increased surface/volume ratio, this canted structure persists to very high fields and contributes to switching field reduction by creating nucleation centers for easy reversal. For a solitary single-domain particle, the coercivity is equal to the switching field, Hc ¼ Hsw < HA. For real nanoassemblies, with a large number of particles, the mean value of the switching field, that is, the coercivity, is usually much lower than that arising from nonuniform magnetization rotation processes. The statistical distribution of individual switching fields (SFD) is quite broad, due to nucleation at defects. The mechanism, responsible for SFD, contributes to the observed FMR linewidth, that is, to dynamic properties. At the same time, the FMR linewidth characterizes the microwave losses in devices, based on nanowire arrays.

21.2.1.2 The role of shape anisotropy: Demagnetizing tensor of nonellipsoidal magnetic elements To discuss the role of interactions in large 2-D arrays of magnetic nanowires, first, the knowledge of the demagnetizing tensor elements is needed. The equilibrium magnetic structure of the nanowire system is governed by the arrangement of the elements’ magnetization according to the energy minimum, following the internal field. For zero magnetocrystalline and magnetoelastic anisotropy, the internal field Hi in an applied

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dc bias field H0 is determined by the demagnetizing field, HD ¼  NM created by the magnetized body, as $

Hi ¼ H0  N M $

(21.1)

where N is the demagnetizing tensor. For magnetic bodies of a shape, different from the ellipsoid of rotation, the demagnetizing tensor elements can be calculated numerically ([4], and references therein). As shown earlier, in the first approximation, the shape of the nanowires can be approximated by a long thin wire, ‘ » 2r, and a corresponding approximation for the internal field can be used. However, to describe and understand the behavior of the magnetization of individual nanowires and that of an interacting array, more precise calculations are necessary. The magnetization is not uniform inside a nanowire (or any other magnetic nanoparticle), because the internal field is inhomogeneous due to shape effects. There is a noncollinear magnetic spin structure at the ends of the wires, even for wires of very large aspect ratio and very high magnetic field, where the material is assumed to be saturated. The demagnetizing field dominates the internal field in individual nanoparticles and interacting nanoparticle arrays. The calculation of the demagnetizing tensor elements for nonellipsoidal bodies is the hardest task in any micromagnetic technique, and significant effort was devoted to this problem from the time when shape and size effects in design of machines and devices, based on magnetic materials, in industrial technology required the knowledge of the DMF for nonellipsoidal shape magnetic materials (see in [4], and references therein). By now the numerical methods for calculating the demagnetizing tensor elements for small particles are well established. To study the magnetic structure and spin-wave excitations in nanowire arrays, the analysis of magnetostatic coupling energy requires the evaluation of the elements of the demagnetizing tensor, or its diagonal terms, the DMF. To obtain a qualitative estimate, a simplified approach is frequently used, where long wires were considered as long ellipsoids and interacting dipoles. However, in the dense arrays of nanowires, this approach is insufficient, especially when it is inconvenient for quantitative analysis of configurational phase transitions to be described in Section 21.2.1.4. In such a case the rigorous method of DMF calculation [5], taking into account the specifics of nanoparticles’ shape, should be performed. The representation of a nanowire as an infinitely long cylinder is also inadequate when considering the configurational phase transition, as infinitely long cylinders do not interact via stray fields; the real aspect ratio of individual nanowire is long, but the finite length of the cylinders should be taken into account. Table 21.1 summarizes the calculated and experimentally obtained DMF for single nanowires and interacting nanowires of actual geometry for a series of nanowire arrays. As shown in Table 21.1, row 4, the calculated values for a single nanowire are close to unity, but even this small difference is important for the analysis of phase equilibrium in the system. The DMF of the nanowire array can be calculated via summation of stray fields of all nanowires in the array, because of the long-range character of dipolar interaction (Table 21.1, row 6).

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Table 21.1 Comparison of experimental and theoretical results for demagnetizing factors for a series of 2500 nm long nanowire samples. SU01

SU02

SU03

2R D Hrk – Hr?

25 65 1926

35 65 1393

45 65 150

I(th) xx (single)

0.99

0.994

0.992

I(exp) xx (interact)

0.81

0.72

0.61

I(th) xx (interact)

0.87

0.75

0.6

Wire diameter—2R; period of the array—D (dimensions in nm). Hrk – Hr? is the anisotropy of the (th) FMR resonance field, due to the shape anisotropy of wires; I(th) xx (single), Ixx (interact) are the calculated normalized demagnetizing factors of an individual wire and for the interacting wire arrays; I(exp) (interact) are the experimental data [6]. xx

In general, there are two techniques of such summation: the real space approach, which was used particularly in Tartakovskaya et al. [6] and the numerical technique of the Fourier transformation of dipolar tensor [7]. Both methods have their advantages and disadvantages. For the array of cylindrical particles, the Fourier transformation simplifies the final calculation routine, but it is hard to apply to finite arrays and/or to disordered arrays. In both of these cases, the real space approach works without any restrictions. In the case of nanowires with rectangular cross section, according to the analytical results of Aharoni for DMF for rectangular prisms [8], the real space approach is the most useful for calculating of DMF in one-dimensional and twodimensional planar nanoparticle arrays, as it was demonstrated in Tartakovskaya [9], and the same approach was applied to nanowire arrays in Tartakovskaya et al. [6].

21.2.1.3 Static dipolar interaction effects in nanowire arrays To understand the interactions and the changes in magnetic structure, caused by the interacting wires in a 2-D array, a more precise calculation of the demagnetizing tensor of the individual wires and of the nanowire system is necessary. Due to the nonvanishing dipolar field of finite-length nanowires, there is always an in-plane component of this dipolar field from neighboring nanowires, coupling the individual 1-D wires into a 2-D system. For a saturated two-dimensional infinite array, the total dipole field acting on one wire is the sum of fields produced by all other wires and can be calculated micromagnetically or written in a simple approximation: HD ¼ 4:2Ms πr 2 L=D3

(21.2)

where r is the radius of the wire, L is the length, and D is the separation of wires. When all the magnetic moments of the wires are aligned perpendicular to the wire axis by an in-plane field, the total field acting on one wire is the sum of the dipole fields and self-demagnetizing field of the wire: H? ¼ 2:1Ms πr 2 L=D3 + 2πMs

(21.3)

Spin waves and electromagnetic waves in magnetic nanowires

619

where the first term is from the dipole fields, which is parallel to the applied field. The total effective anisotropy field is Hk ¼ 2πMs  6:3Ms πr 2 L=D3 + HA

(21.4)

As the wire length L increases, when L ¼ Lc ¼ 2D3/(6.3r2), then Hk decreases linearly to zero. It is assumed that HA ¼ 0 for polycrystalline materials. When L > Lc, Hk becomes negative, that is, there is a reorientation from easy axis to easy plane behavior, as it is discussed in Section 21.2.1.1. In this situation for strongly coupled wires, the magnetic easy axis reorients from parallel to perpendicular to the wire axis [6, 10, 11]. Recognizing the importance of the long-range magnetostatic dipolar interactions in the array anisotropy, that is, configurational anisotropy in the magnetic properties of 2-D nanoparticle arrays, there were several early experimental and numerical works, dealing with the topic. In Encinas-Oropesa et al. [12] the dipolar interaction between wires was modeled by a mean-field approach and measured on nanowire samples 50–250 nm diameter, with corresponding porosity P from 4% to 38%. Linear dependence of the effective field from shape anisotropy and dipolar coupling on porosity was predicted and observed, as Heff ¼ 2 πMs ð1  3PÞ

(21.5)

The importance of magnetostatic interactions in dense nanowire arrays was demonstrated by Clime et al. [13], by calculating the in-plane and out-of-plane interaction fields, using a hybrid numerical-analytical method, and showing that the high field behavior of such systems is dominated by the magnetostatic effect. Analytical calculations or numerical simulation is limited by computing power to relatively small size samples and first-order approximations. However, an analytical approach was published [14] pointing out the fact that first-order approximations are valid only when the wires are far apart, as was shown in an early work for interaction in small-particle systems [15]. The dipolar interactions were numerically modeled by micromagnetic simulations (NMAG package) on finite and infinite periodic systems. It was shown that the finite-size models fail to capture some of the important details of the system [16].

21.2.1.4 Configurational phase transitions in arrays of nanowires There is an interesting question about the existence of a proximity effect in smallparticle magnetic systems, similar to the percolation limit in conductivity. The case of electrical percolation is well-known for metal-nonmetal composites. At percolation volume fraction a transition occurs from positive to negative values of the real part of the composite dielectric constant, the signature of a metal. The composite conductivity changes sharply at the point of the electrical percolation threshold. A deeper analysis of the dielectric properties of metal-insulator composite, in which the metal component is supposed to obey the Drude law, that is, a complex dielectric function, and the insulator component is represented by a dielectric constant, reveals the

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existence of an optical threshold, depending on photon energy. It was observed that the optical threshold is different from the electrical percolation threshold. For magnetic nanoparticles in an insulating matrix, similar electrical and optical percolation effects are expected. While electrical percolation takes place when the first continuous path opens for conduction, magnetic percolation is rather a proximity effect, due to longrange dipolar interactions between particles. The transition from noninteracting uniaxial wires to a weakly interacting system of nanowires, coupled by long-range dipolar forces, is easy to understand and calculate, based on the approach, described earlier. However, a strongly interacting dipolar system can’t be described by the usual far-field dipole approximation, and near-field effects should be included in the description of this state. At a high density of wires, the interaction becomes very strong, comparable with the exchange coupling, leading to an in-plane thinfilm-like behavior, as observed and discussed in Pardavi-Horvath et al. [10] and Tartakovskaya [9]. Transitions between ground states of saturated nanowire systems with different kinds of dominating anisotropies are called configurational reorientation phase transitions. Changes of the easy axis in magnetic crystals with temperature or pressure are well-known as microscopic spin reorientation phase transitions and are caused by competing anisotropies, that is, spin-orbit coupling effects. There is a physical difference between macroscopic CRT of a large system of nanowires, which are described here, and the classical microscopic spin reorientation phase transition. The Landau theory of phase transitions was developed to describe the microscopic phase transitions; however, the formalism of the original Landau theory can be applied to describe the change of easy axis in interacting nanostructured systems. Spin reorientation phase transitions in ferromagnets were described in Landau and Lifshitz [17] in the following way. An expansion of the thermodynamic potential Φ by the order parameter η, Φ ¼ K1 η2 + K2 η4

(21.6)

can be interpreted in terms of magnetic media, if we assume η ¼ sin θ , where θ is the angle between the average magnetization vector and the easy anisotropy axis, and K1 and K2 are the temperature-dependent magnetocrystalline anisotropy constants. Usually, K1 » K2, however, in the vicinity of the phase transition, where K1 changes its sign at some temperature point and K2 and its sign determine the order of the phase transition (first or the second one). If K2 ffi 0, the next term of the expansion should be considered. The formalism, developed for macroscopic configurational reorientation phase transitions, as described later, is based on the formal analogy between the classical formalisms of phase transitions and reorientations in magnetic nanostructures. Returning now to the arrays of magnetic nanowires, one can utilize the same formula (21.6), but now the magnetic energy plays the role of thermodynamic potential, while the dominating contribution to the anisotropy parameters, especially in soft magnetic materials, are the dipolar, magnetostatic forces, leading to the shape anisotropy. The first anisotropy constant K1 is proportional to the effective DMF of the wire

Spin waves and electromagnetic waves in magnetic nanowires

4500

621

SU_0100 PHr SU_0200 PHr

Resonance field, H (Oe)

SU_0300 PHr

–30

3500

2500

1500

0

30

60

90

120

Angle, theta (Deg.)

Fig. 21.2 Out-of-plane angular dependence of the FMR resonance field for a permalloy nanowire series (SU), demonstrating the transition from the uniaxial “wire” anisotropy in samples SU-01 and SU-02 with Hr(0°) > Hr(90°) toward the “easy plane” anisotropy for SU-03, where Hr(0°) > Hr(90°) [10].

in an interacting array. In the transition point the DMF approaches zero. FMR measurements, shown in Fig. 21.2, unambiguously demonstrate that the transition from easy axis to easy plane behavior of dense nanowire array takes place due to the dipolar interactions between wires [10]. However, it is not a phase transition in a thermodynamic sense (a detailed discussion about difference of phase transitions in bulk magnets and in arrays of magnetic nanoparticles is given in Tartakovskaya [9]). Because of its dependence on geometrical configuration of nanowires in array, such a reorientation transitions in arrays of nanoparticles called a configurational reorientation transition. Here the second-order anisotropy parameter, K2, also has a dipolar origin; it can be calculated taking into account the nonuniformities of the magnetic structure, due to the nonuniform internal field near the top and bottom edges of nanowires. The sign of K2 determines the order of the configurational phase transition in the same way, as the corresponding temperature-dependent parameter of magnetocrystalline anisotropy determines the order of the phase transition in the classical Landau’s theory [6, 9, 18]. Similar phenomena can be investigated in one- and two-dimensional arrays of planar nanodots and nanostripes, based on the general approach to the calculation of DMF, described in Section 21.2.1.2. For instance, in one-dimensional array of ferromagnetic stripes, the CRT between in-plane and out-of-plane ground states takes place due to the competition between shape anisotropy (which is reduced compared with a nonpatterned film) and out-of-plane magnetocrystalline anisotropy [19] or a combination of anisotropies of magnetocrystalline and magnetoelastic origin [20].

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Magnetic Nano- and Microwires

On the contrary, in two-dimensional arrays of ferromagnetic wires, the CRT can arise in magnetically soft material a result of the competition between anisotropies of the same dipolar origin: that is, intrawire and interwire shape anisotropies [7]. The other difference of CRT in two-dimensional arrays of circular nanowires and one-dimensional array of stripes is that stripes can be treated as infinite in this context, while such CRT exists in two-dimensional arrays of circular nanowires of finite length. As it was shown in Tartakovskaya et al. [6] by calculating the effective magnetostatic energy taking into account the nonuniform micromagnetic state at the end of the wires, the CRT in long ferromagnetic nanowires due to the competition between different kinds of dipolar interactions is a first-order magnetic phase transition in the Landau sense. A quite different reorientation transition results, if an external magnetic field is applied to the nanowire sample, perpendicular to the wire axis. In this case the competition between the Zeeman interaction and shape anisotropy leads to a transition of the second order [21], with the usual square-root dependence of the order parameter near the phase transition point, θ ∝ (2  2H0/2πMS)1/2, where the order parameter θ is the angle of the magnetization with respect to the wire axis. This has a special influence on the SW spectra. The field dependence of frequencies of the dipolarexchange modes in a nanowire arrays is quite different in some cases, when the external field is applied parallel or perpendicular to the wire. As it was observed in several BLS and FMR experiments [12, 22–24], the spin-wave dispersion behavior is more or less monotonic in parallel external field, whereas in perpendicular field it has a spike, when the value of this field approaches the shape anisotropy field of the infinite nanowire, H0 ¼ 2πMS, as shown in Fig. 21.3. This well-defined minimum of the dispersion relation corresponds to the softening of modes in the vicinity of the secondorder reorientation phase transition. Additionally, a special splitting of spin-wave mode arises, due to the violation of the circular symmetry [21].

21.3

Magnetic nanowires in electromagnetic fields

When a magnetized body, in our case, a magnetic nanowire sample is placed into a high-frequency (GHz range and above) electromagnetic (EM) field, it interacts with the EMW. EMWs are transmitted, reflected, absorbed, and diffracted by the magnetic nanowire material. This interaction depends very strongly on microscopic material parameters and on the shape and size of the wires, the array configuration, the EMW frequency, and power. There are subtle, microscopic reactions to the EMW by exciting the precessional motion of electronic spins and generating spin waves, that is, magnon excitations. Depending on the shape and size of the magnetic nanoparticle, higher-order modes may contribute to the high-frequency magnetic behavior. The spin-wave spectra of magnetic nanowire system are described in Section 21.3.1. On the other hand, there is a macroscopic interaction of a magnetic nanowire system with an incident EMW, resulting in changes by propagation and diffraction effects of EMW in the material response. This approach is a fertile field of applications of

Spin waves and electromagnetic waves in magnetic nanowires

623

35

Spin wave frequency (GHz)

30 25 20 15 10 5 0

0

2000

6000

4000

8000

10000

H (Oe)

Fig. 21.3 The dependence of the theoretical () and experimental BLS values (●) of resonance frequencies as a function of magnetic field applied normally to the axis of the Ni nanorods with low aspect ratio and R ¼ 10 nm. The change of the character of the dispersion takes place near 2800 Oe. A softening of the lowest mode near the reorientation transition of the second order is also shown. A.A. Stashkevich, Y. Rossigne, P. Djemia, S.M. Che’rif, P.R. Evans, A.P. Murphy, W.R. Hendren, R. Atkinson, R.J. Pollard, A.V. Zayats, G. Chaboussant, F. Ott, Spin-wave modes in Ni nanorod arrays studied by Brillouin light scattering, Phys. Rev. B 80 (14) (2009) 144406. Copyright American Physical Society, 2009.

nanowire systems in microwave devices for communication, information technology, and biomedical purposes. Measuring FMR absorption and/or BLS provides the most convenient way to extract information about microscopic characteristics and changes in the magnetic behavior of small-particle systems. The 2-D periodicity of nanowire materials provides an extra degree of freedom to evaluate intrinsic parameters. FMR and S scattering parameter measurements were an early tool to characterize the microwave properties of magnetic nanowire systems, and they remain the most important experimental methods in our days, as it is demonstrated by Goglio et al. [25] who reported FMR data on Fe, Ni, and Fe-Ni nanowire materials up to 40 GHz. Aslam et al. [26] performed a systematic study of the composition dependence of the magnetization dynamics of Fe-Co nanowires. Raposo et al. [27] studied the interaction properties of Ni-Fe nanowires. In the following a brief reiteration of terms will introduce the topic of FMR in interacting nanowire systems and the response of such systems to excitation by harmonic electromagnetic fields in the form of development of spin-wave modes and effects of EMW propagation.

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Magnetic Nano- and Microwires

When a magnetic field applied to a magnetic material changes, then in response the magnetization reacts with a change, relaxing in a short time to the new equilibrium position. This process is accompanied by an irreversible flow of energy to the magnetization and via some intermediate stages, to the crystal lattice, where is it dissipated as heat. This energy loss dampens the motion of the magnetization, as described by the Landau-Lifshitz-Gilbert phenomenological equation of motion of the magnetization: dM=dt ¼ γ ðM x Hi Þ  ðα=M ÞðM x dM=dtÞ

(21.7)

where γ ¼ ge/2m is the gyromagnetic ratio and α is the Gilbert damping constant. The first term gives the torque that the magnetization M experiences in an internal magnetic field, Hi, and the second term, often neglected, is a damping torque, which acts to oppose the rotation of M. In ferromagnetic materials the internal field arises from the anisotropy fields, including magnetocrystalline, magnetoelastic, and shape anisotropies. In FMR the sample is magnetized by a strong steady field H0, and a small amplitude rf field h0 of frequency ω0 is applied perpendicular to H0. The internal field Hi exerts a restoring torque on M. As a response the total magnetization is precessing uniformly about the direction of the internal field, and energy is absorbed strongly from the small transverse rf field when its frequency is equal to the precessional frequency (usually in the gigahertz range), that is, at FMR. The power absorbed by the magnetization from the rf field is given by (½)χ00 h20, where χ 00 is the dissipative part of the magnetic susceptibility of the material. In FMR experiments, ω0 ¼ const, and H0 is swept through the resonance, that is, the maximum in χ 00 at the FMR field of Hr ¼ ω/γ. The shape of the material plays a very important role in FMR. The internal field contains the demagnetizing field, determined by the shape of the specimen. The shape effects are expressed through the demagnetizing tensor components, or DMF for an ellipsoid of rotation, Nx, Ny, and Nz. For nonellipsoidal shapes the demagnetizing tensor elements are local, and only micromagnetic calculations can take it into account when calculating the inhomogeneous internal field and magnetization distribution in nonellipsoidal single-domain elements. In the first approximation, for a very long thin wire, magnetized along its axis (M ¼ Mzz), Nz ¼ 0, and Nx ¼ Ny ¼ 1/2. The shape anisotropy and the role of demagnetizing tensor elements in the establishment of equilibrium magnetic structures were discussed in Section 21.2. Due to the magnetocrystalline anisotropy and anisotropic demagnetizing field, there is a strong angular dependence of the FMR. Measurements of this angular dependence is a convenient tool to extract the interaction field in nanostructures [28]. Although high-aspect ratio nanowires seem to be simple model materials, the study of the high-frequency phenomena in nanowires offers significant insight into their sophisticated magnetism. Due to the high shape anisotropy, the wires should show a uniform magnetization reversal and a rectangular hysteresis loop, except a rounding at the reversal due to statistical distribution of switching fields. As a result, there is a significant magnetic remanence when reducing the field to H ¼ 0 from saturation.

Spin waves and electromagnetic waves in magnetic nanowires

625

It was shown that arrays of Fe-Ni nanowires exhibit resonance frequency tunability at zero-field (natural) FMR absorption, opening the road to unbiased microwave devices [29, 30]. The FMR signal for a single, infinitely long nanowire would be the Kittel textbook example of the microwave response of an ideal quasiinfinitely long material. For a large number of noninteracting identical wires, the signal intensity would increase proportional to the number of wires, or the filling factor. But many identical particles exist only in textbooks. Particles in a real system have a statistical distribution of switching fields, each of which has a slightly different resonance field, leading to a broadened FMR line, with the position of the FMR resonance field, Hr, hardly changing. Upon decreasing the separation between particles, due to long-range dipolar interactions, each particle feels the interaction field from its neighbors. The internal field, Hi, of the system changes, and for particles on a periodic lattice of saturated particles, Hi and Hr will change by the interaction field. The linewidth is becoming more wide from overlapping slightly different FMR spectra. Decreasing the separation further the dipolar interaction is increasing; the particles are no more individuals—they are a coupled system, where the linewidth is expected to narrow (similar to classical exchange narrowing). The resonance field at this point is changing because the demagnetizing fields of the system are different from that of a single small element. The demagnetizing field at strong interaction should approach that of the thin film. The change of the FMR field is a direct measure of the dominating DMF of an element or the system. For strongly interacting system the shape of the sample, and not of the wires, will determine the internal field [10, 11]. Collective excitations can be studied on such a system both theoretically and experimentally by FMR, as FMR is much more sensitive to the interaction effects, than static measurements.

21.3.1 SWs in magnetic nanowires The motivation to study spin waves (SWs) in magnetic nanostructures has different aspects. SWs play a significant role in changing and tuning of such technologically important processes, as the spin reorientation transition and the formation of reversal modes, which influence the hysteretic properties and stability of magnetic nanostructures [31]. Practically, all of the parameters of a magnetic material (including exchange stiffness, magnetocrystalline anisotropy, and shape anisotropy, as well as magnetostriction, surface, and interface effects) influence the measurable properties of SWs. From this point of view, investigations of SW in nanostructures play a significant role as a way of a testing novel magnetic materials. With respect to long cylindrical magnetic nanowires, in addition to all of the previously mentioned details, we should underline the special significance of the investigations of magnetization dynamics from the point of view of fundamental physics; particularly, the cross stimulation between the theory of SW in nanowires and experiment in this field can be noted as an example. Following the FMR results by Ebels et al. [32] and Encinas-Oropesa et al. [12], rigorous analytical theory was developed promptly by Arias and Mills [33]. The corresponding exact solution for dipolar-exchange

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Magnetic Nano- and Microwires

SW in nanowires became an absolutely unique event in the case of magnetic nanostructures, where spatial confinements lead to strong inhomogeneities of the demagnetizing field and, consequently, to large calculation difficulties. This theoretical achievement encouraged the subsequent experimental and theoretical investigations, which for the last 15 years created a comprehensive picture of magnetization dynamics in both individual nanowires and nanowire arrays [34, 35]. In the following, first, the basic theoretical and experimental results describing SW dynamics in isolated nanowires with circular symmetry conserved will be covered. Then the SW dynamics in interacting arrays of nanowires, and generalization to the cases when circular symmetry is violated by noncylindrical shape of nanowires or by arbitrary direction of applied field will be described.

21.3.1.1 Dipolar-exchange spin-wave modes of individual cylindrical nanowires When fresh FMR results stimulated the development of a new theory on spin dynamics in nanowires, a definite theoretical basis already existed. The main ideas of SW formalism in magnetic films and multilayers and established calculations of magnetostatic modes in ferromagnetic cylinders were naturally applied to nanowires. Taking into account the dominant interactions, all the variety of magnetic films and multilayers are divided into three classes [36]: thick, thin, and ultrathin. For thick films (more than a few hundred nanometers), the theory implements a continuum model where the dipolar field dominates, while exchange interactions could be neglected. It is the same domain of small dimensions, where the theory of magnetostatic SW in particles with cylindrical shape was developed [37]. In ultrathin films (less than 100 atomic layers), a discrete model is more appropriate. The so-called thin films with thickness of a few tens of nanometers lay between these two limiting cases: the continuum model is still valid; however, both dipolar and exchange interactions should be taken into account [38]. It is evident that magnetic nanowires with radii from ten to hundred nanometers belong to this last dimension range. Such nanowires were grown for a long time. Electrodeposited Ni wires in polymer templates were probed by the FMR technique, which results are shown in Fig. 21.4 [12, 32]. A special double peak of absorption derivative spectra was observed for nanowires with r ¼ 40 nm, while for nanowires with much larger or smaller radii this additional peak vanished (or it was much less pronounced). This strange behavior was not observed previously and could not been explained in the frame of magnetostatic theory. Here the theoretical approach to calculate dipolar-exchange SW in nanowires, given by Arias and Mills [33], will be reviewed, as it provides a good understanding of the possible generalization of the theory and unavoidable restrictions of it. Following [33], we consider an infinitely long uniform cylindrical nanowire embedded in the external magnetic field of a strength H0, directed along the nanowire. In such a case the average magnetization is parallel to the nanowire (to z-axis), while the linearized SW mode has two perpendicular components, mx and my. For the following analysis, we should note an important consequence of the infinite length and uniformity of the

Spin waves and electromagnetic waves in magnetic nanowires

627

Absorption derivative

34.4 GHz 500 nm 270 nm 80 nm 35 nm

4

6

8 Ho (kOe)

10

12

Fig. 21.4 FMR spectra in parallel applied field for Ni nanowire arrays of different diameters as indicated in the figure. The arrows indicate the approximate position of the resonance field for the different absorption peaks. U. Ebels, J.-L. Duvail, P.E. Wigen, L. Piraux, L.D. Buda, K. Ounadjela, Ferromagnetic resonance studies of Ni nanowire arrays, Phys. Rev. B 6414 (2001) 144421. Copyright American Physical Society, 2001.

nanowire: it gives the possibility to write the spatial dependence of SW components and of the corresponding magnetic potential ΦM as mα ðx, y, zÞ ¼ mα ðx, yÞexp ðikzÞ ΦM ðx, y, zÞ ¼ ΦM ðx, yÞexp ðikzÞ

(21.8)

Here, k is a wave vector along the nanowire. In finite cylinders, where the dependencies like Eq. (21.8) are not valid, the theory of dipolar-exchange SW is usually based on the use of a basic set of functions followed by the implementation of the Ritz method of finding the approximate solutions, like in Kakazei et al. [39]. The SW components obey the Landau-Lifshitz equations: iΩmx ¼ ðH0  Dr2 Þmy + MS ∂ΦM =∂y iΩmy ¼ ðH0  Dr2 Þmx + MS ∂ΦM =∂x

(21.9)

where Ω is a frequency of SW and the parameters of the magnetic material of the wire are the saturation magnetization MS and an exchange stiffness D. The magnetic potential ΦM inside the wire obeys the magnetostatic condition   ∂mx ∂my r ΦM  4π + ¼0 ∂x ∂y 2

(21.10)

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Magnetic Nano- and Microwires

while outside the magnetic material ΦM satisfies the Laplace’s equation. The set of differential equations (21.8), (21.9) can be reduced to one equation of higher order for the magnetic potential: 

    ∂ 2 ΦM Dr2  H0 Dr2  H0  4πMS r2 ΦM + 4πMS Dr2  H0 ¼0 ∂z2

(21.11)

Such a symmetrical form of the equation for the magnetic potential is obtained due to the unperturbed circular symmetry of the given geometry of the nanowire. As we will see later, this equation cannot lead to the exact solution in the case when this symmetry is violated. Using Eq. (21.8) a solution of Eq. (21.11) can be written in cylindrical coordinates as ΦM ðx, y, zÞ ¼ Jm ðKρÞexp ðimϕ + ikzÞ

(21.12)

where Jm(Kρ)are Bessel function of the first kind. Inserting Eq. (21.12) into Eq. (21.11), we find a bicubic equation by the radial wave vector K:  3  2    D2 K 2 + k2 + Dð2H0 + 4πMS Þ K 2 + k2 + H0 ðH0 + 4πMS Þ  4πMS Dk2 K 2 + k2 4πMS H0 k2 ¼ 0

(21.13)

The magnetic potential inside the wire can be represented as a linear combination of three linearly independent solutions (which correspond to the roots of Eq. (21.13) of the form Eq. (21.12)), while outside there exists one independent solution of Laplace’s equation in the form of the modified Bessel function. The usual way to find a dispersion relation for the SW is to apply the appropriate boundary conditions, the number of which should be the same as the number of solutions, that is, in our case, it is four. Two of these boundary conditions stem from general magnetostatics: they are defined by the continuity of the radial component of the induction and the tangential component of the dipolar field. The other two boundary conditions should define the magnetization behavior on the side surface of the cylinder. In the article, described earlier, Arias and Mills considered the most general case of mixed boundary conditions (medium pinning), not specifying their origin. They proved that the boundary conditions lead to the special effects in nanowires, in particular, to the hybridization of the uniform FMR mode with the exchange modes with small wave vector, which strongly distinguishes the SW dynamics in nanowires and thin films. In such a case, some exchange modes can lie below the FMR mode in wires (Fig. 21.5), while in films they always lie above. The situation when the frequencies of FMR mode and one of exchange modes are close can be realized in wires for specific values of radii. In such a way, this theory explains the satellite of the FMR peak, observed in the experiments (Fig. 21.6). The agreement between the double-peaked FMR signal and the explanation of this shape by dipolar-exchange SW theory highlighted the validity of the theoretical

Spin waves and electromagnetic waves in magnetic nanowires

629

1

W /4p M

0.8 0.6 0.4 0.2 0 –2

–1

0 log (kR)

1

2

Fig. 21.5 Theoretically calculated spin-wave frequencies for a nanowire in parallel field as a function of ln(kR), where k is a wave vector along the nanowire. The hybridization arises due to the dipolar-exchange origin of excitations. R. Arias, D.L. Mills, Theory of spin excitations and the microwave response od cylindrical ferromagnetic nanowires, Phys. Rev. B 63 (2001) 134439. Copyright American Physical Society, 2001.

0.04



R = 80 nm

0.00

dH Gauss–1

–0.04

–0.08

Ho (kg) 6

7

8

9

10

Fig. 21.6 Theoretical calculations of absorption derivatives of FMR spectra in parallel applied field for Ni nanowire arrays of R ¼ 80 nm. The shape of the double peak and the positions of the resonance fields agree with the experimental results of Fig. 21.4. R. Arias, D.L. Mills, Theory of spin excitations and the microwave response od cylindrical ferromagnetic nanowires, Phys. Rev. B 63 (2001) 134439. Copyright American Physical Society, 2001.

approach, but could not be considered as a comprehensive approval of the theory. In early 2000th Ni nanowire samples were fabricated by electrochemical deposition of ferromagnetic material (Ni) into hexagonally structured alumina membrane, as shown in Fig. 21.7 [40]. These new, improved samples were investigated by BLS technique. These experiments quantitatively proved Arias’s and Mills’s theory [24]. The dependence of the measured SW frequencies of the wires on the applied field and wire radius

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Fig. 21.7 Top-view SEM micrographs of 1-μm-thick Ni-filled alumina membranes (A) before thinning, (B) 50 nm, and (C) 100 nm underneath the initial surface [40].

demonstrated the predicted quantization of SW in the confined nanowires’ geometry and unambiguously showed the validity of dipolar-exchange approach, while previous theories, where exchange was ignored, turned out to be insufficient in this case. In a theoretical fitting of the experimental data [24], the approximate formalism based on general dipolar-exchange theory was used. It employed the concept of weak or strong pinning of magnetic modes on the surfaces of nanowires. In the absence of an external field, it gives a possibility to write the expression for the frequency of the bulk standing modes (i.e., when k ≪ K) in the simple analytical form:

  1=2 am 2 am 2 Ω¼ D D + 4πMS R R

(21.14)

Spin waves and electromagnetic waves in magnetic nanowires

631

Fig. 21.8 Variation of bulk spin-wave frequencies with nanowire radius in zero applied magnetic field. The experimental data points, for each radius, correspond to the three values of the azimuthal quantum number m, which corresponds to the quantized value of radial spin-wave vector. The solid curves represent the fits of the experimental data by the theory (see text). The ♦ denotes the frequency of the bulk spin wave for bulk Ni. Z.K. Wang, M.H. Kuok, S.C. Ng, D.J. Lockwood, M.G. Cottam, K. Nielsch, R.B. Wehrspohn, U. G€osele, Spin-wave quantization in ferromagnetic nickel nanowires, Phys. Rev. Lett. 89 (2) (2002) 27201. Copyright American Physical Society, 2002.

where the parameters am correspond to the antinodes or nodes of the SW functions on the surfaces and, in such a case, the ratios am/R play the role of radial wave vectors of SW. A good agreement between the experimental data and the calculations by formula (21.14) was achieved under the assumption of weak pinning on the surfaces, as shown in Fig. 21.8. It is appropriate to mention here about the pinning in confined systems. A great deal of attention was paid to the question of dipolar boundary conditions in ferromagnetic nanostructures. It was shown that in thin nanoelements with rectangular cross section [41] and in planar disks [42] the inhomogeneity of dipolar field near the sharp angles on side leads to the strong pinning of SW. However, in infinite cylindrical wires, such strong dipolar inhomogeneities can be avoided, so only the surface anisotropy, which is not large on smooth surfaces, can be the source of the pinning. It explains why the weak pinning conditions used in Wang et al. [24] for fitting the data in circular nanowires is a physically reasonable choice. Summarizing the quantized dipolar-exchange SW modes in long isolated cylindrical nanowires were investigated in early 2000th, and remarkable agreement between theoretical results and experimental data (both of FMR and BLS techniques) was achieved. From the short overview earlier, it is evident that it was possible to find exact analytical solutions, and it is due to the unperturbed circular symmetry of the problem. The future extension of the initial investigations should answer the following questions: How does the dipolar interaction between nanowires influence the dynamical behavior of the magnetization for cases of arrays of saturated or nonsaturated wires? What happens if the cross section of the nanowire is not completely circular? How can the violation of the circular symmetry by the external field with nonzero perpendicular component change the results?

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The attempts to generalize the theory by taking into account all of these physical conditions simultaneously demonstrated how complicated this task is [43–46]. Further, we clarify these questions in more details.

21.3.1.2 Collective spin-wave modes in arrays of interacting nanowires The influence of dipolar interaction between nanowires on the SW modes should be much more pronounced, than between thin magnetic layers in multilayers. Actually the static dipolar interaction between parallel, saturated infinite nanowires is equal to zero in the same way as interaction between ideal films in a multilayer. However, there is a remarkable difference of the interaction between SW modes in thin films in a multilayer and the nanowires in an array. The dipolar field generated by SW in a film decreases with a distance of its surface exponentially, while the dipolar field from the uniform mode of ferromagnetic cylinder falls of inversely with the square of the distance [47]. For instance, a clear influence of dipolar interaction of the lowest mode was observed in FMR experiments with samples of Ni nanowire arrays with different porosity [12]. The theory of collective SW in arrays of ferromagnetic nanowires, based on a multiple scattering approach, was developed by Arias and Mills [47]. The formalism demonstrated the advantages of real space approach, which is applicable to any ordered or even disordered nanowire array. An explicit expression for spin-wave frequencies was found for collective SW of a nanowire pair. Numerical calculations of SW dispersion in a linear array of ferromagnetic cylinders show hybridization between collective modes and the standing wave modes of individual cylinders. Results of this theory were experimentally proved by BLS in hexagonal arrays of permalloy [48] and FeCo [49] nanowires. An excellent agreement between the theory and experiment was achieved. It was shown that in 2-D arrays with usual magnetic and geometrical parameters only the lowest frequency SW mode can be influenced by the interwire interaction. The reduction of the frequency of the collective SW, in comparing with the frequency of the lowest mode of an individual wire, was demonstrated by the investigation of BLS data in arrays with various interwire spacing as shown in Fig. 21.9. A comprehensive study, both experimentally and by micromagnetic simulation, of a densely packed array of permalloy nanowires with wire diameter much greater than the exchange length demonstrated new features of SW dynamics in nanowires [34]. Particularly the tunneling of end modes was observed, which is more efficient in arrays with smaller distance between nanowires, due to the common action of static and dynamic parts of the dipolar field in an array. We consider an individual nanowire or an array of nanowires, magnetized in the same direction. But in a real case the external field might not be strong enough to saturate the sample. The shape anisotropy of a long individual nanowire forces the average magnetization to lie along the nanowire. The individual wires are bistable, having two possible ground states, up and down. As it was mentioned, static dipolar interaction between parallel circular nanowires with homogeneous magnetization and

Spin waves and electromagnetic waves in magnetic nanowires

633

Frequency (GHz)

80

60

40

20

50 30 40 Interwire separation (nm)

Fig. 21.9 Frequencies of the spin waves in Fe48Co52 nanowire arrays as a function of interwire separation, at a longitudinal magnetic field of 0.6 T. Experimental data are marked by dots with error bars. Calculated frequencies of the two lowest-energy collective spin-wave modes are represented by solid lines. Corresponding predicted frequencies for the isolated single nanowire are shown as horizontal dashed lines [49].

infinite length is impossible. But nanowires never happen to be infinitely long, and so, they interact via stray fields arising due to poles on their end surfaces. If we consider only two wires, the energetically favorable is opposite alignment, as it allows flux closure through the wires (Fig. 21.10). However, if the interacting nanowires are arranged on a hexagonal lattice, they cannot establish such an antiferromagnetic order where the closest neighbors to every nanowire are oppositely aligned (Fig. 21.10). In such a case, it is convenient to consider a nonsaturated array of axially magnetized interacting nanowires as an aggregate of two oppositely magnetized wire populations, without specifying the magnetic order of the whole array (the same concerns disordered arrays of parallel wires). The calculation of the effective permeability tensor on the basis of a Maxwell Garnett homogenization procedure [50] was done in this representation. Further, the calculation results were compared and found to agree well with the data of broadband microstrip line measurements of interacting amorphous CoFeB nanowires [51]. The investigation of unsaturated arrays has a strong motivation: to develop micromagnetic devices, which could operate in low magnetic fields. As a result, attention was paid to FMR studies of magnetic excitations of such materials. As it was observed, in both dilute [52] and dense [30] arrays, the FMR signal strongly depends on the remanent state of the sample. It was shown that the Kittel formula, which is usually applied to FMR data in thin films, is insufficient in the present case. Nanowires magnetized in two different directions have different resonance frequencies, so two FMR peaks arise due to the presence of two (up and down) populations of wires. Double FMR resonance was observed both in parallel [53] and perpendicular [54] external static field. It was proven that the origin of double-peak shape in such a

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Magnetic Nano- and Microwires

?

Spin up Spin down

(A)

(B)

Fig. 21.10 (A) Dipolar magnetostatic interactions illustrated on a cross section of a nanowire array. On the left side the energetically favored antiparallel orientation is shown, which allows flux closure through the wires. On the right a parallel orientation is sketched, which requires to quench the flux lines in between the wires. (B) Top view of a nanowire array as an illustration of the geometric frustration in a hexagonal arrangement [174].

case is just the existence of two unsaturated populations: above the saturation, one peak vanished, and only one FMR peak was observed. An influence of magnetocrystalline anisotropy on microwave properties was probed by magnetometry and BLS in cobalt, permalloy, and nickel arrays of unsaturated nanowires [55]. In De La Torre Medina et al. [56], the FMR absorption properties of two bistable nanowire arrays (30-nm-diameter Co55Fe45 and 40-nm-diameter Ni83Fe17) were investigated. The in-field measurements over the full hysteresis cycle demonstrated that the two SW branches that correspond to the double FMR peak have different dispersion relations, dependently on the magnetic configuration (Fig. 21.11), while the intensity of every FMR peak is proportional to the fraction of the correspondent nanowire population. The FMR first-order reversal curve (FORC) technique allowed measuring of the dipolar interaction field and its dependence on the magnetic configuration of the bistable array of nanowires, and the results were explained on the basis of analytical mean-field theory [56].

21.3.1.3 Spin waves in ferromagnetic nanowires with noncircular cross section Interesting attempts to apply Yablonovitch’s ideas related to photonic crystals to magnetic superlattices were made in the context of two-dimensional arrays of long nanocylinders [57, 58], but for the present time, planar arrays of nanoparticles are considered as the most suitable material for magnonics application [59, 60]. Magnetic dynamics in planar structures like arrays of ferromagnetic nanostripes with modulated width was investigated by Piao et al. [61], Xiong and Adeyeye [62], and Xiong et al. [63]. A large domain of SW investigations in planar arrays of dots, antidots, and nanostripes is beyond the scope of the present review (see, e.g., [64, 65] and references therein). The significant progress in fabrication and investigation of

Spin waves and electromagnetic waves in magnetic nanowires

635

45

f (GHz)

40 35 30 25

(A)

20

–4

–2

0

2

4

18

f (GHz)

16 14 12 10

(B)

–2

–1

0

H (kOe)

1

2

Fig. 21.11 Dispersion relations of two spin-wave branches measured over the full hysteresis cycle in (A) CoFe and (B) NiFe nanowires. Empty (filled) symbols correspond to the descending (ascending) part of the cycle. Included also are the dispersion relations of the saturated array of interacting nanowires (dash-dotted line), the single nanowire (dotted line), and the calculated resonance frequencies (continuous line). J. De La Torre Medina, L. Piraux, A. Encinas, Tunable zero field ferromagnetic resonance in arrays of bistable magnetic nanowires, Appl. Phys. Lett. 96 (2010) 042504. https://doi.org/10. 1063/1.3295706. Copyright American Physical Society, 2010.

thin magnetic nanowires with rectangular cross section (which also are called nanostripes or strips) were the driving force to attempt the theoretical investigation of nanowires with noncircular cross sections. A calculation of SW dispersion of linear chain of parallel cylindrical nanowires in Arias and Mills [47] was an attempt to approach this physical situation. For a quantitative fitting of the experimental data, the more exact description of the shape of the wires was a challenging task. However, to develop the exact theory of dipolar-exchange modes in nanostripes, based on the example of circular nanowires, seemed unfeasible. The method of calculations of SW frequencies and wave vectors was presented for nanowires in the magnetostatic limit, with exchange interaction neglected. The key to the solution was to apply the

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Magnetic Nano- and Microwires

extinction theorem. The special attractiveness of the this approach is that it does not use a basic set as functions, and as a result, it can be applied to wires of arbitrary cross section. In the first article on the subject [66], the application of the general results to the wires with square, rectangular, and ellipsoidal cross section was demonstrated. In the next article [67], the generalization of the approach to the case of large width/height ratio was presented, which allowed to apply the calculations to long, thin nanostripes. It is interesting that this problem was solved at the same time in another way: the rectangular cross section was approximated by an ellipsoid; then the exact solution for dipolar-exchange SW was found by the use the Mathieu basic set of functions [68].

21.3.1.4 Magnetic structure and dynamics of multilayered nanowires and magnetic nanotubes To make devices with controlled properties, recently nanomaterials of new geometries were proposed, like multilayered nanowires (MNW) and nanotubes. The attractive common feature of MNW and nanotubes is the presence of extra dimensions, which makes their properties more easily tunable, compared with nanowires. Fabrication of the first MNW was stimulated by the interest in giant magnetoresistance, observed in multilayers, and was reported almost simultaneously by several research groups [69–71]. These novel materials, consisting of alternating layers of Co and Cu, were made by deposition into disordered array of polymer pores. Later a variety of materials were used for MNW fabrication [72–74]. A remarkable progress was achieved in electrochemical synthesis of large-area highly ordered NM by using of ordered templates like nanoporous anodic alumina. Due to this technique the magnetic segments in MNW not only became ordered along the wires but also are arranged into a planar ordered matrix, that is, it is an ordered crystal in all three directions. Together with extensive investigation as a new magnetoresistive material [75–77], MNW were recognized as a perfect model structure that opens new possibilities for investigations of interactions between nanoparticles. Depending on varying parameters, like the aspect ratio of magnetic segments and its relation to the exchange length, the distance between the wires, and the height of nonmagnetic components, MNW can combine properties of planar magnetic nanodisks or elongated magnetic nanorods; moreover, it can be investigated how the properties of arrays of such nanoparticles, due to the dipolar interaction between them, are related to the geometries of long wires or multilayered thin films. Three different micromagnetic states, in-plane flower, outof-plane flower, and vortex states, were recognized in Ni/Cu MNW by the analysis of hysteresis loops, depending on the different aspect ratios of magnetic components [78]. The corresponding phase diagram, which shows what micromagnetic state can be realized in the system with various diameter/exchange length and aspect ratios, qualitatively agreed with the phase diagrams of two-dimensional arrays of elongated cylinders [79] and planar disks [80]. Configurational phase transition and reversal modes in CoNi/Cu multilayered nanowires were investigated by Tang et al. [81]. It was shown that, by varying the thickness of magnetic segment while keeping the thickness of Cu constant, the easy

Spin waves and electromagnetic waves in magnetic nanowires

637

axis can be turned from the direction perpendicular to nanowire to parallel to nanowire, while the magnetization reversal will be of different types: from coherent rotation in disk-shaped magnetic segments to a combination of coherent rotation and a curling mode in a rod-shaped ones. Targeted investigations of dipolar interactions between magnetic segments in CoCu/Cu MN by FMR and magnetometry were presented in De La Torre Medina et al. [82]. It can be observed from the FMR spectra (Fig. 21.12) that the resonance field of the multilayered sample is higher than in the sample of pure Co nonlayered sample. A comparison of experimental results and model calculations allowed to develop a comprehensive anisotropy diagram (Fig. 21.13) that describes the direction of the easy axis (parallel or perpendicular to the wire axis), depending on the aspect ratio of magnetic segments, exchange length, and the thickness of Cu. This anisotropy was proved to be only of dipolar origin. Special attention was paid recently to the magnetostrictive iron-gallium alloys (FeGa), named galfenol (for a review on galfenol alloy, see [83]), because of their anticipated use in actuators, transducers, and magnetic sensors. The first fabrication of highly ordered galfenol nanowires, deposited into nanoporous anodic alumina membranes, was reported in McGary and Stadler [84]. Then a comprehensive study of galfenol-based magnetostrictive MNW was undertaken [85]. MNW samples with different diameters of wires, aspect ratio of galfenol segments, and different thickness of nonmagnetic Cu were grown to investigate magnetization reversal properties (Fig. 21.14).

40

f (GHz)

30

20 Co CoCu CoCu(70)/Cu(4) CoCu(30)/Cu(4) CoCu(17)/Cu(4)

10 (b) 0

0

2

4

6

8

10

H (kOe)

Fig. 21.12 Dispersion relations for samples for arrays of nonlayered and multilayered nanowires. The type of the sample and the layer thickness in nanometers are indicated in the figure. J. De La Torre Medina, M. Darques, T. Blon, L. Piraux, Effects of layering of the magnetostatic interactions in microstructures of CoxCu1-x/Cu nanowires, Phys. Rev. B 77 (1) (2008) 014417. Copyright American Physical Society, 2008.

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1.0 Heff > 0

0.8

rl = I/d

Heff < 0 0.6 0.4 0.2 0.0

Heff = 0

0

1

2

3 4 rg = g/d

5

6

7

Fig. 21.13 Anisotropy diagram as a function of the aspect ratio of the magnetic rl ¼ l/d and nonmagnetic rg ¼ g/d layers. The continuous line shows the values of rl and rg for which the effective anisotropy is zero. Above this curve the effective field is positive, while below it is negative. The easy magnetization axis is represented by the white arrow. J. De La Torre Medina, M. Darques, T. Blon, L. Piraux, Effects of layering of the magnetostatic interactions in microstructures of CoxCu1-x/Cu nanowires, Phys. Rev. B 77 (1) (2008) 014417. Copyright American Physical Society, 2008.

Fig. 21.14 Galfenol-based 100-nm-diameter multilayered nanowires. A: [>25 ARFeGa], B: [4.0 ARFeGa/0.5 ARCu]29, C: [3.0 ARFeGa/3.0 ARCu]8, D: [1.0 ARFeGa/1.0 ARCu]15, E: [0.5 AR(FeGa)/5.0 ARCu]50. Here, AR ¼ aspect ratio ¼ length/diameter. The subscripts denote number of [FeGa/Cu] bilayers [85].

Magnetocrystalline anisotropy was excluded due to the randomization of crystalline easy axes of the polycrystalline segments, so the magnetic properties of such MNW can be treated as the results of the interplay between intrasegment exchange, dipolar interaction, and Zeeman energy. Comprehensive study of hysteretic properties and magnetization reversal mechanisms in FeGa/Cu MNW and comparison with individual multilayered nanowires with measurements on interacting MNW in arrays

Spin waves and electromagnetic waves in magnetic nanowires

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[86, 87] conclusively show the importance of dipolar interactions both between segments in nanowire and between nanowires in the array. Depending on the size of the magnetic segments, multilayer nanowires can present very interesting vortex structure from rigid vortex in discs to specific vortices of elongated cylinders [88], which makes the theoretical investigation of dipolar interaction between vortices convenient [89]. Recent investigations of segmented magnetic nanowires demonstrate the application of variety of novel experimental techniques. Multisegmented metallic nanowires with magnetic/nonmagnetic constituents became promising new structures for spintronic devices. In Grutter et al. [90], polarization-analyzed small-angle neutron scattering (PASANS) was used to observe a 3-D picture of magnetic ordering in high-density arrays of segmented FeGa/Cu nanowires. Due to small dimensions and disk-like shape, a single-domain, weakly inhomogeneous in-plane magnetic configuration forms inside every magnetic segment. The geometrical parameters of the sample exclude the exchange interactions between magnetic segments, while magnetocrystalline anisotropy of galfenol is excluded via procedure of randomization mentioned earlier. Different kinds of dipolar forces (intersegment inside the same wire and interwire ones) compete with Zeeman interaction and establish a highly tunable magnetic structure, as observed by PASANS, shown in Fig. 21.15. For the theoretical explanation of such noncollinear structure, a new model, which takes into account specific shape anisotropy of segments and all kinds of interactions mentioned earlier, was developed. Good agreement between theoretical calculations and experimental data confirms applicability of chosen approach to the variety of ordered magnetic nanostructures. Remnant magnetic states formed in individual Co/Cu nanowires after the application of the saturation field of different directions, that is, along or perpendicularly with respect to the wire’s axis, were investigated in Reyes et al. [91] by off-axis electron holography experiments in combination with micromagnetic simulations. The minima of the system’s magnetic energy are defined by the common action of magnetocrystalline anisotropy, shape anisotropy of dipolar origin, and the intersegment dipolar interaction. Which minimum will be “chosen” by the system in the remnant state is determined by the direction of the saturation field. When the saturation field was applied parallel to the wire, a remnant state became a magnetic vortex in each Co segment, with the cores of the vortices pointing along the wire axis. Due to the intersegment dipolar interaction, vortices often show an alternatively clockwise and anticlockwise orientation. If the field was applied perpendicularly to the wire, homogeneous in-plane magnetic configuration arises in every Co segment, with antiparallel order between segments. The third additional, single-domain-like, state where the magnetizations of all Co segments point uniformly in the wire axis direction was observed also in few samples due to statistical variations of the wire’s parameters. Multilayered nanowires with two different magnetic constituents can be used to investigate the motion of propagating domain walls. In Berganza et al. [92] the magnetic structure and magnetization switching in isolated CoNi/Ni nanowires with long segments were investigated by high-resolution magnetic force microscopy (MFM). Different values of magnetocrystalline anisotropies (MCA) of Ni and CoNi determine the variety of observed magnetic structures. The MCA of Ni is much smaller than the

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Magnetic Nano- and Microwires

Fig. 21.15 (A) Schematic view of the relevant ferromagnetic (FM) and antiferromagnetic (AFM) dipolar interwire (dashed) and intersegment (solid) magnetic interactions alongside schematics of the proposed magnetic structure under (B) zero, (C) small (7.5 mT), (D) moderate (700 mT), and (E) strong (1500 mT) magnetic field applied along the nanowire axis. From A.J. Grutter, K.L. Krycka, E.V. Tartakovskaya, J.A. Borchers, K.S.M. Reddy, E. Ortega, A. Ponce, B.J.H. Stadler, Complex three-dimensional magnetic ordering in segmented nanowire arrays, ACS Nano 11 (2017) 8311–8319.

shape anisotropy of dipolar origin in the present geometry. Due to this, Ni segments proved to be single domain with axial magnetization direction. Contrary to this, CoNi has the MCA comparable with corresponding shape anisotropy. The competition of two anisotropies of different origin can give two different magnetic configurations in CoNi segments—a single vortex state or a complex multivortex magnetic configuration—while switching between these configurations is possible by the application of a low in-plane magnetic field. These results of MFM imaging were confirmed by micromagnetic simulations. Investigation of spin-wave dynamics in cylindrical MNW was performed only theoretically. Spin-wave spectra including damping were obtained by solving the

Spin waves and electromagnetic waves in magnetic nanowires

641

Landau-Lifshitz-Gilbert equation in the “effective medium” approximation. The results could be used in constructing magnonic crystals [93]. A special case of the nanowire geometry is the case of nanotubes. Recently the interest to the theoretical and experimental investigation of ferromagnetic nanotubes is intensified (see details of fabrication in Chapter 7 of this book). There are several advantages of the tubular shape that make the potential application of magnetic nanotubes even more widespread than that of the homogeneous nanowires. For example, changing the wall thickness gives an additional varying geometrical parameter, which allows tuning of the magnetic properties more effectively. It was shown that dipolar fields in arrays of Ni nanotubes can be controlled by varying the thickness of the tube wall. It was also shown that the dipolar interaction between nanotubes is weaker than between solid nanowires, observation of which gives the possibility to decrease the undesirable effect of dipolar interaction in dense arrays [94]. Basic magnetic properties can be manipulated in a nanotube due to its large surface fraction. The temperature dependence of the magnetization also can be changed. It was shown in Sharif et al. [95] that reduced dimensionality and enhanced surface effects lead to a deviation from Bloch’s law, M ¼ (1  BTλ), where the exponent λ decreases considerably compared with both nanowire and bulk. Manipulating of domain wall motion is quite important for applications (see Part II of this book). It is known that Walker breakdown exists in bulk materials, which is a unavoidable restriction on domain wall velocity, V. However, as it was shown in Yan et al. [96] by numerically solving the Landau-Lifshitz-Gilbert equation, the Walker breakdown does not exist in nanotubes: although above the critical velocity of the vortex domain wall, the slope of the curve V(H) suddenly decreases, while after this change V(H) continues to increase (Fig. 21.16). It was shown that this striking feature of domain wall motion in nanotubes is connected with magnon emission by the domain wall, which is a process similar to the Cherenkov emission of photons. Dispersion of SW, associated with the domain wall, was calculated numerically, and the result on dispersion, as shown in Fig. 21.17, is in agreement with analytical calculations [97].

Fig. 21.16 Domain wall velocity as a function of field applied along a permalloy nanotube. Dots: simulations. The dashed line corresponds to the velocity at the Walker limit. The solid lines above and below this dashed line are linear fits to the data of two distinct regions, Walker regime (below) and magnonic regime (above) [96].

Magnetic Nano- and Microwires

w (k) (1011 Hz)

642

4

2

0

0

1

k (108/m)

2

3

Fig. 21.17 Spin-wave dispersion relation of a saturated permalloy nanotube obtained from numerical calculations. The line is a guide to the eye [96].

In a nanotube the magnetization singularity on the axis, which exists in wires, is “cut out.” It permits to provide a comprehensive and rigorous theoretical investigations of technologically important properties of nanotubes. The magnetic configurations and reversal in thick nanotubes, with opposite chirality of end domains [98], and thin ferromagnetic nanotubes, with the same chirality of end domains [99], were investigated recently both by micromagnetic simulation and analytical approach. A theoretical procedure was proposed to control the propagation and chirality switching of a vortex domain wall in nanotubes by magnetic field pulses [100]. Theoretical and experimental investigation of SW dynamics in single nanotubes is a challenging task. If the magnetization and external field are directed along the tube’s axis, only the approximate theoretical solution for SW dispersion is known in the limit of infinitely long and extremely thin nanotube, that is, when the inner radius approaches the external radius,r ! R [101]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 

  H0 1 H0 1 2 2 Ω ¼ MS +J k + 2 +J k + 2 1+ R R MS MS 2

(21.15)

where k is a wave vector along nanotube’s axis, J is an exchange integral, and H0 is an axially aligned magnetic field. This theoretical result turned out to be helpful for the explanation of chirality-dependent spectra of SW in nanotubes, received by micromagnetic simulation, as it will be reviewed in the following paragraph. However, a quantitative agreement between theoretical and experimental results of SW dynamics in nanotubes still was not achieved.

21.3.1.5 Effects of curvature and torsion on the magnetic dynamics in nanowires and nanotubes Chiral properties give one more possibility to tune magnetic parameters of magnetic nanowires and nanotubes for applications in novel magnonic communication and logic devices. In solid magnetic nanowires, new 3-D structures with artificially fabricated curved geometries became a new and promising task. In nanotubes the

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influence on spin dynamics of the chirality of magnetic order is under investigation. We consider here both kinds of the origins of chiral properties and their influence on spin-wave dynamics in the nanotubes and nanowire-based nanostructures. The usual source of stabilization of chiral magnetic order in magnetic materials is the Dzyaloshinskii-Moriya interaction (DMI) arising in crystals of certain asymmetry. The influence of chiral properties on magnetic dynamics was investigated previously in magnetic films, both experimentally [102] and theoretically [103]. It was shown that due to the DMI the spin-wave dispersion relation is asymmetric with respect to wave vector inversion. So, the result of investigation of SW dynamics in nanotubes with taking into account the chirality of magnetic structure of such objects was actually predictable. However, special importance of such investigations is that chiral properties of nanotubes are much more tunable than “usual” DMI and corresponding magnetic orders in crystals. This gives additional possibilities of tailoring SW dynamics by geometry of nanostructures for practical usage. Nonreciprocal SW propagations in such case mean that SWs running in opposite directions along nanotube with the same modulus of wave vector have different oscillation frequencies, f(kz) 6¼ f( kz). This effect was studied in infinitely long nanotube with the vortex magnetic state induced by a circular external magnetic field [104, 105]. Effect of asymmetry of SW dispersion in finite-length soft magnetic nanotube was investigated by micromagnetic simulations in Yang et al. [106]. Chiral properties of the system are originated from vortex like structures that are known as arising at the ends of the nanotubes due to the interplay between dipolar and exchange interactions. Relative chirality orientations of the vortex like structures at both ends of the nanotube can be of two different kinds, parallel rotation of magnetization in vortices and antiparallel. Due to geometrical confinement of the nanotube along its axis, a variety of standing SW modes were observed. Effect of asymmetry of SW dispersion was manifested only in the case of antiparallel orientation of the vortices. Physical reason of asymmetry of SW dispersion was explained via simple phenomenological model that employs the previously developed [101] analytical approach for SW dispersion in infinite nanotubes (see previous paragraph, formula (21.15)). Lateral confinement leads to arising of discrete (quantized) values of axial wave numbers kn depending from pinning conditions at nanotube’s edges. There are new, functional 3-D magnetic nanostructures like nanohelices, which are artificially curved magnetic nanowires. New methods of fabrication like 3-D nanoprinting of magnetic materials by focused electron beam-induced deposition (FEBID) open new possibilities to design such complex magnetic nanostructures [107] and motivate new theoretical and experimental investigations, focused on magnetostatics and dynamics in curved nanowires [108–110]. General theoretical approach [111] leads to the conclusion that in wires and shells the curvature of their shape gives an additional contribution to the magnetic energy functional: effective magnetic anisotropy and effective Dzyaloshinskii-like interaction. This theory was applied to the magnetic helix wire, where common action of curvature, torsion, magnetic anisotropy, and the effective DMI leads to specific magnetic structures: quasitangential distribution (analog of the vortex state in rings) and onion state for the cases of weak and strong curvature effects, respectively [112].

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The dispersion relation for SW with wave vector q along the wire contains the “usual contribution,” quadratic with respect to q, (invariant with respect to the transformation q !q) , and a linear one with respect to q antisymmetrical part, originated from the effective DMI. Such dispersion function achieves its minimum at finite value q0, and corresponding gap depends on the anisotropy constant and on the curvature and the torsion of the wire. Localization of domain walls [113] and of SWs [114] in a local bend of the nanowire is one more effect recently described theoretically and by computer simulation. It has been shown that the local curvature in nanowire plays a role of pinning potential for magnetic excitations. Dispersion of SWs was found in the case of boxlike curvature and for sharp bend. In the last case, three methods give good agreement of the results: analytics; numerical solution of Landau-Lifshitz equation, which in this case comes down to eigenvalue problem for generalized Schr€odinger equation; and spinlattice simulation.

21.4

Interactions of EMWs with nanowires

21.4.1 EMW propagation in nanostructures Nanowire systems combine the properties of high-magnetization metals with dielectric properties of the matrix. Due to the resulting electrical and magnetic properties and the flexibility to control the shape and size of such systems, the research into the interaction of the nanowire systems with EMW is motivated by the high-frequency applications of the nanowire composites. The key to these application lies in the fact that the diameter of these nanowires is much less than the skin depth, and as a consequence, the electromagnetic field completely penetrates the wires. The individual metallic wires are separated from each other by a nonmetallic, insulating matrix of the alumina substrate. Thus the whole structure behaves as an insulator, that is, a low-loss material. At the same time the large magnetization of the metallic ferromagnet (Fe, Ni, Fe-Ni, and Co) ensures the high-frequency operation. The large aspect ratio and the parallel arrangement of the wires create a favorable uniaxial shape anisotropy, and a tunable remanent state, such that self-biased devices can be built without the need for bulky and expensive permanent bias magnets. To have a high magnetization, these nanowire structures are dense. As a result the influence of the dipolar interactions must be taken into account in the design. Numerous publications deal with the modeling of the dipolar field for nanowire structures, as described in Section 21.2 of this chapter.

21.4.1.1 EMW interactions with nanowire structures For successful application of electromagnetic phenomena, the understanding of the details of the EMW propagation in these devices is required. As early as in 1887, Lord Raleigh discussed wave propagation in periodic lattices. He developed the mathematical formalism to describe the wave-material interaction. From there many fruitful applications followed, one of them was the formalism of Bloch waves

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for the description of electrons in a crystal. In a review by Elachi on waves in active and passive periodic structures, the theory of propagation in unbounded and bounded periodic media and a wide variety of applications were discussed [115]. In the 1960s the main emphasis was on the exact solution of EMW equations in periodic, sinusoidal, and laminated media. With the arrival of periodic nanostructures and potential applications, it was necessary to develop theories of the interaction of these nanostructures with EMW and describe the propagation and diffraction effects in geometrically confined (bounded) structures. It should be kept in mind that EMW treatments include optical frequencies (light propagation) and X-rays. Some aspect of optical and plasmonic effects in nanowire metamaterials are briefly reviewed in Section 21.4.2. With the advent of periodic magnetic nanostructures, the task of EM analysis became even more complex. The characteristic size of the elements, 2r, of these one-, two-, and three-dimensional (1-D, 2-D, and 3-D) periodic structures is much below the wavelength λ of the EM wave in the material, 2r < λ. Still the EM field over any of the elements is rapidly changing, making it necessary to include propagation effects. Moreover, the periodicity of the nanoelement lattice Λ (wires, layers, and spherical particles) is not commensurate with the rf field periodicity, Λ 6¼ nλ, as illustrated in Fig. 21.18. To describe the EMW propagation in magnetic nanostructures, it is necessary to solve Maxwell’s equations together with the Landau-Lifshitz equation rigorously and with proper electrodynamic boundary conditions, including the possible nonlinear response of the system. At the same time, to model very long, parallel nanowires is a relatively simple task among the periodic magnetic nanostructures. Usually, there is an easy axis of magnetization, coinciding with the wire long axis and with the direction of the applied bias dc magnetic field. An often neglected aspect of the high-frequency behavior of small particles is that for the nanowire array geometry there is a large microwave magnetic field, generated outside the wire of radius r, by the precession of the magnetization. For a spin wave with finite wave vector k, this field falls off as exp(2kr)/r1/2 far from the wire. For long-wavelength modes, this field has a very long range. In the limit of zero wave vector, the field created by the precession of the magnetization falls off inversely with the square of the distance from the center of the wire. The existence of this large field outside the nanowire is a substantial change with respect to the thin-film case. For the

λ EMW

2r L

Fig. 21.18 Incommensurate size relationship between the EM wavelength λ, propagating in a 2r diameter nanowire array of periodicity Λ, length ‘, where λ > ‘ > Λ, resulting an inhomogeneous rf field distribution over the nanowires.

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thin film, for the uniform precession mode, excited in an FMR experiment, the macroscopic magnetic field created by precession of the magnetization is completely confined to the film. For a spin wave whose wave vector k is parallel to the film surfaces, there is a macroscopic field outside with the spatial variation  exp(2kz), if the film surfaces are parallel to the xy plane, but the prefactor—which controls the strength of this field—scales as 4πMS(kd), where d is the film thickness. This field is thus very weak in the thin-film limit or whenever the wavelength of the spin wave is large compared with the film thickness. On the other side the existence of large rf demagnetizing fields outside the nanowire has interesting implications. They contribute to the interactions between the nanowires, so one is led to explore the collective excitations of the nanowire array. By changing the ratio of the interwire separation to the range of interaction, spin waves of variable wavelength k could be excited. In the case of nanowire arrays, the nature of their collective excitations due to dynamic demagnetizing effects is a most interesting topic, and it was discussed in Section 21.3.1. This dynamic contribution to the interactions at high frequencies will affect the “proximity effects,” as described earlier. The increased interaction strength is expected to reduce the threshold wire separation for the onset of magnetic percolation. This effect has a very strong shape dependence, as the dynamic demagnetizing fields are very sensitive to the shape. Several approaches were developed to treat the EMW interaction with nanowire structures; however, all those methods have to face some numerical complications: the devices, based on magnetic nanostructures, are of finite size, roughly scaled with half of the operating microwave wavelength, that is, the size can be several millimeters, too big for a usual micromagnetic simulation, while the nanometer size elements require fine discretization. Another significant problem is to deal with the statistical distribution of the huge number of particles in any device-size geometry. The question how to take into account the statistical distribution of particle size, shape, and separation in calculating the interactions in a nanowire (or other nanostructured magnetic system) is mostly open. Likewise the temperature dependence is usually ignored; however, these nanoparticle devices have to operate in a feasible wide temperature range, where the stability of such small magnetic particles against thermal fluctuation is uncertain. Most of the theoretical modeling and/or numerical simulation of FMR and spin dynamics in an applied rf field is based on the Landau-Lifshitz (L-L) formalism, including damping, and usually neglecting exchange interactions. Either the “magnetic” approach deals with the system as an assembly of isolated spins, or the magnetic particles are treated as macrospins on a regular array. An early review on the simulations of magnetodynamics in nanometric systems was published by Schmool [116]. Micromagnetic calculations were performed by several authors. It is based on solving the L-L equation with damping included. The most timeconsuming part of the calculations is to find the demagnetizing tensor elements, necessary to obtain the equilibrium magnetic structures. The nanostructure chapter of Schmool [116] is mostly about nanodots, but the methodology is adaptable for other geometries, like to wires. Rivkin [117] applied translational symmetry

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principles to analyze an infinite array of magnetic dots and got good agreement with experiments. Another way to model composite magnetic materials is using an effective media approach. Calculating the permeability tensor and the permittivity, the EMW propagation in a nanocomposite of magnetic particles embedded in a dielectric matrix can be analyzed by using the Maxwell Garnett formula. The task is to calculate microscopic parameters and from there to determine the macroscopic behavior of the nanostructure. Hillion analyzed the propagation of a harmonic plane wave in a medium, consisting of magnetic particles in a periodic dielectric matrix and described the resulting TE and TM modes [118]. Boucher and Menard [50] applied a similar, general model to obtain the effective permeability and permittivity tensors of ferromagnetic nanowire arrays, based on a Maxwell Garnett homogenization procedure. It incorporates the effects of the geometric parameters of the array, the shape of the wires, and their intrinsic dielectric and magnetic properties. Effective external tensors, which include the dynamic dipolar dielectric and magnetic interwire interactions, are then derived to provide a link between the effective susceptibilities of the array and those measured in microwave cavity experiments. The authors of Wang et al. [119] and Liberal et al. [120] applied a numerical homogenization technique to create an effective medium and extract the dispersive and nonreciprocal permeability tensor of a magnetic nanowire system. Using the resulting parameters, they analyzed the performance of microwave devices (isolators and phase shifters) by full-wave simulations. Applying a multilevel fast multipole algorithm (MLFMA), the scattering properties of a 3-D array of deformed nanowires revealed the field enhancement for the forward scattering of deformed nanowires as compared with the nondeformed case [121]. Approximate electrodynamic boundary conditions were established and applied for nanowire systems to be able to describe the behavior of the electric and magnetic field at the edges of finite-size arrays, an important problem in device design in Lisenkov et al. [122]. It is possible to solve the electrodynamic problem for 3-D nanostructures rigorously, by applying a mathematical technique, developed for MMIC microwave devices, based on the decomposition into autonomous blocks with virtual Floquet channels (FAB) [123]. A rigorous electrodynamic model was built [124], based on the solution of the nonlinear 3-D propagation-diffraction boundary problem, to describe the interaction of EMW with anisotropic 3-D nanostructures. The model was applied to solve the size and shape effects in nanowire arrays up to photonic frequencies. The model solves the nonlinear Maxwell’s equations (21.16), (21.17) with electrodynamic boundary conditions, together with the L-L equation, including the exchange term Eqs. (21.18), (21.19): !

curl H ðtÞ ¼ ε0 ε

!

! ∂ E ðtÞ + σ E ðtÞ ∂t !

∂ B ðtÞ curl E ðtÞ ¼  ∂t !

(21.16)

(21.17)

648

Magnetic Nano- and Microwires ! h ! ! i  ! ! ! ∂ M ðt Þ ¼ γ ð M ðtÞ  H ðtÞ + H q ðtÞ + ωr χ 0 H ðtÞ M ðtÞ ∂t !

(21.18)

!

H q ðtÞ ¼ qΔ M ðtÞ

(21.19)

where E and H are the EM fields, M is the magnetization, Hq is the effective exchange field, σ is the electrical conductivity, γ is the gyromagnetic ratio, ωr ¼ αγH0 is the relaxation frequency, χ 0 is the susceptibility, q ¼ 2A/μ0M0 is the exchange constant, and H0 and M0 are the bias magnetic field and the saturation magnetization. A 3-D periodic array of nanowires is assumed, as shown in Fig. 21.19, with a, b, and c periodicity along axes x, y, and z. The 3-D magnetic nanocomposite is divided, by using the decomposition of the elementary cell of array, onto FABs (2r, diameter, and l, length of nanowire). The cells are in the form of rectangular parallelepipeds with virtual Floquet channels (FAB). A monochromatic homogeneous plane EMW (TEM wave; fields E ¼ Ex0, H ¼ Hy0; wave vector k; frequency ω) is incident on the input cross section S1 of a 2-D periodic array of metallic magnetic nanowires, embedded in a nonmagnetic, dielectric matrix, having relative permittivity εr ¼ 5 and relative magnetic permeability μr ¼ 1. A bias magnetic field H0 ¼ H0y0 is applied normal to the propagation direction z (Fig. 21.19A, along the axis of nanowires (Fig. 21.19B). For the calculations the model of the elementary cell of the 2-D periodic array is considered, where each cell contains one ferromagnetic nanowire a

H0

c

a

H0 y

b

H

x (A)

E

z 0

c

V0

l

k

r

(B)

b

ev

V

S2 S1

(C)

Fig. 21.19 Geometry of the magnetic nanowire array for the EM diffraction problem: (A) direction of incident plane EMW of wave vector k; (B) 2-D array of nanowires, periodicity a, b, c; 2r, wire diameter; l, wire length; the bias magnetic field H0 is normal to k; (C) model of the cell of the array using the autonomous blocks with Floquet channels (FAB) with input cross sections S1, S2. a ¼ 3.5 r; b ¼ 1.25 l, c ¼ 2r. G. Makeeva, M. Pardavi-Horvath, O. Golovanov, Size and shape effects in the diffraction of electromagnetic waves on magnetic nanowire arrays at photonic frequencies, PIERS 2009 Moscow Proceedings, The EM Academy, Cambridge, 2009, 1888–1891 http://piers.org/ piersproceedings/piers2k9MoscowProc.php, reproduced courtesy of The Electromagnetics Academy.

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(Fig. 21.19C) as an autonomous block with Floquet channels (FAB). The values of parameters used in the calculations for magnetic nanowires are 2r ¼ 50 nm and l ¼ 500 nm of Fe (4πM0 ¼ 21,580 kG; exchange constant A ¼ 2.2  109 Oe cm2; Gilbert damping α ¼ 0.0023; conductivity σ ¼ 1.03  105 Sm1). The periodicity of the array is a ¼ b ¼ 256 nm, с ¼ 550 nm.

21.4.1.2 Shape and size effects in the EMW propagation in nanoarrays Using the computational algorithm for calculating of FAB conductivity matrix Y [125], the complex wave number Γ 0 of the fundamental mode of the clockwise and counterclockwise polarized modes and quasiextraordinary EMWs in 3-D arrays of magnetic nanowires were determined from the characteristic equation [126]. The main results of the calculation of the real and imaginary parts of complex wave number Γ 0 of the fundamental mode of quasiextraordinary EMW (k ¼ ky0), depending on bias magnetic field H0 ¼ H0y, at f ¼ 26 GHz for the parallel orientation of the bias magnetic field H0 to the axis of magnetic nanowires are shown in Fig. 21.20. For the case presented in Fig. 21.20, the separation of the magnetic nanowires of diameter 2r ¼ 25 nm is large (a ¼ b ¼ 256 nm, с ¼ 550 nm), and as for lower-density arrays, the model of noninteracting nanowires is applicable. The wires behave as a thin long cylinder in a bias magnetic field H0. Respectively, for longitudinal orientation of the magnetic field H0, the DMF are Nx ¼ Ny ¼ 2π and Nz ¼ 0, and in this case the eigenfrequency of the FMR of the array is [127]

Fig. 21.20 Calculated real and imaginary parts of the complex wave number Γ 0 (propagation constant) of the fundamental mode of quasiextraordinary EMW (k ¼ kz0) depending on bias magnetic field H0 ¼ H0y0 in a 3-D array of Fe nanowires (2r ¼ 25 nm, ‘ ¼ 500 nm). f ¼ 26 GHz; — Re Γ 0; ——— Im Γ 0. Courtesy of G. Makeeva.

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ω0 ¼ γ ðH0 + 2πM0 Þ

(21.20)

Upon reducing the periodicity of the magnetic nanowires to a < 340 nm, the system becomes strongly coupled, and the exchange interaction plays the dominant role. For the diameter of the nanowires 2r ¼ 25 nm at separation a ¼ b ¼ 67 nm, the array behaves as an effective quasicontinuum, that is, a thin magnetic film with DMF Nx ¼ 0, Ny ¼ 4π, and Nz ¼ 0, and the eigenfrequency of the FMR for high-density arrays becomes ðω0 =γ Þ2 ¼ H0 ðH0 + 4πM0 Þ

(21.21)

The increase of the resonance field with decreasing separation as given in Eq. (21.21) is illustrated in Fig. 21.21. The calculated bias magnetic field H0 dependence of the imaginary part of the effective permeability Im μ of a 3-D Co80Ni20 nanowire array (4πM0 ¼ 15,356 kG; α ¼ 0.005; σ ¼ 1.0  107 S m1; A ¼ 1.50  109 Oe cm2) is shown in Fig. 21.21, for variable periodicity of the arrays: H0 ¼ H0y0 at f ¼ 26 GHz. There is an interesting and important consequence of the strong interaction in nanoparticle arrays. By a similar model, the full nonlinear propagation effects were studied on a regular 3-D array of Fe nanospheres, assuming that a wave packet is propagating in the nanostructure, and in the arrays the propagating EMW is a superposition of inhomogeneous plane waves. The dependence of the propagation constants Γ 0 of the EMWs on the lattice period of nanoparticles on a 3-D periodic array of r ¼ 150 nm Fe nanospheres was calculated for the fundamental mode of clockwise and counterclockwise polarized EMWs and for the ordinary and extraordinary EMWs, propagating in the array at f ¼ 30 GHz and shown in Fig. 21.22 [124]. 35

Imm1 a = b = 67 nm

30 25

a = b = 76 nm a = b = 87 nm

20 15 10

a = b =150 nm

5 a = b =340 nm 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 4.0 9.0 9.5 H0, kOe

(A)

Fig. 21.21 Bias magnetic field H0 dependence of the imaginary part of the effective permeability of a 3-D array of magnetic nanowires of Co80Ni20 (4πM0 ¼ 15,356 kG; α ¼ 0.005; σ ¼ 1.0  107 Sm1; A ¼ 1.5  109 Oe cm2) for variable periodicity of the arrays with 2r ¼ 25 nm; ‘ ¼ 500 nm; H0 ¼ H0y0; f ¼ 26 GHz. Courtesy of G. Makeeva.

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n 2p f e0 m0 15 1

10

3

5 0 –5i

0.05

0.1

0.15

0.2

0.25

0.3

0.35

r a

2

–10i –15i

4

Fig. 21.22 Propagation constants of EMWs in a 3-D magnetic nanosphere array depending on r/a, the ratio of the Fe nanosphere radius r ¼ 150 nm to the periodicity a: 1, 2—clockwise and counterclockwise polarized EMWs; curves 3, 4—ordinary and extraordinary EMW; H0 ¼ 1000 Oe; f ¼ 30 GHz. G. Makeeva, M. Pardavi-Horvath, O. Golovanov, Electrodynamic analysis of nonlinear propagation of electromagnetic waves in gyromagnetic nanostructured media at microwave frequencies, PIERS 2009 Moscow Proceedings, The EM Academy, Cambridge, 2009, 1883–1887, http://piers.org/piersproceedings/piers2k9MoscowProc.php, reproduced courtesy of The Electromagnetics Academy).

Fig. 21.22 demonstrates that, due to the change of the character of the propagation of EMWs with decreasing separation r/a, the propagation constants change significantly on the interval 0.1 < r/a < 0.35. The propagation constants of counterclockwise polarized and extraordinary modes become imaginary for r/a > 0.25, and consequently, these waves are not propagating waves. As it is well-known, EMW do not propagate in metals—that is, the nanoarray behaves like a metal, reaching the percolation length for conduction without direct metallic contact between the particles! Upon further reducing the separation of nanospheres to r/a > 0.25 (transition to the range of exchange length), the exchange interaction in the system of strongly coupled magnetic nanoparticles plays a dominant role, and the magnetic nanoarray starts to behave like a quasibulk continuum [128]. As for the case of ferromagnetic metals, from the four normal modes, there are two, counterclockwise polarized and ordinary modes with real propagation constants, having a large phase velocity, propagating in the gyromagnetic nanostructured media as the separation r/a of magnetic nanoparticles 0.25 < r/a < 0.35 approaches the exchange length. There is an interaction range where extremely-low-loss nanocomposites can be designed. Using the values of the computed propagation constants and the dispersion relations, the complex diagonal μ1 and off-diagonal μ2 components of the effective permeability tensor and the effective permittivity ε of the periodic magnetic nanosystems, depending on r/a, were calculated [129]. For weakly interacting nanospheres the

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magnetic losses are about two orders smaller, while the nonreciprocity is doubled as compared with YIG. Upon reducing the separation of nanospheres to r/a ¼ 0.35, the imaginary part of the permeability increases due to an additional loss mechanism by the excitation of spin waves in the system. The method of decomposition approach by autonomous blocks with virtual Floquet channels is capable to calculate the propagation and diffraction of EMWs in 3-D complex nanostructured systems, containing dielectric, conducting, and magnetic materials. Carbon (graphene) nanotubes (CNT) can be functionalized by inserting magnetic metal nanoparticles into the CNT, thus creating MFCNTs, for highfrequency shields in extreme environments. It was shown by Avramchuk [130] that the mechanism of the EMW-MCNT interaction in the X-band (8–12 GHz) is mainly by reflection, while in Ka band (26–40 GHz) it is by absorption. For the case of a Co-Ni MCNT, the FAB scattering matrix R was determined by solving the 3-D diffraction boundary problem for the full Maxwell’s equations with electrodynamic boundary conditions, complemented by the Landau-Lifshitz equation with the exchange term, using the Galerkin’s projection method [131, 132]. The decomposition scheme is shown in Fig. 21.23. The real and imaginary parts of the complex wave number Γ 0 (H0) ¼ Re Γ 0 + Im Γ 0 of the fundamental modes (ky) of clockwise and counter clockwise polarized EMWs (k ¼ ky0) and quasiextraordinary EMWs (k ¼ kz0), were determined, depending on bias magnetic field H0 ¼ H0y at f ¼ 26 GHz and at 1 THz. The MCNT material is a nonlinear, anisotropic, 3-D periodic array of CNTs (2r ¼ 25 nm; ‘ ¼ 500 nm; d ¼ 3 nm; σ ¼ 2.5 S m1; εr ¼ 62); the magnetic nanoparticles are Co80Ni20 (2R ¼ 19 nm; 4πM0 ¼ 15.356 kG;

H0 H0

2r 3 1 4

y by

c

d

l

2 3

bz

k bx x

z

b a 0

b

d

Fig. 21.23 The decomposition model for a 3-D periodic array of MCNT. The bias magnetic field is along the y-axis, a—direction of wave vector k of propagating EMW; b—periodic 3-D array of MFCNTs, bias magnetic field H0; c—a cell of array: 1—CNT; 2r, MFCNT diameter; ‘, MFCNT length; d, CNT wall thickness; 2—magnetic nanoparticles; d, decomposition of cell of array onto FABs (3, 4—types of FABs) [132].

Spin waves and electromagnetic waves in magnetic nanowires

20

Γ/k0

653

2r Magnetic nanoparticles

15

c

l

k Ho

10

1

b

a

5 2

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0 H0, kOe

Fig. 21.24 Stop band in EM propagation in a 3-D CNT, filled with magnetic particles (MCNT). Real and imaginary parts of complex wave number Γ 0 of the fundamental mode of clockwise (1) and counterclockwise (2) polarized EMWs (k ¼ ky0), depending on bias magnetic field H0 ¼ H0y0 CNT: (2r ¼ 25 nm; l ¼ 500 nm; Δ ¼ 3 nm; σ ¼ 2.5  m1  m1; εr ¼ 62); Co80Ni20 (2R ¼ 19 nm; 4πM0 ¼ 15.356 kG; α ¼ 0.005; σ ¼1.0  107 S m1; A ¼ 1.5*109 Oe cm2) for filling factor N ¼ 20. The periodicity of array is a ¼ b ¼ 76nm, с ¼ 525nm; f ¼ 26 GHz; — Re Γ 0; - - - Im Γ 0.

A ¼ 1.5  109 Oe cm2; α ¼ 0.005; σ ¼ 1.0  107 S m1). The number of particles in a CNT was varied from N ¼ 20 to 5. Fig. 21.24 illustrates the bias field dependence of the real and imaginary part of the fundamental mode of clockwise and counterclockwise polarized EMWs for a 3-D array of MCNTs, each filled with 20 magnetic nanoparticles particles. The existence of a stop gap in the bias field dependence for propagation of clockwise polarized EMWs is a surprising result, indicating the metamaterial character of the composite MCNT [132].

21.4.1.3 EMW scattering in nanowires at THz frequencies In the gigahertz frequency range, the EM wavelength is still much larger than the size of the particles, and any treatment has to take care of propagation and magnetostatic effects, as before. What happens when the EMW frequency is getting comparable with the characteristic nanometer size of a particle or a 2-D or 3-D nanoarray? Indeed, this frequency range is in the terahertz, where many intriguing new phenomena are taking place, potential magnetophotonic bandgap effects are anticipated, and applications can be expected to be developed. In the terahertz frequency range, the penetration (skin) depth δ1 of EMW in conducting (metallic) media can be close to the nanowire diameter δ1  2r, leading to interesting resonance and spin-wave effects. The normal skin effect is present, if the mean free path of conductivity electrons is smaller than the penetration depth δ1 of the EM field in a metal. For a ferromagnetic metal nanowire, the skin depth is given by  1=2 1 ¼ δ1 ¼ δ= jμeff j + μ00eff k0

rffiffiffiffiffiffiffiffi 1=2 ω  = jμeff j + μ00eff 2πσ

(21.22)

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where δ is the skin depth in a metal, ω is the frequency, σ is the electrical conductivity 0 of the metal, and μeff ¼ μeff  iμ00eff is the effective scalar permeability [127] determined by the components of the high-frequency dynamic permeability tensor μ ^ [5]. The effective scalar permeability μeff depends on the dynamic permeability μ ^, and it is a function of frequency ω, the orientation of the bias magnetic field H0, and the rf magnetic field H of the exciting EM wave. The skin depth in metallic ferromagnetic wires δ1 (Eq. 21.22) depends not only on the conductivity but also on the magnetization, and at THz frequencies, it can become on the order of the size 2r  10 nm (nanowire diameter) due to the large values of permeability (jμeff j + μ00eff ≫ 1). If the nanowire diameter is larger than the penetration depth of the EM field 2r > δ1, there is an inhomogeneous distribution of the magnetization and the rf magnetic field across the nanowire. This results in the modification of the spectrum of excited collective exchange SW modes in such ferromagnetic nanowire arrays at terahertz frequencies. To get an insight into the terahertz behavior of a 2-D nanowire arrays, the diffraction of EMW on arrays of ferromagnetic metallic (iron) nanowires was modeled at terahertz frequencies at electromagnetic accuracy by solving the 3-D diffraction problem for Maxwell’s equations with electrodynamic boundary conditions, complemented by the Landau-Lifshitz equation including the exchange term. Using the computational algorithm based on the decomposition approach by autonomous blocks with Floquet channels (FABs), as in Golovanov [123] and Golovanov and Makeeva [125], the scattering parameters of the S matrix of 2-D magnetic nanowire arrays, depending on the bias magnetic field H0, were calculated for nanowire diameters 10 < 2r < 60 nm at a frequency of 30 THz [133]. The geometry of the wire array and the parameters are the same as in Fig. 21.19. The transmission jT21 j and reflection jR11 j coefficients were calculated at a bias field of 75 Oe for 2-D periodic arrays of nanowires. The scattering parameters depend very strongly on the nanowire diameter in the 10  2r  60 nm wire diameter range. There is an optimal geometry where there is a transmission maximum jT21 jmax at 2r ¼ 27 nm, coinciding with the minimum of the reflection j R11 jmin. Upon increasing the wire diameter, the reflection is increasing to jR11 j ¼ 1, while the transmission is gradually decreasing. When in the array of metallic nanowires the diameter of the nanowires 2r is larger than the skin depth δ in a metal at a frequency f ¼ 30 THz, the EMW are damped with the attenuation coefficient k00  1/δ, [127]: rffiffiffiffiffiffiffiffi 1=2 pffiffiffiffiffiffiffiffiffi 2πσ  ¼ μeffR =δ ¼ 1=δ1 k ¼ k0 jμeff j + μ00eff ω 00

(21.23)

where k0 is the wave number of EMW in free space, μeff ¼ μ0eff  μ00eff is the effective scalar permeability, μeffR ¼ jμeff j + μ00eff, and δ1 is the skin depth in the ferromagnetic metallic medium. According to Eq. (21.23) the minimum of the absorption of EMW is determined by the maximum of the skin depth δ1 ¼ δ(μeffR)1/2. This maximum is realized when the effective permeability μeffR has a minimum (i.e., the real part of the effective

Spin waves and electromagnetic waves in magnetic nanowires

655

permeability μ0eff ¼ 0 and the imaginary part of the effective permeability μ00eff ! 0), that is, in the antiresonance point. The frequency of the ferromagnetic antiresonance (FMAR) in an unbounded ferromagnetic metallic medium is ωar ¼ ωH + ωM

(21.24)

where ωar is the frequency of antiresonance; ωH ¼ γH0 and ωM ¼ 4πMs. Applying the condition for antiresonance in Eq. (21.24) to a finite-size ferromagnetic ellipsoid, the resonance condition becomes ωar ¼ ω0 + ωM

(21.25)

where ω0 is the eigenfrequency of the FMR of the magnetized ellipsoid, determined by the internal magnetic field H0int: $

H0 int ¼ H0  N M 0

(21.26)

$

where N is the demagnetizing tensor (introduced in Section 21.2.1.2). Upon increasing the bias magnetic field H0, several types of exchange spin-wave modes are excited in the ferromagnetic metal (iron) nanowire array at f ¼ 30 THz. These are radial surface spin-wave modes with complex wave numbers, having hyperbolic distributions of the rf magnetization, satisfying the boundary conditions on the surface of the nanowire, and depending on H0. When the standing spin-wave resonance conditions in magnetic nanoarrays are satisfied, the frequencies of the radial antiresonance modes become [127] ωaresn ¼ ω0n + ωM

(21.27)

where ω0n are the eigenfrequencies of the standing spin-wave resonances of exchange spin-wave modes of order n in a thin ferromagnetic nanowire, determined by the internal magnetic field H0int. The maxima of the transmission coefficients jT21 j are located in the antiresonance points as given by Eq. (21.27). When the skin depth δ1 ¼ δ(μeffR)1/2 near these antiresonance points increases, the transmitted EMW through the magnetic nanoarray are also increasing. The effective permeability depends on the interactions in the array through the separation and the diameter of the nanowires; thus the transmission can be influenced by array geometry. The maxima of j T21 j can be tuned by the bias magnetic field H0 for different wire diameter/periodicity ratios. Upon increasing the wire diameter, the maximum of the transmission coefficient is moving to lower fields, and the transmission at the maximum is decreasing. The rf magnetization profiles of the exchange SW modes are numerically simulated taking into account the skin depth of the EMW in a ferromagnetic metal (Fe) at terahertz frequencies [28, 134]. The EMW having a wave vector k ¼ kz0 and fields E ¼ Ex0, H ¼ Hy0, is incident on a 2-D array of ferromagnetic metallic nanowires, biased by

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Magnetic Nano- and Microwires

a magnetic field H0 ? k. The wire axis is y0, the wire diameter 10  2r  25 nm, and length ‘ ¼ 300 nm. If the nanowire diameter is somewhat larger than the penetration depth of the EM field, there is an inhomogeneous distribution of the magnetization and the magnetic field close to the nanowire surface. This results in the modification of the spectrum of excited collective exchange SW modes in such ferromagnetic arrays at terahertz frequencies. The rf magnetization profiles of these modes are numerically simulated for iron nanowires and shown in Fig. 21.25 [28, 134]. Fig. 21.25A shows the bias field dependence of the transmission spectra for different nanowire diameters. The extrema correspond to specific SW mode excitations. The transmission maxima are the FMAR points [133]. The minima of transmission curves correspond to the minima of the skin depth, that is, when the effective permeability μ00effR has a maximum, that is, to the FMR. At standing SW resonances μ00effR and the absorption of power increases, and δ collapses.

|T21| 2

0.8 0.6 1

0.4

a

4

3

b c d

0.2 0

(A)

100

10

1000 1

H0, Oe

10,000

Mj /Mjmax

0.8 0.6

4

0.4

3

0.2 –5

–4

–3

–2

–1 0 –0.2 –0.4 –0.6 –0.8

(B)

1

2

3

4

5 r. nm

1 2

–1

Fig. 21.25 (A) Bias field dependence of the transmission coefficient at 30 THz for different wire diameters, a–d corresponding to 2 r ¼ 10, 15, 20, and 25 nm; (B) magnetization profiles of radial surface exchange SW modes across a nanowire in a 2r ¼ 10 nm iron nanowire array, for different values of the bias field H0. Points 1–4 in (A) correspond to curves in (B): 1, 2 n ¼ 2; 3, 4 n ¼ 1. H0 ¼ H0y0 [28, 134].

Spin waves and electromagnetic waves in magnetic nanowires

657

Upon increasing the bias field H0, the second-order SW modes n ¼ 2 and then n ¼ 1 (k ¼ 0) modes are excited by the spatially uniform TEM wave. However, these modes are influenced by the skin effect. These are radial surface SW modes with complex wave numbers, having a hyperbolic distribution of the rf magnetic field, as shown by curves 1–4 in Fig. 21.25B. The distribution of the rf magnetization component Mφ for the second order (n ¼ 2) and then n ¼ 1 SW modes, depending on the coordinate r across the nanowire, are shown at the field points 1–4 marked in part A of the figure for the 10-nm wire array.

21.4.2 High-frequency applications of magnetic nanowires Nanowire systems attract increased attention from the microwave device community, as there is a definite need to replace the traditional ferrite-based, nonreciprocal passive and active microwave devices with better materials. Ferrites are the most important materials to create low-loss, high-frequency devices, like isolators, circulators, and phase shifters. However, the low magnetization of the oxides prevents their use at frequencies in the upper gigahertz (millimeter-wave) range. Due to their unique electromagnetic (EM) properties, magnetic nanowire systems are good candidates to replace the ferrites. The unique flexibility of controlling several parameters (shape, size, composition, and topology) offers the possibility to extend the applications into the realm of plasmonics and photonics.

21.4.2.1 Microwave devices based on magnetic nanowires To illustrate the interest in the EM properties related to applications of nanowire structures, a representative list of the publications dealing with the design of nanowirebased microwave devices is briefly proposed here. An overview of theory and applications of magnetic nanowire metamaterials in microwave devices is given in Carignan [135]. The concept and experiments on several nanowire-based nonreciprocal microwave devices are reported by Kuanr et al. [136]. One of the fundamental requirements toward microwave devices in communication is the nonreciprocity of EM propagation, that is, isolation of signals. Nonreciprocity is based on the Faraday effect. Recently, millimeter-wave Faraday effect was directly measured at 61.25 GHz for a Ni nanowire structure [137]. In an early paper of Caloz and Itoh [138], the idea of using these man-made metamaterials for high-frequency electronics was raised and later reviewed, based on recent developments of the field [139, 140]. An unbiased microwave circulator on nanowire membranes was proposed by Saib et al. [141] and Darques et al. [29, 142]. A 3-mm device was fabricated with the wire’s height 50% in an alumina template with porosity of P ¼ 20%, as seen in Fig. 21.26 [29]. It exhibits circulation at 29.2 GHz with insertion losses as low as 6 dB and isolation of 35 dB. The modeled and measured transmission and reflection S parameters of the device are shown in Fig. 21.27 [29]. A similar circulator concept was presented by Marson et al. [143, 144]. An integrated self-biased planar microwave circuit on nanowire substrates was proposed by Carignan et al. [145], Carreo´n-Gonza´lez et al. [146],

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Magnetic Nano- and Microwires

Fig. 21.26 (A) Scanning electron microscope top view of an alumina membrane having pore diameter of 50 nm and porosity 20%; (B) sketch of the microwave circulator, where a metallic disc with three ports is evaporated on one face of the membrane: only the region below the disc is filled beforehand with nanowires; (C) top photograph of an actual device having one port connected to a 50 Ω load [29].

Fig. 21.27 Measured forward S12 and reverse S21 transmission for a 3-mm circulator based on NiFe nanowires embedded into an alumina template (porosity P ¼ 20%), filled with NiFe nanowires over 50% of substrate height [29].

Spin waves and electromagnetic waves in magnetic nanowires

659

and Wang et al. [147]. Unbiased planar microwave circulators working in the X-band (8–12 GHz) for airborne applications were designed and built from ferromagnetic nanowire arrays. It was shown that it is possible to precisely control the circulation frequency. Numerical optimization of the device parameters (nanowire type, quality of alumina membrane, and geometrical dimensions) meets the commercial requirements and confirms the potential of the nanowired substrates for on-board systems in aerospace applications [140]. A coplanar circulator with Co nanowires, according to simulations, can achieve 30 dB isolation and 15 dB bandwidth at 28.7 GHz was reported by Zhou et al. [148]. It is illustrated in Figs. 21.28 and 21.29.

Port3

Port2

G1 G3

G2 R

A

L1

B

C W1

(A)

Port1

Cu plate Porous AAO Co nanowires Signal line Cu ground Cu trench Duroid substrate

(B)

Fig. 21.28 Self-biased coplanar circulator based on Co nanowires on Duroid board (A) top view, (B) B-C cross section [148]. 0 S(1,1) S(2,1) S(3,1)

Coefficient (dB)

45 mm –10

–20

–30

20

22

24

26

28

30

32

34

Frequency (GHz)

Fig. 21.29 HFSS simulation of the S parameters for the Co-nanowire circulator in Fig. 21.28 with 45-μm-long Co nanowires.

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Magnetic Nano- and Microwires

Magnetic nanowire-based band-stop filters and phase shifters were designed and fabricated by Sharma et al. [149, 150] and Hamoir et al. [151]. Nanowires were applied to design tunable microwave devices [152]. A slow-wave liquid crystal phase shifter was realized, with a ground plane made from a nanowire array in Jost et al. [153]. In Li et al. [154] a 20-GHz noise suppressor was made from nanowires. Magnetic nanowire substrates are used for compact, configurable microwave bandpass or band-stop filters for signal processing, as modeled and measured in Ramı´rez-Villegas et al. [155]. Magnetic nanowire-based transmission properties of tunable terahertz filters for intensity modulators and spatial light modulators were investigated in Xiang et al. [156]. Monolithic microwave integrated circuits (MMIC) play a very important role in the development of microwave technology. Microstrip transmission lines were made on magnetic nanowire substrates, and band-stop filters and phase shifters were investigated over a range of magnetic fields, for the 10–40 GHz range in Aslam et al. [157]. New technological processes are developed to produce complex, monolithic microwave integrated nanowire devices, and the feasibility of the process is demonstrated by creating a microstrip bandgap filter and substrate integrated waveguide, based on multiple nanowire components [158]. A substrate integrated waveguide isolator made by this technique is described in Van Kerckhoven et al. [159]. Magnetic nanowires are good candidates as EMW absorbers. Electromagnetic interference (EMI) from mobile communication devices and radar systems is a serious social and technical concern and a problem for developers. To reduce or eliminate EMI, microwave absorbers have to be included into the device design. These strong absorbers have to be wideband, operating in the low gigahertz frequency range of Wi-Fi, Bluetooth, and WLAN, besides being lightweight and thin. For impedance matching, both dielectric and magnetic properties have to controlled. The EM absorption properties of microwave materials are characterized by reflection losses. For magnetic materials the complex permeability defines the absorption properties. FeNi3 nanowires encapsulated in graphite shells are proposed as excellent absorbers up to 18 GHz in Sun et al. [160]. Another version of nanowire-based EMI absorbers is a dielectric hybrid structure of a SiC dielectric core with Fe3O4 shell nanowires, demonstrating a reflection loss of 51 dB in the 2–18 GHz band [161]. Fe/TiO2 core-shell nanowires show excellent microwave absorption properties in the 10.8–14.9 GHz range [162]. Tunable Fe nanowires made by a template-free approach are demonstrated with 27-dB reflection loss at 3.68 GHz in Shen et al. [163]. Graphene-BiFeO3 nanowire composites were synthesized, and microwave absorption properties were measured together with electronic band structure calculations. A composite of 97 wt% BFO and 3% Graphene exhibited a 29-dB (99.75%) reflection loss at 10.68 GHz in Moitra et al. [164].

21.4.2.2 Magnetic nanowire metamaterials: Photonics and plasmonics The application of magnetic nanowires, due to their peculiar electromagnetic properties, is being extended into higher energies, into the optical spectrum. Magnetic nanowires, embedded into a periodic dielectric structure, thus can be treated as

Spin waves and electromagnetic waves in magnetic nanowires

661

metamaterials. The plasmonic and optical properties of magnetic nanowire structures and their applications are based on the metamaterial properties. Metamaterials are engineered periodic, composite media, not found in nature. They are designed to have specific properties, unique to the spatial arrangement of the components. Depending on the ratio of the operating wavelength to the characteristic periodicity of the medium, electromagnetic metamaterials can be classified as (1) magnetophotonic crystals (MPC), when the periodicity is on the order of the wavelength, imitating nature-made crystals by having band structures, forbidden bandgaps, and allowed defect states in the gap. In the long-wavelength limiting case of (2) the double-negative index (NIM) media, the characteristic size is much smaller than that of the operating microwave EMW. The interest to metamaterials is motivated by not only their fascinating properties but also the potential applications in information technology, communication, space, security and defense, health and environmental industry, as sensors, new type of laser and LED cavities, superlensing, and cloaking. All this is made possible by controlling EMW propagation in metamaterials. An early and novel application of magnetic nanowire structures, as metamaterials, was reported by Saib et al. [165]. They proposed, designed, and realized a class of microwave MPC materials, using a magnetic nanowire array, showing a bandgap at the FMR frequency of the system. In a search for tunable microwave devices, the interaction of (microwave) photons with a magnetic metamaterial was described theoretically by Lisenkov et al. [166], and it was shown that such a structure inside a waveguide creates an MPC with a nonpropagating bandgap, which can be controlled magnetically. Optical properties of magnetic nanowire metamaterials can be studied by Brillouin scattering of phonons on magnons, as in Stashkevich et al. [167], where anomalous polarization conversion and large enhancement of the degree of circular polarization were observed and interpreted in the Co nanowire medium. Extension into the optical wavelength range made possible to create nanowirebased “optical metacages” for optical radiation shielding in Mirzaei et al. [168]. Magnetic nanowire arrays, due to their photonic band structure, behave as a low-loss 2-D isotropic optical negative index metamaterials, as it was shown theoretically and numerically for a core-shell metal-dielectric Ag-Si nanostructure in Abujetas et al. [169]. If combining this concept with a magnetic metal component, a similar magneto-optical metamaterial could be created. Magneto-optical metamaterials can be incorporated into plasmonic nanowire devices. Plasmons are quasiparticles of coupling of photons and surface plasmons, that is, electron oscillation waves at metal-dielectric interfaces. In a spin-plasmonic device, photons couple to electron spin states, offering a unique possibility to apply for highly integrated photonic communications, imaging devices, and sensors. A recent review by Vavassori P. on magnetoplasmonics can be found in the 2017 magnetism roadmap [170]. In the case of strongly anisotropic media, the electric and magnetic tensor components have opposite signs, and thus hyperbolic metamaterials can be realized, as it was shown experimentally in Kruk et al. [171]. The associated wavelength, in principle, can be infinitely small. The hyperbolic devices show multiple functions for sensing

662

Magnetic Nano- and Microwires

and imaging, for example, diffraction-free, negative refraction and enhanced plasmon resonance effects. These specific properties are also required to fabricate integrated optical metacircuits for quantum information applications. Magnetic nanowire systems might be a good candidate for such hyperbolic metamaterial devices. Such a metamaterial of polaritonic (plasmonic) nanowires operating at 10–50 THz was described theoretically and modeled numerically by PL3 [172]. One example of magnetoplasmonic material is made of gold-core Co-shell nanowire arrays. The magnetooptic response is greatly enhanced by the plasmonic nature of gold, and the response is tunable by external magnetic field, as shown in Toal et al. [173].

21.5

Conclusions and future trends

The development of the technology of the preparation of stable, reproducible, well-controlled two-dimensional arrays of magnetic nanowires opened the path to test theoretical predictions, related to fundamental properties and basic physics of small magnetic particles, and to investigate magnetic and electromagnetic properties of low-dimensional systems in general. By changing the size and shape of the elements, fundamental magnetization processes can be studied. Changing the separation, one can study the transition from single noninteracting nanoelements through an assembly of weakly interacting elements and finally examine the length scales for strong interactions and watch how thin-film properties develop from individual nanoparticles. Magnetic nanowires are serious contenders for novel microwave devices, based on the propagation of EMW in these nanostructures. Passive nonreciprocal microwave devices on magnetic nanowire substrates are being developed. The results of modeling the electrodynamic diffraction problem show how the properties of these flexible systems can be tuned up to the terahertz frequency range. The theoretical and experimental study of the microscopic effects in the high-frequency behavior of magnetic nanowires led to the description and understanding of spin-wave phenomena in confined geometries. There is a macroscopic interaction of a magnetic nanowire system with an incident EMW, resulting in novel propagation and diffraction effects of EMW in the material response. By controlling the configuration alone, one can control the interactions among nanowires, creating metamaterials with designer electrical and magnetic properties. This approach is a fertile field for potential applications of nanowire systems in microwave devices for communication, information technology, and biomedical purposes. To make devices with controlled properties, recently, nanomaterials of new geometries were proposed, like multilayered nanowires and nanotubes. The attractive common feature of these structures is the presence of extra degrees of freedom, which makes their properties more easily tunable. With the technology moving into the terahertz range, microwave ferrites have to be replaced. Metallic ferromagnet-based nanowires, embedded in an insulating matrix, offer a solution. Magnetic nanowire platforms are moving into the realm of photonics and spin plasmonics. New devices, based on the flexibility of the nanowire systems,

Spin waves and electromagnetic waves in magnetic nanowires

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are proposed. The theoretical and modeling background is already developed. However, there are some open questions for theory and technology. For example, one has to take into account and control the statistical distribution of particle size, shape, and separation in calculating and controlling the interactions in a magnetic nanowire (or other nanostructured) system. Likewise the temperature dependence is usually ignored; however, these nanoparticle devices have to operate in a reasonable wide temperature range, where the stability of such small magnetic particles against thermal fluctuations is uncertain. The field is relatively young, and probably, there are laboratories already working on the solution of such practical problems. For newcomers, there are plenty of opportunities to contribute to the development of novel devices and applications based on magnetic nanowire platforms.

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[162] Y. Li, H. Cheng, N. Wang, Y. Zhou, T. Li, Magnetic and microwave absorption properties of Fe/TiO2 nanocomposites prepared by template electrodeposition, J. Alloys Compd. 763 (2018) 421–429. [163] J. Shen, Y. Yao, Y. Liu, J. Leng, Tunable hierarchical Fe nanowires with a facile template-free approach for enhanced microwave absorption performance, J. Mater. Chem. C 4 (32) (2016) 7614–7621. [164] D. Moitra, S. Dhole, B.K. Ghosh, M. Chandel, R.K. Jani, M.K. Patra, S.R. Vadera, N.N. Ghosh, Synthesis and microwave absorption properties of BiFeO3 nanowireRGO nanocomposite and first-principles calculations for insight of electromagnetic properties and electronic structures, J. Phys. Chem. C 121 (39) (2017) 21290–21304. [165] A. Saib, D. Vanhoenacker-Janvier, I. Huynen, A. Encinas, L. Piraux, E. Ferain, R. Legras, Magnetic photonic band-gap material at microwave frequencies based on ferromagnetic nanowires, Appl. Phys. Lett. 83 (12) (2003) 2378–2380. [166] I. Lisenkov, V. Tyberkevych, L. Levin-Pompetzki, E. Bankowski, T. Meitzler, S. Nikitov, A. Slavin, Interaction of microwave photons with nanostructured magnetic metasurfaces, Phys. Rev. Appl. 5 (6) (2016) 064005. [167] A.A. Stashkevich, Y. Roussigne, A.N. Poddubny, S.M. Cherif, Y. Zheng, F. Vidal, I.V. Yagupov, A.P. Slobozhanyuk, P.A. Belov, Y.S. Kivshar, Anomalous polarization conversion in arrays of ultrathin ferromagnetic nanowires, Phys. Rev. B 92 (21) (2015) 214436. [168] A. Mirzaei, A.E. Miroshnichenko, I.V. Shadrivov, Y.S. Kivshar, Optical metacages, Phys. Rev. Lett. 115 (21) (2015) 215501. [169] D.R. Abujetas, R. Paniagua-Domı´nguez, M. Nieto-Vesperinas, J.A. Sa´nchez-Gil, Photonic band structure and effective medium properties of doubly-resonant core-shell metallo-dielectric nanowire arrays: low-loss, isotropic optical negative-index behavior, J. Opt. 17 (12) (2015) 125104. [170] D. Sander, S.O. Valenzuela, D. Makarov, C.H. Marrows, E.E. Fullerton, P. Fischer, J. McCord, P. Vavassori, S. Mangin, P. Pirro, B. Hillebrands, The 2017 magnetism roadmap, J. Phys. D. Appl. Phys. 50 (36) (2017) 363001. [171] S.S. Kruk, Z.J. Wong, E. Pshenay-Severin, K. O’brien, D.N. Neshev, Y.S. Kivshar, X. Zhang, Magnetic hyperbolic optical metamaterials, Nat. Commun. 7 (2016) 11329. [172] M.S. Mirmoosa, S.Y. Kosulnikov, C.R. Simovski, Magnetic hyperbolic metamaterial of high-index nanowires, Phys. Rev. B 94 (7) (2016) 075138. [173] B. Toal, M. McMillen, A. Murphy, W. Hendren, M. Arredondo, R. Pollard, Optical and magneto-optical properties of gold core cobalt shell magnetoplasmonic nanowire arrays, Nanoscale 6 (21) (2014) 12905–12911. [174] H. Schl€orb, V. Haehnel, M.S. Khatri, A. Srivastav, A. Kumar, L. Schultz, S. Fahler, Magnetic nanowires by electrodeposition within templates, Phys. Status Solidi B 247 (10) (2010) 2364–2379.

Part Three Sensing, thermoelectric, robotics, biomedical and microwave applications

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Template-assisted electrodeposited magnetic nanowires and their properties for applications

22

Joseph Uma, Jung Jin Parkb, Alison Flataub, Wen Zhoua, Yali Zhanga, Rhonda Franklina, Kotha Sai Madhukar Reddya, Liwen Tana, Anirudh Sharmaa, Sang-Yeob Sunga, Jia Zoua, Bethanie Stadlera a University of Minnesota, Electrical and Computer Engineering, Minneapolis, MN, United States, bAerospace Engineering/Materials Science and Engineering, University of Maryland, College Park, MD, United States

22.1

Introduction

Magnetic nanowires (NWs) can be fabricated by template-assisted electrochemical deposition, which is a relatively simple and cost-effective process compared with lithographical processes. Typical nanowire templates are porous anodic aluminum oxide (AAO, made by anodization of aluminum foils or films, Fig. 22.1A) or track-etched polycarbonate (TEPC, made by ion bombardment and pore widening through polycarbonate membranes). AAO is preferred to make dense arrays of NWs because TEPC membranes usually have larger interpore distances (¼ smaller pore density or porosity). Pore diameters in AAO can be controlled at the nanometer scale (10–200 nm) with interpore distances about double the diameter in each case, and the pore lengths can be as long as the thickness of the membrane that is typically micrometer scale. Fig. 22.1B shows a cross section of 34-μm-long Ni NWs in a 50-μm-thick AAO template. This means that devices with high aspect ratios (see NWs after AAO etching [Fig. 22.1C] and high-density arrays [Fig. 22.1D]) can be realized. On the other hand the large interpore distances in TEPC enable fundamental research on magnetic properties of nearly single NWs because magnetostatic interactions between nanowires will be smaller than those in AAO. The templates provide insulation and passivation for NW devices, such as hard disk drive (HDD) recording media, bit-patterned media, magnetic random access memory (MRAM), and array read sensors. In other applications, the templates can be etched to release NWs into solutions, a feature that is desirable in many biomedical applications, such as nanoparticles for hyperthermia cancer therapy, nanowarming cryopreserved tissues and organs, cell separation from assays, and magnetic resonance imaging (MRI) contrast imaging. As stated earlier, a range of the applications using magnetic NWs is broad, so the required magnetic properties are also varied. This Magnetic Nano- and Microwires. https://doi.org/10.1016/B978-0-08-102832-2.00022-0 Copyright © 2020 Elsevier Ltd. All rights reserved.

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Fig. 22.1 SEM images of (A) top view of AAO with 40-nm-diameter pores, (B) cross-sectional view of AAO with 34-μm-long Ni NWs grown inside, (C) freestanding Fe NWs after AAO removal, and (D) SEM backscattered image of cross-sectional view of Fe/Au multilayered NWs grown on Cu seed layer.

chapter will review magnetic NWs with a focus on the important and applicationdriven magnetic properties, which have been achieved at the University of Minnesota.

22.2

Fe-Ga alloy nanowires used in tactile sensors

When magnetostrictive materials are magnetized with a magnetic field, their shape or dimensions change. In reverse, when the materials are deformed, their magnetization orientation changes. This phenomenon is a good property for microelectromechanical and nanoelectromechanical sensors and devices. Specifically, tactile sensors have been made from magnetic nanowires that are synthesized such that they are oriented vertically to thin film giant magnetoresistive (GMR) layers, similar to Fig. 22.1D. When the nanowires are used as a sort of artificial skin to touch something, the deformation

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causes a magnetic response that the GMR sensor detects. In fact, many biological species sense touch using cilia in a similar fashion. Three nanowire features are required for tactile sensors: 90-degree rotation of magnetic moments in response to applied stress, ductile mechanical properties, and low shape anisotropy. When these criteria are met, slight touches will yield large changes in magnetization (without breaking nanowires) because the stress-induced rotation of magnetic moments will dominate the magnetic response. Terfenol-D (TbxDy1xFe2) is known for high magnetostriction (2000 ppm), but it is brittle and therefore difficult to machine [1, 2]. Fe-Ga alloys (Fe1xGax, 10 < x < 40), also known as Galfenol, has reasonable magnetostriction (400 ppm) [1, 3] with excellent ductility and strength [3]. Many aspects of Fe-Ga have been extensively studied: fabrication [1, 3–9], magnetostriction [4–6, 10], mechanical properties [11], and magnetic properties [3, 12–17]. In high-aspect ratio NWs, magnetic domains remain aligned along the NW axis due to shape anistropy, so Fe-Ga alloy nanowires do not show magnetostriction under applied magnetic fields or bending loads [3, 14]. To rotate the domains away from the NW axis, a field larger than the shape anisotropy is required that is equal to 2πMs (10,000 Oe for Fe-Ga), where Ms is the saturation magnetization. Another approach is to reduce the aspect ratio of NWs using segmentation by nonmagnetic materials, for example, Au as in Fig. 22.1D or Cu within NWs [3, 14–17]. Magnetization reversal was investigated using Fe-Ga/Cu multilayered NWs composed of alternating 300-nm-long Fe-Ga and 70-nm-long Cu segments with 150 nm diameter [14]. First, an individual Fe-Ga NW with an aspect ratio of 2 was simulated with no applied field to predict the magnetic configuration at remanence (Fig. 22.2A). Two vortex modes were seen separated by a single domain wall in the middle of the NW. Fig. 22.2B shows that the stray fields from the ends and the middle of the NW are visible but very small. It is important to note that truly “zero” magnetic fields are unlikely, so this state is rare. Next, an individual Fe-Ga/Cu multilayered NW was characterized by magnetic force microscopy (MFM) with the field applied. In Fig. 22.3 a 550-Oe magnetic field was applied to the NW axis at different angles, showing that all of the domains were rotated and aligned in the field direction. This means that the shape anisotropy was reduced and controlled by the low aspect ratio of the Fe-Ga layers (¼ 2). This result is promising in that magnetization is certainly able to rotate, and therefore magnetostriction may dominate over shape anisotropy (which is 55) and B (ARFe-Ga 0.1/ARCu 0.3) with 35 nm diameter and C (ARFe-Ga > 25) and D (ARFe-Ga 0.5/ARCu 5) with 100 nm diameter, AR ¼ aspect ratio ¼ length/diameter [3, 17].

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90 degrees. This indicates that small-diameter, long nanowires likely reverse their magnetizations by coherent rotation when the reverse field is applied perpendicular to the nanowire axes. Sample B shows increasing coercivity at 0–60 degrees and decreasing coercivity at 60–90 degrees with minimum at 90 degrees, meaning that segmented nanowires experienced two competing modes (contributions of CR and V were comparable with each other): V mode at 0–60 degrees and CR mode at 60–90 degrees, resulting in a peak at 60 degrees [3, 17]. This also means that the shape anisotropy of sample B was decreased relatively compared with that of sample A at high field angles as expected due to the segmentation. Compared with sample A, the range of Hc values of sample B is narrow. But the structure of sample B is still not magnetically isotropic with the field angle [17]. This also suggests that higher aspect ratios of Cu layers would yield more magnetically isotropic samples [17]. Next, the NW diameter was increased (sample C), and the coercivity had a distinct peak at about 70 degrees, which again indicates a competition of V and CR modes. Finally, sample D was made with Fe-Ga segments similar in length to diameter and with large separation (long Cu). This sample shows constant Hc regardless of the field angle, which indicates that the magnetization is able to rotate toward the applied field for all field angles [3]. In other words, Cu layers are thick enough to hinder dipole interactions between Fe-Ga layers. With reduced dipole interactions between Fe-Ga layers and reduced shape anisotropy within the Fe-Ga segments, this design can be used as the elements of magnetostrictive sensors or actuators, such as pressure sensor elements in [16]. Co NW arrays as well were studied and demonstrated for flow and vibration sensing [20]. There are also other studies on magnetic properties of NWs, such as magnetic configurations in individual Fe-Ga/Cu multilayered NW [13] and multiple NWs in arrays [12, 13], magnetic reversal mechanisms of Co/Cu multilayered NWs in arrays [21], and electron holography of Co NWs for magnetization and crystalline orientations [22].

22.3

Co/Cu multilayered nanowires for CPP-GMR structures

Giant magnetoresistance (GMR) is a phenomenon that occurs in structures composed of two ferromagnetic metals that are separated by a nonmagnetic metal. If the ferromagnetic metals are parallel- (or antiparallel-)magnetized, the resistance of the structure becomes low (or high) due to the weak (or strong) scattering caused by spin-dependent mean free paths. In measuring the resistance, the current can be applied perpendicular to the plane (CPP) or in the plane (CIP) of the “sandwich” structure. As the importance of high areal densities of HDD and MRAM increases, CPP-GMR structures have potential for higher areal densities [23] and larger GMR effects [21] than CIP-GMR structures. These applications are worth discussing before continuing the discussion on the nanowires themselves. HDDs need GMR read sensors that are on the same scale as the magnetic bits that store data. Conventional GMR sensors are made via vacuum deposition of thin film layers that must be etched into the required size. The etching process causes damage to the sidewalls of the device, and as the device size scales ever

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smaller, the sidewalls make up an ever larger percent of the total sample. The next steps in fabrication of conventional hard drive read sensors is a coating of aluminum oxide (or other insulator) and side deposition of a magnetic hard bias material. Since the insulator and hard bias are only needed on the sides of the sensor, chemomechanical polishing is used to remove the top layers before subsequent shielding is deposited. In contrast, GMR structures can be made inside AAO, so they are already the required size and are surrounded by an insulator. Therefore, larger (easier) features can be etched in the AAO around the sensor so the etching does not touch the sensor itself. Then, a mask for hard bias layers completes the process. For GMR read sensors, the switching current density should be very high to prevent switching except by the media itself. For MRAM and STT-RAM, the switching current densities should be low such that the magnetic data can be entered at low energy consumption. In this case, GMR structures inside AAO form a very-high-density (up to 2 terabit/in.2) array. Growing multilayered NWs using relatively simple template-assisted electrodeposition can directly yield various nanometer-sized CPP-GMR structures with high density and AAO insulation. AAO usually has pore diameters in the range of 2.5–500 nm and a pore density range of 107–1012 [24]. GMR structures grown in AAO have shown better MR performance compared with CPP-GMR structures made by conventional vacuum deposition and etching techniques that might damage the sidewalls and interfaces between ferromagnetic and nonmagnetic layers [25–27]. A popular GMR structure involves Co/Cu multilayers. Interestingly, Co can be grown with various crystallographic orientations by varying pH during electrochemical deposition [23]. Specifically, Co/Cu multilayered NWs grown using pH 3.4 and 5.2 electrolytes have c-axis in plane (c-axis perpendicular to the nanowire axis) and c-axis out of plane (c-axis parallel to the nanowire axis), respectively [23]. Generally the concentration of hydrogen reduced during the electrodeposition increases as the pH of the electrolyte decreases. The Co crystalline structures are considerably affected by the concentration of hydrogen absorbed and desorbed during deposition, leading to different types of Co crystalline structure [28]. More information on the relationship between pH of the electrolyte and Co crystallographic orientations is shown in Darques et al. [28, 29]. More information on the crystalline structure by X-ray diffraction of Co/Cu NWs is given in Tan et al. [23]. One example of device engineering using pH involves controlling the sample’s two easy axes. Fig. 22.5A shows the hysteresis loops of Co with an in-plane Co c-axis (pH 3.4) and an in-plane shape anisotropy (i.e., low aspect ratio of 0.083 from 60 nm diameter and 5 nm thickness) [23]. Fig. 22.5B, however, shows the hysteresis loops of Co where the c-axis was grown perpendicular (pH 5.2) to the same shape anisotropy. The in-plane anisotropy is obvious in Fig. 22.5A, but Fig. 22.5B shows almost identical loops because the NWs have two competing easy axes, so the same reversal behavior is observed irrespective of the saturation direction [23]. Then, how do different crystallographic orientations in Co/Cu NWs affect the magnetoresistance (MR)? MR was measured using an AC transport system, applying a 10 μA bias the top to bottom layers of NWs [23]. MR is calculated h current through i by MR ¼

RðHÞRðHsat Þ RðHsat Þ

 100%, where Hsat is the saturated field [23]. As shown in

Fig. 22.6A and B, Co/Cu NWs with c-axis in plane (pH 3.4) have 0.08% MR

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Fig. 22.5 Magnetic hysteresis loops of Co/Cu multilayered NWs at room temperature. (A) Co 5 nm/Cu 10 nm NWs grown at pH 3.4, (B) Co 5 nm/Cu 5 nm NWs grown at pH 5.2 [23]. 8 H parallel to NWs Magnetization direction

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6 4 2

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0 –6

–4

–2

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2

4

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(D) Co/Cu (pH 5.2) at 0 Oe (remanence)

Fig. 22.6 Magnetoresistance of Co/Cu multilayered NWs at room temperature. (A) Co 5 nm/Cu 10 nm NWs grown at pH 3.4, (B) Co 5 nm/Cu 5 nm NWs grown at pH 5.2, and possible magnetization configuration at remanence of two adjacent Co/Cu NWs grown at pH 3.4 (C) and at pH 5.2 (D) [23].

difference when the field is applied parallel to NWs versus perpendicular. However, Co/Cu NWs with c-axis out of plane (pH 5.2) have 1.33% MR angular variation due to the two easy axes [23]. When Co/Cu NWs have in-plane c-axis, the remnant magnetizations are likely to stay along in plane because both Co shape anisotropy and c-axis are in plane (Fig. 22.6C) [23]. However, when Co/Cu NWs have out-of-plane c-axis,

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the remnant magnetizations remain in plane after in-plane saturation (following shape anisotropy) but perpendicular to the plane after perpendicular saturation (following the c-axis) (Fig. 22.6D). In this out-of-plane orientation, it is more favorable for the magnetizations to form domains due to dipole fields between the Co layers. By this research, we found that hysteresis loops and MR measurements provide different but complementary information on multilayered Co/Cu NWs and also that isotropic MR properties were suitable for further CPP-GMR researches [16, 25–27, 30, 31]. Specifically, switching current densities can be lowered in Co/Cu multilayered NWs for low-energy MRAM [30, 31], and switching current densities can be increased in Co/Cu/Co trilayered structures for read head sensors with improved head stability [25–27]. It is noted that smaller numbers of layers in structures yield higher switching current densities and less read sensor noise [25]. In addition, while researching nanometer-sized CPP-GMR for high areal densities, low-resistivity NWs were also explored [25, 27], which could be crucial for interconnects in the near future.

22.4

Nanowires used for biomedical applications

Biomedical applications include cell labeling and identification, MRI contrast agents, and nanowarming agents. For cell labeling, the goal is to have a multitude of labels that can be coated to tag specific biomarkers. These labels should then have a physical mechanism to identify their presence in an assay or other location so that the medical personnel know the cell of interest is present. Possible magnetisms of identification in assay form will be discussed in the succeeding text. In tissue, the label could be identified by MRI if the imaging contrast is high enough. Another biomedical application is nanoheating, which relies on inductive hysteretic losses as either NWs or nanoparticles (NPs) are exposed to an external alternating magnetic field. For a decade or so, there has been interest in using this phenomenon to kill cancer by hypothermia therapy. The idea is to coat the nanostructure such that it adheres best to a tumor and/or tumor cells. The heating would then be local to the tumor for maximum cell death, while surrounding healthy tissue is left cool. A new use of hysteretic losses is nanowarming of cryopreserved tissues and organs. In this application, nanostructures suspended inside a cryoprotectorant (i.e., a biocompatible antifreeze) would be frozen inside and outside of the tissue or organ such that rapid, uniform rewarming is possible. To avoid detrimental crystallization during warming, the heating rate should be well above 50°C/min. So far, NWs are not approved by the US Food and Drug Administration, but a few NPs of iron oxide have been approved [32, 33]. Magnetic NWs may become approved with further research because they do have several advantages over NPs of comparable volume. First, NWs have higher surface-to-volume ratio that enables loading of more coatings for drug delivery [34, 35]. Second, multiple functionalities are possible with multilayered NWs with “barcoded” surface chemistries [36]. Third, nanowires can be mechanically rotated for torque-induced dynamic cell therapy using a relatively weak external field due to the larger magnetic moment and higher aspect ratio [34, 35, 37, 38]. In the near future, more magnetic nanomaterials and

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nanotechnologies could be used in diagnosis and treatment of illnesses like cancer. Then, techniques for cell labeling and identification also should be necessary; otherwise, both unhealthy and healthy cells would be destroyed during treatment. Sharma et al. [35, 38, 39], Shore et al. [40], and Zhou et al. [41] show how to label cells with NWs and how to separate these cells from unlabeled ones. NWs can also be selectively functionalized using antibodies to label different cell types so that each type of NW will match a specific cell type [38]. For example, Ni and Co NWs can be coated with different antibodies so that Ni NWs will label cancer cells only and Co NWs will label immune cells only. Then, if magnetic NWs are used for this cell labeling and identification, what kinds of magnetic properties can be utilized for cell identification? Magnetization hysteresis (MH) loops could be used to count labels if one kind of NW is used to label cells [35]. For example, when Ni nanowires (3 μm long with 100 nm diameter) were coated with Arg-Gly-Asp (RGD), osteosarcoma (bone cancer) cells were shown to self-disperse the NWs throughout the cell population by integrin-mediated responses with low cytotoxicity [35]. To show the potential of hysteresis loop quantification, 300-μL samples with varied concentrations of dispersed Ni NWs in deionized (DI) water were measured, and a reliable linear relationship between magnetic moment and Ni NW volume was clearly demonstrated, as shown in Fig. 22.7. However, it would be hard to distinguish two or more kinds of NWs in a single assay since MH loops simply show the net moment of the entire sample and the loops vary with orientation. Another proposed “read out” technique was Hc versus applied field angle, as in Fig. 22.4. Although this technique can distinguish which one of many types of NWs might be present in an assay, it does not delineate mixtures of NWs very well. First-order reversal curve (FORC) measurements, however, were efficiently used to distinguish two different types of NWs [39]. In FORC, if a sample is saturated at a high positive field, then the magnetic moment of a sample is measured as the magnetic field (H) is swept from a progressively lower reversal field (Hr) back to saturation. The measurements from each reversal field are compiled into a family of curves from 2 which the FORC distribution ρðH, H Þ ¼  1 δ MFORC ðH, Hr Þ is calculated and plotted in a r

2

δHδHr

r plane of Hu ¼ H +2Hr (as y-axis) and Hc ¼ HH 2 (as x-axis), where Hu is the interaction field and Hc is the coercivity [39, 42]. As one example, FORC measurements were used to estimate the ratios of two kinds of Ni NWs from mixtures [39]. Here, high coercivity labels (at low field angles) were engineered by ensuring coherent rotation as the reversal mechanism because, at 5 μm long and 18 nm diameter, their diameters were too small to have a vortex core [39]. Low coercivity labels were engineered by ensuring the diameters were large enough to support a vortex wall reversal mechanism (6 μm long and 100 nm diameter) [39]. Four mixtures were studied with NW ratios of 100 nm:18 nm equal to 1:0, 1:23, 1:115, and 0:1, and these four samples were measured by FORC as in Fig. 22.8 [39]. The end samples 100 nm:18 nm ¼ 1:0 and 100 nm:18 nm ¼ 0:1 were found to have coercivities of 370 and 730 Oe, respectively, and these FORC curves were used in a regression analysis to estimate the ratios of the other two mixtures [39]. The estimated

Magnetic moment (*10^(–4) emu)

Template-assisted electrodeposited magnetic nanowires and their properties

(A)

3

685

NW 300 µL : Dl water 0 µL (A) NW 150 µL : Dl water 150 µL (B) NW 100 µL : Dl water 200 uL (C)

2 1 0 –1 –2 –3 –6000

–4000

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250 200 150

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Fig. 22.7 (A) MH loops obtained for 100-μL samples from Ni NWs 300 μL with DI water 0 μL, Ni NW 150 μL with DI water 150 μL, and Ni NW 100 μL with DI water 200 μL. (B) Moment versus Ni NW volume relationship in cell culture medium [35].

ratios were compared with the known ratios, and the regression analysis was linear with the expected ratios for samples b and c in Fig. 22.8: ρb ¼ 1.2ρa100 nm +0.75ρd18 nm and ρc ¼ 0.28ρa100 nm +0.81ρd18 nm, respectively, as shown in Fig. 22.8E [39]. To get the accurate ratios between the numbers of 100 and 18 nm diameter NWs, their volume ratio 2

πð50 nmÞ 6 μm 100 nm should be considered and used for further analysis, which is Vol Vol18 nm ¼ πð9 nmÞ2 5 μm ¼ 37

(meaning that the volume of one of 6-μm-long and 100-nm-diameter NW is equal to that of thirty seven of 5-μm-long and 18-nm-diameter NWs) since the magnetic moment is proportional to the volume [39]. For sample b, 100 nm:18 nm ¼ 1:23 sample, if the ratio of the intensities of two types of NWs is ρa100 nm/ρd18 nm 5 1.2/0.75 (since ρb ¼ 1.2ρa100 nm +0.75ρd18 nm),

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Fig. 22.8 FORC measurements by VSM for NW ratios of 100 nm:18 nm equal to (A) 1:0, (B) 1:23, (C) 1:115, and (D) 0:1 with insets of FORC analysis (ρ vs. Hc vs. Hu). (E) Integrated intensity of horizontal slices across each FORC with each intensity formulas based on the regression analysis on the right [39]. 1:2 nm 100 nm 100 nm then numerical ratio of the two types of NWs is NN100 ¼ ρa  Vol ρd Vol18 nm ¼ 0:75  37 ¼ 18 nm 8 1 185  23

18 nm

[39]. For sample c, 100 nm:18 nm ¼ 1:115 sample, if the ratio of the intensities is ρa100 nm/ρd18 nm 5 0.28/0.81 (since ρc ¼ 0.28ρa100 nm + 0.81ρd18 nm), then Vol100 nm 0:28 28 1 nm 100 nm ¼ ρa concentration ratio is NN100 ρd18 nm  Vol18 nm ¼ 0:81  37 ¼ 2997  107 [39]. This value 18 nm is within 7% error of the known value, which is very good for such a large difference in NW concentrations between these two types of NWs with such different volumes. Therefore, FORC analysis can be a good way to distinguish mixtures of more than two NWs. Ferromagnetic resonance (FMR) measurements are another way to distinguish between different types of NW labels [41]. FMR occurs when an AC magnetic field is applied to a magnetic material at a frequency (f ) that matches the material’s natural electron precession frequency. This precession frequency can be altered using a DC magnetic field, according to the well-known Kittel equation [43]:

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 2    ω ¼ Hcosðθ  θH Þ + Heff cos 2θ Hcosðθ  θH Þ + Heff cos 2 θ , γ where ω is the angular frequency (ω ¼ 2πf ), γ is the gyromagnetic ratio (γ ¼ gμℏB , where g is the g-factor of the material, μB is the Bohr magneton, and ℏ is the reduced Planck constant), H is the applied DC field, θ is the angle between magnetization direction and NW axis, and θH is the angle between the applied DC field and NW axis as in Fig. 22.9. Heff is the effective field that the NWs experience, and it is a combination of the shape anisotropy field of the nanowire itself and the magnetostatic interaction field resulting from nearby nanowires. These two fields dominate over others, such as magnetocrystalline anisotropy field and magnetoelastic anisotropy field [29]. Therefore, Heff is expressed as Heff ¼ 2πMs(1  3P), where Ms is the saturation magnetization and P is the porosity of the template in which NWs are grown [29]. In addition to dimensional factors, ferromagnetic materials such as Fe, Co, and Ni have different g-factors and saturation magnetizations, so NWs can easily be designed to show many unique FMR behaviors. For FMR measurements, radio frequency (RF) signal is applied to a coplanar waveguide on which the nanowire samples are placed. If the frequency of the RF signal matches the resonance frequency of the nanowires, an intensity reduction occurs in the S21 magnitude, which is a ratio of the output voltage and input voltage of RF signals. To verify the cell labeling using FMR and emulate the mixture of different types of NWs, three types of NWs (18-μm-long Ni, 15.5-μm-long Co, and 10.5-μm-long Fe NWs with 40 nm diameter) were measured individually and in stacks (Ni-Co-Fe and Ni-Fe-Co from bottom to top) [41]. Fig. 22.10A shows FMR measurements for individual NW types as dashed lines and two measurements of stacked types as solid lines. The Ni-Co-Fe stacks had three absorption peaks for S21, representing each individual type of NW. However, when the Co nanowires were further from the coplanar waveguide, as in the Ni-Fe-Co stack, only the absorption peaks of Fe and Ni were obvious. Zhou et al. [41] also provides

Fig. 22.9 Relative orientations of magnetization (M) and external DC magnetic field (H) for single NW.

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Fig. 22.10 (A) FMR measured for three types of NWs (Fe, Co, and Ni NWs) as dashed lines and stacks of Ni-Co-Fe and Ni-Fe-Co as solid lines. (B and C) FMR raw data and fitting curves as solid lines and contribution of each type of NWs as dashed lines from (B) Ni-Fe-Co stack and (C) Ni-Co-Fe stack [41].

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an algorithm detecting multiple NW types and showing the contribution of each to the overall S21 magnitude shape as in Fig. 22.10B and C. This FMR measurement technique shows its ability to detect three different types of NWs. Theoretically, this technique can detect many types of NWs as long as NWs are made of different elements or alloys since they have all different FMR signals. For cell labeling and identification using two or more types of NWs, FORC and FMR techniques have strong potential as described earlier and even stronger if they are used together. Furthermore, these techniques can be combined with fluorescence techniques because these three techniques are orthogonal (noninterfering). NWs can also be used as MRI contrast agents. For example, Fe/Au multilayered NWs exhibited a contrast performance comparable with commercial Fe oxide nanoparticles and showed multiple functionalization capabilities, because of the multilayered structure, which could enable specific tissue imaging and therapy [44]. Furthermore, after systematically controlling the electrodeposition of Co-Fe alloy NWs [19], Au-tipped Co-Fe alloy NWs were studied as nanowarming agents for cryopreservation [45]. Their very high saturation magnetization led to nanowarming rates of 1000°C/min, 20 times faster than critical warming rate (50°C/min) for common cryoprotective agents. This fast rate means that the preserved sample (e.g., tissues or organs) could be heated without crystallization, and the nanoscale dimensions of the nanowires would enable them to be evenly dispersed (e.g., in blood vessels) for uniform heating. The nanowires were shown to have low cytotoxicity [38, 45]. Although Au is a literally a “gold standard” for biocompatibility, electroplated Au often involves cyanide electrolyte chemistry. For biorelated applications, a thiosulfate-sulfite electrolyte, which is a noncyanide solution, was recently proven successful for nanowire deposition [46]. Galfenol has also been studied for remote microactuators for cell biology studies and intracellular applications [47].

22.5

Long-range ordered porous AAO fabricated by double imprinting with line-patterned stamps

NWs are frequently grown inside porous AAO by electrodeposition, but the pores of AAO typically have only short-range order as naturally formed during conventional two-step anodization of aluminum (Al), usually in the range of 20 times the interpore distance [48]. It would be great to make long-range ordered AAO so that NWs or devices can be grown densely at a known location and pinpointed easily for applications, such as HDD recording media, bit-patterned media, and MRAM. Conveniently, long-range ordered AAO is possible with imprinting methods [6, 25, 26, 49] in which stamps with hexagonally ordered arrays of nanopillars (as in Fig. 22.11A) are pressed into Al foils or Al films and the resulting dents guide the pore growth during subsequent anodization. Fig. 22.11B shows pores formed in an AAO (oxide) grown using imprinted Al and the pores naturally occurring in unimprinted areas during anodization [49]. The stamps were conventionally made by e-beam lithography, which is an expensive and very slow process to make billions of nanopillars over large areas (square millimeter scale).

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Fig. 22.11 SEM image of (A) stamp with nanopillars and (B) nanopores from imprinted (left) and unimprinted (right) areas [49].

Therefore, line-patterned Si stamps have been developed (simpler patterns and even commercially available for cheap prices) [27]. Although Si stamps with line patterns are brittle for imprinting, Ni can be electrodeposited onto Si masters such that the deposited Ni films (with duplicate line patterns) can be used as stamps for imprinting. The original Si stamps can be reused to make many Ni stamps. These Ni stamps then are imprinted, rotated 60 degrees, and imprinted again on the Al precursors to make double-imprinted areas from which the nanopores will grow during anodization as in Fig. 22.12. Fig. 22.13A shows the line pattern on a Ni stamp, and the inset shows the whole Ni sample, which is 1  1 cm. The pattern looks blue since the line pattern period demonstrated here is 278 nm, and blue light is diffracted off the Ni stamp at specific viewing angles. When the Al precursor is double imprinted with this Ni stamp, the imprint pattern is shown in Fig. 22.13B. After the anodization of the imprinted Al, the resulting oxide contains pores with long-range order as in Fig. 22.13C. The inset of Fig. 22.13C shows AAO with the patterned region that has ordered pores over 1 cm diameter (blue area) surrounded by natural short-range ordered pores in the region that was not imprinted (gray area). The techniques to make large-scale long-range ordered AAO are interesting to template-assisted fabrications for nanostructured materials. Double imprinting could be the simplest and most cost-effective method. The AAO made by this method shows the potential for the future high-density recording systems requiring long-range ordered devices separated from each other by insulating material to eliminate cross talk, such as 3-D magnetic memory.

22.6

Conclusions

Magnetic nanowires (NWs) grown by template-assisted electrochemical deposition have gathered great attention for nanometer-sized features made by relatively simple and low-cost processes and unique magnetic properties due to their high aspect ratios

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Fig. 22.12 Schematic diagram of fabrication of long-range ordered AAO by double imprinting with line-patterned Ni stamps replicated from Si master stamps. Red dots are shown for pore locations that will result if double-imprinted Al is oxidized via anodization to form AAO templates.

and high densities. Thus there are many applications using magnetic NWs and many magnetic properties that need to be engineered depending on these applications. To get the magnetostriction effect from Fe-Ga alloy NWs, the shape anisotropy should be negligible by using nonmagnetic materials such as Cu to segment the NWs. Fe-Ga/Cu and Co/Cu multilayered NWs and Co NWs were all studied as individual NWs and as arrays to determine each of their magnetic switching mechanisms. For Co/Cu multilayered NW-based CPP-GMR structures for future MRAM and read head sensors, high or low magnetoresistance is achieved using either a trilayer or Co/Cu NWs with in-plane shape anisotropy due to low aspect ratio and in-plane c-axis of Co, respectively. In cell labeling and identification, FORC analysis has been used to estimate the proportion of two different types of NWs in mixtures and FMR

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Fig. 22.13 SEM images of (A) top view of line pattern on Ni stamp with inset showing the Ni stamp, (B) top view of double-imprinted Al precursor, (C) AAO by anodization after double imprinting with inset showing the ordered nanoporous region by double imprinting (seen as a blue area) and short-range ordered region naturally formed in nonimprinted region (seen as a gray area) [27].

measurements to detect three types of NWs. Combining these two techniques and fluorescence techniques could yield a powerful tool for cell analysis. Fe/Au multilayered NWs and Au-tipped Co-Fe alloy NWs show promise as MRI contrast and nanowarming agents, respectively. Finally, NWs or devices made by templateassisted electrodeposition can be enhanced in terms of material quality and magnetic property if long-range ordered AAO is used. A simple and cost-effective double imprinting technique using line-patterned Ni stamps is demonstrated and could be

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applied to future memory devices, such as 3-D magnetic memory that requires wellordered and densely close-packed monodisperse nanodevices with insulation between them.

References [1] P.D. McGary, B.J.H. Stadler, Electrochemical deposition of Fe1 xGax nanowire arrays, J. Appl. Phys. 97 (10) (2005) 10R503. [2] B. Stadler, et al., Galfenol thin films and nanowires, Sensors 18 (8) (2018) 2643. [3] K.S.M. Reddy, J.J. Park, S.-M. Na, M.M. Maqableh, A.B. Flatau, B.J.H. Stadler, Electrochemical synthesis of magnetostrictive Fe-Ga/Cu multilayered nanowire arrays with tailored magnetic response, Adv. Funct. Mater. 21 (24) (2011) 4677–4683. [4] E.C. Estrine, W.P. Robbins, M.M. Maqableh, B.J.H. Stadler, Electrodeposition and characterization of magnetostrictive galfenol (FeGa) thin films for use in microelectromechanical systems, J. Appl. Phys. 113 (17) (2013) 17A937. [5] E.C. Estrine, M. Hein, W.P. Robbins, B.J.H. Stadler, Composition and crystallinity in electrochemically deposited magnetostrictive galfenol (FeGa), J. Appl. Phys. 115 (17) (2014) 17A918. [6] P.D. McGary, L. Tan, J. Zou, B.J.H. Stadler, P.R. Downey, A.B. Flatau, Magnetic nanowires for acoustic sensors (invited), J. Appl. Phys. 99 (8) (2006) 08B310. [7] P.D. McGary, K.S.M. Reddy, G.D. Haugstad, B.J.H. Stadler, Combinatorial electrodeposition of magnetostrictive Fe1 xGax, J. Electrochem. Soc. 157 (12) (2010) D656. [8] K.S.M. Reddy, E.C. Estrine, D.-H. Lim, W.H. Smyrl, B.J.H. Stadler, Controlled electrochemical deposition of magnetostrictive Fe1 xGax alloys, Electrochem. Commun. 18 (2012) 127–130. [9] K.S.M. Reddy, M.M. Maqableh, B.J.H. Stadler, Epitaxial Fe(1 x)Gax/GaAs structures via electrochemistry for spintronics applications, J. Appl. Phys. 111 (7) (2012) 07E502. [10] J.J. Park, E.C. Estrine, S. Madhukar Reddy, B.J.H. Stadler, A.B. Flatau, Technique for measurement of magnetostriction in an individual nanowire using atomic force microscopy, J. Appl. Phys. 115 (17) (2014) 17A919. [11] P.R. Downey, A.B. Flatau, P.D. McGary, B.J.H. Stadler, Effect of magnetic field on the mechanical properties of magnetostrictive iron-gallium nanowires, J. Appl. Phys. 103 (7) (2008) 07D305. [12] A.J. Grutter, et al., Complex three-dimensional magnetic ordering in segmented nanowire arrays, ACS Nano 11 (8) (2017) 8311–8319. [13] E. Ortega, S.M. Reddy, I. Betancourt, S. Roughani, B.J.H. Stadler, A. Ponce, Magnetic ordering in 45 nm-diameter multisegmented FeGa/Cu nanowires: single nanowires and arrays, J. Mater. Chem. C 5 (30) (2017) 7546–7552. [14] J.J. Park, M. Reddy, C. Mudivarthi, P.R. Downey, B.J.H. Stadler, A.B. Flatau, Characterization of the magnetic properties of multilayer magnetostrictive iron-gallium nanowires, J. Appl. Phys. 107 (9) (2010) 09A954. [15] J.J. Park, M. Reddy, B.J.H. Stadler, A.B. Flatau, Hysteresis measurement of individual multilayered Fe-Ga/Cu nanowires using magnetic force microscopy, J. Appl. Phys. 113 (17) (2013) 17A331. [16] J.J. Park, K.S.M. Reddy, B. Stadler, A. Flatau, Magnetostrictive Fe–Ga/Cu nanowires array with GMR sensor for sensing applied pressure, IEEE Sensors J. 17 (7) (2017) 2015–2020.

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[17] K.S.M. Reddy, J. Jin Park, M.M. Maqableh, A.B. Flatau, B.J.H. Stadler, Magnetization reversal mechanisms in 35-nm diameter Fe1 xGax/Cu multilayered nanowires, J. Appl. Phys. 111 (7) (2012) 07A920. [18] L. Sun, Y. Hao, C.-L. Chien, P.C. Searson, Tuning the properties of magnetic nanowires, IBM J. Res. Dev. 49 (1) (2005) 79–102. [19] A. Ghemes, et al., Controlled electrodeposition and magnetic properties of Co35Fe65 nanowires with high saturation magnetization, J. Electrochem. Soc. 164 (2) (2017) D13–D22. [20] M.A. Hein, et al., fabrication of bioinspired inorganic nanocilia sensors, IEEE Trans. Magn. 49 (1) (2013) 191–196. [21] L. Tan, B.J.H. Stadler, Fabrication and magnetic behavior of Co/Cu multilayered nanowires, J. Mater. Res. 21 (11) (2006) 2870–2875. [22] J. Cantu-Valle, et al., Mapping the magnetic and crystal structure in cobalt nanowires, J. Appl. Phys. 118 (2) (2015) 024302. [23] L. Tan, P.D. McGary, B.J.H. Stadler, Controlling the angular response of magnetoresistance in Co/Cu multilayered nanowires using Co crystallographic orientation, J. Appl. Phys. 103 (7) (2008) 07B504. [24] InRedox LLC, Data Sheet with Detailed Specifications for Standard Products. [25] M.M. Maqableh, et al., Low-resistivity 10 nm diameter magnetic sensors, Nano Lett. 12 (8) (2012) 4102–4109. [26] M.M. Maqableh, et al., CPP GMR through nanowires, IEEE Trans. Magn. 48 (5) (2012) 1744–1750. [27] S.-Y. Sung, M.M. Maqableh, X. Huang, K. Sai Madhukar Reddy, R.H. Victora, B.J. H. Stadler, Metallic 10 nm diameter magnetic sensors and large-scale ordered arrays, IEEE Trans. Magn. 50 (11) (2014) 1–5. [28] M. Darques, L. Piraux, A. Encinas, Influence of the diameter and growth conditions on the magnetic properties of cobalt nanowires, IEEE Trans. Magn. 41 (10) (2005) 3415–3417. [29] M. Darques, A. Encinas, L. Vila, L. Piraux, Controlled changes in the microstructure and magnetic anisotropy in arrays of electrodeposited Co nanowires induced by the solution pH, J. Phys. D: Appl. Phys. 37 (10) (2004) 1411–1416. [30] S. Herna´ndez, L. Tan, B.J.H. Stadler, R.H. Victora, Micromagnetic calculation of spin transfer torque in Co/Cu multilayer nanowires, J. Appl. Phys. 109 (7) (2011) 07C916. [31] X. Huang, L. Tan, H. Cho, B.J.H. Stadler, Magnetoresistance and spin transfer torque in electrodeposited Co/Cu multilayered nanowire arrays with small diameters, J. Appl. Phys. 105 (7) (2009) 07D128. [32] B.D. Plouffe, S.K. Murthy, L.H. Lewis, Fundamentals and application of magnetic particles in cell isolation and enrichment: a review, Rep. Prog. Phys. 78 (1) (2015) 016601. [33] C.L. Ventola, Progress in nanomedicine: approved and investigational nanodrugs, Pharm. Ther. 42 (12) (2017) 742. [34] K.M. Pondman, et al., Au coated Ni nanowires with tuneable dimensions for biomedical applications, J. Mater. Chem. B 1 (44) (2013) 6129. [35] A. Sharma, et al., Inducing cells to disperse nickel nanowires via integrin-mediated responses, Nanotechnology 26 (13) (2015) 135102. [36] A.K. Salem, P.C. Searson, K.W. Leong, Multifunctional nanorods for gene delivery, Nat. Mater. 2 (10) (2003) 668–671. [37] A. Hultgren, M. Tanase, C.S. Chen, G.J. Meyer, D.H. Reich, Cell manipulation using magnetic nanowires, J. Appl. Phys. 93 (10) (2003) 7554–7556. [38] A. Sharma, Y. Zhu, S. Thor, F. Zhou, B. Stadler, A. Hubel, Magnetic barcode nanowires for osteosarcoma cell control, detection and separation, IEEE Trans. Magn. 49 (1) (2013) 453–456.

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[39] A. Sharma, et al., Alignment of collagen matrices using magnetic nanowires and magnetic barcode readout using first order reversal curves (FORC) (invited), J. Magn. Magn. Mater. 459 (2018) 176–181. [40] D.E. Shore, T. Dileepan, J.F. Modiano, M.K. Jenkins, B.J.H. Stadler, Enrichment and quantification of epitope-specific CD4 + T lymphocytes using ferromagnetic iron-gold and nickel nanowires, Sci. Rep. 8 (1) (2018). [41] W. Zhou, et al., Development of a biolabeling system using ferromagnetic nanowires, IEEE J. Electromagn. RF Microw. Med. Biol. 3 (2) (2019) 134–142. [42] C.-I. Dobrota˘, A. Stancu, What does a first-order reversal curve diagram really mean? A study case: array of ferromagnetic nanowires, J. Appl. Phys. 113 (4) (2013) 043928. [43] M. Sharma, S. Pathak, M. Sharm, FMR measurements of magnetic nanostructures, in: O. Yalc¸ın (Ed.), Ferromagnetic Resonance – Theory and Applications, IntechOpen, London, 2013. [44] D. Shore, et al., Electrodeposited Fe and Fe–Au nanowires as MRI contrast agents, Chem. Commun. 52 (85) (2016) 12634–12637. [45] D. Shore, et al., Nanowarming using Au-tipped Co35Fe65 ferromagnetic nanowires, Nanoscale 11 (31) (2019) 14607–14615. [46] E.C. Estrine, S. Riemer, V. Venkatasamy, B.J.H. Stadler, I. Tabakovic, Mechanism and stability study of gold electrodeposition from thiosulfate-sulfite solution, J. Electrochem. Soc. 161 (12) (2014) D687–D696. [47] C. Vargas-Estevez, et al., Study of Galfenol direct cytotoxicity and remote microactuation in cells, Biomaterials 139 (2017) 67–74. [48] J. Choi, K. Nielsch, M. Reiche, R.B. Wehrspohn, U. G€ osele, Fabrication of monodomain alumina pore arrays with an interpore distance smaller than the lattice constant of the imprint stamp, J. Vac. Sci. Technol. B: Microelectron. Nanometer Struct. 21 (2) (2003) 763. [49] J. Zou, X. Qi, L. Tan, B.J.H. Stadler, Large-scale ordering of porous Si using anodic aluminum oxide grown by directed self-assembly, Appl. Phys. Lett. 89 (9) (2006) 093106.

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Nanowire transducers for biomedical applications

23

Jose E. Pereza,b, Ju€rgen Koselb a Division of Biological and Environmental Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia, bDivision of Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

23.1

Introduction

Magnetic nanostructures have in recent years been used in a diverse array of biomedical applications. This is attributed to the ability of magnetic nanostructures to be externally manipulated or detected, which confers them with unique niches in areas such as cancer therapy, cell capture and labeling, cell interrogation, and the study of cellular functions, among others. Magnetic nanoparticles (NPs) are some of the most widely utilized nanostructures for this purpose, as their size, shape, composition, and biocompatibility can be finely tuned by the fabrication method [1]. Extensive research in the area has yielded a plethora of different shapes and conformations of NPs, including spherical, cubical, coreshell, and multicore, among others, with promising applications in areas such as cell labeling for magnetic resonance imaging (MRI) [2], drug delivery [3] hyperthermia as a cancer therapy [4], biosensors [5], and magnetic particle imaging [6]. Magnetic nanowires (NWs), so far, have been exploited for biomedical applications less than NPs. However, their unique properties and the ability to easily tailor their length and diameter to the desired parameters make them very potent materials for many biomedical tasks. Magnetic NWs possess a large shape anisotropy, due to the high aspect ratios, which enables their efficient manipulation by magnetic fields, like exerting large torques [7]. Depending on the fabrication method employed and the precursor materials, the magnetic properties of NWs can be finely modulated [8]. Additionally, NWs have a higher magnetization value per unit of volume compared to NPs, a quality that in turn yields higher forces [9]. Furthermore, the native and tunable few nanometers thick oxide layer present in metals [10, 11] provides a surface for biofunctionalization that can be used to regulate biocompatibility and to enhance cell targeting [12–14]. Fe NWs are specifically relevant for biomedical applications due to their extremely low cytotoxicity [10]. The behavior of a magnetic NW in an alternating magnetic field is dependent on its magnetization. In the case of Fe and Ni NWs, the orientation of the magnetization is mostly determined by their shape anisotropy, which dominates over the weak magnetocrystalline anisotropy, orienting the easy axis parallel to the axis of the wire. Magnetic Nano- and Microwires. https://doi.org/10.1016/B978-0-08-102832-2.00023-2 Copyright © 2020 Elsevier Ltd. All rights reserved.

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As a result, the NWs have magnetic properties of a single domain particle; hence, they are permanently magnetized, even when no field is applied [15]. When placed under a magnetic field, a magnetic NW will generate a torque as the magnetic moment along its easy axis tries to align to the direction of the applied field. The generated forces are the basis for the biomedical applications, which utilize vibrations or rotations of the NWs. The magnetic torque of a NW can be calculated from τm ¼ m  B ¼ μ0 m  H ¼ μ0 m H sin θ ,

(23.1)

where m is the magnetic moment of the NW, B is the magnetic field, μ0 is the permeability of free space, H is the strength of the applied magnetic field, and θ is the angle between m and H. The magnetic moment m is equal to m ¼ MV, where M is the magnetization of the NW and V ¼ πr2l (when approximated to the volume of a cylinder). Further, the magnetization M is about equal to the saturation magnetization MS, in case of a Fe NWs. Thus, the torque of a single Fe NW can be estimated by τm ¼ μ0 MS πr 2 l H sin θ:

(23.2)

It should be noted that the maximum torque is obtained at sin θ ¼ 90°, which corresponds to a magnetic field applied in a direction perpendicular to the axis of the NW. Since the saturation magnetization MS is equal to the remanent magnetization MR, Fe NWs can generate large torques using low-amplitude magnetic fields, while this would require the application of a magnetic field strong enough to saturate the material, in the case of NPs. All of the qualities mentioned have made magnetic NWs attractive in biomedical applications such as targeted drug delivery, cell eradication, stem cell stimulation, or as MRI contrast agents. These applications typically require the use of static (0 Hz) or low-frequency (1 Hz–100 kHz) magnetic fields at low amplitudes, which show negligible energy absorption by body tissue and are thus deemed safe [16]. For such fields, the permeability of human tissue is the same as that of air, and shows no attenuation of the external magnetic field intensity [16].

23.2

Biocompatible magnetic nanowires

Given the required interfacing between NWs and biological tissue, their biocompatibility must be evaluated within a biological setting prior to their desired application. Here, the most commonly used methods for the fabrication of magnetic NWs for biomedical applications will be presented, followed by an overview of the biocompatibility and toxicological data available in the literature. Magnetic NWs that are intended for use in biomedical applications are mostly fabricated using electrodeposition into nanoporous templates, a well-defined and widely used method [17]. It is a highly flexible process that allows tuning of various important parameters, such as the material, length, and diameter.

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The fabrication process of electrodeposition into nanoporous templates has been described in detail before. Briefly, it begins with a nanoporous template. Alumina is the material of choice, given how controlled the formation of nanopores is [17], and how easily it can be modified to create thicker or thinner pores, hence NWs. The nanoporous template formation is carried out by a two-step anodization of highly pure aluminum in an electrolytic bath. The electrolyte utilized determines the pore sizes, which allows varying the diameter of the NW [18]. The first anodization of the aluminum, performed under specific current, temperature and time conditions, creates a disordered nanoporous alumina film, which upon growth creates domains with ordered pores at the interface with the aluminum. Wet etching of the disordered nanoporous film reveals these domains, on top of which a second anodization is then performed, in order to create a new, ordered nanoporous film. This is followed by electrodeposition of the desired material into the ordered template, whereby pulsed or DC deposition is commonly used. It should be mentioned that the length of the NW is determined by the electrodeposition parameters and the electrolyte composition. Many examples of magnetic NW fabrication exist in the literature, each with their own anodization and electrodeposition parameters. They include cobalt [19], nickel [20], and iron (Fe) NWs [21]. Further modifications to the fabrication protocol have additionally shown a plethora of different NW configurations, including multisegmented [22–24], alloy [25], and core-shell NWs [10, 26]. Further, the method can be adapted to create substrates with embedded NWs to produce ultra-high-gradient magnetic fields [27], nanostructured cell scaffolds [28], or for molecular sensing [29]. It is important to highlight this flexibility in fabrication, given that all these parameters affect the NW biocompatibility and are the basis of the finetuning of their magnetic properties, which are crucial depending on the desired application. The use of magnetic NWs in the context of biomedical applications has first focused on Ni, and later on Fe and its derivatives, due to their excellent biocompatibility. Ivanov and colleagues recently proposed a simple post-fabrication step that produces highly biocompatible Fe-Fe3O4 core-shell NWs [10]. To achieve this, Fe NWs were oxidized in an oven at 150°C, resulting in NWs with an Fe core and a Fe3O4 shell that acts as a passivation layer, which can further be biofunctionalized for the desired application (Fig. 23.1). Control over the thickness of the Fe3O4 shell via the annealing time allows finetuning the magnetic properties of the NWs. Specifically, lower remanent magnetizations can be achieved with thicker shells, an attractive feature for bioapplications, where magnetostatic interactions between NWs can lead to undesirable effects (inhomogeneous dispersion in solution). The biocompatibility of Fe NWs has been mostly documented in in vitro models, with little to no data found in the literature concerning in vivo studies. Nonetheless, in vitro models, though lacking a full response and biodistribution and excretion scenario, are relevant in providing valuable preliminary cytotoxicological information. Additionally, they are easily and readily implemented. In one of the first in vitro biocompatibility studies with Fe NWs, they were interfaced with HeLa cells at different concentrations and incubation times [30]. The oriented Fe-Ga as it has greater saturation magnetostriction than < 110> oriented Fe-Ga;  300 ppm versus 150 ppm, respectively. However, thin films and nanowires generally exhibit a texture in the growth direction, thus characterization data from both orientations are useful. Additionally, the saturation magnetization along < 100> and < 110> directions are the same, so both have potential for similar magnetization changes in energy harvesting and sensor applications (albeit in response to different stress fields). In Fig. 25.4A and B magnetostriction for the no-load cases are well below the saturation magnetostriction values achieved with 15–30 MPa of compressive stress applied to the rod; 215 versus 300 ppm for the oriented rod, and 15 versus 145 ppm for the oriented rod. Compressive stress helps to align domains perpendicular to the sample axis, which promotes 90 degrees rotation of moments when a field is applied along the sample axis. This is evident in Fig. 25.3, where,   for example, a compressive stress of roughly 12.5 MPa has rotated most 010 and [010] domains into the antiparallel [100] directions by the middle row. This value of 12.5 MPa was estimated using 57–59 GPa as the elastic modulus for Fe81Ga19 [8, 9] and that the sample [010] length was compressed by  216 με. It is fairly common to maximize magnetostriction in actuators by incorporating compressive springs or using stress annealing or field annealing of the magnetostrictive

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Fig. 25.4 Macroscale Fe-Ga data: actuator and sensor characterization curves for a rod of oriented Fe81Ga19 single crystal and for a rod of < 110> oriented Fe82Ga18 single crystal. Both rods were  25.4 mm long and 6.35 mm in diameter. (A) actuator data showing magnetostriction versus applied field for compressive loads of 0–80 MPa. (B) actuator data showing magnetostriction versus applied field for compressive loads of 0–80 MPa. (C) sensor data showing magnetic induction versus compressive stress for DC applied magnetic fields of 0–891 Oe. (D) sensor data showing magnetic induction versus compressive stress for DC applied magnetic fields of 0–891 Oe. The legend in (C) and the arrow in (D) apply to the data in both (C) and (D). The arrow in (D) points in the direction of increasing applied magnetic field. Reprinted from J. Atulasimha, A.B. Flatau, Experimental actuation and sensing behavior of iron-gallium alloys, J. Intel. Mater. Syst. Struct. 19 (2008). https://doi.org/10.1177/ 1045389X07086538.

component in a manner that applies just enough stress to rotate most moments perpendicular to a rod axis [11, 12]. The compressive stress is particularly helpful for the oriented rod, as without a mechanical load, magnetocrystalline anisotropy energy will promote alignment of two-thirds of the domains along the < 100> directions that are oriented at 45 degrees to the cylinder axis. When a field is applied along the sample length, those moments can only rotate 45 degrees producing only half the change in lattice spacing

Fe-Ga Nanowires for actuation and sensing applications

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that can be achieved with a 90 degrees moment rotation from the [100] direction perpendicular to the rod axis. For additional information on this, see the modeling in Chapter 3 by Atulasimha [8], where Atulasimha gives the equations for maximum achievable magnetostriction with and without any compressive stress as λ110,under

load

¼ 3=4 λ100 + 3=4 λ111

(25.1)

and λ110,no load ¼ 1=4 λ100 + 3=4 λ111

(25.2)

Eq. (25.1) assumes that all moments are perpendicular to the rod axis and Eq. (25.2) assumes that roughly one-third of magnetic moments are oriented along each of the three < 100 > magnetic easy axes. Plugging in the constants for Fe82Ga18: 3/2 λ100 ¼ 300 and 3/2 λ111 ¼  10 [8] into Eq. (25.1) gives saturation magnetostriction values for λ110,under load ¼ 145 ppm, which is very close to the measured value. Plugging these constant into Eq. (25.2) gives λ110,no load ¼ 43 ppm, which is roughly three times higher than the measured value. This suggests that this sample had fewer than one-third of the moments aligned along the direction perpendicular to the rod axis prior to field application along the rod axis (the [110] direction). These responses are used to create actuators, sensors, and energy harvesters. The unique mechanical attributes of these alloys allow designers to use Fe-Ga alloys in unimorph actuators to produce bending deflections, and in bending sensor applications such as the one shown in Fig. 25.5 to measure bending deflections. Here, a bending sensor made from a thin strip of Goss textured rolled sheet Fe-Ga (roughly 150  6.35  0.4 mm3) is shown as it goes from being straight to bending roughly

Fig. 25.5 Bending sensor made from a thin strip of Goss textured Galfenol (Fe-Ga) rolled sheet. (A) Rolled sheet is cantilevered from a 3D printed holder containing a small magnet and a Hall effect sensor. (B and C) As the tip of the Fe-Ga sample is deflected to the left, bending-induced changes in stress near the holder produce changes in sample magnetization above the Hall effect sensor. The Hall effect sensor measures the change in magnetization, giving rise to the 800mV change in the Hall effect sensor output in the display as the Galfenol rolled sheet is bent 90-degrees to the left.

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90 degrees. A small biasing magnet in the holder (Fig. 25.5A) provides a constant magnetic bias that promotes alignment of magnetic domains near the holder along the [110] width of the strip. (We have also operated this sensor with the bias magnet positioned to promote magnetization along the length of the strip, and both magnetization orientations are effective for the design of a bending sensor. One can optimize sensor performance by matching biasing field to changes in stress for either bias magnet configuration.) Here, as the strip is deflected to the left, compressive and tensile stress on the left and right sides of the strip, respectively, increase in proportion to the deflection angle. Compressive stress keeps magnetic domains near the magnet aligned with the biasing field from the magnet (i.e., along the width of the strip). Tensile stress induces 90-degree rotation of magnetic moments away from their initial direction toward the [100] easy axis along the length of the strip. This produces an increase in magnetization measured along the length of the strip and a decrease in the magnetization measured along the width of the strip. The change in magnetization with beam deflection is easily measured by a Hall effect sensor located in the holder (Fig. 25.5). The strength, position, and orientation of the biasing magnet can be tuned to maximize the device sensitivity to the stresses in the beam during 90-degree deflections. The simplicity with which Fe-Ga alloy can be used as bending sensors, combined with the alloy mechanical properties being suitable for many loading cycles and that the alloys appeared to be a good candidate for fabrication at the nanoscale using electrochemical deposition methods, motivated the initial studies into if it would be feasible to use Fe-Ga nanowires for the development of cilia-like bending sensors at the nanoscale [13, 14].

25.2

Modeling and micromagnetics simulations of Fe-Ga nanowires

Sun et al. [15] presented a comprehensive review of single-component and twocomponent multilayer magnetic nanowires. Their review discusses the roles of exchange energy, Zeeman energy, magnetoelastic energy, magnetocrystalline anisotropy energy, and magnetostatic shape anisotropy energy (demagnetization energy) in magnetic domain formation and magnetic hysteresis loops. As the size of Fe-Ga samples decreases to the nanoscale, shape anisotropy plays a particularly significant role in enabling the use of these alloys for actuation and sensing. Taking an approach similar to that presented by Sun et al. [15], Downey [13] performed a number of analytical simulations of Fe-Ga nanowires. Downey [13] estimated the critical radius below which an Fe-Ga prolate spheroid (as a representation of an Fe-Ga nanowire) would form only a single domain. Downey used an iterative solution approach to find the aspect ratio for a given nanowire diameter below which multiple domains form. He used the lattice parameter for Fe-Ga, a0 ¼ 0.286 nm, the exchange constant, A ¼ 1  1011 J/m, and the magnetic saturation, Ms ¼ 1456  103 A/m. He first solved for the demagnetization factor Nx, along the nanowire length as a function of the nanowire aspect ratio, c/a (length/diameter), as shown in Fig. 25.6A, and then solved for the critical radius associated with that demagnetization factor, below which only single domains form (shown in

Fig. 25.6 (A) Demagnetization factors associated with Fe-Ga nanowires modeled as a prolate spheroid for aspect ratios of 1–100. (B) The radius below which single magnetic domains will form. (C) Graph that combines information from (A) and (B), showing nanowire (NW) diameter for a given aspect ratio above which multiple magnetic domains will form. This holds for Fe-Ga NW and for Fe-Ga segments in Fe-Ga/Cu NW (sketches at lower right) if the Fe-Ga magnetic segments do not interact significantly. Reprinted from P.R. Downey, Characterization of bending magnetostriction in iron-gallium alloys for nanowire sensor applications, PhD Dissertation, University of Maryland, 2008.

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Fig. 25.6B). (Note that Nx + Ny + Nz ¼ 1, where Ny and Nz, which are also shown in Fig. 25.6A, are the demagnetization factors associated with magnetization perpendicular to the length of the prolate spheroid/nanowire. Due to symmetry, Ny and Nz are equal.) In Fig. 25.6C the information in Fig. 25.6A and B is combined into a single line that identifies the minimum nanowire diameter for a given aspect ratio, or equivalently the maximum aspect ratio for a given nanowire diameter, that will promote the formation of multiple domains. The line in Fig. 25.6C holds as long as the nanowire is not subjected to a mechanical load or magnetic field. This analysis holds for both Fe-Ga nanowires and Fe-Ga segments of multilayer nanowires like the Fe-Ga/Cu nanowire depicted in Fig. 25.6C, where a nonmagnetic layer separates the Fe-Ga segments from one another by a distance that is large enough to minimize interaction of Fe-Ga segments. In Fig. 25.6C, note that a nanowire with a 100-nm diameter will form multiple domains if it has an aspect ratio of lattice sketch between Fig. 25.11C and E illustrates that a < 110 > texture places one of the three Fe-Ga < 100> magnetic easy axes in a plane perpendicular to the NW axis, which should be useful for both sensing and actuation. Finally, while the < 110> texture was most dominant, some NWs, for example, Fig. 25.11F, did exhibit other crystallographic orientations. The Lorentz TEM image of the Fe-Ga nanowire on a carbon mesh (Fig. 25.11G) shows a single axial domain oriented to the left as one continuous bright stripe running along the NW length that is uniformly slightly closer to the bottom of the Fe-Ga NW. If multiple domain segments were present, this line would jump between the top and bottom sides of the nanowire at each domain boundary. Rotating the nanowire in an

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Fig. 25.11 Fe-Ga nanowire microstructure and magnetic domain analysis. (A) TEM image of nanowire. (B) HRTEM image showing a crystalline growth; inset: SAED pattern with arrow shown pointing along the nanowire axis. (C and D) EBSD analysis showing inverse pole figure (IPF) and the corresponding 110 pole figure. (E) IPF of a different section of the same nanowire showing a low-angle grain boundary defect. (F) IPF of less common structure. (G) TEM image of nanowires on a carbon mesh indicating a single axial domain oriented to the left. (H and I) TEM images of nanowire rotated in a magnetic field with a single magnetic domain that flipped from left to right when field along the wire reached 110 Oe with no intermediate state (highlighted stripe on top side of NW in (G) and on bottom side of NW in (H)). (A)–(F) Reprinted from S.M. Reddy, J.J. Park, S.M. Na, M.M. Maqableh, A.B. Flatau, B.J.H. Stadler, Electrochemical synthesis of magnetostrictive Fe-Ga/cu multilayered nanowire arrays with tailored magnetic response, Adv. Funct. Mater. 21 (24) (2011) 4677–4683 and (G)–(I) Reprinted from P.R. Downey, Characterization of bending magnetostriction in iron-gallium alloys for nanowire sensor applications, PhD Dissertation, University of Maryland, 2008.

applied field caused the magnetization to flip 180 degrees, that is, to the right and back to the left in Fig. 25.11H and I as the field along the length of the nanowire reached 110 Oe, with no intermediate state [13]. This experimentally observed singledomain behavior is the expected response for magnetic nanowires with an aspect ratio of greater than 10 [15]. The 110 Oe field at which the magnetization vector flipped is consistent with the Magpar micromagnetic finite element method simulations of M-H curves for a high aspect ratio Fe-Ga NW, presented in Fig. 25.8, where the coercive magnetic field was estimated to be 138 Oe.

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Fig. 25.12 Microstructure analysis of multilayered nanowires. (A) SEM-EDS compositional map of (330 nm Fe-Ga)/(90 nm Cu) multilayered nanowires. (B) TEM image showing Fe-Ga segment in a different nanowire. (C) HRTEM image showing crystalline growth of Fe-Ga segment. (Inset) SAED pattern of the Fe-Ga segment with arrow shown pointing along the nanowire axis. Corroborating XRD results, the TEM results showed the Fe-Ga segments had a preferential (110) texture along the nanowire length. (D) XRD results in the upper panel and JCPDS data for Fe (JCPDS file #00-006-0696) and Cu (JCPDS file #00-004-0836) are shown in the bottom panel of (D). Images reprinted from S.M. Reddy, J.J. Park, S.M. Na, M.M. Maqableh, A.B. Flatau, B.J.H. Stadler, Electrochemical synthesis of magnetostrictive Fe-Ga/cu multilayered nanowire arrays with tailored magnetic response, Adv. Funct. Mater. 21 (24) (2011) 4677–4683.

Compositional and structural analysis of Fe-Ga/Cu two-component multilayered nanowires (Fig. 25.12A–D) and in particular the XRD results in Fig. 25.12D, indicate that the Fe-Ga segments of the electrodeposited nanowire also had a preferential (110) texture along the length of the nanowire [26]. The compositional map of the Fe-Ga/Cu nanowires in Fig. 25.12A shows that the interfaces between the Fe-Ga and Cu segments are well defined. Fig. 25.12B shows the TEM image of a different multilayered nanowire (Fe-Ga segment is within dotted box), and Fig. 25.12C shows a high-resolution TEM (HRTEM) image of the Fe-Ga segment. A SAED pattern (inset of Fig. 25.12C) of this particular Fe-Ga segment showed that the direction was tilted away from the nanowire axis and the < 112 > direction was close to the nanowire axis. Fig. 25.12D shows XRD results, where the upper trace, labeled A, is a single-component Fe-Ga nanowire, and the lower trace, labeled B, is a twocomponent Fe-Ga/Cu multilayer nanowire. Both exhibit strong (110) Fe-Ga peaks, and noticeable (211) Fe-Ga peaks. Trace B also exhibits the expected (111) Cu peak. Fe-Ga segments with the (211) orientation were present to a small extent in all the nanowire structures. Analysis of the reciprocal distances in SAED patterns showed that the (110) interplanar spacing in the multilayered nanowire was 4% larger than that in a uniform Fe-Ga nanowire, very likely due to co-deposition of Cu within the Fe-Ga segments. TEM diffraction studies on Cu segments (not shown) indicated a slight polycrystallinity, supporting XRD observations shown in Fig. 25.12D.

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Single component Fe-Ga NWs grown by McGary [14] underwent resonance testing and tension testing in an SEM to identify NW elastic moduli. These initial studies helped establish successful modulus characterization test protocols (presented in Downey et al. [16]) that were validated using multiwalled carbon nanotubes [13]. The measured Fe-Ga moduli values, however, were both inconsistent and lower than that expected for NWs with a < 110> texture like that shown in Fig. 25.11C–E, ranging from 44 to 98 GPa. Subsequent energy-dispersive X-ray spectroscopy (EDS) of NWs used in these initial studies determined two issues. The gallium content in the NWs was below 4 at.% Ga, rather than the sought after compositions of 17–21 at. % Ga. Also, it was discovered that the copper that was briefly sputtered onto the backside of the AAO template in order to create an electrode for the electrochemical deposition process had resulted in every nanowire having a copper stub at that end that was up to 5-μm long [13]. McGary made modifications to the NW fabrication process that provided substantial improvements. Four consecutive measurements of the composition of different NWs from the batch used for subsequent modulus characterization tests had gallium content of 16.52, 18.84, 19.20, and 17.49 at.%. The modulus of these NWs was obtained using resonance testing, which required electron beam-induced deposition (EBID or nanowelding) of only one end of a NW to an AFM tip at a somewhat arbitrary angle, and could be readily accomplished. This was done by positioning an AFM probe tip adjacent to the free end of one of the longer NWs in an array. After EBID welding, the welded NW was broken free from the array using the piezo stepper motor holding the AFM probe tip to move the welded NW away from the array. Tension testing required EBID of the two ends of a NW to two different AFM probe tips at an angle of 90 degrees to each AFM probe cantilever, which was quite a timeconsuming process, and was subject to additional errors introduced by the presence of copper from the electrode end of the NWs. Resonance testing was performed using the experimental setup shown Fig. 25.13A. (This student-built x-y-z manipulation stage was inspired by a visit to the lab of Professor R. Ruoff and D. Dinkin at Northwestern University and is fully described by Downey [13].) It included a piezoelectric bimorph on the x–y stage, which was used to excite the mechanical resonance of NWs attached to the tip of an AFM probe that was installed in a probe holder on the bimorph (Fig. 25.13B–D). The experiment itself simply consists of driving the piezoelectric bimorph with a low-voltage (typically up to 500 mV) swept sine signal and observing the NW vibration. When the NW is excited near its fundamental resonance, the nanowire appears as a blurred cone bounded by its maximum amplitude of vibration (Fig. 25.13E) while the rest of the micrograph is still. This is due to the frequency of oscillation being much higher than the raster scan rate of the electron beam. The beam spot is placed on the free end of the cantilevered nanowire, corresponding to a bright light intensity which is output to the monitor as a voltage. During the frequency sweep, this point will remain unchanged except for the frequency region across which the NW is strongly vibrating, which, due to the blurred cone effect, will be darker than the stationary wire (Fig. 25.13E). As a result, the measured spot brightness will decrease. When inverted, the recorded voltage, which is proportional to the spot light intensity, has a peak at the NW resonant

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Fig. 25.13 Resonance testing experiment for determination of elastic modulus of Fe-Ga NW, inspired by a visit to Professor R. Ruoff’s lab at Northwestern University. (A) x–y–z manipulation stage installed in the SEM sample holder. (B) Image of AFM probe installed in holder on piezoelectric bimorph that in turn is installed on the manipulator x–y stage. (C) Image of AFM probe cantilever. (D) Image of AFM tip showing the nanoweld (the bright spot) used to rigidly attach Fe-Ga NW to the AFM tip. (E) Image of NW and AFM tip when piezoelectric bimorph is driven at/near NW fundamental resonant frequency. Note that at the crosshair location, shown in yellow, NW motion causes a decrease in light intensity. (F) Typical resonance testing output, showing inverse spot intensity of light versus frequency for a location near the free end of a cantilevered NW as the piezoelectric bimorph was swept through frequencies that excited the NW first bending mode. Images reprinted from P.R. Downey, Characterization of bending magnetostriction in irongallium alloys for nanowire sensor applications, PhD Dissertation, University of Maryland, 2008.

frequency (Fig. 25.13F). The measured resonant frequency, NW length, and diameter were used in an Euler-Bernoulli bending-beam model to back out the elastic modulus associated with resonance of the fundamental vibration mode of a cantilevered beam (NW). The elastic modulus obtained from these tests was between 158 and 181 GPa, which is consistent with the value of 160 GPa in the literature for bulk Galfenol with a [110] crystal texture [13]. Magnetic reversal mechanisms in Fe-Ga/Cu multilayer NWs were studied using both 100-nm diameter NWs [26] and 35-nm diameter NWs [27]. Fe-Ga and Cu segment aspect ratios were varied with the goal of influencing shape anisotropy and intrawire magnetostatic interaction, respectively, and are summarized in Fig. 25.14A and B. As previously discussed, shape anisotropy promotes domain alignment along the NW axis (blue trace in Fig. 25.8). Intrawire magnetostatic interactions also promote alignment of domains along the NW axis. When ferromagnetic segments are close enough

Fig. 25.14 Magnetization versus applied field hysteresis loops for arrays of 100-nm and 35-nm diameter Fe-Ga and Fe-Ga/Cu nanowires (NW). (A and B) Summary information on the structure of NWs. (C and F) M-H curves generated in VSM from arrays of NW with field applied parallel to the NW axis. (D and G) M-H curves generated in VSM from arrays of NW with field applied perpendicular to the NW axis. (E and H) Coercivity as a function of applied field angle as NW arrays were rotated with respect to the NW axis. Images reprinted from S.M. Reddy, J.J. Park, S.M. Na, M.M. Maqableh, A.B. Flatau, B.J.H. Stadler, Electrochemical synthesis of magnetostrictive Fe-Ga/cu multilayered nanowire arrays with tailored magnetic response, Adv. Funct. Mater. 21 (24) (2011) 4677–4683 and K.S.M. Reddy, J. Jin Park, M.M. Maqableh, A.B. Flatau, B.J.H. Stadler, Magnetization reversal mechanisms in 35-nm diameter Fe1-xGax/Cu multilayered nanowires, J. Appl. Phys. 111 (2012) 07A920.

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to one another they provide a preferred path for flux emanating from the north (north seeking) pole of one segment into the south pole of the nearest ferromagnetic segment, as if forming a virtual chain of uniaxially aligned bar magnets, so as to minimize magnetostatic energy in the wire. M-H curves from arrays of the NW in Fig. 25.14A and B were acquired in a VSM with applied fields parallel (Fig. 25.14C and F) and perpendicular (Fig. 25.14D and G) to the NW axis. Data are also presented on the magnitude of coercive field as a function of the angle between the applied field and the NW axis, θ, for angles of 30 to +210 degrees (Fig. 25.14E and H). Trends in the angular dependence of coercivity, Hc,θ, have been associated with four common NW magnetization reversal modes: coherent rotation (CR), transverse wall propagation (T) (or localized CR), curling (C), and vortex wall propagation (V) (or localized C). The first two reversal modes, CR and T, are associated with a decrease in Hc,θ in the 0–90 degree range, with a minimum at 90 degrees. The other two reversal modes, C and V, are associated with an increase in Hc,θ in the 0–90 degree range, with a peak at 90 degrees [27–30]. For structures A–D and a–e (shown in Fig. 25.14A and B), the M-H curves in a perpendicular field (shown in Fig. 25.14D and G) are sheared to the right relative to the M-H curves in a parallel field (shown in Fig. 25.14C and F), which are more upright. This indicates that the magnetic easy axis is parallel to the NW axis for these structures. The only exception to this was structure E, for which the M-H curve and Hc, k and Hc, ? values are nominally the same for both parallel and perpendicular field directions (Fig. 25.14C and D), as are Hc,0° and Hc,90° values in Fig. 25.14E. In fact, the coercivity value in Fig. 25.14E for structure E is the only one that is fairly constant for all angles. This indicates that the magnetic behavior of structure E is isotropic and the ease of rotating the orientation of magnetic domains in these structures will be insensitive to applied field direction. Zeeman energy from the applied field readily dominates the energies that determine direction of the segment magnetization vector. Energies that normally promote domain formation along the NW axis are low in structure E relative to the other structures in Fig. 25.14 because of (1) a low Fe-Ga aspect ratio (low shape anisotropy energy) and (2) a high Cu segment aspect ratio (low intrawire magnetostatic energy). Structures c, d, and e in the 35-nm diameter NWs (Fig. 25.14B) all have an Fe-Ga segment aspect ratio of 0.1, while the Cu segment aspect ratios are 0.1, 0.2 and 0.3, respectively. This increase in span between ferromagnetic segments leads to a pronounced reduction in the Hc,θ variability as a function of the angle of the applied field (Fig. 25.14H). This reduction in Hc,θ variability is consistent with a decrease in intrawire magnetostatic interactions due to the doubling and tripling of the Cu segment length in structures d and e relative to structure c. Although the Fe-Ga segments of structure E in the 100-nm diameter NW have somewhat greater shape anisotropy than the Fe-Ga segments of structures c, d, and e in the 35-nm diameter NWs (Fe-Ga aspect ratio of 0.5 vs. 0.1), it is the Cu segment aspect ratio of 5 that appears to preclude intrawire magnetostatic interaction in structure E, leading to almost no variability Hc,θ with applied field angle and allowing Zeeman energy to drive magnetization rotation and reversal.

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Trends in the magnitude of coercive field as a function of the angle between the applied field and the NW axis, Hc,θ in Fig. 25.14E and H are considered next. Combinations of reversal modes have been observed when different modes satisfy energy minimization at different angles. Lavin et al. [29] presented an analytical model with supporting experimental data for a 50-nm diameter, 4-mm long cobalt NW illustrating a transition from a vortex mode reversal for the applied field direction at an angle of less than 56 degrees (increasing Hc,θ with increasing angle) to a coherent rotation reversal for angles greater than 56 degrees (decreasing Hc,θ with increasing angle). In Fig. 25.14E, structure A experienced a similar competition between the same two modes, as did all of the 35-nm diameter NWs, structures a–e (Fig. 25.14H). The Hc,θ values initially increased as the angle θ increases from 0 degree to 60 70 degrees, indicating reversal by V mode in the low angular regime. In the angular regime from 60 70 degrees to 90 degrees, Hc,θ decreased to a minimum at an angle of 90 degrees, indicating a contribution from CR mode reversal. In contrast, the 100-nm diameter two-component NWs, structures B–E (Fig. 25.14A), exhibited the well-defined bell-shaped curves in Fig. 25.14E with a peak at 90 degrees that one would expect for a pure vortex/curling reversal mode [15]. These results demonstrate the opportunity for using control of the aspect ratios of both Fe-Ga and Cu segments to tailor both the switching fields and the reversal mechanisms over a wide range of values in Fe-Ga/Cu multilayered nanowires. This will be useful as an aid in identifying candidate two-component NW geometries for magnetostrictive NW-based sensor and actuator applications. Va´zquez et al. [31] examined the influence of interwire interactions on NW. They showed that coercivity, Hc and the ratio of remanence magnetization to saturation magnetization, Mr/Ms, both decrease when interwire interactions increase. Their data has a 40%–50% drop in both Hc and Mr/Ms (parallel to the NW) when the ratio of NW diameter to interwire spacing increases above 0.5. Park et al. [32] examined interwire magnetostatic interactions in Fe-Ga/Cu NWs by obtaining M-H curves from an array of NW and an individual NW from that array (Fig. 25.15). The NWs had a 150-nm diameter, an aspect ratio of about 3 for both Fe-Ga and Cu segments and an interwire spacing within the hexagonal array of 200 nm. Thus, the array was well above the 0.5 ratio where the increase in interwire interactions has been observed [31, 33]. The shapes of the M-H curves in Fig. 25.15, with field parallel and perpendicular to the NW axis, indicate the magnetic easy axis is along the nanowire axis in both the single NW and the NW array. The coercivity values in the single NW and in the array are similar, 125 and 150 Oe, respectively, and there is a 40% drop in Mr/Ms in going from a single NW (Mr/Ms ¼ 0.5) to the array (Mr/Ms ¼ 0.2) for field parallel to the NW. That Hc remains essentially constant in Fig. 25.15 is not consistent with published trends [31, 33]. These two published studies focused on single component NW, and it is unknown if that difference or slightly larger NW diameters may contribute to the difference in observations. The 40% drop in Mr/Ms, in going from the single NW to the array, however, is consistent with the experimental trends observed by Va´zquez et al. [31], which show a 43% drop in Mr/Ms as NW diameter to interwire spacing ratio increased from 0.3 to 0.75.

Fe-Ga Nanowires for actuation and sensing applications

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Single FeGa/Cu NW Field parallel to NW axis Field perpendicular to NW axis FeGa/Cu NW array Field parallel to NW axis Field perpendicular to NW axis

0

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Fig. 25.15 Magnetization versus applied field hysteresis loops for Fe-Ga/Cu multilayer NW. Data gathered in an MFM in fields of 850 Oe are shown for the field applied parallel (blue) and perpendicular (red) for a single NW. Data gathered in a VSM in field of 2500 Oe are shown for the field applied parallel (solid line) and perpendicular (dashed line) for a NW array [13].

25.5

Actuation using Fe-Ga/Cu nanowires

Magnetostriction in Fe-Ga/Cu NWs, and hence the ability to use these NWs for actuation and to produce forces or motion, requires the ability to control and produce 90-degree rotation of magnetic domains (recall discussion of Figs. 25.2 and 25.4). Park et al. [18] presented the MFM images in Fig. 25.16 of a 150-nm diameter Fe-Ga/Cu NW showing the ability to control NW magnetic domain orientations in the low aspect ratio Fe-Ga segments of the Fe-Ga/Cu NW (i.e., low enough aspect ratio for a given diameter to form multiple domains; Fig. 25.6). The Fe-Ga segments in this 150-nm diameter NW had an aspect ratio of 2 (a length of 300 nm) and the Cu segments had an aspect ratio of 0.5 (a length of 70 nm). The experimental set up in Fig. 25.16A was used to acquire MFM images as a magnetic field was applied and as the NW rotated to new orientations relative to the applied field. Fig. 25.16B is an MFM image acquired using a 50-nm thick CoCr alloy coated MFM tip, and applying a 15-Oe field along the length of the NW (from top right toward lower left). NW interactions with the MFM tip cause the tip and thus the cantilever to oscillate differently than in the absence of a magnetic field. A repulsive magnetic force will cause the resonant oscillation curve to shift to a higher frequency, resulting in an increase in phase shift (bright contrast). An attractive magnetic force causes the resonance curve to shift to a lower frequency, resulting in a decrease in phase shift (dark contrast). In-plane magnetization that does not

Veeco 3100 MFM with electromagnet

(A)

Regular moment MFM tip plus 15 Oe axial field

(B) Micromagnetic simulations and MFM from Berganza et al. (2017) Mx

(4a)

1

Simulation 1

0 −1 Ni

CoNi

CoNi

(4c)

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(4e) (C) Fig. 25.16 (A) MFM with electromagnet and 150-nm diameter Fe-Ga/Cu NW with segment aspect ratios of 2 and 0.5, respectively. (B) MFM image of the Fe-Ga/Cu NW subjected to a 15-Oe field from electromagnet applied along the NW length. White arrows included to draw the eye to two vortex-pair or axial domain structures; green arrows point to vortex domains with uniform chirality along the NW axis, and yellow arrows point to Cu segments between the vortex domains. (C) Micromagnetic simulations and MFM image of vortex domains with uniform chirality along the NW axis in a 120-nm diameter Co-Ni/Ni NW that resemble the domains the green arrows point to in (B). (A) Reprinted from P.R. Downey, Characterization of bending magnetostriction in iron-gallium alloys for nanowire sensor applications, PhD Dissertation, University of Maryland, 2008. (B) Reprinted from J.J. Park, M. Reddy, C. Mudivarthi, P.R. Downey, B.J.H. Stadler, A.B. Flatau, Characterization of the magnetic properties of multilayer magnetostrictive iron-gallium nanowires, J. Appl. Phys. 107 (9) (2010) 09A954. Images in (C) reprinted with permission from E. Berganza, M. Jaafar, C. Bran, J. A. Ferna´ndez-Rolda´n, O. Chubykalo-Fesenko, M. Va´zquez, A. Asenjo, Multisegmented nanowires: a step towards the control of the domain wall configuration, Sci. Rep. 7 (2017) 11576, doi:https://doi.org/10.1038/s41598-017-11902-w.

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interact with the tip shows up as a neutral contrast. A dark contrast was produced along most of the NW length, indicating that the NW largely interacted with the MFM tip via attractive magnetic force. Upon further inspection, however, several regions of neutral and bright contrast can also be found. The white arrows with brackets near the upper end of the NW in Fig. 25.16B have been included to draw the eye to a region of two pairs of bright-dark contrasts, that based on dimensions are the domain structures in two Fe-Ga segments separated by one Cu segment. The most likely interpretation of these bright-dark pairs is that they are formed by axial domains (aligned along the length of the NW) in the Fe-Ga segments, like domains found in high aspect ratio NWs. Axial domains will flip 180 degrees in response to application of a saturating magnetic field along the NW axis, and will not contribute to NW magnetostriction. The green arrows point to what are likely vortex domain structures in the Fe-Ga segments. These are Fe-Ga segments in which bright contrasts, evident on the low/right-hand side of the NW, are located under and adjacent to a region of dark contrast. These domains are very similar to vortex domains discussed by Berganza et al. [34]. Each Fe-Ga segment has a single domain of one chirality, and adjacent segments have the same chirality. Berganza et al. [34] presented micromagnetic simulations and MFM images of CoNi segments from a 120-nm diameter CoNi/Ni NW (segment aspect ratios of 18 and 7, respectively), and their Fig. 4A, C, and E showing this are included here as Fig. 25.16C. These vortex structures will rotate 90 degrees in response to a sufficiently strong magnetic field applied along the NW axis and produce magnetostriction. The yellow arrows in Fig. 25.16B point to regions of neutral contrast that based on NW dimensions are likely the non-ferromagnetic Cu segments between the vortex domain structures in the Fe-Ga segments. That vortex pairs (or axial domains) can form at the end of a NW while vortex domains form along most of the rest of the NW is specifically addressed by Berganza et al. They noted that nanowires with multiple crystalline orientations can display different MFM domain structures (in their discussion of Fig. S4). Given this insight and the inverse pole figures in Fig. 25.11E and F which show that some of the Fe-Ga/ Cu NWs exhibit more than one crystalline orientation within the same NW, variation in the orientation of the magnetic domains in a NW at or near remanence is not unexpected. The MFM images in Figs. 25.17 and 25.18 are from additional testing done on the NW in Fig. 25.16B. In Fig. 25.17A–D, the field applied along the length of the NW was increased. As the field increases, so does the number of domain structures within Fe-Ga segments that rotate to become aligned with the applied field. In the 550-Oe field, all segments are aligned with the applied field. Again, referring back to discussion of Figs. 25.2 and 25.4, it is 90-degree rotation of magnetic moments that is expected to produce magnetostriction, and that likely did occur as vortex domains disappeared and domains aligned with the applied field along the length of the NW formed. Unfortunately, the resolution of the AFM/MFM used for these images was insufficient to be useful for measuring magnetostriction.

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Fig. 25.17 MFM images of 150-nm diameter Fe-Ga/Cu (aspect ratio 2/0.5) with regular as an increasing magnetic field is applied along the length of the NW in indicated direction. Applied field strength was: (A) 150 Oe; (B) 300 Oe; (C) 450 Oe; and (D) 550 Oe. Reprinted from J.J. Park, M. Reddy, C. Mudivarthi, P.R. Downey, B.J.H. Stadler, A.B. Flatau, Characterization of the magnetic properties of multilayer magnetostrictive iron-gallium nanowires, J. Appl. Phys. 107 (9) (2010) 09A954.

Fig. 25.18 MFM images of 150-nm diameter Fe-Ga/Cu (aspect ratio 2/0.5) as NW was rotated between poles of the electromagnet. Applied field was 550 Oe. Direction of field is indicated with white arrows. The direction of the applied field, relative to the length of the NW was: (A) 0; (B) 55; (C) 105; and (D) 180 degrees. The yellow arrows in (C) point to nonmagnetic Cu segments (aspect ratio 0.5) between Fe-Ga segments (aspect ratio 3). Reprinted from J.J. Park, M. Reddy, C. Mudivarthi, P.R. Downey, B.J.H. Stadler, A.B. Flatau, Characterization of the magnetic properties of multilayer magnetostrictive iron-gallium nanowires, J. Appl. Phys. 107 (9) (2010) 09A954.

The MFM images in Fig. 25.18 show the response of domains to a 550-Oe field as the NW was rotated in the space between the poles of the electromagnet. The angle of the applied field relative to the NW is indicated in each figure. The 550-Oe applied field is a saturating field, and it is large enough that the domains rotate to align with the applied field. Of particular interest in this set of images is Fig. 25.18C, where three distinct Cu segments are visible as the thin region of neutral contrast, indicated with yellow arrows, located between of the Fe-Ga segments. The Fe-Ga segments are evident as domains that are magnetized in the direction of the applied field, almost perpendicular to the NW length. Magnetostriction (change in length per unit length) is challenging to measure directly in NWs. For a magnetostriction value of λ ¼ 150 microstrain (με) or parts

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per million (ppm), the length change in a 10-μm long multilayered [110]-textured Fe-Ga/Cu NW would be actuator characterization data in Fig. 25.4B. Park et al. [35] presented a technique for measuring magnetostriction λ in Fe-Ga/Cu NWs that takes advantage of the relatively large lateral deflections experienced in buckled beams for relatively small changes axial length. This is done by placing NWs on a grid structure and using EBID to weld both ends of the NW to the grid surface. A magnetic field large enough to produce magnetostriction is applied along the NW length, producing a compressive stress in the NW that now has its ends rigidly attached to a grid so they cannot move. If this compressive stress exceeds the NW buckling stress, the NW will buckle. Welding the ends of the NW adjacent to flat surfaces promoted having the buckling deflection in an upward direction, away from the grid surface. As a brief review of buckling mechanics, buckling generally occurs when a load is applied to a slender beam that causes the ends of a beam to move toward one another by a small distance Δ (“small” relative to the beam length L). Once the applied load produces a compressive stress in the beam that exceeds the beam critical buckling  stress, σ cr ¼ π 2 EI AL2 (where E ¼ elastic modulus, I ¼ area moment of inertia, A ¼ cross sectional area, L ¼ beam length) the beam starts to buckle. When the applied load causes stresses that are greater than the critical buckling load but below the beam yield stress, the buckling that occurs is due to elastic deformations, and the buckled beam will return to its original length and shape when the load is removed. If the load increases enough to cause stresses in the beam that exceed the beam yield stress but not the beam ultimate stress, the buckled beam will undergo plastic deformation (initial shape will not be recovered upon load removal). If the load increases enough to cause stresses that exceed the beam ultimate stress, structural failure occurs and the buckled beam will break. Examples of operating in the elastic buckling regime include the work on motion amplification in an elastically buckled piezo bender by Clingman [36, 37] and the use of buckling to measure small changes in the length of micro-columns in micro-mechanical devices mentioned by Leckie and Dal Bello [38]. Park modifies the approach suggested by Leckie and Dal Bello for the measurement of magnetostriction. It is common to model the shape of a deformed buckled beam using trigonometric functions. Leckie and Dal Bello [38] showed that with the use of trigonometric functions and the assumption that L + Δ  L, an estimate of the length Δ is associated with a midspan deflection h and beam length L. This allows an estimate of Δ to be calculated by using measured values of L and h in Eq. (25.3). The elegant and counterintuitive approach used by Park et al. [35] uses almost the same equations, but Park keeps the span between the ends of the buckled beam rigidly fixed. Using the assumption that L + λL  L, Eq. (25.3) also applies to deflections of buckled NWs. The difference between the models is that for traditional buckling scenarios, an externally applied mechanical load produces the internal beam stress that exceeds the beam critical buckling stress, while for the NW case, an externally applied magnetic field

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causes the magnetostriction that produces the internal beam stress that exceeds the NW critical buckling stress. Also, in traditional buckling scenarios, the beam length is fixed and the span between the ends of the beam decreases by length Δ, while for the NW case the span between the ends of the NW is fixed and the length of the NW increases by length λL. Δ h2 π 2 λ¼ ¼ 2 L 4Leff

(25.3)

The term Leff, is introduced in Eq. (25.3) to address different beam boundary conditions. For pinned-pinned boundary conditions Leff ¼ L, the full pre-deflected beam length, and for fixed-fixed boundary conditions, Leff ¼ 0.5 L [38]. Different effective lengths are provided for other boundary conditions, but given the observed shapes of buckled NWs, bounding estimates of λ using the fixed-fixed and pinned-pinned boundary conditions due to the EBID welds for determining an effective NW length in Eq. (25.3) was taken as a reasonable first approximation. To measure L and h, values which are needed to estimate λ for a NW, a profile of the height of the buckled NW is measured. The measurement process is depicted in Figs. 25.19 and 25.20. Fig. 25.20A shows an SEM image of an Fe-Ga/Cu NW with both ends of the NW welded to the flat surface of a grid. In Fig. 25.19A and B height data at one location along the length of the NW are shown. Fig. 25.19C–E shows AFM images of the NW with a white line to provide an estimate of the cross section location along the length of the NW that corresponds to where the height data in Fig. 25.19A and B were obtained. This was one of the 10–15 cross section locations along the NW length where the AFM probe was moved perpendicular to the NW to get data on the height of the NW as a function of position along the NW length. In Fig. 25.19A and B, height data obtained with no field applied to the NW is shown in green. In Fig. 25.19A height data obtained with a 1000-Oe field applied along the NW length is shown in red. In Fig. 25.19B, height data obtained with a 1000-Oe field applied perpendicular to the NW, pointing vertically down into the grid onto which the NW is attached is shown in purple. MFM images of the NW from these three conditions are shown in Fig. 25.19C–E. Sketches below each MFM image depict the corresponding domain alignments/configurations. Starting from the field-off state, only the field along the NW length caused 90 degrees rotation of the magnetic moments, hence producing magnetostriction. Section height analysis was repeated at 10 different locations along the length of this NW to get the buckled NW height profile data shown in Fig. 25.20B and C. The NW diameter with the field off in Fig. 25.19C and D appears to be 25% smaller than that for the two cases with the field on, for which the NW diameters are similar. While AFM probe-tip shape and width artifacts preclude interpreting the span between the left and right rise in height above the baseline as an exact measurement of NW diameter, it was expected that all three cases would be similar to one another; not just the two cases with the field parallel and perpendicular to the NW. Efforts to minimize interaction of the applied field with the measurement process

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Fig. 25.19 (A) Cross section height data detected by AFM probe as probe moved perpendicular to the NW length. Green trace is with no field applied and red is with a field of 1 kOe applied along the length of the NW. (B) Cross section height data detected by AFM probe as probe moved perpendicular to the NW length. Green trace is with no field applied and purple is with a field of 1 kOe applied perpendicular to the length of the NW, and vertically down into the plane of the grid. (C) MFM image of NW with no field applied. Sketch below image indicates vortex domains in the Fe-Ga segments that are indicated in the MFM image. (D) MFM image of NW with a 1 kOe field applied along the NW length. Sketch below image indicates uniaxial domains in the Fe-Ga segments that are aligned with the applied field and are indicated in the MFM image. (E) MFM image of NW with a 1 kOe field applied perpendicular to the NW length, and downward toward the plane of the grid. Sketch below image indicates parallel domains in the Fe-Ga segments that aligned with the applied field as indicated in the MFM image. An estimate of the location along the length of the NW where the cross-section height data in (A, B) were obtained is indicated with white lines in (C–E).

included removing a small spring from the AFM tip probe holder that was made of a ferromagnetic material and fixing the probe in place with beeswax. The magnetostriction value of the Fe-Ga/Cu multilayer NW was estimated by comparing the areas “A1” from the experimental result (Fig. 25.20B and C) with “A2” from the schematic drawing of the area under the analytical solution for deflections of a buckled beam (Fig. 25.20D and E). The area between the starting and buckled height is assigned to A1, which in this example was 75.4 nm2, and was measured over a span of the NW length, L ¼ 4.75 μm. The formulas for the shape of a buckled

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Fig. 25.20 (A) SEM images of Fe-Ga/Cu NW (150-nm diameter, aspect ratio of 3 in both Fe-Ga and Cu segments) showing the length over which 10 height scans perpendicular to the length were conducted, and the 4.75-μm span over which the NW buckled when a 1000-Oe field was applied along the NW length. The EBID welds attaching the NW to the grid surface are located in the two areas enclosed by small yellow boxes. (B) Graphs showing maximum height from section analyses as a function of position along length of NW. Field-off height data shown in black, field-on data shown in red. The area between the two traces is identified as area “A1.” (C) The height data from (B) replotted to show the difference in height, Δ, between field-off and field-on plotted on the vertical axis, and position along the buckled portion of the NW length plotted along the horizontal axis. The area under the upper curve is also “A1.” (D and E) Sketches of analytical model of buckled beam deflections for the two most likely boundary conditions and area “A2” which is equated to area “A1” to estimate NW magnetostriction. In (D), subscript ff indicates fixed-fixed boundary conditions, and in (E), subscript pp indicates pinned-pinned boundary conditions. Adapted from J.J. Park, E.C. Estrine, S. Madhukar Reddy, B.J.H. Stadler, A.B. Flatau, Technique for measurement of magnetostriction in an individual nanowire using atomic force microscopy, J. Appl. Phys. 115 (17) (2014).

beam are u(x) ¼ hpp sin (πx/L) for a pinned-pinned beam (Fig. 25.20E), and u(x) ¼ hff(cos(2πx/L)  1) for a fixed-fixed beam (Fig. 25.20D). The function u(x) is the extent of buckling displacement as a function of position x along the length of the NW. Integrating these two shape functions from 0 to L one gets the area A2pp for a pinned-pinned boundary conditions and area A2ff for fixed-fixed boundary conditions as a function of their midspan heights, hpp and hff, respectively. By equating these areas to A1, one gets midspan height estimates of hpp ¼ 24.9 nm and hff ¼ 15.9 nm.

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In turn, using Eq. (25.3), the corresponding strains in the 4.75-μm long NW were 68 and 110 ppm, respectively. As the NW were composed of Fe-Ga and Cu segments with the same length (aspect ratio), the estimated magnetostriction of the Fe-Ga segments are λpp ¼ 136 ppm using the pinned-pinned model assumptions, and λff ¼ 220 ppm using the fixed-fixed model assumptions. Looking at the shape of the measured height profile in Fig. 25.19C, particularly near the welded ends, the pinned-pinned boundary conditions in Fig. 25.19E are a better match to the height profile than the fixed-fixed boundary conditions in Fig. 25.19D. Additionally, based on Eq. (25.1), a maximum magnetostriction value of 150 ppm is expected for a sample of textured bulk Fe-Ga that has some stress applied to promote 90-degree moment rotation. It is highly likely this magnetostriction value was also achieved in this nanowire.

25.6

Sensing using Fe-Ga/Cu nanowires

The use of arrays of Fe-Ga/Cu NW for the detection of pressure and force has been demonstrated by Park et al. [4]. In this work, an array of Fe-Ga/Cu NW is pressed against a giant magnetoresistance (GMR) sensor that is used to detect the changes in the magnetic state of the array as the NW are elastically deformed. In parallel with experimental work, simulations of the effect of NW deformation on the magnetic behavior of the NW were performed using a two-step analysis approach like that presented by Aimon et al. [39]. The commercial finite element software COMSOL Multiphysics was used to simulate the changing stress and strain field within a single NW as it was pressed against the GMR sensor. The simulation introduced displacements of the free end of the NW upward, toward the fixed end of the NW to match step size increments used in the experiments. The simulated NW was a 10-μm long, 150-nm diameter NW, with 50-nm long segments of both Fe-Ga and Cu (aspect ratios of  0.3). The NW magnetoelastic anisotropy was calculated from the full strain tensor and the magnetoelastic coefficients of the NW. The associated magnetic anisotropy field was imported into the public domain micromagnetics program OOMMF (Object Oriented Micro Magnetics Framework) which was used to simulate the magnetic domains in the NW and subsequently the stray fields surrounding the NW. For the micromagnetic simulations, values for the cubic anisotropy constant K1 ¼ 1.75  104 J m3 and saturation magnetization Ms ¼ 1321  103 A m1 were assumed based on literature values for bulk Fe-Ga alloys [40]. In OOMMF, 2.5-nm cubic cells were used to simulate the NW. This length was chosen to be in the range of the exchange length of the NW. The OOMMF simulations were performed both with no compression and for strain states in the NW predicted by COMSOL. Additional details are in Park et al. [4]. COMSOL simulation results are shown Fig. 25.21A–D and OOMMF simulation results are shown in Fig. 25.21E–H. Fig. 25.21A shows an outline of the undeformed NW and uses colors to indicate the strain state along the length of the NW when the upper end of the NW is fixed and the lower end is subjected to a 100-nm vertical displacement. Fig. 25.21B illustrates the aspect ratio of the Fe-Ga and Cu segments

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Fig. 25.21 Structural and micromagnetic modeling of a 150-nm diameter Fe-Ga/Cu NW with segment aspect ratios of 0.3 in both Fe-Ga and Cu segments. (A) FEM analysis of deformed nanowire illustrating strain induced when a NW is compressed 100 nm in the +z direction (indicated by the arrow). The box outline indicates region of the 6 segments used to provide OOMMF input strains used in the micromagnetic calculations. (B) Modeled NW indicating discrete Fe-Ga and Cu segments. (C and D) Cross section of COMSOL FEM strain analysis of bottom Fe-Ga segment of NW (C) without strain; (D) with strain induced by 100-nm vertical tip displacement. (E and F) Cross section of magnetization vectors in bottom Fe-Ga segment from OOMMF simulations: (E) without strain; and (F) with strain induced by 100-nm vertical tip displacement. (G and F) Vertical slice showing magnetization vectors in bottom three Fe-Ga segments from OOMMF simulations: (G) without strain; and (H) with strain induced by 100-nm vertical tip displacement. (I and J) Cross section and vertical slice of magnetization vectors in bottom Fe-Ga segment with stray fields surrounding the segment from OOMMF simulations: (I) without strain; and (J) with strain induced by 100-nm vertical tip displacement. From J.J. Park, K.S.M. Reddy, B. Stadler, A. Flatau, Magnetostrictive Fe-Ga/Cu nanowires array with GMR sensor for sensing applied pressure, IEEE Sensors J. 17 (2017) 2015–2020.

that comprise the modeled NW. Strains in a cross section of the bottom Fe-Ga segment before and during compression are shown in Fig. 25.21C and D, respectively. In Fig. 25.21C, the entire cross section is green, which indicates no strain in the cross section of the undeformed NW. In Fig. 25.21D, green indicates no strain only at the neutral axis of the deformed NW. Tension on the left side of the neutral axis is indicated in yellow and red and compression on the right side of the neutral axis is indicated in turquoise and blue. Also shown are the corresponding OOMMF results for the NW under no stress in Fig. 25.21E, G, and I and for the compressed NW in Fig. 25.21F, H, and J. The images in Fig. 25.21E and F show magnetization vectors in a cross section of the lower Fe-Ga segment, as if viewed from above the NW. Fig. 25.21G and H shows magnetization

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vectors in a vertical cross section of the bottom three Fe-Ga segments. The strains imported into OOMMF to predict the magnetization state in these Fe-Ga segments are from the six segments enclosed by the small rectangle at the base of the NW in Fig. 25.21A. The three blank sections above the Fe-Ga segments in Fig. 25.21G and H are where the Cu segments were located. The magnetization vectors in Fig. 25.21E and G combine to indicate the formation of a single vortex structure (i.e., not a vortex pair of opposite chirality) in the Fe-Ga segments under no stress. It is interesting to note that the chirality of the bottom two segments is opposite that of the third segment. In contrast, the images in Fig. 25.21F and H of the strained NW depict distinct regions of in-plane and out-of-plane magnetic vector alignments. The portion of the NW cross section that is in tension has magnetization vectors that are aligned along the length of the NW, pointing both up (blue) and down (red), suggesting the formation of antiparallel domains to minimize magnetostatic energy. The portion of the NW cross section that is in compression has groups of in-plane magnetization vectors that are aligned to the left, right, and at 45 degrees in the plane of the cross section. This also suggests the formation of multiple domains to minimize internal energy. In Fig. 25.21I and J, simulations of the stray fields surrounding the bottom NW segment have been added to the images of magnetization vectors in the bottom Fe-Ga segment. There are significantly more stray fields surrounding the strained NW than for the NW with no strain, with most of the stray fields in the area surrounding the side of the Fe-Ga segment under compression. Pressure sensing experiments were performed inside an SEM using the nanomanipulator in Fig. 25.13A to press an array of NWs against a GMR sensor that was attached to one side of the nanomanipulator stage [4]. An array of 150-nm diameter, Fe-Ga/Cu NW, with aspect ratios of 0.3 in both the Fe-Ga and Cu segments, was attached on the other side of the stage for one set of experiments. The experiment was also repeated using an array of Cu NW. The NW array and the GMR sensor were aligned in parallel and then the NW array was moved with a piezoelectric stepper motor to bring the ends of the NWs into contact with the GMR sensor. The stepper motor moved the NW array in either fine or course steps, using step sizes of 100 nm or 1 μm, respectively. Contact of the NW with the GMR sensor produced the changes in GMR resistance shown in Fig. 25.22. The percent change in GMR resistance due to a change in the in-plane magnetic field with time was recorded as the stepper motor moved the NW array into and out of contact with the GMR sensor. The black traces in Fig. 25.22 are from the array of the Fe-Ga/Cu NWs, and the orange trace is from the array of Cu NWs, as the NW arrays were pressed into the GMR sensor. When the Cu NW array was pressed against the GMR surface, GMR resistance remained at a constant value. This indicates that the GMR free magnetic layer was not affected by physical contact or direct application of force to the GMR sensor from the array of Cu NWs as it was pressed against the GMR surface. When the Fe-Ga/Cu NW array was pressed against the GMR surface, GMR resistance decreased, remained at a fixed level until the NW array was retracted, upon which the GMR resistance returned to a value at or near the baseline value.

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Fig. 25.22 (A) GMR% changes as Fe-Ga NWs and Cu NWs pressurized upon GMR sensor. Arrows (#, ") indicate press and retract of NW array, respectively. (Top right) Schematic illustration of Cu and Fe-Ga NW bending on GMR surface and direction of GMR magnetization changes. (B) GMR% changes as Fe-Ga NWs and Cu NWs pressurized upon GMR sensor with fine (R3) and coarse (R4) piezo motor steps. (Bottom right) Schematic illustration of degree of Fe-Ga NW bending on GMR surface with different force level and GMR magnetization changes. From J.J. Park, K.S.M. Reddy, B. Stadler, A. Flatau, Magnetostrictive Fe-Ga/Cu nanowires array with GMR sensor for sensing applied pressure, IEEE Sensors J. 17 (2017) 2015–2020.

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This suggests that the magnetic moments in the plane of the GMR sensor from either the NW itself (in the portion of the NW experiencing compressive strain) and/or from the in-plane stray fields outside of the NW produced a reorientation of the GMR free layer. When the NW array was retracted, the GMR free layer stopped being influenced by in-plane magnetization fields, suggesting NW domains spontaneously returned to their vortex structure state with little or no stray fields. The observed GMR% value change is about 0.4% in trace R2 in Fig. 25.22A. Using the GMR calibration data by Park et al. [4] this corresponds to the GMR experiencing an in-plane field of approximately 450 Oe. This magnitude of response was repeatable as the array was moved back and forth, into and out of contact with the GMR surface using the fine stepper-motor control increments (up and down arrows shown at the top of Fig. 25.22A). Some combination of the in-plane magnetization vectors in the compressed portions of the NWs and in-plane stray fields surrounding the NWs produced this 450-Oe field during GMR contact. In Fig. 25.22B the stepper-motor control increments were alternated between the fine step size (100 nm; indicated with the label R3) and the course steps (1 μm; indicated with R4). Here the change in GMR resistance in response to the fine step size varied slightly more than in Fig. 25.22A, but was still fairly consistent, however, responses to the course step size while always larger than for the fine step were varied in magnitude. To get an estimate of the performance of the NW array as a pressure sensor, FEM analysis was repeated for the two different nanomanipulator step sizes. Results also show that a single NW with this aspect ratio can be easily deformed by the displacements the nanomanipulator applies, and gave the expected result of generally similar strain distributions of two different magnitudes. The sensitivity to applied pressure of this specific NW array and GMR sensor was estimated to be 1–4 mΩ/kPa [4]. The OOMMF simulations indicate that because of the similarity in strain distributions for the different size steps, the resultant NW magnetizations will be similar. Another way of stating this is that the fine step size appears to rotate most of the moments that can be rotated, and therefore a 10-fold increase in the displacement of the NW array toward the GMR sensor does not produce a 10-fold increase in moment rotation. This leads one to speculate that the difference in GMR resistance data in response to the fine and course step sizes may be due to the distribution of NW lengths rather than the difference in force applied to the individual NWs. This was examined next. To improve the understanding of what takes place in the NWs as the array is pressed against the GMR surface, and to better understand the responses in Fig. 25.23B, attempts were made to obtain an image of the NW during contact. This was best achieved by carefully breaking a NW array to expose a side view of a center region of the array and aligning it with a similarly exposed side view of a portion of a Si wafer. The two images in Fig. 25.23A and B show the NWs and Si wafer initially separated by several micrometers and the NW pressed against the Si wafer surface (the scale bar at lower right of both images is 10-μm long). These images show that the Si wafer surface is not flat, and the tips of some of the NWs are clumped together, so even if their lengths were initially quite similar, the effective contact surface is not flat. As a result, when the stepper motor moves the NW array against the wafer, the percent of the NW array that come into contact with the wafer will increase in an undefined

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Fig. 25.23 SEM images of NW array and Si wafer using nanomanipulator (A) before contact and (B) after contact. The yellow boxes enclose the same regions of the NW array and the Si wafer. From J.J. Park, K.S.M. Reddy, B. Stadler, A. Flatau, Magnetostrictive Fe-Ga/Cu nanowires array with GMR sensor for sensing applied pressure, IEEE Sensors J. 17 (2017) 2015–2020.

manner until all NWs in the array are in contact with the wafer. NWs initially in contact with the wafer may experience stress-induced changes in magnetization state while other NWs are not yet experiencing any stress. Additionally, as the contact force increases some NWs appear to respond by bending while others appear to buckle. These images suggest the need for additional studies to understand how to best use Fe-Ga/Cu NWs and NW arrays for sensing pressure and/or force. Proposed future studies include determining if pressure application to a single multilayer NW, mounted on an AFM probe tip as in Fig. 25.13E (for resonance tests), would produce a sufficient field change to be detected by a GMR sensor. This sensing application is one for which the use of shape anisotropy might beneficial as an alternative to the use of a biasing magnet for tuning the interaction between applied mechanical loads and magnetic states. Instead of using Fe-Ga/Cu NW with Fe-Ga segments that form vortex domains, shape anisotropy could be tuned using different NW lengths and/or diameters. One would target uniaxial domain structures that are close to dimensions that would promote multiple domains, so that in the presence of targeted compressive and bending stresses, domains perpendicular to the NW axis will become energetically favorable. These new domains will also create in-plane stray fields, both of which would be detected by the GMR sensor.

25.7

Closing remarks

An introduction to magnetostrictive Fe-Ga alloys and their use at the nanoscale is provided in this article. Attributes that make this alloy useful for the design of macroscale actuators and sensors are summarized. Modeling is used to provide insights into

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design considerations that are particularly important for using nanowires made from these alloys in nanoscale devices. The role of shape anisotropy in motivating the use of two-component Fe-Ga/Cu multilayer nanowires rather than single component Fe-Ga nanowires is explained. A review of electrodeposition techniques that have been used to grow Fe-Ga and Fe-Ga/Cu nanowires, and results from structural, compositional, and magnetic characterization study are presented. The final two sections of the article provided a review of investigations into the use of Fe-Ga/Cu nanowires as an actuator and as a sensor. The actuation results appear quite promising. Magnetostriction of 140–150 ppm was observed in a 5-μm long section of a 150-nm diameter Fe-Ga/Cu nanowire. The ends of a nanowire were nanowelded to a grid so they could not move, a saturating magnetic field was applied along the length of the nanowire, and deformation due to buckling of the nanowire produced by magnetostriction was used to estimate the value of the magnetostriction produced. This experiment was performed in an AFM open to the room environment. The estimated magnetostriction of 150 ppm at the nanoscale is the same as observed at the macroscale. The sensor results also are promising; however, they identify a need for more work before nanowire sensor tests can be successful in a standard room environment. For these tests, an array of Fe-Ga/Cu nanowires was pressed against a GMR sensor and changes in GMR resistance were detected. This experiment was performed inside the vacuum chamber of an SEM. A sensitivity of 1–4 mΩ/kPa was estimated using measured and modeled behaviors, averaged over the full NW array. A challenge with this result was that the gap in the contact area between the GMR substrate and the tips of the NWs within the array was not uniform. As the NWs and GMR sensor came into contact, some NWs would start to be compressed while others were not yet in contact with the GMR sensor. A linear relationship between resistance change and applied pressure was not realized. Magnetostriction for use in design of Fe-Ga/Cu nanowire-based actuators was demonstrated. Pressure sensing using Fe-Ga/Cu nanowires was also demonstrated, although some of the challenges that remain were identified and discussed. It is hoped that this review stimulates further investigation into the unique opportunities these magnetostrictive alloys provide for producing and detecting motion and force at the nanoscale.

References [1] V. Apicella, M.A. Caponero, D. Davino, C. Visone, A magnetostrictive biased magnetic field sensor with geometrically controlled full-scale range, Sensor Actuat. A Phys. (2018). [2] T. Ueno, S. Yamada, Performance of energy harvester using iron–gallium alloy in free vibration. IEEE Trans. Magn. 47 (10) (2011) 2407–2409, https://doi.org/10.1109/ TMAG.2011.2158303. [3] W.J. Fischer, S. Sauer, U. Marschner, B. Adolphi, C. Wenzel, B. Jettkant, B. Clasbrummel, Galfenol resonant sensor for indirect wireless osteosynthesis plate bending measurements, IEEE Sensors (2009) 611–616, https://doi.org/10.1109/ICSENS.2009.5398320.

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