Magnetic Domains

Alex HubertRudolf Schäfer

Magnetic DomainsThe Analysisof Magnetic Microstructures

With 400 Figures, 4 in Colour

Professor Dr. Alex Hubert†

Universität Erlangen-NürnbergInstitut für WerkstoffwissenschaftenLehrstuhl Werkstoffe der ElektrotechnikMartensstr. 7, 91058 Erlangen, Germany

Dr. Rudolf SchäferLeibniz-Institut für Festkörper- und WerkstoffforschungHelmholtzstr. 20, 01069 Dresden, Germany, E-mail: r.schaefer@ifw-dresden.de

Library of Congress Cataloging-in-Publication DataHubert,Alex,1938– Magnetic domains: the analysis of magnetic microstructures / Alex Hubert; RudolfSchäfer, p. cm. Includes biliographical references and indexes.ISBN 978-3-540-64108-7 (hardcover: alk. paper)1. Magnetic materials. 2. Domain structure. I. Schäfer, Rudolf. II. Title.QC765.H83 1998 538’.4-dc21 98-16905

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Preface

Magnetic domains are the elements of the microstructure of magnetic mate-rials that link the basic physical properties of a material with its macroscopicproperties and applications. The analysis of magnetization curves requires anunderstanding of the underlying domains. In recent years there has been arising interest in domain analysis, probably due to the increasing perfectionof materials and miniaturization of devices. In small samples, measurable do-main effects can arise, which tend to average out in larger samples. This bookis intended to serve as a reference for all those who are confronted with thefascinating world of magnetic domains. Naturally, domain pictures form animportant part of this book.

After a historical introduction (Chap. 1) emphasis is laid on a thoroughdiscussion of domain observation techniques (Chap. 2) and on domain theory(Chap. 3). No domain analysis is possible without knowledge of the relevantmaterial parameters. Their measurement is reviewed in Chap. 4. The detaileddiscussion of the physical mechanisms and the behaviour of the main typesof observed domain patterns in Chap. 5 is subdivided according to the crys-tal symmetry of the samples, according to the relative strength of induced orcrystal anisotropies, and according to the sample dimensions. The last chapterdeals with the technical relevance of magnetic domains, which is different forthe different fields in which magnetic materials are applied. The discussionreaches from soft magnetic materials forming the core of electrical machinesthat would not work without the hidden action of magnetic domains, to mag-netic sensor elements used in magnetic recording systems, which may sufferfrom domain-related noise effects.

We emphasize in this book (in Chaps. 5, 6) the analysis of actual magne-tization processes, in particular of those that lead to discontinuities and irre-versibilities in the magnetization curve. This approach is adequate for severalimportant application areas, such as magnetic sensors and transformer mate-rials. In other areas, such as those of bulk polycrystalline soft-magnetic ma-terials, the link between observed elementary processes and possible technicaldescriptions of the hysteresis phenomena is difficult to establish. Formal de-

VIII Preface

scriptions of hysteresis phenomena—shortly introduced by reference to actualtextbooks in the last section of Chap. 6—use abstract concepts of magneticdomains in their justification. Whether the discrepancy between the simpli-fied assumptions of hysteresis theories and the complexities of actual domainbehaviour is technically relevant remains an interesting open question.

Previous books and reviews touching on the subject of magnetic domainsand magnetization processes are listed separately in alphabetic order at theend of the reference list. The few recent works among these cover important,but on the whole rather specialized aspects. One of the aims of this book isto collect the knowledge available from the investigations of many magneticmaterials. Although the technical areas of electrical steels, of high frequencycores, of permanent magnets, or of computer storage media have little incommon, the magnetic microstructures in most of these materials obey thesame rules and differ only in quantitative, not in qualitative aspects.

Despite the preference of many magneticians for the old Gaussian systemof units, standard SI units are used throughout. A reference to the old systemis made only occasionally. But we prefer to reduce equations to dimensionlessunits anyway in all practical examples, so that results get a broader rangeof applicability, and at the same time the mathematical expressions becomeindependent of the unit system.

We take one liberty, however. The magnetic dipole density of a materialis given by the vector field J(r) (measured in Tesla or Vs/m2). Its officialname is magnetic polarization, but very often and also in this book, it iscalled simply magnetization, although in the strict SI system this name isreserved to a quantity M(r) that is measured in A/m and related to J byM = J/μ0. We will never use the latter quantity, so no confusion is possible.Our choice agrees with the recommendation from P.C. Scholten1, except thathe would prefer to use the abbreviation M instead of J , which we think mightlead to confusion2. Anyway, in most cases we use the reduced unit vector ofmagnetization direction only, and we abbreviate this vector field by m(r).

The careful reader will discover quite a few original contributions in thisbook, not published elsewhere. They are intended to improve the grasp ofotherwise abstract concepts and check their applicability. They also help tofill gaps in the published material. Such gaps may be too small or unimportantto justify an independent scientific publication, but they may form a seriousobstacle for newcomers in their attempt to enter the field.

A few hints to the reader: Up to five text organization levels are used.Three of them (chapter, section, subsection) are systematically numbered ina decimal classification. The sections appear in the right-hand page headers tofacilitate orientation. A further subdivision is often needed. It is marked andreferred to within the same subsection by “(A), (B) etc.”. In references from

1 P.C. Scholten: “Which SI?”, J. Magn. Magn. Mat. 149, 57–59 (1995).2 The fundamental material equation is thus expressed in this book in the form

B = μ0H + J .

Preface IX

other subsections a small capital letter (like in “Sect. 3.4.1A”) is appended.Where needed, an informal fifth level, marked “(i), (ii) etc.”, is added. Paren-thesis with numbers like “(3.1)” always refer to equations, while brackets like“[212]” indicate citations. Parenthesis with lower case letters like “(a)” referto parts of a figure mentioned in the same paragraph.

Some passages in Chap. 3 on domain theory are addressed more to theinterested specialist than to the general reader. They are marked by smallerprint. All references are collected in the order of occurrence after the textpart. Orientation within the references section is supported by an authorindex that comprises all authors and coauthors of all cited references (explicittext references are listed first). In addition to the regular, numbered andcaptioned figures we offer numbered “sketches”, that are intended to facilitatereading, but which need no further explanation (thanks to John Chapman forsuggesting this tool). The few tables are numbered and treated as if they werefigures, which makes it easier to find them.

Erlangen, Dresden, Alex HubertJuly 1998 Rudolf Schafer

Acknowledgements

The authors want to thank all the many colleagues who helped with comments andcriticism by reading the manuscript, who permitted the reproduction of their beau-tiful domain images, who supplied valuable samples for domain imaging, and whoassisted in sample preparation, calculations, graphics, and editing of the manuscript.

Most valuable was the help of S. Roth, who completely and critically readthe manuscript, and of W. Andra, who carefully studied most of the text, sup-plying us with many suggestions based on his long experience. Portions of thebook were read by L. Abelmann, S. Arai, B.E. Argyle, M. Bijke, A. Bogdanov,S.H. Charap, K. Fabian, R.P. Ferrier, F.J. Friedlaender, P. Gornert, H. Groen-land, M. Haast, U. Hartmann, H. Hopke, F.B. Humphrey, L. Jahn, E. Jager,V. Kambersky, M.H. Kryder, H. Lichte, J.D. Livingston, C. Lodder, J. McCord,K.-H. Muller, M. Muller, F. Oehme, I.B. Puchalska, W. Rave, A.A. Thiele andP. Trouilloud. Their criticism and suggestions are gratefully acknowledged. We alsoreceived good advice on various subjects by D. Berkov, H. Fujiwara, O. Gutfleisch,R. Hilzinger, H. Huneus and K.H. Wiederkehr.

For illustration we were permitted to show beautiful domain images and com-putational results by L. Abelmann, H. Aitlamine, S. Arai, B.E. Argyle, J. Baruchel,L. Belliard, J.E.L. Bishop, H. Bruckl, R. Celotta, J.N. Chapman, R.W. DeBlois,D.B. Dove, T. Eimuller, A. Fernandez, E. Feldtkeller, J. Fidler, P. Fischer,R. Fromter, J.M. Garcia, K. Goto, R. Grossinger, F.B. Humphrey, J.P. Jakubovics,A. Johnston, K. Kirk, K. Koike, S. Libovicky, C.-J.F. Lin, J.D. Livingston,J.C. Lodder, F. Matthes, R. Mattheis, Y. Matsuo, S. McVitie, J. Miles, J. Mil-tat, K.-H. Muller, T. Nozawa, H.-P. Oepen, L. Pogany, S. Porthun, I.B. Puchalska,D. Rugar, C.M. Schneider, M. Schneider, C. Schwink, J. Simsova, R. Szymczack,A. Thiaville, S.L. Tomlinson, A. Tonomura, S. Tsukahara, J. Unguris, P.A.P. Wend-hausen, Y. Zheng, J.-G. Zhu, L. Zimmermann, and E. Zueco. Thanks to all of themfor their support.

Major contributions to this book either in computations or in experiments aredue to A.N. Bogdanov, K. Fabian, J. McCord, W. Rave and M. Ruhrig. We areespecially grateful for their help.

Special thanks are due to the many collegues from industry and research labora-tories who supplied us with valuable samples for domain imaging: S. Arai, B.E. Ar-gyle, W. Bartsch, R. Celotta, W. Ernst, M. Freitag, P. Gornert, B. Grieb, H. Grimm,

XII Acknowledgements

P. Grunberg, W. Grunberger, A. Handstein, G. Henninger, G. Herzer, W.K. Ho,V. Hoffmann, A. Hutten, F.B. Humphrey, L. Jahn, J.C.S. Kools, S.S.P. Parkin,J. Petzold, B. Pfeifer, T. Plaskett, Ch. Polak, S. Roth, M. Schneider, R. Schreiber,R. Thielsch, W. Tolksdorf, J. Wecker, U. Wende, J. Yamasaki, and K. Zaveta.

We are grateful to the collaborators at the University of Erlangen-Nurnberg, inparticular K. Boockmann, H. Brendel, G. Cuntze, R. Fichtner, W. Grimm, P. Loffler,H. Maier, J. McCord, M. Neudecker, J. Pfannenmuller, K. Ramstock, K. Reber,F. Schmidt, L. Wenzel, S. Winkler and many others, to whom we owe thanks fordomain pictures which were taken with the digitally enhanced Kerr-imaging systemdeveloped there together with P. Doslik. Valuable assistance, for example in samplecharacterization and manipulation, was given by B. de Boer, O. de Haas, A. Burke,D. Eckert, J. Fischer, G. Große, D. Hinz, N. Mattern, S. Roth, U. Schlafer andA. Teresiak, all collaboraters at IFW Dresden.

In literature research we received help in particular by E. Dietrich, BarbaraHubert, B. Richter and H. Wagner-Schlenkrich. We appreciate the support offeredby the Institutes of Erlangen and Dresden by allowing the use of hardware under theresponsibilities of G. Muller, K.-H. Muller, L. Schultz, K. Wetzig and A. Winnacker.

The first author is much indebted to the Magnetics Technology Center ofCarnegie Mellon University, Pittsburgh, where an initial part of the manuscript waswritten during a sabbatical, supported by M. Kryder. Another part was preparedduring a stay in the Institute for Physical High Technology in Jena (the famousformer Magnet-Institut), supported by P. Gornert. And finally a sabbatical couldbe spent at the Institute for Solid-State and Material Research (IFW) in Dresden,generously supported by L. Schultz.

Extremely valuable technical assistance was contributed by W. Habel (Erlangen)in photography and graphics, by the first author’s daughter Birgit in the generationof the indices and the final production of the manuscript, and by S. Schinnerling(Dresden) in sample preparation.

Last but not least both authors would like to thank their wives HeidemarieHubert and Renate Schafer for their support, patience and encouragement.

The original manuscript of this book was written on the Apple MacintoshR©

based on the NisusR© Writer text system, with ample use of the ExpressionistR© for-mula editor, the EndNoteR© literature organizer, the drawing routines MacDrawR©

and ClarisDrawR©, and the Adobe PhotoShopR© Image Processing System. For thepresent printing, which is identical with the first edition, the manuscript was trans-ferred into LaTeX. Friedemann Queisser did an excellent job in doing this. For finalediting the help of F. Kurth, B. Lange and J. McCord is highly appreciated. Any sug-gestions or error reports from the readers are most welcome. Updating informationis collected at the website:

http://www.ifw-dresden.de/∼schaefer/magnetic-domains.html

Prof. Dr. Alex Hubert†

Former address:Institut fur Werkstoffwissenschaftender Universitat Erlangen-NurnbergLehrstuhl Werkstoffe der ElektrotechnikMartensstr. 7, 91058 Erlangen, Germany

Dr. Rudolf SchaferLeibniz Institute for Solid Stateand Materials Research DresdenHelmholtzstr. 20, 01069 DresdenGermanyE-mail: r.schaefer@ifw-dresden.de

Table of Contents

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIX

Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXVII

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 What are Magnetic Domains? . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 History of the Domain Concept . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 The Domain Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Towards an Understanding of Domains . . . . . . . . . . . . . 31.2.3 Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Micromagnetics and Domain Theory . . . . . . . . . . . . . . . . . . . . . 8

2 Domain Observation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Bitter Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 General Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Contrast Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.3 The Importance of Agglomeration Phenomena in

Colloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.4 Visible and Invisible Features . . . . . . . . . . . . . . . . . . . . . 182.2.5 Special Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Magneto-Optical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.1 Magneto-Optical Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2 The Geometry of the Magneto-Optical Rotation Effects 252.3.3 Magneto-Optical Contrast in Kerr Microscopy . . . . . . 282.3.4 Interference and Enhancement by Dielectric Coatings 302.3.5 Kerr Microscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.6 The Illumination Path . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

XIV Table of Contents

2.3.7 Digital Contrast Enhancement and Image Processing 362.3.8 Quantitative Kerr Microscopy . . . . . . . . . . . . . . . . . . . . . 392.3.9 Dynamic Domain Imaging . . . . . . . . . . . . . . . . . . . . . . . . 402.3.10 Laser Scanning Optical Microscopy . . . . . . . . . . . . . . . . 422.3.11 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.3.12 Other Magneto-Optical Effects . . . . . . . . . . . . . . . . . . . . 442.3.13 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4 Transmission Electron Microscopy (TEM) . . . . . . . . . . . . . . . . 492.4.1 Fundamentals of Magnetic Contrast in TEM . . . . . . . . 492.4.2 Conventional Lorentz Microscopy . . . . . . . . . . . . . . . . . . 502.4.3 Differential Phase Microscopy . . . . . . . . . . . . . . . . . . . . . 542.4.4 Electron Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4.5 Special Procedures in Lorentz Microscopy . . . . . . . . . . 592.4.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.5 Electron Reflection and Scattering Methods . . . . . . . . . . . . . . . 612.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.5.2 Type I or Secondary Electron Contrast . . . . . . . . . . . . . 622.5.3 Type II or Backscattering Contrast . . . . . . . . . . . . . . . . 632.5.4 Electron Polarization Analysis . . . . . . . . . . . . . . . . . . . . 672.5.5 Other Electron Scattering and Reflection Methods . . . 692.5.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.6 Mechanical Microscanning Techniques . . . . . . . . . . . . . . . . . . . . 712.6.1 Magnetic Force Microscopy (MFM) . . . . . . . . . . . . . . . . 712.6.2 Near-Field Optical Scanning Microscopy . . . . . . . . . . . 782.6.3 Other Magnetic Scanning Methods . . . . . . . . . . . . . . . . 80

2.7 X-ray, Neutron and Other Methods . . . . . . . . . . . . . . . . . . . . . . 812.7.1 X-ray Topography of Magnetic Domains . . . . . . . . . . . . 822.7.2 Neutron Topography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.7.3 Domain Imaging Based on X-Ray Spectroscopy . . . . . 862.7.4 Magnetotactic Bacteria . . . . . . . . . . . . . . . . . . . . . . . . . . 892.7.5 Domain-Induced Surface Profile . . . . . . . . . . . . . . . . . . . 892.7.6 Domain Observation in the Bulk? . . . . . . . . . . . . . . . . . 90

2.8 Integral Methods Supporting Domain Analysis . . . . . . . . . . . . 922.9 Comparison of Domain Observation Methods . . . . . . . . . . . . . 95

3 Domain Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.1 The Purpose of Domain Theory . . . . . . . . . . . . . . . . . . . . . . . . . 993.2 Energetics of a Ferromagnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.2.2 Exchange Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.2.3 Anisotropy Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.2.4 External Field (Zeeman) Energy . . . . . . . . . . . . . . . . . . 1093.2.5 Stray Field Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.2.6 Magneto-Elastic Interactions and Magnetostriction . . 1253.2.7 The Micromagnetic Equations . . . . . . . . . . . . . . . . . . . . 138

Table of Contents XV

3.2.8 Review of the Energy Terms of a Ferromagnet . . . . . . 1443.3 The Origin of Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

3.3.1 Global Arguments for Large Samples . . . . . . . . . . . . . . 1453.3.2 High-Anisotropy Particles . . . . . . . . . . . . . . . . . . . . . . . . 1513.3.3 Ideally Soft Magnetic Materials . . . . . . . . . . . . . . . . . . . 1553.3.4 The Effect of Anisotropy in Soft Magnetic Materials . 1653.3.5 Resume: The Absence and Presence of Domains . . . . . 168

3.4 Phase Theory of Domains in Large Samples . . . . . . . . . . . . . . . 1703.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713.4.2 The Fundamental Equations of Phase Theory . . . . . . . 1713.4.3 The Analysis of Cubic Crystals as an Example . . . . . . 1783.4.4 Field-Induced Critical Points . . . . . . . . . . . . . . . . . . . . . 1823.4.5 Quasi-Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

3.5 Small Particle Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1883.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1883.5.2 Uniform Single-Domain Switching . . . . . . . . . . . . . . . . . 1903.5.3 Classifying Switching and Nucleation Processes . . . . . 1933.5.4 Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1963.5.5 Numeric Evaluation in the General Case . . . . . . . . . . . 1983.5.6 Continuous Nucleation (Second-Order Transitions) . . 200

3.6 Domain Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2013.6.1 The Structure and Energy of Infinite Planar Walls . . . 2013.6.2 Generalized Walls in Uniaxial Materials . . . . . . . . . . . . 2123.6.3 Walls in Cubic Materials . . . . . . . . . . . . . . . . . . . . . . . . . 2173.6.4 Domain Walls in Thin Films . . . . . . . . . . . . . . . . . . . . . . 2233.6.5 Substructures of Walls—Bloch Lines and Bloch

Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2413.6.6 Wall Dynamics: Gyrotropic Domain Wall Motion . . . . 2543.6.7 Wall Friction and Disaccommodation-Dominated

Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2643.6.8 Eddy-Current-Dominated Wall Motion . . . . . . . . . . . . . 2683.6.9 Resume: Wall Damping Phenomena and Losses. . . . . . 271

3.7 Theoretical Analysis of Characteristic Domains . . . . . . . . . . . . 2733.7.1 Flux Collection Schemes on Slightly Misoriented

Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2743.7.2 Dense Stripe Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 2793.7.3 Domains in High-Anisotropy Perpendicular Films . . . 2873.7.4 Closure Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2963.7.5 Domain Refinement (Domain Branching) . . . . . . . . . . . 3113.7.6 The Neel Block as a Classical Example . . . . . . . . . . . . . 328

3.8 Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

XVI Table of Contents

4 Material Parameters for Domain Analysis . . . . . . . . . . . . . . . 3374.1 Intrinsic Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 3374.2 Mechanical Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

4.2.1 Torque Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3394.2.2 Field Gradient Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 342

4.3 Magnetic Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3434.3.1 Overview: Methods and Measurable Quantities . . . . . . 3434.3.2 Magnetometric Methods . . . . . . . . . . . . . . . . . . . . . . . . . 3444.3.3 Inductive Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 3444.3.4 Optical Magnetometers . . . . . . . . . . . . . . . . . . . . . . . . . . 3484.3.5 Evaluation of Magnetization Curves . . . . . . . . . . . . . . . 349

4.4 Resonance Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3534.4.1 Overview: Types of Resonance . . . . . . . . . . . . . . . . . . . . 3534.4.2 Theory of Ferromagnetic Resonance . . . . . . . . . . . . . . . 3544.4.3 Spin Wave Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3554.4.4 Light Scattering Experiments . . . . . . . . . . . . . . . . . . . . . 3564.4.5 Wall and Domain Resonance Effects . . . . . . . . . . . . . . . 356

4.5 Magnetostriction Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 3574.5.1 Indirect Magnetostriction Measurements . . . . . . . . . . . 3574.5.2 Direct Measurements: General Procedures . . . . . . . . . . 3584.5.3 Techniques of Elongation Measurements . . . . . . . . . . . . 359

4.6 Domain Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3624.6.1 Suitable Domain Patterns . . . . . . . . . . . . . . . . . . . . . . . . 3624.6.2 Band Domain Width in Bubble Materials . . . . . . . . . . 3624.6.3 Surface Domain Width in Bulk Uniaxial Crystals . . . . 3644.6.4 Internal Stress Measurement by Domain Experiments 3654.6.5 Stripe Domain Nucleation and Annihilation . . . . . . . . . 3654.6.6 Domain Wall Experiments in Soft Magnetic Materials 3674.6.7 Measurements of Domain Wall Dynamics . . . . . . . . . . . 368

4.7 Thermal Evaluation of the Exchange Constant . . . . . . . . . . . . 3684.7.1 The Curie Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3694.7.2 Molecular Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 3694.7.3 Low Temperature Variation of Magnetization . . . . . . . 370

4.8 Theoretical Guidelines for Material Constants . . . . . . . . . . . . . 3704.8.1 Temperature Dependence of Anisotropy Parameters . 3704.8.2 Mixing Laws for Magnetic Insulators . . . . . . . . . . . . . . . 3714.8.3 Empirical Rules for Alloy Systems . . . . . . . . . . . . . . . . . 372

5 Domain Observation and Interpretation . . . . . . . . . . . . . . . . . 3735.1 Classification of Materials and Domains . . . . . . . . . . . . . . . . . . 373

5.1.1 Crystal and Magnetic Symmetry . . . . . . . . . . . . . . . . . . 3735.1.2 Reduced Material Parameters . . . . . . . . . . . . . . . . . . . . . 3745.1.3 Size, Dimension and Surface Orientation . . . . . . . . . . . 3755.1.4 Further Aspects and Synopsis . . . . . . . . . . . . . . . . . . . . . 376

5.2 Bulk High-Anisotropy Uniaxial Materials . . . . . . . . . . . . . . . . . 378

Table of Contents XVII

5.2.1 Branched Domain Patterns . . . . . . . . . . . . . . . . . . . . . . . 3785.2.2 Applied Field Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3835.2.3 Polycrystalline Permanent Magnet Materials . . . . . . . . 386

5.3 Bulk Cubic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3885.3.1 Surfaces with Two Easy Axes . . . . . . . . . . . . . . . . . . . . . 3885.3.2 Crystals with One Easy Axis in the Surface . . . . . . . . . 3915.3.3 Stress Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3955.3.4 Slightly Misoriented Surfaces . . . . . . . . . . . . . . . . . . . . . 3985.3.5 Strong Misorientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

5.4 Amorphous and Nanocrystalline Ribbons . . . . . . . . . . . . . . . . . 4105.4.1 The As-Quenched State of Amorphous Ribbons . . . . . 4115.4.2 Ordered Domain States of Metallic Glasses . . . . . . . . . 4135.4.3 Wall Studies on Metallic Glasses . . . . . . . . . . . . . . . . . . 4195.4.4 Domains on Soft-Magnetic Nanocrystalline Ribbons . 421

5.5 Magnetic Films with Low Anisotropy . . . . . . . . . . . . . . . . . . . . 4225.5.1 Overview: Classification of Magnetic Films . . . . . . . . . 4225.5.2 Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4235.5.3 Thick Films with In-Plane Anisotropy . . . . . . . . . . . . . 4325.5.4 Thin and Thick Film Elements . . . . . . . . . . . . . . . . . . . . 4375.5.5 Thick Films with Weak Out-of-Plane Anisotropy . . . . 4505.5.6 Cubic Single-Crystal Films and Platelets . . . . . . . . . . . 4545.5.7 Double Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

5.6 Films with Strong Perpendicular Anisotropy . . . . . . . . . . . . . . 4725.6.1 Extended Domain Patterns . . . . . . . . . . . . . . . . . . . . . . . 4725.6.2 Localized Domains (Bubbles) . . . . . . . . . . . . . . . . . . . . . 4795.6.3 Domain Wall Investigations in Perpendicular Films . . 481

5.7 Particles, Needles and Wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4835.7.1 Observations on High-Anisotropy Uniaxial Particles . 4835.7.2 Very Small Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4845.7.3 Whiskers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4885.7.4 Magnetic Wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

5.8 How Many Different Domain Patterns? . . . . . . . . . . . . . . . . . . . 491

6 The Relevance of Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4936.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4936.2 Bulk Soft-Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

6.2.1 Electrical Steels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4956.2.2 High-Permeability Alloys . . . . . . . . . . . . . . . . . . . . . . . . . 5116.2.3 Amorphous and Nanocrystalline Alloys . . . . . . . . . . . . . 5136.2.4 Spinel Ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

6.3 Permanent Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5196.3.1 Nucleation-Type Sintered Magnets . . . . . . . . . . . . . . . . 5216.3.2 Pinning-Type Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . 5246.3.3 Small-Particle and Nanocrystalline Magnets . . . . . . . . 525

6.4 Recording Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

XVIII Table of Contents

6.4.1 Longitudinal Recording . . . . . . . . . . . . . . . . . . . . . . . . . . 5306.4.2 Perpendicular Recording . . . . . . . . . . . . . . . . . . . . . . . . . 5376.4.3 Magneto-Optical Recording . . . . . . . . . . . . . . . . . . . . . . . 539

6.5 Thin-Film Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5436.5.1 Inductive Thin-Film Heads in Magnetic Recording . . . 5436.5.2 Magnetoresistive Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 5476.5.3 Thin-Film Memories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551

6.6 Domain Propagation Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5566.6.1 Bubble Memories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5576.6.2 Domain Shift Register Devices . . . . . . . . . . . . . . . . . . . . 5626.6.3 Magneto-Optical Display and Sensor Devices . . . . . . . 563

6.7 Domains and Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

Textbooks and Review Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641

Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673

Symbols

Sorting order (example):τ(tau) • ϑ(theta) • Θ(Theta) • t • t1 • T • T1

A

α Quasi dislocationdensity 131

αG Gilbert dampingparameter 141

αLL Landau-Lifshitzdamping parameter 141

αs Analyser anglerelative to s-axis 27

αw Wall dampingfactor 268

a Vector, direction 107aG Gap width 549aL Lattice constant 102A Exchange stiffness

constant 102Aϕ Vector potential

component 161Atot Total amplitude

after analyser 27Az Vector potential

component 233AK Kerr amplitude

after analyser 28AN Regular amplitude

after analyser 28

B

β Domain angle 301βw Wall mobility 262b Branching ratio 319

b(n) Code for branchingpattern 319

bc Partial closure ratio 301bi Stripe domain

coefficients 282bs Spacer layer

thickness 461B Magnetic flux

density (induction) 11B0 Integrated in-plane

induction 51B1 Voigt effect mate-

rial constant 24Bm Maximum

induction 272BS Brillouin function 369Bx Function

(abbreviation) 119

XX Symbols

C

χ Susceptibility 14χ⊥ Transverse

susceptibility 192c Elastic tensor 131c0 Core-tail boundary

in Neel wall 228ci Stripe domain

coefficients 282

c(1)kp Constant in the

kinetic potential 254cn Normal

m-component 307cw Domain magnetiza-

tion product 307C2 Cubic shear modulus

= 12(c1111 − c1122) 127

C3 Cubic shear modulus 127= c1212

Cbl Bilinear couplingconstant 104

Cbq Biquadratic cou-pling constant 104

Cmo Magneto-opticalcontrast 29

Cp Internal closureenergy coefficients 322

Cs Closure energycoefficient 277

D

δ Normalized andreduced length 286

δik Kronecker’s symbol 128δMD Reduced

multi-domain size 169δSD Reduced single

domain size 169δsp Reduced spreading

area depth 308Δ Bloch wall width

parameter 168Δd Exchange length 156Δκ Reduced excess

anisotropy 306Δκd Excess stray field

energy 307

ΔΦ Stress-inducedmagnetizationrotation 223

ΔΘ Wall magnetizationpath length 306

ΔK Excess anisotropyenergy 306

d Dissipation dyadic 256d0 Equilibrium bubble

diameter 295dbc Collapse bubble

diameter 295dbs Bubble diameter

at stripout 295dcr Critical depth 241df Deflection 361dsp Spreading area 306

depthdMR Thickness of

MR-strip 549D Thickness,

diameter 51D Dielectric displace-

ment vector 24Dcr Critical thickness 167

D(SH)i Second harmonic

amplitude 46Ds Branching onset

thickness 315DSD Single domain 153

diameter

E

ε Isotropic dielectricconstant 24

ε(2) Second orderdielectric tensor 46

ε Elastic strain tensor 125ε0 Free magneto-

strictive tensor 127εA Wall-related

integral 262εcl Reduced total 308

closure energyεcp Compensating

deformation tensor 131εd Reduced stray field

energy density 152

Symbols XXI

εe Compatibledeformation tensor 130

εi Energies of inter-stitial positions 266

εtot Reduced totalenergy 152

η Magnetizationangle 241

ηcr Critical field angle 184ηh Field orientation

angle 146ηG Grain orientation

angle 522e0 Energy for

free deformation 127ecoupl Exchange interface

coupling energy 104ed Stray field energy

density 111eel Elastic energy

density (e. d.) 127ei Unit vectors 280eme Magneto-elastic

coupling e. d. 128ems Magnetostrictive

self e. d. 130ered Reduced inter-

action energy 137erel Energy of the

elastically relaxedstate 135

es Surfaceanisotropy e. d. 107

etot Total e. d. 146ex Exchange

stiffness e. d. 102eK Anisotropy e. d. 339eKc Cubic

anisotropy e. d. 106eKi Induced

anisotropy e. d. 107eKo Orthorhombic

anisotropy e. d. 107eKu Uniaxial

anisotropy e. d. 106eXA Exchange

anisotropy e. d. 109E Complete elliptic

integral 295

Es Young modulus 361E Electrical field of

light wave 24E0 Primary electron

energy 64Ed Stray field energy 110Einter Interaction energy 73Eself Self energy 115Etot Total energy 211Ex Exchange stiffness

energy 101EH External field

energy 109EK Anisotropy energy 301

F

f Frequency 272fd Dynamic reaction

force 256f s Static force 256fw Reduced wall area 152fL Lagrangian

parameter 139F Function

(abbreviation) 115F Interface-related

elastic tensor 133Fan Generalized aniso-

tropy functional 138Fi Internal stray field

energy coefficient 314Fikm Integral functions 113Finc Number of incident

photons 29Find Induced anisotropy

coefficient 107F m Magnetic force 339Fn Domain multi-

plication factor 319Fw Wall area 152F L Lorentz force 49FRR Rhodes-Rowlands

function 116

G

γ Gyromagneticfactor 140

γ180 180◦-wall energy 277

XXII Symbols

γG Gilbert gyro-magnetic factor 141

γLL Landau-Lifshitzgyromagnetic factor 141

γw Specific wall energy 152γz Zigzag wall energy 331g Anisotropy

functional 146g Gyrotropic vector 256gL Lande’s

gyromagnetic ratio 369gL Reciprocal lattice

vector 84G Generalized aniso- 204

tropy functionalGϑ Partial derivative 210G∞ Generalized

anisotropy energydensity in domains 207

Gik Integral functions 119Gind Induced anisotropy

coefficient 107Gm Branching speed

limit 322Gs Shear modulus 128

H

h Reduced field 146hb0 Bubble lattice

saturation field 291hbc Reduced bubble

collapse field 291hbs Reduced bubble

stripout field 291hcr Reduced critical

field 184hs0 Stripe pattern

saturation field 291htr Stripe bubble

lattice transition 291H Magnetic field 11H0 Ferromagnetic

resonance field 356Hann Annealing field 414Hb Bias (perpen-

dicular) field 294Hc Coercivity 198Hcd Dynamic coercivity 264

Hcr Bloch-Neeltransition field 241

Hcs Static coercivity 264Hd Stray or demag-

netizing field 110Hdif Diffusion-related

effective field 266Heff Effective field 140Hex External field 109H in Internal field 174Hn Spin wave

resonance field 356Hp Peak velocity field 258Hres Resonance field 354HE Eddy current field 268HK Anisotropy field 192HLor Lorentz field 351HXA Exchange aniso-

tropy field 109

I

I Relative imageintensity 28

I0 Relative back-ground intensity 28

J

j Current density 548J⊥ Perpendicular

magnetizationcomponent 347

J Average magnet-ization 506

J Magneticpolarization(= “magnet-ization”) 11

J1 Bessel function ofthe first kind 293

Jn Magnetizationanisotropycoefficient 341

Js Saturationmagnetization 24

Jsr Saturationremanence curve 532

Jvr Virgin or iso-thermal remanencecurve 532

Symbols XXIII

JRE Magnetization ofrare earth sub-lattice 540

JTM Magnetization oftransition metalsublattice 539

K

κ Anisotropy con-stant ratio 213

κ1 Root in stripedomain analysis 282

κc Reduced cubicanisotropy coeffi-cient 178

κcp Reduced couplingcoefficient 467

k Boltzmannconstant 14

k Electron propa-gation vector 64

k1 Absorption index ofmagnetic medium 31

K Complete ellipticintegral 295

K Anisotropy tensor 107K Anisotropy

coefficient (a. c.) 111Kc Cubic a. c. 105Kd Stray field energy

coefficient 111Kq Combined a. c. 178Ks Surface a. c. 107Ku Uniaxial a. c. 106KXA Exchange a. c. 109

L

λ Magnetostrictionconstant 208

λ100 Cubic magneto-striction constant 127

λA Hexagonal magne-tostriction constant 286

λc Reduced charac-teristic length 288

λlon Longitudinalmagnetic charge 427

λs Isotropic saturationmagnetostriction 128

λsample Volume charge ofsample 73

λtrans Transverse mag-netic charge 427

λv Magnetic volumecharge density 110

Λ Reduced length 308l Antiferromagnetic

unit vector 263lc Characteristic

length 288ld Decay length 549lp Effective particle

size 152lMD Reduced multi-

domain length 169lSD Reduced single

domain length 152L Length 152Lbs Bubble deforma-

tion function 296Lx Function

(abbreviation) 113LSD Particle length at

single domain limit 155

M

μ Spatial frequency 281μ8 Permeability

for 8A/cm 496μ∗ Rotational perme-

ability tensor 121μ0 Vacuum

permeability 11μB Bohr magneton 369m Reduced magnet-

ization 124m∗ Doring mass 261m Reduced magneti-

zation vector 24mB

0 Centre vector ofBloch wall 307

mN0 Centre vector of

Neel wall 307m Spatial average

over m 172me Electron mass 63mϕ Magnetization

component 248

XXIV Symbols

M In-plane magnet-ization modulus 233

Mi Domain wallcounting factor 319

Ms Gaussian-systemmagnetization 111

Mξ Partial derivative 233

N

ν Poisson ratio 361ν1 Real part of root 282n Branching genera-

tions number 319n Normal, perpen-

dicular direction 64n1 Refractive index 31nph Number of phases 171N Demagnetizing factor 112N Demagnetizing

tensor 112Nen Energy-related

demagnetizing factor 116Nij Molecular field

parameters 369Nshot Optical shot noise 29

O

ω Surface/bulk do-main width ratio 314

ω Lattice rotationtensor 125

ωb Reduced basicdomain width 318

ωcp Compensatinglattice rotation 131

ωopt Optimum reduceddomain width 318

ωres Resonancefrequency 354

ωL Light frequency 46Ω Angular defect 136Ωw Wall angle 224

P

ϕ Magnetizationangle 103

ϕ∗ Switching mag-netization angle 191

ϕ0 Equilibrium mag-netization angle 193

ϕap Optical amplitudepenetrationfunction 32

ϕe Electron phase 51ϕp Azimuth angle at

peak velocity 258ϕ∞ Asymptotic mag-

netization angle 204Φd Stray field potential 110Φq Anisotropy type

angle 179Φtip Tip scalar potential 73ϕK Kerr rotation 28Ψ Wall orientation

angle 132Ψp Polarizer angle

relative to p-axis 27p Reduced period 289p Elastic lattice

distortion 125pc Colloid particle

concentration 14pcp Compensating

distortion 131pe Compatible mag-

netostrictive dis-tortion 130

pkin Kinetic potential 254popt Optimum domain

period 331P Period 124Px Function

(abbreviation) 113

Q

q Numericalcoefficient 197

qc Particle magneticmoment 14

qe Electron charge 49Q Reduced anisotropy

material constant 111Qcr Maximum Q-value

for stripe domains 283QV Magneto-optical

parameter 24

Symbols XXV

R

ρ Reduced particleradius 197

ρc Reduced criticalradius 198

ρms Relative magneto-strictive selfenergy 218

r Position vector 11ra Elastic anisotropy

ratio 358rp Parallel reflectivity

(intensity) 31rSN Signal to noise ratio 29R Radius 162Rc Critical radius 198Rp Parallel amplitude

reflectioncoefficient 27

RF Faraday amplitude 25RK Kerr amplitude 25RN Regularly reflected

amplitude 25

S

σ Conductivity 268σ0 Quasi-plastic stress

tensor 128

σbal Balancing stresstensor 131

σ Average stress tensor 208σcp Compensating

stress tensor 131σex External stress

tensor 138

σform Stress tensor ofthe form effect 359

σp Planar stress 450σs Magnetic surface

charge density 110σsample Sample surface

charge 73σu Uniaxial stress 450s Derivative function 233s Inverse magneto-

elastic tensor 133sb Bubble spacing 292sb Electron beam

direction 51

sch Charge pattern set 118smo Relative magneto-

optical signal 28S Shearing function 233Sbc Function for bubble

collapse 295Sbs Function for bubble

stripout 296Sc Kittel’s Coefficient 301Sd Effective stray field

stiffness parameter 462Se Elementary spin

quantum number 369Smo Absolute magneto-

optical signal 29Sr Revolution number 480Sy Electron deflection

signal by y-field 62

T

τrel Relaxation time 266ϑ Magnetization

angle 103ϑ0 Outer angle

of incidence 25ϑs Surface mis-

orientation angle 143ϑ∞ Asymptotic mag-

netization angle 227Θ Angular variable 107Θc Characteristic

temperature 370t Tangential unit

vector in a wall 206T Temperature 14T Magnetic torque 172Tc Curie temperature 102Tcomp Compensation

temperature 539Tint Intrinsic torque 339Tm Mechanical torque 339Tp Parallel trans-

mission coefficient 31

U

u Bubble diameterfunction 294

u Lattice displace-ment vector 140

XXVI Symbols

ui Stripe domaineigensolutioncoefficients 282

V

v Domain volume 211v Wall velocity 256v Average wall

velocity 259vc Critical wall

velocity 255ve Electron velocity 49vi Phase volumes 93vp Peak velocity 258vLor Lorentz-force-

induced electronvelocity 26

V Volume 339

W

w Domain magnetiza-tion differencevector 206

wr Relative domainwidth 291

W Width 124

W Interaction matrix 117Wb Bulk domain width 314Wcr Critical stripe

width 282WF Wall width based

on wall flux 205WJ Jakubovics’ wall

width 205WL Lilley’s wall width 154Wm Wall width based

on m-profile 205Wopt Optimum domain

width 300Ws Surface domain

width 314WSD Particle width at

single-domain limit 155

X

ξ Reduced/transformedspatial variable 203

Z

Z Number of equi-valent bubbles in lattice 292

Acronyms

AMR Anisotropic Magneto-resistance 94

CEMS Conversion ElectronMossbauer Spectroscopy 94

CIP Current In Plane 555CPP Current Perpendicular

to Plane 555DPC Differential Phase

Contrast ElectronMicroscopy 54

ESD Elongated SingleDomain 386

FFT Fast Fourier Transform 117GMR Giant Magneto-

resistance 548HDDR Hydrogenation-

Disproportionation-Desorption-Recombination 527

LEED Low-Energy ElectronDiffraction 69

MD Multi-Domain State 168ME Metal Evaporated 535MFM Magnetic Force

Microscopy/Microscope 71ML Monolayers 433MO Magneto-Optical 539

MOKE Magneto-Optical KerrEffect 23

MR Magnetoresistance 547MRAM Magnetic

Random-Access Memory 554PEEM Photo Emission Electron

Microscope 63PSD Pseudo Single-Domain

State 487SD Single Domain State 168SEM Scanning Electron

Microscopy/Microscope 61SEMPA Scanning Electron

Microscopy withPolarization Analysis 68

SH Second Harmonic 46SPLEEM Spin-Polarized

Low-Energy ElectronMicroscopy 69

SQUID SuperconductingQuantum InterferometerDevice 347

TD Two Domain State 169TEM Transmission Electron

Microscopy/Microscope 49VSM Vibrating Sample

Magnetometer 337