Nanomagnetism: Ultrathin Films, Multilayers and Nanostructures

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Nanomagnetism Ultrathin Films, Multilayers and Nanostructures

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Series: Contemporary Concepts of Condensed Matter Science Series Editors: E. Burstein, M.L. Cohen, D.L. Mills and P.J. Stiles

Nanomagnetism Ultrathin Films, Multilayers and Nanostructures

D.L. Mills Department of Physics and Astronomy University of California Irvine, CA 92697, USA

J.A.C. Bland Cavendish Laboratory University of Cambridge Cambridge CB3 0HE, UK

Amsterdam – Boston – Heidelberg – London – New York – Oxford Paris – San Diego – San Francisco – Singapore – Sydney – Tokyo iii

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r 2006 Elsevier B.V. All rights reserved. This work is protected under copyright by Elsevier B.V., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier’s Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: [email protected]. Requests may also be completed on-line via the Elsevier homepage (http:// www.elsevier.com/locate/permissions). In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 20 7631 5555; fax: (+44) 20 7631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of the Publisher is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier’s Rights Department, at the fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher 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. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2006 Library of Congress Cataloging in Publication Data A catalog record is available from the Library of Congress. British Library Cataloguing in Publication Data Nanomagnetism : ultrathin films, multilayers and patterned media. (Contemporary concepts of condensed matter science) 1. Magnetic films 2. Nanostructured materials 3. Ferromagnetic materials 4. Thin films - Magnetic properties I. Mills, D.L. II. Bland, J.A.C. (J. Antony C.), 1958-538.3 ISBN-10: 0444516808 ISBN-10: 0-444-51680-8 ISBN-13: 978-0-444-51680-0 ISSN: 1572-0934 (series)

∞ The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

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CONTENTS LIST OF CONTRIBUTORS

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SERIES PREFACE

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VOLUME PREFACE

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1. THE FIELD OF NANOMAGNETISM J. A. C. Bland and D. L. Mills

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2. FUNDAMENTAL PROPERTIES OF MAGNETIC NANOSTRUCTURES: A SURVEY Ruqian Wu

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3. EXCHANGE COUPLING IN MAGNETIC MULTILAYERS M. D. Stiles

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4. STATIC, DYNAMIC, AND THERMAL PROPERTIES OF MAGNETIC MULTILAYERS AND NANOSTRUCTURES R. E. Camley

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5. EXCHANGE ANISOTROPY A. E. Berkowitz and R. H. Kodama

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6. SPIN TRANSPORT IN MAGNETIC MULTILAYERS AND TUNNEL JUNCTIONS A. Fert, A. Barthe´le´my and F. Petroff

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7. ELECTRICAL SPIN INJECTION AND TRANSPORT IN SEMICONDUCTORS B. T. Jonker and M. E. Flatte´

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Contents

8. CURRENT-INDUCED SWITCHING OF MAGNETIZATION D. M. Edwards and J. Mathon

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AUTHOR INDEX

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SUBJECT INDEX

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LIST OF CONTRIBUTORS A. Barthe´le´my

Unite´ Mixte de Physique CNRS/Thales, Domain de Corbesville, 91404 Orsay, France

A. E. Berkowitz

Center for Magnetic Recording, University of California, La Jolla, CA 92093-0401, USA

J. A. C. Bland

Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK

R. E. Camley

Department of Physics, University of Colorado, Colorado Springs, CO 80933-7150, USA

D. M. Edwards

Department of Mathematics, Imperial College of Science, Technology and Medicine, London SW7 282, UK

A. Fert

Unite´ Mixte de Physique CNRS/Thales, Domain de Corbesville, 91404 Orsay, France

M. E. Flatte´

Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242-1479, USA

B. T. Jonker

Materials Science and Technology Division, Naval Research Laboratory, Washington, DC 20375-5343, USA

R. H. Kodama

Department of Physics, University of Illinois at Chicago, Chicago, IL 69607, USA

J. Mathon

Department of Mathematics, City University, London EC1V 0HB, UK

D. L. Mills

Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA

F. Petroff

Unite´ Mixte de Physique CNRS/Thales, Corbesville, 91404 Orsay, France

M. D. Stiles

Electron Physics Group, National Institute of Standards and Technology, Gaithersburg, MD 20899-8412, USA

R. Wu

Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA

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Domain

de


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SERIES PREFACE CONTEMPORARY CONCEPTS OF CONDENSED MATTER SCIENCE Board of Editors E. Burstein, University of Pennsylvania M. L. Cohen, University of California at Berkeley D. L. Mills, University of California at Irvine P.J. Stiles, North Carolina State University We introduce a new series of volumes devoted to the exposition of the concepts underlying the key experimental, theoretical and computational developments at the advancing frontiers of exciting, rapidly evolving sub-fields of condensed matter science. The term ‘‘condensed matter science’’ is central, because the boundaries between condensed matter physics, condensed matter chemistry and material science are disappearing. The overall goal of each volume in the series is to provide the reader with an intuitively clear discussion of the underlying concepts and insights that are the ‘‘driving force’’ for the high profile major developments at the advancing frontiers of the sub-field, while providing only the theoretical, experimental and computational detail, data, and results that would be needed for the reader to gain a conceptual understanding of the subject. Each section of a given volume will be devoted to a major development at the advancing frontiers of the sub-field. This will provide an opportunity for those in other areas of research, as well as those in the same area, to have access to the concepts underlying the major developments at the advancing frontiers of the sub-field. Each volume (250 printed pages) is to have a Preface written by the volumeeditor(s), that includes an Overview that highlights the exciting theoretical and experimental advances of the sub-field and their underlying concepts. It also provides an outline and brief summary of the major topics selected by the volumeeditor(s) and authored by key scientists recruited by the volume-editor(s), that highlight the most significant developments of the sub-field. The chapters are selfcontained—it should not be necessary to go to other sources to follow the presentation or the underlying science. The list of references will include the titles. The volume-editor(s) will interact closely with the chapter contributors to insure that the level and presentation of the material conform to the objective of the series. The volumes in this series will emphasize clear writing whose goal is to describe and to elucidate key developments in the sub-field focusing on the underlying concepts,. The model for this is a well-presented colloquium (not a seminar!) ix

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Series Preface

directed to those outside the specialized sub-field, which invites the audience to ‘‘come think with the speaker’’ and which avoids in-depth experimental, theoretical, and computational details. The overall goal of the series is ‘‘comprehension’’ rather than ‘‘comprehensive’’ and the goal of each volume is to provide an ‘‘overview, rather than a review’’ of the highlights of the sub-field. The audience for these volumes will have wide-ranging backgrounds and disparate interests (academic, industrial, and administrative). The ‘‘unique’’ approach of focusing on the underlying concepts should appeal to the entire community of condensed matter scientists, including graduate students and postdoctoral fellows. Further the audience will certainly include people not in the condensed science community that seek understanding of the sub-field.

VOLUME PREFACE During the past two decades, we have witnessed marvelous advances in our ability to synthesize nanoscale structures of all sorts, as well as the development of novel experimental methods that allow us to explore their physical properties. This is exciting for two reasons. First, new forms of matter with no counterpart in nature have been fabricated, and these have unique physical properties not found in bulk materials. This is so because either a large fraction of their atomic constituents reside in surface or interface sites of low symmetry, or their physical size is so small they are completely quantum dominated. Second, we have now realized nanostructures that open new avenues for the development of very small devices. This has already had a remarkable, qualitative impact on technology, as we will see from remarks in the next paragraph. It is our view that soon nanoscience and the nanotechnology derived from it will have an impact on human affairs comparable to the industrial revolution, when combined with the submicron technology of the past 10–15 years. We can appreciate this from the remarkable influence of modern information technology on our lives. This volume is devoted to the exposition of new structures and the associated physics in an important area of modern nanoscience. This is the field of nanomagnetism, where very small-scale structures such as ultrathin (few atomic layer) films of ferromagnetic material, often incorporated into superlattices or multilayer structures have been found to have magnetic and transport properties qualitatively and dramatically different than realized in bulk magnetic matter. More recently, we have seen a new generation of studies of the magnetism of patterned arrays ranging from micron-sized discs and wires down to dimers and single atoms adsorbed on substrates. Physicists have been intrigued by the new phenomena uncovered as these novel materials have been fabricated and their unique properties explored, materials scientists continue to present us with new structures, and by the time of this writing we have witnessed the enormous impact of ultra high-density magnetic data storage on computer technology. Here it is the remarkable phenomenon of giant magnetoresistance (GMR) of magnetic multilayers that has been exploited to increase the capacity of hard discs by over a factor of a hundred in a small number of years. Other exciting applications are envisioned, through the use of the systems and new concepts discussed in this volume. In this regard, we have in hand as well unique effects such as giant tunneling magnetoresistance (TMR), the phenomenon of the spin blockade, and other fascinating new effects that operate only in nanoscale magnetic systems. New methodologies developed by both theorists and experimentalists drive the field. Examples are the use of spin sensitive atomic force microscopy, and the development of large-scale computer simulations of real structures. The material xi

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in this volume is directed toward a broad audience of readers with backgrounds in condensed matter science who may not be experts in the field of nanomagnetism. It is our hope as well that the pedagogical nature of the discussions will also provide some experts with deeper insights in to the fundamental physics associated with areas of the field that have not been the focus of their own research. We comment next on the material discussed in the various chapters. The first chapter, written by the undersigned volume editors, contains a broad overview. We discuss the unique and special aspects of the magnetism of ultrathin ferromagnetic films. Spin ordering is a long-ranged phenomenon, and the excitations which control both the response characteristics and thermodynamics of ultrathin films are influenced by long ranged couplings as well. Thus, as we shrink magnetic structures down to nanometer length scales, we find fundamental differences in all aspects of their physics. In addition, a large fraction of the moment bearing ions sit in interface or surface sites, with qualitative consequences for both their magnetic and chemical properties. Ultrathin films are the ‘‘building blocks’’ of the magnetic multilayers, spin valves and related structures that have been explored and discussed intensively in recent years. We also introduce materials with lateral structure (nanodot, nanodisc and nanowire arrays), and then turn our attention to the experimental methods, which have proved central to the elucidation of the properties of very small magnetic structures. The nature of the magnetic moments found in ultrasmall structures can differ dramatically from their bulk counterparts, by virtue of the fact that a large fraction reside on surfaces, at interfaces as noted above. These also can be affected by the chemisorption of selected molecules. It follows that one realizes magnetic anisotropies one or two orders larger than that found in the bulk, and their strength and character are subject to design. Thus, we have spin engineering. R. Wu provides us with a broad survey of the ground state properties of diverse nanoscale magnetic structures. It is impressive to see the success of modern density functional theory in its ability to provide reliable quantitative accounts of the properties of these often-complex systems with low symmetry. When ultrathin ferromagnetic films are assembled into multilayers or superlattices, weak interactions of exchange character act between the constituent films. These are mediated by the spin polarization induced in nonferromagnetic spacer layers inserted between them. These weak exchange couplings, tunable both in sign and magnitude by varying structural details, lead us to magnetic entities whose underlying structure can be manipulated by very modest or weak applied fields, in contrast to bulk materials whose magnetic ions are tightly coupled by very strong interatomic exchange. This key property allows us to fabricate new, artificial materials whose magnetic properties differ qualitatively from those of bulk materials, and which can be varied over a wide range by design. M.D. Stiles provides us with discussions of the physical origin and nature of these interfilm exchange couplings. Arrays of ultrathin ferromagnetic films coupled by the exchange interactions just discussed display a rich variety of spin structures, where the total (macroscopic) magnetic moment of each ultrathin film plays the role of a large, and consequently fully classical ‘‘spin.’’ R.E. Camley discusses examples of

Volume Preface

xiii

these structures, and the collective excitations they support. We have here the opportunity to develop new materials with microwave response tunable by the application of very modest magnetic fields. When one wishes to exploit magnetic multilayers in devices, the signal detected has its origin in the rotation of the magnetization vector in one selected film relative to that in a neighboring film, and the resulting influence on properties of the structure such as its electrical resistance, as discussed in the next paragraph. A commonly used structure is the ‘‘spin valve,’’ which consists of two ferromagnetic films separated by a nonmagnetic spacer which gives rise to the weak interfilm interactions described in the previous paragraph. A question is then how one may use an applied magnetic field to rotate the magnetization of one of the two films, while the second remains pinned in place. A phenomenon referred to as ‘‘exchange bias,’’ discovered many decades ago, allows one to selectively ‘‘pin’’ the magnetization of one layer. A.E. Berkowitz and R.H. Kodama introduce us to this central topic, whose origin is only very recently appreciated. Transport properties of magnetic multilayers have excited many researchers, since A. Fert and his colleagues reported the discovery of the astonishing phenomenon of GMR in 1988. In parallel with their work, P. Grunberg and collaborators observed this phenomenon as well at very close to the same time. The origin of GMR has stimulated efforts by many experimentalists and theorists for some years now, since it is not only a spectacular physical phenomenon, but has provided us with the basis for ultra high-density magnetic storage and its enormous impact on computer technology. A. Fert, A. Barthelemy and F. Petroff present us with a discussion of this most important phenomenon in their chapter, and cover tunneling magnetoresistance as well. Physicists, materials scientists and engineers actively discuss a new field called ‘‘spintronics,’’ wherein it is the spin of the electron rather than its charge that is exploited and manipulated. This has led to the exploration of new physics that must be understood before the field can prosper. As we have known for decades, we can inject electrons from semiconductors into metals. But if the electrons are highly spin polarized, is the spin polarization preserved or destroyed in the injection process, and if it is the latter how do we make structures that preserve spin polarization? This is a central issue that is fundamental to the new spintronics, and B.T. Jonker and M.E. Flatte provide us with an exposition of where we are at the moment, from both the experimental and theoretical perspectives. The injection of spin polarized electron currents into thin ferromagnetic films can also be used to manipulate and reorient the magnetization of the latter. D.M. Edwards and J. Mathon provide us with an understanding of the physics associated with this important process where the physics is again fascinating, and potential applications most important. As volume editors, we are very pleased indeed that leading figures in the field of nanomagnetism have come forward to provide us with their insights in the chapters summarized above. It is our hope that this volume will stimulate readers to acquire an active interest in this field, which continues to grow in size and impact. D. L. Mills and J. A. C. Bland


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Chapter 1 THE FIELD OF NANOMAGNETISM J. A. C. Bland and D. L. Mills ABSTRACT In this introductory chapter we introduce the field of nanomagnetism, briefly surveying the underlying key theoretical concepts and the experimental methods which have led to its recent rapid development and which, together, have led to key technological advances in computer read-head technology, sensors and magnetic memory. We begin by considering the magnetic length scales (e.g. spin wave length, exchange length), which define ‘‘small’’ in this context and introduce the concept of ‘‘spin engineering’’ that can be considered analogous to that of ‘‘band gap engineering’’, which has proved such a powerful concept in semiconductor physics. We describe the physics of ferromagnetism in the ultrathin film limit and consider, for example, the effect of magnetic anisotropies on the magnetic properties, such as magnetic ordering at finite temperature, the spin reorientation transition and spin wave excitations as well as interlayer coupling. We next describe the physics of magnetic nanostructures, describing the effect of finite size on the spin configurations and magnetization reversal processes in small elements. Finally, we review experimental techniques for highly sensitive magnetic measurements of nanostructures, including both static and dynamic techniques, imaging techniques with spin sensitivity and magnetoresistance measurements. Keywords: nanostructures, magnetic elements, arrays, dipole interactions, experimental techniques, BLS, MOKE, FMR, MFM, SPEELS, STM, Lorentz microscopy, spin - STM, PEEM, SEMPA, AFM, MFM, XMCD, magnetic particles, superparamagnetism, ferromagnetism, antiferromagnetism, dipolar effects, elements, anisotropies, giant magnetoresistance, tunnel magnetoresistance, magnetic random access memory, read heads, spin engineering, interlayer coupling, dynamic measurements, spin waves, magnetoresistance, spin reorientation transition, magnetic ordering, magnetic reading, fast magnetic switching, magnetic writing

Contemporary Concepts of Condensed Matter Science Nanomagnetism: ultrathin films, multilayers and nanostructures Copyright r 2006 by Elsevier B.V. All rights of reproduction in any form reserved ISSN: 1572-0934/doi:10.1016/S1572-0934(05)01001-2

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J. A. C. Bland and D. L. Mills

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1. INTRODUCTION Since the earliest days of condensed matter physics, the study of magnetically ordered materials has played a central role in establishing the fundamental principles and concepts of the field. An early example is provided by Bloch’s introduction of spin waves into physics, and his subsequent derivation of the T3/2 law for the initial decrease with increasing temperature of the magnetization of ferromagnets from the saturation value appropriate to zero temperature. This introduced the key notion that low-lying collective excitations dominate the thermodynamics of solid materials at low temperatures, one of the central differences between extended crystals with long-ranged order and small systems such as molecular entities. In the 1970s and 1980s, by virtue of the appearance of the scaling hypotheses and the related renormalization group method, theorists achieved a deep understanding of the previously challenging phenomenon of the thermodynamic phase transition. Experimental studies of the diverse forms of magnetism found in solids allowed experimentalists to verify the predictions of these theories with impressive quantitative accuracy. Perhaps the reason for the fundamental role played by magnetism in the development of fundamental concepts is that wide classes of materials can be described very well by Hamiltonians that are rather simple in structure, with few parameters in them. Thus, the theorist can apply sophisticated methodologies to these Hamiltonians, while the experimentalist can obtain an account of data with model forms that contain very few parameters. This allows quantitative tests of theory. In the current era, there is great excitement in the study of exotic quantum phenomena such as those realized in spin ladder compounds, Kagome lattices and frustrated systems. Magnetism is again playing the key role in the development of deep new fundamental concepts in condensed matter physics in this exciting new area. This volume is devoted to the topic of nanomagnetism and addresses the new physics and response characteristics encountered in magnetic materials when one or more of its linear dimensions is very small. We shall see that we encounter physics in this regime, qualitatively different from that realized in bulk magnetic materials. Our first task in this chapter is to define the sense in which the word ‘‘small’’ is used in the opening sentence of this paragraph. Broadly speaking, the new physics has its origins in two features one encounters as one or more linear dimensions of a sample are reduced. First, when one enters the nanometer regime, a large fraction of the magnetic moment bearing species resides in the surface or interface sites. For instance, if we have an ultrathin magnetic film five atomic layers in thickness, 40 percent of the magnetic ions reside in such sites. The key aspects of magnetism are very sensitive to local site symmetry, and thus the magnetism in such a film will differ qualitatively from a bulk crystal made with the same atomic constituents. One may vary magnetic properties of the ions that reside in such sites by varying the substrate or crystal face on which the film is grown and by chemisorbing material on the outer surface. One can thus create new ‘‘designer materials’’ with properties and response characteristics controlled by such features. We then have spin engineering, an

The Field of Nanomagnetism

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analog to the bandgap engineering of the semiconductor realm. In addition, an ultrathin film grown epitaxially on a substrate will have a lattice constant controlled by that of the underlying substrate and hence different from the corresponding bulk material. Also, the spacing between parallel atomic planes will differ as well, by virtue of expansion or contraction of the lattice constant parallel to the substrate surface. Hence, the site symmetries in the center of the film differ from the bulk material as well as from those on the surface or at an interface. We know very well that the fundamental interactions that control the nature of the ordered state are highly sensitive to both local geometry and interior spacings. New crystal structures not found in bulk matter are also realized in magnetic nanostructures. One may grow stable, thick epitaxial films of fcc cobalt on suitable substrates, for example, while the bulk form of this element is hcp. Second, bulk magnetic matter is characterized by fundamental length scales. The lattice constant, of course, is the smallest such length scale, but there are two other important lengths that play a central role. Each of these can be substantially larger than the lattice constant in the materials of current interest, as we shall see. When one or more sample dimensions are small or comparable to one of these fundamental lengths, we again enter a domain where the response characteristics, thermodynamic properties and other key aspects of the magnetism differ qualitatively from that found in bulk materials. One such length is the wavelength lT of thermally excited spin waves. Perhaps it is more relevant to discuss their wavevector kT ¼ 2p=lT : If L is any linear dimension of the sample of interest, then when kTLr1, we are in a regime where the fundamental magnetic properties of the material differ from those in the bulk and are affected by sample size. Since the time of Bloch, we know the dispersion relation ~ ¼ Dk2 for wavelengths long of spin waves in a ferromagnet is given by _oðkÞ compared to a lattice constant, where the parameter D is the exchange stiffness. Thus, kT ðkB T=DÞ1=2 ; with kB being Boltzmann’s constant. We shall be interested primarily (but not exclusively) in the 3D transition metal ferromagnets in this volume. For ferromagnetic Fe, D 300 meV A2 : Then at room temperature for Fe, we have a critical crossover length LT ¼ 1=kT 0:3 nm (nanometers), whereas if we cool down to liquid helium (He) temperature, LT 3 nm. If, then, we have an ultrathin film some two or three atomic layers in thickness, and we cool from room temperature to liquid He temperature, we cross over from a regime where the thermodynamics is three dimensional (room temperature) to a quasi-two-dimensional regime (He temperature). Similarly, if at He temperatures, we increase the thickness of the film, we can study the crossover from two to three-dimensional physics. Thus, ultrathin films (and other nanomagnetic structures) offer us the possibility of studying statistical mechanics as we make the transition from threedimensional physics to the physics of a lower dimensional world [1]. Another fundamental length is the exchange length, which for our purposes here we take to be Lex ¼ ðD=mB H A Þ1=2 ; where mB is the Bohr magneton and HA is a measure of the anisotropy field in the material. This may range from roughly 1 to 10 kG, or equivalently from 0.1 to 1 T in typical materials. The exchange length is the thickness of a domain wall of Bloch character in a bulk ferromagnet; for a Neel

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wall, one replaces HA by 4pMS where MS is the saturation magnetization. The magnitude of 4pMS is comparable to that of the anisotropy field. Typically, then, the exchange length lies in the range of 5–10 nm. In a ferromagnet, the cost in energy to create a non-uniform state characterized by a length scale small compared to Lex is very large. Thus, in ferromagnetic films whose thickness is small compared to this length, the magnetization is uniform in the direction normal to the surfaces. Such a film’s domain structure and hysteresis loop differ qualitatively from a bulk ferromagnet with its complex domain structure in the form of a three-dimensional network. Thus, in this volume our attention is directed toward the magnetism found in samples sufficiently small that a large fraction of the moment bearing ions reside in surface or interface sites, whose geometrical structure is controlled by the substrate on which it is grown or other aspects of the local environment, or whose linear dimensions are comparable to or smaller than the length scales discussed above. We conclude from the comments above that one or more of these conditions are satisfied when at least one dimension of the sample of interest is in the nanometer range. In Section 2, we focus on the specific case of the ultrathin ferromagnetic film to provide the reader with an introduction to the new physics encountered in the realm of small-scale magnetism. Nanoscale magnetism, as we shall see at various points in this volume, already has had a dramatic effect on computer technology and we expect more applications of nanoscale magnetic structures in the near future. Patterned media comprised of nanoscale elements are envisioned to be of increasing importance. Thus, this has become a very active research area in the past few years. Section 3 provides the reader with an introduction to this rapidly evolving area. The experimental study of nanoscale magnetic systems presents a challenge, since one wishes to probe a small number of magnetic moments located on a macroscopic substrate. We review the principal experimental methodologies in Section 4.

2. THE ULTRATHIN FERROMAGNETIC FILM In this section, we focus our attention on the unique aspects of ferromagnetism in ultrathin (few atomic layers) ferromagnetic films. Here we shall encounter dramatic and qualitative effects upon reducing one dimension of the sample to nanometerlength scales. It is the case as well that ultrathin ferromagnetic films such as explored here are the building blocks of many of the structures discussed later in the volume. In this brief chapter, our discussion will be somewhat schematic and sketchy. The interested reader may wish to consult two volumes [1] each of which contains chapters that explore in detail particular properties of ultrathin magnetic films and the means of probing them experimentally. A chapter written by one of the present authors may be viewed as an elaboration of the issues explored in the present section [2]. In few-atomic layer films, as discussed in Section 1, a large fraction of the magnetic moment bearing ions reside in sites of low symmetry, in the surface of the film


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or at the interface between the film and the substrate upon which it is grown. This has a strong, qualitative influence of a key property of ferromagnets, their magnetic anisotropy. We begin by introducing the reader to the concept of magnetic anisotropy and its role. Suppose we begin by considering the simplest description of a crystalline array of magnetic moments provided by the well-known Heisenberg model, wherein pairs of spins interact by means of the exchange interaction whose ~2 : The total magnetic energy is found by summing ~1 S form may be written as J 12 S such interactions over all spins in the crystal. If the ground state is ferromagnetic, the only case considered in our brief remarks here, the energy of the spin array is left unchanged by a rigid rotation of all the spins in the system. Hence, the spontaneous magnetization in the ferromagnetic state may point in any direction it desires. It is not tied to the crystalline axes in any manner, for example. In real materials, there are terms in the Hamiltonian that select out particular directions as ‘‘easy axes’’ along which the magnetization prefers to be directed. One refers to such terms as the anisotropy energy. There are two interactions which enter centrally: the long-ranged magnetic dipolar couplings between the atomic magnetic moments, and spin–orbit interactions of the moment bearing electrons which allow the moments to sense the local crystalline axes, as we shall see below. We consider the role of magnetic dipolar interactions first. The magnetic field seen by a given moment with origin in a distant neighbor falls off as 1/r3, with r being the distance between them. Thus, the total magnetic dipole interaction energy of the system is proportional to Z Z 3 3 d rð1=r Þ drr2 ð1=r3 Þ (1) V

V

where the integral is over the volume of the sample. The integral clearly diverges for a truly infinitely extended sample, and thus we must take due account of the finite sample size to evaluate this energy. When this is done, the overall shape of the sample controls the direction of the easy axis, if only dipolar anisotropy is present. For this reason, in the literature on magnetism, dipolar anisotropy is referred to often as ‘‘shape anisotropy.’’ We may appreciate the point just made for a ferromagnetic film, viewed in ~ S be canted out of the plane of macroscopic terminology. Let the magnetization M the film by the angle y. If z^ is the normal to the film surfaces, it is a matter of elementary magnetostatics to show that inside the film, one has a demagnetizing ~dip ¼ ^z4pM S sin y: This is antiparallel to the component of magnetization field H normal to the surface, and thus any canted, out-of-plane state of the magnetization is energetically unfavorable. Stated otherwise, the energy per unit volume associated with a state where the magnetization is canted out of plane is given by, again from elementary magnetostatics, þ2pM 2S sin2 y and the state of lowest energy has the magnetization in plane, with y ¼ 0: Thus, we conclude that if dipolar interactions are the only source of anisotropy in a ferromagnetic film, the shape anisotropy forces the magnetization to always lie parallel to the film surfaces. One may wonder if such macroscopic considerations apply to few atomic layer films. In this limit, one


6

resorts to a lattice description, and interestingly, the macroscopic results prove rather accurate even close to the monolayer level [3]. As remarked above, the spin–orbit interaction allows a magnetic moment to sense the symmetry of its local neighborhood through interaction of the spins with crystalline electric fields associated with neighboring ions. This competes with shape anisotropy, and the resulting orientation of the magnetization is controlled by the one that is dominant. We may appreciate the role of spin–orbit interactions through a simple schematic argument. We introduce into the Hamiltonian the spin–orbit ~ with L ~ S; ~ being the orbital angular interaction, which we may write as lL momentum. The dependence of the energy of a local magnetic moment in a crystal as a function of its spin orientation may be found by the spin Hamiltonian method [4]. This approach notes that in the absence of spin–orbit coupling, which is a weak perturbation to the internal crystalline field for the 3d transition metal ions of primary interest to us, the wave function in the ground state is a product of the orbital wave function and the spin wave function. One generates an effective spin Hamiltonian through perturbation theory in which the coordinates of the electron are integrated out in the various matrix elements, but the spin operators are left standing to operate on the ground-state multiplet. Only even orders of perturbation ~ in any effective theory enter, since time-reversal symmetry forbids odd powers of S spin Hamiltonian. Thus, second-order perturbation theory leads to an effective spin Hamiltonian of the form X ð2Þ H ð2Þ K ab Sa Sb (1a) SO ¼ a;b

where K ð2Þ ab

¼l

2

X o0jLa n4onjLb j04 n

E0 En

(1b)

Local site symmetry controls the structure of the right-hand side of Eq. (1a). Suppose we consider a bulk ferromagnet wherein each ion is positioned at a site of cubic ð2Þ symmetry, as in the case of Fe and Ni. Then the second-rank tensor K ð2Þ ab ¼ K dab ; 2 2 2 so the right-hand side of Eq. (1a) is proportional to S x þ S y þ Sz ¼ SðS þ 1Þ: There is thus no dependence on the direction of the spin, and we must turn to fourth-order terms, which for a cubic site have the form X ð4Þ 4 4 4 2 2 2 2 2 2 H ð4Þ ½K ð4Þ (2a) a fS x þ S y þ S z g þ K b fS x S y þ S y S z þ S z S x g SO ¼ a:::d

This may be rearranged to read 2 2 2 2 2 2 ð4Þ 2 ð4Þ H ð4Þ SO ¼ K a fSðS þ 1Þg þ DK fS x S y þ S y S z þ S z S x g ð4Þ

K ð4Þ b

2K ð4Þ a : ð4Þ

ð4Þ

(2b)

where DK ¼ If DK 40; the magnetization will align along a [100] direction, whereas if DK 40; the magnetization will align along a [111] direction. Even though the anisotropy energy per ion is very small, the order of 0.01 meV for bulk Fe and Ni, the moments are rigidly coupled together by the very strong exchange interactions, so the energy per unit volume associated with reorientation


7

of the magnetization is substantial, the order of 106 erg/cm3. A prediction of the sign of DK(4) is challenging to theorists, since this depends on subtle details of the electronic structure, as one can appreciate from the complexity of the fourth-order perturbation theory. In bulk Fe, the [100] directions are the easy axes, while in bulk Ni, the [111] directions play this role. To return to our example of the film, for thick films of cubic ferromagnets where the bulk crystalline structure is realized throughout the film, the strong shape anisotropy drives the magnetization into the plane of the film, and the much weaker spin–orbit anisotropy controls its orientation in the plane. Now let us turn to an ultrathin film, where a very large fraction of the moment bearing ions reside in surface or interface sites. In such sites, cubic symmetry is strongly broken, and the normal to the surface, the z^ direction, is different from the ð2Þ 2 2 2 x^ and y^ directions. The form in Eq. (1a) then becomes K ð2Þ a fS x þ S y g þ K b S z ð2Þ 2 ð2Þ 2 ð2Þ ð2Þ or K a SðS þ 1Þ þ DK Sz where DK ¼ K a K b : Thus, the low site symmetry realized at surface and interface sites activates the second-order terms in the spin– orbit energy, which are ‘‘silent’’ at cubic sites. The perturbation theoretic arguments sketched above tell us that the quadratic terms are larger than the quartic terms by roughly ðDE=lÞ2 ; with DE being a measure of energy level splittings in the d shell produced by crystalline electric fields. For the 3d transition metal ions, very roughly l0.1 eV whereas DE1 eV, so the second-order terms can be two orders of magnitude larger than the quartic terms. Hence, in an ultrathin film, the spin–orbit anisotropy activated by the low-symmetry sites at surfaces and interfaces can be two orders of magnitude larger than that found in bulk materials synthesized from the same atomic species. This is thus an enormous effect. In fact, the surface and interface anisotropy can often overwhelm the dipolar anisotropy discussed above. Thus, in films where DK ð2Þ o0; we can have films with magnetization perpendicular to the surface, quite in contrast to the conventional orientation. The sensitivity of the sign and magnitude of DK(2) to local environment means one can engineer ultrathin forms of magnetic materials to have anisotropy suited to a particular purpose by varying the substrate upon which the film is grown, or adsorbing material on its outer surface. It remains the case that in the materials of interest here, surface anisotropy remains small compared to the very strong interspin exchange couplings that drive the ferromagnetic order. Hence, as one probes anisotropy by using external magnetic fields to reorient the magnetization, it precesses rigidly. A consequence is that the classic signature of surface or interface anisotropy is its strength that varies inversely with the thickness of the film [5]. Thus, it is common in very thin films for surface anisotropy to dominate in the few-atomic layer limits, whereas as film thickness increases, one realizes a transition to shape anisotropy as the dominant source. One can then realize, with increasing film thickness, the interesting phenomenon of the spin reorientation transition (SRT). Given an ultrathin film where surface anisotropy orients the magnetization normal to the surfaces, as the thickness increases and one enters the regime where dipolar anisotropy dominates, the magnetization reorients parallel to the surface. In a particular film with fixed thickness, the reorientation transition can be realized in the form of a


8

thermodynamic phase transition driven by the difference in the temperature dependencies of the surface and shape anisotropies [6]. So far, in our discussion of ultrathin films, we have described two sources of anisotropy, the shape or dipolar anisotropy, and that associated with surface or interface sites where the low site symmetry activates the quadratic terms in the spin– orbit energy that are inoperative in bulk cubic materials. There is one more very important source of anisotropy in ultrathin films we must discuss, since it also plays an important role in films from materials whose bulk form is cubic. High-quality epitaxial films are grown typically on substrates whose lattice constant is close to that of the bulk crystalline form of the material from which the film is fabricated. However, the mismatch in lattice constant can still be appreciable, in the range of a few percent. If, for example, a film of a nominally cubic material is grown on a substrate whose lattice constant is a bit larger than that of the bulk material, then epitaxial growth requires the film to have an expanded unit cell in the plane. It follows that the spacing between the layers will contract somewhat, so the unit cell even in the center of the film will be tetragonal rather than cubic. This will then activate the quadratic terms in Eq. (1a), to produce uniaxial anisotropy similar to that found at the surface or interface sites. One may view the origin of this third form of ultrathin film anisotropy as having its origin in the phenomenon of magnetostriction, which is the dependence of the anisotropy energy on strain. Imagine assembling an ideal film in free space, where every atom is present in a site appropriate to the bulk crystalline form. Now bring such a film into contact with a substrate whose lattice constant is slightly mismatched. One must strain the ideal film for it to fit onto the substrate with the required perfect lattice match. The coefficients in Eq. (1a) may be regarded as a function of the strain tensor egd: X ð2Þ K ð2Þ ð@K ð2Þ ab ðfegd gÞ ¼ K dab þ ab =@egd Þ0 egd gd

The second terms lead to a contribution to the anisotropy energy with a form similar to that realized at the surface and interface sites. Since the strain is modest, the anisotropy energy per layer from this source will be considerably smaller than that from the interfaces, where the change in symmetry is so dramatic. In films that contain several layers, however, the magnetostrictive anisotropy associated with each layer can sum up to be substantial. An interesting example is the case of Ni films on Cu(100) [7]. Such films may be grown to appreciable thicknesses, in the tetragonally distorted shape. When the films are very thin, the interfacial anisotropy favors magnetization in plane, as does the shape anisotropy, which is rather weak here because of the small saturation magnetization of Ni. However, the magnetostrictive component favors out-of-plane alignment. Hence, when thickness is increased, the films undergo a reorientation transition from magnetization parallel to the surfaces to the perpendicular state, a behavior quite the opposite of that discussed earlier. The anisotropy energy controls the orientation of the magnetization in the ground state of the ultrathin film, as we have seen, and also the response of the film to applied DC or very low-frequency external magnetic fields. As remarked earlier, since the strength of the effective anisotropy field seen by any one particular spin is


9

small compared to the interatomic exchange fields responsible for ferromagnetic alignment of the spins, the spins remain locked into ferromagnetic alignment in response to low-frequency applied magnetic fields, and the macroscopic magnet~ S precesses or rotates in response to such a field. The ability to ization vector M create ultrathin films with desired orientation of the moment in the ground state or desired response to external fields which vary slowly in time allows one to create ‘‘designer materials’’, particularly when this feature is combined with the ability to synthesize magnetic multilayers with the effective interfilm exchange couplings described in the M. Stiles chapter in the present volume. In the literature on ultrathin film and multilayer magnetism, the nature of the anisotropy realized in any structure and the ability to control it through design, or ‘‘spin engineering’’ of materials is a central issue. The reason is that a central role is played by magnetic nanostructures in current device technology, as various authors in this volume discuss. Such devices function by either switching the magnetization or rotating it through application of external magnetic field pulses of a suitable nature. The ability to do this with applied magnetic fields of magnitude suitable for device applications is thus of great significance, and as we have seen, the orientation of the magnetization is controlled by the various forms of anisotropy. As the magnetization is rotated from one easy direction to another, it must surmount a barrier associated with the ‘‘hard axes’’ along which the magnetic anisotropy is a maximum. The height of these barriers is a crucial issue. In the next section of this chapter, we introduce the reader to a number of issues of importance to device applications, and other means of structuring materials to achieve desired response characteristics. The above discussion explores only ground-state properties of the film, in the sense that the magnetization adiabatically follows the slowly varying fields mentioned in the previous paragraphs. The response of the film to an external probe of finite frequency is controlled by its spectrum of elementary excitations, the spin waves that enter the theory of ferromagnetism [8]. These are characterized by a ~ which describes the frequency of the wave as a function of dispersion relation OðkÞ; ~ which in the case of our film, will lie in the appropriate twoits wave vector k; dimensional Brillouin zone. The dispersion relation and character of these modes is influenced by the interatomic exchange interactions, and the anisotropy energies discussed above. The excitation spectrum of the ultrathin ferromagnetic film has unique features not found in bulk magnetic matter, and these influence the response of the film to microwave fields in a fundamental manner. As the spins precess after a spin wave is excited, their motion generates timedependent magnetic dipole fields, which react back on the spin system. These have a qualitative effect on the dispersion relation, in the limit of long-wavelength or small-wave vector, the dispersion relation depends on the direction of propa~ S : In bulk ferromaggation of the wave relative to the magnetization vector M ~ ¼ g½ðH 0 þ Dk2 Þ nets, the effect is well known, and the dispersion relation is OðkÞ 2 1=2 2 ðH 0 þ 4pM S sin yk þ Dk Þ ; where the term in 4pMS has its origin in the dipolar fields generated by the spin motions. Here H0 is an applied DC magnetic field, D is the exchange stiffness with origin in the interatomic exchange couplings and yk is the angle between the wave vector and the magnetization [9]. We have ignored

10


anisotropy, the focus of our earlier discussion, but this may be incorporated into H0 as an effective field. In the ultrathin ferromagnets, the long-ranged dipolar fields generated by the spin motions have the most peculiar influence on the dispersion relation at long wavelengths. They produce a term linear in wave vector at long wavelengths, with coefficient whose sign depends on yk. Thus, we have directions of propagation where the initial slope of the dispersion curve is negative. The influence of exchange, through terms such as the Dk2 in the formula quoted above, dominate at larger wave vectors, to produce a rather strange dispersion relation in which the minimum spin wave frequency occurs at a finite wave vector km(yk), for those directions for which the linear term in the dispersion relation is negative. This peculiar feature of spin waves in the ultrathin ferromagnets has the most important consequence. When one considers the damping of long-wavelength spin motions in the film, an extrinsic damping mechanism referred to as two-magnon damping is activated by this feature of the dispersion relation [10]. In real films, under a variety of circumstances, this can be the dominant source of damping. Thus, consideration of its role becomes most important in the design of devices, which exploit the rapid response of the magnetization. We refer the reader to a review that introduces this mechanism and describes several striking consequences of its influence on the response characteristics of ultrathin films [10]. As we move out to short wavelengths, interatomic exchange dominates and controls the dispersion relation and properties of the spin waves, again as one can see from the dispersion relation quoted earlier for waves in bulk materials. As we move into this domain, it is now clear that an issue ignored so far in this chapter asserts itself most strongly: the films of interest are metallic ferromagnets, or itinerant ferromagnets to employ another term. The moment bearing electrons move in energy bands of considerable width, approximately 4 eV for the d bands of the 3d metals. If we set this issue aside, and view our film as N layers of magnetic moments each of which is described as interacting with neighbors through Heisenberg ex~2 ; then one can see easily that for each wave vector k~ ~1 S change of the form J 12 S in the two-dimensional Brillouin zone, we have N spin wave frequencies, and hence N branches to the dispersion curve. In such a picture, each mode is undamped, with infinite lifetime. The lowest lying such branch, for small values of k~ is the lowfrequency mode discussed above, relevant to the microwave response of the film. However, theory of spin waves in truly itinerant ultrathin ferromagnetic films predicts that throughout much of the Brillouin zone, the spin wave modes are heavily damped by virtue of coupling to the spectrum of electronic excitations in the film/ substrate complex, to the point that only a single rather broad feature is evident in calculated frequency spectra [11] for a given wave vector. This feature moves with k~ in a manner quite similar to a single branch of a dispersion curve. Very interesting recent experiments provide us with our first data on short-wavelength spin waves in ultrathin films [12] through the method of spin-polarized electron energy loss spectroscopy (SPEELS), and this is precisely the behavior found. Calculations directed toward the system studied in the experiments provide a very good account of the data [13]. As metallic ferromagnetic structures of nanoscale dimensions are incorporated into devices, the strong damping revealed by the studies just cited will


11

become an issue, whenever spin motions with strong spatial gradients are excited. The issue will thus emerge as one of broad interest in the coming years. It is well known again since the time of Bloch that at finite temperatures, spin waves are present as thermal excitations, and drive the thermal fluctuations found in the system. The magnetization MS(T) then falls below the value MS(0) appropriate to the absolute zero of temperature. We may thus write M S ðTÞ=M S ð0Þ ¼ 1 DðTÞ where in bulk ferromagnets, we have the famous result of Bloch, wherein D(T)T3/2 [14]. Straightforward application of Bloch’s ideas to the ultrathin films leads to a disaster. In temperature units, the anisotropies and Zeeman energies discussed above are negligibly small, so the dispersion relation of the thermally excited spin waves is described very well by ignoring these small terms. Hence, we ~ ¼ Dk2 for calculating thermal properties of the ferromagnet well may take OðkÞ below the Curie temperature. In spin wave theory we have 8 9 <X = ~ =N DðTÞ ¼ nðkÞ : ~ ; k

~ ¼ 1=½expð_OðkÞÞ ~ 1 is the where N is the number of spins in the system and nðkÞ Bose–Einstein occupation factor for the thermally excited spin waves. When the sum is converted to an integral for a two-dimensional film, a logarithmic divergence is found in the integral as k~ ! 0; so that at any small but finite temperature, one finds DðTÞ ¼ 1: Thermal fluctuations associated with very long- wavelength spin waves are so violent in the two-dimensional world that at any finite temperature, no matter how small, they disrupt the long-ranged order associated with ferromagnetism. The above argument of a ‘‘hand waving’’ nature is thus not rigorous. However, the same conclusion follows from a rigorous theorem in statistical mechanics known as the Mermin Wagner theorem [15]. The theorem states that long-ranged ferromagnetic order is absent in two dimensions, for any spin system described by an underlying Hamiltonian, which is form invariant under a rigid body rotation of all the spins. The theorem applies to the Heisenberg ferromagnet in two dimensions, and also equally well to itinerant materials as well, so long as the Hamiltionian contains only the kinetic energy term for the electron and the Coulomb interaction between them. Despite the Mermin Wagner theorem, it is an experimental fact that ultrathin films display ferromagnetism at finite temperature, many right down to the monolayer level. The Curie temperature may be depressed relative to that appropriate to the bulk material, but such materials are perfectly good ferromagnets so far as experiment is concerned. From the perspective of theory, the day is saved by the various forms of anisotropy discussed in the first portion of the present section. Quite clearly, if easy axes exist either by virtue of shape anisotropy or spin–orbit induced anisotropy, the underlying Hamiltonian is no longer form invariant under rigid rotation of the spins, and the Mermin Wagner theorem is violated. However, one’s first reaction is that the energies involved with these anisotropies are so very small compared to kBT that they cannot possibly result in Curie temperatures in the range observed for ultrathin films, which can be an appreciable fraction of the bulk

12


Curie temperature. In the 3d ferromagnets, Curie temperatures range from 500 to 1000 K or more. However, theory shows that even such small terms in the Hamiltonian can indeed produce substantial Curie temperatures. It has been shown [16]1 that the Curie temperature T ð2Þ C of a two-dimensional film subject to weak uniaxial anisotropy is related to the Curie temperature T ð3Þ C of its three-dimensional ð3Þ 2 counterpart through the relation T ð2Þ C ¼ 2T C = lnðp J=KÞ where J is the strength of the interspin exchange coupling, and K that of the uniaxial anisotropy. Even if ð2Þ K=J 0:01; we find T ð2Þ C 0:3T C ! Thus, very tiny anisotropies can drive the Curie temperature of the ultrathin films to substantial values. The two-dimensional Heisenberg ferromagnet is the most intriguing system. While thermal fluctuations break up its long-ranged order at any finite temperature, it really wants to order and even very weak symmetry breaking interactions such as the anisotropies discussed above can drive the Curie temperature of a monolayer film of 3d magnetic ions up to the vicinity of room temperature. Thus, anisotropy again enters as a central feature of ultrathin films, not only because of its role in devices, but because it is the key to the commonly observed ferromagnetic order in these systems. One may inquire how thick the film must be before it makes a transition from the two-dimensional behavior, just discussed, to quasi-three-dimensional magnetism. This has been addressed in Monte Carlo calculations of the Curie temperature as a function of film thickness [17], where it is found that by the time one has 10 layers or so, the Curie temperature evolves from its two-dimensional anisotropy-driven value to that appropriate to the three-dimensional limit. Thus, ultrathin ferromagnetic films are ideal systems for the study of the transition from the statistical mechanics of the two-dimensional world to that of the three-dimensional world we see around us. One may then ask how an experimentalist can interrogate properties of an ultrathin film as a function of its thickness, to examine the transition from two to three dimensions. The answer lies in the critical exponents of statistical mechanics. In the two-dimensional Heisenberg ferromagnet, theory [16 ] shows that the critical exponents should be those of the two-dimensional Ising model, even though the ratio K/J may be very small compared to unity. Thus, the critical exponent b should be found to have the Ising value of 0.125 ¼ 1/8 in the quasi-two-dimensional regime, whereas when the film becomes thick enough to display quasi-three-dimensional order, this exponent should rise to the three-dimensional value close to 1/3. In very interesting- studies of critical exponents in ultrathin Ni films, Yi and Baberschke [18] find precisely this behavior, and the crossover between two- and three-dimensional behavior indeed occurs at the few atomic layers’ level. This completes our brief discussion of the physics of ferromagnetism in ultrathin ferromagnetic films. We see that we encounter magnetism that differs qualitatively from that found in bulk magnetic matter. Furthermore, since the key property of anisotropy is controlled by atoms in surface or interface sites, one can fabricate ultrathin films with desired characteristics for device applications by ‘‘spin engineering’’, i.e. by varying the nature or crystal face on which the film is grown, by adsorbing material on the outer surface, and so on. In magnetic multilayers, nearby ultrathin films interact via weak exchange interactions mediated by the non-magnetic spacer layer which can be placed between them, as discussed in this volume by Stiles.


13

This also occurs between nanoclusters or single spin-polarized atoms on appropriate metal surfaces. The strength and sign of this coupling can also be engineered through the choice of spacer material and its thickness, as we see from his discussion. We now have a very rich spectrum of new materials in hand with properties found nowhere in nature. Our focus here has been on the purely magnetic properties and response characteristic of ultrathin structures. Their transport properties also are spectacular, as we see from the discussion of giant magnetoresistance (GMR) presented by A. Fert and his colleagues in this volume. Applications of GMR to computer technology have had revolutionary consequences in the past 15 years. While GMR-based read heads have played the dominant role to date, we now see the tunnel magnetoresistance (TMR) heads beginning to be developed and we envision further applications such as magnetic random access memory (MRAM) becoming widespread in the very near future. The underlying physics is described in this volume in the chapter just mentioned. Beyond GMR and TMR we look to the field of spintronics, where the prospect of controlling the efficiency of electron spin injection and detection processes at the interfaces of heterogeneous materials offers, in principle at least, the prospect of achieving infinite magnetoresistance ratios – see the chapter by M. Flatte and B. Jonker. In applications of nanomagnetic structures, one manipulates the magnetization of a specific nanoelement. For instance, in an MRAM, a zero or one is represented by the orientation of the magnetization in very small structure. The act of writing consists of orienting the magnetization in the desired direction. Thus, the physics of magnetization reversals in nanoscale structures emerges as a central topic. Patterned arrays of nanosized magnetic elements have been intensively studied in recent years, with new applications in mind. One encounters fascinating physics in these structures as well. We turn next to a discussion of this area.

3. MAGNETIC NANOSTRUCTURES: SPIN CONFIGURATIONS AND MAGNETIZATION REVERSAL Nanoscale magnetic elements (typically 1–100 nm lateral size) and arrays are objects of intense current interest, in large measure due to their immense potential for applications in nanotechnology. These applications include memory arrays (e.g. MRAM), high-density storage media (the so-called patterned media), hard disc read heads, logic devices or miniaturized field sensors [19,20]. Since the effective fields which control the magnetic switching process (both static and dynamic) can be substantially controlled via the size, shape or density of the elements within the array, it is possible to precisely tailor the switching process for specific applications (e.g. fast writing, fast reading). Thus, we have one more feature that may be exploited in contemporary spin engineering. In general, the equilibrium magnetic states and reversal mechanisms are strongly determined by the interplay of magnetic anisotropies with the dipole fields which depend on the physical size and shape of the element, as we saw in Section 2. As the lateral size approaches 1 nm, the change in electronic structure significantly influences all key parameters such as the exchange constant,

14


magnetic anisotropy field (e.g. magnetocrystalline, strain, surface, atomic step, or edge induced) and spin polarization (and hence effective magnetization) as we have already noted. On this scale, the magnetic switching is profoundly modified in comparison with that of macroscopic elements, giving rise to fundamentally new phenomena. While in large structures the switching process is often well defined and fully repeatable, upon decreasing the lateral size of the element, the effective energy barrier for inducing switching (e.g. that associated with wall depinning or domain nucleation) is reduced, to the point where thermal fluctuations can induce stochastic switching [21]. Thermal effects start to become important in a single particle of volume V, and effective anisotropy Keff when K eff V 25kB T [22]. More subtly, even in micron-size elements, it is possible for a very small region of spins to control the magnetic switching, e.g. through the depinning of domain walls [23]. In this case, thermal fluctuations can influence the entire macroscopic switching process. One of the key challenges for device applications such as MRAM is in achieving sufficient control of the effective dipole field strength (which generally increases the effective switching field as the element size is reduced) and its dispersion (within an array) for a given element size. Spin engineering in multilayer elements offers a powerful strategy for controlling the dipole contribution to the switching field: in the case of a double ferromagnetic layer structure, for example, the two ferromagnetic layers can partially compensate each other’s external dipole field so they reduce the switching field without increasing the element size. This approach is already used in MRAM elements: for example, artificial antiferromagnets (comprising two exchange coupled magnetic layers separated by a thin non-magnetic layer) are used to exchange bias the hard layer (see the chapter by A. Berkowitz in this volume for a discussion of exchange bias); a double-layer structure can also be used to replace the single free layer (which responds to the external field) to achieve more reproducible switching. There is also intense current interest for applications in achieving controllable sub-nanosecond switching for ultrafast high-density writing/reading of data and therefore the dynamical switching processes in nanostructures have become a huge research field in themselves. Rapid advances in lithographic (e.g. optical – including X-ray, focused ion beam and e-beam) techniques and pattern transfer already make it possible to create extremely well-defined structures on the nanometer scale. Combined with advances in novel magnetic measurement methods, it is now possible to study the magnetic reversal process of isolated nanostructures corresponding to 100–1000 atomic spins (e.g. spin-polarized scanning tunneling microscopy (STM), ferromagnetic resonance (FMR), nano-SQUID and magneto-optical techniques) and techniques are being developed to combine high spatial resolution with fast temporal resolution to study switching events (e.g. pump-probe optical techniques) with sub-nanosecond resolution. At the same time, advances in computational techniques make it possible to accurately model the static spin configurations and magnetic reversal processes of nanostructures up to the few microns’ scale, provided the relevant magnetic parameters are known. However, a key challenge is satisfying the need to know magnetic parameters accurately as the nanometer size is reached and to include a proper description of defects, strain and chemically induced effects. For this reason, there is an intense ongoing research effort in studying well-defined nanostructures


15

and comparing the experimentally observed magnetic switching behavior (both static and dynamic) with that predicted computationally. A large research effort is now being made to find the geometries that provide the simplest, fastest and most reproducible switching mechanism, which are essential prerequisites for device applications [24–28]. To control the magnetic switching precisely, one needs first to have a well-defined and reproducible remanent state (i.e. the magnetic state which develops on removing the applied field – ideally the single-domain state) and, second, the switching process itself must be simple and reproducible. In analogy with the ultrathin film limit, for a sufficiently small element – corresponding to typically a few nanometers’ lateral size in transition metals – exchange interactions give rise to a single-domain state, For larger structures, different geometries have been studied for this purpose, from simple circular discs [26,29] to more advanced needle-shaped elements [30]. In such topologically simple elements, the aim is to obtain a singledomain state and switching by coherent rotation of the magnetization [31]. However, due to the demagnetizing field induced by dipole interactions, the magnetization direction will almost always change close to the borders to form edge domains [25,29]. The magnetic configuration in these elements, therefore, is defined by the shape of the edges and is very sensitive to shape fluctuations and edge roughness. Indeed, the edge inhomogeneity in the magnetization will often dominate the magnetic switching properties; acting as nucleation sites for what is, in general, a complicated nucleation propagation switching process, where the magnetization passes through inhomogeneous intermediate states. The complex switching mechanism can lead to different remanent states depending upon the applied field history and hence influence the switching itself. One possible way to overcome these complications is to use circular elements. Circular structures have historically been of technological interest: for example, early computers used ferrite 3D rings as memory elements in a precursor to modern RAM. The perpendicular anisotropy circular disc manifests in particularly simple states: in addition to the uniform state (macrospin) stabilized in sufficient small discs, the so-called bubble bi-domain states appear for appropriate values of disc diameter and anisotropy strength [32]. In a disc with in-plane anisotropy on the other hand, one may realize the most intriguing fundamental state referred to as the vortex state, where the (in-plane) magnetization circulates in a closed loop. Vortices appear widely throughout physics and so the vortex state is of particular fundamental interest in magnetism. Magnetic force microscopy (MFM) has been used successfully to demonstrate the existence of such a state in discs with thicknesses much less than the diameters [33,34]. The magnetic flux is then closed in the element with the consequence that the edge roughness and edge domains play a minor role. The zero stray field in this state will also favor high-density storage for possible use in patterned storage media. A curious feature revealed by such experiments is that the sign of the magnetization singularity at the vortex core is uncorrelated with the vortex circulation. However, the vortex is only stable in discs for diameters typically above some 100 nm [26], depending on the thickness and the material, which limits the density achievable. Beneath this size, the exchange energy dominates and the vortex cannot form, while at very large diameters, a multidomain state again

16


occurs. Moreover, the vortex formation is complex and hard to control [26,28] in disc elements. As a solution to this problem, the vortex state can be made more stable if the highly energetic vortex core is removed by using the high-symmetry ring element [35], which has also been proposed for use in MRAM [19,36] and as a basis for nanoparticle sensors [20]. Apart from the improved stability, the vortex state in ring structures is also almost completely stray field free, whereas in discs, the magnetization in the vortex core points out of the plane of the film, creating a stray field. Circular elements represent only a sub-class – albeit an important one – of a veritable ‘zoo’ of magnetic elements that have been intensively studied using both magnetic imaging techniques such as MFM and computationally with the development of reliable micromagnetic codes. A key technological and fundamental issue is the effect of the dipolar coupling between elements on the reversal processes and switching fields. The challenge for studies of dipolar coupling is to define the element shape with sufficient precision to obtain meaningful results since any variation in edge definition or element shape can lead to a large variation in the interaction strength. This occurs even in the uniformly magnetized state since the dipole field at the edge varies according to the local geometry of the element boundary. In addition, the local spin configuration at the edge can change due to this variation, profoundly altering the resulting dipole field. The field due to the magnetostatic poles – equivalent to charges – created by the spin configuration in neighboring element (interelement magnetostatic coupling) contributes to the effective magnetostatic field. This term is negligible for large spacings, but becomes important for narrow spacings and so influences the switching field. With a view to using magnetic elements in applications, it is important to probe the minimum spacing where the elements can be switched independently (to obtain e.g. the maximum achievable storage density), while from a fundamental viewpoint, it is of particular interest to study the influence of domainwall coupling on distinct switching mechanisms [37]. In contrast to the interaction of single-domain particles and quasi-single-domain states in discs or squares, which have been widely investigated so far [38–41], in the case of rings, the reversal takes place by localized domain-wall processes and so the interaction between these welldefined domain walls governs the collective reversal process [42], and only manifests at small separations. A fundamental as well as practical issue of great current interest is that of the ultimate limits to spin reversal in a magnetic element. For a sufficiently small element, coherent switching of the macrospin (single-domain limit) is in principle achievable under conditions of precessional motion and so the switching time is determined by the applied field’s amplitude. In practice, precessional switching times appear to be limited (no matter how large the magnetic field pulse is) to an intrinsic time of the order of picoseconds, determined by thermal spin excitations [43], thus, significantly limiting the writing times in magnetic storage media. One intriguing possible solution to this problem has recently been proposed by Kimel et al. [44]. Whereas in a ferromagnetic material the Larmor precession frequency scales with the effective field amplitude Heff, in an antiferromagnet, this frequency scales as (HAHex)1/2 where Hex is the exchange field, HA is the effective anisotropy


17

field and HexbHA. Consequently, switching times are much faster in antiferromagnetic materials. Kimel demonstrated that the effective anisotropy in an antiferromagnetic insulator such as the orthorhombic ferrites could be rapidly changed using a fast (picosecond) light pulse, which locally heats the material and induces a near-901 orientation of the spins in appropriate material. If a ferromagnetic element were to be exchange coupled to such an antiferromagnet, it can be expected that the ferromagnet can be made to precess spontaneously in the modified field created by the reoriented antiferromagnet, so offering a new route to rapid switching in magnetic elements.

4. EXPERIMENTAL TECHNIQUES Measurements of the magnetic properties of ultrathin structures are highly demanding in view of the small amount of magnetic material involved. A monolayer (ML) of Fe in a sample of area 1 cm2 has a total magnetic moment of less than 105 emu, for example, which corresponds to a signal close to the detection limit of a conventional vibrating sample magnetometer. It is therefore necessary to find methods that can probe such small signals routinely. The main challenge in using conventional magnetometric techniques in UHV for in situ measurements is the difficulty of meeting the sample–environment requirements (in particular, the ability to anneal the sample for cleaning single-crystal substrates). SQUID systems designed for the measurement of the magnetic moment of thin films can achieve sensitivities and accuracies that rival any other magnetometric technique, as exemplified by the pioneering work of Gradmann and coworkers [45] using a UHVcompatible SQUID. The advantage of in situ SQUID measurements for the case of overlayers grown on a solid substrate is that the relative bulk and magnetic thin film contributions to the total measured moment can be estimated by making thicknessdependent measurements using the same substrate. Wernsdorfer and coworkers [46,47] use micro-SQUID junctions to detect the switching fields of individual magnetic nanoparticles and have achieved sensitivities corresponding to 100–1000 spins. It is only relatively recently that the extraordinary sensitivity of the magnetooptic Kerr effect (MOKE) in probing the magnetic properties of magnetic films has been fully appreciated and now its use is widespread in the study of thin and ultrathin films [48–52]. A particular advantage of surface MOKE, pioneered by Bader and colleagues at Argonne [48], is that a polarized light beam can be readily passed through the windows of a UHV chamber and the reflected beam can also be easily analyzed using an external polarizer. Thus, in contrast with the relative complexity of polarized electron beam techniques, the MOKE can be readily adapted to the sensitive probing of the magnetic properties of ultrathin (typically 1–30 monolayer thickness for the case of Fe) films in situ. In such studies, the saturation and coercive fields can be accurately obtained as a function of the applied field orientation but the loop amplitude is frequently not analyzed. Many of these studies are made at fixed wavelength using a laser source, although a number of groups have now carried out spectroscopic measurements in which the Kerr

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rotation and ellipticity are studied as a function of wavelength. It is possible to obtain high-quality measurements using extremely simple arrangements. The most straightforward of these is based on DC detection using a high-quality photodiode and an intensity-stabilized laser source. Either the Kerr rotation can be measured by setting an analyzing polarizer close to extinction or the ellipticity can be measured by inserting a quarter-wave plate between the sample and analyzer. A great number of important insights into the origin of magnetism in ultrathin films have been obtained from in situ magnetic measurements using MOKE. Here we will briefly discuss just two illustrative examples of the use of MOKE to demonstrate how surface chemical effects can profoundly modify magnetic properties. A striking time-dependent evolution of the magnetic anisotropy of the Co/ Cu(1 1 0) system was first reported by Buckley et al. [53] using in situ MOKE. For a given deposition of Co in the thickness range 5–40 monolayers (ML), the adsorption of residual CO gas was found to cause the magnetic easy axis to switch 901 from the [0 0 1] direction to the [1–10] direction over a period of time which is dependent on the thickness of the initial Co film. This experimental study reinforces the arguments made above, where it was argued that in ultrathin materials where a large fraction of the atoms reside in surface and interface sites, the key property of anisotropy is subject to manipulation or modification through changes in the local site symmetry. We thus describe this study in some detail here to reinforce the point. Dosing experiments reveal that only a small fraction of an ML of CO is required in order to switch the easy axis of a 15ML Co film. The effect is chemically specific with oxygen, argon and hydrogen being unable to switch the easy axis. It was found that a 1 ML Cu overlayer deposited before the CO induced easy axis switch will stop the switch from occurring. This is thought to be the result of the Cu atoms occupying the specific adsorption sites preferred by the CO molecules. In support of this view, STM measurements reveal that the 3D Co growth proceeds via the formation of elongated Co island structures preferentially oriented along the [0 0 1] direction. This creates CO additional adsorption sites at the island edges. It is therefore likely that the effects seen are associated with CO preferentially adsorbing/desorbing at these specific sites. The question then arises as to whether or not it is possible to reverse the CO gas-induced 901 easy axis switch with Cu overlayers. It is found that by depositing sub-monolayer coverages of Cu onto the Co/Cu(1 1 0) system after the easy axis has been switched 901 from the [0 0 1] to the [1–10] direction by the residual CO in the UHV chamber, it is possible to fully reverse the effect of the CO, thereby switching the easy axis back to the [0 0 1] direction. Recent X-ray magnetic circular dichroism (XMCD) results [54] indicate that the effect of CO chemisorption on continuous ultrathin Co(1 1 1) films with perpendicular anisotropy is to suppress the Co orbital moment (but not the spin moment) within a restricted range of Co thickness. This results from a change in the Co d band filling due to electron back donation with the CO molecule and so is specific to CO. The effect of a change in the orbital moment is to profoundly modify (in this case to switch off) the surface anisotropy field so causing the abrupt switch in spin orientation. It is reasonable to infer that in the case of the Co island structures grown on Cu(1 1 0), a similar process is likely to be occurring at the edge of the islands,


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with the CO effectively switching off the edge-induced anisotropy. It should be recognized that the changes in effective field associated with the adsorption of CO are substantial since they compete with the dipolar fields. Electronic structure calculations [55] for CO/Co and CO/Ni [56] indicate that CO also induces a reduction in the Co or Ni moment according to the specific absorption site, but complete suppression of the spin moment is not predicted in agreement with the XMCD studies. These remarkable results illustrate the atomic-scale origin of magnetic properties in nanostructures and the striking results which can accompany subtle chemical changes which are not only site specific but involve a very small number of adsorption sites (as is clear from the low coverages involved). Such site and chemically selective effects are unique to the magnetism of nanoscale structures. Strain effects are also of great importance in the magnetism of nanostructures and chemisorption can have spectacular consequences for the magnetic anisotropy via the adsorbate-induced change in strain. Sander et al. [57] found a reversible switching of the easy axis of magnetization for Ni on Cu(0 0 1) from in-plane to outof-plane using MOKE by changing the partial pressure of hydrogen in the gas phase around the sample. A quantitative low-energy electron diffraction study of the diffracted intensity versus electron energy shows that the hydrogen-induced spin reorientation transition is accompanied by changes of the tetragonal distortion of the topmost Ni layer upon hydrogen adsorption. Surprisingly, the orientation switch perpendicular to the surface comes with a relaxation, i.e. reduction of the film’s tetragonal distortion rather than its amplification i.e. the clean Ni surface shows a strong inward relaxation, whereas upon H-coverage, the layer relaxes outward. Correspondingly, one has a large surface anisotropy in the first case, with magnetization in-plane, and a small surface anisotropy in the second case, with the magnetization out-of-plane. This behavior suggests that a structurally driven unquenching and quenching of the orbital moment occurs. This structural change is limited mainly to the top layer, and the average film structure does not change significantly. In support of this view, a surface X-ray diffraction study of Fe–Ni–W(1 1 0) did not detect any sizeable change of the average film structure upon Fe coverage and SRT from in-plane to out-of-plane [58]. We have here the second most interesting example of the chemical modification of the basic magnetic properties of ultrathin structures. In bulk magnetism, of course, the basic interactions are frozen into the crystal structure, and cannot be manipulated in this manner. MOKE techniques are also compatible with ultrafast magnetic switching measurements since the time resolution is determined by the response of the photodetector. This approach has been widely used in ultrafast (nanosecond and femtosecond range) measurements of the time-dependent reversal process in ferromagnetic elements as described above. Several groups have developed optical pump probe methods that are used in conjunction with strip lines, which generate RF field pulses at the sample position. The primary probe is used to close a photoconductive switch, which allows the field pulse to travel to the vicinity of the sample. A secondary pulse is then used to probe the magneto-optical response of the sample as a function of the time delay with respect to the initial applied field pulse. Alternately, a signal generator is used to generate the field pulse and the response is again

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monitored optically. The sample can also be placed in an external field and in this way FMR can be observed. In studies of the dynamical reversal processes, the separate vectorial components of the magnetization are directly measured in response to a finite-duration field pulse. In such measurements the sample is either combined with the strip line structure or the strip line is attached to a glass substrate which is then placed over the sample and light allowed to pass through the substrate to probe the magneto-optical signal. By either scanning the surface focusing position or using imaging optics in combination with a video camera and signal modulation, it is also possible to create movies of the spatially resolved magnetization across the sample with ultrafast time resolution. Such approaches have been used to investigate inhomogeneous modes of magnetic reversal in ferromagnetic elements and to compare the experimental results with the results of dynamic simulations. For recent reviews, see Choi and Freeman [59] and Hicken et al. [60]. RF techniques using FMR [61] and Brillouin light scattering (BLS) [62,63] have provided an important role in the quantitative determination of magnetic anisotropies and in the study of dynamical properties. In FMR the uniform mode of precession (gigahertz range) – the zero wavevector (infinite wave length) mode corresponding to the Larmor precession of the total sample moment in the effective field of the sample – is induced by an RF field applied in combination with an orthogonal static field of a strength sufficient to saturate the sample. The sample is usually placed in a resonant cavity connected via a waveguide to a Klystron source and the RF transmission strength measured as a function of the applied field strength (fixed-frequency FMR) or at fixed field as a function of frequency (swept field FMR). The resonant frequency observed at the transmission minimum is a direct measure of the effective local field (made up of the external field, dipole, effective anisotropy and exchange contributions) and by varying the applied field strength and orientation, the effective anisotropy field can be obtained directly. The line width provides a direct measure of a, the Gilbert damping constant. This parameter, which controls the strength of the damping term in the classic Landau–Lifshitz–Gilbert equation of motion for the magnetization (reference) enters importantly in the design of devices which exploit the dynamic response of the magnetization. However, at present it is largely treated as an empirical parameter with virtually no ab initio calculations available. In exchange-coupled trilayer structures two modes arise corresponding to the in-phase acoustic and anti-phase optical coupled uniform modes of the individual ferromagnetic layers. The optical mode provides a direct and accurate measure of the interlayer coupling strength [62]. Extremely high sensitivity can be achieved with single ML detection routine for samples of around 1 cm2 area. The technique has been ingeniously adapted to allow high-sensitivity spatially resolved measurements [64] by using frequency modulation techniques to probe the FMR signal via the mechanical resonance of a cantilever as in MFM. BLS uses a tandem Fabry–Perot interferometer to measure the inelastic light spectrum with very high contrast extremely close to the elastic beam. Typically, the free spectral range can be varied in the range 1–100 GHz making it ideal for the investigation of spin wave modes of thin film structures (both exchange type and dipole dominated). By focusing the incident beam it is possible to achieve high spatial


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resolution (around 1 mm) also. The use of a tandem Fabry–Perot interferometer makes extremely good rejection possible so that typically count rates of 0.01 Hz can be achieved within less than 1 GHz of the elastic beam. The technique can be readily combined with a UHV MBE chamber to allow in situ measurements during film growth. A number of groups have adopted this approach that has been shown to provide a powerful probe of the evolving dynamical properties [62,63]. Mode frequencies are identified from the increased photon energy gain or loss at a specific frequency 7o, with the intensity ratio termed the Stokes/anti-Stokes ratio. The inplane wavevector is varied by changing the angle of incidence and using a backscattering geometry so that ky ¼ 2k sin y, where y is the incidence angle and k the wavevector of the incident light. Owing to the relatively long wavelength of the Ar ion laser radiation typically used in BLS, the wavevector can only be varied up to around 105 cm1. However, this wavevector range is sufficient to observe dispersion effects, which characterize the modes and is particularly important for the study of nanostructures as we have already seen. In addition to the so-called Damon Eshbach modes localized at the film surfaces, the so-called exchange modes can be observed in which the film thickness corresponds to one or more half wavelengths of the spin wave spanning the depth of the film. In this case the exchange contributes to the effective field as in FMR and the spin wave amplitude varies across the thickness of the film. The mode frequency depends on the surface boundary conditions (usually the surface is assumed to correspond to a pure anti-node). Each exchange mode gives rise to a manifold of modes according to the value of the in-plane wavevector associated with it. However, the precise boundary conditions appropriate for nonellipsoidal nanostructures remain an open question [65]. The uniform mode of a single-domain magnetic element or particle [66] can be observed by both FMR and BLS. For example, the mode spectrum of single-crystal Fe nanoclusters have been studied using BLS [67] measurements during film growth. With increasing effective thickness, ferromagnetic (FM) order develops from the low-coverage superparamagnetic (SP) state. In the FM regime the spin wave peak line width broadens very close to dc, clearly indicating the influence of spin fluctuations in the vicinity of the phase transition. In the SP regime, where the Fe forms well-separated clusters, collective spin wave modes are still observed but with a frequency close to that of the uniform cluster mode oU By comparing the mode frequency calculated using the demagnetizing fields estimated for the average cluster dimensions determined from STM with that observed experimentally, it was possible to directly attribute the measured frequency in the SP phase to the uniform precessional modes of the clusters. It is found that the intercluster dipole coupling influences the propagation of the modes giving rise to phase coherence between the clusters (as evidenced by the observation of the Stokes/anti-Stokes ratio), but does not significantly change the measured frequency in this case. BLS techniques have been successfully used to study the way in which the spin excitations present in a thin film are modified in a magnetic element [68]. Mathieu et al. [69] were the first to report observations of discrete standing spin wave modes in laterally confined structures by this technique. The authors investigate micronsize wires and observe these modes when the transferred wave vectors are small (as

22


compared to the wire dimensions) and oriented perpendicular to the wires. Such confined modes correspond to a hierarchy of dynamic states of increasing energy in analogy with the particle in a box mode found in quantum mechanics; in the case of spin waves the boundary conditions at the element edge are crucial in determining the mode structure (these again depend critically on e.g. edge anisotropies) Jorzick et al. [70] also investigated micron-size long stripes and rectangular elements with BLS. Due to the shape of these magnetic structures, the internal magnetic field is inhomogeneous (in contrast to the case of elliptical or circular dots) near the edges and there are even regions where H ¼ 0. These inhomogeneities create a potential well for propagating spin waves and consequently, modes were observed that are strongly spatially localized in these areas. A more general theoretical description of these modes has been given by Yu et al. [71]. Jorzick et al. [72] investigated micronsize circular magnetic dot arrays. Similar to the case of the wires, discrete standing spin wave modes are observed due to the in-plane confinement. In addition to that, a shift in spin wave frequency is observed for identical dot dimensions when the dot separation is varied. This finding is attributed to dipolar interdot coupling. The strength of this coupling is found to be stronger for modes with lower frequencies. Similar work on cylindrical dots has also been carried out in Gubbiotti’s group [73,74]. They investigate smaller dots (minimum radius 100 nm) than those investigated by Jorzick et al. but basically observe the same spin dynamic behavior as Jorzick et al., i.e. discrete laterally confined modes. As an alternative to BLS, Buess et al. [75] have successfully used time-resolved Kerr imaging to determine the modes of circular elements. Again, a hierarchy of modes is found, analogous to the vibrational modes of a drum, and excellent agreement between the observed and calculated spatial variations and frequencies is found. Extended antidot (i.e. hole) arrays offer the possibility of studying the effect of a periodic modulation of the local demagnetizing field on the transmission of extended spin wave modes. In the case of micron-scale square antidot arrays, for example, McPhail et al. [76] have shown that the long-wavelength (Damon Eshbach) singlefrequency mode found in a continuous film (derived from the uniform mode observed in FMR) is replaced by a broadened double mode according to the antidot size and wave propagation direction. Thus, we encounter particularly interesting mode structures in such patterned materials since the local effective field (which determines the mode frequency) varies according to the demagnetizing field spatial distribution, but the exchange field remains continuous. In contrast to the mode confinement seen in isolated elements, the two principal long-wavelength modes propagate almost independently across several repeats of the unit cell with distinct frequencies defined by the extremal regions of the demagnetizing field map. This behavior is believed to be a consequence of the specific size and separation of the antidots used in the experiment and, in principle, the degree of coupling between the principal modes can be expected to be modified according to the anti-dot dimensions. Demokritov et al. [77] described how spin waves propagate through a region of an inhomogeneous magnetic field. Spin waves are excited in a magnetized YIG film using a strip-line antenna and their propagation observed using time- and space-resolved BLS. Using a thin conductor mounted across the film, they create a


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local inhomogeneity of the magnetic field. If there are allowed spin wave states (defined by the frequency of the incoming wave and the dispersion relation) within the region of the inhomogeneous field, spin waves are transmitted through it and only change their k-vector depending on the local field. However, if there are no allowed states, the inhomogeneity acts as a tunnel barrier for spin waves. As in the case of quantum-mechanical tunneling of a small particle, the wave is partially reflected and transmitted (by tunneling), with the coefficients for each process depending on the tunnel barrier height and width. From the dependence of the transmission coefficient on the barrier width (which is non-exponential), the authors conclude that the spin waves are dominated by the long-range dipole–dipole interaction. SPEELS is capable of determining the entire surface spin wave dispersion throughout the Brillouin zone and has been successfully applied to the study of ultrathin fcc Co thin films [12]. However, in the case of Fe films, less well-defined modes are observed, possibly due to the surface modes entering the Stoner continuum. Several powerful high-sensitivity imaging techniques have been developed or advanced recently: for example, Atomic force microscopy (AFM) and MFM [78], scanning tunneling microscopy (STM) [79], spin-polarized STM [80], scanning Kerr microscopy, scanning electron microscopy with polarisation analysis (SEMPA) [81], photoelectron emission microscopy (PEEM) [82] and Lorentz microscopy [83]. The imaging of surfaces and of the magnetic configurations have provided valuable information about the surface morphology and the possible magnetic states either at remanence or during the switching of the magnetization process, respectively. This has led to a better understanding of both static and dynamic processes in magnetic films and elements. Probably, the most exhaustive review of magnetic domain imaging to date is that by Hubert and Scha¨fer [84]. Using X-ray optics, high resolution (around 100 nm) can be achieved with XMCD for studying magnetic domains with element specificity and the technique is unique in its ability to probe antiferromagnetic (AF) domains [85]. While there is insufficient space to describe all of these imaging methods in detail, we shall briefly discuss spin-polarized STM which is already making a profound impact on the field having taken almost 20 years to successfully develop. A successful approach to spin STM has been demonstrated by the Hamburg group and used for several studies of magnetic films and surfaces [21]. In their approach, a clean tungsten tip is coated in UHV with an ultrathin ferromagnetic film (e.g. Fe). The spin polarization of the tunneling current is then determined by the magnetic orientation of the magnetic atom or atoms at the tip. The method has also been shown to be compatible with magnetic field-dependent measurements [86]. Using I–V measurements, it is possible to correlate measurements of the electronic structure with the magnetic and physical information provided by STM [21,87]. When a spin split surface state is present, this provides a powerful means of distinguishing surface polarization effects from those of the bulk. Ravlic et al. [88] have described the correlation of structural, local, electronic and magnetic properties of Fe/Cr(0 0 1) using this method. Kawagoe et al. [89] have also used this approach to probe topological antiferromagnetic order on ultrathin Cr(0 0 1) film surfaces. One of the surprising results to have been obtained from the spin STM


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studies is the finding that in ultrathin Fe/W(1 1 0) films the domain walls are extremely narrow, indicating a very substantially reduced surface exchange parameter [21]. The Halle group [90] has developed an alternative approach to spin STM using a magnetically soft ferromagnetic tunneling tip. The tip magnetization can then be modulated at high frequency (which exceeds the bandwidth of the feedback system for the STM stabilization) using a miniature Helmholtz coil so that spin information can be probed using the AC contribution to the tunneling signal. This method has been applied to the study of magnetic domains in ultrathin Co films and again it is found that extremely narrow (2 nm) domain walls arise. Magnetoresistance measurement techniques have also been very effectively applied to the study of reversal processes in nanostructures and have the advantage over conventional techniques that the sensitivity does not decrease with the size of the sample (for a recent review on magnetoresistance, see [91]). Following the pioneering work of Berger [92,93] on current-induced magnetic switching, the huge research activity on magnetoresistance effects has led to a renewal of interest in the investigation of current-induced domain-wall motion [94–96]. This effect shows potential for novel memory and logic devices based on domain-wall propagation or magnetic reversal in nanoelements, as it could simplify designs by eliminating magnetic field-generating circuits as well as lowering the power requirements for switching. While field-induced domain-wall motion is well established, currentinduced domain-wall motion still lacks a thorough understanding. Several effects occur when large electrical currents flow across a domain wall, the most prominent ones being the action of the field created by the current itself (the so-called Oersted field) and the spin-transfer mechanism, also known as spin torque effect [97]. Since the original work by Berger, a number of different theories have been proposed that treat the interactions between the spin-polarized current and the magnetization, but the appropriate form of the spin-transfer contribution still is the subject of much debate. It is clear that this subject will continue to see an intense interplay between theory and experiment as our basic understanding of this fascinating effect develops – see the chapter by Mathon and Edwards which discusses fundamental models of current-induced switching.

NOTES 1. The final result in Ref. [16] for the two-dimensional ordering temperature should be multiplied by a factor of two, as pointed out in Ref. [17].

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Chapter 2 FUNDAMENTAL PROPERTIES OF MAGNETIC NANOSTRUCTURES: A SURVEY Ruqian Wu 1. INTRODUCTION New breakthroughs in materials often have great impact on technological progress and can influence economies or even societies (e.g., the Bronze to Iron Age). While recent improvements in magnetic materials have been central to the notable development of much smaller motors and more efficient generators, it is in the technology of magnetic-data storage where huge improvements in materials have revolutionized the industry. The giant magnetoresistance (GMR) read-head [1–3], for example, has increased the recording density of computer hard discs by a factor 4100 and non-volatile magnetic random access memories (MRAM) are starting to be utilized in computer and communication devices. There is currently widespread belief that much more progress can be made as materials and devices are developed to operate at the nanometer length scale [4]. Applications such as spin filtering, sensing, and ultra-high density magnetic and magneto-optic recording are a few of the items on the horizon. It is known however, that the magnetism of ultra-small entities is not stable due to super-paramagnetism. Thermal fluctuations, the major obstacle for further size reduction of magnetic devices [5], have become an urgent issue, and ways to enhance anisotropy and the pinning of magnetic moments need attention. Simultaneously, the new field of spintronics is progressing [6–9]; here the spin and charge of the electron, rather than its charge alone, are manipulated. Highly spin polarized diluted magnetic semiconductors (DMS) form seamless hybrid structures on base semiconductors with minimal interfacial scattering and may hence provide high thermal stability and high spin injection rate. It is of great interest nowadays to make direct electrical or optical switching of magnetization in DMS for further integration of logic, display, and storage functions [10–16]. Nevertheless, the exploitation of DMS at ambient temperature requires better Contemporary Concepts of Condensed Matter Science Nanomagnetism: ultrathin films, multilayers and nanostructures Copyright r 2006 by Elsevier B.V. All rights of reproduction in any form reserved ISSN: 1572-0934/doi:10.1016/S1572-0934(05)01002-4

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understanding in the mechanism of ferromagnetism and growth control. Furthermore, much needs to be done to understand new functional oxides and alloys for the exploitation of their exceptional properties. The application of innovative experimental tools and the desire to grow nanostructures and novel materials in a controlled manner have created an urgent need for highly reliable, robust, and predictive theoretical models and tools to achieve a quantitative understanding of their growth dynamics and the size, shape, and process dependences of their physical properties. Much of the basis of theoretical tools is founded on density-functional theory (DFT) [17] and related new methods being developed specifically for magnetic materials. It has been increasingly recognized in many fields of materials science that state-of-the-art ab initio electronic structure calculations based on the density-functional theory have been enormously successful, in both explaining existing phenomena and, more importantly, in predicting the properties of new systems. For example, the prediction of enhanced magnetic moments with lowered coordination number at clean metal surfaces and interfaces [18,19] has stimulated both theoretical and experimental investigations for new magnetic systems and phenomena in man-made nanostructures. Synergistic applications of theory and experiment, as have been demonstrated repeatedly in many areas of materials science, become a ‘‘must’’ to further advance our microscopic understanding in nanomagnetism. Although magnetism is probably one of the oldest branches of solid-state physics, studies of nanomagnetism are extremely vigorous and most problems are still very challenging. In nano- and subnano-structures, a large fraction of atoms are exposed to vacuum, which introduces new physics of a fundamental nature. For instance, the orbital magnetic moments can be larger than the spin counterpart. ‘‘Nonmagnetic’’ elements exhibit strong spin polarization, which, combined with their strong spin–orbit-coupling strengths, may enhance the magnetocrystalline anisotropy energy to an unanticipated level. Complex non-collinear magnetic structures are usually formed to compensate the exchange, magnetic anisotropy, and magnetostatic energies. The quantized magnetic transport through ultra-small entities offers new opportunities for fundamental studies and exploitations in new spintronics devices as well. Here we discuss these important magnetic quantities starting from the very basic concepts.

2. ENHANCEMENT OF MAGNETIZATION Magnetic materials are those in which atoms display spin polarization at finite temperature, measured in terms of spin and orbital magnetic moments. It is known that the magnetization stems from the alignment of electronic spins. As stated in Hund’s first rule, most free atoms adopt electronic configurations that maximize the spin moment, oS4, so as to gain the exchange energies. Nevertheless, very few solid materials in nature are magnetic, primarily because of the effects of crystalline fields along with the orbital hybridizations. The Stoner criterion, In(EF)41, is widely used to identify the instability toward spin polarization in itinerant magnetic

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systems. Here I and n(EF) denote the repulsive interaction between electrons with opposite spins and the value of density of states (DOS) at the Fermi level in the nonmagnetic state, respectively. While the value of I is rather stable [20] for a given element, n(EF) typically increases in nanostructures where the numbers of the nearest neighbors are reduced. Consequently, the Stoner criterion is satisfied in many small systems, even though the constituents are traditionally ‘‘nonmagnetic’’. The band narrowing triggers ‘‘revival’’ of magnetization of 4d and 5d elements and strong enhancement in magnetization of 3d elements in reduced dimensions. An additional contribution to magnetization is the orbital magnetic moment, oLz4, that stems from the relativistic spin–orbit coupling (SOC). To the order (v/c)2, the SOC Hamiltonian is expressed as * * _2 @V * * LS ¼ xLS (1) 2 2 2m c r@r In magnetic solids such as the bulk Fe, Co and Ni, HSOC is much weaker than the crystal fields and thus their orbital magnetic moments are mostly quenched (o 0.1 mB). Intriguingly, the strength of HSOC becomes comparable to the bandwidth in nanostructures, especially when magnetized heavy elements are involved. This not only produces large oLz4, but also leads to formation of peculiar magnetic structures. Nowadays, both spin and orbital magnetic moments can be accurately determined through modern DFT calculations by solving the Kohn–Sham equation 1 2 2r þ V ext þ V c þ V ex c ¼ c (2)

H SOC ¼

Here, Vext and Vc are external potential and the Coulomb potential among electrons. Vex is the exchange-correlation potential that takes into account all the complexities of many-body interactions. The local spin density approximation (LSDA) and the more advanced generalized gradient approximation are adopted in practical calculations for Vex. The HSOC term is usually neglected to reduce the computational costs by treating two spin channels separately. Nevertheless, it becomes essential to invoke HSOC self-consistently for studies of magnetic properties of nanostructures. To demonstrate the concepts elaborated above, the spin and orbital magnetic moments for selected elements are presented in Fig. 1, obtained through all-electron full potential linearized augmented plane wave (FLAPW) calculations [21–23]. Enhancements of both spin and orbital magnetic moments in low-dimensional systems are obvious. Similar data have been reported in the literature for free and supported 3d, 4d, and 5d monolayers [24–30] and more recently for nanochains and wires. It is interesting to note that the spin magnetic moments of 4d and 5d elements follow those of their 3d counterparts well. In particular, the orbital magnetic moments can be larger than the spin counterparts in Ir and Pt wires. Nevertheless, magnetization of 4d and 5d atoms is very sensitive to the change in environments because of the large spatial extension of 4d and 5d wave functions. Strong hybridization occurs even with traditionally ‘‘inert’’ substrates such as C(0 0 0 1), on which most 4d and 5d elements are magnetically ‘‘dead’’ except Ru and Rh [31]. It was predicted that

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Fig. 1. The calculated spin and orbital magnetic moments of selected 3d, 4d, and 5d elements in their bulks, monolayers, and monatomic wires with an interatomic distance of 5.4 a.u.

Ru and Rh monolayers may possess large spin magnetic moments on MgO(0 0 1) (1.95 and 1.21mB) [32]. The strong magnetic enhancement predicted in DFT calculations has been examined extensively, using various experimental techniques [33,34]. While the magnetic enhancements of 3d films are mostly verified, corroboration for 4d and 5d magnetism is still controversial mostly due to the experimental impossibility to grow a perfectly uniform film. Using spin polarized secondary electron spectroscopy, Pfandzelter et al. [35] found an evidence for 4d ferromagnetism in Ru/C(0 0 0 1). In contrast, Chado et al. reported the absence of ferromagnetic order in ultrathin Rh deposited on gold [36]. A negative conclusion was also drawn by Beckmann et al. [37], using the anomalous Hall effect and weak localization, but they found a single Ru atom on Au or Ag surfaces possess a small fluctuating magnetic moments of 0.4mB.


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Nevertheless, sizeable magnetic moments have been observed in 4d clusters [38,39], for which the magnetic properties are easy to tailor through tunable parameters such as cluster size, alloying elements, geometry, and surroundings. Transition metal clusters, either in the freestanding form or on inert substrates, have been examined through DFT calculations from 1980s [40–46]. For example, Reddy et al. [47] found Rh13 has a magnetic moment as large as 21mB. Intermixing of small 4d and 5d clusters on Ag(0 0 1) with the substrate atoms can lead to an unexpected enhancement of the local moments [48]. Experimentally, Taniyama et al. [49] conducted dc magnetic measurements as a function of Pd particle size show that the magnetization increases rapidly with decreasing particle size. Clear evidence of the ferromagnetism of gas-evaporated Pd fine particles with a clean surface was also reported by Shinohara et al. [50]; they estimated that the magnetic moment of surface Pd atoms is 0.7570.31mB/atom. Using a molecular beam deflection technique, Cox et al. [51] found high magnetic moments, as high as 0.8mB per atom, in Rh969 clusters. More recently, very small thiol-capped gold clusters were also found to exhibit a localized permanent magnetism, in contrast to the metallic diamagnetism characteristic of bulk Au [52], primarily because of the creation of 5d-holes around the surface Au atoms through charge transfer to the capping molecules. Peculiar magnetic properties have also been found in clusters of 3d elements. Billas et al. found that for cobalt and nickel, convergence toward the bulk is fast while for iron clusters the magnetization stays well above the bulk value up to cluster sizes of several hundred atoms [53]. For Nin clusters (n ¼ 5–740), Apsel et al. [54] found magnetization minima for clusters with closed geometrical shells and maxima for relatively open clusters. Large orbital magnetic moments are also predicted for many clusters, including Ni [55]. One of the most powerful techniques for the quantitative determination of magnetic moments nowadays is soft X-ray magnetic circular dichroism (XMCD) [56]. Specially, XMCD provides element selectivity and resolves the spin and orbit magnetic moments separately through the powerful sum rules [57,58]. For transitions from p3/2 (the L3 edge) and p1/2 (the L2 edge) core levels, the sum rule can be stated as R R hLz i L3 sm ðEÞdE þ L2 sm ðEÞdE R R ¼ 2N h L3 st ðEÞdE þ L2 st ðEÞ dE R R (3) oS z 4 þ 7hT z i L3 sm ðEÞdE 2 L2 sm ðEÞdE R R ¼ 3N h L3 st ðEÞdE þ L2 st ðEÞdE Here sm and st are the difference and sum of the absorption spectra for the left- and right-circularly polarized soft X-ray beams. Tz is the magnetic dipole operator (Tz ¼ S (13cos2 y)/2). The number of valence holes, Nh, is usually obtained from integration over the unoccupied DOS from DFT calculations. The validity and applicability of these sum rules have been carefully examined [59,60], including the effects of technical aspects [61]. XMCD was applied to many low dimension magnetic systems and many interesting results have been obtained. For example, small

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magnetic moments on Cu atoms induced through contact with Fe or Co layers were uniquely measured through XMCD [62,63]. In addition, XMCD also allows exploration of element-specific hysteresis and complex magnetic ordering in various nanostructures [64]. More recently, XMCD was applied to detect smallinduced magnetization on semiconductor sites in DMS [65]. Our recent studies found that it is possible to furthermore use XMCD for the identification of the distribution of magnetic impurities in DMS [66]. Depth-resolved XMCD techniques were also developed recently to observe the magnetic coupling profiles in thin films [67]. Spin-polarized scanning tunneling microscope (SP-STM) is another powerful tool for studies of nanomagnetism, especially for properties in the ultra small scale down to a single atom [68–71]. Even non-spin polarized tips can detect magnetic signals: the presence of spin–split electronic states through scanning tunneling spectroscopy (STS) of electronic resonances [72]. In STS, the differential conductance (dI/dV) spectrum gives a measure of the local density of states (LDOS), which can be directly compared to results of DFT calculations. For 3d transition monomers on NiAl(1 1 0), for example, both DFT calculations and STS measurements reveal a pseudo gap for the bare NiAl(1 1 0) surface, which leads to a depletion zone in LDOS with a sudden onset at 2.5 eV. In the pseudo-band gap region, the sp-states of Mn, Fe, and Co monomers show remarkable resonances from which the exchange splittings can be extracted. The agreement in the spectroscopic features and exchange splittings (0.8, 0.5, and 0.2 eV for Mn, Fe, and Co respectively) given in Figs. 2a and b suggests the validity of DFT calculations for other

Fig. 2.

The calculated l-projected density of states and the measured dI/dV curves for Mn, Fe, and Co monomers on NiAl(1 1 0) (from Ref. [72]).


35

properties of such very small systems. The magnetic moments for Mn, Fe, and Co monomers on NiAl(1 1 0) are 4.4, 3.2, and 1.9mB, respectively.

3. MAGNETIC ANISOTROPY The magnetic anisotropy energy (MAE) determines the orientation of magnetization. In the bulk samples of cubic symmetry, the amplitude of MAE is merely a few or a few tenths meV/atom. In less than three dimensions, the MAE is strongly enhanced by a factor of 100–1000, with the leading term called uniaxial anisotropy (MAE ¼ Ksin2 y+K2sin4 y, where y is the polar angle away from the surface normal). For many thin films, K is positive and is sufficiently large to overcome the tendency of in-plane anisotropy caused by the magnetostatic interactions. Consequently, the easy axes are perpendicular to the film planes, a feature that is desired for the design of next generation high-density perpendicular magnetic recording media [73]. Studies on how to tailor MAE in bulk materials, alloys, and small entities through manipulating atomic structure, step edges, size, shape, and chemical environment are vigorous and fruitful [74–78]. Intriguingly, it was found recently [79] that bcc nickel, epitaxially grown on GaAs(0 0 1), has a positive MAE, opposite in sign from that in the natural fcc Ni. In principle, MAE is associated with two factors: the spin–orbit coupling (this contribution is called ‘‘magnetocrystalline anisotropy,’’ EMCA) and the magnetostatic dipole–dipole interaction (this contribution is called ‘‘shape anisotropy’’) [80–84]. The uniaxial MAE for magnetic films, K, can be determined through DFT calculations with satisfactory accuracy [82]; some of the calculations even gave atom-resolved contributions [85,86]. Fig. 3 illustrates theoretical and experimental results of MAEs for O/Ni/Cu(0 0 1) [87], plotted as a function of the inverse of the film thickness. Well-characterized experimental data allow unambiguous separation for contributions from different regions, namely, the y-intercept for bulk and the slope for surface/interface. It is interesting to note that the presence of O strongly

Fig. 3.

The experimental and theoretical results of magnetic anisotropy energies for Ni/Cu(0 0 1), Cu/Ni/Cu(0 0 1), and O/Ni/Cu(0 0 1).

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alters the surface anisotropy energy: contributions from the bulk Ni, the Cu/Ni and O/Ni interfaces are 27, 59, and 17 meV/atom, respectively. This leads to a shift of the spin-reorientation transition (SRT) from 10 Ni-MLs in the vacuum/Ni/Cu(0 0 1) to 5 Ni-MLs in O/Ni/Cu(0 0 1), as given by the intercept between K(1/d) line with the horizontal line of 2pM2 in Fig. 3a. The latter describes the shape anisotropy, which favors parallel alignment of the magnetization in the film. Similar phenomena have also been observed for Fe, Co, and Ni films covered by Cu, hydrogen or CO [88–91]. Theoretical calculations reproduced the trend of experimental data very well. Analyses in electronic structures furthermore attributed the O-induced change in EMCA to the new surface state with the dxz feature caused by the O adlayer. It is known that magnetism in wires or dots is unstable due to super-paramagnetism [92], driven by thermal fluctuation. How to restrain thermal fluctuations, the major obstacle for further bit reduction of both longitudinal and perpendicular recording media today [93], becomes an urgent issue for fundamental magnetism studies. Investigations of MAEs of magnetic clusters including molecular magnets are getting broad attention [94–97]. Using a new micro-Superconducting Quantum Interference Device (SQUID) setup, Jamet et al. studied magnetic anisotropy in a 3-nm cobalt cluster embedded in a niobium matrix and found the dominating role of the cluster surface [98,99]. Gambardella et al. reported observation of giant MAEs of Co particles grown on Pt(1 1 1) [100]. Ultra-small monatomic chains, the smallest possible magnetic recording units, have been grown on vicinal substrates [101] or built by STM tip manipulations [72,102]. If one can increases their blocking temperature, Tb, to room temperature, in principle one may realize a recording density up to 100–1000 (TB)/in2, a thousand times denser than the best hard drives that are available today. Monte Carlo N1 P* ! S i S iþ1 þ simulations based on a classical model Hamiltonian, H ¼ J i¼1 N P S 2i;z indicate that this requires stringent conditions, e.g., J ¼ 3202420 meV K i¼1

and K ¼ 30250 meV=atom: Such a large K is exceptionally large but is found possible through model calculations. As listed in Table 1, 3d–5d, trimers, especially FeOsFe and FeIrFe, have large EMCA up to 108 meV. Calculations for longer FemIrn chains confirmed that EMCA scales almost linearly with the length. These results Table 1. The calculated K, spin and orbital magnetic moments distributed in different atoms. The easy axis for positive (negative) MAE is along (perpendicular to) the chain. System and magnetic ordering Fe–Ta–Fe (m k m) Fe–W–Fe (m k m) Fe–Re–Fe (m m m) Fe–Os–Fe (m m m) Fe–Ir–Fe (m m m) Fe–Pt–Fe (m m m)

MS (mB) 3d

ML (mB) 5d

3d

5d

MAE (meV)

2.99 2.93 3.22 3.32 3.36 3.34

1.96 2.85 2.94 3.21 1.58 0.69

0.16 0.10 0.03 0.60 0.00 0.20

0.30 0.13 0.04 0.57 1.53 0.34

20.5 30.8 31.2 75.0 108.5 26.6


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indicate that FemIrn chains are promising for exploitation in magnetic recording and spintronics devices. To this end, more theoretical and experimental investigations for magnetic chains and wires should be conducted in the near future [103–107].

4. MAGNETIC ORDERING The switch from antiferromagnetic (AFM) to ferromagnetic (FM) states in the spinvalve structures, which serve as the cores in GMR reading heads, has been studied thoroughly in the past decade [108]. Complex in-plane magnetic orderings and peculiar surface alloys have been investigated both theoretically and experimentally. Because of their exotic structural and magnetic properties, Mn, Cr, and fcc Fe are involved in most such systems. Mn and Cr have complex AFM magnetic phases in their bulks and show a strong tendency to adopt AFM ordering. For example, Mn takes on the c(2 2) magnetic configuration for the Mn monolayer on Fe(0 0 1) and Co(0 0 1) [109–112]. However, Mn and Cu forms a order c(2 2) FM surface alloy [113]. Mn trimers on NiAl(1 1 0), Mn512 clusters are also found to be ferromagnets through DFT and experimental studies [114–116]. The discovery of FM ordering in Mn-doped semiconductors has stimulated many endeavors to explore innovative spintronics materials for simultaneous exploitation of electron charge and spin. The d-doped DMS with doping profile approximately a d-function along the growth direction [117,118], circumvent complexities in 3D random doping cases and hence provide ideal model systems for comprehensive theoretical examinations. We investigated magnetic switching between the FM and AFM phases when the external carrier density r is manipulated for d-doped (Ga,Mn)As [119]. As shown in Fig. 4, a strong FM ground state can be found for thin GaAs spacers (d ¼ 5.56 and 11.31 A˚). The large magnitude of DEFMAFM, e.g., 35 meV/cell even for d ¼ 11.31 A˚, demonstrates the effectiveness of semiconductor spacers to mediate the exchange interaction between magnetic impurities. The carrier induced AFM-FM phase transition occurs for thick spacers. When, d ¼ 16.96 A˚, for example, the AFM configuration is the ground state without external carriers, and is further stabilized by electron doping. In contrast, the FM state gradually prevails in the hole doping side, especially when r41.0 1020 e/a.u.3 This result is in good accordance with experimental observations made by Ohno et al. [13,16] who found that both magnetization and coercive force of (Ga,Mn)As and (In,Mn)As can be manipulated by applying an electric field in the gated structure that subsequently change the effective carrier (hole) density in the DMS. This electrical control of the magnetization, previously inaccessible in other magnetic materials, opens up new opportunities for novel device applications. In reduced sizes, some interesting non-collinear magnetic patterns can be formed in order to ease the demagnetization energy [120]. For example, FM nanodots typically form curling magnetic structures [121–123]. The magnetization close to the center of the dot (Bloch point) turns up along the perpendicular orientation to the curling plane, as displayed in Fig. 5. Experimentally, the vortex core was inferred by magnetic force microscopy (MFM) in which the magnetization near the center

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Fig. 4. DEFMAFM (in meV per unit cell) versus the external carrier density r (in 1020 e/a.u.3) for the d-doped (Ga,Ms)As. The open symbols represent uniform carrier distributions, while the asterisks and crosses represent non-uniform carrier distributions.

Fig. 5.

In-plane and out-of-plan components of magnetization and the calculated magnetic patterns of Fe nanodots.


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Fig. 6. Direction of magnetization in the free Fe(1 1 0) monolayer and the bulk bcc Fe through non-collinear DFT calculations by Nakamura et al. The inset displays the domain structure of Fe/W(1 1 0) observed by Elmer et al.

assumes a perpendicular orientation, and the core structures were observed in a 4 or 5 nm in radius by SP-STM. Through DFT calculations for rod geometry, Nakamura et al. [124] reproduced the curling configuration and found that the formation energy of the vortex core, Evc, is 3 meV/atom. Interest in magnetic domain walls (DW) has been greatly increased in both basic and applied research – including the DW with a few nm widths in quantum spin interfaces and the role of the DW in the exchange bias and nanoscale geometrical structures. The DW, is a transition region within which the magnetization orientation changes from one easy axis to another, is known to be determined by a competition between the exchange energy and the MAE – as revealed in phenomenological continuum micromagnetic calculations. The exchange energy tends to favor a slow variation of the magnetization while the anisotropy energy favors a rapid change from one easy axis to another, which leads to a stable DW width of the order of 10–100 nm in the bulk. Pratzer et al. [125] studied Fe stripes on stepped W(1 1 0) and found extremely narrow DW, as thin as 6 A˚. Through the FLAPW calculations for a free Fe(1 1 0) monolayer with the freedom of non-collinear magnetization, Nakamura et al. [126] found that the moments rotate rapidly within approximately three lattice constants without any discontinuous changes, as shown in Fig. 6. The domain wall is only 6–8 A˚ thin in the simulations. The other important feature from STM observation is that domain walls in Fe strips are oriented along a certain crystallographic direction, regardless of the orientation of the wires. Monte Carlo simulations on a discrete lattice [127] found that the wall orientation is mainly determined by the atomic lattice and the resulting strength of an effective exchange interaction. The magnetic anisotropy and the magnetostatic energy play a minor role for the wall orientation in that system.

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5. MAGNETIC TRANSPORT A significant concern in nanoscience is the transport and magnetic transport properties through nano-entities. The use of molecules or molecular wires as elements in electrical circuits, for example, is an appealing idea. Molecules can have length scales smaller than 1 nm, and hence we would gain orders of magnitude in integration levels if we could assemble them onto a chip. Extensive experimental work has been done to measure junction transport utilizing mechanical break junctions, in the presence of a gating field. Some of these results feature molecular structures containing a single cobalt atom that acts, effectively, as an atomic quantum dot or atomic gate within the molecular structure. As mentioned above, the very narrow domain wall that forms in nanostripes and nanocontacts also has significant consequence on magnetic transport and leads to large room-temperature magnetoresistance (4100% at B ¼ 100 G) [128–131] . Magnetic transport through wires and molecules is still an unresolved issue and currently is under active exploration [132]. Here we discuss ballistic electron transport through monatomic chains suspended between contacting wires. The electrical conduction (G) is quantized according to the Landauer–Buttiker equation [133–135] in the linear response regime: G¼

G0 X T ns 2 ns

(4)

where G0 is the conductance quantum and is equal to 2e2/h or 12.9 kO1. Tns, the transmission coefficient for the nth channel and electron spin s, is either 1 or 0 for monovalent metals corresponding to an open or closed channel. Consequently, conductance measurements for metal nanocontacts display flat plateaus and abrupt drops during the elongation caused by atomic rearrangements. Quantum transport in atomic sized nanowires of various metals formed by mechanical means has been widely studied by several groups. For non-magnetic materials, (e.g., Au [136] and Ag [137]) one would expect that the two spin channels are degenerate; and G is thereby quantized with a unit of G0. If the wire is spontaneously magnetic, the unit for conductance jumps should be reduced to G0/2 (half a conductance quantum) even at room temperature, as long as the characteristic time for electrons conduction is longer than that of the spin rotation. This behavior is demonstrated in Fig. 7a by the plateau in the conductance histograms: in Fig. 7b by the pronounced peak at G0/2 for the count in the number of plateaus in different values of conductance of a Co wire [138]. Intriguingly, magnetic transport behavior was also found in Cu nanocontacts, formed by tapping macroscopic Cu wires together in air at room temperature [139]. With a small field of 5 mT, the G0/2 mode can be clearly seen and is highly repeatable [140]. In contrast, the G0/2 mode was not found in an experiment for Cu junctions done in ultrahigh vacuum [141]. First principles calculations revealed that 1D pure Cu nanowires are nonmagnetic within a wide range of Cu–Cu distances (4.4–6.0 a.u.). However, an alternating CuO chain has sizable magnetic moments in


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Fig. 7. Conductance measurements of Co wires at room temperature and without external magnetic field. (a) Typical electrical transport curves showing conductance plateaus. (b) Global histogram exhibiting the statistical conductance behavior of a sequence of nanowire generation (from Ref. [138]).

both Cu and O atoms when the contact is slightly stretched. The magnetic signal is hence not from the intrinsic properties of Cu wires but rather it is associated with the oxidation of Cu wires. Besides, the CuO wires display the half-metallic feature with the Fermi level lying in a wide band gap of the minority spin. In principle, this is potentially very important to make high efficient spin injection into nanodevices. Since most transition metal elements become magnetic in monatomic wires, one should observe conductance quantum of G0/2 as in Fig. 7b even for 4d and 5d contacts. Transport measurements thus provide a unique way to examine the status of magnetization and spin dynamics for different elements in the form of entities of a few atoms. Rodrigues found this behavior in suspended chains of Pd and Pt [138]. Nevertheless, this is not a settled issue yet since others reported featureless conductance histograms for Pd monatomic contacts in the ultra high vacuum environment [142]. Untiedt et al. [143] further denied the presence of quantization of

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conductance in Fe, Co, Ni, and Pt contacts and attributed the observed fractional conductance quantization to contamination. On the other hand, conductance plateaus at G0/2 were observed from carbon nanotubes, and the reason for this unexpected phenomenon is still unclear [144]. The electron conductance at finite temperature through ultra small entities will require understanding of strong correlations, another fertile ground for fundamental research [145,146].

6. SUMMARY AND FUTURE OUTLOOK Nanomagnetism is a cutting-edge research field, and may potentially revolutionize storage, sensing, spintronics, and optoelectronic technologies. In the near future demands on speed and memory capacity are expected to increase further in computing and communicating technologies and can be met only by further miniaturizing highly functionalized devices. Furthermore, as we have seen, one encounters new and fascinating fundamental physics in these studies. To this end, it is obvious that explorations in the nano- and subnano-scales are both challenging and rewarding. The field is evolving very rapidly and it is very difficult to make meaningful predictions for what will happen in the next 5 years or even shorter. We think several of the following directions deserve particular attention. On the side of theory, it is necessary to develop viable theories and robust software packages for the simulations of magnetic transport (I/V curves), the excitation spectrum and spin dynamics [147]. Accurate description for the excited states will be important since nanostructured ‘‘metals’’ have discrete energy spectra with gaps. The failure of DFT description for the Cr dimers is a good example in this aspect. The development and implementation of time-dependent DFT, dynamic mean field theory or other approaches to treat the correlation effects become crucial. Theoretical investigations of nanoentities usually involve ‘‘excessively large’’ systems for DFT calculations, especially when the environment such as substrates and chemical ligands has to be considered. Furthermore, it is still extremely tough to simulate the growth and post-growth molecular dynamics from first principles. Efficient simulation tools for multi-scale structural modeling, including structural response properties, structural optimization, and quantum dynamical simulations for nanomagnets are considered necessary. Currently, it is still very difficult to fabricate nanostructures in a controlled manner, and many observations may not explore intrinsic properties. Development or refinement of experimental procedures will be essential to study intrinsic properties. This type of research is increasingly dependent upon collaborative efforts among different disciplines, as well as effective partnerships between theory and experiment. Magnetic doping in nanotubes, nanowires, or junctions of semiconductor, for example, can introduce magnetic currents for spintronics manipulations. Detection of single-atom magnetization, spin dynamics and electronic spectrum using SP-STM will be very important to reveal new physics that is peculiar in nanostructures. For example, Heinrich et al. demonstrated the ability to measure the energy required to flip the spin of single adsorbed atoms [148]. Using STM tip


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manipulations, 1D magnetic chains can be built atom-by-atom, and measured hysteresis of single atom in small magnetic entities [149]. One would expect groundbreaking technical innovations as well as significant advances in science from these explorations under well-controlled conditions. On the other hand, for the exploitation of nanostructure in devices, it is equally important to search proper environments such as ligand molecules, hollowed materials, substrates and capping materials, which ideally allow the growth of self-assembled nanostructures and also provide other means to tune the magnetic properties through controlling the hybridization and charge transfer.

ACKNOWLEDGMENTS The author is indebted to simulative discussions with D.L. Mills and Prof. W. Ho at the University of California, Irvine. Collaborations with Profs. J.S. Hong, A.J. Freeman, K. Baberschke, J.A.C. Bland, and D.S. Wang on topics discussed above are highly appreciated. Research work on nanomagnetism was supported by the fund of UC Irvine and also by the Department of Energy.

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Chapter 3 EXCHANGE COUPLING IN MAGNETIC MULTILAYERS M. D. Stiles ABSTRACT In magnetic multilayers, the magnetizations of two ferromagnetic layers separated by a non-magnetic spacer layer are coupled by the electrons in the spacer layer. This coupling oscillates in sign as a function of the thickness of the spacer layer. Extensive research done on these systems has led to a simple model for this coupling and remarkable agreement between predictions of the model and measurements of the coupling. The model predicts that the periods of the coupling are determined by geometrical properties of the Fermi surface belonging to the spacer layer material. The oscillatory coupling is an instance of oscillations in metals caused by the existence of a Fermi surface. Keywords: magnetic multilayers, exchange coupling, quantum well, spin-dependent reflection, Fermi surface, biquadratic coupling, critical spanning vector, interface electronic structure PACS: 73.21.Ac, 75.30.Et, 75.70.Cn, 73.20.At, 72.25.Ba, 71.18.+y

1. INTRODUCTION Magnetic multilayers exhibit dramatic oscillations in their magnetic coupling as a function of the thickness of a metallic separation layer. These systems, see Fig. 1, consist of a sequence of thin films grown on top of each other, each film ranging in thickness from tenths of nanometers to tens of nanometers. Typically, two thin films of ferromagnetic material are separated from each other by a thin film of a non-magnetic material, referred to as a spacer layer. The magnetization directions of the ferromagnetic layers are coupled to each other through an exchange interaction. The sign of this coupling oscillates as the thickness of the spacer layer is Contemporary Concepts of Condensed Matter Science Nanomagnetism: ultrathin films, multilayers and nanostructures Copyright r 2006 by Elsevier B.V. All rights of reproduction in any form reserved ISSN: 1572-0934/doi:10.1016/S1572-0934(05)01003-6

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M. D. Stiles

52 1 cm

1 cm

Au cap 2 nm 1 nm 1 nm

ping la

yer

Fe Au spa cer lay er Fe Ag buff er laye r

GaAs s

ubstrate

Fig. 1. Typical magnetic multilayer. Here, two Fe layers are separated by an Au spacer layer. An Au capping is grown on top to protect the multilayer from corrosion. The multilayer was grown on a GaAs substrate with a buffer layer of Ag to promote better growth of the Fe and Au films.

varied, with the best multilayer samples showing up to 30 periods of oscillation. This chapter gives a pedagogical description of the origin of the coupling and describes a few key measurements and calculations. In metals, the sharp cut-off in occupancy at the Fermi surface causes oscillatory phenomena. The oscillation of the induced magnetization as a function of applied magnetic field, called the de Haas-van Alphen (dHvA) effect [1], is a well-known example. Many other properties oscillate as well. These oscillations arise from fieldinduced oscillations in the density of states. The cross-sectional areas of the Fermi surface, a geometrical property, determine the oscillation periods. Spatially localized disturbances in metals lead to another type of oscillation. The oscillation of the electron density near surfaces of metals, known as Friedel oscillations, is an example. Another example is the oscillation in the interaction between two localized magnetic impurities in a metallic host. As the separation between the impurities is increased, the interaction between them oscillates between favoring parallel alignment and antiparallel alignment of the magnetic moments. This coupling of the moments is known as the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction [2]. A different geometrical property of the Fermi surface determines the spatial periods of the oscillations between the localized disturbances; in this case, a vector spanning the Fermi surface from one side to the other. Interlayer exchange coupling is a particularly dramatic instance of this type of oscillation. Observation of coupling in magnetic multilayers requires high-quality thin films. Early studies [3] were plagued by a number of extrinsic sources of coupling due to defects. One source of such coupling is the presence of pinholes, regions where the non-magnetic layer was not continuous. Pinholes give direct contact between the ferromagnetic layers, leading to coupling favoring parallel alignment of the magnetization directions, referred to as ferromagnetic coupling. In addition, correlated roughness of the films causes ferromagnetic ‘‘orange peel’’ coupling [4]. In 1986,

Exchange Coupling in Magnetic Multilayers

53

Gru¨nberg et al. [5] demonstrated antiferromagnetic coupling between the magnetizations of two Fe layers separated by Cr and two groups demonstrated coupling in rare earth multilayers [6,7]. The observation of antiparallel alignment of the magnetizations meant that the magnetic layers had been grown well enough that the coupling due to any extrinsic coupling was much smaller than the intrinsic coupling. Once high-quality multilayers could be grown, interest in them started to explode. Two years later, Gru¨nberg’s group and Fert’s group discovered the giant magnetoresistance (GMR) effect [8,9]. Magnetoresistance refers to the dependence of a sample’s resistivity on an applied magnetic field. The field dependence may be indirect, for example, the resistance can depend on the magnetization direction, which can depend on the applied field. This is true for GMR; the resistance depends on the relative orientation of the magnetizations of the ferromagnetic layers. If the magnetizations of two layers are antiparallel for zero applied field because the layers are antiferromagnetically coupled, an applied field can overcome the coupling and bring the films into parallel alignment. This change in alignment leads to a change in resistance – GMR. It was immediately realized that the GMR effect could be a sensitive magnetic field detector, particularly for read heads in magnetic disk drives. In fact, read heads based on the GMR effect are now used in essentially all disk drives. To optimize the performance of the read heads, magnetic multilayers with different materials, layer thickness, growth conditions, and other parameters were studied. In 1990 [10], Parkin discovered oscillatory behavior of the dependence of the GMR on the thickness of the non-magnetic spacer layer. These oscillations were not due to variations in the transport properties, but rather variations in the coupling between the ferromagnetic layers. For some thicknesses the coupling was ferromagnetic, favoring parallel alignment of the magnetization directions. For these thicknesses, there was no change in the relative alignment of the magnetizations when a magnetic field was applied and hence the magnetoresistance was zero. The oscillations in the coupling as a function of the separation between two magnetic objects immediately brought to mind the RKKY interaction between magnetic impurities. There was one big puzzle however. The oscillation period, approximately 1.0 nm, was much longer than was expected from estimates based on the analogy with the RKKY interaction. The answer to the puzzle, as is described in detail below, is that it is possible to make much more quantitative comparisons between theory and experiment for magnetic multilayers than has been done for magnetic impurities. The rest of this article is devoted to describing what goes into this comparison. In the following section, I describe a simple model for the physics of the interlayer exchange coupling. Despite its simplicity, this model allows quantitative comparison between theory and experiment. I give this comparison as well as a comparison with more sophisticated models. Section 3 describes the experimental techniques that are used to prepare magnetic multilayers, characterize them, and measure the coupling. I discuss how the presence of defects affects the comparison between theory and experiment. Not only do these defects modify the interlayer exchange coupling, but they can also create effective couplings with different forms. Section 4 describes the most common such effective interaction – biquadratic coupling.

M. D. Stiles

54

2. QUANTUM WELL MODEL 2.1. Model for Transition Metal Ferromagnetism Developing a simple model for interlayer exchange coupling requires starting with a simple model for the electronic structure of ferromagnets. Unfortunately, the electronic structure of transition metal ferromagnets is quite complicated. Ferromagnetism in transition metals is driven by atomic-like exchange and correlation effects in the partially filled d-electron shells. The atomic-like effects suggest a localized description of this part of the electronic structure. However, the d orbitals are strongly hybridized with d orbitals on neighboring atoms and also with the s–p orbitals on neighboring atoms. The strong hybridization suggests an itinerant description of the electronic structure. Reconciling these aspects of the physics is an ongoing area of research, and the resulting models are not simple [11]. Simplifying the model requires approximations that favor one aspect of the physics over the other. In this article, I adopt a description favoring the itinerant aspects because the interlayer exchange coupling depends strongly on the properties of the electrons at the Fermi surface and a realistic description of the Fermi surface requires treating the itinerant nature of the d electrons. The local-spin-density approximation (LSDA) [12] accurately describes the itinerant aspects of the electronic structure while treating the atomic-like exchange and correlation effects in mean field theory. That is, all of the complicated electron– electron interactions are lumped into a local potential that depends on the local density and the local spin density. This approximation was derived for computing the ground state properties of materials. For transition metal ferromagnets, it works quite accurately for properties like the cohesive energy, equilibrium lattice constant, and the magnetic moment [13]. Formally, it is not justified for computing the electronic structure, but it is a good combination of simplicity and accuracy even for this case. A material becomes ferromagnetic when it is energetically favorable for a majority of electrons to align their spins parallel to one another. The electrons then split into those with spins parallel to the majority of the other spins (majority) and those antiparallel (minority). Spin–orbit coupling mixes these states, but is weak enough to ignore to a first approximation. Ignoring the spin–orbit coupling leads to a description of ferromagnets in terms of two separate sets of electrons, majority and minority, which have different properties. This description also holds in the non-magnetic layers in magnetic multilayers provided the magnetizations in the different ferromagnetic layers are all collinear with each other. When the magnetizations are not collinear, spin currents flow in the non-magnetic layers and exert torques on the ferromagnetic layers, as is described below. Figure 2 shows a band structure for Co in a face-centered cubic (fcc) structure calculated using the LSDA, highlighting the importance of the itinerant aspects of the electronic structure. The d-bands have a width in energy, approximately 5 eV, that is large compared to the exchange splitting between the majority and minority bands, about 2 eV. In addition, the hybridization between the d-bands and the s–p


55 Majority Fermi Surface

5

Majority bands Minority bands

"d" band width

Energy (eV)

0

Minority Fermi Surface unhybridized s-p band

-5

Exchange Splitting

s-d hybridization -10 L

Γ

X

Fig. 2. The band structure and Fermi surface of fcc Co. The solid (dashed) curves give the majority (minority) bands along two high-symmetry directions through the Brillouin zone center, G. The long dashed curve shows the sp band if it were not hybridized with the d bands. The bars to the right of the bands show the width of the d bands and the shift between the majority and minority bands. The arrows in the band structure plots give the width of the gap due to the hybridization between the sp and d bands of the same symmetry along the chosen directions. The Fermi surfaces are reproduced with permission from Choy et al. [14].

band is large enough to lead to a gap of about 3 eV where the bands would cross if they were not hybridized. Finally, Fig. 2 shows images of the Fermi surface for majority and minority electrons. Clearly, these are quite different, and the two sets of electrons have very different properties. There are two simplified models for the electronic structure of ferromagnets that have been used extensively in studies of magnetic multilayers. Each emphasizes different aspects. Both include free-electron-like bands. In the spin–split free-electron model, the free-electron-like bands for majority and minority electrons are shifted in energy relative to each other by a constant exchange shift. This model ignores the d electrons completely, but the Fermi surfaces for majority and minority electrons are different. I use this model for pedagogical purposes in much of the rest of this section. The other model, called the s–d model, emphasizes the atomic-like aspects of the d-orbitals by ignoring the d–d hybridization and treating the orbitals as completely localized. This model was originally used to treat isolated magnetic impurities in a non-magnetic host. For ferromagnets and magnetic multilayers, the model for isolated impurities is generalized to treat a dense set of such impurities. In this model, the s–p electrons are weakly hybridized with the d-electrons. In some limits, it is possible to map the s–d model onto a local moment model in which the d-electrons ~i ; where i labels the site. The hybridform a local moment with a fixed total spin S ization between the d and s electrons becomes an exchange interaction between the

M. D. Stiles

56

~i ~ local moment and the s–p electron spin ~ s with the form J S s: The s–d model and the local moment model have been used both to study interlayer exchange coupling and extensively to study transport in magnetic multilayers, i.e. the GMR, see the chapter by Fert in this volume. As long as the scattering rates for minority and majority electrons are different, these models capture much of the essential physics. However, care must be taken when the details of the band structure are important. 2.2. Spin-Polarized Quantum Well States In this section, I describe the properties of a magnetic multilayer using a spin–split free-electron model as described above [15]. In this model, interfaces are simply potential steps. Below, I generalize the results to more realistic models of the electronic structure. An interface between two materials acts on the electrons like a potential step; electrons that strike it have a transmission probability reduced from one. For a free electron going down a simple potential step of height V, the transmission probability is q 2k 2 T step ¼ (1) k kþq qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here, the incident wave vector is k ¼ 2mE=_2 and the transmitted wave vector qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is q ¼ 2mðE þ V Þ=_2 : The first factor accounts for the change in velocity on going over the step. The transmission probability is plotted in Fig. 3(a). If another step is introduced, the electron undergoes multiple reflection inside the quantum well that is formed (see Fig. 3(b)). If the steps are separated by a distance D, the transmission probability is 2 4eiqD kq T well ¼ (2) 2 2 ðk þ qÞ ei2qD ðk qÞ Note that the transmission probability goes to one whenever 2qD ¼ 2np; where n is an integer. At a fixed thickness, there is a series of transmission resonances at the energies E n ¼ ½_2 =ð2mÞ ðnp=DÞ2 V ; for integer n. At a fixed energy, there are 1

1 R

T(well)

T(step)

R T

T

D (a)

0 0

Fig. 3.

(b)

0 1 Energy

2

0

1 Energy

Electron transmission probabilities for a step and a quantum well.

2


57

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi resonances for D ¼ 2np=ð2qÞ with q ¼ 2mðE V Þ=_2 Þ; these resonances are separated by a fixed increment in thickness. At the transmission resonances, the electrons have an increased probability to be in the quantum well; in other words, there is a peak in the density of states in the quantum well at the energy of the resonance. These peaks in the density of states are seen in magnetic multilayers using photoemission and inverse photoemission, for reviews see Refs. [16–19]. Photoemission is a technique in which photons of a particular energy, generally ultraviolet light or X-rays, are incident on a surface. When the photons are absorbed by the material, they excite electrons which can leave the surface and be measured. The energy of the electron in the solid can be inferred from the photon energy and the energy of the photoemitted electron. Peaks are observed in the photoemission spectrum where there is a large density of states in the material. Photoemission probes the near-surface region because the escape depth of the photoemitted electrons is on the order of a nanometer. In order to see the density of states peaks in the non-magnetic spacer layer, the top magnetic layer needs to be stripped off (or never deposited in the first place). In other words, the quantum well states studied in photoemission are not exactly the same quantum well states that are present in the complete magnetic multilayer. None the less, there is very good correspondence between these states and the related states in magnetic multilayers. Fig. 4 shows the photoemission intensity as a function of energy and spacer layer thickness. This figure shows the fixed spacing between peaks as a function of thickness and the quadratic variation of the peaks as a function of energy. There are some differences between what would be expected for a free electron model and what is observed. To understand how the free electron model generalizes to real materials, it is instructive to rewrite the transmission probability in terms of the transmission 0.0

E-EF (eV)

-0.5

-1.0

-1.5 10

20 30 Cu Thickness (Monolayers)

40

Fig. 4. Photoemission from a Cu overlayer on Co. White indicates high photoemission intensity and black low intensity. Reproduced with permission from Qiu and Smith [19].

M. D. Stiles

58

E

Unoccupied states

Antiparallel Alignment

EF

EF

e M Ferromagnet M

M

k

k Spacer

Parallel Alignment

M

M

M

M

EF e M

M

Fig. 5. Quantum wells used to compute interlayer exchange coupling. On the left, the two panels give typical band structures for free electron models of interlayer exchange coupling. On the right, the four panels give the quantum wells for spin up and spin down electrons for parallel and antiparallel alignment of the magnetization. The shaded regions designate the occupied states.

probability for a step and the reflection amplitude at an isolated step R ¼ ðk qÞ=ðk þ qÞ: This substitution emphasizes the contribution made by multiple reflection inside the quantum well. One round trip through the well has the amplitude ei2qD R2 from reflecting from each step and propagating both directions through the well. The transmission probability is 2 2 1 X n iqD iqD i2qD 2 1 ðe R Þ (3) T well ¼ T step e 1ei2qD 2 ¼ T step e R n¼0 The second form shows explicitly the coherent multiple scattering in the well. The basic physics of quantum well states in real materials is captured by replacing the wave vector for propagating through the spacer layer, q, by the appropriate value from the real band structure and by replacing the reflection amplitude and transmission probability by the values calculated for a realistic interface. If one of the materials is ferromagnetic, the potential step for the majority and minority electrons is different. In a multilayer with two magnetic layers, there are four possible quantum wells formed depending on the relative alignment of the magnetizations, see Fig. 5. The quantum well states for each of these are different because the potential steps, and hence reflection probability, is different for each quantum well. However, at a particular energy, like the Fermi energy, the quantum well states in all of the wells have the same periodicity as a function of the thickness of the spacer layer, because the periodicity only depends on the wavelength of the electron in the spacer layer at that energy. 2.3. Interlayer Exchange Coupling The interlayer exchange coupling can be described in terms of an energy that ^ i as depends on the magnetization directions of the two layers, m ^2 ^1 m E ¼ JAm

(4)


59

where A is the area of the two films, and Jo0 gives antiferromagnetic coupling favoring antiparallel alignment. The form of the coupling is called bilinear in contrast to the biquadratic coupling described in Section 4. For interlayer exchange coupling of the form given in Eq. (4), the strength of the exchange interaction is determined by the difference in energy between the quantum well with parallel magnetizations and that with antiparallel magnetizations. Computing the exchange interaction simply requires computing the energies of the quantum wells given in Fig. 5. For large spacer layer thicknesses, the result is _vF Re ðR" R" þ R# R# R" R# R# R" Þei2kF D þ OðD2 Þ 2pD _vF jR" R# j2 cosð2kF D þ fÞ ð5Þ 2pD Here, kF is the Fermi wave vector in the spacer layer, vF ¼ _kF =me the Fermi velocity, me the electron mass, and R" and R# the reflection amplitudes for a majority and minority electron to reflect from the interface. Expanding the square of the reflection amplitudes gives four terms, one for each of the quantum wells in Fig. 5. The exchange coupling oscillates in sign with a period p/kF, the oscillation decays like D1, and the amplitude of the oscillation is determined by the spin dependence of the reflection amplitudes. Physically, these oscillations arise because the quantum well resonances cross through the Fermi level as the thickness of the spacer layer is increased, see Fig. 4. Each time the thickness increases by p/kF, another resonance crosses through the Fermi level. The resonances for each of the quantum wells in Fig. 5 are all different, but they all have the same period because the period is determined by the Fermi wave vector in the spacer layer, and is independent of the properties of the magnetic material. J¼

2.3.1. Critical spanning vectors The expression for the interlayer exchange coupling, Eq. (5), is based on a onedimensional model. Generalizing to realistic three-dimensional systems is straightforward if the growth of the multilayer is coherent. For coherent growth, the inplane lattice of all of the layers is the same, so the in-plane momentum (wave vector) is conserved when the electrons scatter from the interfaces. In this case, electrons with different values of in-plane wave vector do not couple, so the quantum well states for different in-plane wave vectors are independent of each other. Including the full three dimensions of the multilayer then simply requires integrating the onedimensional result over the in-plane wave vector Z 2 h i J _ d K ~ ~ ~ R# ðKÞÞ ~ 2 ei2kz ðKÞD ¼ v ð KÞRe ðR ð KÞ (6) þ OðD3 Þ F " A 2pD ð2pÞ2 This integral is simple in the limit of large D; there is a contribution whenever the ~ is constant as a function of K: ~ The vector spacer layer spanning vector, 2kz ðKÞ; connecting one sheet of the Fermi surface to another at this in-plane wave vector is called a critical spanning vector. In general, the oscillating contributions from

M. D. Stiles

60 Free Electron Fermi Surface

Copper Fermi Surface kmax

kx 0

q

c

k011 0

q q

0

kz

-kmax −π/d

L

q'L

S

0

k100

π/d

Fig. 6. Slices through a free electron and the Cu Fermi surfaces. For the spherical free electron Fermi surface, there is one critical spanning vector qc ¼ 2kF : For the Cu Fermi surface, there are two inequivalent critical spanning vectors, qL which is closely related to the free electron critical spanning vector, and qS which occurs in the necks of the Fermi surface. The Cu Fermi surface is shown within the Brillouin zone appropriate for Cu(001) multilayers. Indicated in gray are parts of the Fermi surface in the extended zone scheme. The critical spanning vector q0 L is equivalent to qL because of the discrete nature of the lattice.

different parts of the Fermi surface are out of phase and tend to cancel each other. ~ the However, when the spacer layer spanning vector is constant as a function of K; contributions from finite region of the Fermi surface near the critical point are in phase. This region gives a contribution to the integral that oscillates in thickness with a period determined by the critical spanning vector. As the thickness D becomes larger, the region of the Fermi surface that contributes in phase gets smaller so that the amplitude of the oscillation decreases with an additional power of D. For free electrons, the critical spanning vector is qcrit ¼ 2kF ; the vector that goes from one side of the sphere to the other, see Fig. 6. For a free electron model, the interlayer exchange coupling is i2 J _vF kF h ~ ~ ¼ R ð 0Þ R ð 0Þ sinð2kF DÞ þ OðD3 Þ " # A 16p2 D2

(7)

The in-plane wave vector dependence of the Fermi velocity and the reflection amplitudes is ignored to lowest order. The oscillation period only depends on the geometry of the spacer layer Fermi surface, but the strength of the oscillation depends on the Fermi surface through vF and kF and also the details of the electronic structure of the ferromagnet through R" and R# : The generalization to realistic materials is illustrated for Cu in Fig. 6. For a free electron model, the only critical spanning vector is one of length 2kF connecting one side of the Fermi sphere to the other. However, for realistic materials, the Fermi surface is more complicated than a sphere; in general, there are several critical spanning vectors and there is a contribution to the coupling from each. For thick spacer layers, the coupling is JðDÞ ¼

h i X _va ka a a 2 iqa? D iwa ? Re ðR R Þ e e þ OðD3 Þ " # 2 D2 16p a

(8)


61

For each critical point, qa? is the critical spanning vector, va? the Fermi velocity, a k the average radius of curvature of the Fermi surface, and eiw a phase that depends on whether the stationary point is a minimum, maximum, or saddle point. Eq. (8) is known as the asymptotic formula. It is valid for thick spacer layers; for thinner spacer layers, the O(D3) represents preasymptotic corrections. Equation (8) shows that the oscillation periods observed in interlayer exchange coupling are determined by geometrical properties of the spacer layer Fermi surface. The first paragraph of this chapter mentioned that the dHvA effect measures the geometry of the Fermi surface. In 1991, Edwards et al. [20] pointed out the analogy between the oscillations in the interlayer exchange coupling and the oscillations seen in the dHvA effect. Also in 1991, Bruno and Chappert [21] showed that it was possible to use the Fermi surfaces determined in dHvA measurements to predict the periods that would be observed in measurements of magnetic multilayers. Table 1 shows the resulting comparison. The agreement between the two sets of periods is quite remarkable. I describe the experimental techniques used to measure the interlayer exchange coupling periods in the next section and discuss some of the possible reasons for the disagreement seen for Cu/Co(100). It is interesting to note that the oscillations in the interlayer exchange coupling and in the dHvA effect come from different critical features of the Fermi surface. In both cases, the oscillations are due to the abrupt change in occupancy at the Fermi energy. In the case of interlayer exchange coupling, there are oscillatory contributions to the coupling at all the energies and in-plane wave vectors of the occupied states. However, all of these oscillations tend to cancel out, except where the occupancy changes at the Fermi surface. Further, the oscillations at the Fermi surface all cancel except at points where two sheets of the Fermi surface become parallel to each other. There are similar oscillations in the de dHvA effect. All these also cancel except at the Fermi energy and where the area of the orbit in reciprocal space is stationary as a function of the wave vector along the magnetic field direction. a

Table 1. Comparison of oscillation periods measured in magnetic multilayers with those expected from the critical spanning extracted from Fermi surfaces measured in de Haas-van Alphen measurements (dHvA).

Ag/Fe(100) Au/Fe(100) Cu/Co(100)

Cr/Fe(100)

Cr/Fe(112)

Period (ML)

Period (ML)

Technique

Reference

5.58 5.7370.05 8.60 8.670.3 5.88 8.070.5 6.06.17 11.1 1271 12.5 14.4 15.4

2.38 2.3770.07 2.51 2.4870.05 2.56 2.6070.05 2.582.77

DHvA SEMPA DHvA SEMPA DHvA MOKE SEMPA DHvA SEMPA MOKE DHvA MOKE

[21] [22] [21] [23] [21] [24] [25] [26,27] [28] [29] [26] [29]

62

M. D. Stiles

In this respect, these oscillatory phenomena are physical examples of a simple mathematical effect. In a Fourier transform, a function is described as a superposition of oscillating functions with different frequencies. An approximate description results when the range of frequencies is finite, i.e. when there is an upper cutoff frequency. Whenever the function being described has a sharp feature, a kink or a step, for example, approximate descriptions with a finite cutoff oscillate close to the location of the sharp feature. These oscillations are known as Gibbs oscillations. For the interlayer exchange coupling and the dHvA effect, the sharp cutoff is the Fermi surface where the occupancy changes from one to zero. Another interesting comparison of the two sets of oscillations is found in the different conditions that are required to observe both. The dHvA oscillations come about when electrons in a magnetic field complete a circular orbit before they scatter. For typical fields, these orbits are large, requiring long mean free paths, that is, low temperatures and very high-quality crystals. As shown above, interlayer exchange coupling requires that electrons complete a round trip within the spacer layer before scattering. This path is much shorter than that required for dHvA oscillations, consistent with the fact that oscillatory interlayer exchange coupling is observed at temperatures even higher than room temperature. Even though the interlayer exchange coupling is less sensitive to the mean free path, it was discovered much later because growing sufficiently good multilayers requires deposition techniques that has only recently been developed. Figure 6 illustrates the resolution to the early doubts that interlayer exchange coupling was related to the RKKY interaction. These doubts were driven by the fact that the period expected from the free electron critical spanning vector is close to 0.3 nm, which is much shorter than the observed period of about 1.0 nm.Taking the actual Fermi surface into account does not immediately help. The Cu Fermi surface is a distorted sphere and the critical spanning vector qL is close to the free electron 2kF. The resolution of this discrepancy derives from the fact that Cu has a periodic lattice and in multilayers has layers of thickness d. Hence, the coupling can only be sampled at discrete values of the thickness nd for integer n. Since the oscillation corresponding to the free electron Fermi surface has a period shorter than d, the oscillation with wave vector qL is indistinguishable from an oscillation with wave vector q0 L ¼ ð2p=dÞ qL : The period associated with q0 L is very close to what is seen experimentally. The equivalence of a discretely sampled rapid oscillation to as lower oscillation is referred to as aliasing. Aliasing is the reason that the Cu(100) critical spanning vectors are labeled the way they are in Fig. 6. The longer critical spanning vector qL gives a longer period oscillation than the shorter critical spanning vector qS gives. Without aliasing, it would be the other way around. 2.3.2. Coupling strength While the critical spanning vectors and other properties of the Fermi surface in Eq. (8) can be extracted from experiment, the reflection amplitudes cannot. For real multilayers, these can be calculated, but the resulting coupling strengths do not agree nearly as well with measured values as the periods do. Some of the disagreements are


63

Coupling Strength (mJ/m2)

due to experimental difficulties, which are discussed in the next section, but some disagreements are due to theoretical difficulties. The asymptotic form is an approximation to the difference in energy between the total energies of multilayers with parallel magnetizations and antiparallel magnetizations. Unfortunately, due to the complexities of the electron–electron interaction, it is currently impossible to compute these total energies without approximation. The best available approximation, as described in the beginning of this section, is the LSDA. This approximation works quite well for magnetic multilayers but with one caveat. The band structure is only approximate, so the Fermi surface deviates from its actual shape. Since the oscillation periods of the interlayer exchange coupling depend on the critical spanning vectors of the spacer layer Fermi surface, the periods computed using the LSDA will be wrong. This means that after a few oscillations, the calculated coupling is out of phase with the experimental coupling. The sign of the coupling may even be wrong. A direct comparison is then misleading because even if the physics is essentially correct, the agreement might be quite poor, see Fig. 7. There, the agreement is made even worse by the effect of disorder on the measured results. In Fig. 7, the calculated and measured periods of the oscillations disagree; the periods extracted from the experiment are 2.4870.05 monolayers and 8.670.3 monolayers and the critical spanning vectors of the theoretical Fermi surface would give periods of 2.65 monolayers and 8.03 monolayers. One way to compensate for the errors in the periods is to fit both theory and experiment to the asymptotic form, and compare the envelopes. The results of fits to the calculations and measurements in Fig. 7 are given in Table 2. The fits to experiment take into account measured thickness fluctuations, as discussed in the next section. Taking into account the thickness fluctuations leads to much better agreement between the strengths in Table 2 than in the raw comparison in Fig. 7. An additional source of disagreement that has not been accounted for is temperature. The calculations are done at zero temperature and the measurements at room temperature. While it is clear that 1.000

0.100

0.010 0.001 0

10 20 Thickness (monolayers)

30

Fig. 7. Calculated [30] and measured [31] coupling strengths for Fe/Au/Fe(100) multilayers. The solid curve is the best fit to the experimental data (symbols) when the measured thickness fluctuations are taken into account. The dashed curve is a linear interpolation between the coupling strengths (symbols) calculated for complete layers.

M. D. Stiles

64

Table 2.

Fit to experiment [31] Asymptotic calculation [34] Fit to full calculation [30]

Comparison of coupling strengths. J s/(1 nm)2

J L/(1 nm)2

1.2970.16 mJ/m2 2.0 mJ/m2 3.4 mJ/m2

0.1870.02 mJ/m2 1.1 mJ/m2 1.1 mJ/m2

accounting for the temperature will improve the comparison, the temperature scale for the correction [32,33] in this system is not known so it cannot be made quantitative. Measuring the coupling at low temperature would be the ideal solution. Possible reasons for any remaining disagreement are discussed in the next section. Also given in Table 2 are calculations done using the asymptotic form. Possible reasons for the imperfect agreement between calculations are discussed below. A difficulty with total energy calculations is that the total energies are typically many orders of magnitude larger than the difference in energy between parallel and antiparallel alignment of the magnetizations. In the LSDA, the density and the potential depend on each other and need to be determined self-consistently. Accurately computing energy differences makes the calculations quite time consuming. One way to simplify the process is to use the force theorem. The force theorem states that if the densities (or potentials) of two configurations are close to each other, the difference in total energies can be approximated by the difference in eigenvalue sums. The eigenvalue sum is the sum of the energies of all of the occupied states. The eigenvalue sum is more easily calculated, is much smaller than the total energy, and the calculation need not be self-consistent. The calculation shown in Fig. 7 used the force theorem. In a real sense, the ‘‘derivation’’ of the asymptotic form for the interlayer exchange coupling is based on the force theorem. The energy was assumed to be given as a sum of the energies of all of the occupied electronic states. It turns out that even for realistic systems such an approach can be a reasonable approximation. Having reduced the interlayer exchange coupling to eigenvalue sums, it is possible to derive the asymptotic formula, Eq. (8), for realistic multilayers [35,36]. This form is derived by ignoring the energy and wave vector dependence of the reflection amplitudes. In the thick limit, including the energy and wave vector dependence gives additional contributions to the coupling that are higher order in 1/D. These contributions are called preasymptotic corrections [37,38]. For Co/Cu(100), where these corrections have been studied in detail, they turn out to be quite important. Asymptotically, the strength of the long period coupling is weak, but for thinner layers, the corrections lead to substantial coupling. For the long period coupling, the wave vector dependence of the majority electron reflection amplitude gives the most important correction. On the other hand, the short period coupling is strong for thick layers, but the energy dependence of the phase of the majority reflection amplitude makes it weaker for thin layers. While Fe/Au(100) multilayers have not been analyzed for preasymptotic corrections, there are indications that they might be important. First, the coupling strengths for the short period coupling calculated using the asymptotic formula disagree with


65

the coupling strength extracted from the full calculation. Second, there are differences between the periods expected from the critical spanning vectors of the Fermi surface and those extracted from Fourier transforms of the calculated coupling [30]. 2.3.3. Torques and spin currents When the magnetizations of two adjacent layers are not collinear, the interlayer exchange coupling exerts a torque on both, given by the negative derivative of the energy, Eq. (4), with respect to the relative angle. Some of the first calculations of the interlayer exchange coupling [39,40] proceeded by directly computing the torque between the magnetization directions. Such calculations proceed analogously to the calculation sketched above, but it is necessary to match non-collinear spin states at the interfaces and to take great care in performing the energy integrals. It is interesting that the torque, which like the energy has contributions from all of the occupied states, can be cast in a form that depends only on the properties of the electrons at the Fermi energy. Since the interlayer exchange coupling is mediated by the electrons in the spacer layer, the torque is as well. The torque is associated with a spin current flowing in the spacer layer carrying angular momentum from one layer to the other. Angular momentum is extracted from one layer, effectively exerting a torque on the magnetization of that layer, and is deposited in the other layer, effectively exerting an opposite torque on the magnetization of second layer. This spin current differs from the spin current of interest in spintronics [41] because it is carried by all of the electrons in the spacer, not the electrons close to the Fermi level. It also exists independent of an applied voltage. Distinguishing spin currents due to quasi-equilibrium interactions and those related to transport [42] is important to understand possible spintronic devices and also current-induced torques [43,44].

3. MEASUREMENT OF INTERLAYER EXCHANGE COUPLING 3.1. Growth and Disorder Growing magnetic multilayers to compare measurements with calculations is quite difficult. Calculations are only tractable for systems that are close to ideal, requiring growth of systems equivalently close to ideal. The first requirement is that the lattices of the different materials need to be compatible. For example, when Co grows on Cu, it grows pseudomorphically, that is, it adopts the fcc structure of Cu, with a lattice constant that is very close to that of Cu. For a review of growth in this system see Ref. [45]. Another pair of metals with identical crystal structures and close lattice constants is Fe and Cr. Both of these pairs of materials can be grown with several different interface orientations. Some of these are shown in Table 1. Unfortunately, these are the only two pairs of materials with such similar crystal structures. The only other pairs that can be grown sufficiently well are Au or Ag and

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Fe, but only in the (0 0 1) interface orientation. It is somewhat surprising that these systems can be grown well because Fe has a body-centered cubic (bcc) structure, while Au and Ag have an fcc structure. In addition, the lattice constants are very different. However, it turns out that if the Fe lattice is rotated by 451 around the interface normal, there is very good in-plane lattice match for each of these pairs [46]. If the starting substrates are sufficiently flat, very good growth can be achieved. However, the presence of steps leads to extended defects through the layer because the Au/Fe growth is not pseudomorphic and the layer thicknesses are quite different. Interpretation of coupling through Cr spacer layers is complicated by the presence of spin density wave antiferromagnetism [47–49]. In bulk Cr, the transition temperature is close to room temperature, the temperature at which most measurements of the coupling are made. In very high-quality multilayers, the antiferromagnetism leads to a short period coupling, which is not well described by the model presented in this paper. However, in addition to the short period coupling associated with the antiferromagnetism, there is a long period oscillatory coupling, which appears to be well described by the model presented here. The properties of this long period coupling are given in Table 1. Unfortunately, the pairs of materials in Table 1 exhausts the systems that are well enough lattice matched to make high-quality comparisons between calculations and measurements. When a material is grown on a substrate that is not so well lattice matched, it assumes the in-plane lattice constant of the substrate on which it is deposited for a couple of monolayers. Then, as the thickness of the deposited layers increases, the strain energy associated with its modified lattice structure becomes too large and the film relaxes by introducing dislocations at the interface. Systems that have been studied in addition to those in Table 1 are reviewed in Refs [50,51]. The same quality of growth has not been achieved in these systems as in the systems with much smaller lattice mismatch. Even when the lattice mismatch is close to zero, real multilayers are still not perfect. The starting substrate is never perfectly flat and the growth is never perfectly layer-by-layer, so there are always variations in the thickness of the layers, typically called thickness fluctuations. Frequently, the lateral length scales of the thickness fluctuations are in an intermediate regime. The flat terraces are large enough that the ideal coupling is reasonably well defined over each terrace. On the other hand, the terraces are small enough that the magnetizations do not vary significantly on that length scale. In this intermediate regime, the comparison between calculated and measured coupling strengths is improved by averaging over measured variations of the spacer layer thickness. If the coupling for an ideal thickness of n layers is J(n) and the probability of having a thickness nd for a nominal deposited thickness of D is P(n,D), the effective coupling strength is X JðDÞ ¼ Pðn; DÞJðnÞ (9) n

If the width of the growth front is measured by scanning tunneling microscopy (STM), X-ray diffraction, or some other technique, theoretical coupling strengths can be averaged to compare with measured strengths. The curve in Fig. 7 that is a fit


67

to the experimental data includes the effect of the measured thickness fluctuations through Eq. (9). The coupling strengths with the effect of thickness fluctuations removed are given in Table 2. When the interlayer exchange coupling has rapid oscillations, the measured coupling is rapidly reduced by the thickness fluctuations. When the coupling strength is reduced by orders of magnitude, it is hard to extract meaningful coupling strengths, even if the growth front has been measured. For this reason, it is desirable to grow the sample in a layer-by-layer mode to keep the growth front as narrow as possible. Since layer-by-layer growth is determined by the competition between nucleation of islands and diffusion of deposited adatoms, it tends to require higher substrate temperatures during growth. Unfortunately, higher growth temperatures tend to promote interdiffusion at the interfaces. Interdiffusion, which gives rise to scattering centers, is more difficult to treat theoretically than thickness fluctuations. It also can be more difficult to measure. Interdiffusion can have an important and counterintuitive effect on the coupling. In the Fe/Cr(100) system, the measured interdiffusion [52–54] is believed [55] to cause the sign reversal in the measured short period coupling [28,56,57]. Extensions of the calculations discussed above for Fe/Au(100) [30], indicate that interdiffusion may be responsible for the difference between the calculated and measured coupling strengths in Table 2. Those values have already been corrected for the measured thickness fluctuations. It is peculiar that the short period coupling strengths agree much better than the long period strengths. Calculations that include interdiffused atoms at the interface indicate that interdiffusion reduces the long period coupling more than the short period. Since the interdiffusion is not measured for this case, it is possible that it explains the remaining discrepancy between theory and experiment. The choice of substrate plays a large role in the quality of the growth. The best measurements are made on substrates of one of the materials in the multilayer. Iron whiskers, which can be extremely flat [58], and copper single crystals give the best results. However, insulating substrates are necessary if the samples are also to be used for transport measurements. For these substrates, great care is required to get really high-quality growth. See Refs [59,60] for descriptions of the complexity of growing an Fe/Au multilayer on a GaAs substrate. One of the key advances that allowed accurate determination of the oscillation periods was the use of wedge-shaped spacer layers, see Fig. 6. Growing wedge samples simply involves moving a shutter between the sample and the evaporator during growth to expose different parts of the sample to different total fluxes. Such samples ensure that all thicknesses are grown under the same conditions because they are grown simultaneously on the same substrate. Wedge samples also make it easier to accurately determine the thickness of the samples. 3.2. Measurement Techniques Measurement of interlayer exchange coupling relies on two broad categories of measurements. One set determines the structure of the multilayer in as much detail as possible. The second determines the magnetic coupling. Common techniques for

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determining the structure of multilayers are reflection high-energy electron diffraction (RHEED), STM, X-ray scattering, and neutron scattering. Common techniques for determining the coupling are magneto-optical Kerr effect (MOKE), scanning electron microscopy with polarization analysis (SEMPA), Brillouin light scattering (BLS), and ferromagnetic resonance (FMR). The technique RHEED [61] is commonly used to determine the quality of a surface during growth. A high-energy electron beam is reflected from the surface at glancing angles. The resulting diffraction pattern is sensitive to the details of the surface, in particular the presence of steps. If the growth is layer-by-layer, there are fewer steps when layers are close to complete and more when the layer is half filled. In this case, the intensity of different spots in the RHEED pattern oscillate with a period of one layer. Since the oscillations have a period of a single layer, RHEED oscillations can be used to determine the total thickness of the film. For samples of uniform thickness, RHEED is used to monitor the thickness film during growth. For wedge samples, it is typically used after growth, when the RHEED beam is scanned along the wedge and the RHEED oscillations are monitored as a function of position to give the thickness at that position, see Fig. 8. Techniques used to measure the coupling fall into two broad classes. In the first class, the magnetic configuration is measured, frequently as a function of applied magnetic field. For example, Parkin et al. [10] first observed oscillatory interlayer exchange coupling using the GMR. Here, the resistance of the film in zero field was compared with the resistance in large field. If the coupling is ferromagnetic, there is no change, and if the coupling is antiferromagnetic, the change can be substantial. A commonly used technique to determine the magnetic configuration is MOKE [62]. The magneto-optic Kerr effect is the dependence of reflected light on the polarization of the light and the magnetization of the surface. Typically, the polarization of the light rotates through a small angle on reflection. Variations in the rotation can be measured to give the variations in the magnetization that cause them. For wedge samples, MOKE can be used in an imaging mode by scanning the focused spot of a laser across the surface or by imaging a wide area of illumination. It is not particularly surface sensitive and has the advantage that it is sensitive to the magnetic state of both layers. Using the sensitivity to both layers, MOKE images [63] have directly identified perpendicular alignment of two layers, see Section 4. MOKE was used to measure the coupling strengths in Fig. 7. An imaging technique that has been used to determine coupling periods is SEMPA [64]. When a high-energy electron beam scatters from a surface, it excites low-energy electrons, which leave the surface. These secondary electrons tend to maintain the polarization they had when in the surface. SEMPA measures the magnetization of a region of the sample’s surface by determining the polarization of the secondary electrons. Since this technique is based on measuring secondary electrons, which have low energy, it is generally not used with an applied field, limiting it to studies of the remnant state. On the other hand, it has greater spatial resolution than optical techniques like MOKE, and can be used on smaller wedges, requiring smaller areas of sample perfection. Since it can only measure the remnant state, SEMPA has not been used to measure coupling strengths, but it has been


69 2 mm

ayer e Overl Au Spacer Layer

Thin F

12 nm

Overlayer Magnetization

Fe Whisker Substrate

Magnetization

RHEED

0

20 40 Au Thickness (ML)

60

Fig. 8. Interlayer exchange coupling in a wedge-shaped Fe/Au/Fe trilayer measured by SEMPA and RHEED [22]. A schematic view of the wedge-shaped sample is shown at the top of the figure. The approximate dimensions give an indication of the very small slope of the wedge. Immediately below is a SEMPA image of the magnetization of the Fe overlayer. White and black indicate parallel and antiparallel alignment to the substrate, and hence ferromagnetic and antiferromagnetic coupling. Below that is a line scan through the image and then a measurement of the RHEED intensity along the wedge. The oscillations are used to determine the thickness of the spacer layer along the wedge. The wedge is slightly curved. The RHEED and the magnetization curves have been corrected for this curvature, but the image has not, hence the variation of the lines connecting the image with the line scan.

used to determine the sign of the coupling for enough oscillations of the coupling to allow high-precision measurements of the periods, see Fig. 8. In addition, the electron beam serves as a very high-resolution source for measuring RHEED so that both measurements can be done in situ. When the magnetic state is measured, the interlayer exchange coupling is inferred from the state rather than directly measured. SEMPA images are analyzed with the assumption that the interlayer exchange coupling dominates other energies so that the magnetization points in the direction of the coupling. The MOKE measurements in Fig. 7 are based on images like those shown for SEMPA in Fig. 8, but in the presence of an external applied field. In this case, it is assumed that the direction of the coupling is determined by the balance of the interlayer exchange coupling and the interaction with the external field. In both cases, magnetic hysteresis is ignored. Both of the analyses are simple examples of a more general approach. Usually, some magnetic property of a sample, like its hysteresis loop, is measured and the exchange coupling is inferred by comparing the measured property with a model. Some or all of the parameters of the model, including the interlayer coupling

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constant, are varied until the predictions of the model agree with the measurement. The reliability of the results depend on the accuracy of the model, in particular whether it includes all of the physics necessary to describe the experiment. The second class of measurements used to determine the coupling is based on determining the curvature of the free energy of the magnetization with respect to small variations in the magnetization direction. Two such techniques are FMR and BLS [65]. FMR is based on finding peaks in the microwave absorption of multilayers. The peaks identify resonance frequencies; in other words, the frequencies of the uniform modes of a layer. The coupling between the magnetizations of different layers gives rise to coupled oscillations. The different in-phase and out-of-phase resonance frequencies then give the size of the coupling. BLS is closely related. When light is scattered from the sample, there is a small amount of scattered light that has gained or lost energy by absorbing or exciting a mode of the system. Peaks in the absorption spectrum identify the modes of the system. The mode frequencies can then be analyzed to give the interlayer exchange coupling.

4. BIQUADRATIC COUPLING In almost all multilayers, there is a contribution to the coupling that favors perpendicular alignment of the magnetizations. In many multilayers, this contribution dominates, leading to actual perpendicular alignment of the magnetizations [63,66]. Phenomenologically, this alignment can be explained by a coupling of the form E ^1 m ^ 2 Þ2 ¼ J 2 ðm (10) A called biquadratic in contrast to the bilinear coupling discussed above. It is called biquadratic because it is quadratic in both of the magnetization directions. The fact that all measured values of J2 are negative, favoring perpendicular orientation of the two magnetizations, shows that biquadratic coupling does not have an intrinsic origin similar to the bilinear coupling, but, as Slonczewski [67,68] showed, has an extrinsic origin due to disorder. Thickness fluctuations lead to variations of the coupling strength on different terraces. To lowest order, the intralayer exchange coupling forces the magnetizations in each layer to be uniform so that the bilinear coupling gets averaged over the growth front as described in Eq. (9). To next order, the magnetization can fluctuate around its average direction. Over each terrace, the magnetization fluctuates in the direction that lowers the energy. The fluctuations lower the energy the most when the two magnetizations are perpendicular to each other. This is the origin of the effective interaction favoring perpendicular alignment between the magnetizations. Consider the simple model shown in Fig. 9, in which the spacer layer consists of parallel strips of width L with alternating thicknesses and hence coupling strengths J n and J n+1. The relative angle of the magnetizations is y ¼ y0+dysin(px/L), where dy is the size of the fluctuations. Over the region from 0 to L, where the coupling is J n, the energy change due to the fluctuations is proportional to J nsin(y0)dy. Over


71

Upper Ferromagnet n

L

n +1

Spacer Layer

Lower Ferromagnet

Fig. 9. Thickness variations and biquadratic coupling. The thickness of the spacer layer varies periodically between n and n+1 layers, with ferromagnetic coupling for thicknesses n+1 and antiferromagnetic for thicknesses n. The heavy arrows show the local rotation in the magnetization direction into the direction that minimizes the coupling for each terrace.

the region from L to 2L, the sine function changes sign and the energy change due to the fluctuations is proportional to J n+1sin(y0)dy. The net coupling energy per area due to the fluctuations is proportional to DJsin(y0)dy, where DJ ¼ J n j nþ1 : Fluctuations in the correct direction lower the energy of the system. The energy gain is balanced by the cost in intralayer exchange energy because the magnetization now varies spatially. Since the intralayer exchange coupling depends on the square of the gradient of the magnetization, for this simple model, it is proportional to (Aex/L)dy2, where Aex is the strength of the exchange interaction. Combining the changes due to the fluctuations for the interlayer exchange coupling and the intralayer exchange and finding the minimum with respect to the amplitude of the fluctuations gives dypsin(y0)DJ/(Aex/L). For this fluctuation amplitude, the change in the energy per area due to the fluctuations gives the strength of the biquadratic coupling J2

ðDJÞ2 L Aex

(11)

While Eq. (11) is quite simple, it qualitatively describes the features of more realistic situations. In real systems, a characteristic length scale L of the arbitrary-shaped terraces replaces the width of the stripes. As this length scale increases, the biquadratic coupling strength increases because the fluctuations can get larger. Also, realistic growth fronts generally consist of more than two thicknesses, which introduces an effective DJ. The coupling increases as the difference in the coupling for the different terraces get larger. The differences tend to be largest when the coupling is oscillating rapidly, that is when there is short period coupling. The coupling gets weaker as the intralayer exchange interaction increases because exchange suppresses the fluctuations in magnetization direction that lower the energy.

5. SUMMARY I have attempted to pedagogically describe interlayer exchange coupling using a simple physical picture that has evolved over the last decade. Spin-dependent reflection from the interfaces in multilayers communicates the magnetic state of one

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ferromagnetic layer through a non-magnetic layer to a neighboring ferromagnetic layer. This communication leads to an exchange interaction between the layers. The exchange interaction has oscillatory contributions with periods determined by the critical spanning vectors of the spacer layer Fermi surface and strengths determined by the spin-dependent reflection at the interfaces. The oscillations in the exchange interaction arise from the spin-dependent quantum well states set up by the reflection. These quantum well states evolve in energy as the thickness of the spacer layer is varied. As these states pass through the Fermi energy, they fill or empty, changing the energy of the multilayer. These changes are periodic because the quantum well states cross the Fermi energy with a period determined by the Fermi surface of the spacer layer material. At critical points of this Fermi surface, many quantum well states have the same period giving a net oscillatory contribution to the energy. The energy difference between the parallel and antiparallel alignment of the magnetizations gives the interlayer exchange coupling. The oscillations in the interlayer exchange coupling are analogous to the oscillations in the dHvA effect. The periods of both are determined by critical geometrical properties of the Fermi surface. In fact, the Fermi surface geometries extracted from dHvA experiments can be used to predict the periods of the coupling. The agreement between these periods and the coupling periods, which have been measured to 3% accuracy in the best cases, provides strong support for our understanding of the coupling. The periods of the oscillatory interlayer exchange coupling are relatively insensitive to the presence of defects in the multilayer, but the coupling strengths are not. The measurements that have been done on defects and the calculations of their effect on the coupling give indications that our understanding of interlayer exchange coupling is correct. Improving our understanding will require measuring the coupling in systems that are as close to perfect as possible and then quantitatively measuring all the defects that remain. The necessary structural characterization will require multiple techniques to measure all of the defects. Then, calculations will need to explicitly treat the measured defects. The biquadratic coupling that is ubiquitous in magnetic multilayers is an example of an effective interaction that arises because of defects and frustration. Interfacial roughness gives rise to fluctuations in the strength of the bilinear coupling. The system can lower its energy by allowing the magnetization of the layers to fluctuate in response to the roughness-induced variations. The system can lower its energy the most when the two magnetizations are perpendicular to each other, giving an effective coupling that favors perpendicular alignment of the magnetizations. For more information on interlayer exchange coupling, there are a number of review articles. The series Ultrathin Magnetic Structures I–IV consists of review articles on the general topic of magnetic multilayers. Chapter 2 of Ultrathin Magnetic Structures II contains four review articles written around 1993 covering various aspects of interlayer exchange coupling [69]. Volume III contains yet another review article, written in 2002. Volume 200 of the Journal of Magnetism and Magnetic Materials consists of a series of review articles covering much of magnetism and includes many related to magnetic multilayers. The article on interlayer


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exchange coupling [50] focuses on the comparison between theory and experiment. Other general review articles on interlayer exchange coupling include [68,70–73]. For a compendium of theoretical and experimental results for specific systems, see [51]. The system Fe/Cr is sufficiently rich that it has generated three review articles on its own [47–49]. Most of the articles above focus on transition metal multilayers; for reviews of rare earth multilayers, see [74,75]. As mentioned above, photoemission studies of quantum well states in magnetic multilayers are reviewed in Refs [16–19]. Biquadratic coupling is reviewed in Ref. [76].

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Chapter 4 STATIC, DYNAMIC, AND THERMAL PROPERTIES OF MAGNETIC MULTILAYERS AND NANOSTRUCTURES R. E. Camley 1. INTRODUCTION Why have magnetic multilayers received so much attention in the last few years? There are many answers: (1) Multilayers represent a new class of materials where surface and interface effects fundamentally influence the properties of the entire material. For example, in a typical bulk material there might be 1015 surface atoms and 1023 atoms in the interior. Thus the ratio of surface to bulk atoms is 108. In contrast, a multilayer made up of films of alternating materials, where each film is five atomic layers thick, has a ratio of surface or interface atoms to bulk atoms of 40%. (2) A rich variety of structures and behaviors are possible. One can combine ferromagnets with nonmagnets, ferromagnets with ferromagnets, ferromagnets with antiferromagnets, ferromagnets with helical magnets, to name only a few possibilities. Each of these structures has unique and potentially useful properties. For example, the giant magnetoresistance effect [1–3]1 originally found in Fe/Cr multilayers is now being used in magnetic memory storage systems. In addition, one can obtain and design unusual behaviors including magnetization that increases with temperature and a thermal hysteresis that is tunable with a small magnetic field. (3) The importance of surface and interface effects may be controlled by the layering pattern. Adding more layers to each film emphasizes bulk effects and conversely thinner films as constituents in the multilayer emphasize interface effects. Contemporary Concepts of Condensed Matter Science Nanomagnetism: ultrathin films, multilayers and nanostructures Copyright r 2006 by Elsevier B.V. All rights of reproduction in any form reserved ISSN: 1572-0934/doi:10.1016/S1572-0934(05)01004-8

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(4) The properties of magnetic multilayers can be substantially modified by small external fields. Because of the tunability discussed above, one can design multilayers that are sensitive to small, convenient magnetic fields. For example spinflop fields in antiferromagnets are often in the 100 kOe range. In contrast, a spin-flop in a magnetic multilayer can be designed to occur in the 100 Oe range. In this chapter, we will explore simple calculational methods for determining the equilibrium spin structures of multilayers and nanostructures at any magnetic field and at any temperature. The results of these calculations will then be compared with experimental measurements. In addition, we will develop simple methods for finding the dynamic behavior of magnetic multilayer structures and again compare the results with available experiments.

2. THEORETICAL TREATMENT OF MAGNETIC MULTILAYERS In bulk systems, one often employs the ‘‘mean field’’ approximation to understand the thermal behavior of magnetic materials. In this method, an individual spin feels an effective magnetic field produced by its near neighbors. From symmetry one expects, for example, that in a ferromagnet all spins will be parallel and equivalent and this assumption is used in calculating the average field. In a layered structure, it is necessary to modify this assumption and a reasonable supposition is that the spins within a single atomic layer are parallel and equivalent for a ferromagnet. For an antiferromagnet, one may assume that all spins on a given sublattice within a single atomic layer are equivalent. The orientation and thermal averaged magnitude of a spin in one atomic layer may be quite different from those in a neighboring layer. As a result of these considerations, the magnetic multilayer may often be treated as an effective one-dimensional system with one spin representing an entire layer of spins. We note that dealing with the system in a layer-by-layer fashion is particularly helpful in multilayer systems because surface/interface variations in exchange and anisotropy may easily be taken into account. A schematic illustration of the layer-by-layer structure is shown in Fig. 1. A spin in layer n, Sn, feels a number of fields which influence its motion. These include: (1) Local exchange fields from within a layer and exchange fields from neighboring layers (2) Local anisotropy fields (3) The external field, H0 (4) Demagnetizing fields caused by dipole motion. We first describe an iterative energy minimization method for the T ¼ 0 case. One starts with an assumed set of values for the angular position of each layer of spins. In many cases, the spins will lie in a plane parallel to the interfaces

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Local uniaxial anisotropy axis Layer n-1

Layer n Exchange coupling Jn,n+1

Layer n+1

Applied magnetic field

Fig. 1. Schematic illustration of the layered structure and interactions for a magnetic multilayer. The top layers have a uniaxial anisotropy. All the layers feel an external field and the exchange fields of the neighboring layers.

(this reduces the demagnetization energy) and for simplicity we will use this assumption. It is also assumed that all spins in each layer are equivalent. Due to exchange coupling, the energy of the spins in any given layer depends on the orientation and magnitude of the spins in its own layer and in the nearby layers. An arbitrary layer n, of spins, can be regarded as having an energy En (neglecting anisotropy) where X J nm Sn Snþm gn mB H 0 S n (1) En ¼ m¼0;1;2

Here Jnm is the effective exchange coupling constant between spins in the nth and mth atomic layers. Interactions betweens spins in the same layer can be accounted for by a Jnn term. In terms of the angle yn that the spins in layer n make with the applied field: X J nm Sn Snþm cosðyn ynþm Þ gn mB Sn H 0 cosðyn Þ (2) En ¼ m¼0;1;2

For simplicity we have neglected anisotropy energy in this example, but it can be easily included, as we will see later. Thus at T ¼ 0 a single variable for each layer, yn, characterizes the magnetic structure. The initial set of angles described above will generally not be a ground state or a stable state of any kind. An iterative procedure can be used to allow the system to relax to a self-consistent stable state. First, the energy in an individual layer, n, is

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minimized by choosing the angle yn generated by the minimization condition @E n =@yn ¼ 0

(3)

This procedure can be viewed as rotating the spins in the nth layer to lie parallel to their effective field, a process which reduces the layer’s energy. We can then find a configuration of the entire structure as follows: a different layer is then randomly chosen and the spins in that layer are rotated to lie in the direction of the local effective field. The process is continued until one has a self-consistent, stable state where all the spins are aligned with their local fields. Since the method only minimizes the energy locally, different initial configurations may lead to different selfconsistent final states. The ground state is, of course, the stable state with the lowest energy for the entire structure. At finite temperatures, both the direction and thermal averaged magnitude of the spins in each layer must be specified. In this case the iteration procedure is slightly different. A layer of spins is first rotated into the direction of the effective field, and then the spin’s thermal averaged magnitude in that direction is found through the use of the Brillouin function hS n i ¼ S n Bsn ðxÞ

(4)

gn mB Sn H n kT

(5)

where x¼

Here Hn is the effective field acting on a spin in layer n and is defined explicitly in Eq. (7). The Brillouin factor is given by x ð2S þ 1Þ ð2S þ 1Þx 1 Bs ðxÞ ¼ (6) coth coth 2S 2S 2S 2S Here oSn4 is the thermal average of the spins in the nth layer in the direction of the effective field and Hn the effective field acting on layer n. The effective field is now given by Hn ¼ H0 þ

X

J nm S nþm gm m¼0;1;2 n B

(7)

Note that in the effective field the spins in the neighboring layers are replaced by their thermal averaged magnitudes. Again the entire operation is iterated through all spins until a self-consistent state emerges. For a particular temperature and applied magnetic field the iteration procedure requires the order of 103 to 107 iterations per layer to reach a converged state. It generally takes a larger number of iterations to find a converged state near a phase transition. To speed up the calculation, one often uses the final state at one temperature to be the initial guess for the spin configuration at a nearby temperature. Of course, the final state may only be a local minimum and it is not guaranteed that the true ground state will always be found by this method. Nonetheless, if one takes


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a sufficient number of different initial sets of angles, the method seems to discover all the low-energy stable states. The stability of a particular state can be checked in a direct way. Once a final state is found, one picks a new spin configuration slightly perturbed from the final state. Variations of a few degrees in each layer are typical. The iterative process then begins with this new configuration and one checks whether the system relaxes back toward the original state or diverges away from it. We note in passing that this mean-field theory is really the simplest possible approximation that takes into account a layer-by-layer variation, spin orientation, and thermal averaged magnitude. Such methods have been used previously in thinfilm calculations [4]. Improvements would include Bethe–Peirels–Weiss methods or Monte Carlo techniques. However these methods involve significantly larger computational investment.

3. EXAMPLES OF MAGNETIC MULTILAYER STRUCTURES Among the most interesting of the magnetic superlattice structures are those that involve some antiferromagnetic coupling. In this case, the exchange energy favors some kind of antiparallel alignment while the Zeeman energy due to an external field is minimized if the magnetic moments are all parallel to the field. As a result, these structures can display a wide set of phase transitions, induced through variations of either the magnetic field or temperature, with the transition temperatures and fields controlled by changing the layering pattern. We, therefore, concentrate on multilayers with antiferromagnetic coupling. One of the ‘simplest’ layered systems with an interesting magnetic structure and a fascinating dependence on layering pattern is the Gd/Y superlattice. Although bulk Y is normally nonmagnetic, in the Gd/Y superlattice the Y layers mediate an effective exchange interaction between neighboring Gd films [5,6]. The application of a magnetic field in this system produces some interesting results. Because of the competition of interface exchange and Zeeman energies, the ground state is not a macroscopic antiferromagnetic state where the ferromagnetic films are strictly antiparallel. Instead, the equilibrium structure resembles a macroscopic spin-flop state that represents a compromise between exchange and Zeeman energies. The resulting configuration is illustrated in Fig. 2. An increase in the external field reduces the canting angle y and changes the net magnetic moment of the structure. Small fields can have large effects because the applied field acts on all spin of a given film, while the effective exchange between films is weak and acts only on the interface layers of the films. In contrast, spins in bulk materials see large exchange fields from nearest neighbors that are very large compared to laboratory Zeeman fields. The magnetization is thus very sensitive to applied fields and also to temperature, which plays a strong role in determining the average interlayer exchange energy. In Fig. 3, we present theoretical and experimental results [6,7] for the magnetization as a function of temperature for Gd/Y superlattices. We see an unusual

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Fig. 2. The spin configurations in the Gd/Y multilayer at low temperatures and high temperatures. Note that at high temperature the thermal averaged magnetic moments in Gd are smaller at the surface than in the bulk. From Ref. [7].

temperature dependence with a magnetization that increases as temperature is increased. This can be understood by looking at the exchange coupling of the Gd spins. The Gd spins on the interior are strongly coupled to each other compared to those on the exterior, which are only weakly coupled on the Y side via the RKKY interaction. When the temperature is increased, the Gd spins in the interior thus retain a larger thermal averaged magnitude than those on the exterior since they see a larger effective field. As a result, the interface exchange energy (which should scale as J1(SGdexterior)2) is rapidly reduced and the competition between exchange and Zeeman energies now favors a state where the Gd moments point more closely along the external field. This is automatically included in the iterative calculation and the results are illustrated schematically in Fig. 2. Thus even though oSGd4 decreases as the temperature increases, the change in orientation of the Gd spins is sufficiently large such that the total magnetization of the structure increases. As T continues to increase the oSGd4 begin to decrease very quickly and the magnetization decreases (more or less as single Gd films). The field dependence of the magnetization is also striking. Here, we have an ordered state that displays a significant change in magnetization as a function of field for very modest fields as can be seen in Fig. 3. We now turn to the influence of the outermost surfaces of the magnetic structure of a finite multilayer system. The surface layer plays a special role in the magnetic structure simply because it is an exchange coupled to other spins on only one side. Thus in the competition between exchange and Zeeman energies, the influence of the exchange energy is reduced and it is easier for an external field to direct the surface spins. As the outermost layer of spins tries to turn toward the field direction, the remaining layers must adjust their orientations as well. Thus surface-induced


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Fig. 3. Magnetization as a function of temperature for different external fields for a Gd/Y multilayer. The top panel shows the theoretical calculation, Ref. [7], and the bottom shows the experimental results from Ref. [6].

phase transitions can occur at magnetic field strengths well below that required for the equivalent bulk phase transition [8–11]. Also the reconstructed state can have surprisingly large penetration-depths into the bulk of the superlattice. As the first example of a surface phase in a superlattice, we consider a finite Fe/ Cr- or Gd/Y-type structure. We assume that the temperature is low enough that all spins have a thermal averaged moment equal to their maximum value. Furthermore, we assume that the exchange coupling between atomic layers is sufficiently strong that the spins within a film are rigidly coupled together. These assumptions are reasonable for Fe/Cr or Co/Ru superlattices at room temperature, for film thicknesses under 10–20 nm.

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For an infinitely extended Fe/Cr-type structure in an external field one expects a canted configuration as discussed earlier. From energy considerations, it is easy to show that, in the absence of any anisotropy, the uniform canting angle is given by the equation cos a0 ¼ H 0 M=4J

(8)

where, M is the magnetization of a film and J the interface exchange energy. In a finite multilayer, however, this uniformly canted state is not stable [11]. As mentioned above, the reason for this is that the outer layers of the finite structure experience only half the exchange coupling of the interior layers and thus are easier to turn toward the applied field. Stable ground state configurations for finite Fe/Cr-type multilayers have been found [11] using the numerical method outlined above. These states are compared to the uniform canting state in Fig. 4. The structure in Fig. 4(b), the low-field case,

Fig. 4. Illustration of bulk and surface spin-flop configurations in an antiferromagnet with no anisotropy. From Ref. [11].


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is quite complex. The outermost spins are twisted into the direction of the field as expected, but the spins of the second layer are actually turned farther away from the field than they would be in the bulk configuration. This alternation continues as one progresses into the bulk of the multilayer, but the amplitude of the deviation decreases with increasing distance from the surface. The main difference between the configurations of Fig. 4(b) and (c) is that there is no alternation in the high-field configuration of Fig. 4(c). The surface twist states illustrated in Fig. 4(b) and (c) reduce the energy of the structure as compared to the uniform canting state by lowering the Zeeman energy. (As a result, the surface twisted states have a slightly higher magnetization than does the uniformly canted state.) However, the twist also results in a small increase of exchange energy. The resulting structure is as usual a compromise between exchange and Zeeman energies. It is worthwhile to compare these nonuniform canted structures discussed here to the very different case of nonuniform canting found in a domain wall. For example, the width of a domain wall in a ferromagnet is governed by the competition between the exchange and anisotropy energies. The exchange energy can be minimized by spreading the canting out over many layers of spins and thus favors a very extended wall. In contrast, the anisotropy energy is minimized when spins point in an easy direction, thus favoring a narrow wall. In the surface twist problem, however, the competition is between exchange and Zeeman energies, and there is also the additional freedom of having two sublattices. This allows new kinds of states to occur when minimizing the total energy. The alternating spin configuration seen in Fig. 4(b), for example, has the exchange energy between layers of spins, which alternately increases and decreases as one moves through the multilayer. This extra degree of freedom also allows the width of the nonuniform canting region to vary in an interesting manner as a function of applied field. At low fields the nonuniform region is very large, for moderate fields it is quite small, and then for larger fields the width of the nonuniform regions is again large. The considerations above apply to microscopic structures as well as multilayers. For example, one can use the same techniques as described above to examine the magnetic field and temperature behavior of ultra-thin antiferromagnetic films. A particularly interesting feature relates to the number of layers in the antiferromagnetic film. For an even number of layers, one of the outer layers must have its magnetization pointing opposite to the applied field and, as a result, a surface phase transition is nucleated at this surface. For an odd number of layers the magnetic moments at both surfaces will point parallel to the external field and no surface transition will be induced, and the change to a spin-flop structure must occur in the bulk. These configurations are illustrated in Fig. 5. Calculations for the fields required to cause a spin-flop have been done for several antiferromagnetic materials. An example is shown in Fig. 6, where the spin-flop field is shown as a function of the number of layers in an ultra-thin film of the antiferromagnet MnF2 [12]. For an even number of layers the spin-flop field is nearly independent of the number of layers. The reason for this is that there is always a surface spin-flop in this system. In contrast, when the number or atomic

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Surface phase transition nucleates here

Odd

Applied Field

Fig. 5. Illustration of the difference between an even and odd number of layers in an antiferromagnetic film. For an even number of layers one of the surface layers always has its moments pointing oppositely to the applied field.

Fig. 6. Spin-flop field in MnF2 for an even and odd number of magnetic layers at T ¼ 0: When the number of layers is even the spin-flop field is smaller because a surface spin-flop occurs. From Ref. [12].

layers is odd, the spin-flop field is significantly larger for thinner films, in part due to the pinning effect of the outer layers. The spin-flop fields for MnF2 calculated above are at T ¼ 0: Finite temperature calculations show that the odd/even behavior persists at finite temperature as well, as can be seen in the phase diagrams of Fig. 7 [12] for and even number of atomic layers (N ¼ 10) and an odd number of layers (N ¼ 11). It is interesting to note that the differences between an odd number of atomic layers and an even number of atomic layers persists in temperature until the phase transition to a paramagnetic state. The even/odd considerations in the microscopic antiferromagnet are all important in the multilayer as well. The details of surface phase transitions in an Fe/ Cr(211) multilayer have been explored both theoretically and experimentally recently [13–15]. In these structures, the unit cell is rectangular in the plane of the surface and provides a uniaxial anisotropy.


Fig. 7.

87

Phase diagrams for an ultrathin antiferromagnetic film of MnF2 with an even (N ¼ 10) and odd (N ¼ 11) number of layers. From Ref. [12].

The even/odd behavior was demonstrated most dramatically in measurements of the magnetization and magnetic susceptibility in the Fe/Cr(211) multilayer system [13] as shown in Fig. 8. As seen in this figure, the magnetization of the 20-Fe film structure shows two rapid changes, one, at low field, associated with the surface spin-flop, and the other at higher field associated with a bulk spin-flop. In contrast, the 21-Fe film structure displays only one transition close to the bulk transition of the 20-Fe film sample. Spin-flop states are generally associated with substantially increased susceptibility. The reason for this is simple. Susceptibility measures the ability of the spin configuration to change due to an external field. In the aligned states only the magnitudes of the magnetic moments can change. In contrast, in the spin-flop state both the magnitude and orientation can change, leading to a large susceptibility. We will see an enhanced susceptibility used as an indication of a canted structure in other systems as well. The behavior of Fe/Cr(211) multilayers with an even number of Fe films has been shown to be particularly interesting [13–15]. If the system starts in the antiferromagnetic state at low field and the field is increased, a surface spin-flop occurs at fields well below that required for the bulk spin-flop. Essentially a domain wall is introduced near the surface. As the field is further increased, the domain wall migrates to the center of the film and then broadens in its spatial extent so that eventually the entire film is in a spin-flop configuration. These theoretical predictions were confirmed in detail in subsequent experimental work with polarized neutron scattering [14]. For example, it was shown that for the particular case measured the domain wall reached the center of the film at an external field about 15% above that necessary to cause a surface spin-flop transition, and that the domain wall extended over the entire film at a field about double that required for the surface spin-flop. We now consider a multilayer system with alternating ferromagnetic films that are antiferromagnetically coupled at the interfaces. An example of such a system is

88

R. E. Camley

Fig. 8. Experimental results on (a) magnetization and (b) susceptibility in Fe/Cr(211) multilayers. For an even number of Fe films (N ¼ 20), the bulk and surface phase transitions each cause a sharp change in magnetization. This results in two peaks in the susceptibility. In contrast, there is only one peak in the susceptibility for an odd number of films. From Ref. [13].

the Fe/Gd multilayer [7,16–20]. There are some key differences between bulk Fe and Gd. The Curie temperature of Fe is 1048 K, while Gd has a Curie temperature of 298 K. In addition, magnetic moment of Fe is 2.2 mB, while in Gd it is 7.2 mB. Thus in the temperature range of 0 to about 300 K, the Gd thermal averaged magnetic moment changes from near 7 mB to 0, while the Fe moment is nearly temperature-independent. This has some interesting consequences, as we shall see below. There are a number of known magnetic states for the Fe/Gd system that are illustrated in Fig. 9. There is a low-temperature aligned-Gd state where the Gd magnetization is aligned with the external field (and the Fe is opposite) and there is a high-temperature aligned-Fe state where the Fe magnetization is aligned with the external field (and Gd is opposite). These configurations are stable when the applied magnetic field is small; in higher magnetic fields there is a twisted or spin-flop like state where the Fe and Gd moments are nonuniform within a film and both canted with respect to the applied field. This system is essentially and artificial ferrimagnet where the effective concentrations of the two magnetic components are controllable by changing the layering pattern. The properties of the different phases and the phase diagram of Fe/Gd have been extensively studied experimentally and theoretically.


89

Fig. 9. Illustration of the three main phases in the Fe/Gd superlattice. In the Gd-aligned phase the thermal averaged moments of the Gd are large and aligned with the external field. In the Fe-aligned phase at higher temperature the Gd moments are small and the Fe moments align with the field. In the twisted phase, the moments are nonuniform within a film and canted with respect to the magnetic field.

The different phases can be easily identified by an M versus H measurement as seen in Fig. 10 [7,21]. The aligned phases are characterized by small changes in M as H is increased. In contrast, the spin-flop or twisted phase shows large changes in M as H is increased. Thus the susceptibility dM/dH is a sensitive tool to characterize the phase with a large susceptibility indicating a twisted or canted state. Figure 11 shows the experimental and theoretical phase diagram [17] for three different layering patterns. In Fig. 11(a) and (b) one sees similar behaviors – at low fields and low temperatures the system is in a Gd-aligned state; as the magnetic field is increased there is a transition to the spin-flop or twisted state, and at high temperatures the structure is in the Fe-aligned state. The critical magnetic field required for the transition from the Gd-aligned state to the twisted state depends on the layering pattern. In Fig. 11(a), the individual films are thinner – thus emphasizing interfacial exchange energy in the balancing of Zeeman and exchange energies. As a result, the critical field required for the transition to the twisted state is large. In Fig. 11(b) the films are thicker; the interfacial exchange becomes less important, and the transition field is smaller by about a factor of 2 when compared to Fig. 11(a). Finally, in Fig. 11(c) the thickness of the Gd films has been reduced sufficiently so that the Gd-aligned state is never stable, i.e. the net magnetic moment in the Gd never becomes larger that the net moment in the Fe. We outline a recent treatment of an interesting phenomenon that occurs in Fe/Gd multilayers – magnetically tunable thermal hysteresis [22]. Thermal hysteresis is well known in physical changes of state such as melting or crystallization. The temperature width for the thermal hysteresis is often only a few Kelvin or less. Many of these systems have a requirement of ultrapure samples since impurities, often at a

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R. E. Camley

Fig. 10. STotal versus applied field for a Fe/Gd multilayer structure. The field is measured in dimensionless units given by h ¼ gmB H=J Fe S and the temperature is given by t ¼ T=T Curie-Fe : At t ¼ 0:25; for example, one sees a slow change in STotal versus h at low fields (Gd-aligned) then a rapid change at higher h (twisted phase), and then a slow variation at even higher h (the Fe-aligned phase). The right panel shows experimental results for Co/Gd, a similar system. From Refs. [16,21].

surface, can act as nucleation sites and initiate the phase transition. In contrast, the Fe/Gd structure is a simple magnetic multilayer system that can be designed to show a thermal hysteresis curve. Calculations show the width, in temperature, of this hysteresis curve can be controlled by an external magnetic field with small fields (200–300 Oe) leading to thermal hysteresis spanning about a hundred degrees Kelvin. Larger fields (600 Oe) can reduce the width of the hysteresis curve to 20 K, and even larger fields can completely eliminate this effect. The physical origin for the thermal hysteresis is simple. One must have two states that are stable at the same temperature. The two states are the Fe-aligned state and the Gd-aligned state. At high temperatures, the system is in the Fe-aligned state; the Gd moments, which are small, are oppositely directed to the field. As the temperature decreases, the Gd moments increase and the net magnetic moment can actually be opposite to the external field if an anisotropy in Fe holds the Fe spins in place. Eventually, however, this configuration becomes unstable and the structure reverses with the Gd moments along the field and the Fe moments antiparallel to the applied field. A similar situation occurs on heating the system from low temperatures. The general situation of a Fe/Gd multilayer is quite complicated and, in fact, shows significant surface effects, as we shall see. However, the general features discussed here should persist if the anisotropy is found in the Fe, in both materials, or only in the Gd. The calculation proceeds in the usual way. A spin in monolayer i feels an effective field composed of the exchange fields from the layers above and


Fig. 11.

91

Theoretical and experimental phase diagrams for Fe/Gd multilayers with different layering patterns. From Ref. [17].

R. E. Camley

92

below, external fields, and anisotropy fields: X ~iþ1 i þ J i;i1 hS ~i1 iÞ þ H z^ þ H a hS z i=S i ~i ¼ H ðJ i;iþ1 hS i

(9)

nn

We use an anisotropy of 50 G in Fe and Ha is taken to be 0 in Gd. All other parameters are given in Ref. [17]. (The reduction parameter of Ref. [17] is 0.75.) The minimum energy state is found using the iterative method described earlier. The results for the Fe magnetization as a function of temperature for an infinite Fe/Gd superlattice are given in Fig. 12. This curve is calculated for a structure with 17 monolayers Gd and 23 monolayers of Fe in each unit cell. This choice of values is intended to match the experimental parameters of Gd 50 A˚/Fe 35 A˚. Periodic boundary conditions are used to simulate the infinite superlattice. A thermal hysteresis spanning 90 K is seen for the lowest-field value. This is reduced to about a 30 K span for a field of 600 Oe. For even higher fields the system does not display hysteresis, but makes a transition from the aligned phases to the spin-flop phase and the magnetization essentially increases linearly as a function of temperature. It is known that the finite size of a Fe/Gd superlattice can have a significant influence on the phase transitions. For example at low temperatures and low fields the system is typically in the Gd-aligned state. If the magnetic field is increased the magnetic structure will eventually change to a spin-flop like state. However, the critical field at which this phase transition takes place depends dramatically on whether the outermost surface is Gd- or Fe-terminated. If the structure has Fe spins on the outside, then in the Gd-aligned state the Fe spins are opposite to the external field and a surface phase transition to the spin-flop state nucleates at the Fe surface

Fe moment (arb units)

17 layers Gd / 23 layers Fe 50 Gauss anisotropy field in Fe Infinite multilayer 1.0 0.5

H = 300 Oe H = 600 Oe H = 1200 Oe

0.0 -0.5 -1.0 60

80 100 120 140 160 180 200 220 240 Temperature (K)

Fig. 12. Theoretical thermal hysteresis curves for an infinite Fe/Gd multilayer with a unit cell of 17 ML Gd/23 ML Fe superlattice for different applied magnetic fields. The hysteresis curves narrow as the applied field is increased. At higher fields (1200 Oe) the system is in a spin-flop phase and does not show substantial hysteresis. From Ref. [22].


93

[10,23,24]. The critical field for this surface transition is about a factor of 5 lower than that for the bulk transition. The thermal hysteresis effect discussed here also shows substantial finite size effects. In Fig. 13, we compare results for an infinitely extended multilayer with a unit cell 17 ML Gd/23 ML Fe with those of a finite multilayer with the same unit cell but with a total of 15 repetitions. The thermal hysteresis for an infinite structure is approximately twice as wide as that for the finite structure. This is due to the surface phase transition discussed above. In a finite structure, composed simply of a repetition of the Fe/Gd unit cell, the spins at one surface will always be opposite to the applied field in an aligned state, and as a result a surface phase transition will nucleate at that surface, reducing the temperature range over which two states are stable. Nonetheless, the main features of the thermal hysteresis remain qualitatively the same regardless of the number of unit cells, and it is important to compare the theoretical results with the results of an experiment. Neutron scattering measurements were taken on a multilayer with composition [Gd 50 A˚/Fe 35 A˚] 15 deposited on a Si wafer with Fe as the top layer of the multilayered stack. The magnetic properties were studied with polarized neutron reflectometry at the Intense Pulsed Neutron Source of Argonne National Laboratory. In these experiments neutrons are incident on the sample. The scattered neutrons are collected and analyzed and I+ represents the intensity of the scattered beam when the magnetic moment of the incident neutrons is parallel to the applied field. I denotes the similar case except that the magnetic moment of the incident neutrons is antiparallel. A convenient þ quantity is the spin asymmetry P ¼ II þ I ; which is basically proportional to the þI 17 layers Gd / 23 layers Fe H = 300 Oe

Fe moment (arb units)

1.0 Infinite Multilayer

0.5

15 repetitions of unit cell

0.0

-0.5

-1.0 80

100

120

140

160

180

200

220

240

260

Temperature (K)

Fig. 13. Theoretical thermal hysteresis curves for a finite and infinite Fe/Gd multilayer. The unit cell is 17 ML Gd/23 ML Fe in both cases. The applied field is 300 Oe. We note that the finite multilayer has a substantially narrower thermal hysteresis curve. From Ref. [22].

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R. E. Camley

Fig. 14. Experimental neutron scattering results for the thermal hysteresis of the asymmetry P. The vertical axis is proportional to the Fe magnetization. The experiments were done on a [Gd 50 A˚/Fe 35 A˚]15 sample. From Ref. [22].

magnetization of the Fe sublattice, and thus it can be immediately compared with the theoretical predictions. Figure 14 shows the spin asymmetry when the sample is cycled in temperature at different fields. For Ho1200 Oe the loop shows hysteresis. The area of the loop decreases as the field is increased; yet the slope of the segments uniting the lower to the upper branch remains the same. Above 900 Oe the loop has closed down almost completely, but with a smoother transition from the lower to the upper branch. The results presented here show the excellent qualitative agreement of theory and experiments, particularly in that the width of the thermal hysteresis decreases as the external field is increased, and in the transition to the spin-flop state at higher fields, a state where the magnetization increases approximately linearly with temperature. It is clear from these examples that the simple self-consistent local mean field theory does a good job of qualitative and quantitative description for magnetic multilayer systems of many different kinds. Additional examples can be found in Ref. [7].

4. THE DYNAMIC RESPONSE OF MAGNETIC MULTILAYERS: COLLECTIVE SPIN WAVE MODES As we have seen, much of the interesting physics that occurs in magnetic multilayers is due to the coupling across interfaces and the effects of having a surface. Unfortunately the nature and strength of interactions across an interface are difficult to measure directly. Even with techniques such as spin-polarized low-energy electron diffraction (SPLEED) and neutron scattering [25,26], it is difficult to obtain unambiguous information on the magnetic structure of surfaces and interfaces. Measurements of the static magnetization of magnetic multilayers give data on the averaged magnetic moment from a relatively large volume of sample, rather than providing detailed information on any of the individual films within the multilayer.


95

Another problem, for example, is that ferromagnetic interlayer exchange coupling cannot always be inferred from static magnetization measurements. One method of investigating surface and interface magnetism, which has proven useful in studies of ferromagnetic films and multilayers, is to probe the spin-wave excitations of a magnetic system [27–30]. In a spin wave, the magnetic moments at each site precess about their individual equilibrium directions. Because the spins are coupled with one another through exchange and dipolar interactions, spin-wave excitations are the eigenmodes of the magnetic system and are characterized by frequency and wavelength which are linked through a dispersion relation. Thus the frequency of a spin-wave may depend quite sensitively on the exchange coupling between spins as well as other effective fields caused by, for example, bulk and surface anisotropies [29] and magneto-elastic effects. These interactions will not only affect the frequency of precession but also the relative phase of precession between spins at neighboring lattice sites. In ferromagnetic systems, such as Fe and Co the lowest spin-wave frequencies are typically of the order of 5–10 GHz. These are long-wavelength excitations and can be studied using ferromagnetic resonance [30] and Brillouin light scattering techniques [27–29]. Higher-energy excitations can be observed by magnetic neutron scattering [31,32]. Spin-wave frequencies in antiferromagnets are typically much higher, existing at several hundred GHz and into the THz range, and thus also influence the optical properties in the far infrared region [33,34]. Excitations in antiferromagnets can be observed with infrared reflectivity [35–37], Raman scattering as well as neutron scattering [32]. In magnetic multilayers, the energies of longwavelength spin-wave excitations will also range from a few GHz to several hundred GHz depending on the exchange interactions between the ferromagnetic films. In this section, we will explore how the dynamic response of the spin system reflects the static configurations, and how the frequencies of the excitations can be used to determine the important magnetic parameters that govern the magnetic structure of the multilayer. Most of the discussion which follows will relate to excitations which can be measured in ferromagnetic resonance. In this case, the spin-excitation wavevector parallel to the surface and interfaces is usually zero. This allows a substantial simplification of the calculations, particularly for the dipole fields. For nonzero wavevectors, one must include the dipole fields in a careful way, and a variety of interesting behaviors result [7,38]. We begin by presenting a simple calculation, which illustrates the spin motion in two ferromagnetically coupled ferromagnetic films. As noted earlier, one cannot find the exchange coupling constant for this magnetic structure through static measurements. However, this is possible for the dynamic measurements. The primary goal of this calculation is to obtain the excitation frequencies of the normal modes of the coupled ferromagnetic films and connect these frequencies to the interfacial coupling between the two films. The energy per unit area is such a system is given by ~1 H ~1 H ~2 H ~1 M ~ þ 1M ~D1 t1 M ~2 H ~ þ 1M ~D2 t2 J I M ~2 E¼ M 2 2

(10)

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96

~D1 the demagnetizing field acting on the film, Here, t1 is the thickness of one film, H ~ and M 1 is its magnetization. Similar definitions hold for the second film. H is the applied field and JI the interfacial exchange coupling constant. This equation contains the Zeeman and demagnetizing energies for each film, and the interface coupling energy. Because we are using a total energy/area at this point, the Zeeman and dynamic demagnetizing energy densities have to be multiplied by the thickness of the appropriate film. For a spatially uniform mode the demagnetizing field in film 1 is given by ~ D1 ¼ 4pM 1y y^ H

(11)

which is due to the uniform distribution of magnetic surface charges on the top and bottom of each film. Similarly, ~ D2 ¼ 4pM 2y y^ H The energy density in film 1 is thus ~ ~ ~1 H ~ 2pM 2 J I M 1 M 2 U1 ¼ M 1y t1

(12)

(13)

~1 From the last equation we can find the effective field acting on M ~2 ! @U 1 ! J IM H eff1 ¼ ¼ H 4pM 1y y^ þ ~ t1 @M 1

(14)

Note that the last term on the right-hand side looks like a thickness-dependent exchange field. A similar equation can be written for the effective field acting on the magnetization in the second film. One now uses the torque equation of motion for each film ~1 dM ~1 H ~ eff1 ¼g M (15) dt ~2 dM ~2 H ~ eff2 ¼g M (16) dt where g is the gyromagnetic ratio. The value of g varies from material to material but is typically around 1.82 107 rad/s Oe or equivalently 2.9 GHz/kOe for the ferromagnetic metals. For the simple problem of two ferromagnets with ferromagnetic coupling the calculation is relatively straightforward. The applied field is set in the z^ direction so we expect the magnetizations in each film to point in this direction as well, with small oscillations in the x and y directions. So ~ 1 ¼ M 1x x^ þ M 1y y^ þ M 1 z^ M

(17)

~ 2 ¼ M 2x x^ þ M 2y y^ þ M 2 z^ M

(18)

where the Mx and My terms are small and have a time dependence exp[iot]. When all the substitutions are made into the equations of motion, we obtain a matrix


97

equation that has the form of an eigenvalue equation. This equation is not, however Hermitian! 0

i og

B B B þ H þ JI M 2 t B 1 B B 0 B B @ Mt22J 1

H þ Jt1I M 2 þ 4pM 1

Jt1I

i og M2JI t2

0

þM 1 Jt1I

0 M1

i og H þ Jt2I M 2

1

C C C 0 C C C H þ Jt2I M 1 þ 4pM 2 C C A o i g

0

1 M 1x BM C B 1y C B C B M 2x C ¼ 0 @ A M 2y

(19) In order to obtain a simple analytic expression, the problem is simplified further. We set M 1 ¼ M 2 ¼ M; and t1 ¼ t2 ¼ t: Also, we introduce the exchange field H ex ¼ J I M=t: With these approximations, two frequencies are found pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o ¼ g HðH þ 4pMÞ (20) and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o ¼ g 4H 2ex þ ð4H þ 8pMÞH ex þ HðH þ 4pMÞ

(21)

Obviously, one mode has a frequency which is independent of the exchange coupling and the other mode frequency depends on this coupling. This is easy to understand in terms of a simple model of coupled pendula as seen in Fig. 15. If we consider two identical independent pendula, they each oscillate with the same frequency. If we couple the two together by a spring, there are two normal modes. One occurs when the two pendula move in phase. In this case, the spring is neither stretched nor compressed and the frequency remains unchanged. The other normal mode had the two pendula moving 1801 out of phase with each other and in this case the spring modifies the frequency of oscillation. The spin motion in the two films also decomposes into an in-phase mode (acoustic mode) and an out-of-phase a)

b)

Fig. 15. Two coupled pendula. In (a) the two pendula move in-phase and the spring is never compressed or stretched, so the frequency of this motion is independent of the spring. In (b) the motion of the two pendula is 1801 out-of-phase and the spring influences the frequency of the mode.

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98

mode (optic mode). The in-phase mode has a frequency that is insensitive to the exchange coupling, but the optic mode frequency can be used to find the interface exchange coupling. As we have seen above, one may obtain the interfacial exchange coupling between two ferromagnetically coupled films by measuring the frequencies of the dynamic modes of the entire structure. This has proven quite useful because it is often difficult to measure the exchange coupling using static measurements alone such as magnetometry. The reason for this is that in a static measurement, the two films always act as a single unit as long as the coupling is ferromagnetic and anisotropies are small. With the dynamic measurements one can determine that value of the exchange coupling, independent of the sign of the coupling [39,40]. We show in Fig. 16 an example of how the dynamic frequencies depend on the interface-coupling constant for two coupled Co films. More complex systems with many layers and arbitrary couplings are treated in a similar manner. It is necessary in this case to find the complex equilibrium positions before obtaining the linearized eigenvalue equation. Nonetheless, this can be done in a straightforward manner, although the algebra can be painful. We will discuss an alternative method in the next section, but first we present some examples using the method outlined above. We now consider two identical ferromagnetic films that are coupled together antiferromagnetically. Each film has an in-plane uniaxial anisotropy associated with the z axis and an external field is applied along z. Even this simple system can have a rich variety of behaviors as we will see. Clearly if the external field is very large both magnetizations will align with the external field. For small external fields 7

Applied Field = 100 Oe

Frequency (GHz)

6

Optic Mode

5 Acoustic Mode 4

3

2 -60

-40

-20

0

20

40

60

Interfacial Exchange Field (Oe)

Fig. 16. Frequencies for two exchange coupled ferromagnetic Co films as a function of the coupling strength. One mode is independent of the coupling, while the other changes smoothly as a function of the coupling.


99

the antiferromagnetic exchange will dominate and the magnetizations will be antiparallel. For intermediate fields the two magnetizations will have a spin-flop configuration. The energy density for the system includes the Zeeman energy density, the interface exchange coupling energy density and the anisotropy energy density: ~1 M ~M ~2 H ~ H ex M ~ 2 =M H a ðM 1z Þ2 =M H a ðM 2z Þ2 =M (22) ~1 H U ¼ M ~ 1 is the magnetization in film, M2 is defined similarly, and M the magnitude Here M of the magnetization. If we assume the equilibrium positions of the spins lie in the plane of the interface we may omit the demagnetizations terms in the calculation that follows. They must be included, however, for the dynamics. If we assume a spin-flop like configuration where M1 is canted away from the z axis by an angle y and M2 is canted away from the z axis by an angle y2 the energy density of film 1 can be written U ¼ MH cos y1 H ex M cosðy1 þ y2 Þ H a Mcos2 y1

(23)

The equilibrium angle y is found by minimizing the energy density dU ¼ 0 ¼ MH sin y1 þ H ex M sinðy1 þ y2 Þ þ H a M cos y1 sin y1 dy1

(24)

We can now assume, by symmetry, that the canting angles for the two magnetizations are equal in magnitude, and solve for the canting angle H ¼ cos y1 2H ex þ H a

(25)

Note that for antiferromagnetic coupling Hex is a negative quantity in this example. If the external field is large (so that the magnetizations are parallel) and then is reduced, the field at which canting can begin occurs when cos y1 ¼ 1 and we obtain H c ¼ ð2H ex þ H a Þ

(26)

As the external field is further lowered a second transition from the spin-flop state to the antiparallel state occurs. In Fig. 17, we show the net magnetic moment as a function of applied field for two antiferromagnetically coupled Co films. The parameters for the system are M ¼ 1:4 kG; H a ¼ 0:057 kOe; and the thickness-dependendent H ex ¼ :058 kOe: If we scan from large applied fields to small fields, we see the two transitions discussed above. However if we start at a small external field in the antiferromagnetic state and increase the field, the system can remain stable in the antiparallel state until a transition to the parallel state occurs at a much higher field. Thus an effective hysteresis is introduced into the system. Similar behaviors have been noted in more complex systems. The dynamical modes of the two-coupled ferromagnetic films may now be calculated as indicated above. We present the results of this calculation in Fig. 18 where we show the dynamic modes which exist when the system is started at a high external field and the field is subsequently reduced. The change of state from one

R. E. Camley

100 3.0

Net Magnetic Moment (arb units)

2.5

2.0

1.5

1.0

0.5

0.0 0.00

0.05

0.10

0.15

0.20

Applied Field (kOe)

Fig. 17. Magnetization versus applied field for two antiferromagnetically coupled Co films. The hysteretic behavior occurs because the system goes through a spin-flop phase as the magnetic field is reduced from large values, but when the system is started at zero field and the field is increased, the system goes directly from a antiparallel state to a parallel state.

configuration to another is very clearly indicated in the dynamical modes. It is interesting to note that the continuous second order transition from the parallel state to the spin-flop state is associated with a clear ‘‘soft mode’’ where the frequency is driven toward 0. In contrast, the first order transition from the spinflop state to the antiparallel state does not show a soft mode, but instead shows a discontinuous change in the frequency. Theoretical [40] and experimental [41] results for cases where an in-plane wave vector is not zero and the dipole fields need to be included carefully are quite similar to the ones shown in Fig. 18. As seen in Fig. 17, the magnetic structure is different if the magnetic field is increased from zero rather than reduced from a high value. The dynamic modes also show this difference in a very clear way. Figure 19 presents the frequency of the two dynamic modes as a function of field when the field is increased from zero. In contrast to Fig. 18, there is only one transition phase transition which leads to a discontinuous jump in the frequency at H ¼ 100 Oe: It is interesting to see the information that is available in dynamic measurements for multilayers with larger numbers of films as well. Figure 20 shows theoretical and experimental results for a Fe/Cr multilayer with a total of 16 Fe films [42]. One expects 16 closely spaced modes, but from Fig. 20 it is seen that only four modes are resolvable in the low-field region where the spin structure is canted. Nonetheless, it is clear that the main features of the dynamic modes are measurable.

Static, Dynamic, and Thermal Properties 6

Antiparallel

101

Spin Flop

Parallel

Frequency (GHz)

5 4

Acoustic Mode

3 Optic Mode

2 1 0 0. 00

0.02

0.04

0.06

0.08

Applied Field (kOe)

Fig. 18. Frequency of spin excitations versus applied magnetic field for two antiferromagnetically coupled Co films. The field is started at a large value and reduced to 0. The parameters are the same as those used for Fig. 17.

Parallel

Antiparallel

Frequency (GHz)

6 Acoustic Mode 4 Optic Mode 2

0 0.00

0.05

0.10

0.15

Applied Field (kOe)

Fig. 19. Frequency of spin excitations versus applied magnetic field for two antiferromagnetically coupled Co films. The field is started at 0 and increased to a large value. The parameters are the same as those used for Fig. 17.

In Fig. 21, we present the theoretical results for magnetization as a function of field (scan from high to low field) for a Co/Cu multilayer with 19 Co films. In this example, the outermost Co film is assumed to have an exchange coupling that is three times larger than the coupling between all other Co films. Such a situation could be easily arranged by changing the thickness of the Cu spacer layer next to the

R. E. Camley

102

Fig. 20. Experimental and theoretical results for frequency versus magnetic field for a Fe/Cr multilayer. q gives the phase shift between neighboring magnetic layers in an infinite multilayer system. From Ref. [42].

Magnetization (arb units)

0.30 0.25 0.20 19 Co layers in a Co/Cu multilayer 0.15

Outer layer exchange increased by factor of 3

0.10 Scan from high field to low field 0.05 0.00 0.0

0.1

0.2

0.3

0.4

Applied Field (kOe)

Fig. 21. Theoretical results for magnetization versus field for a Co/Cu multilayer with 19 Co films. The thickness of the Cu is chosen so the Co layers have antiferromagnetic exchange coupling between each other. The outer Co layer experiences an antiferromagnetic exchange three times larger than that in the bulk. Some phase changes are easy to see as evidenced by sharp changes in the magnetization.

outermost Co film. Figure 21, which plots magnetization as a function of field, shows a behavior that indicates a number of different phases. This is evident in a theoretical plot, but noise might easily obscure the existence of the different phases in an experimental measurement. Figure 22 shows the behavior of the dynamic


103

19 Co layers in a Co/Cu multilayer 12

Frequency (GHz)

10

Outer layer exchange increased by factor of 3

8 6 4 2

0.0

0.1

0.2

0.3

0.4

Applied Field (kOe)

Fig. 22. Mode frequencies as a function of field for Co/Cu multilayer in Fig. 21. The soft mode near H ¼ 0:33 kOe is due to a surface spin-flop of the outermost Co moment. The different phases are separated by the vertical dashed lines. The bulk spin-flop region exists in the range 0.09 oH o 0.17.

modes in the same structure. Even at high field, where the magnetic moments in all the films are aligned with the external field, one can see that one mode is far away from the continuum of bulk modes. This indicates an impurity-type mode, in this case associated with the outermost film. As the external field is reduced, the impurity mode goes soft (H ¼ 0:33 kOe) as a surface phase transition takes place. This transition is virtually undetectable in the magnetic moment calculation of Fig. 21 because the outer Co film provides only a small portion of the net magnetic moment of the structure.

5. A SINGLE COMPUTATIONAL METHOD THAT PROVIDES STATIC AND DYNAMIC RESULTS Up to now we have discussed equilibrium configurations obtained by the energy minimization method. This method is easy to implement and generally gives good results. However, some of the assumptions that are often used in association with this method can produce incorrect results. An example of this occurred in the study of the behavior of ferromagnetic/antiferromagnetic structures. If one grows a thin ferromagnetic film on an antiferromagnet and cools the system in the presence of a magnetic field, one often finds that the hysteresis curve associated with the ferromagnet is both shifted in field and the coercivity is substantially widened in field. The shift of the hysteresis curve is known as the exchange bias effect and has been studied for many years [43].

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Zero Field

Small reverse field

Fig. 23. Schematic illustration of spin configurations in a ferromagnet/antiferromagnet structure with a spin-compensated antiferromagnetic surface. The ferromagnet is represented by the top two layers.

A recent calculation, based on energy minimization, indicated a mechanism for the exchange bias [44]. This calculation deal with a ferromagnet that was coupled to an antiferromagnet with a compensated surface, i.e. the surface had an equal number of spins from each sublattice. The magnetic structure, in low field, is shown schematically in Fig. 23. When the field is reversed, as is shown in Fig. 23, a domain wall or twist is built up in the antiferromagnet. The building up of the domain wall prevents the ferromagnet from simply reversing at low-reverse field. This produces the exchange bias shift. If the external magnetic field is returned to its original direction, the twist was expected to simply unwind and the system proceeds reversibly back to its original state. The mechanism above would explain the exchange bias effect, but not the enhanced coercivity. Unfortunately, some limitations of the energy minimization calculation obscured some of the physics. In the energy minimization calculation, it is assumed that the magnetic moments in the ferromagnet and antiferromagnet lie in the plane of the sample. Later calculations [45,46] showed that as the domain wall was ‘‘wound up’’ in the reversal process, the spin system would become unstable due to out-of-plane fluctuations. The spin moments would briefly rise up out of plane, and then settle down in a new configuration – effectively unwinding the wall a bit. In a compensated system, this leads to enhanced coercivity, but not to any exchange bias. In a system with some lack of compensation, one can find both exchange bias and enhanced coercivity. The calculations that indicated the instability were based on the Landau–Lifshitz–Gilbert (LLG) equation. This equation is an analog of the classical equation dL=dt ¼ torque and is given by ! ~ ~ dM a d M ~ H ~ eff Þ ~ ¼ gðM M (27) dt M dt where the first term on the right-hand side represents the torque on a magnetic moment due to an effective field Heff, and the second term is a phenomenological damping term which allows relaxation of the magnetization toward the equilibrium direction (Heff) and keeps the length of the magnetization vector a constant in time. The use of the LLG equations can give information on both the static and dynamic properties of magnetic multilayers. In this method one writes a set of LLG


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equations for each layer in the multilayer. ~i dM ~i H ~i Þ a ¼ gðM Mi dt

~ ~ i dM i M dt

! (28)

Layer i is coupled to the other layers through the effective field acting on layer i, Hi, as we have seen earlier. One picks an initial spin configuration and integrates the coupled set of equations forward in time using a differential equation solver such as the fourth order Runge–Kutta method. To find the equilibrium position a large value of the damping constant is used and one integrates forward in time until the Mi values do not change significantly. Again, there is no guarantee that this is the ground state rather than some metastable state so care must be taken to choose an initial configuration that is close to the expected ground state. In contrast to the energy minimization method where it is often assumed that the spin moments lie in the plane of the film, the LLG calculation allows for complete three-dimensional motion of the spins. It is important, therefore, that the demagnetizing fields acting on each film be included in the effective field, because these fields provide the physical mechanism that encourages the equilibrium spin positions to be in the plane of the film. To find the dynamic modes, the damping is then reduced substantially and the system is given a small perturbation. Again the system is allowed to evolve in time and one finds mx(t), my(t), and mz(t) for each layer. The Fourier transform of the time-dependent magnetizations then gives the spin-wave frequencies. The total effective field is a sum of exchange, anisotropy, dipolar, and external fields. We assume that the main contribution to the dipole field is the demagnetizing field of a thin layer acting on itself; thus, 4pMy, where y is the direction perpendicular to the film surface. This approximation works well in the thin film or long-wavelength limit. The method outlined above has been used to study a number of different systems. These include ferromagnetic/antiferromagnetic structures, coupled ferromagnetic films, and magnetic nanostructures. In addition to finding spin-wave frequencies, this method can also find the response of the structure to a driving field. If the driving field is oscillatory in time, one essentially models ferromagnetic resonance, but this method can also find the response of the spin system to a single field pulse. An early example of the use of the dynamical calculation outlined above came in the study of the Fe/Cr(211) system [13–15] which has already been discussed in the previous section. Here theoretical calculations involving the time-integration of the LLG equations provided information on the static phase diagram, the hysteresis curves and the real and imaginary parts of the low frequency dynamic susceptibility [14]. As we have discussed above, there were also some differences between the results for this system found by the energy minimization method and the results found through the integration of the LLG equations. The differences in this case were small, and the theoretical calculations were in reasonable agreement with the measured hysteresis curves and with low-frequency susceptibility measurements [14]. As an additional example, we review the behavior of the exchange-spring system where a ferromagnet with a large anisotropy SmCo is coupled to a ferromagnet

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with a weak anisotropy, Fe. We emphasize that a single calculational scheme and a single set of parameters is able to describe both the static properties (hysteresis curves) and the dynamic properties (spin-wave frequencies) with reasonable accuracy [47]. The parameters for the calculation involve the exchange coupling between layers in the Fe and between layers in the SmCo and at the interface between the two materials. In addition, one needs the magnetization and anisotropy for each material. Except for the interface coupling, all of these parameters are well established for single films of each material. The equilibrium spin structure depends on the direction of the applied field as is illustrated in Fig. 24. For fields aligned with the anisotropy axis, one has a simple configuration where the moments in both ferromagnets line up with the external field. When the field is reversed (Fig. 24b), the moments in the SmCo are held in place by the strong anisotropy in that material. The exchange coupling between the Fe and the SmCo then fixes the Fe moments as well. As the reversed field is increased the outermost Fe moments try to line up with the external field, creating a domain wall in the Fe (Fig. 24c). Eventually the external field is so large that the SmCo moments also switches and lines up with the applied field. The hysteresis curve with the field along the easy axis for 100 A˚ Fe film on SmCo is shown in Fig. 25. The symbols are experimental data and the full line is the result of the calculations using the coupled LLG equations. As the field is reversed from positive to negative values, the magnetization stays constant until about 1.5 kOe due to the pinning of the Fe moments. As the reversed field is further increased in magnitude, the twist seen in Fig. 24(c) occurs and the magnetization is gradually reduced. In the theoretical calculations this twist extends out to a value of about 17 kOe, however, the experiments show the SmCo film switches at a much lower field of 6 kOe. It is assumed that the reason for this is that the model only allows switching to occur by coherent rotation, but that in the sample domain nucleation and domain (a)

(b)

(c)

(d)

Fe

SmCo H

Fig. 24. Schematic illustration of spin configurations in the Fe/SmCo exchange-spring system. In (a) the moments are all aligned with the external field H. In (b) the external field has been reversed but is small. The outermost Fe spins begin to turn toward the field. In (c) the external field is reversed and larger and the Fe moments are nearly completely reversed near the top of the film. In (d) the external field is now so large that all the moments, both in Fe and SmCo line up with the reversed field.


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Fig. 25. Experimental and theoretical results for magnetization in an applied field for a Fe/ SmCo structure. The main difference between the calculated and measured behavior is in the reversal field of the SmCo. If this reversal is aided by domain wall motion the calculation will overestimate the field required for reversal. From Ref. [47].

wall motion may contribute substantially to the switching field for the SmCo. Except for the switching field of the SmCo film, excellent agreement is obtained, and this allows one to obtain a reasonable value for the interface coupling constant. Using the same numerical method, one can now find the frequencies of the dynamic modes as described above. In Fig. 26, we present frequency as a function of applied field for the 100 A˚ Fe/200 A˚ SmCo sample. Having found the interface coupling from the static measurements, there are no adjustable parameters in this calculation. The agreement between experiment and theory is quite good, with the agreement in the region of 5 kOe o H o 0 kOe confirming the picture of a domain wall-like twist in the Fe. The model discussed here has substantial predictive power. One can use the same parameters to calculate the excitation frequencies for structures with different Fethicknesses. Figure 27 shows the theoretical and experimental results for frequency at H ¼ 0 as a function of the thickness of the Fe film. Again the agreement between theory and experiment is excellent and shows a dramatic increase in frequency as the thickness of the Fe is reduced. Similar structures and the resulting frequency range of 20–70 GHz may be used to create new signal-processing devices at high frequencies [48]. This same method has been employed in a number of calculations for the static and dynamic properties of nanostructures. A recent study, for example, examined

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Frequency (GHz)

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-6

-4

-2 0 2 Applied Field (kOe)

4

Fig. 26. Frequency versus applied field for a 10 nm-Fe film on a 20 nm-SmCo film. The region – 5 kOe o Ho 0 spans the twisted state seen in 23 (c) and (d). The filled circles are for measurements with the magnetic field parallel to the easy axis and the open circles represent measurements with the field perpendicular to the easy axis. From Ref. [47].

Fig. 27.

Theoretical (solid line) and experimental (dots) frequencies found at H ¼ 0 as a function of Fe film thickness for Fe/SmCo structures. From Ref. [47].

the properties of Permalloy dots with a diameter of 800 nm and a thickness of 60 nm [49]. For this size of dot, both a vortex magnetization state and a saturated state are possible, with the vortex state favored at low-applied fields and the saturated state occurring in higher magnetic fields. The dynamics of the structure are predicted by using the LLG equations to first find the equilibrium spin structure. This is done with a large value of a to ensure relaxation to the ground state. The value of a is then reduced and the system is allowed to evolve in time and the Fourier transform


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Fig. 28. Fourier transform of the time evolution of one component of the magnetization for a nanodot in the vortex state and in a saturated state. The time evolution is shown in the inset for each case. From Ref. [49].

is calculated as discussed above. The results are shown in Fig. 28. It is immediately obvious that the vortex state has excitations at two frequencies, while the saturated state shows an excitation at only one frequency. These calculations were complemented by Brillouin light scattering measurements. As seen in Fig. 29, these measurements find two modes at low fields where the vortex state is stable and only one mode at high fields where the saturated state is stable. A recent study points out the possibility of studying localized modes in nonuniform magnetic structures. The static and dynamic behavior in a rectangular bar of a ferromagnetic material was explored in Refs. [50,51]. Because of demagnetizing fields the entire structure is not in the saturated state even for large applied fields along the long direction of the bar. Nonetheless, the excitations of the system may be found using the methods outlined above. This calculation used a sophisticated method to calculate the dipole fields and as a result one may find the normal modes of the system without any artificial imposition of ‘‘pinned’’ or ‘‘unpinned’’ spins as a boundary condition at the surface of the nanostructures. As an example a ‘‘photograph’’ of one component of the magnetization is shown in Fig. 30. This endmode excitation in this example has a frequency of 31 GHz, which is considerably lower than the lowest bulk mode at 52 GHz. The frequency difference is due, in part, to the nonuniform demagnetizing field that is largest near the ends of the bar, thus showing the ability of these calculations to allow for local variations.

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Fig. 29. Theoretical calculation (solid lines and dashed lines) and experimental measurements (symbols) for dynamic mode frequencies in a nanodot. At low fields two modes are observed, a result found in the vortex state. At high fields only one mode is found in the saturated state. From Ref. [49].

Fig. 30.

A ‘‘snapshot’’ of the behavior of one component of magnetization in a rectangular nanobar. The picture shows a well-defined end-mode. From Ref. [51].


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6. SUMMARY It is clear that to understand magnetic multilayers and nanostructures both the static configurations and the spin dynamics must be examined. As we have seen, the theoretical methods discussed above produce results which agree remarkably well with the data and allow us to understand the rich physics contained in magnetic multilayers, both in their ground state properties and in their dynamical response to probes. The techniques outlined here give simple methods to obtain complex magnetic structures as a function of magnetic field and temperature. All the parameters that characterize the magnetic material – anisotropy, exchange coupling, magnetization, surface effects, impurities, dipolar interactions, and magnetic structure – play a role in determining the frequency of the allowed spin waves. As a result, measurements of the dynamic properties are very useful in determining the fundamental parameters of magnetic structures. The techniques discussed here have already been applied to a large range of problems. In addition to the study of equilibrium states in nanostructures, there have also been investigations into magnetization reversal in ultra-small objects. Dots [52,53], dot arrays [54], and anti-dot arrays have been examined, and the influence of magnetic pinholes in the spacer layer between ferromagnetic films on the exchange coupling between the films as a function of temperature has been studied [55]. Differences in the dynamics between granular systems and multilayers have been examined [56]. Using the LLG equations, one can also study the nonlinear dynamic behavior of multilayers and other systems [57]. As we have seen in this chapter, magnetic multilayer systems have a rich and varied set of physical properties, many of which have no counterpart in bulk magnetism. Because the importance of interfacial interactions can be controlled by layering pattern or film thicknesses, one can make ‘‘designer materials’’ with magnetic phase diagrams and dynamic response subject to design. In addition, surprisingly simple theoretical approaches can be used to make quantitative predictions which relate the phase diagrams and the dynamic response to specific material parameters.

NOTES 1. See also the chapter in this book by A. Fert.

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Chapter 5 EXCHANGE ANISOTROPY A. E. Berkowitz and R. H. Kodama 1. INTRODUCTION The phenomenon of exchange anisotropy, which is responsible for the shift of hysteresis loops along the H-axis, commonly known as ‘‘exchange bias’’, was discovered by Meiklejohn and Bean [1,2] about 50 years before the publication of the present volume. Since that time, upwards of several thousand papers have been published on that general topic. This extensive worldwide activity is principally due to the fact that virtually every computer containing a magnetic disk for information storage ‘‘reads’’ the stored bits with a sensing device that utilizes exchange bias. It is not an exaggeration to infer that computers would not have achieved their present impressive performance levels without the use of exchange bias. This technological application was initially described in 1978 [3]. The bulk of the literature on exchange anisotropy has appeared since that date, and represents efforts to characterize, model, and optimize the phenomenon. However, the initial papers by Meiklejohn and Bean [1,2], and a subsequent review by Meiklejohn [4] described most of the general concepts currently used to explain the origins and characteristics of exchange anisotropy. The most recent significant contributions to the investigation of exchange anisotropy have utilized the enormous increases over the past five decades in characterization capabilities and in computing power. The technological application of exchange anisotropy involves the interfacial magnetic exchange coupling of thin films of antiferromagnets (AFM) with ferromagnetic (FM) or ferrimagnetic films. Therefore, reliable modeling of exchange anisotropy requires detailed atomic-level information about the microstructure and magnetic states at the interface. The techniques and instrumentation to provide this information are presently available with rapidly improving performance. The computer power to deal rigorously with this information is also available, thanks, in large measure, to the exchange anisotropy phenomenon itself. The objectives of this chapter are to: present the experimental features of ex change anisotropy, and the characterization of the materials and structures involved; Contemporary Concepts of Condensed Matter Science Nanomagnetism: ultrathin films, multilayers and nanostructures Copyright r 2006 by Elsevier B.V. All rights of reproduction in any form reserved ISSN: 1572-0934/doi:10.1016/S1572-0934(05)01005-X

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discuss the development of models to explain the phenomenology; describe the applications of exchange bias; and note those aspects of exchange anisotropy which can benefit from more detailed investigation. A report on a subject with as long a history and as vast a literature as exchange anisotropy, must adopt some selectivity in the work discussed. In the present case, in addition to a necessarily subjective evaluation of the significance of the various investigations, an effort was made to consider those experiments in which the materials characterization was most complete, and the data reliability well supported; and those models in which the connection to experiment was reasonably transparent. Only those coupled systems in which one component is AFM will be considered in this chapter, although the term ‘‘exchange anisotropy’’ has been applied to coupled systems in which both components are either ferromagnetic or ferrimagnetic. Previous reviews of exchange anisotropy include Refs. [5–8]. Ref. [6] is a perceptive review of the early work on coupled films in general, as well as covering exchange anisotropy. It also offers useful insight into the contributions of Neél, which are often not immediately transparent in the original papers. A substantial fraction of the literature on the subject up to 1999 is noted in Ref. [7], with comprehensive data summaries. Ref. [8] focuses on materials and modeling details. Refs. [9,10] are recent topical reviews on modeling.

2. MEIKLEJOHN AND BEAN’S RESEARCH In spite of the prominence of the topic, it is not generally recognized that the discovery of exchange anisotropy was a serendipitous event. Below is W.H. Meiklejohn’s succinct description of how he and C.P. Bean happened on this event. Here it is in a nutshell. Exchange anisotropy was discovered during an attempt to demonstrate the predicted coercive force of hexagonal-close-packed cobalt, based on its crystalline magnetic anisotropy. Single domain particles of cobalt were produced by the electro-deposition of cobalt into mercury. These particles of cobalt were removed from the mercury by oxidizing the surface of the particles. The particles were placed in alcohol. The mixture was placed in a strong magnetic field to align the ‘‘c’’ axes of the particles. The mixture was then cooled to liquid nitrogen temperature to solidify the alcohol in order to hold the particles. Magnetic measurements yielded a shifted hysteresis loop and a sinY torque curve. X-ray diffraction studies showed that the oxide was CoO (antiferromagnetic) and the cobalt structure was face- centered-cubic. When the material was cooled in a zero magnetic field, and when the oxide was removed by treatment in a hydrogen atmosphere, the hysteresis loop was symmetrical. Hence, the effect was attributed to an exchange coupling between the antiferromagnetic CoO and the ferromagnetic cobalt and called exchange anisotropy. Bill Meiklejohn, 13 July 2004

Figure 1 shows the behavior described above by Meiklejohn on a compact of oxide-coated Co particles with diameters 10–100 nm. The strongly shifted loop in Fig. 1(a) was measured at 77 K after cooling the sample from room temperature to 77 K in a field of 10 kOe. Meiklejohn and Bean (M–B) showed that if a unidirectional

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Fig. 1. (a) Hysteresis loops at 77 K of oxide-coated Co particles. Solid curve measured after cooling sample in 10 kOe. Dashed curve measured after cooling in zero field. (b) Torque curves of oxide-coated Co particles cooled to 77 K in 20 kOe and measured in 7.5 kOe. y is the angle between the cooling field and the measuring field. The two curves are for increasing and decreasing y. (From Ref. [2].)

anisotropy, Ku, was added to the free energy density expression, F, for a single domain particle with uniaxial anisotropy, K1, aligned with its easy axis in the direction of the field, H, applied anti-parallel to the particle’s magnetization, Ms, i.e., F ¼ HM s cos Y K u cos Y þ K 1 sin2 Y

(1)


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where Y is the angle between the easy direction and the direction of Ms, solutions to Eq. (1) could be expressed in terms of H0 ¼ H K u =M s

(2)

yielding a hysteresis loop displaced by Ku/Ms. Unidirectional anisotropy is clearly indicated in the torque curves in Fig. 1(b), which were measured on oxide-coated Co particles cooled in 20 kOe to 77 K, and measured at that temperature in a field of 7.5 kOe applied with clockwise (CW) and counterclockwise (CCW) rotations. M–B described the establishment of the unidirectional anisotropy by first noting that the AFM CoO coating on the FM particles magnetically ordered with a Neél temperature, TN, below 300 K. For CoO, the spins on (1 1 1) planes are uniformly magnetized, with adjacent (1 1 1) planes anti-parallel, in a compensated AFM configuration. Since the oxide coated the Co particles’ surfaces, some interfaces could consist of uncompensated (1 1 1) AFM planes, while some others could have unequal numbers of anti-parallel AFM moments, i.e., uncompensated spins [11]. As the magnetically polarized FM was cooled through TN, an exchange coupling could be established at the interface between the polarized FM spins and the uncompensated AFM spins, with a component in the FM magnetization direction. Since the bulk of the AFM spins are compensated, there is negligible direct interaction of the AFM with the applied field. At low temperatures, if the magnetocrystalline anisotropy energy of the AFM material were higher than the interfacial exchange coupling energy, the AFM spin lattice would exert an interfacial torque on the FM, equivalent to an effective field in the initial direction of the FM magnetization. This effective field would result in a shift of the hysteresis loop along the field axis in a direction opposite to the initial FM magnetization direction, as in the solid curve in Fig. 1(a). When the coated particles are cooled in zero field, with random Co magnetizations, a symmetric hysteresis loop results, as shown by the dashed loop in Fig. 1(a). If the AFM coating is too thin on some of the particles, the net AFM anisotropy energy can be less than the interfacial FM–AFM exchange energy, and the FM core can now irreversibly switch (by a discontinuous rotation of the Neél axis) the AFM coating. This is the explanation for the rotational hysteresis apparent in the torque curves in Fig. 1(b). Rotational hysteresis is half of the integrated energy difference between torque curves measured with 3601 CW and CCW rotation of the magnetic field. It indicates the presence of irreversible magnetization changes. For an FM sample, if the field is high enough for saturation, the resulting torque curves will be reversible, i.e., identical for both rotation directions. M–B measured the rotational hysteresis in the coated particle compacts, and found that the rotational hysteresis extrapolated to finite values for infinite fields. This was ascribed to the existence of very thin CoO coatings on a fraction of the particles in the compacts. The same behavior would be noted if the AFM coating simply had a low intrinsic anisotropy compared to the interfacial FM–AFM exchange interaction. Meiklejohn described in detail [4] how the relative values of KFM, KAFM, and Ku determined the magnitude and field-dependence of the rotational hysteresis. Ku/Ms, the loop displacement, is currently called the exchange bias, HE. When FM and AFM thin films are exchange-coupled with an interfacial coupling energy of Ds per unit area (often

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referred to as Ji), H E ¼ Ds=M FM tFM

(3)

if K AFM tAFM Ds; with K and t referring to the uniaxial anisotropy and film thickness, respectively. This estimate implies that the applied field is aligned with the anisotropy axis of the AFM. Investigations of fine FM particles with AFM coatings have continued to the present. However, they have not provided significantly more insight into the basic nature of exchange anisotropy than the original explanations of M–B. The principal reason is the great difficulty in characterizing the FM–AFM interface in these particles. Another problem is that, as with the M–B particles, there is usually a distribution of particle dimensions, although progress is being made on that issue [12]. Thus, we will concentrate on thin film exchange couples for the remainder of this chapter. Reliable characterization and control of these structures is easier than for particles, and the bulk of the literature on exchange anisotropy deals with films.

3. EARLY THIN FILM RESEARCH In the two decades following M–B’s discovery, much of the thin film research on exchange anisotropy utilized superficially oxidized films of Co, Ni, and Permalloy (81Ni–19Fe), which exhibited shifted hysteresis loops at low temperatures. CoO and NiO are two-sublattice AFM materials, as described above for CoO. The AFM oxide on Permalloy was not well identified. In all these samples, it was assumed that the AFM consisted of small grains which were exchange-coupled with the FM. The work with these samples, although incompletely characterized, demonstrated several additional features of exchange anisotropy, as well as providing some very useful insights into modeling the phenomenon. It was observed [13,14] that in oxidized Co films at low temperatures, both the displacement of the loop, HE, and the coercive force, HC, decrease with increasing number of field cycles. The same behavior was found in bilayers of Ni80Fe20/ NixFeyMnz, formed by evaporating a film of Ni80Fe20 onto a film of Mn at 3501C, which produced a layer with NixFeyMnz alloys whose TN ranged from 150 to 2501C. This property, called ‘‘training’’, is found in almost all polycrystalline exchange coupled films. It is also generally observed that the largest decrease comes after the first cycle. Neél [15,16] proposed a model which qualitatively explained ‘‘training’’, the proportionality between the initial large training decrease and the rotational hysteresis, the smaller training effects for subsequent cycles, HE, unidirectional torque curves, and high field rotational hysteresis. His model treats the behavior of an ensemble of interacting AFM grains and their response to the exchange field due to the FM film. The training is shown to occur after the system is prepared in a thermoremanent state by cooling under the influence of the exchange field. This exchange field is assumed small compared to the field required to switch the AFM grain (equivalent to K AFM tAFM Ds) Neél’s modeling is succinctly described in Yelon’s review [6], and, in detail, in the translation of the original paper

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[16]. A basic feature of Neél’s modeling is to treat the AFM grains as entities possessing moments due to unequal numbers of spins in the antiparallel sublattices, i.e., uncompensated spins. This feature has been an important component of several later models, as discussed below. It was also noted during this period that HE vanished at temperatures below the nominal TN of the corresponding bulk AFM. The temperature at which HE vanished was termed the blocking temperature, TB. This behavior was associated with thermally activated fluctuations of the moments of the small AFM grains. This is the superparamagnetism of AFM fine particles discussed by Neél [17], in which the total energy of an AFM particle, defined by the product of the particle’s volume and anisotropy energy density, defines a relaxation time as a function of temperature. Another manifestation of AFM superparamagnetism was the frequency dependence of HE and HC observed by Fulcomer and Charap [18]. The same authors [19] presented a model that included thermally activated fluctuations of the AFM grains, which provided a basis for accounting for TB, as well as the temperature and frequency dependence of HE and HC in oxidized films. Their model considered the oxide layer to consist of non-interacting AFM grains with a distribution of grain sizes. They utilized Neél’s concept that the spin plane of the AFM grain at the interface with the FM film could consist of varying fractions of uncompensated spins, due to, e.g., roughness or simply unequal numbers of spins in the two sublattices. Although their model necessarily included various assumptions and fitting parameters, they were able to get very reasonable agreement with the temperature dependence of HE for Schlenker’s [20] data on oxidized Co and Ni films. Qualitative agreement with the temperature dependence of HC was also demonstrated, with the irreversibility arising from the discontinuous rotation of the Neél axes of those AFM grains with relatively low anisotropy energy by virtue of their exchange coupling to the reversing FM film.

4. INTRODUCTION TO MORE RECENT RESEARCH The pace of research quickened markedly with the description by Hemstead, Krongelb, and Thompson [3] in 1976 of how HE might be used to bias the sensors of stored information in computer hard disk drives. Before considering the principal accomplishments of the more recent work, it is useful to summarize the established aspects of exchange anisotropy at that time. Exchange anisotropy is the result of exchange coupling at the interface between FM and AFM systems. Interfacial AFM uncompensated spins are required for the existence of this phenomenon. The principal manifestations are a unidirectional anisotropy, most directly measured by torque curves, and a shift of the hysteresis loop along the field axis, HE. If Ds is the exchange coupling energy per unit area of the FM–AFM interface, and KFM, KAFM, tFM, and tAFM are the respective anisotropy energy densities and thicknesses of the FM and AFM films, and if is assumed that KFM KAFM, it was suggested that [4]: If K AFM tAFM Ds; there will be a loop shift, HE, a unidirectional torque, and rotational hysteresis will vanish at some finite field.

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If K AFM tAFM Ds; there will be no loop shift or unidirectional torque, and rotational hysteresis will be roughly constant at high fields. If K AFM tAFM and Ds have similar values, or, in the case of polycrystalline films, the AFM grains have a distribution of ratios of K AFM tAFM =Ds; including values both 4 and o 1, there will be a loop shift, HE, a component of torque that is uniaxial, and the rotational hysteresis will be roughly constant at high fields. Contributions to HC can result from the discontinuous switching of the spin system of AFM grains with KAFM tAFMpDs, and its temperature dependence in that case is governed by relaxation times associated with superparamagnetic behavior of the AFM grains. HE vanishes at a blocking temperature, TB, which can be significantly less than TN if KAFM tAFM is comparable to kTN. For polycrystalline FM–AFM exchange couples, both HE and HC can decrease with repeated cycling of the applied field. This is known as ‘‘training’’. All of these experimental features of exchange anisotropy have been reproduced in many different single crystal and polycrystalline thin film exchange-coupled systems in subsequent investigations. The qualitative suggested underlying mechanisms have also been generally accepted. Subsequent modeling has focused on the detailed nature of the interfacial exchange interaction, and how it might contribute to the unidirectional anisotropy, hysteresis, and relaxation phenomena. An important issue was the fact that if the interfacial coupling energy, Ds, in Eq. (3) represented exchange coupling of all interfacial FM and AFM spins, its value, and the corresponding value of HE, would be several orders of magnitude larger than the values actually observed. In the following sections, we will first summarize the different types of AFM systems used, and discuss the most informative observations from those FM–AFM systems, which have been most extensively investigated with the more conventional magnetization techniques. More details of these studies can be found in Refs. [7,8]. A major development has been the introduction of several new experimental techniques which permit element-specific magnetic imaging of surfaces and buried layers of interfacial atomic and spin structures. These will be discussed next. Then, those models that are most transparently associated with the available experimental observations will be considered. In the final section, the applications of exchange biasing will be described.

5. ANTIFERROMAGNETIC SYSTEMS 5.1. AFM Oxides Most studies of thin film exchange couples with oxide AFMs have involved CoO, NiO, and alloys of these two, CoxNi(1x)O. All these monoxides are fcc above TN, with a slight distortion below TN [21]; a rhombohedral contraction along /1 1 1S. For CoO, this distortion is combined with a tetragonal contraction along /1 0 0S. Bulk TN varies linearly with x for the CoxNi1xO alloy [22] , from 293 K for CoO to 525 K for NiO. TN for polycrystalline thin film monoxides decreases with

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decreasing film thickness [23]. The monoxides order below TN with parallel spins on (1 1 1) planes, with antiparallel spin directions on adjacent planes. This spin sublattice arrangement is due to the superexchange interaction between next-nearestneighboring cations along /1 0 0S via the intervening oxygen atom. An AFM domain is identified by its Neél axis, the direction along which the spins are either parallel or antiparallel. With four (1 1 1) planes and the non-cubic distortions below TN, a complex domain structure exists in the monoxides [24–26]. Few measurements have been done of anisotropy constants, primarily due to the difficulty in preparing single domain samples. Spins in NiO can be rotated more easily within the uniformly magnetized (1 1 1) planes than out of these planes. Reported out-of(1 1 1) plane values for K are 1–5 106 erg/cm3 [27–29], while in-plane K values are roughly 103 erg/cm [3,30]. The uniaxial anisotropy value for rotation away from the /1 1 7S easy axis [31] for CoO was measured to be KE2.3 108 erg/cm3 by inelastic neutron scattering [32]. Thus CoO has a much higher anisotropy with a lower TN, as compared to NiO. K varies linearly with x for the alloys CoxNi1xO [33]. The exchange-coupled bilayers studied have included polycrystalline and single crystal films, with various textures and orientations. A comprehensive example of these investigations is by Michel et al. [34] who studied epitaxial and polycrystalline NiO–NiFe bilayers. They found that the (0 0 1)-oriented bilayers exhibited a finite HE which was smaller than that of the polycrystalline ones. Thus uncompensated spins were present even on nominally compensated (0 0 1) interfacial surfaces. As noted in Section 3, these could arise from roughness or defects in these planes. The random fields arising from interfacial roughness or defects are the basis for several models discussed in Section 7. The complexity of the interfacial spin distribution when the AFM is polycrystalline, including the presence of uncompensated spins due to monatomic steps in compensated planes, grain boundaries, and defects such as dislocations, is schematically depicted in Fig. 2. The investigations of Takano et al. [35,36] on polycrystalline CoO–Ni81Fe19 definitively established the correlation between the interfacial uncompensated AFM spins and HE. They first measured the thermoremanent magnetization (TRM) of a series of CoO/MgO multilayers after field-cooling them from T4TN. The TRM was that of the uncompensated spins. Its temperature dependence was similar to that of bulk CoO, as determined from neutron diffraction [37]. The uncompensated AFM spins constituting the TRM were shown to be interfacial and represented 1% of a monolayer of interfacial CoO spins. The temperature dependence of the TRM was the same as that of the CoO–Ni81Fe19 bilayers after field-cooling, as shown in Fig. 3. Therefore, it was concluded that the interfacial uncompensated AFM spins were responsible for the undirectional anisotropy, as had been suggested in the initial work of M–B. Furthermore, the low density of uncompensated spins resolved the issue of the magnitude of Ds noted above. They also demonstrated that HEpL1, where L is the diameter of the interfacial AFM grains. This finding provided a structural basis for the quantitatively predictive model for HE discussed in Section 7. TRM of polycrystalline AFM films or multilayers is a generally useful method to determine the magnitude and temperature dependence of the uncompensated spins present [38].

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123 INTERFACIAL STRESS

FM LAYER

INTERFACIAL ROUGHNESS

AFM

GRAIN BOUNDARY DISORDER

Fig. 2.

CRYSTALLOGRAPHIC ORIENTATION

Schematic illustration of the complexity of an FM/AFM interface which can produce uncompensated AFM spins even on nominally compensated surfaces.

Fig. 3. HE(T) of a Ni81Fe19/CoO(10 m) bilayer, and the TRM of a [CoO(10.3 nm)/ MgO(3.0 nm)]15 multilayer, normalized at 70 K. The overlap for T450 K strongly suggests that the interfacial uncompensated spins responsible for the TRM play a dominant role in the HE mechanism. (From Ref. [36].) Reprinted with permission from the Journal of Applied Physics r 1998, American Institute of Physics.

Takano [39] measured HE(T) of Ni81Fe19 (300 A˚)–CoO(t), with t ¼ 15, 25, 50, and 100 A˚, as shown in Fig. 4. He determined from analysis of TEM plan view micrographs that the CoO grain diameter increased roughly as the square root of the film thickness. Thus the increase in HE(50 K) with decreasing CoO film thickness (for tX25 A˚) is due to the increase in uncompensated spins with decreasing

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Fig. 4.


HE(T) of Ni81Fe19(300 A˚)-CoO bilayers for CoO thicknesses p100 A˚. (From Ref. [40].)

CoO grain diameter, as discussed above. Below 25 A˚, an increasing fraction of CoO grains are too small to possess sufficient magnetic anisotropy energy to pin the Ni81Fe19 film. In Fig. 5, TN and TB are shown as functions of the AFM thicknesses for bilayers of Ni81Fe19(300 A˚) with CoO and NiO [39]. The TN(t) of the CoO and NiO were obtained from heat capacity measurements [40]. Both CoO and NiO reach bulk TN for 100 A˚ thick layers. However, TB for CoO reaches bulk TN for 100 A˚, but TB of NiO does not reach bulk TN for much greater thicknesses due to its much lower magnetocrystalline anisotropy. Prados et al. [41] observed a reversal of the sign of HE near the AFM TB in fieldcooled polycrystalline bilayers of CoO and NiO with permalloy. They consider that the uncompensated spins responsible for HE are located primarily on the boundaries of the AFM crystallites. They propose that the ordering temperature of these spins, TN(gb), is lower than that of the crystallites’ cores, TN(c). Therefore, when TN(gb)oToTN(c), HE reverses sign since the net direction of the new stable uncompensated spins reverses, as a consequence of the antiparallel arrangement of the nearest–neighbor cations. A positive HE was also observed just below TB of 186 K in a Co–CoO bilayer by Radu et al. [42]. These authors ascribed this behavior to the existence of some interfacial regions with a superexchange interaction, resulting in antiparallel coupling of FM and AFM spins, instead of the usually inferred direct exchange [43]. In another observation of positive HE just below TB of a Co–CoO bilayer [44], the suggested mechanism is the distribution of TB values among the CoO crystallites, as discussed for Ni81Fe19-NiO bilayers [45]. Accordingly, some CoO crystallites with lowest TB reversibly switch their net magnetizations into the negative (opposite to the cooling field) direction when the field is negative due to exchange coupling with the FM. This creates a more disordered interface than is present after field cooling in the positive direction, with increased pinning sites, and a consequent higher HC in the positive direction. More detailed interfacial spin

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Fig. 5. TN (from heat capacity measurements of CoO/MgO and NiO/MgO multilayers) [41] and TB (of bilayers of Ni81Fe19(300 A˚) with CoO and NiO) as functions of the AFM thicknesses. (From Ref. [40].)

distribution studies are required to establish the validity of each of these explanations for positive HE in the region just below TB. An innovative approach to increasing HE in epitaxially grown Co/CoO bilayers was proposed by Milteńyi et al. [46] and further modeled [47] and described [48] by them. They diluted the CoO with non-magnetic inclusions of Mg and excess O. For both types of dilution, HE, after field cooling to low temperatures, increased by factors of 3 or 4 for certain concentrations of the diluents. In very brief terms, the explanation offered is that the presence of non-magnetic atoms in the CoO film volume facilitated the formation of low-energy AFM domain walls, since these walls included these non-magnetic atoms. These walls provided net AFM magnetization at the interface, resulting in higher HE. This explanation awaits explicit confirmation of the existence of these walls. However, there is no doubt that this


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approach is very effective in increasing HE in these epitaxially grown films. It is very likely to stimulate a variety of efforts to increase interfacial uncompensated spin density by introducing non-magnetic defects. An important issue that is particularly relevant with the monoxide AFM materials is the likelihood of chemical reactions at the interface. A study of this phenomenon in NiO–NiFe bilayers [49] demonstrated that the existence of two thermodynamically favorable interfacial reactions were possible, producing FeO and Fe2O3, respectively. The thickness of the reaction zone was 1.0–1.5 nm. The possibility of such chemical changes again emphasizes the vital importance of detailed atomic-level interfacial information.

5.2. Metallic AFM The TN’s and anisotropies of metallic AFM materials are generally higher than those of the AFM oxides. For these reasons, they are preferred for most applications, and their behavior in thin film FM–AFM exchange couples has been extensively investigated. However, with a few important exceptions, these investigations have not provided much additional insight into the mechanisms of exchange anisotropy than have resulted from the studies with the monoxide AFMs. For this reason, metallic AFMs will be treated only briefly. Refs. [7,8] may be consulted for more details. The metallic AFM materials have generally been Mn alloys. The paper of Hempstead et al. [3] was seminal not only for initiating the applications of HE, but also for emphasizing the virtues of Mn-based AFM alloys for that purpose. They reported that stable g-FeMn could be deposited on Ni80Fe20 at room temperature, producing a significant HE without cooling from above TN. They noted that Mn alloys with Ni, Rh, and Pt are useful g-phase AFM materials for producing unidirectional properties, and they stressed the importance of the interfacial conditions. The salient properties of some bulk Mn-based AFM alloys are listed below. However, it is not often established how the properties of the thin films compared with those of the corresponding bulk systems. Fe–Mn: The chemically disordered, AFM g-phase is fcc and extends from 30 to 55 at.% Mn at room temperature [50]. TN increases from 425 to 525 K in this range [50]. Most use is made of alloys with X50% Mn to achieve higher TB. The atoms at the (0, 0, 0), (0, 1/2, 1/2), (1/2, 0, 1/2), and (1/2, 1/2, 0) positions form a tetrahedron, and the spins on these atoms are directed along the four /1 1 1S directions toward the center of this tetrahedron. Although this system was initially technologically important, other Mn-based AFMs have shown higher TB’s and better corrosion resistance. Ni–Mn: The chemically ordered fct Y-phase of Ni-Mn extends from 43 to 53 at.% Mn [51]. The Mn atoms, with moments E3.8mB, and the Ni atoms, with o0.2mB, are alternately placed on (0 0 2) planes. The nearest-neighbor Mn atoms are coupled AFM, with the next-nearest-neighbors FM. Above 1050 K, the Y-phase transforms to a non-magnetic bcc phase [51]. Lin et al. [52] reported much

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higher Ds and TB, and superior corrosion resistance, for ordered Ni50Mn50 than for Fe46Mn53 films coupled with Ni81Fe19. Annealing the Ni50Mn50 to produce the ordered fct phase resulted in a higher HC (47.4 Oe) and Ds (0.27 erg/cm2) for that bilayer than for the similarly annealed sample with Fe46Mn53 (HC ¼ 5.2 Oe, Ds ¼ 0.09 erg/cm2). It is plausible that the higher Ds of the ordered Ni50Mn50 sample was responsible for the higher HC, but this possibility was not discussed in the paper. Pt–Mn: Ordered Pt-Mn alloys are AFM from 41 to 66 at.% Mn, with a maximum TN of 975 K at the equiatomic composition [53,54]. Farrow et al. [55] used a variety of substrates and underlayers, combined with annealing, to promote the ordered structure in polycrystalline and epitaxial exchange couples of MnxPt(1x) with Ni81Fe19. Their work illustrates the difficulties in achieving ordered structures in Mn-based AFMs in order to utilize the higher TB, Ds, and corrosion resistance. Ir–Mn: From 14 to 31 at.% Ir, the alloy is a nominally disordered fcc AFM, which minimizes the annealing requirement for applications. TN varies from 600 to 750 K with increasing Ir [56]. The average spins on each (0 0 2) plane are aligned parallel along the c-axis, with alternating signs on neighboring (0 0 2) planes [57]. Imakita et al. [58] reported on Mn73Ir27–Co70Fe30 bilayers in which the Mn73Ir27 was deposited at substrate temperatures up to 2001C, the Co70Fe30 was deposited after cooling to room temperature, and the bilayers were annealed at temperatures up to 4001C for 1 h in 1 kOe. They observed superlattice lines in o1 1 14 textured Mn73Ir27 deposited at temperatures X1001C, indicating the presence of ordered Mn3Ir. Unusually high maximum values of Ds ¼ 1.3 erg/cm2 were achieved when the Mn73Ir27 was deposited at 1701C and the bilayer was annealed at 3201C. TB under these conditions was 3601C, almost 1001C higher than for bilayers in which the Mn73Ir27 was deposited at room temperature. It was suggested that these superior properties were due to the increased presence of ordered Mn3Ir. Tong et al. [59], using Ir–Mn focused on the fact that exchange anisotropy can be established in FM–AFM bilayers deposited in a field below TN, without cooling through TN. This has long been recognized [3], but analysis of the implications of this fact for the mechanisms governing exchange anisotropy have been neglected. These authors investigated trilayer structures consisting of CoFe films sandwiching an IrMn film. The thicknesses of the CoFe films ranged from 2.0 to 5.0 nm, and the IrMn thicknesses were from 2.0 to 7.0 nm. It was found that even when the CoFe and IrMn films were each only of the order of one grain thick, HE of the upper CoFe film could be set in arbitrary directions with respect to the preset direction of HE of the lower CoFe film. The direction of HE for the top CoFe film depended only on the direction of the bias field applied during its deposition. Deposition was carried out at ambient temperature, below the blocking temperature, TB ¼ 2251C. The very small thickness of the IrMn films makes it unlikely that domain walls parallel to the interface exist in the AFM. This result led Tong et al. to suggest that in their polycrystalline AFM films, exchange interactions with some grains could accommodate the upper HE orientation, whereas others could favor the lower HE direction, although the mechanism was unclear. This behavior is corroborated by

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an investigation [60] of the issues raised by the fact that exchange anisotropy does not require field-cooling through TN. 5.3. AFM Fluorides The Fe2+ ions in FeF2 form a body-centered tetragonal structure [61] with the ions at the unit cell center ordering antiferromagnetically with those at the corners [62]. Nogue´s et al. [63] observed an unusual behavior of HE in bilayers of FeF2 (E90 nm)/Fe(E13 nm) grown on (0 0 1) MgO. The sign and magnitude of HE depended on the magnitude of the field applied while cooling below TN ( ¼ 78.4 K) to 10 K. After cooling in a positive field, HE is increased from 200 to +200 Oe monotonically as the cooling fields increased up to 70 kOe, and the sign of HE remained constant until TN was reached. The HE behavior varied with the deposition temperature, which influenced interfacial roughness. Similar dependence of HE on the magnitude of the cooling field was observed with MnF2/Fe bilayers. Both AFM fluorides have similar crystal and spin structures with uniaxial anisotropies along the c-axis. In brief, the suggested model is that higher cooling fields begin polarizing AFM spins in the field direction above TB, and that an AFM coupling between the interfacial AFM and FM spins produces a positive HE once the exchange couple is cooled below the blocking lower temperature. The authors suggested that a certain minimum field is required to create a positive HE, since the Zeeman interaction with the interfacial AFM spins is in competition with the interfacial exchange [64]. It was found that increased roughness, makes the HE less positive, purportedly because it reduces the AFM nature of the interfacial coupling. While it is clear that the change of sign of HE is due to the interaction of the applied field with the AFM, the precise mechanism is not well established. A one-dimensional model utilizing a partial domain wall in the AFM was developed to explain the exchange anisotropy behavior in Fe–MnF2 bilayers [65]. We note the variety of proposed explanations for positive HE discussed earlier for the AFM monoxides and here for the AFM fluorides. It is not yet clear how similar these cases are, particularly because the cooling field dependence of HE was not reported for the AFM monoxides, and because the positive HE in the AFM fluorides appears to persist to zero temperature, while only occurring in a range just below TB in the AFM monoxides. Perhaps further experimental and theoretical work will shed light on possible connections between positive HE in these different systems.

6. PROBING SPIN STRUCTURES 6.1. Neutron Diffraction Neutron diffraction [66] has been an extremely useful technique for determining spin distributions in FM and AFM materials in general, and is now extensively used for studying magnetic nanostructures [67]. The particular relevance to exchange anisotropy is due to the facts that wide-angle neutron diffraction provides

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information on domain sizes and ordering temperatures of FM and AFM [68] layers, while polarized neutron reflectivity (PNR) [69] probes the profiles of buried interfacial structures with high resolution. Some neutron investigations have focused on the interfacial spin structure. PNR studies on Co/LaFeO3 bilayers [70] indicate that a net magnetization is present in the AFM near the interface, and it is constant during field cycling, and is antiferromagnetically coupled to the Co. This finding affirms the dominant role of interfacial uncompensated spins in Ni81Fe19/CoO bilayers [35,36], although in this system the exchange coupling seems to be of the direct Heisenberg type [43]. Neutron diffraction studies of epitaxial Fe3O4/CoO bilayers and multilayers [71] found dramatically different dependencies of TB and TN on the CoO thickness, as shown in Fig. 6. TB progressively decreases for tCoOo50 A˚, while TN monotonically increases in this range, reaching values well above TN for bulk CoO, possibly due to magnetic proximity effects. Although an explicit model for this remarkable behavior is not yet available, it is a dramatic demonstration of the difference between TB and TN. Using neutron diffraction, Fe3O4/CoO superlattices were shown to exhibit perpendicular interfacial coupling [72]. However, the complexity of the possible domain configurations did not permit the conclusion that HE was causally related to the perpendicular coupling as suggested by Koon [73]. Neutron diffraction investigations on Fe3O4/NiO superlatices [74] showed that the NiO domain sizes, after cooling in zero field, shrink in increasing applied fields, while the sizes of the smaller domains that are present after field-cooling in 5 T are independent of the applied field. The nature of the FM reversal process has been examined with neutrons. Gierlings et al. [75], using PNR, found that in [Co/CoO/Au] multilayers, rotation was the mechanism in increasing fields (parallel to the cooling field direction) while

Fig. 6. The measured CoO ordering temperatures, TN (upper curve), and blocking temperatures, TB (lower curve) versus the thickness of the CoO layer for the Fe3O4/CoO system. The divergence of the two curves indicates that the measured reduction of TB is not due to a reduction of the ordering temperature at low CoO thickness. (From Ref. [71].)

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domain-wall motion was operative in decreasing fields. A study of Co/CoO(1 1 1) textured bilayers [76] showed the same two reversal mechanisms for the first cycle as in Ref. [75], but all other reversals were dominated by domain rotation processes. Interfacial magnetic domains strongly coupled to the CoO were also observed in this work. In both these studies, the CoO was prepared by oxidation of the CoO with a thickness of 25 A˚ and TBE186 K, as compared to TN ¼ 293 K. The use of such thin films with TB TN may introduce different issues in the behavior of these samples, as compared with thicker CoO, e.g., if interfacial intermixing is present. Thickness of the AFM was not an issue in studies of bilayers of Fe with MnFe2 (50 nm) and FeF2 (90 nm) [77]. PNR generally showed that in decreasing fields, reversal of the Fe film proceeded by rotation, while domain-wall motion was operative for increasing fields. There was no indication of this asymmetry in reversal mechanisms in the measured hysteresis loops, demonstrating the valuable contribution of PNR. It was also observed that the reversal mechanism was sensitive to the direction of the cooling field with respect to the AFM anisotropy axis. Another PNR study on Fe/MnF2 exchange coupled bilayers [78] indicated that the magnitude of HE was larger when the FM reversal was asymmetric than when the same mechanism was operative on both sides of the hysteresis loop. Thus, the neutron studies have provided detailed information on specific systems, while emphasizing the complex varieties of behavior that are dependent on the microstructure and composition. 6.2. Linear and Circular Magnetic Dichroism The availability of tunable, narrow-band, polarized X-rays from synchrotron sources has enabled the development of several magnetic imaging techniques that rely on element-specific core-level excitations [79]. Magnetic dichroism refers to the dependence of core-level spectrum on the relative orientation of the X-ray polarization and the magnetization direction. Circularly polarized X-rays tuned to the absorption edge of a particular element permit determination of the net FM moment, and is called X-ray magnetic circular dichroism (XMCD). Linearly polarized X-rays allows the magnetic order and the spin axis in AFM materials to be determined (XMLD). Thus both the FM and AFM layer may be examined with these methods, although the top layer must be thin enough for the photoelectrons generated from the core excitations to reach the detectors. Bilayers with metallic AFMs: Both XMCD and XMLD were used to investigate FeMn/Co bilayers with Co thicknesses up to 1.7 nm [80]. The Co and FeMn spin structures were found to be aligned, consistent with direct exchange coupling. Interfacial uncompensated Fe spins were present (i.e., observed with XMCD), and it was suggested that they play a key role in the generation of HE. Both these findings were confirmed in an XMCD study of epitaxial Co/FeMn, with the thicknesses of the top FeMn ranging up to 13 monolayers [81]. A net moment on the AFM layer was found in an XMCD study of Co/Ir25Mn75 bilayers with Ir25Mn75 thicknesses of 7.5–27.0 nm [82]. However, the possibility that the net AFM moment was due to spin canting could not be excluded. Additional confirmation of interfacial uncompensated spins was obtained in XMCD investigations of Co(2 nm)/Ir20Mn80(20 nm) and

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Co90Fe10 (t ¼ 1, 2, and 3 nm)/PtMn(25 nm) [83]. These combinations and thicknesses are prominent candidates for applications. It was determined that the uncompensated pinned spins accounted for 4% of an interfacial monolayer. It was suggested that these uncompensated spins were associated with the grain boundaries of the AFM grains. It is important to emphasize the significance of these studies with metallic AFM films. They all provided evidence of uncompensated AFM interfacial spins. With one exception [81], the bilayers were polycrystalline. Thus, they confirm the initial M–B [11] assumption of the role of uncompensated spins in establishing HE. They also make contact with the experimental results and quantitative modeling of Takano et al. [35,36] of polycrystalline Ni81Fe19/CoO bilayers, namely, the AFM crystallites are AFM domains, and the uncompensated spin density is related to the grain sizes and orientations, and to roughness. Another issue is the fact that the bilayers with metallic AFM films are generally annealed to optimize HE. This almost certainly promotes additional interfacial mixing, which can increase interfacial uncompensated AFM spin density. Bilayers with NiO: A number of dichroism studies have been made with NiO layers since, as with the metallic AFM layers discussed above, TN and TB are above room temperature. NiO has a smaller magnetocrystalline anisotropy than other AFM materials, which is reflected in its behavior in bilayers. Co(3 nm)/NiO(50 nm) bilayers examined with XMCD [83] exhibited uncompensated interfacial AFM spins with a density of 4% of a monolayer, as did the bilayers with metallic AFM films in that study. XMCD-generated hysteresis loops of a Co(3 nm)/NiO(50 nm) bilayer [84] also exhibited interfacial uncompensated Ni spins, and showed that a portion of the Ni spins rotated with the Co layer. A 2.5 nm Co film on a freshly cleaved NiO(0 0 1) surface was investigated with XMLD [85] in fields up to 0.8 T. A planar interfacial wall in the NiO was found as the field direction was varied. Fitting the data included the assumption of uniaxial NiO anisotropy and that the rotation was in-plane and reversible. The estimated interfacial exchange energy was 1% of that of Ni, which is consistent with the general finding that Ds is a few percent of complete interfacial FM–AFM coupling. Zhu et al. [86] used XMLD to investigate bilayers with 2.0 nm of Co84Fe16 on both /1 1 1S textured and untextured NiO. They found a preference for in-plane Ni moments lying along the fieldcooling direction, with no significant difference between textured and untextured films, consistent with similar values of HE. Field-cooling the textured bilayers induced growth of those domains with magnetic axes closest to field direction. They noted the agreement of their findings to those of Takano et al. [35,36]. 6.3. X-ray Photoelectron Emission Microscopy X-ray photoemission microscopy [87] (XPEEM) uses the dichroism of the photoelectrons generated by a focussed X-ray beam having circular (XMCD) or linear (XMLD) polarization in order to provide domain images and spin orientation information from FM or AFM materials. The same thickness constraint on the top layer as with XMCD and XMLD is thus present, and the lateral resolution is


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o50 nm. The utilization of XPEEM to image AFM domains was demonstrated with epitaxial LaFeO3 [88]. The relative orientations of the linear X-ray polarization and the AFM spin direction produced contrast changes in the domain images. Both linear and circular dichroic effects were used in an XPEEM investigation of a bilayer of Co(1.2 nm)/LaFeO3(200 nm) [89]. The magnetic alignment of domains in both the Co and LaFeO3 were correlated. The conclusion was: ‘‘The magnetic alignment of the Co domains, which exhibit an in-plane axis, must therefore be caused by a coupling to uncompensated spins at the LaFeO3 surface with an in-plane component parallel to the projection of the AFM axis.’’ This conclusion provides significant support for an explicit model of the interfacial spin structure in a bilayer exhibiting exchange anisotropy. The ability to image small concentrations of interfacial spins in FM/AFM bilayers with XPEEM was demonstrated in studies of 1.5 nm Co on a cleaved NiO(0 0 1) bulk crystal [90,91]. Figure 7 shows XPEEM images of the NiO and Co domains, as well as FM Ni spins (1 monolayer) at the interface, i.e., XMCD shows the exchange coupling of the Co and NiO at their interface. It also demonstrates the remarkable sensitivity of XPEEM to small spin concentrations. An (0 0 1)-oriented Co(1.2 nm)/LaFeO3(40 nm) epitaxial bilayer, prepared in zero field with no subsequent field cooling, was examined with XPEEM [92]. The (0 0 1) surface of LaFeO3 is compensated with Neél axes along out-of-plane /1 1 0S directions, with in-plane projections along /1 0 0S or /0 1 0S. The Co was thus constrained by exchange coupling to the LaFeO3 and by magnetostatic energy considerations to lie along [7100] or [7010] directions. The two classes of Co domains were imaged using XMCD and XPEEM, and it was observed that each class could be switched with fields along their respective easy axes, while the other class remained undisturbed. Furthermore, HE was measured for more than 1200 individual Co domains as functions of domain area for areas40.1 mm2. Analysis of the data showed an increase of the width of the HE distribution with decreasing NiO XMLD

Ni XMCD

Co XMCD

NiO(001) surface

Co/NiO interface

Co metal

Fig. 7. AFM domains (left) and FM domains (right) in Co(1.5 nm)/bulk NiO(0 0 1). The Ni interface magnetic structure is shown in the center. FM Ni spins (1 monolayer thickness) at the Ni/Co interface show XMCD linking the FM Co layer to the AFM NiO. The FM domain structure of the interface layer is identical to Co domain structure. (Taken from work reported in Refs. [87,90,91]. Courtesy of H. Ohldag).

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domain area, and an inverse dependence of HE on domain diameter, in agreement with the findings of Takano et al. [35,36] on polycrystalline CoO, and with the stochastic modeling of Malozemoff [93]. It is quite clear that the XPEEM, with its magnetic imaging properties of element specificity, and high resolution and sensitivity, will continue to provide the type of nano-scale information required for reliable modeling of exchange anisotropy. 6.4. Mo¨ssbauer Spectroscopy The Mo¨ssbauer effect involves the recoil-free resonant absorption or emission of g-rays by the nuclei of atoms in solids [94]. The nuclear spectrum is split by the dipolar and electric fields associated with the atom. This splitting can be interpreted as reflecting the chemical and magnetic state (valence and magnetic moment) of the atom and its environment. It is particularly fortunate for research in magnetic materials that the 57Fe isotope, which occurs as 2% of natural Fe, can exhibit Mo¨ssbauer spectra when excited with g-rays from 57Co. For the study of films on substrates which would absorb the g-rays, the electrons emitted as the excited nucleus returns to its ground state can be monitored to give the Mo¨ssbauer spectra. This technique is called Conversion Electron Mo¨ssbauer spectroscopy (CEMS). If information at a specific level of a thin film structure is desired, a thin layer enriched with 57Fe is deposited at that location, insuring that the resulting spectrum will primarily reflect the chemical and magnetic states at that site. CEMS was used to study interfacial roughness in Ni80Fe20/Fe50Mn50 (1 1 1) bilayers [95]. Interfacial probes of 57Fe50Mn50 and Ni80 57Fe20 were inserted for that purpose. The conclusion was that the interfacial intermixed zone in which the Mo¨ssbauer spectra deviated significantly from the bulk spectra was only two monolayers wide. Thus the magnetic moments in the Ni80Fe20 very close to the interface in this bilayer had bulk values. This sharp interface was undoubtedly a consequence of the careful preparation of this bilayer by molecular beam epitaxy (MBE). Sputtered Ni/NiO and Fe/NiO bilayers were examined with CEMS, using interfacial probe layers of various thicknesses of 57Fe [39]. Very complex chemical and magnetic interfaces were observed. Figs. 8 and 9 show, schematically, the general features of the Fe interfacial environments for the two types of bilayers (for details of this extensive investigation, the original Thesis [39] can be consulted). More than 97% of the 57Fe atoms were magnetically ordered, and the metallic and ionic 57Fe moments lay in film plane. In the as-deposited state, the interfacial coupling between the interfacial metal and oxide environments appears to be strong and collinear. The dominant metallic environment clearly indicated the presence of stress, with its potential role on exchange coupling. 6.5. X-ray Absorption Spectroscopy Using the synchrotron-based L-edge X-ray absorption spectroscopy (XAS), with its element and chemical state specificity, Regan et al. [96] examined the interfacial properties of bilayers in which the AFM was either CoO or NiO, and the FM was


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Ni

Dilute

57 Fe

in Ni Metal

57

Fe/Ni alloy & strained 57Fe due to epitaxy with oxide 57

Ni x Fe yOz

Oxide

NiO

Fig. 8. Schematic of interfacial environments of Ni/57Fe/NiO films. The metallic 57Fe/Ni alloy and strained 57Fe environments possibly constitute an 57Fe–Ni distribution, and not necessarily line compounds. The interfaces between environments are not smooth, as depicted in this figure. (From Ref. [39].)

α-Fe

α-57Fe Metal 57Fe

Strained due to epitaxy with oxide Nix57FeyOz

Oxide

NiO

Fig. 9. Schematic of interfacial environments of Fe/57Fe/NiO films. The metallic Fe atoms at the interface are stressed due to epitaxy with the oxide. For 57Fe depositions 40.4 nm, the dominant metal environment is a-Fe with bulk properties. The interfaces between environments are not smooth, as depicted in this figure. (From Ref. [39].)

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Fe, Co, or Ni. For all samples, a metal(oxide) layer adjacent to an oxide(metal) layer was partially oxidized(reduced). The oxidized or reduced interfacial regions were 2–4 monolayers thick. It was shown, by an appropriate series of samples, that the oxidation potential of the adjacent cations determined the amount of oxidation/ reduction. Thus, all the interfacial probes indicate redox reactions when the AFM is an oxide, in addition to the atomic mixing which also is present when the AFM is metallic. The robust nature of exchange anisotropy under these complex interfacial conditions is an issue to be resolved. However, it is certainly plausible to consider that the density of uncompensated interfacial AFM spins is actually enhanced by the redox reactions, or by intermixing. 6.6. Some Comments on Spin-Probe Findings Data from these various spin-probe techniques are presently being reported at an increasing rate, and these data will certainly clarify the microstructure and magnetic behavior at the interfaces of exchange-coupled bilayers. It is already clear, as described above, that the AFM interface bears a net moment (i.e., interfacial uncompensated spins). However, in some cases it is found by XMCD that some nominal AFM spins switch along with the FM magnetization. These atoms may simply be exchange-coupled to interfacial FM atoms (in either an FM or AFM sense) with a net combined moment, and thus rotate along with the FM magnetization by virtue of weaker exchange-coupling with the rest of the AFM system. It is expected that such spins would not create an exchange bias, but rather contribute to the overall moment of the FM layer. Those interfacial AFM spins that do not switch with the FM are expected to contribute to an exchange bias. Several mechanisms by which AFM atoms could switch along the FM can be suggested. One is certainly the redox reactions, discussed above, when the AFM is an oxide. Also there is likely to be some degree of interfacial mixing of FM with AFM atoms. A net moment can arise in this case from alloying with metallic FM and AFM layers, or from the formation of a ferrimagnetic interfacial layer if the AFM is an oxide. We can look forward to a satisfactory resolution of these very basic system-dependent microstructural/magnetic issues with the spin-probe techniques currently available, and those under development.

7. THEORY 7.1. Interfacial Uncompensated Spins (IUS) As already discussed, a wealth of experimental data has established interfacial uncompensated spins (IUS) as the basis for exchange anisotropy. A few general comments are in order. This term itself is potentially misleading, since it implies that there are some particular spins that have a special, ‘‘uncompensated’’ character. In fact, the ‘‘uncompensated’’ nature is a property of a finite collection of spins,

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namely the net moment of the collection. The bulk magnetic order of an AFM drives this moment (as a fraction of the total number of spins) to zero for an infinite collection of spins, but it is almost always non-zero for a finite collection. One of the key difficulties in applying this concept to a physical situation is identifying which spins constitute a meaningful collection, in the context of exchange anisotropy. At least for a polycrystalline system having small, single-domain AFM grains, this can be described specifically. The relevant collection consists of the AFM spins in one grain that are near enough to the FM/AFM interface to be substantially exchange coupled to FM spins. The net moment of these spins aligns itself upon field cooling to create an effective field acting on the FM. If one considers the termination of a crystal of AFM material at a planar, free surface, there is an obvious classification of crystallographic orientations into ‘‘uncompensated’’ and ‘‘compensated’’ planes, having a net surface moment and zero surface moment, respectively. Although this distinction is frequently made in the exchange anisotropy literature, it is probably not as important a distinction as it seems. This is because for an interface to be perfectly compensated or uncompensated it would have to be perfectly flat, perfectly oriented, and atomically smooth. Consequently, any real interface is partially uncompensated to an extent that is largely random, but can have well-defined, reproducible average values over a macroscopic interface. The relevant counting of such interfacial, uncompensated spins (IUS) is done over grain or domain areas in the AFM or FM film. However, the detailed relationship between the IUS (and how the counting should be done) and magnetic properties such as coercivity (HC) and loop shift (HE) depends on intrinsic material properties and the reversal process. It is not the case that crystal orientation is unimportant to exchange anisotropy, but that differences in magnetic properties stemming from strain or defect structures are perhaps more important than whether the orientation of an interfacial plane is nominally ‘‘uncompensated’’ or ‘‘compensated’’. The ‘‘modern era’’ of exchange anisotropy theory can be said to have started with the work of Malozemoff [93]. He took the random nature of the IUS and used it to derive an equilibrium domain configuration in the AFM film, based on interfacial exchange energy. Considering a single crystalline, single domain AFM, he took the view that the interfacial spins are locally somewhat random due to atomic-scale roughness, but are globally compensated. He pointed out that the total interfacial exchange energy can be lowered if the AFM breaks up into domains, where the IUS of each domain is aligned with the FM magnetization. Further, he suggested that these domains ‘‘freeze-in’’ as the sample is cooled below the Neél temperature. While this elegant theory highlighted important concepts in the phenomenon, we believe it has limited applicability to real materials, primarily because it treats a highly idealized AFM film. Domain structures in real AFM films are strongly influenced by grain or defect structures as well as the stress state of the film (via magnetoelastic energies). These factors need to be included in addition to the interfacial exchange energy to determine an equilibrium AFM domain configuration, particularly in the epitaxial or quasi-single-crystalline materials that this model considers.

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Koon [73] reported atomistic, classical spin Hamiltonian simulations at zero temperature for idealized FM/AFM interfaces. He argued that a loop shift could occur in the absence of IUS due to a subtle effect in an interfacial domain wall. However, subsequent work by Schulthess and Butler [97] and Stiles and McMichael [98] failed to reproduce the result, particularly if AFM spins were allowed to rotate out of the plane of the interface. Schulthess and Butler found that the ‘‘spin flop’’ state for an ideally compensated interface could contribute to coercivity, but IUS were necessary to simulate a loop shift. Takano et al. [35] directly linked IUS to exchange bias by showing the correlation between the uncompensated moment measured in CoO/MgO multilayers and HE for similarly prepared Ni81Fe19/CoO bilayers. Using experimentally determined distributions in grain size and orientation, and a reasonable model for interface roughness, they were able to predict approximate values of HE as a function of AFM film thickness/grain size. The application of the model was limited by lack of direct information about interface roughness and the microscopic interface exchange. These two parameters have never been definitively determined on any system, and this remains one of the key challenges for experiment and theory. To emphasize the difficulty of this challenge, it is important to know not only the location and chemical environment of interfacial magnetic atoms, but also how their moments are coupled to the bulk of the FM and AFM films. For example, it is conceivable for all or part of the interfacial monolayer of the AFM film to effectively become part of the FM film, or likewise for the interfacial monolayer of the FM film to become part of the AFM film. Important progress has been made in recent years, most notably with Mo¨ssbauer [39,95] and X-ray and neutron scattering techniques (see Section 6), but the information is far from complete. Due to the current impossibility of magnetically imaging a buried interface on an atomic scale, it is likely that progress will be made by a combination of structural characterization, magnetic imaging/scattering, and ab-initio electronic structure calculation (see Schulthess and Butler [99]). We believe that real progress on this point would require substantial and collaborative effort between experiment and theory that has so far been lacking. For the most part, theoretical work has taken the microscopic, interfacial spin structure and exchange interactions as a (poorly known) input parameter, and used various approaches to compute its macroscopic consequences, e.g., HE, HC, etc. Even treatments that are microscopic in nature are hampered by lack of microscopic data on structure and magnetism, as already mentioned, when applied to real materials. Apart from determination of atomic-level magnetic properties of interfaces as outlined above, there are many issues for exchange anisotropy theory not yet discussed, namely: (1) finite temperature, (2) AFM grain coupling, (3) reversal processes, and (4) magnetoelastic effects. Next, we will examine these issues. (1) Finite temperature. The earliest approach to finite temperature is due to Fulcomer and Charap (F–C) [18,19]. By its nature, their model is most applicable to polycrystalline systems, because it assumes negligible interactions between AFM grains. The F–C model is arguably the most quantitatively successful treatment of

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finite temperature, although they relied on distributions of IUS density and anisotropy energy that are essentially fit parameters for HE versus temperature, and were unable to fit the details of the temperature dependence of HC. Stiles and McMichael [100] investigated finite temperature within a modified F–C model, where the AFM grains were assumed to reverse in an inhomogeneous mode in contrast to the uniform rotation assumed by F–C. They were able to simulate, among other data, the ‘‘rotatable anisotropy’’ observed by FMR in terms of AFM grains relaxing via thermally activated reversal in the time scale of field changes. Nowak et al. [47] have carried out Monte Carlo simulations on diluted Ising models for AFM layers, successfully predicting the influence of bulk defects on exchange bias and its temperature dependence. However, it is not clear how appropriate their physical model is for materials with low bulk defect density, although in principle the model could be modified to treat realistic grain/twin boundaries via a nonrandom distribution of defects. Their quantitative discrepancy with experiments [48] on samples with variable bulk defect density highlights the difficulty, although the work illustrates important qualitative features and alternative routes toward increasing HE experimentally. In general, the Monte Carlo approach may not be the best choice because of difficulties in simulating slow dynamics on a calibrated time scale, and ‘‘signal-to-noise’’ problems can be an issue for large arrays of spins due to computational limitations. Some authors have considered ‘‘local mean field’’ models, but such models do not treat thermally activated effects properly. The present authors have recently developed [101] an extension of the thermal activation formalism used by F–C, for a system with finite FM–FM and AFM–AFM exchange, accounting for the complexities of FM reversal processes as well as FM–AFM random fields. (2) AFM grain coupling. In general, we should expect a net AFM–AFM coupling, since any general argument that would say this coupling is zero, would say that FM–AFM interfacial coupling is zero as well. However, it is reasonable to expect that this coupling is ‘‘weak’’ compared to bulk exchange, and F–C-type models have assumed that this coupling is negligible. Fujiwara et al. [102] used a zero temperature model to investigate the influence of coupling between AFM grains. They argue in particular, that the training effect hinges on the significance of such coupling. They simulated arrays of interacting grains and compared their results to experimentally measured torque and M–H curves. Key results were the necessity of AFM–AFM coupling and the anisotropic distribution of AFM grain easy axes within the film plane, presumably induced by field cooling. Kai et al. [103] reported a first-principles electronic structure calculation of direct exchange coupling across a fairly crude model of a grain boundary in the AFM NiMn, showing a fairly complex dependence of the exchange on interface structure. (3) Reversal processes. Almost every theoretical consideration of exchange bias assumes a uniformly magnetized FM film. There are some experimental situations where this is completely appropriate, such as high-field torque, or particulate systems (such as the Meiklejohn–Bean experiment). There are also systems that are thought to behave in a quasi-single-domain manner under some conditions

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(i.e., certain range of field angles with respect to the easy axis), but this is often inferred from indirect evidence. A notable exception to the above generalization is the work of Zhang [104], who used the random field concept to derive the pinning of an idealized FM domain wall under the influence of FM–AFM coupling at zero temperature. He was able to successfully predict the FM film thickness dependence of the average coercivity. In later work, Li and Zhang [105,106] used a zero-temperature, Landau–Lifshitz–Gilbert equation integration method to study a macroscopic model of FM/AFM bilayers. They found a variety of results that are largely verified by experiment, including the scaling of HE and HC with key structural parameters like film thickness, grain size, and texture. They also find the significant qualitative result that the AFM domains postulated by Malozemoff for a single crystal AFM film are not frozen during a hysteresis cycle, even at zero temperature. Zhang’s important work is unique in its realistic treatment of the problem on a macroscopic level, by considering domain processes in both FM and AFM layers, and by its contact with experimentally accessible parameters. Various experimental studies mentioned in earlier sections infer magnetization reversal by rotational processes or domain wall motion. It becomes clear considering the results of Zhang et al. and our own simulations [101] that such characterizations are not always mutually exclusive. In particular, if a rotational process is inferred, for example by a component of the net moment perpendicular to the applied field, it does not rule out domain wall motion or non-uniform reversal processes. A comprehensive theoretical picture should include treatment of domains in the FM film, although the temptation may be to rely on a uniform rotation model for the FM (or even a completely one-dimensional model as is common in the literature) in light of experimental evidence of rotation under certain conditions. (4) Magnetoelastic effects. Some AFM materials (e.g., NiO, CoO) are known to have strong magnetoelastic coupling. Given the substantial effect of magnetostriction on the magnetic properties of FM thin films, it would be remarkable if magnetoelastic effects were not significant in all AFM materials. For example, the domain configuration in NiO and CoO crystals is very sensitive to pressure. This has been utilized in neutron investigations of bulk CoO [107], where a uniaxial pressure was applied to obtain a nearly twin-free crystal. Researchers of NiO crystals [30] noted that the distribution of twins could be strongly, irreversibly modified by the light touch of a finger. Twins are almost omnipresent in these materials, since they undergo magnetoelastic distortions upon magnetic ordering. The resultant twin configuration, even if the specimen is a single crystal above the ordering temperature, undoubtedly depends on the global stress state of the system. This includes considerations of interface structure, differential thermal expansion of substrate and films, magnetostriction of the FM film, and deposition-induced stress. To further complicate matters, the question of the bulk AFM spin state in thin films is not settled. For example, theoretical considerations of NiO and CoO nanoparticles [108–110] indicate that the bulk spin state is sensitive to the finite size of grains, as well as to minute structural distortions. These systems are predicted to have distinctly non-bulk-like spin configurations due to their finite size.

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7.2. Outlook and Current Work We believe that the difficulty in treating magnetoelastic effects is a serious impediment to a full, quantitative description of epitaxial or quasi-single-crystalline materials. Even if magnetoelastic energies were known to be weak in a particular material, the problem of formation of AFM domains and motion of AFM domain walls is still not understood. Perhaps Malozemoff’s model is applicable to some material systems, but the main premise of domain formation is unproven by experiment. In addition to knowledge of the equilibrium domain structure, it would be of importance to consider the ‘‘freeze-in’’ process itself, particularly in light of the possibility to set the bias direction well below TB. For example, considering field-cooling through TN/TB: Does a well-defined AFM order set in first, then the domains flip or nucleate (as an activated process) into the appropriate direction? or Does the AFM order set in with domains already in the appropriate direction? Furthermore, the understanding of domain wall pinning and motion in the AFM would be important to treat finite temperature and/or training effects, but there has been very little theoretical or experimental work in this area. We argue that the situation is somewhat better for polycrystalline materials. Surely, magnetoelastic effects still need to be considered, but it is reasonable to suppose that the main effect is a stress-induced anisotropy on each AFM grain. The anisotropy of each grain will have contributions from bulk magnetocrystalline, stress, and surface/interface anisotropies. While it may be difficult to calculate the net anisotropy from first principles, it should have some well-defined value that in principle can be measured experimentally, either by inference from exchange anisotropy (as done by F–C) or by direct imaging of AFM grain switching (e.g., by XPEEM). The main advantages of polycrystalline materials with respect to theoretical treatment is that there are fixed regions (AFM grains) where IUS are to be counted, and the activation energy barriers are more easily defined. A case where epitaxial systems could be quite interesting is if the films are subdivided/patterned into very small regions, such as by lithography. It may be possible, by appropriate choice of substrate and deposition process (establishing an appropriate state of stress) to create truly single-domain AFM islands coupled to single-domain FM islands. It is expected the regions would need to be rather small and have a great degree of structural perfection to eliminate domains in both FM and AFM layers. For example, Baltz et al. [110] estimated a domain wall width in IrMn of 25 nm, so the critical size for this material would be smaller than this. They also point out the difficulty in patterning such small islands via post-processing without causing substantial structural damage (for this reason they advocated deposition on pre-patterned substrates). While there have been a number of studies of patterned exchange couples, they are typically polycrystalline, multidomain materials so the theoretical interpretation is not necessarily simpler than for macroscopic samples. The problems of determining atomic-level magnetic properties of interfaces, as well as the proper treatment of domain formation and evolution in AFM films are still open questions that are far from satisfactory solution, but at least the former has

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begun to be seriously attacked experimentally. Within the phenomenology of IUS, detailed simulation of exchange anisotropy effects is possible, at least for polycrystalline systems. Recently, we have developed [101] a simulation technique that addresses some of the limitations of earlier work. We represent each FM grain with a single macroscopic spin, and each AFM grain with two macroscopic spins. The FM–AFM coupling is given some randomness to represent IUS, and finite values of FM–FM and AFM–AFM coupling are included in a large array of grains. In this way, we can simulate reverse domain nucleation and growth in the FM film, and any other cooperative effects. An energy surface ‘‘walking’’ technique is used to search for saddle points in configuration space that represent activation barriers in the system. Once the local energy surface is characterized, a Arrhenius–Neél approach is used to determine whether a particular transition occurs in the experimental time scale. In this way, finite temperature effects, with possible slow-dynamics, are simulated, including training and resetting of bias at temperatures below TN.

8. APPLICATIONS OF EXCHANGE ANISOTROPY All modern computers use exchange-bias in the magnetic thin film sensors that ‘‘read’’ the ‘‘bits’’ stored on the rewriteable magnetic hard disk. Figure 10 is a schematic view of an integrated read-write head addressing the recording medium in a computer disk drive. The recording medium is a magnetic film, 10–20 nm thick, with high HC and remanence, BR. It is supported on a flat disk that rotates at rates from 7200 to 15 000 rpm, and is protected with a diamond-like carbon coating, 5–10 nm thick, and covered with 1 nm of a lubricant. The read-write structure is mounted on a lever arm (‘‘slider’’) that is aerodynamically suspended 10–20 nm (‘‘d’’ in Fig. 10) above the lubricant layer. The radial position of the readwrite head is precisely controlled by the slider. Pulsed currents in the winding of the high permeability write element produce the magnetic fields to write the bits. The write element selectively magnetizes small regions in the desired binary code pattern

Fig. 10.

Illustration of a ‘‘Read-Write’’ head addressing the magnetic storage film on a hard disk in a computer hard disk drive. See text for details.

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of 1’s and 0’s on a ‘‘track’’ determined by the radial position of the head as the storage disk rotates under it. The 1’s and 0’s are distinguished by whether or not there is stray flux emanating from regularly spaced locations on a track. Stray flux is present when adjacent bits are magnetized in opposite directions. Since the magnetic film is composed of grains with random shapes and positions (‘‘D’’ in the inset), the region separating oppositely magnetized bits has a finite width (as indicated in the inset by ‘‘a’’), which is a source of noise. As depicted in Fig. 10, the read sensor is located between the left-hand pole of the write element and another high-permeability film. The read sensor is thereby shielded from exposure to the flux from bits other than the one being read. The arrow in the read sensor (visible through the cut-out in the write element pole) indicates the neutral direction of the magnetization of the sensing film which rotates within the white sector angle depending on the magnitude and direction (up or down) of the stray flux from oppositely magnetized adjacent bits. The basic sensing phenomenon is Giant Magnetoresistance (GMR). The integrated multilayer thin film sensor structure utilizing GMR is called a spin valve. When the stray flux from a bit produces a change in the magnetization direction of the sensing film in the spin valve, the resistance of the spin valve changes, as monitored by a bias current, and this resistance change is the read signal. GMR was discovered in 1988 [111], and the first implementation of GMR in a spin valve, using exchange bias, was reported in 1991 [112]. Only a brief qualitative account of the phenomenon will be presented here; excellent reviews of GMR [113,114] and its implementation in exchange-biased spin-valves [115–117] can be consulted for detailed discussions of the physics and structural issues. GMR was discovered in thin film multilayers (ML) of alternating FM and non-magnetic (NM) metallic layers. Cross-sections of these multilayers are illustrated in Fig. 11, with arrows indicating the magnetization directions of the FM films. When successive FM films have anti-parallel magnetizations, the resistance measured parallel to the ML planes, is maximum. When the magnetizations are parallel, the resistance is minimum. When the NM layers are thin enough, exchange coupling exists between successive FM layers, with a sign that oscillates between FM and AFM with increasing NM thickness [118]. This oscillatory behavior is present for a number of NM spacer materials with thicknesses up to 15 monolayers, and with periods varying with the NM material [113,114,119]. As illustrated in Fig. 11, for a ML in zero field with NM spacers of the correct thickness, the AFM-coupled FM layers have anti-parallel magnetizations, and the resistance is maximum. When an applied field produces parallel magnetizations, resistance is a minimum. Figure 12 shows the normalized GMR behavior at 4 K for the AFM-coupled Fe/Cr multilayers in which GMR was discovered [111]. Figure 13 illustrates the simplest thin film structure in which GMR spin-valve sensors employ exchange-bias. One FM film is exchange-biased by an AFM film such that its magnetization is ‘‘pinned’’ in the direction indicated by the open dashed arrow. A NM conducting film, usually Cu, separates the pinned film from the ‘‘free’’ FM film. The NM film is chosen thick enough to minimize exchange coupling of the two FM films, of the type discussed above for the Fe/Cr multilayers.

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H=0

H = H(saturation)

Fig. 11. Schematic of cross-section of multilayer of alternating FM (white) and non-magnetic (dark) metallic films. When successive FM films are magnetized anti-parallel, the inplane resistance is maximum. Resistance is minimum with parallel magnetization of FM films.

Fig. 12.

Normalized GMR behavior in Fe/Cr multilayers with various Cr thicknesses. (Taken from Ref. [111].)

The free film has an easy axis (the dashed arrow) determined by its anisotropy field direction and any residual exchange coupling with the pinned film. The magnetization of the free film rotates, as indicated by the solid arrows, in response to the presence of stray flux from the bits on the storage disk. This creates a resistance change, as measured with a bias current in the plane of the films. The pinned magnetization direction and the easy axis of the free layer are crossed, as indicated in Fig. 13, to produce a linear response. An important technological objective is to make the sensor as thin as possible, so that the distance between the shields in the shields in the read sensor is compatible with high bit densities. The AFM and FM


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R TO C M M ED DU E F N AF N E N O FR PI C M N FM

Fig. 13. Simplest spin-valve utilizing GMR. The pinned FM film is biased with an AFM film. Open arrow shows pinned direction, which is crossed with the easy axis (dashed arrow) of the free FM film.

films are o8 nm, and the Cu spacer is the order of 2 nm. The value of HE must, of course, exceed the magnitude of the stray-flux fields being sensed. These fields are of order 100 Oe. Thus the combination of Ds and the thickness of the pinned FM film must produce an adequate HE. The initial commercial spin-valves of this type used NiO as the AFM pinning material. Since NiO is an insulator, it did not shunt any of the bias current used for sensing the resistance change. Minimizing the bias current reduces the possibility of heating the sensor to temperatures close to, or exceeding TB. Another advantage of NiO was that it served as a specular reflector of the transport electrons [120], thereby increasing the S/N ratio. However, the TB of NiO is lower than those of the Mn-based AFM materials, and it was replaced by Ni–Mn, and finally, by Pt–Mn. Various combinations of AFM and FM films have supplemented the basic structure in Fig. 13 [116,117]. One problem with the simple sensor design illustrated in Fig. 13 is the magnetostatic interaction of the pinned FM layer with the free FM layer, with a resulting degradation of performance. Another is the tendency of the magnetization of the pinned FM layer to curl at the edge adjacent to the recording medium due to its demagnetizing field [121]. This problem is minimized by replacing the single pinned FM layer with a pair of FM layers exchange-coupled in an AFM configuration by an intermediate conducting film, typically Ru or Cu, utilizing the GMR phenomenon discussed above [122–124]. This sensor structure is shown in Fig. 14, and is known as the anti-parallel pinned (A-P pinned) or synthetic spin valve. Since the net moment from the AFM-coupled FM films is much smaller than that of a single FM film, the pinning strength due to coupling with the AFM (HE) is increased, and the

Exchange Anisotropy

FM

C

EE

FR

N

M

C

O

N

-2

D U

G PL U

O C

FM

IN

-1 FM

AF

M

N

TO

M

R

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Fig. 14. Spin-valve using a ‘‘synthetic’’ AFM. FM-1 and FM-2 are exchange coupled with vertical anti-parallel magnetizations by the thin intervening non-magnetic conductor. FM-1 is pinned vertically as shown, with an AFM film. The easy axis of the free FM film is crossed with the axis of FM-2.

interaction with the free film and curling at the edge of the pinned film are minimized. PtMn is probably the most frequently used AFM in this application. Spin polarized tunneling is another magnetoresistive effect. The basic structure is two conducting FM films separated by an insulator whose thickness is of the order of 1 nm. A bias voltage applied to the FM films can produce electron transport by a tunneling mechanism, with a conductance that depends on the relative orientation of the magnetizations of the FM electrodes [113,125]. In 1995, a significantly large tunnel magnetoresistance at room temperature was demonstrated [126]. Since that time, there has been a significant effort to develop magnetic-tunnel-junction-based non-volatile magnetic random access memories (MRAM) to replace the current semiconductor-based memories [127,128]. As illustrated in Fig. 15, the basic cell is an AFM-exchange-biased FM film separated by a non-magnetic insulating layer from a free FM film. The free layer magnetization is switched either parallel or anti-parallel to that of the biased film for maximum or minimum conductance, respectively. An A-P pinned structure, such as illustrated in Fig. 14 for the spin-valve, is also under investigation for biasing the magnetic tunnel junctions [129]. The reports of magnetic tunnel junction resistance ratios far in excess of 100% [130,131] will certainly increase the prospects of commercial applications in MRAM and spin valves. Other applications of the biasing properties of AFM materials and A-P pinning are under active consideration. A magnetic tunnel junction transistor has been described [117] in which the emitter is biased with Ir–Mn. Tondra et al. [132] have reported a picotesla field sensor based on a biased magnetic tunnel junction. It is likely that the very attractive and economical field sensing capabilities of biased spin valves and magnetic tunnel junctions will find an increasing number of applications in the future.


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R

M

IN

FM EE

FR

ED NN

SU

PI

AF

M

FM

O

T LA

N

Fig. 15. Magnetic tunnel junction in which one FM film is biased, as shown, by an AFM film. The free FM film is switched either parallel or anti-parallel to the biased film, producing maximum or minimum conductance, respectively, of the spin-polarized electrons tunneling across the non-magnetic insulator.

ACKNOWLEDGMENTS We are especially grateful for the stimulating collaborations with our present and former colleagues at CMRR and the Physics Dept. in UCSD – Matt Carey, Ken Takano, Fred Parker, Fred Spada, Frances Hellman, and YunJun Tang. The investigations of Gareth Thomas, David J. Smith, and Wei Cao have helped us to appreciate the central importance of microstructure in exchange anisotropy. The neutron studies of Julie Borchers, Ross Irwin, Chuck Majkrzak, and Gian Felcher have kept us aware of the variety and complexity of interfacial interactions. We are, of course, delighted that Bill Meiklejohn agreed to describe the birth pangs of exchange anisotropy. RHK acknowledges the support of the State of Illinois, Petroleum Research Fund, and ORNL Center for Computational Sciences. AEB acknowledges the support of CMRR. It seems entirely appropriate to dedicate this chapter to Charlie Bean, a wonderful scientist and associate, and hope he would have approved our presentation.

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Chapter 6 SPIN TRANSPORT IN MAGNETIC MULTILAYERS AND TUNNEL JUNCTIONS A. Fert, A. Barthe´le´my and F. Petroff ABSTRACT Spintronics, at the interface between magnetism and electronics, includes a broad class of phenomena based on the influence of the spin on the electronic transport in ferromagnetic materials. This influence has been known for a long time in bulk materials but its exploitation in magnetic nanostructures came only in the eighties. The giant magnetoresistance (GMR) of the magnetic multilayers, discovered in 1988, opened the way. Nowadays, the GMR is used in various devices and, in particular, in the read heads of the hard disc drives (HDD) of our computers. The tunnelling magnetoresistance, an effect similar to the GMR but observed in magnetic tunnel junctions, is now applied in some HDD and a new type of magnetic memory called magnetic random access memory. Several promising new directions of research are emerging today. In this chapter, after having presented the fundamentals of conduction in ferromagnetic conductors, we describe the basic physics of spin transport in magnetic multilayers and tunnel junctions.

1. INTRODUCTION Spintronics today includes a broad class of phenomena exploiting the influence of the spin on electron transport in magnetic nanostructures. Its development followed the discovery of the giant magnetoresistance (GMR) of the magnetic multilayers in 1988 [1,2]. A magnetic multilayer is a film composed of alternate ferromagnetic and nonmagnetic layers, for example Fe and Cr. The resistance of such a multilayer is lowest when the magnetic moments of the ferromagnetic layers are aligned, and highest when they are antiparallel (AP). As the relative change of resistance can be Contemporary Concepts of Condensed Matter Science Nanomagnetism: ultrathin films, multilayers and nanostructures Copyright r 2006 by Elsevier B.V. All rights of reproduction in any form reserved ISSN: 1572-0934/doi:10.1016/S1572-0934(05)01006-1

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as high as 220%, this effect has been called GMR. The first GMR experiments were performed in the current in plane (CIP) geometry. After 1991, experiments in the current perpendicular to the plane of the layers (CPP) geometry have also been performed and have led to the analysis of the physics of spin accumulation. The GMR has important applications, in particular for the read heads of hard discs. The discovery of GMR triggered an extensive research effort on spin transport in magnetic nanostructures and other interesting effects turned out rapidly. One of the most important is the tunnelling magnetoresistance (TMR) of the magnetic tunnel junctions (MTJ). A MTJ is composed of two ferromagnetic conducting layers separated by an ultra-thin insulating layer. As in GMR, the resistance of the junction is lower when the moments of the ferromagnetic layers are parallel. The first TMR experiment (at low temperature) actually dates back to 1975 [3] but it is only in 1995 [4] that the observation of large and reproducible effects kicked off the research effort on MTJ. The magnetic random access memory (MRAM), a new type of memory for computers, will be an important application of TMR. The physical origin of GMR, TMR and most spintronic effects is the influence of the spin on the conduction or tunnelling properties for the electrons of a ferromagnetic conductor. The different conduction properties of the majority and minority spin electrons in a ferromagnetic metal was first suggested by Mott [5]. The so-called ‘‘two-current conduction’’ was then confirmed, quantified and modelled around 1970 [6–9], long time before the discovery of GMR. The spin polarization of the conduction results from the spin splitting of the energy band in a ferromagnet. The spin polarization of the electrons tunnelling from a ferromagnetic metal across an insulator also results from the spin splitting of the energy band but can be quite different from the spin polarization of the conduction inside the ferromagnetic metal. All the sections of the chapter are focused on the basic ideas and key concepts involved in the physics of spin transport, GMR and TMR. For specific details we refer the reader to more technical review papers. For GMR, the most recent are those by Barthe´le´my et al. [10] and by Tsymbal and Pettifor [11]. For TMR we refer to Tsymbal et al. [12]. The book ‘‘Spin dependent transport in magnetic nanostructures’’ [13] includes chapters on GMR by Shinjo and by Levy and Mertig, on TMR by Miyazaki and by Maekawa et al., and on the applications of magnetic nanostructures by Parkin.

2. SPIN-DEPENDENT CONDUCTION IN FERROMAGNETIC METALS The origin of GMR and spintronics is the influence of the electron spin on the electronic transport in ferromagnetic conductors. This is a consequence of the spin s of the energy bands in the ferromagnetic state, as this has been first suggested by Mott [5]. The spin dependence of the conduction in ferromagnetic metals and alloys has been experimentally demonstrated and quantitatively described around 1970, by Fert and Campbell [6,7] for series of iron- and nickel-based alloys, and then for a great variety of systems, see Loegel and Gautier [8], Dorleijn and Miedema [9] and

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for a review Campbell and Fert [14]. These experimental results could be accounted in the ‘‘two-current model’’ of the conduction in ferromagnetic metals [6,7,14]. In the low-temperature limit (T T c ) of the ‘‘two-current model’’, when the spin-flip scattering of the conduction electrons by magnons is frozen out, the spinmixing rate is much smaller than the momentum relaxation rate and there is conduction in independent parallel channels by the spinm (majority) and spink (minority) electrons. The resistivity of the ferromagnet (FM) is then expressed as r¼

r" r# r" þ r#

(1)

where rm (rk) are the resistivities of the spinm (k) channels. In a given channel s, the parameter rs includes the contributions from s, d and s–p or s–d hybridized electrons and from various scattering processes. The asymmetry between the two channels is characterized by spin asymmetry coefficients: r# r"

(2)

ðr# r" Þ a 1 ¼ ðr# þ r" Þ a þ 1

(3)

a¼ or alternatively, b¼

There are several origins for the difference between rm and rk. Schematically, the resistivity rs can be written as a function of the number ns, effective mass ms, relaxation time ts and density of states (DOS) at the Fermi level Ns(EF) of spin s electrons in the following way: rs ¼

ms ns e 2 t s

(4)

with, for a single type of scattering potential characterizefd by its matrix elements Vs and in the Born approximation, 2 t1 s jV s j N s ðE F Þ

(5)

There are intrinsic origins of the spin dependence of rs that are related to the spin dependence of ns, ms, or Ns(EF). In transition metals (TM), the most important of these intrinsic origins comes from the proportionality of the relaxation rate to the DOS, Ns(EF), in Eq. (5). In first approximation, it can be said that a major part of the current is carried by light electrons of s character and that these electrons are more strongly scattered when they can be scattered into heavy states of the d band for which the DOS is large. In Ni, Co and alloys like NiFe or CoFe, the dm band is below the Fermi level (i.e. Ndm(EF) ¼ 0, whereas Ndm(EF)6¼0), s–d scattering exists only for the spin k s electrons and there will be a general intrinsic tendency for stronger scattering and higher resistivity in the spin k channel. In other words, the spin k d states at the Fermi level contribute weakly to the conduction but strongly to the scattering rate, Eq. (5), of the electrons of predominant s character.


156

Fig. 1.

Resistivities, rm and rk, induced by 1% of several types of impurity in the spink and spinmchannels of Ni [7,14].

On the other hand, there are also extrinsic origins of the spin dependence of rs related to the spin dependence of the impurity or defect potential Vs. In Fig. 1, we show the spinm and spink resistivities induced by 1% of several impurities in Ni. For Co and Fe, the ratio a of rk to rm is larger then 1 (b40) and can be as large as 20, which is consistent with the general tendency described above. In contrast, for Cr (or also V), a is smaller then 1 (bo0). This can be explained by the electronic structure of Cr impurities in Ni. The spinm d levels of Cr are at an energy above the spinm d band of Ni. This prevents the hybridization of the dm states of Cr with the dm band of Ni. It is replaced by an hybridization with the sm band of Ni to form a virtual bound state at an energy close to the Fermi level. This leads to a strong scattering in the spinm channel and explains the higher mobility in the spink channel for nickel with chromium impurities. In multilayers, similar arguments will explain the opposite spin asymmetries of the scattering at, for example, Co/Cu and Co/Cr interfaces. The ratio a has been derived from resistivity measurements for many Ni-, Coand Fe-based alloys (see the review by Campbell and Fert [14] for extensive tables). The experimental data have been first explained by qualitative electronic structure arguments and then confirmed by ab initio numerical calculations [15]. Away from the low-temperature limit, it is necessary to take into account the transfer of momentum between the two channels of conduction by spin-flip electron–magnon scattering. Spinm (spink) electrons are scattered to spink (spinm) states by annihilating (creating) a magnon and the transfer of momentum from the fast to the slow channel – the so-called spin-mixing effect – tends to equalize the two currents. In this case, the general expression of the resistivity is r¼

r" r# þ r"# ðr" þ r# Þ r" þ r# þ 4r"#

where rmk is the spin–mixing resistivity term [7].

(6)


157

3. GMR: EXPERIMENTAL SURVEY The GMR was discovered in 1988 by Baibich et al. [1] on Fe/Cr (0 0 1) superlattices and by Binash et al. [2] on Fe/Cr/Fe (0 0 1) trilayers, in both cases on samples grown by molecular beam epitaxy (MBE). In Fig. 2, we show the variation of the resistance as a function of the magnetic field for Fe/Cr superlattices at T ¼ 4.2 K. The resistance drops as the configuration of the magnetizations in neighbour Fe layers goes from AP to parallel. Arrows indicate the saturation field Hs, that is the field required to overcome the antiferromagnetic (AF) interlayer coupling between the Fe layers and align the magnetizations of consecutive layers. It turns out that aligning the magnetizations reduces significantly the resistivity of Fe/Cr superlattices, this is the so-called GMR. The magnetoresistance ratio, defined as the ratio of

Fig. 2. Magnetoresistance curves at T ¼ 4.2 K for Fe(0 0 1)/Cr(0 0 1) superlattices. The layer thicknesses are indicated on the curves. The sketches in the bottom of the figure represent the antiparallel (parallel) configurations of the magnetizations in consecutive Fe layers at zero (high) field. From Baibich et al. [1].

158


the resistivity change to the resistivity in the parallel configuration r rP (7) MR ¼ AP rP reaches 79% at 4.2 K, for the sample with 9 A˚ thick Cr layers shown in Fig. 2 (and still 20% at room temperature (RT)). Note in Fig. 2 an alternate definition of the GMR is used where the denominator is the resistance at zero field. A record MR ratio of 220% has been obtained in 1994 by Schad et al. again on Fe/Cr multilayers. As indicated above, the results of Baibich et al. [1] and Binash et al. [2] were obtained on samples epitaxially grown by MBE. In 1990, Parkin et al. [16] obtained similar GMR effects with polycrystalline Fe/Cr, Co/Ru and Co/Cr multilayers deposited by sputtering. They explored very broad thickness ranges and found the oscillatory variation of the magnetoresistance, which reflects the oscillations of the interlayer exchange coupling as a function of the spacer thickness. GMR effects exist in the thickness ranges where the coupling is AF and vanishes when the coupling is ferromagnetic, as shown in Fig. 3 for Fe/Cr. Exchange coupling in magnetic multilayers is reviewed in this volume by M. D. Stiles. A more extensively studied system, also displaying GMR oscillations, is Co/Cu. It was found in 1991 by the teams of Fert and Parkin [17,18]. As shown in Fig. 4, the variation of the MR ratio as a function of the Cu thickness exhibits three welldefined maxima associated with three ranges of AF coupling. The height of the maxima is a decreasing function of the Cu thickness. The oscillatory behaviour disappears in the thickness range where the exchange coupling becomes weaker than the coercive fields. The magnetic arrangement at low field is then approximately random and the GMR is due to the difference in resistance of the random and parallel magnetic configurations. The GMR vanishes when the Cu thickness

Fig. 3. GMR ratio of (Fe (2 nm)/Cr) multilayers at T ¼ 4.5 K as a function of the thickness of the Cr layers. Different symbols correspond to different deposition temperatures. From Parkin et al. [16].


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Fig. 4. MR ratio of (Co (1.5 nm)/Cu) multilayers as a function of the thickness of Cu layers at T ¼ 4.2 K (black circles) and T ¼ 300 K (open squares). The solid lines are guides for the eyes. The dashed line is the envelop of the MR peaks. From Mosca et al. [17].

becomes larger than the electron mean free path (MFP) in Cu, as will be discussed in Section 4. Co/Cu became an archetypal multilayer for GMR. It is also the only system with Fe/Cr for which the GMR can exceed 200%. GMR requires that an AP configuration of the magnetizations in the multilayers can be switched into parallel by applying a magnetic field, but AF interlayer exchange is not the only way to obtain an AP configuration. GMR effects can also be obtained with multilayers combining hard and soft magnetic layers [17–21]. As the switching of the magnetizations of the hard and soft magnetic layers occurs at different fields, there is a field range in which they are AP and the resistance is higher. The best-known structure in which interlayer exchange is not used to obtain an AF configuration and GMR, is the spin valve structure, introduced in 1991 [22,23] and now used in most applications of GMR. A spin valve structure, in its simplest form shown in Fig. 5, consists of a magnetically soft layer separated by a nonmagnetic layer from a second magnetic layer, which has its magnetization pinned by an exchange biasing interaction with an antiferromagnetic (FeMn) or ferrimagnetic layer. The operation of the spin valve can be understood from the magnetization and magnetoresistance curves shown in Fig. 6. One of the permalloy layers has its magnetization pinned by the FeMn in the negative direction. When the magnetic field is increased from negative to positive values, the magnetization of the free layer reverses in a small field range close to H ¼ 0; whereas the magnetization of the pinned layer remains fixed in the negative direction. Consequently, the resistance increases steeply in this small field range. Magnetic multilayers of the spin valve type are used in most applications of GMR, in particular in the read heads of hard discs. More details about spin valves and related applications can be found in the excellent review by Coehoorn [24]. All the measurements described above were performed with the electrical current parallel to the plane of the layers, what is called the current in plane or CIP


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Cap layer AF

F (pinned) NM F (free) Buffer layer substrate

Fig. 5.

Typical layer structure of a spin valve.

geometry. GMR effects can also be obtained with the current perpendicular to the plane of the layers, that is in the so-called CPP geometry. The first measurements in the CPP geometry have been performed at Michigan State University (MSU) by Pratt et al. [25] by sandwiching multilayers between two superconducting strips of Nb to produce uniform current density between the strips through the multilayer and measuring the voltage between the Nb strips. This technique, with detection of the very small signal by SQUID, has been applied at MSU for extensive series of multilayers. A review of this type of measurements has been given by Bass and Pratt [26]. An example of experimental curve and a comparison between the amplitude of the CIP and CPP-GMR for a series of Co/Cu multilayers can be seen in Fig. 7. The GMR is definitely larger in the CPP geometry. In addition, as will be discussed in Section 3, the CPP-GMR subsists for much thicker layers than for CIPGMR. This is due to additional spin accumulation effects. We will show that the scaling length of the CIP-GMR is the electron MFP, whereas the scaling length of the CPP-GMR is the much longer spin diffusion length (SDL). Consequently, sizable CPP-GMR effects can even be observed for individual layer thicknesses in the micron range. It can also be seen in Fig. 7 that, except for samples with strong AF coupling (filled circles in Fig. 7c) the resistance of the virgin state (H0) is larger than the resistance maximum (HP) obtained by cycling the magnetic field. This difference has been generally observed in the measurements of the MSU group on uncoupled samples and indicates that the magnetic configuration is different for the virgin state and the ‘‘peak’’ states. The MSU group has shown by neutron reflectivity experiments that the virgin state of their samples exhibits a strong AP cor-


Fig. 6.

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Magnetization (a) and magnetoresistance (b) loops of a spin valve [NiFe (6 nm)/Cu (2.2 nm)/NiFe (4 nm)/FeMn (7 nm)] at RT. From Dieny et al. [22].

relation of magnetic domains across the spacer layers, so that the GMR measured at H0 is characteristic of the difference between the resistances of the AP and P configurations [27]. The technique developed at MSU for CPP-GMR measurements has the advantage of a relatively direct sample preparation without the use of advanced patterning. However, measurements can be performed only below the critical temperature of Nb. Several other methods without superconducting contacts have been developed for measurements at higher temperatures and also to avoid the difficulty of measuring very low signals. Pillars (mesas) have been patterned in multilayers by photolithography and have been used to obtain the first data on the temperature dependence of the CPP-GMR [28,29]. However, the quantitative interpretation of such experiments is not at all straightforward. First, because the measured signal includes significant and not well-known contributions from contacts and lead resistances and secondly, with the aspect ratio obtained by photolithography, the

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Fig. 7. (a) and (b) CPP-GMR and CIP-GMR curve for a [Co (6 nm)/Cu (6 nm)] 60 multilayer. The field of the maximum resistance in the virgin state (as-prepared sample) of the multilayer, is designated by H0, the saturation field (P configuration) by HS and the field of the local maxima in the MR cycles (at bout the coercive filed) by HPk; (c) CIP and CPP-GMR as a function of the Cu thickness for a series of [Co(1.5 nm)/Cu(t)] N multilayers. Different symbols distinguish the results obtained for the CIP-GMR, the CPP-GMR at HPk and the CPP-GMR at H0. From Bass and Pratt [26].

current lines are not strictly perpendicular to the planes of the layers. More recently, higher aspect ratios have been obtained with submicronic pillars fabricated by e-beam lithography. Pillar-shaped F/N/F trilayers have been used by several groups for experiments of current-induced magnetic switching [30]. These experiments are not generally aimed at GMR studies but rather the GMR is used to detect the current-induced magnetic switching of one of the magnetic layers. Finally, CPP- or current at an angle to plane- (CAP) GMR experiments have also been performed on multilayers deposited on V-grooved substrates [31,32]. The CPP-GMR has also been studied in multilayered nanowires electrodeposited into the pores of nuclear track-etched polycarbonate membranes [33], as illustrated in Fig. 8a. A review of this type of experiments can be found in Ref. [34]. The very high aspect ratio, with a length of 20–40 mm for a diameter in the range 30–100 nm, guarantees that the current is perpendicular to the layers. Also, the relatively large resistance cannot be affected by spurious contact resistances. On the other hand, it


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Fig. 8. (a) Schematic representation of an array of multilayered nanowires elecrodeposited in an insulating polymer matrix. From Piraux et al. [33]; (b) GMR ratio versus applied field parallel to the layers at T ¼ 77 K for NiFe(12 nm)/Cu(4 nm) (solid lines) and Co(10 nm)/ Cu(5 nm) (dashed lines) multilayers. From Fert and Piraux [34].

is not generally possible to know the number of nanowires that are contacted in parallel (even if Voegeli et al. [35] succeeded in determining this number in some experiments). Consequently, the measurements can only determine the GMR ratio and not the absolute value of the resistance of the individual wires. Most experiments on multilayered nanowires have been performed on Co/Cu and NiFe/Cu systems. Except in the earliest experiments, the GMR ratios obtained with nanowires are very similar to those measured with conventional multilayers in the CPP geometry, at least for layers thicker than a few nanometer. An example of experimental GMR curve for nanowires is shown in Fig. 8b. GMR effects can be obtained not only in multilayers but also in granular systems [36,37]. Although the physics in both type of systems is very similar, we will limit our review to multilayers. We simply mention that some structures are really intermediate between multilayers and granular alloys, e.g. multilayers with discontinuous magnetic layers that can be viewed as layers of flat islands [38], and also ‘‘hybrid structures’’ combining continuous magnetic layers and layers of magnetic clusters, in with large GMR effects can be obtained at very low field [39].

4. MODELS OF GMR AND DISCUSSION 4.1. Physics of GMR In Fig. 9 we illustrate the mechanism of GMR in a simple picture of free electrons with spin-dependent scattering by the defects and impurities of the magnetic layers and by the roughness of their interfaces. In a given magnetic layer the scattering


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Fe Cr Fe +

-

(a) AntiParallel configuration

Fe Cr Fe +

(b) Parallel configuration

Fig. 9. Schematic picture of the GMR mechanism. The electron trajectories between two scatterings are represented by straight lines and the scatterings by abrupt changes of direction.+and – are for spins sz ¼ +1/2 and sz ¼ 1/2, respectively. The arrows represent the majority and minority spin direction in the magnetic layers.

probability is thus different for the majority and minority spin electrons. We also assume that the electron MFP, for both spin directions, is much larger than the individual layer thickness, so that the scattering must be averaged over a large number of layers. In the parallel (P) configuration, the electrons of the spin+ and spin channels are respectively majority and minority spin electrons in all the magnetic layers, and this gives different resistances r+ and r for the two channels. The final resistance is rP ¼

rþ r rþ þ r

(8)

If, for example, r+ is much smaller than r, the current is shorted by the spin+ channel of fast electrons and rP rþ (throughout the paper, our notation is+and for the spin directions corresponding to sz ¼ +1/2 and 1/2, z being an absolute axis, and m and k for the majority and minority spin directions inside a ferromagnet). In the AP configuration, electrons of both channels are alternatively majority and minority spin electrons and the shorting by one of the channels disappears. The resistance is then ðrþ þ r Þ=2 for both channels and the final resistance is rAP ¼

rþ þ r 4

(9)


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This gives a GMR ratio of GMR ¼

rAP rP ðr rþ Þ2 ¼ rP 4rþ r

(10)

The above simple picture holds for the CIP and CPP geometries and would predict the same GMR ratio in both geometries. To improve the description, we must take into account first the finite length of the electron MFP, and second the following additional effects which, in particular, explain the difference between the CIP- and CPP-GMR: (i) The interfaces introduce not only diffusive scattering due to their roughness but also specular reflections. The effect of the specular reflections can be described in CIP by a partial channelling of the electrons within the layers (see 4.2.4), and, in CPP, by the introduction of interface resistances (see 4.3.1). In some conditions, the interference between specular reflections at successive interfaces can also build superlattice Bloch functions. (ii) In the CPP-geometry, there are spin accumulation effects (see 4.3) that introduce spin-dependent diffusion currents. As we will see later, the scaling length governing the thickness dependence of the GMR becomes the spin diffusion length (SDL) for the CPP geometry instead of the MFP in CIP. To illustrate the several types of spin-dependent interactions acting on the conduction electrons, we sketch in Fig. 10 the potential landscape they feel in a multilayer. Fig. 10a and b are for the spin+ and spin– electrons in the parallel (P) Bulk scattering potentials

Interface scattering potentials

a) Spin + channel in the parallel configuration Intrinsic potential

b)

Spin – channel in the parallel configuration c)

N

F

N

F

N

Spin + or spin – channel in the antiparallel configuration

Fig. 10. Potential landscape seen by spin+ and spin conduction electrons in the P and AP configurations. The intrinsic potential is represented by a periodic array of steps (Kronig– Penney-like potential); the bulk and interface scattering potentials are represented by spikes.

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configuration and (c) is for any spin directions in the AP configuration. The potential can be separated into (i) the intrinsic potential of the perfect multilayered structure, represented by a Kronig–Penney potential in Fig. 10, and (ii) the scattering (extrinsic) potentials due to defects (atomic disorder, impurities and interface roughness) and represented by spikes. Both the intrinsic and extrinsic potentials are spin dependent. The spin dependence of the intrinsic potential, related to the exchange splitting of the energy bands in a ferromagnetic metal, is represented in Fig. 10 by the spin-dependent steps of the Kronig–Penney potential. These steps give rise to specular reflections that are spin dependent. In other words, these reflections express the spin-dependent mismatch of the wave functions of the ferromagnetic and nonmagnetic metals at their interfaces. The scattering potentials (spikes in Fig. 10) are also spin dependent within the magnetic layers and at their interfaces. The description of the respective roles of the spin-dependent intrinsic and extrinsic potentials is different in what is called the superlattice and layer-by-layer picture. The supperlattice picture holds for the limit in which the MFP is much longer than the period of the multiplayer. In this limit, the interferences between the Bragg-like specular reflections at the interfaces build the Bloch functions of the artificial superlattice. These Bloch functions are different for the P and AP magnetic configurations and, in the P configuration, are different for the two spin channels. In Fig. 11 we show an example of the Fermi surfaces derived from ab initio calculations by Schep et al. [40] for the P and AP magnetic configurations of a Co/Cu superlattice. On the other hand, the relaxation time of this superlattice Bloch states is controlled by the scattering potentials. In this regime the GMR can be described by the scattering of the configuration dependent Bloch waves of the superlattice by configuration-dependent distributions of scattering potentials. The above superlattice approach is valid only if the MFP is much longer than the multilayer period, which is the condition for coherent interferences between the

Fig. 11. Projections inside the first Brillouin zone of different Fermi surfaces for a (1 1 1) oriented Co3/Cu3 superlattice on a plane parallel to the interfaces. (a) Majority spin and (b) minority spin in the P configuration, (c) either spin in the AP configuration. From Schep et al. [40].


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Bragg-like reflections at successive interfaces. For real multilayers, in addition to the scattering by the structural disorder inside the layers, there is a significant probability for the electrons to be scattered by interface roughness at each interface roughness, so that the MFP cannot be many times longer than the layer thickness. Consequently the condition for coherent interferences is rarely satisfied. This is confirmed by the absence of most effects expected from a superlattice situation, oscillations of the conductance as a function of the layer thickness for example. A more realistic approach is the so-called ‘‘layer-by-layer’’ approach, in which one takes into account the specular reflections at the interfaces but not the interference between the reflections at successive interfaces.

4.2. Models of CIP-GMR 4.2.1. Free-Electron Semi-Classical Models of CIP-GMR The first model of GMR was the semi-classical free-electron model of Camley and Barnas [41,42]. This model is based on the picture of free electrons scattered by a distribution of spin-dependent scatterers that depends on the magnetic configuration of the multilayer. Specular reflections by the interfaces can also be taken into account. Let us call z an axis perpendicular to the layers and x the axis of the current. In this geometry, the electric field E is a vector oriented along x and independent of z. On the other hand, except in the limit where the MFP is much longer than the layer thickness, the current density depends on z. In the model of Camley and Barnas, the current-related departure from the Fermi–Dirac distribution f0 at a position z, g ðz; vz Þ is derived from a Boltzmann equation of the type: @g g eE @f 0 þ ¼0 @z t vz mvz @vx

(11)

where t is the relaxation time of electrons of spins+or in the layer one considers. Equation (11) is the condition for a steady electron distribution, the first term expressing the contribution due to the diffusion current generated by the zdependent electron distribution, while the second and third ones describe the effect of electron scattering and electric field. The general solution of this type of equation is of the form: eEt @f 0 z 4ðoÞ g ðvz ; zÞ ¼ 1 þ A4ðoÞ exp (12) t jvz j m @vx where 4(o) refer to whether the z component of the electron velocity is positive or are determined by matching boundary conditions negative. The coefficients A4ðoÞ at the interfaces, that is by assuming that a proportion Ts of the spin s electrons are transmitted without scattering, while the proportion (1Ts) is diffusely scattered. A probability Rs of specular reflection can also be introduced into the model, with now (1TsRs) for the probability of diffusive scattering. For a finite number of layers, a probability of specular reflection at the outer boundaries is also introduced. The major success of this model was to predict the thickness dependence of

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the GMR. In particular, we see from the exponential dependence on z in Eq. (12) that the influence of the spin-dependent scattering in adjacent magnetic layers on the electron distribution function in a nonmagnetic layer does not extend further than about the MFP ls ¼ tsvF. Consequently, the resistance becomes independent of the relative orientation of consecutive magnetic layers when they are apart by more than the MFP. The semi-classical approach of Camley and Barnas [41,42] has been further developed in a large number of papers, and extensively applied to the interpretation of experimental data [42–46]. Analytical expressions of the GMR have been derived by Barthe´le´my and Fert [47] in the simple case where only the interface scattering is spin dependent. It was found that, in the limit of thick nonmagnetic layers, the GMR ratio vanishes asymptotically as exp(tN/lN), where lN(tN) is the MFP (thickness) of the spacer. In contrast the GMR ratio decreases as lF/tF when the thickness of the magnetic layers, tF, becomes much larger than the MFP lF in the magnetic layers. This reflects that the conduction in a magnetic layer is affected by the orientation of the neighbour magnetic layers only within the depth lF in which the electron retains the knowledge of its linear momentum. Phenomenological simple expressions have also been derived by Dieny et al. [43]. In Fig. 12, we show the variation of the GMR ratio of Co/Cu multilayers as a function of the thickness of the Cu layers [48]. Below about 5 nm, the intrinsic thickness dependence of the GMR cannot be separated from what is indirectly due to the oscillations of the interlayer coupling. Above 5 nm, the interlayer coupling is

Fig. 12. GMR versus Cu layer thickness for several series of [Co (1.1 nm/Cu(tCu)]N multilayers at T ¼ 295 K (a and b) and T ¼ 4.2 K (c and d). The number of Co/Cu periods is N ¼ 20 (solid circles) and N ¼ 6 (open and closed squares). The GMR for N ¼ 6 in (a) and (c) has been scaled by a factor 1.6 for comparison with N ¼ 20. From Parkin et al. [48].


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negligible and the GMR always results from the switching from random to parallel orientations. At 295 K, the experimental GMR ratio varies approximately as exp(tCu/l)/tCu with 32 nm for the MFP in Cu (Fig. 12b). This corresponds to the expected exponential drop when tCu/l tends to infinity, with an additional factor 1/tCu coming from the progressive dilution of the spin–dependent scattering as the thickness of the Cu layers increases. At 4.2 K (Fig. 12d), the variation is roughly proportional to 1/tCu. This means that, with a longer MFP at low temperature, the exponential regime is not reached in the experimental thickness range and only the variation as 1/tCu subsists. In Fig. 13, we give an example of the variation of the GMR as a function of the thickness of FM in FM/Cu/NiFe trilayers. The solid lines represent the fit with a phenomenological expression derived from a semi-classical approach. At first the GMR increases as thickening the FM layer increases the proportion of spin-dependent scattering. It then decreases when an increasing part of the FM layer, outside a depth of the order of the MFP, contributes to the conduction but not to the GMR. Although much experimental data can be fitted by using the semi-classical free-electron model, the extracted parameters should be treated with caution. For example, Camley and Barnas [41] have found that they could account for the temperature dependence of experimental data by assuming an unrealistic increase of the MFP from 18 nm at RT to 600 nm at 4.2 K. Initially, it has been often debated whether GMR was mainly due to ‘‘bulk’’ or ‘‘interface’’ scattering. As a matter of fact, both contribute, the proportion bulk/ interface increasing with the thickness of the magnetic layers. This is particularly clear in CPP-GMR, which allows a quantitative separation between the bulk and interface contributions, as discussed in Section 4.3.2. Although a quantitative separation is harder in CIP-GMR, a qualitative information on the respective contributions of bulk and interface scattering was given by experiments on multilayers

Fig. 13. MR at RT of FM(dFM)/Cu (2.2 nm)/Ni80Fe20 (5 nm)/FeMn (8 nm) spin valves versus thickness, dFM, of the free FM layer, with FM ¼ Co, Ni80Fe20 or Ni. From Dieny et al. [43].

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Fig. 14. MR of Co/Cu/Co/FeMn spin valves in which the Co layers have been doped with a submonolayer of various impurities placed at a distance x from the Co/Cu interfaces. For each type of impurity, the MR is plotted as a function of the distance x. The graph for Co can be regarded as the control experiment indicating the level of the MR (4%) without impurities. From Marrows and Hickey [50].

‘‘doped’’ with additional interfaces or additional impurities. For example, the GMR is increased by inserting an ultra-thin Co layer between the NiFe and Cu layers of NiFe/Cu multilayers, which shows the enhancement of the GMR by the introduction of a Co/Cu interface [49]. In contrast, other experiments have shown the significant effect of the scattering of impurities introduced inside magnetic layers. Impurity effects are shown in the experimental results of Marrows and Hickey [50] of Fig. 14. A sub-monolayer of various elements (d-doping) is inserted in one of the Co layers of a Co/Cu/Co spin valve at the distance x from the Cu/Co interface. Even when the d-doping if far (2 nm) from the Cu/Co interface, the MR ratio (approximately 4% without doping, see control experiment) can be enhanced by a factor 1.5 (Ni or Cu doping) or reduced by a factor of 2 (Cr or Mn doping). If we admit that the conduction is predominantly by spinm electrons in the Co/Cu system, we expect the MR can be enhanced (reduced) by doping with impurities of small (large) scattering cross sections for the spinm direction. The reduction of GMR by Cr and Mn is in good agreement with the a values 0.3 and 0.8 found for dilute Cr and Mn impurities in bulk Co [14]. There are no data for Ni or Cu impurities in bulk Co, but there are several arguments to predict aNi 1 and equally aCu has been found larger than unity in CPP-GMR experiments [51], which agrees with the enhancement of GMR by doping Co with Ni or Cu. The smaller enhancement of the GMR when the Cu impurities are placed close to the Co/Cu interface is probably due to additional effects on the interface electronic structure.

4.2.2. Free-Electron Quantum-Mechanical Models of CIP-GMR The first quantum mechanical model of the GMR was introduced by Levy et al. [52] and uses the Kubo formalism. The starting point is the quantum-mechanical equation of motion of the density matrix ri (the equivalent of the distribution function in


171

the semi-classical theory): i_

dri ¼ ½H t ; ri dt

(13)

where the Hamiltonian Ht includes the time independent Hamiltonian of the unperturbed system and a time dependent perturbation due to the electric field E, starting from zero at t ¼ 1 and growing adiabatically to E at t ¼ 0 H t ¼ H þ eE ret

(14)

The model of Levy et al. [52] uses the Kubo formalism to calculate the conductivity of the free electrons scattered by a distribution of spin-dependent potentials of the form: ^ s^ dð~ ^ ¼ v þ jM V ð~ r; sÞ r ~ ra Þ

(15)

^ is a unit vector in the where the operator s^ represents the Pauli spin matrix, and M direction of the magnetization. The scattering centres at position ~ ra are randomly distributed in the layers and in the plane of the interfaces, with v ¼ vN, j ¼ 0 in the nonmagnetic layers, v ¼ vM, j ¼ jM in the magnetic layers, v ¼ vS, j ¼ jS at the interfaces. The model of Levy et al. [52,53] uses the Kubo formalism in momentum space. Other models [54] have been worked out with the Kubo formalism in real space to describe the same problem of free electrons scattered by spin-dependent potentials of the form of Eq. (15). The differences between these various approaches have been discussed by Zhang and Butler [55]. All the above models assume free electrons scattered by bulk and interface spin-dependent scattering potentials and ignore the intrinsic spin-dependent potential of the multilayer. Extensions including also a spin-dependent intrinsic potential of the Kronig–Penney-type have been developed by Zhang and Levy [56] and Bulka and Barnas [57]. This type of model predicts oscillations of the GMR as a function of the layer thickness due to quantumsize effects (superlattice effects). Such quantum-size effects, which distinguish the quantum free-electron models from the free-electron semi-classical models, have not yet been observed experimentally in the apparently not perfect enough multilayers. Apart from the quantum-size effects, the variation of the CIP-GMR as a function of the layer thickness predicted by quantum free-electron models qualitatively reproduces the semi-classical results and can be fitted correctly with the experimental variations. However, in both types of model, quantitative predictions of the amplitude of the GMR are difficult because one poorly knows the parameters controlling this amplitude, that is the MFPs and the interface boundary conditions in the semi-classical models, and, in quantum models, the parameters characterizing the spin-dependent potentials (bulk and interface scattering potentials, and, in some models, the Kronig–Penney type periodic potential). For a realistic comparison with experimental results and quantitative predictions, it is necessary to replace the free-electron picture by an accurate description of the spin-polarized band structure.

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4.2.3. Models of CIP-GMR Based on Realistic Band Structure Calculations A tight-binding model is the simplest way to describe the band structure of a metal. This was the first approach for the GMR in terms of band structure, with s-band [58] or d-band [59] tight-binding models. But the importance of the aspects of the GMR related to the s–d hybridization and s–d scattering led rapidly to the development of multiband models, multiband tight-binding models [60] and, more recently, ab initio models describing the band structure of the multilayer without any empirical parameter. The common approximation of these ab initio models is the local density approximation (LDA) and its extension to magnetic systems known as local spin density approximation (LSDA) [61]. A general review of all the ab initio models worked out for the interpretation of the GMR is not in the scope of this chapter. First we will not consider the purely ballistic models that are those based on the assumption that the sample dimensions are much smaller than the MFP. In the ballistic regime, the conductance is not affected by scattering but is determined entirely by the band structure. Actually, the ballistic regime of the GMR does not seem to have been observed in experiments. In contrast, we have seen that the GMR is strongly influenced by impurity or defect scattering. We will only consider models worked out for the diffusive regime. The complication in the diffusive regime of the GMR is that an ab initio calculation of the wave functions of the multilayer is not sufficient. One must also know the scattering of these wave functions by defects (disorder and impurities) to introduce it into transport equations. This necessarily implies some modelling of the scattering. The choice is between empirical parameters describing the scattering and a somewhat arbitrary assumption on the predominant defects scattering the electrons in the multilayer. In a first class of model, the relaxation times are free parameters and are incorporated with accurate calculations of the band structure into semi-classical transport equations. For example, to calculate the GMR of Fe/Cr multilayer, Zahn et al. [62] developed a linear combination of atomic orbitals (LCAO) calculation of the band structure and suppose that the scattering is mainly by Cr impurities in the Fe layers. In the P configuration, they take the ratio tm/tk ¼ 0.11 found for the scattering by diluted Cr impurities in Fe. In the AP configuration, they suppose a complete averaging of the relaxation rates in the structure, that is 1/tAP ¼ 1/tm+1/ tk. This leads to GMR ratios (600% in CIP) much higher than found in experiments on Fe/Cr multilayers. Similar calculations for Co/Cu, either with relaxation times calculated for Fe, Ni or Cu impurities at the interfaces [63] or for interdiffused Co/Cu interfaces [64] have also led to much too high values of the GMR. These highly overestimated values are a direct consequence of the nearly identical electronic structure of Co and Cu in one of the spin channel. This is illustrated in Fig. 15 where is shown the number of valence electrons in each channel for Co/Cu and Fe/Cr. In Co/Cu, the number of spinm electrons in Co and Cu are extremely close, whereas there is a strong mismatch between these numbers for spink. The Co and Cu layers, as well as the Co atoms and the diffused Cu impurities in Co, will appear very similar to the spinm electrons which will be very weakly scattered. In


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Fig. 15. Calculated number of valence electrons for the (1 1 1) Co/Cu/Co (a) and (0 0 1) Fe/Cr/Fe trilayers (b) for the majority- (m) and minority-spin (k) electrons with good match for malority spins in (a) and for minority spins in (b). From Butler et al. [65].

contrast, for Fe/Cr, an extremely small scattering is expected for the spink electrons. This explains the very high GMR found in the calculations. Several explanations have been proposed to account for the discrepancy between theory and experiments. It has been first suggested by Butler et al. [64] that the spin–orbit coupling, which is not considered in the calculations, can mix the spinm and spink states and reduce the contrast between the two channels. The second explanation is that, in the real multilayers, the scattering is not only due to interdiffusion and impurity effects but also to a great variety of defects (structural defects, grain boundaries, etc.) that, on average, scatter in a similar way both spin directions. The spin–orbit coupling is explicitly taken into account in the fully relativistic calculation of Blaas et al. [66] for Co/Cu/Co spin valves. Their calculation is based on screened Korringa–Kohn–Rostoker method for layered systems, a coherent potential approximation (CPA) to treat the intermixing at interface and a Kubo– Greenwood transport formalism. The GMR calculated with this relativistic approach is more in the range of the experimental results than what was found in previous calculations, which is probably related to Butler’s suggestion that the GMR is overestimated when the spin–orbit is not taken into account. However the model is not yet predictive since the GMR still depends strongly on the assumption on the defects scattering the electrons. Blaas et al. [66] describe how the GMR is sensitive to the interdiffusion profile at the Co/Cu interface and the presence of impurities, so that the GMR cannot be a priori predicted since the defects in the multilayer are in general poorly known. A general conclusion of the above review of theoretical models is therefore that the GMR, at least the CIP-GMR, cannot be really quantitatively predicted. A precise prediction would require the various defects and impurities at the origin of the scattering to be known whereas the knowledge of the experimentalists on these defects is very poor. However, if the recent theoretical developments are still unable to predict the GMR ratio, they have put forward important semi-quantitative aspects, as we will discuss in the next paragraph on quantum channelling.

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4.2.4. Quantum Channelling in CIP-GMR The concept of ‘‘quantum channelling’’, demonstrated by ab initio calculations of several types [67,68], allows us to understand what is important to add to the simple free-electron picture described in Sections 3.2.1 and 3.2.2. In a multilayer, depending on the mismatch between the wave functions of the different metals at their interfaces and on the resulting reflections, the electron eigenstates show a periodic modulation as illustrated in Fig. 16. Fig. 16a represents the local DOS at the Fermi energy calculated by Zahn et al. [68] for the successive atomic planes of a Co9/ Cu7(1 0 0) multilayer in a P configuration. Some of the states, corresponding to the dark part of the DOS in Fig. 16a and to Fig. 16b, are free electron-like (delocalized), that is extending more or less equally throughout a Co/Cu bilayer period. These states exist for the majority spin direction but practically not for the minority one. The other states, most of the states for the minority spin direction, are quantum well-like, that is concentrated either in the Co layers (light grey part in Fig. 16a and c) or in the Cu layers (light grey and Fig. 16c). More surprisingly, there are ‘‘interface’’ states characterized by a high DOS at the interface and a much smaller DOS within the layers (white and Fig. 16c). These different states have also different averaged Fermi velocities. Due to this quantum channelling, only the electrons in extended states will probe all the spin-dependent scattering centres equally in the multilayer (as they do in a simple free-electron picture), whereas others will be affected mainly by the scattering in a single Co layer, or in a Cu layer, or at an interface. This leads to a picture where the GMR is predominantly described in terms of an interplay between the spin-dependent quantum channelling by the intrinsic potential of the multilayer and the spin-dependent scattering by extrinsic potentials. The quantum channelling is the only channelling effect when the layers are thinner than the MFP. If not, the scatterings are averaged only within a fraction of the thickness of the order of the MFP, which can be described as a ‘‘classical channelling’’ adding up to the quantum channelling. The free-electron models described in Sections 3.2.1 and 3.2.2 can, in principle, reproduce quantum channelling effects by introducing phenomenological reflection/ transmission coefficients at the interfaces as discussed by several authors [41,42,44]. However, to reproduce the situation illustrated by Fig. 14, the number of free parameters is so large that an analysis of experimental results in this way is not really meaningful. The importance of quantum channelling is also put forward by calculations of Stiles [69] for the transmission probabilities across various interfaces. In a layerby-layer approach, generalized Bloch states for a layer are computed in an linearized-augmented-plane-wave method and the generalized Bloch states of the two materials are matched together at the interface to obtain the reflection and transmission amplitudes. Fig. 17 shows the results of the calculation for a Co/Cu(0 0 1) interface and for various points on the Fermi surface. One sees that the minority spin electrons are poorly transmitted by the Co/Cu interface, which is at the origin of the channelling within Cu discussed above. Although the discussion has been limited so far to the CIP-GMR, we can conclude this part by some comments on the difference between CIP and CPP. In CIP


175

Fig. 16. (a) Local DOS at EF in states/spin Ry for Co9Cu7(1 0 0) multilayer in the P configuration (9 and 7 indicate the number of atomic layers). The different shaded areas correspond to the weights of four types of eigenstates (b–e) probability amplitudes for suitably chosen values of k: (b) representative extended (majority), (c) Co quantum well (minority), (d) Cu quantum well (minority) and (e) interface (minority). From Zahn et al. [68].

geometry, the interface specular reflections contribute to the GMR by channelling the current in some of the layers. In CPP geometry, the same current goes through all the layers and the interface reflections play a completely different role. Their effect on the connection between the wave functions of the two materials can be described by the introduction of spin-dependent interface resistances. The role of

176


Fig. 17. The spin-dependent transmission probabilities for the Co/Cu(0 0 1) interface are shown for various points on the Fermi surface projected onto the interface Brillouin zone. The colour scale for the transmission is at the top. The top (bottom) two panels how the transmission for electrons from Cu (Co) into Co (Cu) (majority and majority spin directions on the left and right, respectively). From Stiles [69].

the scattering potentials is also different. In CIP-GMR, due to the channelling of the current in some of the layers, the scattering potentials of different layers can be weighted very differently. On the contrary, for CPP-GMR, the electrons probe successively the scattering potentials of all the layers. 4.3. Models of Spin Accumulation and CPP-GMR 4.3.1. Concepts of Interface Resistance and Spin Accumulation While the studies focused on the CIP-geometry during the first decade of research on GMR, the CPP-GMR has now become a more important subject of research. The CPP-GMR experiments have been particularly important for the understanding of


177

the spin accumulation effects, which are also involved in several new developments of spintronics such as spin transfer and spin injection into semiconductors. The CPP-GMR is now also important from the point of view of applications since, with the TMR, it will probably replace the CIP-GMR in next generations of read heads of hard discs and several other devices. There are significant differences in the physics of CIP- and CPP-GMR. The most important are related to the concepts of interface resistance and spin accumulation. (i) Interface resistance. As already described at the end of the preceding paragraph, the specular reflections at the interfaces play different roles in CIP and CPP. In CIP-geometry, the specular reflections conserve the electron momentum in the current direction, so that they do not contribute directly to the resistivity but indirectly by channelling partly the current inside some of the layers. In CPP, there is a direct contribution of the specular reflections to the resistivity. This contribution can be generally expressed by a spin-dependent interface resistance [70–74], which introduces a spin-dependent voltage drop between the two sides of an interface. The usual notation for the interface resistance is rb"ð#Þ ¼ 2rb ½1 ðþÞg

(16)

where g is the interface spin asymmetry coefficient. The interface resistance has been first calculated in models where the interface is described by the abrupt step of a Kronig–Penney potential [70] and then by several types of more realistic calculations for various couples of magnetic and nonmagnetic metals [71]. The interface resistance has also been calculated for rough interfaces [72,73] and it has been shown that it can be increased or reduced by roughness. The interface resistance is expected to be independent of the thickness of the layers when there is no coherent transmission between successive interfaces (no quantum well-effects), that is when the layer thickness is larger than the MFP. If not, it can depend on the ratio of the thickness to the MFP [73,74], although such a dependence have not been really observed in experiments up to now. (ii) Spin accumulation. The second important difference between the CIP- and CPP-GMR is due to the spin accumulation effects generated by spin transport perpendicularly to an interface. Suppose an interface between ferromagnetic and nonmagnetic semi-infinite layers, as illustrated by Fig. 18a. A current of electrons is going from left to right. If, for example, the conductivity in the ferromagnetic material is larger for the majority spin electrons, the incoming current is carried, in larger part, by the spin+channel, while the outgoing flux is carried equally by the two channels. In the central region, due to the unbalance between the ingoing and outgoing fluxes, there is accumulation of spin+electrons and depletion of spin electrons, which corresponds to an out of equilibrium electron distribution char acterized by a splitting of the chemical potentials E+ F and EF (Fig. 18b). A steady state is reached when the number of spin flips generated by this out equilibrium electron distribution is just what is needed to balance the ingoing and outgoing fluxes in the two channels. The spin accumulation diffuses and spreads far away from the central region, the characteristic length of its exponential decrease being


178

(a)

zone

of

spin

accumulation

µ+ , µ-

Ω .m2 x1.1015

F

2,5 2,0 1,5 1,0 0,5 0,0 -0,5 -1,0 -1,5

N

∆µ =

chemical potentials

EF,+ −EF, -

(b) E F,+ E F,-

-250

0

250

P=(j+-j-)/j

0,5

500

750

current polarization

0,4

1000

(c)

0,3 0,2 0,1 Cu

Co

0,0 -250

0

250 z (nm)

500

750

1000

Fig. 18. (a) Sketch illustrating the balance between the incoming and outgoing electron fluxes in the spin-up (orange) and -down (blue) channels at an interface between a ferromagnetic (F) and a normal (N) metal. (b) Spin-up (EF,+) and -down (EF,) chemical potentials, and spin accumulation, Dm ¼ EF,+EF,, calculated as a function of the distance from a Co/Cu interface and for unit current density. (c) Spin polarization of the current calculated as a function of the distance from a Co/Cu interface. The calculations of (b) and (c) are performed with experimental values of resistivity, spin asymmetry coefficients b and SDL for Co and Cu.

the SDL of the corresponding material. The gradient of the spin-dependent chemical potentials gives rise to spin-dependent pseudo-electric fields that slow down the faster electrons, accelerate the slower ones and, on the whole, increase the effective resistivity in the spin accumulation zone. The spin accumulation have been introduced in the models of the CPP-GMR by Valet and Fert [75]. It contributes significantly to the GMR and, since the spin accumulation and especially the interaction between spin accumulations at successive interfaces extends over distances of the order of the SDL, the CPP-GMR subsists for layers as thick as the SDL (which is much larger than the MFP).


179

4.3.2. The Valet–Fert Model of CPP-GMR Most of the experimental results of CPP-GMR have been analysed in the Valet– Fert (VF) model [75] based on a Boltzmann equation treatment of the transport. To describe the spin accumulation effects, the electron distribution function introduced into the Boltzmann equation takes into account a spin splitting of the chemical potential (dependent on the coordinate x on the current axis). Accordingly, the electric field is replaced by a spin-dependent electric field related to the gradient of the global electro-chemical potential. A spin-lattice relaxation term describing the relaxation of spin accumulation by spin-flip scattering is also introduced in addition to the usual relaxation terms for the momentum. In the limit where the momentum relaxation times, tm and tk, are much shorter than the spin-lattice relaxation times, tsf, a simple solution can be found by taking the terms of first order in tm(k)/tsf [75]. In this solution, the equations governing longitudinal spin accumulation and current in a given layer can be written as j þðÞ ¼

1 @mþðÞ erþðÞ @x

(17)

jþ þ j ¼ je

(18)

@ðj þ j Þ 2eNðE F ÞDm ¼ @x tsf

(19)

where je is the total current density, j+ (j), m+ (m) and r+ (r) the spin+(spin ) current density, electro-chemical potential and resistivity, Dm ¼ mþ m is the spin accumulation, e ¼ jej; NðE F Þ is the DOS at the Fermi level for a given spin direction, and tsf (tFsf or tN sf ) is the spin relaxation time. The usual notation for the resistivity in the majority or minority channel of a ferromagnetic layer is rm(k) ¼ 2r*F[1 (+)b] in a ferromagnetic layer and r+() ¼ 2r N in a nonmagnetic layer. b is the bulk spin asymmetry coefficient and g in Eq. (16) is the interface spin asymmetry coefficient. Equation (17) is a pseudo-Ohm law relating the current to the gradient of the electro-chemical potential. Equation (18) expresses the charge conservation and Eq. (19) expresses the spin conservation, that is the balance between spin injection proportional to the gradient of the spin current and spin relaxation proportional to the spin accumulation. By combining Eqs. (17)–(19), one finds that the variation with x of the electro-chemical potentials is governed by the equation: @2 mþðÞ mþðÞ ¼ 2 @z2 l sf l sf is the SDL in the layer one considers, l Fsf or l N sf ; that is h i1=2 N

1=2 1 ; lN l Fsf ¼ lFsf =3 l1 " þ l# sf ¼ lsf l=6

(20)

(21)

N where lFsf ¼ tFsf vF ; lN sf ¼ tsf vF ; l" ¼ t" vF ; l# ¼ t# vF and l ¼ tvF in F and N, respectively, in a free-electron model with the same number of spinm and spink


180

conduction electrons in the magnetic layers and an equal number of electrons in the nonmagnetic layers (a more general expression is given by Fert and Lee [76]). The spin-lattice relaxation time is due only to the spin–orbit part of the scattering potentials and is in general much longer than the momentum relaxation time, so that the SDL is much longer than the MFP, typically by an order of magnitude. The solutions of the above equations for the electro-chemical potentials m+ and m and then, from Eq. (17), for the currents j+ and j, are combinations of exponential functions of z, exp(z/lsf) and exp(z/lsf). These solutions in adjacent layers are linked by the condition of continuity of j+ and j and by the boundary condition expressing the discontinuity of m+ and m induced by the interface resistances xþ

x

0 0 0 mþðÞ ¼ rbþðÞ j xþðÞ mþðÞ

(22)

where rbþðÞ is the interface resistance given by Eq. (16) for the majority and minority channels. Finally, the resistances of the multilayer in the P and AP configurations are derived from the variation of the electro-chemical potential throughout the multiplayer and the GMR of a periodic multilayer can be expressed as a funcF tion of rF ; rN ; rb ; b, g, tN =l N sf and tF =l sf ; where tN (tF) are the thickness of the nonmagnetic (ferromagnetic) layers. The length scale governing the variation with thickness is the SDL, in contrast with the situation in CIP where the scaling length of the thickness dependence is the MFP. We however point out that, in principle, the approach of the VF model, with interface resistances independent on the layer thickness, is not valid for layers thinner then the MFP. As a function of the layer thickness, several regimes are therefore expected: (i) a regime with ‘‘MFP effects’’ for layers thinner than the MFP, in which the contribution of the interfaces cannot be described by an interface resistance independent of the thickness. (ii) The regime corresponding to practically all the experimental results and described by the VF model, that is, in principle, for layers thicker than the MFP. In this regime, the VF model predicts that the SDL is the unique scaling length and the MR can be F expressed as a function of tN =l N sf and tF =l sf : We do not reproduce here the general expression of the resistance in the P or AP configurations of a periodic multilayer, Eqs. (40)–(42) of Valet and Fert [75], but F only the expressions obtained in several simple limits: (i) tN l N sf and tF l sf (long SDL regime). In this limit, the equations of the GMR are equivalent to those derived by the phenomenological resistor series model of Lee et al. [77] and can be written down in the following different ways:

½ðRAP RP ÞRAP 1=2 ¼ N brF tF þ 2grb (23) or equivalently AP 1=2 r tF þ 2rb R RP r tN ¼ F þ N AP brF tF þ 2grb brF tF þ 2grb R

(24)

RAP ¼ N rF tF þ rN tN þ 2rb

(25)

with


181

RP and RAP are the resistances of a unit area of the multilayer and N the number of periods. In the initial VF model, expressions of the type of Eqs. (23)–(25) were derived in the simple case where RP and RAP are the resistances of P and AP configurations. It has however been shown by Fert et al. [78] that, in the long SDL limit, the resistance is the same for a true AP configuration and for a state with zero net magnetization in a set of layers included within the SDL. Equations (23–25) therefore hold for both AP and random configurations. The equivalence of the AP and random configurations in a similar limit has also been demonstrated by Zhang and Levy [56] (ii) tF l Fsf with tN l N sf : The MR ratio is given by ðRAP RP Þ 2pb2 l Fsf ¼ P R ð1 b2 ÞtF

(26)

where p is the proportion of AP configurations between consecutive magnetic layers (p ¼ 1 for a perfect AP configuration). Equation (26) expresses that only a depth N l Fsf on both sides of a magnetic layer is ‘‘active’’ in this limit. (iii) tN l sf with F N tF l sf : The MR ratio is expected to decrease as exp tN =l sf : Outside the above limits, the GMR of a periodic multilayer can be calculated from the general expression of the VF article [75]. For a nonperiodic multilayer, the GMR can be calculated by solving a set of equations of the type of Eqs. (17)–(19) with boundary conditions similar to Eq. (16). In the initial version of the VF model [75], the spin relaxation controlling the variation of the current spin polarization is introduced only inside the layers and the SDL reflects only the spin-flip part of the ‘‘bulk’’ scattering. However, interface scattering should also induced spin flips and contribute to the spin relaxation. This interface spin relaxation can be introduced in several ways, either with boundary conditions introducing a discontinuity of the spin current as in the extended version of the model by Fert and Lee [76], or by inserting an artificial interface layer I having a thickness tI and a very short SDL l Isf : The latter is the calculation technique extensively used by the MSU group. The interface spin memory loss is then characterized by the parameter d ¼ tI =l Isf ; which has been derived for series of interfaces [26,79]. 4.3.3. Interpretation of Experimental Results on CPP-GMR Most experimental results have been accounted for with the simple expressions of the long SDL regime of the VF model. Equation (23) has been extensively used at MSU [26] for the interpretation of experimental data on several types of multilayers, as illustrated by Fig. 19. For samples with a constant total thickness L, variable N and tN ¼ tF ¼ L/2N, the second member of Eq. (23) is the sum of a constant term brF L=2 and a term 2Ngrb : This corresponds to the linear variation as a function of N in Fig. 19a, with a slope proportional to grb and an intercept with the vertical axis proportional to brF : The experimental points for two nonmagnetic metals having very different MFP, Ag (rN ¼ 1 mO cm) and an alloy of Ag and Sn

182


Fig. 19. For a fit with Eq. (23), A[(RAP–RP)RAP]1/2 is plotted versus the total number N of bilayers in Ag/Co and AgSn(4%)/Co multilayers with a constant total thickness, L ¼ 720 nm. A is the sample area, RAP is the resistance in the virgin state, RP is the resistance in the high-field parallel configuration. (a) tCo ¼ tAg ¼ tAgSn ¼ L/2N (the intercept of the straight line with the vertical axis gives b and its slope gives g) and (b) tCo ¼ 6 and 2 nm with tAg varying. The data for the Ag and AgSn layers (respective resistivities 1 and 18 mO cm) are on the same line in (a), as expected from Eq. (23). From Pratt et al. [79].

(rN ¼ 18.5 mO cm), lie on the same line. This means that, even for layers of only 6 nm at the maximum value of N, MFP effects are not observed. For the samples of Fig. 19b, with constant L and constant tF, Eq. (23) predicts a proportionality to N with a slope depending on the value of tF, which is well observed throughout the Ag thickness range between 5 and 18 nm. Other experiments have also shown that the data for true AP configurations and for random orientations lie equally on the same straight lines. Deviations from Eq. (23) are expected when the SDL becomes shorter than the layer thickness. In Fig. 20, we show the results of Yang et al. [80] on Co/Ag multilayers in which the SDL of the Ag layers has been reduced to a few nanometers by doping the Ag layers with paramagnetic impurities (Mn) or impurities with strong spin–orbit coupling (Sn). The strong reduction of GMR and the deviations from a linear variation are quantitatively accounted for by the general VF expressions of the GMR (solid curves) with the shortened SDL indicated on the curves. Equation (24) has been used for the interpretation of experimental results in the long SDL regime of the CPP-GMR in multilayered nanowires. Examples of linear variations of (DR/RAP)1/2 as a function of tN in samples with constant tF are shown in Fig. 21a. From Eq. (24), the vertical coordinate of the crossing point of straight lines for different values of tF is b1, so that the value of b for Co can be


183

Fig. 20. A[(RAP–RP)RAP]1/2 (same notation as in Fig. 19) is plotted versus the total number N of bilayers in Ag/Co, AgSn(4%)/Co, AgMn(9%)/Co and AgPt(6%)/Co multilayers with a constant total thickness, L ¼ 720 nm. The solid and dotted-dashed curves correspond to a fit to the equations of the VF model with the SDL indicated on the curves. From Bass et al. [80].

directly derived from the crossing point of the two lines in Fig. 21a. Deviations from Eq. (24) are expected for values of tN approaching or exceeding the SDL in the nonmagnetic layers. These deviations begin to appear in Fig. 21b where experimental data are plotted for a broader range of tN than in Fig. 21a. The observed upturns for values of tN above approximately 150 nm can be accounted by an exponential variation with l Cu sf ¼ 140 nm: On the other hand, in the regime of Eq. (26), for values of tF above the SDL in the magnetic layers, one expects a variation of the MR as l Fsf =tF : This variation is illustrated by the data in Fig. 22 that led to the determination of the SDL in Co, 59 nm at 77 K and 38 nm at 300 K. It is interesting to note that the CPP-GMR can be observed for thickness up to about 1 mm. In the most general case, that is for tF (tN) in the range of l Fsf ðl N sf Þ or just above, the expressions for the MR derived from the VF model are more complex and cannot be reduced to simple exponential or linear variations. An example of fit of experimental data in the general case is shown in Fig. 23 for Ni80Fe20/Cu/Ni80Fe20 spin valves at 4.2 K [81]. This fit leads to a very short SDL in Ni80Fe20 (permalloy or Py), l Py sf ¼ 5:5 nm; in agreement with the result found for Py/Cu nanowires, 3:3 nmol Py sf o5:3 nm [82]. In Fig. 23, there is saturation of DR for Py layers thicker than about 3l Py sf ; in a thickness range where the resistance R is predominantly due to Py layers and is proportional to 1/tPy, so that DR/R exhibits the variation as 1/tPy expected from Eq. (26).

184


Fig. 21. Plot of (DR/RAP)1/2 as a function of tCu for Co(tCo)/Cu(tCu) multilayered nanowires (a) in the long SDL regime, i.e. for tCuo120 nm and at 77 K, with the two straight lines corresponding to two different Co thicknesses and (b) in an extended thickness range for tCo ¼ 8 nm, at two different temperatures. Corresponding MR values (in percent) are also given on the right-hand side scale. From Fert and Piraux [34].

Most of the recent analyses of experiments take into account the interface spin memory loss. The parameter d, which characterizes the relative spin loss at an interface has been derived by specific measurements in samples in which, for example, a thin layer of vanadium or tungsten is inserted inside the Cu layer of a spin valve [83]. One finds, for example, that the spin loss is very small (d ¼ 0.44%) for the interface of Cu with Ag, somewhat larger (d ¼ 7–25%) for the interface of Cu with 3d metals (including ferromagnetic metals) and definitely larger for heavy elements like W. The spin loss turns out to be primarily related to the spin–orbit contrast at the interface. Moreover, at an interface with a magnetic layer, there are probably additional effects due to spin transfer when there is some departure from a perfectly collinear magnetic configuration. Taking into account the interface spin


185

Fig. 22. Linear variation of the inverse of the magnetoresistance versus Co thickness for Co(tCo)/Cu(8 nm) multilayered nanowires at T ¼ 77 K (filled symbols) and T ¼ 300 K (open symbols). The circles refer to data obtained on a Co(435 nm)/Cu(15 nm) sample. From Fert and Piraux [34].

Fig. 23. ADR versus tNiFe for Ni80Fe20(tNiFe)/Cu(20 nm)/Ni80Fe20(tNiFe) exchange-biased spin-valves. A is the area of the sample and DR the resistance change between P and AP configurations. The dots represent experimental data, the dashed curve is calculated in the ¼ 1 and the solid curve is for l NiFe ¼ 5:5 nm: From Steenwyck et al. [81]. VF model for l NiFe sf sf

memory loss in the interpretation of experiments with the VF model lead to small changes in the spin asymmetry coefficients b and g derived from the analysis. All the interpretations described above are based on analyses in which the interface resistances are independent of the layer thickness. However, as discussed in Section 3.2, a constant value of the interface resistance is expected only when the

186


layer thickness is larger than the electron MFP (diffusive transport) and, more generally, deviations from the diffusive models can be predicted for ballistic propagation between two interfaces [71,74]. In particular, the GMR should not depend only on the ratio of the thickness to the SDL but also on its ratio to the MFP when this ratio is smaller than unity. MFP effects have been searched for in Co/Cu multilayers including Co layers of different thickness [84]. If only the bulk spin flip is taken into account, the same GMR is expected for different configurations of thin and thick Co layers within a total thickness of the order of the bulk SDL, 60 nm in Co. The different GMR found for different configurations was ascribed to MFP effects. However the interface spin memory loss was forgotten in this conclusion. Eid et al. [85] pointed out that the effective spin loss length through a set of thin and thick Co layers, due to the spin losses at the interfaces, was strongly reduced below the 60 nm found in thick single Co layers. They could explain the experimental results by these interface effects. Other experiments on samples in which the MFP was reduced to very small values by doping showed that the MFP could not be involved in the experimental observations [85]. MFP effects have been searched for recently by Chiang et al. [86] who looked accurately at the dependence of the CPPGMR on the thickness of the nonmagnetic layers in Co/Cu/Co, Co/Ag/Co and Co/Au/Co spin valves. They found that MFP effects, at the borderline of uncertainty, could not be completely ruled out. It can be concluded that the observation of ballistic MFP effects in CPP-GMR is still a pending issue. 4.3.4. Physical Data from the Interpretation of CPP-GMR Experiments The interpretation of CPP-GMR experiments leads to the determination of a complete set of parameters: resistivities rN and rF and spin asymmetry coefficient b of the resistivity of the magnetic material, interface resistance rF=N and its spin asymF metry coefficient g, SDLs l N sf and l sf ; and finally, if the interface spin memory loss is analysed, its characteristic coefficient d. The systems that have been the most investigated have been Co/Cu and Py/Cu. Data obtained by several groups on these two systems are presented in Table 1. For Co/Cu and Py/Cu, both the bulk and interface asymmetry coefficients, b and g, are positive, as expected from theory (cf. Fig. 15). The large value of b for Py is also in agreement with the strongly spindependent scattering by Fe impurities in Ni (cf. Fig. 1). The wider dispersion of b for Co results probably from the influence of the sample preparation on the nature of the defects that scatter the electrons in a pure metal. There is only a small dispersion of the values of the spin asymmetry coefficient g for both the Co/Cu and Py/Cu interfaces. This seems to show that the predominant contribution to the interface resistance comes from the intrinsic specular reflections of the perfect interfaces. The positive sign found for b and g in Co/Cu and Py/Cu is not a general rule and negative signs can be found in other systems or by doping Co with selected impurities. A spectacular inversion of GMR can be obtained in multilayers with spin asymmetries of opposite signs in consecutive magnetic layers and interesting effects are also obtained with competing interface and bulk spin asymmetries [51,89]. The explanation of the inversion is sketched in Fig. 24. A typical example with these two


187

Table 1. Best-fit parameters derived from CPP-GMR measurements at low temperature (4.2–77 K) on Co/Cu, Py/Cu and Co/Cu/Py/Cu multilayers.

rCu ð108 OmÞ rCo ð108 OmÞ bCo rCo=Cu ð1015 Om2 Þ gCo=Cu rPy ð108 OmÞ bPy rPy=Cu ð1015 Om2 Þ gPy=Cu l Cu sf ðnmÞ l Co sf ðnmÞ l Py sf ðnmÞ tCo (nm) tPy (nm)

MSU (multilayers)

Eindhoven (grooved substrates)

0.6 7.5 0.46 0.51

0.36 5.7 0.27 0.2

3.1 1872 0.3670.02 0.370.15

1.3–3.3 51–57 0.4670.05 0.3–1.1

0.77 15.9

0.52

0.8570.15 26.3

0.5570.07

Louvain–Orsay (nanowires)

0.73 0.54

0.870.1

0.70

0.870.1 140720 59718 4.371

5.5 19 6.5

13.5

Lausanne (nanowires)

7.9

Note: The last two lines indicate also the threshold thickness t* above which the contribution from bulk scattering to the square root of the GMR exceeds the interface contribution, i.e. t ¼ 2grb =brF ; according to Eq. (23). Source: From Bass and Pratt [26], Fert and Piraux [34], Oepts et al. [87], Doudin et al. [88].

Fig. 24. Mechanism of inverse GMR: with opposite spin asymmetries in the left and right magnetic layers, the current is shorted in the AP magnetic configuration by the electrons that are successively majority and minority spin electrons in consecutive magnetic layers, and the GMR is inverse, that is RAPoRP.

188


effects is presented in Fig. 25 for Ni0.95Cr0.05/Cu/Co/Cu multilayers in which the Ni layers have been doped with 5 at.% of Cr impurities. Cr impurities in Ni scatter more strongly the majority spin electrons [14,15], which make the bulk scattering spin asymmetry coefficient bNiCr negative. For thick enough layers the global spin asymmetry of the NiCr layers is also negative, whereas bCo and gCo/cu are positive for the Co layers. This explains the inverse GMR seen in the inset of Fig. 25 for tNiCr ¼ 7 nm and for tNiCr41.8 nm on the variation of the GMR with tCu in the figure itself [51]. The GMR in Fig. 25 is normal for tNiCro1.8 nm because, with gNiCr/Cu positive, the GMR is reversed only when the bulk contribution in NiCr with negative spin asymmetry overcomes the interface one with positive spin asymmetry. The change from normal to inverse GMR occurs at tNi Cr ¼ 1:8 nm; which is the compensation thickness between bulk and interface scatterings. A similar behaviour has been observed in a large number of multilayers of the type A/ Cu/Co/Cu, where A can be Ni doped with Cr impurities, Co doped with Cr and Fe doped with Co or V. The inversion of the GMR above a compensation thickness tA is always a signature of the competition between a positive gA/Cu and a negative bA. Negative values of b have been found for Ni95Cr5 (0.13), Fe70Cr30 (0.28),

Fig. 25. Variation of the CPP-GMR as a function of the thickness of the Ni0.95Cr0.05 layers in NiCr/Cu/Co/Cu multilayers. The competition between the bulk scattering (characterized by bNiCr o 0) and interface scattering (with gNiCr/Cu40) leads to a compensation thickness tNi Cr of zero GMR. Below tNi Cr ; the interface scattering is predominant, the global spin asymmetry of the NiCr layer is positive as that of the Co layer and the GMR is normal (see left insert). Above the compensation thickness, the bulk scattering in NiCr becomes predominant, the global asymmetry of the NiCr layers is negative and the GMR is inverse (see right insert). From Vouille et al. [51].


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Fe85V15 (0.11) and Co90Cr10 (0.12), while positive values have been found for Co90Fe10 (+0.65), Ni80Fe20 (+0.73) and Ni70Cu30 (+0.19). The sign of b is in agreement with that previously found for bulk alloys by experiments [14] or calculations [15]. The absolute value of b is generally smaller in multilayers which is likely due to the existence of structural defects that introduce an additional contribution to the scattering. Negative values of the interface spin asymmetry coefficient g had been found for the interface of Co and Fe with Cr. The variation of the GMR of [Co, t/Cr, 0.9 nm/Py, 7 nm/Cr, 0.9 nm]N multilayers with the thickness t of the Co layers is shown in Fig. 26. Due to the predominant contribution from the negative gCo/Cr for thin Co layers, the global spin asymmetries of the Co and Py layers are of opposite signs and the GMR is inverse at small values of tCo (the Py layer is thick with a predominant contribution to its spin asymmetry from its positive bPy). For tCo above 4 nm, the contribution from the positive bCo become predominant and the GMR is normal. It turns out that the sign of both b and g can be predicted, at least in first approximation, by simple arguments concerning the matching of the wave functions in different metals. For example, the positive b for Ni doped with Cu and the positive g of the Co/Cu or Ni(Fe)/Cu interfaces expresses the good match between the wave functions of Cu with those of NiCo or Ni(Fe) for the majority spin direction. In the same way, the negative sign of b for Co or Fe doped with Cr and g for the Co/Cr and Fe/Cr interfaces corresponds to a good match between the wave functions of Cr with those of NiCo or Fe for the minority spin direction (cf. Fig. 15). The SDL, according to Eq. (21), is related to the spin-lattice relaxation time and results from the spin–orbit part of the scattering by defects or impurities [76]. The long SDL of Cu, approaching the micron range, is related to the almost pure s character of the conduction electrons. As this has been seen in the preceding paragraph, the SDL in Cu can be considerably shortened by impurities like Pt, which bring the strong spin–orbit coupling of 5d states. The SDL in Cu (or Ag) is definitely longer than in Fe-, Co- or Ni-based layers in which the conduction electrons have an s mixed s–d character. The SDL in 3d metals and alloys varies from about 60 nm in pure Co to a few nanometers in disordered alloys in which both the d character of the conduction band and the strong scattering by disorder contributes to the spin relaxation (around 4 nm in Ni97Cr3, 6 nm in Ni84Fe16, 12 nm in Co90Fe10 [26]. It must be finally pointed out that the spin-flips induced by electron–magnon scattering contribute to the spin mixing of the two-conduction channel and not (or contributes indirectly and weakly) to the SDL. This is because the electron–magnon collisions conserve the total spin of the electron system and cannot contribute to the relaxation of the spin accumulation to the lattice [76]. The interface spin relaxation, characterized by the spin memory loss coefficient d, is also controlled by spin–orbit effects, as it turns out for the large values of d found for interfaces with Pd, Pt or W. For the interfaces of ferromagnetic layers, an additional contribution can come from the absorption (spin transfer) of transverse spin current when there are departures from collinear configurations of the magnetizations in successive layers.

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Fig. 26. GMR of (Co(t)/Cr (0.9 nm)/Py (7 nm)/Cr (0.9 nm))N multilayers. (a) Variation of the MR as a function of the Co thickness. The inversion and zero GMR at t ¼ 4 nm results from the competition between negative interface and positive bulk spin asymmetries, i.e. gCo/Cro0 and bCo40. (b) Inverse GMR for t ¼ 1.5 nm. (c) Normal GMR for t ¼ 8 nm. From Vouille et al. [51].

5. INFLUENCE OF TEMPERATURE ON GMR Several effects contribute to the variation of the GMR with temperature: (i) Inelastic scattering processes by phonons or magnons add up to elastic scattering processes by imperfections. As the spin asymmetry is generally different for different scattering processes, the GMR is necessarily changed. For example, if the inelastic scatterings are spin independent, they contribute to R but not to DR and the GMR ratio is reduced. (ii) Electron–magnon scattering with spin-flip transfers momentum between the spinm and spink currents. This is the so-called spin mixing characterized by the


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spin–mixing resistivity rmk in ferromagnetic materials [7,14]. Spin mixing equalizes the two currents and reduces the GMR. As the global scattering rate increases with temperature, the scaling lengths governing the dependence of the GMR on thickness, MFP in CIP and SDL in CPP, are shortened and the variation of the GMR with thickness is more pronounced. Including all these effects into a theoretical model is a difficult task and necessarily involves a relatively high number of adjustable parameters. Again this is in the CPP-geometry that the modelling of the temperature dependence is easier and the interpretation of experimental results can be more quantitative. When spin mixing is neglected, the CPP-resistance and GMR in the long SDL limit are still given by Eqs. (22)–(25) where now rF ; rN ; rb ; b and g are replaced by temperature-dependent parameters rF ðTÞ; rN ðTÞ; rb ðTÞ; b(T) and g(T), taking into account both the elastic and inelastic contributions to the scattering. The experimental results of Oepts et al. [87] on Co/Cu multilayers deposited on grooved substrates have been analysed in this way. This leads to the temperature dependence of the resistivity and interface parameters shown in Fig. 27. The interface resistance parameters rb ðTÞ and its spin asymmetry coefficient g(T), are weakly temperature dependent. This means that, up to RT, the main contribution to the interface resistances still comes from elastic

Fig. 27. Temperature dependence of parameters derived from CPP-GMR experiments on Co/Cu multilayers: (a) interface spin asymmetry coefficient g, (b) mean resistance of the Co/Cu interface, (c) bulk spin asymmetry coefficient b in Co and (d) spin-up and -down Co resistivities and their mean value rCo : From Oepts et al. [87].


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scattering (specular and diffuse), with a relatively small contribution from inelastic scattering. The increase of the resistivities rF ðTÞ and rN ðTÞ by more than a factor of 2 between 4.2 and 300 K reflects the additional inelastic scattering by phonons and magnons. However, the spin asymmetry coefficient b of the Co layers decreases only by about 8% between 4.2 K and RT. A small variation of b has also been found in multilayered nanowires [34]. These results suggest that the spin asymmetry is not very different for elastic and inelastic scattering in Co, in agreement with results on bulk Co alloys [8]. Taking into account the spin mixing leads to a renormalization and reduction of the coefficients b and g [34,90,91]. It turns out that the temperature dependence of b can be reasonably accounted for by this renormalization [91]. Spin mixing can also be taken into account outside of the long SDL limit. For example, in the limit of very thick magnetic layers (with thin nonmagnetic layers), Eq. (26) becomes DR ¼ RP

2pb2eff ðTÞ rF ðTÞ rF ðTÞþr"#

l Fsf ðTÞ 2 tF ðTÞ beff ðTÞ

(27)

Equation (27) has been used by Piraux et al. [34,91] to analyse the RT data on Co/Cu multilayered nanowires and derive the SDL in Co at 300 K, l Co sf ð300 KÞE38712 nm. As pointed out in the preceding paragraph, the moderate reduction of the SDL between 77 K (59 nm) and 300 K indicates that the SDL is mainly controlled by elastic spin–orbit scattering or imperfections and has no important contribution from electron–magnon scattering.

6. ANGULAR DEPENDENCE OF GMR In most theoretical descriptions of the GMR, only P and AP orientations of the magnetizations in consecutive layers are considered and only a few publications have addressed the question of the variation with the angle between the magnetizations. In CIP geometry and in the simple case with only spin-dependent scattering (no potential barriers between the layers), the resistance is found to vary as [53]: g RðgÞ ¼ Rp þ DR sin2 (28) 2 where g is the angle between the magnetizations of consecutive magnetic layers. Several theoretical models have predicted departures from this simple variation but, on the experimental side, there are very few experimental results. In conventional multilayers, the angle g varies as the field H is applied, but there is generally some uncertainty in the relation between g and H. It can also occur that the angle g is not uniformly distributed over the sample, and this makes a precise determination of the angular dependence difficult. The best results seem to have been obtained in structures in which the magnetization of some of the layers can be rotated, while the magnetization of the neighbour layers is pinned by their coercivity or by their


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interaction with an AF layer. Measurements have been performed on (NiFe/Cu/ NiFe/FeMn) spin valves [92] and on multilayers alternating soft and hard magnetic layers, NiFe/Cu/Co/Cu and NiFe/Ag/Co/Ag [93,94]. None of these measurements could bring clear results on the deviation from Eq. (28). The angular dependence of the CPP-GMR is an interesting problem as it involves parameters (the so-called mixing conductance coefficients) that are also important in the theory of the magnetization reversal by spin transfer. In the experiments performed in the CPP-geometry [95,96] the deviations from Eq. (28) seem to be a little more pronounced than in CIP but, due to the uncertainty of the measurements, a quantitative determination of the involved parameters is still elusive.

7. BASICS OF SPIN-DEPENDENT TUNNELLING The spin splitting of the energy band in a ferromagnetic conductor gives rise to a spin dependence not only of the conduction but also of the tunnelling probability through an insulating layer. This is the origin of the TMR in a MTJ. A MTJ is composed of two ferromagnetic metal layers sandwiching a very thin insulating barrier that electrons can tunnel through. Spin-dependent tunnelling between two ferromagnetic layers was first observed and interpreted in 1975 [3] in a MTJ with an amorphous Ge semiconductor layer separating Co and Fe ferromagnetic electrodes. But it is only in the mid-1990s after the observation of a large effect at RT [4], that a tremendous research has been performed in this field. As for GMR, the resistance was found to depend on the relative orientation of the magnetizations of the two ferromagnetic electrodes. The spin dependence of the tunnelling from a ferromagnetic conductor can also be observed when the electrons tunnel to a superconductor. Extensive experiments on Al/Al2O3/FM tunnel junctions have been performed by Meservey and Tedrow [97,98] in the early 1970s. This led to the determination of the spin polarization of the electrons tunnelling from various types of FM through amorphous alumina. A clear and extensive review is presented in [99]. We will describe here the concepts behind the pioneering experiments of Meservey and Tedrow. When an Al/Al2O3/FM junction is placed in a magnetic field, there is a spin splitting of the quasiparticle DOS of the superconducting Al, as illustrated in Fig. 28 where one sees the different energy levels of the gap for the spin-up and -down directions. This leads to unequal DOS for the spin-up and -down directions in the superconductor and allows for a determination of the spin-up and -down contributions to the tunnelling current from or to the FM. The tunnelling current of spin-up or -down electrons at energy E flowing from electrode 1 (superconductor) to electrode 2 (FM) when the applied bias to the junction is V, can be written as 2 I 1!2 ðV ; EÞ / D1 ðEÞf ðEÞjM j D2 ðE þ eV Þð1 f ðE þ eV ÞÞ

(29) D i ðEÞ

where V is the voltage applied to electrode 2 with respect to electrode 1, is the DOS of electrode i at energy E for the+or – absolute spin directions, f is the Fermi– Dirac function and M is the matrix element of the transition. Similarly, the tunnel


194 (a)

Barrier

Superconductor

FM

∆ eV EF

Superconductor

σ1 dI/dV

-∆

Barrier FM

σ3 σ 4

(b)

σ2

V

∆+µ0µBH ∆-µ0µBH -∆+µ0µBH

eV

-∆-µ0µBH

EF

Fig. 28. Illustration of tunnelling experiments of Meservey and Tedrow type in FM/I/ superconductor tunnel junctions [99]. (a) Quasi-particule DOS of the superconductor at zero field (top left) and spin split by a magnetic field H (bottom left). The splitting of the sharp peaks at D 7 m0mBH is exploited to determine the spin polarization of the tunnelling from the ferromagnet. (b) Bias dependence of the differential conductance of a FM/I/Superconductor tunnel junction (solid line) and decomposition into spin-up (dotted) and -down (dashed) contributions. The spin polarization of FM is derived from the four conductance maxima by using Eq. (31).

current from electrode 2 to electrode 1 is 2 I 2!1 ðV ; EÞ / D1 ðEÞð1 f ðEÞÞjM j D2 ðE þ eV Þf ðE þ eV Þ

(30)

Since the energies involved in superconductivity are of the order of a few meV, whereas the barrier height is of the order of eV, Meservey and Tedrow interpreted their data by considering a constant tunnelling matrix. The total current is then given by integration of (I1-2–I2-1) over all energies Z þ1 IðV Þ / jM j2 D1 ðEÞD2 ðE þ eV Þ½ f ðEÞ f ðE þ eV Þ dE (31) 1

which leads, considering constant the DOS of the FM, to the conductance Z þ1 dI ebðEþeV Þ / jM j2 D2 D1 ðEÞ

2 dE dV 1 1 þ ebðEþeV Þ

(32)

Considering a BCS expression of.the DOS of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi the superconductor with singularities at E ¼ 7D and D1 ¼ Re jE j 2 E 2 D2 in the absence of field, the conductance will present sharp peaks at eV ¼ 7D when no magnetic field is


195

applied, as shown in the top left of Fig. 28a. Applying a magnetic field spin splits the DOS of the superconductor as represented in the bottom left of Fig. 28a. Considering that a proportion a, respectively (1a) comes from the majority (minority) bands of the FM, Meservey and Tedrow expressed the conductance G at voltages, –Vh, V+h, Vh, V+h (with h ¼ m0mBH/e), as a function of the unsplit DOS conductance g: s1 ¼ GðV hÞ ¼ agðV Þ þ ð1 aÞgðV 2hÞ

(33a)

s2 ¼ GðV þ hÞ ¼ agðV þ 2hÞ þ ð1 aÞgðV Þ

(33b)

s3 ¼ GðV hÞ ¼ agðV Þ þ ð1 aÞgðV 2hÞ

(33c)

s4 ¼ GðV þ hÞ ¼ agðV þ 2hÞ þ ð1 aÞgðV Þ

(33d)

the spin polarization P is then derived from P¼

D" D# ðs4 s2 Þ ðs1 s3 Þ ¼ 2a 1 ¼ ðs4 s2 Þ þ ðs1 s3 Þ D" þ D#

(34)

In the above calculation, one assumes that the DOS has a polarization P and the tunnelling probability is proportional to the DOS. It is not necessary to suppose this proportionality between DOS and tunnelling probability. Then the coefficient P derived from the experiments represents an effective spin polarization of the tunnelling probability from the FM through a given barrier. A more accurate analysis can also take into account spin–orbit effects in the superconductor [99]. Values of the spin polarization of TM derived by Meservey and Tedrow with this analysis are reported in the second column of Table 2 for TM/Al2O3/Al junctions. Results obtained more recently with a similar technique but still with an amorphous alumina barrier are shown in the third column. Finally, P derived with the same technique but with other insulating materials for the barrier are show in the fourth column. The first conclusion that can be derived from the data of Table 2 is that the spin polarization P of the tunnelling probability from a given FM can be quite different from the polarization of the conduction in the same ferromagnet. In permalloy (Ni80Fe20) for example, we know from Table 1 that the polarization b of the conductivity is 0.73 (73%), whereas the polarization P of the tunnelling probability through alumina is only 48% (or 32% in earlier experiments). In the same way, the polarization of the tunnelling is generally different from the polarization derived from Andreev reflection experiments as it clearly turns out when one considers that the spin polarization on the Fermi surface is not averaged in the same way for tunnelling and Andreev reflection [106]. In addition, as discussed in Section 10, the spin polarization of the tunnelling depends definitely on the electronic bondings at the metal/insulator interface and on the electronic structure of the insulating material, that is on parameters that are not at all involved in metallic conduction or Andreev reflection. For a recent discussion about the application of Andreev reflection to spin-polarization measurements we refer to Woods et al. [107]. The second clear conclusion is that P is not the spin polarization of the global DOS at the Fermi level of the FM but represents the specific spin polarization of the


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Table 2. Spin polarizations (P) of the tunnelling probability from various ferromagnets extracted from Meservey and Tedrow-type experiments. FM

Ni Co Fe Ni80Fe20 Ni40Fe60 Co50Fe50 Co70Fe30 Co84Fe16 CrO2 La0.7Sr0.3MnO3 SrRuO3

P (%) for Al2O3 barrier (early experiments) from [99]

P (%) new values for Al2O3 barrier (recent experiments) from [100,101]

23 35 40 32

33 42 45 48 55 55

Other barriers

74 [102] (MgO barrier)

85 [102] (MgO barrier) 55 490% [103] 72 [104] (SrTiO3 barrier) 9.5 [105] (SrTiO3 barrier)

Note: The second column corresponds to the early experiments of Meservey and Tedrow on TM/Al2O3/Al junctions [99]. Larger values of P have been recently obtained due to the improvement of the quality of the MTJ and are reported in column three [100,101]. Few experiments have been performed to determine P with other barriers than Al2O3. Some data are reported in the fourth column. Note that P obtained with Al2O3 is always of positive sign.

tunnelling probability. A large negative spin polarization of the DOS is indeed expected in Ni or Co, for which the majority spin d band is below the Fermi level, so that the DOS at EF should be definitely smaller for the majority spin direction. The positive values of the spin polarization obtained from tunnelling experiments for 3d ferromagnetic metals have been ascribed to the predominant contribution from s-like electrons to tunnelling. This was in agreement with results of Gadzuk [108] and Politzer and Cutler [109] who explained the positive spin polarization in Ni field emission experiments by a tunnelling probability of s electrons 102103 larger than of d electrons. On the specific example of Ni and Fe, Stearns [110] proposed that the main contribution to tunnelling comes from free electron-like dispersive bands of the electronic structure and is proportional to the DOS of these bands at EF. With this assumption, Stearns could calculate a positive spin polarization of 45% and 10% for Fe and Ni, respectively. However we will see that such a selection of a certain part of the total DOS depends on the insulator used for the barrier. As discussed in the next paragraph, different barriers can lead to different selections, therefore to different polarizations and even to different signs of the spin polarization for the same ferromagnetic electrode (see Section 9.3). This change of sign has been sometimes described as a crossover between s and d tunnelling. The third conclusion derived from the data of Table 2 is that the spin polarization depends not only on the ferromagnetic electrodes but also on the insulating material


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of the junction. For Fe, replacing the amorphous Al2O3 barrier by an epitaxial MgO (0 0 1) barrier increases the spin polarization from about 45% to 74% [102]. The important role of the barrier material in tunnelling experiments turns out in various TMR results showing the dependence of the amplitude and even the sign of the spin polarization on the insulator. Here we limit our discussion at this stage and we will come back to this point in Sections 9.3 and 10 for the theoretical interpretations that have been worked out more recently.

8. JULLIE`RE’S PIONEERING TMR EXPERIMENT AND MODEL In 1975, Jullie`re reported on the first measurements of a TMR in Fe/oxidized amorphous Ge/Co MTJ [3] and proposed their interpretation in term of the spin polarization of the tunnelling from a ferromagnet. At zero bias voltage, he obtained a 14% variation of the conductance between the AP and P states. This TMR rapidly decreases and vanishes at increasing bias, as represented on Fig. 29. Jullie`re interpreted his results with assumptions similar to those of the analysis of Meservey and Tedrow. A first assumption of the model is the conservation of the spin during the tunnelling process, i.e. spin-up and -down electrons carry the current in parallel in two independent channels. A second assumption is that the tunnelling matrix is independent of the energy and spin direction of the electron. The tunnelling current and the conductance are then proportional to the product of the DOS of the two electrodes in the spin channel one considers. For the conductances GP and GAP of

Fig. 29. Conductance variation as a function of the bias applied to a Fe/Ge oxidized/Co tunnel junction at T ¼ 4.2 K. DG is the difference of conductance between the AP and P configurations of the magnetizations of the two electrodes. A TMR effect of 14% is observed near zero bias but the effect disappears rapidly and vanishes for a bias around 6 mV. From Jullie`re [3].


198

the P and AP configurations, this leads to G P ¼ G "" þ G## / D1" D2" þ D1# D2#

(35a)

G AP ¼ G "# þ G #" / D1" D2# þ D1" D2#

(35b)

where D1m(k) and D2m(k) are the DOS of the two FM electrodes at the Fermi level. This leads to Jullie`re’s expression of the TMR ratio: TMR ¼

RAP RP GP G AP 2P1 P2 ¼ ¼ RP G AP 1 P1 P2

(36)

with Pi ¼

Di" Di# Di" þ Di#

(37)

The model of Jullie`re has been widely used to correlate the TMR ratio with the spin polarizations derived from Meservey–Tedrow-type experiments with a general good agreement in the case of amorphous Al2O3 barriers. However, as pointed out in the preceding paragraph, in both cases, the coefficient P is not necessarily the spin polarization of the DOS but, more generally, can represent an effective spin polarization of the tunnelling probability.

9. TMR: EXPERIMENTAL SURVEY 9.1. TMR with Transition Metal Electrodes and Alumina Barrier During two decades after the observation of Jullie`re, experiments on MTJ were performed by several groups [111–115] but only small TMR effects were observed at relatively low temperatures. For example, with Ni/NiO/Co tunnel junctions and at 4 K, Maekawa and Ga¨fvert [111] obtained a TMR that was clearly correlated with the magnetization reversal of the ferromagnetic electrodes but amounted at only 2%. The turning point was in 1995 when Moodera et al. at MIT [4], and Miyazaki and Tezuka in Sendai, found large and reproducible TMR effects on MTJ with an amorphous alumina barrier, respectively, for CoFe/Al2O3/Co at MIT and Fe/Al2O3/Fe in Sendai. In addition, these large TMR effects were observed not only at low temperature but also at RT. A typical TMR curve is shown in Fig. 30. One of the keys of the progress was the preparation of pinhole-free barriers of amorphous alumina with smooth interfaces. The results of 1995 triggered an extensive research on MTJ combining an amorphous Al2O3 barrier and ferromagnetic transition metal electrodes, Co, Fe, Ni and various alloys of these metals. A considerable effort has been devoted to improving the TMR ratio. Annealing MTJ in certain conditions was shown by Sousa et al. [116] to nearly double the TMR as compared to the as-grown state. This improvement was ascribed to a more homogeneous oxygen distribution within the amorphous Al2O3 barrier and sharper interfaces. The progressive improvement of the


199

Fig. 30. First observation of a large TMR at RT. The tunnel junction is of the type CoFe/ Al2O3/Co. The arrows indicate the orientation of the CoFe and Co magnetizations. From Moodera et al. [4].

quality of the junctions led to TMR ratios at RT of 50% for Co75Fe25 electrodes [117] and 70% for amorphous CoFeB electrodes [118]. Spin valve-type MTJ of submicronic lateral dimensions with different aspect ratios have been fabricated by e-beam lithography and extensively studied for the development of MRAM and field sensors for read heads. For these types of applications, the challenge is not only obtaining a high TMR ratio but also relatively low junction resistance area products (RA), around 103–104 O mm2 for the first generation MRAM and below 10 O mm2 in read head sensors. Several Al oxidation techniques have been developed to minimize the RA. A typical example [119] is a tunnel barrier obtained by in situ natural oxidation of a 0.7 nm Al layer with an RA of about 10 O mm2 while maintaining a TMR ratio around 20%. For MTJ with an alumina barrier and ferromagnetic TM or alloy electrodes, the spin polarizations derived by introducing the maximum TMR obtained at low temperature into Jullie`re’s expression is generally in approximate agreement with the spin polarizations derived from experiments of tunnelling through amorphous Al2O3 between the same FM and superconducting Al (listed in Table 2 for some of them). However, owing to the broad scattering of the spin polarizations derived in MTJ prepared differently, it cannot be concluded that the spin polarizations found for MTJ with alumina and those derived from tunnelling through alumina to superconducting Al are strictly the same. Actually, it is now clearly known from experiments with other barriers (see Section 9.3) that the spin polarization depends on each specific electrode/barrier combination and that a ferromagnetic metal has not a unique spin polarization, in contrast to what has been long assumed. For the same ferromagnetic metal, a dependence on the crystallographic orientation is also expected. This was first demonstrated by Yuasa et al. [120] in semi-epitaxial MTJ combining single-crystal Fe electrodes with an amorphous Al2O3 barrier as shown in Fig. 31 for Fe[(1 0 0), (1 1 0) or (2 1 1)]/Al2O3/CoFe. This dependence on the crystallographic orientation of the bottom electrode has been attributed to the anisotropy of the spin polarization on the Fermi surface but a slightly different

200


Fig. 31. Dependence of the TMR ratio as a function of the thickness of the Al2O3 barrier in Fe/Al2O3/CoFe tunnel junctions for different crystalline orientations of the bottom Fe layer. From Yuasa et al. [120].

Fig. 32. TMR ratio as a function of the Cu thickness for Co(0 0 1)/Cu(0 0 1)/Al2O3/Ni80Fe20 tunnel junctions at T ¼ 2 and 300 K. The oscillations (with a period of 11.4 A˚ in good agreement with the Cu Fermi surface) are due to reflections at the Co/Cu interface leading to spin-polarized quantum well states within the Cu layer. From Yuasa et al. [123].

growth mode of the amorphous Al2O3 barrier for the different crystallographic orientations of the Fe bottom electrode could not be excluded. Semi-epitaxial MTJ were also used to probe predicted resonant tunnelling effects [121,122]. A perfectly grown Cu(0 0 1) layer at the interface between Co and Al2O3 in Co(0 0 1)/Cu(0 0 1)/Al2O3/Ni80Fe20 tunnel junctions allowed Yuasa et al. [123] to evidence oscillations of the TMR as a function of Cu thickness due to resonant tunnelling as shown in Fig. 32. This effect was interpreted in terms of spin-polarized


201

quantum well states within the Cu layer due to spin-polarized reflections at the Co/Cu interface. The oscillation period was found in good agreement with the geometrical properties of Cu Fermi surface (see the chapter on exchange coupling by M. D. Stiles). 9.2. Search for Highly Spin-Polarized Ferromagnets In MTJ with an amorphous alumina barrier and transition metal electrodes, which have been extensively studied after 1995, the spin polarization of the tunnelling does not exceed 60% and it motivated the search for ferromagnetic conductors with higher spin polarizations. In particular, it launched the research on half-metals, a name which designates ferromagnets of metallic character for one of spin direction and insulating for the other. A 100% spin-polarized tunnelling current can be expected from a half-metallic electrode, at least if the half-metallic character is conserved in the atomic layers close to the electrode/barrier interface. Several materials have been predicted to be half-metallic, oxides like CrO2, Fe3O4, manganites (La0.7Sr0.3MnO3, La0.7Ca0.3MnO3) and several double perovskites of the Sr2FeMoO6 type, Heusler alloys like NiMnSb or Co2MnSi, and some magnetic semiconductors. For a synthetic review on half-metals and magnetic semiconductors we refer the reader to [124]. Large TMR effects have been obtained in epitaxial MTJ with manganites [125–128]. In Fig. 33, we give an example of the large TMR (80% at 77 K) observed with La0.7Ca0.3MnO3 electrodes by Jo et al. [128]. However this TMR vanishes below RT. The highest TMR (1800% at low temperature, which corresponds to a spin polarization of +95%) has been observed

Fig. 33. TMR curve for a La0.7Ca0.3MnO3/NdGaO3/La0.7Ca0.3MnO3 tunnel junction at T ¼ 77 K. A spin polarization of 86% can be deduced from the large TMR ratio. From Jo et al. [128].

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by Bowen et al. [127] on La0.7Sr0.3MnO3/SrTiO3/La0.7Sr0.3MnO3 tunnel junctions. The TMR is reduced to 30% at 250 K and vanishes at about 300 K, which turns out to be an effective Curie temperature at the La0.7Sr0.3MnO3/SrTiO3 interfaces (compared to 350–370 K for bulk La0.7Sr0.3MnO3). Large TMR effects have also been observed in the same type of tunnel junctions with a TiO2 barrier replacing SrTiO3 [129]. A spin polarization of 90% at low temperature was also found for electrodes of the double perovskite Sr2FeMoO6 [130]. MTJ with Fe3O4 electrodes [131–134] have led to strongly scattered TMR results, which is generally ascribed to the difficulty in controlling the structure and the stochiometry at the interfaces. For Fe3O4/CoCr2O4/La0.7Sr0.3MnO3 MTJ, Hu and Suzuki [133] have observed an inverse TMR (25% at 60 K, 0.5% at RT), which is consistent with the theoretically predicted negative spin polarization of Fe3O4. In contrast, with Co/Al2O3/Fe3O4 MTJ, Seneor et al. [132] have found a normal TMR (43% at 4.2 K, 13% at RT) corresponding to a positive spin polarization of Fe3O4. For CrO2, a high-spin polarization has been demonstrated in Andreev reflection experiments [103] but the TMR results have been always quite modest. The TMR of CrO2/ CrOx–AlOxAl2O3/Co MTJ [135] is inverse (–25% at 5 K) and vanishes around RT. Heusler compounds have also been introduced into MTJ. Recently Schmalhorst et al. [136] have studied MTJ combining electrodes of the Heusler alloy Co2MnSi and CoFe with an alumina barrier. They found a TMR value of 95% at 20 K, which corresponds to a spin polarization of 65.5% for Co2MnSi. The TMR drops to about 40% at RT. Large TMR effects have also been found with magnetic semiconductors like GaMnAs but only at very low temperature [137]. Finding halfmetals conserving a high-spin polarization at RT is still a challenge. 9.3. Dependence of the TMR on the Barrier and Electrode/Barrier Interface Besides the extensive research on MTJ with an amorphous alumina barrier, efforts have been also devoted to study alternative insulating materials. As pointed out in Section 9.1, applications of TMR for MRAM or sensors require a relatively small RA product and one of the first motivations was to find insulators with lower barrier heights than alumina and thus smaller resistances. MTJ with AlOxNy [138,139], Ga2O3 [140], ZrO2 [141] and ZrAlOx [142], barriers have shown TMR of nearly the same amplitude as with Al2O3 and a reduced RA. Moreover some of the experiments on MTJ with alternative barriers have been definitely useful to show the important role of the insulator in the physics of the spin polarization. De Teresa et al. [143,144] showed that, for a given ferromagnetic electrode, a change of the barrier material can even change the sign of the spin polarization of the electrons tunnelling from this electrode. They studied La0.7Sr0.3MnO3/I/Co tunnel junctions where I is a SrTiO3, Ce0.85La0.15O3 or Al2O3 layer or also a SrTiO3/Al2O3 double barrier. The La0.7Sr0.3MnO3 manganite is a positively spin-polarized halfmetal (see Section 9.2). It collects only electrons that are spin polarized in its majority spin direction that is majority (minority) spin electrons of the cobalt electrode for the P (AP) magnetic configuration of the MTJ. This leads to normal TMR (RAP4RP) if PCo is positive, or inverse TMR (RP4RAP) if PCo is negative (in the same way as in


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multilayers, the GMR is normal or inverse depending on the relative sign of the spin asymmetry in successive layers, see Section 3.3.4). With an Al2O3 barrier (Fig. 34C) the TMR is normal, in agreement with the positive spin polarization commonly observed for cobalt and other ferromagnetic TM with an alumina barrier. On the contrary, when SrTiO3 or Ce0.85La0.15O3 is used for the barrier, an inverse TMR effect (30% at low bias) is found, see Fig. 34A and B, which corresponds to a negative spin polarization for Co. Furthermore, when a thin Al2O3 layer is inserted between SrTiO3 and Co (Fig. 34D), a normal TMR is retrieved. This demonstrates the crucial role played by the electronic structure of the insulating material and, more particularly, by the local electronic structure at the electrode/barrier interface. Roughly speaking, it can be said that different barriers select electrons of different character, either positively spin-polarized s-electrons or negatively spin-polarized d-electrons. As described in Section 10, recent theoretical works [145–151] have clearly put forward the role of the interfacial bonding and the importance of the symmetry in determining the hybridization between the Bloch states of the metal and the most slowly decaying complex Bloch states in the insulator. Depending on the

Fig. 34. TMR curves obtained at small bias voltage (10 mV) for La0.7Sr0.3MnO3/I/Co tunnel junctions. I is SrTiO3 (A), Ce0.85La0.15O3 (B), Al2O3 (C) and SrTiO3/Al2O3 (D). From De Teresa et al. [143,144].

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barrier material, symmetry rules can select either spin direction and it can lead to different signs of the spin polarization for the same ferromagnetic metal. Actually, the spin polarization can depend not only on the barrier material but also on the interfacial termination and on local chemical modifications. Other examples of inversion of the spin polarization related to the choice of the insulator or to some change in the interface chemistry have been found in recent experiments. A negative spin polarization has also been found for Co with a TiO2 barrier [152] and for Ni with SrTiO3 [153]. Sun et al. [154] have found that chemical instabilities can change the sign of the spin polarization of Fe and Co with a SrTiO3 barrier. The results of Sharma et al. [155] for permalloy with a Ta2O5 barrier are more complex with different signs at different bias. The crucial role played by the interface in spin-dependent tunnelling is further evidenced by experiments in which ultra-thin layers are inserted at the interfaces, known as ‘‘dusting experiments’’. Although some early results have been blurred by roughness effects, the most recent results on high quality samples have clearly demonstrated that the spin polarization is controlled by a couple of atomic layers at the interfaces. In the case of a flat Cu layer inserted at the interface between Co and Al2O3, LeClair et al. [156] have found a decrease of the TMR as exp(tCu/x) with x ¼ 0.26 nm, which corresponds approximately to a monolayer of Cu. An even more rapid decrease is observed when Cr is used for dusting the interface [157] due to the strong modification of the DOS associated to the creation of virtual bound states on the Cr sites. The important role of the insulator in determining the spin polarization and the TMR has been clearly emphasized by the extremely high TMR revealed in 2003– 2004 by experiments on MTJ with single-crystalline or textured MgO(0 0 1) barriers. Pioneering work by Bowen et al. [158] on single-crystalline Fe/MgO/FeCo(0 0 1) MTJ had already found a TMR of 60% at low temperature and shown that TMR ratios as high as with alumina could be found with an MgO barrier. Even larger TMR have been found recently by several groups [102,159–162]. With single-crystal Fe/MgO/Fe(0 0 1) MTJ fabricated by MBE, Yuasa et al. [161] have found a TMR amounting to 250% at low temperature and 180% at RT. For sputter-deposited Co84Fe16/MgO/Co84Fe16 polycrystalline MTJ with MgO barriers highly (1 0 0) textured, Parkin et al. [102] found a TMR of 300% at low temperature (220% at RT) after thermal annealing. Moreover, with CoFeB/MgO/CoFeB MTJ grown by sputtering in which the MgO barrier is (0 0 1) textured but CoFeB amorphous, Djayaprawira et al. [162] found a TMR of 294% at low temperature and 230% at RT. These results demonstrate the very high-spin polarizations that can be obtained by coherent tunnelling through single crystal or highly textured barriers and the strict spin selection by symmetry rules at well-defined interfaces. The physics involved in the determination of the spin polarization in coherent tunnelling is described in Section 10. Remarkably, it even turns out from the results with amorphous electrodes that the crystalline character is more important for the barrier than for the electrodes. In Fig. 35, we give an example of experimental TMR curves obtained for (0 0 1) Fe/MgO/Fe MTJ and the variation of the TMR ratio with the thickness of MgO [161]. The oscillations as a function of the MgO thickness can


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Fig. 35. Resistance versus magnetic field curves (left) and TMR oscillations as a function of MgO thickness (right) measured at T ¼ 20 and 300 K in (0 0 1)Fe/MgO/Fe tunnel junctions. From Yuasa et al. [161].

be ascribed, in coherent tunnelling, to the interference between evanescent waves of different symmetry (different metal-induced gap states (MIGS), see Section 10). As also described in Section 10, the increase of the TMR with the MgO thickness is predicted by theory and attributed to a progressive blockade of a minority spin channel with a nonzero momentum component in the layer plane. Although the TMR observed in MTJ with (0 0 1) MgO tunnel barriers is extremely high, it is still smaller than predicted by theory and some features of the experimental results indicate that it is still influenced (and probably reduced) by structural defects that gives hope for further improvement of the TMR. 9.4. TMR: Bias Voltage Dependence Most MTJ exhibit a significant decrease of the TMR when the bias voltage applied to the junction increases. A criterion of merit is often the voltage V1/2 at which the TMR is divided by a factor of two. In the first experiment of Jullie`re, V1/2 was around 3 mV. After the observation of room temperature TMR in 1995 [4], more systematic studies of the bias voltage dependence have been carried out. For junctions with an alumina barrier, values of V1/2 higher than 500 mV are now routinely obtained [163–165]. However, clear correlations of V1/2 to the nature of the ferromagnetic metal and to the amplitude of TMR appear difficult to find [166]. By optimizing both the oxidation time of Al and the annealing process, Ahn et al. [165] could increase V1/2 up to 650 mV with a symmetric bias dependence when the oxidation of the barrier is complete without oxidation of the bottom electrode. An obvious way to reduce the bias voltage TMR dependence is also to design double or multiple junctions in order to reduce accordingly the voltage applied to each junction, as first demonstrated by Montaigne et al. [167]. Although mastering the bias voltage dependence of the TMR is highly relevant from a technological point of view, a detailed understanding of its physical origin is still lacking. Part of the drop is supposed to be related to the sharp structure of the

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conductance of tunnel junctions at low bias (V o 150 mV), called the zero-bias anomaly (ZBA). The ZBA is generally ascribed to inelastic processes. LeClair et al. [157] have also found a strong influence of impurities inserted at the Co/Al2O3 interfaces on the ZBA of Co/Al2O3/Co tunnel junctions. In spin-polarized scanning tunnelling microscopy experiments between a Co(0001) surface and an amorphous CoFeSiB tip through a vacuum barrier [168], a very flat bias dependence with no ZBA was obtained, which led the authors to the conclusion that the ZBA is related to scattering of electrons at defects in the barrier. However, these measurements were performed at RT where generally the ZBA is no more observed or strongly attenuated. Inelastic scattering by magnons [169,170] or by defects within the barrier [171] has been proposed for the interpretation of the variation of the TMR at low bias. By flipping the spin of the electrons, electron–magnon scattering makes that the spin-up (spin-down) electrons can tunnel into spin-down (spin-up) states. This is equivalent to a reduction of the asymmetry between the spin-up and -down DOS. Zhang et al. [170] predict a linear decrease of the TMR with the bias voltage if the scattering is by interface magnons, which is in agreement with the variation observed in many junctions. Lu¨ et al. [172] have also obtained a good fit of the bias dependence of the TMR up to 0.2 eV by considering assisted tunnelling by magnons and phonons. For scattering by defects or impurities within the barrier, one expects a variation with the bias voltage V as V p ; where p depends on the number of successive scatterings. By doping the barrier with Si, Ni or Fe impurities in Co/ Al2O3/NiFe tunnel junctions, Jansen and Moodera [173] did not find any clear variation of the ZBA and either a large decrease or a small increase of the TMR for Ni or Fe impurities, respectively. The energy dependence of the density and symmetry of the Bloch states in the ferromagnetic electrodes also impacts strongly on the bias dependence of the TMR. Electrons tunnelling at different bias voltages probe different energy ranges and information on the electronic structure can be obtained from an accurate analysis of the bias dependence of the conductance [174]. De Teresa et al. [143,144] have interpreted the opposite signs of the cobalt spin polarization in La2/3Sr1/3MnO3/ SrTiO3/Co and La2/3Sr1/3MnO3/Al2O3/Co junctions, as well as the very different bias dependences of the TMR presented in Fig. 36, in terms of the d- or s-character (respectively) of the electrons tunnelling through the barrier to the Co electrode. The monotonous bias dependence observed with alumina (Fig. 36b) is consistent with the energy dependence of the DOS of sp character in Co. On the other hand, for a SrTiO3 barrier (Fig. 36a), the TMR presents a maximum at negative bias (that is when electrons tunnel to Co) that can be related to a maximum in the DOS of d character in Co. In the case of Co/Al2O3/Co tunnel junctions, Xiang et al. [175] have found a very flat energy dependence of the DOS in agreement with the s-character part of the Co DOS. More recently, Tiusan et al. [176] reported on the bias dependence of (0 0 1)Fe/MgO/Fe and (0 0 1)Pd/Fe/MgO/Fe epitaxial tunnel junctions. The inversion of the TMR at+0.2 V for Fe/MgO/Fe (S1 in Fig. 37), can be ascribed to the existence of a sharp resonant interface state at the Fe/MgO interface. When Pd is inserted below the bottom Fe electrode, the coupling of this interface state with Bloch states of the electrode is strongly affected and the inversion disappears (S2 in Fig. 37). These


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Fig. 36. Bias voltage dependence of the TMR for La0.7Sr0.3MnO3/SrTiO3/Co (a) and La0.7Sr0.3MnO3/Al2O3/Co tunnel junctions. From De Teresa et al. [143,144].

Fig. 37. TMR versus bias voltage V for (0 0 1)Fe/MgO/Fe and (0 0 1)Pd/Fe/MgO/Fe tunnel junctions. The labels S1 and S2 are explained in the text. From Tiusan et al. [176].

features have not been observed in other recent studies on (0 0 1)Fe/MgO/Fe [102,158,161], which seems to show that MTJ fabricated in different groups present different defects and that the fabrication of even more perfect structures can be considered. It also paves the way to a control of the TMR bias voltage dependence by engineering the electronic structure of the layers and interfaces.

9.5. TMR: Temperature Dependence The TMR decreases with temperature. The reduction between low and RT can be as large as 25%, even in epitaxial tunnel junctions with a very high TMR ratio. The temperature dependence of the conductance and TMR of NiFe/Al2O3/Co and Co/Al2O3/Co tunnel junctions has been analysed by Shang et al. [177]. They assume that the spin polarization of the tunnelling electrons follows a classical Bloch-T3/2 law, P(T) ¼ P0(1-aT3/2). They also take into account the existence of additional

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thermally excited conduction channels derived from an analysis of the junction conductance and assume these additional channels to be unpolarized. A good agreement is obtained with a value of the parameter a somewhat larger than for bulk ferromagnetic metals that is consistent with the softening of exchange at an interface. An analysis of the temperature dependence of the TMR has been also performed by Garcia et al. [178] for La2/3Sr1/3MnO3-based tunnel junctions. The variation with temperature of the tunnelling spin polarization turns out to follow a Bloch law similar to that of the magnetization of the bulk LSMO with however a lower Curie temperature (300 K instead of 360 K) and a larger coefficient a. The enhancement of a with respect to its value in bulk LSMO is similar to what is observed by Shang et al. [177] for the spin polarization of permalloy and much smaller than observed at the surface of LSMO. 9.6. Spin Filtering by Ferromagnetic Barriers All the TMR results described above have been obtained for MTJ in which the electrodes are ferromagnetic whereas the insulating barrier is nonmagnetic. Interesting results have also been reported with ferromagnetic insulators in which the spin splitting of conduction band leads to different barrier heights for spinm and spink electrons. Even for electrons tunnelling from a nonmagnetic electrode, the tunnelling probability through a ferromagnetic barrier is expected to be different for the spinm and spink directions. Furthermore, the exponential dependence of the transmission on the barrier height can lead to very large spin polarizations, so that a (nonmagnetic electrode/ferromagnetic barrier) structure can be equivalent to an artificial half-metal for highly spin-polarized injection. Early experiments in this

Fig. 38. Schematic representation of the spin filter composed of a nonmagnetic electrode (M), a ferromagnetic insulating barrier (EuS) and a ferromagnetic counter-electrode (FM). In the P configuration of the magnetizations of EuS and FM, a large current flows due to both the large majority spin current emerging from the barrier and the large DOS at the Fermi level for the majority spins in the counter-electrode. In the AP configuration, the current is smaller. From LeClair et al. [180].


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field were performed by Moodera et al. [179] with spin filtering by the ferromagnetic semiconductor EuS. A spin filter efficiency of 90% was demonstrated using a superconducting Al layer to analyse the spin polarization. The spin filtering concept has been extended recently by LeClair et al. [180] by replacing the Al layer by a ferromagnetic counter-electrode to detect the spin polarization of electrons tunnelling from a nonmagnetic metal as represented on Fig. 38. The resistance depends on the relative orientation of the magnetizations in the barrier and the counter-electrode and Leclair et al. [180] observed a TMR larger than 100% at 2 K. However the Curie temperature of EuS is only 16.8 K which limits its use to for spin filtering to low temperatures. Similar spin filtering effects have now been obtained with other types of ferromagnetic insulators having higher Curie temperatures than EuS such as BiMnO3 [181]. If materials for efficient spin filtering at RT can be found, this can be of high interest for example to inject spins into semiconductors. Very high magnetoresistance effects have also been predicted for devices associating two spin filters [182].

10. MODELS OF TMR The huge experimental activity on TMR has stimulated a considerable development of the theory of spin-dependent tunnelling. In this section, we will first survey the simple models extending Jullie`re’s model (see Section 8) and then review the main progress in the last years using first-principles-based electronic structure calculations. We refer the interested reader to a number of review papers focusing on the theoretical aspects of TMR and spin-dependent tunnelling [12,13,100,183,184].

10.1. Free-Electron Models As indicated in Section 8, a model widely used to describe spin-dependent tunnelling was first derived by Jullie`re [3]. A more rigorous treatment was done by Slonczewski in 1989 [185] within the free-electron framework. He considered a rectangular barrier potential of height VB separating two similar ferromagnetic electrodes described by exchange split parabolic bands and derived the following expression for the spin polarization P: 2 k" k# k k" k# P¼ (38) k" þ k# k2 þ k" k# and minority Fermi momenta for the two ferrowhere km(k) represent the majority qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi magnetic electrodes and k ¼ ð2m=_2 ÞðV B E F Þ is the decay constant of the wave function in the barrier region for kjj ¼ 0. Thus P is found to depend on the barrier height. In the limit of infinite barrier height, the spin polarization P in Eq. (38) reduces to that of Eq. (37) derived by Jullie`re. However, in the general case, P depends strongly on the potential height and can even change sign for sufficiently low barriers. In 1997, Bratkovsky [186] extended the Slonczewski model by taking

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into account the electron effective mass, m*, within the barrier. The incidence on P as expressed in Eq. (38) is to weigh the kmkk factors by (m*)2. Zhang and Levy [183], in reviewing the various models for TMR, underlined the role played by the barrier potential profile in determining the amplitude of TMR. They showed that, within the free-electron model, the TMR is highly sensitive to the profile of the potential barrier. This issue was further addressed in a free-electron model by Montaigne et al. [187] who showed that asymmetries in the TMR(V) curves could be related to the shape of the barrier. Montaigne et al. could also qualitatively reproduce other experimental features such as the decrease of the TMR with voltage and the oscillations of TMR at high voltages. Although the simple models based on a free-electron approach are instructive and physically transparent, they miss essential features such as complex bands in the barrier, interface resonance states and chemical bonding effects that have to be taken into account in realistic systems as described in the next section.

10.2. Bonding at the Ferromagnet/Insulator Interface The electronic bonding at the ferromagnet/insulator interface determines the transmission across the interface and the spin polarization of this transmission. Tsymbal and Pettifor [188] have shown that a sss bonding can transmit a positive polarization by selecting states of s character in a transition metal for which the DOS has globally a negative spin polarization. This is because, at the Fermi level, the s component of the DOS in a metal like Co is reduced for the minority spin direction by s–d hybridization so that the s DOS has a positive polarization. Such a selection of a part of the DOS is admitted as a qualitative explanation of the different signs of the spin polarization in the total DOS of the metal and in the tunnelling current, as observed for a majority of systems in Table 2. In contrast, a predominance of the sds bonding is expected to transmit the negative polarization of the d band. But many types of different situations can be found. With a layer of oxygen deposited on the surface of Fe, Tsymbal et al. [145] predicted an inversion of the spin polarization at the Fermi level between the negatively polarized DOS in the interfacial Fe layer and the positively polarized evanescent wave in vacuum. In this case, they explained the inversion by the hybridization of the Fe 3d orbitals with the O 2p orbitals and the strong exchange splitting of the antibonding oxygen states. At the Co/SrTiO3 interface, Oleinik et al. [149] predicted than an exchange coupling mediated by oxygen between Co and Ti atoms induces a magnetic moment of 0.2mB on the interfacial Ti aligned AP with the moment of the Co layer. This is important to understand the opposite spin polarizations of the electrons tunnelling from Co through Al2O3 and SrTiO3, respectively [143,144]. However, an explanation of the spin polarization of the tunnelling current by interfacial bonding effects must be completed by an additional consideration on the mechanism of the transmission through the barrier to understand the global transmission between the electrodes. Before discussing this issue, we present in Fig. 39 a result of first-principle electronic structure calculations by Oleinik et al. [147] for a


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Fig. 39. Calculated atomic structure and local DOS for majority-spin electrons (top panels) and minority-spin electrons (bottom panels) for a Co/Al2O3/Co tunnel junction. After Oleinik et al. [147]).

Co/Al2O3/Co junction, which shows the complexity of the problem. The interface bonding induces a negative spin polarization of the local DOS on the O and Al atoms close to the interface but the spin polarization of the local DOS becomes positive on interiors atoms in Al2O3. The physics of the propagation inside the barrier and the role of the electronic structure of the insulator are discussed in the next section.

10.3. First-Principle Calculations of TMR and Symmetry Effects Ab initio methods based on density functional theory within the LSDA for the electronic structure provide the basis for an accurate calculation of the TMR. This leads to a multi-band description of the electronic structure, which takes into account the symmetry and the spin polarization of the electronic states in the ferromagnets as well as the interfacial localized states and the symmetry of the evanescent states (MIGS) in the insulator. The importance of the symmetry of the MIGS in the tunnelling barrier has been clearly emphasized by Mavropoulos et al. [148]. Near the surface or interface of a crystalline insulator, the MIGS are Bloch waves with a complex wave vector k ¼ q+ik, the imaginary part describing

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the decay of the wave function. The energy bands corresponding to complex k in the gap region are called complex bands. For a given insulator, there are several complex bands of different symmetry with different values of k at a given energy, as illustrated for ZnSe in Fig. 40. MIGS with different values of k decay differently in the barrier. Mavropoulos et al. [148] calculate the complex band structure for a wave vector perpendicular to the interface in several insulators and find that, for a broad class of insulators, MIGS belonging to the identity representation D1 should have a minimum decay. It favours tunnelling when, for the spin channel one considers, both ferromagnetic electrodes have Bloch states of the same D1 symmetry that can be coupled with the slowly decaying MIGS. The important role of the symmetry match between the metal and insulator wave functions appears also clearly in several recent ab initio calculations [146,150,151]. Junctions with a MgO barrier have been particularly investigated, for example by Mathon and Umerski [146], and Butler et al. [150,151]. We will focus here on a discussion of the results obtained by Butler et al. [151] for Fe(1 0 0)/MgO(1 0 0)/ Fe(1 0 0) junctions. The electronic structure is calculated by using the layer Korringa–Kohn–Rostoker technique and the conductance is expressed as a function of the transmission coefficients by the Landauer conductance formula. Understanding

Fig. 40. Dispersion curves, E ¼ E(kz, kjj ¼ 0) with kz ¼ q+ik, of the complex band structure of ZnSe (k is the decay parameter of an evanescent MIGS). E(k) is plotted as a function of k for q ¼ 0 (left panel) and q ¼ p/a (right panel). At a given energy in the gap above the valance band of ZnSe, the smallest value of k, corresponding to the slowest decay, is found a the intercept with the dispersion curve #1 (symmetry D1). From Mavropoulos et al. [148].


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the physics of the transmission through the insulator is simpler when one first supposes that the tunnelling current is predominantly carried by states of wave vector perpendicular to the interfaces (i.e. kjj ¼ 0) as expected for a thick enough barrier. Fig. 41 shows the layer-resolved tunnelling DOS (TDOS) in MgO for the individual MIGS induced by each of the incident Fe(1 0 0) Bloch states with kjj ¼ 0. This is shown for the majority and minority spin directions in the P (top) and AP (bottom) configurations of the structure. As evident from these figures, the TDOS decays much more slowly for the D1 states (spd character) and, from the symmetry of the Bloch states with kjj ¼ 0 at the Fermi energy of Fe, these D1 states exist only for incident electrons of the majority spin direction. In first approximation, the large TMR of the junctions can be explained as follows. The low resistance in the P configuration is due to the transmission by slowly decaying D1 states that are efficiently coupled at both interfaces with Bloch states of Fe of the same symmetry. In contrast, for the AP configuration, an incident Bloch state of majority spin in Fe is efficiently coupled with a D1 state in MgO but this D1 state cannot be coupled with a minority spin itinerant state in the right electrode (one sees on the figure that its TDOS decays inside Fe as it would occur for a nonitinerant interface state). Similarly, none of the minority spin Bloch states of the left electrode have the right symmetry to be bonded with a D1 state in MgO.

Fig. 41. TDOS for kjj ¼ 0 for Fe(1 0 0)/8 MgO/Fe(1 0 0). The four panels show the TDOS for majority (upper left), minority (upper right), and AP alignment of the moments in the two Fe electrodes (lower panels). Each curve is labelled by the symmetry of the incident Bloch state in the left Fe electrode. From Butler et al. [151].

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The physics becomes a little more complex when, for very thin MgO layers, states with kjj 6¼ 0 contribute to the tunnelling current. As Fig. 42 shows, the conductance is predominantly due to states with kjj close to zero for majority spin incident electrons and to peaks at kjj 6¼ 0 for the minority spin direction. These peaks come from interfacial resonance states [151] and their contribution to the minority spin current reduces the TMR expected from the single channel kjj ¼ 0. However as the MgO thickness increases, the conductance coming from the peaks at kjj 6¼ 0 decreases dramatically and, for example in junctions with 12 MgO layers in Fig. 42, becomes extremely small. An increase of TMR with MgO thickness has been therefore predicted [151] and recently observed in experiments [161]. This behaviour is of course expected for perfect structures. It was shown by Levy et al. [189] that interface roughness can blur the symmetry rules governing the coupling between Bloch functions of the metals and MIGS and mix itinerant and interfacial states. As for GMR, a quantitative prediction is difficult without a precise information on structural defects. This probably explains that, even for the recently investigated epitaxial junctions with MgO barriers, the TMR ratios are generally smaller than predicted theoretically and also broadly scattered.

Fig. 42. Majority (left panel) and minority (right panel) conductances for 4, 8 and 12 layers of MgO. Units for kx and ky are inverse Bohr radii. From Butler et al. [151].


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In summary, in epitaxial junctions and if we neglect the influence of defects, the symmetry and spin selection by the interfacial bonding with the most slowly decaying evanescent states plays an essential role in TMR. Slowly decaying states of D1 type (spd character) are expected not only for MgO [146,151] but also for several other types of insulators [148]. In contrast, slowly decaying states of d character are expected when states of d character are close in energy on both sides of the gap of the insulator. As shown in the next section, this picture for perfect structures must be qualified in the presence of defects.

10.4. Models for Disordered Junctions Real tunnel junctions, even epitaxial junctions, contain several types of defects: interface roughness, interdiffused interfaces, impurities or vacancies, stacking faults, etc. Various effects related to these imperfections have been predicted. For example, localized interface states are not coupled with itinerant states of the metal and do not contribute to the tunnelling for perfect interfaces, but, as shown by Levy et al. [189], interfacial disorder mixes the two types of states and provide additional conduction channels. This affects the TMR if the itinerant and interface states have different spin polarizations. Interfacial disorder also breaks the symmetry of the system, mixes Bloch waves of different symmetries and blurs the symmetry rules. This occurs not only with interfacial disorder but also with bulk defects. It has been shown that by introducing some disorder within 10 ML of the electrode adjacent the interface can reduce significantly the spin polarization of the tunnelling [12]. The role of the interface structure has been clearly shown in the experiments of LeClair et al. [156,157] as discussed in Section 9.3. Disorder inside the barrier can also affect the TMR. Disorder can broaden the conduction and valence bands of the insulator and lower the barrier height. Defects and impurities can also lead to the formation of localized states in the gap of the insulator. If the energy of these states is close to the Fermi energy, it introduces a channel of impurity-assisted tunnelling [190]. The conductance per spin of a channel at EF is given by G¼

4e2 G1 G2 h ðE F E i Þ2 þ ðG1 þ G2 Þ2

(39)

where Ei is the energy of the impurity state, and G1 =_ and G2 =_ the leak rates of an electron from the impurity to the left and right and left electrodes. The conductance should be averaged over the energies and the position-dependent values of the leak rates G1 =_ and G2 =_: Tsymbal et al. [190] have shown that depending on the distribution of the impurity energy levels, this can lead to complex variations of the TMR with the bias voltage and inversions of the TMR. They explained in this way that both positive and negative TMR can be observed in Ni/NiO/Co nanowire junctions [191]. Results of the same type, with complex bias dependence and inversion of the TMR, have been obtained by Garcia et al. [192] on MnAs/GaAs/MnAs and MnAs/AlAs/MnAs tunnel junctions. Vedyayev et al. [193] and Jansen and Lodder

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[194] studied the influence of magnetic impurities and found that, in some conditions, the TMR can be enhanced with respect to the situation of direct tunnelling. In tunnel junctions with amorphous barriers like alumina, the situation is even more complicated due to multiple resonant scatterings by several localized states and the resulting interferences. Tsymbal and Pettifor [195] found that the current flows through a few regions corresponding to highly conducting channels induced by the local disorder in agreement with local measurements of the tunnel conductance of an alumina layer by Da Costa et al. [196]. The situation with an amorphous barrier is therefore hardly accessible to theory. We can however point out the work of Tsymbal and Pettifor [195] who predict a reduction of the TMR by disorder but with still a TMR ratio given by an expression of Jullie`re’s type.

11. APPLICATIONS OF GMR AND TMR A review of the multiple applications of GMR and TMR is not in the scope of this chapter. For a general review on the applications of magnetic nanostructures, we refer the reader to the chapter by Parkin in the book ‘‘Spin dependent transport in magnetic nanostructures’’ [13] and, for an article devoted to the physics and technical aspects of the spin valve structures, to the excellent review by Coehoorn [24]. Most spin valves are based on the Co/Cu system, which has been found by Mosca et al. [17] and Parkin et al. [18] in the earliest years of GMR. The main improvements have been obtained by improving the design of the spin valve structures. What has been done is, for the free layer, combining Co or CoFe with a magnetically soft layer like Py, improving the pinning of the second layer with an artificial AF structure, proceeding from a simple to a dual spin valve design, increasing the specular character of the reflections on the outer interfaces. GMR values of more than 24% at RT have been reached with dual structures [197]. For the CPP-GMR, the best results are obtained by doping Ni or Co with Fe impurities to enhance the bulk spin asymmetry coefficient b by impurity scattering. For TMR, most present applications are using junctions with NiFe or CoFe electrodes and a barrier of amorphous alumina. But one knows now that high TMR can be obtained with other types of barrier like epitaxially grown MgO. As discussed before, TMR ratios of more than 1000% with some magnetic oxides have been obtained at low temperature. Therefore, an important challenge now is to find half-metallic ferromagnets working at RT.

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Chapter 7 ELECTRICAL SPIN INJECTION AND TRANSPORT IN SEMICONDUCTORS B. T. Jonker and M. E. Flatte´ 1. INTRODUCTION The operation of every semiconductor device to date is based upon the motion of charge. The absence or presence of an electron is the functional basis of every device or circuit, ranging from those found in the cheapest wristwatch to the mightiest supercomputer. The remarkable advance of the performance to cost ratio which has come to typify modern electronics can be attributed largely to size scaling. This trend was noted years ago [1], and is commonly referred to as Moore’s Law, which specifically projects that the number of transistors per unit area will approximately double every 18 months. History has proven this projection to be remarkably accurate (Fig. 1a). However, simple reduction of length scale cannot continue indefinitely. Eventually, this trend would push the thickness of the SiO2 gate oxide layer, a key component of CMOS technology, below 1 nm (Fig. 1b), where it fails to perform its intended role as an effective insulator, and macroscopic scaling laws are no longer valid. Scaling has thus confronted scientists with the challenge of finding alternatives to the gate oxide layer or other solutions. The thought of abandoning SiO2, or the unwavering dependence of function based on charge motion is indeed unsettling, and represents a watershed for future electronic technology. A variety of new materials and avenues for device operation are currently being explored. Several represent sophisticated extensions of existing practice, while others are true paradigm shifts requiring development of entirely new concepts and operating principles. Spintronics, or the use of spin angular momentum as the basic functional unit rather than charge, is an example of the latter [2–4]. Like charge, spin is an intrinsic fundamental property of an electron. This fact is well appreciated by the magnetic recording industry, but until recently has been largely unrecognized by the semiconductor device community. Whereas magnetic recording media is based upon the macroscopic manifestation of electron spin, i.e., the static magnetization, this giant industry was recently revolutionized by a remarkable phenomenon, giant Contemporary Concepts of Condensed Matter Science Nanomagnetism: ultrathin films, multilayers and nanostructures Copyright r 2006 by Elsevier B.V. All rights of reproduction in any form reserved ISSN: 1572-0934/doi:10.1016/S1572-0934(05)01007-3

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Fig. 1. The two faces of Moore’s Law (a) the number of transistors per unit area approximately doubles every 18 months due to reduction in feature size, which (b) forces a corresponding reduction in the thickness of the SiO2 layer used as the gate insulator due to scaling rules. The projection indicates that this will require sub-nanometer thicknesses by 2005, at which point SiO2 fails to function effectively at existing CMOS voltage levels. Reprinted by permission of Intel Corporation, Copyright Intel Corporation. See http:// www.intel.com/technology/silicon/mooreslaw/index.htm

magnetoresistance (GMR), resulting from electron spin current rather than simple charge current. Discovered in 1988 [5,6], GMR refers to the change in resistance to spin-polarized current flow between two ferromagnetic metals (FM1 and FM2) separated by a non-magnetic spacer metal when the magnetization of FM1 is parallel (low resistance) or antiparallel (high resistance) to that of FM2. A detailed description

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of GMR is included in Chapter 6, and reviews may be found in several references [7,8]. This new property, derived from the spin of the electron rather than its charge, is the basis for a new generation of magnetic read heads that were introduced in 1998, and are found in virtually every hard disk drive manufactured thereafter. The use of the spin degree of freedom in semiconductors promises many new avenues and opportunities that are simply inaccessible in metal-based structures. This is due to the characteristics for which semiconductors are so well known: the existence of a band gap and the accompanying optical properties on which a vast optoelectronic industry is based, the ability to readily control carrier concentrations and transport characteristics via doping, gate voltages and band offsets, and the ability to tune many parameters (band gap, lattice constant/strain, band offsets) over a significant range in binary and ternary compounds. Carrier spin coupled with the traditional band gap engineering of modern semiconductor electronics offers new functionality and performance [9–12] as well as an avenue to circumvent the dielectric breakdown and capacitive limits which are major near-term concerns in existing electronics. Two broad regimes of spin-dependent device operation may be distinguished: one, in which the net spin polarization is the key parameter (i.e., there are more spins oriented in a given direction than in the opposite direction in either current or number density), and a second, in which spin-phase coherence is important. This article will focus on the former. The latter is relevant to other avenues, including the development of spin-based quantum computation or cryptography, which rely on controlled entanglement of wave functions. 1.1. Basic Requirements for Semiconductor Spintronics A semiconductor-based spintronics technology has at least four essential requirements for implementation [13]: (i) efficient electrical injection of spin-polarized carriers from an appropriate contact into the device heterostructure, (ii) adequate spin diffusion lengths and lifetimes within the semiconductor host, (iii) effective control and manipulation of the spin system to implement the desired function, and (iv) efficient detection of the spin system to determine the output. Spin storage may also be identified as a separate requirement, although it is implicitly included here under spin lifetimes, or may be accomplished in the form of the remanent magnetization which accompanies ferromagnetic (FM) order. Research has provided very encouraging progress in several of these areas. Spin diffusion lengths of many microns [14,15] and spin lifetimes 4100 ns [16,17] have been reported for electrons in optically pumped GaAs, providing new motivation for utilizing semiconductors as hosts for spin. Since sub-micron length scales and high-frequency operation (4100 MHz) are the norm, these results demonstrate that a spin-polarized mode of operation is certainly feasible for every modern transport device. A number of successful methods have been demonstrated to manipulate the spin system [18–20], including gate voltage-induced spin precession. Although

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practical electronic spin detection remains a challenge, several methods have been demonstrated or are being explored to probe the state of the spin system, including GMR-like behavior at a Schottky contact [21], the emission of circularly polarized radiation resulting from the recombination of spin-polarized carriers [14], Faraday rotation [15,16], the so-called spin–Hall effect [22–26], or spin–split features in tunneling spectra [27]. Spin injection from a scanning tunneling microscope tip in ultra-high vacuum was reported in 1992 and recently revisited [28–30]. However, an efficient and practical means of electrical spin injection has been unavailable until recently, and this lack has been a critical issue severely hampering progress. Electrical spin injection via a discrete contact is highly desirable, since it provides a very simple and direct means of implementing spin injection compatible with existing device fabrication technology in which the contact area defines the spin source. In this chapter, we will discuss the factors relevant to spin injection and transport in semiconductors, review the concepts necessary, and illustrate these by way of example to provide an overview of the current state-of-the-art.

2. MATERIAL PROPERTIES INFLUENCING SPIN INJECTION 2.1. Coupling between Light and Electron Spin, and Optical Spin Excitation The earliest method of generating a highly polarized population of spin-polarized carriers within a semiconductor is by optical excitation using polarized light, often referred to as ‘‘optical pumping’’ [31]. Absorption of a photon whose energy is equal to or greater than the band gap promotes an electron from the valence band (VB) to the conduction band (CB). If the light is circularly polarized, the electron and hole must share the angular momentum transferred in a manner that satisfies conservation laws. In a direct gap, zincblende semiconductor like GaAs, the CB is slike in character and two-fold spin degenerate, and the electron can occupy states with values of spin angular momentum mj ¼ 1=2: The VB is p-like in character and four-fold degenerate in bulk material, so that the hole can occupy states with values of angular momentum mj ¼ 1=2; 73/2, corresponding to ‘‘light hole’’ (LH) and ‘‘heavy hole’’ (HH) states, respectively. Interband transitions at the zone center (k== ¼ 0), which satisfy conservation of angular momentum satisfy the selection rule Dmj ¼ 1; reflecting absorption of the photon’s original angular momentum (Fig. 2a) [31,32]. The probability of a transition involving a light or heavy hole state is weighted by the square of the corresponding matrix element connecting it to the appropriate electron state, so that HH transitions are three times more likely than LH transitions. Thus, absorption of photons with angular momentum +1 produces three ‘‘spin-down’’ (mj ¼ 1=2) electrons for every one ‘‘spin-up’’ (mj ¼ þ1=2) electron, resulting in an electron population with a spin polarization of 50%. In an AlGaAs/GaAs quantum well (QW), quantum confinement lifts the light/ heavy hole degeneracy, and the HH states define the VB edge, as shown in Fig. 2b.


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Fig. 2. Optical transitions allowed due to absorption or emission of circularly polarized light for (a) bulk material and (b) a structure in which the light/heavy hole band degeneracy is lifted, as in a QW.

In principle, circularly polarized light whose energy is just sufficient to excite these states (and not the LH states) can produce an electron population which is 100% spin polarized. These high levels of polarization are easier to achieve at the subpicosecond time scale. Over longer time scales, imperfections in the material or intrinsic spin relaxation mechanisms reduce the peak spin polarization. 2.2. Spin Lifetimes Optical investigations of semiconductors using these techniques to generate spinpolarized carriers have provided considerable insight into spin lifetimes and relaxation mechanisms. In the following paragraphs, we briefly summarize the principal spin relaxation mechanisms. A more detailed discussion may be found in several references [17,31,33]. Elliot–Yafet (EY) spin scattering refers to processes in which an electron scatters from a specific defect within the semiconductor in a manner that causes the electron to flip its spin. These defects may be structural defects such as misfit dislocations, grain boundaries, or local impurities, or they may be dynamic defects such as phonons. The defects themselves are assumed to be non-magnetic, as expected for most sources of scattering in non-magnetic semiconductor materials. In the absence of spin–orbit interaction, these defects would not be able to flip the spin orientation of carriers. The spin–orbit interaction, however, causes the wave functions of states not to factorize into a single product of an orbital wave function and a spinor. The correlation between the spin component of the wave function and the orbital component of the wave function differs for electronic states with different momenta,

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and thus non-magnetic scattering from one state with momentum k1 to another state with momentum k2 will lead to a small probability of a spin flip. This happens even when the materials are inversion symmetric, and the dispersion relations are doubly degenerate at each momentum k, because it is still not possible to factorize states at a given momentum into an orbital and a spin part. The EY process [33] tends to dominate for inversion-symmetric materials, such as simple metals. It is expected that the EY process will dominate also near room temperature in inversion-symmetric semiconductors such as silicon and germanium. As the non-magnetic scattering rate increases (and thus the mobility decreases), the spin relaxation rate via the EY process increases. A detailed experimental study and model calculations of EY spin scattering due to interface defects may be found in Ref. [34]. In this study, spin-polarized electrons were electrically injected from a ZnMnSe layer into an AlGaAs/GaAs heterostructure. A linear correlation was found between the electron spin polarization in the target semiconductor, GaAs, and the number density of misfit dislocations at the ZnMnSe/AlGaAs spin-injecting interface which formed due to lattice mismatch. Thus, the EY process can be the spin relaxation process limiting spin-injection efficiency from a magnetic semiconductor into a non-magnetic semiconductor. D’yakonov–Perel’ (DP) spin scattering results from the motion of a carrier through an inversion-asymmetric material. As it moves, the combination of spin– orbit interaction and the inversion asymmetry of the material causes it to feel an effective magnetic field. This effective magnetic field can be used for coherent spin manipulation, as described later. In an ensemble of carriers with varying momenta, such as is found in an electron gas, these magnetic fields point in varying directions for different carriers. For an electron gas in equilibrium, the average magnetic field vanishes, but the variance of the magnetic field does not. As carriers scatter from momentum state to momentum state, they feel a time-dependent effective fluctuating field. Considering the effect of this fluctuating field in the standard fashion [33], D’yakonov and Perel’ derived an effective spin relaxation rate[35]. One way to visualize the effect of this fluctuating effective time-dependent magnetic field is as a random walk of the spin around the sphere. Each step is the precession of the spin in the coherent effective magnetic field associated with a single momentum state. The precession angle can be determined from the precession rate (the effective magnetic field) and the amount of time the carrier stays in that orbital state, which is the momentum scattering time. The mean rotation angle of a spin from its initial orientation is proportional to the step size a, and also to the square root of the number of steps N. If the momentum scattering time is half as long, the step size becomes half as large, and the number of steps becomes twice as big. The mean rotation angle, however, changes as ð2NÞð1=2Þ ða=2Þ ¼ N ð1=2Þ a=2ð1=2Þ : Thus, the effective spin relaxation rate decreases as the momentum scattering time (and thus the mobility) decreases. This dependence on the mobility is the opposite of the dependence of the EY spin relaxation rate on the mobility. DP scattering is the predominant spin relaxation mechanism at higher temperatures (470 K). The spin relaxation time resulting from the DP mechanism, tDP s , has a characteristic temperature dependence which depends on the degree of spatial


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confinement of the electron, i.e., its ability to move about in the lattice. In the bulk (3D), this dependence has generally been expressed in the semiphenomenological form [36] 1 (1) tsDP-3D tp ðTÞT 3 while in a two-dimensional system such as a quantum well it is expressed as [35] 1 (2) tsDP-2D tp ðTÞT where tp(T) is the momentum relaxation time. There is continued debate on the temperature dependence of tp(T), but it is often expressed as tp(T)T0.5 for QW (2D). Recent work indicates that the temperature dependence of tp is similar in both bulk and QWs [36], although it may exhibit a stronger dependence in the bulk [37]. If the dominant scattering mechanism changes with temperature, unusual temperature dependences of ts are possible [38]. A higher degree of spatial confinement (as in a quantum dot) prevents electron motion in the lattice and suppresses the accompanying DP spin scattering, which can result in spin relaxation times which are considerably longer than those in bulk or QW configurations [39]. When the DP process is quenched, the limiting process may be one-phonon scattering events connecting spin-up and spin-down levels mixed via the spin–orbit interaction [40,41]. Recent measurements of spin lifetimes T1 in quantum dots yield times as long as 20 msec at a magnetic field of 4 T and a temperature of 1 K [42]. Bir–Aronov–Pikus (BAP) spin scattering results from electron–hole interaction processes. Mutual spin–flip scattering is common among carriers in a semiconductor. If a spin-up electron scatters from a spin-down electron, and in the process the electrons exchange spins, then the electron gas’ spin polarization is unchanged. With multiple types of carriers, such as when both electrons and holes are present, a spinup electron and a spin-down hole can exchange spin orientation. In this process, the total spin polarization of the combined electron–hole plasma still does not change. In most inorganic semiconductors, however, the symmetries of the conduction electron band and the valence hole band are very different. The CB commonly consists primarily of antibonding s states, and the VB primarily of bonding p states. The resulting additional degeneracy in the VB provides many final states for a scattered hole that have very different spin–orbit structure. Only a few scattering events are required to fully randomize a hole’s spin, in a process most similar to the EY process described above. The BAP process recognizes the potential for coupling between the electron spin polarization and the hole spin polarization. When the hole spin relaxation time is shorter than the time for electrons and holes to mutually spin flip, then the electron spin relaxation time is effectively given by the time required to transfer spin polarization to the holes (which then rapidly relax). The effectiveness of this process severely limits electron spin relaxation times in optically generated populations in undoped semiconductors. The process of optically generating spin-polarized electrons requires the simultaneous generation of holes. In n-doped semiconductors these holes can recombine with the original background of unpolarized electrons, leaving behind a long-lived spin memory [43]. In undoped semiconductors this is not possible, and

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the persistent presence of the holes rapidly depolarizes the electrons as well. Studies of the BAP process have also examined the influence of the electron–hole attractive interaction, and exciton formation, on spin relaxation rates. Hyperfine interactions between the nuclear and carrier spin systems can also lead to a reduction of carrier spin polarization. In most III–V semiconductors, all the nuclei have non-zero spin, and can engage in mutual spin–flip processes with electrons and holes in a process analogous to the BAP process described above. A key difference is the short-range, weak coupling that characterizes the hyperfine interaction. The nuclear–electron interaction is generally described by a Fermi contact Hamiltonian, whereas the electron–hole interaction is a long-range Coulomb interaction. Typical nuclear spin relaxation times in III–V semiconductors range from a few milliseconds to several minutes [31], and can be modified by altering the overlap of electronic wave functions with nuclei of various types [44]. In many semiconductors, the nuclear polarization can be enhanced dramatically by optical pumping [45]. In this process spin-polarized electrons transfer their polarization to the nuclear system [46], putting the nuclear spin system in a kinetic equilibrium with the electronic spin system. This dynamic nuclear polarization process can yield nuclear spin polarizations in excess of 90%. If the vast majority of nuclei are spin polarized, then an electron spin with the same orientation cannot undergo a mutual spin–flip process. This has been proposed as a method of lengthening the electron spin lifetimes at low temperatures in semiconductor quantum dots [47]. 2.3. Spin Currents versus Charge Currents Spin currents differ fundamentally from charge currents in semiconductor systems. Perhaps the most straightforward difference is the mere existence of a spin lifetime. No such lifetime exists for electron charge; as a rigorously conserved quantity, any electrical charge entering an element must leave the same element. The finite spin lifetime in semiconductor materials dramatically indicates that electron spin is not a conserved quantity. Non-conserved currents, however, are common in semiconductor materials in other contexts [48]. Perhaps the most closely related type of non-conserved current is the current associated with a specific carrier type during conditions of current transport by more than one band. In semiconductors, it is common for current to be carried simultaneously by both conduction electrons (referred to here simply as electrons), and valence holes (referred to here simply as holes). The classic bipolar semiconductor device, a p–n diode (Fig. 3), has a fixed

Fig. 3. Schematic diagram of a p–n diode. In this structure, the charge current density is the same at every point, but the specific entities carrying that current can be electrons or holes, depending on location. Holes are indicated with light color and electrons with dark color.


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current through the device at all points. In the leftmost region the current is carried principally by holes, and in the rightmost region the current is carried principally by electrons; these regions are referred to as drift regions, and the carriers are majority carriers (i.e., of the same type as the equilibrium carriers in the region). The conversion of current carriers from holes to electrons occurs in the middle, in which the current is carried by minority carriers (holes on the right, electrons on the left). Due to charge conservation, the charge current is the same at every point in the device, but the current carried by a specific carrier type depends on the position in the device. Conversion from electron current to hole current is described by a carrier lifetime (not a charge lifetime), which is determined by the probability of an electron and hole to recombine in a charge-conserving process. Another vital difference distinguishes the interaction between two charges and that between two spins. A charge–charge interaction force is large and long-range compared to a spin–spin interaction force. The addition of a small excess concentration of charges (such as a charge accumulation layer) has a corresponding selfconsistent field from the Poisson equation that can be quite large. For example, a space charge layer of 1012 cm2 positive charges, separated by 1 mm from the same number of negative charges, produces a potential energy shift of 10 V! This key characteristic is vital for the proper performance of field-effect transistors. The interaction between two spins, from electron exchange and correlation, is much weaker and short-range. Generation of two similar layers of spin accumulation, of opposite spin orientation and separated by 1 mm, would require essentially no energy at all. There do exist inhomogeneous carrier distributions in semiconductors that are not spin polarized, and yet do not require large energy costs. These distributions are associated with spatially dependent concentrations of oppositely charged carriers. For example, an accumulation of excess electrons and an equal number of excess holes would have a very small interaction energy from charge–charge interactions. These types of distortions are often called ‘‘charge polarization packets’’, and a schematic of one is shown in Fig. 4a below. When they move through a material under the influence of a small electric field, the excess electrons and the excess holes cannot separate very far. If they did, then large electric fields would be produced forcing them back together. The process of movement is shown in Fig. 4(b and c); the electric field pushes the electrons in one direction and the holes in another (b), then the self-consistent space-charge field pulls the two together and the carrier with the higher conductivity moves to the location of the carriers with the lower conductivity. We thus obtain the well known, but somewhat counterintuitive result that the motion of a charge polarization packet is determined by the mobility of the carrier with the lower conductivity [48]. The motion of spin-polarized packets is also influenced by space–charge fields [49]. The two different carrier types that constitute the packet are no longer electrons and holes – instead they are spin-up and spin-down versions of the same carrier type. The electron version of the spin packet is shown in Fig. 4d, and does not involve any excess holes. There is a marked distinction between the response of these ‘‘unipolar’’ packets to an electric field. Instead of the two components of the

236 B. T. Jonker and M. E. Flatte´

Fig. 4. Schematics of the motion of charge-polarization packets and spin-polarization packets in semiconductors. (a) Excess electrons and holes in a charge-polarization packet balance to produce no excess charge density. (b) Under the influence of an electric field, the holes and electrons attempt to move in different directions, generating a space-charge field. (c) The high-conductivity carriers move to the location of the low-conductivity carriers, eliminating the space-charge field. (d) Excess spin-up electrons in a spin packet are balanced by missing spindown electrons in an n-doped semiconductor, so that once again there is no excess charge density. (e) Both spin-up and spin-down electrons move to the right under the influence of an electric field, generating a much smaller space-charge field, and resulting motion to the right (f) of the packet.


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packet being pulled in opposite directions, as in Fig. 4b, the two components are pulled in the same direction, as shown in Fig. 4e. If the mobilities of spin-up and spin-down carriers are the same, then no space-charge field is established, and the packet moves seamlessly through the material. As mobilities typically depend on carrier density, however, a difference in the mobility of the spin-up and spin-down carriers can be expected. In this case, the pulling of the carriers in one direction will be slightly different (as shown in Fig. 4e), a space-charge field will again be generated, which brings the two disturbances together (Fig. 4f). In a semiconductor with electrons and holes two different types of spin packets can be made, characterized by two different mobilites: an electron spin packet, with high mobility, and a hole spin packet, with low mobility. Only one type of charge packet can be made, and its mobility is determined by that of the low-mobility carriers. Spin packets thus provide greater flexibility in packet motion than charge packets. These packets are not just elementary disturbances of an equilibrium semiconductor, they also can directly influence the performance of devices. Shown below in Fig. 5 are (a) a bipolar junction transistor and (b) a unipolar spin transistor [50]. The unipolar spin transistor is a spin-based analog of the bipolar junction transistor. Just as a bipolar junction transistor has an emitter and collector with majority carriers of one type (as shown in Fig. 5a, n-type), and a base with majority carriers of another type (p-type), the emitter and collector of a unipolar spin transistor have majority spin-down carriers and the base has majority spin-up carriers. The switch-on speed of (a) is determined by the charge polarization packet transport speed, and of (b) by the spin packet transport speed. The role of the electron–electron interaction on the motion of spin and charge packets has been described at a simple level here – in a more complete description, the additional electron interaction effects due to the exchange energy and the correlation energy might be included. These do not qualitatively affect the conclusions described above for typical electron temperatures and densities, although they can become quantitatively important at small densities and small temperatures. The influence of these effects has been considered in detail in Ref. [51]. Electron spin packets have been generated experimentally by optical illumination of n-doped GaAs with circularly polarized light [16]. In such materials, the hole

Fig. 5. Shown on the left (a) is the propagation of a charge-polarization packet in a bipolar junction transistor structure. The speed of propagation of this packet can determine the switch-on time for the transistor. Shown on the right (b) is the propagation of a spinpolarization packet in a unipolar spin transistor structure.

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recombination lifetime is very short – typically of the order of a few picoseconds. The hole spin relaxation time, however, is even shorter – typically of the order of 1 psec. Thus, initially the optically generated packet consists of an equal number of spin-polarized holes and spin-polarized electrons. Shortly after its creation, however, the holes have lost their spin polarization. When they recombine with the electrons, they reduce the excess spin-polarized electrons, and also remove some of the background electrons, leaving behind a spin packet. Small electric fields have been used to drag these packets around in GaAs [15], and the characteristic mobility of the spin packet is the electron mobility. 2.4. Drift Effects on Spin Currents A further unusual aspect of semiconductor carrier transport is the nature of drift currents. The typical expression for the conductivity s in a semiconductor is s ¼ nem; where n is the carrier density, e the electron charge, and m the mobility. The dependence of the conductivity on the carrier density has some surprising effects on the motion of inhomogeneous regions of carrier density [52]. If there is a point source of spin-polarized current into a semiconductor, from a point contact or perhaps from optical illumination, then that disturbance will decay away from the injection point over a length scale given by the diffusion constant and the spin relaxation time, l diffusion ¼ ðDT1 Þð1=2Þ ; as shown in Fig. 6a. If, however, an electric field is simultaneously applied to the material (Fig. 6b), then the inhomogeneously spin-polarized region will distort, and downstream of the injection point, the decay length will be dramatically enhanced, while upstream of the injection point, the decay length will be suppressed. This phenomenon is well known from minority carrier injection into semiconductors [48], such as when minority electrons are injected into a p-doped material, or minority holes injected into an n-doped material. Observations of the evolution of injected minority carriers formed the basis of the Haynes–Shockley experiments determining the mobility and lifetimes of minority carriers in germanium [53]. The two new lengths, an upstream spin diffusion length and a downstream spin diffusion length [54], are related to the zero-field diffusion length, the electric field E, and the temperature T according to the following

Fig. 6. (a) When spin-polarized carriers are injected into a semiconductor in the absence of an electric field, the disturbance decays away in all directions with the same length scale. (b) With an electric field, the disturbance is spread more in one direction than the other.


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expressions: l upstream ¼ l diffusion

n

l downstream ¼ l diffusion

o 2 ð1=2Þ 1 þ eEl diffusion =2kb TÞ þ eEl diffusion =2kb T

n

2 oð1=2Þ 1 þ eEl diffusion =2kb T eEl diffusion =2kb T

(3a) (3b)

where kb is the Boltzmann constant, when the carriers are non-degenerate. If the carriers are highly degenerate, as they would be in a metal, then the Fermi energy should be used instead of kbT, and the difference between the upstream and downstream spin diffusion lengths is negligible.

3. ELECTRICAL SPIN INJECTION INTO SEMICONDUCTORS FROM MAGNETIC MATERIALS 3.1. Detection of Spin-Polarized Carriers: The Spin-LED In order to study and develop electrical spin injection from a discrete contact into a semiconductor, one must first have a reliable means of detecting the presence of spin-polarized carriers therein and of quantifying the spin polarization. A simple light-emitting diode (LED) structure provides a powerful platform for this purpose. In a normal LED, electrons and holes recombine in the vicinity of a p–n junction or QW to produce light when a forward bias current flows. This light is unpolarized, because all carrier spin states are equally populated. However, if electrical injection produces a spin-polarized carrier population within the semiconductor, the same selection rules discussed above for optical pumping also describe the radiative recombination pathways allowed. Inspection of Fig. 2 reveals that if spin-polarized carriers radiatively recombine, the light emitted will be circularly polarized. A simple analysis based on these selection rules provides a quantitative and model independent measure of the spin polarization of the carriers participating.

Fig. 7. Schematic of spin-LEDs showing relative orientation of electron spins and light propagation direction appropriate for deducing the spin polarization of the electrons participating in the radiative recombination process for (a) surface-emitting and (b) edge-emitting geometries. The holes are assumed to be unpolarized in these examples.


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A schematic of such a spin-polarized LED (spin-LED) [55] is shown in Fig. 7. As an example, we consider the injection of spin-polarized electrons from a magnetic contact layer, and the subsequent recombination of those electrons with unpolarized holes supplied from the substrate. Assuming radiative recombination occurs in a bulk-like region (degenerate light/heavy hole bands), a general expression for the degree of circular polarization Pcirc follows directly from Fig. 2a, and can be written in terms of the relative populations of the electron spin states n " ðmj ¼ þ1=2Þ and n # ðmj ¼ 1=2Þ; where 0 n 1; n " þn #¼ 1: Pcirc ¼ ½IðsþÞ IðsÞ=½IðsþÞ þ IðsÞ ¼ 0:5ðn # n "Þ=ðn # þn "Þ

ð4Þ

I(s+) and I(s) are the intensities of the positive and negative helicity components of the electroluminescence (EL) intensity, respectively. The optical polarization is directly related to the spin polarization of the electron population, Pspin ¼ ðn # n "Þ= ðn # þn "Þ; and a measure of Pcirc provides a direct measure of Pspin. Pspin ¼ 2Pcirc

(5)

In many semiconductor structures, the degeneracy of the light and heavy hole bands is lifted by various effects, including strain and quantum confinement. For example, the LH/HH band splitting is typically several meV even in shallow QWs, and is much larger than the thermal energy at low temperature (0.36 meV at 4.2 K). Since the radiative recombination process strongly favors the lowest energy channel, only the lowest lying levels participate in this case, as illustrated in Fig. 2b for the HH states. Pcirc is calculated as before, and by inspection yields: Pcirc ¼ ðn # n "Þ=ðn # þn "Þ ¼ Pspin

(6)

Thus the spin-LED serves as a polarization transducer, effectively converting carrier spin polarization, which is difficult to measure by any other method, to an optical polarization which can be easily and accurately measured using standard optical spectroscopy techniques. Note that this approach measures the spin polarization of the carrier population in the semiconductor (electron or hole) achieved by electrical injection, and not the spin polarization of the injected current. Detailed knowledge of the transport mechanism and of the spin scattering/lifetimes are necessary to connect the two. A number of conditions facilitate application of the spin-LED approach: (i) The analysis of the measurements is considerably easier if the experimental geometry is oriented conveniently for the optical selection rules [31,32]. The hole spin, electron spin, and the optical emission/analysis axes (or projections thereof) must be co-linear. The hole states in a QW have preferred orientations due to quantum confinement and reduced symmetry [56,57]: the k ¼ 0 HH orbital angular momentum is oriented entirely along the growth direction (z-axis), whereas the k ¼ 0 LH orbital angular momentum has non-zero projections in all three directions. Therefore, the simplest analysis of the HH exciton requires that the injected electron spin must be along the z-axis and the


(ii)

(iii)

(iv)

(v)

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optical measurement must also be performed along the same axis, meaning that surface- rather than edge-emitted light should be analyzed (Fig. 7a) [58]. Thus, the Faraday rather than the Voigt geometry must be used. If the injected spins are oriented in-plane, then an edge-emission geometry (Fig. 7b) may be utilized to analyze the LH exciton in a QW, or the HH exciton if the radiative recombination region is bulk-like. Note, however, that the selection rules may be different for edge and surface emission for the same idealized structure [59]. The radiative recombination region should not be highly strained. Strain modifies the selection rules, compromising the simple quantitative relationship between Pcirc and Pspin if radiative recombination occurs in a strained region of the semiconductor. Strain may produce optical polarization in the absence of carrier spin polarization, or reduce Pcirc below that expected from the corresponding Pspin. While no system is perfect due to lattice mismatch, thermal expansion coefficients and contact issues, the AlxGa1xAs/GaAs-QW system comes very close due to the very small variation of lattice constant with Al concentration. In contrast, the lattice constant varies rapidly with In concentration and strain is almost unavoidable in the GaAs/InxGa1xAs-QW system. Other factors complicate the use of InGaAs in spin-LEDs – the introduction of In, with its stronger spin–orbit interaction, reduces spin lifetimes [60], and the relatively large g-factor introduces significant magnetic field-dependent effects, which complicate interpretation of the EL polarization [61]. The origin of the EL must be correctly identified to determine its spatial origin within the structure, and to confirm that it derives from recombination processes for which the selection rules are relevant [62], i.e., spin-conserving processes such as free exciton or free electron recombination. In practice, this means that the EL must be spectroscopically resolved and standard analyses applied to assist in the identification. Note that bound exciton and impurity-related emission typically involve non-spin-conserving processes, and therefore cannot be used. The VB degeneracy must be correctly identified, as described above. In a QW, the LH/HH splitting can be readily calculated for many materials [63], and should be compared to the measurement temperature to determine their relative contributions. One should bear in mind that the carrier spin polarization, Pspin, detected by this technique is that averaged over the radiative lifetime, tr, of the carriers involved. Essentially, one is taking a snapshot of the spin system with tr as the shutter speed. If the spin lifetime, ts, is much longer than tr, this would provide an accurate meaure of Pspin. This is rarely the case, however, and therefore the result measured more typically represents a lower bound for the carrier spin polarization achieved by electrical injection. If tr and ts are known, a simple algebraic relationship may be applied to obtain a more accurate measure of the initial carrier spin polarization, Po, that exists at the moment the electrically injected carriers enter the region of radiative recombination (e.g. the QW). Po is given by (see p. 26 of Ref. [59]) P0 ¼ Pspin ð1 þ tr =ts Þ

(7)

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where Pspin is the value measured experimentally as described above. This provides a first-order correction for what is effectively the efficiency of the radiative recombination process to serve as a spin detector in the particular structure and material utilized. The value of tr/ts can be determined by a simple photoluminescence (PL) measurement using near band edge circularly polarized excitation. Assuming that both LH and HH states are excited, the intial spin polarization is P0 ¼ 0:5 (see Fig. 2). One measures the circular polarization of the PL, PPL, and uses Eq. (7) to solve for tr/ts: 0:5 ¼ PPL ð1 þ tr =ts Þ: 3.2. Magnetic Semiconductors: Paramagnetic or Semimagnetic Materials Two classes of materials have been utilized as contacts for electrical spin injection into inorganic semiconductors: semiconductors and metals [13]. The ideal material on which to base a semiconductor spintronics technology is a semiconductor which is very clean (high mobility) and simultaneously FM above room temperature, i.e., exhibits a spin-polarized band structure with corresponding carrier polarization, and a macroscopic magnetic moment. This is certainly true from the perspective of electrical spin injection – this would enable design of a spin-injecting contact interface facilitated by known principles of CB and VB offsets, lattice match, interface structure and materials compatibility. Such materials, referred to as ‘‘ferromagnetic semiconductors’’ or ‘‘FMS’’, have in fact been studied for decades, but their properties currently fall short of what we might hope for. They will be discussed in the next section. Pronounced magnetic behavior can be introduced into some semiconductors by adding a small amount of certain magnetic impurities. These materials are referred to as ‘‘diluted magnetic semiconductors’’ (DMS), because the material is formed by diluting a host semiconductor lattice with a magnetic impurity, most typically Mn, to form an alloy. Strong exchange interactions between the magnetic impurity and the host carriers lead to magnetic behavior. Some of these materials are truly FM, while others are paramagnetic, yet still exhibit very useful and highly flexible magnetic properties. These non-FM compounds are commonly based on the II–VI or IV–VI semiconductors, where there is a high degree of solubility for the magnetic element. Classic examples include Zn1xMnxSe, Cd1xMnxTe, and Pb1xMnxSe [64,65]. These materials are often called ‘‘semimagnetic semiconductors’’ (SMS) because their paramagnetic behavior and Zeeman splitting of the CB and VB is dramatically enhanced over what might be expected from the magnetic ions alone. This enhancement originates from very large exchange interactions between the s- and p-like carriers of the CB and VB of the host, and the d electrons of the substitutional magnetic impurity. This sp–d exchange leads to a tremendous amplification of the Zeeman splitting of the band edges in an applied magnetic field, and produces amplified magneto-optic properties such as giant Faraday rotation as well as other field-dependent effects. If this Zeeman splitting is parametrized by DE ¼ guB H; then the value of the host g-factor is increased from the value of 2 typical of most hosts to a value of 100 or greater [64]. This pronounced behavior is limited to low


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temperatures (To25 K) and relatively high applied magnetic fields (H40.5 T), making them less attractive for device fabrication. Nevertheless, bulk crystals have found application as magnetically tunable optical polarizers operating at room temperature [66], where the long optical path length compensates for the severe reduction in g-factor and magnetization at higher temperatures. The giant Zeeman splitting was utilized in some of the first demonstrations of all electrical spin injection into a non-magnetic semiconductor by Fiederling et al. [67] and Jonker et al. [68]. In each case, an n-doped ZnSe-based SMS (ZnBeMnSe or ZnMnSe) was used as the spin-injecting contact on an AlyGa1yAs/GaAs QW heterostructure, which served as the spin detector. Zn1xMnxSe was chosen as the contact [68] because it forms high-quality epitaxial films on GaAs due to a close lattice match, and the CB offset can be tailored to facilitate electron flow from the Zn1xMnxSe into the AlyGa1yAs/GaAs structure by suitable choice of the Mn and Al concentrations [68]. A flat band diagram of the resultant structure is shown in Fig. 8. When a magnetic field is applied, the CB and VB edges of the Zn1xMnxSe split into spin-resolved bands, creating a spin-polarized carrier population. The resultant polarization may be varied simply by varying the applied field, a very useful handle for experimental studies. At sufficiently high fields, the spin splitting (DE ¼ guB H) is much larger than the measurement temperature (5 K), so that the material is essentially 100% spin polarized. Under appropriate bias, electrons flow from the n-ZnMnSe into the AlGaAs/ GaAs structure. To determine whether spin-polarized electrons are successfully injected across the II–VI/III–V heterointerface, the EL resulting from their radiative recombination in the GaAs QW with unpolarized holes injected from the p-type GaAs substrate is measured using standard optical spectroscopic techniques. If the electrons reach the QW with a net spin polarization, the EL is circularly polarized per the transitions shown in Fig. 2, as discussed for the spin-LED in a preceeding section. The quantum selection rules then provide a direct, quantitative measure of the electron spin polarization in the QW, Pspin ¼ Pcirc (Eq. 6). To utilize this approach, surface emitting spin-LEDs are fabricated using conventional photolithographic processing techniques. Examples of such devices are

Fig. 8. Schematic of the flat band diagram for injection of spin-polarized electrons from the Zeeman split CB of a Zn1xMnxSe epitaxial layer into an AlyGa1yAs/GaAs QW structure.

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Fig. 9. Surface-emitting, LEDs used to quantify electrical spin injection. (a) Cross-section of device illustrating the various layers and fabrication steps, (b) photograph of the final processed devices. The darker central area labeled ‘‘LED mesa’’ is the active region containing the ZnMnSe spin-injecting layer and the GaAs QW.

shown in Fig. 9 in cross-section (Fig. 9a) and following processing (Fig. 9b). A more detailed discussion of the design and fabrication may be found in Ref. [68]. The top ohmic metal contact to the ZnMnSe is a bullseye pattern to simultaneously provide uniform current distribution and optical transparency for the surface emitted EL. EL spectra obtained for selected values of applied magnetic field are shown in Fig. 10, analyzed for positive (s+) and negative (s) helicity. At zero field, the two components are identical, as expected, since no spin polarization yet exists in the ZnMnSe. The energy of the emission confirms that the radiative recombination is occuring in the GaAs QW via the ground state exciton (note that other tests are applied to confirm this identification – see Ref. [62]). As the magnetic field increases, the ZnMnSe bands split into spin states due to the giant Zeeman effect, and spinpolarized electrons are injected across the II–VI/III–V interface into the AlGaAs/ GaAs LED structure. The corresponding spectra exhibit a large difference in intensity between the s+ and s components, demonstrating that the spin-polarized electrons successfully reach the GaAs QW. Note that the QW emission energy is far below the band gap of the ZnMnSe (2.8 eV), so that any Faraday rotation contribution to the EL polarization due to transmission through the ZnMnSe is negligible. The circular polarization, Pcirc ¼ ½IðsþÞ2IðsÞ=½IðsþÞ þ IðsÞ; increases with applied field as the ZnMnSe polarization increases, and saturates at a value of 80% at 4 T. The spin polarization of the electron population in the QW, Pspin, is therefore 80% at the time of radiative recombination. Due to the radiative and spin lifetime effects summarized in the discussion of Eq. (7), this value is a lower bound for the actual spin polarization, Po, that exists at the moment the electrons enter the GaAs QW. Independent PL measurements using circularly polarized excitation [59] provide a value of tr =ts ¼ 0:25 0:05: Equation (7) then gives P0 100%; indicating that a well-ordered ZnMnSe/AlGaAs interface is essentially transparent to spin transport. Thus, an all-electrical process using a


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Fig. 10. EL spectra obtained from a ZnMnSe-based spin-LED for selected values of applied magnetic field, analyzed for positive (s+) and negative (s) helicity light. The large difference in intensity between the two components is a signature of electrical spin injection.

discrete contact can produce a carrier spin polarization within the semiconductor that equals or exceeds that achieved via optical pumping. In the work just described, the samples were grown in connected molecular beam epitaxy (MBE) chambers, so that the entire spin-LED structure was grown under pristine vacuum conditions. Subsequent work has demonstrated that similar results can be obtained for spin injection across an air-exposed interface, illustrating the robustness of the spin-injection process [69]. Further studies have correlated the spin-injection efficiency at the ZnMnSe/AlGaAs interface with the interface defect density [34], and demonstrated that strong spin injection persists even when the misfit dislocation density reaches values which greatly exceed those required and routinely achieved for conventional devices such as III–V based diode lasers and field effect transistors. This is especially reassuring, since interface misfit dislocations are a generic defect routinely encountered in heteroepitaxial device structures. 3.3. Magnetic Semiconductors: Ferromagnetic Materials FMS are materials that simultaneously exhibit semiconducting properties and spontaneous long-range FM order. The co-existence of these properties in a single material

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provides fertile ground for fundamental studies, and offers exciting possiblities for a broad range of applications. Such materials have, in fact, been studied for decades – classic examples include the europium chalcogenides (e.g. EuO, S, and Se) and the chalcogenide spinels [70,71]. Simple transport experiments have demonstrated that the exchange splitting of the band edges below the Curie temperature could be used as a spin-dependent potential barrier, which selectively passes one spin component while blocking the other, leading to current with a net spin polarization. This ‘‘spin filter effect’’ was demonstrated in EuS- and EuSe-based single barrier heterostructures in 1967 by Esaki et al. [72], and more recently by Moodera et al. [73] However, device applications languished due to low Curie temperatures and the inability to incorporate these materials in thin film form with mainstream semiconductor device materials. In these ‘‘classic’’ FMS compounds, the magnetic element forms a significant fraction of the atomic constituents. Interest in FMSs was rekindled with the discovery of spontaneous FM order in DMS alloys such as In1xMnxAs in 1989 [74] and Ga1xMnxAs in 1996 [75–77], when FM properties were realized in semiconductor hosts already widely recognized for device applications. Non-equilibrium growth at relatively low substrate temperatures by MBE permits incorporation of Mn at levels well above the solubility limit of the host lattice. Since Mn acts as both the magnetic element and an acceptor, they are p-type. After much research effort [78–80], these new FMS materials now exhibit Curie temperatures up to 50 [81] and 250 K [82], respectively. Although their optical and electronic properties are not nearly as clean and controllable as their non-magnetic hosts, these materials have been vigorously studied for their potential in future spin-dependent semiconductor device technologies. For example, Ga1xMnxAs has been used as a source of spinpolarized holes in LEDs [83], resonant tunneling diodes [77], and for current-induced magnetization switching [84]. Electric field control of FM order has recently been reported in In1xMnxAs [85], MnxGe1x [86], and Ga1xMnxAs [87] heterostructures, demonstrating one of the highly unusual properties of these materials and portending a host of new applications. Theory has predicted that FM order should be stabilized in a wide variety of semiconductor hosts when diluted or alloyed with Mn at a concentration of order 5% and sufficiently high hole densities [88,89]. Subsequent work has indicated that other magnetic atoms should produce similar effects. This has stimulated a groundswell of research activity to synthesize these and other compounds, with the goal of achieving technologically attractive materials. To that end, the ‘‘ideal’’ FMS should have many of the following attributes: A Curie temperature, Tc, well above room temperature to permit operation at temperatures typically encountered in modern electronics. A Tc4400 K is therefore highly desirable, although certain specialized applications (e.g. cooled infrared detectors) may permit Tc’s as low as 100 K. (ii) The material should be thermally stable at temperatures typically encountered in device processing schedules. Even if the FMS is deposited late in the growth sequence, permitting acceptably low deposition temperatures, subsequent processing may require significantly higher temperatures. (i)


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(iii) The material should be thermally stable at temperatures typically required to deposit the other layers of the device heterostructure. This varies considerably with the materials system, deposition techniques, and whether the FMS layer is grown early or late in the growth sequence. Growth temperatures routinely employed to obtain high-quality GaAs or Si layers by MBE, for example, are 550–600 C and 600–700 C, respectively, although considerable latitude is possible. (iv) The FMS should ideally be compatible with materials already used in existing device technology, e.g. Si, GaAs, or InP, to facilitate acceptance and incorporation in ‘‘mainstream’’ electronics. This means in part that the material can be epitaxially grown on a readily available substrate, forms a stable interface (i.e. is chemcially compatible), and exhibits band offsets suitable for device design. (v) It should be possible to dope the FMS both n- and p-type, independent of the magnetic doping, enabling independent control over the electrical and magnetic properties, and providing the maximum flexibility in device design and operation. Note that many of the ‘‘new’’ diluted FMS compounds derive their FM character from hole-mediated exchange, and the magnetic impurity conveniently acts as an acceptor. While this may seem advantageous at first glance, it imposes significant limitations on device design. In addition, there is considerable interest in developing an n-type FMS to take advantage of the high electron mobility and the corresponding range of higher operating frequencies. (vi) The FMS should exhibit reasonable magnetic properties suitable for the application envisioned. One obvious characteristic is a low coercive field, important for applications which require switching (Hco200 Oe). But perhaps a more important and often overlooked characteristic is a significant remanent magnetization, i.e. the magnetization retained at zero field. The non-volatile behavior essential for many applications (memory, optical isolators, reprogrammable logic) is predicated upon a remanence that is a substantial fraction of the saturation magnetization. (vii) The material should exhibit reasonable transport and optical properties, e.g. good mobilities if it is part of the active transport channel, and clean luminescence or absorption if it is part of the active optical element. The above list is neither exhaustive nor intended to be an absolute yardstick by which to gauge the merits of a particular material – performance and intended application define the latter. No FMS material currently studied meets all of these criteria. It is useful, however, to keep these items in mind as we continue to develop new materials and explore the new functionality these FMS offer. The principal obstacle in the efforts to synthesize new FMS materials has been the formation of unwanted phases due to the relatively low solubility of the magnetic atoms (Mn,Cr, Fe, Co) in most of the hosts considered (e.g. the group III-As, III-Sb, and III-N families). In many cases, nano- and micro-scale precipitates of known FM bulk phases form, which are exceedingly difficult to detect by the usual structural probes of X-ray diffraction and electron microscopy, but are all too readily detected with standard magnetometry measurements. A major challenge

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facing the experimentalist is to utilize characterization techniques which discriminate against the potential presence of such precipitates, clearly distinguish the contribution of the FMS material from that of the precipitates, and directly probe the key characteristics expected for the FMS itself. Electrical spin injection from the FMS provides one such litmus test, since it is enabled by the spin polarization of either the CB or VB of the FMS. In the following paragraphs, we briefly summarize the results of spin-LED experiments which demonstrate electrical spin injection of holes and electrons from two very different FMS materials, p-type Ga1xMnxAs, and n-type CdCr2Se4. The first studies of electrical injection of spin-polarized holes from a diluted FMS, p-GaMnAs, were reported for resonant tunneling diode structures [77,90]. These were based on the appearance of peaks in the transport data, which were attributed to the VB exchange splitting of the GaMnAs emitter layer. More direct evidence was afforded by spin-LED experiments (which incorporate explicit polarization analysis) by Ohno et al. [83] in 1999. They observed a 1% polarization of the EL (above an 8% background), which correlated with the GaMnAs magnetization and was attributed to spin-polarized hole injection from the GaMnAs emitter into the InGaAs QW where the radiative recombination occurred. Since they used an edge-emission geometry to take advantage of the in-plane easy axis of the GaMnAs magnetization, it was not possible to rigorously quantify the degree of injected hole spin polarization achieved. Recent calculations have treated this geometry, and found that the orbital coherence of the hole must be taken into account to interpret the polarized EL [91]. Subsequent work [92,93] utilized similar sample structures (Fig. 11a, inset), but analyzed surface emission in the Faraday geometry rather than edge emission,

Fig. 11. (a) Spectrally resolved EL intensity along the growth direction for several bias currents, I. Inset shows device schematic and EL collection geometries. (b) Temperature dependence of the relative changes in the energy-integrated (gray shaded area in (a)) polarization DP as a function of out-of-plane magnetic field. When To62 K, polarization saturates at H?2.5 kOe. Inset shows M(T), indicating that the polarization is proportional to the magnetic moment. From Ref. [92].


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Fig. 12. (a) Schematic band structure of an Esaki diode/LED electrical electron spin-injection device. (b) Relative polarization of EL DPEL as a function of magnetic field perpendicular to the sample plane at 78 K (surface emission from the backside of the wafer). Applied voltage of 1.6 V and device current of 9 mA. From Ref. [95].

permitting more quantitative interpretation of the data. The EL spectra at T ¼ 5 K and the field dependence of the circular polarization for selected temperatures are shown in Fig. 11. The field dependence tracks the out-of-plane (hard axis) magnetization of the GaMnAs (Fig. 11b), and the saturation value depends upon the thickness, d, of the undoped GaAs spacer layer between the GaMnAs injector and the InGaAs QW. The circular polarization is 4% for d ¼ 70 nm; and increases to 7% for d ¼ 20 nm; atttributed to the hole spin diffusion length effects in the GaAs spacer. These data indicate that the spin polarization of the QW hole population is at least 7% at the measurement temperature. The polarization decreases with increasing temperature, and disappears by 62 K, the Curie temperature of the GaMnAs injector. In Ref. [93], use of small mesas to force the easy axis of the magnetic contact out of plane permitted the demonstration of hole spin injection without an applied magnetic field in the surface emission geometry. A p-type GaMnAs contact layer can also be used to inject spin-polarized electrons if it is incorporated in a p+-n+ junction to form a tunnel diode, or an Esaki diode [94–97]. A schematic diagram is shown in Fig. 12a [95], where a GaAs/ InGaAs QW LED is used to detect the spin polarization of the injected electrons. Under reverse bias, spin-polarized electrons tunnel from the VB of the GaMnAs through the narrow depletion width of the degenerate p+-n+ junction to the CB of the n+-GaAs, and radiatively recombine in the InGaAs QW with unpolarized holes from the p-GaAs subtrate. The circular polarization of the surface-emitted EL (Faraday geometry) is summarized in Fig. 12b as a function of applied magnetic field. The optical polarization again tracks the out-of-plane magnetization of the GaMnAs, and saturates at a value of 6% [95]. More recent work using a structure that incorporates an Al0.7Ga0.3As/GaAs QW and a doping profile optimized for spin transport by numerical simulations reported a measured circular polarization of 20% [97]. Calculations appropriate for the oblique Hanle geometry employed yield a remarkable value of 60% for the steady state electron spin polarization in the wide GaAs QW (100 nm) at 4.5 K. This polarization decreases rapidly with temperature and applied bias due to the magnetization and band bending of the GaMnAs [97].

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Interest in using electrons rather than holes in a semiconductor spintronic device is due to the fact that electrons exhibit remarkably long spin lifetimes and spin scattering lengths in materials such as Si [98], GaAs [14–16,99], and GaN [100], while spin lifetimes for holes are much shorter due to stronger spin–orbit interactions. In addition, electron transport is the basis for high frequency, low power device operation, while holes typically have much lower mobilities. Most of the ‘‘new’’ diluted FMS are p-type due to the acceptor character of the magnetic impurity. Thus, there is keen interest in developing an n-type FMS which is compatible with existing mainstream semiconductor materials. Several groups have explored new materials doped with Mn, Cr, or Co, which are n-type and exhibit a magnetic signature when examined by magnetometry or a magnetic spectroscopy [101–106]. Measurements which more directly probe the exchange-split character of the host band structure have yet to be performed, however. Recent work has successfully incorporated epilayers of n-type CdCr2Se4, a classic FMS, with GaAs-based heterostructures by MBE growth [107], and demonstrated electrical injection of spin-polarized electrons into GaAs using the spin-LED approach [108]. CdCr2Se4 is a direct gap (E g ¼ 1:3 eV) chalcogenide spinel with the AB2X4 structure (56 atoms per cubic unit cell) and a lattice constant of 10.721 A˚.It is reasonably lattice matched to technologically important materials such as Si and GaP (1.7% tensile mismatch), and to GaAs (5.2% tensile mismatch), assuming one uses an effective lattice constant of ao/2 for the CdCr2Se4. Single crystal epilayers grown on both GaP(001) and GaAs(001) substrates are n-type (n1018 cm3), FM with the easy magnetization axis in plane (along the GaAs [110]), and have a Curie temperature of 132 K [107]. Surface-emitting spin-LED structures consisting of 200 nm n-CdCr2Se4/50 nm n-ZnSe/50 nm n-Al0.1Ga0.9As/20 nm GaAs QW/50 nm p-Al0.3Ga0.7As were grown on a p-doped GaAs(001) buffer layer and substrate. A favorable band alignment ensures there is no barrier for electron transport from the CdCr2Se4. The ZnSe layer was inserted to improve the interface structure and reduce the large CB offset between the CdCr2Se4 and Al0.1Ga0.9As (650 meV) [109] into two smaller increments in an effort to minimize spin scattering that might accompany energy relaxation. The EL spectra obtained in the Faraday geometry are shown in Fig. 13a, and are dominated by the HH exciton with a full width at half maximum of 7 meV.The magnetic field applied along the surface normal rotates the magnetization (and electron spin) out of plane so that the quantum selection rule analysis can be applied. The spectrum at 0.5 T exhibits a pronounced circular polarization Pcirc ¼ 6%; where Pcirc is defined as before. This demonstrates a corresponding electron spin polarization in the GaAs QW (Pcirc ¼ Pspin ) with a predominance of mj ¼ þ1=2 electrons, as confirmed by simple inspection of the allowed radiative transitions (Fig. 2). The field dependence of Pcirc is shown in Fig. 13b for two LEDs which exhibit saturation polarizations of 6% (best case) and 4% (typical). Pcirc clearly mirrors the field dependence of the magnetization obtained by superconducting quantum interference device (SQUID) magnetometry and shown by the dashed line (these

Electrical Spin Injection and Transport in Semiconductors Fig. 13. Spin injection from an n-type FMS, CdCr2Se4, into an AlGaAs/GaAs QW. (a) EL data at zero and 0.5 T applied field, analyzed for positive and negative helicity circular polarization (Faraday geometry). (b) Magnetic field dependence of Pcirc (circles and squares), PL from a reference sample (triangles), and out-of-plane magnetization of the CdCr2Se4 film (dashed line). (c) Photograph of one device under operation obtained with infrared sensitive camera.

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data have been inverted and scaled to the 4% EL data), confirming that the electrons participating in the radiative recombination in the GaAs QW originate from the CdCr2Se4. Note that the sign of Pcirc is opposite to that observed for spinpolarized electron injection from ZnMnSe (Fig. 10). The mj ¼ þ1=2 electrons injected from the CdCr2Se4 are in fact minority spin electrons, since their magnetic moment (m ¼ mj mB ) is opposite to the magnetization of the film [110]. Density functional theory calculations within the local density approximation have shown that minority spin states dominate the bottom of the CdCr2Se4 CB [108]. The value of 6% for Pspin is significantly smaller than obtained using ZnMnSe contacts. This is attributed to two factors related to the spin-injecting interface. First, interface structural defects have been shown to reduce spin-injection efficiency [34], and are likely to form due to the mismatch in lattice parameter and crystal symmetry between the chalcogenide spinel and zincblende materials. Second, the spin-polarized electrons are injected from the CdCr2Se4 into the III–V LED structure with high kinetic energy due to the large CB offset mentioned earlier [109]. This energy must be dissipated before they reach the GaAs QW ground state. Such large energy relaxation is likely to proceed by non-spin-conserving mechanisms, resulting in much lower spin polarizations for electrons ultimately forming the QW exciton and contributing to the EL. The large lattice mismatch and CB offset are specific to the choice of GaAs substrate, and may be reduced by growth on a material such as GaP. Control experiments were performed to insure that the EL polarization and field dependence observed is not due to some other effect (e.g. magnetic dichroism arising from transmission through the CdCr2Se4). Linearly polarized optical excitation (780 nm) was used to produce carriers with zero net carrier spin polarization in the GaAs QW, so that the surface-emitted PL is unpolarized and provides an internal reference. Any measured circular polarization in the PL is then due to magnetic dichroism resulting from transmission through the CdCr2Se4, or some other spurious effect. As the data of Fig. 13b show, this effect, if present, is much smaller than that attributed to electrical spin injection. Thus, the CdCr2Se4 CB is indeed spin polarized, and electrical injection is dominated by minority spin electrons. A great deal of research effort continues to address development of FMS materials and explore their technological applications. Additional information, overviews, and exciting new developments on FMS may be found in Refs [106,111–114]. 3.4. Ferromagnetic Metals In contrast with magnetic semiconductors, FM metals enjoy a well-developed materials technology due to decades of research and development driven in large part by the recording industry. They offer many of the properties desired for a practical spin-injecting contact material: high Curie temperatures are the norm, a source of electrons rather than holes, low coercive fields, and fast switching times. In addition, metallization is a standard process in any semiconductor device fabrication line, so that the use of an FM metallization could easily be incorporated into existing processing schedules.


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Theoretical work has explored the band structure at the FM metal/semiconductor interface in an effort to elucidate the basic factors which impact spin injection [115–118]. This work has emphasized the importance of matching the symmetries as well as the energies of the bands between metal and semiconductor to optimize spininjection efficiency. For the case of (001) oriented interfaces between Fe films and a zincblende semiconductor such as ZnSe or GaAs, for example, the CB of the semiconductor is well matched in symmetry and energy to the majority spin bands of the Fe, while a poor match exists to the Fe minority spin bands. Hence, ignoring the band bending (Schottky barrier formation) which is likely to extend 100 nm into the semiconductor, these results predict that band structure considerations provide a strong spin filtering at the interface with a marked preference for majority spin injection in the ballistic limit. Wunnicke et al. [118] simulated the effect of a Schottky barrier by introducing a potential step between the Fe and the semiconductor, and found that the current remained strongly polarized, but the polarization was reduced for thicker barriers. Similar results were obtained for epitaxial Fe/MgO tunnel barriers by Butler et al. [119]. A number of groups have attempted to inject spin-polarized carriers from an FM metal contact into a semiconductor without detailed analysis of the effect of the barrier. Several used a change in voltage or resistance (expected to arise due to spin accumulation or transport) as an indication of spin injection. Measured effects on the order of 0.11% have been reported [120–122]. An estimate of actual injection efficiency can be extracted from a particular transport model based on assumptions believed appropriate for a given experiment. Such small effects, however, make it difficult to either unambiguously confirm spin injection or successfully implement new device concepts. In addition, some have argued that this measured change in resistance or voltage attributed to spin injection may be compromised by possible contributions from anisotropic magnetoresistance or a local Hall effect [123–125]. These latter effects can easily result in a contribution of a few percent to the measured signal, and care must be taken in the experimental design to eliminate their role. Zhu et al. [126] utilized optical rather than electrical detection in a spin-LED structure consisting of an Fe Schottky contact to a GaAs/In0.2Ga0.8As QW LED detector. They were unable to observe any clear difference in intensity when they analyzed the EL for positive (s+) or negative (s) helicity polarization as described above for magnetic semiconductor contacts. However, by examining the high- and low-energy tails of the EL peak (which they attributed to LH and HH contributions, respectively), and using pulsed current injection and lock-in detection techniques, they detected a signal which they attributed to electrical spin injection from the reverse-biased Fe Schottky contact. They concluded that an injected spin polarization of 2% had been achieved, and found that this signal was independent of temperature from 25 to 300 K. 3.4.1. Conductivity mismatch One of the fundamental obstacles and unanticipated challenges to utilizing an FM metal as a spin-injecting contact on a semiconductor is the very large difference in

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conductivity between the two materials. This issue is commonly referred to in the literature as the problem of ‘‘conductivity mismatch’’, and can readily be understood in the context of the canonical two channel model typically used to describe spin-polarized current flow in FM metals [127]. In this model, spin-up or ‘‘majority spin’’ [110] electrons flow in one channel, while spin-down or ‘‘minority spin’’ electrons flow in the other. Using the analogy of water flow through a hose for electrical resistance, the relative conductivities of these channels in each material can be represented by the diameter of the hose, as shown in Fig. 14a. In the metal, the diameter of both spin-up and spin-down hoses is very large (high conductivity), while in the semiconductor, the diameter is very small (low conductivity). The problem of conductivity mismatch is thus reduced to the intuitive physical picture of attempting to effectively transfer water from a sewer pipe to a drinking straw. This picture provides rather surprising insight into efforts to transfer spinpolarized carriers between these two materials. In the FM metal, the carriers are partially spin polarized (by definition), so that one of the pipes is nearly full, while the other may be nearly empty. In the non-polarized, low carrier density semiconductor, the two spin channels are of equal diameter and partially full. If one now considers the flow of water (spin-polarized current) from the pipes (metal) to the drinking straws (semiconductor), it is immediately apparent that the comparatively small conductivity of the semiconductor limits current flow. While one drop of water may be transferred from the majority spin sewer pipe of the metal to the corresponding majority spin drinking straw of the semiconductor, an identical amount will flow in the minority spin channel, even though the metal pipe may be

Fig. 14. (a) Schematic of the two channel model of spin transport illustrating the conductivity mismatch issue, which must be considered for spin injection from a FM metal into a semiconductor. (b) Equivalent resistor model from Ref. [12] (due to A. Petukhov).


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nearly empty, due to the limited conductivity of the semiconductor. This results in zero spin polarization in the semiconductor. It is apparent that this will be the case regardless of how nearly empty the metal minority spin band may be, due to the very limited ability of the semiconductor bands to accept current flow. This is the essence of the conductivity mismatch issue. Only in the limiting case of a 100% polarized FM metal (minority channel completely empty) will current flow produce a significant spin polarization in the semiconductor. Alternatively, the conductivity of the FM metal and semiconductor must be closely matched. No FM metal meets either of these criteria. While half metallic materials offer 100% spin polarization in principle [128,129], defects such as antisites or interface structure rapidly suppress this value [130]. Model calculations by several groups [125,131–135] have addressed this issue and provide a quantitative treatment. An equivalent resistor circuit explicitly incorporates the conductivities of the FM metal, interface, and semiconductor [12,131] as shown in Fig. 14b, and permits discussion of spin-injection efficiency in terms of these physical parameters in the classical diffusive transport regime. The resistances representing the FM metal and semiconductor are given by LF/sF and Ls/ss, where L and s are the spin diffusion lengths and conductivities, respectively, with sF ¼ s " þs # summing the two FM spin channels. The interface conductivity S ¼ S " þS # is assumed to be spin dependent. With some algebra and effort, the spin-injection coefficient g can be shown to be: I" I# g¼ I Ds DS ¼ rF þ rc ð rF þ rS þ rc Þ ð8aÞ sF S where rF ¼ LF

rc ¼

sF 4s" s#

(8b)

S 4S" S#

(8c)

Ls ss

(8d)

rs ¼

are the effective resistances of the ferromagnet, contact interface, and semiconductor, respectively, and Ds ¼ s " s # and DS ¼ S " S # : For an FM metal, rF/(rF+rs+rc) oo 1, and the contribution of the first term in Eq. (8a) is negligible. The second term is significant only if DSa0 and rc rF ; rs : Thus, two criteria must be satisfied for significant spin injection to occur across an FM metal/semiconductor interface: the interface resistance (1) must be spin selective and (2) must dominate the series resistance in the near-interface region.

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3.4.2. Tunnel barrier-based spin injection A tunnel barrier inserted between the FM metal and semiconductor satisfies both criteria [125,132], solving the seemingly insurmountable obstacle of conductivity mismatch. A detailed theoretical treatment was provided by Rashba [132]. The first criterion is satisfied quite naturally by the fact that the FM metal exhibits an imbalance in the spin-resolved density of states (DOS) by definition, and the tunneling probability depends explicitly on the DOS of the source contact. Such a tunnel barrier can be realized experimentally either by tailoring the band bending and carrier depletion which typically occurs at a semiconductor interface, or by explicitly incorporating a discrete insulating layer such as Al2O3 within the structure. In the following paragraphs, we briefly summarize examples of each. 3.4.3. Tailored Schottky tunnel barriers The band bending, which occurs at the interface between most metals and semiconductors due to charge transfer leading to Schottky barrier formation produces a built-in potential barrier. In an n-type semiconductor, the interface region is depleted of electrons and the CB bends up, forming a pseudotriangular barrier. The depletion width associated with the Schottky barrier depends upon the doping level of the semiconductor, and is generally far too large to allow tunneling to occur. For example, in n-GaAs, the depletion width is on the order of 100 nm for n 1017 cm3, and 40 nm for n1018 cm3 [136]. However, this width can be tailored by the doping profile used at the semiconductor surface [137]. Heavily doping the surface region can reduce the depletion width to a few nanometers, so that electron tunneling from the metal to the semicondutor becomes a highly probable process. This approach was utilized to achieve large electrical spin injection from Fe epilayers into AlGaAs/GaAs QW LED structures [138,139]. A schematic of the flat band diagram is shown in Fig. 15a. The n-type doping profile of the surface AlGaAs was designed by solving Poisson’s equation with several criteria in mind: (a) minimize

Fig. 15. Design of an Fe Schottky barrier and spin-LED using a doping profile to facilitate tunneling of spin-polarized electrons from the Fe through the Schottky tunnel barrier. (a) Flat band diagram illustrating desired carrier flow. (b) Poisson equation solution corresponding to doping profile described in the text.


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the Schottky barrier depletion width to facilitate tunneling, (b) use a minimum amount of dopant and heavily doped regions to accomplish this, since high n-doping is associated with stronger spin scattering and short spin lifetimes [16,17], and (c) avoid formation of an electron ‘‘puddle’’ or accumulation region (EF above the CB edge), which may either dilute the polarization of the electrons injected from the Fe contact or contribute to spin scattering. The LED structure consisted of the following: p-GaAs(001) substrate/p+ GaAs buffer/25 nm p(1018)-Al0.3Ga0.7As/25 nm Al0.3Ga0.7As/10 nm GaAs QW/10 nm n(1016)-Al0.1Ga0.9As/15 nm n-transition (10161019) Al0.1Ga0.9As/45 nm 19 Al0.1Ga0.9As/15 nm n(10 )-Al0.1Ga0.9As/10 nm Fe(001) contact layer. The n- and p-doping levels are shown in parentheses in units of cm3 of Si and Be, respectively. The corresponding graphical solution to the Poisson equation for the Fe/AlGaAs spin-LED is shown in Fig. 15b. Details of the growth may be found elsewhere [138,139]. A higher Al concentration was used in the Al0.3Ga0.7As barrier on the pdoped side to form a higher barrier in an effort to improve electron capture and resultant EL in the GaAs QW. Surface-emitting spin-LEDs fabricated from these structures show strongly polarized EL, demonstrating successful electrical spin injection from the FM Fe contact. Under appropriate bias, spin-polarized electrons tunnel from the Fe through the tailored Schottky barrier and reach the GaAs QW, where they radiatively recombine with unpolarized holes from the substrate. The EL data are shown in Fig. 16a, analyzed for positive/negative helicity circular polarization (s+/s) for selected values of applied magnetic field (Faraday geometry). The raw EL spectra exhibit a clean peak at an energy corresponding to the GaAs QW ground state exciton. Since the easy magnetization axis of the Fe is in-plane, the electron and HH spins are orthogonal, and the EL data at zero field show no polarization. A field is necessary to pull the magnetization (and electron spin orientation) out-of-plane so that the quantum selection rule analysis can be reliably applied, and the resulting spectra exhibit a high degree of circular polarization. The field dependence of the polarization is summarized in Fig. 16b, together with the out-of-plane magnetization of the Fe film (dashed line, scaled to the EL polarization data) obtained by independent magnetometry measurements. The EL polarization clearly tracks the Fe magnetization data, saturating at a value of Pcirc ¼ Pspin ¼ 32% at an applied field of B ¼ 4pM 2:2 T at which the Fe magnetization saturates. The sign of Pcirc confirms that Fe majority spin electrons dominate, consistent with earlier work on Fe/Al2O3/Al tunnel structures [140]. Control experiments rule out any significant contributions due to dichroism as the emitted EL passes through the Fe film. This was determined by independent PL measurements on undoped Fe/AlGaAs/GaAs QW test structures. Linearly polarized laser excitation creates unpolarized electron and holes in the AlGaAs/GaAs, which emit unpolarized light when they recombine in the GaAs QW. This light passes through the Fe film, and any circular polarization measured is therefore due to a combination of dichroism in the Fe, field-induced splittings in the QW, or other background effects. The solid triangles in Fig. 16b summarize these measurments, and show that the sum of any such contributions is o1%.

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Fig. 16. EL data from an Fe Schottky tunnel barrier spin-LED. (a) EL spectra for selected values of applied magnetic field, analyzed for positive and negative helicity circular polarization. (b) Magnetic field dependence of Pcirc ¼ Pspin : The dashed line shows the out-ofplane Fe magnetization obtained with SQUID magnetometry and scaled to fit the EL data. The triangles indicate the measured background contribution, including dichroism, using PL from an undoped reference sample.

The transport process providing such successful spin injection can be identified by examining the current–voltage (I–V) characteristics of the tailored Fe/AlGaAs Schottky contact. The observation of phonon signatures and a pronounced zerobias anomaly in the conductance spectra as well as a detailed analysis using the Rowell criteria (characteristics that the I–V data should exhibit if tunneling is occurring [141,142]) convincingly show that tunneling is the dominant transport mechanism [139]. Thus, the raw EL data reveal that an electron spin polarization of 32% (lower bound) is achieved in the GaAs QW by electrical injection from the reverse-biased Fe Schottky contact. Since this value is averaged over the radiative lifetime and thus limited by spin lifetime effects, the injected spin polarization is likely to be significantly higher. Independent PL measurements can again be used to determine the value of the ratio tr/ts, and Eq. (7) employed to determine the initial spin polarization Po that exists at the moment the electrically injected electrons enter the GaAs QW, as described previously for the case of spin injection from ZnMnSe films. These measurements yield tr =ts ¼ 0:78 0:05; resulting in a value P0 ¼ 57%: It is interesting to note that this value exceeds the nominal spin polarization of bulk Fe (45%), clearly demonstrating that the bulk spin polarization of FM metals does not represent a limit to the spin polarization that can be achieved via electrical injection. Effects such as spin filtering at the Fe/AlGaAs interface and spin accumulation in the GaAs QW enable the generation of highly polarized carrier populations, and can be exploited in the design and operation of semiconductor spintronic devices.


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Higher values of Pspin can be expected upon optimization of the heterostructure by more sophisticated modeling than described above, and several groups have modeled the effects of the various device parameters on spin injection [143–145]. A Schottky barrier forms naturally at many FM metal/semiconductor interfaces, and avoids the use of an additional discrete insulating layer and the accompanying challenges of achieving a flat, pinhole-free barrier. Schottky contacts are also routine ingredients in semiconductor technology, and thus offer a ready pathway for the integration of spin injection and transport into semiconductor devices. 3.4.4. Discrete layers as tunnel barriers Discrete insulating layers may also be used to form a tunnel barrier between the FM metal and the semiconductor. These provide the more familiar and well-studied rectangular potential barrier (Fig. 17a), but also introduce an additional interface into the structure. Spin-polarized tunneling has been well studied in metal/Al2O3/ metal heterostructures. Careful measurements using a superconductor such as Al for one contact have provided quantitative information on the spin polarization of the tunneling current from several FM metals [146–148]. Several groups have utilized Al2O3 barriers in FM metal/Al2O3/AlGaAs-GaAs spin-LED structures with great success [149–152]. van‘t Erve et al. [152] obtained a value of Pspin ¼ 40% at 5 K for samples which utilized a lightly n-doped (1016 cm3) Al0.1Ga0.9As layer adjacent to the Al2O3. The field dependences of Pcirc and Pspin from such devices are shown in Fig. 17b), and exhibit the same characteristics discussed for Fig. 16b. Note that in this case a wide (50 nm) undoped GaAs region was used rather than a 10 nm

Fig. 17. EL data from an Fe/Al2O3-based spin-LED. (a) Schematic of the flat band diagram. (b) Magnetic field dependence of Pcirc and Pspin (Pspin ¼ 2Pcirc due to the use of a wide GaAs recombination region). The dashed line shows the out-of-plane Fe magnetization obtained with SQUID magnetometry and scaled to fit the EL data.

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GaAs QW, so that Pspin ¼ 2 Pcirc due to the HH/LH degeneracy (Eq. 5). However, they reported lower operating efficiency (higher bias voltages and currents) than for the Schottky barrier-based spin-LEDs of Refs [138,139]. This was subsequently remedied by including the same surface doping profile shown in Fig. 15a before the growth of the Al2O3 layer, so that high Pspin was achieved at bias conditions comparable to those utilized for the Schottky tunnel barrier devices. Applying the same procedure described in the preceeding section (see Eq. 7) yields Po 70%, a value which again significantly exceeds the spin polarization of bulk Fe. 3.5. Temperature Dependence of Spin Polarization The temperature dependence of the EL polarization measured in the Faraday geometry at H ¼ 3 T is summarized in Fig. 18 for two surface-emitting spin-LEDs, one utilizing a tailored Schottky barrier [139] and the other a discrete Al2O3 tunnel barrier [152] with an Fe contact. Both are based upon an Al0.1Ga0.9As/50 nm GaAs/ Al0.3Ga0.7As QW LED grown in the same MBE chamber, facilitating comparison of their characteristics. The behavior is remarkably similar despite the differences between the physical structure of the spin-injecting contacts: the former is entirely single crystal, while the latter incorporates an amorphous oxide tunnel barrier, a polycrystalline Fe film, and an additional interface in the contact region. This suggests that the temperature dependence observed is due to that of the AlGaAs/GaAs

Fig. 18.

Temperature dependence of Pcirc and Pspin for Schottky- and Al2O3-based surfaceemitting spin-LEDs (Faraday geometry).


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detector rather than to the spin-injection process itself. The temperature dependence of Fig. 18 is, in fact, very similar to that often observed for the electron spin lifetime in GaAs QW [31,153]. Previous work has shown that the electron spin relaxation in a GaAs QW generally occurs more rapidly with increasing temperature, suppressing the measured circular polarization – the GaAs(001) QW is simply an imperfect spin detector, with a strong temperature dependence of its own. As noted earlier, the optical polarization measured depends upon the values of the spin and radiative carrier lifetimes. Both have been extensively studied, vary with temperature, and depend upon the physical parameters of the structure and the ‘‘quality’’ of the sample material. Despite this limitation, the data indicate that an electrically injected QW electron spin polarization of at least 3–5% (lower bound) persists at room temperature after decreasing by a factor of 2 between 5 and 100 K. Recent work by Motsnyi et al. [151] on FeCo/Al2O3/Al0.2Ga0.8As/GaAs/Al0.3Ga0.7As spin-LEDs reported measured values for the GaAs electron spin polarization of 14–21% at 80 K, and 6% at 300 K, in good agreement with the data of Fig. 18 (note that no corrections for spin lifetimes have been included in the values for Pspin quoted here). In addition, by using the oblique Hanle effect [154], they were able to experimentally determine the electron lifetime information necessary to correct for the effects of the radiative and spin lifetimes of the GaAs itself, as discussed above and in Section 2. When they correct the measured values for this ‘‘instrument response function’’ of the GaAs detector, they obtain a value of Pspin ðcorrectedÞ ¼ 16% at 300 K (this quantity is denoted by ‘‘P’’ in their notation). These results are very encouraging, and highlight the importance of experimentally determining the electron lifetime information and including it in the data analysis. Magnesium oxide (MgO) is another metal oxide which has produced very favorable results in FM metal/MgO/FM metal spin-dependent tunneling structures [155,156]. Very recent work has reported the use of an MgO tunnel barrier in CoFe/ MgO/GaAs/Al0.08Ga0.92As/GaAs/Al0.08Ga0.92As spin-LED samples [157]. Surfaceemitting devices exhibited a measured circular polarization (Pcirc ¼ Pspin ) of 30– 35% at 5 K, very similar to the work reported previously [138,139,152]. At low temperatures, the GaAs QW spin detector is in a well-defined state in which the radiative recombination is excitonic, permitting more reliable comparison of the spin-injection process itself between spin-LEDs based on tailored Schottky, Al2O3, and MgO tunnel barriers. The authors observe an initial decrease in Pcirc up to temperatures of 60 K. However, between 60 and 100 K, Pcirc increases significantly from 20% to 45–55%, in contrast to the behavior shown in Fig. 18. This is attributed to the different temperature dependence of the electron spin and radiative lifetimes exhibited by the GaAs material rather than increased spin-injection efficiency, and again underscores the importance of these lifetime parameters in the interpretation of data and the design of future devices. Optical pumping experiments have revealed significantly longer electron spin lifetimes in (110)-oriented GaAs QW structures, and that the lifetime increased with increasing temperature [158], in contrast to the case for the (001)-oriented structures. This was attributed to suppression of the D’yakonov–Perel scattering mechanism.

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Initial efforts to utilize this orientation in Fe Schottky/AlGaAs/GaAs(110) spinLED structures produced lower measured polarizations, and a temperature dependence similar to that shown in Fig. 18 for the (001)-oriented LED structures [159]. This was attributed to several factors: (a) higher dopant compensation for (110) growth [160] limited the maximum n-doping achieved to 5 1018 cm3 rather than the target 1 1019 cm3 used in the (001)-oriented structures to control the Schottky barrier depletion width. Consequently, the effective tunnel barrier width was larger for the (110) structure, significantly reducing the probability for tunneling; (b) MBE growth in the (110) orientation is more challenging and less developed, resulting in lower material quality than for the familiar (001) orientation. This is likely to result in stronger spin scattering; (c) the lower polarization observed is also consistent with theoretical calculations in the ballistic limit which suggest that poor matching of the transport band symmetries across the Fe/AlGaAs(110) interface should inhibit spin injection [161].

3.6. Drift Effects on Spin Injection The sensitivity of the conductivity to the carrier density, explored in Section 2, also enhances spin injection from a magnetic contact into a semiconductor. If a spinpolarized distribution is injected into the semiconductor (Fig. 19a), then the conductivity increases for the majority spin carrier, and decreases for the minority spin carrier (Fig. 19b). This, however, permits a larger current of majority spins to flow into the semiconductor, and reduces the current of minority spins. This feedback effect can lead to orders of magnitude enhancement in the spin-injection efficiency [52]. This type of spin-injection enhancement can also be used with a tunnel barrier to dramatically increase the spin-injection efficiency through the tunnel barrier. Thus, less resistive tunnel barriers are required, which would permit spin-injection to be achieved in much lower power devices. This phenomenon has been predicted theoretically, but has yet to be taken advantage of directly in spin-LED design.

Fig. 19. (a) A spin-polarized current is forced into an unpolarized semiconductor from a magnetic contact through the application of an electric field. Initially, the conductivity for the two spin directions is the same, but as the spins of one orientation accumulate the conductivity becomes larger for the majority spin direction. (b) That permits more majority spins to flow in response to the electric field, and the spin polarization increases further. This feedback effect can enhance spin injection by many orders of magnitude.


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4. ELECTRICAL SPIN INJECTION INTO SEMICONDUCTORS FROM NON-MAGNETIC MATERIALS The results presented up to this point emphasize the use of magnetic materials to generate spin-polarized distributions, as these materials naturally manifest profound differences in the electronic structure between spin-up and spin-down carriers. It is these differences that are vital for spin injection, not the magnetic fields generated by the magnetic materials. Thus, if it were possible to generate distinct spin-up and spin-down electronic structures (dispersion relations or wave functions) without a macroscopic magnetization, it would then be possible to achieve spinpolarized currents without magnetic materials. One of the sources of the significant differences in spin-up and spin-down electronic structure is the spontaneous breaking of time-reversal symmetry in a magnetic material. Time-reversal symmetry places strong restrictions on the permissible dispersion relations of spin-up and spin-down carriers. Yet, differences between spin-up and spin-down electronic structure are still possible in non-magnetic materials without violating time-reversal symmetry. The restrictions on the dispersion relations originate from the Kramers degeneracy [162], which requires the energy of a spin-up state to be the same as the energy of a spin-down state with an equal and opposite momentum. This does not, however, require spin-up and spin-down states at the same momentum to be the same. A spin–split dispersion relation that satisfies timereversal symmetry is shown in Fig. 20a. If a material, in addition to being timereversal symmetric, was also inversion symmetric, then spin-up states of equal and opposite momentum would have the same energy, and (with Kramers degeneracy) the dispersion relations would be everywhere spin-degenerate. Most compound semiconductors, however, such as III–V and II–VI materials, are not inversion symmetric. The spin splittings at a given momentum are quite small for a material such as GaAs, and are typically less than 1 meV for occupied states at room temperature [31]. For materials with a larger spin–orbit interaction, such as GaSb or InSb, the spin splittings can be considerably larger [31]. For a material in equilibrium, the electronic energy states are occupied according to their energy only. For every spinup state of a given momentum that is occupied, there is also a spin-down state of opposite momentum with the same energy that is occupied. The result is no spin

Fig. 20. (a) In a non-magnetic semiconductor that is not inversion symmetric, such as a III–V semiconductor, the electronic energies are different for spin-up and spin-down carriers of the same momentum. However, as a spin-up carrier has the same energy as a spin-down carrier of opposite momentum, without current flow the spin polarization vanishes. (b) When current flows, the differing chemical potentials for right-moving and left-moving carriers yields a nonzero spin polarization in the material.

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polarization in equilibrium. If, however, there is a differing occupation of states depending on whether they are moving to the right, or to the left (as shown in Fig. 20b), then more states of spin-up can be occupied than spin-down. In these cases, the time-reversal symmetry is broken by the presence of the current, and the result is a spin-polarized carrier distribution in a non-magnetic semiconductor. Several variations on this approach have been considered. Generation of spinpolarized current distributions in the bulk has been predicted, although the size of the polarization is expected to be small [163]. To amplify the spin polarization, moderately spin-polarized distributions in contact regions can be used as carrier sources in tunneling devices [164]. These tunneling regions can be designed to be highly spin–orbit selective. Thus, carriers with one momentum can be filtered to be entirely spin-up, and carriers with the opposite momentum can be filtered to be entirely spin-down. The range of possible tunneling energies can be precisely controlled, so that it is possible for a moderate current flowing in the contact, generating a small asymmetry between right- and left-moving chemical potentials, to produce a 100% spin-polarized injected current through the device. Such spin-injection devices have also been proposed [165] as contact regions for non-magnetic spin-field-effect-transistors (spin-FETs) [9]. Here, they could replace the magnetic contacts of the original design with purely non-magnetic elements. The manipulation strategy for spin-FETs never required magnetic elements – the plan for manipulation was to use the effect of an electric field to modify the propagation properties of the spin-polarized carriers in the channel. Some recent designs have suggested [165] using the electric field dependence of the electron spin lifetime [166,167] to switch the current through the device on or off. One proposed device structure is shown above in Fig. 21. The constituent materials are InAs, GaSb, and AlSb, as these provide access to very large spin splittings as well as large VB offsets. The spin splittings in the VB are always considerably larger than those in the CB, as the VBs have a strong p-orbital character. The large

Fig. 21. Schematic diagram for a non-magnetic spin field effect transistor, from Ref. [165]. Initially, poorly polarized carriers would be injected through a highly efficient spin filter on the left into an InAs channel. The spin lifetime would be controlled through an applied gate voltage, and spins would be preferentially collected on the right through another spin filter.


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VB offsets permit electrons to tunnel through spin–split VB states on their way into the InAs channel. The larger spin–orbit coupling in InAs and AlSb (relative to GaAs) permits more dramatic tuning of the electron spin relaxation times with applied electric fields. Coherent manipulation of spin polarization in non-magnetic semiconductors has also been demonstrated experimentally [168]. As previously predicted theoretically [169], the presence of the spin splittings shown in Fig. 20 can be considered as k-dependent effective magnetic fields that coherently rotate the spin orientation. By dragging spin packets around with electric fields, the spin orientation of these packets can be directly controlled. An alternating electric field was even used in Ref. [168] to drive electron spin resonance. Such coherent spin manipulation in nonmagnetic semiconductors opens up access to additional device scenarios for manipulating spin in the absence of applied magnetic fields and magnetic materials.

5. SUMMARY AND OUTLOOK Over the past several years, several significant milestones have been reached in the science and technology of electrically generating spin-polarized populations in nonmagnetic semiconductors. Clear evidence for spin injection from semimagnetic semiconductors, from FMS, and from FM metals into non-magnetic semiconductors has been reported by several groups around the world, and reasonable initial agreement with theoretical calculations has been obtained. All-electrical generation of spin-polarized populations in the absence of any magnetic material has also been achieved experimentally. Several alternate theoretical strategies for spin injection that promise higher efficiency, higher temperature operation, or other advantages, have been proposed but are yet to be thoroughly explored. We see two vital and complementary directions that should be pursued over the next several years. The first is one of optimization. Although clear evidence of high-efficiency spin injection has been obtained at low temperature, further design work for the junction properties should lead to high-efficiency spin injection at room temperature and above. There is no fundamental reason why this should not be achievable. In the near term, this might be easier to realize with FM metal injectors, although major improvements in FMS properties may lead to similar results with those materials. For other uses, such as magneto-optical elements and devices with tunable magnetic properties, such advances in FMS material properties are highly desirable. The second direction concerns some of the revolutionary possibilities in spin injection, including spin injection across a ballistic metal–semiconductor interface, spin injection with very low resistance barriers through use of drift phenomena, and spin injection through tunneling structures incorporating all non-magnetic elements. The first two could provide ways to reduce the resistance of the spin-injecting junction, a key technological issue as the power consumption of semiconductor devices becomes more and more problematic in system design. The last permits the elimination of magnetic materials from the device altogether. As fringe magnetic fields are a problem in the high-density stacking of magnetic


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elements, the elimination of magnetic materials (and the corresponding fringe fields) would permit much higher density circuitry than otherwise possible. Thus, the field of electrical spin injection into semiconductors remains a challenging and exciting area of current research, both fundamental and applied. Many similar issues confront the much less-developed area of electrical spin detection in non-magnetic semiconductors – an area that takes on even more importance as the issues of spin injection become solved, and that will be critical for an all-electrical semiconductor spintronic technology.

ACKNOWLEDGMENTS This work was supported by the Defense Advanced Research Projects Agency, the Office of Naval Research, and the Army Research Office. The authors gratefully acknowledge the many contributions of their collaborators and colleagues, and the support of their home institutions.

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Chapter 8 CURRENT-INDUCED SWITCHING OF MAGNETIZATION D. M. Edwards and J. Mathon 1. INTRODUCTION Conventional electronics utilizes the fact that electrons (holes) carry charge but makes no use of the spin angular momentum. This is justified for macroscopic samples even in the presence of magnetic components since the spin of charge carriers flips over a characteristic distance lsf (spin diffusion length), which is of the order of several tens of nanometers. This is much shorter than the dimensions of conventional electronics components. It follows that spin memory is lost in macroscopic samples and, therefore, the spin angular momentum plays no role in transport of charge. However, when the dimensions of the sample are shorter than lsf the spin is conserved and, therefore, charge current flows in two independent spin channels, which can be regarded as two wires connected in parallel [1]. When the sample contains magnetic metal (semiconductor) components, then the two spin channels are inequivalent. This is because the numbers of charge carriers with up and down spin are unequal in a ferromagnet and, even more importantly, up- and down-spin carriers are scattered at different rates at nonmagnet/magnet interfaces and in bulk ferromagnets. This has interesting and highly exploitable consequences. Because of different scattering rates for up- and down-spin carriers the total resistance of a magnetic nanostructure depends on the magnetic configuration of all its magnetic components. This in turn can be altered by an applied magnetic field and, therefore, the resistance of a magnetic nanostructure can be influenced by an applied magnetic field. The effect is known as the giant magnetoresistance (GMR). When two magnetic electrodes are separated by an insulating tunneling barrier a similar effect, called tunneling magnetoresistance (TMR), can occur. The GMR effect was discovered nearly 20 years ago [2] and a large TMR effect was first observed in 1995 [3]. The discovery of the GMR effect marks the beginning of the era of spintronics. Both GMR and TMR effects have been thoroughly explored over the last 10 years and have found many applications [4]. Contemporary Concepts of Condensed Matter Science Nanomagnetism: ultrathin films, multilayers and nanostructures Copyright r 2006 by Elsevier B.V. All rights of reproduction in any form reserved ISSN: 1572-0934/doi:10.1016/S1572-0934(05)01008-5

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D. M. Edwards and J. Mathon

274

As discussed above, it is now well established that by altering the magnetic configuration of a magnetic multilayer we can influence the charge current flowing in it. However, it was recognized only much more recently by Slonczewski [5,6] that, conversely, by passing a strong charge current one can alter the magnetic state of a magnetic multilayer. Early related theoretical work is due to Berger [7]. The layer structure in which this effect is expected to occur is shown schematically in Fig. 1, where p and m are unit vectors in the direction of the magnetization. The structure consists of a thick (semi-infinite) left magnetic layer (polarizing magnet), a nonmagnetic metallic spacer layer, a thin second magnet (switching magnet) and a semi-infinite lead. It is assumed that the magnetization of the polarizing magnet is pinned (for example by a strong anisotropy field) in a particular direction y. Slonczewski argued that the left magnet will then spin polarize the current passing through it and the resultant spin current flowing through the spacer can be partially absorbed by the switching magnet. The rate of change of the total spin, given by the difference between the spin current entering the switching magnet and that leaving the magnet, is equal to the torque exerted on the magnetic moment of the switching magnet. If the charge current, and the associated spin current, is strong enough the spin-transfer torque can cause total reversal of the switching magnet moment. This effect, which is called current-induced switching of magnetization, is the subject of the present review. On the most elementary level, one can simply assume that the polarizing magnet produces a spin current that gets partially or fully absorbed by the switching magnet and explore the consequences of the resultant torque acting on the magnetization of the switching magnet. This can be done using a phenomenological Landau–Lifshitz (LL) equation with an appropriate spin-transfer torque term. To treat correctly the dynamics of current-induced switching of magnetization, it is also necessary to include in the LL equation the usual Gilbert damping term. We shall refer to the LL equation with the Gilbert damping term as the LLG equation and begin our review of the current-induced switching of magnetization with this phenomenological treatment. We shall also use this phenomenological description to discuss relevant experiments. However, the phenomenological approach leaves many questions unanswered. In particular, to understand and optimize the switching effect, we need to know the z

z α

m

P θ φ

Y

Y

n-1 n x

x current

Fig. 1.

Schematic picture of a magnetic layer structure for current-induced switching (magnetic layers are darker, nonmagnetic layers lighter).

Current-Induced Switching of Magnetization

275

magnitude and direction of the spin-transfer torque for any specific combination of nonmagnetic and magnetic layer materials. We also require the dependences of the spin-transfer torque on the thicknesses of both the nonmagnetic and magnetic layers. Finally, to describe correctly the switching effect, we also need to know the detailed dependence of the spin-transfer torque on the angle between the magnetizations of the polarizing and switching magnets. To answer all these questions we need to calculate microscopically the spin current entering and leaving the switching magnet, i.e. the torque acting on it. The most direct microscopic approach is the original calculation of Slonczewski [5] for a simple parabolic band model of a magnetic multilayer. He calculated the spin current from the one-electron wave functions assuming that the magnetizations of the polarizing and switching magnets are kept at a given fixed angle. This type of calculation corresponds to a scattering experiment. An incoming electron with a given spin orientation (determined by the polarizing magnet) is scattered off an exchange field of the switching magnet, which is not parallel to the spin orientation of the incident electron. Calculations based on this idea are designed to tell us how much of the spin angular momentum of the incident electron is absorbed by the switching magnet. The reaction of the switching magnet to the absorbed spin angular momentum is ignored at this stage and is determined separately in a second independent calculation using the phenomenological LLG equation. In the way originally described by Slonczewski, the method was applicable without any approximations only to ferromagnets with a very large exchange splitting of up- and down-spin bands. However, we shall show that this restriction can be relaxed using the transfer matrix method to match the electron wave functions across all the interfaces. The principal limitation of this method is that it is not easily generalizable to a realistic band structure. On a more fundamental level, there are two questions that need to be addressed. The first concerns the self-consistency of all scattering-type calculations. In the one-electron scattering problem we are solving, the incoming electron moves in the local exchange field of the switching magnet, which is at an angle to its spin. In a steady state, the self-consistently determined local exchange field must be parallel to the total local spin, and that includes the spin of the incoming electron. We shall address the general question of self-consistency in the calculation of the spin-transfer torque in Section 3. The second question that needs to be addressed concerns the evaluation of the spin current from one-electron wave functions. One requires a general method, which links the current obtained from the spin current operator to an applied bias. In the analogous problem of charge transport such a method was developed by Landauer [8]. He showed that the conductance of a system sandwiched between two reservoirs is given by its total quantum mechanical transmission coefficient. It is quite straigthforward to generalize Landauer’s method to transmission of spin [9,10]. However, care has to be taken when applying such a Landauer-like formula to a magnetic-layer system. The Landauer formula for ordinary charge current gives us the conductance of the whole system. This is all that is needed since the charge current is conserved and, therefore, can be calculated anywhere in the structure. This is the reason why in the Landauer method one evaluates only the

276


total transmission coefficient for the whole structure. However, in the case of a magnetic layer structure, the spin current is not conserved since it may get partially or even fully absorbed by the magnetic layers. It follows that to calculate the spintransfer torques acting on various parts of a layer structure, we require the local spin currents entering and leaving each magnetic layer. The generalization of the Landauer formula to such a situation will be described in Section 4. We shall show that when such a generalized Landauer formula is applied to a simple parabolic band model, it gives results for local spin currents that are in complete agreement with the results obtained from the rigorous Keldysh formalism, which will be explained in Section 5. Finally, in Section 5 we shall describe the application of the Keldysh formalism to current-induced switching of magnetization. In this approach, one no longer regards the calculation of the spin current as a scattering problem. One adopts instead the point of view that, for any given applied bias (charge current), the magnetization of the switching magnet will reach a steady state in which the spin-transfer torque acting on the switching magnet is exactly compensated by the torques due to anisotropy and applied magnetic fields. Hence we set ourselves the task of calculating the local steady-state spin current. The method for calculating such a steadystate current was developed by Keldysh [11,12]. It is assumed that, initially, the multilayer is separated by a cleavage plane into two independent left- and righthand parts so that charge carriers cannot move between them. When a bias is applied to such a system no current flows and the system remains in equilibrium but with unequal chemical potentials mL and mR for the left- and right-hand parts. Next, carrier hopping across the cleavage plane is adiabatically turned on and the system evolves toward a steady state in which both charge and spin currents flow. The Keldysh method [11] provides a rigorous prescription for calculating all the steadystate properties of the connected nonequilibrium system from the known properties of the equilibrium cleaved system. In particular, to calculate the spin-transfer torque, it is only necessary to move the cleavage plane from immediately before to immediately after the switching magnet and the difference between the incoming and outgoing spin currents obtained by the Keldysh method is the required torque. It is straightforward to implement the method for a fully realistic band structure [13]. While the Keldysh method gives us all the possible steady states and the corresponding spin-transfer torques of a magnetic multilayer under an applied bias, there remains a question of the dynamical stability of the steady state. If one is merely using the Keldysh method to determine the spin-transfer torque for any particular orientation of the switching magnet magnetization, the stability of the corresponding steady state is immaterial. However, more generally, we can regard the current-induced switching of magnetization as a loss of stability of the steady state. One can argue that, when an applied bias (charge current) reaches a critical value, a steady state that the system has reached via a sequence of other steady states becomes unstable and the system seeks out a new stable steady state in which the magnetization is switched to the opposite direction [13]. The stability of steady states cannot be decided within the microscopic Keldysh formalism but this poses


277

no problem since the stability of any particular state can be decided using the phenomenological LLG equation. Provided the microscopically determined spintransfer torques are used as an input into the LLG equation, this approach should yield an essentially first-principle modelling of current-induced switching of magnetization.

2. PHENOMENOLOGICAL TREATMENT OF CURRENTINDUCED SWITCHING OF MAGNETIZATION In this section, we explore the consequences of the spin-transfer torque acting on a switching magnet using a phenomenological Landau–Lifshitz equation with Gilbert damping (LLG equation). This is essentially a generalization of the approach used originally by Slonczewski [5] and Sun [14]. We assume that there is a polarizing magnet whose magnetization is pinned in the (x,z)-plane in the direction of a unit vector p, which is at a general fixed angle y to the z-axis, as shown in Fig. 1. The pinning of the magnetization of the polarizing magnet can be due to its large coercivity (thick magnet) or a strong uniaxial anisotropy. The role of the polarizing magnet is to produce a stream of spin-polarized electrons, i.e. spin current, that is going to exert a torque on the magnetization of the switching magnet whose magnetization lies in the general direction of a unit vector m. The orientation of the vector m is defined by the polar angles a, f shown in Fig. 1. There is a thin nonmagnetic metallic layer inserted between the two magnets whose role is merely to separate magnetically the two magnetic layers and allow a strong charge current to pass. The total thickness of the whole trilayer sandwiched between two nonmagnetic leads must be smaller than the spin diffusion length lsf so that there are no spin flips due to impurities or spin-orbit coupling. A typical junction in which current-induced switching is studied experimentally [15] is shown schematically in Fig. 2. The thickness of the polarizing magnet is 40 nm, that of the switching magnet 2.5 nm and the nonmagnetic spacer is 6 nm thick. The materials for the two magnets and the spacer are cobalt and copper, respectively, which are the

Cu

Cu

V+

Au Cu

Co Co

V-

CURRENT

Fig. 2.

Schematic picture of a junction in which current-induced switching is studied experimentally.


278

most commonly used. The junction cross section is oval-shaped with dimensions 60 nm 130 nm: A small diameter is necessary so that the torque due to the Oersted field generated by a charge current of 107–108 A/cm2, required for current-induced switching, is much smaller than the spin-transfer torque we are interested in. The aim of most experiments is to determine the orientation of the switching magnet moment as a function of the current (applied bias) in the junction. Sudden jumps of the magnetization direction, i.e. current-induced switching, are of particular interest. The orientation of the switching magnet moment m relative to that of the polarizing magnet p, which is fixed, is determined by measuring the resistance of the junction. Because of the GMR effect, the resistance of the junction is higher when the magnetizations of the two magnets are antiparallel than when they are parallel. In other words, what is observed are hysteresis loops of resistance vs. current. A typical experimental hysteresis loop of this type [16] is reproduced in Fig. 3. It can be seen from Fig. 3 that, for any given current, the switching magnet moment is stationary (the junction resistance has a well-defined value), i.e. the system is in a steady state. This holds everywhere on the hysteresis loop except for the two discontinuities where current-induced switching occurs. As indicated by the arrows, jumps from the parallel (P) to antiparallel (AP) configurations of the magnetizations, and from AP to P configurations, occur at different currents. It follows that in order to interpret experiments, which exhibit such hysteretic behavior, the first task of the theory is to determine from the LLG equation all the possible steady states and then investigate their dynamical stability. At the point of instability the system seeks out a new steady state, i.e. a discontinuous transition to a new steady state with the switched magnetization occurs. We have tacitly assumed that there is always a stable steady state available for the system to jump to. There is now experimental evidence that this is not always the case. In the absence of any stable steady state the switching magnet moment must remain permanently in a

Resistance (ohms)

0.348

0.347

AP

0.346 P 0.345

Fig. 3.

-20

-10 0 10 Injected DC current (mA)

20

Resistance vs. current hysteresis loop (after Grollier et al. [16]).


279

time-dependent state. This interesting case is implicit in the phenomenological LLG treatment and we shall discuss it in detail later. In describing the switching magnet by a unique unit vector m, we assume that it remains uniformly magnetized during the switching process. This is only strictly true when the exchange stiffness of the switching magnet is infinitely large. It is generally a good approximation as long as the switching magnet is small enough to remain single domain, so that switching occurs purely by rotation of the magnetization as in the Stoner–Wohlfarth theory [17] of field switching. This seems to be the case in many experiments [15,16,18,19]. Before we can apply the LLG equation to study the time evolution of the unit vector m in the direction of the magnetization of the switching magnet, we need to determine all the contributions to the torque acting on the switching magnet. First of all, there is a torque due to the uniaxial in-plane and easy-plane (shape) anisotropies. The easy-plane shape anisotropy torque arises because the switching magnet is a thin layer typically only a few nanometers thick. The in-plane uniaxial anisotropy is usually also a shape anisotropy arising from an elongated cross section of the switching magnet [15]. We take the uniaxial anisotropy axis of the switching magnet to be parallel to the z-axis of the coordinate system shown in Fig. 1. Since the switching magnet lies in the (x,z)-plane, we can write the total anisotropy field as H A ¼ H u þ H p;

(1)

H u ¼ H u0 ðm ez Þez ;

(2)

H p ¼ H p0 ðm ey Þey :

(3)

where Hu and Hp are given by

Here, ex, ey, ez are unit vectors in the direction of the axes shown in Fig. 1. If we write the energy of the switching magnet in the anisotropy field as HA /StotS, where /StotS is the total spin angular momentum of the switching magnet, then Hu0, Hp0, which measure the strengths of the uniaxial and easy-plane anisotropies have dimensions of frequency. These quantities may be converted to a field in tesla by multiplying them by _=2mB ¼ 5:69 1012 : The next problem is to choose the correct phenomenological form of the spintransfer torque T st. Without loss of generality, the total spin-transfer torque T st may be written as the sum of the two components in the directions of the vectors m p and m ðp mÞ; where p is a unit vector in the direction of the magnetization of the polarizing magnet. Thus T st ¼ T ? þ T k ;

(4)

T ? ¼ ðgex þ g? eV b Þðm pÞ;

(5)

T k ¼ gk eV b m ðp mÞ;

(6)

where


280

Vb is the applied bias and e is the electronic charge (a negative quantity). It is more usual in experimental papers to relate spin-transfer torque to current rather than bias. However, in theoretical work, based on Landauer or Keldysh approach, bias is more natural. In practice, the resistance of the systems considered is rather constant (the GMR ratio is only a few percent) so that bias and current are in a constant ratio. The bias-independent term gex in the perpendicular component of the torque corresponds to the usual zero-bias oscillatory exchange coupling of two magnetic layers separated by a nonmagnetic spacer [13]. The amplitude of this oscillatory torque tends to zero with increasing thickness of the spacer layer and, in what follows, we shall assume that the spacer is thick enough for this term to be negligible. We assume here implicitly that the terms proportional to the bias remain finite for an arbitrary thickness of the nonmagnetic spacer layer. This is confirmed by microscopic calculations [5,6,13] in the ballistic limit. To a very good approximation, as discussed in Section 3, the spin-transfer torque depends only on the angle c between the magnetizations of the polarizing and switching magnets p, m and not on the orientations of p and m relative to the coordinate axes. It follows that the coefficients g? and g|| we have introduced in Eqs (5) and (6) are functions only of c. The modulus of both vector products in Eqs (5) and (6) is equal to sin c and microscopic calculations [5,6,13] show that the sin c factor accounts for most of the angular dependence of T? and T||. In fact, to a good approximation g? and g|| may sometimes be regarded as constant parameters which fully determine the phenomenological spin-transfer torque. We are now ready to study the time evolution of the unit vector m in the direction of the switching magnet moment. The LLG equation takes the usual form dm dm þ gm ¼ C; dt dt

(7)

where the reduced total torque C acting on the switching magnet is given by C¼

ðH A þ H ext Þ hS tot i þ T ? þ T k : jhS tot ij

(8)

Here, Hext is an external field, in the same frequency units as HA, and g is the Gilbert damping parameter. Following Sun [14], Eq. (7) may be written more conveniently as dm ¼ C gm C: 1 þ g2 dt

(9)

It is also useful to measure the strengths of all the torques in units of the strength of the uniaxial anisotropy [14]. We shall, therefore, write the total reduced torque C in the form C ¼ H u0 ðm ez Þm ez hp ðm ey Þm ey þ vk ðcÞm ðp mÞ þ ½v? ðcÞ þ hext m p ; ð10Þ where the relative strength of the easy-plane anisotropy hp ¼ Hp0/Hu0 and vjj ðcÞ ¼ vgjj ðcÞ; v? ðcÞ ¼ vg? ðcÞ measure the strengths of the torques T|| and T ? : The


281

reduced bias is defined by v ¼ eV b =ðjhS tot ijH u0 Þ and has the opposite sign from the bias voltage since e is negative. Thus positive v implies a flow of electrons from the polarizing to the switching magnet. The last contribution to the torque in Eq. (10) is due to the external field Hext with hext ¼ H ext =H u0 : The external field is taken in the direction of the magnetization of the polarizing magnet, as is the case in most experimental situations. It follows from Eq. (7) that in a steady state C ¼ 0. We shall defer the discussion of how steady-state solutions m ¼ m0 ¼ (sin a0 cos f0, sin a0 sin f0, cos a0) of this equation are obtained in general. (The polar angles a, f are defined in Fig. 1.) We shall first consider some cases of experimental importance where the steady-state solutions are trivial and the important physics is concerned entirely with their stability. To discuss stability, we linearize Eq. (9), using Eq. (10), about a steadystate solution m ¼ m0. Thus m ¼ m0 þ xea þ Zef ;

(11)

where ea, ef are unit vectors in the direction that m moves when a and f are increased independently. The linearized equation may be written in the form dx dZ ¼ Ax þ BZ; ¼ Cx þ DZ: (12) dt dt Following Sun [14], we have introduced the natural dimensionless time variable t ¼ tH u0 =ð1 þ g2 Þ: The conditions for the steady state to be stable are F ¼ A þ D 0;

G ¼ AD BC 0

(13)

excluding F ¼ G ¼ 0 [20]. For simplicity we give these conditions explicitly only for the case where either v0 k ðc0 Þ ¼ v0 ? ðc0 Þ ¼ 0; with c0 ¼ cos1 ðp m0 Þ; or m0 ¼ p: The case m0 ¼ p is very common experimentally as is discussed below. The stability condition G 0 may be written Q2 v2k þ ðQh þ cos 2a0 ÞðQh þ cos2 a0 Þ þ hp fQhð1 3sin2 f0 sin2 a0 Þ þ cos 2a0 ð1 2sin2 a0 sin2 f0 Þg h2p sin2 a0 sin2 f0 ð1 2sin2 f0 sin2 a0 Þ 0;

ð14Þ

where v|| ¼ v||(c0), h ¼ v? ðc0 Þ þ hext and Q ¼ cos c0 : The condition F 0 takes the form 2ðvk þ ghÞQ gðcos 2a0 þ cos2 a0 Þ ghp ð1 3sin2 f0 sin2 a0 Þ 0:

(15)

We now discuss several interesting examples, the first of these relating to experiments of Grollier et al. [18] and others. In these experiments the magnetization of the polarizing magnet, the uniaxial anisotropy axis and the external field are all collinear (along the in-plane z axis in our convention). In this case, the equation C ¼ 0, with C given by Eq. (10), shows immediately that possible steady states are given by m0 ¼ pða0 ¼ 0; pÞ; corresponding to the switching magnet moment along the z axis. These are the only solutions when hp ¼ 0: For hp a0; other steady-state solutions may exist but in the parameter regime which has been investigated they are always unstable [13]. We shall assume this is always the case and concentrate on the solutions m0 ¼ p: In the state of parallel magnetization (P) m0 ¼ p, we have


282

v|| ¼ vg||(0), h ¼ vg?(0)+hext, a0 ¼ 0 and Q ¼ 1: The stability conditions (14), (15) become ½gk ð0Þ2 v2 þ ðvg? ð0Þ þ hext þ 1Þ2 þ hp ½vg? ð0Þ þ hext þ 1 0 1 gk ð0Þv þ g vg? ð0Þ þ hext þ 1 þ hp 0: 2

(16)

(17)

In the state of antiparallel magnetization (AP) m0 ¼ p; we have v|| ¼ vg||(p), h ¼ vg?(p)+hext, a0 ¼ p and Q ¼ 1: The stability conditions for the AP state are thus ½gk ðpÞ2 v2 þ ðvg? ðpÞ hext þ 1Þ2 þ hp ½vg? ðpÞ hext þ 1 0 1 gk ðpÞv þ g vg? ðpÞ þ hext 1 hp 0: 2

(18)

(19)

In the regime of low external field (hextE1, i.e. HextEHu0) we have H p hext (hpE100). Eqs (16) and (18) may then be approximated by vg? ð0Þ þ hext þ 140

(20)

vg? ðpÞ þ hext 1o0:

(21)

Equation (20) corresponds to P stability and (21) to AP stability. It is convenient to define scalar quantities T?, T|| by T? ¼ g?(c) sin c, T|| ¼ g||(c) sin c, these being scalar components of spin-transfer torque in units of eVb (cf. Eqs (5) and (6)). Then gi ð0Þ ¼ ½dT i =dcc¼0 and gi ðpÞ ¼ ½dT i =dcc¼p ; i ¼ ?,||. Model calculations [13] show that both g? and g|| can be of either sign, although positive values are more common. Also, there is no general rule about the relative magnitude of gi(0) and gi(p). We now illustrate the consequences of the above stability conditions by considering two limiting cases. We first consider the case g? ðcÞ ¼ 0; gjj ðcÞ40; as assumed by Grollier et al. [18] in the analysis of their data. In Fig. 4 we plot the regions of P and AP stability, deduced from Eqs (17), (19)–(21), in the v hext plane. Grollier et al. plot current instead of bias but this should not change the form of the figure. Theirs is rather more complicated, owing to a less transparent stability analysis with unnecessary approximations. The only approximations made above, to obtain Eqs (20) and (21), can easily be removed, which results in the critical field lines hext ¼ 1 acquiring a very slight curvature given by hext 1 þ ½vgjj ðpÞ2 =hp and hext 1 ½vgjj ð0Þ2 =hp : The critical biases in the figure are given by g 1 þ 12 hp hext g 1 þ 12 hp þ hext ; vP!AP ¼ : (22) vAP!P ¼ gk ðpÞ gk ð0Þ A downward slope from left to right of the corresponding lines in Fig. 4 is not shown there. Since the damping parameter g is small (g 0:01), this downward slope of the critical bias lines is also small. From Fig. 4 we can deduce the behavior of resistance vs. bias in the external field regimes |hext|o1 and |hext|41.


283

v

BOTH UNSTABLE

P STABLE vAP

−1

P

hext

1 vP

Fig. 4.

AP

BOTH UNSTABLE

AP STABLE

Bias-field stability diagram for g? ðcÞ ¼ 0; gjj ðcÞ40: A small downward slope of the lines VAP-P, VP-AP (see Eq. (22)) is not shown.

vP

AP

(a)

(b)

R

R AP

AP

P

P

v vAP

P

* vP

AP

* v vAP

P

Fig. 5. (a) Hysteresis loop of resistance vs. bias for |hext|o1 (b) Reversible behavior (no hysteresis) for hexto1 (upper curve) and hext41 (lower curve). The dashed lines represent hypothetical behavior of average resistance in regions of Fig. 4 marked ‘‘both unstable’’ where no steady states exist.

Consider first the case |hext|o1. Suppose we start in the AP state with a bias v ¼ 0; which is gradually increased to vAP-P. At this point, the AP state becomes unstable and the system switches to the P state as v increases further. On reducing v the hysteresis loop is completed via a switch back to the AP state at the negative bias vP-AP. The hysteresis loop is shown in Fig. 5(a). The increase in resistance R between the P and AP states is the same as would be produced by varying the applied field in a GMR experiment. Now consider the case hexto1. Starting again in the AP state at v ¼ 0 we see from Fig. 4 that, on increasing v to vAP-P, the AP state becomes unstable but there is no stable P state to switch to. This point is marked by an asterisk in Fig. 5(b). For v4vAP-P, the moment of the switching magnet is in a persistently time-dependent state. However, if v is now decreased below vAP-P the system homes in on the stable AP state and the overall behavior is

284


reversible, i.e. no switching and no hysteresis occur. When hext 41; similar behavior, now involving the P state, occurs at negative bias, as shown in Fig. 5(b). The dashed curves in Fig. 5(b) show a hypothetical time-averaged resistance in the regions of time-dependent magnetization. As discussed later, time-resolved measurements of resistance suggest that several different types of dynamics can occur in these regions. It is clear from Fig. 5(a) that the jump AP-P always occurs for positive reduced bias v, which corresponds to a flow of electrons from the polarizing to the switching magnet. This result depends on the assumption that gjj 40; if gjj o0; it is easy to see that the sense of the hysteresis loop is reversed and the jump P-AP occurs for positive v. To our knowledge this reverse jump has never been observed, although gjj o0 can occur in principle and is predicted theoretically [13] for the Co/Cu/ Co(111) system with a switching magnet consisting of a single atomic plane of Co. It follows from Eq. (22) that jvP!AP =vAP!P j ¼ jgjj ðpÞ=gjj ð0Þj in zero external field. Experimentally this ratio, essentially the same as the ratio of critical currents, may be considerably less than 1 (e.g. o0.5 [15]), greater than 1 (e.g. E2 [19]) or close to 1 [16]. Usually, the field dependence of the critical current is found to be stronger than that predicted by Eq. (22) [15,16]. We now discuss the reversible behavior shown in Fig. 5(b), which occurs for |hext|41. The transition from hysteretic to reversible behavior at a critical external field seems to have been first seen in pillar structures by Katine et al. [21]. Curves similar to the lower one in Fig. 5(b) are reported with |vP-AP| increasing with increasing hext, as expected from Eq. (22). Plots of the differential resistance dV/dI show a peak near the point of maximum gradient of the dashed curve. Similar behavior has been reported by several groups [22–24]. It is particularly clear in the work of Kiselev et al. [22] that the transition from hysteretic behavior (as in Fig. 5(a)) to reversible behavior with peaks in dV/dI occurs at the coercive field 600 Oe of the switching layer (hext ¼ 1). The important point about the peaks in dV/dI is that for a given sign of hext they only occur for one sign of the bias. This clearly shows that this effect is due to spin-transfer and not to Oersted fields. Myers et al. [25] show a current-field stability diagram similar to the bias-field one of Fig. 4 with a critical field of 1500 Oe.They examine the time dependence of the resistance at room temperature with the field and current adjusted so that the system is in the ‘‘both unstable’’ region in the fourth quadrant of Fig. 4 but very close to its top lefthand corner. They observe telegraph-noise-type switching between approximately P and AP states with slow switching times in the range 0.1–10 s. Similar telegraph noise with faster switching times was observed by Urazhdin et al. [23] at current and field close to a peak in dV/dI. In the region of P and AP instability Kiselev et al. [22] and Pufall et al. [24] report various types of dynamics of precessional type and random telegraph switching type in the microwave GHz regime. Kiselev et al. [22] propose that systems of the sort considered here might serve as nanoscale microwave sources or oscillators, tunable by current and field over a wide frequency range. We now return to the stability conditions (17), (19)–(21) and consider the case of g? ðcÞa0 but hext ¼ 0: The conditions of stability of the P state may be written


285

approximately, remembering that g 1; hp 1; as 1 vgk ð0Þ4 ghp : 2

vg? ð0Þ4 1;

(23)

The conditions for stability of the AP state are vg? ðpÞo1;

1 vgk ðpÞo ghp : 2

(24)

In Fig. 6 we plot the regions of P and AP stability, assuming g? ð0Þ ¼ g? ðpÞ ¼ g? and gjj ð0Þ ¼ gjj ðpÞ ¼ gjj for simplicity. We also put r ¼ g? =gjj : For r40 we find the normal hysteresis loop as in Fig. 5(a) if we plot R against vg|| (valid for either sign of g||). In Fig. 7 we plot the hysteresis loops for the cases rc oro0 and rorc ; where rc ¼ 2=ðghp Þ is the value of r at the point X in Fig. 6. The points labeled by asterisks have the same significance as in Fig. 5(b). If in Fig. 7(a) we increase vg|| beyond its value indicated by the right-hand asterisk we move into the ‘‘both unstable’’ region where the magnetization direction of the switching magnet is perpetually in a time-dependent state. Thus, negative r introduces behavior in zero applied field which is similar to that found when the applied field exceeds the

vg BOTH UNSTABLE

P STABLE

X 1 γh 2 p 1 γh 2 p

BOTH UNSTABLE

Fig. 6.

r =g g

AP STABLE

vg = -1

Stability diagram for hext ¼ 0:

(a)

(b)

R

R

* 1 2

vg =1

γ hp

Fig. 7.

AP

AP

P

P

1 2

γh p

*

*

vg

1 2

γ hp

* 1 2

γ hp

Hysteresis loop for (a) rc oro0; (b) rorc :

vg


286

coercive field of the switching magnet for r ¼ 0: This behavior was predicted by Edwards et al. [13], in particular for a Co/Cu/Co(111) system with the switching magnet consisting of a Co monolayer. Zimmler et al. [26] use methods similar to the ones described here to analyze their data on a Co/Cu/Co nanopillar and deduce that gjj 40; r ¼ g? =gjj 0:2: It would be interesting to carry out time-resolved resistance measurements in this system at large current density (corresponding to vg? o 1) and zero external field. So far we have considered the low-field regime (HextE coercive field of the switching magnet) with both magnetizations and the external field in-plane. There is another class of experiments in which a high field, greater than the demagnetizing field (42T), is applied perpendicular to the plane of the layers. The magnetization of the polarizing magnet is then also perpendicular to the plane. This is the situation in the early experiments where a point contact was employed to inject high current densities into magnetic multilayers [27–29]. In this high-field regime a peak in the differential resistance dV/dI at a critical current was interpreted as the onset of current-induced excitation of spin waves in which the spin-transfer torque leads to uniform precession of the magnetization [6,27,28]. No hysteretic magnetization reversal was observed and it seemed that the effect of spin-polarized current on the magnetization is quite different in the low- and high-field regimes. Recently, however, O¨zyilmaz et al. [30] have studied Co/Cu/Co nanopillars (E100 nm in diameter) at T ¼ 4:2 K for large applied fields perpendicular to the layers. They observed hysteretic magnetization reversal and interpreted their results using the Landau– Lifshitz equation. We now give a similar discussion within the framework of this section. Following O¨zyilmaz et al., we neglect the uniaxial anisotropy term in Eq. (10) for the reduced torque C while retaining Hu0 as a scalar factor. Hence C ¼ H u0 f½hext þ v? ðcÞ hp cos cm p þ vk ðcÞm ðp mÞg;

(25)

where p is the unit vector perpendicular to the plane. When vjj ðcÞa0 the only possible steady-state solutions of C ¼ 0 are m0 ¼ p: On linearizing Eq. (9) about m0 as before, we find that the condition G 0 is always satisfied. The second stability condition F o0 becomes fvk ðc0 Þ þ g½v? ðc0 Þ þ hext hp g cos c0 40;

(26)

1

where c0 ¼ cos (m0 p). Applying this to the P state (c0 ¼ 0) and the AP state (c0 ¼ p) we obtain the conditions v4g

ðhp hext Þ gð0Þ

vo g

ðhp þ hext Þ ; gðpÞ

(27)

(28)

where the first condition applies to the P stability and the second to the AP stability. Here, gðcÞ ¼ gjj ðcÞ þ gg? ðcÞ: The corresponding stability diagram is shown in Fig. 8, where we have assumed gðpÞ4gð0Þ40 for definiteness. The boundary lines


287

v P STABLE

- hp

hp

hc

hext

vAP vP

P AP

AP STABLE

Fig. 8.

Bias-field stability diagram for large external field (hext4hp) perpendicular to the layers.

cross at hext ¼ hc, where hc ¼ hp ½gðpÞ þ gð0Þ=½gðpÞ gð0Þ: This analysis is only valid for fields larger than the demagnetizing field (hext4hp) and we see from the figure that for hext4hc hysteretic switching occurs. This takes place for only one sign of the bias (current) and the critical biases (currents) increase linearly with hext as does the width of the hysteresis loop |vP-APvAP-P|. This accords with the observations of O¨zyilmaz et al. The critical currents are not larger than those in the low-field or zero-field regimes (cf. Eqs (27) and (28) with Eq. (22)) and yet the magnetization of the switching magnet can be switched against a very large external field. However, in this case the AP state is only stabilized by maintaining the current. The experiments on spin transfer discussed above have mainly been carried out at constant temperature, typically 4.2 K or room temperature. The effect on currentdriven switching of varying the temperature has recently been studied by several groups [23,25,31]. The standard Neél–Brown theory of thermal switching [32] does not apply because the Slonczewski in-plane torque is not derivable from an energy function. Li and Zhang [33] have generalized the standard stochastic Landau– Lifshitz equation, which includes white noise in the effective applied field, to include spin-transfer torque. In this way they have successfully interpreted some of the experimental data. A full discussion of this work is outside the scope of the present review. However, it should be pointed out that in addition to the classical effect of white noise there is an intrinsic temperature dependence of quantum origin. This arises from the Fermi distribution functions which appear in expressions for the spin-transfer torque (see Eqs (60) and (61)). So far we have discussed steady-state solutions of the LLG equation (9). It is important to study the magnetization dynamics of the switching layer in the situation during the jumps AP-P and P-AP of the hysteresis curve in zero external field, and secondly under conditions where only time-dependent solutions are possible, for example in the regions of sufficiently strong current and external field marked ‘‘both unstable’’ in Fig. 4. The first situation has been studied by Sun [14], assuming single-domain behavior of the switching magnet, and by Miltat et al. [34]


288

with more general micromagnetic configurations. Both situations have been considered by Li and Zhang [35]. In the second case they find precessional states, and the possibility of ‘‘telegraph noise’’ at room temperature, as seen experimentally in Refs. [22,24]. Switching times (AP-P and P-AP) are estimated to be of the order of 1 ns. Micromagnetic simulations [34] indicate that the Oersted field cannot be completely ignored for typical pillars with diameter of the order of 100 nm. Finally, in this section, we briefly discuss some practical considerations, which may ultimately decide whether current-induced switching is useful in spintronics. Sharp switching, with nearly rectangular hysteresis loops, is obviously desirable and this demands single-domain behavior. In experiments on nanopillars of circular cross section [21] multidomain behavior was observed with the switching transition spread over a range of current. Subsequently, the same group [15] found sharp switching in pillars whose cross section was an elongated hexagon, which introduces strong uniaxial in-plane shape anisotropy. It was known from earlier magnetization studies of nanomagnet arrays [36] that such shape anisotropy can result in singledomain behavior. A complex switching transition need not necessarily indicate multidomain behavior. It could also arise from a marked departure of T?(c) and/or T||(c) from sinusoidal behavior, such as occurs near c ¼ p in calculations for Co/Cu/Co(111) with two atomic planes of Co in the switching magnet (see Fig. 13(b)). In the calculations of the corresponding hysteresis loops (Fig. 16) the torques were approximated by sine curves but an accurate treatment would certainly complicate the AP-P transition which occurs at negative bias in Fig. 16(b). Studies of this effect are planned. The critical current density for switching is clearly an important parameter. From Eq. (22) the critical reduced bias for the P-AP transition is to a good approximation given by ghp =½2gjj ð0Þ: Using the definitions of reduced quantities given after Eq. (10), we may write the actual critical bias in volts as V P!AP ¼

MgM s H d ; 2gk ð0Þjej

(29)

where M is the number of atomic planes in the switching magnet, Ms is the average moment (J/T) of the switching magnet per atomic plane per unit area, and Hd ¼ _Hp0/(2mB) is the easy-plane anisotropy field in tesla. As expressed earlier gjj ð0Þ ¼ ðdT jj =dcÞc¼0 where the torque T|| is per unit area in units of eVb. (The calculated torques in Figs 13 and 14 of Section 5 are per surface atom so that if these are used to determine g||(0) in Eq. (29) Ms must be taken per surface atom.) An obvious way to reduce the critical bias, and hence the critical current, is to reduce M, the thickness of the switching magnet. Calculations show [13] (see also Fig. 14) that g|| does not decrease with M and may, in fact, increase for small values such as M ¼ 2: Careful design of the device might also increase g||(0) beyond the values (o0.01 per surface atom), which seem to be obtainable in simple trilayers [13]. Jiang et al. [37,38] have studied various structures in which the polarizing magnet is pinned by an adjacent antiferromagnet (exchange biasing) and in which a thin Ru layer is incorporated between the switching layer and the lead. Critical current densities of 2 106 Acm2 have been obtained which are substantially


289

lower than those in Co/Cu/Co trilayers. Such structures can quite easily be investigated theoretically by the methods of Section 5. Decreasing the magnetization Ms, and hence the demagnetizing field (pHd), would be favorable but g|| then tends to decrease also [13]. A possible way of decreasing Hd without decreasing local magnetic moment in the system is to use a synthetic ferrimagnet as the switching magnet [39]. The Gilbert damping factor g is another crucial parameter but it is uncertain whether this can be decreased significantly. However, the work of Capelle and Gyorffy [40] is an interesting theoretical development. The search for structures with critical current densities low enough for use in spintronic devices (105 Acm2 perhaps) [41] is an enterprise where experiment and quantitative calculations [13] should complement each other fruitfully.

3. ORIGIN OF SPIN-TRANSFER TORQUE 3.1. General Principles To put the phenomenological treatment of Section 2 on a first-principle quantitative basis we must discuss the quantum-mechanical origin of the spin-transfer torques (Eqs (5) and (6)) in a steady state. For this purpose it is convenient to describe the magnetic and nonmagnetic layers of Fig. 1 by tight-binding models, in general multiorbital with s, p and d orbitals, whose one-electron parameters are fitted to first-principle bulk band structure [42]. However, we note that any Hamiltonian that can be written in a localized basis can be used, for example, linear-muffin-tinorbital (LMTO) tight-binding method implemented for a layer system [43] satisfies this requirement. The Hamiltonian is, therefore, of the form H ¼ H 0 þ H int þ H anis þ H imp ; where the one-electron hopping term H 0 is given by X X H0 ¼ tmm;nn ðkk Þcykk mms ckk nns ;

(30)

(31)

kk s mm;nn

where cykk mms creates an electron in a Bloch state, with in-plane wave vector k|| and spin s, formed from a given atomic orbital m in plane m. Hint is an on-site interaction between electrons in d orbitals which leads to an exchange splitting of the bands in the ferromagnets and is neglected in the spacer and lead. Diagonal disorder is included in Himp, which is of the general form X X H imp ¼ V nm ðkk k0k Þcyk0 nms ckk nms : (32) k

kk; k0k n;m;s

Finally, Hanis contains anisotropy fields in the switching magnet and is given by X Sn H A; (33) H anis ¼ n

where Sn is the operator of the total spin angular momentum of plane n and HA is given by Eqs (1)–(3) with the unit vector m in the direction of Sn hS n i; where /SnS


290

is the thermal average of Sn. We assume here that the anisotropy fields Hu0, Hp0 are uniform throughout the switching magnet but we could generalize to include, for example, a surface anisotropy. In our tight-binding description, the spin angular momentum operator Sn is given by

T 1 X y Sn ¼ _ ckk nm" ; cykk nm# s ckk nm" ; ckk nm# (34) 2 km k

and the corresponding operator for spin angular momentum current between planes n 1 and n is

T iX j n1 ¼ tðkk Þnn;n1m cykk nn" ; cykk nn# s ckk n1m" ; ckk n1m# þ h:c. (35) 2 k mn k

Here, r ¼ ðsx ; sy ; sz Þ; where the components are Pauli matrices. Equation (35) yields the charge current operator if 12s is replaced by a unit matrix multiplied by the electronic charge e=_; where e is the electronic charge (negative). All currents flow in the y direction, perpendicular to the layers, and the components of the vector j correspond to transport of x, y, and z components of spin. The rate of change of Sn in the switching magnet is given by dS n ¼ ½S n ; H 0 þ ½S n ; H anis : (36) dt This result holds since the spin operator commutes with the interaction Hamiltonian Hint and also with Himp. It is straightforward to show that i_

½S n ; H 0 ¼ i_ðj n1 j n Þ;

(37)

½S n ; H anis ¼ i_ðH A S n Þ:

(38)

and

On taking the thermal average, Eq. (36) becomes

dS n ¼ j n1 j n H A hS n i: dt

(39)

When this equation is summed over all planes in the switching magnet we have d hS tot i ¼ T st H A hS tot i; dt where the total spin-transfer torque is given by

T st ¼ j spacer j lead :

(40)

(41)

Here /jspacerS and /jleadS are spin currents in the spacer and lead, respectively, and /StotS is the total spin angular momentum of the switching magnet. Equation (40) is equivalent to Eq. (7), for zero external field, in the absence of damping. Equation (41) shows how T st, required for the phenomenological treatment of


291

Section 2, is to be determined from the calculated spin currents in the spacer and lead. As discussed in Section 2 the magnetization of a single-domain sample is essentially uniform and the spin-transfer torque T st depends on the angle c between the magnetizations of the polarizing and switching magnets. To consider time-dependent solutions of Eq. (7) it is necessary to calculate T st for arbitrary angle c and for this purpose HA can be neglected. To reduce the calculation of the spin-transfer torque to effectively a one-electron problem, we replace Hint by a self-consistent exchange field term Sn S n Dn ; where the exchange field Dn should be determined self-consistently in the spirit of an unrestricted Hartree–Fock (HF), or local spin density (LSD), approximation. The essential selfconsistency condition in any HF or LSD calculation is that the local moment /SnS in a steady state is in the same direction as Dn. Thus we require Dn S n ¼ 0

(42)

for each atomic plane of the switching magnet. It is useful to consider first the situation when there is no applied bias and the polarizing and switching magnets are separated by a spacer, which is so thick that the zero-bias oscillatory exchange coupling [44] between the two magnets is negligible. In that case we have two independent magnets and the self-consistent exchange field in every atomic plane of the switching magnet is parallel to its total magnetization, which is uniform and assumed to be along the z axis. Referring to Fig. 1, the self-consistent solution therefore corresponds to uniform exchange fields in the polarizing and switching magnets, which are at an assumed angle c ¼ y with respect to one another. When a bias Vb is applied, with a uniform exchange field D ¼ Dez in the switching magnet imposed, the calculated local moments /SnS will deviate from the z direction so that the solution is not self-consistent. To prepare a self-consistent state with D and all /SnS ¼ /SS in the z direction it is necessary to apply fictitious constraining fields Hn of magnitude proportional to Vb. The local field for plane n is thus D+Hn but to calculate the spin currents in the spacer and lead, and hence T st from Eq. (41), the fields Hn, of the order of Vb, may be neglected compared with D. Although the fictitious constraining fields Hn need therefore never be calculated it is st interesting to see that they are in fact

related to T . For the constrained self_ consistent steady state ðhS n i ¼ hS i; S n ¼ 0Þ in the presence of the constraining fields, with HA neglected as discussed above, it follows from Eq. (39) that

j n1 j n ¼ ðD þ H n Þ hS i ¼ H n hS i; (43) where the local field D þ H n replaces HA. On summing over all atomic planes n in the switching magnet we have

X H n hS i: (44) T st ¼ j spacer j lead ¼ n

Thus, as expected, in the prepared state with a given angle c between the magnetizations of the magnetic layers the spin-transfer torque is balanced by the total torque due to the constraining fields.

292


The main conclusion of this section is that the spin-transfer torque for a given angle c between magnetizations may be calculated using uniform exchange fields making the same angle with one another. Such calculations are described in Sections 4 and 5. The use of this spin-transfer torque in the LLG equation of Section 2 completes what we shall call the ‘‘standard model’’ (SM). It underlies the original work of Slonczewski [5] and most subsequent work. The spin-transfer torque calculated in this way should be appropriate even for time-dependent solutions of the LLG equation. This is based on the reasonable assumption that the time for the electronic system to attain a ‘‘constrained steady state’’ with given c is short compared with the time-scale (E1 ns) of the macroscopic motion of the switching magnet moment.

3.2. Discussion of Previous Work and SM Concepts We have established that the SM is a valid approximation for the calculation of the total spin-transfer torque. Before proceeding to the implementation of the SM in Sections 4 and 5, we need to discuss two concepts which arise in the SM and are widely used by the community working in this field. The first concept is so-called spin accumulation in the switching magnet. The second concept is that of a decoherence length which is supposed to control the spatial variation of the spin current within the switching magnet. To understand how these concepts arise within the SM we need to examine P carefully the role of self-consistency. When Hint in Eq. (30) is replaced by S n Dn ; n which no longer commutes with Sn, Eq. (39) becomes

d hS n i ¼ j n1 j n ðH A þ Dn Þ hS n i: (45) dt When the condition (42) is satisfied at all times this reduces to Eq. (39), showing that self-consistency restores the spin-rotational symmetry of the electron–electron interactions, which is broken by an arbitrary exchange field. The unrestricted HF type of self-consistency considered here will correctly describe exchange interaction within the magnet at the level of the random phase approximation. Previous authors [5,6,45] using the SM, with uniform exchange field D and with HA neglected, do not consider the role of self-consistency and the requirement of constraining fields to enforce it in the steady state. As pointed out earlier the nonself-consistent local moments /SnSNSC deviate from the direction of D. It follows from Eq. (45) that in the steady state they satisfy

(46) j n1 NSC j n NSC ¼ D hS n iNSC : The transverse components of /SnSNSC (perpendicular to D) represent small deviations of the switching magnet moment from the direction of its exchange field. These spin components are referred to by previous authors [46,47] as ‘‘spin accumulation’’. Although this is not done by the proponents of the spin accumulation concept, both the local spin /SnSNSC and local spin-transfer torque hj n1 iNSC hj n iNSC


293

in Eq. (46) can be calculated microscopically within the SM by the Keldysh method described in Section 5. One then finds [13] that Eq. (46) indeed holds in every atomic plane n of the switching magnet. This clarifies how spin accumulation arises in the SM model. However, in a self-consistent treatment such deviations do not occur because the exchange field is always in the direction of the local moment. Thus transverse spin accumulation in the ferromagnet is a nonphysical concept. Another misleading result arising from nonself-consistency in the SM concerns the spatial variation of transverse spin current through the switching magnet. To see this clearly we consider directly true quantum-mechanical steady states in the presence of the anisotropy field HA. Neither fictitious constraining fields nor the LLG equation are now required. The self-consistency conditions (42) determine the magnetization in the switching magnet completely for a given bias Vb. In the steady state, Eq. (39) becomes

(47) j n1 j n ¼ H A hS n i: Clearly the longitudinal component of the spin current (parallel to the magnetization) is conserved whereas both transverse components change by a constant amount on crossing each atomic plane. (In the nonmagnetic layers all components of spin current are conserved since HA ¼ 0 in these layers.) Spin-transfer torque is therefore applied uniformly across the switching magnet to balance, in the steady state, the torque due to the anisotropy field which is assumed to be uniform. However, it is almost universally believed [6,45] that the transverse spin current decays rapidly on entering the ferromagnet so that spin-transfer torque is only applied close to the interface between the spacer and the switching magnet. It is true that the local spin current calculated in the SM by the methods discussed in Sections 4 and 5 does not vary linearly across the switching magnet. Its behavior is shown schematically as curve B in Fig. 9 where it is contrasted with the linear behavior A of the true self-consistent solution. To understand the discrepancy between curves A and B we must recognize that the assumption made so far of a completely uniform magnetization in the switching magnet is not strictly true. In the true selfconsistent solution for a given bias the exchange field Dn on each plane n deviates slightly from the constant value D. The local spin /SnS varies similarly since /SnS

<Jspacer> A B C SPACER

FERROMAGNET

<Jlead> LEAD

Fig. 9. Schematic plot of a transverse component of spin current showing the true selfconsistent behavior (A), a schematic SM result (B), and the contribution associated with exchange stiffness (C).

294


and Dn are in the same direction. It must be stressed that these self-consistent deviations of /SnS from a uniform spin direction are completely different from those of the /SnSNSC in the nonself-consistent SM. The self-consistent deviations produce internal transverse spin currents, proportional to the bias, corresponding to exchange coupling between noncollinear moments in adjacent atomic planes of the magnet. The profile of this internal spin current is plotted schematically in Fig. 9 as curve C. When the contributions of the SM spin current and internal exchange stiffness spin current are combined, the total spin current will drop linearly across the switching magnet (curve A in Fig. 9). Finally, we must ask whether deviations of the local moment /SnS from uniformity, which are necessary to produce the required compensating exchange stiffness torque, are going to influence the spin current determined within the SM. The answer is no because the exchange stiffness of a magnet such as Co is very large and small deviations from uniformity are thus sufficient to produce the required compensating internal torques. These small deviations have negligible effect on the spin current. We conclude that the spatial variation of the transverse spin current in the switching magnet determined in the SM is quite incorrect and unphysical. However, since the compensating internal spin current which must be added to the SM spin current to restore the correct balance between local spin-transfer and anisotropy torques, vanishes at both surfaces of the switching magnet, the spin currents at the interfaces between the switching magnet and spacer/right lead are correctly calculated in the SM. Moreover, since the spin current is a continuous function of position and is conserved in the nonmagnetic layers, its values at the interfaces with the spacer and right lead are equal to the values of the spin current in the spacer and right lead, respectively. This explains why the total spin-transfer torque /jspacerS/jleadS calculated in the SM remains a very good approximation to the total spin-transfer torque determined self-consistently. A concept that is directly related to transverse spin accumulation in the SM is that of a so-called spin decoherence length lc over which the spin accumulation is supposed to decay [46,47]. It is clear from Eq. (46) that the spatial variation of transverse spin accumulation in the SM is directly related to the spatial variation of the transverse spin current in the switching magnet determined in the SM. We have already demonstrated that the calculated spatial dependence of the SM spin current within the magnet is unphysical. It follows that the spatial dependence of the SM spin accumulation also has no physical significance. However, since the concept of a spin decoherence length is so widely used by the community working in this field, it merits a further investigation. There is only one situation where this concept could be justified. If the SM spin current were to decay rapidly on entering the ferromagnet, one could argue that for thicknesses of the switching magnet greater than this characteristic decay length no spin current would emerge in the right lead. The total spin-transfer torque would be then determined only by the spin current in the spacer. It is widely believed [6,45,47] that precisely this scenario is generally valid. To check this idea, we have calculated the transverse spin current in the spacer and in the lead as a function of the switching magnet thickness. Our calculations within the SM are for a single-orbital tight-binding model and were made using the


295

Keldysh method described in Section 5. We have chosen the tight-binding parameters of our model so that the polarizing magnet is a semi-infinite strong magnet, i.e. the minority-spin band is empty and we set E "F ¼ 0:12W ; where W is the bandwidth, to make the Fermi level lie near the bottom of the majority-spin band. It follows that electrons passing through the polarizing magnet are 100% spin polarized. The parameters of the switching magnet were chosen so that the Fermi level intersects both the majority- and minority-spin bands but the minority-spin band is almost empty (E "F ¼ 0:07W and E #F ¼ 0:017W ). This choice of parameters provides a reasonable model of a ferromagnet such as Co. In fact, it is seen in Section 5 that the relative magnitude of the spin currents in the spacer and lead for a Co/Cu/Co system (Fig. 15) is similar to that obtained for this simple model (Fig. 10). We varied the thickness of the switching magnet between 1 and 20 atomic planes. The dependences of the in-plane J|| and out-of-plane J? spin currents in the spacer on the number of atomic planes M in the switching magnet are shown in Fig. 10(a). The corresponding dependences of J|| and J? in the right lead on the switching magnet thickness are shown in Fig. 10(b). The spin currents in Fig. 10 are per surface atom and in units of eVb, where Vb is the applied bias. Clearly the spin current does not decay rapidly in the switching magnet. The in-plane component J|| of the spin current in the right lead oscillates with a large amplitude that is comparable to the magnitude of the spin current in the spacer. The out-of-plane component J? not only does not decay but may even grow for certain thicknesses of the switching magnet so that the spin current can change its sign on passing through the switching magnet. These results are confirmed by fully realistic calculations for Co/ Cu/Co(111) which are described in Section 5 (see Fig. 15). From our results, we conclude that, in general, the transverse spin current does not decay rapidly in the switching magnet and the concept of a decoherence length cannot be justified. The simple model described here, with completely polarized electrons entering the spacer from the polarizing magnet and with low electron densities so that the occupied bands are essentially parabolic, is similar to the parabolic band model used by Stiles and Zangwill [45]. However, their results are qualitatively different

(a)

(b)

0.004

J

0.002

J

0.001 0

0.003 spin current

spin current

lead

J

0.004

spacer

0.003

spacer

0.002

Jlead

0.001 0 -0.001

-0.001

0

5

10

15

switching magnet thickness M

20

0

5 10 15 switching magnet thickness M

20

Fig. 10. Dependences of the in-plane J|| and out-of-plane J? spin currents in the spacer (a) and in the right lead (b) on the number of atomic planes M in the switching magnet.

296


and feature a puzzling discontinuity of the spin current at the spacer switching magnet interface. Zhang et al. [46] proposed a phenomenological theory based on the concept of transverse spin accumulation and an associated length-scale lJ, the diffusive analogue of the decoherence length lc. Their expressions for the coefficients v|| and v? in Eq. (10), which measure the strengths of the torques T|| and T?, depend primarily on lJ. We have argued that spin accumulation in a steady state is unphysical so we regard this theory as ill-founded. The theory of Heide et al. [48] is also phenomenological and these authors consider only the torque T? of Eq. (5). The coefficient in that equation is the sum of the usual interlayer exchange interaction which exists for zero bias and the term proportional to bias which Heide et al. call the nonequilibrium exchange interaction (NEXI). The torque T? is clearly equivalent to one due to an effective field. When only this torque is present the stability criteria of Section 2 are equivalent to the conditions for a minimum of an energy function whose gradient yields the effective field, just as in the Stoner– Wohlfarth theory of field switching [17]. As soon as the in-plane torque T|| is introduced no energy function exists and the full dynamical stability criteria of Section 2 must be applied. Heide et al. do not calculate NEXI microscopically but it is automatically included in the microscopic calculations described in Sections 4 and 5.

4. GENERALIZED LANDAUER METHOD FOR CALCULATING THE SPIN CURRENT The method of choice for calculating the spin-transfer torque is the Keldysh formalism [11,12] which is specifically designed for calculating the transport properties of a nonequilibrium system in its steady state. This approach, which can be implemented for a fully realistic band structure [13], is described in Section 5. However, the Keldysh method, which is formulated in terms of one-electron Green functions, is less transparent than the original calculation of Slonczewski [5,6] for a simple parabolic band. Although it would be difficult to generalize his calculation to a realistic band structure, Slonczewki’s method, which relies on simple matching of electron wave functions across the layer structure, is much more intuitive. We shall, therefore, first describe his original calculation and then generalize and reformulate his wave function method in the spirit of the Landauer scattering theory so that we can make direct contact with the Keldysh formalism of Section 5. We consider a geometry with two leads, one before the polarizing magnet and one after the switching magnet. Slonczewski evaluated the spin current from one-electron wave functions. Writing a two-component wave function as f ¼ (fmfk)T, where fm,k are the wave functions for m and k spin projections, we can derive an expression for the spin current by considering the rate of change of the spin angular momentum d/dt(fy(1/ 2)_rf). This is determined only by the kinetic energy operator and takes the form of


297

a continuity equation, where the relevant components J|| and J? of the spin current flowing in the y direction are given by 2 0 _ Jk ¼ Im ðfn" f0# fn" f# Þ; 2m

0 _2 J? ¼ Re ðfn" f0# fn" f# Þ; 2m

(48)

where || and ? correspond to the x and y components of the spin current, referring to the system of coordinates in Fig. 1, and f0s ¼ @fs =@y: Electrons in the spacer and in the leads see spin-independent potentials Vsp and Vl, respectively, but the HF potential in the polarizing and switching ferromagnets ex have spin-dependent components V ex p and V sw given by ! cos c sin c 1 0 ex V ex ¼ ðD =2Þ ¼ ðD =2Þ (49) V p sw p sw sin c cos c 0 1 where Dp and Dsw are the exchange splittings between the majority- and minorityspin bands in the polarizing and switching magnets. The magnetization of the switching magnet is taken in the z direction and c ¼ y is the angle between the magnetizations of the two magnets. To evaluate the spin current from Eq. (48) it is necessary to solve the Schro¨dinger equation by matching wave functions and their derivatives across all the interfaces. In general, this can only be done numerically. However, the problem can be solved analytically in the situation when the polarizing and switching magnets have infinitely large exchange splittings Dp-N, DswN (half-metallic ferromagnet) and the potentials in the majority-spin bands and in the spacer and leads have the same value (perfect matching). This is the case considered first by Slonczewski in his original calculation [5]. In this case the current flowing in the lead following the polarizing magnet is completely spin polarized in the z direction so that J ? ¼ J jj ¼ 0 in this lead. Slonczewski showed that the torque T||, which is equal to the spin current J|| in the spacer, is given by _ tanðc=2Þ ðcharge currentÞ: (50) T jj ¼ 2jej It should be noted that the torque T|| goes to zero for c-p since the charge current for a half metallic magnet contains a factor 1 þ cos c: It turns out that T? is strictly zero. These results obtained by Slonczewski are only valid for this rather special model. It will be seen that, in general, T? is nonzero for ferromagnets with a finite exchange splitting and can be comparable in magnitude to T||. The interesting result that T ? ¼ 0 for this model may be traced to an effective reflection symmetry of the system about a plane at the center of the spacer. Although the system with ferromagnets having different thicknesses appears asymmetric the infinite exchange splitting makes it equivalent to a symmetric system with semi-infinite magnets. More generally, one can show [13] that the J? component of the spin current in the spacer always vanishes for a system with reflection symmetry. In general, however,

298


the J? spin current in the lead is nonzero so that T ? a0 even for a symmetric system. The result T ? ¼ 0 for the above model is, therefore, a very special one due to the artifact of a very large exchange splitting in the ferromagnets. To determine the local spin current in the more realistic case of ferromagnets with a finite exchange splitting, it is necessary not only to match numerically the wave functions across all the interfaces but also to specify the correct boundary conditions in the left and right leads. In the analogous problem of charge transport this was done by Landauer [8]. He showed in the ballistic limit that the conductance of a system sandwiched between two reservoirs is given by its total quantum mechanical transmission coefficient. Since the charge current is conserved everywhere the total transmission coefficient (conductance) gives us all the information about transport of charge. Similarly, if we are only interested in the spin current due to a single nonmagnet/magnet interface the concept of conductance remains valid and can be easily generalized to include spin-dependent scattering from the local exchange potential. One then defines the so-called mixing conductance [10]. However, since spin current is not conserved, we need to calculate it locally within the structure. In that case, the knowledge of the transmission coefficient for an isolated interface is not sufficient since we also need to take into account all the internal multiple reflections of carriers from all the interfaces. Their contribution to the local spin current is essential. To determine the local spin current without any approximations, the Landauer method needs to be applied to the whole magnetic layer structure. This was done in Ref. [9]. To explain the method, we consider, in the spirit of Landauer, a structure consisting of two magnets separated by a nonmagnetic spacer layer, the magnets being connected to left and right reservoirs by nonmagnetic leads. An infinitesimal bias Vb is applied between the left and right reservoirs so that their electron distributions are characterized by Fermi functions f ðo mL Þ and f ðo mR Þ with mL mR ¼ eV b : Since we assume that Vb is infinitesimal, the one-electron states of the system can be calculated from the Schro¨dinger equation neglecting the effect of Vb. We take the global spin quantization axis to be the z axis and thus classify the electron spin projections as m or k. Electrons of either spin orientation (m, k) are incident on the magnetic trilayer both from the left and right reservoirs. To determine the spin current from Eq. (48), we therefore need to solve four independent one-electron scattering problems. The first problem corresponds to an m-spin electron incident from the left which is partially reflected both to the m-spin and k-spin channels in the left lead and partially transmitted to the m-spin and k-spin channels in the right lead. Similarly, a k-spin electron incident from the left reservoir is reflected and transmitted to both m- and k-spin channels in the left and right leads, respectively. Finally, both m- and k-spin electrons incident from the right reservoir are similarly reflected and transmitted to both spin channels in the right and left leads, respectively. If j Li ðs; o; kk Þ is the spin current component iði ¼ jj; ?Þ due to an electron of spin r incident from the left reservoir with an energy o and parallel wave vector k||, and j R i ðs; o; kk Þ is the corresponding contribution due to an electron incident from the right reservoir, we can write the total spin current in any part of the structure by summing over the


299

contributions from all the electrons incident on the magnetic structure X Z J tot ¼ do f ðo mL Þ J Li "; o; kk þ J Li #; o; kk i kk

R þf ðo mR Þ J R ; i "; o; kk þ J i #; o; kk

ð51Þ

where J iLðRÞ ¼ j iLðRÞ ðdk? =doÞ was used to convert the sum over k? into an energy integral. Using the identities f R ¼ ð1=2Þ½f R þ f L þ ð1=2Þ½f R f L and f L ¼ ð1=2Þ½f R þ f L ð1=2Þ½f R f L ; where fL and fR denote the Fermi functions in the left and right reservoirs, we can finally write the total spin current as X Z tot do J Li s; o; kk þ J R fL þ fR Ji ¼ i s; o; kk kk ;s

þ J Li ðs; o; kk Þ J R i ðs; o; kk Þ ðf L f R Þ:

ð52Þ

It is straightforward to evaluate the spin current from Eq. (52) for the parabolic band model using, for example, the transfer matrix method to determine the electron wave functions. However, it would be difficult to implement such a calculation for a realistic band structure and, to our knowledge, no such calculation has ever been attempted. Nevertheless, the Landauer formula (52) is very useful for discussing some general properties of the spin current. In the absence of bias, only the terms proportional to the sum of the Fermi functions remain. It is clear on physical grounds that J tot k ¼ 0 both in the spacer and tot in the leads. We also have J tot ¼ 0 in the leads but J ? ? a0 in the spacer. The term J tot in the spacer determines the oscillatory exchange coupling between the two ? ferromagnets [44]. It is easy to verify that the results for the oscillatory exchange coupling obtained from Eq. (52) are in complete agreement with previous parabolic-band model calculations of this effect [49–51]. We note that the result J tot i ¼0 in the leads in the absence of bias shows explicitly that the contribution of the spin current transmitted through the magnetic structure must be exactly compensated by the contribution due to electrons reflected from the structure. Direct numerical evaluation of Eq. (52) for a parabolic band model confirms this result. Since the oscillatory exchange coupling tends to zero for a thick spacer layer, we shall now concentrate on the bias-induced term which is proportional to the difference between the Fermi functions and remains nonzero [5,6,13] even for an infinitely thick spacer (in the ballistic limit). This is indeed the term that determines the spin-transfer torques responsible for the current-induced switching of the magnetization discussed in the phenomenological section. In the linear-response case of small bias, which we are considering, the Fermi functions in Eq. (52) are expanded to first order in Vb. Hence the energy integral is avoided, being equivalent to multiplying the integrand by eVb and evaluating it at the common zero-bias chemical potential m0. Consequently, upon changing the direction of the bias (current), the spin-transfer torque vector changes its direction but not its magnitude. It is clear from the Landauer formulation described above that this important symmetry property of the spin-transfer torque holds only within a theory which includes not only the transmission but also the reflection of the spin current from all the

300


interfaces. Treatments that consider only the transmission of the spin current through the nonmagnet/magnet interface (see e.g. [45]) are thus insufficient. In this connection we now mention a recent theory of Weinberger et al. [52] in which the only coupling between the magnetic layers considered explicitly is the zero-bias interlayer exchange coupling (the term proportional to the sum of the Fermi functions in Eq. (52)). This tends to zero for sufficiently thick spacer layers and is neglected in other theories. The torque associated with flow of current, which twists the magnetization away from its equilibrium direction is not considered explicitly. However the integral over the twist angle of the equilibrium zero-bias effect is mysteriously identified with the Joule heat produced during the time required to accomplish the rotation. The torque due to a current I is therefore proportional to I2 and does not change sign when the current is reversed. This is in complete disagreement not only with the Landauer theory described above but also with all the experimental data discussed in Section 2. We shall conclude this section by giving some examples of spin-transfer torques obtained from the Landauer formula (52) for a parabolic-band model. The matching of wave functions, which is required in the calculation of the spin current, was done numerically using the standard transfer matrix method. The results we present here are for a trilayer sandwiched between two semi-infinite nonmagnetic leads. It is assumed that electrons in a nonmagnetic spacer separating two magnets of a finite thickness see the same potential as in the leads, i.e. V sp ¼ V l ¼ V 0 : As in Slonczewski’s calculation described earlier, we have determined the ratios of the spin-transfer torque components T||, T? to the particle current Ið_ ¼ e ¼ 1Þ: To demonstrate the existence of a component T? of the spin-transfer torque, we have deliberately broken the reflection symmetry of the trilayer by choosing different thicknesses of the polarizing and switching magnets. We also chose the polarizing magnet potentials so that the minority-spin band is empty (strong ferromagnet) and the exchange splitting Dp is large. On the other hand, the switching magnet potentials were chosen so that it is a weak ferromagnet (the Fermi level lies in both the majority- and minority-spin bands). The dependence of the ratios T||/I, T?/I on the thickness t of the spacer layer, measured in units of the switching magnet thickness, are shown in Fig. 11. The angle between the polarizing and switching magnet orientations is c ¼ p=2: It can be seen that in this particular example of a system with broken reflection symmetry, T? and T|| are comparable in magnitude but have opposite signs. They both oscillate about constant values as a function of the spacer thickness and remain nonzero even for an infinite spacer thickness (in the ballistic limit assumed here). The dependence of T||/I, T?/I on the angle c is shown in Fig. 12 for a fixed spacer thickness t ¼ 20a; where a is the switching magnet thickness. Since the particle current I depends also on c and we wish to illustrate the net angular dependence of the spin-transfer torques, we have used in Fig. 12 a constant value of the normalizing particle current I corresponding to c ¼ p=2: The angular dependence of T? and T|| is dominated by a sin c factor but we shall see that in fully realistic calculations deviations from the sinusoidal dependence occur. Although the Landauer formula (52) evaluated for a parabolic band model gives qualitatively correct results, the model itself is too simple to predict the correct


301

1.5

spin-transfer torque

1

T I

0.5 0 -0.5 -1 -1.5

0

10 20 spacer thickness t

30

The dependence of the ratios T||/I, T?/I on the thickness t of the spacer layer, measured in units of the switching magnet thickness a.

spin-transfer torque

Fig. 11.

T I

T I

0.5

0

-0.5

T I 0

1

2

3

angle ψ

Fig. 12.

The dependence of T||/I, T?/I on the angle c between the magnetizations of the polarizing and switching magnets.

magnitudes, relative sign and angular dependencies of the spin-transfer torques T||, T? for real systems. We shall therefore explain in Section 5 the general Keldysh formalism for calculating the spin-transfer torques and conclude this review by presenting the results of fully realistic calculations based on this formalism for the experimentally most important Co/Cu system.

5. KELDYSH FORMALISM FOR FULLY REALISTIC CALCULATIONS OF THE SPIN-TRANSFER TORQUE In this section we show how to calculate the local spin and charge currents flowing in the direction perpendicular to the layers of an arbitrary magnetic layer structure.


302

The structure we consider is shown schematically in Fig. 1. It consists of a thick (semi-infinite) left magnetic layer (polarizing magnet), a nonmagnetic metallic spacer layer of N atomic planes, a thin switching magnet of M atomic planes, and a semi-infinite lead. The broken line between the atomic planes n 1 and n represents a cleavage plane separating the system into two independent parts so that charge carriers cannot move between the two surface planes n 1 and n. It will be seen that our ability to cleave the whole system in this way is essential for the implementation of the Keldysh formalism. This can be easily done within a tight-binding parametrization of the band structure by simply switching off the matrix of hopping integrals tnn;n1m between atomic orbitals n, m localized in planes n 1 and n. We shall, therefore, adopt the tight-binding description with the Hamiltonian defined by Eqs (30)–(34) bearing in mind that the same method can be applied, for example, also to LMTO tight-binding band structure implemented for a layer system [43]. To use the Keldysh formalism [11,12,53] to calculate the charge or spin currents flowing between the planes n 1 and n, we consider an initial state at time t ¼ 1 in which the hopping integral tnn;n1m between planes n 1 and n is switched off. Then both sides of the system are in equilibrium but with different chemical potentials mL on the left and mR on the right, where mL mR ¼ eV b : The interplane hopping is then turned on adiabatically and the system evolves to a steady state. The cleavage plane, across which the hopping is initially switched off, may be taken in either the spacer or in one of the magnets or in the lead. In principle, the Keldysh method is valid for arbitrary bias Vb but here we restrict ourselves to small bias corresponding to linear response. This is always reasonable for a metallic system. For larger bias, which might occur with a semiconductor or insulating spacer, electrons would be injected into the right part of the system far above the Fermi level and many-body processes neglected here would be important. Following Keldysh [11,12], we define a two-time matrix D E y 0 0 Gþ (53) RL ðt; t Þ ¼ i cL ðt ÞcR ðtÞ ; where R ðn; n; s0 Þ and L ðn 1; m; sÞ; and we suppress the k|| label. The thermal average in Eq. (53) is calculated for the steady state of the coupled system. The matrix G þ RL has dimensions 2m 2m where m is the number of orbitals on each atomic site, and is written so that the m m upper diagonal block contains matrix elements between m spin orbitals and the m m lower diagonal block relates to k spin. 2m 2m hopping matrices tLR and tRL are written similarly and in this case only the diagonal blocks are nonzero. If we denote tLR by t, then tRL ¼ ty. We also generalize the definition of r so that its components are now direct products of the 2 2 Pauli matrices sx, sy, sz and the m m unit matrix. The thermal average of the spin current operator, given by Eq. (35), may now be expressed as

1 X þ y j n1 ¼ Tr GRL ðt; tÞt G þ LR ðt; tÞt r : 2 k k

(54)


303

Introducing the Fourier transform G+(o) of G+(t,t0 ), which is a function of t t0 ; we have Z

1X do þ y Tr G RL ðoÞt Gþ j n1 ¼ (55) LR ðoÞt r : 2 k 2p k

The charge current is given by Eq. (55) with 12s replaced by the unit matrix multiplied by e/_. Similarly, the total spin angular momentum on atomic planes on either side of the cleavage plane, in the nonequilibrium state, is given by Z 1 X do þ hS n1 i ¼ i_ Tr G LL ðoÞr (56) 2 2p k k

1 X hS n i ¼ i_ 2 k k

Z

do þ Tr G RR ðoÞr : 2p

(57)

Following Keldysh [11,12], we now write 1 a r (58) Gþ AB ðoÞ ¼ ðF AB þ G AB G AB Þ; 2 where the suffices A and B are either R or L. FAB(o) is the Fourier transform of D E F AB ðt; t0 Þ ¼ i ½cA ðtÞ; cyB ðt0 Þ (59) and Ga, Gr are the usual advanced and retarded Green functions [54]. Note that in [11] and [12] the definitions of Ga and Gr are interchanged and that in the Green function matrix defined by these authors G+ and G should be interchanged. Charge and spin current, and spin density, are related by Eqs (54)–(58) to the quantities Ga, Gr, and FAB. The latter are calculated for the coupled system by starting with decoupled left and right systems, each in equilibrium, and turning on the hopping between planes L and R as a perturbation. Hence, we express Ga, Gr and FAB in terms of retarded surface Green functions gLgLL, gRgRR for the decoupled equilibrium system. It is then straightforward to show that the spin current between planes n 1 and n can be written as the sum hj n1 i ¼ hj n1 i1 þ hj n1 i2 ; where the two contributions to the spin current /j n S1 and /j n S2 are given by Z

1 X j n1 1 ¼ (60) do ReTrfðB AÞrg f ðo mL Þ þ f ðo mR Þ 4p k k

Z

1 X 1 j n1 2 ¼ do ReTr gL tABgyR ty AB þ ðA þ BÞ r 2p k 2 k f ðo mL Þ f ðo mR Þ : y

1

gyR ty gyL t1 ;

ð61Þ

Here, A ¼ ½1 gR t gL t ; B ¼ ½1 and f(om) is the Fermi function with chemical potential m and mL mR ¼ eV b : In the linear-response case of small


304

bias which we are considering, the Fermi functions in Eq. (61) are expanded to first order in Vb. Hence the energy integral is avoided, being equivalent to multiplying the integrand by eVb and evaluating it at the common zero-bias chemical potential m0. Following the method outlined for obtaining Eqs (60) and (61), similar expressions in terms of retarded surface Green functions may be obtained for the nonequilibrium spin angular momentum on atomic plane n. Writing again hS n i ¼ hS n i1 þ hS n i2 ; we obtain Z _ X hS n i 1 ¼ do Im TrfAgR rg f ðo mL Þ þ f ðo mR Þ (62) 4p k k

h S n i2 ¼

_ X 2p k k

Z do Im Tr

1 A BgyR r f ðo mL Þ f ðo mR Þ : 2

(63)

To obtain /Sn1S defined by Eq. (56), we must interchange L and R, and t and ty, everywhere in Eqs (62) and (63). It can be seen that Eqs (60)–(63) which determine the spin and charge currents and the transport spin densities all depend on just two quantities, i.e. the surface retarded one-electron Green functions for a system cleaved between two neighboring atomic planes. The surface Green functions can be determined without any approximations by the standard adlayer method (see e.g. [42,44]) for a fully realistic band structure. We first note that there is a close correspondence between Eqs (60) and (61) and the generalized Landauer formula (52). The first term in Eq. (52) corresponds to the zero-bias spin current /jn1S1 given by Eq. (60). When the cleavage plane is taken in the spacer, the spin current /jn1S1 determines the oscillatory exchange coupling between the two magnets and it is easy to verify that the formula for the exchange coupling obtained from Eq. (60) is equivalent to the formula used in previous total energy calculations of this effect [42,44]. When the cleavage plane is taken in the switching magnet with a nonuniform magnetization, Eq. (60) yields the local internal exchange stiffness torque discussed in Section 3. Finally, the contribution to the transport spin current given by Eq. (61) clearly corresponds to the second term in the Landauer formula (52), which is proportional to the bias in the linearresponse limit. Placing the cleavage plane first between any two neighboring atomic planes in the spacer and then between any two neighboring planes in the lead, we obtain from Eq. (61) the total spin-transfer torque T st of Eq. (41). Although a rigorous proof is lacking of complete equivalence between the Landauer formula (52) and the formulas for the spin current obtained from the Keldysh formalism, we have verified by evaluating the spin current both from Eq. (52) and from the Keldysh formulas for a simple exactly solvable model that both formulations give identical results. We note, however, that the Keldysh formulation is superior not only because it can be easily implemented for a fully realistic band structure but also because it provides explicit formulas (62), (63) for transport spin densities anywhere in the structure.


305

We are now ready to discuss the application of the Keldysh formalism to real systems. Because it allows us to calculate the spin-transfer torque acting on every atomic plane of the switching magnet and, at the same time, to obtain the local transport spin /SnS, Eqs (60)–(63) can be used together with Eq. (47) to determine fully self-consistently the steady-state spin-transfer torque. This has been done by Edwards et al. [13] for a single-orbital tight-binding model to test the validity of the SM. It was found that, for realistic values of the parameters, deviations of the SM from a fully self-consistent solution are negligible. These results underpin our qualitative discussion of Section 3. It follows that one can use the SM approximation of uniform exchange fields in the polarizing and switching magnets, which make an assumed angle c, to determine the spin-transfer torque. The microscopically determined spin-transfer torque can be then used as a realistic input into the LLG equation so that the results obtained in the phenomenological Section 2 can be applied to real systems. This program was implemented by Edwards et al. [13] for the experimentally most important Co/Cu system, and we conclude this review by reproducing some of their results. Referring to Fig. 1, the system considered by Edwards et al. [13] consits of a semiinfinite slab of Co (polarizing magnet), the spacer of 20 atomic planes of Cu, the switching magnet containing M atomic planes of Co with M ¼ 1 and 2 and the lead which is semi-infinite Cu. The spacer thickness of 20 atomic planes of Cu was chosen so that the contribution of the oscillatory exchange coupling term is so small that it can be neglected. The spin currents in the right lead and in the spacer were determined from Eq. (61) using the tight-binding parametrization of an ab initio band structure of Co and Cu (see Ref. [42,44]). Figs. 13(a), (b) shows the angular dependences of T||, T? for the cases M ¼ 1 and M ¼ 2; respectively. For the monolayer switching magnet, the torques T? and T|| are equal in magnitude and they have opposite sign. However, for M ¼ 2; the torques have the same sign and T? is somewhat smaller than T||. A negative sign of the ratio of the two torque

(b)

(a)

0.003

T

0.002

0.004

T

0.002

T

torque

torque

0.001 0 −0.001 −0.002 −0.003

0

T 0

1

2 angle ψ

3

0

1

2

3

angle ψ

Fig. 13. Dependence of the spin-transfer torque T|| and T? for Co/Cu/Co(111) on the angle c. The torques per surface atom are in units of eVb. (a) is for M ¼ 1; and (b) for M ¼ 2 monolayers of Co in the switching magnet.


306 0.005 0.004 0.003 torque

in-plane out-of-plane

T

0.002 0.001 0

T

-0.001 -0.002 -0.003

1

2

3 4 5 6 7 8 switching Co thickness M

9

10

Fig. 14. Dependence of the spin-transfer torque T|| and T? for Co/Cu/Co(111) on the thickness of the switching magnet M for c ¼ p=3: The torques are in units of eVb.

components has important and unexpected consequences for hysteresis loops as already discussed in the phenomenological section. It can be seen that the angular dependence of both T? and T|| is dominated by a sin c factor but distortions from this dependence are clearly visible. In particular, the slopes of the angular dependences of the torques T? and T|| at c ¼ 0 and c ¼ p are quite different. As pointed out in the phenomenological section, this is important in the discussion of the stability of steady states and leads to quite different magnitudes of the critical biases VP-AP and VAP-P. In Fig. 14 we reproduce the dependence of T? and T|| on the thickness of the Co switching magnet. It can be seen that the out-of-plane torque T? becomes smaller than T|| for thicker switching magnets. This is the expected behavior since the polarizing magnet is semi-infinite Co, so that as the switching Co magnet becomes thicker we approach the limit of a symmetric junction for which the J? component of the spin current vanishes in the spacer. However, T? is by no means negligible (27% of T||) even for a typical experimental thickness of the switching Co layer of ten atomic planes. It is also interesting that beyond the monolayer thickness, the ratio of the two torques is positive with the exception of M ¼ 4: To clarify the behavior of the total spin-transfer torque, we show in Fig. 15 the dependence of the in-plane J|| and out-of-plane J? components of the spin current on the switching Co thickness M. The solid line in Fig. 15(a) corresponds to J|| in the spacer and the broken line denotes J|| in the lead. The corresponding dependences of J? in the spacer and lead are shown in Fig. 15(b). It can be seen that the spin current in the lead is not small and its contribution to the total spin-transfer torque is particularly important for the out-of plane component. As already explained, this is because the spin current J? in the spacer vanishes for a symmetric junction. The microscopically calculated spin-transfer torques for Co/Cu/Co(111) were used by Edwards et al. [13] as an input into the phenomenological LLG equation. The LLG equation was first solved numerically to determine all the steady states and then the stability discussion outlined in the phenomenological section was


307

(a)

(b) 0.003

0.003

J

0.001 0 -0.001 -0.002

0.002

spacer

J 1

2

lead

3

J

0.001 spin current

spin current

0.002

0 -0.001 -0.002

J

-0.003 4 5 6 7 8 9 10 11 12 number of Co planes

spacer

-0.004

0

1

2

lead

3

4 5 6 7 8 9 10 11 12 number of Co planes

Fig. 15. Dependence of the spin current J|| (a) and J? (b) for Co/Cu/Co(111) on the thickness of the switching magnet M. Solid lines denote the spin currents in the spacer and broken lines those in the lead. All spin currents are for c ¼ p=3 and are measured in units of eVb.

applied to determine the critical bias for which instabilities occur. Finally, the ballistic resistance of the structure was evaluated from the real-space Kubo formula at every point of the steady state path. Such a calculation for the realistic Co/Cu system then gives hysteresis loops of the resistance vs. bias, which can be compared with the observed hysteresis loops. The LLG equation was solved including a strong easy-plane anisotropy with hp ¼ 100: If we take H u0 ¼ 1:86 109 s1 corresponding to a uniaxial anisotropy field of about 0.01 T, this value of hp corresponds to the shape anisotropy for a magnetization of 1:6 106 A=m; similar to that of Co [14]. Also a realistic value [14] of the Gilbert damping parameter g ¼ 0:01 was used. Finally, referring to the geometry of Fig. 1, two different values of the angle y were employed in these calculations: y ¼ 2 rad and y ¼ 3 rad, the latter value being close to the value of p which is realized in most experiments. We first reproduce in Fig. 16 the hysteresis loops for the case of Co switching magnet consisting of two atomic planes. We recall that the ratio r ¼ T?/T?E0.65 deduced from Fig. 13 is positive in this case. Fig. 16(a) shows the hysteresis loop for y ¼ 2 rad and Fig. 16(b) that for y ¼ 3 rad. The hysteresis loop for y ¼ 3 rad shown in Fig. 16(b) is an illustration of the stability scenario with r40 discussed in the phenomenological section. It is rather interesting that the critical bias for switching is E0.2 mV both for y ¼ 2 and y ¼ 3 rad. When this bias is converted to the current density using the calculated ballistic resistance of the junction, it is found [13] that the critical current for switching is E107 A/cm2, which is in very good agreement with experiment [15]. The hysteresis loops for the case of the Co switching magnet consisting of a single atomic plane are reproduced in Fig. 17. The values of hp, g, Hu0, and y are the same as in the previous example. However, the ratio r 1; is now negative and the hysteresis loops in Fig. 17 illustrate the interesting behavior discussed in the phenomenological section when the system subjected to a bias higher than a critical bias moves to the ‘‘both unstable’’ region shown in Fig. 6. As in Fig. 7, the points on the


308

0

1.8 1.75 1.7

E’

A’

1.65

D

B

C -4

-2

0

2

-4

2

F

1.9

-15

G 1.85

Ωm )

(b)

1.85

Resistance (x 10

Resistance (x 10

-15

2

Ωm )

(a)

1.8 1.75 1.7 1.65

4

-2

Bias (x 10 V)

0

2

-4

Bias (x 10 V)

Fig. 16. Resistance of the Co/Cu/Co(111) junction as a function of applied bias with M ¼ 2 monolayers of Co in the switching magnet. (a) is for y ¼ 2 rad and (b) is for y ¼ 3 rad.

(a)

(b)

1.8

2

Ωm )

0

1.7

A’

D

C* B 1.65

-3

-2

E’

-15

1.75

Resistance (x 10

Resistance (x 10

-15

2

Ωm )

F *G

-1

0

1 -4

Bias (x 10 V)

2

3

1.85 1.8

*

1.75 1.7

*

1.65 1.6

-3

-2

-1

0

1

2

3

-4

Bias (x 10 V)

Fig. 17. Resistance of the Co/Cu/Co(111) junction as a function of applied current, with M ¼ 1 monolayer of Co in the switching magnet. (a) is for y ¼ 2 rad and (b) is for y ¼ 3 rad.

hysteresis loops shown in Fig. 17 corresponding to the critical bias are labeled by asterisks. We conclude that the current-induced switching of magnetization is one of the most topical and rapidly developing areas of spintronics. In particular, there is rich new physics associated with the observed nonhysteretic (time-dependent) motion of the switching magnet moment. The steady-state (hysteretic) regime can now be modelled by essentially first-principle calculations of the type described here. We also argued in Section 3 that the spin-transfer torque calculated in the ‘‘constrained steady state’’, i.e. within the SM, should be a good approximation for time-dependent solutions of the LLG equation. It follows that the torques obtained by the Keldysh or Landauer methods can also be used to discuss the nonhysteretic regime. An interesting outstanding problem is the effect of impurities/interfacial roughness on current-induced switching. All the theoretical results discussed in Sections 4 and 5 are valid in the ballistic limit. In principle, imperfections could be included in


309

the Keldysh formalism using the lateral supercell method but, to our knowledge, it has not been done. We are grateful to our colleagues A. Umerski and F. Federici for their contributions to our work on this topic and to the Engineering and Physical Sciences Research Council, UK for financial support.

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312

AUTHOR INDEX aan de Stegge, J. 61, 67 Abarra, E.N. 122 Abe, E. 29, 37 Abe, S. 288 Abraha, K. 95 Abraham, D.W. 145 Adachi, H. 202 Adachi, T. 261 Adeyeye, A.O. 15–16, 288 Agrait, N. 40, 42 Ahn, S.J. 205 A˚kerman, J. 145 Akiba, N. 230, 248 Akinaga, H. 259 Akiyama, T. 39 Albani, L. 15 Albert, F.J. 277, 279, 284, 287–288, 307 Albrecht, J.D. 259 Albuquerque, G. 287–288 Al-Haj Darwish, M. 186 Allenspach, R. 61 Almeida, S. 64 Alonso, J.A. 33 Altarelli, M. 33 Altman, R.A. 205 Alvarado, S.F. 230 Ambrose, T.F. 246 Amemiya, K. 34, 36 Amemiya, T. 246 Anane, A. 201–202, 204 Anders, S. 132 Ando, K. 145, 199–200, 204–205, 207, 214, 250, 261 Ando, Y. 199, 205 Andre, A. 22 Anguita, J.V. 204, 207 Anil Kumar, P.S. 10, 23 Anisimov, A.N. 35 Ansermet, J.Ph. 163 Antel, W.J. 130 Anthony, T.C. 202

Apsel, S.E. 33 Aremiya, K. 18 Arenholz, E. 131 Arias, R. 21 Arrott, A.S. 68, 70 Arvanitis, D. 34 Asamitsu, A. 229 Asano, Y. 172 Astalos, R.J. 104, 107 Awschalom, D.D. 29, 229–230, 233, 237–238, 248–250, 257, 265 Azevedo, A. 100 Baberschke, K. 8, 12, 18, 34–35, 64 Back, C.H. 22 Bader, S.D. 16–17, 29, 35, 61, 86–88, 93, 105–110, 128 Bagrets, A. 215 Bagrets, D. 215 Baibich, M.N. 29, 53, 77, 142–143, 153, 157–158, 228 Bailleul, M. 21–22 Ballentine, C.A. 35 Baltz, V. 139–140 Bander, M. 12, 24 Bansmann, J. 33 Baquero, R. 31 Barbara, B. 17 Bardasis, A. 56 Barnas, J. 77, 98, 159, 167–169, 171, 174, 177, 200 Barnes, C.H.W. 253 Barra, A.L. 36 Barradas, N.P. 198–199, 202 Bartenlian, B. 15–16, 21–22 Barthe´le´my, A. 73, 153, 168, 170, 186, 188, 190, 193, 201–204, 206–210, 215, 229 Bass, J. 160–162, 180–187, 189, 284, 286–287 Bastard, G. 241 Bateson, R.D. 95

313

314 Battle, P.D. 121 Baudelet, F. 34 Bauer, E. 16 Bauer, G.E.W. 24, 166, 177, 186, 275, 292, 294, 298 Bauer, M. 16 Bauer, Ph. 88–89, 91–92 Baumgart, P. 168–169 Baxter, D.V. 184 Beach, R.S. 144 Bean, C.P. 115–117 Beauvillain, P. 159 Beckmann, H. 32 Beech, R.S. 115, 145 Belanger, R.M. 95 Belliard, L. 192 Bellouard, C. 204, 206–207 Ben Youssef, J. 204, 278–279, 281–282, 284 Bennemann, K. 33 Bennett, B.R. 243–245, 253 Benoit, A. 17 Benson, H. 6, 18 Berger, L. 24, 65, 274 Bergman, A. 33 Bergmann, G. 32 Berkov, D.V. 22 Berkovits, V.L. 261 Berkowitz, A.E. 115–116, 121–126, 128–129, 131, 133, 137, 139, 144, 153, 163, 216 Berry, J.J. 246 Bertotti, G. 14 Bertram, H.N. 144 Beschoten, B. 125, 138, 250 Besse, M. 202 Bettini, J. 40–41 Beyeres, R.B. 145 Bhadra, R. 158–159, 216 Bibes, M. 201–202, 204, 208–209, 215 Biercuk, M.J. 42 Billas, I.M.L. 33 Binasch, G. 53, 77, 153, 157–158, 228 Binder, J. 63–65, 67, 172, 174–175 Binns, C. 33 Birge, N.O. 284, 287 Bischof, A. 61 Bischoff, M.M.J. 37 Blaas, C. 173 Blackman, J.A. 31 Blamire, M.G. 201

Author Index Blanco, J.R. 31 Bland, J.A.C. 1, 14–16, 18, 21–22, 36–37, 40, 95, 230 Bleloch, A. 15 Blu¨gel, S. 31 Blomquist, P. 64 Blondel, A. 163, 187 Bloomfield, L.A. 33 Blundell, S.J. 95 Bobo, J.F. 34 Bode, M. 14, 23–24, 34, 37, 39 Bode, P.J. 37 Boeve, H. 205 Bohr, J. 73 Bonet, E. 284, 287 Borchers, J. 53 Borchers, J.A. 128–129, 161, 186 Borghs, G. 205 Borghs, S. 253, 255–256 Bouen, M. 215 Boulenc, P. 24 Bounnak, S. 13, 16 Bouzehouane, K. 202, 204, 209 Bowen, M. 201–202, 204, 207–208 Bozec, D. 186 Brandbyge, M. 40 Brataas, A. 275, 292, 294, 298 Bratkovsky, A.M. 209 Bu¨rgler, D.E. 67 Brinkman, W.F. 258 Briones, F. 204, 207 Brodsky, M.B. 29, 53, 273 Brookes, N.B. 57, 73 Broto, J. 142–143 Broto, J.M. 29, 53, 77, 153, 157–158, 228 Brouwer, P.W. 275, 298 Brown, P. 145 Brown, S.L. 145 Brown, W.F. 287 Bruno, P. 40, 61, 64, 73, 206 Bruyere, J.C. 52 Bucchigano, J. 145 Bucher, J.P. 15, 32 Buckley, M.E. 18 Buda, L.D. 16, 111 Buess, M. 22 Buettner, O. 16 Buhrman, R.A. 230, 277, 279, 284, 286–288, 307

Author Index Bulka, B.R. 171 Bullock, D.W. 230 Butcher, B. 145 Butler, W.H. 104, 137–138, 171–174, 203, 209, 212–215, 253, 255 Byers, J.M. 230, 235 Cable, J.W. 53, 81 Cabria, I. 31 Cai, J.W. 139 Cambril, E. 16, 21–22 Camley, R. 106–108 Camley, R.E. 77, 81–89, 91–95, 104, 107, 109–111, 159, 167–169, 174 Campbell, I.A. 154–156, 191, 273 Campion, R.P. 246 Caneschi, A. 36 Canet, F. 204 Cao, W. 122–123, 129, 131, 133, 137 Capelle, K. 289 Carbone, C. 36 Carey, M.J. 122, 131, 144, 163, 284, 288 Carlotti, G. 15, 20–22 Caroli, C. 276, 296, 302–303 Carr, D.M. 241 Carra, P. 33 Carrey, J. 202 Carric- o, A.S. 83–87 Carre´te´ro, C. 202, 204 Castano, F.J. 14 Castro, M. 33 Cebollada, A. 204, 207 Celinski, Z. 70, 130–131 Celotta, R.J. 23, 61, 63–64, 66–67, 69, 73 Chaban, E. 33 Chado, I. 32 Chai, C.L. 126 Chaiken, A. 122 Chakarian, V. 34 Chambers, S.A. 250, 252 Chang, C.M. 33 Chang, Y. 57, 73 Chang, Y.-C. 61 Chantrell, R.W. 111 Chapman, J.N. 23 Chappert, C. 15–16, 21–22, 61, 159 Charap, S.H. 120, 137 Chatelain, A. 33

315 Chaussy, J. 193 Chazelas, A. 153, 157–158 Chazelas, J. 53, 77, 142–143, 228 Cheetham, A.K. 121 Chein, C.L. 73 Chen, C.T. 33–34 Chen, L.J. 31 Chen, P.J. 144, 216 Chen, W. 34, 286 Chen, Y. 16 Cheng, S.F. 13, 16, 161 Cheong, H.D. 243–244 Cherifi, K. 88–89, 91–92 Cherifi, S.H.S. 16 Chesman, C. 100 Cheung, C.L. 34 Chiang, W.C. 186, 286 Chiba, D. 29, 37, 246 Chien, C.L. 139, 163 Choi, B.C. 36, 37 Choi, H.J. 57, 73 Choi, Y. 93 Chou, M.Y. 33 Chow, H. 83 Chuang, D.S. 35 Chun, S.H. 246 Cibert, J. 246 Cochran, J.F. 68, 70, 95 Coehoorn, R. 61, 133, 137, 142, 205 Coey, J.M.D. 201 Coffrey, K.R. 163 Collins, M.F. 95 Combescot, R. 276, 296, 302–303 Conover, M.J. 61 Contour, J.-P. 201–204, 206–208, 210, 215 Cornet, A. 204, 207 Costa, A.T. 10, 37, 42 Costa-Kra¨mer, J.L. 204, 207 Cowburn, R.P. 15–16, 288 Cox, A.J. 33 Creuzet, G. 29, 53, 142–143, 153, 157–158, 228 Creuzet, P. 77 Crommie, M.F. 34 Crooker, S.A. 240 Cros, V. 24, 202, 204, 207, 278–279, 281–282, 284 Crowell, P.A. 241 Cui, Z. 15–16 Cullen, J.R. 65, 299

316 Cunningham, J.E. 53 Curson, N.J. 22 Cutler, P.H. 196 da Costa, V. 145, 216 da Silva, M.F. 198 Dahlberg, E.D. 124 d’Albuquerque e Castro, J. 64, 289, 291, 299, 304–305 Dallmeyer, A. 36 Dalta, R. 195 D’Amico, I. 237 Dan Dahlberg, E. 128 Darrouzet, J. 16 Dartyge, E. 34 Das, B. 229, 264 Das Sarma, S. 65, 227, 229 Datta, S. 40, 229, 264 Daughton, J.M. 145, 199, 229 Dauguet, P. 193 Dave, R.W. 145 Davies, A. 67 de Aguiar, F.M. 77, 100, 111 De Boeck, J. 205, 246 De Crescenzi, M. 15 de Groot, C. 202 de Groot, R.A. 255 de Heer, W.A. 33 de Jonge, W.J.M. 187, 191, 204–206, 208–209, 215, 259 de Teresa, J.M. 202–204, 206–207, 210 Decanini, D. 15–16, 21–22 Dederichs, P.H. 31, 33, 156, 174–175, 188–189, 203, 211–212, 214–215, 253, 262 DeHerrera, M. 145 Dekker, D.M.T. 41 Delin, A. 31 Demangeat, C. 37 Demidov, V.E. 22 Demokritov, S.O. 16, 21–22, 66–68, 70, 73 Dender, D. 129 Deng, J. 33 Despres, J.F. 162–163 Dey, S. 127 Di Fabrizio, E. 15 Di Salvo, F.J. 81, 83 Dieny, B. 139–140, 142, 159, 161, 168–169, 171, 193, 200, 215

Author Index Dietl, T. 29, 37, 246, 252 Dimitrov, D.V. 139 Dimopoulos, T. 145, 216 Ding, H.F. 206 Ding, Z. 230 Djayaprawira, D.D. 204 Djukic, D. 41 Donahue, M.J. 109 Dong, Z. 202 Donzelli, O. 15 Dorantes-Da´vila, J. 33, 37 Dorensen, M.R. 40 Dorleijn, J.W.F. 154 Doudin, B. 163, 187, 215 Dravid, V.P. 201 Drchal, V. 177 Dresselhaus, J. 37 Dreysse´, H. 19, 37, 67 Drouet, M. 201 Drovosekov, A.B. 100, 102 Dsofsky, M.S. 195 Du, J. 206 Dubois, S. 183, 192 Duffy, D.M. 31 Dufour, C. 88–89, 91–92 Dunlap, B.I. 33 Dunn, J.H. 34 Dupas, C. 159 Dura, J.A. 129–130, 161 Durand, O. 202, 204 Durlam, M. 145 Dutta, B. 253, 255–256 Duvail, J.L. 168, 181, 183 D’yakonov, M.I. 230, 232–233, 261 Dynes, R.C. 258 Dzhioev, R.I. 229, 231, 233, 257 Eames, P. 124 Ebels, U. 111 Eberhardt, W. 36 Ebert, H. 31, 37 Edelstein, V.M. 230 Edmonds, K.W. 246 Edwards, D.M. 61, 64, 273, 276, 280–282, 284, 286, 288–289, 291, 293, 296–297, 299, 304–307 Egelhoff, W.F. 144, 195, 216, 230 Eid, K. 186

Author Index Eigler, D.M. 29, 42 Elliott, R.J. 296 Emery, C. 230 Emley, N.C. 279, 284, 288 Emmert, J.W. 33 Engel, B.N. 145 Epstein, R. 29, 248–249 Erickson, R.A. 122, 128 Erickson, R.P. 12, 24, 299 Eriksson, O. 31 Erskine, J.L. 17 Erwin, R.W. 53, 73, 129 Erwin, S.C. 229, 246, 252, 254–255 Esaki, L. 246 Eschrig, H. 172 Escorcia-Aparicio, E.J. 35, 57, 73 Etienne, P. 29, 53, 77, 142–143, 153, 157–158, 228 Etzkorn, M. 10, 23, 124, 130 Fabian, J. 65, 227, 229 Faini, G. 24, 204, 278–279, 281–282, 284 Falicov, L.M. 168, 174 Falk, D.S. 56 Farle, M. 35 Farley, N.R.S. 246 Farrow, R.F.C. 65, 127, 250, 252 Faure-Vincent, J. 204, 206–207 Fayfield, R.T. 229 Federici, F. 276, 280–282, 284, 286, 288–289, 293, 296–297, 299, 305–307 Feher, G. 250 Feiner, L.F. 129 Feiven, S.A. 95 Felcher, G.P. 93, 128–129 Ferain, E. 162–163 Fermon, C. 21–22, 201 Ferrand, D. 246, 255 Ferrell, R.A. 56 Ferrer, S. 19 Fert, A. 24, 29, 53, 73, 77, 142–143, 153–159, 162–163, 168, 170–171, 177–181, 183–193, 201–210, 215–216, 228, 254–255, 273, 278–279, 281–282, 284, 292, 294, 296 Fiederling, R. 241 Filip, A.T. 253, 255–256 Fink, A. 115, 145, 161 Fink, J. 199

317 Fishman, R.S. 66, 73 Fitzsimmons, M.R. 128–130 Flatte´, M.E. 227, 229, 233–235, 237, 255, 264 Flynn, C.P. 53 Folkerts, W. 67 Fompeyrine, J. 129, 132 Fontaine, A. 34 Fontana, R.E. 24 Fontcuberta, J. 209 Foxon, C.T. 246 Frahm, R. 33 Frank, A. 21 Freeman, A.J. 30–31, 33, 35, 37, 39 Freitas, P.P. 198–199, 202 Freyss, M. 67 Friederich, A. 29, 53, 77, 142–143, 228 Friederich J. 153, 157–158 Fritsche, H. 129–130 Fruchart, O. 15 Fujii, A. 41 Fujimori, H. 88 Fujiwara, H. 138, 255 Fukamichi, K. 16 Fukushima, A. 145, 204–205, 207, 214, 261 Fukuyama, H. 40 Fulcomer, E. 120, 137 Fulde, P. 54, 193 Fulghum, D.B. 111 Fullenbaum, M.S. 56 Fullerton, E.E. 61, 86–88, 105–108, 132 Fulthorpe, B.D. 130 Funada, S. 127 Furdyna, J.K. 128, 242 Furis, M. 241, 244–245 Fusil, S. 202, 209 Fuss, A. 159, 167–168, 174 Gadzuk, J.W. 196 Ga¨fvert, U. 198 Gaines, J.M. 129 Gajek, M. 209 Gall, H.Le. 109 Gallagher, B.L. 246 Gallagher, W.J. 145, 205 Gambardella, P. 36

318 Gandit, P. 193 Gansewinkel, R.M.J. 162 Garanin, D.A. 37 Garcı´ a, N. 40 Garcia, A.E. 31 Garcia, V. 208, 215 Gardelis, S. 253 Gautier, F. 154, 192 Ge, Q. 19 Geballe, T.H. 196, 209 Gehring, G.A. 121 Geick, R. 95 George, J.M. 162–163, 183, 204, 278–279, 281–282, 284 Ge´rardin, O. 109 Gerardino, A. 15 Gere, E.A. 250 Ghosh, K. 202 Giacomoni, L. 200 Giaever, I. 206 Gibbs, D. 73 Gibson, G.A. 209, 246 Gider, S. 127 Gierlings, M. 125, 129–130 Giesbers, J.B. 161 Gijs, M.A.M. 24, 161–162, 187, 191 Gill, H. 144 Gillies, M.F. 205 Gillingham, D.M. 21, 40 Giorgetti, C. 34 Girgis, E. 205 Givord, D. 15 Glazman, L. 42, 234 Gomez, R.D. 15 Gondo, Y. 198 Gong, G.Q. 201 Gonzalez, J.C. 40 Gonzalez-Roble, V. 31 Gorman, G. 126 Gorn, N.L. 22 Gossard, A.C. 230, 261, 265 Grabs, P. 241 Gradmann, U. 17, 32 Grandit, P. 193 Gredig, T. 124 Greene, R.L. 202 Gregg, J.F. 230 Grimsditch, M. 16, 86–87, 105–110

Author Index Gru¨nberg, P. 29, 53, 66–68, 70, 73, 77, 98, 157–159, 167–168, 174, 228, 273 Grollier, J. 24, 204, 278–279, 281–282, 284 Grundler, D. 253, 262 Grutter, P. 23 Gruyters, M. 129–130 Grynkewich, G. 145 Gu, E. 36 Gubbiotti, G. 15, 20–22 Guëll, F. 204, 207 Guinea, F. 206 Guirado-Lo´pez, R.A. 33 Gu¨ntherodt, G. 20–21, 125, 138 Gu¨ntherodt, H.-J. 67 Gupta, A. 201 Gupta, J.A. 29, 42, 248–249 Gurney, B.A. 142, 144, 159, 161, 168–169, 193 Gu¨rtler, C.M. 22 Guslienko, K.Yu. 16, 21–22, 108–110 Guyen Van Dau, F.N. 29 Gyorffy, B.L. 289, 300 Ha, J.K. 208–209 Haas, C. 246 Hadjipanayis, G.C. 139 Hagele, D. 229–230, 250 Hahn, W. 93 Halkias, C.C. 249 Hall, K.C. 264 Hammar, P.R. 253 Hammel, P.C. 20 Hammer, L. 19 Hamzic, A. 24, 278–279, 281–282, 284 Hamzio, A. 204 Han, X.F. 199, 206 Hanbicki, A.T. 232, 241, 244–246, 250, 252, 256–258, 260–261 Hansen, J.B. 253 Hansen, M.H. 128 Hao, X. 209, 246 Harp, G.R. 130 Hartmann, C. 16 Hartmann, U. 16 Hase, T.P.A. 130 Hashimoto, Y. 246 Haskel, D. 93 Hassdorf, R. 15, 37

Author Index Hasselbach, K. 17 Hathaway, K.B. 65, 299 Haug, T. 22 Hayashi, T. 229, 246 Haynes, J.R. 238 Hayward, T. 14 Hedin, L. 172 Hehn, M. 15, 145, 204, 206–207, 210 Heide, C. 214–215, 296 Heinrich, A.J. 42 Heinrich, B. 20, 67–68, 70, 95 Heinz, K. 19 Hellberg, C.S. 246 Hellman, F. 122, 124–125 Hemstead, R.D. 115, 120, 122, 126–127 Henk, J. 206 Henry, Y. 145 Hergert, W. 31 Hernando, A. 124–125 Heyderman, L.J. 16 Hicken, R.J. 20 Hickey, B.J. 170, 186 Higo, Y. 202 Hihara, T. 119 Hillebrands, B. 16, 20–22 Hillebrecht, F.U. 130, 132 Himpsel, F.J. 32, 57, 73 Hiramoto, M. 202 Hirasawa, M. 29 Hirohata, A. 288 Hirsch, J.E. 230 Hjortstam, O. 31, 34 Ho, G.H. 33 Ho, W. 34, 36 Hoellinger, R. 111 Hofer, R. 67 Hoffmann, A. 128–130 Hoffmann, B. 16 Hohenberg, P. 30 Hollinger, R. 22 Holody, P. 163, 181–183, 186, 193 Hong, J.S. 35, 37 Hong, M. 53, 73, 81, 83 Hood, R.Q. 168, 174 Hope, S. 36 Hopster, H. 8, 18 Hosoito, N. 192 Hoving, B.H. 253, 255–256 Hoving, W. 61, 67

319 Howard, J.K. 126, 163 Howson, M.A. 186 Hsu, S.Y. 161, 170, 183, 185–186, 188, 190 Hu, C.-M. 253 Hu, G. 202 Huang, J.L. 34 Huang, Y.Y. 93 Hubert, A. 23, 68, 70 Hughes, B. 145, 196–197, 204, 207, 261 Hughes, N.D. 20 Huhne, T. 31 Huisman, P.E. 162 Humbert, P. 168–169 Hutchings, M.T. 122 Hu¨tten, A. 163, 202 Hwang, C. 126 Hylton, T.L. 163 Ibach, H. 10, 23 Ibarra, M.R. 204, 207 Ichikawa, T. 89–90 Idzerda, Y.U. 33–34 Igel, T. 67 Ijiri, Y. 129 Ilegems, M. 256 Imagawa, T. 124 Imakita, K.-I. 127 Imry, Y. 40 Inomata, K. 288 Inoue, J. 172 Ionescu, A. 21 Isakovic, A.F. 241 Ishakawa, Y. 126 Ito, T. 39 Itoh, H. 172 Itskos, G. 232, 241, 244–245, 252, 256–257, 260–261 Ivanov, P.G. 129, 196, 202 Iye, Y. 29, 246 Jaccarino, V. 95 Jacobs, I.S. 116 Jacobsen, K.W. 40 Jacquet, E. 202, 208, 215 Jaffres, H. 204, 255, 278–279, 281–282, 284 Jager, K. 67 Jamet, M. 36 Jamneala, T. 34

320 Jamorski, C. 33 Janak, J.F. 31, 54 Janesky, J. 145 Jansen, A.G.M. 286 Jansen, R. 206–208, 216 Jedema, F.J. 253, 255–256 Jenkins, S.J. 19 Jensen, A. 253 Jensen, M.R.F. 95 Jiang, J.S. 93, 163 Jiang, S. 106–108 Jiang, X. 142, 144–145 Jiang, Y. 288 Jin, I. 202 Jo, M.-H. 201 Johansson, B. 31 Johnson, A.D. 95 Johnson, L.E. 122 Johnson, M. 202, 253 Johnson, M.T. 35, 61, 67, 133, 137, 162 Johnson, P.D. 57, 73 Johnston-Halperin, E. 29, 248–249 Jones, B.A. 73 Jonker, B.T. 227, 229–230, 232, 241–246, 250, 252, 254–257, 260–261 Jordan, D.W. 281 Jorzick, J. 21–22 Jouguelet, E. 204, 206–207 Ju, G. 16 Judy, J.H. 144, 153, 216 Jullie`re, M. 154, 193, 197, 209 Jungblut, R. 133, 137 Jungblut, R.M. 187, 191 Jungwirth, T. 29 Kabos, P. 100 Kachkachi, H. 37 Kadar, G. 127 Kai, T. 138 Kaiser, C. 142, 144–145, 196–197, 204, 207, 261 Kaka, S. 284, 288 Kalevich, V.K. 265 Kamiguchi, Y. 88 Kammerer, S. 202 Kao, C.-C. 131 Ka¨mper, K.-P. 8, 18

Author Index Kaper, H.G. 109–110 Karimov, O.Z. 264 Karpeev, D.A. 109–110 Kashuba, A.B. 16 Katayama, T. 199–200 Katine, J. 284, 288 Katine, J.A. 277, 279, 284, 286, 288, 307 Kato, T. 205 Kato, Y. 29, 248–249, 265 Kato, Y.K. 230 Katsumoto, S. 29, 246 Kawagoe, T. 23 Kawai, T. 250 Kawakami, R.K. 35, 37, 57, 73 Kay, E. 94 Keavney, D.J. 34 Keffer, F. 83 Keldysh, L.V. 276, 296, 302–303 Keller, J. 125, 138 Kelley, M.H. 23, 161 Kelly, P.J. 166, 177, 186 Kent, A.D. 15, 286 Kern, K. 36 Khaetskii, A.V. 233–234 Khan, D.C. 122 Khanna, S.N. 33 Kholin, D.I. 100, 102 Kienle, P. 33 Kikkawa, J.M. 229–230, 233, 237–238, 250, 257 Killinger, A. 111 Kim, K.S. 31 Kimel, A.V. 16 Kimura, M. 243 Kinder, L.R. 145, 154, 193, 198–199, 205, 259, 273 King, D.A. 19 King, F. 16 Kioseoglou, G. 232, 241, 243–245, 250, 252, 256–257, 260–261 Kioussis, N. 31 Kirilyuk, A. 16 Kirk, W.P. 262 Kirschner, J. 10, 19, 23–24, 36–37, 39, 67, 70, 206 Kiselev, S.I. 284, 288 Kitagawa, S. 34, 36 Kittel, C. 9, 18, 52 Kiwi, M. 116

Author Index Klaüi, M. 15–16 Klaua, M. 67 Kleiber, M. 34 Kleinman, D.A. 261 Kling, A. 198 Knickelbein, M.B. 37 Knowles, T.P.J. 22 Kobayashi, S. 246 Koch, R.H. 286–287 Kodama, R.H. 115, 122–123, 128–129, 131, 133, 137, 139 Kohda, M. 249 Kohlhepp, J.T. 204, 206, 208–209, 215 Kohn, W. 30, 54 Kohno, H. 24 Koike, K. 23 Kolmogorov, A.N. 100, 102 Koltsov, D.K. 15–16, 288 Komineas, S. 15 Kondoh, H. 122 Konno, T.J. 119 Koo, H. 15 Kools, J.C.S. 142, 144 Koon, N.C. 104, 129, 137 Korenev, V.L. 265 Kortright, J.B. 128 Koshihara, S. 29 Kostial, H. 253 Kostylev, M.P. 22 Kosubek, E. 35, 64 Kovalev, A.A. 292, 294 Kowalewski, M. 70 Kramers, H.A. 263 Krause, M. 19 Kravtsov, E.A. 168 Krebs, J.J. 161 Kreines, N.M. 100, 102 Krey, U. 22, 111 Krivorotov, I.N. 124, 128, 284, 288 Kra¨mer, C. 21 Kreń, E. 126–127 Krongelb, S. 115, 120, 122, 126–127 Kroutvar, M. 233 Ku, K.C. 246 Kubetzka, A. 23, 39 Kubota, H. 199, 205 Kudrnovsky, J. 177 Kumar, S. 86–87, 105 Kurokawa, S. 41

321 Kurosawa, K. 122, 139 Kurt, H. 186 Kwo, J. 53, 73, 81, 83 Labella, V.P. 230 Lacroix, C. 200 Lampel, G. 234 Landauer, R. 275, 298 Landis, S. 139–140 Landry, G. 206 Lang, J.C. 93 Lang, P. 33 Lau, W.H. 233, 264 Lauter, H.J. 95 Lazarovits, B. 37 Le Gall, H. 204, 278–279, 281–282, 284 Leaf, G.K. 109–110 Lebib, A. 16 Leclair, P. 198, 204, 206, 208–209, 215 LeClair, P.R. 154, 209, 215 Lecoeur, P. 201 Lederman, D. 128 Lee, B. 29, 61 Lee, C.P. 264 Lee, D.R. 93 Lee, H.J. 34, 36 Lee, S.F. 160, 180–183, 189, 193 Lee, S.-H. 129 Lefakis, H. 168–169 Legras, R. 162–163 Leighton, C. 128, 130 Lemaire, R. 139 Lemaitre, Y. 201–202 Lenczowski, S.K.J. 161–162 Lepage, J.G. 83, 93 Leroy, C. 162–163 Levy, P.M. 170–171, 173, 181, 192, 206, 209–210, 214–215, 292, 294, 296 Li, C.H. 262 Li, E. 202 Li, X.W. 201, 205 Li, X.Z. 215 Li, Z. 139, 202, 287–288 Liberati, M. 131 Lieber, C.M. 34 Lin, H.-J. 33

322 Lin, T. 126 Lind, D.M. 129, 202 Lindner, J. 35, 64 Linfield, E.H. 253 Lininghton, I. 40 Liu, C. 17 Liu, F. 127 Liu, K. 130 Liu, S.S. 264 Locatelli, A. 16 Locquet, J.-P. 129, 132 Lodder, J.C. 216 Loegel, B. 154, 192 Loewenhaupt, M. 93 Loloee, R. 158–161, 163, 170, 180–183, 185–186, 188, 190, 193, 216 Lopez-Diaz, L. 15–16 Loss, D. 234 Lottis, D.K. 168 Louderback, J.G. 33 Louie, R.N. 286 Lu, B. 16 Lu¨, C. 206 Lu, Y. 145, 201, 205 Lucena, M.A. 100 Lundqvist, B. 172 Lu¨ning, J. 132–133 Lupke, G. 250, 252 Luthi, B. 95 Lutz, C.P. 29, 42 Lyanda-Geller, Y. 233 Lyonnet, R. 202–203, 206–207, 210 Maat, S. 131 MacDonald, A.H. 29 MacLaren, J.M. 172–174, 203, 212–215, 253 Madhavan, V. 34 Maehara, H. 204 Maekawa, S. 172, 198 Mahan, G.D. 303 Mailly, D. 17 Maiti, K. 36 Majkrzak, C.F. 53, 73, 81, 83, 129–130, 161 Malagoli, M.C. 36 Malezemoff, A.P. 133, 136 Malinowski, A. 233

Author Index Malkinski, L. 130–131 Mamin, H.-J. 23 Manago, T. 259 Mangin, Ph. 88 Mankey, G.J. 32 Marchal, G. 88–89, 91–92 Marcus, C.M. 42 Marks, R.F. 127 Marley, A. 205–206 Marley, A.C. 127 Marrows, C.H. 170 Martin, J. 42 Martin, R.W. 240 Martı´ nez Boubeta, C. 204, 207 Mason, N. 42 Massenet, O. 52, 119 Masumoto, T. 243 Materlik, G. 33 Mathieu, C. 16, 21 Mathon, G. 145, 196, 209 Mathon, J. 61, 64, 172, 203, 212, 215, 273, 276, 280–282, 284, 286, 288–289, 291, 293, 296–297, 299, 304–307 Mathur, N.D. 201 Matsuda, H. 202 Matsukawa, N. 202 Matsukura, F. 29, 37, 230, 246, 248–249, 261 Matsumoto, Y. 250, 252 Matsumura, D. 18, 34, 36 Matthes, F. 130–131 Mattson, J.E. 61, 86–88, 105, 246, 250 Mauri, D. 94, 126, 142, 159, 161, 193 Maurice, J.-L. 202 Mavropoulos, Ph. 203, 211–212, 215, 253, 262 May, F. 34 Mazin, I.I. 195 McGee, N.W.E. 61, 67 McMichael, R.D. 137–138, 144, 153, 216 McPhail, S. 22 McWhan, D.B. 53, 81 Medvedkin, G.A. 250 Meigs, G. 33 Meijer, F. 229 Meiklejohn, W.H. 115, 117–118, 120 Meisinger, F. 67

Author Index Mekata, M. 127 Mermin, N.D. 11–12, 36 Mertig, I. 63–65, 67, 156, 172, 174–175, 188–189 Mescheriakov, V.F. 100, 102 Meservey, R. 145, 154, 193–196, 198–199, 205, 209, 246, 257, 259, 273 Metin, S. 144, 168–169 Metlushko, V. 16 Meyer, W. 19 Meyerheim, H.L. 19 Meyers, E.B. 284, 288 Mibu, K. 24, 192 Michel, R.P. 122 Midzor, M. 20 Miedema, A.R. 154 Miliayev, M.A. 100, 102 Miller, M.M. 13, 16 Miller, R.C. 261 Millman, J. 249 Mills, D.L. 1, 4, 6, 10, 12, 18, 21, 24, 37, 42, 56, 64, 83, 86–88, 105 Miloslavsky, L. 127 Miltat, J. 287–288 Milteńy, P. 125, 138 Minowa, T. 41 Mirwald-Schulz, B. 35 Mitchell, J.R. 163 Mitsumori, Y. 29 Mitsuoka, K. 124 Miura, M. 122, 139 Miyake, K. 24 Miyazaki, T. 199, 205 Molenkamp, L.W. 241, 255, 259 Monchesky, T.L. 16, 67 Monsma, D.J. 196, 259, 286 Montaigne, F. 202–207, 210 Montmory, R. 52, 119 Montone, A. 124–125 Monzon, F.G. 253 Moodera, J.S. 145, 154, 193, 196, 198–199, 202, 205–209, 246, 255, 259, 273 Moog, E.R. 17 Moore, G. 227 Moore, J.R. 20 Moore, T.A. 14 Moran, T.J. 128 Moraru, I. 184

323 More, N. 29, 53, 68, 158 Morel, R. 163, 193 Morelloń, L. 204, 207 Morgenstern, M. 37 Moriya, R. 29 Morpurgo, A.F. 229 Moruzzi, V.L. 54 Mosca, D.H. 158–159, 216 Mosler, R. 68, 70 Motokawa, M. 88, 95 Motsnyi, V.F. 259, 261 Mott, N.F. 154 Mpondo, F.E. 170, 186, 188, 190 Mryasov, O.N. 154, 209, 215 Mueller, S. 19 Mu¨ller, A. 16 Munekata, H. 29, 246 Muniz, R.B. 10, 37, 42, 61, 64, 289, 291, 299, 304–305 Munõz, M. 40 Murielle Villeret, M. 289, 304–305 Myers, E.B. 275, 279, 284, 286–287, 298 Myers, R.C. 230, 265 Myrtle, K. 67, 70 Nadgorny, B. 195 Nagaev, E.L. 246 Nagahama, T. 145, 200, 204–205, 207, 214, 261 Nagai, M. 204 Nagy, E. 126 Nagy, I. 126 Nakajima, R. 33 Nakamura, K. 39 Narishige, S. 124 Narita, Y. 229 Nassar, J. 201, 205 Nasu, S. 24 Natali, M. 16 Nautiyal, T. 31 Nawate, M. 89–90 Nazarov, Yu.V. 233, 275, 298 Nazmul, A.M. 37, 246 Neél, L. 52, 119–120 Neumann, A.C. 246

324 Nguyen Van Dau, F. 53, 142–143, 153, 157–158, 205, 228 Nicholson, D.M.C. 173–174 Nickel, J.H. 202, 204 Nilius, N. 36 Nishikawa, Y. 241 Nitta, J. 229, 253 Nizzoli, F. 15, 22 Noertemann, F.C. 83–84 Nogue´s, J. 116, 121, 126, 128, 130 Nolting, F. 23, 131–133 Nonas, B. 31 Nordland, W.A. 261 Nordman, C. 199 Novosad, V. 16, 108–110 Nowak, J. 15–16, 198, 206–208, 259 Nowak, U. 125, 138 Nozaki, T. 288 Nozieres, J.P. 15, 276, 296, 302–303 O’Brien, W.L. 35 Ochiai, T. 288 Odagawa, A. 202 Odom, T.W. 34 O’Donovan, K.V. 129, 186 Oepen, H.P. 39 Oepts, W. 187, 191, 205 Oestreich, M. 250 Ogale, S.B. 202 Oguri, A. 172 O’Handlet, R.C. 35 Ohldag, H. 23, 131–133 Ohnishi, H. 40 Ohno, H. 29, 37, 230, 246, 248–249, 252, 261 Ohno, Y. 29, 37, 230, 246, 248–249, 261 Ohta, E. 33 Ohta, T. 18, 34, 36 Ohtani, K. 29, 37, 230, 248 Oiwa, A. 29, 246 Okuno, T. 15, 22, 37 Oleinik, I.I. 203, 210–211 Olesberg, J.T. 233 Oliver, W.F. 230 Omiya, T. 29, 37 Ono, T. 15, 24, 37, 162, 192 Onodera, K. 243 Oogane, M. 199

Author Index Opitz, J. 63–65, 67 Orgassa, D. 255 Ortega, J.E. 32 Ossau, W. 241 O’Sullivan, E.J. 145 Otani, Y. 16, 108–110 Ouazi, S. 19 Ounadjela, K. 15–16, 111, 145, 162–163, 183, 216 Overhauser, A.W. 234 O¨zyilmaz, B. 286 Paccard, D. 119 Padmore, H.A. 23, 132 Paget, D. 234 Pai, S.P. 202 Pailloux, F. 204 Pa´l, L. 126–127 Palmstrom, C.J. 241 Pan, W. 19 Panchula, A. 142, 144–145, 196–197, 204, 207, 261 Pang, Y. 29, 53, 273 Papanicolaou, N. 15 Papanikolaou, N. 203, 211–212, 215 Pappas, D.P. 8, 18, 230 Park, W. 184 Park, Y.D. 232, 241, 243–246, 250, 252 Parker, F.T. 124, 129, 163 Parker, J.S. 196, 202 Parker, M. 144 Parker, M.A. 163 Parker, T.J. 95 Parkin, S.S.P. 15, 24, 29, 53, 68, 93, 127, 131, 142, 144–145, 158–159, 161, 168, 170, 193, 196–197, 204–207, 216, 259, 261, 287 Parlebas, J.C. 37 Pastor, G.M. 33, 37 Patton, C.E. 100 Pearson, J. 16, 35 Pearton, S.J. 252 Peiro´, F. 204, 207 Pellegrin, E. 33 Penfold, J. 95 Peng, D.L. 119 Pereira, L.G. 168 Perel’, V.I. 230, 232–233, 261 Perjeru, F. 130

Author Index Persson, M. 34, 36 Pescia, D. 22 Petroff, F. 29, 53, 73, 77, 142–143, 153, 157–159, 193, 202, 204–205, 207, 216, 228 Petrou, A. 229, 232, 241, 243–245, 252, 254–257, 260–261 Pettifor, D.G. 177, 186, 203, 210–211, 216 Petukhov, A.G. 229, 232, 245, 252, 254–255 Pfandzelter, R. 32, 67 Ph. Ansermet, J. 187 Phan, M.S. 61 Pick, S. 19, 37 Pickett, W.E. 255 Piecuch, M. 34 Pierce, D.T. 23, 61, 63–64, 66–67, 69, 73, 230 Pietambaram, S. 145 Pietzsch, O. 23, 37, 39 Pina, E. 124–125 Pinarbasi, M. 144 Piraux, L. 162–163, 183–185, 187, 192 Pisarev, R.V. 16 Pizzini, S. 34 Plaskett, T.S. 198 Platow, W. 35 Ploog, K.H. 253 Poelsema, B. 36 Pohm, A.V. 229 Pokhil, T. 15–16 Politzer, B.A. 196 Popescu, R. 19 Popova, E. 204 Portner, D. 186 Potashnik, S.J. 246 Poulopoulos, P. 64 Powell, C.J. 144, 216 Pradolini, M.J. 129–130 Prados, C. 124–125 Prange, R.E. 56 Pratt, W.P. 158–162, 180–187, 189, 216, 284, 287 Pratzer, M. 39 Prejbeanu, I.L. 16, 111 Prinz, G.A. 13, 16, 29, 161, 227 Pufall, M.R. 284, 288 Purcell, S.T. 61, 67 Pustilnik, M. 42

325 Qasba, A. 129 Qian, C. 127 Qian, Z. 199 Qiu, Z.Q. 17, 35, 57, 73 Radu, F. 124, 130 Ralph, D.C. 275, 277, 279, 284, 286–288, 298, 307 Ramesh, R. 202 Ramos, A.R. 199 Ramsteiner, M. 253 Raoux, S. 132 Rashba, E.I. 65, 255–256, 259 Rasing, Th. 16 Rau, C. 32 Rauluszkiewicz, J. 198 Ravlic, R. 23, 34 Reddy, B.V. 33 Redepenning, J. 215 Reed, S.A. 128 Regan, T.J. 132–133 Rego, L.G.C. 40 Reilly, A. 170, 186, 188, 190 Reinders, A. 133, 137, 162, 187, 191 Reinecke, T.L. 233 Reiss, G. 202 Remer, L. 95 Ren, Y.H. 250, 252 Renard, D. 159 Renard, J.P. 159 Renaud, P. 230 Rennert, P. 31 Respaud, M. 36 Rezende, S.M. 100, 111 Rho, T.H. 31 Rhyne, J.J. 53, 73 Rice, P.M. 145, 196–197, 204, 207, 261 Richter, M. 172 Ricodeau, J.A. 127 Riedling, S. 16 Riegel, D. 129–130 Rippard, W.H. 284, 288 Ritchie, D.A. 253 Ritz, C. 186 Rizzo, N.D. 145 Robach, O. 19 Robinson, A.M. 299 Robson, M. 202 Rocha, A.R. 40

326 Roche, K.P. 29, 53, 68, 145, 142, 144–145, 205, 158–159, 204, 216 Rodmacq, B. 139–140 Rodrigues, V. 40–41 Romashev, L.N. 100, 102 Rooks, M.J. 145, 286–287 Roos, B. 16 Ross, C.A. 14 Rotenberg, E. 57, 73 Roth, W.L. 122 Rothman, J. 15–16 Roukes, M.L. 20, 253 Rousseaux, F. 15–16, 21–22 Rowell, J.M. 258 Ruderman, M.A. 52 Ru¨dt, C. 64 Ruediger, U. 15 Rugar, D. 23 Ru¨hrig, M. 68, 70 Russek, S.E. 284, 288 Ryzhanova, N. 171, 193, 200 Rzhevski, A. 130–131 Sabirianov, I.F. 215 Safarov, V.I. 234, 261 Saint-James, D. 276, 296, 302–303 Saito, S. 122, 139 Sajieddine, M. 88–89, 91–92, 204 Sakai, A. 41 Sakakima, H. 202 Salahub, D.R. 33 Salamo, G.J. 230 Salamon, M.B. 53 Samant, M.G. 142, 144–145, 196–197, 204, 207, 261 Samarth, N. 233, 242, 246 Samuelson, E.J. 122 Sańchez-Hanke, C. 131 Sander, D. 19, 35 Sanders, J. 195 Sankey, J.C. 284, 287–288 Sanvito, S. 201 Sapoval, B. 234 Sato, T. 33, 199–200 Sauer, Ch. 133, 137 Sauer, H. 95 Saunders, R.W. 95 Saurenbach, F. 53, 77, 153, 157–158, 228

Author Index Schaller, D.M. 67 Scheinfein, M.R. 22–23, 132 Schelten, J. 205 Schep, K.M. 166, 177, 186 Scheuerlein, R.E. 145 Scheurer, F. 32 Scha¨fer, R. 23, 68, 70 Schiffer, P. 246 Schlenker, C. 119–120 Schmalhorst, J. 202 Schmidt, C.M. 67 Schmidt, G. 241, 255, 259 Schmitte, T. 124 Schmool, D.S. 20 Schneider, C.M. 130–131 Schoelkopf, R.J. 284, 288 Scholl, A. 23, 131–132 Scholl, D. 94 Scho¨nherr, H.-P. 253 Schreiber, R. 29, 53, 273 Schroeder, P.A. 158–160, 163, 170, 180–183, 186, 188, 190, 193, 216 Schuhl, A. 204–207, 210 Schuller, I.K. 116, 121, 126, 128, 130 Schulthess, T.C. 104, 128, 137–138, 203, 212–215, 253, 255 Schultz, B.D. 241 Schulz, B. 8, 18 Schumann, F.O. 18 Schu¨tz, G. 33 Schweizer, J. 139 Sears, R. 131 Seck, M. 286 Seneor, P. 202–203, 206–207, 210 Seo, J.W. 129, 132 Serga, A.A. 22 Sessoli, R. 36 Setsune, K. 202 Sette, F. 33 Seve, L. 131 Shackleton, C. 95 Sham, L.J. 54 Shang, C.H. 207–208 Sharma, M. 202, 204 Sharma, P. 250 Shatz, S. 186 Shen, A. 230, 248 Shen, J. 37

Author Index Shi, J. 173 Shi, X. 127 Shigeto, K. 15, 37 Shilton, J.M. 22 Shima, H. 16 Shimizu, H. 202 Shinjo, T. 15, 22, 24, 37, 159, 162, 192 Shinohara, T. 33 Shockley, W. 238 Shutol, Y. 246 Siebrecht, R. 130 Siebrecht, S. 124 Siegmann, H.C. 16, 94 Siegwart, H. 129, 132 Sievers, A.J. 122 Silva, P.C. 40–41 Silva, T.J. 284, 288 Silver, R.N. 255 Sinha, S.K. 53, 128 Sinkovic, B. 131 Slack, G.A. 122 Slater, R.D. 186 Slaughter, J.M. 145, 160 Slavin, A.N. 21–22 Slonczewski, J.C. 65, 70, 73, 162, 209, 274–275, 277, 280, 286, 292–294, 296–297, 299 Slupinski, T. 29, 246 So´lyom, J. 127 Smith, C.G. 253 Smith, D.J. 122 Smith, D.L. 255, 259 Smith, D.S. 128 Smith, K. 145 Smith, N.V. 33, 57, 73 Smith, P. 281 Smith, R.A. 234–235, 238 Smits, B.A. 259 Smorchkova, I.P. 233 Snoeck, E. 202 Soares, J.C. 198–199, 202 Soares, V. 198 Soeya, S. 124 Sokolov, A. 215 Sommers, C. 173 Søndergard, E. 21 Song, D. 15–16 Sort, J. 139–140 Soulen,, R.J. 195

327 Sousa, R.C. 198 Sowers, C.H. 61, 106–108 Sowers, H. 29, 53, 253 Spada, F.E. 163 Spanos, G. 246 Speaks, R. 15 Speriosu, V.S. 142, 159, 161, 168–169, 193 Squires, G.L. 128 Srajer, G. 93 Srikanth, H. 195 Stamm, C. 16, 132–133 Stamps, R.L. 81–89, 94–95, 98, 100, 116 Staud, N. 126 Stearns, M.B. 196 Steenwyck, S.D. 183, 185 Steenwyk, S. 184 Steierl, G. 32 Steinmu¨ller, S.J. 21 Stepanyuk, V.S. 31, 33 Steren, L.B. 163, 193 Sto¨hr, J. 16, 23, 33, 35, 131–133 Stiles, M.D. 51, 61, 64, 66, 73, 137–138, 142, 144, 153, 174, 176, 216, 230, 292–295, 300 Stiles, P.J. 246 Stoeffler, D. 67 Stoner, E.C. 15, 279, 296 Stout, J.W. 128 Strand, J. 241 Strom, V. 21 Stroscio, J.A. 67 Stroud, R.M. 232, 245, 252 Suezawa, Y. 198 Sugahara, S. 246 Sugita, Y. 192 Sullivan, J.M. 246 Sumiyama, K. 119 Sun, B. 250, 252 Sun, J.J. 145, 198 Sun, J.Z. 201, 204, 277, 280–281, 286–287, 289, 307 Suran, G. 17 Suzuki, Y. 23, 145, 199–200, 202, 204–205, 207, 214, 261 Swagten, H.J.M. 204, 206, 208–209, 215, 259 Szabo´, P. 126–127 Sze, S.M. 256 Szunyogh, L. 35, 37, 173, 214–215, 300

328 Tabata, H. 250 Tackeuchi, A. 241 Takagi, H. 29 Takahashi, F. 198 Takahashi, M. 127 Takaki, H. 127 Takamura, K. 249 Takanashi, K. 88 Takano, K. 116, 121–126, 129, 131, 133, 137, 144, 153, 216 Takayanagi, H. 229, 253 Takeda, M. 41 Takeda, Y. 39 Takeuchi, M. 202 Tamura, E. 199–200 Tanaka, M. 202, 229, 246 Tang, C. 144 Tang, H.X. 253 Tang, Y.J. 122, 128 Taniyama, T. 33 Tanner, B.K. 130 Tarnoćzi, T. 127 Tatara, G. 24, 40 Taylor, J.A. 115, 145 Tedrow, P.M. 193–196, 257, 259 Tehrani, S. 145 Teitelman, M. 64 Terauchi, R. 261 Tezuka, N. 288 Thiaville, A. 287–288 Thibado, P.M. 230 Thole, B.T. 33 Thomas, G. 122–123, 129, 131, 133, 137, 163 Thomas, L. 15, 17 Thompson, D.A. 115, 120, 122, 126–127 Tian, C.S. 35 Tifrea, I. 234 Tilley, D.R. 95 Tinkham, M. 122 Tischer, M. 34 Tiusa, C. 145 Tiusan, C. 204, 206–207, 216 Todd, N.K. 201 Tondra, M. 145 Toney, M.F. 128, 132 Tong, H.C. 127 Tong, L.-N. 130–131 Tosatti, E. 31 Tricker, D.M. 15–16, 288

Author Index Trigui, F. 159 Trouilloud, P.L. 145 Trygg, J. 31, 34 Tse, D.H.Y. 14 Tselepi, M. 21 Tsetseris, L. 61 Tsoi, M. 24, 186, 286–287 Tsoi, V. 286 Tsunashima, S. 89–90 Tsunekawa, K. 204 Tsunoda, M. 127 Tsvetkov, A. 16 Tsymbal, E.Yu. 154, 177, 186, 203, 209–211, 215–216 Tudosa, I. 16 Tulchinsky, D. 161 Turek, I. 177 Uchiyama, S. 89–90 Ueda, K. 250 Ugarte, D. 40–41 Uiberacker, C. 35 Umbayashi, H. 126 Umerski, A. 172, 203, 212, 215, 276, 280–282, 284, 286, 288–289, 293, 296–297, 299, 304–307 Unguris, J. 16, 23, 61, 63–64, 66–67, 69, 73, 161 Untiedt, C. 41 Urano, C. 29 Urazhdin, S. 284, 287 Urban, R. 67 Usadel, K.D. 125, 138 Ustinov, V.V. 100, 102, 168 Valet, T. 178–181, 254 Van Dau, N.F. 77 van de Veerdonk, R.J.M. 162, 204–206, 215, 259 Van de Vin, C.H. 208–209 van den Berg, H.A.M. 145 van der Laan, G. 33 van der Zaag, P.J. 129 Van Dijken, S. 36 Van Dorpe, P. 249 van Ek, J. 253 van Gansweinkel, R.M.J. 187, 191 van Kampen, M. 259

Author Index van Laar, B. 139 van Ruitenbeek, J.M. 40–42 van Wees, B.J. 253, 255–256 van‘t Erve, O.M.J. 259–261 Varela, M. 209 Vaure`s, A. 24, 202–203, 205–207, 210 Vavassori, P. 108–110 Vavra, W. 161 Vaz, C.A.F. 15–16 Vedmedenko, E.Y. 39 Vedyayev, A. 171, 193, 200, 215 Veillet, P. 159 Velu, E. 159 Venkatesan, A. 230 Venkatesan, T. 202 Venus, D. 67 Verheijen, M.A. 129 Vernes, A. 300 Vettier, C. 53, 81 Vignale, G. 229, 237 Villeret, M. 64, 172, 291, 299, 304–305 Vinter, B. 230, 240 Viret, M. 201 Voegeli, B. 163 Vohl, M. 98 Vollmer, R. 10, 23, 36 von Bergmann, K. 39 von Molnar, S. 246 Vorob’ev, L.E. 264 Voskoboynikov, A. 264 Vouille, C. 170, 186, 188, 190, 287–288 Vukadinovic, N. 109 Wachowiak, A. 37 Wada, O. 241 Wagner, H. 11–12 Wagner, W. 33 Waintal, X. 275, 298 Wallis, T.M. 36 Wang, C.T. 122 Wang, D. 115, 145, 199 Wang, D.S. 37 Wang, J. 202 Wang, K.S. 214–215 Wang, K.Y. 246 Wang, R. 261 Wang, R.W. 86–88, 105 Wang, S.X. 202, 204

329 Wang, X. 33, 203, 212, 253 Wang, Y.Y. 201 Wanger, H. 36 Warner, R.A. 145 Wassermeier, M. 253 Wastlbauer, G. 21 Waszczak, J.V. 53, 81, 83 Watanabe, N. 204 Watts, S.M. 196, 202 Weber, W. 61 Weht, R. 31 Wei, P. 202 Weinberger, P. 35, 37, 173, 300 Weisbuch, C. 230, 240 Weiss, D. 22 Welland, M.E. 15–16, 288 Weller, D. 16, 29, 35, 128 Wernsdorfer, W. 15, 17 Westerholt, K. 124, 130 White, R.L. 132 Wiebe, J. 37 Wiesendanger, R. 23, 34, 37, 39 Wilczynski, M. 200 Wildberger, K. 31, 33 Wilhelm, W. 33 Wilhoit, D.R. 142, 159, 161, 193 Wilkins, S.B. 130 Wilks, R. 20 Williams, A.R. 54 Willis, R.F. 32 Wills, J.M. 31, 34 Wilson, A. 246 Wingreen, N.S. 34 Winter, H. 67 Wiser, N. 186 Wohlfarth, E.P. 15, 279, 296 Wolf, J.A. 67–68, 70 Wolf, R.M. 129 Wolf, S. 128 Wolf, S.A. 227 Wolfe, J.H. 57, 73 Wong, T.M. 145, 154, 193, 198–199, 205, 273 Wongsam, M.A. 111 Woods, G.T. 195 Woods, L.M. 233 Worledge, D.C. 196, 209 Wa¨ppling, R. 64 Wu, J. 20 Wu, M.W. 206

330 Wu, R. 29 Wu, R.Q. 30–35, 37 Wulfhekel, W. 24, 206 Wunnicke, O. 253, 262 Wyder, P. 286 Xia, K. 177 Xiang, X.H. 206 Xiao, G. 201, 205 Xiao, J.M. 126 Xiao, J.Q. 163, 206 Xin, Y. 202 Xiong, P. 196, 202 Yacoby, A. 40, 42 Yafet, Y. 53, 73, 81 Yamagata, S. 204 Yamaguchi, A. 24 Yamamori, H. 199–200 Yamamoto, H. 159 Yamamoto, Y. 33 Yamamuro, S. 119 Yamanouchi, M. 29, 37, 246 Yamaoka, T. 127 Yanagi, S. 246 Yang, M.J. 253 Yang, Q. 181–183 Yang, S.-H. 145, 196–197, 204, 207, 261 Yashar, P. 130 Yelon, A. 116, 119 Yeyati, A.L. 40, 42 Yi, Y. 12 Yokoyama, T. 18, 34, 36 York, B.R. 168 You, C.Y. 29 Young, A.P. 163 Young, A.T. 131 Young, D.K. 29, 248–249 Yu, G.H. 126 Yu, J. 15

Author Index Yu, Z.G. 255 Yuan, S.W. 144 Yuasa, S. 145, 199–200, 204–205, 207, 214, 261 Zabel, H. 66, 73, 124, 130 Zahn, P. 63–65, 67, 172, 174–175 Zak, J. 17 Zangwill, A. 292–295, 300 Zeller, R. 31, 33, 156, 174–175, 188–189, 253, 262 Zeper, W.B. 67 Zhang, K. 138 Zhang, S. 139, 163, 170–171, 181, 192, 206, 209–210, 230, 287–288, 292, 294, 296 Zhang, S.F. 214–215 Zhang, X.G. 171–174, 203, 209, 212–215, 253 Zhang, Z. 20 Zhang, Z.G. 199 Zhao, H.B. 250, 252 Zhao, T. 138 Zhao, Y.-W. 40 Zheng, Y. 13, 15–16 Zhou, T.C. 262 Zhou, X.C. 262 Zhu, F.W. 126 Zhu, H.J. 253 Zhu, J.G. 13, 15–16 Zhu, T. 206 Zhu, W. 131 Zilberman, P.E. 296 Zimmler, M.A. 286 Zink, B.L. 122 Zinn, W. 53, 66, 73, 77, 153, 157–159, 167–168, 174, 228 Zivieri, R. 22 Zutic´, I. 65, 227, 229, 252

SUBJECT INDEX 4d and 5d clusters, 33 4d and 5d, 31–33, 41 4d ferromagnetism, 32 AFM, 1, 23 Al2O3, 256–257, 259–261 aliasing, 62 AlSb, 264–265 anisotropies, 1, 8, 11–13, 20, 22 antiferromagnetic fluorides, 116, 121, 128–129 antiferromagnetic oxides, 116, 121 antiferromagnetism, 1 applications, 116, 121, 126–127, 131, 141, 145 arrays, 1, 13, 22 asymptotic form, 61, 63–64 asymptotic formula, 61, 64 biquadratic coupling, 51, 53, 59, 70–73 Bir–Aronov–Pikus, 233–234 blocking temperature, 120–121, 127, 129 Brillouin light scattering (BLS), 1, 20–22, 68, 70 CdCr2Se4, 248, 250–252 charge current, 228 charge polarization packets, 235–236 circular polarization, 240, 242, 244, 249–252, 257–258, 261 circularly polarized, 230–231, 237, 239, 242–244 CMOS, 227–228 Co, 54–55, 57, 61, 64–65 Co/Cu/Co trilayers, 289 coercive field, 247, 252 coherent growth, 59 conductance quantum, 40–41 conductivity mismatch, 253–256 conversion electron Mo¨ssbauer spectroscopy, 133 coupling strength, 62–68, 70–72 critical bias, 282, 287–288, 306–308

critical current, 284, 286–289, 307 critical spanning vector, 51, 59–63, 65, 72 Cu, 57, 60–62, 64–65 Cu/Co(100), 61 current-induced switching of magnetization, 274, 276–277, 308 d-doped DMS, 37 D’yakonov–Perel, 232–233, 248–249 de Haas-van Alphen (dHvA) effect, 52 density of states, 52, 57 density-functional theory, 30 dichroism, 252, 257–258 diluted magnetic semiconductors, 242 dipolar effects, 1 dipole interactions, 1, 15 dynamic measurements, 1 dynamical magnetization, 94–95 eigenvalue sum, 64 electroluminescence, 240–241, 243–245, 248–253, 257–260 electronic spins, 30 electronic structure, 51, 54–55 elements, 1, 4, 6, 13–17, 19–20, 22–23 Elliot–Yafet, 231–233 Esaki diode, 249 exchange and correlation effects, 54 exchange anisotropy, 115–116, 119–121, 127–128, 132–133, 135–137, 141, 146 exchange bias, 115–116, 118, 135, 137–138, 142 exchange coupling, 280, 294, 299–300, 304–305 exchange interaction, 51, 55, 59, 71–72 exchange-correlation potential, 31 experimental techniques, 1, 17 Faraday rotation, 230, 242, 244 Faraday, 230, 241–242, 244, 248–251, 257, 260

331

332 fast magnetic switching, 1, 19 Fe/Au(100), 64, 67 Fe/Cr(100), 67 Fermi surface, 51–52, 54–55, 59–63, 72 Fermi velocity, 60–61 Fermi wave vector, 59 ferromagnet, 154–155, 164, 194–195, 197, 210 ferromagnetic resonance (FMR), 1, 14, 20–22, 68, 95, 105 ferromagnetic semiconductors, 242 ferromagnetism, 1, 4, 9, 11–12 FLAPW, 31, 39 force theorem, 64 free-electron model, 56 Friedel oscillations, 52 GaSb, 263–264 generalized gradient approximation, 31 giant magnetoresistance (GMR), 53, 56, 68, 142, 227–230, 153 giant, 1, 13 Gibbs oscillations, 62 Gilbert damping, 274, 280, 289, 307 growth front, 66–67, 70–71 half metallic, 255 Haynes–Shockley, 238 heavy hole, 230, 240–242, 250, 253, 257, 260 hybridization, 54–55 hyperfine interactions, 234 hysteresis loop, resistance vs current, 278, 283, 307 hysteresis loops, 278, 285, 288, 306–308 InAs, 264–265 in-plane, 279, 281, 286–289, 295–296, 306 interdiffusion, 67 interfacial coupling energy, 118, 121 interlayer coupling, 1, 20 interlayer exchange coupling, 52, 54, 56, 58–62, 64–65, 67, 69–73 inversion symmetric, 232, 263 iron whiskers, 67 itinerant electrons, 54 Keldysh method, 276, 293, 295–296 Kohn–Sham equation, 31 Kramers degeneracy, 263

Subject Index Landa–Lifshit–Gilbert equation, 139 Landauer method, 275, 296, 308 Landauer–Buttiker equation, 40 Landau–Lifshitz equation, 274, 277, 286–287 layer-by-layer growth, 67 light hole, 230–231, 240–242, 253, 260 local density of states, 34 Local Mean Field Theory, 94 local mean field, 138 local moment model, 55–56 local-spin-density approximation (LSDA), 31, 54 Lorentz microscopy, 1, 23 Magnesium oxide (MgO), 261 magnetic anisotropy energy, 35 magnetic clusters, 36 magnetic domain walls, 39 magnetic elements, 1, 13, 16–17 magnetic enhancement, 32 magnetic multilayers, 51–57, 59, 61, 63, 65, 67, 69, 71–73, 77–78, 94–95, 104, 111, 153, 158–159 magnetic nanostructures, 153–154, 216 magnetic ordering, 1 magnetic particles, 1 magnetic random access memory, 13 magnetic reading, 1 magnetic transport, 30, 40, 42 magnetic tunnel junctions, 153 magnetic writing, 1 magnetocrystalline anisotropy, 30, 35 magneto-optical Kerr effect (MOKE), 1, 17–19, 68 magnetoresistance, 1, 13, 24 magnetostatic interactions, 35 majority carriers, 235, 237 mechanisms, 231, 252 metallic antiferromagnets, 126, 130–131, 135 MFM, 1, 15–16, 20, 23 misfit dislocations, 231–232, 245 molecular magnets, 36 Monte Carlo, 138 Moore’s Law, 227–228 nanopillars, 286, 288 nanostructures, 1, 3, 9, 13–14, 19, 21, 24

Subject Index Neél axis, 118, 122 neutron diffraction, 122, 128–129 non-collinear magnetic patterns, 37 nuclear polarization, 234 optical excitation, 230, 252 optical polarization, 240–241, 249, 261 optical pumping, 230, 234, 239, 245, 261 optical selection rules, 240 orange peel coupling, 52 orbital magnetic moment, 30–33, 36 out-of-plane, 295, 306 PEEM, 1, 23 period, 53, 59–62, 64, 66–68, 71–72 phonon signatures, 258 photoemission, 57, 73 pinholes, 52 point contacts, 286 Poisson’s equation, 235, 256–257 polarization transducer, 240 polarized neutron reflectivity, 129 preasymptotic correction, 61, 64 pseudomorphic, 66 quantitative basis, 289 quantum confinement, 230, 240 quantum dot, 233–234 quantum selection rules, 243 quantum well state, 56–58, 72–73 quantum well, 51, 54, 56–59, 72–73 radiative lifetime, 241, 258, 261 radiative recombination, 239–244, 248, 252, 261 rare earth multilayers, 53, 73 read heads, 1, 13, 53 redox reactions, 135 reflection amplitude, 58–59, 62, 64 reflection high-energy electron diffraction (RHEED), 68–69 remanent magnetization, 229, 247 resonances, 56–57, 59 rotatable anisotropy, 138 Rowell criteria, 258 Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, 52–53, 62

333 scanning electron microscopy with polarization analysis (SEMPA), 1, 23, 68 scanning tunneling microscopy (STM), 1, 14, 18, 21, 23–24, 66, 68 Schottky barrier, 253, 256–257, 259–260, 262 s–d model, 55–56 selection rule, 230, 239–241, 243, 250, 257 self-consistency, 275, 292–293 semiconductor spintronics, 229, 242 semimagnetic semiconductors, 242, 265 shifted hysteresis loops, 119 soft X-ray magnetic circular dichroism, 33 space charge layer, 235 spacer layer, 51–53, 57–61, 63, 65–67, 69–72 Sp–d exchange, 242 SPEELS, 1, 10, 23 spin – STM, 23–24 spin accumulation, 154, 160, 165, 176–179, 189, 235, 253, 258, 292–294, 296 spin current, 234, 238, 274–277, 290–291, 293–300, 302–307 spin decoherence length, 294 spin density wave antiferromagnetism, 66 spin diffusion length, 229, 238–239, 249, 255 spin engineering, 1–2, 9, 12–14 spin flop, 137 spin lifetimes, 229, 231, 233–234, 241, 250, 257, 261 spin packets, 237, 265 spin polarization, 154, 178, 181, 193–199, 201–204, 206–211, 215, 229–234, 236, 238–246, 248–250, 252–253, 255, 258–265 spin polarized tunneling, 145 spin relaxation, 231–234, 238, 261, 265 spin reorientation transition, 1, 7 spin scattering, 231–233, 240, 250, 257, 262 spin transistor, 237 spin transport, 153–154, 177 spin valve, 142, 144–145 spin waves, 1–3, 10–11, 22–23, 111 spin-field-effect-transistors, 264 spin-LED, 239–241, 243, 245, 248, 250, 253, 256–262 spin-orbit coupling, 31, 35, 54 spin-polarized scanning tunneling microscope, 34 spin-reorientation transition, 36

Subject Index

334 spin-transfer torque, 274–280, 282, 286–287, 289–294, 296, 299–301, 304–306, 308 spintronics, 153–154, 177 stability, 276–278, 281–287, 296, 306–307 standard model, 292 steady state solutions, 281, 286–287 Stoner criterion, 30–31 strain, 66 sum rules, 33 super-paramagnetism, 1, 29, 36 telegraph noise, 288 temperature, 62–64, 66 terrace, 66, 70–71 theory, 135–137 thermally activated reversal, 138 thermoremanent magnetization, 122–123 thickness fluctuations, 63, 66–67, 70 time-reversal symmetry, 263–264 torque curves, 118–120 torque, 65 total energy calculations, 64 training, 119, 121, 138, 140–141 transition metal ferromagnets, 54 transmission probability, 56–58

tunnel barrier, 253, 256, 258–262 tunnelling magnetoresistance (TMR), 1, 13, 154 uncompensated spins, 118, 120, 122–124, 129, 131–132, 135 uniaxial anisotropy, 35 unidirectional anisotropy, 118 unipolar spin transistor, 237 Voigt geometry, 241 wedge-shaped spacer layer, 67 x-ray absorption microscopy, 133 X-ray diffraction, 66 x-ray magnetic circular dichroism (XMCD), 1, 18–19, 23, 130 x-ray magnetic linear dichroism, 130–132 x-ray photoemission microscopy, 131 Zeeman splitting, 242–243 Zn1 xMnxSe, Cd1 xMnxTe, Pb1 xMnxSe, 242–243 ZnMnSe/AlGaAs interface, 244–245

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